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.

Development of modularity in the neural activity of childrenʼs brains

“…8 as the fitness for modularity, f ( m ), in the brain and use quasispecies theory [46] to predict how modularity develops with age. The average modularity, M ( t ) = 〈 m ( t )〉, changes with age according to…”

“…$$\frac{dM}{dt}=Nfalse\langle mffalse[pfalse(tfalse),mfalse]false\rangle -NMfalse\langle ffalse\rangle -\mu M$$ where μ is the rate of mutations in the connection matrix [46]. Following the bottom arrow of Fig.…”

“…9b, is a qualitative prediction for the developing human brain. Due to mutation [46], the observed modularity in Fig. 9b is below the value that maximizes the fitness, M * ( t ), defined by f [ p ( t ) ,M * ( t )] ≥ f [ p ( t ), M ] ∀ M .…”

“…9b is below the value that maximizes the fitness, M * ( t ), defined by f [ p ( t ) ,M * ( t )] ≥ f [ p ( t ), M ] ∀ M . We calculate the modularity in the adiabatic limit, M ∞ ( t ), from the steady-state average fitness derived from a solution of the quasispecies theory [46]: $$\begin{array}{c}left2pt{f}_{\text{pop}}=\underset{\xi }{max}\{f[p(t),\xi ]-\mu C[(N-L)L/N^2][2+(N/L-2)\xi -2\sqrt{(1-\xi )(1+(N/L-1)\xi )}]\}\\ =f[p(t),M^{\infty }(t)]\end{array}$$ where C is the average number of connections in the connection matrix, per row.…”

“…Here, the fitness is 10× the overlap in Fig. 8, and the rate of mutation is μ = 0.1 [46]. Also shown is the adiabatic approximation to the modularity, M ∞ (dashed) as well as the modularity that maximizes the fitness, M * (dotted).…”