Gene regulatory networks

We employed a simple model for gene expression dynamics based on a sigmoidal function to study the evolution of the genotype–phenotype map. There are dynamics of protein expression level for each protein (gene) i. This model comprises a network consisting of genes that mutually activate or inhibit each other’s expression. In this model, there are M genes whose gene expression level, (i = 1, 2, …, M), corresponding to the host genotype l at time t is described by (1) where J_ij denotes the matrix representing the influence of gene j on the expression of gene i. When j activates (represses) the expression i, it takes the form J_ij = 1(−1), while J_ij = 0 if j does not influence i. The sigmoidal function f(z) is given by (2) with β = 25; this implies that f(z) closely resembles a step function. Furthermore, θ_i represents the threshold of the input required for the expression of gene i. Therefore, whether the gene(protein) is expressed (f(z) ≈ 1) or not (f(z) ≈ 0) depends on whether the sum of inputs from the expression of other genes(proteins) is larger than θ_i. In Eq (1), the value θ_i is selected randomly from a uniform distribution between 0.05 and 0.3. The term representing the interaction with other genes is scaled by , where α denotes the path density, that is, the fraction of nonzero values of J_ij. We adopted this scaling so that x_i is comparable with the order of θ_i regardless of total gene number M(= 64). (Because the connection is sparse, many of J_ij = 0, and the average number of inputs for gene is αM. We normalized the interaction with other genes by the square root of the path number αM so that the variance of J_ij is independent of αM.) Moreover, there are l_out(= 8) “output genes” i = 1, 2, …, l_out, which determine the fitness as defined below. To eliminate the potential for the output genes to influence others, the summation is taken only for j > l_out. The term η^(l)(t) denotes a Gaussian white noise originating from molecular fluctuations in chemical reactions.

Here, the noise is applied to the expression of each gene, and it is not correlated across genes. The value of σ represents the strength of this noise (In the case where the gene expression of x is negative, we set x to zero.) For constant f(z), Eq (1) defines an Ornstein-Uhlenbeck process, and to give the equilibrium variance of σ²/(2γ). Initially, all gene expression levels are set smaller than the threshold θ_i. Furthermore, ϵ denotes the spontaneous expression level, which is smaller than θ_i. I_j denotes the external input for l_inp(= 8) “input genes,” with I_j = 1 for j = M − l_inp + 1, M − l_inp + 2, …, M, and I_j = 0 for j ≤ M − l_inp. The input genes are always/constitutively expressed. In order for x_i to be expressed, x_i must receive external or internal inputs from other x_j values to advance beyond θ_i.

Initially, we select J_ij at random to obtain J_ij = 1(−1) with a probability of 0.15 so that α = 0.3. We set the initial number of host cells as N and total number is constant at N = 300 throughout all simulations. Here, we set M = 64, l_out = 8, and l_inp = 8. The model parameters used in this study are set as in Table 1.

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Table 1. Model parameters and variable definitions.

https://doi.org/10.1371/journal.pcbi.1008694.t001

Raw fitness and reproduction

The raw fitness of the host alone, i.e., in the absence of parasites, is determined by the fraction of output genes that are expressed as (3) where x_j denote the expression of the output genes and μ₀ denotes the basal fitness. Therefore, the state with all output genes expressed (x_j = 1, for j = 1, 2, …, l_out) is an optimal phenotype for achieving maximal growth.

We introduce parasite attacks into the model. There are parasite genotypes that are coded as with i_j = 0 or 1. For example, for l_p = 3, there are 2³ = 8 genotypes coded by 000, 001, …, 111. Each parasite attacks hosts that possess the corresponding target gene expression, , that is, each parasite attacks the host whose gene expression pattern, x_j (1 ≤ j ≤ l_p), matches (). Here, we term the l_p genes as “target genes.” The target condition for the attack of each parasite is given by s(x_i) = 1 for x_i > 0.5 and s(x_i) = 0 otherwise (as the threshold function is close to a step function, x_i takes either ∼0 or ∼1). For instance, a host whose expression pattern (s(x₁), s(x₂), s(x₃)) ≈ (0, 1, 0) is attacked by the parasite (i₁, i₂, i₃) = (0, 1, 0). Let m () be the parasite type whose matches with host for i = 1, …, l_p. The growth rate of each host genotype l decreases owing to the effect of the parasite m as (4) here P_m denotes the population density of the m-th parasite that matches with the host gene expression for (i = 1, …, l_p) as mentioned and c denotes the strength coefficient of the attack by the parasite. The evolution of hosts is modelled by an individual-based approach with a fixed population size akin of the Moran process. Each host has its own genotype (gene regulatory networks) and the growth rate determined by the gene expression levels calculated by numerical simulations in Eq (1). The volume of the host cells grows according to (5)

When the volume exceeds a threshold value of 2 (v(0) set to 1), the host cell is divided into two parts. The total number of hosts N is set to be constant over time, and thus, whenever a cell divides, another cell is randomly selected and removed from the original population.

Here, after division, the initial expression of x_i is reset to take a random value smaller than the threshold θ_i. The division time of the host is determined by the individual growth rate that ranges between 500 and 1,000 time units. At this time scale, the host gene expression reaches a steady state. The gene expression of hosts reaches a steady state after around 200 time units. Mutation is introduced in the division process and added to the network J_ij with a low mutation rate m = 0.03. An (i, j) path in the network matrix J_ij is selected at random and its value is changed to one of the others to preserve the total path number. For example, if J_ij = 1(−1) and J_kj = 0, then they are changed to J_kj = 1(−1) and J_ij = 0. Here, we randomly select a gene k ≠ i, i ≠ j, k ≠ j. Then, the path j → i is replaced by j → k to represent the mutation of a path in gene regulatory networks. We prohibited direct connections between the “input” and “output” genes. Initially, J_ij is selected randomly with ±1 with the rate α = 0.3, and α remains constant throughout the simulation. (However, the result is independent of this choice.)

It should be noted that once cells settle down to different fixed-point attractors, no more switching occurs in time. The on/off difference in the gene by cells occurs due to the developmental noise from the initial condition, which subsequently results in phenotypic variance in the population.

Parasite population dynamics

Unlike host phenotype (gene expression levels), which is partially stochastic, the parasites’ phenotype is determined uniquely by its genotype. Hence parasites of a given genotype share a common phenotype and parasite population dynamics can be modelled by deterministic equations (instead of individual-based simulations).

The population of each parasite P_i increases in proportion to the host population with the corresponding phenotype, where the host density H_i to be attacked by the parasite strain i is determined by the host population that satisfies s(x_j) = i_j. Further, there is a mutation from other parasite types . Because parasite types are represented by a binary string of length l_p, they are represented in a l_p-dimensional hypercube space. Each distinct genotype is represented by a vertex on the cube. Mutations in the parasite genotypes occur by flipping each gene i_l = 0 ↔ i_l = 1 at the mutation rate D_p. This is represented by diffusion in the hypercube (e.g., 010 ↔ 000) determined by a diffusion constant D_p. Therefore, the population dynamics is expressed as (6) where κ_ij takes unity only if the binary string of j can change to that of i via a single mutation, otherwise it equals zero. We assume competition within all the parasite genotypes, such that the total number of populations is assumed to be fixed, to avoid the divergence of parasite populations. Accordingly, we normalize the density P_i such that . Thus, the reduction term is introduced ensuring that and remain satisfied. We computed the population dynamics of parasites with a given generation time T_p (s). We set the initial density P_i to take the same value for each type, i.e., (The result below, however, is independent of this initial distribution). The generation time of parasites is shorter than that of hosts. Therefore, the fittest type would quickly become dominant.

Phenotypic variances attributed to noise and mutation

The model includes a noise component that allows gene expression level to vary even among individuals sharing the same gene regulatory network. We define the phenotypic variance as the variance of the phenotype over the entire host population with different genotypes. Here, the variation of to give arises both from noise and from network mutations. In addition, we compute the isogenic phenotypic variance of the gene expression level within each host genotype l. Because of the noise term ση_i in Eq (1), the expression level x_i for the same host genotype (network) varies between individuals and this variance is positive. We define two phenotypic variances to distinguish between these sources: V_ip(i) and V_g(i)[23, 25]. The variance is defined as the variance of gene expression levels in an isogenic population, i.e., the phenotypic variance of x_i caused by noise in the gene expression dynamics within the clonal population of hosts, i.e., sharing the common J_ij. In contrast, is defined as the phenotypic variance due to genetic mutation, as computed by the variance of within a population of initial genotype i in which each individual receives one random mutation, where is the average over the clonal distribution of host l. Details describing the calculation of variances V_ip and V_g are provided in the Methods section. Here, as we are interested in the variance of the expression of output genes, we term the average variances of the output genes (i = 1, 2, …, 8) as V_ip, V_g, and V_p, thereby omitting the l notation. In addition, we define the average variances of the target genes corresponding to parasite attacks as and .

Note that developmental noise (the ση term in Eq (1)) has no significant effect on host fitness once gene expression dynamics have reached a stable state with all x_i close to 0 or 1. This is because host growth depends only on mean gene expression levels and parasite attack depends on rounded values (whether x_i is larger or smaller than 0.5). However, noise is important, because when amplified in the GRN, it may cause the initial dynamics to converge to a different stable state (i.e., noise-induced switches in gene expression). Regarding phenotypic variances, we discuss two types of robustness: mutational robustness and developmental robustness. Mutational robustness describes the “ability” of a GNR to prevent changes in gene-expression patterns due to mutations in the network structure, and developmental robustness describes the ability to prevent changes in expression patterns due to noise-induced fluctuations in gene expression level during the initial network dynamics.

Parasite interaction accelerates host diversity

We started simulations with N host cells of an initial gene-regulatory network created by creating random paths with probability α = 0.3. We took 50 replicated populations to obtain the statistics. The initial condition for each cell is given by a state in which only the input genes are expressed: all other genes are initialized with expression levels drawn uniformly between 0 and 0.05. This same initial condition was also applied to the daughter cells after each cell division. In the following results, gene regulatory networks (GRNs) that went extinct as a result of parasite interactions are excluded immediately from the sample at the initial time t = 0.

Fig 1 shows the population dynamics and fitness values of the host in three parasite environments (c = 0, 2, and 4) in the absence of noise (σ = 0). In the absence of parasite interaction, i.e., at c = 0, the host population is dominated by GRNs that generate a phenotype expressing all target genes, such as type “111” (black line), which dominate the host population (see Fig 1A). Then, if sufficient interactions with the parasites occur, the host population evolves into multiple groups with different phenotypes (Fig 1B and 1C). This is explained by the selection pressure to avoid parasite attacks. When the host population is concentrated on the fittest type “11111111,” which includes the target gene expressions “111,” the parasite population is also concentrated on the corresponding type matching the target “111,” which results in suppression of the population of the fittest host types by the interaction with the parasite. Consequently, other host types with less preferred/less fit target patterns have more chances to survive. In short, parasites induce negative frequency-dependent selection pressure, and the effective fitness (Eq (4)) varies over time because of the change in the distribution of parasite genotypes. Thus, multiple phenotypes and genotypes can coexist within a host population. Note that in the absence of noise, the expression dynamics of isogenic individuals always converge to the same pattern.

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Fig 1. Dynamics of host phenotypes under interaction with parasites.

The populations of hosts H_i corresponding to the parasite type () are plotted against the generation time. The number of parasite types is N_p = 2³ = 8, which corresponds to the on/off expression of the three target genes of the host. The population of each host type i for H_i represents {s(x₁), s(x₂), s(x₃)} = “000,” “001,”…, and “111” is indicated by different colors. The interaction strength is c = 0 (A), 2 (B), and 4 (C). The noise level is σ = 0.0. In the absence of parasites (c = 0), the population is dominated by the “111” type in which all output gene expression levels are turned on, whereas other phenotypic types with low growth rates (“110,” “101,” and “011”) also coexist at c = 2 and 4. In addition, N = 300, M = 64, l_inp = 8, l_out = 8, and l_p = 3 (see Table 1).

https://doi.org/10.1371/journal.pcbi.1008694.g001

Next, we computed the phenotypic variance V_p that is the variance of output gene expression levels over the host population with polymorphic genotypes (i.e., network J_ij) and plotted it as a function of the interaction strength c to determine the extent to which host–parasite interactions enhance phenotypic diversity (Fig 2). Independent of the level of developmental noise, Fig 2 shows a marked increase in the phenotypic variance at a threshold interaction strength c = c_t ≈ 1. This diversification indicates that multiple phenotypes coexist within the population, as shown in Fig 1B and 1C. In contrast, below the transition point c_t, the variance decreases and the host population is concentrated on the fittest phenotype, as shown in Fig 1A.

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Fig 2. Total phenotypic variance V_p of the output genes against the parasite interaction strength.

The variance V_p of the output genes plotted against the parasite interaction strength c (bars represent standard deviation). V_p is computed from the expression levels of the output genes over the host population (N = 300 individuals) and plotted as the average V_p over 2500–3000 generations. Each color represents a different noise strength: σ = 0.03 (blue), 0.02 (orange), 0.01 (black), and 0.001 (red).

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This critical interaction strength c_t ≈ 1 follows from the fitness function in Eq (4). The host growth rate is maximized when the expression level of all output genes is close to x_j = 1. If the parasite population is concentrated on the type that attacks this fittest phenotype, the population growth is reduced to cP_m ∼ c, with P_m as the population fraction of the parasite type m(= 1, …, l_p). To allow the phenotype to escape from the parasite attack, one target gene should be switched off, which reduces the raw growth rate by one unit. Then, the raw fitness loss needs to be smaller than the gain by the escape c. Since, P_m ≤ 1, the total fitness gain is positive if c ≳ 1. Indeed, Fig 2 shows that, for c > c_t, the variance increases substantially. When the interaction is strong (e.g., c = 4), the number of coexisting host genotypes approaches the maximal number 2^lp = 8, which corresponds to the total number of parasite types. (see Fig 1C).

In our model, phenotypic variance can have two different origins: genetic variation and noise-induced phenotypic variation. These correspond to two strategies to sustain phenotypic diversity. We investigated how the choice of each of the two adaptive strategies depends on c and σ, by computing the noise-induced isogenic phenotypic variation, V_ip, and the variance induced by mutations, V_g.

Evolution of robustness and phenotypic fluctuation

Fig 3 shows evolutionary time courses for the variance components V_ip and V_g (for both output and target genes) for various combinations of c and σ. The temporal averages of V_ip and V_g after evolution of the system reaches a stable state are shown in Figs 4 and 5, respectively. These results are independent of the choice of the initial network. Initial values of V_g are high because random networks are not canalized and are highly sensitive to mutation. In addition to the critical interaction strength c_t, we can see (e.g., from Figs 4 and 5 against c and σ) the existence of a critical noise level σ_c ≈ 0.02, above which V_ip increases markedly. Based on these two thresholds, behaviors of host gene expression states can be classified into four regimes:

Regime I: Noninteracting, nonrobust (σ < σc ∼ 0.02 and c < ct = 1)

1. When the noise is below the threshold σ_c = 0.02 and the interaction is weak (c < c_t = 1), V_ip is low, whereas V_g is maintained at a high level (Fig 4). The phenotype varies considerably by mutation, which indicates that mutational robustness does not evolve. (A large V_g implies that the phenotypes are not particularly robust against mutations.) Host phenotypes (i.e., gene expression) fluctuate around the fittest type, as shown in Fig 1A, where most of the target genes are expressed, whereas the variances are sustained at moderate values because of genetic variation in the network structure.

Regime II: Noninteracting, robust (σ > σc and c < ct)

1. The evolution of developmental robustness against the noise in gene expression dynamics occurs when the noise is above a threshold of σ_c = 0.02 and the interaction is weak (c < c_t = 1). In this region, the strength of the interaction is weaker than at the transition point c_t = 1, which means that the attack by the parasite is not sufficient to cause phenotypic diversification (Fig 2). Fig 4 shows that the values of both V_ip and V_g are low relative to the case with σ < σ_c and c < c_t. The phenotype is not significantly changed by genetic variation. V_ip and V_g both decrease while satisfying the inequality V_ip > V_g (see Fig A in S1 Appendix), and this leads to the evolution of robustness to both noise and mutation. This supports previous studies [17, 23, 25]. The host population loses both genetic and phenotypic inhomogeneity, which indicates that the output genes are all switched on and exhibit minimal variation with respect to the fittest type.

Regime III: Interaction-induced genetic diversification (σ < σc ∼ 0.02 and c > ct)

1. The evolution of genetic diversification occurs when the noise level is below the threshold σ < σ_c and the interaction strength exceeds the transition point c_t. V_g remains at high levels, whereas V_ip remains small, as shown in Fig 5. Phenotypic diversity is generated by genetic diversity. For each given genotype, the generated phenotype is unique and exhibits almost no variation. Multiple groups with different genotypes coexist because of interactions with diversified parasites. The stronger the interaction, the more phenotypes arise, which leads to a larger V_g and higher genetic diversity in the population. For moderate c > c_t, the genotypes with the fittest type “111,” coexists with genotypes having one target gene switched off (“110,” “101,” “011,” etc.), whereas for c = 4, even more host types appear, leading to coexistence, including expression patterns where only one target gene switched on (“100,” “010,” “001,” etc.)

Regime IV: Interaction-induced phenotypic variation (σ > σc and c > ct)

1. As exemplified by the case of c = 3 and σ = 0.02, the phenotypic variation of the output gene expression (V_ip) increases in the initial stage of evolution (up to 500 generations; see Fig 3). Both V_ip and V_g maintain high values, while satisfying V_ip > V_g. Host phenotypes are diversified without resorting to genetic diversification. Here, isogenic hosts can have more than one phenotype. Gene expression levels are diversified by noise, which implies the phenotypic variation of the isogenic population. Variances in expression levels are highest for target genes, whereas those of the other output genes are maintained at high levels (see Fig B in S1 Appendix). Accordingly, the fitness is reduced.

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Fig 3. Evolutionary time course of the average phenotypic variations V_ip and V_g.

The time course of the average phenotypic variations V_ip and V_g (i.e., and ). The plot is from the first generation rather than 0th generation. Note that host types that went extinct during the simulation have been excluded. This explains the different initial variance levels between parameter combinations. (A) Average V_ip and V_g over all output genes are plotted against the generation for different values of the noise level σ and interaction strength c, as indicated by different colors. Red: Example of evolution with no host–parasite interactions. Hosts are occupied by individuals with the fittest phenotype, thereby losing phenotypic diversity (σ = 0.02). Both the isogenic phenotype variation V_ip and the genetic variance V_g decrease. Orange: Case with c = 2 and σ = 0.01, which shows that V_ip decreases, whereas V_g maintains high values, implying the evolution of genotypic diversity. Green: c = 2 and σ = 0.03. Both V_ip and V_g increase. Blue: c = 4 and σ = 0.05. V_ip decreases slightly, whereas V_g maintains low values. (B) and for the target genes. The line colors represent the equivalent conditions as in (A).

https://doi.org/10.1371/journal.pcbi.1008694.g003

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Fig 4. Dependencies of V_ip and V_g on the noise level σ and interaction strength c.

V_ip and V_g for all output genes upon noise level σ (horizontal axis) and interaction strength c (vertical axis). The variance values for each parameter are displayed using color maps. The variance is computed by taking the average of 2500–3000 generations. The color maps indicate that the host–parasite interaction enhances V_ip and V_g. Parameter values of σ and c for each square are shown at their left and bottom edge (e.g., the bottom-leftmost square is the results for σ = 0.001 and c = 0) Each parameter regime is categorized into one of four regimes based on the values of V_ip and V_g (see the text for details).

https://doi.org/10.1371/journal.pcbi.1008694.g004

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Fig 5. Transition of V_g and V_ip with noise strength.

Dependence of the average variances V_ip (red) and V_g (blue) for output genes (solid lines) and (red) and (blue) for target genes (dotted lines) upon the noise level σ for a strong host–parasite interaction (c = 3). Bars represent the σ standard deviation. exceeds at σ_c ≈ 0.02. Based on the data in Fig 4, the phenotypic diversification occurs for all parameter values here, but the main origin of the phenotypic variances changes from the mutation to the noise at σ ≈ σ_c.

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Transition between Regime III and IV against noise strength

We now explore the transition between the last two regimes. Fig 5 shows the dependence of the variances V_ip and V_g of the output genes on the noise level σ for c = 3. Below the threshold noise level σ_c, V_g exceeds V_ip, indicating that there is little robustness against mutation; the diversity of the target gene expression levels is based primarily on genetic variation. In contrast, when the noise level exceeds σ_c, V_ip > V_g and the diversification of gene expression patterns is achieved over isogenic phenotypic variation, i.e., due to noise.

Note that, in Fig 5 V_g decreases slightly with σ once σ exceeds the critical value σ_c. This reflects previous results for c = 0 (i.e., in the absence of parasites), which showed that mutational robustness decreases as a correlated response to the evolution of developmental robustness against increasing noise levels (Fig A in S1 Appendix). This is because, for a larger noise strength, the acquisition of developmental robustness to noise is needed which also leads to an increase in robustness to genetic mutations. These factors have been investigated in detail in previous studies [23]. When c is nonzero, V_g is larger than the case for c = 0, but its decrease against σ for the case with c = 0 still remains, so that V_g shows slight decrease against σ.

Evolution of the noise-induced on/off switches

As mentioned above, once expression levels of host target genes have settled down close to one of the fixed points x = 0 or 1, continued variation due to the noise term in Eq (1) has no effect on the interaction with the parasite. The noise is important, however, for the initial network dynamics that determine which of these fixed points is approached in the first place. In order to separate the contributions of noise-induced on/off-switches from those of the noise itself, we define a variance which only reflects the former. More precisely, is the variance of the on/off variable z_i = tanhβ(x_i − 0.5) with β = 100 and defined as the variance of z_i, computed in the same manner as V_ip(i).

Fig 6 shows the dependence of the variances (output genes) and (target genes) on the noise level σ for c = 3. The key difference to Fig 5 is that for σ > 0.02, remains essentially constant for both target and output genes, suggesting that the variance due to noise-induced switches cannot be increased arbitrarily, and that the continued increase of V_ip seen in Fig 5 is merely a reflection of the increase in white noise. The reason that does not increase much beyond σ > 0.02 will be because the level of phenotypic variation reached by the noise, in conjunction with genetic variation, is sufficient to produce the maximal phenotypic variation. Indeed, V_p with c = 3 in Fig 2 is the same regardless of the noise strength. Evolutionary time courses for both V_ip and are shown in Fig E in S1 Appendix. For target genes, both measures increase from their initial value and then stay at a high level, with no evidence for evolution of developmental robustness. For non-target output genes, in contrast, V_ip and decrease over time relative to the values in the initial random networks. This shows that the decrease in the variance has not been observed in the target gene implying that the developmental robustness has not evolved. Also, as discussed in the previous section, V_g decreases with σ, but not more than the case with c = 0, suggesting that there exists some coupling between the and . On the other hand, the variance of the non-target output gene () decreases compared to the initial network, and the developmental robustness evolves in correlation with mutational robustness (V_g). At around the noise level (σ = σ_c ∼ 0.02), the variances and show remarkable increase. Albeit less significant as compared to , also increases. Although only the on/off of the target gene is relevant to protect against the parasite attack, the noise-induced switch is increased for other genes, due to the coupling in GRN. In the figure, is of the same level beyond σ = 0.02. The required switching frequency is of the same level, once at σ ∼ σ_c it occurs. (It increases with the interaction strength c, though).

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Fig 6. Transition of with noise strength.

Dependence of the average variances for output genes (solid lines) and for target genes (dotted lines) due to noise-induced switches plotted against the noise level σ for a strong host–parasite interaction (c = 3). We used the function z_i = tanhβ(x_i − 0.5) with β = 100 and defined as the variance of z_i. Bars represent standard deviation.

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Dependencies of the variances on genetic change and noise

In the absence of parasites, the variances V_ip and V_g decrease as the host phenotype adapts to the environment. This indicates the evolution of robustness to mutation and noise (regime II). Under the pressure of the host–parasite interactions, the gene expression dynamics evolved to diversify the phenotypes in the manner described above. Here, we discuss the increase in V_ip and V_g through evolution.

First, we plot the evolutionary time course of V_ip and V_g starting from the initial random networks J_ij setting with parameters c = 3, σ = 0.03 in Fig 7A. The variances and of target genes both increase rapidly because of interactions with parasites. (GRNs that went extinct due to parasite infections of the 0th generation are excluded. Therefore, the variance V_ip tends to be larger at c = 3 initially.) As shown in Fig 7A(b), the two variance terms exhibit considerably closer correlation between V_ip and V_g for the expression of the target gene. Moreover, both variances maintain large values, and the host can escape parasite attacks by producing a different phenotype via mutation. In contrast, Fig 7A(a) shows that the variances of other output genes decreased and developmental robustness increased for those phenotypes that were not attacked by parasites. This decrease, however, is much smaller than that observed in the absence of the host–parasite interactions (see Fig F in S1 Appendix). The decrease implies that the expression of target and non-target output genes is partially decoupled, but the reduction of the decrease suggests some coupling. The evolution of robustness and phenotypic fluctuation occurs in the same GRN’s.

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Fig 7. Time course of the variances (V_ip, V_g) after switching interaction strength.

(A) The time course of the variances (V_ip, V_g) for output genes (a) and the variance (, ) for target genes (b) plotted over generations for c = 3 when starting with initial random networks (circle; dotted lines). The variances are computed from the isogenic variance over 100 iterations. The time course over generations is plotted for the variances of gene expression. The plots cover 3500 generations. We set the noise level to σ = 0.03 > σ_c. The two variances of the non-target output genes decrease, while V_g < V_ip is maintained throughout the evolutionary course (a). Conversely, under host–parasite interactions, and show a correlated increase (b). (a) shows that the variances of other output genes not attacked by parasites decreased. However, this decrease is much smaller than that corresponding to the absence of host–parasite interactions. (B) The time course of the variances (, ) for target genes (a) and variance (V_ip, V_g) for output genes (b) over generations for σ = 0.03 when switching interactions c from 0 to 3 (asterisks; bold lines). For the first 2500 generations, the evolution of the host was modeled without parasites (c = 0). Up to this generation, the evolution of developmental and mutational robustness was complete V_ip and V_g decreases. Then, we introduced the interaction with the parasites by changing c from 0 to 3. The variances were computed from the isogenic variance over 20 iterations. The time course of (V_ip, V_g) and () over generations after this switch is plotted for the output and target genes. The color bar indicates the number of generations since the switch. The variances and V_ip both increase because of interactions with the parasites, which results in the evolution of phenotypic variation. Although this increase is prominent for target genes (b), those for other output genes also show a slight increase up to 2000 generations (a). Variances and finally reach the same values as those in (A). Other model parameters: N = 300, M = 64, l_inp = 8, l_out = 8, and l_p = 3.

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Coping with parasites by phenotypic variation

Next, we investigated whether high variability in phenotype (i.e., phenotypic fluctuation) can evolve in response to the introduction of parasites, even after developmental robustness has evolved. To investigate this issue, we first set c = 0 and let the system evolve, and once the host had adapted to the environment and acquired robustness (i.e., after V_ip and V_g had decreased from their high initial values), we switched on the interaction by increasing c from 0 to 3. The time course of the variance is shown in Fig 7B. After switching the interaction strength, the host growth rate decreases temporarily before increase in phenotypic variation evolves (see Fig G in S1 Appendix). Then, both the variances and increase. Thus, notwithstanding the prior evolution of developmental robustness, phenotypic fluctuations can evolve in response to parasite infection. The phenotypic fluctuations (V_ip) in the host population show a notable increase within 2000 generations. In addition to the target genes, the fluctuations in the expression level of the output genes also increase. Hence, the output and target genes are not completely decoupled.

A comparison of Fig 7A and 7B shows that the final variances are nearly identical. This means that the variances in both target and output genes after evolution are independent of the initial conditions. To increase the phenotypic variance of the target genes, variances of other gene expressions need to be maintained to a certain degree because of gene interactions through the GRN. This trend of a weak increase in the phenotypic variance for all genes holds for c > c_t (see Figs C and D in S1 Appendix).

Trade-off between growth and tolerance

We examined how the increase in the phenotypic variances caused by parasite interactions influences the growth rate of a single host. Parasites concentrate their attacks on the hosts with the most frequent phenotype in the population. Fig 8 plots the relationship between V_ip and the raw fitness, and it shows the trade-off between the two. Individuals exhibiting high plasticity tend to have lower growth rates because of the uncertainty in the expression of output genes; however, they are attacked less by the parasites. As the variance V_ip is larger, the parasite attacks are reduced. Consequently, the host genotypes with larger V_ip have a larger chance of survival, even if their growth rate as a single cell is lower. In contrast, in region III with large V_g, host populations are adapted to parasite infection through genetic variation. In this case, low phenotypic fluctuations and high raw fitness are compatible. Here, target genes and non-target output genes are well decoupled and raw fitness tends not to decrease even if the expression of target genes is variable (Fig 8 σ = 0.01 and σ = 0.001 case). Therefore, the fitness costs are higher for the variance by noise (V_ip) than that by the genetic variation (V_g). The lack of fluctuations, however, makes the host population more susceptible to parasite infections when genetic variation is suddenly reduced because of extinction.

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Fig 8. Trade-off between phenotypic variation (V_ip) and fitness.

Trade-off between isogenic phenotypic variation (V_ip) of the output genes and raw fitness . Each value is computed by the average from 2500–3000 generations. Each color represents a different interaction strength: c = 0 (black), c = 1 (green), c = 2 (red), c = 3 (blue), and c = 4 (yellow). Each marker shape represents a different noise strength: σ = 0.05(⋄), σ = 0.04(⋆), σ = 0.03(⊳), σ = 0.02(∘), σ = 0.01(△), and σ = 0.001(⊲).

https://doi.org/10.1371/journal.pcbi.1008694.g008