Spontaneous evolution of modularity and network motifs

Abstract

Biological networks have an inherent simplicity: they are modular with a design that can be separated into units that perform almost independently. Furthermore, they show reuse of recurring patterns termed network motifs. Little is known about the evolutionary origin of these properties. Current models of biological evolution typically produce networks that are highly nonmodular and lack understandable motifs. Here, we suggest a possible explanation for the origin of modularity and network motifs in biology. We use standard evolutionary algorithms to evolve networks. A key feature in this study is evolution under an environment (evolutionary goal) that changes in a modular fashion. That is, we repeatedly switch between several goals, each made of a different combination of subgoals. We find that such "modularly varying goals" lead to the spontaneous evolution of modular network structure and network motifs. The resulting networks rapidly evolve to satisfy each of the different goals. Such switching between related goals may represent biological evolution in a changing environment that requires different combinations of a set of basic biological functions. The present study may shed light on the evolutionary forces that promote structural simplicity in biological networks and offers ways to improve the evolutionary design of engineered systems.

Fig. 1.

Evolution of electronic circuits toward fixed and modularly varying goals. (a) Fitness as a function of generations under a fixed-goal G1 as defined in the text. Red indicates the best circuit in the population; gray indicates mean fitness. (b) Fitness as a function of generations under modularly varying goals evolution toward goals G1 and G2. The goal was switched every 20 generations. Fitness is shown in 20 generations resolution, just before every goal switch event. (Inset) Zooms into fitness around two switching events. The fitness drops and then recovers after each switch of the goal. (c) Significance of all three-node subgraphs compared with randomized networks. Mean Z-score ± SE is shown for ≈100 networks. Red circles indicate networks evolved under modularly varying goals (MVG); black squares indicate networks evolved under fixed-goal evolution to G1 and G2. The absolute significance of all subgraphs in corresponding random networks for both types of networks is <0.2. (d) Significance of four-node subgraphs. The four-node subgraphs displayed were selected as follows: for each type of network the 10 subgraphs with highest absolute Z-scores were selected. The 12 subgraphs shown are the union of the selected subgraph sets for the different networks.

Fig. 2.

Electronic circuits evolved under a fixed goal and modularly varying goals. (a) A typical circuit evolved by fixed-goal evolution toward goal G1. Each gate in the circuit represents a NAND gate. Similar nonmodular solutions were found for goal G2. (b) Circuits evolved with modularly varying goals evolution. Connections that are rewired when the goal is switched are marked in red. (c) Another example of a circuit evolved with modularly varying goals evolution. The circuit shows feedbacks between two of the three lower NAND gates. (d) Modular structure of the circuits evolved under modularly varying goals. The circuits are composed of two XOR modules that input into a third module that implements an AND/OR function, depending on the goal. Each XOR module is composed of six nodes (two input nodes and four internal gates) and display two feed-forward loops and one diamond network motif.

Fig. 3.

An initially modular circuit rapidly loses modularity under fixed-goal conditions. Each experiment started from a population of identical modular circuits that had perfect fitness for goal G2 = (X XOR Y) OR (W XOR Z). At generation zero, the population was placed under a fixed-goal evolution with the same goal G2, with a selection pressure for small circuit size (a fitness penalty of 0.2 for every additional gate above the 10th gate). Mean modularity measure (±SE) vs. generations of best-fitness circuits is shown. Statistics are for 20 independent experiments, with four different initial modular circuits that satisfy G2.

Fig. 4.

An electronic circuit with six inputs and three outputs evolved under modularly varying goals, with the following four goals: G1 = (A OR B; A AND C; B AND C); G2 = (A AND B; A OR C, B AND C); G3 = (A AND B; A AND C; B OR C); and G4 = (A AND B; A AND C; B AND C), where A = X XOR Y, B = Z XOR W, C = Q XOR R. The goal was changed every 20 generations in the order G1, G4, G2, G4, G3, G4, G1, G4, etc. Perfect solutions that can rapidly switch between the goals evolved within 1.2 × 10⁵ ± 8 × 10⁴ generations. a, b, c, and d correspond to the networks found sequentially under goals G1, G4, G2, and G4, respectively. The lines and gates in red represent changes upon adaptation to each new goal.

Fig. 5.

Neural networks evolved for a pattern recognition task using fixed goals and modularly varying goals. (a) The aim is to recognize objects in the left and right sides of a 4-pixel-by-2-pixel retina. A left (or right) object exists if the four left (or right) pixels match one of the patterns of a predefined set. A left object is defined by three or more black pixels or one or two black pixels in the left column only. A right object is defined in a similar way, with one or two black pixels in the right column only. The goal is to identify when objects exist at both sides of the retina (L AND R). (b) A network evolved under fixed-goal evolution. Black/red lines represent positive/negative weights. Thick lines are double weights. (c) A network evolved under modularly varying goals evolution with two goals: L AND R, and L OR R. The network can rapidly adapt each time the goal is switched, by changing a single threshold of the lowest neuron (from t = 2 at the first goal to t = 1 at the second goal). In b and c, n indicates the number of neurons (nodes) in the network. (d) Four node patterns and their significance in the evolved networks. Shown are Z-scores ± SE for ≈50 networks. Strong motifs in the modularly varying goal networks are the bifan and diamond (patterns 3 and 5). Subgraphs were selected as in Fig. 1d.(e) Modular structure of the neuronal network evolved under modularly varying goals. Two distinct modules each monitor one side of the retina, and a third module processes their outputs.