MAKING EVOLUTION RIGOROUS- THE ERROR THRESHOLD

ABSTRACT. In a series of influential papers Eigen and his coworkers introduced the quasispecies model as a first-order approximation to (Darwinian) evolution and applied it to self-reproducing molecules such as RNA or DNA in an attempt to explain the origin of genetic information which gives rise to life. It was argued by Eigen that an important property in such a model of evolution is the existence of an error threshold: A rate of error during the reproduction phase below which genetic information is intact and above which it disappears. Besides the insights the quasispecies model has provided on the emergence of life, perhaps its most powerful impact has been on the study of viruses where error threshold phenomena has been leveraged to design drug strategies that attempt to mutate the virus to death. While the existence of error thresholds for specific settings has been verified by computer simulations, and has been the basis for the design of mutagenic drugs, a mathematical proof of this phenomena has remained elusive. The trouble is that one can construct pathological examples for which no non-trivial error threshold can exist. In this paper we present a proof of existence of a sharp error threshold in the quasispecies model for a large set of biologically relevant evolutionary parameters. Our analysis benefits from viewing the quasispecies model as an evolutionary process on the hypercube which permits the use of simple yet powerful ideas from linear algebra and Fourier analysis. FIGURE 1. Simulation results for a single peak model on binary strings of length 20 where the curves shown correspond to the relative concentration of strings of the given Hamming weight at the steady state. The illustration implies a phase transition when the error rate is about 0.11. From [Eig02]