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Mathematical modeling of evolution. Solved and open problems

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Abstract

Evolution is a highly complex multilevel process and mathematical modeling of evolutionary phenomenon requires proper abstraction and radical reduction to essential features. Examples are natural selection, Mendel’s laws of inheritance, optimization by mutation and selection, and neutral evolution. An attempt is made to describe the roots of evolutionary theory in mathematical terms. Evolution can be studied in vitro outside cells with polynucleotide molecules. Replication and mutation are visualized as chemical reactions that can be resolved, analyzed, and modeled at the molecular level, and straightforward extension eventually results in a theory of evolution based upon biochemical kinetics. Error propagation in replication commonly results in an error threshold that provides an upper bound for mutation rates. Appearance and sharpness of the error threshold depend on the fitness landscape, being the distribution of fitness values in genotype or sequence space. In molecular terms, fitness landscapes are the results of two consecutive mappings from sequences into structures and from structures into the (nonnegative) real numbers. Some properties of genotype–phenotype maps are illustrated well by means of sequence–structure relations of RNA molecules. Neutrality in the sense that many RNA sequences form the same (coarse grained) structure is one of these properties, and characteristic for such mappings. Evolution cannot be fully understood without considering fluctuations—each mutant originates form a single copy, after all. The existence of neutral sets of genotypes called neutral networks, in particular, necessitates stochastic modeling, which is introduced here by simulation of molecular evolution in a kind of flowreactor.

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Notes

  1. There is also a third stationary state defined by ϕ = 0. For strictly positive fitness values, f i  > 0 ∀ i = 1, 2, …, n, this condition can only be fulfilled by x i  = 0 ∀ i = 1, 2, …, n, which is identical to state (i). If some f i values are zero—corresponding to lethal variants—the respective variables vanish in the infinite time limit because of dx i /dt = −ϕ(t) x i with ϕ(t) > 0.

  2. All chromosomes are autosomes except the sexual chromosomes X and Y.

  3. In case matrix A is not symmetric, the dynamical system (10) may show more complex dynamics like oscillations, deterministic chaos, etc.

  4. Protein synthesis in vivo is regulated by a complex network controlling gene activity called gene expression. The network involves regulation of transcription (DNA → RNA), post-transcriptional modification and maturation of the messenger-RNA, its translation into protein, and post-translational modification before the protein unfolds its function.

  5. Qβ-replicase is an enzyme consisting of four subunits. Three subunits are host proteins involved in translation, the ribosomal protein S1 and the elongation factors Ef-Tu and Ef-Ts. The fourth subunit is a virus-specific protein encoded by the viral RNA.

  6. Standard amplification of single stranded DNA by means of the polymerase chain reaction (PCR) is a frequently used technique for replication that circumvents isothermal duplex dissociation by means of a temperature program: Single stranded DNA is completed to a double helical duplex by means of a polymerase from Thermophilus aquaticus (Taq), the duplex is dissociated into single stands at higher temperature, and cooling of single strands completes the cycle (see also Cahill et al. 1991).

  7. A square non-negative matrix T = {t ij ; i, j = 1, …, n; t ij  ≥ 0} is called primitive if there exists a positive integer m such that T m is strictly positive: T m > 0 which implies T m = {t (m) ij ; i, j = 1, …, n; t (m) ij > 0}.

  8. The Hamming distance d H ij between two strings, X i and X j of equal length counts the number of positions in which the two end-to-end aligned strings differ (Hamming 1986).

  9. It can be proven by means of a recursion that the eigenvalues of the matrix \(\tilde{{W}}\) fulfill the relation \(\lambda^{n-1}\left(\lambda-\kappa^{-\ell} \sum\nolimits_{i=1}^nf_i\right)=0.\)

  10. A sharp transition from the structured quasi-species to the uniform distribution is found for the single-peak landscape and some related landscapes only (see “Fitness landscapes and error thresholds” section.

  11. The data obtained from biomolecules suggest a high degree of ruggedness for the landscapes derived for structures and functions: nearby sequences may lead to identical or very different structures. By the same token functions like fitness values may be the same or very different for close by lying genotypes. Ruggedness is an intrinsic property of mapping from biopolymer sequences into structures or functions.

  12. Compatibility means that the sequence can form the structure but not necessarily as the minimum free energy structure.

  13. It is important to stress two facts about relay series: (i) The same shape may appear two or more times in a given relay series series. Then, it was extinct between two consecutive appearances. (ii) A relay series is not a genealogy which is the full recording of parent-offspring relations a time-ordered series of genotypes.

  14. Identical means here that everything was kept unchanged in the computer experiments except the seeds for the random number generator.

  15. Efficiency of evolutionary optimization is measured by average and best fitness values obtained in populations after a predefined number of generations.

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  1. Institut für Theoretische Chemmie, Universität Wien, Währingerstraße 17, 1090, Wien, Austria

    Peter Schuster

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Schuster, P. Mathematical modeling of evolution. Solved and open problems. Theory Biosci. 130, 71–89 (2011). https://doi.org/10.1007/s12064-010-0110-z

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