Calculation Of The Relative Density And The Crossing Time Through The Fitness Barrier In An Asymmetric Sharply-Peaked Landscape

We have calculated the relative density and crossing time through the fitness barrier by switching on an asymmetric sharply-peaked landscape, from the initial state which is the quasispecies in a sharply-peaked landscape. It is found that the increment of the relative density with the reversal sequence is a linearly increasing function of time unless a new stationary state in an asymmetric sharply-peaked landscape is reached. It is also found that the relative density with the reversal sequence at the new stationary state$X_L^*$is in inverse proportion to the asymmetric parameter when the asymmetric parameter is greater than the saturation asymmetric parameter. We have derived the approximate formulae for the relaxation time, the saturation asymmetric parameter, and the relative density with the reversal sequence$X_L^*$, which nicely fit computer simulation results. It is found that the crossing time diverges at the critical fitness parameter in the asymmetric sharply-peaked landscape, in contrast with the symmetric sharply-peaked landscape where the crossing time scales as a power law in the fitness parameter. It is also found that the critical fitness parameter decreases as the asymmetric parameter and sequence length increase.