Stochastic dynamics of a self-activating gene

For simplicity, we considered two biochemical events (protein synthesis or degradation) that either increase or decrease the number of proteins A by one molecule: (1) (2)

The synthesis rate f(A), which describes the probability per unit time that protein synthesis occurs and that A is increased by one, is a nonlinear function of activator A: (3) where k (number of proteins/unit time) represent the maximum synthesis rate, α is the basal synthesis rate relative to k, K_D (number of proteins) is related to the protein-DNA dissociation constant, and n_H is the degree of molecular cooperativity (i.e. Hill coefficient). The degradation rate γ ⋅ A, which describes the probability per unit time that protein degradation occurs and that A is decreased by one, is a linear function of activator A where γ (1/unit time) is the protein degradation rate constant. With no loss of generality, we defined the unit of time as τ = t ⋅ γ, which allows us to substitute time t with a time-dimensionless variable (see Methods). We used the Gillespie algorithm, a kinetic Monte Carlo method that explicitly simulates the probabilistic dynamics of a defined set of biochemical events [39], to simulate the stochastic dynamics of our gene circuit (see Methods).

Evolutionary model

For simplicity, we evolved a haploid, asexual population with non-overlapping generations in a fluctuating environment (Fig 2). The underlying biophysical parameters depend on protein stability, protein-protein, protein-RNA, and protein-DNA interactions, which can change through genetic mutations. For simplicity, we allowed mutations on the maximum synthesis rate (k), Hill coefficient (n_H) and DNA dissociation constant (K_D) during our evolutionary simulations, while keeping basal activity (α = 0.25) and degradation rate (rescaled, γ = 1) fixed. The set of variable parameters {k, n_H, K_D} are the genotype (θ). The environment fluctuated periodically with frequency ν between LOW (selects for optimal phenotype A^(L) = 20 proteins) and HIGH (selects for optimal phenotype A^(H) = 80 proteins). The number of generations spent in a constant environment (“epoch”) was equal to 1/ν generations. The selective environment switched to the alternative state at the end of an epoch. Starting from an isogenic population, we ran each evolutionary simulation for 10,000 generations. We implemented the following algorithm each generation:

1. Simulate stochastic gene dynamics of a self-activating gene in each cell given its biophysical parameters (i.e. genotype) and initial protein level inherited from its parent in the previous generation for 4 units of time (the cell “life span”).
2. Evaluate the fitness of each cell i based on the protein level at the end of its life span (A_i, phenotype). The individual fitness function (ω_i) is: (4) where E = {L, H} is the current environment, A^(E) is the optimal phenotype for each environment, and ϵ² = 0.2 ⋅ A^(E) is the width of the Lorentzian function. We define the population fitness (w) as the average over all cells.
3. Select the next generation using tournament selection [40, 41], where s_t cells are chosen randomly from the population. The cell with highest fitness within the chosen cohort is cloned into the new population. This “tournament” is repeated N times with replacement to create a new population. The tournament size (s_t) modulates the selection pressure, where small s_t is weak selection (e.g. for s_t = 1, there is no selection pressure and only genetic drift because any randomly selected individual is the tournament winner). Increasing s_t leads to stronger selection and a faster selective sweep of fitter cells (e.g. for s_t = N, only the fittest individual in the entire population will be cloned into the next generation).
4. Allow random mutations with a fixed probability (u) in each cloned cell. If a mutation occurs, the parameter values in the cell genotype are updated as follows: (5) (6) (7) where M is the maximum fold change (mutation step-size), r ∼ U(0, 1), ϕ₁ ∼ U(−1, 1), and ϕ₂ ∼ U(0, 2π) are uniformly distributed random values between 0 and 1, -1 and 1, and 0 and 2π, respectively (see Methods). This spherically symmetric 3D mutation scheme permits co-variation in biophysical parameters in a single mutational step, with the “distance” to the parental genotype uniformly distributed (in the logarithmic scale) in the range defined by the mutation step-size. Unlike additive mutation, multiplicative mutation better reflects how mutations affect thermodynamics of protein stability, protein-DNA and protein-protein interactions [42, 43]. All mutated parameters were constrained to lie within a physiological range that is typical for a bacterial transcription factor (see Methods).
5. If the evolutionary simulation is at the end of an epoch, then change to other environment; otherwise keep the same environment. Return to Step 1 to simulate next generation.

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Fig 2. Evolutionary model.

The environment fluctuates periodically with frequency ν. The total number of generations spent in a constant environment (epoch) has the same length (1/ν) and each environment (HIGH or LOW) selects for a different distribution of protein levels (phenotypes). Each generation, we simulated the stochastic protein dynamics of a self-activating gene in each cell across a population of size N. At the end of each simulation, the population phenotypes varied because gene expression is stochastic and because cells can have different underlying biochemical parameters (genotypes). The current environment in each generation assigned a fitness (ω_i) to each cell (i) based on its final protein level. We used tournament selection (where s_t determines the strength of selection) to determine the next generation of cells according to their fitness. Each cell in the next generation was mutated with probability u, where the current set of biophysical parameters were multiplied or divided up to a maximum step-size of M.

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Populations adapt to the fluctuating environment

The evolutionary model was simulated over a range of population size (N = [100, 10000]), selection pressure (s_t = [3, 250]), environmental fluctuation frequency (ν = [0.01, 0.1]), mutation rate (u = [0.01, 0.1]), and mutation step-size (M = [1.1, 5]). We restricted these parameters to regions where our simulations were feasible and where epigenetic switching and genetic adaptation were competitive with one another. For example, we only considered uN ≥ 1; otherwise, mutations were too infrequent for genetic adaptation to compete with epigenetic switching. We verified that our results presented below were robust to alternative assumptions, such as different models of selection, non-periodic environmental fluctuations, a Moran model of reproduction, and alternative mutation schemes (see Methods).

To control for the possibility that bistability could be selected for reasons other than epigenetic switching [28], we also ran a parallel deterministic simulation (CONTROL) of the expression of the auto-activating gene in Step 1 (see Methods). Given that there is no biochemical noise in our CONTROL simulations, cells in the bistable region display hysteresis and stay in the stable state closest to the inherited parental state and never stochastically switch to the other state (i.e. no epigenetic switching).

All populations started with a monostable genotype θ₀ that was adapted to one environment but not the other. These populations all evolved to a higher fitness solution. Initially, populations had a slightly higher geometric mean fitness per cycle because it accrued benefits by being adapted to the HIGH environment despite being maladapted to the LOW environment (Fig 3A). We consider geometric mean fitness per cycle rather than fitness per generation (w) because it better reflects the long-term growth of fitter phenotypes in natural populations [44], but analogous results are obtained considering the arithmetic mean instead. During the LOW epoch that follows the HIGH epoch, the populations shifted towards higher w^(L) values. The epoch was too short and mutation too weak for the populations to perfectly adapt to the new environment before it changed again. In all cases, we observed that the evolutionary dynamics in early epochs were dominated by noisy genetic adaptation of a population maladapted to at least one of the environments, even if this implied decreasing W_cycle. The “no-response” behavior, i.e. being adapted to one environment and “ignoring” the alternative state, is not a stable solution for this system. Consistent with previous work, this illustrates the importance of considering the full population dynamics in the adaptation process and not only the long-term average fitness [45, 46].

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Fig 3. Evolutionary trajectory of a population adapting to a fast fluctuating environment displays selection for high nonlinearity.

The initial population started from a non-optimal genotype (θ₀) where k = 80, n_H = 1, K_D = 10 with evolutionary parameters N = 10000, ν = 0.1, s_t = 40, u = 0.03, and M = 1.1 both under (A) the original model and (B) in the absence of biochemical noise (CONTROL). The average K_D (〈K_D〉_cycle) versus the average Hill coefficient (〈n_H〉_cycle) are shown for each environmental cycle simulated, colored given the geometric mean of the population fitness per environmental cycle (), and the marker size corresponds to the average percentage of bistable genotypes in the population (see legend). The traces of other nine replicas are shown in gray as a reference. Each cycle spans a LOW (dark green) and HIGH (light green) epoch and there are 500 environmental cycles over 10,000 generations for each of these simulations. The population fitness w per generation for the first and last cycles, as well as the cycle with the minimum and maximum W_cycle are shown in the bottom for comparison, and colored according to their W_cycle value. In both simulations (A-B), the population evolved to higher nonlinearity values 〈n_H〉_cycle and W_cycle.

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Sub-optimal populations eventually increased the nonlinearity in gene expression 〈n_H〉_cycle. For example, in a fast-fluctuating environment (ν = 0.1) with small mutation step size (M = 1.1), the population evolved to a bistable genotype θ_B that used epigenetic switching (Fig 3A); nevertheless, high n_H values appear in the population before transitioning from a monostable to a bistable genotype (e.g. 〈n_H〉 ∈ [2, 3.3] and less than 10% bistable individuals per generation). In this case, the bistable genotype was a global optimum in W_cycle and adaptation occurred in a few generations after each environmental change due to epigenetic switching. We verified that this bistable genotype was globally optimal by re-running evolutionary simulations for different initial genotypes (θ₀) and for more generations. The CONTROL simulations, where epigenetic switching was not available, also evolved high nonlinearity in gene expression 〈n_H〉_cycle, but kept monostable genotypes (Fig 3B).

Selecting for genetic potential

Meyers et al. [38] have shown that populations in variable environments can stably evolve to higher genetic potential, i.e. genotypes with a higher sensitivity to mutations. To explore this possibility in our model, we took advantage of the defined mechanistic model, where for a given steady state A_* (i.e. deterministic solution of the gene expression system, ), and each value of K_D and n_H, the synthesis rate value k_* is uniquely defined as follows: (8)

Then, the optimal synthesis rate can be calculated for each environment E with an optimal phenotype (Fig 4A and 4B).

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Fig 4. Genetic potential increases around the bistable region at high nonlinearity.

(A) The f(A) function for some solution genotypes are shown (see row and column titles) exemplifying cases where (1) both and are monostable (i.e. the associated f(A) and γ ⋅ A intersect only once; all examples with n_H = 1, and {n_H = 16, K_D = 1}), (2) a bistable solution genotype with ({n_H = 16, K_D = 50}, f(A) and γ ⋅ A intersect both in A = 20 and A = 80, with an unstable steady state between these steady states), and (3) only is bistable, and not a solution for the LOW environment ({n_H = 16, K_D = 120}). (B) Solution genotypes per environment were calculated (Eq 8). (C) The maximum genetic potential for different values of K_D and n_H (i.e. ) is shown in the colormap. As reference, the bistable solution genotypes for each environment are delimited by the green lines: as dark green, and as light green. Additionally, we show as an example the population average K_D and n_H per cycle for ten simulations over 10000 generations with k = 80, n_H = 1, and K_D = 10 as the initial genotype (red circle, θ₀), and evolutionary parameters N = 10000, u = 0.03, M = 1.1, ν = 0.10, and s_t = 40 (black lines), and the equivalent CONTROL simulations (i.e. without biochemical noise; gray lines). The colorbar shows the one-mutation distance corresponding to each value of M.

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For each solution genotype, , we calculate the genetic potential as the minimum genetic distance to adapt after an environmental transition, i.e. to reach any solution genotype in the alternative environment: (9) (10) When (i.e. for both environments), bistability arises and adaptation can occur with a noise-induced epimutation (Fig 4). If there is no noise (e.g. deterministic CONTROL), then the bistable region exhibits hysteresis and the system can only switch when genetic adaptation forces the genotype to leave the bistable region (usually through K_D mutations).

We observed that starting from an arbitrary initial genotype θ₀, populations always migrated to a region of higher genetic potential, until the solution to the alternative state is reachable in just one genetic mutation, if this is available. This occurred regardless of the specific evolutionary conditions (e.g. selection pressure s_t or environmental fluctuation frequency ν), or the absence of noise (i.e. CONTROL simulations). In the gene expression model described here (Eqs (1) and (2)), the genetic potential increases around the bistable region, which occurs at high nonlinearity n_H (Fig 4C). We also analyzed the double negative feedback loop (“toggle switch”), an alternative system with the ability to display bistability and epigenetic switching. The genetic potential of the toggle switch also increases with higher nonlinearity and particularly around the bistable region (S2 Fig). This supports the generality of our observations, as each of these systems (positive versus double negative feedback loops) produce different bifurcation types [47]. These results show that the initial selection for high genetic potential can drive the population to a genotypic space where bistability is available, suggesting a potential mechanism for the initial emergence of epigenetic switches in fluctuating environments.

Notably, the region where the alternative solution is reachable in one genetic mutation depends directly on the mutation step-size M (Eqs (9) and (10); Fig 4C). Congruently, CONTROL simulations with higher M values showed a higher diversity, with no obvious selection pressure once the population is inside this optimal genetic potential region (S3 Fig). Nevertheless, simulations with biochemical noise resulted in more constraint genotype distributions regardless of the specific mutation step-size.

Trade-off between adaptation time and phenotypic robustness

In all evolutionary conditions tested here, the optimal W_cycle was obtained in the high nonlinearity regime. Nevertheless, the average Hill coefficient increased as the environmental fluctuation frequency decreased (Fig 5; S3 Fig). In order to explore the evolutionary advantage of the distinct Hill coefficient values here, we took a closer look to the population fitness.

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Fig 5. Trade-off between minimizing adaptation time and maximizing phenotypic robustness correlates with nonlinearity.

Contour plot of the bi-variate histogram of the average Hill coefficient per epoch 〈n_H〉_epoch and the population fitness at the (A) first generation w_+1g and (B) ninth generation w_+9g after an environmental transition are shown grouped by the environmental fluctuation frequency (ν). (C) An analogous plot but grouped by the mutation step-size value (M; three example values) shows that the observed trend is related to the ν value. In all cases, the last 100 epochs of ten independent replicas were considered, each simulation running for 10,000 generations with evolutionary parameters N = 10000, s_t = 40, u = 0.03 and k = 80, n_H = 6 and K_D = 45 as the initial genotype (θ₁; this initial genotype sped up evolutionary simulations by being closer to final selected genotypes in all simulations). In each group, the shade area shows the bins with at least 1% of cases, and at least 5% of cases in the darker area.

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The population fitness is always low and maladapted when the environment first changes. Selection favors those genotypes which produce phenotypes that better match the selection pressure in the new environment. In this first phase (the “adaptation phase”), the population fitness one generation after each environmental change (w_+1g) display higher values for faster environmental fluctuations (i.e. large ν; Fig 5A). However, all populations eventually reach a higher fitness and become adapted to the new environment. In this second phase (the “constant phase”), purifying selection maintains the optimal phenotype against perturbations from gene expression noise, mutations, and/or epimutation. The population fitness nine generations after each environmental change (w_+9g) display higher values for slowly fluctuating environments (i.e. small ν; Fig 5B). The population spends proportionally more time in the constant phase as the epoch length increases (i.e. ν decreases), which favors those strategies with robust phenotypes (i.e. minimize frequent, maladapted phenotypes that arise from biochemical noise and epigenetic switching). Noteworthy, higher mutation step size (M) favors a slightly higher w_+1g, while slightly decreasing w_+9g, but with very similar distributions between groups and no obvious correlation with the selected Hill coefficient (n_H; Fig 5C).

These results indicate a trade-off in the evolutionary process between minimizing the adaptation time during the adaptation phase and increasing the robustness of the phenotype during the constant phase. Further, the distribution of Hill coefficient values (〈n_H〉_epoch) as a function of epoch length suggests that natural selection tunes this trade-off via this biophysical parameter. We additionally corroborated the relationship between Hill coefficient value and the expected fitness in an infinite, isogenic population (E[w^(E)]); we observed that in the region of interest in the biophysical parameter space (k ≈ 80), E[w^(E)] increases as n_H increases regardless of the value of K_D (S4 Fig). In the following section, we explore the relationship between this fitness trade-off and the selected adaptation strategy.

Lineage analysis shows the selection of epigenetic switching in fast fluctuating environments

In order to determine the selected adaptation strategy for each evolutionary condition, it was informative to analyze the genealogy (i.e. lineages) of each population (see Methods). We tracked the evolutionary history of each individual in the population to identify those lineages that persisted with or without mutations over one full environmental cycle (i.e. LOW epoch + HIGH epoch; S5 Fig). More than one lineage can persist over a cycle, but fewer than expected from coalescent theory because our population is evolving under selection and faces a selective sweep at each environmental transition [48]. If a particular adaptation strategy is successful, then we expect those lineages using that strategy to have a higher fitness and a larger number of progeny. The weight of each persisting lineage is proportional to the number of progeny at the end of the cycle.

An epigenetic switch can adapt with no mutation; thus, lineages with a bistable genotype and no mutations during a cycle were classified as following an epigenetic switching (ES) strategy. On the other hand, those lineages that had at least one monostable genotype and accumulated mutations during a cycle were classified as following a genetic adaptation (GA) strategy. Lineages with only bistable genotypes that accumulated mutations during a cycle were classified as following a hybrid bistable adaptation (BA) strategy. Although some of these mutations can be neutral, we found that most of them modulated the DNA dissociation constant (K_D) and directly affected the rate of epigenetic switching (i.e. epimutation; Fig 6A). Thus, the mutation can be adaptive in the hybrid BA strategy, although the circuit remained bistable and ultimately adapted through epigenetic switching.

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Fig 6. Each adaptation strategy is favored under different evolutionary conditions and the transition between selected strategies is gradual.

(A) Illustration of epigenetic switching (ES), bistable adaptation (BA), and genetic adaptation (GA) strategies and underlying genotypes with fixed synthesis rate (k = 80). The bistable region of genotypic space is highlighted in pink. The phenotype stationary distribution ρ(A, τ = ∞) for each genotype (θ) is shown in the inset, both in linear and logarithmic scale. BA is bistable, as seen in logarithmic scale, but appears effectively monostable in linear scale. This arises because K_D evolves each epoch to favor one mode over the other by decreasing the relative rates of epigenetic switching between the largest and smallest mode. (B) Each colormap shows the fraction of parental lineages using specific adaptation strategy (ES, BA, or GA) averaged over all cycles and ten independent replica simulations for the corresponding mutation step-size (M) and environmental fluctuation frequency (ν). Each simulation ran for 10,000 generations with evolutionary parameters N = 10000, s_t = 40, u = 0.03 and k = 80, n_H = 6 and K_D = 45 as the initial genotype (θ₁). (C) Results of the CONTROL simulations where gene expression dynamics are deterministic and no stochastic epigenetic switching can occur. All lineages exhibited GA and neither bistable strategy (ES or BA) was ever selected. (D) The corresponding bistable fraction (〈f_B〉_sim) averaged over all cycles and ten independent replica simulations for the stochastic simulations (top) and deterministic CONTROL (bottom; same simulations than B-C, respectively).

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Our lineage analysis demonstrated two propositions: (1) epigenetic switching (ES) effectively allows for adaptation to fluctuating environments, outcompeting genetic adaptation, with lineages persisting for multiple environmental cycles without mutations (S5 Fig); and (2) distinct strategies were favored in different evolutionary conditions (Fig 6B). Consistently with previous works, ES was the preferred strategy when the environment fluctuated frequently (i.e. high ν values); in slowly fluctuating environments, GA was favored if the mutation step-size (M) was large enough, otherwise BA was the dominant adaptation strategy. These trends persisted over a range of different evolutionary parameters, although the boundaries shifted. For example, increasing the selection pressure (s_t) or the mutation rate (u) shifted boundaries to favor GA, whereas increasing population size (N) favored ES (S6 Fig).

Noteworthy, the fraction of bistable genotypes (〈f_B〉_sim) as a function of evolutionary parameters did not reflect the favored adaptation strategy (Fig 6D). The 〈f_B〉_sim never fell to zero even when GA was the optimal lineage strategy (e.g. low ν, high M). These results arise because of the increased seeding of genetic mutants (which is facilitated by higher M) from the monostable to bistable subpopulation. Conversely, we expect more cases of neutral or nearly neutral mutations in the bistable region for smaller mutational step-sizes (M). It is exactly in this regime where the fraction of ES parental lineages decreased whereas BA increased. However, most lineages displaying BA in this regime also persisted into the next cycle without accumulating any mutations and, thus, automatically switched to ES (S7 Fig).

The dominant adaptation strategy trend (Fig 6B) highly correlates with the observed fitness trade-off (Fig 5). Moreover, we observed a mixture of strategies across lineages, and the transition between preferred adaptation strategies as a function of ν and M was gradual. This is expected if the described trade-off is the driving selection force. If each adaptation strategy favors distinct aspects of the trade-off, whenever these fitness costs have similar values, genetic drift will dominate during the selection process. As expected, the simulations in regimes with co-dominant strategies showed high temporal variation in the fraction of adaptation strategies each evolutionary cycle (S7 Fig).

Finally, in our CONTROL simulations with deterministic dynamics (where no stochastic epigenetic switching can occur even if the system is bistable), none of the parental lineages exhibit ES or BA (Fig 6C). This corroborates that the selection of ES as adaptation strategy requires the presence of biochemical noise. In the following section, we show that, in addition of generating epimutations, biochemical noise is also an important fitness component which must be optimized.

The role of the noise load

The observed trade-off suggests that the cost of biochemical noise on the population plays a fundamental role in the evolutionary dynamics. For each genotype, we define the associated noise load as the expected decline in fitness by the end of the cell life span: (11) given an optimal initial phenotype (i.e., ρ(A = A^(E), τ = 0) = 1 and ; see Methods). Noteworthy, corresponds to the estimated fitness for an isogenic population after one generation in a given environment excluding the cost of genetic drift (i.e. assuming infinite population) and mutation (i.e. assuming perfect selection), then exposing the effect of the biochemical noise. In our system, we observed a significant noise load for all solution genotypes for both environments (as defined in Eq 8; Fig 7A), with an expected fitness decay of more than 60% (, depending on the specific genotype θ). Moreover, around and within the bistable region –where the genetic potential is higher– the minimum noise load for each environment occurs at higher n_H values (Fig 7A), especially when the two environments are considered (Fig 7B). Interestingly, the noise load is lower in both environments right at the center of the bistable region, where the same genotype has steady state solutions optimal in both environments (i.e. ).

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Fig 7. Increasing nonlinearity reduces the noise load in the bistable region, decreasing also the epimutation rate.

(A) The noise load (Eq 11) is calculated for the solution genotypes for each environmental state, LOW and HIGH. (B) The maximum noise load comparing in both environmental states for each value of n_H and K_D (with depending on the environment being considered). (C) The probability of an individual with a solution genotype (or ) and initial phenotype A^(L) (or A^(H)) has a good fitness score ω in the alternative environment at the end of the generation, i.e. ω^(H)(A_τ=4) ≥ 0.5 (or ω^(L)(A_τ=4) ≥ 0.5), which is considered as a LOW → HIGH “epimutation” (or HIGH → LOW “epimutation”). (D) The minimum probability of “epimutation” comparing in both directions, LOW → HIGH and HIGH → LOW, for each value of n_H and K_D (with depending on the environment being considered). Values less or equal to 10⁻⁶ are omitted. Same range of values and bistable region lines as in Fig 4.

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How can the fitness be maximized in both environments in a bistable system? Even if the associated stationary phenotype distribution (i.e. at long time scales, ρ(A, τ = ∞)) is bimodal and well-balanced (i.e. ∼50% of chance of being around each mode), the genotypes with high nonlinearity can display strong memory, and retain the parental phenotype for multiple generations. Increasing the memory on bistable systems increases the expected fitness once the individual displays the optimal phenotype. But this memory also directly decreases the epimutation rate (Fig 7C and 7D). Consequently, there is a fundamental trade-off between minimizing noise load and maximizing adaptation potential in bistable genotypes. Moreover, the balance between these properties depends once more on the Hill coefficient value. To further explore this theory, we run some experiments where the environment was stopped after 1,000 generations (S8 Fig). As expected, populations in constant environment still can display bistable genotypes with very high nonlinearity, resulting in low noise load and high population fitness (S8 Fig).

Similarly, this trade-off also affects the genetic adaptation strategy. In the optimal solution space, the population requires to effectively cross the bistable region by genetic mutations (usually mutations on K_D value for our system) to adapt genetically. The width of the bistable region increases as n_H increases, reducing the probability of a beneficial mutation, and then the genetic potential.

Selection for higher genetic potential allows the emergence of epigenetic switching

Starting from an arbitrary initial condition and different evolutionary conditions, all populations evolved to a similar region of the genotypic space with higher fitness (see examples in Fig 3). These final genotypes often had 20 < K_D < 80, and large n_H (S3 Fig). To help understand the forces that select for large nonlinearity, we calculated the genetic potential for the distinct genotypes available in our gene expression model (Fig 4). We observed that both in the presence or absence of biochemical noise, all populations initially selected for high genetic potential, i.e. genotypes more sensitive to genetic mutation, allowing for efficient adaptation after each environmental transition. In our gene expression model, the genetic potential increases around the bistable region (Fig 4), which can explain the de novo selection or emergence of epigenetic switching starting from a non-regulated (constitutive) gene.

Nonlinearity modulates the trade-off between adaptation potential and phenotypic robustness

The average population fitness was mostly determined by the environmental fluctuation frequency (ν), where fitness decreased as the epoch length decreased. This agrees with previous theoretical work, which showed that the evolutionary dynamics are governed by environmental dynamics [16, 31]. In our simulations, the correlation between fitness and epoch length arises from the balance between two competing fitness costs: reducing the time required to adapt every time the environment changes (adaptation phase; Fig 5A) and increasing the phenotypic robustness when the environment is fixed (constant phase; Fig 5B). Our work suggests that the trade-off between adaptation time and phenotypic robustness is mostly modulated by the nonlinearity level (Fig 5).

To fully understand this relationship, we estimated the noise load, defined here as the expected decay in the fitness arising just from the biochemical noise, for the distinct genotypes available in our gene expression model (Eq 11; Fig 7A and 7B). We show that around the region where the genetic potential is higher, the noise load is lower for higher Hill coefficient values (n_H) regardless of the specific environmental state (LOW or HIGH), or whether the genotype is bistable. Nevertheless, in this region of the genotypic space, a fundamental trade-off exists between minimizing the noise load in a population (Fig 7A and 7B) and maximizing the adaptation potential, either through genetic adaptation (i.e. genetic potential; Fig 4) or epigenetic switching (i.e. epimutation probability; Fig 7C and 7D).

Epigenetic switches are superior in fast fluctuating environments

Previous theoretical work established that optimal long-term growth occurs when the phenotype switching rate matches the environmental switching rate [16–19]. The phenotype could switch either due to genetic adaptation with a rate that depends on the mutation rate and mutation step-size, or due to epigenetic switching (epimutation) with a rate determined by the underlying molecular system. In the natural world, epimutation rates are often faster than genetic mutation rates [12], which suggests that fast fluctuating environments might select for epigenetic switching (ES) over genetic adaptation (GA). Previous models did not integrate and evaluate these two competing processes in a population of cells evolving in a fluctuating environment.

Our simulations confirmed that ES consistently emerges in fast fluctuating environments (Fig 6). In agreement with Soyer and colleagues [28, 37], our populations evolved to genotypes with high nonlinearity (Fig 5; S3 Fig). However, we found that bistable genotypes (ES and BA) were only favored in the presence of biochemical noise, and in CONTROL simulations only GA was observed (Fig 6C). Moreover, the trade-off highlighted above reliably predicts the conditions under which an epigenetic switch was able to outcompete genetic adaptation. The ES strategy is selected at high ν (when the population spends proportionally more time in the adaptation phase) because it has a faster adaptation time (Fig 5A) at the cost of lower phenotypic robustness (Fig 5B). Nevertheless, the genetic potential map (Fig 4) shows that GA can potentially adapt quickly (especially for larger mutation step-size M), but these trajectories lie in noisy regions of the genotypic space, resulting in an even higher noise load (Fig 7). Therefore, we conclude that ES is selected over GA in fast fluctuating environments because this strategy minimizes adaptation time through epimutation while still reducing the noise load in the system.

Lineages reveal hidden selection forces & epigenetic switching persistence

All evolved bistable and monostable genotypes were relatively close to each other (Fig 6A). Our simulations had a relatively high mutation rate (u) such that bistable genotypes could mutate to monostable genotypes, and vice versa. The elevated rate of seeding between these subpopulations made it challenging to distinguish whether ES (bistable) was being selected for. It has been shown previously that individual history or genealogy can efficiently reveal hidden selection forces. For example, Kussell and colleagues [34, 49] demonstrated that selective pressures on a population, such as those imposed by a fluctuating environment, can be efficiently quantified by measurements on the surviving lineages. More recently, Cerulus et al. [50] used life-history traits of cellular growth to show that high single-cell variance in growth rate can be beneficial for the population, and that this benefit depends on the epigenetic inheritance of the growth rate between mother and daughter cells.

To this end, we analyzed the strategy of lineages across multiple cycles during our simulations. Lineage analysis demonstrated that apparent coexistence of bistable and monostable subpopulations was a transient phenomenon, and one type of strategy was typically dominant across lineages (Fig 6B and 6C). Moreover, the persistence of well-adapted lineages without genetic mutations effectively proved that epigenetic switching can outcompete genetic adaptation in fluctuating environments. Our analysis suggests that population snapshots (e.g. bimodal versus unimodal distribution of phenotypes) can miss the contribution of epigenetic switching (Fig 6D). Future experimental studies on the evolution of epigenetic switches might consider analyzing lineages using time-lapse microscopy, as done by Balaban et al. [13].

Model limitations and future directions

We verified that our observations were robust to many alternative model assumptions (S9 Fig). Nevertheless, our simple model of a haploid, asexual population [51] omits some features of the evolutionary process. For example, variable population size, diploid genetics, sexual reproduction and linkage disequilibrium could all affect the evolutionary dynamics and selection of epigenetic switches in a fluctuating environment. Our model also fixed the mutation rate (u) and step-size (M), which imposes a mutational load when the population is adapted to a constant environment. Future simulations could allow natural selection to mutate and tune these parameters, which might favor genetic adaptation over epigenetic switching [52]. Our model also did not consider the case where mutations (e.g. adaptive mutation) or biophysical parameters (e.g. phenotypic plasticity) directly respond to changes in the environment. Last, we assumed that mutations continuously increase or decrease the biochemical parameters. This overlooked an important class of mutations, such as indels (i.e. rapid loss or abrupt change of function), gene duplication, and gene recruitment, which could abruptly change the topology of the gene network.

We considered the simplest genetic circuit that can exhibit epigenetic switching. However, alternative gene regulatory networks could generate different dynamics and phenotypes that are even better adapted to the fluctuating environment. For example, adding a negative feedback loop could reduce gene expression noise [53, 54] or generate oscillations [55, 56]. An oscillatory gene circuit (e.g. circadian clock) might anticipate and respond to an environment that fluctuates regularly (e.g. day/night). Future research will explore more complicated gene regulatory circuits to understand the specific environmental dynamics and evolutionary conditions that favor oscillation versus epigenetic switching in the context of genetic adaptation. This should be of broad relevance to evolutionary biologists and systems biologists.

Biophysical parameters

The maximum synthesis rate k and the degradation rate γ depend on time. With no loss of generality, we reduced the number of free parameters in our model by substituting time t with a time-dimensionless variable τ = t ⋅ γ. Many proteins in bacteria are not actively degraded and are diluted through cell growth and division. Thus, τ and k are in units of cell cycle time. With no loss of generality, we used τ = 4 in all our stochastic simulations. All mutated parameters were constrained to lie within a physiological range (10⁻² ≤ k ≤ 10³, 10⁻² ≤ n_H ≤ 16, 10⁻² ≤ K_D ≤ 120) for a bacterium such as E. coli. The number of molecules for a transcription factor ranges between 0 − 10³ proteins per bacterium or concentration range 0 − 10³ nM for a bacterial volume of 1 fL [57, 58]. The DNA dissociation constant (K_D) has a similar range to the underlying transcription factors [58, 59].

Gillespie algorithm

The propensity or probability rate (r_j) of chemical reaction j occurring during the next interval dt is related to the rates of mass-action chemical kinetics in a constant chemical reactor volume V. In our simple biochemical network, the propensity of protein synthesis is f(A) and the propensity of protein degradation is γ ⋅ A. Each step, given the current number of chemical species, Gillespie’s direct algorithm first calculates the propensities r_j and then calculates when the next reaction occurs and which one occurred. The waiting time of the next reaction is drawn from an exponential distribution with parameter Σ_jr_j, where the cumulative distribution function of any reaction occurring before time τ_* is . Given that a reaction has occurred at time τ_*, the probability that the event is reaction i is equal to r_i/(Σ_j r_j). Thus, for each iteration, two random numbers are required to determine τ_* and i as drawn from the probability distributions. The random numbers are generated using the “Minimal” random number generator of Park and Miller with Bays-Durham shuffle and added safeguards [60].

Deterministic CONTROL simulations

The dynamic of the mean concentration (A) obeys a first-order ordinary differential equation: (12)

The cell life span was assumed to be long enough for steady state (where Eq 12 equals 0) to be reached before selection. We used numerical methods to calculate the steady state solution (phenotype) for any genotype and initial protein level inherited from the parent. All Gillespie, deterministic, and evolution simulations were implemented in C++ (https://github.com/mgschiavon/EpiEvoDynamics/releases), and all the analyses and figures were done using MATLAB.

Alternative model assumptions

We tested the robustness of our results to alternative choices and assumptions in the evolutionary model.

Lineage analysis

An unfit mutant could arise at the end of a cycle or a fit genotype might go extinct due to genetic drift and gene expression noise. To obtain insights on the evolutionary stability of different strategies, we focused our analysis on lineages that have persisted –with or without mutations– through at least one full environmental cycle (i.e. LOW epoch + HIGH epoch). We maintained genealogical records and traced the lineage and ancestral genotype of all cells over 2 cycles (S5 Fig). If any genotype between the 1-cycle and 2-cycle ancestors was monostable, then we classified the evolutionary strategy of that lineage as genetic adaptation (GA). If all genotypes between the 1-cycle and 2-cycle ancestors were bistable, then we classified the evolutionary strategy as either epigenetic switching (ES) or bistable adaptation (BA). The lineage was BA only if the 1-cycle bistable ancestor had accumulated at least one mutation since the 2-cycle bistable ancestor. At the end of each cycle, we calculated the fraction of surviving cells whose ancestors had one of these 3 strategies (GA, ES, BA) and averaged over all cycles (Fig 6). Each individual ancestral lineage was counted regardless of whether it was shared with other individuals or not.

To determine the transition rate between evolutionary strategies, we analyzed lineages back to the 3-cycle ancestor. We calculated the statistics of transitions between adaptation strategies by comparing strategies in 1-cycle and 2-cycle ancestors (current adaptation strategy) versus 2-cycle and 3-cycle ancestors (previous adaptation strategy) and averaged over all cycles (S7 Fig).

Noise load and “epimutation” probability

The Finite State Projection (FSP) method [62] allows the estimation of a probability distribution ρ(A, τ) at any specific time τ_f having an initial distribution ρ(A, 0) and state reaction matrix B(θ): (13)

Assuming an individual with genotype θ was selected in the previous generation with the optimal phenotype for the environment E_S, i.e. ρ(A = A^(ES), 0) = 1, ρ(A, τ = 4) reflects the phenotype probability distribution of the individual at the end of the following generation. This phenotype distribution can be used to calculate the expected fitness of a genotype θ with perfect selection and no genetic mutation, isolating the effect of biochemical noise. Then, we define the genotype’s noise load as the expected decay on fitness at the end of the cell life span (Eq 11; Fig 7A and 7B). Additionally, the probability that an individual selected in one environment E_S displays high fitness in the alternative environment E_A (e.g. ) was used as an approximation of the epimutation probability of epigenetic switches (Fig 7C and 7D).