Model of cell division, growth, and survival

We begin by considering a population of individual mature cells of total mass E. Each mature cell has a mass M, such that the total number of mature cells is E ∕ M. The cells may then undergo n ≥ 0 rounds of binary fission. Each resulting daughter cell then has mass , while the total number of daughter cells in the population is . Following this fission phase, each daughter cell is subjected to an extrinsic mass-dependent mortality, such that larger daughter cells are more likely to survive into the next growth cycle (see upper half of Fig 1). While many choices for such a mass-dependent function are possible, in this paper we focus primarily on the Vance survival function [65], defined as

(1)

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Fig 1. Schematic for model life cycle.

Cells (green circles) are characterized by different genotypes (orange and blue nuclei) that control the non-recombining traits and (respectively the daughter cell mass and fusion rate of the genotype). Daughter cell mass is a compound parameter that depends both on the mature cell mass and the number of cell divisions () and is treated as a single continuous parameter. Following release of the maternal cell wall, daughter cells are exposed to each other. Should genotypes in the population evolve non-zero fusion rates (), binary cell fusion between cells is possible (see Eq (3)). With a finite time-window for cell fusion, T, only a fraction of cells will be fused at the end of this fusion period. Following this both fused and unfused cells are exposed to a mass-dependent survival function (see Eq (1)) under which larger fused cells experience a survival benefit as a result of their increased cytoplasmic volume. Surviving fused cells can grow larger due to their increased size at the beginning of the growth period. The mathematical model assumes that the size of daughter cell size of each genotype is conserved (see Eq (4)). This comes with the implicit assumption the fused cells undergo an additional round of cell division that allows for segregation of nuclei in daughter cells. Costs to cell fusion can arise due to failed cell fusion, hindered growth of the fused cell and failed segregation, as described in the main text. These periods during which these costs can manifest are highlighted in red and captured mathematically by the parameter 1 ≥ C ≥ 0.

https://doi.org/10.1371/journal.pcbi.1012418.g001

Here the parameter β describes the magnitude of the mortality process (i.e. the harshness of the environment); for a given value of m, an increase in β decreases the survival probability of daughter cells. We therefore refer to β as the environmental harshness parameter, with high β corresponding to harsh environments in which survival is difficult, and low β corresponding to more benign environments in which even cells of modest mass have a high probability of surviving. We note that Eq (1) is both mathematically simple (being defined by only one parameter, β) and has the biologically reasonably property of capturing the principle of a minimum cell size (see also Fig 7) and is thus a common assumption in the literature [66–68]. Surviving cells then go on to grow to mass M and seed the next generation, which will again consist of E ∕ M cells. In this way, E can be understood as setting the carrying capacity of the population in terms of its total mass.

We now wish to incorporate evolutionary dynamics to the simple picture described above. We note that the mass of daughter cells, , is dependent both on the mature cell size M and the number of cell divisions, n. For simplicity we therefore treat m as a genetically determined continuous trait, and explore its evolution using simulations. We note that this treatment is biologically equivalent to the evolving the cell sizer mechanism in Chlamydomonas, which operates to ensure that daughter cells have a uniform size [53].

Suppose that we have S genotypes in the population, such that each genotype produces daughter cells of mass for i ∈ S. We further suppose that initially the frequency of the genotype is , such that the number of mature cells of the type at the beginning of a growth cycle is . These mature cells then produce daughter cells, of which a fraction survive to the next growth cycle. Renormalizing by the total number of surviving cells, the frequency of type i cells that survive, is

(2)

Since each of these surviving cells grows to a size , the next growth cycle begins with mature cells.

Note that to increase the number of daughter cells they produce () mature cells can grow to smaller sizes (reduced , which increases the number of mature cells in the population) or increase their number of cell-divisions (increased ) . However by decreasing and increasing , individuals also produce smaller daughter cells that are more vulnerable to extrinsic mortality. The size of daughter cells is thus subject to a quality-quantity trade-off and over multiple iterations of the deterministic cycles described above, certain genotypes may come to dominate the population. Mutants are introduced stochastically at the start of a growth cycle with average rate μ (the number of growth cycles between successive mutations is taken from a geometric distribution parameterized by rate μ). Mutant genotypes are also chosen stochastically; ancestral genotypes are chosen with probability , and mutants either increase or decrease the daughter cell mass trait of their ancestor (i.e. ), each with an unbiased 50% probability. The parameter δm thus captures the size of mutational steps in trait space.

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Fig 2. Selection for cell fusion as an alternative to increased cell size in response to a harsh environment.

Stochastic simulations of evolutionary trajectories when the system is subject to a switch from the benign environment (, green region) to the harsh environment (, orange region) at growth cycle 500. Panel A illustrates the case where the fusion rate is held at α = 0, representing the scenario where the physiological machinery for fusion has not evolved. Panel B illustrates the case where fusion rate is also subject to evolution. Remaining model and simulation parameters are given in Sect 4 of S1 Appendix and the initial condition is  ( m ( 0 ) , α ( 0 ) ) = ( 1 . 16 , 0 ) .

https://doi.org/10.1371/journal.pcbi.1012418.g002

Fig 2 summarizes the outcome of such evolutionary dynamics. Fig 2A shows that the population evolves towards an evolutionarily stable strategy (ESS) in m for a given environment. Should the environment suddenly become harsher (via an increase in β) the population evolves towards a new ESS, in which daughter cells are larger (i.e. daughter cells evolve to become larger to withstand more adverse conditions).

Model additionally incorporating binary cell fusion

We now modify the model to allow for the possibility of binary cell fusion following the cell fission described above (see Fig 1). Daughter cells may now fuse to form a binucleated cell (e.g. a dikaryon [69], in which the cytoplasm of the contributing cells are mixed but their nuclei or nucleoids remain distinct [38]) or remain a mononucleated cell (with a single nucleus or nucleoid). The rate of cell fusion is given by α. We assume that this second trait is genetically linked to the daughter cell mass trait, such that each of the S genotypes is now defined by a trait pair . Cell fusions occur as a result of mass-action kinetics. For simplicity, we assume that fusions between distinct genotypes occur at their average fusion rate, . Denoting by the number of unfused cells of genotype i and the number of fused cells resulting from daughter cells of genotype i and j we have

(3)

with . These fusion dynamics are allowed to run for a fixed period T. When α = 0 all cells remain mononucleated. Rates α > 0 can be biologically understood as the evolution of any traits the promote cell fusion, such as the loss of the cell wall in the unicellular red algae Galdieria [70] or the formation of a mating structure in the unicellular green algae Chlamydomonas [71]. For α > 0, some proportion of daughter cells will have fused in the period T, leaving some cells unfused; the total number of unfused cells of type i is then and the total number of fused cells with i and j nuclei are .

We assume that cells that remain unfused at the end of the fusion period T can still survive and subsequently grow vegetatively. Essentially we are assuming that at early stages of this evolutionary trajectory daughter cells are not overly specialized for strict cell fusion (e.g. full gametic differentiation). This is akin to asexual reproduction in cell wall-less Galdieria [70] and gametic de-differentiation in Chlamydomonas following exposure nitrogen compounds that can be used for growth [58,72]. For simplicity, we further assume that any potential dedifferentiation is costless. Unfused cells again survive according to Eq (1) at a mass-dependent rate. The final number of unfused cells that survive of type i is therefore given by .

Fused cells will receive a survival advantage from their increased mass, . As these fused cells enter the growth period with a larger inital immature cell size, we also assume that fused cells can grow to a larger mature size, . In order for this mature cell to produce daughter cells of the same size as unfused cells (e.g. controlled by a sizer mechanism as in Chlamydomonas [53]) we implicitly assume an additional round of cell division that allows for the production of mononucleated progeny through vegetative segregation [73] (or alternatively through plasmid segregation machinery [74]), as illustrated in Fig 1.

In practice, the evolution of the machinery required for cell-fusion (and subsequent segregation) is likely to come with costs. These costs include: (i) cell-fusion failure [75] in the initial fusion phase; (ii) selfish extra-genomic elements in the cytoplasm [15], cytoplasmic conflict [16,17] and maintenance of a binucleated cell [76] hindering growth in the growth phase; (iii) the possibility of binucleated cells failing to form mononucleated progeny (i.e. failed segregation) in the vegetative segregation phase. These phases with potential costs are highlighted in red in Fig 1. We account for these costs with a single parameter 1 ≥ C ≥ 0. For (i) costs arising solely from cell fusion failure, fused cells survive to be exposed to the mass-dependent survival phase. For (ii) costs arising solely from inhibited growth, mature cells would reach a size . For (iii) costs arising solely from failed segregation, only surviving mature cells would successfully segregate, with the remainder dying.

Analogously to Eq (4) we can now write down the change in frequency of type i the frequency of type i cells that survive, is

(4)

We now explore the coevolution of daughter cell mass, m, and fusion rate, α. We assume that mutations in α occur independently of m but at the same fixed rate μ, where μ is measured in units of (see also Sect 2.5 of S1 Appendix and [50]). Analogously to mutations in daughter cell mass, ancestral genotypes are chosen proportional to their frequency, , with mutant genotypes inheriting their parental genotype plus or minus some trait deviation, δα; .

In Fig 2B, we see that in the benign environment, α remains at zero, and the population evolves towards an ESS in m as in Fig 2A. However now when the population is introduced to a harsher environment, the evolutionary dynamics differ from those in Fig 2A (where α was held artificially at zero). Rather than cells evolving to be larger, we see a different response emerging; selection for binary cell fusion (α > 0).

The result above is in some sense surprising. Despite the presence of additional survival costs associated with binary cell-fusion, selection for non-zero fusion rates (rather than increased daughter cell size) persists in the harsh environment. We explain the emergence of this behaviour mathematically in the Mathematical analysis and results section.

Switching environments with phenotypic plasticity

We have seen in Fig 2B that selection for cell fusion can evolve in response to harsh environmental conditions. We are particularly interested in exploring when such cell fusion can evolve as a plastic stress response to changing environmental conditions, as observed in the sexual reproduction of unicellular organisms such as Tetrahymena [19,20], C. reinhardtii [21] and S. pombe [22]. We recall that while the core model in a fixed environment (described above) has been analysed in [50], the possibility of phenotypic plasticity was not considered.

We model environmental change as switching between two environments and . If , then is the harsher environment in which cells have a lower survival probability (see Eq (1)). For clarity, we will keep this convention for the remainder of the paper. We will also allow for phenotypic plasticity such that the population can evolve different strategies in different environments. The population’s evolutionary state is now described by four traits; the daughter cell mass in environments 1 and 2 ( and ) and fusion rate in these environments ( and ). We assume that these traits are all linked, so that recombination between the traits is not possible. For simplicity we initially assume that any cost of phenotypic switching or environmental sensing is negligible and that this plastic switching is instantaneous upon detection of the change in environmental conditions.

Implementation of numerical simulations

The stochastic simulations of the evolutionary trajectories are also implemented using a Gillespie algorithm [77] where successive mutations and environmental switching events occur randomly with geometrically distributed waiting times. The rates of mutations μ and environmental switching λ are measured in units of . In the simulations, multiple traits coexist under a mutation-selection balance (see S1 Appendix and [50] and [78] for more detail), which allows us to account for variations in selection strengths in simulations of our evolutionary trajectories.

As in [50], environmental switching is modeled as a discrete stochastic telegraph process, with the time spent in each environment distributed geometrically, the discrete analogue of an exponential distribution since the time spent in each environment is measured in the number of discrete generations. The population spends an average of in environment 1 and in environment 2, where is the transition rate from environment i to j.

Fixed environment

Adopting the classical assumptions of adaptive dynamics [79,80], we obtain equations for the selective gradient on traits m and α, which we write as and respectively (see Sect 1 of S1 Appendix and [50]). We find that along the zero-fusion boundary, the selection gradient can be negative. While this makes sense mathematically (it is equivalent to the additional production of cells during the fusion window, see Eq (3)) it makes no sense biologically (the population’s cell mass is not conserved over the cell division and fusion period). We therefore introduce a biologically realistic boundary at α = 0. The equations for the evolutionary dynamics of m and α are then given by

(5)

with

(6)

In total we find three possible attractors for the evolutionary dynamics (see Sect 2 of S1 Appendix), which we outline below.

We begin by considering the dynamics along the α = 0 boundary. For α = 0, the equation for the evolutionary dynamics of m simplifies to

(7)

while the inequality simplifies to

(8)

These equations, together with the condition α ≥ 0, suggest that a fixed point exists on the α = 0 boundary at

(9)

Meanwhile for large α we find an additional diverging trajectory. In the limit α → ∞, we find

(10)

These equations suggest that an attracting manifold m = β ∕ 4 exists in the limit α → ∞

(11)

The third and final potential evolutionary attractor is found by solving and for arbitrary m and α. We obtain

(12)

Unlike the previous evolutionary attractors (see Eqs (9–11)), this fixed point is a function of the cost to cell-fusion C. It is therefore useful to focus on the behaviour of this fixed point as C is increased in order to characterize the behaviour of the population in different parameter regimes, as illustrated in Fig 3.

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Fig 3. Fusion costs govern the basin of attraction for the zero-fusion state in a fixed environment.

Each panel gives the phase portrait for the co-evolutionary dynamics of  ( m , α )  in a fixed environment (see (5)). (A) High fusion rates are the only evolutionary outcome when costs to cell fusion are low, with an unstable fixed point at (open red circle) and selection for increasing α on the invariant manifold m = 1 ∕ β (blue arrow). Purple arrow illustrates the point on the m = 0 boundary when selection for α becomes positive (see Eq (8)). (B) Under low-intermediate fusion costs, a third saddle fixed point emerges (see Eq (12)), undergoing a bifurcation with and rendering it stable (solid red disk). There are now two evolutionary outcomes, depending on initial conditions; trajectories either increase in α along the invariant manifold, , or are attracted to the fixed point. (C) For intermediate to high fusion costs, the separatrix of the saddle fixed point (red arrows) crosses the origin, rendering the entire m = 0 boundary within the basin of attraction of the fixed point. While for low initial cell masses the fusion rate may initially increase, ultimately the population will evolve to a low fusion state. (D) Under high fusion costs the saddle fixed point crosses the invariant manifold m = 1 ∕ β, reversing selection for the fusion rate such that  d α ∕ d τ is now negative along this line. In this scenario all initial conditions evolve towards the fixed point. Coloured dashed lines show the average over multiple stochastic realizations of the simulated model (see also Fig A of S1 Appendix) with initial conditions m ( 0 ) = 0 . 02 (blue), m ( 0 ) = 0 . 5 (orange), m ( 0 ) = 1 . 75 (green) and all in an initial state without fusion (α ( 0 ) = 0). Each simulated trajectory is run for growth cycles and remaining parameters are given in Sect 4 of S1 Appendix.

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When C = 0, the fixed point given in Eq (12) yields a negative, and therefore biologically nonphysical, value of . The remaining biologically relevant evolutionary attractors are an unstable saddle at the fixed point along the α = 0 boundary (see Eq (9)) and a diverging trajectory towards α → ∞ along the manifold (see Eq (11)), as illustrated in Fig 3A. As C is increased, at the fixed point given in Eq (12) also increases, eventually crossing the α = 0 boundary when (see expressions for in Eqs (9) and (12)). At this point the fixed point given by Eq (12) crosses the fixed point along the α = 0 boundary and a bifurcation occurs, yielding a saddle fixed point on the interior and rendering the fixed point at the boundary stable (see Fig 3B). At this stage the basin of attraction for the stable fixed point on the α = 0 boundary only contains larger values of m for initial conditions with α ( 0 ) = 0.

As the cost to cell fusion, C, becomes larger still, the basin of attraction for the stable fixed point on the α = 0 boundary grows to encompass the entire α = 0 boundary. We denote the critical value of C at which this occurs as . At this stage, should be sufficiently small, an initial increase in fusion can be observed but the population is ultimately attracted to a zero-fusion state at long times, as illustrated in Fig 3C. Finally as C is increased to very large values, the interior fixed point crosses the diverging trajectory with α → ∞ when (see expressions for in Eqs (11) and (12)). This results in another transcritical bifurcation, following which selection for α along the m = β ∕ 4 manifold becomes negative, and the fixed point given in Eq (12) becomes biologically nonphysical (see Fig 3D). This bifurcation then leaves the zero fusion state as the only stable fixed point.

The scenarios described above for are qualitatively similar, in that they feature a single evolutionary endpoint for all trajectories with initial conditions (see Fig 3C–3D). In order to determine analytically, we need to obtain the value of C for which the separatrix, marking the boundary of the basin of attraction for the fixed point , intersects the origin point  ( m , α ) = ( 0 , 0 ) . We can obtain a linear approximation for this separatrix using the stable directions of the fixed interior saddle fixed point given in Eq (12) (see red arrows in Fig 3B–3C). In Sect 2.4 of S1 Appendix we show this is given approximately by the solution to

(13)

The root of this equation can be understood as being governed by the compound parameter ET (see Fig 4). When ET is small (i.e. the density of cells entering the fusion pool is low or the period allowed for fusion short), the root approaches the solution to . Meanwhile when ET is large the root approaches the solution to . These limits to the solution for are precisely the bifurcation points identified previously.

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Fig 4. Critical cost above which cell fusion becomes impossible to evolve increases with cell density and time available for cell fusion.

Blue line shows numerical solutions of Eq (13). For values of , obligate cell fusion remains a potential evolutionary outcome (see Eq (14)). The value of is bounded above and below by the bifurcation points identified in the main text (black dashed lines, see Eq (15)).

https://doi.org/10.1371/journal.pcbi.1012418.g004

Summarizing the results above, we therefore have three possible evolutionary scenarios starting from initial conditions . Which of these scenarios we observe depends on the cost to cell fusion, C. When C is low, the only evolutionary endpoint is the invariant manifold at high fusion rate. In this scenario, obligate fusion is the only evolutionary outcome, irrespective of . For intermediate costs to cell fusion, there are two possible evolutionary outcomes, depending on the initial daughter cell mass : if is small, selection acts to increase fusion rate and obligate fusion is the ESS; if conversely if is large, the state of no cell fusion is the ESS. Finally when costs to fusion are high selection for decreased fusion rate acts regardless of the initial value of . Mathematically

(14)

where

(15)

We note that it has been previously shown in [50] that the first of these conditions has additional consequences in this model for the evolution of suppressed fertilization in eukaryotic macrogametes. Should the costs to fertilization (cell fusion) exceed , evolutionary branching can occur at large values of the fusion rate α. However this branching happens at a later evolutionary stage than the focus of this study, the early emergence of binary cell fusion.

We conclude this section by addressing the key biological result that arises from this analysis; cell fusion is uniformly selected for even under moderately high costs (with a fraction of up to C ≈ 0 . 39 of fused cells failing to survive) and can even be selected for under higher costs (up to C ≈ 0 . 86) given necessary initial conditions and parameters. In the context of the evolution of early binary cell fusion, this provides a surprising nascent advantage to cell fusion. This advantage could even help compensate for other short-term costs arising from the later evolution of sex and recombination. The selective advantage experienced by fusing cells comes from their increased cytoplasmic volume, which leads to increased survival probabilities.

Switching environments with phenotypic plasticity

We now consider the case where evolution acts on the same traits as before, but where the environment is subject to change and the population can evolve a specific response to each environment. We find that under the assumption of costless phenotypic plasticity, the evolutionary dynamics in each environment are decoupled (see Sect 3.2 of S1 Appendix);

(16)

where and retain the functional form in (5) and we again restrict the dynamics to the biologically reasonable α ≥ 0. Note that the key effect of environmental switching here is to moderate the rate of evolution, with each trait pair evolving at a rate proportional to the time the population is exposed to environment i.

While the dynamics of the population in each environment is decoupled, the evolutionary trajectories in each environment are coupled by the initial trait values for the population in each environment, which we assume are the same;

(17)

i.e. the population begins in a phenotypically undifferentiated state. As addressed, we are particularly interested in the case where cell fusion emerges as a plastic response, being selected for in one environment and against in the other. Considering the scenarios summarized in Eq (14), this restricts our attention to the case of intermediate costs to cell fusion, , for which both the zero fusion and the obligate fusion states are potential evolutionary outcomes.

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Fig 5. Selection for cell fusion as an alternative to increased cell size in response to a harsh environment.

Illustrative phase portrait for co-evolutionary dynamics of in a switching environment with phenotypic switching that exhibits facultative binary cell fusion. In both environment 1 (panel A) and environment 2 (panel B) the cost to cell fusion is C = 0 . 6, purple circles represent the initial condition , and orange circles represent the initial condition (see Eq (20)). The red shaded regions shows the approximate region of the basin of attraction for the stable zero fusion fixed points ( (see red disk, panel (A)) and ( (see red disk, panel (B)). Red arrows indicate the intersection of these basins with the α = 0 boundary. Environmental parameters are and making environment 1 the more benign environment, in which the population typically spends a proportion of its time.

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As Eqs (16) are only coupled through their shared initial conditions, and , the choice of these initial conditions is an important consideration. Since we are interested in the initial evolution of binary cell fusion, it is natural to assume that the population evolves from a state of zero fusion, . Deciding on a plausible initial daughter cell mass, , takes more thought. We will consider two parsimonious choices.

One parsimonious choice would be that the population is already adapted to either environment 1 or environment 2 and that is given by an evolutionary fixed point in one of these environments (this is the situation illustrated in Fig 2). Suppose that the population was first exposed to environment 1 (with ), and that in this environment there was no selective pressure to evolve cell fusion (see Eq (9)). The population would then initially reside at the zero-fusion fixed point of environment 1,

(18)

as illustrated by the purple circles in Fig 5). In order for selection for cell fusion to act in environment 2, must lie outside the basin of attraction of the stable zero-fusion fixed point in environment 2. In general, the conditions required for this scenario to hold must be obtained numerically. However when ET is large (e.g. in high-density environments), the condition simplifies analytically to

(19)

i.e. the second environment must be sufficiently harsher than the first environment in order for obligate cell fusion to evolve as a stress response. This is illustrated in Fig 6A.

An alternative choice for the initial cell mass assumes that the population has evolved under exposure to both environment 1 and environment 2, but has not yet evolved phenotypically plastic responses. We have shown in [50] (see also Sect 3.1 of S1 Appendix) that for α = 0, the ESS for daughter cell cell mass in such a scenario is a bet-hedging strategy;

(20)

Note that this is intuitively equivalent to the α = 0 evolutionary strategy in environment 1 (weighted by the probability of being in environment 1) plus the evolutionary strategy in environment 2 (weighted by the probability of being in environment 2), as illustrated by the orange circles in Fig 5. We have shown in [50] that Eq (20) is an accurate approximation if the timescale of environmental switching is faster than the evolutionary timescale (.

We must now determine whether the initial condition lies in the basins of attraction of in environment 1 and in environment 2. Again this in general requires a numerical approach. However we show in Sect 3.2.2 of S1 Appendix that analytic progress can be made in the limit of large ET. We find that in the cost region in which a facultative response is possible (), is within the basin of attraction of in environment 1 if

(21)

and is within the basin of attraction of in environment 2 if

(22)

If only one of these conditions holds, then facultative cell fusion can be observed in one environment but not the other. Further, if facultative cell fusion does occur, it will always occur in response to the harsher of the two environments.

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Fig 6. Conditions on cell fusion cost and deterioration in environment required to observe cell fusion as a facultative stress response.

Regions are calculated in the limit of large ET and show when fusion is selected for in both environments (blue, ) and in neither environment (red, e.g. ) and when fusion is selected for in only the harsher environment (purple and pink). On the interval the evolutionary outcome depends on the ratio of environmental parameters and on the initial evolutionary conditions. In panel (A) the population is initially adapted to environment 1, and the condition for facultative cell fusion in environment 2 is given in Eq (19). In panels (B-D) the population has initially adapted a bet-hedging strategy to the two environments. Cell fusion remains at zero in environment 1 but evolves in environment 2 if Eq (21) hold but Eq (22) is violated (see purple regions). Conversely cell fusion evolves in environment 1 but is held at zero in environment 2 if Eq (21) is violated but Eq (22) holds (see pink regions).

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In Fig 6A-6D we illustrate how the possible evolutionary outcomes vary as (the probability of being in environment 1) is reduced (note that the plot for tends to Fig 6A, see Eq (20)). We see that as the time spent in environment 1 is reduced, so too is the parameter regime in which one would expect to see facultative cell fusion in response to a harsher environment 2. However we also see the emergence of a region in which environment 2 is more benign () and in which by symmetry facultative cell fusion can emerge in response to the harsher environment 1 (see pink regions in Fig 6B-6D). In Figs E-G of S1 Appendix, we show that these analytic results are good predictors of the dynamics obtained via numerical simulation.

In summary, for both scenarios described above, we see how facultative binary cell fusion can evolve as a response to harsh environmental conditions that lower the survival probability of daughter cells. Such a response is possible if costs to cell fusion are intermediate and if there is an appreciable increase in environmental harshness, β, between the environments. We have also seen that regular exposure to both environments before the evolution of cell fusion has taken place (such that the population has previously adapted a bet-hedging strategy in daughter cells mass) may reduce the parameter region over which we expect to see cell fusion as a stress response. However we note that in many sexually reproducing unicellular eukaryotes, the environmental conditions that trigger sexual reproduction (and cell fusion during plasmogamy) are very rare; sexual reproduction is estimated to be triggered approximately once in every 2000 generations in Saccharomyces paradoxus and once in every 770 generations in C. reinhardtii [81,82]. These values would be equivalent to in our model, which is the regime in which facultative binary cell fusion occupies the broadest region of parameter space.

Alternative survival functions: the importance of minimum cell size

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Fig 7. Various choices for the survival function.

The blue line illustrates the Vance survival function given in Eq (1) for β = 1. We note that this function implicitly captures the feature of a minimum viable cell size as which the survival of cells drops precipitously (see vertical dashed blue line at m = β ∕ 4). The green dashed line shows the survival function given in Eq (23) with γ = 1 and . Here cell survival decays linearly with mass for small m. The orange dashed line shows Eq (23) with γ = 0 . 75 and .

https://doi.org/10.1371/journal.pcbi.1012418.g007

Throughout the mathematical analysis above, we have considered one specific form of survival function, namely the Vance survival function Eq (1). In order to illustrate the robustness of our results, we explore the effect of alternative survival functions in Sect 5 of S1 Appendix. These functions are illustrated in Fig 7. We show that an important feature of the Vance survival function is that it implicitly accounts for very small cells having a low survival probability (the growth of Eq (1) is sub-linear when m < β ∕ 2 as Eq (1) is convex in this region). Conversely the function 1 −  exp ⁡  ( − γm )  is linear in m when m is small. Consequently we find that the evolutionary dynamics generated by the survival function 1 −  exp ⁡  ( − γm )  lead to selection for the production of infinitely many (and infinitely small) daughter cells. In order to correct for this behavior, we can modify the function to account for a minimum viable cell size;

(23)

The incorporation of this minimum cell size allows us to identify analogous parameter regimes as identified when using the Vance survival function. In particular we once again find that binary cell fusion can be selected for as a stress response should one environment be sufficiently harsher. The exact parameters at which this occurs are dependent on the choice of and the fusion cost, C. However one notable point of departure is the maximum value for C under which cell fusion can be selections for. Whereas we have shown that the Vance survival function can allow selection for cell fusion up to C ≈ 0 . 86, using Eq (23) leads to a maximum cell fusion cost of C = 0 . 5.