Ising quantum chain is equivalent to a model of biological evolution

TLDR: A problem of biochemical physics that may be mapped exactly onto a quantum chain is presented that can be solved without approximation and describes mutation and selection as going on in parallel.

Abstract: One-dimensional systems, and quantum chains in particular, have long been important tools to understand, at least approximately, various physical situations, and there is even a recipe “how to reduce practically any problem to one dimension” [1]. As a complement, we present a problem of biochemical physics that may be mapped exactly onto a quantum chain. Selected examples can then be solved without approximation. In the theory of (molecular) biological evolution, various sequence space models are well established, the best known being Kauffman’s adaptive walk [2] and Eigen’s quasispecies model [3]. Whereas the former describes a hill-climbing process of a genetically homogeneous population in tunably rugged fitness landscapes, the latter includes the genetic structure of the population due to the balance between mutation and selection. For equal fitness landscapes, the quasispecies model is thus more difficult to treat than the corresponding adaptive walk. Some progress was made in [4] through the identification of the quasispecies model with a specific, anisotropic 2D Ising model: The mutation-selection matrix is equivalent to the row transfer matrix, with the mutation probability as a temperaturelike parameter, and error thresholds corresponding to phase transitions. This equivalence was exploited to treat simple fitness landscapes as well as spin-glass Hamiltonians with methods from statistical mechanics [5–7]. Of these results, most are approximate or numerical, and the few exact ones in [5] are of limited value as the order parameter was not calculated correctly. The quasispecies model assumes mutations to originate as replication errors on the occasion of reproduction events. An alternative was introduced in [8] and describes mutation and selection as going on in parallel; we would like to abbreviate it as para-muse (parallel mutation selection) model. In subsequent investigations [9,10], this model turned out to be both more powerful and structurally simpler than the quasispecies model. Which is the more appropriate one from the biological point of view amounts to the question