Developmental model

To study how stability, robustness and autoregulation change during evolution, we use a standard model for GRN. In one generation, we assume that that the phenotype of an organism S(t) develops over time t, starting from an initial phenotype S(t = 0), under the influence of a gene-interaction network W. In general, phenotypes are thought of as expression levels of the genes of the organism at time t. Thus, they are vectors of dimension N, , with binary entry values , where N is the number of genes of the organism.

Phenotypes S(t) change by the action of a gene-interaction network that drives their development, and is represented by an matrix, W, whose elements, w_ij, denote the effect on gene i of the product of gene j. These interaction weights w_ij are nonzero and binary, . Thus, all genes either repress or activate each other's expression.

In this study, we assume that size of the gene interaction network is N = 10 genes. The matrix W is not necessarily symmetric. Diagonal elements, w_ii, represent autoregulation, i.e., the action of the i^th gene on itself.

Each network W determines the dynamics of the phenotype S(t) in a series of development steps. The repeated application of such development steps on a phenotype results in deterministic, discrete-time dynamics of S(t), modeled by the set of nonlinear coupled difference equations:(1)where sgn(0) = 1. This spin glass or neural network-type model [22] represents a subclass of Random Boolean Networks [24] known as Random Threshold Networks [25].

When simulating development, the network is updated synchronously, that is, only values of s_i from time step t are used for the calculation of s_i (t+1) (see [26]–[28] for asynchronous updates.) We refer to Equation (1) as the development process (see [23], [29] for model illustration, biological motivation and assumptions).

The development process can be extended to include sparse networks G. Sparse networks are used to model gene interactions in which only a fraction of the genes repress or activate a fraction of all the other genes, in contrast to fully connected networks W, where all genes have some effect on all other genes.

Let G denote an interaction network represented by a , N = 10 square matrix whose entries, g_ij, take the values of {−1,0,1}. The parameter c, the density of the network, determines the proportion of non-zero matrix elements. When simulating sparse networks (see Results), we chose c = 0.2 (due to similarity to the biological networks in Table 1) and a regular, directed graph topology, where all genes have degree 2. This means that all genes in a network are regulated by two genes and also regulate two other genes.

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Table 1. Most studied eukaryotic transcription factor networks (including yeast, flies and mammals) show values of p ranging from 0.76 to 1.

https://doi.org/10.1371/journal.pcbi.1003916.t001

Random and stable networks

To generate a random network W_r the matrix-elements w_ij are sampled from {−1,1} with equal probability (0.5 per element and entry). Additionally, we can generate stable networks W_s with a pre-selection procedure. In this procedure, a random network W_r and random initial state pair are first generated. This pair undergoes the development process. If no fixed point is attained, a new pair is sampled and developed. This step is repeated until some (W, S(0))- pair generates a fixed point. The final network W_s, is a stable network. This notion of stability refers to an individual level stability, which differs from a notion of population level stability that will be introduced below.

In this study, we refer to stability as the property of a network, while strictly speaking, it is a property of a W, S(0) pair. However, we have previously shown [30] that the network is by far the most important determinant of stability. If a network is stable/unstable with a random initial state, it most likely remains stable/unstable with any other initial state. For this reason, we classify networks as stable or unstable, even if we just solve Equation (1) for one possible initial state.

Evolved and non-evolved networks

Two types of simulation experiments are our primary focus in this study. First, experiments in which a population of organisms undergoes the development process only, which we refer to as non-evolved. Secondly, experiments with multiple generations (evolved), where after each development process (one generation) the composition of the population of organisms is additionally altered by evolutionary mechanisms. The development process is completed after all organisms have reached some stage of development: either a fixed point or a cycle. We implemented standard evolutionary mechanisms, such as selection, mutation and recombination. After these evolutionary forces have acted on the population, a new development process starts in the next generation with identical initial phenotypes for each organism.

In this study, we set the population size to n = 500 across all experiments (unless otherwise noted). This population size remains constant during evolutionary simulations (Wright-Fisher model with sampling with replacement).

Selection

To study how selection affects evolving populations, we implemented different types of selection or selection models. The mutation and recombination mechanisms applied were the same for all evolved populations.

Selection mechanisms modify the number of copies of one specific network within the population depending on the fitness of the phenotype that specific network has generated through development. In a selection mechanism, one phenotypic state can be marked as the optimal state, with the highest possible fitness. If such an optimal state, S^opt(∞) is specified, the fitness of a network with attractor S(∞) is given by:(2)where d is the normalized Hamming distance and σ>0 determines selection strength [23].

Small values of σ imply strong selection against deviations from the optimal state. Large values minimize the fitness difference between phenotypes. The Hamming distance d corresponds to the number of differing expression states of individual genes between two phenotypic states [31], subsequently normalized to the interval [0,1] in this study.

Equation (2) is valid under the assumption that S^opt(∞) and S(∞) are attractors with identical, optimal cycle lengths l_opt. The cycle length of the attractor with highest fitness is denoted with l_opt. The fitness of attractors of length l≠l_opt depends on the selection model. We use attractor length and cycle size or period as synonyms.

Selection models

We implemented selection models similar to those used by other authors [23], [29] and also introduced new ones. In these models, the fitness of a developed organism depends on two parameters: selection strength, σ, and optimal period, l_opt.

Selection model 1 (selecting for stability): l_opt = 1, fitness(l≠l_opt) = 0

 σ = 0.1 ‘target’ model

 Selects for fixed points and an optimal gene expression state. Fitness is given by Equation (2) for fixed points and is 0 for cycles.

 σ = ∞ ‘no target’ model

 Fitness is 1 for all fixed points and 0 for cycles.

Selection model 2 (selecting against stability): l_opt>1, fitness(l≠l_opt) = 0

 σ = ∞, l_opt = 2,3,…,7 (cycles)

 We generalize the ‘no target’ model to select for cycles. Fitness is 1 for cycles of length l = l_opt and 0 otherwise, including fixed points. We try different l_opt>1.

Selection model 3 (neutral for stability):

 σ = 0.1, fitness = max(fitness(S)) for all S in S(∞), any l (S represents any state in the attractor S(∞))

 We generalize the ‘target’ model to not require stability. When l = 1, we have the ‘target’ model as a special case and fitness is given by Equation (2). When l>1, fitness is the maximum fitness given by Equation (2) for all states in the cycle. The attractor S(∞) can be a fixed point or a cycle.

Selection model 4 (random sampling):

 σ = ∞, fitness = 1, any l

 No selection. We take this as the null model.

 For each selection model, we generated between z = 100 and z = 300 independent populations (depending on the model). Specifically: z = 200 for the ‘target’ model; z = 300 for the ‘no target’ model; z = 200 for Selection model 2; z = 100 for Selection models 3 and 4. We denote such an aggregation of populations as a set of populations and z as its set size. Each evolved population has a different initial state, but all individuals within the same population have the same initial state.

Mutation

Mutations randomly change the sign of w_ij at a rate μ = 0.1 per network per generation. All matrix entries, w_ij, including diagonal elements, w_ii, have equal probability of changing sign, namely μ/N² = 0.001 per generation. For sparse networks we use a probability for changing sign of μ/(c N²).

Recombination

To model recombination we follow the methods in [23], where full chromosome segregation (no crossover) is implemented. The two offspring of a randomly chosen pair of recombinant parents are generated by randomly taking half the rows from each parent matrix. This procedure is performed on the entire population.

Population metrics

We define a population-level stability (henceforth referred to as stability if not otherwise stated) as the fraction of networks that are stable (individual-level) in a given population [30]:(3)where n_f≤n is the number of times the attractor is a fixed point, and n is population size, that is, the number of network matrices. Stability takes values between 0 and 1.

Similarly, we define the robustness of a population as the fraction of all possible mutated networks in a population that reach the same fixed point attractor as their un-mutated originals [23], conditional on the fact that the attractor did not become a limit-cycle. Specifically, we estimate robustness by looping through the population of networks and mutating every element of each network matrix W (changing the sign of w_ij for binary matrices), thus generating N² single-mutants per network. Then, the networks undergo the development process starting from the same initial phenotypes as their originals, and are further analyzed.

For a single network, we define individual-level viability as the fraction of single-mutants that attain a fixed point:(4)where n_fixed<N² is the number of times the N² single-mutants have still generated a fixed point. With this metric, we can now define individual-level robustness:(5)and n₌≤n_fixed is the number of times the same attractor state as the one attained by the un-mutated original is reached starting with identical initial conditions (i.e., the mutant has the same phenotype as its wildtype). The population-level viability and robustness measures are computed from the averages of all networks in the entire set of populations.

Both robustness and viability take values between 0 and 1. In normalizing robustness by n_fixed instead of N², we attempt to decouple the effects of stability and robustness. In the vast majority of cases, mutations that change the stability of a network do not affect its robustness score. Exceptions to this are the rare occasions when n_fixed = 0 (robustness is not defined), or when n_fixed is low (robustness can only take a few specific values).

In an extension of the fraction of activating connections-statistic [32], we found it useful to measure properties of diagonal and off-diagonal elements of a matrix W separately, thereby decoupling the effects of direct autoregulation and off-diagonal regulation. For a single network matrix W, we define:(6)where N⁺_p and N⁺_q are the number of positive diagonal and off-diagonal elements of W, respectively. Both p and q are always positive and take values between 0 and 1. We call p the sign of autoregulation, because autoregulation is predominantly positive when p>0.5 (we call this positive autoregulation), and mostly negative when p<0.5 (we call this negative autoregulation).

The metrics p and q measure direct regulatory influence. However, network dynamics can also be affected by long-range interactions. To assess the role of such long-range regulation, we introduce a metric of indirect positive autoregulation r, which measures the fraction of autoregulatory paths over two genes that are positive (i.e., gene A activates gene B, which activates gene A). For a single network W, we define:(7)where N⁺_r is the number of positive off-diagonal elements of WW^T (W^T is the transpose of W). Because WW^T is symmetrical, it suffices to count the fraction of positive entries in either of the triangles of the matrix.

We also define metrics to assess the population-average of gene interaction strengths:(8)where n⁺_ij is the number of positive elements in position i,j across all n networks in a population, abs is the absolute value function, and t₁ and t₂ are two different evolutionary time points. Here, o_ij, is referred to as average positive ij- interaction strength, and measures how much gene i activates gene j on average, whereas the conservation statistics between population-averaged gene interaction strengths measures how much these interaction strengths are maintained over time. In evolutionary experiments, conservation, p, q, and r are averaged over all individuals and all populations.

The code utilized in this paper can be downloaded from https://github.com/rpinho/phd.

Individual- and population-level stability and autoregulation are correlated in Boolean GRNs

To study the relationship between the sign of autoregulation (p) and stability during development, we devised two experiments. First, for each p = 0, 0.1, …, 1, a pair consisting of one random network and one random initial condition was sampled (RNRC setup; see Methods). Equation (1) was then evaluated for each pair: if the attractor was a fixed point, the network was considered stable. Instead, if the solution to Equation (1) was a limit cycle, the network was considered unstable. This process was repeated n = 10⁵ times for each p.

Figure 1A shows that individual-level stability is strongly associated with p. Stable networks have significantly higher values of p than unstable networks (median of 0.9 compared to 0.4 for unstable networks; Mann-Whitney U p-value ∼0, Figure 1A).

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Figure 1. Positive autoregulation favors stability.

A. Boxplots of the sign of autoregulation (p in Equation (6)) for two network classes: stable (fixed points) and unstable (cycles). Whiskers are 1.5 times the inter-quartile range (the difference between the first and third quartiles). All outliers are represented. B. Histogram of the values of p for stable matrices, i.e., stability (Equation (3)) as a function of p. The dotted line is an exponential function of p. Networks are random and non-evolved. Panels A and B represent the same data in different form. Equation (1) is solved n = 10⁵ times for each p = 0, 0.1,…, 1 (11 bins).

https://doi.org/10.1371/journal.pcbi.1003916.g001

This positive association was also observed for population-level stability in a second experiment. We subdivided the networks generated in the first experiment into populations of identical p, and measured the average population-level stability for each p. We observed that the fraction of stable networks increases rapidly with higher values of p (Figure 1B). These results also indicate that p and stability are strongly associated.

Evolution of autoregulation when selecting for stability is non-linear

We next studied how p changes when explicitly selecting for and against individual-level stability in evolutionary simulations. To this end, we founded sets of populations with random networks and the same initial state for each population (RNIC setup; see Methods) [23], [29]. The average p was set to p = 0.5 at generation 0. We then evolved all populations under the six different selection models (including mutation and recombination) described in the Methods. For all of these selection models, we followed the evolution of the sign of autoregulation p over 10⁶–10⁷ generations (until equilibrium was attained).

Consistent with the observations for non-evolving networks, positive autoregulation is strongly favored during evolution, both under the ‘target’ and ‘no target’ models (Figure 2A). However, the evolution of p follows a complex, non-linear pattern. After a sharp initial increase over the first ∼50 generations, p reaches its maximum when population-level stability is above 95% (stability-metric not shown in the Figure), and starts to decay slowly to a stable evolutionary equilibrium of p∼0.8 from t∼10³ generations for both models (Figure 2A). At the peak, a fraction of up to p∼0.95, or 19/20 surviving matrices show positive autoregulation for all genes under the ‘no target’ selection model.

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Figure 2. Stability selects for positive autoregulation.

Time evolution of p (Equation (6)) on the vertical axis for different selection models. A. Selecting for stability: only fixed points are allowed to survive. ‘no target’ means all fixed points have fitness of 1 and all cycles have fitness of 0. ‘target’ means selection for a specific fixed point, with fitness given by Equation (2) as a function of the hamming distance from the target phenotype. B. Selecting against stability: only cycles of specific length are allowed to survive. Shown are selection for cycles of size 2 and 3. C. Neutral for stability: fitness does not depend on stability. ‘no target’ means random sampling. ‘target’ means selection for a specific fixed point, with fitness given by the maximum value of Equation (2) applied to all points of the attractor as explained in Methods (allows cycles of all sizes including fixed points). Sample size is n∼10⁵ evolved networks per generation.

https://doi.org/10.1371/journal.pcbi.1003916.g002

Interestingly, selecting for cycles of length l = 2 (i.e. against individual-level stability), has the opposite effect on evolving networks: p decreases sharply down to ∼0.3 (Figure 2B), leading to negative autoregulation. A similar, but less pronounced pattern is observed when selecting for longer cycles with lengths l>2 (Figures 2B and S2).

As expected, a neutral model with no selection for individual-level stability or a specific target produces random networks, with values of p centered on p∼0.5 (Figure 2C). However, mean values of p<0.5 also evolve when not selecting for any particular attractor length, but still selecting for a specific target (Figure 2C).

Thus, the selection for stability leads to positive autoregulation.

Autoregulatory motifs are highly conserved over time

These results suggest that selection may act over direct autoregulatory motifs (i.e. the diagonal elements of the GRN matrix) to promote individual-level stability. If this is the case, positive diagonal elements should be overrepresented across evolved populations relative to off-diagonal elements, since selection for individual-level stability could be achieved by maximizing autoregulation p.

To test this hypothesis, we calculated the average interaction weights in populations evolved from a RNIC setup (the number of populations, z, was 100, and the population size n = 500). The metric is the average value of the w_ij entry of individual networks W, taken across a set of populations (including individuals within the populations) as well as evolutionary time. Extreme values of indicate that the matrix element i,j is identical across individuals, different populations and different generations, whereas a neutral value of means the matrix element i,j fluctuates randomly in individuals and populations and is not conserved over time.

We found that, in our evolution experiments, the averaged diagonal elements attained higher values than the off-diagonal elements, consistent with stronger selection for positive autoregulation acting on the diagonal elements (Figure S3). This is further supported by the observation that selecting for individual-level stability leads to positive values of , whereas selecting against individual-level stability leads to negative values. Selecting neither for nor against stability, but still selecting for a specific target, also yields negative .

To increase our confidence that the value of p = 0.8 emerging under the no-target model is maintained by selection for stability, we compared the effects of diagonal and off-diagonal mutations on individual-level stability in the evolved networks. We hypothesized that since single mutations in diagonal elements could result in unstable networks more often than in off-diagonal elements, networks carrying diagonal entry mutations would therefore be weeded out at higher rates. Because p and stability are correlated, p would then be maintained at a high value.

In a first approach, we sampled n = 10⁵ stable networks at t = 10⁶ generations (equilibrium reached) evolved under the ‘no-target’ model. The overall fraction of networks that survive after acquiring single mutations (viability in Equation (4)) is high (95%). Consistent with our hypothesized maintenance mechanism, the population-level viability of networks is significantly higher for off-diagonal elements (median of 0.97 compared to 0.93 for diagonal; Mann-Whitney U p-value∼0, Figure 3A). Intriguingly, the difference in viability is even higher for t = 48 generations, when p is close to maximum (Figure 3B), and the evolutionary dynamic has not yet reached equilibrium. These results suggest that a mutation of a diagonal element is more likely to lead to a cycle than mutation of off-diagonal elements.

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Figure 3. Changing the sign of an autoregulatory element is more likely to kill a network than a random mutation.

A. Boxplots of average viability (Equation (4)) for diagonal and off-diagonal elements of the network matrices at evolutionary equilibrium (t = 10⁶ generations). B. Same as (A) but at t = 48, when p (Equation (6)) is largest. C. Heat map of average viability for all matrix elements at t = 10⁶ generations. Darker blue (off-diagonal elements) means higher viability after mutation than lighter blue (diagonal elements). D. Same as (C) but at t = 48. Sample size is n∼10⁵ evolved networks per generation.

https://doi.org/10.1371/journal.pcbi.1003916.g003

In a second approach, we studied the conservation of values in diagonal versus non-diagonal elements between two given time points. To this end, we sampled n = 10⁵ evolved stable networks at t₁ = 10⁵ and t₂ = 10⁶ generations from the ‘no-target’ and the random models. Subsequently, we counted the fraction of positive elements o_ij, (Equation (8)) across all networks and for each position in the gene regulatory matrix (w_ij) for both t₁ and t₂, and estimated conservation values as in Equation (8).

For the no-target model, we found that diagonal entries w_ii with positive sign are significantly more conserved over time than off-diagonal matrix elements (medians of 0.86 and 0.70, respectively; Mann Whitney U p-value∼0, Figure 4A). These diagonal elements under the no-target model are also more conserved when compared to diagonal elements evolved under the random model (medians of 0.86 and 0.68 respectively; Mann-Whitney U p-value∼0, Figure 4A). We found no significant differences in conservation of the off-diagonal entries between the ‘no-target’ and the random models (one-sided Mann-Whitney U p-value∼0.1, Figure 4A). This finding provides further evidence that positive autoregulation is maintained by selection for stability.

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Figure 4. Autoregulation is conserved over time.

A. Boxplots of average conservation (Equation (8), t₁ = 10⁵ and t₂ = 10⁶ generations) for diagonal and off-diagonal elements of random and evolved ‘no target’ networks. B. Heat map of average conservation for all matrix elements under the ‘no-target’ model. Darker blue (diagonal elements) means higher conservation than lighter blue (off-diagonal elements). C. Same as (B) but for random networks. Sample size is n∼10⁵ evolved networks per generation.

https://doi.org/10.1371/journal.pcbi.1003916.g004

Stability, robustness and autoregulation coevolve

The time course of p displays an intriguing complexity. After the stability metric reaches its maximum and ceases to change, p keeps evolving and decreases to a lower equilibrium value (Figure 2A). To investigate this behavior, we asked which other network parameters may also affect the evolution of p. Since Wagner [23] has previously shown that during network evolution robustness is also (indirectly) selected for when selecting for stability, we studied how this robustness compares with p over the course of evolution.

Intriguingly, during the simulation experiments under the no-target model, we observed that robustness (Equation (5)) increases with time and appears to coevolve with p, reaching its maximum at the same time point at which p reaches equilibrium (Figure 5). This association, however, is also not linear: shortly after stability has reached equilibrium, robustness still increases despite the fact that p has started decreasing.

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Figure 5. Positive/negative regulation evolves non-linearly with robustness.

Coevolution of stability (Equation (3)), robustness (Equation (5)), p and q (Equation (6)) under selection for stability (‘no-target’ model). Black line (p) is the same as in Figure 2A. White and grey regions serve as guides to the eye signaling different phases of evolution. From t = 1 to t∼48, p reaches its maximum and stability reaches 0.96. From t∼48 to t∼10³, p approaches equilibrium and q reaches its maximum. From t∼10³ to t∼10⁵, robustness and q approach equilibrium. From t∼10⁵ onwards, all represented quantities are at evolutionary equilibrium. Sample size is n∼10⁵ evolved networks per generation.

https://doi.org/10.1371/journal.pcbi.1003916.g005

Robustness and autoregulation are associated, but this relationship is dynamic during evolution

Because robustness and p reach equilibrium at about the same time, we hypothesized that p could be adapting under indirect selection for robustness. In that case, the equilibrium value of p∼0.8 would favor higher robustness (or perhaps: “maximize robustness”).

To test this, we generated groups of 10⁵ random stable networks for p = {0.1,0.2,0.3,…,1}, and calculated the robustness for each group. Robustness was assessed after running one single development process for each network in the RNRC setup. Only stable networks were considered for this analysis.

Surprisingly, we found that, similarly to stability, robustness is also positively associated with p and does not have a maximum at an intermediate p∼0.8 (Figure 6A). Contrary to our hypothesis, robustness of stable, non-evolved networks is maximized by p = 1 (see Figure S4 for random networks not pre-selected for stability). This positive association was also observed when representing the data differently: stable networks binned by increasing average values of robustness also show increasing p (Figure 6B).

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Figure 6. Maximization of robustness during evolution determines the equilibrium values of p.

The relationship between robustness and autoregulation changes during evolution. A. Boxplots of robustness (Equation (5)) for each p = 0, 0.1,…, 1 (11 bins) for stable, non-evolved networks (q = 0.5). B. The inverse of (A), i.e., boxplots of p (Equation (6)) for 5 bins of increasing robustness, each of size 0.2, from 0 to 1. C. Same as (A) but for evolved networks, at t = 48, when p is largest. At this stage of evolution, there are no significant samples of networks with p<0.6. D. Same as (C) but at evolutionary equilibrium (t = 10⁶ generations); there are no significant samples of evolved networks with p<0.3. q = 0.5 (Equation (6)) for panels A and B but q is not constant or controlled for in panels C and D. (It varies with p in a non-linear fashion in evolved networks.) Sample size is n = 10⁴ networks for each p (total n∼10⁵) for (A) and (B). Sample size of evolved networks varies with p but total size is n∼10⁵ for (C) and (D). Robustness is measured by Equation (5) with N² = 100 mutations per network.

https://doi.org/10.1371/journal.pcbi.1003916.g006

This general positive association is inconsistent with the hypothesized relationship between p and robustness after stability reaches its maximum. However, a more in-depth analysis of robustness of evolving networks at different time points reveals a completely different picture to the situation in non-evolved matrices (Figures 6C and D).

At both early (t = 48, p_max) and late (t = 10⁶, p_equilibrium) evolutionary stages, robustness is maximal in evolving matrices with values of p<1. At early time points, when p has reached its maximum, the relationship between p and robustness is fully inverted compared to non-evolved networks, with lower p having significantly higher robustness, thus suggesting strong selection for lower p to increase robustness (Figure 6C). Strikingly, at equilibrium values, robustness is non-monotonic in p and is maximal for p∼0.7–0.8, coinciding with the equilibrium value of p (Figure 6D). Therefore, these results strongly suggest that, as we hypothesized, it is the maximization of robustness during evolution that determines the equilibrium value of p.

Evolved networks are a distinct subset of stable networks

The above results may seem contradictory: whereas p and robustness show a positive association in stable networks generated completely at random (i.e. for all non-evolved stable networks), this association is non-monotonic for stable networks selected by evolution (i.e. evolved networks), where robustness is maximized for more intermediate values of p (0.7–0.8). The solution to this apparent paradox might lie in the fact that evolved networks constitute only a subset of all stable networks.

Remarkably, the average robustness for the subset of evolved networks is twice as high as that of non-evolved networks with similar p (compare Figures 6A and 6D), suggesting that the relationship between p and robustness is modulated by other matrix characteristics on which selection can act.

In order to investigate this possibility, we studied the evolution of the sign of off-diagonal elements (q). There are more off-diagonal than diagonal elements; thus the former offer many more targets for mutation. However, a mutation in the off-diagonal has a smaller effect on q than a mutation on the diagonal has on p. For this reason, q may seem to evolve at lower rates than p. More importantly, off-diagonal elements represent regulation of other genes and can form larger and more complex motifs than autoregulatory loops of size one. For this reason, they are harder to study and to interpret. However, under a random model, it is clear that the expected average value of q equals 0.5.

The evolution of q occurs over a much smaller range than that of p, with values spanning from 0.5 to ∼0.63. However, it also shows a non-linear pattern of co-evolution with the other variables (Figure 5). q increases up to its maximum approximately until p stabilizes, and then it starts to slowly decrease to its equilibrium value at t∼10⁵. It reaches equilibrium around the same time as robustness.

These co-evolutionary dynamics suggests that stability and robustness may not only depend on p, but also on q. Therefore, although robustness is maximized by high p for stable networks with q = 0.5, the same is not necessarily true when q>0.5. This is the case at q>0.65, for which robustness is higher for p = 0.7–0.8 than for p = 1 (Figure S5). Interestingly, these values correspond closely to (p = 0.8, q = 0.63) of evolved networks at generation ∼1000, when both stability and p reach their equilibrium values.

Engineering super-robust networks: robustness is not fully maximized in evolution

A prediction from our results is that certain combinations of p and q are more likely to provide stable networks.

To test this, we combined the off-diagonal elements (determining q) of stable networks with low p and a q similar to that of equilibrium (q = 0.53 or 0.54), with diagonal elements of high p (p = 0.9 or 1.0). We call these networks “engineered”. As before, we generated groups of 10⁵ random stable networks for different values of (p,q) and calculated the robustness for each group. Robustness was assessed after running one single development process for each network in the RNRC setup. Only stable networks were considered for this analysis.

The engineered networks resulted in extremely stable and robust networks (Figure S6A); importantly, randomly sampled stable networks with the same average p and q are not nearly as stable and robust to mutations (Figure S6B). These observations support the idea that features in the topology of off-diagonal elements of these matrices (i.e., how genes regulate one another) may buffer the destabilizing effects of mutations.

These findings also show that it is possible to engineer networks more robust than those evolved under selection for stability. Thus, robustness is not fully maximized during evolution. In fact, when the founding populations (t = 0) have a mean of q = 0.9 (rather than being normally distributed around 0.5, see Methods), robustness decreases throughout evolution (Figure S7).

Positive autoregulatory motifs of length two are also associated with stability and robustness

Direct regulatory influence of genes on one another can explain qualitatively why p and q are being maximized at the beginning of evolution experiments. Their subsequent decline below the maximum values is related to constraints imposed by the indirect selection on robustness. To further elucidate how these constraints operate, we investigated how long-range interactions embedded in the matrix of direct influences (direct interaction, i.e., W) of an organism could contribute to the settling of p and q below their maximum values.

To this end, we repeated the evolutionary experiments tracking the measure for length-2 autoregulatory interaction r, which measures the frequency of networks that contain self-reinforcing interaction loops (gene A activates gene B, which reactivates A, see Methods). We found that r is maximized early and attains values above 0.5 throughout evolution, indicating positive long-range autoregulation (Figure S8). Additionally, r lags behind the evolution of p and q, adapting to selective pressures at lower rates.

To understand why engineered networks are more robust to mutations than random stable networks with the same values of p and q, we measured r for networks similar to the ones shown in Figure S6. We find that r is larger for the engineered networks (Figure S9), which explains why engineered networks are more robust for the same values of p and q.