Text S2 Supporting Results. More results exploring the dependence of stability on parameters of the model. More parameters are also explored. Sample sizes and transient times are analyzed too. (PDF) Figure S1 Stability decreases with N even for scale-free biological topologies. Average stability (Equation (2)) is plotted for different network sizes N and topology. SKT~2. Ki~2 for every gene, i, in the regular network. The Poisson network draws the degree distributions from a Poisson distribution with mean and variance equal to 2. exp-pow stands for exponential in-degree distribution, and power-law out-degree distribution, both with mean 2. Other parameters as in Figure 1. (EPS) Figure S2 Stability decreases with N in spite of topology for K~4. Same as Figure S1 for SK T~4. In this case we do not represent the biological topology, since biological networks usually have SK T*2 [32]. (EPS) Figure S3 Path length to equilibrium grows rapidly with N. Represented is average transient time (i.e. the number of time steps until the network reaches an attractor) as a function of network size N and for different degrees K . Only fixed points are considered. Equation (1) is solved up to n~108 times with parameters as in Figure 1. Although coefficients of a least-squares fit are shown in the figure legend for each K, these regression lines are presented here only as a qualitative description. The estimates are employed for the extrapolation of transient times of larger networks, used to impose a cut-off on the convergence time of Equation (1). Error bars are big but are not shown to avoid cluttering the figure. (EPS) Figure S4 Sample size decreases with increasing N. Shown is the number of samples from which the results presented in Figure 1 are drawn. Increase of convergence time with N, as depicted in Figure S3, limits sample size. (EPS) Figure S5 The effect of network degree on stability seems to depend on network size. Represented is ½stabilityðK ~K ’Þ{stabilityðK~NÞ for different K’~2,4,6 and N. It seems that the difference in stability between sparse networks and the densest one has a maximum for an intermediate Nw10. After an initial increase, it goes down with increasing N, where stability(K ~N)?0, and thus the difference is reduced to stability(K ~K ’). This basically represents the difference between the different plots in Figure 1. (EPS) Figure S6 Stability is higher than 0:5 for the f0,1g map with K~1. The f0,1g map produces the only case where reaching a fixed point is actually more likely than reaching a cycle for any network size. This happens, however, for the uninteresting case of K ~1 regular networks, where each gene receives input from only one other gene, or itself. Equation (1) is solved up to n~108 times for each N and K with wij *N (0,1), si*f0,1g, f (x)~H(x) and H(0)~1 (Text S1). (EPS) Figure S7 The f0,1g map has more stable and less unstable matrices than f{1,1g. Equation (1) is solved for N~4, c~1, wij *f{1,1g and full enumeration of the network Most Networks in Wagner’s Model Are Cycling 2N2 ~65,536 and state ð2N ~16Þ spaces. Other parameters are as in Figure 4. (EPS) Figure S8 The f{1,1g map and real matrices allow for faster discovery of novel phenotypes. Shown is the number of samples needed to reach all 1024 fixed point attractors of networks of size N~10, for different c and maps (a) or types of regulatory interactions (either binary or real) (b). Equation (1) is solved up to n~109 times for each c. For c~0:1 and 0:2, the f0,1g map reaches the maximum sample size before the discovery of all stable phenotypes. The results shown are for one run only. Other parameters are as in Figure 4 (a) and Figure 5 (b). (EPS) Figure S9 Stability is not monotonic in a. Although still low, the probability of fixed points has a maximum for 0:7vav0:8. Stability also seems slightly higher for a~100 than a~10. Equation (1) is solved n~106 times with wij *N (0,1), N~10, c~1, si[½{1,1 and sigmoidal f (x)~z(x; a), varying a (Text S1). Other sigmoid parameters are as in Figure 6. The dashed line is a guide to the eye. (EPS) Figure S10 Probability of attractor length decays slower than a power-law. Shown is the attractor period distribution for the two different maps and networks of size and density N~K~10. The f{1,1g map has an antisymmetric property where cycles of even length are overrepresented at least 2-fold (Figure S11; [12,31]). For that reason, even and odd periods are analyzed separately. Also shown is a least-squares fit for the f0,1g map. Equation (1) is solved with parameters as in Figure 4. (EPS) Figure S11 Cycle size distribution for the f{1,1g map. Represented is the attractor size distribution, as in Figure S10, but just for the f{1,1g map with the odd and even length cycles taken together. Note how cycles of even length are overrepresented at least 2-fold. (EPS) Figure S12 Different conventions for f (0) result in similar stability profiles. Stability is shown for different N and definitions of f (0) (Equation (1)). Networks are regular and binary, wij *f{1,1g. random means we choose f (0)~1 or {1 with equal probability. f (0)~S(t{1) means si(t)Df (0)~si(t{1). si*{1,0,1 for the latter and also for f (0)~0. (EPS) Figure S13 The sign and sigmoid functions are very similar for a§10. The parameter a controls the steepness of the lim sigmoid function, z(x; a), where z(x; a)~sgn(x) (Text S1). a?? (EPS) Table S1 List of all model variants and corresponding figures and references. Most of the evolutionary studies done with this model vary in the parameters used in Equation (1). We estimate the dependence of stability measured by Equation (2) on most of these parameters. (PDF) Table S2 Limits on transient times as a function of N and K. Table entries are the range of N values for which each T is used. The time it takes for Equation (1) to reach an attractor grows with N (Figure S3). To be able to produce Figure 1, a time limit T (N; K )v? is enforced for large or dense networks. (PDF) PLoS ONE | www.plosone.org 7 April 2012 | Volume 7 | Issue 4 | e34285