Bacterial adaptation to antibiotic combinations depends on the joint inhibitory effects of the two drugs (drug interaction [DI]) and how resistance to one drug impacts resistance to the other (collateral effects [CE]). Here we model these evolutionary dynamics on two-dimensional phenotype spaces that leverage scaling relations between the drug-response surfaces of drug-sensitive (ancestral) and drug-resistant (mutant) populations. We show that evolved resistance to the component drugs – and in turn, the adaptation of growth rate – is governed by a Price equation whose covariance terms encode geometric features of both the two-drug-response surface (DI) in ancestral cells and the correlations between resistance levels to those drugs (CE). Within this framework, mean evolutionary trajectories reduce to a type of weighted gradient dynamics, with the drug interaction dictating the shape of the underlying landscape and the collateral effects constraining the motion on those landscapes. We also demonstrate how constraints on available mutational pathways can be incorporated into the framework, adding a third key driver of evolution. Our results clarify the complex relationship between drug interactions and collateral effects in multidrug environments and illustrate how specific dosage combinations can shift the weighting of these two effects, leading to different and temporally explicit selective outcomes.
antibiotic resistanceprice equationdrug interactionscollateral sensitivityResearch organismNonehttp://dx.doi.org/10.13039/100000002National Institutes of Health1R35GM124875WoodKevin Bhttp://dx.doi.org/10.13039/100000001National Science Foundation1553028WoodKevin Bhttp://dx.doi.org/10.13039/501100007472Fundação Luso-Americana para o Desenvolvimento274/2016GjiniEridahttp://dx.doi.org/10.13039/501100000348Calouste Gulbenkian FoundationGjiniEridahttp://dx.doi.org/10.13039/100006129FCTUIDB/04621/2020GjiniEridahttp://dx.doi.org/10.13039/100006129FCTUIDP/04621/2020GjiniEridaThe funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.Author impact statementA simple mathematical model reveals that antibiotic interactions and collateral effects of evolution are inseparable drivers of multidrug resistance linked by the well-known Price equation from evolutionary theory.Introduction
Understanding and predicting evolutionary dynamics is an ongoing challenge across all fields of biology. Microbial populations offer relatively simple model systems for investigating adaptation on multiple length scales, from the molecular to the population level, on timescales ranging from a few generations to decades. Yet even these simplest of systems exhibit rich and often counterintuitive evolutionary dynamics. Despite enormous progress, both theoretical and experimental, predicting evolution remains exceedingly difficult, in part because it is challenging to identify the phenotypes, selective gradients, and environmental factors shaping adaptation. In turn, controlling those dynamics – for example, by judicious manipulation of environmental conditions – is often impossible. These challenges represent open theoretical questions but also underlie practical public health threats – exemplified by the rapid rise of antibiotic resistance (Davies and Davies, 2010; Levy and Marshall, 2004) – where evolutionary dynamics are fundamental to the challenge, and perhaps, to the solution.
In recent years, evolution-based strategies for impeding drug resistance have gained significant attention. These approaches have identified a number of different factors that could modulate resistance evolution, including spatial heterogeneity (Zhang et al., 2011; Baym et al., 2016a; Greulich et al., 2012; Hermsen et al., 2012; Moreno-Gamez et al., 2015; Gokhale et al., 2018; De Jong and Wood, 2018; Santos-Lopez et al., 2019); competitive (Read et al., 2011; Hansen et al., 2017; Hansen et al., 2020), cooperative (Meredith et al., 2015b; Artemova et al., 2015; Sorg et al., 2016; Tan et al., 2012; Karslake et al., 2016; Yurtsev et al., 2016; Hallinen et al., 2020), or metabolic (Adamowicz et al., 2018; Adamowicz et al., 2020) interactions between bacterial cells; synergy with the immune system, especially in the context of adaptive treatment (Gjini and Brito, 2016); epistasis between resistance mutations (Trindade et al., 2009; Borrell et al., 2013; Lukačišinová et al., 2020); plasmid dynamics (Lopatkin et al., 2016; Lopatkin et al., 2017; Cooper et al., 2017); precise tuning of drug doses (Lipsitch and Levin, 1997; Yoshida et al., 2017; Meredith et al., 2015a; Nichol et al., 2015; Fuentes-Hernandez et al., 2015; Coates et al., 2018; Iram et al., 2021); cycling or mixing drugs at the hospital level (Bergstrom et al., 2004; Beardmore et al., 2017); and statistical correlations between resistance profiles for different drugs (Imamovic and Sommer, 2013; Kim et al., 2014; Pál et al., 2015; Barbosa et al., 2017; Rodriguez de Evgrafov et al., 2015; Nichol et al., 2019; Podnecky et al., 2018; Imamovic et al., 2018; Barbosa et al., 2019; Rosenkilde et al., 2019; Apjok et al., 2019; Maltas and Wood, 2019; Maltas et al., 2020; Hernando-Amado et al., 2020; Roemhild et al., 2020; Ardell and Kryazhimskiy, 2020).
Drug combinations are a particularly promising approach for slowing resistance (Baym et al., 2016b), but the evolutionary impacts of combination therapy remain difficult to predict, especially in a clinical setting (Podolsky, 2015; Woods and Read, 2015). Antibiotics are said to interact when the combined effect of the drugs is greater than (synergy) or less than (antagonism) expected based on the effects of the drugs alone (Greco et al., 1995). These interactions may be leveraged to improve treatments – for example, by offering enhanced antimicrobial effects at reduced concentrations. But these interactions can also accelerate, reduce, or even reverse the evolution of resistance (Chait et al., 2007; Michel et al., 2008; Hegreness et al., 2008; Pena-Miller et al., 2013; Dean et al., 2020), leading to tradeoffs between short-term inhibitory effects and long-term evolutionary potential (Torella et al., 2010). In addition, resistance to one drug may be associated with modulated resistance to other drugs. This cross-resistance (or collateral sensitivity) between drugs in a combination has also been shown to significantly modulate resistance evolution (Barbosa et al., 2018; Rodriguez de Evgrafov et al., 2015; Munck et al., 2014).
Collateral effects (Pál et al., 2015; Roemhild et al., 2020) and drug interactions (Bollenbach et al., 2009; Chevereau et al., 2015; Lukačišin and Bollenbach, 2019; Chevereau et al., 2015), even in isolation, reflect interactions – between genetic loci, between competing evolutionary trajectories, between chemical stressors – that are often poorly understood at a mechanistic or molecular level. Yet adaptation to a drug combination may often reflect both phenomena, with the pleiotropic effects that couple resistance to individual drugs constraining, or constrained by, the interactions that occur when those drugs are used simultaneously. In addition, the underlying genotype space is high-dimensional and potentially rugged, rendering the genotypic trajectories prohibitively complex (de Visser and Krug, 2014).
In this work, we attempt to navigate these obstacles by modeling evolutionary dynamics on lower-dimensional phenotype spaces that leverage scaling relations between the drug-response surfaces of ancestral and mutant populations. Our approach is inspired by the fact that multiobjective evolutionary optimization may occur on surprisingly low-dimensional phenotypic spaces (Shoval et al., 2012; Hart et al., 2015). To develop a similarly coarse-grained picture of multidrug resistance, we associate selectable resistance traits with changes in effective drug concentrations, formalizing the geometric rescaling assumptions originally pioneered in Chait et al., 2007; Hegreness et al., 2008; Michel et al., 2008 and connecting evolutionary dynamics with a simple measurable property of ancestral populations. We show that evolved resistance to the component drugs – and in turn, the adaptation of growth rate – is governed by a Price equation whose covariance terms encode geometric features of both (1) the two-drug-response surface in ancestral populations (the drug interaction) and (2) the correlations between resistance levels to those drugs (collateral effects). In addition, we show how evolutionary trajectories within this framework reduce to a type of weighted gradient dynamics on the two-drug landscape, with the drug interaction dictating the shape of the underlying landscape and the collateral effects constraining the motion on those landscapes, leading to deviations from a simple gradient descent. We also illustrate two straightforward extensions of the basic framework, allowing us to investigate the effects of both constrained mutational pathways and sequential multidrug treatments. Our results clarify the complex relationship between drug interactions and collateral effects in multidrug environments and illustrate how specific dosage combinations can shift the weighting of these two effects, leading to different selective outcomes even when the available genetic routes to resistance are unchanged.
Results
Our goal is to understand evolutionary dynamics of a cellular population in the presence of two drugs, drug 1 and drug 2. These dynamics reflect a potentially complex interplay between drug interactions and collateral evolutionary tradeoffs, and our aim is to formalize these effects in a simple model. To do so, we assume that the per capita growth rate of the ancestral population is given by a function G(x,y), where x and y are the concentrations of drugs 1 and 2, respectively. We limit our analysis to two-drug combinations, but it could be extended to higher-order drug combinations, though this would require empirical or theoretical estimates for higher-dimensional drug-response surfaces (see, e.g., Wood et al., 2012; Zimmer et al., 2016; Zimmer et al., 2017; Russ and Kishony, 2018; Tekin et al., 2016; Tekin et al., 2017; Tekin et al., 2018). At this stage, we do not specify the functional form of G(x,y), though we assume that this function can be derived from pharmacodynamic or mechanistic considerations (Engelstädter, 2014; Bollenbach et al., 2009; Wood and Cluzel, 2012) or otherwise estimated from experimental data (Greco et al., 1995; Wood et al., 2014). In classical pharmacology models (Loewe, 1953; Greco et al., 1995), the shape of these surfaces – specifically, the convexity of the corresponding contours of constant growth (‘isoboles’) – determines the type of drug interaction, with linear isoboles corresponding to additive drug pairs. In this framework, deviations from linearity indicate synergy (concave up) or antagonism (concave down). While there are multiple conventions for assigning geometric features of the response surface to an interactions type – and there has been considerable debate about the appropriate null model for additive interactions (Greco et al., 1995) – the response surfaces contain complete information about the phenotypic response. The manner in which this response data is mapped to a qualitative interaction type – and therefore used to label the interaction as synergistic, for example – is somewhat subjective, though in what follows we adopt the isobole-based labeling scheme because it is more directly related to the geometry of the response surface than competing models (e.g., Bliss independence; Greco et al., 1995).
Resistance as a continuous trait and rescaling in a simple model
The primary assumption of the model is that the phenotypic response (e.g., growth rate) of drug-resistant mutants, which may be present in the initial population or arise through mutation, corresponds to a simple rescaling of the growth rate function G(x,y) for the ancestral population. As we will see, this scheme provides an explicit link between a cell’s level of antibiotic resistance and its fitness in a given multidrug environment. Specifically, we assume that the growth rate (gi) of mutant i is given bygi=G(αix,βiy),where αi and βi are rescaling parameters that reflect the physiological effects of mutations on the growth rate. In some cases – for example, resistance due to efflux pumps (Wood and Cluzel, 2012) or drug degrading enzymes (Yurtsev et al., 2013) – this effective concentration change corresponds to a physical change in intracellular drug concentration. More generally, though, this hypothesis assumes that resistant cells growing in external drug concentration x behave similarly to wild-type (drug-sensitive) cells experiencing a reduced effective concentration αx. Similar rescaling arguments were originally proposed in Chait et al., 2007, where they were used to predict correlations between the rate of resistance evolution and the type of drug interaction. These arguments have also been used to describe fitness tradeoffs during adaptation (Das et al., 2020) and to account for more general changes in the dose-response curves, though in many cases the original two-parameter rescaling was sufficient to describe the growth surface in mutants (Wood et al., 2014).
When only a single drug is used, this rescaling leads to a simple relationship between the characteristic inhibitory concentrations – for example, the half-maximal inhibitory concentration (IC50) or the minimum inhibitory concentration (MIC) – of the ancestral (sensitive) and mutant (resistant) populations. In what follows, we refer to these reference concentrations as Kij, where i labels the cell type and, when there is more than one drug, j labels the drug. Conceptually, this means that dose-response curves for both populations have the same basic shape, with resistance (or sensitivity) in the mutant corresponding only to a shape-preserving rescaling of the drug concentration (D→D/Ki; Figure 1A). In the presence of two drugs, the dose-response curves become dose-response surfaces, and rescaling now corresponds to a shape-preserving rescaling of the contours of constant growth. There are now two scaling parameters, one for each drug, and in general they are not equal. For example, in Figure 1B, the mutant shows increased sensitivity to drug 1 (α≡KWT1/KMut1>1) and increased resistance to drug 2 ( β≡KWT2/KMut2<1), where superscripts label the drug (1 or 2) and subscripts label the cell type (wild type, WT; mutant, Mut).
Drug resistance as a rescaling of effective drug concentration.
The fundamental assumption of our model is that drug-resistant mutants exhibit phenotypes identical to those of the ancestral (‘wild type’) cells but at rescaled effective drug concentration. (A) Left panel: schematic dose-response curves for an ancestral strain (blue) and a resistant mutant (red). Half-maximal inhibitory concentrations (KWT,KMut), which provide a measure of resistance, correspond to the drug concentrations where inhibition is half maximal. Fitness cost of resistance is represented as a decrease in drug-free growth. Right panel: dose-response curves for both cell types collapse onto a single functional form, similar to those in Chait et al., 2007; Michel et al., 2008; Wood and Cluzel, 2012; Wood et al., 2014. (B) Left panel: in the presence of two drugs, growth is represented by a surface; the thick blue curve represents the isogrowth contour at half-maximal inhibition; it intersects the axes at the half-maximal inhibitory concentrations for each individual drug. Right panel: isogrowth contours for ancestral (WT) and mutant cells. In this example, the mutant exhibits increases resistance to drug 2 and an increased sensitivity to drug 1, each of which corresponds to a rescaling of drug concentration for that drug. These rescalings are quantified with scaling constants α≡KWT1/KMut1 and β≡KWT2/KMut2, where the superscripts indicate the drug (1 or 2). (C) Scaling factors for two different mutants (red square and red circle) are shown. The ancestral cells correspond to scaling constants α=β=1. Mutant 1 exhibits increased sensitivity to drug 1 (α>1) and increased resistance to drug 2 (β<1). Mutant 2 exhibits increased resistance to both drugs (α<β<1), with higher resistance to drug 1. (D) Scaling parameters describe the relative change in effective drug concentration experienced by each mutant. While scaling parameters for a given mutant are fixed, the effects of those mutations on growth depend on the external environment (i.e., the drug dosage applied). This schematic shows the effective drug concentrations experienced by WT cells (blue circles) and the two different mutants (red circles and red squares) from panel (C) under two different external conditions (open and closed shapes). True dosage 1 (2) corresponds to higher external concentrations of drug 1 (2). The concentrations are superimposed on a contour plot of the two drug surface (similar to panel B). Right panel: resulting growth of mutants and WT strains at dosage 1 (bottom) and dosage 2 (top). Because the dosages are chosen along a contour of constant growth, the WT exhibits the same growth at both dosages. However, the growth of the mutants depends on the dose, with mutant 1 growing faster (slower) than mutant 2 under dosage 2 (dosage 1). A key simplifying feature of these evolutionary dynamics is that the selective regime (drug concentration) and phenotype (effective drug concentration) have same units.
Detailed selection dynamics associated to <xref ref-type="fig" rid="fig2">Figure 2</xref>.
Here we visualize the selection going on in a hypothetical bacterial population under the two-drug dosage-selective regime (x0,y0) and growth landscape shown in Figure 2, where the collateral effects, represented in the available mutants (αi,βi), are estimated from data (Figure 3—figure supplement 1C). The size of the circles in rescaling factor trait space corresponds to the relative frequency of each subpopulation, calculated at each timepoint.
The power of this rescaling approach is that it directly links growth of the mutant populations to measurable properties of the ancestral population (the two-drug-response surface) via traits of the mutants (the rescaling parameters). Each mutant is characterized by a pair of scaling parameters, (αi,βi), which one might think of as a type of coarse-grained genotype (Figure 1C). When paired with the ancestral growth surface, these traits fully determine the per capita growth rate of the mutant at any external dosage combination (x,y) via Equation 1. While the scaling parameters are intrinsic properties of each mutant, they contribute to the phenotype (growth) in a context-dependent manner, leading to selection dynamics that depend in predictable ways on the external environment (Figure 1).
We assume a finite set of M subpopulations (mutants), (i=1,…M), with each subpopulation corresponding to a single pair of scaling parameters. For simplicity, initially we assume each of these mutants is present in the original population at low frequency and neglect mutations that could give rise to new phenotypes, though later we show that it is straightforward to incorporate them into the same framework. We do not specify the mechanisms by which this standing variation is initially generated or maintained; instead, we simply assume that such standing variation exists, and our goal is to predict how selection will act on this variation for different choices of external (true) drug concentrations. As we will see, statistical properties of this variation combine with the local geometry of the response surface to determine the selection dynamics of these traits.
Population dynamics of scaling parameters
The mean resistance trait to drug 1, which in our case is the scaling parameter α¯(t)≡∑i=1Mαifi(t), evolves according todα¯dt=∑i=1Mαidfi(t)dt,where fi(t) is the population frequency of mutant i at time t in the population. Assuming that each subpopulation grows exponentially at the per capita growth rate (dni/dt=gini, with ni the abundance of mutant i and gi the per capita growth rate given by Equation 1), the frequency fi(t) changes according todfidt=ddt(ni∑ini)=fi(gi−g¯),where g¯=∑i=1Mfigi is the (time-dependent) mean value of gi across all M subpopulations (mutants). Combining Equations 2 and 3, we havedα¯dt=Covx(α,g),where Cov(α,g)𝒙≡∑i=1Mαifi(gi-g¯) is the covariance between the scaling parameters αi and the corresponding mutant growth rates gi. The subscript 𝒙 is a reminder that the growth rates gi and g¯ that appear in the covariance sum depend on the external (true) drug concentration 𝒙≡(x,y). An identical derivation leads to an analogous equation for the scaling parameter with respect to drug 2, mean susceptibility to drug 2, β¯, relative to the original population, and the full dynamics are therefore described bydα¯dt=Covx(α,g),dβ¯dt=Covx(β,g).
To complete the model described by Equation 5, one must specify the external (true) concentration of each drug (𝒙); a finite set of scaling parameter pairs αi,βi corresponding to all ‘available’ mutations; and an initial condition (α¯(0),β¯(0)) for the mean scaling parameters. When combined with the external drug concentrations, the scaling parameters directly determine the effective drug concentrations (D1eff,D2eff) experienced by each mutant according toD1eff=αix,D2eff=βiy.
We note that drug concentrations can be above or below the MIC contour, with higher concentrations leading to population collapse (G<0) and lower concentrations to population growth (G>0) in the ancestral (drug sensitive) cells. However selection dynamics remain the same in both cases as selection depends only on differences in growth rates between different subpopulations.
Equation 5 is an example of the well-known Price equation from evolutionary biology (Price, 1970; Price, 1972; Frank, 1995; Lehtonen et al., 2020), which says that the change in the (population) mean value of a trait is governed by the covariance of traits and fitness. In general, fitness can be difficult to measure and, in some cases, even difficult to define. However, the rescaling assumption of our model replaces fitness with g, which can be directly inferred from measurable properties (the two-drug-response surface) in the ancestral population. In what follows, we will sometimes casually refer to Equation 5 as a ‘model,’ but it is important to note that the Price equation is not, in and of itself, a mathematical model in the traditional sense. Instead, it is a simple mathematical statement describing the statistical relationship between variables, which are themselves defined in some underlying model. In this case, the mathematical model consists of a collection of exponentially growing populations whose per capita growth rates are linked by scaling relationships. Equation 5– the Price equation – does not include additional assumptions, mechanistic or otherwise, but merely captures statistical relationships between the model variables. We will see, however, that these relationships provide conceptual insight into the interplay between collateral effects and drug interactions.
Equation 5 encodes deceptively rich dynamics that depend on both the interaction between drugs and the collateral effects of resistance. First, it is important to note that αs and βs vary together in pairs, and the evolution of these two traits is not independent. As a result, constraints on the joint co-occurrence of αi and βi among the mutant subpopulations can significantly impact the dynamics. These constraints correspond to correlations between resistance levels to different drugs – that is, to cross-resistance (when pairs of scaling parameters simultaneously increase resistance to both drugs) or to collateral sensitivity (when one scaling parameter leads to increased resistance and the other to increased sensitivity). In addition, g contains information about the dose-response surface and, therefore, about the interaction between drugs. The evolution of the scaling parameters is not determined solely by the drug interaction or by the collateral effects, but instead by both – quantified by the covariance between these rescaled trait values and the ancestral dose-response surface.
As an example, we integrated the model numerically to determine the dynamics of the mean scaling parameters and the mean growth rate for a population exposed to a fixed concentration of two drugs whose growth surface has been fully specified (Figure 2A). The dynamics can be thought of as motion on the two-dimensional response surface; if the initial population is dominated by the ancestral cells, the mean scaling parameters are approximately 1, and the trajectory therefore starts near the point representing the true drug concentration (in this case, (x0,y0)), where growth is typically small (Figure 2A). Over time, the mean traits evolve, tracing out a trajectory in the space of scaling parameters (Figure 2B). When the external concentration of drug is specified, these dynamics also correspond to a trajectory through the space of effective drug concentrations, which, in turn, can be linked with an average growth rate through the drug-response surface (Figure 2B). The model therefore describes both the dynamics of the scaling parameters, which describe how resistance to each drug changes over time (Figure 2C), and the dynamics of growth rate adaptation in the population as a whole (Figure 2D).
Using the rescaling framework for predicting resistance evolution to two drugs in a population of bacterial cells.
(A) Growth landscape (per capita growth rate relative to untreated cells) as a function of drug concentration for two drugs (drug 1, tigecycline; drug 2, ciprofloxacin; concentrations measured in μg/mL) based on measurements in Dean et al., 2020. We consider evolution of resistance in a population exposed to a fixed external drug concentration (x0,y0)=(0.03,0.05) (white asterisk). (B) Growth landscape from (A) with axes rescaled to reflect scaling parameters (α,β). The point (x0,y0) in drug concentration space now corresponds to the point (1,1) in scaling parameter space; intuitively, a strain characterized by scaling parameters of unity experiences an effective drug concentration equal to the true external concentration. We consider a population that is primarily ancestral cells (fraction 0.99), with the remaining fraction uniformly distributed between an empirically measured collection of mutants (each corresponding to a single pair of scaling parameters, white circles). At time 0, the population is primarily ancestral cells and is therefore centered very near (1,1) (gray circle). Over time, the mean value of each scaling parameter decreases as the population becomes increasingly resistant to each drug (black curve). (C) The two-dimensional mean scaling parameter trajectory from (B) plotted as two time series, one for α (red) and one for β (blue). For the detailed selection dynamics, see Figure 1—video 1. (D) The mean fitness of the population during evolution is expected to increase and can be computed from the selection dynamics by numerically integrating the model at each time step.
Selection dynamics depend on drug interaction and collateral effects
This rescaling model indicates that selection is determined by both the drug interaction and the collateral effects, consistent with previous experimental findings. As a simple example, consider the case of a fixed drug interaction but different qualitative types of collateral effects – that is, different statistical relationships between resistance to drug 1 (via α) and resistance to drug 2 (via β). In Figure 3A, we consider cases where resistance is primarily to drug 2 (black), primarily to drug 1 (cyan), strongly positively correlated (cross-resistance, pink), and strongly negatively correlated (collateral sensitivity, green). Using the same drug interaction surface as in Figure 2, we find that a mixture of both drugs leads to significantly different trajectories in the space of scaling parameters (Figure 3B) and, in turn, significantly different rates of growth adaptation. In this example, cross-resistance (pink) leads to rapid increases in resistance to both drugs (rapid decreases in scaling parameters, Figure 3C, D) and the fastest growth adaptation (Figure 3E). By contrast, if resistance is limited primarily to one drug (cyan or black), growth adaptation is slowed (Figure 3E) – intuitively, purely horizontal or purely vertical motion in the space of scaling parameters leads to only a modest change in growth because the underlying response surface is relatively flat in those directions, meaning that the rescaled concentration, in each case, lies near the original contour (Figure 3). As a result of the contour shapes (drug interaction), resistance to both drugs can develop at approximately the same rate (for example), even when collateral structure suggests resistance to one drug will dominate (e.g., cyan case in Figure 3). We note that the dynamics will depend on the (true) external drug concentrations (x0,y0), even if the rescaling parameters remain identical, because a given rescaling transformation will lead to different effective drug concentrations, and therefore different growth rates, for different values of (x0,y0).
Different collateral profiles drive different evolutionary dynamics under the same treatment.
We simulated random collateral profiles for susceptibilities to two drugs and use them to predict phenotypic trajectories of a bacterial population. For the two-drug landscape, we chose one of the experimentally measured surfaces from Dean et al., 2020; Figure 3—figure supplement 1C, corresponding to the drug combination TGC-CIP (drug 1-drug 2). (A) Four special profiles include predominant variation in β (black), predominant variation in α (cyan), positive correlation between αs and βs (pink), and negative correlation between the susceptibilities to the two drugs (green). These were generated from a bivariate normal distribution with mean (1,1) and covariance matrices Σ1=(0.010.020.020.4), Σ2=(0.50.020.020.01), Σ3=(0.30.210.210.3), Σ4=(0.3-0.15-0.150.3). (B) The trajectories in mean α-β space following treatment (x0,y0), with x0=0.5xmax=0.025 and y0=0.5ymax=0.25, corresponding to different collateral profiles. (C) The dynamics of mean susceptibility to drug 1 α¯(t) for the four cases. (D) The dynamics of mean susceptibility to drug 2 β¯(t) for the four cases. (E) The dynamics of mean growth rate for the four cases. For underlying heterogeneity, we drew 100 random αi and βi as shown in (A) and initialized dynamics at ancestor frequency 0.99 and the remaining 1% evenly distributed among available mutants. It is clear that each collateral structure in terms of the available (αi,βi) leads to different final evolutionary dynamics under the same two-drug treatment. In this particular case, the fastest increase in resistance to two drugs and increase in growth rate occurs for the collateral resistance (positive correlation) case. The time course of the detailed selective dynamics in these four cases is depicted in Figure 3—videos 1–4.
Response surfaces and scaling parameters for three drug pairs.
Growth rate surfaces (left) and scaling parameters α and β for three drug combinations: ampicillin (AMP)-streptomycin (STR) (A); ceftriaxone (CRO)-ciprofloxacin (CIP) (B); and ciprofloxacin (CIP)-tigecycline (TGC) (C). Large color-coded circles on the left panels indicate external concentrations at which selection took place. Circles on the right panels indicate all observed mutants (αi,βi) that emerged, pooled across conditions. Data from Dean et al., 2020. Growth rate is measured in normalized units where growth of the drug-free ancestral cells (approximately 1 hr−1) is set to 1.
Evolutionary dynamics for collateral effects in <xref ref-type="fig" rid="fig3">Figure 3A</xref>.
Trajectory of mean traits in rescaling factor space (α,β) for case A in Figure 3 where the randomly generated available standing variation is higher in β (resistance to drug 2) and much lower in α (resistance to drug 1). The selective regime in drug dosage is (x0,y0)=0.5(xmax,ymax). The size of the circles at t=0,... represents the relative frequencies of different mutant subpopulations available at the beginning and changing over time.
Evolutionary dynamics for collateral effects in <xref ref-type="fig" rid="fig3">Figure 3B</xref>.
Trajectory of mean traits in rescaling factor space (α,β) for case A in Figure 3 where the randomly generated available standing variation is lower in β and higher in α. The selective regime in drug dosage is (x0,y0)=0.5(xmax,ymax). The dynamic size of the circles represents the relative frequencies of different mutant subpopulations available at the beginning and changing over time.
Evolutionary dynamics for collateral effects in <xref ref-type="fig" rid="fig3">Figure 3C</xref>.
Trajectory of mean traits in rescaling factor space (α,β) for case A in Figure 3 where the randomly generated available standing variation in β correlates positively with that in α (collateral resistance between the two drugs TGC-CIP). The selective regime in drug dosage is (x0,y0)=0.5(xmax,ymax). The dynamic size of the circles represents the changing relative frequencies of different mutant subpopulations.
Evolutionary dynamics for collateral effects in <xref ref-type="fig" rid="fig3">Figure 3B</xref>.
Trajectory of mean traits in rescaling factor space (α,β) for case A in Figure 3 where the randomly generated available standing variation in β correlates negatively with that in α (collateral sensitivity between TGC and CIP). The selective regime in drug dosage is (x0,y0)=0.5(xmax,ymax). The size of the circles represents the relative frequencies of different mutant subpopulations available at the beginning.
When both collateral effects and drug interactions vary, the dynamics can be considerably more complex, and the dominant driver of adaptation can be drug interaction, collateral effects, or a combination of both. Previous studies support this picture as adaptation has been observed to be driven primarily by drug interactions (Chait et al., 2007; Michel et al., 2008; Hegreness et al., 2008), primarily by collateral effects (Munck et al., 2014; Barbosa et al., 2018), or by combinations of both (Baym et al., 2016b; Barbosa et al., 2018; Dean et al., 2020). Figure 4 shows schematic examples of growth rate adaptation for different types of collateral effects (rows, ranging from cross-resistance [top] to collateral sensitivity [bottom]) and drug interactions (columns, ranging from synergy [left] to antagonism [right]). The growth adaptation may also depend sensitively on the external environment – that is, on the true external drug concentration (blue, cyan, and red). In the absence of both drug interactions (linear isoboles) and collateral effects (uncorrelated scaling parameters), adaptation is slower when the drugs are used in combination than when they are used alone, consistent with the fact that adaptation to multiple stressors is expected to take longer than adaptation to each stressor alone (Figure 4; middle row, middle column). As in Figure 3, modulating collateral effects with drug interaction fixed can have dramatic impacts on adaptation (Figure 4, columns). On the other hand, modulating the drug interaction in the absence of collateral effects will also significantly impact adaptation, with synergistic interactions leading to accelerated adaptation relative to other types of drug interaction (Figure 4, middle row; compare green curves across row). Similar interaction-driven adaptation has been observed in multiple bacterial species (Chait et al., 2007; Michel et al., 2008; Hegreness et al., 2008; Dean et al., 2020).
Adaptation depends on drug interactions and collateral effects of resistance.
Drug interactions (columns) and collateral effects (rows) modulate the rate of growth adaptation (main nine panels). Evolution takes place at one of three dosage combinations (blue, drug 2 only; cyan, drug combination; red, drug 1 only) along a contour of constant growth in the ancestral growth response surface (bottom panels). Drug interactions are characterized as synergistic (left), additive (center), or antagonistic (right) based on the curvature of the isogrowth contours. Collateral effects describe the relationship between the resistance to drug 1 and the resistance to drug 2 in an ensemble of potential mutants. These resistance levels can be positively correlated (top row, leading to collateral resistance), uncorrelated (center row), or negatively correlated (third row, leading to collateral sensitivity). Growth adaptation (main nine panels) is characterized by growth rate over time, with dashed lines representing evolution in single drugs and solid lines indicating evolution in the drug combination. In this example, response surfaces are generated with a classical pharmacodynamic model (symmetric in the two drugs) that extends Loewe additivity by including a single-drug interaction index that can be tuned to create surfaces with different combination indices (Greco et al., 1995). The initial population consists of primarily ancestral cells (α=β=1) along with a subpopulation 10−2 mutants uniformly distributed among the different phenotypes. Scaling parameters are sampled from a bivariate normal distribution with equal variances (σα=σβ) and correlation ranging from 0.6 (top row) to −0.6 (bottom row). See also Figure 4—figure supplement 1 for gradient approximation to selection dynamics.
Illustration of the local growth gradient approximation to the Price equation.
(A, B) Comparison between numerical solutions to the full Price equation (dotted line), the local growth gradient approximation (solid line), and the trajectory of pure (unweighted) gradient descent (dashed line) for mean resistance evolution to two drugs (α¯(t),β¯(t)). The dotted and solid lines are almost indistinguishable. In (A) and (B), we assume two growth landscapes respectively (drug combination 1 vs. drug combination 3, see Figure 3—figure supplement 1A, C), external concentrations (x0,y0)=0.4×(xmax,ymax), and initial trait distributions following the empirically derived (αi,βi). The large black dot represents the initial value (α¯(0),β¯(0)) under initial ancestor frequency of 90% and the remaining 10% spread uniformly among available mutants (denoted via the small black asterisks in the 2-D trait space). (C, D) Root mean square error between mean trait evolution calculated with the full iteration of the Price equation and the gradient approximation (for the two cases in A and B, respectively), varying ancestor initial frequency. Here for each p∈(0.5,1) (the initial frequency of the ancestor subpopulation with scaling parameters α=β=1), we simulated 20 stochastic realizations of the evolutionary process under drug dosage (x0,y0), starting from different random frequencies for the remaining 1-p proportion of the population composed of available mutants.
Selection as weighted gradient dynamics on the ancestral response surface
To gain intuition about the dynamics in the presence of both collateral effects and drug interactions, we consider an approximately monomorphic population where scaling parameters are initially narrowly distributed around their mean values. In this case, we can Taylor expand the function gi=G(αix0,βiy0) (corresponding to the growth of mutant i) around the mean values α¯, β¯, leading togi≈G(α¯x0,β¯y0)+∂xG(x,y)|α¯x0,β¯y0(αi−α¯)x0+∂yG(x,y)|α¯x0,β¯y0(βi−β¯)y0,where we have neglected terms quadratic and higher. In this regime, gi is a linear function of the scaling parameters, and the covariances can therefore be written asCovx0(α,g)=σααx0∂xG+σαβy0∂yG,Covy0(β,g)=σαβx0∂xG+σββy0∂yG,where σuv=uv¯-u¯v¯ and we have used the fact that g¯≡∑ifiG(αix0,βix0)=G(α¯x0,β¯y0) to first order. This is a type of weak-selection approximation: the trait that is evolving may have a very strong effect on fitness, but if there is only minor variation in such trait in the population, there will only be minor differences in fitness. Equation 5 for the rate of change in mean traits therefore reduces todα¯dt=Σ∇Gwhere 𝜶¯ is a vector of mean scaling factors with components α¯ and β¯, 𝚺 is a covariance-like matrix given byΣ=(σαασαβσαβσββ)and ∇G is a weighted gradient of the function G(x,y) evaluated at the mean scaling parameters,∇G=(x0∂xG(x,y)y0∂yG(x,y))|x=x0α¯,y=y0β¯=(∂αG(x0α,y0β)∂βG(x0α,y0β))|α=α¯,β=β¯.
Equation 9 provides an accurate description of the full dynamics across a wide range of conditions (Figure 4—figure supplement 1) and has a surprisingly simple interpretation. Adaptation dynamics are driven by a type of weighted gradient dynamics on the response surface, with the weighting determined by the correlation coefficients describing resistance levels to the two drugs. In the absence of collateral effects – for example, when 𝚺 is proportional to the identity matrix – the scaling parameters trace out trajectories of steepest ascent on the two-drug-response surface. That is, in the absence of constraints on the available scaling parameters, adaptation follows the gradient of the response surface to rapidly achieve increased fitness, and because the response surface defines the type of drug interaction, that interaction governs the rate of adaptation. On the other hand, collateral effects introduce off-diagonal elements of 𝚺, biasing trajectories away from pure gradient dynamics to account for the constraints of collateral evolution.
Model predicts experimentally observed adaptation of growth and resistance
Our model makes testable predictions for adaptation dynamics of both the population growth rate and the population-averaged resistance levels to each drug (i.e., the mean scaling parameters) for a given drug-response surface, a given set of available mutants, and a specific combination of (external) drug dosages. To compare predictions of the model with experiment, we solved Equation 5 for the 11 different dosage combinations of tigecycline (TGC) and ciprofloxacin (CIP) used to select drug-resistance mutants in Dean et al., 2020. We assumed that the initial population was dominated by ancestral cells (α=β=1) but also included a subpopulation of resistant mutants whose scaling parameters were uniformly sampled from those measured across multiple days of the evolution (see Materials and methods). The model predicts different trajectories for different external doses (selective regimes: red to blue, Figure 5, top panel), leading to dramatically different changes in resistance (IC50) and growth rate adaptation (Figure 5, bottom panels), similar to those observed in experiment. Specifically, the model predicts (and experiments confirm) a dramatic decrease in CIP resistance as TGC concentration is increased along a contour of constant growth. As TGC eclipses acritical concentration of approximately 0.025 g/mL, selection for both CIP resistance and increased growth is eliminated. We note that comparisons between the model and experiment involve no adjustable parameters, and while the model captures the qualitative trends, it slightly but systematically underestimates the growth across TGC concentrations – perhaps suggesting additional evolutionary dynamics not captured by the model. Intuitively, the experimental results can be explained by the strongly antagonistic interaction between the two drugs, which reverses the selection for resistance to CIP at sufficiently high TGC concentrations (compare CIP resistance at TGC = 0 and TGC = 0.04 g/mL, which both involve CIP ≈ 0.2 g/mL; similar results have been seen in other species and for other drug combinations; Chait et al., 2007). We also compared model predictions with experimental adaptation to two additional drug pairs (Figure 5—figure supplements 1 and 2) and again found that the model captures the qualitative features of both resistance changes to the component drugs and growth adaptation.
Drug rescaling describes adaptation dynamics of growth and drug resistance in <italic>Enterococcus faecalis</italic> under different selective regimes.
(A) Experimentally measured growth response surface for tigecycline (TGC) and ciprofloxacin (CIP). Circles represent 11 different adaptation conditions, each corresponding to a specific dosage combination (x0,y0). Solid lines show 100 simulated adaptation trajectories (i.e., changes in effective drug concentration over time: mean D1eff D2eff, for a total time T) predicted from the full rescaling model. Black xs indicate the mean (across all trajectories) value of the effective drug concentration at the end of the simulation. In each case, the set of available mutants – and hence, the set of possible scaling parameters αi and βi – is probabilistically determined by uniformly sampling approximately 10 scaling parameter pairs from the ensemble experimentally measured across parallel evolution experiments involving this drug pair (see Materials and methods). (B) Half-maximal inhibitory concentration (IC50, normalized by that of the ancestral strain) for populations adapted for a fixed time period T≈72 hr in each of the 11 conditions in the top panel. The solid curve is the mean trajectory from the rescaling model (normalized IC50 corresponds to the reciprocal of the corresponding scaling constant) for 100 simulated samples of the scaling parameter pairs; shaded region is ±1 standard deviation over all trajectories. Small circles are experimental observations from individual populations; larger markers are average experimental results over all populations adapted to a given condition. Note that drug 1 (TGC) concentration (x-axis) here is just a proxy for the different conditions as you move left to right along the contour in panel (A). (C) Change in population growth rate (gm-ga, where gm is per capita growth of ancestral strain and ga for the mutant; all growth rates are normalized to ancestral growth in the absence of drugs) for populations adapted in each condition. Solid curve is the mean trajectory across samplings, with shaded region ±1 standard deviation. Small circles are results from individual populations; larger markers are averages over all populations adapted to a given condition. Experimental data from Dean et al., 2020. See also Figure 5—figure supplements 1–5.
Model captures qualitative features of experimental evolution in combinations of ampicillin (AMP) and streptomycin (STR).
Top panel: experimentally measured growth response surface for AMP and STR. Circles represent four different adaptation conditions, each corresponding to a specific dosage combination (x0,y0). Solid lines show adaptation trajectories (i.e., changes in effective drug concentration over time) predicted from the Price equation over an interval t = 48 hr. In each case, the set of available mutants – and hence, the set of possible scaling parameters α and β – is determined by sampling from experimentally measured changes in resistance (half-maximal inhibitory concentration IC50) for each drug in the full ensemble of mutants arising from adaptation to these drugs (see Figure 3—figure supplement 1). Bottom panels: comparing results from the model (left column) and experiment (right panel). Quantities include relative change in IC50 for each drug (top two rows) as well as overall change in population growth (bottom row); in each case, results are shown for each of the four conditions in the top panel. For the experiments, small markers are results from individual populations; larger markers are averages over all populations adapted to a given condition. For the model, small markers are results from each subsampling of the experimental scaling parameters (approximately 10 per run); larger markers are averages over all subsamples. Experimental data from Dean et al., 2020.
Model captures qualitative features of experimental evolution in combinations of ceftriaxone (CRO) and ciprofloxacin (CIP).
Top panel: experimentally measured growth response surface for CRO and CIP. Circles represent four different adaptation conditions, each corresponding to a specific dosage combination (x0,y0). Solid lines show adaptation trajectories (i.e., changes in effective drug concentration over time) predicted from the Price equation over an interval t = 48 hr. In each case, the set of available mutants – and hence, the set of possible scaling parameters α and β – is determined by sampling from experimentally measured changes in resistance (half-maximal inhibitory concentration IC50) for each drug in the full ensemble of mutants arising from adaptation to these drugs (see Figure 3—figure supplement 1). Bottom panels: comparing results from the model (left column) and experiment (right panel). Quantities include relative change in IC50 for each drug (top two rows) as well as overall change in population growth (bottom row); in each case, results are shown for each of the four conditions in the top panel. For the experiments, small markers are results from individual populations; larger markers are averages over all populations adapted to a given condition. For the model, small markers are results from each subsampling of the experimental scaling parameters; larger markers are averages over all subsamples. Experimental data from Dean et al., 2020.
Resampling ensemble of scaling parameters does not dramatically impact dynamics.
(A) Experimentally measured growth response surface for tigecycline (TGC) and ciprofloxacin (CIP) from Dean et al., 2020. Circles represent 11 different adaptation conditions, each corresponding to a specific dosage combination (x0,y0). Solid lines show the mean adaptation trajectories (i.e., changes in effective drug concentration over time) predicted from the Price equation. In each case, the set of available mutants – and hence, the set of possible scaling parameters α and β – is determined by sub-sampling M pairs from experimentally measured scaling parameters (from a total of 72 pairs). Different curves are averages over 100 realizations (samples) of M=72 (darkest), M=36, M=24, M=12, M=7, M=4, and M=2 (lightest) αi-βi pairs. In each realization, the ancestor frequency (α=β=1) was fixed at 99% and the remaining 1% was randomly spread among the available mutants. (B) Predicted change in IC50 for populations adapted in each of the 11 conditions in (A). (C) Predicted change in population growth rate for populations adapted in each condition in (A).
Different estimates of total time lead to similar qualitative dynamics.
(A) Experimentally measured growth response surface for tigecycline (TGC) and ciprofloxacin (CIP) from Dean et al., 2020. Circles represent 11 different adaptation conditions, each corresponding to a specific dosage combination (x0,y0). Solid lines show the mean adaptation trajectories (i.e., changes in effective drug concentration over time) predicted from the Price equation. In each case, the set of available mutants – and hence, the set of possible scaling parameters α and β – is determined by sampling M=12 pairs (αi,βi) from experimentally measured scaling parameters (from a total of 72 pairs). Different curves are averages over 100 realizations (samples) for total simulation times ranging from T=12 to T=84 hr. In each realization, the ancestor frequency (α=β=1) was fixed at 99% and the remaining 1% was randomly spread among the available mutants. (B) Predicted change in IC50 for simulated populations adapted in each of the 11 conditions in (A). (C) Predicted change in population growth rate for populations adapted in each condition in (A). In panels (B) and (C), the horizontal axis (TGC) is the external concentration of tigecycline (μg/mL) used in each selecting condition in (A).
Different estimates of initial resistant fraction lead to similar qualitative dynamics.
(A) Experimentally measured growth response surface for tigecycline (TGC) and ciprofloxacin (CIP) from Dean et al., 2020. Circles represent 11 different adaptation conditions, each corresponding to a specific 2-drug dosage combination (x0,y0). Solid lines show the mean adaptation trajectories (i.e., changes in effective drug concentration over time) predicted from the Price equation. In each case, the set of available mutants – and hence, the set of possible scaling parameters α and β – is determined by sampling M=12 pairs from experimentally measured scaling parameters (from a total of 72 pairs). Different curves are averages over 100 realizations (samples) for initial mutant fractions ranging from p=10-1 to p=10-6 hr. In each realization, the dynamics were simulated for a total time of T=72 hr. (B) Predicted change in IC50 for simulated populations adapted in each of the 11 conditions in (A). (C) Predicted change in population growth rate for populations adapted in each condition in (A). In panels (B) and (C), the horizontal axis (TGC) is the external concentration of tigecycline (μg/mL) used in each selecting condition in (A).
Effects of mutations
While we have focused on selection dynamics, the model can be extended to include mutation events linking different phenotypes (i.e., linking different pairs of scaling parameters). These mutational pathways may be necessary to capture certain evolutionary features – for example, historical contingencies between sequentially acquired mutations (Barbosa et al., 2017; Card et al., 2019; Das et al., 2020). To incorporate mutational structure and processes, we modify the equation for each subpopulation ni todnidt=gini−μni+μ∑jMmjinjwhere μ is the mutation rate and mji is the probability that strain j mutates to strain i, given that there is a mutation in strain j. We will refer to the matrix formed by the parameters mji as the mutation matrix as it contains information about allowed mutational trajectories linking different phenotypes (Day and Gandon, 2006). The Price equation for the mean traits (the mean scaling parameters) now becomesdα¯dt=Covx(α,g)−μ(α¯−α¯m),dβ¯dt=Covx(β,g)−μ(β¯−β¯m),where α¯m and β¯m are the mean trait values in all mutants that arise, and are given byα¯m=∑ijαimjifj,β¯m=∑ijβimjifj.ll
When the mutation matrix has a particularly simple structure, the effects of mutation will be similar to those of selection. For example, when mutations occur with uniform probability between the ancestral strain and each mutant, the evolutionary dynamics are qualitatively similar to those in the absence of mutation (Figure 6—figure supplement 1, compare to Figure 4). On the other hand, certain mutational structures can lead to new behavior. As an example, we consider a toy model consisting of four phenotypes defined by the level of resistance (sensitive, S, or resistant, R) to one of two drugs. The phenotypes are designated SS (α=1,β=1), RS (α=1/5,β=1), SR (α=1,β=1/5), or RR (α=1/5,β=1/5). Note that these phenotypes do not assume, a priori, any particular relationships between the underlying genotypes. For example, each phenotype could correspond to a single mutation – in which case the RR phenotype would be said to exhibit strong cross-resistance – or alternatively, the phenotypes could correspond to single drug mutants (SR and RS) and double mutants (RR), which implies a particular sequence in which the mutations can be acquired. These two situations would lead to significant differences in the expected structure of the mutational matrix and, as we will see, in the evolutionary dynamics.
For simplicity, we consider two different mutational structures: one corresponding to direct and uniform pathways between the ancestor (SS) and all mutant phenotypes (SR, RS, and RR), and the second corresponding to sequential pathways from ancestor to single mutants (SR, RS) and then from single mutants to double mutants (Figure 6). To illustrate the impact of mutational structure, we consider two drugs that interact antagonistically and choose the external drug concentrations such that D1>D2 (Figure 6C). While both mutational structures lead eventually to a population of double-resistant phenotypes (RR), the sequential pathway leads to slower adaptation of growth as the population evolves first toward the most fit single mutant (in this case, RS because the drug combination contains a higher concentration of drug 1) before eventually arriving at RR (Figure 6C–E). The specific trajectory will of course depend on both the drug interaction (the shape of the growth contours) and the specific dosage combination (see Figure 6—figure supplement 2).
(A) Evolutionary dynamics reflect a combination of collateral effects, drug interactions, and mutational structure. (B) Phenotypes are defined by relative resistance (R) or sensitivity (S) to each of two drugs, leading to four possible phenotypes: SS (α=1,β=1), RS (α=1/5,β=1), SR (α=1,β=1/5), or RR (α=1/5,β=1/5). Mutations can occur via direct paths (B), where each mutant phenotype is accessible directly from the fully sensitive ancestor, or via sequential paths (C), where the doubly resistant mutant (RR) is only accessible from single mutants (RS or SR). Entries of the mutation matrix mji (heatmaps) reflect the probability of mutating from state j (rows) to state i (columns) given that a mutation occurs in state j. (D) Main panel: contour plot of growth surface in ancestral (SS) cells. For a specific choice of drug dosages, SS cells experience the true dosage combination (D1≈2.6,D2≈0.6; blue circle, SS) while single mutants (black x, RS; and black square, SR) and double-resistant mutants (RR) experience decreased effect concentration of one or more drugs. Dashed lines show mean trajectories (α¯(t),β¯(t)) for direct (red) and sequential (blue) mutational structures. Inset: mean growth rate for direct (red) and sequential (blue) mutational structures. (E, F) Population frequencies for single mutants (panel D; xs are RS, squares are SR) and double mutants (panel E) under direct (red) or sequential (blue) mutational structures. See also Figure 6—figure supplements 1–4.
Adaptation in the presence of mutation depends on drug interaction and collateral effects of resistance.
Similar to Figure 4, but with initial population composed entirely of ancestral cells and an assumed mutation rate μ=10-2. The mutation matrix allows mutations to occur with equal probability from the WT to any of the available mutants. Drug interaction (columns) and collateral effects (rows) modulate the rate of growth adaptation (main nine panels). Evolution takes place at one of three dosage combinations (blue, drug 2 only; cyan, drug combination; red, drug 1 only) along a contour of constant growth in the ancestral growth response surface (bottom panels). Drug interactions are characterized as synergistic (left), additive (center), or antagonistic (right) based on the curvature of the isogrowth contours. Collateral effects describe the relationship between the resistance to drug 1 and the resistance to drug 2 in an ensemble of potential mutants. These resistance levels can be correlated (top row, leading to collateral resistance), uncorrelated (center row), or anticorrelated (third row, leading to collateral sensitivity). Growth adaptation (main nine panels) is characterized by growth rate over time, with dashed lines representing evolution in single drugs and solid lines indicating evolution in the drug combination. In this example, response surfaces are generated with a classical pharmacodynamic model that extends Loewe additivity by including a single-drug interaction index that can be tuned to create surfaces with different combination indices (Greco et al., 1995).
Mutational structure modulates evolutionary dynamics for additive and synergistic drug combinations.
Top and bottom panels are identical to Figure 6 but with additive (top) and synergistic (bottom) drug interactions. Phenotypes are defined by resistance (R) or sensitivity (S) to each of two drugs, leading to four possible phenotypes: SS (α=1,β=1), RS (α=1/5,β=1), SR (α=1,β=1/5), or RR (α=1/5,β=1/5). Mutations can occur via direct paths, where each mutant phenotype is accessible directly from the fully sensitive ancestor, or via sequential paths, where the doubly resistant mutant (RR) is only accessible from single mutants (RS or SR). Main left panel: contour plot of growth surface in ancestral (SS) cells. For a specific choice of drug dosages, SS cells experience the true dosage combination (D1≈2.6,D2≈0.6; blue circle, SS) while single mutants (black x, RS; and black square, SR) and double-resistant mutants (RR) experience decreased effect concentration of one or more drugs. Dashed lines show mean trajectories (α¯(t),β¯(t)) for direct (red) and sequential (blue) mutational structures. Inset: mean growth rate for direct (red) and sequential (blue) mutational structures. Right panels: population frequencies for single mutants (upper panel; xs are RS, squares are SR) and double mutants (lower panel) under direct (red) or sequential (blue) mutational structures.
Uniform vs. distance-based mutation and evolutionary dynamics for 4 collateral structures.
We generated the same four collateral structures (A–D) for correlations between αi and βi as in Figure 3 and used the same growth surface for drug interactions. Instead of setting initial frequencies as non-zero, here we started simulations with 100% WT, at trait (1,1) and allowed other subpopulations to arise via mutation. We used a mutation rate μ=10-4. Mutations can occur with uniform rates (M1) or via distance-based scalings (M2), where mutant phenotype i is accessible from all mutants j but with higher probability if j is closer in the space of scaling parameters (mji∼e-dij, with dij the Euclidean distance between pairs of scaling parameters in (α,β) space). We summarize 10 stochastic realizations for each collateral structure. Solid lines show mean system trajectories for uniform mutation, whereas dashed lines show mean trajectories of mean traits (α¯(t),β¯(t)) for distance-based mutation.
Stochastic realizations of collateral structures and evolutionary dynamics under two mutation models.
We show here 10 stochastic realizations for each generated collateral structure (like in Figure 6—figure supplement 3, Figure 3) for correlations between αi and βi and used the same growth surface for drug interactions. We started simulations with 100% WT, at trait (1,1) and allowed 100 other subpopulations to arise via mutation. We used a mutation rate μ=10-4. Mutations can occur with uniform rates (M1) (A–C) or via distance-based scalings (M2) (D–F), where mutant phenotype i is accessible from all mutants j but with higher probability if j is closer mji∼e-dij, and the distance metric being, for example, the Euclidean distance in (α,β) space. (A, D) show mean resistance evolution to drug 1. (B, E) show mean resistance evolution to drug 2. (C, F) show mean growth rate dynamics. Different colors indicate different collateral structures, and different lines depict different sampled mutants (αi,βi) from the same distribution.
There are other types of mutational constraints that can be implemented in the formalism, including explicit functional dependencies between different antibiotic resistance phenotypes. As an example, we assumed a distance-based mutation matrix where the mutation probability between different strains depends on the Euclidean distance between their scaling parameter pairs (αi,βi) according to mji∼e-dji, where dji=(αi-αj)2+(βi-βj)2. Intuitively, this structure means that mutations are more likely between strains with similar rescaling parameters (and therefore similar levels of resistance to the component drugs). As expected, this constrained mutational structure leads to slower growth rate adaptation than a model with a simple uniform mutation model (Figure 6—figure supplement 3). These effects are further compounded by statistical properties of the collateral structures (e.g., positive or negative correlations between the possible αs and βs; Figure 3), the specific (stochastic) realization of those scaling parameters (Figure 6—figure supplement 4), and the precise shape of two-drug growth surface. Note, however, that these dynamics depend fundamentally on the global mutation rate μ, which modulates the relative balance between selection and mutation in a given environment.
Evolutionary dynamics under temporal sequences of drug combinations
Evolutionary dynamics in the presence of multiple drugs can be extremely complex. While past work has focused primarily on the use of either temporal sequences or (simultaneous) combinations of antibiotics, more complex scenarios are possible, in principle, but difficult to analyze even in theoretical models. The simplicity of the Price equation framework allows us to investigate evolutionary dynamics in response to a more complex scenario: temporal sequences of antibiotic combinations. As proof of principle, we numerically studied time-dependent therapies consisting of two sequential regimes (treatment A and treatment B; Figure 7) characterized by different dosage combinations of tigecycline (drug 1) and ciprofloxacin (drug 2) (the growth surface was measured in Dean et al., 2020). The two dosage combinations were chosen to lie along a contour of constant growth (Figure 7A), meaning that the net inhibitory effects of A and B are the same when applied to ancestral cells. Using experimental estimates for α,β (Figure 3—figure supplement 1C), we find that both the resistance levels to the two drugs and the growth rate increase during treatment, as one might expect. However, the dynamics of these changes depend on both the relative duration of each treatment and total treatment length (Figure 7—figure supplements 1 and 2). For example, consider a treatment of total length T consisting of an initial period of treatment A followed by a final period of treatment B. If we vary the relative length of the two epochs while keeping the total treatment length fixed, we find that the resistance to each drug and the growth rate increase monotonically as the fraction of time in B increases (Figure 7H–J). This is a consequence of the interplay between the distribution of available mutants and the shape of the growth surface under these particular two drugs; the mutants tend to feature higher levels of resistance to drug 2 than drug 1, yet the benefits of this resistance – that is, the increase in growth rate due to rescaling the concentration of drug 2 – are favored much more so under condition B than condition A (where rescaling tends to produce effective drug concentrations that lie near the original growth contour). It is notable, however, that effects (both resistance levels and final growth) change nonlinearly as the fraction of time in B is increased – that is, even in this simple model, the effects of a two-epoch (A then B) treatment cannot be inferred as a simple linear interpolation between the effects of A-only and B-only treatments.
Sequential multidrug treatments lead to different evolutionary outcomes under different schedules.
The model framework can be used to predict phenotypic evolution and growth dynamics in a bacterial population under consecutive treatments involving different dosage combinations of the same two drugs. (A) Treatments A and B (asterisks) are chosen to lie along a contour of constant growth in the space of drug concentrations. Two-drug growth surface (top panel) and the collection of scaling parameter pairs are based on data from Dean et al., 2020 for tigecycline (drug 1) and ciprofloxacin (drug 2). The same set of scaling parameters leads to different effective drug concentrations when exposed to treatment A (bottom-left panel; each mutant is represented by a single point representing the effective concentration of each drug) and treatment B (bottom-right panel). (B-G) Time series of drug concentrations (B, E), mean scaling parameters (C, F), and mean growth rate (D, G) for sequential treatments (treatment A followed by treatment B). Panels (B–D) correspond to a short epoch (10 time units) of A followed by a longer (20 time units) epoch of B. Panels (E–G) correspond to a long epoch of A followed by a shorter epoch of B. (H–J) Changes in resistance to drug 1 (1/α¯) (H), resistance to drug 2 (1/β¯) (I), and growth rate (J) for treatments of a fixed length (T=10 time units) but varying the fraction of time spent in each treatment. Each treatment consisting of one epoch of A (for time t1) followed by one epoch of B (for time T-t1). See also Figure 7—figure supplement 1 (for detailed temporal trajectories) and Figure 7—figure supplement 2 (for longer total therapies with saturating dynamics). In all panels, we assumed mutants were characterized by scaling parameter pairs (αi and βi) measured experimentally (Figure 3—figure supplement 1). Initial populations consisted of primarily ancestral cells (fraction 0.99), while the remaining population (fraction 0.01) was uniformly distributed among all available mutants.
Detailed temporal trajectories of population mean traits and growth rate under sequential two-drug therapies.
We combined sequentially treatments A and B in Figure 7 in different proportions amounting to a total T=10 hours of therapy. Treatment A combines drugs 1 and 2 in the concentrations (x0,y0)=(0.036,0.25) and B in the concentrations (x0,y0)=(0.02,0.35). Panels show time series of mean growth rate (A) and mean scaling parameters (B-C).
Detailed temporal trajectories of population mean traits and growth rate under sequential two-drug therapies for longer total duration.
We combined sequentially two-drug treatments: treatment A (x0,y0)=(0.036,0.25) and B (x0,y0)=(0.02,0.35) in Figure 7 in different proportions amounting to a total duration of T=30 hr of therapy. As time of therapy is increased (compare to Figure 7—figure supplement 1) and selection proceeds further to select high-fitness mutants, the space of available mutants is exhausted and evolutionary trajectories saturate. Panels show time series of mean growth rate (A) and mean scaling parameters (B-C).
Discussion
Antibiotic resistance is a growing threat to modern medicine and public health. Multidrug therapies are a promising approach to both increase efficacy of treatment and prevent evolution of resistance, but the two effects can be coupled in counterintuitive ways through the drug interactions and collateral effects linking the component drugs. Our results provide a unified framework for incorporating both drug interactions and collateral effects to predict phenotypic adaptation on coarse-grained fitness landscapes that are measurable features of the ancestral population (Table 1). Special cases of the model reproduce previous experimental results that appear, on the surface, to be contradictory; indeed, adaptation can be driven primarily by drug interactions, primarily by collateral effects, or by a combination of both, and the balance of these effects can be shifted using tunable properties of the system (i.e., the ratio of drug dosages). Our model was inspired by rescaling arguments that were originally introduced in Chait et al., 2007 and have since been shown to capture phenotypic features of dose-response surfaces or adaptation in multiple bacterial species (Michel et al., 2008; Hegreness et al., 2008; Torella et al., 2010; Wood and Cluzel, 2012; Wood et al., 2014; Dean et al., 2020; Das et al., 2020). Our results complement these studies by showing how similar rescaling assumptions, when formalized in a population dynamics model, lead to testable predictions for the dynamics of both growth adaptation and phenotype (resistance) evolution. Importantly, the model also has a simple intuitive explanation, with evolutionary trajectories driven by weighted gradient dynamics on two-dimensional landscapes, where the local geometry of the landscape reflects the drug interaction and collateral effects constrain the direction of motion. We have also illustrated how de novo mutation can be integrated in the same framework, provided the mutational structure among a pool of possible phenotypes is known or if specific assumptions are made regarding functional links and constraints for shifts between different resistance levels. Mutation at constant rate is indeed a classic version of the complete Price equation. However, more complex mutational effects – including stress-dependent modulation of mutation rate (Kohanski et al., 2010; Vasse et al., 2020) – could be included as a more flexible tunable term in Equation 5.
Different biological properties correspond to different features of the model.
Biological property
Model property
Scaling parameters (ancestral)
α=β=1
Resistance to drug 1*
Inverse scaling parameter α-1
Resistance to drug 2*
Inverse scaling parameter β-1
Synergistic interaction
Locally convex isoboles in 2-drug growth surface
Antagonistic interaction
Locally concave isoboles in 2-drug growth surface
Cross-resistance
Positively correlated scaling parameters (α,β)
Collateral sensitivity
Negatively correlated scaling parameters (α,β)
External concentration (x,y)
Effective drug concentration (αx,βy)
Fitness cost
Prefactor g→(1-γ)g
Resistance-dependent fitness cost
γ≡γ(α,β)
De novo mutation
Mutation matrix mji
*Fold change in minimum inhibitory concentration or similar inhibitory concentration.
It is important to keep in mind several limitations of our approach. The primary rescaling assumption of the model is that growth of a drug-resistant mutant is well approximated by growth of the ancestral strain at a new ‘effective’ drug concentration – one that differs from the true external concentration. This approximation has considerable empirical support (Chait et al., 2007; Wood and Cluzel, 2012; Wood et al., 2014; Das et al., 2020) but is not expected to always hold; indeed, there are examples where mutations lead to more complex nonlinear transformations of the drug-response surface (Wood et al., 2014; Munck et al., 2014). In addition, it is possible that selection can act on some other feature of the dose-response curve characterizing single-drug effects – modulating, for example, its steepness (rather than merely its scale). While these effects could in principle be incorporated into our model – for example, by assuming transformations of the ancestral surface, perhaps occurring on a slower timescale, that go beyond simple rescaling – we have not focused on those cases. For simplicity, we have also neglected a number of features that may impact microbial evolution. For example, we have assumed that different subpopulations grow exponentially, neglecting potential interactions including clonal interference (Gerrish and Lenski, 1998), intercellular (Koch et al., 2014; Hansen et al., 2017; Hansen et al., 2020) or intra-lineage (Ogbunugafor and Eppstein, 2017) competition, and cooperation (Yurtsev et al., 2013; Sorg et al., 2016; Estrela and Brown, 2018; Frost et al., 2018; Hallinen et al., 2020), as well as potential effects of demographic noise and population extinction (Coates et al., 2018). These complexities could also be incorporated in our model, perhaps at the expense of some intuitive interpretations (e.g., weighted gradient dynamics) that currently apply. In addition, we have not explicitly included a fitness cost of resistance (Andersson and Hughes, 2010) – that is, we assume that growth rates of mutants and ancestral cells are identical in the absence of drug. This assumption could be relaxed by including a prefactor to the growth function, gi→(1-γi(αi,βi))gi, where γi(αi,βi) is the cost of resistance, which in general depends on the scaling parameters (if not, it can be easily incorporated as a constant). While such fitness costs have traditionally been seen as essential for reversing resistance (with, e.g., drug-free ‘holidays’; Dunai et al., 2019), our results underscore the idea that reversing adaptation relies on differential fitness along a multidimensional continuum of environments, not merely binary (drug/no drug) conditions. Our results indicate resistance and bacterial growth can be significantly constrained by optimal tuning of multidrug environments, even in the absence of fitness cost. Finally, our model deals only with heritable resistance and therefore may not capture phenotypic affects associated with, for example, transient resistance (El Meouche and Dunlop, 2018) or cellular hysteresis (Roemhild et al., 2018).
Our goal was to strike a balance between analytical tractability and generality vs. biological realism and system specificity. But we stress that the predictions of this model do not come for free; they depend, for example, on properties of the dose-response surfaces, the collection of scaling parameters, and the specific mutational structure. In many cases, these features can be determined empirically; in other cases, some or all of these properties may be unknown. The evolutionary predictions of the model will ultimately depend on these inputs, and it is difficult to draw general conclusions (e.g., 'synergistic combinations always accelerate resistance') that apply across all classes of drug combinations, collateral effects, and mutational structures. But we believe the framework is valuable precisely because it generates testable predictions in situations that might otherwise seem intractably complex.
Our approach also has pedagogical value as it connects evolutionary effects of drug combinations and collateral effects with well-established concepts in evolutionary biology and population genetics (Price, 1970; Price, 1972; Day and Gandon, 2006). Approximations similar to Equation 9 have been derived previously in quantitative genetics models (Abrams et al., 1993; Taylor, 1996) and other applications of the Price equation (Lehtonen, 2018). While the gradient approximation does not require that the population be monomorphic with rare mutants or a particular form for the phenotype distribution, it does require that the majority of trait values (here scaling parameters) are contained in a regime where G(x,y) is approximately linear; in our case, that linearity arises by Taylor expansion and neglecting higher-order deviations from the population mean. More generally, the direction of evolutionary change in our model is determined by the gradient of the fitness function ∇G evaluated at the mean trait (α¯,β¯); when the gradient vanishes, this point corresponds to a singular point (Waxman and Gavrilets, 2005) or an evolutionarily singular strategy (Geritz et al., 1998). Whether such point may be reached (convergence stability) and how much variance in trait values can be maintained around such point (whether evolutionarily stable [ESS]) depend on other features, such as higher-order derivatives (Otto and Day, 2011; Eshel et al., 1997; Lehtonen, 2018; Smith, 1982; Parker and Smith, 1990). In the case of drug interactions, the fitness landscape in (α,β) space will typically have a single maximum at (0, 0) corresponding to effective drug concentrations of zero. However, whether that point is reachable in general or within a given time frame will depend on the available mutants preexisting at very low frequencies in the population, or on the speed and biases in the mutational process itself, if such mutants are to be generated de novo during treatment. In principle, the model also allows for long-term coexistence between different strains; in that case, the rescaled effective drug concentrations experienced by both strains would fall along a single contour of constant growth. Hence, while variance in the population growth will necessarily decrease over time, variance in the traits (scaling parameters) themselves can change non-monotonically.
The framework is sufficiently flexible to integrate different strands of empirical data, and our results underscore the need for quantitative phenotype data tracking resistance to multiple drugs simultaneously, especially when drug combinations are potentially driving selection dynamics. At an epidemiological level, the dominant approach in describing resistance has been to use fixed breakpoints in MIC and track the percentage of isolates with MIC above (drug-resistant) or below (drug-sensitive) such breakpoint (e.g., ECDC, 2019; Chang et al., 2015). By missing or decoupling patterns of co-occurrence between MICs to different drugs across isolates, this approach remains incomplete for mechanistic predictions. Our framework suggests that going beyond such binary description towards a more continuous and multidimensional phenotype characterization of drug resistance is possible, with applications not just in microbiology but also in the evolutionary epidemiology of drug resistance (Day and Gandon, 2012; Day et al., 2020). In the long run, these advances may yield better and more precise predictions of resistance evolution at multiple scales, and, in turn, optimized treatments that balance short-term inhibitory effects of a drug cocktail with its inseparable, longer-term evolutionary consequences.
Perhaps most importantly, our approach provides a low-dimensional approximation to the high-dimensional dynamics governing the evolution of resistance. In contrast to the classical genotype-centric approaches to resistance, our model uses rescaling arguments to connect measurable traits of resistant cells (scaling parameters) to environment-dependent phenotypes (growth). This rescaling dramatically reduces the complexity of the problem as the two-drug-response surfaces – and, effectively, the fitness landscape – can be estimated from only single-drug dose-response curves. Such coarse-grained models can help extract simplifying principles from otherwise intractable complexity (Shoval et al., 2012; Hart et al., 2015). In many ways, the classical Price equation performs a similar function, revealing links between trait-fitness covariance and selection that, at a mathematical level, are already embedded in simple models of population growth. In the case of multidrug resistance, this formalism reveals that drug interactions and collateral effects are not independent features of resistance evolution, and neither, alone, can provide a complete picture. Instead, they are coupled through local geometry of the two-drug-response surface, and we show how specific dosage combinations can shift the weighting of these two effects, providing a framework for systematic optimization of time-dependent multidrug therapies.
Materials and methodsEstimating scaling parameters from experimental dose-response curves
The scaling parameters for a given mutant can be directly estimated by comparing single-drug dose-response curves of the mutant and ancestral populations. To do so, we estimate the half-maximal inhibitory concentration (Ki) for each population by fitting the normalized dose-response curve to a Hill-like function gi(d)=(1+(d/Ki)h)-1, with gi(d) the relative growth at concentration d and h a Hill coefficient measuring the steepness of the dose-response curve using nonlinear least-squares fitting. The scaling parameter for each drug is then estimated as the ratio of the Ki parameters for the ancestral and mutant populations. For example, an increase in resistance corresponds to an increase in Ki for the mutant population relative to that of the ancestor, yielding a scaling parameter of less than 1. Estimates for the scaling parameters for the three drug combinations used here are shown in Figure 3—figure supplement 1 (from data in Dean et al., 2020).
While it is straightforward to estimate the scaling parameters for any particular isolate, it is not clear a priori which isolates are present at time 0 of any given evolution experiment. To compare predictions of our model with lab evolution experiments, we first estimated scaling parameters for all isolates collected during lab evolution experiments in each drug pair (Dean et al., 2020). This ensemble includes 50–100 isolates per drug combination and includes isolates collected at different timepoints during the evolution (after days 1, 2, or 3) as well as isolates selected in different dosage combinations Figure 3—figure supplement 1. We then randomly sampled from this ensemble to generate low-level standing diversity (on average, approximately 10 distinct pairs of scaling parameters) at time 0 for each simulation of the model, and we repeated this subsampling 100 times to generate an ensemble of evolutionary trajectories for each condition.
The results of the simulation can, in principle, depend on how these scaling parameters are sampled. While the qualitative differences between simulations do not depend heavily on this choice of subsampling in the data used here (Figure 5—figure supplement 3), one can imagine scenarios where details of the subsampling significantly impact the outcome. Similarly, precise comparison with experiment requires accurate estimates for the total evolutionary time and for the initial frequency of all resistant mutants, though for these data the qualitative results do not depend sensitively on these choices (Figure 5—figure supplement 4 and Figure 5—figure supplement 5). We stress that these are not fundamental limitations of the model itself, but instead arise because we do not have a precise measure of the standing variation characterizing these particular experiments. In principle, a more accurate ensemble of scaling parameters could be inferred from cleverly designed fluctuation tests (Luria and Delbrück, 1943) or, ideally, from high-throughput, single-cell phenotypic studies (Baltekin et al., 2017). At a theoretical level, subsampling could also be modulated to simulate the effects of different effective population sizes, with standing diversity expected to be significantly larger for large populations.
Acknowledgements
This work was supported by Fundação Luso-Americana para o Desenvolvimento (FLAD/NSF grant-274/2016 to EG) and Instituto Gulbenkian de Ciência, the National Science Foundation (NSF No. 1553028 to KW), and the National Institutes of Health (NIH No. 1R35GM124875 to KW). The Center for Stochastic and Computational Mathematics is supported by FCT via UIDB/04621/2020 and UIDP/04621/2020.
Additional informationCompeting interests
No competing interests declared
Author contributions
Conceptualization, Formal analysis, Investigation, Methodology, Writing - original draft, Writing - review and editing
Conceptualization, Formal analysis, Investigation, Methodology, Writing - original draft, Writing - review and editing
Additional filesData availability
Data used in this paper was taken from a public repository: Dean, Ziah; Maltas, Jeff; Wood, Kevin (2020), Antibiotic interactions shape short-term evolution of resistance in Enterococcus faecalis, Dryad, Dataset, https://doi.org/10.5061/dryad.j3tx95x92 There are no restrictions on any new results.
The following previously published dataset was used:
ZiahDMaltasJWoodK2020Data from: Antibiotic interactions shape short-term evolution of resistance in Enterococcus faecalisDryad Digital Repository10.5061/dryad.j3tx95x92
ReferencesAbramsPAHaradaYMatsudaH1993On the relationship between quantitative genetic and ess modelsEvolution4798298510.2307/2410204AdamowiczEMFlynnJHunterRCHarcombeWR2018Cross-feeding modulates antibiotic tolerance in bacterial communitiesThe ISME Journal122723273510.1038/s41396-018-0212-z29991761AdamowiczEMMuzaMChacónJMHarcombeWR2020Cross-feeding modulates the rate and mechanism of antibiotic resistance evolution in a model microbial community of Escherichia coli and Salmonella entericaPLOS Pathogens16e100870010.1371/journal.ppat.100870032687537AnderssonDIHughesD2010Antibiotic resistance and its cost: is it possible to reverse resistance?Nature Reviews Microbiology826027110.1038/nrmicro231920208551ApjokGBorossGNyergesÁFeketeGLázárVPappBPálCCsörgőB2019Limited evolutionary conservation of the phenotypic effects of antibiotic resistance mutationsMolecular Biology and Evolution361601161110.1093/molbev/msz10931058961ArdellSMKryazhimskiyS2020The population genetics of pleiotropy, and the evolution of collateral resistance and sensitivity in BacteriabioRxiv10.1101/2020.08.25.267484ArtemovaTGerardinYDudleyCVegaNMGoreJ2015Isolated cell behavior drives the evolution of antibiotic resistanceMolecular Systems Biology1182210.15252/msb.2014588826227664BaltekinÖBoucharinATanoEAnderssonDIElfJ2017Antibiotic susceptibility testing in less than 30 min using direct single-cell imagingPNAS1149170917510.1073/pnas.170855811428790187BarbosaCTreboscVKemmerCRosenstielPBeardmoreRSchulenburgHJansenG2017Alternative evolutionary paths to bacterial antibiotic resistance cause distinct collateral effectsMolecular Biology and Evolution342229224410.1093/molbev/msx15828541480BarbosaCBeardmoreRSchulenburgHJansenG2018Antibiotic combination efficacy (ACE) networks for a Pseudomonas aeruginosa modelPLOS Biology16e200435610.1371/journal.pbio.200435629708964BarbosaCRömhildRRosenstielPSchulenburgH2019Evolutionary stability of collateral sensitivity to antibiotics in the model pathogen Pseudomonas aeruginosaeLife8e5148110.7554/eLife.5148131658946BaymMLiebermanTDKelsicEDChaitRGrossRYelinIKishonyR2016aSpatiotemporal microbial evolution on antibiotic landscapesScience3531147115110.1126/science.aag082227609891BaymMStoneLKKishonyR2016bMultidrug evolutionary strategies to reverse antibiotic resistanceScience351aad329210.1126/science.aad329226722002BeardmoreREPeña-MillerRGoriFIredellJ2017Antibiotic cycling and antibiotic mixing: which one best mitigates antibiotic resistance?Molecular Biology and Evolution34msw29281710.1093/molbev/msw29228096304BergstromCTLoMLipsitchM2004Ecological theory suggests that antimicrobial cycling will not reduce antimicrobial resistance in hospitalsPNAS101132851329010.1073/pnas.040229810115308772BollenbachTQuanSChaitRKishonyR2009Nonoptimal microbial response to antibiotics underlies suppressive drug interactionsCell13970771810.1016/j.cell.2009.10.02519914165BorrellSTeoYGiardinaFStreicherEMKlopperMFeldmannJMüllerBVictorTCGagneuxS2013Epistasis between antibiotic resistance mutations drives the evolution of extensively drug-resistant tuberculosisEvolution, Medicine, and Public Health2013657410.1093/emph/eot00324481187CardKJLaBarTGomezJBLenskiRE2019Historical contingency in the evolution of antibiotic resistance after decades of relaxed selectionPLOS Biology17e300039710.1371/journal.pbio.300039731644535ChaitRCraneyAKishonyR2007Antibiotic interactions that select against resistanceNature44666867110.1038/nature0568517410176ChangHHCohenTGradYHHanageWPO'BrienTFLipsitchM2015Origin and proliferation of multiple-drug resistance in bacterial pathogensMicrobiology and Molecular Biology Reviews7910111610.1128/MMBR.00039-1425652543ChevereauGDraveckáMBaturTGuvenekAAyhanDHToprakEBollenbachT2015Quantifying the determinants of evolutionary dynamics leading to drug resistancePLOS Biology13e100229910.1371/journal.pbio.100229926581035CoatesJParkBRLeDŞimşekEChaudhryWKimM2018Antibiotic-induced population fluctuations and stochastic clearance of BacteriaeLife7e3297610.7554/eLife.32976CooperRMTsimringLHastyJ2017Inter-species population dynamics enhance microbial horizontal gene transfer and spread of antibiotic resistanceeLife6e2595010.7554/eLife.2595029091031DasSGDireitoSOLWaclawBAllenRJKrugJ2020Predictable properties of fitness landscapes induced by adaptational tradeoffseLife9e5515510.7554/eLife.55155DaviesJDaviesD2010Origins and evolution of antibiotic resistanceMicrobiology and Molecular Biology Reviews7441743310.1128/MMBR.00016-1020805405DayTParsonsTLambertAGandonS2020The price equation and evolutionary epidemiologyPhilosophical Transactions of the Royal Society B: Biological Sciences3752019035710.1098/rstb.2019.0357DayTGandonS2006Insights from price's equation into evolutionaryDisease Evolution: Models, Concepts, and Data Analyses712310.1090/dimacs/071/02DayTGandonS2012The evolutionary epidemiology of multilocus drug resistanceEvolution661582159710.1111/j.1558-5646.2011.01533.xDe JongMGWoodKB2018Tuning spatial profiles of selection pressure to modulate the evolution of drug resistancePhysical Review Letters12023810210.1103/PhysRevLett.120.23810229932692de VisserJAKrugJ2014Empirical fitness landscapes and the predictability of evolutionNature Reviews Genetics1548049010.1038/nrg374424913663DeanZMaltasJWoodKB2020Antibiotic interactions shape short-term evolution of resistance in E. faecalisPLOS Pathogens16e100827810.1371/journal.ppat.100827832119717DunaiASpohnRFarkasZLázárVGyörkeiÁdámApjokGBorossGSzappanosBGrézalGFaragóABodaiLPappBPálC2019Rapid decline of bacterial drug-resistance in an antibiotic-free environment through phenotypic reversioneLife8e4708810.7554/eLife.47088ECDC2019Surveillance of Antimicrobial Resistance in Europe 2018ECDCEl MeoucheIDunlopMJ2018Heterogeneity in efflux pump expression predisposes antibiotic-resistant cells to mutationScience36268669010.1126/science.aar798130409883EngelstädterJ2014Fitness landscapes emerging from pharmacodynamic functions in the evolution of multidrug resistanceJournal of Evolutionary Biology2784085310.1111/jeb.1235524720850EshelIMotroUSansoneE1997Continuous stability and evolutionary convergenceJournal of Theoretical Biology18533334310.1006/jtbi.1996.0312EstrelaSBrownSP2018Community interactions and spatial structure shape selection on antibiotic resistant lineagesPLOS Computational Biology14e100617910.1371/journal.pcbi.100617929927925FrankSA1995George price's contributions to evolutionary geneticsJournal of Theoretical Biology17537338810.1006/jtbi.1995.01487475081FrostISmithWPJMitriSMillanASDavitYOsborneJMPitt-FrancisJMMacLeanRCFosterKR2018Cooperation, competition and antibiotic resistance in bacterial coloniesThe ISME Journal121582159310.1038/s41396-018-0090-429563570Fuentes-HernandezAPlucainJGoriFPena-MillerRRedingCJansenGSchulenburgHGudeljIBeardmoreR2015Using a sequential regimen to eliminate Bacteria at sublethal antibiotic dosagesPLOS Biology13e100210410.1371/journal.pbio.100210425853342GeritzSAHKisdiÉ.MeszénaGMetzJAJ1998Evolutionarily singular strategies and the adaptive growth and branching of the evolutionary treeEvolutionary Ecology12355710.1023/A:1006554906681GerrishPJLenskiRE1998The fate of competing beneficial mutations in an asexual populationGenetica102-10312714410.1023/A:10170678165519720276GjiniEBritoPH2016Integrating antimicrobial therapy with host immunity to fight Drug-Resistant infections: classical vs. adaptive treatmentPLOS Computational Biology12e100485710.1371/journal.pcbi.100485727078624GokhaleSConwillARanjanTGoreJ2018Migration alters oscillatory dynamics and promotes survival in connected bacterial populationsNature Communications911010.1038/s41467-018-07703-yGrecoWRBravoGParsonsJC1995The search for synergy: a critical review from a response surface perspectivePharmacological Reviews473313857568331GreulichPWaclawBAllenRJ2012Mutational pathway determines whether drug gradients accelerate evolution of drug-resistant cellsPhysical Review Letters10908810110.1103/PhysRevLett.109.08810123002776HallinenKMKarslakeJWoodKB2020Delayed antibiotic exposure induces population collapse in enterococcal communities with drug-resistant subpopulationseLife9e5281310.7554/eLife.5281332207406HansenEWoodsRJReadAF2017How to use a chemotherapeutic agent when resistance to it threatens the patientPLOS Biology15e200111010.1371/journal.pbio.2001110HansenEKarslakeJWoodsRJReadAFWoodKB2020Antibiotics can be used to contain drug-resistant Bacteria by maintaining sufficiently large sensitive populationsPLOS Biology18e300071310.1371/journal.pbio.300071332413038HartYSheftelHHausserJSzekelyPBen-MosheNBKoremYTendlerAMayoAEAlonU2015Inferring biological tasks using pareto analysis of high-dimensional dataNature Methods1223323510.1038/nmeth.325425622107HegrenessMShoreshNDamianDHartlDKishonyR2008Accelerated evolution of resistance in multidrug environmentsPNAS105139771398110.1073/pnas.080596510518779569HermsenRDerisJBHwaT2012On the rapidity of antibiotic resistance evolution facilitated by a concentration gradientPNAS109107751078010.1073/pnas.1117716109Hernando-AmadoSSanz-GarcíaFMartínezJL2020Rapid and robust evolution of collateral sensitivity in Pseudomonas aeruginosa antibiotic-resistant mutantsScience Advances6eaba549310.1126/sciadv.aba5493ImamovicLEllabaanMMHDantas MachadoAMCitterioLWulffTMolinSKrogh JohansenHSommerMOA2018Drug-Driven phenotypic convergence supports rational treatment strategies of chronic infectionsCell17212113410.1016/j.cell.2017.12.01229307490ImamovicLSommerMO2013Use of collateral sensitivity networks to design drug cycling protocols that avoid resistance developmentScience Translational Medicine5204ra13210.1126/scitranslmed.300660924068739IramSDolsonEChielJPeleskoJKrishnanNGüngörÖKuznets-SpeckBDeffnerSIlkerEScottJGHinczewskiM2021Controlling the speed and trajectory of evolution with counterdiabatic drivingNature Physics1713514210.1038/s41567-020-0989-3KarslakeJMaltasJBrummPWoodKB2016Population density modulates drug inhibition and gives rise to potential bistability of treatment outcomes for bacterial infectionsPLOS Computational Biology12e100509810.1371/journal.pcbi.100509827764095KimSLiebermanTDKishonyR2014Alternating antibiotic treatments constrain evolutionary paths to multidrug resistancePNAS111144941449910.1073/pnas.1409800111KochGYepesAFörstnerKUWermserCStengelSTModamioJOhlsenKFosterKRLopezD2014Evolution of resistance to a last-resort antibiotic in Staphylococcus aureus via bacterial competitionCell1581060107110.1016/j.cell.2014.06.04625171407KohanskiMADePristoMACollinsJJ2010Sublethal antibiotic treatment leads to multidrug resistance via radical-induced mutagenesisMolecular Cell3731132010.1016/j.molcel.2010.01.00320159551LehtonenJ2018The price equation, gradient dynamics, and continuous trait game theoryThe American Naturalist19114615310.1086/69489129244566LehtonenJOkashaSHelanteräH2020Fifty years of the price equationPhilosophical Transactions of the Royal Society B: Biological Sciences3752019035010.1098/rstb.2019.0350LevySBMarshallB2004Antibacterial resistance worldwide: causes, challenges and responsesNature Medicine10S122S12910.1038/nm114515577930LipsitchMLevinBR1997The population dynamics of antimicrobial chemotherapyAntimicrobial Agents and Chemotherapy4136337310.1128/AAC.41.2.363LoeweS1953The problem of synergism and antagonism of combined drugsArzneimittel-Forschung328529013081480LopatkinAJHuangSSmithRPSrimaniJKSysoevaTABewickSKarigDKYouL2016Antibiotics as a selective driver for conjugation dynamicsNature Microbiology11604410.1038/nmicrobiol.2016.4427572835LopatkinAJMeredithHRSrimaniJKPfeifferCDurrettRYouL2017Persistence and reversal of plasmid-mediated antibiotic resistanceNature Communications8168910.1038/s41467-017-01532-129162798LukačišinMBollenbachT2019Emergent gene expression responses to drug combinations predict Higher-Order drug interactionsCell Systems942343310.1016/j.cels.2019.10.00431734160LukačišinováMFernandoBBollenbachT2020Highly parallel lab evolution reveals that epistasis can curb the evolution of antibiotic resistanceNature Communications11310510.1038/s41467-020-16932-z32561723LuriaSEDelbrückM1943Mutations of Bacteria from virus sensitivity to virus resistanceGenetics2849151110.1093/genetics/28.6.49117247100MaltasJKrasnickBWoodKB2020Using selection by nonantibiotic stressors to sensitize Bacteria to antibioticsMolecular Biology and Evolution371394140610.1093/molbev/msz30331851309MaltasJWoodKB2019Pervasive and diverse collateral sensitivity profiles inform optimal strategies to limit antibiotic resistancePLOS Biology17e300051510.1371/journal.pbio.300051531652256MeredithHRLopatkinAJAndersonDJYouL2015aBacterial temporal dynamics enable optimal design of antibiotic treatmentPLOS Computational Biology11e100420110.1371/journal.pcbi.100420125905796MeredithHRSrimaniJKLeeAJLopatkinAJYouL2015bCollective antibiotic tolerance: mechanisms, dynamics and interventionNature Chemical Biology1118218810.1038/nchembio.175425689336MichelJBYehPJChaitRMoelleringRCKishonyR2008Drug interactions modulate the potential for evolution of resistancePNAS105149181492310.1073/pnas.080094410518815368Moreno-GamezSHillALRosenbloomDIPetrovDANowakMAPenningsPS2015Imperfect drug penetration leads to spatial monotherapy and rapid evolution of multidrug resistancePNAS112E2874E288310.1073/pnas.142418411226038564MunckCGumpertHKWallinAIWangHHSommerMO2014Prediction of resistance development against drug combinations by collateral responses to component drugsScience Translational Medicine6262ra15610.1126/scitranslmed.300994025391482NicholDJeavonsPFletcherAGBonomoRAMainiPKPaulJLGatenbyRAAndersonARScottJG2015Steering evolution with sequential therapy to prevent the emergence of bacterial antibiotic resistancePLOS Computational Biology11e100449310.1371/journal.pcbi.100449326360300NicholDRutterJBryantCHujerAMLekSAdamsMDJeavonsPAndersonARABonomoRAScottJG2019Antibiotic collateral sensitivity is contingent on the repeatability of evolutionNature Communications1033410.1038/s41467-018-08098-630659188OgbunugaforCBEppsteinMJ2017Competition along trajectories governs adaptation rates towards antimicrobial resistanceNature Ecology & Evolution1000710.1038/s41559-016-0007OttoSPDayT2011A Biologist's Guide to Mathematical Modeling in Ecology and EvolutionPrinceton University Press10.1111/j.1442-9993.2007.01856.xPálCPappBLázárV2015Collateral sensitivity of antibiotic-resistant microbesTrends in Microbiology2340140710.1016/j.tim.2015.02.00925818802ParkerGASmithJM1990Optimality theory in evolutionary biologyNature348273310.1038/348027a0Pena-MillerRLaehnemannDJansenGFuentes-HernandezARosenstielPSchulenburgHBeardmoreR2013When the most potent combination of antibiotics selects for the greatest bacterial load: the smile-frown transitionPLOS Biology11e100154010.1371/journal.pbio.100154023630452PodneckyNLFredheimEGAKloosJSørumVPrimicerioRRobertsAPRozenDESamuelsenØJohnsenPJ2018Conserved collateral antibiotic susceptibility networks in diverse clinical strains of Escherichia coliNature Communications9367310.1038/s41467-018-06143-y30202004PodolskySH2015The Antibiotic Era: Reform, Resistance, and the Pursuit of a Rational TherapeuticsJohns Hopkins University Press10.3201/eid2106.150212PriceGR1970Selection and covarianceNature22752052110.1038/227520a05428476PriceGR1972Extension of covariance selection mathematicsAnnals of Human Genetics3548549010.1111/j.1469-1809.1957.tb01874.x5073694ReadAFDayTHuijbenS2011The evolution of drug resistance and the curious orthodoxy of aggressive chemotherapyPNAS108 Suppl 2108711087710.1073/pnas.110029910821690376Rodriguez de EvgrafovMGumpertHMunckCThomsenTTSommerMO2015Collateral resistance and sensitivity modulate evolution of High-Level resistance to drug combination treatment in Staphylococcus aureusMolecular Biology and Evolution321175118510.1093/molbev/msv00625618457RoemhildRGokhaleCSDirksenPBlakeCRosenstielPTraulsenAAnderssonDISchulenburgH2018Cellular hysteresis as a principle to maximize the efficacy of antibiotic therapyPNAS1159767977210.1073/pnas.181000411530209218RoemhildRLinkeviciusMAnderssonDI2020Molecular mechanisms of collateral sensitivity to the antibiotic nitrofurantoinPLOS Biology18e300061210.1371/journal.pbio.300061231986134RosenkildeCEHMunckCPorseALinkeviciusMAnderssonDISommerMOA2019Collateral sensitivity constrains resistance evolution of the CTX-M-15 β-lactamaseNature Communications1011010.1038/s41467-019-08529-y30728359RussDKishonyR2018Additivity of inhibitory effects in multidrug combinationsNature Microbiology31339134510.1038/s41564-018-0252-130323252Santos-LopezAMarshallCWScribnerMRSnyderDJCooperVS2019Evolutionary pathways to antibiotic resistance are dependent upon environmental structure and bacterial lifestyleeLife8e4761210.7554/eLife.4761231516122ShovalOSheftelHShinarGHartYRamoteOMayoADekelEKavanaghKAlonU2012Evolutionary trade-offs, pareto optimality, and the geometry of phenotype spaceScience3361157116010.1126/science.121740522539553SmithJM1982Evolution and the Theory of GamesCambridge University Press10.1017/CBO9780511806292SorgRALinLvan DoornGSSorgMOlsonJNizetVVeeningJW2016Collective resistance in microbial communities by intracellular antibiotic deactivationPLOS Biology14e200063110.1371/journal.pbio.200063128027306TanCSmithRPSrimaniJKRiccioneKAPrasadaSKuehnMYouL2012The inoculum effect and band-pass bacterial response to periodic antibiotic treatmentMolecular Systems Biology861710.1038/msb.2012.4923047527TaylorPD1996The selection differential in quantitative genetics and ess modelsEvolution502106211010.1111/j.1558-5646.1996.tb03598.xTekinEBepplerCWhiteCMaoZSavageVMYehPJ2016Enhanced identification of synergistic and antagonistic emergent interactions among three or more drugsJournal of the Royal Society Interface132016033210.1098/rsif.2016.0332TekinESavageVMYehPJ2017Measuring higher-order drug interactions: a review of recent approachesCurrent Opinion in Systems Biology4162310.1016/j.coisb.2017.05.015TekinEYehPJSavageVM2018General form for interaction measures and framework for deriving Higher-Order emergent effectsFrontiers in Ecology and Evolution616610.3389/fevo.2018.00166TorellaJPChaitRKishonyR2010Optimal drug synergy in antimicrobial treatmentsPLOS Computational Biology6e100079610.1371/journal.pcbi.100079620532210TrindadeSSousaAXavierKBDionisioFFerreiraMGGordoI2009Positive epistasis drives the acquisition of multidrug resistancePLOS Genetics5e100057810.1371/journal.pgen.100057819629166VasseMBonhoefferSFrenoyA2020Ecological effects of stress drive bacterial evolvability under sub-inhibitory antibiotic treatmentsbioRxiv10.1101/2020.06.30.181099WaxmanDGavriletsS2005Issues of terminology, gradient dynamics and the ease of sympatric speciation in adaptive dynamicsJournal of Evolutionary Biology181214121910.1111/j.1420-9101.2005.00953.x16135117WoodKNishidaSSontagEDCluzelP2012Mechanism-independent method for predicting response to multidrug combinations in BacteriaPNAS109122541225910.1073/pnas.120128110922773816WoodKBWoodKCNishidaSCluzelP2014Uncovering scaling laws to infer multidrug response of resistant microbes and Cancer cellsCell Reports61073108410.1016/j.celrep.2014.02.00724613352WoodKBCluzelP2012Trade-offs between drug toxicity and benefit in the multi-antibiotic resistance system underlie optimal growth of E. coliBMC Systems Biology64810.1186/1752-0509-6-4822631053WoodsRJReadAF2015Clinical management of resistance evolution in a bacterial infection: a case studyEvolution, Medicine, and Public Health201528128810.1093/emph/eov02526454762YoshidaMReyesSGTsudaSHorinouchiTFurusawaCCroninL2017Time-programmable drug dosing allows the manipulation, suppression and reversal of antibiotic drug resistance in vitroNature Communications81558910.1038/ncomms1558928593940YurtsevEAChaoHXDattaMSArtemovaTGoreJ2013Bacterial cheating drives the population dynamics of cooperative antibiotic resistance plasmidsMolecular Systems Biology968310.1038/msb.2013.3923917989YurtsevEAConwillAGoreJ2016Oscillatory dynamics in a bacterial cross-protection mutualismPNAS1136236624110.1073/pnas.152331711327194723ZhangQLambertGLiaoDKimHRobinKTungCKPourmandNAustinRH2011Acceleration of emergence of bacterial antibiotic resistance in connected microenvironmentsScience3331764176710.1126/science.120874721940899ZimmerAKatzirIDekelEMayoAEAlonU2016Prediction of multidimensional drug dose responses based on measurements of drug pairsPNAS113104421044710.1073/pnas.160630111327562164ZimmerATendlerAKatzirIMayoAAlonU2017Prediction of drug cocktail effects when the number of measurements is limitedPLOS Biology15e200251810.1371/journal.pbio.20025182907320110.7554/eLife.64851.sa1Decision letterOgbunugaforC BrandonReviewing EditorYale UniversityUnited States
In the interests of transparency, eLife publishes the most substantive revision requests and the accompanying author responses.
Acceptance summary:
Your manuscript offered a provocative integration of the Price equation with fundamental theory in population genetics, in the context of a relevant biomedical problem. We found the approach creative and rigorous, and is the type of cross-disciplinary theoretical work that evolutionary medicine could benefit from.
Decision letter after peer review:
Thank you for submitting your article "Price equation captures the role of drug interactions and collateral effects in the evolution of multidrug resistance" for consideration by eLife. Your article has been reviewed by 3 peer reviewers, one fo whom is a member of our Board of Reviewing Editors, and the evaluation has been overseen by George Perry as the Senior Editor. The reviewers have opted to remain anonymous.
Summary:
I thank the authors for submitting a compelling and well-executed study. The reviewers and reviewing editor agree that the manuscript offers an interesting take, is technically sound and provides a provocative synergy between modern questions in the evolution of antibiotic resistance, and classical theory in the form of the Price equation.
Some of the discussion between the reviewers involved the question of whether the paper's main argument, as outlined in the draft, was too strongly bound to the usage of the Price equation. That is, that the main result can be couched in terms of the Price equation may or may not mean that the finding is a true contribution to modern examinations of the evolution of antibiotic resistance, a field that is increasingly sophisticated in its use of both theory and technology.
I especially applaud the author's transparency: the issue is not that they oversold their result. To the contrary-the limitations were responsibly articulated. A lingering question is whether the result, as reported, is anchored to a slice of reality that is too thin to be broadly useful. This is an admittedly subjective area of discussion, but was a theme from the reviews and discussions.
In order to assuage some of the concerns, I have provided a list of recommended changes, collated from the formal; reviews and further exchanges with the reviewers, that could make the manuscript stronger.
I thank the authors for their hard work. I want to especially acknowledge that we are all working in times that are challenging for many reasons. I congratulate the authors on producing a strong piece of science under the circumstances.
Essential Revisions:
1) Please propose some treatment of the setting in the result that includes mutation and other stochastic processes. Several reviewers made mention of the "model" that was used. Some had specific issues with the notion that the rescaling in the form of the Price was truly a model at all. Others had the related concern that the paper, as it stands, applies to the very strict circumstance: that of a small population of equal numbers of every potential mutant in the model. For example, the authors state:
"We assume a finite set of M subpopulations (mutants), (i = 1…M), with each subpopulation corresponding to a single pair of scaling parameters. For simplicity, we assume each of these mutants is initially present in the population at low frequency and neglect mutations that could give rise to new phenotypes, though it is straightforward to incorporate them into the same framework"
If it is indeed "straightforward," then it would be great to see this explored, with new mutations and other stochastic processes. We recognize that the strength of the study is in its elegance re-scaling, however, there are concerns that the use case is small enough that it limits its applicability.
2) Somewhat relatedly, the reviewers had some issues with use of the term "model" to describe the Price equation. While the mathematics and framing themselves are quite responsible, there were some particular issues raised about the framing. I want to especially point you towards the comments from reviewer #2, who explained issues with usage of the term "model" and "prediction" throughout.
3) There are several issues with figures, figure labels, captions and descriptions that require the author's attention. As this paper does hinge on an understanding of the particulars, it is especially important that figures clearly communicate the results. Please see the individual reviewer's comments for specifics.Reviewer #1:
This study utilizes the price equation in order to describe evolution of drug resistance, recognizing the cutting-edge problem of predicting the evolution of resistance in light of collateral effects and drug-drug interactions.
The manuscript offers a formalism that cleverly integrates theory across paradigms, from quantitative genetics to the evolution of antibiotic resistance. As an intellectual exercise, this is courageous, the type of integrative thinking necessary in the modern iteration of evolutionary medicine.
I am not, however, of the mindset that the Price equation is so important a framing that its invocation/application to a different paradigm (antibiotic resistance) is especially interesting by itself. Said differently, I am not personally intrigued by the a study that is successful at connecting antibiotic resistance to the Price equation. Alternatively, I am much much more intrigued by the possibility that the Price equation might add a richer and more nuanced overall perspective to cutting-edge problems in the evolution of antibiotic resistance.
And I can say that, on the balance, the manuscript was successful in offering a new take on relevant problem.
Also, this study carries out its methods carefully, and its formalisms are well-written. The essential idea is that the vagaries of drug interactions and collateral effects can be collapsed into a formalism that considers both when predicting the trajectory of drug resistance, is a fascinating and important one.
The study is especially transparent (refreshingly so) about the limitations of its scope. And while this transparency is a legitimate positive of this story, I did find myself concerned about the small number of cases that theory proposed in this study applies to. For example, the Price equation as applied in the study applies to selection on standing variation, and not for the step-wise acquisition of resistance that fitness landscapes are often constructed to study. The authors argue that this type of nuance can be engineered into the formalism, which is not obvious to me (I'm not suggesting that it isn't true, only that it isn't obvious).
But that this study has a relatively thin application space does not make the study irrelevant. For the outlined problem space, the arguments appear sound.
I must say that given the current state of the problem of antimicrobial resistance in 2021, sometimes featuring complex, highly dynamic, epistastic fitness seascapes, for example, I was hoping for framework that could be used for fitness landscapes as they are currently constructed and studied.
On the balance, however, I found the study to be well-constructed, written, and the formalisms to be correct. The main concerns regarding the scope might be a matter of preference. And even I must recognize that responsible theoretical biology often requires a narrow scope in order to generate formalisms. The idea here is that from this contribution, the community can then extend the study to other such systems and data sets.
I would like the authors to consider, and possibly cite, if they are relevant, more recent papers by Pamela Yeh on this topic:
Tekin, Elif, Pamela J. Yeh, and Van M. Savage. "General form for interaction measures and framework for deriving higher-order emergent effects." Frontiers in Ecology and Evolution 6 (2018): 166.
Tekin, Elif, Van M. Savage, and Pamela J. Yeh. "Measuring higher-order drug interactions: A review of recent approaches." Current Opinion in Systems Biology 4 (2017): 16-23.
Tekin, Elif, et al., "Enhanced identification of synergistic and antagonistic emergent interactions among three or more drugs." Journal of The Royal Society Interface 13.119 (2016): 20160332.
There are some small issues with the figures that need to be resolved:
I found Figure 6 to be complicated. I would suggest the authors re-generate it with the goal of explaining it to someone not well-versed in the system.
I had some issues with the legends in Figures 4 and 5. Ensure that they describe the information in the figures properly.Reviewer #2:
This paper examines the interaction between pharmacological and evolutionary factors that mediate the emergence of resistance to combination antimicrobial therapy. The authors introduce and demonstrate the utility of a mathematical equation that describes how both (a) the pharmacological interaction between two drugs (e.g. synergy vs antagonism) and (b) the interaction between the fitness effects of mutations that confer resistance to one drug or the other (or both), jointly determine the direction of selection in the presence of two-drug therapy. They show that the equation they describe follows the same form as a two-trait version of the classical "Price equation" in population genetics. They also examine a "weak selection" limit (when evolution proceeds by selection of a sequence of strains each with only a slight change in resistance level from the previous). In this limit their equation has an even more intuitive physical meaning as movement along an weighted version of the gradient of the drug's dose-response curve, where the weighting reflects the nature of the cross-resistance profile. Many examples of the implications of this equation for different drug interactions and different fitness landscapes are presented. The results of this paper unite findings from quite a few previous studies in a common framework. For example, the authors show how to understand (i) when resistance to one drug in a combination will be more strongly selected than resistance to the other and how this direction of selection can change based on the relative levels of two drugs in a combination – even if total inhibition is the same, (ii) when drug antagonism (vs synergy) is morel likely to slow down the rate of adaptation, and (iii) how the relative ordering and frequency of cycling between two drugs in a combination impacts the rate of resistance emergence.
It is important for readers to understand that the Price equation – and thus by extension the result in this paper – is not really a mathematical "model" that includes assumptions about the underlying biological processes and makes predictions about potential experimental results. Instead, the Price equation is a mathematical truth that simply involves decomposing certain statistical relationships between variables and expressing them in an intuitive way. Roughly, in this case the Price equation describes how the average level of resistance in a population of microbes will change over time depending on the frequencies of different strains in the population and their resistance levels, including the correlation between resistance to each drug. Thus, this paper is primarily purely theoretical/mathematical work.
The authors do apply their equation to some real-life drug profiles and fitness landscapes using dose-response curves for two-drug antibiotic combinations to calculate the direction of selection for different dose combinations. Later in the paper, they do also apply their equation as a "model" to predict entire evolutionary trajectories, by additionally assuming that they can define a priori all the available resistance mutations and ignore any additionally generated mutations. Then, they show that the level of resistance selected after a specific time as predicted by their equations agrees relatively well with the results of in vitro experiments containing these same mutations. This is actually quite a surprising and nice result.
Overall this paper provides a useful new framework for understanding drug resistance evolution and provides multiple examples of insights offered by this framework. They provide a nice connection between the mainly mathematical nature of their results and scenarios involving antibiotic treatment and resistance evolution in the lab. The introduction of this paper is very comprehensive and summarizes a huge body of work on the evolution of drug resistance. The discussion is equally thorough and mentions most of the caveats I describe here.
Although the authors of the paper focus on bacterial infections and antibiotic resistance, I think the results are much more general, and likely also apply to multi-drug therapy to viral infections (e.g. HIV, HCV), parasitic infections (e.g. malaria), and even anti-cancer therapy.
While not a weakness that is specific to this paper, it is important for readers to understand that a Price equation only describes the act of selection acting on existing genetic variation in a population, and does not describe the act of generating that variation (i.e. mutations). Thus, while this approach can be helpful for understanding the direction and speed of selection in one vs two-drug combinations of different types, it cannot be used to understand the overall risk of resistance emergence, since the rate at which mutations are generated may also differ. For example, it often takes two separate mutations to generate resistance to two drugs with different mechanisms of action, so resistance to two-drug therapy is often slowed by the rate of generating resistant mutations. In addition, the rate at which mutations are generated during therapy depends on the population size over time, which might differ along different selective paths described by the Price equation for different drug combinations. Therefore, there are some important results from previous work related to factors that accelerate or inhibit multi-drug resistance which cannot be understood in the framework presented in this paper.
One assumption that the authors use in order to get a simplified Price-equation-like result is that the effect of a drug resistance mutation on the dose-dependent effect of a drug on microbial growth can be described by a single parameter, which just shifts the drug concentration to a lower value. This is equivalent to saying that a mutation simply alters the IC50 of the drug. However, many studies have shown that in general mutations have at least one and often two other effects on dose-response curves: firstly, they have a fitness cost, meaning that the maximum fitness in the absence of drug is lower, and secondly, they can also alter the slope of the dose response curve. It is unclear how these effects change the equation the authors have derived.
The impact of the authors approach on studying resistance evolution in any particular system depends on the regime that system is in. If resistance tends to evolve by selection slowly acting on an extremely diverse population containing strains with many different resistance levels, then this approach will have the most utility. Alternatively, if resistance evolution tends to be limited by rare mutational events that produce strains with large differences in resistance levels from the parental strain, then this approach will have less use.
To summarize, the most important caveat for readers to understand is that the mathematical expressions that come out of this paper should not be said to be able to "predict" evolution, since they say nothing about the availability of possible resistance mutations or the rate at which they could be generated.
Concerns and suggested changes/clarifications:
– It is a bit misleading to call the Price Equation a mathematical model. It's not a model in the way that word is generally used in biology – it doesn't descibe a mechanism or make any assumptions, and it's not subject to experimental verification. It's just a mathematical statement about a statistical relationship among variables, based on the definition of those variables and basic laws of calculus. The authors should be very clear about this throughout the paper.
– The parts of the paper that are more like a "model" are the parts where you use the Price Equation to describe an evolutionary trajectory, assuming a particular distribution of mutants from standing genetic variation and ignoring additional mutations, or, when you assume a weak selection regime.
– Overall the authors should be more careful about the use of the word "model" and claims about "predictions" throughout.
– I'm not sure about Equation 2 – 5 requires the assumption that the effect of a mutant can be described by a rescaling of drug concentrations. Instead, I think you could replace the α parameters with another parameter r for example that simply means "resistance level" but does not specify how the "resistance level" changes the shape of the dose-response curve.
– Equation 5 actually is only half of the regular Price equation. It is the "selection" component but is missing the "transmission" component. The authors should be clearer about what processes relevant to the evolution of drug resistance they are ignoring that allow them to set this second term to zero.
Comments on Figures and results:
Figure 2 – What are the units of the axes on all these graphs?
a) I don't understand this statement in the caption: "We then simulated an external condition where the combination drug concentration is held constant at 0.6 of the maximal concentration of drug 1 and 0.1 of the maximal concentration of drug 2.". There is no such thing as a "maximal" drug concentration … you can always keep adding more and more drug. It would be clearer just to state the drug level used in absolute terms. Or maybe you mean the "highest tested drug level"?
Figure 3 – Some of the results shown here are a little confusing and hard to understand. In the black scenario, there are few available resistant strains with lower α, so we expect selection to proceed in the direction of Drug 2/β.. that agrees with the results in B. But for the blue case where, despite lots of available α but little available β, the trajectory in B seems relatively symmetrical in both drugs … why? Also I was surprised that the case of collateral sensitivity (green) wasn't the worst case … but I realized it's a bit misleading because the four scenarios don't only differ in the nature of the correlations in resistance levels, but also in the magnitude of the variance in α or β and hence the best possible available mutation. I think that it would be a fairer comparison if for the green and magenta scenarios the variance was narrower around the lines of perfect correlation/anticorrelation.
Figure 4 – This is nice figure that really summarizes the combined effects of drug interactions and cross-resistance profile. However this figure also summarizes the major limitations of this approach. The early slopes of each growth rate vs time curve do show how these factors interact to determine selection strength. However, the fact that the maximum growth rate varies by curve is really just an artifact of the arbitrary and limited available mutants the authors have created for selection to act on. In reality, mutations are continually and sequentially generated over time, and this rate of generation of mutants also impacts the increase in growth rate over time and might remove any "maximum" resistance level that is achievable.
Figure 5 – Are these predicted trajectories using the weak selection limit or the full equations without this additional assumption?
– I'm assuming there was some fixed time limit that selection was allowed to proceed for in these experiments and that it was the same in all cases?
– B: is the y axis absolute IC50 or change in IC50? The axes titles just say IC50 but the figure caption says "relative change in IC50". Also it would be helpful if the baseline IC50 (ie for the wild type strain) where shown on these graph as well. Finally, I don't understand how the results for the TGC IC50 agree with the results in A. From B, the IC50 looks totally flat with TGC concentration. But from A, it looks like there is a large increase in TGC resistance (increase in TGC IC50) for high TGC concentrations, but little TGC resistance for low TGC concentrations.
– C: I don't understand what the y axis is here. If this is the increase in growth rate, then why is it highest for low TGC concentrations, yet both panels in B show no change in IC50 to either drug at low TGC concentrations? And any idea what could be causing the systematic difference between the predicted and observed changes in growth rate?
Figure 6 – This figure was quite confusing, it needs more annotation on the figure itself (not just written in the caption), and some panels are unnecessarily small (with unnecessary white space between them). For panels H and K, are there better axis limits that could be used so that 95% of the figure is not all one color? Also I found it is hard to see in this figure any of the differences between the outcomes of the two different orderings that are described in the text.
Reviewer #3:
In this work, Gjini and Wood create a model to simulate evolutionary dynamics of antibiotic resistance arising as a response to drug combinations. They demonstrate that evolution to these drug combinations is a variation of the Price equation and show that some evolutionary paths experimentally correspond to a weighted gradient descent on the growth landscape.
The work appears mathematically sound, though we have concerns about the scope and generality of the results. In particular this work appears to rely fundamentally on the assumption that drug resistance acts as a linear rescaling of the two-drug dose response curve. We believe this work would be strengthened by making it more clear what, beyond the mathematical description of the rescaling hypothesis they cite extensively, this work adds to the literature. Since the model appears to also require an a priori description of the resistance characteristics of every possible accessible mutant, which typically is not known, it isn't clear the utility of the model.
– This paper models evolution in the presence of physiological interactions between antibiotics but not evolutionary interactions. The fundamental assumption that evolution of resistance scales the surface but does not change its shape, which is central to the findings, is true in some cases, though violated for many mutations (the authors cite numerous papers on cross-resistance and collateral sensitivity that are examples of this).
– While the authors describe it as "straightforward" to include discrete mutational events instead of a small population of equal numbers of every potential mutant in the model, it is not clear that this is the case. For example, a population exposed to an above-MIC concentration of a bacteriostatic drug will no longer generate additional mutants, and so the standing diversity before drug application matters critically. As it stands the model is really about differential growth and not evolution per se, since neither mutation nor complete extinction can occur.
– It isn't clear the experiments are representative of nontrivial behavior predicted by the model. For example in figure 5 no adaptation to TGC was seen, so it seems the only real evolution was rescaling CIP, which wouldn't yield the suppression-induced selection against resistance they then discuss. These results could be entire an artifact of the selection being entirely on CIP and not at all on TGC. The evolution experiments in Figures S3 and S4 seem much more compelling, though.
– If it is indeed straightforward to add mutational events, this would make the model appreciably more realistic.
10.7554/eLife.64851.sa2Author response
Essential Revisions:
1) Please propose some treatment of the setting in the result that includes mutation and other stochastic processes. Several reviewers made mention of the "model" that was used. Some had specific issues with the notion that the rescaling in the form of the Price was truly a model at all. Others had the related concern that the paper, as it stands, applies to the very strict circumstance: that of a small population of equal numbers of every potential mutant in the model. For example, the authors state:
"We assume a finite set of M subpopulations (mutants), (i = 1…M), with each subpopulation corresponding to a single pair of scaling parameters. For simplicity, we assume each of these mutants is initially present in the population at low frequency and neglect mutations that could give rise to new phenotypes, though it is straightforward to incorporate them into the same framework"
If it is indeed "straightforward," then it would be great to see this explored, with new mutations and other stochastic processes. We recognize that the strength of the study is in its elegance re-scaling, however, there are concerns that the use case is small enough that it limits its applicability.
In the revised manuscript, we include a new section (including 1 additional figure, Figure 6, and several associated supplemental figures) that incorporates mutation. Under these conditions, the Price Equation includes an additional term that depends on a mutation matrix that describes the probability of mutation between any pairs of strains (Equation 13). Under some conditions (e.g. when mutations occur with uniform probability between the ancestor cells and each mutant), the effects of mutation are qualitatively similar to those of selection (Figure 4 and Figure 6 figure supplement 1). On the other hand, mutational matrices can be chosen to reflect known features of the mutation process or fitness landscapes. As examples, we consider several illustrative cases, including (1) a case where mutations can only be acquired sequentially (Figure 6, Figure 6 figure supplement 2) and (2) a case where mutations occur in a distance-dependent manner in the space of rescaling parameters, in essence making mutations between strains with similar scaling parameters more likely than those with vastly different scaling parameters (Figure 6 figure supplements 3-4).
Similar to the case with selection only, it is difficult to draw general conclusions (e.g. “synergistic combinations always accelerate resistance”) that apply across all classes of drug combinations, collateral effects, and mutational structures. The relationship between these features and the evolutionary dynamics is complex, but as these examples illustrate, the framework can be extended to incorporate these features and produce testable predictions. In the discussion, we discuss this point as follows:
“Our goal was to strike a balance between analytical tractability and generality vs biological realism and system specificity. We stress that the predictions of this model do not come for free; they depend, for example, on properties of the dose response surfaces, the collection of scaling parameters, and the specific mutational structure. In many cases, these features can be determined empirically; in other cases, some or all of these properties may be unknown. The evolutionary predictions of the model will ultimately depend on these inputs, and it is difficult to draw general conclusions (e.g. “synergistic combinations always accelerate resistance”) that apply across all classes of drug combinations, collateral effects, and mutational structures. But we believe the framework is valuable precisely because it generates testable predictions in situations that might otherwise seem intractably complex.”
2) Somewhat relatedly, the reviewers had some issues with use of the term "model" to describe the Price equation. While the mathematics and framing themselves are quite responsible, there were some particular issues raised about the framing. I want to especially point you towards the comments from reviewer #2, who explained issues with usage of the term "model" and "prediction" throughout.
We thank the reviewers for their comments, and we have taken additional care to discuss these important, but subtle, distinctions in the revised manuscript. We agree that the manuscript is not merely about the Price Equation, which (as the reviewers point out) is really just an algebraic restatement of generic statistical features of this (and similar) models. Our underlying model formalizes the concept of concentration rescaling as a general description of resistance, a continuous trait that evolves in a manner that depends on the multi-component environment, and that model could indeed be analyzed in the absence of the Price Equation connection. At the same time, we believe the connection to the Price Equation has conceptual value, not only because it links our work with a classical result in evolutionary biology, but also because it provides a convenient form for disentangling the effects of collateral resistance and drug interactions (at least in the stated limits). We now include the following statement in the Results, just after the brief derivation of the Price Equation for our model:
“In what follows, we will sometimes casually refer to Equation 5 as a "model", but it is important to note that the Price Equation is not, in and of itself, a mathematical model in the traditional sense. Instead, it is a simple mathematical statement describing the statistical relationship between variables, which are themselves defined in some underlying model. In this case, the mathematical model consists of a collection of exponentially growing populations whose per capita growth rates are linked by scaling relationships. Equation 5 – the Price Equation – does not include additional assumptions, mechanistic or otherwise, but merely captures statistical relationships between those model variables. We will see, however, that these relationships provide conceptual insight into the interplay between collateral effects and drug interactions.”
In addition, we now consistently refer to the “model” rather than the “Price Equation” when describing our simulations and results (e.g. in the figure captions).
3) There are several issues with figures, figure labels, captions and descriptions that require the author's attention. As this paper does hinge on an understanding of the particulars, it is especially important that figures clearly communicate the results. Please see the individual reviewer's comments for specifics.
Thank you for these suggestions. The particulars indeed make this a challenging story to tell, and we agree that some parts of the initial presentation added unnecessary confusion. We have revised all the figures (and captions) in question in an effort to clarify, and in some cases simplify, the presentation. Most notably, we have completely remade Figure 7 (previously Figure 6) while correcting minor errors and moving additional details to the SI.
Reviewer #1:
This study utilizes the price equation in order to describe evolution of drug resistance, recognizing the cutting-edge problem of predicting the evolution of resistance in light of collateral effects and drug-drug interactions.
The manuscript offers a formalism that cleverly integrates theory across paradigms, from quantitative genetics to the evolution of antibiotic resistance. As an intellectual exercise, this is courageous, the type of integrative thinking necessary in the modern iteration of evolutionary medicine.
We thank the reviewer for the positive evaluation.
I am not, however, of the mindset that the Price equation is so important a framing that its invocation/application to a different paradigm (antibiotic resistance) is especially interesting by itself. Said differently, I am not personally intrigued by the a study that is successful at connecting antibiotic resistance to the Price equation. Alternatively, I am much much more intrigued by the possibility that the Price equation might add a richer and more nuanced overall perspective to cutting-edge problems in the evolution of antibiotic resistance.
The reviewer makes a good point. We agree that the manuscript is not merely about the Price Equation, but instead about formalizing the concept of concentration rescaling as a general description of resistance as a continuous trait that manifests itself in environmentally-dependent ways. Note that this is quite distinct from previous applications of the Price Equation, which typically cast resistance as a binary trait (0 or 1) and where the primary goal is predicting mean trajectories for the frequency of resistance in a population. In addition, we believe the connection to the Price Equation within our rescaling model has conceptual value, not only because it links our work with a classical result in evolutionary biology, but also because it provides a convenient form for disentangling the effects of collateral resistance and drug interactions (at least in the stated limits).
And I can say that, on the balance, the manuscript was successful in offering a new take on relevant problem.
Also, this study carries out its methods carefully, and its formalisms are well-written. The essential idea is that the vagaries of drug interactions and collateral effects can be collapsed into a formalism that considers both when predicting the trajectory of drug resistance, is a fascinating and important one.
The study is especially transparent (refreshingly so) about the limitations of its scope. And while this transparency is a legitimate positive of this story, I did find myself concerned about the small number of cases that theory proposed in this study applies to. For example, the Price equation as applied in the study applies to selection on standing variation, and not for the step-wise acquisition of resistance that fitness landscapes are often constructed to study. The authors argue that this type of nuance can be engineered into the formalism, which is not obvious to me (I'm not suggesting that it isn't true, only that it isn't obvious).
But that this study has a relatively thin application space does not make the study irrelevant. For the outlined problem space, the arguments appear sound.
I must say that given the current state of the problem of antimicrobial resistance in 2021, sometimes featuring complex, highly dynamic, epistastic fitness seascapes, for example, I was hoping for framework that could be used for fitness landscapes as they are currently constructed and studied.
On the balance, however, I found the study to be well-constructed, written, and the formalisms to be correct. The main concerns regarding the scope might be a matter of preference. And even I must recognize that responsible theoretical biology often requires a narrow scope in order to generate formalisms. The idea here is that from this contribution, the community can then extend the study to other such systems and data sets.
We thank the reviewer for recognizing our efforts to be transparent, for highlighting the value of the formalism – and also recognizing the difficulty of developing theoretical biology with broad scope – and for the generally positive appraisal. The revised version extends the formalism to include mutations, which allows us to incorporate empirical constraints on the mutational pathways linking different phenotypes (similar, though not identical, to the information contained in fitness landscapes). The trade-off with this added complexity is, of course, the need for additional assumptions, and while we have focused on simple illustrative examples (e.g. sequential mutations, Figure 6 and Figure 6 figure supplement 2, and distance-dependent mutations in parameter space, Figure 6 figure supplements 3-4) we believe they illustrate that the approach is flexible and amenable to new empirical constraints as they become available. We discuss these points in the revised manuscript, and (most notably) we also added the following to the discussion:
“Our goal was to strike a balance between analytical tractability and generality vs biological realism and system specificity. We stress that the predictions of this model do not come for free; they depend, for example, on properties of the dose response surfaces, the collection of scaling parameters, and the specific mutational structure. In many cases, these features can be determined empirically; in other cases, some or all of these properties may be unknown. The evolutionary predictions of the model will ultimately depend on these inputs, and it is difficult to draw general conclusions (e.g. “synergistic combinations always accelerate resistance”) that apply across all classes of drug combinations, collateral effects, and mutational structures. But we believe the framework is valuable precisely because it generates testable predictions in situations that might otherwise seem intractably complex.”
I would like the authors to consider, and possibly cite, if they are relevant, more recent papers by Pamela Yeh on this topic:
Tekin, Elif, Pamela J. Yeh, and Van M. Savage. "General form for interaction measures and framework for deriving higher-order emergent effects." Frontiers in Ecology and Evolution 6 (2018): 166.
Tekin, Elif, Van M. Savage, and Pamela J. Yeh. "Measuring higher-order drug interactions: A review of recent approaches." Current Opinion in Systems Biology 4 (2017): 16-23.
Tekin, Elif, et al., "Enhanced identification of synergistic and antagonistic emergent interactions among three or more drugs." Journal of The Royal Society Interface 13.119 (2016): 20160332.
Thank you for these suggestions. We have cited all 3 papers, which focus on higher-order (more than 2) drug interactions and therefore may be useful for extending our framework to higher dimensions.
There are some small issues with the figures that need to be resolved:
I found Figure 6 to be complicated. I would suggest the authors re-generate it with the goal of explaining it to someone not well-versed in the system.
We have simplified and remade Figure 6 (now Figure 7), which we agree was overly complicated in the initial draft.
I had some issues with the legends in Figures 4 and 5. Ensure that they describe the information in the figures properly.
We have clarified the legends of Figures 4 and 5, which were indeed confusing in the initial draft.
Reviewer #2:
This paper examines the interaction between pharmacological and evolutionary factors that mediate the emergence of resistance to combination antimicrobial therapy. The authors introduce and demonstrate the utility of a mathematical equation that describes how both (a) the pharmacological interaction between two drugs (e.g. synergy vs antagonism) and (b) the interaction between the fitness effects of mutations that confer resistance to one drug or the other (or both), jointly determine the direction of selection in the presence of two-drug therapy. They show that the equation they describe follows the same form as a two-trait version of the classical "Price equation" in population genetics. They also examine a "weak selection" limit (when evolution proceeds by selection of a sequence of strains each with only a slight change in resistance level from the previous). In this limit their equation has an even more intuitive physical meaning as movement along an weighted version of the gradient of the drug's dose-response curve, where the weighting reflects the nature of the cross-resistance profile. Many examples of the implications of this equation for different drug interactions and different fitness landscapes are presented. The results of this paper unite findings from quite a few previous studies in a common framework. For example, the authors show how to understand (i) when resistance to one drug in a combination will be more strongly selected than resistance to the other and how this direction of selection can change based on the relative levels of two drugs in a combination – even if total inhibition is the same, (ii) when drug antagonism (vs synergy) is morel likely to slow down the rate of adaptation, and (iii) how the relative ordering and frequency of cycling between two drugs in a combination impacts the rate of resistance emergence.
We thank the reviewer for the concise summary and for recognizing the intuitive value of the framework and its potential to unify findings from several previous studies.
It is important for readers to understand that the Price equation – and thus by extension the result in this paper – is not really a mathematical "model" that includes assumptions about the underlying biological processes and makes predictions about potential experimental results. Instead, the Price equation is a mathematical truth that simply involves decomposing certain statistical relationships between variables and expressing them in an intuitive way. Roughly, in this case the Price equation describes how the average level of resistance in a population of microbes will change over time depending on the frequencies of different strains in the population and their resistance levels, including the correlation between resistance to each drug. Thus, this paper is primarily purely theoretical/mathematical work.
We agree with this point, and we have now been careful to clarify this distinction. Most notably, we added the following to the Results section, just after the introduction of the Price equation (copied from response to Evaluation Summary, above):
“In what follows, we will sometimes casually refer to Equation 5 as a "model", but it is important to note that the Price Equation is not, in and of itself, a mathematical model in the traditional sense. Instead, it is a simple mathematical statement describing the statistical relationship between variables, which are themselves defined in some underlying model. In this case, the mathematical model consists of a collection of exponentially growing populations whose per capita growth rates are linked by scaling relationships. Equation 5 – the Price Equation – does not include additional assumptions, mechanistic or otherwise, but merely captures statistical relationships between those model variables. We will see, however, that these relationships provide conceptual insight into the interplay between collateral effects and drug interactions.”
In addition, we now consistently refer to the “model” rather than the “Price Equation” when describing our simulations and results (e.g. in the figure captions).
The authors do apply their equation to some real-life drug profiles and fitness landscapes using dose-response curves for two-drug antibiotic combinations to calculate the direction of selection for different dose combinations. Later in the paper, they do also apply their equation as a "model" to predict entire evolutionary trajectories, by additionally assuming that they can define a priori all the available resistance mutations and ignore any additionally generated mutations. Then, they show that the level of resistance selected after a specific time as predicted by their equations agrees relatively well with the results of in vitro experiments containing these same mutations. This is actually quite a surprising and nice result.
Overall this paper provides a useful new framework for understanding drug resistance evolution and provides multiple examples of insights offered by this framework. They provide a nice connection between the mainly mathematical nature of their results and scenarios involving antibiotic treatment and resistance evolution in the lab. The introduction of this paper is very comprehensive and summarizes a huge body of work on the evolution of drug resistance. The discussion is equally thorough and mentions most of the caveats I describe here.
Although the authors of the paper focus on bacterial infections and antibiotic resistance, I think the results are much more general, and likely also apply to multi-drug therapy to viral infections (e.g. HIV, HCV), parasitic infections (e.g. malaria), and even anti-cancer therapy.
While not a weakness that is specific to this paper, it is important for readers to understand that a Price equation only describes the act of selection acting on existing genetic variation in a population, and does not describe the act of generating that variation (i.e. mutations). Thus, while this approach can be helpful for understanding the direction and speed of selection in one vs two-drug combinations of different types, it cannot be used to understand the overall risk of resistance emergence, since the rate at which mutations are generated may also differ. For example, it often takes two separate mutations to generate resistance to two drugs with different mechanisms of action, so resistance to two-drug therapy is often slowed by the rate of generating resistant mutations. In addition, the rate at which mutations are generated during therapy depends on the population size over time, which might differ along different selective paths described by the Price equation for different drug combinations. Therefore, there are some important results from previous work related to factors that accelerate or inhibit multi-drug resistance which cannot be understood in the framework presented in this paper.
Thank you for these points. We have now included additional discussion of the limitations of the model, potential applications and, more importantly, we now illustrate several examples where mutational dynamics are included. We copy below our response to Point 1 in the Evaluation Summary (above):
In the revised manuscript, we include a new section (including 1 additional figure, Figure 6, and several SI figures, S8-S10) that incorporates mutation. Under these conditions, the Price Equation includes an additional term that depends on a mutation matrix that describes the probability of mutation between any pairs of strains (Equation 13). Under some conditions (e.g. when mutations occur with uniform probability between the ancestor cells and each mutant), the effects of mutation are qualitatively similar to those of selection (Figure 4 and Figure 6 figure supplement 1). On the other hand, mutational matrices can be chosen to reflect known features of the mutation process or fitness landscapes. As examples, we consider several illustrative cases, including (1) a case where mutations can only be acquired sequentially (Figure 6, Figure 6 figure supplement 2) and (2) a case where mutations occur in a distance-dependent manner in the space of rescaling parameters, in essence making mutations between strains with similar scaling parameters more likely than those with vastly different scaling parameters (Figure 6 figure supplements 3-4).
Similar to the case with selection only, it is difficult to draw general conclusions (e.g. “synergistic combinations always accelerate resistance”) that apply across all classes of drug combinations, collateral effects, and mutational structures. The relationship between these features and the evolutionary dynamics is complex, but as these examples illustrate, the framework can be extended to incorporate these features and produce testable predictions. In the discussion, we discuss this point as follows:
“Our goal was to strike a balance between analytical tractability and generality vs biological realism and system specificity. We stress that the predictions of this model do not come for free; they depend, for example, on properties of the dose response surfaces, the collection of scaling parameters, and the specific mutational structure. In many cases, these features can be determined empirically; in other cases, some or all of these properties may be unknown. The evolutionary predictions of the model will ultimately depend on these inputs, and it is difficult to draw general conclusions (e.g. “synergistic combinations always accelerate resistance”) that apply across all classes of drug combinations, collateral effects, and mutational structures. But we believe the framework is valuable precisely because it generates testable predictions in situations that might otherwise seem intractably complex.”
One assumption that the authors use in order to get a simplified Price-equation-like result is that the effect of a drug resistance mutation on the dose-dependent effect of a drug on microbial growth can be described by a single parameter, which just shifts the drug concentration to a lower value. This is equivalent to saying that a mutation simply alters the IC50 of the drug. However, many studies have shown that in general mutations have at least one and often two other effects on dose-response curves: firstly, they have a fitness cost, meaning that the maximum fitness in the absence of drug is lower, and secondly, they can also alter the slope of the dose response curve. It is unclear how these effects change the equation the authors have derived.
Thank you for this suggestion. We now include additional discussion of this point in the discussion.
The impact of the authors approach on studying resistance evolution in any particular system depends on the regime that system is in. If resistance tends to evolve by selection slowly acting on an extremely diverse population containing strains with many different resistance levels, then this approach will have the most utility. Alternatively, if resistance evolution tends to be limited by rare mutational events that produce strains with large differences in resistance levels from the parental strain, then this approach will have less use.
To summarize, the most important caveat for readers to understand is that the mathematical expressions that come out of this paper should not be said to be able to "predict" evolution, since they say nothing about the availability of possible resistance mutations or the rate at which they could be generated.
We agree with these points. We have added brief discussions of these points in the new section on mutations as well as in the discussion (as we have noted in the previous responses).
Concerns and suggested changes/clarifications:
– It is a bit misleading to call the Price Equation a mathematical model. It's not a model in the way that word is generally used in biology – it doesn't descibe a mechanism or make any assumptions, and it's not subject to experimental verification. It's just a mathematical statement about a statistical relationship among variables, based on the definition of those variables and basic laws of calculus. The authors should be very clear about this throughout the paper.
– The parts of the paper that are more like a "model" are the parts where you use the Price Equation to describe an evolutionary trajectory, assuming a particular distribution of mutants from standing genetic variation and ignoring additional mutations, or, when you assume a weak selection regime
– Overall the authors should be more careful about the use of the word "model" and claims about "predictions" throughout
Thank you for these important suggestions. In the revision, we have been careful to distinguish between the assumptions of our model (e.g. concentration rescaling, statistical features of collateral effects, or particular mutational structures) and the Price Equation, which the reviewer rightly points out is simply a restatement of statistical relationships emerging from the underlying model. We have also been more careful in our use of “prediction”, which can indeed be misleading in this context. Please see also our responses to point 2 in the Evaluation Summary above.
– I'm not sure about Equation 2 – 5 requires the assumption that the effect of a mutant can be described by a rescaling of drug concentrations. Instead, I think you could replace the α parameters with another parameter r for example that simply means "resistance level" but does not specify how the "resistance level" changes the shape of the dose-response curve.
The reviewer is correct that selection need not occur only on the drug concentration, but can instead alter any feature of the dose response curve, and we now discuss this briefly in the discussion.
– Equation 5 actually is only half of the regular Price equation. It is the "selection" component but is missing the "transmission" component. The authors should be clearer about what processes relevant to the evolution of drug resistance they are ignoring that allow them to set this second term to zero.
As described above, the revised manuscript now includes a new analysis including the “transmission” (in this case, mutation) terms. Please see our response to point 1 in the Evaluation Summary above.
Comments on Figures and results:
Figure 2 – What are the units of the axes on all these graphs?
a) I don't understand this statement in the caption: "We then simulated an external condition where the combination drug concentration is held constant at 0.6 of the maximal concentration of drug 1 and 0.1 of the maximal concentration of drug 2.". There is no such thing as a "maximal" drug concentration … you can always keep adding more and more drug. It would be clearer just to state the drug level used in absolute terms. Or maybe you mean the "highest tested drug level"?
Thank you for the suggestion. We have now rewritten the caption for clarity.
Figure 3 – Some of the results shown here are a little confusing and hard to understand. In the black scenario, there are few available resistant strains with lower α, so we expect selection to proceed in the direction of Drug 2/β.. that agrees with the results in B. But for the blue case where, despite lots of available α but little available β, the trajectory in B seems relatively symmetrical in both drugs … why? Also I was surprised that the case of collateral sensitivity (green) wasn't the worst case … but I realized it's a bit misleading because the four scenarios don't only differ in the nature of the correlations in resistance levels, but also in the magnitude of the variance in α or β and hence the best possible available mutation. I think that it would be a fairer comparison if for the green and magenta scenarios the variance was narrower around the lines of perfect correlation/anticorrelation.
Indeed these results can be counterintuitive. In short, they arise from a combination of the collateral effects and the drug interaction. While the reviewer is correct that the four different cases are not a precise “apples to apples” comparison, the seemingly strange results do make sense in light of the rescaling hypothesis (and are actually not driven by the somewhat smaller differences in correlation parameters generating the collateral profiles). For example, the black trajectory in B is not dominated by resistance to drug 2, which is what one would expect from the collateral effects alone. The reason is that additional resistance to drug 2 does not offer significant growth advantages at this dosage combination due to the drug interactions. Intuitively, one can see this by noting that a rescaling of drug 2 concentration would correspond to moving downward on the contour plot. But moving downward at this location would not change growth dramatically because the rescaled concentration would fall near the same contour as the original drug concentration (i.e. the growth contour itself is nearly vertical at this location). We have added a brief discussion of this point in the main text and also included the specific covariance parameters in the caption.
Figure 4 – This is nice figure that really summarizes the combined effects of drug interactions and cross-resistance profile. However this figure also summarizes the major limitations of this approach. The early slopes of each growth rate vs time curve do show how these factors interact to determine selection strength. However, the fact that the maximum growth rate varies by curve is really just an artifact of the arbitrary and limited available mutants the authors have created for selection to act on. In reality, mutations are continually and sequentially generated over time, and this rate of generation of mutants also impacts the increase in growth rate over time and might remove any "maximum" resistance level that is achievable.
The reviewer makes a good point: the steady state behavior in the no-mutation limit is likely to be determined by growth of a single mutant (or strictly, a series of mutants that fall along a single growth contour), and it is therefore highly dependent on the extremes in the set of scaling parameters. In practice, new dynamics are likely to arise on these long time scales, and the model is expected to fail as mutational processes (and potentially a breakdown of the scaling assumption, in general) become increasingly relevant. The upshot is that the no-mutation model is useful over a limited time horizon, though fortunately, that timescale is similar to that observed in many lab evolution experiments aimed at understanding drug interactions and collateral effects. Perhaps more importantly, we have also added a mutational process to the model in the revised manuscript. While the set of available mutants must still be assigned up front, different mutational structures can be chosen to mimic longer-time processes (e.g. by making mutation rate dependent on the distance between mutants in scaling parameter space, the evolutionary trajectory may proceed through increasingly fit phenotypes). We have also included new supplemental figures that illustrate the connection between the time of observation and potential saturation of the dynamics, though in a slightly different context (Figure 5 figure supplement 4; Figure 7 figure supplements 1-2).
Figure 5 – Are these predicted trajectories using the weak selection limit or the full equations without this additional assumption?
– I'm assuming there was some fixed time limit that selection was allowed to proceed for in these experiments and that it was the same in all cases?
– B: is the y axis absolute IC50 or change in IC50? The axes titles just say IC50 but the figure caption says "relative change in IC50". Also it would be helpful if the baseline IC50 (ie for the wild type strain) where shown on these graph as well.
We have clarified the caption to note that (1) these simulations are from the full equations and (2) these experiments were performed for a fixed time period of 72 hours, and (3) IC50 for each drug is measured in units of the ancestral strain IC50 (so they are indeed normalized IC50s). The wild type IC50 is therefore 1 for each drug (this was not clear in the original version because we didn’t properly indicate that the IC50 is normalized).
Finally, I don't understand how the results for the TGC IC50 agree with the results in A. From B, the IC50 looks totally flat with TGC concentration. But from A, it looks like there is a large increase in TGC resistance (increase in TGC IC50) for high TGC concentrations, but little TGC resistance for low TGC concentrations.
We have now revised the caption to hopefully clarify these points. In panel A, we show a collection of 100 trajectories at each condition, with each trajectory based on a uniformly sampled collection of approximately 10 parameter pairs. The black x marks indicate the mean value (across all trajectories for a given dosage combination) at the end of the simulation. In B, we plot these final-time average values (normalized IC50 corresponds to the reciprocal of the corresponding scaling constant) as a function of TGC concentration in the selecting environment (which here is just a proxy for the different external dosage combinations, the different conditions, as you move left to right on the contour). The discrepancy you note at high TGC concentrations arises from the fact that the trajectories with large increases in TGC are relatively rare and therefore contribute little to the trajectory average.
– C: I don't understand what the y axis is here. If this is the increase in growth rate, then why is it highest for low TGC concentrations, yet both panels in B show no change in IC50 to either drug at low TGC concentrations?
The axis shows increase in growth rate, which we have clarified in the caption. At low TGC concentrations, panel B shows that the change in TGC IC50 is minimal (TCG scaling parameter near 1) while the change in CIP IC50 is larger (it increases by a factor of roughly 3-4). In terms of rescaling, this trend makes intuitive sense: a vertical rescaling (increased resistance to CIP) leads to an increase in growth for conditions at low TGC, where the rescaling moves the effective concentration into a region of higher growth. But for higher TGC concentrations, such a vertical concentration rescaling would not be expected to have much effect on growth because a vertical shift in effective drug concentration does not lead to large deviations from the original growth contour (and in fact experiments in this regime did not select for large increases in CIP resistance). In words: TGC has a strongly suppressive effect on CIP (strongly antagonistic drug interaction). As a result, low TGC concentrations select for CIP resistance while higher TGC concentrations do not, even when the CIP concentrations are similar.
And any idea what could be causing the systematic difference between the predicted and observed changes in growth rate?
This is an interesting question, and we don’t know the answer. One potential explanation is that the time chosen for the simulation (72 hours) does not match the true time over which selection was taking place in the experiment. The experiment was performed for 72 hours, but various environmental variables (fluctuations in temperature, density-dependent changes in growth rate, etc) could mean that the evolution occurred, effectively, for more or less than 72 hours. The upshot is that the total number of generations may not match perfectly between model and experiment. We did not attempt to tune this parameter to fit the data, but indeed we can decrease the systematic error by choosing a different value of T (but we prefer to show the model without fitted parameters). It is also possible that the systematic error arises from other features, such as uncertainty in the estimates of the wild-type growth surface. We have now explicitly noted and briefly discussed the systematic discrepancy in the main text.
Figure 6 – This figure was quite confusing, it needs more annotation on the figure itself (not just written in the caption), and some panels are unnecessarily small (with unnecessary white space between them). For panels H and K, are there better axis limits that could be used so that 95% of the figure is not all one color? Also I found it is hard to see in this figure any of the differences between the outcomes of the two different orderings that are described in the text.
We have replaced this figure (now Figure 7) with a simplified version. Our hope is to highlight one way that the framework can be extended to make predictions in an otherwise intractably complex scenario. However, the previous figure was unnecessarily confusing. In the new figure, we choose a simpler example than in the previous version, and we focus on a smaller set of the dynamics.
Reviewer #3:
In this work, Gjini and Wood create a model to simulate evolutionary dynamics of antibiotic resistance arising as a response to drug combinations. They demonstrate that evolution to these drug combinations is a variation of the Price equation and show that some evolutionary paths experimentally correspond to a weighted gradient descent on the growth landscape.
The work appears mathematically sound, though we have concerns about the scope and generality of the results. In particular this work appears to rely fundamentally on the assumption that drug resistance acts as a linear rescaling of the two-drug dose response curve. We believe this work would be strengthened by making it more clear what, beyond the mathematical description of the rescaling hypothesis they cite extensively, this work adds to the literature. Since the model appears to also require an a priori description of the resistance characteristics of every possible accessible mutant, which typically is not known, it isn't clear the utility of the model.
Thank you for these suggestions. In our view, there are two main advances. First, we incorporate multiple features of the evolutionary process (collateral effects, drug interaction, and, in the new version, mutational structure) into a model that makes testable predictions. As with all models, the outputs are only as good as the inputs, and we can not use the model to predict resistance evolution without those inputs (e.g. the drug interaction and statistical properties of the collateral effects). However, when those properties are known, or when we can make reasonable assumptions about them, the model allows us to make predictions that would not otherwise be possible and to systematically investigate the impacts of different phenomena (e.g. collateral effects vs drug interactions). Second, we believe the approach has considerable conceptual value. It provides a direct link to the Price Equation, which connects the work with classical approaches in population genetics. Perhaps more importantly, the Price Equation formalism allows us to mathematically disentangle effects of collateral resistance and drug interactions, at least in the stated limits, in a way that highlights how the geometry of the response surface interacts with statistical properties of the available mutants.
We also note that in the new version, we now incorporate mutation between different phenotypes, which significantly extends the scope of the model (see Figure 6 and Figure 6 figure supplements).
Please also see our responses to points 1 and 2 in the Evaluation Summary (above).
We also copy below a (somewhat related) response to a suggestion from Reviewer 1:
We thank the reviewer for recognizing our efforts to be transparent, for highlighting the value of the formalism, and also recognizing the difficulty of developing theoretical biology with broad scope, and for the generally positive appraisal. The revised version extends the formalism to include mutations, which allows us to incorporate empirical constraints on the mutational pathways linking different phenotypes (similar, though not identical, to the information contained in fitness landscapes). The trade-off with this added complexity is, of course, the need for additional assumptions, and while we have focused on simple illustrative examples (e.g. sequential mutations, Figure 6, and distance-dependent mutations in parameter space, Figure 6 figure supplements 3-4) we believe they illustrate that the approach is flexible and amenable to new empirical constraints as they become available. We discuss these points in the revised manuscript, and (most notably) we also added the following to the discussion:
“Our goal was to strike a balance between analytical tractability and generality vs biological realism and system specificity. We stress that the predictions of this model do not come for free; they depend, for example, on properties of the dose response surfaces, the collection of scaling parameters, and the specific mutational structure. In many cases, these features can be determined empirically; in other cases, some or all of these properties may be unknown. The evolutionary predictions of the model will ultimately depend on these inputs, and it is difficult to draw general conclusions (e.g. “synergistic combinations always accelerate resistance”) that apply across all classes of drug combinations, collateral effects, and mutational structures. But we believe the framework is valuable precisely because it generates testable predictions in situations that might otherwise seem intractably complex.“
– This paper models evolution in the presence of physiological interactions between antibiotics but not evolutionary interactions. The fundamental assumption that evolution of resistance scales the surface but does not change its shape, which is central to the findings, is true in some cases, though violated for many mutations (the authors cite numerous papers on cross-resistance and collateral sensitivity that are examples of this).
We agree that the rescaling assumption will not always hold, and we discuss these limitations extensively in the discussion. We also discuss the fact that the model does not incorporate interactions between cells (e.g. cooperation and competition), it is an avenue for future work, we hope! We would like to point out, however, that cross-resistance and collateral sensitivity in and of themselves are not indications that the scaling assumption is violated (testing this would require measuring the full dose response surface in the mutants and determining whether a simple rescaling of the two axes is sufficient to link the two surfaces, that is, whether the shape of the surface changes, not merely the axes scales). There is considerable experimental support suggesting the scaling assumption provides a reasonable approximation, at least for conditions of a typical lab evolution experiment, but indeed the assumption will not always hold, and we do note that important limitation.
– While the authors describe it as "straightforward" to include discrete mutational events instead of a small population of equal numbers of every potential mutant in the model, it is not clear that this is the case. For example, a population exposed to an above-MIC concentration of a bacteriostatic drug will no longer generate additional mutants, and so the standing diversity before drug application matters critically. As it stands the model is really about differential growth and not evolution per se, since neither mutation nor complete extinction can occur.
We have incorporated mutation in the revised manuscript (see response to point 1 in the Evaluation Summary, above). But the reviewer makes an important point: the model accounts only for differences in growth rates between different strains. It does not account for changes in total population size, though these effects would be particularly important as the population nears extinction. The model also neglects demographic stochasticity, which could play a significant role in some regimes. We’ve added a short discussion of these points in the discussion.
– It isn't clear the experiments are representative of nontrivial behavior predicted by the model. For example in figure 5 no adaptation to TGC was seen, so it seems the only real evolution was rescaling CIP, which wouldn't yield the suppression-induced selection against resistance they then discuss. These results could be entire an artifact of the selection being entirely on CIP and not at all on TGC. The evolution experiments in Figures S3 and S4 seem much more compelling, though.
We found this example quite compelling, and at first we were puzzled by your comment. But then we realized that there was a bug in the code that plotted the experimental results, which obscured the most interesting features of the experimental data (they were still present in the model). Correcting this plotting error also led to an improved agreement between experiment and model. Thank you for your comment, which helped us avoid a silly mistake (the bug was apparently introduced during an internal revision of the figure prior to submission; it wasn’t there in our original internal drafts but showed up in the version we submitted, and we simply missed it).
With that bug now fixed, hopefully it is clear why we find the example compelling. Selection for CIP resistance occurs only for sufficiently low concentrations of TGC, even in cases where inhibitory concentrations of CIP are present. For example, evolution in the far left condition (TGC~0, CIP~0.2) leads to a 2-3 fold increase in CIP IC50 (panel B) and significant growth adaptation (panel C). However, evolution at the 3rd condition from the right (TGC~0.04, CIP~0.2) shows very little resistance to CIP, despite the fact that a similar concentration of CIP was used. The fourth condition from the right is even more striking, as there the concentration of CIP is even higher (about 0.35) yet little CIP resistance is observed.
– If it is indeed straightforward to add mutational events, this would make the model appreciably more realistic.
We have now incorporated mutation in a new section of the manuscript. Please see response to point 1 in the Evaluation Summary (above).