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Berríos-Caro E, Galla T, Constable GWA. Switching environments, synchronous sex, and the evolution of mating types. Theor Popul Biol 2021; 138:28-42. [PMID: 33639174 DOI: 10.1016/j.tpb.2021.02.001] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/31/2020] [Revised: 02/11/2021] [Accepted: 02/12/2021] [Indexed: 01/31/2023]
Abstract
While facultative sex is common in sexually reproducing species, for reasons of tractability most mathematical models assume that such sex is asynchronous in the population. In this paper, we develop a model of switching environments to instead capture the effect of an entire population transitioning synchronously between sexual and asexual modes of reproduction. We use this model to investigate the evolution of the number of self-incompatible mating types in finite populations, which empirically can range from two to thousands. When environmental switching is fast, we recover the results of earlier studies that implicitly assumed populations were engaged in asynchronous sexual reproduction. However when the environment switches slowly, we see deviations from previous asynchronous theory, including a lower number of mating types at equilibrium and bimodality in the stationary distribution of mating types. We provide analytic approximations for both the fast and slow switching regimes, as well as a numerical scheme based on the Kolmogorov equations for the system to quickly evaluate the model dynamics at intermediate parameters. Our approach exploits properties of integer partitions in number theory. We also demonstrate how additional biological processes such as selective sweeps can be accounted for in this switching environment framework, showing that beneficial mutations can further erode mating type diversity in synchronous facultatively sexual populations.
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Affiliation(s)
- Ernesto Berríos-Caro
- Theoretical Physics, Department of Physics and Astronomy, School of Natural Sciences, Faculty of Science and Engineering, The University of Manchester, Manchester M13 9PL, United Kingdom.
| | - Tobias Galla
- Instituto de Física Interdisciplinar y Sistemas Complejos IFISC (CSIC-UIB), 07122 Palma de Mallorca, Spain; Theoretical Physics, Department of Physics and Astronomy, School of Natural Sciences, Faculty of Science and Engineering, The University of Manchester, Manchester M13 9PL, United Kingdom
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2
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Abraham JS, Somasundaram S, Maurya S, Gupta R, Makhija S, Toteja R. Characterization of Euplotes lynni nov. spec., E. indica nov. spec. and description of E. aediculatus and E. woodruffi (Ciliophora, Euplotidae) using an integrative approach. Eur J Protistol 2021; 79:125779. [PMID: 33706203 DOI: 10.1016/j.ejop.2021.125779] [Citation(s) in RCA: 7] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/14/2020] [Revised: 01/14/2021] [Accepted: 01/23/2021] [Indexed: 11/26/2022]
Abstract
Four species belonging to the genus Euplotes have been investigated, namely: E. lynni nov. spec., E. indica nov. spec., E. aediculatus, and E. woodruffi. All populations are from India and were investigated using morphological and molecular markers. The phylogenetic relationships were inferred from small subunit ribosomal rRNA gene (SSU rRNA), internal transcribed spacer (ITS) region, and mitochondrial cytochrome c oxidase subunit I (COI) gene. Predicted secondary structure models for two new species using the hypervariable region of the SSU rRNA gene and ITS2 region support the distinctness of both species. Morphological characters were subjected to principal component analysis (PCA) and genetic variations were studied in-depth to analyze the relatedness of the two new species with their congeners. An integrative approach combining morphological features, molecular analysis, and ecological characteristics was carried out to understand the phylogenetic position of the reported species within the different clades of the genus Euplotes.
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Affiliation(s)
- Jeeva Susan Abraham
- Ciliate Biology Laboratory, Acharya Narendra Dev College, University of Delhi, Govindpuri, Kalkaji, New Delhi 110019, India
| | - Sripoorna Somasundaram
- Ciliate Biology Laboratory, Acharya Narendra Dev College, University of Delhi, Govindpuri, Kalkaji, New Delhi 110019, India
| | - Swati Maurya
- Ciliate Biology Laboratory, Acharya Narendra Dev College, University of Delhi, Govindpuri, Kalkaji, New Delhi 110019, India
| | - Renu Gupta
- Maitreyi College, University of Delhi, Bapu Dham, Chanakyapuri, New Delhi 110021, India.
| | - Seema Makhija
- Ciliate Biology Laboratory, Acharya Narendra Dev College, University of Delhi, Govindpuri, Kalkaji, New Delhi 110019, India
| | - Ravi Toteja
- Ciliate Biology Laboratory, Acharya Narendra Dev College, University of Delhi, Govindpuri, Kalkaji, New Delhi 110019, India.
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3
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Wang G, Chen K, Zhang J, Deng S, Xiong J, He X, Fu Y, Miao W. Drivers of Mating Type Composition in Tetrahymena thermophila. Genome Biol Evol 2020; 12:2328-2343. [PMID: 32946549 PMCID: PMC7846192 DOI: 10.1093/gbe/evaa197] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Accepted: 09/15/2020] [Indexed: 11/23/2022] Open
Abstract
Sex offers advantages even in primarily asexual species. Some ciliates appear to utilize such reproductive strategy with many mating types. However, the factors determining the composition of mating types in the unicellular ciliate Tetrahymena thermophila are poorly understood, and this is further complicated by non-Mendelian determination of mating type in the offspring. We therefore developed a novel population genetics model to predict how various factors influence the dynamics of mating type composition, including natural selection. The model predicted either the coexistence of all seven mating types or fixation of a single mating type in a population, depending on parameter combinations, irrespective of natural selection. To understand what factor(s) may be more influential and to test the validity of theoretical prediction, five replicate populations were maintained in laboratory such that several factors could be controlled or measured. Whole-genome sequencing was used to identify newly arising mutations and determine mating type composition. Strikingly, all populations were found to be driven by strong selection on newly arising beneficial mutations to fixation of their carrying mating types, and the trajectories of speed to fixation agreed well with our theoretical predictions. This study illustrates the evolutionary strategies that T. thermophila can utilize to optimize population fitness.
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Affiliation(s)
- Guangying Wang
- Key Laboratory of Aquatic Biodiversity and Conservation, Institute of Hydrobiology, Chinese Academy of Sciences, Wuhan, Hubei, China.,University of Chinese Academy of Sciences, Beijing, China
| | - Kai Chen
- Key Laboratory of Aquatic Biodiversity and Conservation, Institute of Hydrobiology, Chinese Academy of Sciences, Wuhan, Hubei, China.,University of Chinese Academy of Sciences, Beijing, China
| | - Jing Zhang
- Key Laboratory of Aquatic Biodiversity and Conservation, Institute of Hydrobiology, Chinese Academy of Sciences, Wuhan, Hubei, China.,University of Chinese Academy of Sciences, Beijing, China
| | - Shanjun Deng
- State Key Laboratory of Biocontrol, School of Life Sciences, Sun Yat-Sen University, Guangzhou, China
| | - Jie Xiong
- Key Laboratory of Aquatic Biodiversity and Conservation, Institute of Hydrobiology, Chinese Academy of Sciences, Wuhan, Hubei, China
| | - Xionglei He
- State Key Laboratory of Biocontrol, School of Life Sciences, Sun Yat-Sen University, Guangzhou, China
| | - Yunxin Fu
- Laboratory for Conservation and Utilization of Bioresources, Yunnan University, Kunming, China.,Department of Biostatistics and Data Science and Human Genetics Center, School of Public Health, The University of Texas Health Science Center, Houston
| | - Wei Miao
- Key Laboratory of Aquatic Biodiversity and Conservation, Institute of Hydrobiology, Chinese Academy of Sciences, Wuhan, Hubei, China.,University of Chinese Academy of Sciences, Beijing, China.,CAS Center for Excellence in Animal Evolution and Genetics, Kunming, China
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Krumbeck Y, Constable GWA, Rogers T. Fitness differences suppress the number of mating types in evolving isogamous species. ROYAL SOCIETY OPEN SCIENCE 2020; 7:192126. [PMID: 32257356 PMCID: PMC7062084 DOI: 10.1098/rsos.192126] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 12/13/2019] [Accepted: 01/31/2020] [Indexed: 06/11/2023]
Abstract
Sexual reproduction is not always synonymous with the existence of two morphologically different sexes; isogamous species produce sex cells of equal size, typically falling into multiple distinct self-incompatible classes, termed mating types. A long-standing open question in evolutionary biology is: what governs the number of these mating types across species? Simple theoretical arguments imply an advantage to rare types, suggesting the number of types should grow consistently; however, empirical observations are very different. While some isogamous species exhibit thousands of mating types, such species are exceedingly rare, and most have fewer than 10. In this paper, we present a mathematical analysis to quantify the role of fitness variation-characterized by different mortality rates-in determining the number of mating types emerging in simple evolutionary models. We predict that the number of mating types decreases as the variance of mortality increases.
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Affiliation(s)
- Yvonne Krumbeck
- Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK
| | | | - Tim Rogers
- Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK
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Czuppon P, Constable GWA. Invasion and Extinction Dynamics of Mating Types Under Facultative Sexual Reproduction. Genetics 2019; 213:567-580. [PMID: 31391266 PMCID: PMC6781889 DOI: 10.1534/genetics.119.302306] [Citation(s) in RCA: 11] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/09/2019] [Accepted: 08/04/2019] [Indexed: 01/08/2023] Open
Abstract
In sexually reproducing isogamous species, syngamy between gametes is generally not indiscriminate, but rather restricted to occurring between complementary self-incompatible mating types. A longstanding question regards the evolutionary pressures that control the number of mating types observed in natural populations, which ranges from two to many thousands. Here, we describe a population genetic null model of this reproductive system, and derive expressions for the stationary probability distribution of the number of mating types, the establishment probability of a newly arising mating type, and the mean time to extinction of a resident type. Our results yield that the average rate of sexual reproduction in a population correlates positively with the expected number of mating types observed. We further show that the low number of mating types predicted in the rare-sex regime is primarily driven by low invasion probabilities of new mating type alleles, with established resident alleles being very stable over long evolutionary periods. Moreover, our model naturally exhibits varying selection strength dependent on the number of resident mating types. This results in higher extinction and lower invasion rates for an increasing number of residents.
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Affiliation(s)
- Peter Czuppon
- Center for Interdisciplinary Research in Biology, CNRS, Collège de France, PSL Research University, 75231 Paris, France
- Institute of Ecology and Environmental Sciences of Paris, Sorbonne Université, UPEC, CNRS, IRD, INRA, 75252 Paris, France
| | - George W A Constable
- Department of Mathematical Sciences, The University of Bath, BA2 7AY, United Kingdom
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Czuppon P, Rogers DW. Evolution of mating types in finite populations: The precarious advantage of being rare. J Evol Biol 2019; 32:1290-1299. [PMID: 31479547 DOI: 10.1111/jeb.13528] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/04/2019] [Revised: 06/28/2019] [Accepted: 08/19/2019] [Indexed: 01/03/2023]
Abstract
Sexually reproducing populations with self-incompatibility bear the cost of limiting potential mates to individuals of a different type. Rare mating types escape this cost since they are unlikely to encounter incompatible partners, leading to the deterministic prediction of continuous invasion by new mutants and an ever-increasing number of types. However, rare types are also at an increased risk of being lost by random drift. Calculating the number of mating types that a population can maintain requires consideration of both the deterministic advantages and the stochastic risks. By comparing the relative importance of selection and drift, we show that a population of size N can maintain a maximum of approximately N1/3 mating types for intermediate population sizes, whereas for large N, we derive a formal estimate. Although the number of mating types in a population is quite stable, the rare-type advantage promotes turnover of types. We derive explicit formulas for both the invasion and turnover probabilities in finite populations.
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Affiliation(s)
- Peter Czuppon
- Department of Evolutionary Theory, Max Planck Institute for Evolutionary Biology, Plön, Germany.,Centre Interdisciplinaire de Recherche en Biologie, CNRS Collège de France, PSL Research University, Paris, France.,Institut d'Ecologie et des Sciences de l'Environnement (IEES), UPEC, CNRS, IRD, INRA, Sorbonne Université, Paris, France
| | - David W Rogers
- Department of Microbial Population Biology, Max Planck Institute for Evolutionary Biology, Plön, Germany
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Hadjivasiliou Z, Pomiankowski A. Evolution of asymmetric gamete signaling and suppressed recombination at the mating type locus. eLife 2019; 8:48239. [PMID: 31464685 PMCID: PMC6715347 DOI: 10.7554/elife.48239] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/07/2019] [Accepted: 07/25/2019] [Indexed: 11/24/2022] Open
Abstract
The two partners required for sexual reproduction are rarely the same. This pattern extends to species which lack sexual dimorphism yet possess self-incompatible gametes determined at mating-type regions of suppressed recombination, likely precursors of sex chromosomes. Here we investigate the role of cellular signaling in the evolution of mating-types. We develop a model of ligand-receptor dynamics, and identify factors that determine the capacity of cells to send and receive signals. The model specifies conditions favoring the evolution of gametes producing ligand and receptor asymmetrically and shows how these are affected by recombination. When the recombination rate evolves, the conditions favoring asymmetric signaling also favor tight linkage of ligand and receptor loci in distinct linkage groups. These results suggest that selection for asymmetric gamete signaling could be the first step in the evolution of non-recombinant mating-type loci, paving the road for the evolution of anisogamy and sexes. Sexual reproduction, from birds to bees, relies on distinct classes of sex cells, known as gametes, fusing together. Most single cell organisms, rather than producing eggs and sperm, have similar sized gametes that fall into distinct ‘mating types’. However, only sex cells belonging to different mating types can fuse together and sexually reproduce. At first glance, it seems illogical that cells from the same mating type cannot reproduce with each other, as this restricts eligible partners within a population and makes finding a mate more difficult. Yet the typical pattern amongst single cell organisms is still two distinct classes of sex cells, just as in birds and bees. How did the obsession with mating between two different types become favored during evolution? One possibility is that cells with different mating types can recognize and communicate with each other more easily. Cells communicate by releasing proteins known as ligands, which bind to specific receptors found on the cell’s surface. Using mathematical modelling, Hadjivasiliou and Pomiankowski showed that natural selection typically favors ‘asymmetric’ signaling, whereby cells evolve to either produce receptor A with ligand B, or have the reverse pattern and produce receptor B with ligand A. These asymmetric mutants are favored because they avoid producing ligands that clog or activate the receptors on their own surface. As a result, different types of cells are better at recognizing each other and mating more quickly. When cells sexually reproduce they exchange genetic material with each other to produce offspring with a combination of genes that differ to their own. However, if the genes coding for ligand and receptor pairs were constantly being ‘swapped’, this could lead to new combinations, and a loss of asymmetric signaling. Hadjivasiliou and Pomiankowski showed that for asymmetric signaling to evolve, natural selection favors the genes encoding these non-compatible ligand and receptor pairs to be closely linked within the genome. This ensures that the mis-matching ligand and receptor are inherited together, preventing cells from producing pairs which can bind to themselves. This study provides an original way to address an evolutionary question which has long puzzled biologists. These findings raise further questions about how gametes evolved to become the sperm and egg, and how factors such as signaling between cells can determine the sex of more complex organisms, such as ourselves.
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Affiliation(s)
- Zena Hadjivasiliou
- Department of Biochemistry, University of Geneva, Geneva, Switzerland.,Centre for Mathematics and Physics in the Life Sciences and Experimental Biology (CoMPLEX), University College London, London, United Kingdom.,Department of Genetics, Evolution and Environment, University College London, London, United Kingdom
| | - Andrew Pomiankowski
- Centre for Mathematics and Physics in the Life Sciences and Experimental Biology (CoMPLEX), University College London, London, United Kingdom.,Department of Genetics, Evolution and Environment, University College London, London, United Kingdom
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The rate of facultative sex governs the number of expected mating types in isogamous species. Nat Ecol Evol 2018; 2:1168-1175. [DOI: 10.1038/s41559-018-0580-9] [Citation(s) in RCA: 21] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/06/2017] [Accepted: 05/16/2018] [Indexed: 01/30/2023]
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Repeated evolution of self-compatibility for reproductive assurance. Nat Commun 2018; 9:1639. [PMID: 29691402 PMCID: PMC5915400 DOI: 10.1038/s41467-018-04054-6] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/15/2017] [Accepted: 03/29/2018] [Indexed: 01/22/2023] Open
Abstract
Sexual reproduction in eukaryotes requires the fusion of two compatible gametes of opposite sexes or mating types. To meet the challenge of finding a mating partner with compatible gametes, evolutionary mechanisms such as hermaphroditism and self-fertilization have repeatedly evolved. Here, by combining the insights from comparative genomics, computer simulations and experimental evolution in fission yeast, we shed light on the conditions promoting separate mating types or self-compatibility by mating-type switching. Analogous to multiple independent transitions between switchers and non-switchers in natural populations mediated by structural genomic changes, novel switching genotypes readily evolved under selection in the experimental populations. Detailed fitness measurements accompanied by computer simulations show the benefits and costs of switching during sexual and asexual reproduction, governing the occurrence of both strategies in nature. Our findings illuminate the trade-off between the benefits of reproductive assurance and its fitness costs under benign conditions facilitating the evolution of self-compatibility. Mating-type switching enables self-compatible reproduction in fungi, but switching ability is variable even within species. Here, the authors find de novo evolution of switching genotypes in experimentally evolved fission yeast populations and show a trade-off between mating success and growth.
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Hadjivasiliou Z, Pomiankowski A. Gamete signalling underlies the evolution of mating types and their number. Philos Trans R Soc Lond B Biol Sci 2017; 371:rstb.2015.0531. [PMID: 27619695 PMCID: PMC5031616 DOI: 10.1098/rstb.2015.0531] [Citation(s) in RCA: 23] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Accepted: 06/11/2016] [Indexed: 01/02/2023] Open
Abstract
The gametes of unicellular eukaryotes are morphologically identical, but are nonetheless divided into distinct mating types. The number of mating types varies enormously and can reach several thousand, yet most species have only two. Why do morphologically identical gametes need to be differentiated into self-incompatible mating types, and why is two the most common number of mating types? In this work, we explore a neglected hypothesis that there is a need for asymmetric signalling interactions between mating partners. Our review shows that isogamous gametes always interact asymmetrically throughout sex and argue that this asymmetry is favoured because it enhances the efficiency of the mating process. We further develop a simple mathematical model that allows us to study the evolution of the number of mating types based on the strength of signalling interactions between gametes. Novel mating types have an advantage as they are compatible with all others and rarely meet their own type. But if existing mating types coevolve to have strong mutual interactions, this restricts the spread of novel types. Similarly, coevolution is likely to drive out less attractive mating types. These countervailing forces specify the number of mating types that are evolutionarily stable. This article is part of the themed issue ‘Weird sex: the underappreciated diversity of sexual reproduction’.
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Affiliation(s)
- Zena Hadjivasiliou
- Centre for Mathematics and Physics in the Life Sciences and Experimental Biology, University College London, Gower Street, London WC1E 6BT, UK Department of Genetics, Evolution and Environment, University College London, Gower Street, London WC1E 6BT, UK
| | - Andrew Pomiankowski
- Centre for Mathematics and Physics in the Life Sciences and Experimental Biology, University College London, Gower Street, London WC1E 6BT, UK Department of Genetics, Evolution and Environment, University College London, Gower Street, London WC1E 6BT, UK
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Wang Y, Lo WC, Chou CS. A modeling study of budding yeast colony formation and its relationship to budding pattern and aging. PLoS Comput Biol 2017; 13:e1005843. [PMID: 29121651 PMCID: PMC5697893 DOI: 10.1371/journal.pcbi.1005843] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/03/2017] [Revised: 11/21/2017] [Accepted: 10/20/2017] [Indexed: 11/23/2022] Open
Abstract
Budding yeast, which undergoes polarized growth during budding and mating, has been a useful model system to study cell polarization. Bud sites are selected differently in haploid and diploid yeast cells: haploid cells bud in an axial manner, while diploid cells bud in a bipolar manner. While previous studies have been focused on the molecular details of the bud site selection and polarity establishment, not much is known about how different budding patterns give rise to different functions at the population level. In this paper, we develop a two-dimensional agent-based model to study budding yeast colonies with cell-type specific biological processes, such as budding, mating, mating type switch, consumption of nutrients, and cell death. The model demonstrates that the axial budding pattern enhances mating probability at an early stage and the bipolar budding pattern improves colony development under nutrient limitation. Our results suggest that the frequency of mating type switch might control the trade-off between diploidization and inbreeding. The effect of cellular aging is also studied through our model. Based on the simulations, colonies initiated by an aged haploid cell show declined mating probability at an early stage and recover as the rejuvenated offsprings become the majority. Colonies initiated with aged diploid cells do not show disadvantage in colony expansion possibly due to the fact that young cells contribute the most to colony expansion. Budding yeast is a model organism in understanding fundamental aspects of eukaryotic cells, such as cell polarization and cell aging. Previously, extensive research has focused on the molecular mechanisms of biological processes in yeast, but many questions regarding yeast budding remain unsolved. For example, how do different budding patterns affect yeast colony growth? How does declined spatial order due to aging impact the colony at the population level? To address these questions, we developed a computational agent-based model, which incorporates key biological processes, the effect of aging, as well as cell-environment interaction. We performed and analyzed a large number of simulations for a variety of situations, and obtained insightful results. We found that axial budding pattern enhances the percentage of diploid cells at early stage and bipolar budding pattern improves colony development under nutrient limitation; the frequency of mating type switch might control the trade-off between diploidization and inbreeding; aging affects the percentage of diploid cells in colonies initiated by a single haploid cell, but does not have much influence in the expansion of colonies initiated by diploid cells. The framework of the model can be extended to study other important systems, such as tissue with stem cell lineage.
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Affiliation(s)
- Yanli Wang
- Department of Mathematics, The Ohio State University, Columbus, Ohio, United States of America
| | - Wing-Cheong Lo
- Department of Mathematics, City University of Hong Kong, Hong Kong, China
| | - Ching-Shan Chou
- Department of Mathematics, The Ohio State University, Columbus, Ohio, United States of America
- * E-mail:
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12
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Nieuwenhuis BPS, Immler S. The evolution of mating-type switching for reproductive assurance. Bioessays 2016; 38:1141-1149. [DOI: 10.1002/bies.201600139] [Citation(s) in RCA: 24] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/19/2022]
Affiliation(s)
| | - Simone Immler
- Department of Evolutionary Biology; Uppsala University; Uppsala Sweden
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