Spontaneous Emergence of Modularity in a Model of Evolving Individuals

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.

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 $T_2=20$, and the severity of environmental change is $p=0.40$. The upper left inset shows the growth of modularity starting from a power-law distributed contact matrix ($\gamma =3$). The upper middle inset shows the improvement in the energy as the modularity grows. The upper right inset shows the improvement of evolvability, or change in energy in one environment, as the modularity grows. The inset in the middle row shows the 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. The lower left inset shows how the energy changes within one environment and between environmental changes ($T_2=20$). The lower right inset shows that emergence of modularity requires both environmental change and horizontal gene transfer. In all cases modularity is measured by Eq. (2), and excess modularity is measured by $\delta M^{\alpha }=M^{\alpha }-M_0$, with $M_0=22$.

Figure 2

The velocity at which modularity grows is positively correlated with the magnitude of environment change, $p$. The frequency of environment change is set at $1/T_2=1/40$. The inset shows the response function of the system $dM/d(t/T_3)$ as a function of the severity of environmental change.

Figure 3

Frequency of environmental change also affects the time evolution of spontaneous modularity. The severity of environment change is $p=0.40$. The upper inset shows the growth of modularity versus real time rather than versus time relative to the number of environmental changes. The lower inset shows the response function of the system $dM/d(t/{10}^5)$ as a function of the frequency of environmental change.

Figure 4

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 $p=0.40$.