{"paper_id":"0d6abfaf-eeef-42da-a73c-e24c6ac57f3d","body_text":"Cell size and selection for stress-induced binary cell fusion\nXiaoyuan Liu1*, Jonathan W. Pitchford 1,2, George W. A. Constable 1,\n1 Department of Mathematics, University of York, York, North Yorkshire, United\nKingdom 2 Department of Biology, University of York, York, North Yorkshire, United\nKingdom\n*xiaoyuan.liu@york.ac.uk (XL)\nAbstract\nIn unicellular organisms, sexual reproduction typically begins with the fusion of two\ncells (plasmogamy) followed by the fusion of their two haploid nuclei (karyogamy) and\nfinally meiosis. Most work on the evolution of sexual reproduction focuses on the\nbenefits of the genetic recombination that takes place during meiosis. However, the\nselection pressures that may have driven the early evolution of binary cell fusion, which\nsets the stage for the evolution of karyogamy by bringing nuclei together in the same\ncell, have seen less attention. In this paper we develop a model for the coevolution of\ncell size and binary cell fusion rate. The model assumes that larger cells experience a\nsurvival advantage from their larger cytoplasmic volume. We find that under favourable\nenvironmental conditions, populations can evolve to produce larger cells that undergo\nobligate binary cell fission. However, under challenging environmental conditions,\npopulations can evolve to subsequently produce smaller cells under binary cell fission\nthat nevertheless retain a survival advantage by fusing with other cells. The model thus\nparsimoniously recaptures the empirical observation that sexual reproduction is\ntypically triggered by adverse environmental conditions in many unicellular eukaryotes\nand draws conceptual links to the literature on the evolution of multicellularity.\nAugust 16, 2024 1/35\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted August 20, 2024. ; https://doi.org/10.1101/2024.08.19.608569doi: bioRxiv preprint \n\nAuthor summary\nSexual reproduction is commonly observed, both in eukaryotic microorganisms and in\nhigher multicellular organisms. Sex has evolved despite numerous apparent costs,\nincluding investment in finding a partner and the energetic requirements of sexual\nreproduction. Binary cell fusion is a process that sets the stage for sexual reproduction\nby bringing nuclei from different cells into contact. Here, we provide a mathematical\nexplanation of the advantage conferred by binary cell fusion due to increased cell mass.\nWe show that when unicellular organisms have the option to invest in either cell fusion\nor cell mass, they can evolve to fuse together as rapidly as possible in the face of\nadverse environments, instead of increasing their mass. These results are consistent with\nthe empirical observation that sexual reproduction is often triggered by environmental\nstress in unicellular eukaryotes. Our results imply an advantage to cell fusion, which\nhelps to shed light on the early evolution of sexual reproduction itself.\nIntroduction 1\nAlthough the details of the early evolution of sexual reproduction in the last common 2\neukaryotic common ancestor (LECA) are shrouded in mystery, it is argued that the 3\nemergence of eukaryotic sex began with the evolution of cell–cell fusion and meiosis [1] 4\nin an archaeal ancestor [2,3]. This step can be further broken down into the evolution of 5\nbinary cell fusion, the one spindle apparatus, homologous pairing and chiasma, and 6\nfinally reduction, division and syngamy [4]. The vast majority of theoretical studies 7\ninvestigating the evolution of sexual reproduction have focused on later stages of this 8\nevolutionary trajectory, namely the conditions that give rise to a selective pressure for 9\ngenetic recombination [5–8]. However, comparatively few studies have investigated the 10\nselective pressures that may have first given rise to binary cell fusion, which may have 11\nfacilitated the evolution of a host of other eukaryotic traits [9], including the homologous 12\npairing and meiotic recombination, by bringing nuclei together in the same cell. 13\nHypotheses for the evolution of binary cell fusion often rely on hybrid fitness 14\nadvantage. It has been suggested that selection for cell–cell fusions might have initially 15\nbeen driven by “selfish” transposons and plasmids [10–12], or negative epistatic 16\nAugust 16, 2024 2/35\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted August 20, 2024. ; https://doi.org/10.1101/2024.08.19.608569doi: bioRxiv preprint \n\ninteractions between mitochondrial mutations [13,14]. However, once a heterokaryotic 17\ncell has been formed (binucleate with nuclei from both parental cells), the advantage of 18\nhybrid vigor and the masking of deleterious mutations could lead to the maintenance of 19\ncell fusion [4]. Such benefits are required to alleviate costs to cell-fusion, which include 20\nselfish extra-genomic elements in the cytoplasm [15] and cytoplasmic conflict [16,17]. 21\nIn these previous studies on the evolution of binary cell fusion, the effect of changing 22\nenvironmental conditions is not considered. However, in many extant unicellular 23\norganisms, binary cell fusion (and the karyogamy and genetic recombination that follow) 24\noccur in response to challenging environmental conditions [18] such as starvation 25\n(Tetrahymena [19,20]) and depleted nitrogen levels ( Chlamydomonas reinhardtii [21] 26\nand Saccharomyces pombe [22]). Meanwhile in benign conditions with abundant 27\nresources these species reproduce asexually using binary cell fission. The mechanisms 28\nthat drive selection for genetic recombination under challenging environmental 29\nconditions are well-studied [23]; recombination can facilitate adaptation to a novel 30\nenvironment [24,25] and evolving to engage in more sex when fitness is low 31\n(fitness-associated sex) can allow an organism to maximise the advantages of sex while 32\nminimising the costs [26,27]. However, this focus on the benefits of recombination leaves 33\nspace to ask whether binary cell fusion itself could be selected for as a stress response, 34\neven in the absence of any genetic advantages. 35\nIn this paper, we do not account for the genetic factors discussed above. Instead, we 36\nfocus only on how the survival advantage associated with increasing cytoplasmic volume 37\nmight select for binary cell fusion; this relies on the physiological advantages conferred 38\nby cell-cell fusion and is independent of the question of the genetic advantages (and 39\ndisadvantages) of sexual reproduction. This alternative perspective offers useful new 40\ninsights that can be compared with empirical observation. 41\nThat size-based processes could play a role in the early evolution of sexual 42\nreproduction has empirical and theoretical support. The “food hypothesis” [28] suggests 43\nthat metabolic uptake could drive horizontal gene transfer in bacteria and archaea, with 44\nDNA molecules providing nutrients for the receiving cell [29,30]. Indeed, horizontal 45\ngene transfer has been shown experimentally to be an important source of carbon and 46\nnutrients in bacteria [31,32]. Binary cell fusion is possible in bacteria (where it has been 47\nshown to come with selective benefits from mixed cytoplasm [33]) and archaea [34,35]. 48\nAugust 16, 2024 3/35\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted August 20, 2024. ; https://doi.org/10.1101/2024.08.19.608569doi: bioRxiv preprint \n\nMeanwhile amongst eukaryotes, the benefits of increasing cytoplasmic volume are 49\nunderstood to be strong enough to drive selection for the sexes themselves [36,37]. That 50\nearly selection for syngamy may have been driven by the survival benefits of larger 51\ncytoplasmic volume (and the concomitant increase in solid food reserves) has been 52\nargued verbally [38], however has not been explored mathematically. In suggesting a 53\nmechanistic hypothesis for the evolution of binary cell fusion, our work has interesting 54\nparallels with [39], where an advantage to cell fusion is identified in terms of shortening 55\nthe cell-cycle. 56\nMoving to consider potential physiological benefits of binary cell fusion naturally 57\nleads to work on the evolution of multicellularity. While multicellularity and binary cell 58\nfusion are clearly biologically distinct, from a modelling perspective they share 59\nsimilarities in that they can involve the “coming together” of cells to produce a larger 60\ncomplex [40]. Multicellularity achieved via aggregation allows organisms to rapidly 61\nadapt to novel environments that favour increased size [41,42]. It has also been 62\nsuggested that the genetic nonuniformity of such aggregates may also make them 63\nwell-suited to resource limited environments [43,44], echoing the hybrid vigor 64\nhypotheses for the evolution of early syngamy [4]. Relatively few theoretical studies 65\nhave investigated the evolution of facultative aggregation in response to changing 66\nenvironments [42]. However in the context of clonal multicellularity (“staying together”) 67\nsuch environments have been considered more extensively [45,46]. In this clonal context, 68\nthe evolutionary dynamics act primarily on fragmentation modes [47,48] (e.g. how a 69\n“parental” multicellular complex divides to form new progeny). Interestingly the same 70\nquality-quantity trade-off arises here [45] as drives selection for the sexes [36] 71\n(anisogamy, gametes of differing sizes); larger daughter cells (or gametes) are more able 72\nto withstand unfavourable environmental conditions, while smaller cells can be 73\nproduced in larger quantities. 74\nIn this paper we adapt the classic Parker-Baker-Smith [36] (PBS [37]) model for the 75\nevolution of sexes in order to investigate the evolution of binary cell fusion. This builds 76\non recent work that investigates how the possibility of parthenogenetic reproduction can 77\ndrive selection for oogamy in eukaryotes [49]. We assume for simplicity that parental 78\ncells undergo a number of cell-divisions. The size of daughter cells is a compound 79\nevolvable trait determined both by the size of parental cells and the number of cell 80\nAugust 16, 2024 4/35\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted August 20, 2024. ; https://doi.org/10.1101/2024.08.19.608569doi: bioRxiv preprint \n\ndivisions. Daughter cells are then introduced to a pool in which they can undergo 81\nbinary cell fusion, with larger fused cells experiencing a survival advantage. Unlike in 82\nthe classic PBS model, the fusion rate is also a trait subject to evolution. In the 83\nfollowing sections we proceed to outline some insights developed from numerical 84\nsimulations before going on to develop analytical results for the model in a fixed 85\nenvironment. Finally we introduce switching environments and show that under plastic 86\nphenotypic responses, facultative binary cell fusion in response to harsh environmental 87\nconditions can evolve. 88\nModel 89\nInsights from Simulations 90\nWe consider a computational model of a haploid population that reproduces via binary 91\ncell fission. This population dynamics proceed as follows: there is an initial growth and 92\nbinary fission phase where a fixed energy budget E can be used for population growth 93\nand binary fission (see Fig 1A). The parameter E thus sets the carrying capacity of the 94\npopulation. This growth phase is followed by an environmentally-induced mortality 95\nphase where growth and fission are suspended and survival depends on cell size. 96\nExplicitly, we assume that cells reach maturity at size M, so that in the absence of 97\nbinary fission ( n = 0 rounds of fission) the total number of mature cells at the end of a 98\ngrowth cycle is E/M (see Fig 1A). If, however, the population undergoes n > 0 rounds 99\nof binary fission, then each resulting daughter cell has mass m = M/2n and the total 100\nnumber of daughter cells is (2 nE)/M. After this growth and fission phase each 101\ndaughter cell is subject to an extrinsic mass-dependent mortality, such that larger 102\ndaughter cells are more likely to survive into the next growth cycle (see Fig 1A). We 103\ndenote this survival function S(m; β); the parameter β describes the magnitude of the 104\nmortality process (i.e. the harshness of the environment). For a given value of m, an 105\nincrease in β decreases the survival probability. 106\nIn an initial investigation, we view the mass of daughter cells, m, as a single trait 107\nsubject to evolution. To increase the number of daughter cells they produce ( E/m) 108\nmature cells can grow to smaller sizes (reduced M, which increases the number of 109\nAugust 16, 2024 5/35\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted August 20, 2024. ; https://doi.org/10.1101/2024.08.19.608569doi: bioRxiv preprint \n\nDivision Survival Growth\nFusion\nPossibility\nof fusion\nfailure\nGrowth /\nVeg. Seg.\n(b) Coevolutionary dynamics of cell mass and cell fusion\n(a) Evolutionary dynamics of cell mass\nFig 1. Schematic for the model dynamics within each growth cycle. Panel (a):\nIllustrative dynamics for the evolutionary dynamics of cell mass alone. Due to energetic\nconstraints genotypes in the population can either produce fewer, larger mature cells or\nmore numerous, smaller cells (see different shades of green). Daughter cells are produced\nfollowing cell division. Their survival is dependent on mass, such that smaller cells are\nmore likely to die (see Eq. (2)). Surviving cells seed the next growth cycle. Panel (b):\nIllustrative dynamics for the coevolutionary dynamics of cell mass and cell fusion rate.\nThe model is similar to that in panel A, but now a fraction of daughter cells are given\nthe opportunity to risk fusing to form binucleated cells; with probability C fusion fails,\nand both daughter cells are lost. However should a fused cell successfully form, it\nexperiences an enhanced survival probability as a result of its larger cytoplasmic volume.\nFollowing growth and vegetative segregation, surviving cells seed the next growth cycle.\nmature cells in the population) or increase their number of cell-divisions (increased n) . 110\nHowever by decreasing M and increasing n, individuals also produce smaller daughter 111\ncells that are more vulnerable to extrinsic mortality. The size of daughter cells is thus 112\nsubject to a quality-quantity trade-off. For simplicity, we model the mass of daughter 113\ncells m as a continuous trait, and explore its evolution using simulations (see 114\nSupporting Information 3.3). 115\nFig 2 summarises the outcome of such evolutionary dynamics. Fig 2A shows that the 116\npopulation evolves towards an evolutionarily stable strategy (ESS) in m for a given 117\nenvironment. Should the environment suddenly become harsher (via an increase in β) 118\nthe population evolves towards a new ESS, in which daughter cells are larger (i.e. 119\ndaughter cells evolve to become larger to withstand more adverse conditions). 120\nWe now modify the model to allow for the possibility of binary cell fusion following 121\nthe cell fission described above. Daughter cells may now fuse to form a binucleated cell 122\n(e.g. a dikaryon [50], in which the cytoplasm of the contributing cells are mixed but 123\nAugust 16, 2024 6/35\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted August 20, 2024. ; https://doi.org/10.1101/2024.08.19.608569doi: bioRxiv preprint \n\ntheir nuclei or nucleoids remain distinct [38]) or remain a mononucleated cell (with a 124\nsingle nucleus or nucleoid). The rate of cell fusion is given by α, such that when α = 0 125\nall cells remain mononucleated, and cell survival into the next growth cycle is calculated 126\nas before. Conversely for α > 0, some proportion of daughter cells will have fused and 127\nin the limit α → ∞, all cells will have fused. These fused cells will receive a survival 128\nadvantage from their increased mass. However they will also pay an additional cost, C, 129\nresulting from factors such as cell-fusion failure [51], selfish extra-genomic elements in 130\nthe cytoplasm [15], cytoplasmic conflict [16,17] and maintenance of a binucleated 131\ncell [52]. Together this means that fused cells survive with a total probability 132\n(1 − C)S(2m; β). Surviving adults divide to form a new growth cycle of mononucleated 133\nhaploid daughter cells, with binucleated parental cells producing mononucleated 134\nprogeny through vegetative segregation [53] (or alternatively through plasmid 135\nsegregation machinery [54]). Note that although we do not account for the possibility of 136\nbinucleated cells failing to form mononucleated progeny (i.e. failed segregation), this 137\ncan be accounted for by their additional survival cost, C (see Fig 1B). 138\nWe now explore the coevolution of daughter cell mass, m, and fusion rate, α. In 139\nFig 2B, we see that in the benign environment, α remains at zero, and the population 140\nevolves towards an ESS in m as in Fig 2A. However now when the population is 141\nintroduced to a harsher environment, the evolutionary dynamics differ from those in 142\nFig 2B (where α was held artificially at zero). Rather than cells evolving to be larger, 143\nwe see a different response emerging; selection for binary cell fusion ( α > 0). 144\nThe result above is in some sense surprising. Despite the presence of additional 145\nsurvival costs associated with binary cell-fusion, selection for non-zero fusion rates 146\n(rather than increased daughter cell size) persists in the harsh environment. We explain 147\nthe emergence of this behaviour mathematically in the Results section. 148\nMathematical Model 149\nOur model takes inspiration from the classic PBS model for the evolution of 150\nanisogamy [36] (the production of sex cells of differing size). However, whereas such 151\nmodels typically consider the binary cell fusion (fertilization) rate a fixed parameter, we 152\nhere treat it as a trait subject to evolution. In doing so our work builds on [49], where a 153\nAugust 16, 2024 7/35\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted August 20, 2024. ; https://doi.org/10.1101/2024.08.19.608569doi: bioRxiv preprint \n\n0 500 1000 1500 2000\n0.0\n0.5\n1.0\n1.5\n2.0\nTim e ( Generat ions,τ)\nCell m ass ( m )\nA Evolut ionary dynam ics of cell m ass\nBenign\nEnvironm ent\nHarsh\nEnvironm ent\n0.0\n0.5\n1.0\n1.5\n2.0Cell m ass ( m )\nCoevolut ionary dynam ics of cell m ass and cell fusion\nBenign\nEnvironm ent\nHarsh\nEnvironm ent\n0 500 1000 1500 2000\n0.0\n0.2\n0.4\n0.6\n0.8\n1.0\nTim e ( Generat ions,τ)\nFusion Rat e (α)\nBenign\nEnvironm ent\nHarsh\nEnvironm ent\nB\nFig 2. Selection for cell fusion as an alternative to increased cell size in\nresponse to a harsh environment . Stochastic simulations of evolutionary\ntrajectories when the system is subject to a switch from the benign environment\n(β1 = 0.5, green region) to the harsh environment (β 2 = 2.2, orange region) at growth\ncycle 500. Panel A illustrates the case where the fusion rate is held at α = 0,\nrepresenting the scenario where the physiological machinery for fusion has not evolved.\nPanel B illustrates the case where fusion rate is also subject to evolution. Remaining\nmodel and simulation parameters are given in Supporting Information 7 and the initial\ncondition is ( m(0), α(0)) = (1.16, 0).\nvery similar model with a different biological motivation was used to investigate the 154\nevolution of anisogamy with parthenogenesis. In order to analyse the dynamics of the 155\nmodel, we use tools from adaptive dynamics [55], assuming that traits are continuous 156\nand that mutations have small effect. 157\nIn addition we will explore the effect of switching environments, another departure 158\nfrom the PBS model. As such it is important to keep track of the hierarchy of 159\nAugust 16, 2024 8/35\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted August 20, 2024. ; https://doi.org/10.1101/2024.08.19.608569doi: bioRxiv preprint \n\ntimescales at play. The shortest timescale is the timescale of a growth cycle (see 160\nSupporting Information 1). The intermediate timescale is that over which the invasion 161\nof a rare mutant (taking place over many growth cycles) can occur (see Supporting 162\nInformation 2.1) and the longest timescale is the evolutionary timescale, representing 163\nthe cumulative effect of multiple mutations and invasions (see Supporting 164\nInformation 3.1). Finally we assume that environmental switching can take place on 165\neither intermediate or long timescales (see Supporting Information 5 and [49]). 166\nDynamics within each growth cycle 167\nA total of (2 nE)/M daughter cells enter a pool in which binary cell fusion can occur. 168\nAfter a finite time window, the resultant cells are subject to a round of mass dependent 169\nmortality, such that cells of larger mass are more likely to survive. The surviving cells 170\nform the basis of the next growth cycle, as illustrated in Fig 1. 171\nF usion Kinetics 172\nWe assume that all daughter cells may fuse with each other, an assumption consistent 173\nwith most models of the early evolution of sexual reproduction, which suppose the 174\nexistence of a “unisexual” early ancestor that mated indiscriminately [56]. Following 175\ngrowth and binary cell fission, the population is comprised of N unfused daughter cells. 176\nFusion between these mononucleated cells occurs at a rate of α, such that the number of 177\nunfused cells, N, is given by the solution to 178\ndN\ndt = −αN 2 , N (0) = 2nE\nM = E\nm . (1)\nAt the end of the fusion window of duration T there are then N(T ) unfused 179\n(mononucleated) cells remaining, and (N (0) − N(T ))/2 fused (binucleated) cells. 180\nSurvival Probability 181\nWe assume that both unfused and fused cells are subject to the same extrinsic 182\nmass-dependent mortality function, S(m; β), while fused cells pay an additional 183\nmass-independent cost C. Many choices for such a function are possible, so long as it is 184\nan increasing function of cell size (which we equivalently refer to as cell mass m). 185\nAugust 16, 2024 9/35\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted August 20, 2024. ; https://doi.org/10.1101/2024.08.19.608569doi: bioRxiv preprint \n\nHowever here we assume that S(m; β) is the Vance survival function [57], a common 186\nassumption in the literature [58 –60]. We thus have that at the end of the fusion window, 187\nthe survival probability of unfused and fused cells are given respectively by 188\nS(mi; β) = exp\n\u0012\n− β\nmi\n\u0013\n,\n(1 − C)S(mj + mk; β) = exp\n\u0012\n− β\nmj + mk\n\u0013\n,\n(2)\nwhere mi is the mass of a particular unfused daughter cell and mj and mk are the 189\nmasses of two daughter cells that have fused. For a given cell mass, increasing β will 190\ndecrease the survival probability. We therefore refer to β as the environmental 191\nharshness parameter, with high β corresponding to harsh environments in which 192\nsurvival is difficult, and low β corresponding to more benign environments in which 193\neven cells of modest mass have a high probability of surviving. 194\nHaving defined how the survival of a cell depends on its mass, we have the necessary 195\ntools to mathematically characterise the fitness of a rare mutant, and whether it can 196\ninvade the resident population. In the following section, we provide mathematical 197\napproximations of the invasion dynamics of such a mutant. 198\nInvasion Dynamics 199\nAdopting the classical assumptions of adaptive dynamics [55,61] (see also Supporting 200\nInformation 3.1 and [49]), we mathematically approximate the invasion dynamics of a 201\nmutant (which occur over discrete growth cycles). Deriving these invasion dynamics 202\nanalytically is only possible when we assume that mutations in m and α occur 203\nindependently. However the evolutionary dynamics we obtain if we consider mutations 204\noccurring in both m and α simultaneously remains identical to those obtained by 205\nassuming that they occur independently (see Supporting Information 4 and [49]). 206\nDenoting by ˆfm the frequency of mutants of size m ± δm in the population where 207\nδm is the mutational stepsize in m, which is assumed to be small and tg the number of 208\ngrowth cycles, we find (see Supporting Information 2) 209\nd ˆfm\ndtg\n= hm(m, α, β, C) ˆfm(1 − ˆfm) , (3)\nAugust 16, 2024 10/35\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted August 20, 2024. ; https://doi.org/10.1101/2024.08.19.608569doi: bioRxiv preprint \n\nwhere hm(m, α, β, C) is a constant that depends on the parameters m, α, β, C (see 210\nEq. (S10)). This constant provides the fitness gradient of the mutant. Similarly, 211\ndenoting by ˆfα the frequency of mutants with fusion rate α ± δα in the population 212\nwhere δα is the mutational stepsize in α, which is again assumed to be small, we find 213\nd ˆfα\ndtg\n= hα(m, α, β, C) ˆfα(1 − ˆfα) (4)\nwhere hα(m, α, β, C) is the fitness gradient of a mutant with fusion rate α + δα. We see 214\nthat in the case of a single mutant, we have frequency-independent selection for mutants 215\nwith different masses and fusion rates. We note that in reality, frequency-dependent 216\ninvasion dynamics can occur when multiple mutants that change both m and α arise in 217\nthe population (see Supporting Information 4 for mathematical analysis), which can in 218\nturn lead to evolutionary branching [62,63]. However, since this branching does not 219\noccur in the regimes we are focusing on in this paper [49], we assume for simplicity that 220\nmutants encounter a monomorphic resident population (trait substitution) for the 221\nremainder of the mathematical analysis. 222\nEvolutionary Dynamics 223\nWe assume that haploid daughter cells are characterised by two genetically 224\nnon-recombining traits mass m and cell fusion rate α. We assume that mutations occur 225\nin m or α independently at a fixed rate µ, where µ is measured in units of 226\n(number of growth cycles) −1 (see Supporting Information 3.3 and [49]). A mutation in 227\nm represents a change in the mass of the daughter cell produced, and a mutation in α 228\nrepresents a change in the fraction of the population that undertakes either one of the 229\nreproductive routes (i.e. binary cell fusion vs strictly binary cell fission). 230\nMutants with a different mass to their ancestor can produce either more or fewer 231\ndaughter cells than their ancestor (see Eq. (1)), which impacts their survival (see 232\nEq. (2)). When mutants have a different fusion rate to their ancestor, although the 233\nnumber of daughter cells produced does not differ from their ancestor, the number of 234\nfused cells at the end of a growth cycle can either increase/decrease, which impacts their 235\nsurvival, since fused cells have greater mass. The survival of fused cells is also 236\ninfluenced by the cost of fusion C (see Eq. (2)). 237\nAugust 16, 2024 11/35\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted August 20, 2024. ; https://doi.org/10.1101/2024.08.19.608569doi: bioRxiv preprint \n\nOur mathematical analysis in the remainder of this section assumes that mutants 238\nencounter a strictly monomorphic population (i.e. that mutations fixate before the 239\nintroduction of a new mutant). However in our numerical simulations, we release this 240\nrestriction and stochastically allow for the coexistence of multiple traits in the 241\npopulation, held under a mutation-selection balance, as described in the subsequent 242\nsection. 243\nFixed Environment 244\nWe first consider the evolutionary dynamics in the case where the environment is fixed 245\n(i.e. when the parameter β, which measures the harshness of the environment (see 246\nEq. (2)), is constant throughout the evolution. Assuming that δm and δα are small 247\n(small mutational step size), we use techniques from adaptive dynamics [49,64] to 248\nobtain equations for the evolutionary dynamics of m and α, which are given by 249\ndm\ndτ = Hm(m, α; β, C)\n= −4m(m − β) + EαT (1 − C)e\nβ\n2m (4m − β)\n4m2(m + EαT (1 − C)e\nβ\n2m )\ndα\ndτ = Hα(m, α; β, C)\n= −m(1 − (1 − C)e\nβ\n2m ) ln(1 + EαT\nm )\n2α(EαT (1 − C)e\nβ\n2m + m)\n(5)\nfor α ≥ 0. This boundary is imposed to prevent α from becoming negative, which is 250\nbiologically unrealistic since it corresponds to an increase in daughter cell numbers 251\nduring the fusion period, as can be seen from Eq. (1). Therefore when α becomes 252\nstrictly decreasing along this boundary α = 0 boundary (i.e [ dα/dτ]|α=0 < 0 in Eq. (5)), 253\nwe introduce a discontinuous change in the dynamics, given by 254\ndm\ndτ = Hm(m, 0; β, C) = β − m\nm2\ndα\ndτ = 0\n(6)\nThe derivations of these equations can be found in Supporting Information 3.1. 255\nAugust 16, 2024 12/35\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted August 20, 2024. ; https://doi.org/10.1101/2024.08.19.608569doi: bioRxiv preprint \n\nSwitching environments with phenotypic plasticity 256\nNow, we consider the case where evolution acts on the same traits as before, but the 257\nenvironment is subject to change. We model environmental change as switching 258\nbetween two environments β1 and β2. If β1 > β 2, then β1 is the harsher environment 259\n(see Eq. (2)). We also allow for phenotypic plasticity such that the population can 260\nevolve different strategies in different environments. The population’s evolutionary state 261\nis now described by four traits; the daughter cell mass in environments 1 and 2 ( m1 and 262\nm2) and fusion rate in these environments (α 1 and α2). 263\nFor simplicity we assume that any cost of phenotypic switching or environmental 264\nsensing is negligible and that this plastic switching is instantaneous upon detection of 265\nthe change in environmental conditions. The evolutionary dynamics in each environment 266\nare then decoupled. However the evolutionary trajectories in each environment are 267\ncoupled by the initial trait values for the population in each environment, which we 268\nassume are the same (i.e. the population begins in a phenotypically undifferentiated 269\nstate). With phenotypic plasticity, the evolutionary dynamics are then given by 270\ndm1\ndτ = P1Hm(m1, α1; β1, C), dm2\ndτ = P2Hm(m2, α2; β2, C)\ndα1\ndτ = P1Hα(m1, α1; β1, C), dα2\ndτ = P2Hα(m2, α2; β2, C)\n(7)\nwith initial conditions 271\nm1(0) = m2(0) = m0, and α1(0) = α2(0) = α0 . (8)\nHere, Hm(m, α; β, C) and Hα(m, α; β, C) retain the functional form in Eq. (5). 272\nAs Eqs. (7) are only coupled through their shared initial conditions, m0 and α0, the 273\nchoice of these initial conditions is an important consideration. Since we are interested 274\nin the initial evolution of binary cell fusion, it is natural to assume that the population 275\nevolves from a state of zero fusion, α0 = 0. Deciding on a plausible initial daughter cell 276\nmass takes more thought. One parsimonious choice would be that the population is 277\nalready adapted to either environment 1 or environment 2 and that m0 is given by an 278\nevolutionary fixed point in one of these environments (this is the situation illustrated in 279\nFig 2). However if the population has been exposed to both the environments before 280\nAugust 16, 2024 13/35\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted August 20, 2024. ; https://doi.org/10.1101/2024.08.19.608569doi: bioRxiv preprint \n\nphenotypic placticity has evolved, it is possible that m0 is instead given by a bet-hedging 281\nstrategy. We explore that such a strategy would take in the following section. 282\nSwitching environments without phenotypic plasticity 283\nWe now consider the case where there is switching between environments but where the 284\npopulation exhibits no phenotypic plasticity. As described in the previous section, we 285\nare particularly concerned with the period before the physiological machinery for cell 286\nfusion has evolved, and so focus on the case where the cell fusion rate is fixed to zero, 287\nα = 0. Evolution then solely acts on the daughter cell mass, m. 288\nAs in [49], environmental switching is modelled as a discrete stochastic telegraph 289\nprocess, with the time spent in each environment distributed geometrically. The 290\npopulation spends an average of τ1 = 1/λ1→2 in environment 1 and τ1 = 1/λ2→1 in 291\nenvironment 2, where λi→j is the transition rate from environment i to j. 292\nThe two switching rates most relevant to our model are when the environment 293\nswitches many times before an invasion can complete, (fast relative to invasion, FRTI) 294\nand when each switching event occurs after multiple invasions have completed, (fast 295\nrelative to evolution, FRTE). More detail of these switching rates are provided in 296\nSupporting Information 5. However in [49], we show that the evolutionary dynamics for 297\nm in both these regimes can be approximated using the same dynamical equations. 298\nUsing adaptive dynamics techniques modified to account for such environmental 299\nswitching [65], we obtain 300\ndm\ndτ = P1Hm(m, 0; β1, C) + P2Hm(m, 0; β2, C) (9)\nwhere Hm(m, α; β, C) retains the functional form in Eq. (5) and P1 = τ1/(τ1 + τ2) and 301\nP2 = τ2/(τ1 + τ2) are the probabilities of finding the population in the two respective 302\nenvironments. We therefore see that in the absence of phenotypic plasticity, the 303\nevolutionary dynamics is the weighted average of the dynamics in the two environments. 304\nObtaining the ESS for Eq. (9) is relatively straightfoward. Substituting for 305\nHm(m, 0; β1, C) and Hm(m, 0; β2, C) using the functional form given in Eq. (5) and 306\nAugust 16, 2024 14/35\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted August 20, 2024. ; https://doi.org/10.1101/2024.08.19.608569doi: bioRxiv preprint \n\nsetting dm/dτ = 0 in Eq. (9), we obtain the ESS 307\nm∗\nBH,α=0 = P1β1 + (1 − P1)β2 . (10)\nThis strategy constitutes a bet-hedging strategy in cell mass when the population has 308\nyet to evolve phenotypic plasticity nor the capacity for cell-cell fusion. In the limits 309\nP1 → 1 and P2 → 1, we can recover the ESS strategies in the two respective 310\nenvironments: 311\nm∗\n1,α=0 = β1 , m ∗\n2,α=0 = β2 , (11)\nwhich can be verified from a consideration of the equations for d m/dτ = 0 in a fixed 312\nenvironment with α = 0 (see Eq. (6)). We can now proceed to analyse how binary cell 313\nfusion can be selected for when the fusion rate α is allowed to increase from zero in the 314\nfollowing Results section. 315\nImplementation of Numerical Simulations 316\nThe stochastic simulations of the evolutionary trajectories are also implemented using a 317\nGillespie algorithm [66] where successive mutations and environmental switching events 318\noccur randomly with geometrically distributed waiting times. The rates of mutations µ 319\nand environmental switching λ are measured in units of (number of growth cycles) −1. 320\nIn the simulations, multiple traits coexist under a mutation-selection balance (see 321\nSupporting Information and [49] and [67] for more detail), which allows us to account 322\nfor variations in selection strengths in simulations of our evolutionary trajectories. 323\nResults 324\nIn this section we proceed to analyse the evolutionary dynamics derived from the 325\nmathematical model and compare our results to numerical simulations of the full 326\nstochastic simulations. 327\nAugust 16, 2024 15/35\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted August 20, 2024. ; https://doi.org/10.1101/2024.08.19.608569doi: bioRxiv preprint \n\nIn a fixed environment the population evolves to either no cell 328\nfusion, or to high levels of cell-fusion, dependent on the cost of 329\ncell fusion 330\nIn Fig 3, we see two possible evolutionary outcomes for the fusion rates in a fixed 331\nenvironment; the population can evolve either a high (technically infinite) fusion rate or 332\nto zero fusion rates. To which of these fusion rates the population is attracted depends 333\nboth on the parameters and the initial conditions. 334\nWhen the costs to cell fusion are low ( C ⪅ 0.39), the only evolutionary fixed point is 335\nthe high fusion rate fixed point (see Fig 3A). In this scenario, obligate fusion is the only 336\nevolutionary outcome. 337\nFor intermediate costs to cell fusion 0 .39 ⪅ C ⪅ 0.86, there are two possible 338\nevolutionary outcomes. The outcome depends on the initial conditions (see Fig 3B). If 339\nthe initial mass on the α = 0 boundary is small, selection acts to increase fusion rate 340\nand obligate fusion is the ESS. However, if the initial mass on the boundary is 341\nsufficiently large, the state of no cell fusion becomes the evolutionarily stable state. 342\nFinally when costs to fusion are extremely high ( C ⪆ 0.86, see Fig 3C), selection for 343\ndecreased fusion rate acts regardless of the initial value of m on the α = 0 boundary, 344\nand a state in which α = 0 (zero fusion rate) is the only evolutionary outcome. Under 345\nthis high cost regime, dα/dt < 0 along the entire line α = 0 and so fusion rate is never 346\nselected to increase given any initial daughter cell mass. 347\nA rigorous mathematical analysis that formalises the arguments above are provided 348\nin [49], which uses a similar model to investigate the evolution of anisogamy with 349\nparthenogenesis. In summary, the set of possible evolutionary attractors ( m∗, α∗), 350\nstarting from an initial condition ( m(0), α(0)) = (m(0), 0), are given by 351\n(m∗, α∗) →\n\n\n\n\n\n\n\n\n\n\n(β/4, ∞) if 1 − 1\n√e > C ≥ 0\n(β/4, ∞) or ( β, 0) if 1 − 1\ne2 > C ≥ 1 − 1√e\n(β, 0) if C ≥ 1 − 1\ne2\n(12)\nwhere we note 1 − e−1/2 ≈ 0.39 and 1 − e−2 ≈ 0.86. While intermediate costs 352\n(0.86 ⪆ C ⪆ 0.39) lead to two potential evolutionary outcomes depending on the initial 353\nAugust 16, 2024 16/35\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted August 20, 2024. ; https://doi.org/10.1101/2024.08.19.608569doi: bioRxiv preprint \n\n0.0 0.5 1.0 1.5 2.0\n0.0\n0.2\n0.4\n0.6\n0.8\n1.0\nDaught . cell m ass ( m )\nFusion Rat e (α)\nLow Fusion Cost\n( C≲0.39)\n0.0 0.5 1.0 1.5 2.0\n0.0\n0.2\n0.4\n0.6\n0.8\n1.0\nDaught . cell m ass(m)\nFusion Rat e (α)\nI nt erm ediat e Fusion Cost\n( 0.39≲C≲0.86)\n0.0 0.5 1.0 1.5 2.0\n0.0\n0.2\n0.4\n0.6\n0.8\n1.0\nDaught . cell mass(m)\nFusion Rat e (α)\nHigh Fusion Cost\n( 0.86≲C)A CB\nFig 3. Phase portraits for the co-evolutionary dynamics in a fixed\nenvironment (see Eq. (5)). High fusion rates are the only evolutionary outcome\nwhen costs to cell fusion are low (panel A), while under intermediate costs (panel B),\nhigh fusion rate and zero fusion rate (obligate asex) are both evolutionary outcomes,\nand under high costs, zero fusion rate becomes the only evolutionary outcome (panel C),\nas summarised analytically in Eq. (12). The red shaded region shows trajectories\nleading to points on the α = 0 boundary for which evolution selects for decreasing\nfusion rate (d α/dτ < 0) and the critical point at which d α/dτ = 0 is marked by the red\narrow (see Supporting Information 6.2). The red circles mark a fixed point in the\nevolutionary dynamics of m (m∗ = β, see Eq. (12)), which may be unstable (open\ncircles) or stable (filled circle) under coevolution with α. The blue circles and arrows\nillustrate the evolutionary fixed point for high fusion rates ((m ∗, α∗) → (β/4, ∞), see\nEq. (12)). Average population trait trajectories, (⟨m⟩(t), ⟨α⟩(t)), from simulation of the\nfull stochastic model are plotted in light gray, and their mean over multiple realisations\nare dashed. Initial conditions: ( m(0), α(0)) = (1.5, 0.6) and ( m(0), α(0)) = (2, 0.1).\nSimulation is run for 1.1 × 107 growth cycles in panel A, 1.24 × 107 growth cycles in\npanel B and 10 7 growth cycles in panel C. Remaining system parameters are given in\nSupporting Information 7.\nconditions, it is the second of these, ( m∗, α∗) = (β, 0), that is arguably the most 354\nrelevant for the evolution of early cell fusion; if evolution had acted on daughter cell size, 355\nm, before the physiological machinery necessary for cell fusion had evolved, the initial 356\ncondition for the co-evolutionary dynamics would be ( m(0), α(0)) = (β, 0), at which the 357\npopulation would be subsequently held by costs to fusion. 358\nIn Fig 3 we also see that our mathematical analysis is a good predictor of the 359\noutcome of stochastic simulations (gray shaded lines). One minor point of departure is 360\nthat at high fusion rates our simulated trajectories begin to diverge from our analytic 361\nprediction. This discrepancy is the result of evolutionary branching in cell mass, which 362\nwe explore in another paper relating to the emergence of size dimorphism in sex 363\ncells [49]. However this branching happens at a later evolutionary stage than the focus 364\nof this study, the early emergence of binary cell fusion. 365\nWe conclude this section by addressing the key biological result that arises from this 366\nAugust 16, 2024 17/35\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted August 20, 2024. ; https://doi.org/10.1101/2024.08.19.608569doi: bioRxiv preprint \n\nanalysis; cell fusion is uniformly selected for even under moderately high costs (with a 367\nfraction of up to C ≈ 0.39 of fused cells failing to survive) and can even be selected for 368\nunder higher costs (up to C ≈ 0.86) given necessary initial conditions. In the context of 369\nthe evolution of early binary cell fusion, this provides a surprising nascent advantage to 370\ncell fusion. This advantage could even help compensate for other short-term costs 371\narising from the later evolution of sex and recombination. The selective advantage 372\nexperienced by fusing cells comes from their increased cytoplasmic volume, which leads 373\nto increased survival probabilities. 374\nIn a switching environment with phenotypic plasticity, binary 375\ncell fusion can evolve as a facultative stress response to harsh 376\nenvironments 377\nHaving considered the case of the evolutionary dynamics in a fixed environment, we now 378\nmove on to consider the evolutionary dynamics of a population exhibiting phenotypic 379\nplasticity in a switching environment (see Eq. (7)). We recall that under the 380\nassumptions of costless and immediate phenotypic switching, the dynamics of ( m1, α1) 381\nand (m2, α2) are decoupled. The evolution of the traits in the respective environments 382\nare coupled however through the initial conditions from which they evolve, which must 383\nbe the same (i.e. a phenotypically undifferentiated state). 384\nWe consider two parsimonious choices for these initial conditions, both beginning in 385\na state without fusion (α 1(0) = α2(0) = 0). In the first scenario, we assume that the 386\npopulation has evolved to a stable non-fusing mass adapted to a single environment (see 387\nEq. (11)) such that m1(0) = m2(0) = m∗\n1,α=0 or m1(0) = m2(0) = m∗\n2,α=0. This is a 388\nsituation in which the alternate environment is in some sense novel and one to which 389\nthe population has not adapted. In the second scenario, we instead assume that the 390\npopulation has evolved to a bet-hedging strategy in mass (optimising the mass of 391\ndaughter cells across the two environments) such that m1(0) = m2(0) = m∗\nBH,α=0 (see 392\nEq. (10)). An illustrative phase portrait is shown in Fig 4. 393\nWe initially consider first scenario in which a population has initially evolved under 394\nenvironment 1 to reach a stable state ( m1, α1) = (β1, 0) (see red disk and surrounding 395\npurple circle, Fig 4A). The population is now exposed to a second, harsher environment 396\nAugust 16, 2024 18/35\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted August 20, 2024. ; https://doi.org/10.1101/2024.08.19.608569doi: bioRxiv preprint \n\n(β2 > β 1) and allowed to evolve a phenotypically plastic response to this new 397\nenvironment. Starting from an initial state ( m2(0), α2(0)) = (β1, 0) (see purple circle, 398\nFig 4B), traits m2 and α2 can evolve to ( m2, α2) = (β2/4, ∞). 399\nSimilarly, we now consider second scenario in which the population has initially 400\nevolved under the switching environments to reach a stable state bet-hedging strategy 401\nin mass ( m1(0), α1(0)) = (m2(0), α2(0)) = (m∗\nα=0, 0) (see orange circles in Fig 4). Upon 402\nthe evolution of phenotypic plasticity, in the benign environment we see the population 403\ntraits relax to a stable state ( m1, α1) = (β1, 0) (no cell fusion, see Fig 4A). However in 404\nthe harsh environment, we see that the bet-hedging strategy in mass becomes unstable, 405\nand the population evolves towards traits ( m2, α2) = (β2/4, ∞) in environment 2 (i.e. 406\nwe observe selection for binary cell fusion, see Fig 4B). 407\n0.0 0.5 1.0 1.5 2.0\n0.0\n0.2\n0.4\n0.6\n0.8\n1.0\nDaught . cell m ass ( m1)\nFusion Rat e (α1)\nBenign Environm ent 1\nA\n0.0 0.5 1.0 1.5 2.0\n0.0\n0.2\n0.4\n0.6\n0.8\n1.0\nDaught . cell m ass ( m2)\nFusion Rat e (α2)\nHarsh Environm ent 2\nB\nFig\n4. Selection for cell fusion as an alternative to increased cell size in\nresponse to a harsh environment . Illustrative phase portrait for co-evolutionary\ndynamics of ( m1, α1, m2, α2) in a switching environment with phenotypic switching that\nexhibits facultative binary cell fusion. In both environment 1 (panel A) and\nenvironment 2 (panel B) the cost to cell fusion is C = 0.6, purple circles represent the\ninitial condition (m 1(0), α1(0)) = (m2(0), α2(0)) = (β1, 0), and orange circles represent\nthe initial condition (m 1(0), α1(0)) = (m2(0), α2(0)) = (m∗\nα=0, 0), with m∗\nα=0 taken\nfrom Eq. (10). Environmental parameters are β1 = 0.5 and β2 = 2.2 making\nenvironment 1 the more benign environment, in which the population typically spends a\nproportion P1 = 0.7 of its time.\nIn both scenarios described above, we see the emergence of facultative binary 408\ncell-fusion as a response to harsh environmental conditions that lower the survival 409\nprobability of daughter cells. However we note that this is only possible if there is an 410\nappreciable increase in environmental harshness, β, between the environments. In Fig 5, 411\nwe summarise the key results over the β1 − β2 parameter plane. Here we assume that 412\nthe cost to cell fusion is intermediate (1 − e−2 > C > 1 − e−1/2, i.e. 0 .86 ⪆ C ⪆ 0.39) 413\nsuch that there are regions on the boundary α = 0 at which increased fusion rates are 414\nAugust\n16, 2024 19/35\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted August 20, 2024. ; https://doi.org/10.1101/2024.08.19.608569doi: bioRxiv preprint \n\nboth selected for and against depending on the value of m (see Supporting 415\nInformation 6.2 and Eq. (12)); this restricts us to the more interesting parameter regime 416\nin which different evolutionary outcomes are possible in each environment. 417\nIn Fig 5, we see that when one of the environments is not appreciably worse than the 418\nother, binary cell fusion does not evolve in either environment. However when the 419\ndifference between the environments grows more substantial, it is possible to evolve cell 420\nfusion in the harsher environment from initial condition 421\n(m1(0), α1(0)) = (m2(0), α2(0)) = (βi, 0) (where the population has first evolved towards 422\nthe evolutionary optimum of the more benign environment). Finally when the difference 423\nbetween the environments is extreme, it is also possible to evolve cell fusion in the 424\nharsher environment from initial condition ( m1(0), α1(0)) = (m2(0), α2(0)) = (m∗\nα=0, 0) 425\n(where the population has first evolved towards a bet-hedging strategy in cell mass). 426\n0 1 2 3 4\n0\n1\n2\n3\n4\nβ1\nβ2\nNo Fusion in either\nenvironment\nFusion evolves in\nenvironment\ngiven m(0)=m1,α=0\nFusion evolves in\nenvironment 2\ngiven m(0)=m1,α=0\nand m(0)=mBH,α=0\n*\n*\n*\n2\n2\nFig 5. Regions in the β1 − β2 plane where binary cell fusion evolves as a\nstress-response to environment 2. The region plot is independent of E and T .\nHere, C = 0.5, P1 = 0.3 and the initial condition is ( m(0), 0). Since C > 1 − 1/√e (see\nEq. (12)), fusion can only evolve in at most one of the two environments. In this case it\nis environment 2 where fusion can evolve since m(0) = m∗\n1,α=0 (see Eq. (11)). A\nnumerical simulation to support this regionplot is shown in Fig S2.\nDiscussion 427\nThe evolution of sexual reproduction and its consequences for the subsequent 428\nevolutionary trajectory of populations is of general importance in biology [7,13,68]. In 429\nthis paper we have illustrated a reversal of the classic two fold cost of sex that appears 430\nin organisms with distinct sexes [69]; in unisexual [56], unicellular organisms, binary cell 431\nAugust 16, 2024 20/35\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted August 20, 2024. ; https://doi.org/10.1101/2024.08.19.608569doi: bioRxiv preprint \n\nfusion can be selected for, even in the presence of substantial costs, due to a survival 432\nbenefit that comes from increased mass. These results allow us to quantitatively assess 433\nthe verbal hypothesis that syngamy evolved allow cells to store more food reserves and 434\nthus increase their survival rate [38]. It is particularly interesting that the benefits 435\nconferred to cell fusion through increased cytoplasmic mass are sufficient to withstand 436\nremarkably high costs; “obligate sexuality” is the only evolutionary outcome with costs 437\nequivalent to a loss of ∼ 39% cells that attempt to fuse, and remains a potential 438\noutcome with costs of up to ∼ 86% of fused cells dying. 439\nPerhaps most interesting is the case of switching environments with phenotypic 440\nplasticity. Here we find under a broad set of biologically reasonable conditions (costs to 441\ncell fusion equivalent to 39% − 86% additional mortality to fused cells and at least 442\nmoderate changes in environmental quality) that high fusion rates are selected for in 443\nharsh environments and zero fusion rates are maintained in benign environments. This 444\nbehaviour parsimoniously recapitulates the empirically observed reproductive strategies 445\nof numerous facultatively sexual species, including C. reinhardtii [21], S. pombe [22] and 446\nD. discoideum [70]. This mechanism, under which cell fusion evolves to increase the 447\nsurvival probability of daughter cells, provides a complementary perspective on the 448\nfrequent evolution of survival structures (resistant to environmental stress) that form 449\nfollowing the formation of a zygote. These include ascospores in fungi [71] and 450\nzygote-specific stress-resistant stress wall in C. reinhardtii [72]. Note that such 451\ncorrelations between sexual reproduction and the formation of survival structures are 452\nnot as easily explained under genetic explanations for the evolution of sexual 453\nreproduction, where engaging in both behaviours at once constitutes a simultaneous 454\n(and therefore potentially costly) change in genotype and temporal dislocation in 455\nenvironment [73,74]. 456\nThe results above are particularly interesting in the case of the evolution of early 457\nbinary cell-fusion as a first step in the evolution of sexual reproduction. While most 458\nstudies focus on the genetic benefits of cell-fusion [75] (including a functionally-diploid 459\ndikaryotic cell [4]), or the genetic benefits of mixed cytoplasm [13,14] (which can also 460\ncome with costs [15–17,40,76–78]), the mechanism at play here is purely physiological. 461\nYet, as addressed above, it naturally captures the empirical observation of binary 462\ncell-fusion as response to challenging environmental conditions, a feature absent in these 463\nAugust 16, 2024 21/35\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted August 20, 2024. ; https://doi.org/10.1101/2024.08.19.608569doi: bioRxiv preprint \n\nearlier models. While the mechanism does not explain the evolution of sexual 464\nreproduction and genetic recombination itself, it does provide a nascent advantage to 465\nbinary cell fusion that sets the stage for the evolution of sex by bringing nuclei from 466\ndifferent cells into contact for prolonged periods. The mechanism also shows the 467\npotential to counter short-term costs associated with the initial formation of a 468\nbinucleated cell. In this way the mechanism could facilitate the transition from 469\nhorizontal gene transfer [79,80] to meiotic recombination, which is advantageous when 470\ngenome sizes increase in length [8]. Conceivably, if genetic recombination is beneficial 471\nfor myriad genetic reasons in the long-term [8,81], it would seem natural that it would 472\nbe instigated when the opportunity arises (i.e. when physiological survival mechanisms 473\nbring nuclei into close contact). We note that it is obviously possible that the first 474\ndiploid cells arose by errors in endomitosis [82 –84] (essentially doubling the chromosome 475\nnumber within a single cell) and that meiosis first evolved in this context. Such a 476\nsequence of events is still compatible with our very general model, which can alleviate 477\nshort-term costs of sex such as the energy involved in finding a partner and undergoing 478\nfusion [69]. In either scenario sexual reproduction may not be only a direct response to 479\nenvironmental variability [85,86], but also to the correlated formation of a survival 480\nstructure. 481\nMore generally, it is interesting to note that the conditions for facultative sexuality 482\n(e.g. harsh environmental conditions) broadly coincide with those for facultative 483\nmulticellularity in both bacteria and eukaryotes, with starvation triggering the 484\nformation of fruiting bodies in myxobacteria [87,88] and flocking in yeast [89,90]. 485\nMeanwhile in C. reinhardtii, the formation of multicellular palmelloids and aggregates 486\nare an alternate stress response to sexual reproduction [91], as are the formation of 487\nfruiting bodies in D. discoideum [92]. In this multicellular context, the sexual behaviour 488\nof D. discoideum is particularly interesting, as once formed, the zygote attracts 489\nhundreds of neighboring cells that are then cannibalised for the provision of a 490\nmacrocyst [93]. These various survival strategies are unified in our model as a 491\nmechanism for the evolution of binary cell fusion. 492\nOne element absent from our model is the fusion of multiple cells, which is likely to 493\nbe selected for under the assumptions implicit in our model. There would clearly be an 494\nupper-limit on the number of fusions selected for, arising from the likely multiplicative 495\nAugust 16, 2024 22/35\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted August 20, 2024. ; https://doi.org/10.1101/2024.08.19.608569doi: bioRxiv preprint \n\neffect of the fusion cost C. However in this context, it is interesting to note that one of 496\nthe hypotheses for the evolution of self-incompatible mating types is as a signal to 497\nprevent the formation of polyploid cells [94]. Such a mechanism could also prevent the 498\nformation of trikaryotic cells should the cost of multiple fusions be too great. Thus, the 499\nmodel neatly preempts the second stage in models for the evolution of eukaryotic sex, 500\nthe regulation of cell–cell fusion [1]. 501\nThe selection pressure for binary cell fusion is dependent on the formation of a 502\nbinucleated cell with an increased survival benefit arising from its larger size. Although 503\nit is reasonable to assume that a single mononuclear cell has lower total resource 504\nrequirements than a multicellular complex of the same size [95], we have not considered 505\nthe detailed energetics of the maintenance of two nuclei [9]. In these respects 506\nincorporating dynamic energy budget theory into the model would be an important 507\nnext step [96] as it would provide a clear distinction between the survival benefits of 508\nfused cells and unfused bicellular complexes. Within our modelling framework, these 509\ntwo structures are broadly similar [40]. However as we have shown, increased cell-cell 510\nattraction can be selected for even in the presence of large costs that one might expect 511\nunder binary cell fusion but not associate with the formation of a bicellular complex. 512\nWe have assumed for simplicity a simple cell division scheme; parental cells undergo 513\nn rounds of symmetric division to produce 2 n daughter cells. In the context of 514\nmulticellularity, switching environments have been shown to promote binary 515\nfragmentation [45]. However non-symmetric modes can be selected for [47] reflecting the 516\ndiverse modes of facultative multicellular life cycles observed in bacteria [97]. It would 517\nbe interesting to incorporate our results into models of cell division that account 518\nexplicitly for growth [98,99] to determine how these results for multicellular organisms 519\ncarry over to the unicellular scenario, and further how they may affect those we have 520\nshown here. 521\nFinally, we have not explicitly modelled any sources of cytoplasmic or genetic 522\nconflict [100], which we have for simplicity included in the fusion cost C. Nevertheless, 523\nsocial conflict does emerge in this model. In a recent paper we have shown how 524\nevolutionary branching can arise, with some individuals producing fewer larger cells and 525\nothers producing more numerous but smaller daughter cells [49]. This branching is 526\ndriven by the same evolutionary forces that drive selection for anisogamy [101], in which 527\nAugust 16, 2024 23/35\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted August 20, 2024. ; https://doi.org/10.1101/2024.08.19.608569doi: bioRxiv preprint \n\ncontext this can be viewed as sexual conflict [102]. That social conflict should arise in 528\nthe formation of multicellular aggregates is well understood [40,88,103]. However these 529\nmodels typically assume cells of fixed size [46,104]. Combining the insights derived from 530\nthe evolution of anisogamy literature with the theory developed in the multicellularity 531\nliterature represents another promising research direction. 532\nAs addressed above, trade-offs between cell fusion rate and mass [105], 533\ncell-energetics, inbreeding, and the possibility of multiple cell-fusion events offer 534\ninteresting avenues to extend this analysis. In addition, we have not accounted for the 535\ndiscrete nature of divisions leading to daughter cells, costs to phenotypic switching, 536\nnon-local trait mutations, or pre-existing mating types. More generally, extending our 537\nmathematical approach leveraging adaptive dynamics to switching environments in 538\nother facultatively sexual populations might prove particularly fruitful [106,107]. 539\nIn this paper we have adapted the classic PBS model [36] in two key ways; allowing 540\nthe fusion rate to evolve and subjecting the population to switching environments. In 541\ndoing so, we have shown its capacity to parsimoniously capture the evolution of obligate 542\nbinary cell fusion, obligate binary cell fission, and stressed induced binary cell fusion in 543\nunicellular organisms. These results offer particularly interesting implications for the 544\nevolution of binary cell-fusion as a precursor to sexual reproduction, as well as 545\nsuggesting common mechanistic links between the evolution of binary cell fusion and 546\nmulticellularity. Moreover, our analysis emphasises the importance of exploring the 547\ncoevolutionary dynamics of a range of evolutionary parameters, and of developing 548\ncomputational and mathematical approaches to elucidate facultative sexual 549\nreproduction. 550\nSupporting information 551\nS1 T ext. Supporting information.pdf 552\nAcknowledgments 553\nThis work has made use of Viking high performance computing service at the University 554\nof York. 555\nAugust 16, 2024 24/35\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted August 20, 2024. ; https://doi.org/10.1101/2024.08.19.608569doi: bioRxiv preprint \n\nAuthor Contributions 556\nConceptualization: Jonathan Pitchford, George Constable 557\nMethodology: Xiaoyuan Liu, George Constable, Jonathan Pitchford 558\nF ormal Analysis: Xiaoyuan Liu, George Constable 559\nSoftware: Xiaoyuan Liu 560\nInvestigation: Xiaoyuan Liu 561\nSupervision: George Constable, Jonathan Pitchford 562\nW riting- Original draft preparation: Xiaoyuan Liu, George Constable 563\nW riting- review & editing: Jonathan Pitchford, George Constable 564\nReferences\n1. Goodenough U, Heitman J. Origins of eukaryotic sexual reproduction. Cold\nSpring Harbor perspectives in biology. 2014;6(3):a016154.\n2. Eme L, Spang A, Lombard J, Stairs CW, Ettema TJ. Archaea and the origin of\neukaryotes. Nature Reviews Microbiology. 2017;15(12):711–723.\n3. 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