Calculation of the Relative Density and the Crossing Time Through the Fitness Barrier in an Asymmetric Multiplicative Landscape

Calculation of the Relative Density and the Crossing Time Through the Fitness Barrier in an Asymmetric Multiplicative Landscape

We have calculated the relative density and the crossing time through the fitness barrier by switching on an asymmetric multiplicative landscape, from the initial state which is the quasi-species in a multiplicative landscape. The relative densities X*d with the Hamming distance d > L/2 at the stationary state in an asymmetric multiplicative landscape are shown to increase as the asymmetric parameter decreases, and the relative densities Xd* become saturated and symmetric with respect to d = L/2 when the asymmetric parameter is smaller than the saturation asymmetric parameter. The crossing time is found to diverge at the critical fitness parameter in the asymmetric multiplicative landscape, in contrast with the symmetric multiplicative landscape where the crossing time scales as a power law in the fitness parameter. The critical fitness parameter is also found to decrease and approach the mutation rate as the asymmetric parameter and the sequence length increase.

We have calculated the relative density and the crossing time through the fitness barrier by switching on an asymmetric multiplicative landscape, from the initial state which is the quasi-species in a multiplicative landscape. The relative densities X*d with the Hamming distance d > L/2 at the stationary state in an asymmetric multiplicative landscape are shown to increase as the asymmetric parameter decreases, and the relative densities Xd* become saturated and symmetric with respect to d = L/2 when the asymmetric parameter is smaller than the saturation asymmetric parameter. The crossing time is found to diverge at the critical fitness parameter in the asymmetric multiplicative landscape, in contrast with the symmetric multiplicative landscape where the crossing time scales as a power law in the fitness parameter. The critical fitness parameter is also found to decrease and approach the mutation rate as the asymmetric parameter and the sequence length increase.