Genetic analyses, which examine the phenotypic effects of mutations both individually and in combination, have been fundamental to our understanding of cellular functions. Such analyses rely on a neutrality function that predicts the expected phenotype for double mutants based on the phenotypes of the two individual non-interacting mutations. In this study, we examine fitness, the most fundamental cellular phenotype, through an analysis of the extensive colony growth rate data available for budding yeast. Our results confirm that the Product neutrality function describes the colony growth rate, or fitness, of a double mutant as the product of the fitnesses of the individual single mutants. This Product neutrality function performs better than Additive or Minimum neutrality functions, supporting its continued use in genetic interaction studies. Furthermore, we explore the mechanistic origins of this neutrality function by analyzing two theoretical models of cell growth. We perform a computational genetic analysis to show that in both models, the Product neutrality function naturally emerges due to the interdependence of cellular processes that maximize growth rates. Thus, our findings provide mechanistic insight into how the Product neutrality function arises and affirm its utility in predicting genetic interactions affecting cell growth and proliferation.
epistasisbudding yeastgenetic interactionssystems biologyquantitative geneticscell growthResearch organismS. cerevisiaehttp://dx.doi.org/10.13039/100000002National Institutes of HealthGM134858SkotheimJan MThe funders had no role in study design, data collection, and interpretation, or the decision to submit the work for publication.Author impact statementCoarse-grained models of cellular growth and high-throughput empirical genetic data support the Product neutrality function in which a double-mutant fitness is the product of single-mutant fitnesses.publishing-routeprcIntroduction
Genetic analysis has been one of the primary methods scientists use to understand how a cell works. One way this is done is through the analysis of genetic interactions, in which the phenotypic effects of mutations are analyzed both as single mutations and then together as double mutations in the same cell. Genes are then said to interact if their combination produces phenotypes that are different from what is predicted from a generic model combining two non-interacting mutations (Phillips, 2008; Beltrao et al., 2010; Costanzo et al., 2019). In other words, a genetic interaction is identified when combining multiple mutations yields something unexpected. Yet, what should we expect when combining mutations in such a complex system as a living cell?
The expected phenotype of a double mutant predicted from the two single mutants’ phenotypes is defined by the neutrality function. In this way, the neutrality function calculates the expected phenotype of a double-mutant strain carrying two non-interacting mutations (Beltrao et al., 2010; Mani et al., 2008). If the double-mutant phenotype deviates significantly from that given by the neutrality function for two specific mutations, they are then said to interact. A lot of care, therefore, needs to be taken in selecting the appropriate neutrality function, which depends on the context and phenotype to be examined (Phillips, 2008). The neutrality function should be defined such that most mutations are categorized as non-interacting. If the neutrality function were not defined this way, the majority of genes would appear to interact, leaving only a few genes with distinct functions. However, decades of cell biological and structural biological analysis have identified specific functions for many genes and their associated proteins. For example, the components of the ribosome or RNA polymerase have the specific task to form these complexes, and metabolic enzymes catalyze specific biochemical reactions. This implies that a judiciously selected neutrality function should predict most double-mutant fitnesses from the single-mutant fitnesses since any two randomly selected genes should be unlikely to interact. Recent technological advances have enabled the screening of the proliferation of single and double genetic mutants at increasingly larger scales in E. coli (Typas et al., 2008; Butland et al., 2008; Babu et al., 2014), fission yeast (Roguev et al., 2008; Dixon et al., 2008), C. elegans (Lehner et al., 2006; Byrne et al., 2007), and human cells (Horlbeck et al., 2018). In budding yeast, Synthetic Genetic Arrays (SGAs) Tong et al., 2001 have generated the largest and most comprehensive such datasets (Baryshnikova et al., 2010b; Costanzo et al., 2010; Costanzo et al., 2016).
The most fundamental phenotype of a cell is its fitness, namely how quickly it grows and proliferates in a given environment. Here, we focus on this property and define fitness as the relative exponential growth rate with respect to that of a wild-type cell. For yeast proliferation, single-mutant fitnesses are usually assumed to combine according to a Product neutrality function, namely that the fitness of the double mutant is the product of the fitnesses of the single mutants (Mani et al., 2008). This neutrality function was shown to better predict double-mutant fitnesses than an Additive neutrality function, where the differences between wild-type and mutant fitnesses were simply added together, and a Minimum neutrality function, where the double-mutant fitness was taken as the lowest fitness of the two single-mutant strains. However, this analysis was based on older, less extensive data, which raises the question of whether this neutrality function remains accurate when the large amount of more recently collected yeast data is also considered. And, if so, then why does the Product neutrality function accurately describe double-mutant fitnesses? In other words, what are the properties of the underlying system controlling cell growth and proliferation that result in a Product neutrality function for mutant fitnesses?
In this paper, we conduct an in-depth analysis of recent yeast double-mutant datasets and show that they support the Product neutrality function. Moreover, we analyze two theoretical models of growth of increasing complexity (Scott et al., 2010; Weiße et al., 2015) and find that the Product neutrality function emerges naturally from both growth models, albeit with small deviations specific to each. Taken together, our work supports the use of the Product neutrality function to model genetic interactions in the regulation of cell growth and gives mechanistic insight into its origin.
ResultsHigh-throughput gene perturbation experiments in budding yeast support a Product neutrality function for double-mutant fitness
To test the general validity of neutrality functions for mutations affecting cell proliferation, we sought to examine the most extensive such dataset. In the SGA dataset, over 20 million single- and double-mutant budding yeast strains were generated. Then, the growth rates of their colonies were measured in SGAs (Tong et al., 2001; Baryshnikova et al., 2010b; Costanzo et al., 2010; Costanzo et al., 2016). Each mutant’s fitness was then defined as this measured growth rate normalized by that of the wild-type strain, enabling a consistent comparison across thousands of genotypes. Next, we use SGA datasets of growth of single- and double-mutant cells for pairs of mutations to test specific neutrality functions.
Here, we followed (Mani et al., 2008) and began our examination using the Product, Additive, and Minimum neutrality functions (see Figure 1A). The Product neutrality function predicts that the fitness of a double mutant is the product of the fitnesses of the two corresponding single mutants, while the Additive neutrality function proposes that the difference between the double-mutant and wild-type fitnesses is the sum of the differences between the two mutant and wild-type fitnesses. The Minimum neutrality function proposes that the fitness of the double mutant is equal to the fitness of the least fit single mutant. Effectively, these neutrality functions express different forms of modularity or independence between cellular processes. The Product model suggests that a mutation’s effect depends on the fitness of the background strain without that mutation, while a mutation’s effect is independent of the fitness of the background strain in an Additive model. The Minimum model suggests that there is some rate-limiting process whose slow time scale dominates the determination of cell growth so that more minor mutations affecting other processes have no additional effect.
High-throughput gene deletion experiments in budding yeast support a Product neutrality function for double-mutant fitness.
(A) Budding yeast mutant fitness is defined as the colony growth rate relative to that of wild-type cells. Schematic illustration of epistasis in growth rate and of different laws proposed in the literature. λ denotes the growth rate, and W the fitness. (B–D) For each double mutant, we plot the residual of the fitness predicted from the indicated model against the fitness of the fittest of the two separate single mutants (maximum single-mutant fitness). Dots indicate the median for 10 equally spaced bins between 0.5 and 1. (E) Box plots for the distributions of the residuals for the three neutrality functions as a function of the maximum single-mutant fitness. A thick line denotes the median, and boxes denote the 25th and 75th percentiles of the distributions. The data plotted here represents a subset of the entire Synthetic Genetic Array (SGA) dataset, corresponding to the Deletion Mutant Array (DMA) at 30°C.Appendix 1—figures 1 and 2 report results for the other subdatasets.
To compare the different neutrality functions with double-mutant fitnesses, we first perform some minor pre-processing of the SGA data (see Methods for details). We note that these data are from combinations of gene deletions, temperature-sensitive alleles, and hypomorphic mutants (see Methods). Moreover, cells were growing quickly on the relatively rich synthetic complete media containing glucose (Baryshnikova et al., 2010a). For these reasons, increasing the growth rate is difficult, and we are in the regime where mutations generally decrease fitness. This contrasts with evolution experiments, where fitness increases very slowly through the gradual accumulation of mutations, which likely exhibit different neutrality functions from those we consider here (Phillips, 2008; Bakerlee et al., 2022; Johnson et al., 2023). Consistent with previous work (Mani et al., 2008), we see that the Product neutrality function better predicts double-mutant fitnesses as a function of the single-mutant fitnesses over a broad range of fitness defects (Figure 1, Appendix 1—figure 1). Indeed, the median residual for the Product neutrality function remains very close to zero even for highly deleterious mutations, while it significantly deviates for the other two. For instance, for a maximum single-mutant fitness of 72%, the median residual is −0.8% for the Product neutrality function, while they are −16.8% and 9.8% for the Minimum and Additive neutrality functions, respectively. For smaller values of the maximum single-mutant fitness, the median residual remains virtually unchanged for the Product neutrality function, while it deviates even further from zero for the other two. However, we note that significant variation around the median residual remains. At a maximum single-mutant fitness of 72%, the interquartile range lies between 8% and 10% depending on the neutrality function considered.
This observation that the Product neutrality function best describes double mutant fitnesses holds for mutations to essential and nonessential genes, and across different temperature conditions (see Methods and Appendix 1—figures 1 and 2). The Minimum neutrality function generally predicts fitnesses that are too high, while the Additive neutrality function generally predicts fitnesses that are too low.
The Product neutrality function describes interactions between genes associated with two distinct biological processes
While the Product neutrality function predicts double-mutant fitnesses better than the other ones we considered, there remains significant variation (residuals) in the data. This suggests that mutations affecting different functional parts of the cell might be following different neutrality functions. To determine whether this is the case, we analyze the distribution of epistasis residuals for pairs of distinct biological processes and their associated genes. We use the Gene Ontology (GO) dataset (Ashburner et al., 2000, Aleksander et al., 2023) and, for each biological process, extract the genes in the SGA dataset that are associated with this process (see Methods for details). In particular, we define inter-process gene pairs as pairs of gene perturbations in two distinct biological processes identified using GO annotations (Figure 2A). Similarly, intra-process pairs are defined as pairs of gene perturbations in the same GO-annotated biological process (Appendix 1—figure 3A). Then, for each pair of GO-defined processes, we compute the residuals for the neutrality functions for all pairs of mutations where one mutation is associated with one process and the second mutation with the other. For each neutrality function and pair of GO processes, we extract the median residual as a function of the largest single-mutant fitness defect (a proxy for mutation severity). This shows that, while imperfect, the Product neutrality function is a good description of typical interactions, while the Additive and Minimum neutrality functions have large, systematic residuals (Figure 2B–D). In general, this result is expected and consistent with the use of this type of genetic analysis to define mutations in genes from different biological processes as not interacting. Moreover, we find that this is not only generally true, but also true for each specific pair of processes that we consider. In other words, we do not find evidence that there are particular pairs of processes whose mutations significantly deviate from the Product neutrality function or more closely follow an Additive or Minimum neutrality function.
The Product neutrality function describes interactions between genes associated with two distinct biological processes.
(A) Schematic illustration of the analysis process. We first select two different Gene Ontology (GO) biological processes and extract the double mutants in the Synthetic Genetic Array (SGA) dataset associated with them. Then, we compute the median residual for each pair of biological processes and each neutrality function. (B–D) Median residual for the Minimum, Product, and Additive neutrality functions as a function of the maximum single-mutant fitness. Each line denotes mutations to a different pair of distinct GO biological processes. The majority of biological process pairs closely follow the Product model.
While mutations associated with different biological processes have fitnesses generally predicted by the Product neutrality function, since they generally do not interact, this may not be the case for mutations associated with the same process. For example, if two mutations break the same protein complex, one would not expect any additional drop in fitness for the double mutant. Consistent with this notion, for gene pairs in the same biological process, we observe more deviations as well as significantly larger residuals (Appendix 1—figure 3B–D). The quantitative comparison of the two types of interactions reveals that large residuals (both positive and negative) are significantly more likely for two mutations categorized as being in the same GO process (Appendix 1—figure 3E, F).
We note that the SGA dataset has already been used to assign a biological process to each gene (Costanzo et al., 2016). For each mutation, a vector of residuals for the Product neutrality function with all other mutations was generated. Then, a Pearson correlation coefficient was calculated for each pair of these vectors. The reasoning was that mutations affecting the same biological process should have similar genetic interaction profiles, which was found to be the case. This then allowed the clustering of groups of correlated mutations, which were named using prior knowledge of many genes in each cluster. That this analysis generally works, that is, the gene clusters have discernible biological meaning, can be viewed as further support of the Product neutrality function.
A bacterial growth model partially supports the Product neutrality function
Having verified empirically that the Product neutrality function is supported by the latest data for cell proliferation, we now turn our attention to its origins. Addressing this question requires some mechanistic model of biosynthesis. However, most mechanistic models of growth apply directly to single cells in rich nutrient conditions, which may not directly apply to the SGA measurements of colony expansion rates. In particular, colony growth has been shown to follow a biphasic pattern (Meunier and Choder, 1999). A first exponential phase is followed by a slower linear phase as the colony expands. Previous modeling and empirical work indicates that this second linear expansion rate reflects the underlying exponential growth of cells in the periphery of the colony (Pirt, 1967; Gray and Kirwan, 1974; Baryshnikova et al., 2010a; Gandhi et al., 2016; Zackrisson et al., 2016; Miller et al., 2022). More precisely, mathematical models show the linear colony-size expansion rate is directly proportional to the square root of the exponential growth rate under non-limiting conditions. Intuitively, this relationship arises because colony growth is dominated by the expansion of the population of cells in an annulus at the colony border that are exposed to rich nutrient conditions. These cells expand at a rate similar to the exponential rate of cells growing in a rich nutrient liquid culture. In contrast, the cells in the interior of the colony experience poor nutrient conditions, grow very slowly, and do not contribute to colony growth.
This intimate relationship between both proliferation rates allows us to explore the origin of the Product neutrality function in mechanistic models of cell growth. Indeed, if colony-based fitnesses follow a Product model, thenWxyc∼WxcWyc⇔λxycλWTc∼λxcλyc(λWTc)2,\begin{document}$$\displaystyle W^{\,c}_{xy}\sim W^{\,c}_{x}W^{\,c}_{y}\Leftrightarrow \frac{\lambda^{c}_{xy}}{\lambda^{c}_{WT}}\sim \frac{\lambda^{c}_{x}\lambda^{c}_{y}}{(\lambda^{c}_{WT})^{2}},$$\end{document}
where the superscript c\begin{document}$c$\end{document} indicates colony-based values for the fitness W\begin{document}$W$\end{document} and the growth rate λ. Taking into account the relationship between single-cell exponential growth rates and colony growth rates, we can writeλc∝λl,\begin{document}$$\displaystyle \lambda^{c}\propto \sqrt{\lambda^{l}},$$\end{document}
where the superscript l\begin{document}$l$\end{document} denotes liquid cultures. Combining these expressions, we obtainλxylλWTl∼λxlλylλWTl2⇒Wxyl∼WxlWyl.\begin{document}$$\displaystyle \frac{\sqrt{\lambda^{l}_{xy}}}{\sqrt{\lambda^{l}_{WT}}}\sim \frac{\sqrt{\lambda^{l}_{x}}\sqrt{\lambda^{l}_{y}}}{\sqrt{\lambda^l_{WT}}^{2}}\Rightarrow W^{\,l}_{xy}\sim W^{\,l}_{x}W^{\,l}_{y}.$$\end{document}
In other words, from the perspective of the Product neutrality function, fitnesses based on colony expansion rates are equivalent to fitnesses based on single-cell exponential growth rates. The prevalence of the Product neutrality model—both in the SGA data and in previous studies on datasets from liquid cultures (Mani et al., 2008; Jasnos and Korona, 2007; Onge et al., 2007)—encourages the exploration of its origin in mechanistic models of cell growth.
While models of entire cells do exist, these models are complex and computationally intensive (Onge et al., 2007; Karr et al., 2012). This makes probing and extracting explanatory information from these models difficult. We therefore sought to analyze simpler, more tractable, lower-dimensional models of cell growth. Coarse-grained models offer an appealing alternative for probing the fundamental principles of metabolism and growth (Scott et al., 2010; Weiße et al., 2015; Roy et al., 2021; Balakrishnan et al., 2022; Chure and Cremer, 2023; Calabrese et al., 2023). Rather than representing as many reactions as possible, they provide an integrated representation of generic processes in the cell. Their simplicity and low dimensionality make them easy to compare with empirical measurements and to examine for potential explanatory relationships.
The reduced, tractable model of cell growth that we will consider first was developed for E. coli (Scott et al., 2010; Scott and Hwa, 2011; Figure 3A). While this model was developed for E. coli bacteria and validated using data from this organism, there is nothing specific to prokaryotes in the model. Experimental measurements in other organisms suggest that the observations leading to this model, including that the cellular ribosome fraction increases with growth rate, are in fact generic and also seen in the yeast S. cerevisiae (Metzl-Raz et al., 2017; Elsemman et al., 2022; Xia et al., 2022). In its simplest form, the model defines growth as resulting from two sets of processes, metabolic and translational, that interact in a linear pathway. The metabolic sector provides precursors that are then assembled into proteins by the translational sector, and the flux through each sector is determined by the amount of proteins in that sector. For optimal growth, in which no proteins are wasted, the flux through the metabolic sector is equal to the flux through the translational sector so thatλ=κtϕt=κnϕn,\begin{document}$$\displaystyle \lambda = \kappa_{t}\phi_{t}= \kappa_{n}\phi_{n},$$\end{document}
A bacterial growth model partially supports the Product neutrality function.
(A) Schematic of the bacterial growth model by Scott and Hwa. Growth rate is defined by the translation flux, which is itself equal to the metabolic flux. The cell partitions its proteome so as to maximize growth rate. (B) Mutations are modeled such that they affect either of the parameters, separately. Values of κt\begin{document}$\kappa_t$\end{document} and κn\begin{document}$\kappa_n$\end{document} in the mutant are indicated with primes and are sampled from a uniform distribution from 0 to their value in wild-type cells. indicates the corresponding growth rate. The analytical expression of the double-mutant fitness consists of the Product model with a perturbation. (C–E) For each sampled double mutant, we plot the residual of the fitness predicted from the indicated model against the fitness of the fittest of the two separate single mutants (maximum single-mutant fitness). Dots indicate the median for 10 equally spaced bins between 0.5 and 1. (F) Box plots for the distributions of the residuals for the three models and the model in (C) as a function of the maximum single-mutant fitness. A thick line denotes the median, and boxes denote the upper and lower quartiles of the data. The analytical model in (B), named Scott–Hwa and shown in red, is exact.
where the flux through translation and metabolic sectors is characterized by the parameters κt\begin{document}$\kappa_{t}$\end{document} and κn\begin{document}$\kappa_{n}$\end{document} multiplying the fraction of the proteome devoted to ribosomes and metabolic proteins, ϕt\begin{document}$\phi_{t}$\end{document} and ϕn\begin{document}$\phi_{n}$\end{document}, respectively. These fluxes directly determine the growth rate of the cell, λ\begin{document}$\lambda$\end{document}. The total proteome is fixed and is partitioned into metabolic, translational, and ‘other’ sectors of the cell so that1=ϕt+ϕn+ϕo\begin{document}$$\displaystyle 1 = \phi_{t}+ \phi_{n}+ \phi_{o}$$\end{document}
where ϕo\begin{document}$\phi_{o}$\end{document} is the fraction of the cell devoted to other housekeeping functions. This set of algebraic equations can be solved for the optimal growth rateλ=(1−ϕo)11/κt+1/κn.\begin{document}$$\displaystyle \lambda =(1-\phi_{o}) \frac{1}{1/\kappa_{t}+ 1/\kappa_{n}}.$$\end{document}
We can then model a mutation as a perturbation to these parameters that decreases the growth rate since these are the types of mutations that dominate the budding yeast data (Figure 3B). Single mutants have either κt\begin{document}$\kappa_{t}$\end{document} or κn\begin{document}$\kappa_{n}$\end{document} perturbed, while double mutants have both parameters perturbed. Given that we are concerned with two non-interacting mutations, we do not consider the cases where both mutations affect the same parameters. Indeed, one expects two mutations affecting the same parameter to interact. These combinations are therefore inappropriate to study the emerging neutrality functions from growth models, and we do not consider them in this paper.
We can also consider mutations to ϕo\begin{document}$\phi_{o}$\end{document}, which could be associated with deleting a gene encoding a protein that is not required for the given growth condition. This would serve to increase the cell growth rate because now a larger fraction of the proteome could be devoted to metabolism and translation. In this case, a mutation to ϕo\begin{document}$\phi_{o}$\end{document} and another to either κt\begin{document}$\kappa_{t}$\end{document} or κn\begin{document}$\kappa_{n}$\end{document} would combine exactly multiplicatively so thatWxy=WxWy.\begin{document}$$\displaystyle W_{xy}= W_{x}W_{y}.$$\end{document}
However, in the generally rich media conditions the SGA experiments were done, there is no evidence that any gene deletion causes an increase in cell growth rate so we do not consider this type of mutation further. We therefore ignore the multiplicative factor (1−ϕo)\begin{document}$(1-\phi_{o})$\end{document} and analyze the following expression for growth rateλ=11/κt+1/κn.\begin{document}$$\displaystyle \lambda=\frac{1}{1/\kappa_{t}+ 1/\kappa_{n}}.$$\end{document}
We can then analytically derive a closed-form solution for the double-mutant fitness as a function of the single-mutant fitnesses (Figure 3C; see SI for details). Under this model, which we call Scott–Hwa in reference to the authors of the initial work, we observe that the Product neutrality function fits the mutational analysis of the Scott–Hwa model better than the Additive or Minimum neutrality functions (Figure 3D-G).
We understand the better performance of the Product neutrality function to arise from a type of feedback regulation that ensures that the flux through all sectors is equal. This effectively makes the mutations to the metabolic and translational sector interdependent, despite their a priori independent functions. In this model, these sectors are coupled because the cell is assumed to have a feedback process to optimize growth rate under any perturbation to its parameters. For instance, in the case of a mutation that decreases the metabolic capacity κn\begin{document}$\kappa_{n}$\end{document}, this feedback drives an increase in the fraction of metabolic proteins at the expense of translational proteins such that the growth rate is maximized under those new parameters. If this feedback were absent, then the growth rates of the double mutant would be significantly lower. In that case, the double-mutant fitness actually follows a Minimum neutrality function (see Appendix 1 and Appendix 1—figure 4). Finally, we also note that the Product neutrality function does not accurately predict model fitnesses well for beneficial mutations—that is, mutations that increase growth rate (see Appendix 1—figure 5). This is because the deviations from the Product neutrality function in the mathematical derivation for the double-mutant fitness in Figure 3 can diverge for beneficial mutations (e.g. Wx=Wy=2\begin{document}$W_{x}= W_{y}= 2$\end{document}). When the Scott–Hwa model’s growth-optimizing feedback operates in the context of beneficial mutations, as one process is made more efficient, proteomic resources are allocated to accelerate other processes in the cell. In this way, improving the efficiency of one process will indirectly benefit other processes, leading to compound effects such that double-mutant fitnesses are higher than any of the three models predicts for beneficial mutations.
We note that in this analysis, we do not aim to replicate the statistics of mutations in the SGA dataset, where mutations to either sector could be statistically rarer or more frequent than the other. Instead, we here aim to analyze how mutations to independent parameters governing cell growth combine considering the simplest model with two sectors and their corresponding parameters.
The Product neutrality function accurately predicts fitness for many pairs of parameters in a more complex cell growth model
While the Scott–Hwa model has proven successful for predicting many aspects of bacterial growth, it remains very simple. Therefore, we sought to explore a more complex model that explicitly incorporates more aspects of biosynthesis. Here, we consider the model of Weiße et al., 2015, which incorporates nutrient intake, transcription, competitive binding between mRNAs and ribosomes, and translation, all of which are mediated by associated enzymes and a limiting cellular ‘energy’ (Figure 4A). The Weiße model decomposes cell growth into multiple steps (see supporting material for a full model description). External nutrients are first imported into the cell and then metabolized into a cellular ‘energy’. Both of these steps are catalyzed by associated transport and metabolic enzymes according to Michaelis–Menten kinetics. Transcription and translation are then activated by this generated ‘energy’, also via Michaelis–Menten kinetics. In particular, the model incorporates different transcription rates for ribosomal and non-ribosomal mRNAs. Different mRNAs then compete for free ribosomes to form a ribosome–mRNA complex. This mRNA competition is modeled using mass action kinetics with specified binding and unbinding rates. Four types of proteins are explicitly modeled as a product of translation: transport proteins, metabolic enzymes, ribosomal proteins, and so-called q-proteins which support housekeeping functions much like the ‘other’ proteins in the Scott–Hwa model.
The Product neutrality function accurately predicts fitness for many pairs of parameters in a more complex cell growth model.
(A) Schematic of the growth model from Weiße et al., 2015. This model includes nutrient intake, metabolism, transcription, and translation. (B) Schematic of the mutational analysis. For each pair of parameters α and β, mutations are modeled such that they affect either of the parameters, separately. Then, the median residual is computed for each neutrality function and they are subsequently reported for every pair of parameters considered. For each parameter pair, we report the mean deviation of the simulated double mutants from (C) the Product neutrality function and (D) the analytical expression of the double-mutant fitness under the Scott–Hwa model. Only parameter pairs corresponding to two different biological processes are considered. Those corresponding to the same process are grayed out. Parameter pairs involving translation (gmax,Kp\begin{document}$g_{max},K_p$\end{document}) are the ones described best by the Scott–Hwa model, while the others are better described by the Product neutrality function.
To perform a mutational analysis of the Weiße model, we first identified the parameters in the model that can reasonably be expected to change through a gene perturbation (see supporting material for details). For instance, we assume that some parameters, such as the maximum nutrient import rate, could be impacted by a gene perturbation, while other parameters, such as the average gene length in the genome, could not. This led to the identification of 9 easily interpreted parameters whose mutation could negatively impact the cell growth rate and correspond to 28 parameter pairs associated with different biological processes. We then performed a similar analysis as we did for the Scott–Hwa model. Namely, we constructed mutants for each pair of parameters by rescaling the values of the original parameters by a number randomly sampled between 0 and 1 (see Methods for details). We then analyzed the statistics of the epistasis coefficients for the neutrality functions that we considered so far (Figure 4B).
Our mutational analysis of the Weiße model revealed several striking observations. First, the Product neutrality function is generally better than the Additive or Minimal neutrality functions at describing the mutational results. However, we observe a range of responses and can identify two key subpopulations of parameter pairs. The subset of parameter pairs involving protein translation follows the Scott–Hwa model very closely (Figure 4D). This is not entirely surprising, as the Weiße model is an extension of the Scott–Hwa model and incorporates a similar global feedback optimizing cell growth and a competition for resources. On the other hand, another subset of parameter pairs follows the Product neutrality function even more closely. These parameter pairs involve mutations to the other sectors, including the transport, metabolism, and transcription sectors (Figure 4C). This raises the question of why some parameter pairs more closely follow the Product neutrality function than others.
Nonlinear kinetics drive deviations from the Product neutrality function in the Weiße model for cell growth
To address the question as to what drives deviations from the Product neutrality function in our genetic analysis of the Weiße model, we took an analytical approach. We examined the dependence of the growth rate on the parameter pairs exhibiting small deviations from the Product neutrality function. To do this, we first extracted a closed form expression that models the growth rate λ and its dependence on two mutated parameters α and β (Figure 5A; see supporting material). While only approximate, this derivation represents the data appropriately for members of this subset of parameters (see Appendix 1—figure 7). There are two striking features in this derivation. First, the growth rate λ has an explicit dependence on the two parameters α and β, when α and β are selected from the subset of parameters governing metabolism and transport sectors. Second, the amplitude of deviation from the Product neutrality function is governed by an additional parameter, γ, which is the inverse of the Michaelis–Menten constant giving the transcription rate as a function of the cellular ‘energy’. Thus, when γ\begin{document}$\gamma$\end{document} is small, transcription is a less efficient process that is then linearly related to the cellular ‘energy’ available. When γ\begin{document}$\gamma$\end{document} is larger, transcription is saturated and performed at a rate unrelated to the available ‘energy’. As we decrease γ\begin{document}$\gamma$\end{document}, we observe that the Product neutrality function is a better and better approximation (Figure 5C). Importantly, the intuition provided by the analytical approximation extends to multiple pairs of parameters (see Appendix 1 and Appendix 1—figure 8). Taken together, our analysis of the Weiße model shows how the Product neutrality function naturally arises for many different parameter pairs and how deviations from it can be driven by nonlinear effects, such as those that can emerge from Michaelis–Menten kinetics.
Nonlinear kinetics drive deviations from the Product neutrality function in the Weiße model.
(A) A subset of parameter pairs we analyzed follows the Product model very closely. We derived an analytical approximation of the growth rate and the double-mutant fitness for these pairs and found that the deviation from the product law is governed by nonlinear kinetics. (B) In the case of the parameter pair (vt,ns)\begin{document}$(v_{t}, n_{s})$\end{document}, we show that the deviation from the Product model is driven by the Michaelis–Menten constant θx\begin{document}$\theta_{x}$\end{document} associated with transcription (see Supporting Information). (C) Tuning the value of γ impacts how good of an approximation the Product neutrality function is for this and other parameter pairs (see text). This analysis validates the analytical approximation and highlights how nonlinear kinetics, in this case Michaelis–Menten kinetics, can drive deviations from the Product neutrality function.
Discussion
Cell growth and proliferation are fundamental to cell biology and have been subject to extensive genetic analysis aiming to understand the underlying regulatory network. Such genetic analysis often aims to identify interactions between mutations through the combination of individual mutations in a double-mutant cell. If the double-mutant proliferates at an unexpected rate, the mutated genes are considered to interact. This, of course, raises the question as to what is the expected rate of proliferation for a cell containing both mutations given the proliferation rate of a cell containing only one of the individual mutations. By analyzing a high-throughput dataset of interactions of gene perturbations in budding yeast (Costanzo et al., 2010; Costanzo et al., 2016), we found that single-mutant fitnesses tend to combine multiplicatively, consistent with earlier work (Mani et al., 2008).
After establishing that the fitness of a double mutant is expected to be approximately the product of the fitness of the individual mutants, namely, the Product neutrality function, we sought to determine if this was also a feature of models of cell growth. If so, then what underlying mechanisms present in these models give rise to this Product neutrality function? Our analysis complements previous, more abstract theoretical attempts at understanding the origin of the Product neutrality function that are not based on any specific model of cell growth (Chiu et al., 2012). Indeed, we found that the Product neutrality function best fits the simulated double-mutant fitnesses despite deviations that depend on the specific parameter pairs and the particular model considered.
That the Product neutrality function fits the budding yeast data and cell growth models better than the Additive and Minimum neutrality functions has important implications for the underlying network controlling cell growth and proliferation. On the one hand, the Minimum neutrality function implies that the double-mutant fitness is set by the most deleterious mutation so that the process this gene is involved in becomes rate limiting for cell growth. Clearly, cell growth does not operate this way, likely because the underlying processes are interconnected. Mutations impairing protein translation impact synthesis of all the proteins in the cell so that other processes, like transcription or surface transport, are also affected. As the cell readjusts its machinery to ensure optimal use of resources, interconnected processes are impacted through a redistribution of cellular resources. On the other hand, the Additive neutrality function implies that a mutation has the same absolute effect on the proliferation rate regardless of the presence of another mutation. This is also clearly not the case as the Additive neutrality function consistently predicts fitnesses below those observed in the data. In the models, this is due in part to growth-supporting feedback that reapportions the proteome. In reality, this may reflect the presence of the general stress response which supports cells in response to genetic or environmental perturbations limiting their growth rate (Gasch et al., 2000). In this way, the Product neutrality function is a reasonable intermediate model between Minimum and Additive that incorporates—albeit approximately—effects such as growth-optimizing feedback and is consistent with the phenomenon of diminishing returns or increasing cost epistasis (Reddy and Desai, 2021). Moreover, our theoretical analysis gives insight into the mechanistic underpinning of the Product neutrality function. In our analyses, a product of the single-mutant growth rates naturally emerges in the analytical treatment of both theoretical models that we consider, albeit with deviation terms that depend on the specific model and simplifying assumptions.
Taken together, our work here constitutes a first step toward understanding the structure of interactions inherent in cell growth models. While we focused on coarse-grained models for their simplicity and mechanistic interpretability, they might be too simple to effectively model large double-mutant datasets and the resulting double-mutant fitness distributions. For instance, it is not possible to differentiate between multiple types of growth rate perturbations impacting the same sector, as they would all be modeled through a limited number of parameters (Metzl-Raz et al., 2017). We therefore expect the combination of high-throughput genetic data with the analysis of larger-scale models, for instance based on Flux Balance Analysis, Metabolic Control Analysis, or whole-cell modeling, to lead to important complementary insights regarding the regulation of cell growth and proliferation (Karr et al., 2012; Oftadeh et al., 2021; Segrè et al., 2005; He et al., 2010; Orth et al., 2010; Kacser and Burns, 1973; Szathmáry, 1993; Dykhuizen et al., 1987; de Vienne et al., 2023; Kryazhimskiy, 2021) We also believe that theoretical exploration of fitness landscapes will shed light on the underlying structure of growth and metabolism networks (Reddy and Desai, 2021; Guo et al., 2019; Boffi et al., 2023). In addition to larger-scale models, we see the refinement of the measurement of cell growth rates as a path forward to a better understanding of its regulation (MacLean, 2010). While we showed here that the Product neutrality function fits the data well for deleterious mutations, we anticipate that there are significant and meaningful deviations that are currently obscured by experimental noise. Similarly, large-scale measurements of the impacts of beneficial mutations will be instrumental in testing the validity of the Product neutrality function in this other regime. From our modeling efforts, we anticipate that such measurements could give important insights into the underlying genetic network regulating growth and proliferation.
MethodsAnalysis of the SGA dataset
The complete SGA dataset was accessed on the cell map webpage. In the SGA genetic interaction dataset, a set of query mutant strains is crossed to an ordered array of mutants.
There are two sets of query mutants. The first one consists of a mix of nonessential deletion mutant strains and of temperature-sensitive alleles of essential genes. The second one is a set of mutants carrying hypomorphic, Decreased Abundance by mRNA Perturbation (DAmP) alleles of essential genes.
There are also two types of arrays. The Deletion Mutant Array (DMA) denotes deletions to a set of nonessential genes, while the Temperature Sensitive Array (TSA) contains a mix of essential and nonessential genes.
Both sets of query mutants are crossed to either type of array, at two different temperature conditions, namely 26 and 30°C. The analysis of Figure 1 reports the analysis of the first set of query mutants crossed to the DMA at 30°C. In Appendix 1—figure 1, we report the same analysis for the first set of query mutants for the other array–temperature combinations. In Appendix 1—figure 2, we report the analysis for the DAmP set of query mutants in the different array–temperature combinations.
For each subdataset—that is, each combination of query mutants, array, and temperature condition—the data is processed in the following steps. First, only deleterious mutations are kept. That is, we remove mutants having a fitness larger than 1. Second, we eliminate mutants where the Additive model predicts a negative fitness, that is, such thatWx+Wy<1,\begin{document}$$\displaystyle W_{x}+ W_{y}\lt 1,$$\end{document}
because in this case the prediction under the Additive neutrality function is negative (Wx+Wy−1<0\begin{document}$W_{x}+ W_{y}- 1 \lt 0$\end{document}). While we could have analyzed these datapoints with the other neutrality functions, we sought to analyze all neutrality functions on the same consistent dataset.
For the sake of clarity, the scatter plots in Figure 1 do not reproduce the entire dataset. Instead, the dataset is binned in 10 bins along the x-axis, and 500 values are sampled at random in that bin. However, the median lines on top of the scatter plots (e.g. Figure 1B–D) as well as the box plots (e.g. Figure 1, Appendix 1—figures 1 and 2) apply to the entire dataset.
Analysis of the GO biological processes
The analysis of GO biological processes is based on the Uniprot database. In this dataset, genes are associated with a series of GO biological processes. To analyze the behavior of the fitness of double mutants associated with different biological processes, we first selected the set of biological processes that were represented by a large enough number of single mutants in the SGA dataset. Arbitrarily, this limit was set at 50. This led to a limited number of 47 biological processes (and a maximum total of 1081 pairs), which we report in the section ‘Analysis of GO biological processes’. Naturally, as genes are potentially associated with multiple GO biological processes, this analysis sometimes leads to pairs of processes with genes in common. In this case, we discarded the pair so that we only consider biological process pairs that do not have any genes in common. This results in 685 pairs of biological processes that do not share any genes.
Mutational analysis of growth models
A mutation is modeled as a perturbation of a parameter that decreases the growth rate. For a given parameter α, we model a perturbation as α′=θα\begin{document}$\alpha' = \theta \alpha$\end{document}, where θ is a random variable uniformly distributed in [0,1]\begin{document}$[0, 1]$\end{document}. When estimating the impact of the parameter γ in the Weiße model (see section ‘Nonlinear kinetics drive deviations from the Product neutrality function in the Weiße model for cell growth’), we perform a mutational analysis as described above for different values of the parameter γ and collect the median residual.
Additional informationCompeting interests
No competing interests declared
Author contributions
Conceptualization, Formal analysis, Investigation, Writing – original draft, Writing – review and editing
Conceptualization, Formal analysis, Supervision, Investigation, Writing – review and editing
Conceptualization, Resources, Supervision, Funding acquisition, Investigation, Writing – original draft, Project administration, Writing – review and editing
Additional filesData availability
The current manuscript is a computational study, so no data have been generated for this manuscript. Modeling code is uploaded to the Skotheimlab Github repository https://github.com/skotheimlab/GrowthModels (copy archived at Fuentes Valenzuela, 2025).
The following previously published dataset was used:
CostanzoMVanderSluisBKochEBaryshnikovaAPonsCTanGWangWUsajMHanchardJLeeSPelechanoVStylesEBillmannMVan LeeuwenJVan DykNLinZKuzminENelsonJPiotrowskiJSrikumarTBahrSChenYDeshpandeRKuratCLiSLiZMattiazzi UsajMOkadaHPascoeNSan LuisBSharifpoorSShuteriqiESimpkinsSSniderJGaradi SureshHTanYZhuHMalod-DogninNJanjicVPrzuljNTroyanskayaOStagljarIXiaTOhyaYGingrasARaughtBBoutrosMSteinmetzLMooreCRosebrockACaudyAMyersCAndrewsBBooneC2025Data from: A global genetic interaction network maps a wiring diagram of cellular functionDryad Digital Repository10.5061/dryad.4291s
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The 47 biological processes that result from the analysis (with their GO code) are given in Appendix 1—table 1.
A.2 Scott–Hwa modelA.2.1 Derivation of the double-mutant fitness
Here, we derive the expression for the double-mutant fitness in the Scott–Hwa model. First of all, the wild-type growth rate λWT\begin{document}$\lambda_{WT}$\end{document} is written asλWT=κtκnκt+κn.\begin{document}$$\displaystyle \lambda_{WT}= \frac{\kappa_{t}\kappa_{n}}{\kappa_{t}+ \kappa_{n}}.$$\end{document}
As we consider only deleterious mutations, we can model a perturbation to parameters asκn′=(1−ε)κn,\begin{document}$$\displaystyle \kappa_{n}'= (1-\varepsilon) \kappa_{n},$$\end{document}κt′=(1−ε)κt,\begin{document}$$\displaystyle \kappa_{t}'= (1-\varepsilon) \kappa_{t},$$\end{document}
where δ,ε∈[0,1]\begin{document}$\delta, \varepsilon \in [0, 1]$\end{document}. For simplicity, we will denote by the x\begin{document}$x$\end{document} subscript mutations affecting κt\begin{document}$\kappa_{t}$\end{document} and the y\begin{document}$y$\end{document} subscript mutations affecting κn\begin{document}$\kappa_{n}$\end{document}. We note the order does not matter. Therefore, the different fitnesses W⋅=λ⋅/λWT\begin{document}$W_{\cdot}= \lambda_{\cdot}/\lambda_{WT}$\end{document} are given byWx=1−ε1−εκt/(κt+κn),\begin{document}$$\displaystyle W_{x}= \frac{1-\varepsilon}{1 - \varepsilon\kappa_{t}/(\kappa_{t}+ \kappa_{n})},$$\end{document}Wy=1−δ1−δκn/(κt+κn),\begin{document}$$\displaystyle W_{y}= \frac{1-\delta}{1 - \delta\kappa_{n}/(\kappa_{t}+ \kappa_{n})},$$\end{document}Wxy=(1−ε)(1−δ)(κt+κn)(1−ε)κt+(1−δ)κn.\begin{document}$$\displaystyle W_{xy}= \frac{(1-\varepsilon) (1-\delta) (\kappa_{t}+ \kappa_{n})}{(1-\varepsilon)\kappa_{t}+ (1-\delta) \kappa_{n}}.$$\end{document}
From there, we can calculate thatWxWy=(1−ε)(1−δ)(κt+κn)(1−ε)κt+(1−δ)κn+εδλWT.\begin{document}$$\displaystyle W_{x}W_{y}= \frac{(1-\varepsilon)(1-\delta)(\kappa_{t}+ \kappa_{n})}{(1-\varepsilon)\kappa_{t}+ (1-\delta)\kappa_{n}+ \varepsilon\delta\lambda_{WT}}.$$\end{document}
This expression is similar to Equation 6. Indeed, we can rewrite it asWxy=WxWy(1+εδλWT(1−ε)κt+(1−δ)κn).\begin{document}$$\displaystyle W_{xy}= W_{x}W_{y}\left(1 + \frac{\varepsilon\delta\lambda_{WT}}{(1-\varepsilon) \kappa_{t}+ (1-\delta)\kappa_{n}}\right).$$\end{document}
We see that the double-mutant fitness Wxy\begin{document}$W_{xy}$\end{document} consists of the product of single-mutant fitnesses and a deviation. We will now express this deviation as a function of single-mutant fitnesses only, showing that the expression does not depend on the value of the parameters κt\begin{document}$\kappa_{t}$\end{document} and κn\begin{document}$\kappa_{n}$\end{document}.
Rearranging Equations 4 and 5, we have1−Wx=εκn(1−ε)κt+κn,\begin{document}$$\displaystyle 1-W_{x}= \frac{\varepsilon \kappa_{n}}{(1-\varepsilon)\kappa_{t}+ \kappa_{n}},$$\end{document}1−Wy=δκt(1−δ)κn+κt,\begin{document}$$\displaystyle 1-W_{y}= \frac{\delta \kappa_{t}}{(1-\delta)\kappa_{n}+ \kappa_{t}},$$\end{document}⇒(1−Wx)(1−Wy)=1κt+κnεδλWT(1−εκt/(κt+κn))(1−δκn/(κt+κn)).\begin{document}$$\displaystyle \Rightarrow \left(1-W_{x}\right)\left(1-W_{y}\right)= \frac{1}{\kappa_{t}+ \kappa_{n}}\frac{\varepsilon\delta\lambda_{WT}}{(1-\varepsilon\kappa_{t}/(\kappa_{t}+ \kappa_{n}))(1-\delta\kappa_{n}/(\kappa_{t}+\kappa_{n}))}.$$\end{document}
which can be rearranged to see thatεδλWT(1−ε)κt+(1−δ)κn=(1−Wx)(1−Wy)(1−εκt/(κt+κn))(1−δκn/(κt+κn))(1−ε)κt+(1−δ)κn\begin{document}$$\displaystyle \frac{\varepsilon\delta\lambda_{WT}}{(1-\varepsilon) \kappa_{t}+ (1-\delta)\kappa_{n}}= (1-W_{x})(1-W_{y}) \frac{(1-\varepsilon\kappa_{t}/(\kappa_{t}+ \kappa_{n}))(1-\delta\kappa_{n}/(\kappa_{t}+\kappa_{n}))}{(1-\varepsilon) \kappa_{t}+ (1-\delta)\kappa_{n}}$$\end{document}=(1−Wx)(1−Wy)1−εκt/(κt+κn)−δκn/(κt+κn)+εδλWT/(κt+κn)(1−ε)κt+(1−δ)κn\begin{document}$$\displaystyle = (1-W_{x})(1-W_{y}) \frac{1 - \varepsilon\kappa_{t}/(\kappa_{t}+ \kappa_{n}) - \delta\kappa_{n}/(\kappa_{t}+\kappa_{n}) + \varepsilon\delta\lambda_{WT}/(\kappa_{t}+ \kappa_{n})}{(1-\varepsilon) \kappa_{t}+ (1-\delta)\kappa_{n}}$$\end{document}=(1−Wx)(1−Wy)(1−ε)κt+(1−δ)κn+εδλWT(1−ε)κt+(1−δ)κn\begin{document}$$\displaystyle =(1-W_{x})(1-W_{y}) \frac{(1-\varepsilon)\kappa_{t}+ (1-\delta)\kappa_{n}+ \varepsilon\delta\lambda_{WT}}{(1-\varepsilon) \kappa_{t}+ (1-\delta)\kappa_{n}}$$\end{document}=(1−Wx)(1−Wy)(1+εδλWT(1−ε)κt+(1−δ)κn).\begin{document}$$\displaystyle = (1-W_{x})(1-W_{y}) \left(1 + \frac{\varepsilon\delta\lambda_{WT}}{(1-\varepsilon) \kappa_{t}+ (1-\delta)\kappa_{n}}\right).$$\end{document}
We see a recursive relationship appear, such thatεδλWT(1−ε)κt+(1−δ)κn=(1−Wx)(1−Wy)(1+(1−Wx)(1−Wy)(1+…))\begin{document}$$\displaystyle \frac{\varepsilon\delta\lambda_{WT}}{(1-\varepsilon) \kappa_{t}+ (1-\delta)\kappa_{n}}= (1-W_{x})(1-W_{y}) \left(1 + (1-W_{x})(1-W_{y}) \left(1 + \dots\right)\right)$$\end{document}=(1−Wx)(1−Wy)+((1−Wx)(1−Wy))2+((1−Wx)(1−Wy))3+…\begin{document}$$\displaystyle = (1-W_{x})(1-W_{y}) + \left((1-W_{x})(1-W_{y})\right)^{2}+ \left((1-W_{x})(1-W_{y})\right)^{3}+\dots$$\end{document}
Therefore, we have from Equation 8 thatWxy=WxWy∑k=0∞((1−Wx)(1−Wy))k\begin{document}$$\displaystyle W_{xy}= W_{x}W_{y}\sum_{k=0}^{\infty}\left((1-W_{x})(1-W_{y})\right)^{k}$$\end{document}=WxWy11−(1−Wx)(1−Wy)\begin{document}$$\displaystyle = W_{x}W_{y}\frac{1}{1-(1-W_{x})(1-W_{y})}$$\end{document}=WxWy(1+(1−Wx)(1−Wy)1−(1−Wx)(1−Wy)),\begin{document}$$\displaystyle = W_{x}W_{y}\left(1 + \frac{(1-W_{x})(1-W_{y})}{1-(1-W_{x})(1-W_{y})}\right),$$\end{document}
from the property of geometric series and noting that (1−Wx)(1−Wy)≤1\begin{document}$(1-W_{x})(1-W_{y}) \leq 1$\end{document}.
A.2.2 Scott–Hwa model with no feedback
To understand the impact of growth rate optimization in the Scott–Hwa model, we consider an alternative model formulation where this feedback is absent. We assume that we still have two different processes (metabolism and translation) and that the flux through both of them has to be equal. However, we do not incorporate the assumption that the proteome has to be partitioned between these two processes. Instead, each process is assumed to have a given amount of proteins or enzymes available at its disposal. This formulation implies thatλ=κtϕt=κnϕn,\begin{document}$$\displaystyle \lambda = \kappa_{t}\phi_{t}= \kappa_{n}\phi_{n},$$\end{document}0≤ϕt≤ϕtmax,\begin{document}$$\displaystyle 0\leq \phi_{t}\leq \phi_{t}^{max},$$\end{document}0≤ϕn≤ϕnmax,\begin{document}$$\displaystyle 0\leq\phi_{n}\leq\phi_{n}^{max},$$\end{document}
where κt\begin{document}$\kappa_{t}$\end{document} and κn\begin{document}$\kappa_{n}$\end{document} are the capacities of the translation and metabolic sector, respectively, as in the Scott–Hwa model, and where ϕt\begin{document}$\phi_{t}$\end{document} and ϕn\begin{document}$\phi_{n}$\end{document} are the normalized concentrations of proteins or enzymes.
If we assume that the cell maximizes its growth rate, we have thatλmaxnf=min(κtϕtmax,κnϕnmax),\begin{document}$$\displaystyle \begin{align}\lambda_{max}^{nf}= \min(\kappa_{t}\phi_{t}^{max}, \kappa_{n}\phi_{n}^{max}),\end{align}$$\end{document}
where the nf\begin{document}$nf$\end{document} superscript stands for no feedback.
Clearly, in this simplified model, one sector will be limiting either because it is not efficient enough, that is, κi\begin{document}$\kappa_{i}$\end{document} is too small, or because it does not have enough resources to deploy to accelerate the process, that is, ϕimax\begin{document}$\phi_{i}^{max}$\end{document} is too small. In this case, we see that the growth rate λ corresponds to the minimum of the potential maximal fluxes in either sector.
Therefore, modeling mutations as affecting either κt\begin{document}$\kappa_{t}$\end{document} or κn\begin{document}$\kappa_{n}$\end{document} but not the protein fractions, we find thatWx=min(κt′ϕtmax,κnϕnmax)λ0,Wy=min(κtϕtmax,κn′ϕnmax)λ0,Wxy=min(κt′ϕtmax,κn′ϕnmax)λ0\begin{document}$$\displaystyle \begin{array}{ll} W_{x}&= \frac{\min(\kappa_{t}'\phi_{t}^{max}, \kappa_{n}\phi_{n}^{max})}{\lambda_{0}},\\ W_{y}&= \frac{\min(\kappa_{t}\phi_{t}^{max}, \kappa_{n}' \phi_{n}^{max})}{\lambda_{0}},\\ W_{xy}&= \frac{\min(\kappa_{t}'\phi_{t}^{max}, \kappa_{n}' \phi_{n}^{max})}{\lambda_{0}}\end{array}$$\end{document}
For the sake of argument, assume that κtϕtmax≤κnϕnmax\begin{document}$\kappa_{t}\phi_{t}^{max}\leq \kappa_{n}\phi_{n}^{max}$\end{document}. Therefore, λ0=κtϕtmax\begin{document}$\lambda_{0}=\kappa_{t}\phi_{t}^{max}$\end{document} and Wx=κt′ϕtmaxλ0=κt′κt\begin{document}$W_{x}= \frac{\kappa_{t}' \phi_{t}^{max}}{\lambda_{0}}= \frac{\kappa_{t}'}{\kappa_{t}}$\end{document}, and Wy=min(1,κn′ϕnmaxκtϕtmax)\begin{document}$W_{y}= \min(1, \frac{\kappa_{n}'\phi_{n}^{max}}{\kappa_{t}\phi_{t}^{max}})$\end{document}.
We can then write thatWxy=min(Wx,κn′ϕnmaxκtϕtmax).\begin{document}$$\displaystyle W_{xy}= \min\left(W_{x}, \frac{\kappa_{n}'\phi_{n}^{max}}{\kappa_{t}\phi_{t}^{max}}\right).$$\end{document}
If κn′ϕnmaxκtϕtmax>1\begin{document}$\frac{\kappa_{n}'\phi_{n}^{max}}{\kappa_{t}\phi_{t}^{max}}\gt 1$\end{document}, then Wxy=Wx≤1\begin{document}$W_{xy}= W_{x}\leq 1$\end{document}. However, if κn′ϕnmaxκtϕtmax≤1\begin{document}$\frac{\kappa_{n}'\phi_{n}^{max}}{\kappa_{t}\phi_{t}^{max}}\leq 1$\end{document}, then Wy=κn′ϕnmaxκtϕtmax\begin{document}$W_{y}= \frac{\kappa_{n}'\phi_{n}^{max}}{\kappa_{t}\phi_{t}^{max}}$\end{document}. Therefore, we have thatWxy=min(Wx,Wy),\begin{document}$$\displaystyle W_{xy}= \min\left(W_{x}, W_{y}\right),$$\end{document}
that is, in the model with no feedback laid out above, double-mutant fitnesses are actually the minimum of the single-mutant fitnesses. Simulation results are reported in Appendix 1—figure 4.
We note that this absence of feedback results in significantly smaller growth rates. In the Scott–Hwa model, we haveλSH=κtκnκt+κn.\begin{document}$$\displaystyle \lambda^{SH}= \frac{\kappa_{t}\kappa_{n}}{\kappa_{t}+\kappa_{n}}.$$\end{document}
This implies that ϕt=κnκt+κn\begin{document}$\phi_{t}= \frac{\kappa_{n}}{\kappa_{t}+\kappa_{n}}$\end{document} and ϕn=κtκt+κn\begin{document}$\phi_{n}= \frac{\kappa_{t}}{\kappa_{t}+\kappa_{n}}$\end{document}.
Let us now consider, in the context of the model in this section, ϕtmax=κnκt+κn\begin{document}$\phi_{t}^{max}= \frac{\kappa_{n}}{\kappa_{t}+\kappa_{n}}$\end{document} and ϕnmax=κtκt+κn\begin{document}$\phi_{n}^{max}= \frac{\kappa_{t}}{\kappa_{t}+\kappa_{n}}$\end{document}. Upon mutation of κt\begin{document}$\kappa_{t}$\end{document} or κn\begin{document}$\kappa_{n}$\end{document}, we haveλnfλSH=min(κt′κnκt+κn,κn′κtκt+κn)κt′κn′κt′+κn′\begin{document}$$\displaystyle \frac{\lambda^{nf}}{\lambda^{SH}}= \frac{\min\left(\kappa_{t}' \frac{\kappa_n}{\kappa_t+\kappa_n}, \kappa_{n}' \frac{\kappa_t}{\kappa_t+\kappa_n}\right)}{ \frac{\kappa_{t}'\kappa_{n}'}{\kappa_{t}'+\kappa_{n}'} }$$\end{document}=min(κn(κt′+κn′)κn′(κt+κn),κt(κt′+κn′)κt′(κt+κn))\begin{document}$$\displaystyle = \min\left(\frac{\kappa_{n}(\kappa_{t}' + \kappa_{n}')}{\kappa_{n}' (\kappa_{t}+ \kappa_{n})}, \frac{\kappa_{t}(\kappa_{t}' + \kappa_{n}')}{\kappa_{t}' (\kappa_{t}+ \kappa_{n})}\right)$$\end{document}=κt′+κn′κt+κnmin(κnκn′,κtκt′)\begin{document}$$\displaystyle = \frac{\kappa_{t}' + \kappa_{n}'}{\kappa_{t}+ \kappa_{n}}\min\left(\frac{\kappa_{n}}{\kappa_{n}'}, \frac{\kappa_{t}}{\kappa_{t}'}\right)$$\end{document}
Assume, without loss of generality, that κtκt′≤κnκn′\begin{document}$\frac{\kappa_{t}}{\kappa_{t}'}\leq \frac{\kappa_{n}}{\kappa_{n}'}$\end{document}, that is κn′κn≤κt′κt\begin{document}$\frac{\kappa_{n}'}{\kappa_{n}}\leq \frac{\kappa_{t}'}{\kappa_{t}}$\end{document}. Then, we haveκt′+κn′κt+κn≤κt′+κt′κnκtκt+κn=κt′κt,\begin{document}$$\displaystyle \frac{\kappa_{t}' + \kappa_{n}'}{\kappa_{t}+ \kappa_{n}}\leq \frac{\kappa_{t}' + \kappa_{t}' \frac{\kappa_n}{\kappa_t}}{\kappa_{t}+ \kappa_{n}}= \frac{\kappa_{t}'}{\kappa_{t}},$$\end{document}
which implies that λnf/λSH≤1\begin{document}$\lambda^{nf}/\lambda^{SH}\leq 1$\end{document}. Therefore, in the absence of feedback, the growth rate is smaller than in the Scott–Hwa model, which is consistent with the interpretation of growth-rate optimization.
A.3 Weiße modelA.3.1 Original model
The original model consists of a system of equations describing the synthesis, degradation, and reaction of a series of molecules and proteins. In particular, external nutrients s\begin{document}$s$\end{document} are converted into internal nutrients si\begin{document}$s_{i}$\end{document}. Those nutrients are then converted into a generic cellular energy a\begin{document}$a$\end{document} that enables transcription and translation. Indeed, mRNAs m\begin{document}$m$\end{document} are transcribed and then bind to ribosomes to form an mRNA–ribosome complex c\begin{document}$c$\end{document} that governs the rate of protein translation. There are four main types of mRNAs and complexes in the model, indicated by a subscript. Each is associated with different processes: transport, metabolism, ribosomal, and q-proteins. This last type of protein denotes housekeeping proteins whose concentration remains approximately constant.si˙=νimp(et,s)−νcat(em,si)−λsi\begin{document}$$\displaystyle \dot{s_i}= \nu_{\text{imp}}(e_{t}, s) - \nu_{\text{cat}}(e_{m}, s_{i}) - \lambda s_{i}$$\end{document}a˙=ns⋅νcat(em,si)−∑x∈{r,t,m,q}nxνx(cx,a)−λa\begin{document}$$\displaystyle \dot{a}= n_{s}\cdot \nu_{\text{cat}}(e_{m}, s_{i}) - \sum_{x \in \{r, t, m, q\}}n_{x}\nu_{x}(c_{x}, a) - \lambda a $$\end{document}r˙=νr(cr,a)−λr+∑x∈{r,t,m,q}(νx(cx,a)−kbrmx+kucx)\begin{document}$$\displaystyle \dot{r}= \nu_{r}(c_{r}, a) - \lambda r + \sum_{x \in \{r, t, m, q\}}\left(\nu_{x}(c_{x}, a) - k_{b}r m_{x}+ k_{u}c_{x}\right)$$\end{document}et˙=νt(ct,a)−λet\begin{document}$$\displaystyle \dot{e_t}= \nu_{t}(c_{t}, a) - \lambda e_{t}$$\end{document}em˙=νm(cm,a)−λem\begin{document}$$\displaystyle \dot{e_m}= \nu_{m}(c_{m}, a) - \lambda e_{m}$$\end{document}q˙=νq(cq,a)−λq\begin{document}$$\displaystyle \dot{q}= \nu_{q}(c_{q}, a) - \lambda q$$\end{document}m˙x=ωx(a)−(λ+dm)mx+νx(cx,a)−kbrmx+kucx\begin{document}$$\displaystyle \dot{m}_{x}= \omega_{x}(a) - (\lambda + d_{m}) m_{x}+ \nu_{x}(c_{x}, a) - k_{b}r m_{x}+ k_{u}c_{x}$$\end{document}c˙x=−λcx+kbrmx−kucx−νx(cx,a)\begin{document}$$\displaystyle \dot{c}_{x}= - \lambda c_{x}+ k_{b}r m_{x}- k_{u}c_{x}- \nu_{x}(c_{x}, a)$$\end{document}
All of these reactions are modulated by parameters reported in Appendix 1—table 2.
The rates governing the system of equations are assumed to follow Michaelis–Menten kinetics as followsνimp(et,s)=etvtsKt+s\begin{document}$$\displaystyle \nu_{\text{imp}}(e_{t}, s)= e_{t}\frac{v_{t}s}{K_{t}+ s}$$\end{document}νcat(em,si)=emvmsiKm+si\begin{document}$$\displaystyle \nu_{\text{cat}}(e_{m}, s_{i})= e_{m}\frac{v_{m}s_{i}}{K_{m}+ s_{i}}$$\end{document}νx(cx,a)∼cxγ(a)nx\begin{document}$$\displaystyle \nu_{x}(c_{x}, a) \sim c_{x}\frac{\gamma(a)}{n_{x}}$$\end{document}γ(a):=γmaxaKγ+a\begin{document}$$\displaystyle \gamma(a):= \frac{\gamma_{max}a}{K_{\gamma}+ a}$$\end{document}ωx(a)=ωxaθx+a\begin{document}$$\displaystyle \omega_{x}(a)= \omega_{x}\frac{a}{\theta_{x}+ a}$$\end{document}
Finally, the growth rate corresponds to total mass of proteins being synthesized at steady state, that isλ=γ(a)∑xcxM.\begin{document}$$\displaystyle \lambda = \frac{\gamma(a) \sum_{x}c_{x}}{M}.$$\end{document}
A.3.2 Isolation of parameters
This model has a total of 21 parameters whose values were set (either from the literature or estimated) in the original model (Table S2 in Weiße et al., 2015; reproduced here under Appendix 1—table 2).
Among them, some represent quantities or parameters that would not reasonably change under a mutation. Therefore, we do not consider the following parameters in our mutation analysis: external nutrients s\begin{document}$s$\end{document}, mRNA degradation rate dm\begin{document}$d_{m}$\end{document}, ribosome length nr\begin{document}$n_{r}$\end{document}, length of non-ribosomal proteins nx\begin{document}$n_{x}$\end{document}, q-autoinhibition Hill coefficient hq\begin{document}$h_{q}$\end{document}, the mRNA–ribosome binding/unbinding rates kb\begin{document}$k_{b}$\end{document} and ku\begin{document}$k_{u}$\end{document}, the total cell mass M\begin{document}$M$\end{document}.
To determine whether a numerical mutational analysis with the remaining 13 parameters could be considered, we computed the impact of a change of these parameters on the growth rate λ (Appendix 1—figure 6). Upon mutation, most parameters do impact the growth rate negatively. However, two parameters vm,Km\begin{document}$v_{m}, K_{m}$\end{document} do not seem to have an impact on the growth rate. We attribute this to the metabolic sector not being limiting in this model, with those parameter values. We, therefore, exclude those two parameters from further analysis. We also exclude the parameters associated with the q-proteins Kq\begin{document}$K_{q}$\end{document} and ωq\begin{document}$\omega_{q}$\end{document} as these are associated mostly with housekeeping functions.
We are therefore left with nine parameters modeling the impact of four different processes. In total, this results in 28 potential combinations of two parameters associated with different processes.
A.3.3 Simplification
To facilitate analytical treatment, we consider a simplified model with only two types of proteins, instead of four: a general protein p\begin{document}$p$\end{document} and ribosomes r\begin{document}$r$\end{document}. We also eliminate the equations corresponding to the competitive binding of mRNAs for ribosomes. Practically, this assumption means that every mRNA binds immediately to a ribosome. While this assumption might not be true in all regimes, we find that this simplifies the analytical treatment and still enables us to understand something concrete about the model.
The simplified model we consider iss˙i=νimp(p,s)−νcat(p,si)−λsi\begin{document}$$\displaystyle \dot{s}_{i}= \nu_{\text{imp}}(p, s) - \nu_{\text{cat}}(p, s_{i}) - \lambda s_{i}$$\end{document}a˙=nsνcat(p,si)−∑xnxνx(cp,a)−λa\begin{document}$$\displaystyle \dot{a}= n_{s}\nu_{\text{cat}}(p, s_{i}) - \sum_{x}n_{x}\nu_{x}(c_{p}, a) - \lambda a$$\end{document}r˙=νr(cr,a)−λr+∑xνx(cx,a)\begin{document}$$\displaystyle \dot{r}= \nu_{r}(c_{r}, a) - \lambda r + \sum_{x}\nu_{x}(c_{x}, a)$$\end{document}p˙=νp(p,a)−λp\begin{document}$$\displaystyle \dot{p}= \nu_{p}(p, a) - \lambda p$$\end{document}c˙r=ωr(a)−νr(cr,a)−λcr\begin{document}$$\displaystyle \dot{c}_{r}= \omega_{r}(a) - \nu_{r}(c_{r}, a) - \lambda c_{r}$$\end{document}c˙p=ωp(a)−νp(cp,a)−λcp\begin{document}$$\displaystyle \dot{c}_{p}= \omega_{p}(a) - \nu_{p}(c_{p}, a) - \lambda c_{p}$$\end{document}
If we assume the dilution fluxes to be negligible in comparison to other fluxes in Equations 42–47, we get thatνimp(p,s)∼νcat(p,si)⇒pvtsKt+s∼pvmsiKm+si\begin{document}$$\displaystyle \nu_{\text{imp}}(p, s)\sim \nu_{\text{cat}}(p, s_{i}) \Rightarrow p v_{t}\frac{s}{K_{t}+ s}\sim p v_{m}\frac{s_{i}}{K_{m}+ s_{i}}$$\end{document}nsνcat(p,si)∼∑xnxνx(cp,a)=(cp+cr)γ(a)\begin{document}$$\displaystyle n_{s}\nu_{\text{cat}}(p, s_{i})\sim \sum_{x}n_{x}\nu_{x}(c_{p}, a) = (c_{p}+ c_{r}) \gamma(a) $$\end{document}ωp(a)∼νp(cp,a)\begin{document}$$\displaystyle \omega_{p}(a)\sim \nu_{p}(c_{p}, a)$$\end{document}ωr(a)∼νr(cr,a)\begin{document}$$\displaystyle \omega_{r}(a)\sim \nu_{r}(c_{r}, a)$$\end{document}
The dilution flux needs to be nonzero for p\begin{document}$p$\end{document}, as its governing equation only contains two terms. We have therefore thatνp(p,a)=λp⇒p∼ωpaθx+a1λ\begin{document}$$\displaystyle \nu_{p}(p, a) = \lambda p \Rightarrow p \sim \omega_{p}\frac{a}{\theta_{x}+ a}\frac{1}{\lambda}$$\end{document}
As the growth rate λ=γ(a)∑xcxM\begin{document}$\lambda = \frac{\gamma(a)\sum_{x}c_{x}}{M}$\end{document}, we haveλ=nsνcatM∼1MnsvtpsKt+s\begin{document}$$\displaystyle \lambda = \frac{n_{s}\nu_{\text{cat}}}{M}\sim \frac{1}{M}n_{s}v_{t}p \frac{s}{K_{t}+ s}$$\end{document}
Injecting Equation 52 into Equation 53, we haveλ∼1MnsvtwpsKt+saθx+a\begin{document}$$\displaystyle \lambda \sim \sqrt{\frac{1}{M}n_{s}v_{t}w_{p}\frac{s}{K_t+s}\frac{a}{\theta_x+a}}$$\end{document}
From the above expression, we see that terms belonging to different sectors combine as a product. However, the energy a\begin{document}$a$\end{document} is not a parameter, but rather a variable in the model. Its value also directly depends on the values of other parameters.
From Equation 49, we have thatλ∼1M(npωpaθx+a+nrωraθr+a)\begin{document}$$\displaystyle \begin{align}\lambda \sim \frac{1}{M}\left(n_{p}\omega_{p}\frac{a}{\theta_{x}+ a}+ n_{r}\omega_{r}\frac{a}{\theta_{r}+ a}\right)\end{align}$$\end{document}
If θx<a<θr\begin{document}$\theta_{x}\lt a \lt \theta_{r}$\end{document}, we can approximate that withλ∼1M(npωp+nrωraθr).\begin{document}$$\displaystyle \lambda\sim \frac{1}{M}\left(n_{p}\omega_{p}+ n_{r}\omega_{r}\frac{a}{\theta_{r}}\right).$$\end{document}
If a≪θx,θr\begin{document}$a \ll \theta_{x}, \theta_{r}$\end{document}, we haveλ∼1M(npωpθx+nrωrθr)a.\begin{document}$$\displaystyle \lambda \sim \frac{1}{M}\left(\frac{n_{p}\omega_{p}}{\theta_{x}}+ \frac{n_{r}\omega_{r}}{\theta_{r}}\right)a.$$\end{document}
In both cases, we can assume that there is a linear relationship between the growth rate λ and the energy vector a\begin{document}$a$\end{document}, such thata∼bλ+c.\begin{document}$$\displaystyle a \sim b \lambda + c.$$\end{document}
We can inject that in the a/(θx+a)\begin{document}$a/(\theta_{x}+a)$\end{document} term in Equation 54, which givesaθx+a∼λ+c/b(θx+c)/b+λ\begin{document}$$\displaystyle \frac{a}{\theta_{x}+ a}\sim \frac{\lambda + c/b}{(\theta_{x}+ c)/b + \lambda}$$\end{document}
Assuming that c/b≪λ\begin{document}$c/b \ll \lambda$\end{document}, we haveaθx+a∼λK′+λ,\begin{document}$$\displaystyle \begin{align}\frac{a}{\theta_{x}+ a}\sim \frac{\lambda}{K' + \lambda},\end{align}$$\end{document}
with K′=(θx+c)/b\begin{document}$K' = (\theta_{x}+ c)/b$\end{document}.
Injecting the above in Equation 54, we get that,λ∼1MnsvtwpsKt+sλK′+λ.\begin{document}$$\displaystyle \lambda \sim \sqrt{\frac{1}{M}n_{s}v_{t}w_{p}\frac{s}{K_t+s}\frac{\lambda}{K' + \lambda}}.$$\end{document}
Therefore, K′\begin{document}$K'$\end{document} acts as a scale of growth rate where deviations from the Product model can appear. If K′≫λ\begin{document}$K' \gg \lambda$\end{document}, thenλ∼1MnsvtwpsKt+s.\begin{document}$$\displaystyle \lambda \sim \frac{1}{M}n_{s}v_{t}w_{p}\frac{s}{K_{t}+s}.$$\end{document}
On the other hand, if K′≪λ\begin{document}$K' \ll \lambda$\end{document}, thenλ∼1MnsvtwpsKt+s.\begin{document}$$\displaystyle \lambda \sim \sqrt{\frac{1}{M}n_{s}v_{t}w_{p}\frac{s}{K_t+s}}.$$\end{document}
In both of these cases, the growth rate λ behaves as the product of multiple parameters. However, if λ∼K′\begin{document}$\lambda \sim K'$\end{document}, and if we denote two parameters in the expression by α, β without loss of generality, we haveλ2(K′+λ)∼αβλ.\begin{document}$$\displaystyle \lambda^{2}(K'+\lambda) \sim \alpha \beta \lambda.$$\end{document}
Either λ=0\begin{document}$\lambda =0$\end{document} (which we will not consider), orλ(K′+λ)=αβ⇔λ2+K′λ−αβ=0⇔λ=−K′+K′2+4αβ2=K′2(−1+1+4αβK′2)\begin{document}$$\displaystyle \begin{array}{ll}\lambda(K'+\lambda)&= \alpha \beta\\ \Leftrightarrow \lambda^{2}+K'\lambda - \alpha\beta&= 0\\ \Leftrightarrow \lambda&= \frac{-K' + \sqrt{K'^{2}+ 4\alpha\beta}}{2}\\&= \frac{K'}{2}\left(-1 + \sqrt{1 + \frac{4\alpha\beta}{K'^{2}}}\right)\end{array}$$\end{document}
We can write, for small x\begin{document}$x$\end{document}, that1+x=1+x2−x28+…\begin{document}$$\displaystyle \sqrt{1+x}= 1 + \frac{x}{2}- \frac{x^{2}}{8}+ \dots$$\end{document}
Assuming that 4αβK′2\begin{document}$\frac{4\alpha\beta}{K'^{2}}$\end{document} is small and keeping terms up to second order, we therefore have thatλ=K′2x2(1−x4),with x=4αβK′2⇒λ∝αβ(1−γαβ),with γ=1/K′2\begin{document}$$\displaystyle \begin{array}{ll}\lambda&= \frac{K'}{2}\frac{x}{2}\left(1 - \frac{x}{4}\right),\ \text{with }x = \frac{4\alpha\beta}{K'^{2}} \\ \Rightarrow \lambda &\propto \alpha\beta \left(1 - \gamma \alpha \beta\right),\ \text{with }\gamma = 1/K'^{2}\end{array}$$\end{document}
A.4 Double-mutant fitness
From there, we can compute an approximation to the double-mutant fitness. Let us assume that λ=αβ(1−γαβ)\begin{document}$\lambda = \alpha \beta \left(1-\gamma \alpha\beta\right)$\end{document} as derived above. To simplify notation, we will rescale the parameters to their WT\begin{document}$WT$\end{document} value, that isα′=α/αWT,β′=β/βWT,γ¯=γαWTβWT.\begin{document}$$\displaystyle \begin{array}{ll}\alpha'&= \alpha/\alpha_{WT},\\ \beta'&= \beta/\beta_{WT},\\ \bar{\gamma}&= \gamma\alpha_{WT}\beta_{WT}.\end{array}$$\end{document}
Under this rescaling, we see thatλWT=αWTβWT(1−γ¯),Wx=α′(1−γ¯α′)1−γ¯,Wy=β′(1−γ¯β′)1−γ¯,Wxy=α′β′1−γ¯α′β′(1−γ¯).\begin{document}$$\displaystyle \begin{array}{ll}\lambda_{WT}&= \alpha_{WT}\beta_{WT}(1-\bar{\gamma}),\\ W_{x}&= \frac{\alpha'(1-\bar{\gamma}\alpha')}{1-\bar{\gamma}},\\ W_{y}&= \frac{\beta'(1-\bar{\gamma}\beta')}{1-\bar{\gamma}},\\ W_{xy}&= \alpha'\beta'\frac{1-\bar{\gamma}\alpha'\beta'}{(1-\bar{\gamma})}.\end{array}$$\end{document}
we can show thatWxy=WxWy−α′β′(1−γ¯)2γ¯(1−α′)(1−β′),=WxWy(1−γ¯1−α′1−γ¯α′1−β′1−γ¯β′).\begin{document}$$\displaystyle \begin{array}{ll}W_{xy}&= W_{x}W_{y}- \frac{\alpha'\beta'}{(1-\bar{\gamma})^{2}}\bar{\gamma}(1-\alpha')(1-\beta'),\\&= W_{x}W_{y}\left(1 - \bar{\gamma}\frac{1-\alpha'}{1-\bar{\gamma}\alpha'}\frac{1-\beta'}{1-\bar{\gamma}\beta'}\right).\end{array}$$\end{document}
If γ‾\begin{document}$\bar{\gamma}$\end{document} is small, then Wx=α′(1−γ¯α′)1−γ¯∼α′\begin{document}$W_{x}= \frac{\alpha'(1-\bar{\gamma}\alpha')}{1-\bar{\gamma}}\sim \alpha'$\end{document} and Wy=β′(1−γ¯β′)1−γ¯∼β′\begin{document}$W_{y}= \frac{\beta'(1-\bar{\gamma}\beta')}{1-\bar{\gamma}}\sim \beta'$\end{document}, and therefore we can write thatWxy∼WxWy(1−γ¯1−Wx1−γ¯Wx1−Wy1−γ¯Wy).\begin{document}$$\displaystyle W_{xy}\sim W_{x}W_{y}\left(1 - \bar{\gamma}\frac{1-W_{x}}{1-\bar{\gamma}W_{x}}\frac{1-W_{y}}{1-\bar{\gamma}W_{y}}\right).$$\end{document}
Double-mutant fitnesses are best described by the Product neutrality function in the Synthetic Genetic Array (SGA) dataset.
Box plots for the distributions of the residuals for the three neutrality functions as a function of the maximum single-mutant fitness. Each plot corresponds to a different subset of the SGA dataset. Namely, they correspond to the first set of query mutants (see Methods) crossed to different types of mutant arrays in different temperature conditions (A) Deletion Mutant Array at 26°C. (B) Temperature Sensitive Array at 26°C. (C) Temperature Sensitive Array at 30°C. Thick line denotes the median, and boxes denote the 25th and 75th percentiles of the distributions. The Product neutrality function models the data consistently better than the others.
Double-mutant fitnesses are best described by the Product neutrality function in the Synthetic Genetic Array (SGA) dataset.
Box plots for the distributions of the residuals for the three neutrality functions as a function of the maximum single-mutant fitness. Each plot corresponds to a different subset of the SGA dataset. Namely, they correspond to the second set of query mutants (DAmP, see Methods) crossed to different types of mutant arrays in different temperature conditions (A) Deletion Mutant Array at 30°C. (B) Temperature Sensitive Array at 26°C. (C) Temperature Sensitive Array at 30°C. Thick line denotes the median, and boxes denote the 25th and 75th percentiles of the distributions. The Product neutrality function models the data consistently better than the others.
Larger deviations from the Product neutrality function characterize gene pairs affecting the same GO biological process.
(A) Schematic illustration of the analysis process for double mutants where both mutations affect the same GO biological process. We first select two different GO biological processes and extract the double mutants in the Synthetic Genetic Array (SGA) dataset associated with them. Then, we compute the median residual for each pair of biological processes and each neutrality function. (B–D) Median residual for the Minimum, Product, and Additive neutrality functions as a function of the maximum single-mutant fitness. Each line denotes mutations to a single GO biological process. We see larger deviations from the Product model than in Figure 2. (E) Histogram of the SGA dataset after extracting pairs affecting either two different (inter) or the same (intra) GO biological process. Large residuals are much more likely when both mutations affect the same GO biological process.
The Scott–Hwa model with no feedback follows a Minimum neutrality function.
Box plots for the distributions of the residuals for the model of Scott–Hwa model with no feedback as a function of the maximum single-mutant fitness. A thick line denotes the median, and boxes denote the upper and lower quartiles of the data. The absence of feedback due to resource competition in the model of Scott–Hwa model with no feedback results in a Minimum neutrality function.
Large deviations from the Product neutrality function characterize beneficial mutations.
Box plots for the distributions of the residuals for the different neutrality functions for beneficial mutations as a function of the minimum single-mutant fitness. Thick lines denote the median, and boxes denote the upper and lower quartiles of the data. Deviations from the Product neutrality function derived in Derivation of the double-mutant fitness can be unbounded.
Eleven parameters exhibit negative impact on growth rate upon mutation in the Weiße model.
Among the initial 21 parameters, 13 were kept as candidates for a mutational analysis. Two of them vm,Km\begin{document}$v_{m}, K_{m}$\end{document}, associated with the metabolic sector, do not have any impact on growth rate upon mutation, likely because this sector is not limiting for growth in that parameter range. The others have a negative impact upon mutation.
Deviations from the Product neutrality function in the Weiße model are captured by the <italic>γ</italic> approximation.
Box plots for the distributions of the residuals for all models considered in this paper, for two example parameter pairs (A: ωq,ns\begin{document}$\omega_{q}, n_{s}$\end{document}, B: Kt,ωe\begin{document}$K_{t}, \omega_{e}$\end{document}). A thick line denotes the median, and boxes denote the upper and lower quartiles of the data. The Gamma model, in purple, denotes the derivation in Figure 5A. It captures the small deviations from the Product neutrality function.
Tuning <italic>γ</italic> impacts how good an approximation the Product neutrality function is for multiple parameter pairs in the Weiße model.
The analysis of Figure 5C, illustrating the impact of γ on a single parameter pair (ns,vt\begin{document}$n_{s}, v_{t}$\end{document}) is here extended to other parameter pairs to demonstrate the validity of the mechanistic interpretation. (A) vt,ωr\begin{document}$v_{t}, \omega_{r}$\end{document}; (B) ωe,ns\begin{document}$\omega_{e}, n_{s}$\end{document}; (C).θr,vt\begin{document}$\theta_{r}, v_{t}$\end{document} In all cases, we see that decreasing γ results in better alignment with the Product model.
GO biological processes used in the analysis.
GO biological process name
GO biological identifier
DNA integration
GO:0015074
DNA recombination
GO:0006310
DNA repair
GO:0006281
Ascospore formation
GO:0030437
Cell cycle
GO:0007049
Cell division
GO:0051301
Cell wall organization
GO:0071555
Cellular response to DNA damage stimulus
GO:0006974
Cellular response to oxidative stress
GO:0034599
Chromatin remodeling
GO:0006338
Chromatin silencing at telomere
GO:0006348
Chromosome segregation
GO:0007059
Cytoplasmic translation
GO:0002181
Endocytosis
GO:0006897
Endoplasmic reticulum to Golgi vesicle-mediated transport
GO:0006888
Fungal-type cell wall organization
GO:0031505
Intracellular protein transport
GO:0006886
Intracellular signal transduction
GO:0035556
mRNA splicing, via spliceosome
GO:0000398
Macroautophagy
GO:0016236
Maturation of SSU-rRNA from tricistronic rRNA transcript
GO:0000462
Meiotic cell cycle
GO:0051321
Mitochondrial translation
GO:0032543
Negative regulation of transcription by RNA polymerase II
GO:0000122
Positive regulation of transcription by RNA polymerase II
GO:0045944
Proteasome-mediated ubiquitin-dependent protein catabolic process
GO:0043161
Protein folding
GO:0006457
Protein import into nucleus
GO:0006606
Protein phosphorylation
GO:0006468
Protein targeting to vacuole
GO:0006623
Protein transport
GO:0015031
Protein ubiquitination
GO:0016567
Pseudohyphal growth
GO:0007124
rRNA methylation
GO:0031167
rRNA processing
GO:0006364
Reciprocal meiotic recombination
GO:0007131
Regulation of transcription by RNA polymerase II
GO:0006357
Regulation of transcription, DNA-templated
GO:0006355
Ribosomal large subunit biogenesis
GO:0042273
Sporulation resulting in formation of a cellular spore
GO:0030435
Transcription by RNA polymerase II
GO:0006366
Transcription elongation from RNA polymerase II promoter
GO:0006368
Translational termination
GO:0006415
Transmembrane transport
GO:0055085
Transposition, RNA-mediated
GO:0032197
Ubiquitin-dependent protein catabolic process
GO:0006511
Vesicle-mediated transport
GO:0016192
Model parameters from <xref ref-type="bibr" rid="bib54">Weiße et al., 2015</xref>, obtained either from the literature or from parameter optimization.
Description
Default value
Unit
s\begin{document}$s$\end{document}
External nutrient
104
[molecs]
dm\begin{document}$d_{m}$\end{document}
mRNA-degradation rate
0.1
[min−1]
ns\begin{document}$n_{s}$\end{document}
Nutrient efficiency
0.5
None
nr\begin{document}$n_{r}$\end{document}
Ribosome length
7459
[aa/molecs]
nx,x∈{t,m,q}\begin{document}$n_{x}, x \in \{t,m,q\}$\end{document}
10.7554/eLife.105265.3.sa0eLife AssessmentBitbolAnne-FlorenceReviewing EditorEcole Polytechnique Federale de Lausanne (EPFL)SwitzerlandIncompleteValuable
The paper addresses the question of gene epistasis and asks what is the correct null model for which we should declare no epistasis. By reanalyzing synthetic gene array datasets regarding single and double-knockout yeast mutants, and considering two theoretical models of cell growth, the authors reach the valuable conclusion that the product function is a good null model. While the justification of some assumptions is incomplete, the results have the potential to be of value to the field of gene epistasis.
Detecting unexpected epistatic interactions between multiple mutations requires a robust null expectation-or neutral function-that predicts the combined effects of multiple mutations on phenotype based on the individual effects of single mutations. This study evaluated the relevance of the product neutrality function, where double-mutant fitness is represented as a multiplicative combination of single-mutant fitness in the absence of epistatic interactions. The authors used a recent large dataset on fitness, specifically yeast colony size, to analyze epistatic interactions.
The study confirmed that the product function outperformed other neutral functions in predicting double-mutant fitness, showing no bias between negative and positive epistatic interactions. Additionally, in the theoretical portion of the study, the authors employed a previously established theoretical model of bacterial cell growth to simulate growth rates of both single- and double-mutants under multiple parameters. The simulations similarly demonstrated that the product function was superior to other functions in predicting the fitness of hypothetical double-mutants. Based on these findings, the authors concluded that the product function is a robust tool for analyzing epistatic interactions in growth fitness and effectively reflects how growth rates depend on the combination of multiple biochemical pathways.
Strength
By leveraging a previously published large dataset of yeast colony sizes for single- and double-knockout mutants, this study validated the relevance of the product function, which has frequently been used in genetics to analyze epistatic interactions. The confirmation that the product function provides a more reliable prediction of double-mutant fitness compared to other neutral functions is valuable for researchers analyzing epistatic interactions, particularly those working with the same dataset.
Notably, this dataset has been previously used in studies exploring epistatic interactions with the product neutrality function. This study's findings affirm the validity of using the product function, which could enhance confidence in the conclusions drawn by those earlier studies. Consequently, both researchers utilizing this dataset and readers of prior research will benefit from the confirmation provided by this study.
Weakness
This study contains several serious problems, primarily stemming from the following issues: ignoring the substantial differences in the mechanisms regulating cell growth between prokaryotes and eukaryotes and adopting an overly specific and unrealistic set of assumptions in the mutation model. Below, the details are discussed.
(1) Misapplication of prokaryotic growth models
The mechanistic origin of the multiplicative model observed in yeast colony fitness is explained using a bacterial cell growth model. However, there is no valid justification for linking these two systems. The bacterial growth model, the Scott-Hwa model, heavily rely on specific molecular mechanisms, such as ppGpp-mediated regulation, which adjusts ribosome expression and activity during translation. In particular, this mechanism is critical to ensure growth-dependency of the fraction of ribosome in proteome in the Scott-Hwa model [https://doi.org/10.1111/j.1462-2920.2010.02357.x; https://doi.org/10.1073/pnas.2201585119]. Yeast cells lack this regulatory mechanism, making it inappropriate to directly apply bacterial growth models to yeast.
The Weiße model is based on a larger set of underlying equations and involves more parameters than the Scott-Hwa model. In the original paper by Weiße et al. (PNAS, 2015), however, the model parameters were fitted solely to experimental data from E. coli, and the model's applicability to yeast was never assessed. In summary, for neither the Scott-Hwa model nor the Weiße model has it been demonstrated that the entire model quantitatively fits experimental data from yeast. A positive correlation between growth rate and RNA/protein ratio, often observed in yeast, supports only a limited portion of either model, and does not constitute validation of the models as a whole.
(2) Overly specific assumptions in the theoretical model
The theoretical model assumes that two mutations affect only independent parameters of specific biochemical processes. However, this overly restrictive assumption weakens the model's validity in explaining the general occurrence of the multiplicative model in mutations. Furthermore, experimental evidence suggests limitations of this approach. For example, in most viable yeast deletion mutants with reduced growth rates, the expression of ribosomal proteins remained largely unchanged, contrary to the predictions of the Scott-Hwa model [https://doi.org/10.7554/eLife.28034]. This discrepancy highlights that the Scott-Hwa model and its derivatives cannot reliably explain mutants' growth rates based on current experimental evidence.
(3) Limited reliability of the mechanistic origin of the multiplicative model
The authors seem to regard growth-optimizing feedback as the mechanistic origin of the multiplicative model. However, the importance of growth-optimizing feedback in explaining product neutrality heavily depends on the very specific framework of the Scott-Hwa model. As I pointed out above, the Scott-Hwa model is a bacterial growth model that considers only a narrowly defined set of biochemical reactions. Using such a narrow model to explore the mechanistic origin of product neutrality observed on a genome-wide scale appears to be inappropriate. Arguments based on either the Scott-Hwa model or the Weiße model fail to account for the generality of product neutrality across diverse genetic perturbations. These models, in their current form, do not explain the broader patterns of product neutrality observed experimentally.
The paper deals with the important question of gene epistasis, focusing on asking what is the correct null model for which we should declare no epistasis.
In the first part, they use the Synthetic Genetic Array dataset to claim that the effects of a double mutation on growth rate is well predicted by the product of the individual effects (much more than e.g. the additive model). The second (main) part shows this is also the prediction of two simple, coarse-grained models for cell growth.
I find the topic interesting, the paper well written, and the approach innovative.
Comments on revisions:
The authors have adequately addressed the comments raised in the review below, and I find that the paper has improved.
10.7554/eLife.105265.3.sa3Author responseFuentes ValenzuelaLucasAuthorStanford UniversityStanfordUnited StatesFrancoisPaulAuthorUniversité de MontréalMontrealCanadaSkotheimJan MAuthorStanford UniversityStanfordUnited States
The following is the authors’ response to the original reviews
Reviewer #1 (Public review):
Summary:
Detecting unexpected epistatic interactions among multiple mutations requires a robust null expectation - or neutral function - that predicts the combined effects of multiple mutations on phenotype, based on the effects of individual mutations. This study assessed the validity of the product neutrality function, where the fitness of double mutants is represented as the multiplicative combination of the fitness of single mutants, in the absence of epistatic interactions. The authors utilized a comprehensive dataset on fitness, specifically measuring yeast colony size, to analyze epistatic interactions.
The study confirmed that the product function outperformed other neutral functions in predicting the fitness of double mutants, showing no bias between negative and positive epistatic interactions. Additionally, in the theoretical portion of the study, the authors applied a wellestablished theoretical model of bacterial cell growth to simulate the growth rates of both single and double mutants under various parameters. The simulations further demonstrated that the product function was superior to other functions in predicting the fitness of hypothetical double mutants. Based on these findings, the authors concluded that the product function is a robust tool for analyzing epistatic interactions in growth fitness and effectively reflects how growth rates depend on the combination of multiple biochemical pathways.
Strengths:
By leveraging a previously published extensive dataset of yeast colony sizes for single- and double-knockout mutants, this study validated the relevance of the product function, commonly used in genetics to analyze epistatic interactions. The finding that the product function provides a more reliable prediction of double-mutant fitness compared to other neutral functions offers significant value for researchers studying epistatic interactions, particularly those using the same dataset.
Notably, this dataset has previously been employed in studies investigating epistatic interactions using the product neutrality function. The current study's findings affirm the validity of the product function, potentially enhancing confidence in the conclusions drawn from those earlier studies. Consequently, both researchers utilizing this dataset and readers of previous research will benefit from the confirmation provided by this study's results.
Weaknesses:
This study exhibits several significant logical flaws, primarily arising from the following issues: a failure to differentiate between distinct phenotypes, instead treating them as identical; an oversight of the substantial differences in the mechanisms regulating cell growth between prokaryotes and eukaryotes; and the adoption of an overly specific and unrealistic set of assumptions in the mutation model. Additionally, the study fails to clearly address its stated objective-investigating the mechanistic origin of the multiplicative model. Although it discusses conditions under which deviations occur, it falls short of achieving its primary goal. Moreover, the paper includes misleading descriptions and unsubstantiated reasoning, presented without proper citations, as if they were widely accepted facts. Readers should consider these issues when evaluating this paper. Further details are discussed below.
(1) Misrepresentation of the dataset and phenotypes
The authors analyze a dataset on the fitness of yeast mutants, describing it as representative of the Malthusian parameter of an exponential growth model. However, they provide no evidence to support this claim. They assert that the growth of colony size in the dataset adheres to exponential growth kinetics; in contrast, it is known to exhibit linear growth over time, as indicated in [Supplementary Note 1 of https://doi.org/10.1038/nmeth.1534]. Consequently, fitness derived from colony size should be recognized as a different metric and phenotype from the Malthusian parameter. Equating these distinct phenotypes and fitness measures constitutes a fundamental error, which significantly compromises the theoretical discussions based on the Malthusian parameter in the study.
The reviewer is correct in pointing out that colony-size measurements are distinct from exponential growth kinetics. We acknowledge that our original text implied that the dataset directly measured the exponential growth rate (Malthusian parameter), when in fact it was measuring yeast colony expansion rates on solid media. Colony growth under these conditions often follows a biphasic pattern in that there is typically an initial microscopic phase where cells can grow exponentially, but as the colony expands further then the growth dynamics become more linear (Meunier and Choder 1999). We have revised our text to state clearly what the experiment measured.
However, while colony size does not exhibit exponential growth kinetics, several studies have argued that the rate of colony expansion is related to the exponential growth rate of cells growing in non-limiting nutrient conditions in liquid culture. This is because colony growth is dominated by cells at the colony boundaries that have access to nutrients and are in exponential growth. Cells in the colony interior lack nutrients and therefore contribute little to colony growth. This has been shown both in theoretical and experimental studies, finding that the linear growth rate of the colony is directly linked to the single-cell exponential growth rate (Pirt 1967; Gray and Kirwan 1974; Korolev et al. 2012; Gandhi et al. 2016; Meunier and Choder 1999). In particular, the above studies suggest that the linear colony growth rate is directly proportional to the square root of the exponential growth rate. Therefore, one would expect that the validity of the product model for one fitness measure implies its validity for the other measure. In addition, colony size was found to be highly correlated with the exponential growth rate of cells in non-limiting nutrients in liquid cultu
re (Baryshnikova et al. 2010; Zackrisson et al. 2016; Miller et al. 2022). For these reasons, we treated the colony size and exponential growth rate as interchangeable in our original manuscript.
However, while colony size does not exhibit exponential growth kinetics, several studies have argued that the rate of colony expansion is related to the exponential growth rate of cells growing in non-limiting nutrient conditions in liquid culture. This is because colony growth is dominated by cells at the colony boundaries that have access to nutrients and are in exponential growth. Cells in the colony interior lack nutrients and therefore contribute little to colony growth. This has been shown both in theoretical and experimental studies, finding that the linear growth rate of the colony is directly linked to the single-cell exponential growth rate (Pirt 1967; Gray and Kirwan 1974; Korolev et al. 2012; Gandhi et al. 2016; Meunier and Choder 1999). In particular, the above studies suggest that the linear colony growth rate is directly proportional to the square root of the exponential growth rate. Therefore, one would expect that the validity of the product model for one fitness measure implies its validity for the other measure. In addition, colony size was found to be highly correlated with the exponential growth rate of cells in non-limiting nutrients in liquid culture (Baryshnikova et al. 2010; Zackrisson et al. 2016; Miller et al. 2022). For these reasons, we treated the colony size and exponential growth rate as interchangeable in our original manuscript.
To address the important point raised by the reviewer, we now explain more clearly in the text what the analyzed data on colony size show and why we believe it is reflective of the exponential growth rate. Finally, we note that our results supporting the product neutrality function are consistent with the work of (Mani et al. 2008), which used smaller datasets based on liquid culture growth rates (Jasnos and Korona 2007; Onge et al. 2007).
The text in Section 2.3 now reads:
“Having verified empirically that the Product neutrality function is supported by the latest data for cell proliferation, we now turn our attention to its origins. Addressing this question requires some mechanistic model of biosynthesis. However, most mechanistic models of growth apply directly to single cells in rich nutrient conditions, which may not directly apply to the SGA measurements of colony expansion rates. In particular, colony growth has been shown to follow a biphasic pattern (Meunier et al. 1999). A first exponential phase is followed by a slower linear phase as the colony expands. Previous modeling and empirical work indicates that this second linear expansion rate reflects the underlying exponential growth of cells in the periphery of the colony (Pirt 1967; Gray et al. 1974; Gandhi et al. 2016; Baryshnikova, Costanzo, S. Dixon, et al. 2010; Zackrisson et al. 2016; Miller et al. 2022). More precisely, mathematical models show the linear colony-size expansion rate is directly proportional to the square root of the exponential growth rate under non-limiting conditions. Intuitively, this relationship arises because colony growth is dominated by the expansion of the population of cells in an annulus at the colony border that are exposed to rich nutrient conditions. These cells expand at a rate similar to the exponential rate of cells growing in a rich nutrient liquid culture. In contrast, the cells in the interior of the colony experience poor nutrient conditions, grow very slowly, and do not contribute to colony growth.
This intimate relationship between both proliferation rates allows us to explore the origin of the Product neutrality function in mechanistic models of cell growth. Indeed, if colony-based fitnesses follow a Product model, thenWxyc∼WxcWyc⇔λxycλWTc∼λxcλyc(λWTc)2\begin{document}$$\displaystyle W_{x y}^{c} \sim W_{x}^{c} W_{y}^{c} \Leftrightarrow \frac{\lambda_{x y}^{c}}{\lambda_{W T}^{c}} \sim \frac{\lambda_{x}^{c} \lambda_{y}^{c}}{\left(\lambda_{W T}^{c}\right)^{2}}$$\end{document}
where the superscript c indicates colony-based values for the fitness W and the growth rate λ. Taking into account the relationship between single-cell exponential growth rates and colony growth rates, we can writeλc∝λl\begin{document}$$\displaystyle \lambda^{c} \propto \sqrt{\lambda^{l}}$$\end{document}
where the superscript l denotes liquid cultures. Combining these expressions, we obtainλxylλWTl∼λxlλylλWTl2⇒Wxyl∼WxlWyl\begin{document}$$\displaystyle \frac{\sqrt{\lambda_{x y}^{l}}}{\sqrt{\lambda_{W T}^{l}}} \sim \frac{\sqrt{\lambda_{x}^{l}} \sqrt{\lambda_{y}^{l}}}{{\sqrt{\lambda_{W T}^{l}}}^{2}} \Rightarrow W_{x y}^{l} \sim W_{x}^{l} W_{y}^{l}$$\end{document}
In other words, from the perspective of the Product neutrality function, fitnesses based on colony expansion rates are equivalent to fitnesses based on single-cell exponential growth rates. The prevalence of the Product neutrality model—both in the SGA data and in previous studies on datasets from liquid cultures (Jasnos et al. 2007; Onge et al. 2007; Mani et al. 2008)—encourages the exploration of its origin in mechanistic models of cell growth.”
(2) Misapplication of prokaryotic growth models
The study attempts to explain the mechanistic origin of the multiplicative model observed in yeast colony fitness using a bacterial cell growth model, particularly the Scott-Hwa model. However, the application of this bacterial model to yeast systems lacks valid justification. The Scott-Hwa model is heavily dependent on specific molecular mechanisms such as ppGppmediated regulation, which plays a crucial role in adjusting ribosome expression and activity during translation. This mechanism is pivotal for ensuring the growth-dependency of the ribosome fraction in the proteome, as described in [https://doi.org/10.1073/pnas.2201585119]. Unlike bacteria, yeast cells do not possess this regulatory mechanism, rendering the direct application of bacterial growth models to yeast inappropriate and potentially misleading. This fundamental difference in regulatory mechanisms undermines the relevance and accuracy of using bacterial models to infer yeast colony growth dynamics.
If the authors intend to apply a growth model with macroscopic variables to yeast double-mutant experimental data, they should avoid simply repurposing a bacterial growth model. Instead, they should develop and rigorously validate a yeast-specific growth model before incorporating it into their study.
There is nothing that is prokaryote specific in the Scott-Hwa model. It does not include the specific ppGpp mechanism to regulate ribosome fraction that does not exist in eukaryotes. The general features of the model, like how the ribosome fraction is proportional to the growth rate have indeed been validated in yeast (Metzl-Raz et al. 2017; Elsemman et al. 2022; Xia et al. 2022). Performing a detailed physiological analysis of budding yeast across varying growth conditions in order to build a more extensive model is beyond the scope of this work. Finally, we note that the Weiße model, which we also analyzed, is also generic and has replicated empirical measurements both from bacteria and yeast (Weiße et al. 2015).
To clarify this point in the text, we have added the following to Section 2.3:
“Experimental measurements in other organisms suggest that the observations leading to this model, including that the cellular ribosome fraction increases with growth rate, are in fact generic and also seen in the yeast S. cerevisiae (Metzl-Raz et al. 2017; Elsemman et al. 2022; Xia et al. 2022).”
(3) Overly specific assumptions in the theoretical model
he theoretical model in question assumes that two mutations affect only independent parameters of specific biochemical processes, an overly restrictive premise that undermines its ability to broadly explain the occurrence of the multiplicative model in mutations. Additionally, experimental evidence highlights significant limitations to this approach. For example, in most viable yeast deletion mutants with reduced growth rates, the expression of ribosomal proteins remains largely unchanged, in direct contradiction to the predictions of the Scott-Hwa model, as indicated in [https://doi.org/10.7554/eLife.28034]. This discrepancy emphasizes that the ScottHwa model and its derivatives do not reliably explain the growth rates of mutants based on current experimental data, suggesting that these models may need to be reevaluated or alternative theories developed to more accurately reflect the complex dynamics of mutant growth.
In the data from the Barkai lab referenced by the reviewer (reproduced below), we see that the ribosomal transcript fraction is in fact proportional to growth rate in response to gene deletions in contradiction to the reviewer’s interpretation. However, it is notable that the ribosomal transcript fraction is a bit higher for a given growth rate if that growth rate is generated by a mutation rather than generated by a suboptimal nutrient condition. We know that the very simple Scott-Hwa model is not a perfect representation of the cell. Nevertheless, it does recapitulate important aspects of growth physiology and therefore we thought it is useful to analyze its response to mutations and compare those responses to the different neutrality functions. We never claimed the Scott-Hwa model was a perfect model and fully agree with the referee’s statement above that “... these models may need to be reevaluated, or alternative theories developed to more accurately reflect the complex dynamics of mutant growth.” Indeed, we say as much in our discussion where we wrote:
“While we focused on coarse-grained models for their simplicity and mechanistic interpretability, they might be too simple to effectively model large double-mutant datasets and the resulting double-mutant fitness distributions. We therefore expect the combination of high throughput genetic data with the analysis of larger-scale models, for instance based on Flux Balance Analysis, Metabolic Control Analysis, or whole-cell modeling, to lead to important complementary insights regarding the regulation of cell growth and proliferation.”
To further clarify this point, we discuss and cite the Barkai lab data for gene deletions see Figure 2 from Metzl-Raz et al. 2017.
(4) Lack of clarity on the mechanistic origin of the multiplicative model
The study falls short of providing a definitive explanation for its primary objective: elucidating the "mechanistic origin" of the multiplicative model. Notably, even in the simplest case involving the Scott-Hwa model, the underlying mechanistic basis remains unexplained, leaving the central research question unresolved. Furthermore, the study does not clearly specify what types of data or models would be required to advance the understanding of the mechanistic origin of the multiplicative model. This omission limits the study's contribution to uncovering the biological principles underlying the observed fitness patterns.”
We appreciate the reviewer’s interest in a more complete mechanistic explanation for the product model of fitness. The primary goal of this study was to explore the validity of the Product model from the perspective of coarse-grained models of cell growth, and to extract mechanistic insights where possible. We view our work as a first step toward a deeper understanding of how double-mutant fitnesses combine, rather than a final, all-encompassing theory. As the referee notes, we are limited by the current state of the field, which has an incomplete understanding of cell growth.
Nonetheless, our analysis does propose concrete, mechanistically informed explanations. For example, we highlight how growth-optimizing feedback—such as cells’ ability to reallocate ribosomes or adjust proteome composition—naturally leads to multiplicative rather than additive or minimal fitness effects. We also link the empirical deviations from pure multiplicative behavior to differences in how specific pathways re-balance under perturbation, and we suggest that a product-like rule emerges when multiple interconnected processes each partially limit cell growth.
In the discussion, we clarify what additional data and models we think will be required to advance this question. Namely, we propose extending our approach through larger-scale, more detailed modeling frameworks – that may include explicit modeling of ppGpp or TOR activities in bacteria or eukaryotic cells, respectively. We also emphasize the importance of refining the measurement of cell growth rates to uncover subtle deviations from the product rule that could yield greater mechanistic insight. By integrating high-throughput genetic data with nextgeneration computational models, it should be possible to hone in on the specific biological principles (e.g., metabolic bottlenecks, resource reallocation) that underlie the multiplicative neutrality function.
Reviewer #2 (Public review):
The paper deals with the important question of gene epistasis, focusing on asking what is the correct null model for which we should declare no epistasis.
In the first part, they use the Synthetic Genetic Array dataset to claim that the effects of a double mutation on growth rate are well predicted by the product of the individual effects (much more than e.g. the additive model). The second (main) part shows this is also the prediction of two simple, coarse-grained models for cell growth.
I find the topic interesting, the paper well-written, and the approach innovative.
One concern I have with the first part is that they claim that:
"In these experiments, the colony area on the plate, a proxy for colony size, followed exponential growth kinetics. The fitness of a mutant strain was determined as the rate of exponential growth normalized to the rate in wild type cells."
There are many works on "range expansions" showing that colonies expand at a constant velocity, the speed of which scales as the square root of the growth rate (these are called "Fisher waves", predicted in the 1940', and there are many experimental works on them, e.g. https://www.pnas.org/doi/epdf/10.1073/pnas.0710150104) If that's the case, the area of the colony should be proportional to growth_rate X time^2 , rather than exp(growth_rate*time), so the fitness they might be using here could be the log(growth_rate) rather than growth_rate itself? That could potentially have a big effect on the results.
We thank the reviewer for their thoughtful remarks. As they rightly pointed out, a large body of literature supports that colonies expand at constant velocity both from a theoretical and experimental standpoint.
As discussed in the answer to the first question of Reviewer 1, this body of work also suggests that the linear expansion rate of the colony front is directly related to the single-cell exponential growth rate of the cells at the periphery. Hence, although the macroscopic colony growth may not be exponential in time, measuring colony size (or radial expansion) across different genotypes still provides a consistent and meaningful proxy for comparing their underlying growth capabilities.
In particular, these studies suggest (consistently with Fisher-wave theory) that the linear growth rate of the colony 𝐾 is proportional to the square root of the exponential growth rate 𝜆. Under the assumption that the product model is valid for a given double mutant and for the exponential growth rate, we would have thatWxy=WxWy⇔λxy=λxλy/λ0\begin{document}$$\displaystyle W_{x y}=W_{x} W_{y} \Leftrightarrow \lambda_{x y}=\lambda_{x} \lambda_{y} / \lambda_{0}$$\end{document}
The associated wave-front velocities would then be predicted to beKxy∝λxy⇔Kxy∝λxλy/λ0⇔Kxy∝KxKy/K0\begin{document}$$\displaystyle K_{x y} \propto \sqrt{ } \lambda_{x y} \Leftrightarrow K_{x y} \propto \sqrt{ } \lambda_{x} \sqrt{ } \lambda_{y} / \sqrt{ } \lambda_{0} \Leftrightarrow K_{x y} \propto K_{x} K_{y} / K_{0}$$\end{document}
In other words, if the product model is valid for fitness measures based on exponential growth rates, it should also be valid for fitness measures based on linear colony growth rates.
We now include this discussion in the revised version of Section 2.3.
Additional comments/questions:
(1) What is the motivation for the model where the effect of two genes is the minimum of the two?
The motivation for the minimal model is the notion that there might be a particular process that is rate-limiting for growth due to a mutation. In this case, a mutation in process X makes it really slow and process Y proceeds in parallel and has plenty of time to finish its job before cell division takes place. In this case, even a mutation to process Y might not slow down growth because there is an excess amount of time for it to be completed. Thus, the double mutant might then be anticipated to have the growth rate associated with the single mutation to process X. We now add a similar description when we introduce the different neutrality functions in Section 2.1.
(2) How seriously should we take the Scott-Hwa model? Should we view it as a toy model to explain the phenomenon or more than that? If the latter, then since the number of categories in the GO analysis is much more than two (47?) in many cases the analysis of the experimental data would take pairs of genes that both affect one process in the Scott-Hwa model - and then the product prediction should presumably fail? The same comment applies to the other coarse-grained model.
From our perspective, models like the Scott-Hwa model constitute the simplest representation of growth based on data that is not trivial. Moreover, the Scott-Hwa model is able to incorporate interactions between two different biological processes. We believe models, like the Scott-Hwa and Weiße models, should be viewed as more than mere toy models because they have been backed up by some empirical data, such as that showing the ribosome fraction increases with growth rate. However, the Scott-Hwa model is inherently limited by its low dimensionality and relative simplicity. We do not claim that such models can provide a full picture of the cell. As argued in the main text, we have chosen to focus on such models because of their tractability and in the hope of extracting general principles. We nonetheless agree with the reviewer that they do not have the capacity to represent interactions between genes in the same biological process. We now note this limitation in the text.
(3) There are many works in the literature discussing additive fitness contributions, including Kaufmann's famous NK model as well as spin-glass-type models (e.g. Guo and Amir, Science Advances 2019, Reddy and Desai, eLife 2021, Boffi et al., eLife 2023) These should be addressed in this context.
We thank the reviewer for pointing out this part of the literature. We do believe these works constitute a relevant body of work tackling the emergence of epistasis patterns from a theoretical grounding, and now reference and discuss them in the text.
(4) The experimental data is for deletions, but it would be interesting to know the theoretical model's prediction for the expected effects of beneficial mutations and how they interact since that's relevant (as mentioned in the paper) for evolutionary experiments. Perhaps in this case the question of additive vs. multiplicative matters less since the fitness effects are much smaller.
This is an interesting question. Since mutations increasing the growth rate generated by gene deletions or other systematic perturbations are rare, we did not focus on them. Of course, as the reviewer notes, in the case of evolution experiments, these fitness enhancing mutations are selected for. To address the reviewer's question, we can first consider the Scott-Hwa model. In this case, the analytical solution remains valid in the case of fitness enhancing mutations so that the fitness of the double mutant will be the product neutrality function multiplied by an additional interaction term (see Figure 3). The mathematical derivation predicts that the double mutant fitness can potentially grow indefinitely. Indeed, the denominator can be equal to zero in some cases. In simulations, we see that the observation for deleterious mutations does not seem to hold for beneficial mutations (new supplementary Figure S5 shown below). Indeed, no model seems to replicate double mutant fitnesses much better than any other. This suggests that the growth-optimizing feedback we discuss in section 2.3 may have compound effects that ultimately make double-mutant fitnesses much larger than any model predicts.
We recognize this may be an important point, and discuss it in detail in the revised section 2.3 as well as in the discussion.
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