Selective advantage for conservative viruses

Theoretical Models of Generalized Quasispecies

Theoretical modeling of quasispecies has progressed in several directions. In this chapter, we review the works of Emmanuel Tannenbaum, who, together with Eugene Shakhnovich at Harvard University and later with colleagues and students at Ben-Gurion University in Beersheva, implemented one of the more useful approaches, by progressively setting up various formulations for the quasispecies model and solving them analytically. Our review will focus on these papers that have explored new models, assumed the relevant mathematical approximations, and proceeded to analytically solve for the steady-state solutions and run stochastic simulations . When applicable, these models were related to real-life problems and situations, including changing environments, presence of chemical mutagens, evolution of cancer and tumor cells , mutations in Escherichia coli, stem cells , chromosomal instability (CIN), propagation of antibiotic drug resistance , dynamics of bacteria with plasmids , DNA proofreading mechanisms, and more.

Detailed analysis of an Eigen quasispecies model in a periodically moving sharp-peak landscape

The Eigen quasispecies model in a periodically moving sharp-peak landscape considered in previous seminal works [M. Nilsson and N. Snoad, Phys. Rev. Lett. 84, 191 (2000)] and [C. Ronnewinkel, in, edited by L. Kallel, B. Naudts, and A. Rogers (Springer-Verlag, Heidelberg, 2001)] is analyzed in greater detail. We show here, through a more rigorous analysis, that results in those papers are qualitatively correct. In particular, we obtain a phase diagram for the existence of a quasispecies with the same shape as in the above cited paper by C. Ronnewinkel, with upper and lower thresholds for the mutation rate between which a quasispecies may survive. A difference is that the upper value is larger and the lower value is smaller than the previously reported ones, so that the range for quasispecies existence is always larger than thought before. The quantitative information provided might also be important in understanding genetic variability in virus populations and has possible applications in antiviral therapies. The results in the quoted papers were obtained by studying the populations only at some few genomes. As we will show, this amounts to diagonalizing a 3×3 matrix. Our work is based instead in a different division of the population allowing a finer control of the populations at various relevant genetic sequences. The existence of a quasispecies will be related to Perron-Frobenius eigenvalues. Although huge matrices of sizes 2(ℓ), where ℓ is the genome length, may seem necessary at a first look, we show that such large sizes are not necessary and easily obtain numerical and analytical results for their eigenvalues.

Optimal mutation rates in dynamic environments: The Eigen model

We consider the Eigen quasispecies model with a dynamic environment. For an environment with sharp-peak fitness in which the most-fit sequence moves by k spin-flips each period T we find an asymptotic stationary state in which the quasispecies population changes regularly according to the regular environmental change. From this stationary state we estimate the maximum and the minimum mutation rates for a quasispecies to survive under the changing environment and calculate the optimum mutation rate that maximizes the population growth. Interestingly we find that the optimum mutation rate in the Eigen model is lower than that in the Crow-Kimura model, and at their optimum mutation rates the corresponding mean fitness in the eigenmodel is lower than that in the Crow-Kimura model, suggesting that the mutation process which occurs in parallel to the replication process as in the Crow-Kimura model gives an adaptive advantage under changing environment.

Maximum, minimum, and optimal mutation rates in dynamic environments

We analyze the dynamics of the parallel mutation-selection quasispecies model with a changing environment. For an environment with the sharp-peak fitness function in which the most fit sequence changes by k spin flips every period T , we find analytical expressions for the minimum and maximum mutation rates for which a quasispecies can survive, valid in the limit of large sequence size. We find an asymptotic solution in which the quasispecies population changes periodically according to the periodic environmental change. In this state we compute the mutation rate that gives the optimal mean fitness over a period. We find that the optimal mutation rate per genome, k/T , is independent of genome size, a relationship which is observed across broad groups of real organisms.