From genes to collective modes: biological constraints shape metabolic evolution

Abstract

Population genetics has been successful in explaining how drift, selection, and mutation shape allele frequencies. However, we still lack an understanding of how biological and evolutionary constraints arising from genotype-phenotype maps shape evolutionary dynamics. To explore this question, we developed a new framework for simulating the evolution of an inherently polygenic and epistatic trait - metabolism - combining population genetics with flux balance analysis. We found that the evolution of metabolic networks exhibits surprisingly simple, reproducible dynamics. We identify evolutionary collective modes (EvCMs) – linear combinations of genes with high, constant selective pressure – as organizers of the long-term dynamics and the origin of this simplicity. EvCMs arise through the interaction of physical constraints, evolvability, and the requirements for growth. We developed a theoretical framework to predict EvCMs from large-scale metabolic models and found quantitative agreement between theory and simulation of E. coli central metabolism. Inspired by these results, we re-analyzed mutational data from Lenski’s long-term evolution experiment and found compelling evidence for the existence of EvCMs. This work suggests that biological constraints encoded in genotype-phenotype maps play an important role in shaping evolutionary dynamics, offering a new perspective on complex trait evolution — one that shifts focus from individual genes to collective modes.

Most features of life today were shaped by evolution. Yet, understanding how complex biological traits evolve remains a central challenge in biology. Population genetics excels at mathematically modeling how mutation, selection, recombination, and genetic drift change gene frequencies in populations over time (1–6). However, the abstract nature of population genetic models often obscures the central role played by biological constraints in evolutionary dynamics (4, 7–10). For this reason, understanding the evolution of higher-level traits such as metabolism, which involve complex biochemical networks that are regulated and controlled by large numbers of interacting genes (11–13), requires new theoretical approaches that explicitly incorporate genotype-to-phenotype maps into evolutionary processes.

To investigate evolution in the presence of realistic genotype-to-phenotype maps that are polygenic and epistatic, we performed evolutionary simulations of metabolic networks using a sequential fixation model where mutations arise in a population one at a time, and either fixate or go extinct depending on the fitness advantage they convey to an organism (1, 14, 15). Our simulations show that evolutionary dynamics are strongly shaped by the metabolic and genetic constraints arising from the genotype-to-phenotype map underlying metabolism. The reason for this is that the growth rate of an organism, and hence the fitness advantage a mutation conveys, is a highly non-local, collective property. Physical conservation laws, such as mass conservation (i.e., flux balance (9)), place strong constraints on the types of mutations that can fix, constraining evolutionary trajectories. A surprising feature of our simulations is that despite this complexity, the evolutionary dynamics of metabolic networks are predictable and can be understood in terms of “evolutionary collective modes” (EvCMs) – linear combination of genes that have high constant selective pressure. The EvCMs organize the long-term evolutionary dynamics and encode the direction of evolution in genetic space. The emergence of EvCMs is especially noteworthy because the selective pressure on individual genes, including those that constitute the EvCM, varies dramatically throughout the course of evolution.

Inspired by these results, we develop a computational framework for predicting the EvCMs of a metabolic network by combining ideas from flux balance analysis, optimization, and evolutionary theory. This framework relates the structure of an EvCM to the expected mutational spectrum of genes, allowing us to make connections with long-term evolution experiments. We compare our theoretical predictions to experimentally observed mutation counts in the Lenski lines (1) and identify signatures of EvCMs, suggesting that the genotype-to-phenotype map stemming from metabolism likely places strong constraints on evolutionary dynamics.

Results

To simulate the evolutionary dynamics of metabolic networks, we consider a clonal population of cells with an effective populations size, $N_e$, grown in a constant environment. We focus on the strong-selection, weak-mutation regime where a single mutant can arise in the population and either fixate or go extinct (Figure 1A, (2)). The probability of fixation depends on the genotypes at time $t$ of the mutant, ${\mathbf{g}}_t^{\prime }$, and parent, ${\mathbf{g}}_t$, through the selection coefficient $s$, defined as the difference in growth rates, $f$, between the mutant and parent strains: $s=f\left({\mathbf{g}}_t^{\prime }\right)-f\left({\mathbf{g}}_t\right)$. From population genetics, we know that the fixation probability is given by Kimura’s formula (14, 16):

$$P_{fix}\left(s,N_e\right)=\frac{1-e^{-2s}}{1-e^{-2s{N}_e}}$$ [1]

Fig. 1.

(A) Evolution can be modeled over various time scales from short to long. Over short times, a single mutation might occur, and the dynamics between the mutant and parent are important. At longer time scales, many mutations can occur. If the mutation rate is low, the population can reach a homogenous state between mutations; the mutant either overtakes the population or goes extinct. This logic allows us to model evolution as a sequential model where the frequency of populations is not tracked and instead the sequence of mutations is of interest. Kimura’s equation describes the probability of a new mutant fixing in this sequential limit. (B) Metabolism serves as a genotype-phenotype map. The genotype of interest is the metabolic genes. The phenotype of interest is the metabolic fluxes. Metabolic fluxes directly influence the growth rate, which can be used in Kimura’s equation. (C) A single step of evolutionary simulations. The initial set of genes $\mathbf{g}$ sets the phenotype $\mathbf{v}$. The population is homogenous (1). A mutation affects one gene, ${\mathbf{g}}_1$, which changes its value to ${\mathbf{g}}_1^{\prime }$. This mutation expands the feasible region of fluxes and a new set of fluxes $\mathbf{v}\prime$ is achieved. Now, a single mutant is present in the population (2). Then, the mutant either fixes in the population (3) or goes extinct, leaving the parental population (1). As mentioned, the probability of fixation is determined by Kimura’s equation. Regardless of the outcome, the population is homogenous again.

This expression arises from the interplay of stochastic (neutral) processes and selection. Selection dominates when the population size is large ($\left|s\right|\gg 1/N_e$) and neutral processes dominate with smaller populations ($\left|s\right|\ll 1/N_e$) (1, 17).

Calculating the selective advantage of a mutation requires a model of metabolism that explicitly relates genotype (genes) to phenotype (metabolic fluxes) and growth rate (Figure 1B). Here, we take advantage of the recent advances in metabolic modeling centered around flux balance analysis (FBA) (8, 9, 18). FBA is a constraint-based modeling approach that predicts metabolic fluxes — the rates at which metabolites flow through biochemical reactions in a metabolic network — by optimizing a biological objective function such as the growth rate. The method works by imposing stoichiometric constraints (mass balance: for internal metabolites, across all reactions the net production of the metabolite equals net consumption) and capacity constraints (bounds on reaction flux based on enzyme kinetics, thermodynamics, or the environment) on the flux distribution. Given these constraints, FBA uses linear programming to find the optimal flux distribution that maximizes the objective function, effectively predicting cellular metabolism without requiring detailed kinetic parameters for every reaction (9).

In mathematical terms, the most common formulation of FBA is a linear program of the form (9)

$$\begin{array}{r}\underset{\mathbf{v}}{\mathrm{m}\mathrm{a}\mathrm{x}}{\mathit{\beta }}^T\mathbf{v}\\ S\mathbf{v}=0\\ l_i\le v_i\le u_i\end{array}$$ [2]

where $\mathbf{v}$ is a vector of metabolic fluxes with each element in the vector representing the flux through a single metabolic reaction in the network, $f={\mathit{\beta }}^T\mathbf{v}$ is the flux-dependent growth rate, and $\mathit{\beta }$ is a vector encoding the contribution of each reaction to growth. Mass conservation of metabolites is imposed through constraints on allowed flux configurations of the form $S\mathbf{v}=0$, where $S$ is the stoichiometric matrix of the metabolic network, which encodes how metabolites are produced and consumed in each reaction. Additional constraints, such as limits on influx of metabolites from the environment and the maximum allowable flux through a reaction due to genetic constraints, are encoded as inequalities, represented generically as $l_i\le v_i\le u_i$.

One major shortcoming of this formulation as a model for metabolism is that there is no explicit dependence of the metabolic fluxes on the genotype. Instead, the dependence on genotype is implicit in the values of the lower- and upper-bound inequality constraints. For example, if an organism is missing the gene that encodes for an enzyme necessary for a particular metabolic reaction, then both the upper and lower bounds for the corresponding metabolic flux are set to zero (9). To rectify this, we draw inspiration from recent FBA models that study the effects of gene knockouts on metabolism by explicitly modeling the effect of genes on metabolism as inequality constraints on allowed metabolic flux configurations (9, 19–21). We assume that the effect of a gene on its associated reaction is to limit the flux, rather than directly set the flux. Fluxes can also be limited due to non-genetic factors. For example, an exchange flux that governs how much of a metabolite is imported may be limited by metabolite concentration in the environment rather than gene expression. Furthermore, in the process of evolution, the constraint on a metabolic flux may shift from being limited by its genotype to the environment.

These considerations lead us to propose the following formulation of FBA, which we take as our model of metabolism:

$$\begin{array}{r}f=\underset{\mathbf{v}}{\mathrm{m}\mathrm{a}\mathrm{x}}{\mathit{\beta }}^T\mathbf{v}\\ S\mathbf{v}=0\\ A\mathbf{v}\le \mathrm{m}\mathrm{i}\mathrm{n}\left(G\mathbf{g},{\mathit{a}}_{\mathit{m}\mathit{a}\mathit{x}}\right)\end{array}$$ [3]

Notice that the inequality constraints now explicitly depend on the genotype of a cell, $\mathbf{g}$, through matrices $A$ and $G$, which allow for relationships between genes and reactions that are not one-to-one. For example, if two genes catalyze the same metabolic reaction, the reaction may be limited by the sum of the maximum allowable fluxes associated with the two genes (i.e., $v_i\le g_a+g_b$). Similarly, multiple genes might need to cooperate to catalyze a reaction. In this case, the flux will be limited by the minimum of the genes’ maximum allowable fluxes. This relationship can be encoded in $A$ (i.e., $v_i\le g_a$ and $v_i\le g_b$). We have also introduced a vector ${\mathit{a}}_{\mathit{max}}$ that encodes non-genetic limitations on metabolic fluxes. In most examples, we will assume that genetic limitations are the primary constraint on fluxes and take ${\mathit{a}}_{\mathit{max}}$ to be infinite. However, the immutable constraints arising from environmental limitations encoded by ${\mathit{a}}_{\mathit{max}}$ will play an important role in the evolutionary dynamics when examining realistic metabolic networks. Finally, the optimal growth rate $f\left(\mathbf{g}\right)$ is an explicit function of genotype through the constraints on metabolic fluxes.

To perform evolutionary simulations (Figure 1C, Methods) of a metabolic network described by $S,A,G$, and $\mathit{\beta }$, we start with a randomly selected value for the genes, $\mathbf{g}$, and solve Equation 3 to obtain the phenotype, $\mathbf{v}$. Then, we mutate $\mathbf{g}$ by adding random effect $∆\mathbf{g}$ (typically a sparse vector altering a single gene) and resolving Equation 3 with $\mathbf{g}+∆\mathbf{g}$ to obtain the mutant phenotype $\mathbf{v}\prime$. We calculate $s={\mathit{\beta }}^T{\mathbf{v}}^{\prime }-{\mathit{\beta }}^T\mathbf{v}$ and then allow the mutant to randomly fixate or go extinct with probability specified by Kimura’s equation (Equation 1). This process is repeated many times to simulate an evolutionary trajectory.

A toy metabolic network for building intuition.

Before considering large, realistic metabolic networks, it is useful to begin our discussion by considering the “toy metabolic network” shown in Figure 2A, consisting of four reactions and two metabolites. Despite its simplicity, this network contains many of the interesting behaviors seen in the evolution of more realistic metabolic models. For this reason, the toy network is extremely useful for building intuition about how evolutionary dynamics are shaped by genotype-to-phenotype maps.

Fig. 2.

(A) Diagram of the toy metabolic network. This network has two metabolites, $X$ and $Y.X$ can be imported into the cell (reaction 1) and consumed for growth (reaction 2). $X$ can also be converted into $Y$ (reaction 3), which can also be consumed for growth (reaction 4). Each reaction has one associated gene that undergoes evolution. During evolutionary simulation, the growth rate (B) and fluxes (C) of the toy metabolic network increase as more mutations are attempted. Using the genes, we can calculate the evolutionary collective mode (EvCM) as the change in genes throughout evolution. We confirmed that evolution is proceeding along the EvCM by examining the cosine similarity between the estimated EvCM (${\mathbf{g}}_{EvCM}$) and the local change in genes. The cosine similarity is high, indicating that the vectors are pointing in the same direction. (E) The selective pressure on individual genes and the EvCM were estimated over a small window of the evolutionary simulation. The selective pressure on the EvCM is nearly constant, while the selective pressure on the individual genes varies consistently across time.

In the toy network, a metabolite $X$ is imported into the cell (reaction 1) and can either be consumed (reaction 2) or converted into a metabolite $Y$ (reaction 3), which is also consumed (reaction 4). The growth rate of the cell is the sum of fluxes through reactions 2 and 4. We also assume a simple genotype-to-phenotype map where each of the four reactions is regulated by a different gene that sets an upper bound for the flux through the reaction (both $A$ and $G$ are identity matrix in Equation 3; SI Appendix Section 4A).

Evolutionary simulations show the emergence of an Evolutionary Collective Mode.

We begin by considering the case where the metabolite $X$ is in excess in the environment and hence all metabolic constraints on fluxes are genetic (i.e., ${\mathit{a}}_{\mathit{m}\mathit{a}\mathit{x}}$ is infinite; we extend below to the case where $X$ is limited in the environment). We simulated the evolution of this toy network and the resulting evolutionary dynamics are shown in Figure 2. As mutations accumulate, the growth rate of the population increases with time (Figure 2B), the population gets fitter, and the flux through all four reactions increases (Figure 2C), as does the upper-bound constraint on the fluxes encoded by the four genes (SI Appendix Figure S1). Since the external metabolite $X$ is always in excess, evolution proceeds indefinitely with continual fitness gains since mutations can always increase the total flux of nutrients into the cell.

One striking aspect of these results is that the ratio of the fluxes through each of the four reactions, as well as the ratio of the upper-bound constraints on each of the four reactions, is approximately constant and fixed to a ratio of 3 : 2 : 1 : 1 (Figure 2D, SI Appendix Figure S1). This indicates that the evolutionary dynamics have a preferred evolutionary direction in both genetic (constraint) and phenotypic space. A natural question one can ask is: how reproducible is this observation across replicates?

Even for this simple toy network, the answer is far from obvious. There is a huge degeneracy in flux space resulting from the fact that the growth rate is just the sum of the fluxes through reactions 2 and 4. For example, routing all the flux from reaction 1 through reactions 3 and 4 results in the same growth rate as routing all the flux from reaction 1 through reaction 2. As a result, it is difficult to predict a priori how the system will evolve if we “replay the tape of evolution” (22) with different sequences of random mutation.

To answer this question, we performed five replicate simulations of evolution in the toy network. We found that, despite the degeneracy discussed above, in all five replicates, both the fluxes and the upper-bound constraints associated with each of the four reactions evolve in an approximately constant 3:2:1:1 ratio (SI Appendix Figure S2). Since in this toy network both $A$ and $G$ are the identity matrix and ${\mathit{a}}_{\mathit{m}\mathit{a}\mathit{x}}$ is infinite, we can directly identify the upper-bound constraints with the genotype vector $\mathbf{g}$. Our simulations indicate that evolution proceeds along a preferred direction in this constraint/gene space, namely in the direction of the vector ${\mathbf{g}}_{EvCM}=[3,2,1,1]$. We call this preferred direction the Evolutionary Collective Mode (EvCM). The existence of the EvCM reflects constraints placed on the evolutionary dynamics by the genotype-to-phenotype map resulting from metabolism. Even though each of the four genes in our network independently regulate a different reaction, they become coupled in the evolutionary dynamics so that mutations occur at a fixed ratio.

The EvCM corresponds to directions of high, constant selective pressure.

To gain more insight into why the EvCM emerges in our evolutionary simulations, it is helpful to think about the selective pressure on genes. Consider a population with genotype ${\mathbf{g}}^{\mathrm{*}}$ that undergoes a mutation ($∆\mathbf{g}$), resulting in a change in the growth rate

$$\mathrm{\Delta }f={\left.\frac{\partial f}{\partial \mathbf{g}}\right|}_{{\mathbf{g}}^{\mathrm{*}}}\mathrm{\Delta }\mathbf{g}$$ [4]

We call $\frac{\partial f}{\partial g_i}$ the selective pressure on gene $i$. Note that in general, the selective pressure will depend strongly on the current genotype ${\mathbf{g}}^{\mathrm{*}}$ since the effect of a mutation depends on the genetic background through the upper-bound constraints on fluxes. For this reason, we expect that the selective pressure on individual genes will vary significantly across the evolutionary dynamics. As shown in Figure 2E, this is indeed the case for all four genes in our toy metabolic network.

Despite this variability, our simulations also show that the growth rate of the population increases approximately linearly with evolutionary time (Figure 2B). These two observations can only be reconciled if the time-averaged selective advantage across all fixed mutations (the change in growth rate averaged over mutations across all genes in a time window) is approximately constant throughout the evolutionary dynamics. Since we have already observed that evolution proceeds preferentially along the EvCM, ${\mathbf{g}}_{EvCM}$, this suggests that the selective pressure along the EvCM, defined as $\mathrm{\Delta }{f}_{EvCM}=\frac{\partial f}{\partial \mathbf{g}}{\mathbf{g}}_{EvCM}$, must be consistently high and constant. Surprisingly, this must be true even though the selective pressure on the individual genes that make up the EvCM fluctuates dramatically across the evolutionary dynamics.

To test if this is indeed the case, we used our evolutionary simulations to calculate the selective pressure on individual genes and in the direction of the EvCM as a function of time. To do so, we exploited the fact that the selective pressure is closely related to the shadow prices associated with metabolic constraints and thus can be calculated using the dual formulation of the linear programming objective in Equation 3 (Methods, (23)). As shown in Figure 2D&E, the selective pressure on the direction of the EvCM is consistently high and constant, independent of the population’s current genotype and evolutionary history. These simulations suggest that the EvCM corresponds to directions in genotype space with high, constant selective pressure. We show below that these ideas can also be used to computationally predict the collective modes of any metabolic network.

Non-genetic constraints lead to evolutionary regimes.

Thus far, in all simulations we have assumed that mutations can always increase the flux through any reaction (i.e., ${\mathit{a}}_{\mathit{max}}$ is infinite in Equation 3). However, we expect that in most evolutionary scenarios some reactions will face immutable constraints. This limit can arise due to environmental limitations (e.g., the external metabolite X in Figure 2A is at a low concentration) or due to an inherent limit on expression levels of proteins.

To understand how the presence of these non-genetic constraints affects evolutionary dynamics, we created two additional toy networks (Figure 3). In the first network, there is a maximum allowable exchange flux through reaction 1, mimicking evolution in an environment where extracellular resources are limited. We assume that the $a_{\mathit{max}}$ for reaction 1 is sufficiently small so that the upper-bound constraint on reaction 1 is fixed and its gene can no longer evolve. In the second network, there is a maximum allowable flux through reaction 3, mimicking a scenario where the enzyme for reaction 3 cannot be produced at higher concentrations. We again assume that the $a_{\mathit{max}}$ for this reaction is sufficiently small that gene 3 can no longer evolve.

Fig. 3.

Variations of the toy network reveal how resource limitations and gene-reaction relationships shape the EvCM. (A) Simulations were performed where reaction 1 is externally limited ($a_{max}\le \mathrm{\infty }$ for reaction 1). Initially, the growth rate increases. However, once the externally limited flux is reached, evolution stops, splitting the simulation into two evolutionary regimes. (B) The cosine similarity between direction of change and the EvCM for each regime. The EvCM for the second regime is 0, so the cosine similarity is 0. As the first regime ends and the second starts, the direction of evolution becomes less aligned with the EvCM of the first regime. (C) Simulations were performed where reaction 3 is limited by the expression of its associated gene ($a_{max}\le \mathrm{\infty }$ for reaction 3). Again, evolution is split into two regimes. However, the reaction 2 pathway can still evolve, so growth rate continues to increase. (D) The cosine similarity between direction of change and the EvCM for each regime. As the first regime ends and the second starts, the direction of evolution becomes less aligned with the EvCM of the first regime and more aligned with the EvCM of the second regime. (E) ${\mathbf{v}}_{EvCM}$ as multiple genes can independently increase the maximum allowable flux of reaction 2 (OR relationship). As more genes are added, the EvCM favors flux through reaction 2 over the reaction 3/4 pathway. (F) ${\mathbf{v}}_{EvCM}$ as multiple genes dependently increase the maximum allowable flux of reaction 2 (AND relationship). As more genes are added, the EvCM favors flux through the reaction 3/4 pathway over reaction 2. (G) Predictions of the EvCM compared to simulation EvCM for metabolic networks discussed to this point. Predictions occur at the level of genes and in terms of the maximum allowable flux.

The evolutionary dynamics of these networks are shown in Figure 3. Initially, both these networks evolve along an EvCM, as in the original toy metabolic network. For both networks, the growth rate increases approximately linearly in time with a fixed slope. However, the presence of immutable non-genetic constraints dramatically changes the evolutionary dynamics of both networks once the maximum allowable flux is reached through the clamped reaction. In the first network, evolution stops entirely (the growth rate plateaus). This is because cells can no longer increase the amount of nutrients they import, making it impossible to keep increasing their growth rate.

The evolutionary dynamics of the second network are much more interesting. In this case, the evolution continues but along a new EvCM where the growth rate also grows linearly in time, but now with a different slope. In this regime, the ratio of fluxes and upper-bound constraints are again constant, but with different numerical values from the first period. We call this behavior where the evolution shifts between different EvCMs the emergence of distinct evolutionary regimes.

To test whether more than two evolutionary regimes can arise during evolution, we performed additional simulations (SI Appendix Section 4B). SI Appendix Figure S3 shows the evolutionary dynamics for the first network (reaction 1 cannot evolve) if we modify the fitness function so that the flux in reaction 2 contributes to growth slightly less than the flux through reaction 4. In this case, we see that there are three distinct evolutionary regimes: an initial EvCM, which transitions to a second EvCM, before evolution stops. Additional evolutionary regimes can emerge if we make our toy network more complicated. For example, if we take the same network and now add a third pathway that contributes to growth (SI Appendix Figure S3), we find that are now four evolutionary regimes. The system now transition between three distinct EvCM before evolution stops.

The emergence of distinct evolutionary regimes reflects that fact that as reactions can no longer evolve, the optimal direction for evolution changes, resulting in the evolutionary dynamics following a new EvCMs. For this reason, we expect evolutionary regimes to emerge in any realistic or sufficiently complex metabolic network.

Evolvability of reactions shapes the structure of the EvCM.

Thus far we have said very little about why the EvCM of our toy network points in the direction it does. What gives rise to the conserved 3:2:1:1 ratio for fluxes and upper-bound constraints seen in our evolutionary simulations? Here, we present evidence suggesting that the EvCM reflects an interplay between biophysical constraints stemming from mass conservation and the “evolvability” of reactions, with reactions that are easier to evolve contributing more to the collective mode. Consider the flux through each reaction in the EvCM. Examining Figure 2A, it is clear that mass balance requires that the flux through reaction 1 must equal the total flux through reactions 2 and 3. Indeed, looking at the EvCM ${\mathbf{v}}_{EvCM}={\mathbf{g}}_{EvCM}=[3,2,1,1]$, we see that the total flux through reaction 1 ($v_1=3$) equals the sum of the flux through reaction 2 ($v_2=2$) and reaction 3 ($v_3=1$). Similarly, the total flux through reaction 3 must equal the flux through reaction 4. However, the 2:1 ratio between the metabolic fluxes through reactions 2 and 3 cannot be explained by mass balance constraints. This suggests the structure of the EvCM also depends on mapping between genes and fluxes/constraints (the genotype-to-phenotype map).

Recall that the growth rate of a cell in our toy network is the sum of the fluxes through reaction 2 and reaction 4. Looking at the structure of our toy network in Figure 2, it is clear that the cell can increase its growth rate in two distinct ways: either by increasing the flux through reaction 2 or increasing the flux through reactions 3 and 4. A single beneficial mutation to gene 2, which sets the upper bound on the flux through reaction 2, is sufficient to increase the flux through reaction 2. In contrast, the flux through reaction 4 is limited by the minimum of upper-bound constraints of reactions 3 and reaction 4. For this reason, on average, it takes twice as many mutations to increase fitness using reaction 3 and 4 compared to reaction 2; both reactions 3 and 4 will need to be mutated to increase the flux through this pathway. This suggests that mutations should fix twice as often in the gene for reaction 2 compared to genes for reactions 3 or 4. Since all genes are equally likely to mutate, this suggests that the maximum allowable flux (and upper-bound constraint) of reaction 2 should grow twice as fast as those of reactions 3 and 4 during the course of evolution.

This analysis suggests that the structure of the metabolic network influences evolvability and the collective mode. If more genes must be mutated for the same fitness benefit, the associated portion of the metabolic network will have smaller contributions to the collective mode. To test this hypothesis, we performed further simulations where we added additional genes to our toy metabolic network. First, we considered a scenario where multiple genes can independently catalyze the same reaction (i.e., independently increase the upper-bound constraint on the corresponding reaction). For example, in E. coli, the isomerization of 3-phosphoglycerate into 2-phosphoglycerate can be catalyzed by gpmM or gpmA (24). When multiple genes are present, we expect that a reaction should be more evolvable, and thus, a mutation that increases the maximum allowable flux for the reaction should be more likely to occur.

To computationally test this, we augmented the toy network by adding additional genes that code for reaction 2 (Figure 3E). We assume genes can independently catalyze the reaction, so the maximum allowable flux for the reaction is the sum of the maximum allowable fluxes associated with each gene. When we examine the collective mode of these metabolic networks, we see that the relative flux through reaction 2 increases as more genes can catalyze it, while the flux through reactions 3 and 4 necessarily decreases (Figure 3E).

A second gene-reaction relationship of interest is when multiple distinct genes must coordinate to catalyze a reaction. For example, in E. coli, the conversion of succinyl-CoA to coenzyme A and succinate in the citric acid cycle requires sucD and sucC, the alpha and beta subunits of succinyl-CoA synthetase, respectively (25). In this setting, we expected the evolvability of the reaction to decrease. If we assume that the subunits must cooperate stoichiometrically, increasing the maximum allowable flux with one gene will not necessarily lead to an increase in the maximum allowable flux of the reaction.

To test this hypothesis, we again augmented the toy network. We assumed that the protein catalyzing reaction 2 had multiple subunits with distinct genes (Figure 3F). Because the genes must cooperate to catalyze the reaction, we assume the maximum allowable flux of the reaction is the minimum of the maximum allowable fluxes associated with the genes. When we examine the collective mode of these metabolic networks, we see that the relative flux through reaction 2 decreases as more subunits are needed to catalyze it, while the flux through reactions 3 and 4 increases (Figure 3F).

Taken together, these results indicate that the structure of the EvCM results from a delicate interplay between biophysical constraints and how evolvable a reaction is. When multiple genes catalyze a reaction independently, evolvability increases, and the reaction is favored in the collective mode. Alternatively, when multiple genes are required, evolvability decreases, and the reaction is disfavored in the collective mode.

Non-identifiability of genes.

With a one-to-one mapping between genes and reactions the EvCM could be easily mapped between genotype and phenotype, but with multiple genes associated with a given reaction it is less clear if a single, preferred direction in phenotype space would also be reflected in a single, preferred direction in genotype space. Mathematically, this might be highlighted when there is degeneracy in the $G$ matrix, as happens when multiple genes independently catalyze a reaction. To examine this, we looked at the variability of the change in genotype compared with the variability of the change in the associated reaction fluxes over relatively short periods of evolution (5 × 10⁴ attempted mutations). We found that when a gene uniquely maps to a reaction, the variability is similar. However, when multiple genes independently contribute to a single reaction, there is greater variability at the level of genes than at the level of fluxes (SI Appendix Figure S4). This result suggests that evolution cannot differentiate among multiple independently contributing genes in this setting, and that the evolutionary dynamics converge to the EvCM as a single direction in phenotypic, but not necessarily genotypic, space.

Predicting the EvCM from the genotype to phenotype map.

We saw above that the EvCM has two key properties – high and constant selective pressure – and that we could make sense of the “preference” for this direction through the notion of evolvability of reactions. Now, we ask if we can formalize these concepts with the goal of predicting the direction of evolution for a given metabolic network. We approach this problem first in the case of a single evolutionary regime, then discuss prediction when non-genetic constraints lead to multiple regimes.

The shadow prices introduced above play a central role in prediction. Recall that shadow prices capture the selective pressure on each constraint and that they shift as the network evolves and thus are a function of genotype; at time $t$ we have shadow prices $\mathbf{\lambda }\left({\mathbf{g}}_t\right)$. These are mapped to selective pressure on genotype through the $G$ matrix. The selective pressure on a direction of genotypic change ${\mathbf{g}}_{EvCM}$ can then be written as $p\left({\mathbf{g}}_{EvCM},{\mathbf{g}}_t\right)=\mathbf{\lambda }{\left({\mathbf{g}}_t\right)}^{T}G{\mathbf{g}}_{EvCM}$. With the property of constant selective pressure in mind, we’re interested in vectors ${\mathbf{g}}_{EvCM}$ for which $p\left({\mathbf{g}}_{EvCM},{\mathbf{g}}_t\right)$ does not vary, even as the genotype ${\mathbf{g}}_t$ changes.

At first glance, it seems challenging to identify such a vector since the genotype continuously evolves (especially in the absence of non-genetic constraints), and it is therefore difficult to say over what set of genotypes the selective pressure should remain constant. However, even when the genotype continues to evolve, the genotype-to-phenotype map defines a limited set of shadow price vectors; that is, the space of selective pressure is limited even when the genotype is not.

For instance, in the toy network of Figure 2A, there are only three possible values of $\mathbf{\lambda }$ (SI Appendix Figure S5). These reflect the different possibilities for limiting reactions under a mass balance constraint: when the import (reaction 1) is limiting, there is selective pressure on that reaction and no other. If the import reaction is not limiting, then there are two scenarios reflecting pressure to increase downstream flux, reflecting whether reaction 3 or 4 is limiting in the bottom branch (SI Appendix Figure S5). Thus, in this case a direction ${\mathbf{g}}_{EvCM}$ with constant pressure over just these three values of $\mathbf{\lambda }$ will have constant pressure over all genotypes and timepoints.

In principle, there could be many vectors with the property of constant selective pressure (in the toy network this includes all vectors that, when interpreted as reaction fluxes, achieve mass balance, though the result is more complicated with non-identity $A$ or $G$), and so to predict the EvCM we need a way to differentiate among this set. To do so, we posit that among this set the EvCM will have the maximum selective pressure. We formalize this as a constrained optimization, searching for the direction of maximal selective pressure, subject to the constraint that this pressure is constant across all values of $\mathbf{\lambda }$ (SI Appendix Section 6). Given the fitness function, $S,G,A$, and $\mathit{\beta }$, we can enumerate the $\mathbf{\lambda }\mathrm{s}$, then solve a quadratic program to arrive at a predicted direction of evolution.

For the toy network, this routine accurately predicts the EvCMs. For the simple case in Figure 2, the prediction is ${\mathbf{g}}_{EvCM}=[3,2,1,1]$, matching simulation outcomes (Figure 3G). Similarly, we can accurately predict how the EvCM changes as independent or dependent genes are added for reaction 2. When multiple independent genes are added, ${\mathbf{v}}_{EvCM}$ favors reaction 2, while when dependent genes are added, ${\mathbf{v}}_{EvCM}$ disfavors reaction 2, consistent with our simulation results.

With non-genetic constraints and different evolutionary regimes, we can also accurately predict the EvCM. To incorporate the effect of non-genetic constraints, we force the non-evolvable genes to be 0 in ${\mathbf{g}}_{EvCM}$. Then, we still have that ${\mathbf{g}}_{EvCM}$ has constant selective pressure over the values of $\mathbf{\lambda }$ reached by evolution (SI Appendix Section 6D). When we use this approach in the presence of regimes, we can accurately predict the EvCM within each regime (Figure 3G).

EvCMs in random metabolic networks.

We next examined whether properties of the toy network – including the presence of an EvCM and our ability to predict the EvCM – generalize to larger, randomly generated metabolic networks. We examined 400 networks with 25–100 metabolites, 50–200 reactions, a randomly generated stoichiometric matrix, and random biomass reactions, while varying gene-reaction relationships from one-to-one to more complex (with non-identity $A$ or $G$; SI Appendix Section 7).

Two small differences from simulations with the toy network are worth pointing out. Positive and negative (reverse direction) fluxes may appear (the formalism introduced above handles this transparently), and conceivably a randomly generated network could fail to evolve. We avoid this scenario in our random selection of metabolic networks (SI Appendix Section 7A) and we focus our analysis here on evolving networks, in which mutations accumulate, the population gets fitter, and where the magnitude of constraints and flux increases.

In the evolving random networks, we consistently observe EvCMs (Figure 4). First, the direction of evolution is constant. We find that, across a given simulation, the changes in flux or genes vary along a single direction (SI Appendix Figure S7). Moreover, the selective advantage of mutations in this direction appears to be constant. We observe this by noting that the population fitness over time can be described by a linear function (with $R^2$ values >0.99 across all random simulations), and by observing that that the shadow price-based calculation of selective pressure on a given network’s direction of evolution is nearly constant (Figure 4B; SI Appendix Figure S6).

Fig. 4.

(A) Random metabolic networks were generated (as described in SI Appendix Section 7) and evolutionary simulations were performed. The growth rate and flux increased in magnitude for each simulation, indicating evolution is occurring. 400 random metabolic networks were simulated. (B) The distribution of the standard deviation of the selective pressure on each network’s EvCM compared to the standard deviation of the selective pressure on random directions for each network. The selective pressure on the EvCM is constant for each network. (C) The distribution of correlations between simulation EvCM and predicted EvCM using the criteria of constant and maximal selective pressure. The simulation EvCM can be accurately predicted.

Second, evolutionary outcomes are consistent across trials. Replicate trials of the same network produce the same outcome (SI Appendix Figure S7; >99% correlation of genes and fluxes), even though many combinations of fluxes could achieve the same final fitness. Thus, a given random network is always driven by the same EvCM.

Third, evolvability shapes the collective mode in random metabolic networks. As before, we find that when more genes can independently catalyze a reaction, there is greater evolvability, and an increasing representation of that reaction in the collective mode (SI Appendix Figure S8). When more dependent genes are required, there is less evolvability, and a decreasing representation in the collective mode (SI Appendix Figure S8). We also observe the effects of gene non-identifiability. We see that the difference in variability of fluxes and genes is positively correlated with the number of independent genes for a reaction, consistent with our earlier results in the toy network (SI Appendix Figure S8). This result again suggests that evolution cannot differentiate among multiple genes that independently contribute to the same constraint.

Finally, we used the procedure above to predict the collective mode of each randomly generated metabolic network. With few exceptions, these predictions are >90% correlated with simulation outcomes at the gene level (SI Appendix Figure S13).

EvCMs of E. coli central metabolism.

We next simulated the evolution of a more realistic network: E. coli central metabolism (Figure 5A). We use the ‘e_coli_core’ model from BiGG Models (18) with slight adjustments (SI Appendix Section 8). This network has 72 metabolites and 95 reactions. The reactions primarily consist of glycolysis, the citric acid cycle, and oxidative phosphorylation. Additionally, each reaction has an associated ‘gene rule’ that relates genes to reactions, leading to 137 genes in the model. We used these gene rules to generate $A$ and $G$ matrices (SI Appendix Section 2). This model assumes that growth is limited by glucose and no other energy sources can be imported into the cell (18). As a result, there is a non-genetic constraint defining a fixed maximum allowable flux on the exchange reaction for glucose.

Fig. 5.

Simulations of the E. coli central metabolism have evolutionary regimes with EvCMs. (A) The E. coli central metabolism broadly consists of the glycolysis pathway, citric acid cycle, and oxidative phosphorylation. The network consists of 95 reactions, but five reactions are highlighted to capture the broad changes in the network. (B) The growth rate of the metabolic network increases as more mutations are attempted. The simulation has six evolutionary regimes, which can be observed partially from changes in the slope of the growth rate (see SI Appendix Figure S10 for replicates). (C) Five replicate simulations were performed, each of which had six evolutionary regimes. The EvCM at the flux level was estimated for each regime, and the correlations of all EvCMs were calculated. Hierarchical clustering was performed on each EvCM, and the EvCMs cluster by regime, as indicated by color along border, showing consistency across replicates. (D) The selective pressure was calculated on individual genes (associated with reactions highlighted in (A)) and the EvCM within regimes 1, 4, and 5. The selective pressure on the individual genes is variable across time, and often zero. In contrast, the selective pressure on the EvCM is nearly constant across time and always greater than zero.

We analyzed the results of 5 simulation trials with up to ~ 2 × 10⁸ attempted mutation. Multiple regimes are evident as periods with different slopes in fitness over time for a given trial (Figure 5B). We found six regimes based on abrupt changes in flux directions (SI Appendix Figure S10), with distinct directions of evolution within each regime. The direction of evolution within each regime and the ordering of regimes was strikingly consistent across replicate trials (SI Appendix Figures S10). Within each regime the direction of evolution evaluated as the ratio of fluxes was constant (SI Appendix Figure S10), as was the shadow price-based selective pressure on these directions (Figure 5, SI Appendix Figure S10). Predictions of the collective mode in each regime at the gene level were >80% correlated with simulation outcomes (SI Appendix Figure S9). In aggregate, these results show that in simulation the central metabolism of E. coli evolves along collective modes.

While this simulation is only an approximation of evolution, we share what occurs biologically in each regime. Identifying specific regimes is challenging, so we speak in broad terms and only focus on the largest and most obvious features. Initially, the flux through glycolysis increases and ethanol is fermented. After a transition period, the largest regime (labeled 4), appears to focus on utilizing the citric acid cycle and oxidative phosphorylation to generate ATP. The last portion of the simulation appears to be further refinements, including switching from using pyruvate formate lyase to pyruvate dehydrogenase for generation of acetyl-CoA. Again, we emphasize these collective modes are likely sensitive to what reactions and genes are included in the metabolic network. However, these results do match our expectations. The proteins in the steps of oxidative phosphorylation require coordination of many genes (26), while the steps of ethanol fermentation generally only have a single or few genes (27). As a result, from the perspective of our previous results, the fermentation pathway is more evolvable than oxidative phosphorylation and should be favored by the EvCM initially. Oxidative phosphorylation is evolved once growth rate can no longer be increased with fermentation.

Evidence of EvCMs in a long-term evolution experiment.

Finally, we turn to experimental data and look for evidence of collective modes in the Lenski lines. The Lenski lines are a long-term evolution experiment where E. coli has been serially cultured for 60,000 generations (1). Twelve initially identical lines have been independently cultured, and fitness increased in all lines throughout the experiment. Six of the lines developed a hypermutator phenotype (1, 28), which makes analysis difficult, so we focus on the normal mutators. While metabolic fluxes (and constraints) were not systematically measured, the structure of EvCMs should be reflected in the mutational spectrum across genes. Metagenomic sequencing has been performed every 500 generations (1), and we looked for evidence of EvCMs in this dataset.

The normal mutator lines display a remarkable consistency in evolutionary dynamics and outcomes. Each of the lines achieved rapid fitness gains during the first ~10,000 generations, followed by a period of slower, but still accumulating fitness gains for the duration of the experiment (Figure 6A). This is reminiscent of a two regime outcome seen in simulation (Figure 3C). Further, the mutational spectra (counts of fixed mutations per gene across all 3,969 genes mutated in the experiment) accumulated by the end of the experiment are 25% correlated on average (SI Appendix Figure S11).

Fig. 6.

The Lenski lines were analyzed to look for evidence of EvCMs. (A) The fitness of the non-mutator lines over the course of the experiment. Within our framework, the fitness trajectories appear consistent with multiple regimes. The end of the first regime was estimated at 2,500 generations. (B) While the mutation counts across the entire experiment are weakly (SI Appendix Figure S11), regulatory-adjusted counted are more correlated across lines, suggesting the Lenski lines may be evolving in a similar manner. (C) The Lenski lines were analyzed for evidence of constant selective pressure. To avoid any regions of overlap between the regimes, only data from beyond generation 10,000 was analyzed (indicated in A). The selective pressure within the three held-out lines was calculated on individual genes and the estimated direction of evolution. The shown genes had the largest average selective pressure across times and lines. The selective pressure on the individual genes is variable across time, and often zero. In contrast, the selective pressure on the EvCM is nearly constant across time and greater than zero. (D) The coefficient of variation of the selective pressure on the estimated EvCM across lines and time compared to 10,000 samples from a null distribution. The null distribution was obtained by shuffling the estimated direction of evolution and recalculating the coefficient of variation of the selective pressure on it (SI Appendix Section 9B).

Since our previous analysis of non-identifiability of genes indicated that a common direction of fluxes can emerge even as directions of evolution vary at the level of genes, we wondered if there might be more in common among the lines than suggested by the baseline correlation. Notably, many of the most mutated genes can act as transcription factors (e.g., malT, nadR, and iclR) (1). We considered that the same change in metabolic phenotype could be achieved through a mutation in a regulator (altering expression levels) or direct mutation in a metabolic enzyme. To capture this idea, we generated adjusted mutation counts for each line, where a mutation in a regulator was propagated to a mutational count in each of its downstream targets, as identified by RegulonDB (29) (SI Appendix Section 9A). With these “regulatory adjusted” mutation counts, the outcomes of evolution in each line appear even more consistent, with an average correlation of 55% among the 3,807 non-regulatory genes (Figure 6B), suggesting a common direction of evolution. (While these increased correlations could be induced by the regulatory-adjustment process, we confirmed this effect was minimal by repeating the procedure with shuffled regulatory and regulated genes (SI Appendix Figure S14)).

The data are limited compared to what we can evaluate in simulation, but nevertheless the larger, second regime is sufficient to test for constant selective pressure over time. To do so, we turn to an empirical measure of the sensitivity of fitness to a mutation. In brief, we use the fact that mutations conferring a stronger advantage should rise in frequency in the population more quickly, allowing us to estimate the fitness sensitivity of each mutation by fitting a logistic curve to an individual mutation’s frequency trajectory (SI Appendix Figure S12). For a given window of time, we can then compute a proxy for the selective pressure on each gene through the accumulated fitness sensitivities of any mutation to that gene over the specified window. The fitness sensitivity to a direction of genotypic change can be calculated though a linear combination of these gene sensitivities.

Using this approach, we evaluated variance in fitness sensitivity across time and lines. We calculate the direction of change from mutation counts accumulated in the second regime (SI Appendix Section 9B). In short, we select the three lines with the largest average correlation and aggregated their regulatory-adjusted mutation counts in the second regime. Using the remaining three lines, we then evaluate fitness sensitivity to this direction over 5,000 generation windows (Figure 6C). Because the measure of selective pressure is imperfect and the data limited, for comparison we repeat the process using random permutations of the mutation count vector to estimate a null expectation (SI Appendix Section 9B). Computing the variance in fitness sensitivity for the true and permuted directions, we find the true direction of evolution has lower variance in fitness sensitivity than 99.5% of the randomly permuted vectors (Figure 6D).

In aggregate, the long-term evolution experiments show signs of being driven by EvCMs. Our results indicate common, reproducible directions of evolution emerge at higher levels of organization, and we find evidence for evolution along a constant selective pressure direction.

Discussion

Our understanding of evolution is best in simple settings: a single gene controls a single phenotype that determines fitness. However, complex traits underlie fitness in every scale and domain of life. Understanding the evolution of such traits remains a central challenge in biology. The dominant approach of population genetics abstracts away the genotype-to-phenotype map, obfuscating or replacing complexity with marginal and linear effects. Here, we established a genotype-to-phenotype-to-fitness map underpinning a model complex trait (flux through a metabolic network), building on the successful and quantitatively tractable framework of flux balance analysis. Surprisingly, we find that explicit representation of this complexity in evolutionary simulation gives rise to simple and reproducible dynamics.

The EvCMs identified in our work bear a conceptual resemblance to the evolutionary sectors identified by Halabi, Rivoire, Leibler, and Ranganathan in protein families (30), where statistical analysis of sequence variation revealed groups of coevolving amino acids under collective rather than individual selection. Just as sectors identify collections of amino acids shaped by shared functional constraints (30, 31), our EvCMs identify collections of genes whose coordinated changes are favored by the structure of the metabolic genotype-phenotype map. The analogy is instructive: in both cases, biological constraints impose low-dimensional collective modes on an otherwise high-dimensional evolutionary space, suggesting that selection on collections — whether amino acids or genes — may be a general feature of the evolution of complex biological systems.

Though our setup is considerably simplified compared to living systems, we expect many of the lessons above will also generalize to more complex settings beyond metabolic networks and proteins. We identify collective modes, rather than individual genes, as the coherent and organizing objects of selection – a result we expect to hold in some general form when genes and phenotype contribute to fitness in a non-local, collective way. This could arise through developmental tradeoffs, common in many systems (e.g., growth versus defense in plants (32), investment in secondary sexual traits in beetles (33, 34), or warm versus cool adaptation in bacteria (35)), or through alternative instances of conservation constraints. In these cases, traits become coupled and evolutionary dynamics are not easily reduced to marginal impacts from individual genes. When collective mode-like dynamics are in play, we also anticipate that evolvability will be a fruitful lens to understand the favored direction among seemingly degenerate outcomes, and that evolutionary dynamics will transition through regimes based on evolvability and non-genetic constraints.

Taking a broader view, a defining feature of our framework is the centrality of phenotypic constraints in evolutionary analysis. Optimality has frequently been used as a lens to understand evolution but has little utility in our setting; without constraints, there is no well-defined optimum to the metabolic network. Subject to fixed constraints, phenotype and fitness may be optimized (with considerable degeneracy), but this teaches only about a relatively static picture of evolution. The constraints, therefore, dictate not only what is possible in relative evolutionary stasis, but also dictate the setting for longer-term dynamics. In light of this, analyzing the dual of evolution’s primal problem (i.e., shadow price vectors and not just phenotype vectors) becomes pivotal.

Taking a longer view, our work begins to clarify how evolution might act on the phenotypic constraints themselves. While we only considered scenarios where the constraints placed on phenotype were fixed, our work suggests that evolvability plays a key role in determining the EvCM. This observation suggests that evolution might act on constraints to increase evolvability. For example, might operons act to increase the evolvability of a collection of genes and be selected for? We believe our framework of constant selective pressure can help to answer this question and others regarding genetic structure.

Other aspects of this work may not generalize so readily. Most notably, despite degeneracy and randomness our simulation outcomes were remarkably consistent across replicate trials. This appears to be mirrored, at least to some extent, in the long-term evolution experiment with E. coli. What is absent from both settings is ecology or significant environmental feedback (though it has been observed that ecology emerged in the Lenski lines despite the experimental design to minimize ecological effects (36, 37)). Such feedback can in principle destabilize evolutionary trajectories (38, 39) so that “replaying the tape of life” would indeed produce new outcomes. Even if through some other mechanism, we do not in general expect outcomes to be as deterministic as demonstrated here. Further, a defining feature of collective modes in this setting – constant selective pressure – likely arises from the linearity of the system and might not manifest the same way in non-convex settings.

Materials and Methods

Further discussion of methods is included in the SI Appendix.

Evolutionary Simulations.

As discussed in the main text, simulations were performed by mutating the genes of a metabolic network, using Equation 3 to calculate a new growth rate, and using Equation 1 to determine if the mutant goes extinct or fixes. Specifically, for a metabolic network described by $S,A,G$, and $\mathit{\beta }$, we start with a randomly selected value for the genes, $\mathbf{g}$. The initial values of $\mathbf{g}$ were independently sampled from a normal distribution with a small variance. The small noise makes it easier to evolve initially as only a few reactions will be rate-limiting. Then, mutation is modeled by adding a random change $∆\mathbf{g}$ to $\mathbf{g}$. To sample $∆\mathbf{g}$, each gene was independently selected to be mutated with equal probability and, of the genes that were mutated, the effect of the mutation was sampled from a normal distribution with mean 0. One gene was mutated on average per attempted mutation. Next, the mutant phenotype ${\mathbf{v}}^{\prime }$ and fitness $f\prime$ are again found using Equation 3. Then, $s={\beta }^T{\mathbf{v}}^{\prime }-{\mathit{\beta }}^T\mathbf{v}$. Equation 1 was used to determine if the mutant fixes or goes extinct. This process is repeated for 10⁶ – 10⁸ steps to simulate an entire evolutionary trajectory.

Dual of FBA and Selective Pressure.

We used the dual of Equation 3 to estimate the selective pressure on a gene. The dual of the Equation 3 is

$$\begin{array}{r}\underset{\mathbf{\lambda },\mathit{\nu }}{\mathrm{m}\mathrm{i}\mathrm{n}}{\mathbf{\lambda }}^{T}G\mathbf{g}\\ A^T\mathbf{\lambda }+S^T\mathit{\nu }=\mathit{\beta }\\ \mathbf{\lambda }\ge 0\end{array}$$ [5]

$\mathbf{\lambda }$ and $\mathit{\nu }$ are referred to as shadow prices and represent the sensitivity of the growth rate to changes in the constraints. Of importance to us, $\mathbf{\lambda }$ determines the sensitivity to a change in the constraints imposed by genes:

$${\mathbf{\lambda }}^{T}G=\frac{\partial f}{\partial \mathbf{g}}$$

As mentioned, $\frac{\partial f}{\partial \mathbf{g}}$ is the selective pressure on a gene. Then, we solve Equation 5 to obtain $\mathbf{\lambda }$ and use this to calculate the selective pressure on a gene. To calculate selective pressure on a direction (e.g., the EvCM), we weight the selective pressure of each gene by its contribution to the direction and sum: ${\mathbf{\lambda }}^{T}G\mathbf{z}$. To calculate selective pressures over evolution, we resolve Equation 5 at each step. As $\mathbf{g}$ varies throughout evolution, so too will the solution to Equation 5 and value of $\mathbf{\lambda }$.

Data, Materials, and Software Availability.

Code to perform simulations and example analysis to generate analysis for figure 2 are included on Github: https://github.com/algo-bio-lab/evolutionary-collective-modes.