Evolving Always-Critical Networks

Abstract

Living beings share several common features at the molecular level, but there are very few large-scale "operating principles" which hold for all (or almost all) organisms. However, biology is subject to a deluge of data, and as such, general concepts such as this would be extremely valuable. One interesting candidate is the "criticality" principle, which claims that biological evolution favors those dynamical regimes that are intermediaries between ordered and disordered states (i.e., "at the edge of chaos"). The reasons why this should be the case and experimental evidence are briefly discussed, observing that gene regulatory networks are indeed often found on, or close to, the critical boundaries. Therefore, assuming that criticality provides an edge, it is important to ascertain whether systems that are critical can further evolve while remaining critical. In order to explore the possibility of achieving such "always-critical" evolution, we resort to simulated evolution, by suitably modifying a genetic algorithm in such a way that the newly-generated individuals are constrained to be critical. It is then shown that these modified genetic algorithms can actually develop critical gene regulatory networks with two interesting (and quite different) features of biological significance, involving, in one case, the average gene activation values and, in the other case, the response to perturbations. These two cases suggest that it is often possible to evolve networks with interesting properties without losing the advantages of criticality. The evolved networks also show some interesting features which are discussed.

Keywords: Boolean models; criticality; edge of chaos; evolving systems; gene regulatory networks; genetic algorithms; random Boolean networks.

Figure A1

(a) Probability occurrence for each Boolean function, in the case of random and evolved RBN-the bars for each measure indicate the standard deviation of the distribution). Boolean functions are ordered by increasing bias. (b) The same of (a), by using the four canalization classes; F fixed functions; C1 canalizing function in one input channel and C2 in two input channels; R, the XOR and NOT XOR functions. (c) Average population bias, and average fitness. In random BN the number of active genes is on average equal to the bias of the Boolean functions: this property seems maintained by the BN evolved through the free GA. (d) Frequency for each Boolean function in the case of random (red line) and evolved RBN: the Boolean functions are categorized by using their bias. In (a) and (b) parts of the figure the variables deviating from independence at level of 0.02 (stronger evidence) are highlighted by red circles, and those deviating at 0.05 level (weaker evidence) by blue circles.

Figure A2

(a) Frequency of each Boolean function, in the case of random and evolved RBN-the bars for each measure indicate the standard deviation of the distribution. Boolean functions are ordered based on the presence of a zero or a one in correspondence of the input consisting of a double “one”. (b) The same of (a), by using the four canalization classes; F fixed functions; C1 canalizing function in one input channel and C2 in two input channels; R, the XOR and NOT XOR functions. (c) A table illustrating the strategy implemented by the balanced GA-the table shows only the trend of growth or decrease of the Boolean functions whose frequency has significantly changed. (d) Probability occurrence for each Boolean function in the case of random (red line) and evolved RBN: the Boolean functions are categorized by using their bias. In (a) and (b) parts of the figure the variables deviating from independence at level of 0.02 (stronger evidence) are highlighted by red circles, and those deviating at 0.05 level (weaker evidence) by blue circles.

Figure 1

Average or maximum values of some indices, as the 15 evolutionary processes of each GA version-free and balanced, respectively GAf and Gab-progress. (a) The maximum of the 15 fitness maxima, and the minimum of the 15 minima, for the two versions of the GA, in cases of populations made of RBNs with k = 2 and initial bias = 0.5, which corresponds, on average, to a critical dynamic regime. (b) The average Derrida coefficient of free GA and balanced GA (RBNs with k = 2); note that the balanced GA succeeds in maintaining the critical dynamic regime. Similar considerations can be made in the case of RBNs with k = 3 (data not shown).

Figure 2

Average or maximum values of some indices, as the 15 evolutionary processes of each GA version-free and balanced, respectively GAf and Gab-progress. (a) The maximum of the 15 fitness maxima, and the minimum of the 15 minima, for the two versions of the GA, in case of populations made of RBNs with k = 2 and initial bias = 0.5, which corresponds on average to a critical dynamic regime. (b) The average Derrida coefficient of free GA and balanced GA (RBNs with k = 2); note that the balanced GA succeeds in maintaining the critical dynamic regime. Similar considerations can be made in the case of RBNs with k = 3 (data not shown).

Figure 3

Average or maximum values of some indices, as the 10 evolutionary processes progress. (a) The maximum of the 10 fitness maxima, and the minimum of the 10 minima, for the two versions of the GA, in case of populations made of RBNs with k = 2 and initial bias = 0.5. (b) The same for RBNs with k = 3 and initial bias = 0.21. It can be noted that in the case of BNs with k = 3, the task is more difficult than with systems with k = 2, with the former systems having a higher connectivity, a fact that leads to many more feedback loops which must be be controlled. (c) The average Derrida coefficient of free GA and balanced GA (RBNs with k = 2). (d) The same for RBNs with k = 3 and initial bias = 0.21. (e) Average population bias of free GA and balanced GA (RBNs with k = 2). (f) The same for RBNs with k = 3 and initial bias = 0.21.