Spontaneous emergence of modularity in a model of evolving individuals and in real networks

We investigate the selective forces that promote the emergence of modularity in nature. We demonstrate the spontaneous emergence of modularity in a population of individuals that evolve in a changing environment. We show that the level of modularity correlates with the rapidity and severity of environmental change. The modularity arises as a synergistic response to the noise in the environment in the presence of horizontal gene transfer. We suggest that the hierarchical structure observed in the natural world may be a broken symmetry state, which generically results from evolution in a changing environment. To support our results, we analyze experimental protein interaction data and show that protein interaction networks became increasingly modular as evolution proceeded over the last four billion years. We also discuss a method to determine the divergence time of a protein.

Figure 1

Spontaneous emergence of excess modularity $M>M_0=22$ from a state with no excess modularity $M=M_0$. The random, symmetric distribution of structural connections is spontaneously broken as the system evolves. Here $T_2=20$, $T_3={10}^4\times T_2$, and the severity of environmental change is $p=0.40$.

Figure 2

Improvement in the energy as time increases and as the modularity grows, as shown by Fig. 1. Here the severity of environmental change is $p=0.4$ and the period of change is $T_2=20$. Here, and in all figures, $T_3={10}^4\times T_2$.

Figure 3

Energy improvement as evolution proceeds within each environment and large energy disruption due to environmental changes. Here the severity of environmental change is $p=0.4$ and the period of change is $T_2=20$.

Figure 4

Improvement of evolvability or evolved improvement of the energy in one environment as the modularity grows. Here the severity of environmental change is $p=0.4$ and the period of change is $T_2=20$.

Figure 5

Spontaneous emergence of excess modularity $M>M_0=22$ from an initial scale-free network $(\gamma =3)$ with $M=M_0$. Here $T_2=20$, and the severity of environmental change is $p=0.40$.

Figure 6

Emergence of modularity as a result of a horizontal gene transfer operator with a Poisson random swap length and uniform random starting position. Shown are data for an average swap length of 10 (○), 20 (◻), 20 (◇), 5 (△), and 40 (▽) with 12, 6, 12, 24, and 3 attempted swaps, respectively, of probability 0.1 per sequence pair. Here $T_2=20$, and the severity of environmental change is $p=0.40$.

Figure 7

Emergence of modularity in this model requires both horizontal gene transfer and a changing environment. Here $T_2=20$. For the case of no horizontal gene transfer with a changing environment, the severity of environmental change is $p=0.40$. For the case of no environmental change with horizontal gene transfer, $p=0$, and the transfer is of fixed position $10k+1$ and fixed length 10 and attempted every $T_2$ steps.

Figure 8

The rate at which modularity grows $dM∕dt$ is positively correlated with the magnitude of environment change $p$. The frequency of environment change is set at $1∕T_2=1∕40$.

Figure 9

The response function of the system $dM∕d(t∕T_3)$ as a function of the severity of environmental change for the data of Fig. 8.

Figure 10

Frequency of environmental change also affects the time evolution of spontaneous modularity. Here $1∕T_2$ is the frequency of environmental change and the severity of the environmental change is $p=0.40$.

Figure 11

Frequency of environmental change affects the time evolution of spontaneous modularity shown in real time $t$. Here $1∕T_2$ is the frequency of the environmental change and the severity of the environmental change is $p=0.40$.

Figure 12

The response function of the system $dM∕d(t∕{10}^5)$ as a function of the frequency of environmental change $(1∕T_2)$ for the data from Fig. 10.

Figure 13

The spontaneous modularity saturates at a steady-state level. If the initial value of the modularity is greater than the steady-state value, the modularity decays with time. Here $T_2=20$, and the severity of environment change is $p=0.40$.

Figure 14

Distribution of conserved sequences with compositional age to find (a) the age of LUCA and (b) the divergence time of fungi.

Figure 15

S. cerevisiae. The average $dN∕dS$ is negatively linearly related to the compositional age.

Figure 16

The degree distribution of the S. cerevisiae domain-domain interaction network.

Figure 17

(Color online) The reordered topological overlap matrix of the E. coli protein interaction network constructed from proteins whose compositional age are larger than 12.8 (a), 12.6 (b), and 12.2 (c). The color reflects the strength of the topological overlap of two nodes (from 0.0 to 1.0), as shown in the color bar in (a). (d) The linear relationship between compositional age and real age. (e), (f) The banded modularity evolution of E. coli and S. cerevisiae, respectively. The lines of different color in (e) and (f) correspond to different band sizes $\left(W\right)$. Modularity grows with time. Banded modularity of a saturated matrix, i.e., a matrix with all elements being 1 except the diagonal ones being 0, is shown in (e) and (f) for comparison. The banded modularity of a saturated network is at its minimum value of 1.

Figure 18

(Color online) Evolution of block modularity of protein interaction network in E. coli (a) and S. cerevisiae (b).

Figure 19

(a) Average number of proteins in a module at different compositional ages, (b) size of network in different compositional age network.

Figure 20

(Color online) Evolution of banded modularity of the domain-domain interaction network in E. coli (a) and S. cerevisiae (b).

Figure 21

Domain interaction network modularity evolution in E. coli (a) and S. cerevisiae (b). The score is the inverse of modularity.

Figure 22

Different definitions of inverse modularity, for E. coli and S. cerevisiae.

Figure 23

(Color online) (a) A random network. (b) E. coli protein interaction network at compositional age 12.2.

Figure 24

Comparison of banded modularity with width 8 between the E. coli network and the random network. The red line is the banded modularity of random network. The black line is the E. coli network with size 36 at compositional age 12.6, size 335 at compositional age 12.2, and size 949 at compositional age 11.8. Error bars are shown in blue.