Characteristic time in quasispecies evolution

Abstract

a b s t r a c tThe time a phenotype takes to achieve a stationary state from an initial condition depends on multiple factors. In particular, it is a function of both its fitness and its mutation rate. We evaluate the average time, referred to as the characteristic time, T c , that the system takes to reach a final steady state of simple models of populations formed by self-replicative sequences. The dependence of T c on the mutation rate and on the fitness landscape is also studied. For simple fitness landscapes, e.g. single peak, the characteristic time can be analytically obtained as a function of the system parameters. In this case, T c for obtaining the quasispecies distribution presents a maximum at a Q-value that depends on the initial conditions and decreases monotonously as the mutation rate tends to zero. For most of the complex landscapes handled in this paper, the characteristic time to achieve the quasispecies distribution picked around the fittest phenotype attains a local minimum for a given mutation rate between 0 and the Q-value at which T c reaches its local maximum. Thus, in these cases, an optimum value for the mutation rate exists that corresponds to the lowest value of the characteristic time for quasispecies evolution.

The Advantage of Arriving First: Characteristic Times in Finite Size Populations of Error-Prone Replicators

“…As before, and . As it can be observed, the curves for large values of are qualitatively similar, all exhibiting a minimum value for approximately the same mutation rate and a relative maximum near the error catastrophe (an extended description of this behavior has been previously presented in [15]). Note that as the amplification factor decreases, the curves move to the left and to higher values of .…”

“…In addition, this fitness degeneracy is partly responsible for the strong discrepancy with the deterministic outcome. The effect of ruggedness on the characteristic time has already been studied in a previous paper [15]. In that paper we studied the characteristic time of a population of replicators in more rugged landscapes, namely multiplicative with two peaks, binary rugged and Kauffman-NK landscapes, and showed that it presents a similar dependence on the mutation rate.…”

“…(2) has eleven differential equations that can be solved numerically using, for instance, a Runge-Kutta method implemented in MATLAB. The characteristic time for the time evolution of the master copy , denoted as , is then computed as in [14], [15] (see also the section Methods). As an example, Fig.…”

“…Many different methods have been suggested to get an estimation of the scale of this intrinsic system dynamics. We recently presented in [15] the characteristic time of a continuos variable as a way of overcoming important deficiencies in previous definitions, particularly for non-linear systems, by taking into account the whole path from the initial condition to the final state of a given trajectory. As discussed in that paper, this characteristic time can be interpreted as: (i) specifically for linear system, as a weighted average of the inverse of the system eigenvalues, (ii) the hypothetical time at which the whole transition occurs and (iii) the mean time of the transition.…”

“…using the molar fraction of each phenotype) the characteristic time has been introduced beforehand [14]. Recently, this approximation has been used to quantify the dependence of the evolutionary time on the mutation rate for different fitness landscapes [15]. We showed that, as a consequence of the trade-off between the searching capabilities and the fixation probabilities of the master sequences, the characteristic time exhibits a minimum for a positive mutation rate lower than the error threshold.…”