Gene-mating dynamic evolution theory: fundamental assumptions, exactly solvable models and analytic solutions
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 tim...
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| Format: | Article |
| Language: | English |
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Springer Science and Business Media LLC
2020
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| Online Access: | https://hdl.handle.net/1721.1/128469 |
| _version_ | 1811086122346872832 |
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| author | Wang, Juven Chen, Jiunn-Wei |
| author2 | Massachusetts Institute of Technology. Department of Physics |
| author_facet | Massachusetts Institute of Technology. Department of Physics Wang, Juven Chen, Jiunn-Wei |
| author_sort | Wang, Juven |
| collection | MIT |
| description | 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. |
| first_indexed | 2024-09-23T13:21:12Z |
| format | Article |
| id | mit-1721.1/128469 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T13:21:12Z |
| publishDate | 2020 |
| publisher | Springer Science and Business Media LLC |
| record_format | dspace |
| spelling | mit-1721.1/1284692022-10-01T14:44:28Z Gene-mating dynamic evolution theory: fundamental assumptions, exactly solvable models and analytic solutions Wang, Juven Chen, Jiunn-Wei Massachusetts Institute of Technology. Department of Physics 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. NSF (Grants DMR-1005541, NSFC-11074140, NSFC-11274192, PHY-1606531, DMS-1607871) 2020-11-12T22:27:30Z 2020-11-12T22:27:30Z 2020-02 2019-07 2020-09-24T21:05:37Z Article http://purl.org/eprint/type/JournalArticle 1431-7613 1611-7530 https://hdl.handle.net/1721.1/128469 Wang, Juven C. and Jiunn-Wei Chen. "Gene-mating dynamic evolution theory: fundamental assumptions, exactly solvable models and analytic solutions." Theory in Biosciences 139, 2 (February 2020): 105–134 © 2020 Springer-Verlag en https://doi.org/10.1007/s12064-020-00309-3 Theory in Biosciences Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. Springer-Verlag GmbH Germany, part of Springer Nature application/pdf Springer Science and Business Media LLC Springer Berlin Heidelberg |
| spellingShingle | Wang, Juven Chen, Jiunn-Wei Gene-mating dynamic evolution theory: fundamental assumptions, exactly solvable models and analytic solutions |
| title | Gene-mating dynamic evolution theory: fundamental assumptions, exactly solvable models and analytic solutions |
| title_full | Gene-mating dynamic evolution theory: fundamental assumptions, exactly solvable models and analytic solutions |
| title_fullStr | Gene-mating dynamic evolution theory: fundamental assumptions, exactly solvable models and analytic solutions |
| title_full_unstemmed | Gene-mating dynamic evolution theory: fundamental assumptions, exactly solvable models and analytic solutions |
| title_short | Gene-mating dynamic evolution theory: fundamental assumptions, exactly solvable models and analytic solutions |
| title_sort | gene mating dynamic evolution theory fundamental assumptions exactly solvable models and analytic solutions |
| url | https://hdl.handle.net/1721.1/128469 |
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