Quasispecies theory for evolution of modularity

Abstract

Biological systems are modular, and this modularity evolves over time and in different environments. A number of observations have been made of increased modularity in biological systems under increased environmental pressure. We here develop a quasispecies theory for the dynamics of modularity in populations of these systems. We show how the steady-state fitness in a randomly changing environment can be computed. We derive a fluctuation dissipation relation for the rate of change of modularity and use it to derive a relationship between rate of environmental changes and rate of growth of modularity. We also find a principle of least action for the evolved modularity at steady state. Finally, we compare our predictions to simulations of protein evolution and find them to be consistent.

Figures

FIG. 1

a) Shown is the fitness of a single evolving system with a given modularity as a function of time. Positive fitness means growth of the system. The environment is repeatedly changed each T = 300 time steps. Shown in b) is the average of these responses during a time 0 to T after each environmental change, averaged over many environmental changes. Shown in c) is the average response function to p = 1 environmental changes, 〈g〉(t). The response function in b) follows from a master response function curve in c), being the t*−T to t*subset where 〈g〉(t*−T) = (1−p)〈g〉(t*). Here p = 0.3 and T = 300. The present theory applies once the curve in c) has been determined.

FIG. 2

Shown is the fitness of an evolving system. a) The fitness of the non-modular (〈g0〉, solid) and block-diagonal (〈g1〉, dashed) system are shown, starting from a random initial configuration. These 〈g0〉 and 〈g1〉 are inputs to the theory. The modular system is taken to be more fit at short time and less fit at long time. b) The evolved, steady-state fitness of a system predicted by the theory in a changing environment (dot dashed), shown for varying T and p = 1. The fitness follows the high-modularity curve at rapid environmental changes, small T, and the low-modularity curve at slow environmental changes, large T. Since p = 1, the function fp=1,T (M) = 〈g(M)〉(t = T). The function 〈g(M)〉 is here taken for simplicity to be (1−M)〈g0〉(t)+M〈g1〉(t). Note the modularity tends to 1 and the fitness to 〈g1〉 for rapid environmental change (small T), and the modularity tends to 0 and the fitness to 〈g0〉 for slow environmental change (large T). The modularity calculated from theory, Eq. (13), is shown (dotted). Also shown is the theoretical result for small M, Eq. (15), to first order in l/L (short dashed). In this example L = 120, l = 10, μ = 0.01, and C = 5.77. For these particular 〈g0〉 and 〈g1〉, the modularity emerges only for environmental changes that occur on a timescale T < tc ≈ 285.

FIG. 3

The phase diagram for emergence of modularity. Below a critical mutation rate, modularity spontaneously emerges. Results are shown for f(M) = kM2/2 (solid), f(M) = kM3/2 (long-dashed), f(M) = kM4/2 (short-dashed), f(M) = kM10/2 (dotted), and f(M) = ekM − kM − 1 (dot-dashed). Results here are shown for l = 10,L = 120.

FIG. 4

Shown is modularity versus time for a population that exhibits spontaneous emergence of modularity. The curves are from theory, Eq. (1), and the data (circles) are from [27]. Two different initial conditions are shown, M(0) = 0 and M(0) = 0.38. In this example the derived underlying fitness function is f(M) = 1.4M − 1.31M2, the mutation rate is μ = 2/346, and the average number of connections is C = 346 × 2/L = 5.77.