Consider a mutation A that perturbs only one rate constant xi⁢j, such that the wildtype value xi⁢j0 changes to xi⁢jA. This mutation can be quantified at the microscopic level by its relative effect δA⁢xi⁢j=xi⁢jA/xi⁢j0-1. If the reaction between metabolites i and j belongs to nested modules μ,ν,…, then mutation A may impact the functions of these modules, which can be quantified by the relative effects δA⁢yμ, δA⁢yν, etc. at each level of the hierarchy.

Consider now another mutation B that only perturbs the rate constant xk⁢ℓ of another reaction. Since mutations A and B perturb distinct enzymes, they by definition do not genetically interact at the microscopic level. However, if both perturbed reactions belong to the metabolic module μ (and, as a consequence, to all higher-level modules which contain μ), they may interact at the level of the function of this module, in the sense that the effect of mutation B on the effective rate yμ may depend on whether mutation A is present or not. Such epistasis between mutations A and B can be quantified at the level μ of the metabolic hierarchy by a number of various epistasis coefficients (Wagner et al., 1998; Hansen and Wagner, 2001; Mani et al., 2008). I will quantify it with the epistasis coefficient(5)εA⁢B⁢yμ=δA⁢B⁢yμ-δA⁢yμ-δB⁢yμ2⁢δA⁢yμ⁢δB⁢yμ,where δA⁢B⁢yμ denotes the effect of the combination of mutations A and B on phenotype yμ relative to the wildtype. Since I only consider two mutations A and B, I will write ε⁢yμ instead of εA⁢B⁢yμ to simplify notations. Note that other epistasis coefficients can always be computed from ε⁢yμ, δA⁢yμ and δB⁢yμ, if necessary. Expressions for epistasis coefficients at other levels of the metabolic hierarchy are analogous.

Assuming that the effects of both individual mutations and their combined effect at the lower-level μ are small, it follows from Equation 4 and Equation 5 that(6)εyν=εyμC+H2C2+o(1),where C=f1′⁢yμ/yν and H=f1′′⁢yμ2/yν are the first- and second-order control coefficients of the lower-level module μ with respect to the flux through the higher-level module ν and o⁢(1) denotes all terms that vanish as the effects of mutations tend to zero (see Materials and methods for details). Note that Equation 6 is a special case of a more general Equation 49 which describes the case when mutations affect multiple enzymes. Equation 6 defines a linear map ϕ with slope 1/C and a fixed point ε¯=-H⁢(2⁢C⁢(1-C))-1, which both depend on the topology of the higher-level module ν and the rate constants x→ν∖μ.

To gain some intuition for how the map ϕ governs the propagation of epistasis from a lower level μ to a higher level ν, suppose that module ν is a linear metabolic pathway. In this case, it is intuitively clear that function f1 is monotonically increasing (i.e. the higher yμ, the more flux can pass through the linear pathway ν) and concave (i.e. as yμ grows, other reactions in ν become increasingly more limiting, such that further gains in yμ yield smaller gains in yν). Indeed, it is easy to show that C=(1+α⁢yμ)-1>0 and H=-2⁢α⁢yμ⁢(1+α⁢yμ)-2<0, where α is a positive constant that depends on other reactions in the pathway (see Materials and methods for details). It then immediately follows that any zero or negative epistasis ε⁢yμ that already arose at the lower level would propagate to negative epistasis ε⁢yν at the level of the linear pathway ν. Moreover, since C<1, the fixed point of the map in Equation 6 is unstable. Therefore, if epistasis ε⁢yμ was already sufficiently large at the lower level, it would induce even larger positive epistasis ε⁢yν at the level of the linear pathway ν. In fact, when module ν is a linear pathway, ε¯=1, so that ε⁢yν>1 whenever ε⁢yμ>1.

The first result of this paper is the following theorem, which shows that the same rules of propagation of epistasis hold not only for a linear pathway but for any module (Figure 2).

Which of the three regimes described above can emerge in metabolic networks and under what circumstances? In other words, if two mutations affect the same module, are there any constraints on epistasis that might arise at the level of the effective rate constant of this module? To address this question, I consider two mutations A and B that affect the rate constants of different single reactions within a given module.

Consider a relatively simple module ν shown in Figure 1A and two mutations A and B that affect the reactions, as shown in Figure 3A. I will now show that the epistasis coefficient ε⁢yν can take values in all three domains described above, depending on the biochemical details of this module. Using the recursive procedure for evaluating yμ described in Materials and methods, it is straightforward to obtain an analytical expression for yν as a function of the rate matrix x→ν, from which ε⁢yμ can also be obtained (see Materials and methods for details). To demonstrate that ε⁢yμ can take values below 0, between 0 and 1, and above 1, it is convenient to keep all of the rate constants fixed except for the rate constant z≡x34 of a reaction that is not affected by mutations A or B, as shown in Figure 3A. Figure 3B then shows how the epistasis coefficient ε⁢yμ varies as a function of z for one particular choice of all other rate constants. When z is small, ε⁢yμ<0. As z increases, it becomes weakly positive (0<ε⁢yμ<1) and eventually strongly positive (ε⁢yμ>1). Thus, in my model, there are no fundamental constraints on the types of epistasis that can emerge between mutations.

This simple example also reveals that not only the value but also the sign of epistasis generically depend on the rates of other reactions in the network, such that other mutations or physiological changes in enzyme expression levels can modulate epistasis sign and strength. In other words, ‘higher-order’ and ‘environmental’ epistasis are generic features of metabolic networks.

Upon closer examination, the toy example in Figure 3 also suggests that the sign of ε⁢yν may depend predictably on the topological relationship between the affected reactions. When z=0, the two reactions affected by mutations are parallel, and epistasis is negative. When z is very large, most of the flux between the I/O metabolites passes through z such that the two reactions affected by mutations become effectively serial, and epistasis is strongly positive. Other toy models show consistent results: epistasis between mutations affecting different reactions in a linear pathway is always positive and epistasis between mutations affecting parallel reactions is negative (see Materials and methods for details). These observation suggest an interesting conjecture. Do mutations affecting parallel reactions always exhibit negative epistasis and do mutations affecting serial reactions always exhibit positive epistasis? In fact, such relationship between sign of epistasis and topology has been previously suggested in the literature (e.g. Dixon et al., 2009; Lehner, 2011).

To formalize and mathematically prove this hypothesis, I first define two reactions as parallel within a given module if there exist at least two distinct simple (i.e. non-self-intersecting) paths that connect the I/O metabolites, such that each path lies within the module and contains only one of the two focal reactions. Analogously, two reactions are serial within a given module if there exists at least one simple path that connects the I/O metabolites, lies within the module and contains both focal reactions.

According to these definitions, two reactions can be simultaneously parallel and serial, as, for example, the reactions affected by mutations A and B in Figure 3A. I call such reaction pairs serial-parallel. I define two reactions to be strictly parallel if they are parallel but not serial (Figure 3C) and I define two reactions to be strictly serial if they are serial but not parallel (Figure 3D). Thus, each pair of reactions within a module can be classified as either strictly parallel, strictly serial or serial-parallel.

The second result of this paper is the following theorem.

Both Theorem 1 and Theorem 2 strictly hold only when the effects of both mutations are infinitesimal. Next, I investigate how these results might change when the mutational effects are finite.

The results of previous sections revealed a relationship between network topology and the ensuing epistasis coefficients in an analytically tractable model. However, the assumptions of this model are most certainly violated in many realistic situations. It is therefore important to know whether the same or similar rules of epistasis emergence and propagation hold beyond the scope of this model. I address this question here by analyzing a computational kinetic model of glycolysis developed by Chassagnole et al., 2002. This model keeps track of the concentrations of 17 metabolites, reactions between which are catalyzed by 18 enzymes (Figure 5A and Figure 5—figure supplement 1; see Materials and methods for details). This model falls far outside of the analytical framework introduced in this paper: some reactions are second-order, reaction kinetics are non-linear, and in several cases the reaction rates are modulated by other metabolites (Chassagnole et al., 2002).

Testing the predictions of the analytical theory in this computational model faces two complications. First, in a non-linear model, modules are no longer fully characterized by their effective rate constants, even at steady state. Instead, each module is described by the flux between its I/O metabolites which non-linearly depends on the concentrations of these metabolites. Consequently, the effects of mutations and epistasis coefficients also become functions of the I/O metabolite concentrations. An epistasis coefficient at the level of module ν can still be evaluated according to Equation 5, with yν now representing the flux through module ν evaluated at a particular concentration of the I/O metabolites. For simplicity, I computationally find the steady state of the full glycolysis network and evaluate the epistasis coefficients only at this steady state, that is, for each module, I keep the concentrations of the I/O metabolites fixed at their steady-state values for the full network (see Materials and methods for details).

The second complication is that some control coefficients are so small that they fall below the threshold of numerical precision. Perturbing such reactions has no detectable effect on flux (Figure 5—figure supplement 2). In the analysis that follows, I ignore such reactions because the epistasis coefficient defined by Equation 5 can only be computed for mutations with non-zero effects on flux. In addition, the control coefficients of some reactions are negative, which implies that an increase in the rate of such reaction decreases the flux through the module (Figure 5—figure supplement 2). I also ignore such reactions because there is no analog for them in the analytical theory presented above. After excluding seven reactions for these reasons, I examine epistasis in 55 pairs of mutations that affect the remaining 11 reactions.

The glycolysis network shown in Figure 5A (see also Figure 5—figure supplement 1) can be naturally partitioned into four modules which I name ‘LG’ (lower glycolysis), ‘UGPP’ (upper glycolysis and pentose phosphate), ‘GPP’ (glycolysis and pentose phosophate), and ‘FULL’. Modules LG and UGPP are non-overlappng and both of them are nested in module GPP which in turn is nested in the FULL module. Thus, at least for some reaction pairs it is possible to calculate epistasis coefficients at three levels of metabolic hierarchy. There are three such pairs, and the results for them are shown in Figure 5B. Epistasis for the remaining pairs of reactions can be evaluated only at one or two levels of the hierarchy because these reactions belong to different modules at the lowest levels or because their individual effects are too small. The results for all reaction pairs are shown in Figure 5—figure supplement 3.

The strongest qualitative prediction of the analytical theory described above is that negative epistasis for a lower-level phenotype cannot turn into positive epistasis for a higher-level phenotype, while the converse is possible. Figure 5B and Figure 5—figure supplement 3 show that the data are consistent with this prediction. Another prediction is that epistasis between strictly parallel reactions should be negative. There is only one pair of reactions that are strictly parallel, those catalyzed by glucose-6-phosphate isomerase (PGI) and 6-phosphogluconate dehydrogenase (PGDH), and indeed the epistasis coefficients between mutations affecting these reactions are negative at all levels of the hierarchy (Figure 5B). Finally, the analytical theory predicts that mutations affecting strictly serial reactions should exhibit strong positive epistasis. There are 36 reaction pairs that are strictly serial. Epistasis is positive between mutations in 33 of them, and it is strongly positive in 17 of them (Figure 5—figure supplement 3). Three pairs of strictly serial reactions (those where one reaction is catalyzed by PK and the other is catalyzed by PGI, PGDH, or PFK) exhibit negative epistasis (Figure 5—figure supplement 3). These results suggest that, although one may not be able to naively extrapolate the rules of emergence and propagation of epistasis derived in the simple analytical model to more complex networks, some generalized versions of these rules may nevertheless hold more broadly.

Simple models help us identify generic phenomena—those that are shared by a large class of systems—which should inform our ‘null’ expectations in empirical studies. Deviations from such null in a given system under examination inform us about potentially interesting peculiarities of this system. The model presented here suggests several generic features of epistasis between genome-wide mutations.

My analysis shows that the value of an epistasis coefficient measured for a higher level phenotype is a result of two contributions (Domingo et al., 2019), propagation and emergence, which correspond to two terms in Equation 6 (or the more general Equation 49). The first term, propagation, shows that if two mutations exhibit epistasis for a lower-level phenotype they also generally exhibit epistasis for a higher-level phenotype. The second contribution comes from the fact that lower-level phenotypes map onto higher-level phenotypes via non-linear functions. This is true even in a simple model with linear kinetics considered here. As a result, even if two mutations exhibit no epistasis at the lower-level phenotype, epistasis must emerge for the higher-level phenotype, as previously pointed out by multiple authors (e.g. Kacser and Burns, 1981; DePristo et al., 2005; Martin et al., 2007; Chiu et al., 2012; Otwinowski et al., 2018; Domingo et al., 2019; Husain and Murugan, 2020).

My analysis shows that the value of an epistasis coefficient for a particular pair of mutations is in large part determined by the topological relationship between reactions affected by them. Since the topology of the metabolic network itself depends on the genotype (which genes are present in the genome) and on the environment (which enzymes are active or not), the topological relationship between two specific reactions might change if, for example, a third mutation knocks out another enzyme or if an enzyme is up- or down-regulated due to an environmental change (see Figure 3). Thus, we should generically expect epistasis between mutations to depend on the environment and on the presence or absence of other mutations in the genome. In other words, G×G×G interactions (higher-oder epistasis) and G×G×E interactions (environmental epistasis) should be common (Snitkin and Segrè, 2011; Flynn et al., 2013; Lindsey et al., 2013; Taylor and Ehrenreich, 2015; Sailer and Harms, 2017a). This fact complicates the interpretation of inter-gene epistasis since mutations in the same pair of genes can exhibit qualitatively different genetic interactions in different strains, organisms and environments, as has been observed (St Onge et al., 2007; Musso et al., 2008; Tischler et al., 2008; Dowell et al., 2010; Heigwer et al., 2018; Li et al., 2019). However, the situation may not be hopeless because the topological relationship between two reactions cannot change arbitrarily after addition or removal of a single reaction. For example, if two reactions are strictly parallel, removing a third reaction does not alter their relationship (see Proposition 5). Thus, comparing matrices of epistasis coefficients measured in different environments or genetic backgrounds could inform us about how the organism rewires its metabolic network in response to these perturbations (St Onge et al., 2007; Musso et al., 2008; Heigwer et al., 2018; Li et al., 2019).

Studies that measure epistasis for fitness-related phenotypes among genome-wide mutations usually find both positive and negative epistases, but the preponderance of positive and negative epistasis varies. Some authors reported a skew toward positive interactions among deleterious mutations (Jasnos and Korona, 2007; He et al., 2010; Johnson et al., 2019), whereas others reported a skew toward negative interactions (Szappanos et al., 2011; Costanzo et al., 2016). Beneficial mutations appear to predominantly exhibit negative epistasis, also known as ‘diminishing returns’ epistasis (e.g. Martin et al., 2007; Khan et al., 2011; Chou et al., 2011; Kryazhimskiy et al., 2014; Schoustra et al., 2016). The reasons for these patterns are currently unclear. Several recent theoretical papers offer possible statistical explanations for them (Martin, 2014; Lyons et al., 2020; Reddy and Desai, 2020). On the other hand, mechanistic predictions for the distribution of epistasis coefficients are not yet available (but see Sanjuán and Nebot, 2008; Macía et al., 2012; Chiu et al., 2012). The present work does not directly address this problem either, but it provides some additional clues.

First, my model shows that the sign of epistasis at least to some extent reflects the topology of the network. Thus, the distribution of epistasis coefficients at high-level phenotypes in real organisms should ultimately depend on the preponderance of different topological relationships between the edges in biological networks. It then seems a priori unlikely that positive and negative interactions would be exactly balanced. Thus, we should expect the distribution of epistasis coefficients to be skewed in one or another direction.

The second observation is that in the metabolic model considered here a positive epistasis coefficient at one level of the hierarchy can turn into a negative one at a higher level, but the reverse is not possible. This bias toward negative epistasis at higher-level phenotypes appears to be even stronger in networks with saturating kinetics (Figure 5 and Figure 5—figure supplement 3).

The third observation is that epistasis among beneficial and deleterious mutations affecting metabolic genes should be identical at the level where they arise, provided that beneficial and deleterious mutations are identically distributed among metabolic reactions. Thus, a stronger skew toward negative epistasis among beneficial mutations at the level of fitness could arise in my model for two mutually non-exclusive reasons. One possibility is that beneficial mutations tend to affect certain special subsets of genes, those that predominantly give rise to negative epistasis. For example, beneficial mutations may for some reason predominantly arise in enzymes that catalyze strictly parallel reactions. Another possibility is that when epistasis between beneficial mutations propagates through the metabolic hierarchy it tends to exhibit a stronger negative bias compared to epistasis between deleterious mutations. Indeed, this phenomenon arises in my model among mutations with large effect (see Figure 4A, Figure 4—figure supplement 1 and Figure 4—figure supplement 2).

Perhaps the most important—and also the most intuitive—conclusion of this work is that we should expect epistasis for high-level phenotypes, such as fitness, to be extremely common. Consider first a unicellular organism growing exponentially. Its fitness is fully determined by its growth rate, which can be thought of as the rate constant of an effective biochemical reaction that converts external nutrients into cells (see Appendix 6 for a simple mathematical model of this statement). In other words, growth rate is the most coarse-grained description of a metabolic network and, as such, it depends on the rate constants of all underlying biochemical reactions. Many previous studies have shown that within-protein epistasis is extremely common (e.g. Lunzer et al., 2005; DePristo et al., 2005; Sailer and Harms, 2017b; Husain and Murugan, 2020). Present work shows that, once epistasis arises at the level of protein activity, it will propagate all the way up the metabolic hierarchy and will manifest itself as epistasis for growth rate. It also suggests that growth rate is a generically non-linear function of the rate constants of the underlying biochemical reactions, such that all mutations that affect growth rate individually would also exhibit pairwise epistasis for growth rate with each other (Kacser and Burns, 1981; DePristo et al., 2005; Martin et al., 2007; Chiu et al., 2012; Otwinowski et al., 2018; Husain and Murugan, 2020).

In more complex organisms and/or in certain variable environments, it may be possible to decompose fitness into multiplicative or additive components, for example, plant’s fitness may be equal to the product of the number of seeds it produces and their germination probability, as pointed out by Chou et al., 2011. Then, mutations that affect different components of fitness would exhibit no epistasis. However, such situations should be considered exceptional, as they require fitness to be decomposable and mutations to be non-pleiotropic.

If epistasis is in fact generic for high-level phenotypes, why do we not observe it more frequently? For example, a recent study that tested almost all pairs of gene knock-out mutations in yeast found genetic interactions for fitness for only about 4% of them (Costanzo et al., 2016). One possibility is that many pairs of mutations exhibit epistasis that is simply too small to detect with current methods. As the precision of fitness measurements improves, we would then expect the fraction of interacting gene pairs to grow. Another possibility is that systematic shifts in the distribution of estimated epistasis coefficients away from zero are taken by researchers as systematic errors rather than real phenomena, and are normalized out. Thus, some epistasis that would otherwise be detectable may be lost during data processing.

If epistasis is indeed as ubiquituous as the present analysis suggests, it would call into question how observations of inter-gene epistasis are interpreted. In particular, contrary to a common belief, a non-zero epistasis coefficient does not necessarily imply any specific functional relationship between the components affected by mutations beyond the fact that both components somehow contribute to the measured phenotype (Boyle et al., 2017). The focus of future research should then be not merely on documenting epistasis but on developing theory and methods for a robust inference of biological relationships from measured epistasis coefficients.

Before proceeding to the detailed proofs of Theorem 1 and Theorem 2, I informally outline some key ideas and the basic logic.

The central object of the theory is a metabolic module. Modules have two key properties. First, a module is somewhat isolated from the rest of the metabolic network, in the sense that all metabolites inside it interact with only two metabolites outside, the I/O metabolites. The second property is that the metabolites within the module are sufficiently connected that each of them individually as well as any subset of them collectively can achieve a quasi-steady state (QSS), given the concentrations of the remaining metabolites. This property is proven in Proposition 1.

When some metabolites are at QSS, they can be effectively removed from the network and replaced with effective reactions among the remaining metabolites. In other words, one can ‘coarse-grain’ the network by removing metabolites. This approach is a standard biochemical-network reduction technique (Segel, 1988); for example, the Briggs-Haldane derivation of the Michaelis-Menten formula is based on this idea. In general, the resulting effective reactions have more complex (non-linear) kinetics than the original reactions. However, when the original reactions are first-order, the effective reactions are also first-order, that is, there is no increase in complexity. In Network coarse graining and an algorithm for evaluating the effective rate constant for an arbitrary module, I formally define the coarse-graining procedure (CGP) that eliminates one or multiple metabolites and replaces them with effective reactions.

CGP is an essential concept in my theory. I use it to compute the QSS concentrations for internal metabolites within a module (Corollary 1) and thereby prove Proposition 1, mentioned above. Since any module μ can achieve a QSS at any concentrations of its I/O metabolites and since any module has only two I/O metabolites, it can be replaced with a single effective reaction (Corollary 2). CGP provides a way to calculate the rate constant yμ of this reaction. In other words, the CGP is an algorithm for evaluating function F⁢(x→μ) in Equation 1 for any module μ.

CGP has an important property: its result does not depend on the order in which metabolites are eliminated. Therefore, in computing the effective rate constant of a module, we can choose any convinient way to eliminate its metabolites. Suppose that one module μ is nested within another module ν as in Figure 1A. A convenient way to compute the effective rate yν of the larger module is to first coarse-grain the smaller module μ, replacing it with an effective rate yμ, and then eliminate all the remaining metabolites in ν. Since effective rates after coarse-graining do not depend on the order of metabolite elimination, yν must depend on the rate constants x→μ only indirectly, through yμ. In other words, all the information about the smaller module μ that is relevant for the performance of the larger module ν is contained in yμ. Therefore, if a mutation or mutations perturb only reactions inside of the smaller module μ, we only need to know their effects on the effective rate constant yμ to completely understand how they will perturb the performance of the larger module ν. Specifically, if we have two such mutations A and B, all the information about them is contained in three numbers, δA⁢yμ, δB⁢yμ and ε⁢yμ. Theorem 1 then describes how epistasis at the level of module μ propagates to epistasis at the level of module ν.

The proof of Theorem 1 proceeds as follows. Let a be the effective reaction with rate constant yμ that represents module μ within the larger module ν, and consider yν as a function of yμ, as in Equation 4. To obtain f1⁢(yμ), it is convenient to first eliminate all metabolites that do not participate in reaction a. No matter what the initial structure of module ν is, such coarse-graining will produce only one of three topologically distinct ‘minimal’ modules, which differ by the location of reaction a with respect to the I/O metabolites of module ν (Figure 7). This implies that the function f1 can belong to three parameteric families, where the parameters are the effective rate constants of reactions other than a in each of the minimal modules. This is the essence of Proposition 2. Then Theorem 1 can be easily proven by explicitly evaluating the first- and second-order control coefficients for each of the three functions and showing that the statements of the theorem hold for all of them, irrespectively of the function’s parameters.

Now consider two reactions a and b with rate constants ξ and η, and imagine the two mutations A and B that affect these reactions. To understand what value of ε⁢yμ will occur, we need to obtain yμ as a function of ξ and η (Equation 3). To do so, it is convenient to first eliminate all metabolites that do not participate in reactions a or b. No matter what the initial structure of module μ is, such coarse-graining will produce only one of nine topologically distinct minimal modules, which differ by the location of reactions a and b with respect to the I/O metabolites of module μ and each other (Figure 8). This implies that the function f2 can belong to nine parameteric families. This is the essence of Proposition 3 and Corollary 3.

How does the topological relationship between reactions a and b translate into epistasis? First, there are only three types of relationships between any pair of reactions in a module: strictly serial, strictly parallel, or serial-parallel (see Figure 3). Second, Proposition 4 and Corollary 4 show that coarse-graining does not alter the strict relationships, that is, if reactions a and b are strictly serial or strictly parallel before coarse-graining they will remain so after coarse-graining. This is important because it implies that to prove Theorem 2 we do not need to consider an infinitely large space of all modules but only a much smaller space of all minimal modules, that is, those that have only those metabolites that participate in the affected reactions a and b. Although the space of all minimal modules is still infinite, the space of their topologies is finite (see Figure 8). For some minimal topologies, the connection between the strictly serial or strictly parallel relationship and epistasis can be established very easily. For example, if reaction a and reaction b both share an I/O metabolite as a substrate (see topological class ℳio,io,IO in Figure 8), then they are always strictly parallel, no matter what the rest of the module looks like. Evaluating the first- and second-order control coefficients for the function f2 that corresponds to this topological class reveals that ε⁢yμ≤0 for any parameter values of f2.

Unfortunately, most topological classes are too broad and include modules where reactions a and b are strictly serial as well as modules where they are strictly parallel or serial-parallel (e.g., class ℳio,io,∅). Consequently, the sign of ε⁢yμ for such modules can change depending on the values of the rate constants. However, since the number of distinct minimal topologies is finite, it is possible to identify all minimal topologies where the reactions a and b are strictly serial or strictly parallel. These topological sub-classes define parametric sub-families of function f2, and we can explicitly calculate ε⁢yμ for all such functions. However, such brute-force approach is extremely cumbersome because the number of distinct minimal topologies is very large.

Fortunately, the following simple and intuitive fact greatly simplifies this problem. If two reactions are strictly serial or strictly parallel, this relationship does not change if a third reaction is removed from the module. This statement is the essence of Proposition 5. However, if the two reactions are serial-parallel, removal of a third reaction can change the relationship to a strictly serial or a strictly parallel one. As a consequence, there exist certain most connected ‘generating’ topologies where the relationship between the focal reactions is strictly parallel or strictly serial, and any other strictly serial minimal topology can be produced from at least one of the generating topologies by removal of reactions. This is the essence of Proposition 6. All generating topologies can be discovered by a simple algorithm provided in Appendix 3. They are listed in Table 2 and Table 3 and shown in Figure 10 through Figure 14. Each generating topology defines a parametric sub-family of function f2, and I explicitly evaluate the first- and second-order control coefficients for all these sub-families (see Proposition 7 and Proposition 8) which essentially completes the proof of Theorem 2.

I begin by outlining the main idea behind the CGP, which is to replace one metabolite internal to a module with a series of effective reactions between metabolites adjacent to it. If the effective rate constants are defined appropriately, the dynamics of all metabolites in the resulting coarse-grained network are the same as in the original network, provided that the eliminated metabolite is at QSS in the original network.

To formalize this idea, suppose that module μ=(Aμ,x→μ) contains m internal metabolites. Let k∈Aμ be the internal metabolites that will be eliminated. Let A{k}=A∖{k} be the reduced metabolite set and let x→{k} be the reduced (n-1)×(n-1) matrix of rate constants defined by (12)xij{k}=xij+xikxkjDk,i,j∈A{k},i≠j,(13)xii{k}=0,i∈A{k},where Dk is given by Equation 11.

Such metabolite elimination has three properties that follow immediately from Equation 12 and Equation 13. First, the rate constants of reactions do not change as long as the eliminated metabolite does not participate in them. Mathematically, xi⁢j{k}=xi⁢j for all metabolites i and j that are not adjacent to the eliminated metabolite k. In particular, this is true for all metabolites external to module μ. Second, because equilibrium constants have the property Ki⁢j=Ki⁢ℓ⁢Kℓ⁢j for any metabolites i,j,ℓ, the rate constants xi⁢j{k} obey Haldane’s relationships. Therefore, the reduced metabolite set A{k} and the reduced rate matrix x→{k} define a new ‘coarse-grained’ metabolic network 𝒩{k}=(A{k},x→{k}). It is easy to show that subnetwork μ after the elimination of metabolite k is still a module. Third, the reaction set of module μ (i.e., its topology) in the coarse-grained network 𝒩{k} depends only on the reaction set of this module in the original network 𝒩, but not on the particular values in the rate matrix x→μ.

Next, I will show that the dynamics of metabolites in the coarse-grained network 𝒩{k} are identical to the dynamics of metabolites in the original network 𝒩 where metabolite k is at QSS. Note that if metabolite k is at QSS in the network 𝒩, its concentration is given by(14)Sk=∑j∈AμIO∪Aμ∖{k}xj⁢k⁢SjDk,which follows from Equation 10. Now, the dynamics of metabolites in the network 𝒩{k} are governed by equations(15)S˙i=∑j∈A{k}xji{k}Sj−Di{k}Si,fori∈A{k},where Di{k}=∑j∈A{k}xi⁢j{k}. As mentioned above, xi⁢j{k}=xi⁢j for all pairs of metabolites where at least one metabolite is external to module μ. Therefore, Equation 15 for the external metabolites are identical to Equation 10 that govern the dynamics of these metabolites in the original network 𝒩. Next, consider the dynamics of the I/O and internal metabolites for module μ in the coarse-grained network 𝒩{k}, that is, those in the set AμIO∪Aμ∖{k}. For any such metabolite i, the sum in the righthand side of Equation 15 can be re-written as(16)∑j∈AμIO∪Aμ∖{k}xji{k}Sj=∑j∈AμIO∪Aμ∖{k}(xji+xjkxkiDk)Sj−xikxkiDkSi=∑j∈AμIO∪Aμ∖{k}xjiSj+xki∑j∈AμIO∪Aμ∖{k}xjkSjDk−xikxkiDkSi.

According to Equation 14, the second term in Equation 16 equals xk⁢i⁢Sk, so that Equation 16 becomes(17)∑j∈AμIO∪Aμ∖{k}xj⁢i{k}⁢Sj=∑j∈AμIO∪Aμxj⁢i⁢Sj-xi⁢k⁢xk⁢iDk⁢Si.

For any metabolite i∈AμIO∪Aμ∖{k}, the second term in the righthand side of Equation 15 can be re-written as(18)Di{k}=∑j∈AμIO∪Aμ∖{k}(xi⁢j+xi⁢k⁢xk⁢jDk)-xi⁢k⁢xk⁢iDk=Di-xi⁢k⁢xk⁢iDk.

Substituting Equation 17 and Equation 18 into Equation 15, we see that Equation 15 is in fact equivalent to Equation 10 for all i∈A∖{k} with Sk given by Equation 14.

Here, I define the CGP for an arbitrary set of internal metabolites by applying metabolite elimination recursively.

Let E⊆Aμ be an arbitrary subset of metabolites internal to module μ in the metabolic network 𝒩 and let nE be the number of elements in E. Let AE=A∖E be the reduced metabolite set after the metabolites have been eliminated. I define the reduced (n-nE)×(n-nE) matrix of rate constants x→E as follows. If nE=1, the matrix x→E is defined by Equation 12 and Equation 13. If nE>1, then I define it recursively. Suppose that all metabolites in E other than some metabolite k∈E have been previously eliminated, such that the coarse-grained network 𝒩E′=(AE′,x→E′) is defined, with the set of eliminated metabolites E′=E∖{k}, AE′=A∖E′, and the known matrix x→E′. Then, I define the matrix x→E through the elimination of metabolite k from 𝒩E′, that is, (19)xi⁢jE=xi⁢jE′+xi⁢kE′⁢xk⁢jE′DkE′,i,j∈AE,i≠j,(20)xi⁢iE=0,i∈AE,with(21)DkE′=∑j∈AE′xk⁢jE′.

I define the the coarse-graining procedure that eliminates the metabolite set E as a mapCGE:𝒩↦𝒩E=(AE,x→E).

As with the elimination of a single metabolite, it is straightforward to show that the rate constants xi⁢jE obey Haldane’s relationships, so that 𝒩E is indeed a metabolic network. CGE maps module μ within the metabolic network 𝒩 onto a subnetwork μ′ within the metabolic network 𝒩E. It is straightforward to show that μ′ is a module. Whenever there is no ambiguity, I will label both the original and the coarse-grained versions of the module by μ. To simplify notations, if the CGP eliminates the entire module μ (i.e., if E=Aμ), I label it CGμ. I label the coarse-grained network that restults from the application of CGμ by 𝒩μ and I label the effective rate of the reaction substituting module μ in network 𝒩μ as yμ.

Intuitively, the result of coarse-graining should not depend on the order in which the metabolites are eliminated. To see this, let us obtain explicit (i.e. not recursive) expressions for xi⁢jE. First, by applying the recursion Equation 19, it is easy to show that elimination of two metabolites E={k,ℓ} yields effective rate constantsxij{k,ℓ}=xij{k}+xiℓ{k}xℓj{k}Dℓ{k}(22)=xij+Dkxiℓxℓj+Dℓxikxkj+xikxkℓxℓj+xiℓxℓkxkjDkDℓ−xkℓxℓk(23)=xij+xiℓxℓjDℓ{k}+xikxkjDk{ℓ}+xikxkℓxℓjDk{ℓ}Dℓ+xiℓxℓkxkjDℓ{k}Dk,i,j∈A∖{k,ℓ},i≠j.

As expected, Equation 22 and Equation 23 are symmetric with respect to the eliminated metabolites k and ℓ. Extrapolating from Equation 23, it is possible to show that for an arbitrary metabolite subset E⊆Aμ that contains nE metabolites,(24)xi⁢jE=xi⁢j+∑L=1nE∑(k1,…,kL)xi⁢k11⁢xk1⁢k2Dk1E∖{k1}⁢⋯⁢xkL-1⁢jDkLE∖{k1,k2,…,kL},i,j∈A∖E,i≠j.

Here, the second sum is taken over all nE!/(nE-L)! ordered lists of metabolites (k1,…,kL) from E. Each list can be thought of as a simple path within E that connects metabolites i and j. The proof of Equation 24 can be found in Appendix 1. As expected, Equation 24 shows that the effective reation rate xi⁢jE does not depend on the order in which metabolites are eliminated. This and other properties of the CGP are listed in Box 1.

One of key building blocks of the proofs of Theorem 1 and Theorem 2 is the fact that modules can be classified into a finite number of topological classes (see below). To arrive at this classification, it will be convenient to define a composition of coarse-graining procedures, as follows. Suppose that CGE1 and CGE2 are two coarse-graining procedures of network 𝒩 for two subsets of metabolites E1⊂Aμ and E2⊂Aμ. If the sets E1 and E2 are non-overlapping, CGE2 is also defined for the coarse-grained network 𝒩E1 which is the result of applying CGE1 to the original network 𝒩. The result of applying CGE2 to the 𝒩E1 is called the composition of coarse-graining procedures CGE1 and CGE2 of the original network 𝒩 and is denoted as CGE1∘CGE2.

As defined above, coarse-graining is a formal procedure, and there is no a priori guarantee that (a) it can in fact be carried out for every set of metabolites and (for example, because a metabolite set does not have a steady-state solution); and (b) it will not distort the dynamics in the rest of the network. The following proposition alleviates both of these concerns and thereby justifies the use of the CGP for any subset of internal metabolites within a module (including the entire module μ). It is straightforward to prove it by induction, using the same logic as in the elimination of a single metabolite.

Consider a linear pathway with I/O metabolites 1 and m and internal metabolites 2,3,…,m-1 (Figure 6A). This labeling of metabolites is more convenient for the linear pathway. To calculate yμ, I will apply recursion Equation 19 and Equation 20. I start by eliminating metabolite 2. After this initial coarse-graining step, the resulting module is still a linear pathway, where two reactions 1↔2↔3 were replaced with a single reaction 1↔3 with the effective rate constant.x13{2}=x12⁢x23x21+x23=(1K12⁢x23+1x12)-1.

All other rate constants remain unchanged. Next, I eliminate metabolite 3. The resulting module is still a linear pathway, where now three reactions 1↔2↔3↔4 were replaced with a single reaction 1↔4 with the effective rate constantx14{2,3}=x13{2}⁢x34x31{2}+x34=(1K13⁢x34+1K12⁢x23+1x12)-1.

All other rate constants remain unchanged. Continuing this process until all internal metabolites are eliminated, I obtain(26)yμ=(∑i=1m-11K1⁢i⁢xi⁢i+1)-1,which is identical to the expression originally obtained by Kacser and Burns, 1973.

Consider two parallel pathways with I/O metabolites 1 and 2 and internal metabolites 3 and 4 (Figure 6B). I obtain the effective rate constant using Equation 22 with i=1, j=2, k=3, ℓ=4. Since x12=x34=0, Equation 22 simplifies to(27)yμ=D3⁢x14⁢x42+D4⁢x13⁢x32D3⁢D4=x14⁢x42x41+x42+x13⁢x32x31+x32.

Thus, the contributions of parallel pathways are simply added.

To obtain the effective rate constant for module μ shown in Figure 1, I again use Equation 22 with i=1, j=2, k=3, ℓ=4.(28)yμ=D3⁢x14⁢x42+D4⁢x13⁢x32+x13⁢x34⁢x42+x14⁢x43⁢x32D3⁢D4-x34⁢x43,with D3=x31+x32+x34 and D4=x41+x42+x43.

The CGP described above allows us to calculate the function F that maps the rate matrix x→μ for an arbitrary module μ onto the module’s effective rate constant yμ. F is a multivariate function of the entire matrix x→μ. However, in many applications, only one or two reactions are varied at a time while all others remain constant, and we want to know how module’s effective parameter yμ depends on these one or two perturbed reactions. I refer to such singled-out reactions as ‘marked’. When yμ is considered as a function of the rate constant ξ of one marked reaction, I write yμ=f1⁢(ξ), as in Equation 2. When yμ is considered as a function of the rate constants ξ and η of two marked reactions, I write yμ=f2⁢(ξ,η) as in Equation 3.

The functional form of F and, as a consequence, the functional forms of f1 and f2 depend only on the topology of module μ (see Property #5 of the CGP in Box 1). In other words, modules with identical topologies have the same functional forms of f1 and f2, such that each topology of module μ defines a parametric family of functions f1 and f2, where all rate constants within module μ other than ξ, or ξ and η, play a role of parameters.

Since the number of possible topologies is infinite, there is an infinite number of functional forms of F. However, the number of parameteric families of functions f1 and f2 is finite, and it turns out to be small. To see this, notice that for any module with a single marked reaction, the CGP can be carried out in two stages. In the first stage, we can eliminate all metabolites that do not participate in the marked reaction. The resulting coarse-grained module is minimal in the sense that it can have at most two internal metabolites. Such minimal modules (and, as a consequence, all modules with one marked reaction) fall into three distinct topological classes, which are specified by the location of the marked reaction with respect to the I/O metabolites, as shown in Figure 7. This implies that there are only three parameteric families of the function f1. The topologies of the three minimal modules are sufficiently simple that the three corresponding parametric functional forms of f1 can be easily computed by applying the coarse-graining Equation 19 or Equation 22. This result is formulated in Proposition 2.

The same logic applies to modules with two marked reactions. CGP that eliminates all metabolites that do not participate in the marked reactions maps all such modules onto respective minimal modules, which can have at most four internal metabolites (see Figure 8). This result is formulated in Proposition 3. Minimal modules (and, as a consequence, all modules with two marked reactions) fall into nine distinct topological classes, which are specified by the locations of the marked reactions. All modules from the same topological class are described by functions f2 from the same parametric family. These families are characterized in Corollary 3.

These topological classifications are extremely useful for the following reason. If we can show that all functions from the same parameteric family (corresponding to a given topological class) have some common property irrespectively of the values of the parameters, it would imply that this property holds for all modules from the corresponding topological class. This logic is a key part of the proofs of both Theorem 1 and Theorem 2.

To formalize this reasoning, consider module μ=(Aμ,x→μ) and let a=ia↔ja and b=ib↔jb be two reactions from its set of reactions Rμ. I call a pair (μ,a) a single-marked module and I call a triplet (μ,a,b) a double-marked module, and I refer to reactions a and b as marked within module μ. The topology of a single-marked module (μ,a) is determined not only by the reaction matrix Rμ, but also by the position of the marked reaction, so I refer to the pair (Rμ,a) as the topology of the single-marked module (μ,a). Similarly, I refer to the triplet (Rμ,a,b) as the topology of the double-marked module (μ,a,b). I denote by x→μ∖a the matrix of rate constants of all reactions in module μ other than reaction a and I denote by x→μ∖{a,b} the matrix of all rate constants in module μ other than reactions a and b. I denote the sets of all single- and double-marked modules by ℳ1 and ℳ2, respectively. To avoid metabolite labeling ambiguities, I adopt the following conventions:

It is easy to see that the set of all single-marked modules ℳ1 can be partitioned into three non-overlapping topological classes depending on the type of the marked reaction a. I denote the classes of all single-marked modules where the marked reaction is bypass, i/o or internal (see Notations and definitions) by ℳb, ℳio and ℳi, respectively (Figure 7). Similarly, the set ℳ2 can be partitioned into nine non-overlapping topological classes according to the types of marked reactions and the type of metabolite that is shared by both of these reactions (I/O, internal, or none). These classes are listed in Table 1 and illustrated in Figure 8.