Ising Quantum Chain is Equivalent to a Model of Biological Evolution

E. Baake, M. Baake, H. Wagner
1997 Physical Review Letters  
A sequence space model which describes the interplay of mutation and selection in molecular evolution is shown to be equivalent to an Ising quantum chain. Three explicit examples with representative fitness landscapes are discussed and exactly solved with methods from statistical mechanics. [S0031-9007(96) PACS numbers: 87.10. + e, 05.50. + q, 64.60.Cn, One-dimensional systems, and quantum chains in particular, have long been important tools to understand, at least approximately, various
more » ... l 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] [6] [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 whether rates of molecular evolution are closer to constant per generation or constant in time-a long-standing, but still unresolved issue [11, 12] . Even in the former situation, however, the parallel version is an excellent approximation. Note that both models are sequence space versions of the muse equations of classical population genetics [13] . In this Letter, we will show that, in the same way as the quasispecies model is equivalent to the row transfer matrix of a 2D Ising model, the para-muse model corresponds to the Hamiltonian of an Ising quantum chain. Methods of statistical mechanics will then be employed to treat three sample fitness landscapes exactly, with emphasis on the correct order parameter. More biological implications will be dealt with elsewhere [14] . Evolution and quantum chain.-In the framework of sequence space models, genetic information (like nucleic acid strings) is identified with points in (binary) sequence space, e.g., ͕21, 1͖ N , where N [ ‫ގ‬ is the (fixed) length of the string considered; so there are n 2 N different sequences (or alleles in the language of genetics) termed A i , i 1, . . . , n. The composition of an (infinite) population of haploid organisms (they carry one copy of every gene only) under the influence of mutation and selection acting independently of each other is described by the following ODE system (for review, see [15] ): Here, x i denotes the relative frequency of A i individuals ͑1 # i # n 2 N ͒, the r i their (Malthusian) fitness (i.e., the difference between reproduction and death rates), and m ij the rate at which A j mutates to A i . When every digit mutates independently at rate m $ 0, we have where d ij : d͑A i , A j ͒ is the Hamming distance between A i and A j (i.e., the number of positions where the two 0031-9007͞97͞78(3)͞559(4)$10.00
doi:10.1103/physrevlett.78.559 fatcat:r2wd5ql2ofabxobelgrimw5clu