We map the Eigen model of biological evolution [Naturwissenschaften 58, 465 (1971)] into a quantum spin model with non-Hermitian Hamiltonian. Based on such a connection, we derive exact relaxation periods for the Eigen model to approach static energy landscape from various initial conditions. We also study a simple case of dynamic fitness function.

We present an exact solution of Eigen's quasispecies model with a general degradation rate and fitness functions, including a square root decrease of fitness with increasing Hamming distance from the wild type. The found behavior of the model with a degradation rate is analogous to a viral quasispecies under attack by the immune system of the host. Our exact solutions also revise the known results of neutral networks in quasispecies theory. To explain the existence of mutants with large Hamming distances from the wild type, we propose three different modifications of the Eigen model: mutation landscape, multiple adjacent mutations, and frequency-dependent fitness in which the steady-state solution shows a multicenter behavior.quasispecies ͉ virus evolution ͉ error threshold M olecular models of biological evolution have attracted much attention in recent decades (1-15). Among them, Eigen's concept of quasispecies plays a fundamental role (1, 2). It describes the evolution of a population consisting of a wild type accompanied by a large number of mutant types in sequence space by a large system of ordinary differential equations. The Eigen model has been found to describe quite well the evolution of viral populations (3) and has deeply changed our view of the process of evolution: adaptation does not wait for better adapted mutants to arise but starts with the selection of the better adapted mutants and then explores by mutation the surrounding sequence space for even better mutants. When the mutation rate surpasses an error threshold, the population gets genetically unstable, and it could be shown that indeed virus populations can be driven to extinction when the error rate is artificially raised beyond the error threshold.To describe the population precisely, we should know the fitness value of each type and the mutation rates to go from one type to another. The experimental efforts to do so are immense. During the last three decades, the model has been investigated numerically as well as analytically for a simple fitness function. Although this sort of data reduction does allow a view on a large population, the fitness functions chosen are too simplistic to explain realistic cases such as a population of RNA virus. In this work, we solve the system of differential equations exactly, assuming uniform degradation rates and fitness functions including a square root decrease of fitness with increasing Hamming distance (HD) from the wild type. Our exact solutions also revise the known theoretical results of neutral networks in quasispecies theory (2). To explain biological systems more realistically (16), we propose three different modifications of the Eigen model: mutation landscape, multiple adjacent mutations, and frequency-dependent fitness in which the steady-state solution shows a multicenter behavior. ModelSeveral excellent reviews (2-5) emphasize the merits of the quasispecies model for the interpretation of virological studies. Let us give a brief description of the quasispecies model as we use it in this ...

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.