Dynamics of the Eigen and the Crow-Kimura models for molecular evolution

Abstract

We introduce an alternative way to study molecular evolution within well-established Hamilton-Jacobi formalism, showing that for a broad class of fitness landscapes it is possible to derive dynamics analytically within the 1N accuracy, where N is the genome length. For a smooth and monotonic fitness function this approach gives two dynamical phases: smooth dynamics and discontinuous dynamics. The latter phase arises naturally with no explicite singular fitness function, counterintuitively. The Hamilton-Jacobi method yields straightforward analytical results for the models that utilize fitness as a function of Hamming distance from a reference genome sequence. We also show the way in which this method gives dynamical phase structure for multipeak fitness.

Evolutionary dynamics on strongly correlated fitness landscapes

“…For D > L/2, the probability that w L−D,0 (D) is a record is given by (22) and w 1,0 (D) is a record by (26). Thus the probability of a record occurring for odd D > L/2 can be expressed as…”

“…We are interested in finding how the new global maximum is reached starting with an initial condition in which all the population is at the sequence that was globally fittest before the environmental change. The exact evolutionary dynamics of average Hamming distance and overlap function has been studied on permutationally invariant [22] and uncorrelated [23] fitness landscapes. Here we will be tracking the evolution of the most populated sequence in time on strongly correlated fitness landscapes.…”

Finite population size effects in quasispecies models with single-peak fitness landscape

“…1 for J = 1.5, 2, 3, 4. One could follow the method used in [18] to solve Eq. (8) and get time evolution of φ(t, x).…”

“…In the infinite population limit the evolution equations are deterministic, and, for molecular evolution models [3]- [8], there are many exact results [8]- [19]. It is possible even to find exact solutions for the steady state and dynamics [15,16,18]. In biology, the populations are often relatively small.…”

Dynamics of the chemical master equation, a strip of chains of equations ind-dimensional space

“…The problem was solved exactly in the infinite system size (the maximal number of particles) limit for D = 1 using quantum mechanics or mapping to the Hamilton-Jacobi equation (HJE) [5], [6], [7], [8], [9]. In [10] an implicit expression was obtained for the dynamics of the population distribution variance, following the HJE method of [11], [12], [13]. To obtain an exact complete solution of CME in dimensions D ≥ 2 is a very hard problem and only a few solutions are known, see [14].…”