Abstract
Fundamental properties of macroscopic gene-mating dynamic evolutionary systems are investigated. A model is studied to describe a large class of systems within population genetics. We focus on a single locus, any number of alleles in a two-gender dioecious population. Our governing equations are time-dependent continuous differential equations labeled by a set of parameters, where each parameter stands for a population percentage carrying certain common genotypes. The full parameter space consists of all allowed parameters of these genotype frequencies. Our equations are uniquely derived from four fundamental assumptions within any population: (1) a closed system; (2) average-and-random mating process (mean-field behavior); (3) Mendelian inheritance; and (4) exponential growth and exponential death. Even though our equations are nonlinear with time-evolutionary dynamics, we have obtained an exact analytic time-dependent solution and an exactly solvable model. Our findings are summarized from phenomenological and mathematical viewpoints. From the phenomenological viewpoint, any initial parameter of genotype frequencies of a closed system will eventually approach a stable fixed point. Under time evolution, we show (1) the monotonic behavior of genotype frequencies, (2) any genotype or allele that appears in the population will never become extinct, (3) the Hardy–Weinberg law and (4) the global stability without chaos in the parameter space. To demonstrate the experimental evidence for our theory, as an example, we show a mapping from the data of blood type genotype frequencies of world ethnic groups to our stable fixed-point solutions. From the mathematical viewpoint, our highly symmetric governing equations result in continuous global stable equilibrium solutions: these solutions altogether consist of a continuous curved manifold as a subspace of the whole parameter space of genotype frequencies. This fixed-point manifold is a global stable attractor known as the Hardy–Weinberg manifold, attracting any initial point in any Euclidean fiber bounded within the genotype frequency space to the fixed point where this fiber is attached. The stable base manifold and its attached fibers form a fiber bundle, which fills in the whole genotype frequency space completely. We can define the genetic distance of two populations as their geodesic distance on the equilibrium manifold. In addition, the modification of our theory under the process of natural selection and mutation is addressed.
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Notes
This model presented here is originated from an independent thought of the first author during his undergrad freshman year. The governing equations and model are derived in 2003. Substantial work is completed, and exact solutions are found in 2006. The manuscript presented here is a late update of our 2006’s work aiming to contribute to the academic literature.
For biologists or physicists who are familiar with the renormalization group procedure, the linear stability analysis here is simply finding the relevancy of small perturbations. For any eigenvalue equal to 0, the perturbation is marginal. For any eigenvalue smaller than 0, the perturbation is irrelevant, namely stable against perturbations. For any eigenvalue larger than 0, the perturbation is relevant, namely unstable against perturbations. See, for instance, Mehran (2007).
The fiber is bounded by the whole simplex of genotype frequency space, which has all coordinates of genotype frequency bounded from 0 to 1. Also, the sum of all genotype frequencies is 1. The Euclidean fiber is meant to emphasize that the fiber is straight as a Euclidean submanifold with Cartesian coordinates, instead of a curved Riemannian curved fiber/submanifold. When we refer to Euclidean fiber, we always mean the fiber bounded within the constrained genotype frequency space (within the constrained simplex).
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Acknowledgements
JW would like to thank Mehran Kardar and Leonid Mirny for showing great interests and encouragements and giving comments. We thank Jeremy England, Hsien-Ching Kao and Matthew Pinson for comments on the manuscript. JW wishes to thank Yih-Yuh Chen and Ning-Ning Pang for comments in 2006 and thank Sze-Bi Hsu for introducing the reference Waltman (1983) in 2007 and mentioning the alternative approach: Hirsch’s monotone flow, for proving the global stability (our independent proof is in “General gene-mating evolution model and theory” section (1)). JW thanks Mehran Kardar, Patrick Lee and Xiao-Gang Wen for encouraging posting the paper. JW is supported by NSF Grant No. DMR-1005541, NSFC 11074140, NSFC 11274192, the BMO Financial Group and the John Templeton Foundation. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research. JW acknowledges the NSF Grant PHY-1606531. This work is also supported by NSF Grant DMS-1607871 “Analysis, Geometry and Mathematical Physics” and Center for Mathematical Sciences and Applications at Harvard University. JWC is supported in part by the MOST, NTU-CTS and the NTU-CASTS of R.O.C.
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Wang, J.C., Chen, JW. Gene-mating dynamic evolution theory: fundamental assumptions, exactly solvable models and analytic solutions. Theory Biosci. 139, 105–134 (2020). https://doi.org/10.1007/s12064-020-00309-3
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DOI: https://doi.org/10.1007/s12064-020-00309-3