Evolution of self-limited cell division of symbionts

Abstract

In mutualism between unicellular hosts and their endosymbionts, symbiont's cell division is often synchronized with its host's, ensuring the permanent relationship between endosymbionts and their hosts. The evolution of synchronized cell division thus has been considered to be an essential step in the evolutionary transition from symbionts to organelles. However, if symbionts would accelerate their cell division without regard for the synchronization with the host, they would proliferate more efficiently. Thus, it is paradoxical that symbionts evolve to limit their own division for synchronized cell division. Here, we theoretically explore the condition for the evolution of self-limited cell division of symbionts, by assuming that symbionts control their division rate and that hosts control symbionts' death rate by intracellular digestion and nutrient supply. Our analysis shows that symbionts can evolve to limit their own cell division. Such evolution occurs if not only symbiont's but also host's benefit through symbiosis is large. Moreover, the coevolution of hosts and symbionts leads to either permanent symbiosis where symbionts proliferate to keep pace with their host, or the arms race between symbionts that behave as lytic parasites and hosts that resist them by rapid digestion.

1. Introduction

Symbiotic partners sometimes evolve to form a single inseparable unit of organism, such as plastids and their hosts [1–3]. The evolutionary process driving symbionts to organelles, which is called ‘symbiogenesis’, is a core of the ‘major transition’ [4] from prokaryotes to eukaryotes. Clarifying this process is the major challenge in evolutionary biology to understand the origin of eukaryotes and organelles.

One of the key steps of symbiogenesis is thought to be the evolution of synchronized cell division between endosymbionts and their unicellular host [2,5–7]. If endosymbionts proliferate faster than their host, they accumulate and impose a heavy burden on the host, which may induce its death or burst. By contrast, if their proliferation is slower than their host's, they are left behind by the host's proliferation and are eventually lost from the host. Therefore, synchronized cell division is necessary for the permanent relationship between endosymbionts and their host, and its evolutionary acquisition is necessary at some point of symbiogenesis. In fact, the synchronized cell divisions have been reported in various mutualistic systems between unicellular hosts and their endosymbionts (e.g. green paramecium and its symbiotic algae [8,9] and trypanosomatid parasite and its symbiotic bacterium [10]).

Because endosymbionts would be able to divide potentially faster than the host, their cell division must be limited by themselves and/or by their hosts for the synchronized cell division. Since excessive accumulation of symbionts in the host cell is detrimental, limiting symbionts' cell division is potentially advantageous for hosts. On the other hand, it is clearly paradoxical that symbionts would self-limit their own cell division because it is the principal component of their fitness—if symbionts accelerate their cell division without regard for synchronization with their host, they would be able to proliferate more efficiently. Evolutionary questions then arise as to whether synchronized cell division cannot be achieved unless a host controls its symbionts to limit their cell division. In other words, we ask whether symbionts can evolve to restrain their own cell division and, if they can, under what condition such an evolution could occur.

These questions still remain unresolved. Conventionally, it has been believed that a host controls the cell division of its symbionts to be synchronized with its own division [7,10,11]. From this perspective, empirical and theoretical studies have focused on how a host controls cell division of its symbionts to keep the within-host population size of symbionts to be constant [12–15]. However, the perspective of symbiont's adaptation towards synchronized cell division, which we study in this paper, has not yet, to our knowledge, been explored in the literature. Even if symbionts' cell division seems to be host-controlled, it is by no means obvious that the host achieves the synchronization by actually forcing its symbionts to limit their cell division, because there is a possibility that synchronized cell division could be an adaptive strategy of symbionts. Therefore, it is important to ask whether a self-limited cell division of symbionts would evolve, and, if it is possible, under which conditions.

Here, we theoretically examine when symbionts evolve to limit their division to synchronize with the host's cell division. We constructed a mathematical model that is inspired by the mutualism between unicellular hosts and their symbionts, such as host ciliate and symbiotic algae. In our model of coevolution between hosts and symbionts, we assume that the symbionts can control their division rates in the host, while the hosts can control the mortality rate of the associated symbionts. In particular, the advantage and disadvantage of limiting cell division for symbionts are naturally implemented through our explicit modelling of symbiont dynamics within a host cell. The interaction between hosts and symbionts in our model is either mutualistic or parasitic; namely, although the symbionts always benefit their host, the host can incur more harm than the amount of benefit it receives if their division rate is too high. Our study can provide insights into the evolutionary pathway towards symbiogenesis.

2. Model

To examine the evolution of self-limited cell division of symbionts, we assume that the cell division rate of symbionts in a host cell is controlled by themselves and hence is regarded as an evolutionary trait of symbionts. We also assume that the death rate of symbionts in a host cell is controlled by hosts, and hence is regarded as an evolutionary trait of hosts. Though we do not specify how hosts control the death of symbionts in our model, such control is possible in various ways. For example, hosts can increase the mortality of symbionts through intracellular digestion or resistance against over-grown symbionts. Conversely, they can also reduce the mortality of symbionts by supplying nutrients and physical protection from their enemies (e.g. symbiotic chlorella within a host cell can escape from the infection of chlorella viruses [16]). Hereafter, we call the host-controlled reduction of symbiont's death rate ‘host generosity’: a generous host tries to keep symbionts within its cell longer and more securely, while an ungenerous host tries to kill its symbionts rapidly.

In order to investigate the joint evolution of the self-limited cell division of symbionts and the host generosity to symbionts, we employed the evolutionary invasion analysis of adaptive dynamics [17], where evolutionary dynamics is considered as a sequential trait substitution by the invasion of mutants in the equilibrium population of residents. Therefore, we first constructed the model of population dynamics of hosts and symbionts and then examined the coevolutionary dynamics through the sequential invasion of a rare mutant into the equilibrium population of the resident.

(a). Population dynamics of symbionts and hosts

As shown in figure 1, we consider population dynamics of two species, a unicellular host and its intracellular symbiont. For modelling symbiont dynamics within a host cell explicitly, we classify the host cells according to the number of symbionts residing in its cell. Here, for simplicity, we assume that a host can keep only up to two symbionts and thus keep track of time evolution of the density x₀ of free-living hosts, the densities x_i of hosts that harbour i symbionts (i = 1, 2; see table 1 for the list of symbols). We also keep track of the density y of free-living symbionts. These densities are assumed to change with time as follows (the hosts that harbour i symbionts (i = 0, 1, 2) are denoted by H_i, and the free-living symbionts are denoted by S in the diagram; see also figure 1 and table 1):

• (i)birth and death of free-living hosts: a free-living host proliferates at a rate B₀ and dies at a mortality rate D₀x, where we assumed that mortality is proportional to the total host density, $x\equiv {\sum }_{i=0}^2{x}_i$, with a proportionality constant D₀;
• (ii)birth and death of free-living symbionts: a free-living symbiont proliferates and dies at a rate b₀ and d₀y, where we also assumed that the mortality is proportional to the density of free-living symbionts, y, with a proportionality constant d₀;
• (iii)infection of free-living symbionts to free-living hosts: a free-living symbiont can infect a free-living host at a rate c;
• (iv)
  birth and death of endosymbionts: after infection, an endogenous symbiont proliferates in a host cell with the division rate b and dies in a host cell at a rate d. This endosymbionts' cell division rate b controlled by symbionts themselves and their death rate d controlled by hosts are two key traits focused in our evolutionary analysis. A host cell is assumed to burst at the moment when the number of endogenous symbionts exceeds a prefixed threshold (two in this model). Therefore, a birth of one of the symbionts in the host harbouring two symbionts results in the burst of the host cell—three symbionts are then released alive with probability p to contribute to the free-living symbionts:

  $$\begin{array}{rl}& H_2\stackrel{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}2bp\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}{\to }3S,\hspace{0.17em}{H}_1\stackrel{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}b\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}{\to }{H}_2,\\ & H_2\stackrel{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}2d\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}{\to }{H}_1,\hspace{0.17em}{H}_1\stackrel{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}d\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}{\to }{H}_0;\hspace{0.17em}\mathrm{and}\end{array}$$
• (v)
  birth and death of symbiotic hosts: a symbiotic host proliferates at a rate B, and through the division of the host cell, each symbiont in the mother cell is randomly redistributed in either of the two daughter cells. For example, the cell division of a host with two endogenous symbionts results in two host cells each with one symbiont with probability 1/2, but with the rest of probability, it produces one host cell with two symbionts and another without symbiont (figure 1). A symbiotic host dies at a rate Dx, where, as in free-living hosts, the mortality is assumed to be proportional to the total host density, x, with a proportionality constant D. When it dies, it releases the endogenous symbionts, and with probability p they survive and contribute to the free-living symbiont population:

  $$\begin{array}{rl}& H_2\stackrel{B}{⟶}\frac{1}{2}\times 2H_1+\frac{1}{2}(H_0+H_2),\hspace{0.17em}{H}_1\stackrel{B}{⟶}{H}_0+H_1,\\ & H_2\stackrel{Dxp}{⟶}2S,\hspace{0.17em}{H}_1\stackrel{Dxp}{⟶}S.\end{array}$$

Figure 1.

Schematic diagram of our model. Grey squares and green circles indicate host and symbiont cells, respectively. Transparent symbols represent the transient states during the division (three symbols on the top) and burst (on the right bottom) of symbiotic hosts. Arrows are the transition of hosts and symbionts. The definitions of parameters are described in the main text. (Online version in colour.)

Table 1.

List of symbols in the model. (Symbols with a hat (^) represent their equilibrium values, a prime (′) represent mutant's values and a dot (.) on top represent their derivatives with respect to evolutionary time.)

symbol	definition
xi	density of hosts having i symbionts inside them (i = 0 means free-living host; $x={\sum }_{i=0}^2{x}_i$)
y	density of free-living symbionts
Hi	symbol for host that harbour i symbionts (i = 0, 1, 2)
S	symbol for free-living symbiont
B0	birth rate of a free-living host
D0	proportionality constant of the density-dependent death rate of a free-living host
B	birth rate of a symbiotic host
D	proportionality constant of the density-dependent death rate of a symbiotic host
b0	birth rate of a free-living symbiont
d0	proportionality constant of the density-dependent death rate of a free-living symbiont
b	birth rate of a symbiont within a host
d	death rate of a symbiont within a host
c	association rate per contact between a free-living host and symbiont
p	probability that a symbiont survives the death of its host
Jmut = F − V	Jacobi matrix of mutant dynamics on a resident population (see equations (2.2) and (2.3))
${\mathcal{R}}_0(b\mathrm{\prime },b)$	basic reproductive number of a symbiont mutant (see equation (2.4))
σ	extent of reproductive coupling (see equation (2.5))
${\mathcal{Q}}_0(d\mathrm{\prime },\hspace{0.17em}d)$	basic reproductive number of a host mutant (see equation (2.6))
g(b)	fitness gradient of symbionts
G(d)	fitness gradient of hosts
θ	parameter that determines the speed of symbiont evolution
Θ	parameter that determines the speed of host evolution

Combining these processes, the host densities x₀, x₁, x₂ and the free-living symbionts density y change with time as

$$\left.\begin{array}{rl}\frac{\mathrm{d}{x}_0}{\mathrm{d}t}=& \hspace{0.17em}(B_0-D_{0}x)x_0-c{x}_{0}y+(d+B)x_1+\frac{B}{2}{x}_2,\\ \frac{\mathrm{d}{x}_1}{\mathrm{d}t}=& \hspace{0.17em}c{x}_{0}y-(b+d+Dx)x_1+(2d+B)x_2,\\ \frac{\mathrm{d}{x}_2}{\mathrm{d}t}=& \hspace{0.17em}b{x}_1-\left(2b+2d+\frac{B}{2}+Dx\right)x_2\\ \mathrm{and}\frac{\mathrm{d}y}{\mathrm{d}t}=& \hspace{0.17em}(b_0-d_{0}y)y-c{x}_{0}y+6b{x}_{2}p+Dx(x_1+2x_2)p.\end{array}\right\}$$ 2.1

We assume that both hosts and symbionts can benefit from their symbiotic interaction through the enhancement of their own survival. We measure the magnitude of these benefits by the reduction of the death rates in the symbiotic state compared to that in the free-living state (D₀/D for hosts and d₀/d for symbionts); the larger these ratios, the greater the reduction in death rates by virtue of symbiotic interaction. It should be noted here, however, that the interaction is not always mutualistic even if hosts can enjoy the benefit of reduced mortality (D₀/D > 1). Indeed, if symbionts divide too rapidly within a host cell, the burst of the host cell occurs very frequently, and its cost for hosts may exceed the benefit of reduced mortality brought by symbiosis—such symbionts are parasitic and harmful to hosts. Therefore, the mode of the symbiotic interaction depends not only on the magnitude of reduction in mortality but also on the burst rate of host cells by overgrown symbionts.

The cell division rate of endogenous symbionts and the host generosity (host-controlled survivorship of endogenous symbionts) focused in our model thus describe the continuum of host–symbiont interaction connecting parasitism and mutualism, and also define a trade-off between horizontal and vertical transmission. If symbionts in a host cell divide faster than their host, symbionts are likely to burst their host cell and thus tend to be transmitted horizontally. On the other hand, if symbionts divide slowly, symbionts tend to remain in their host cell and thus are likely to be transmitted vertically (through the host's cell division). Therefore, symbionts face the trade-off between horizontal and vertical transmission, and they can choose either of transmission modes by changing their division rate within a host cell.

(b). Coevolutionary dynamics of symbionts and hosts

We analysed coevolutionary dynamics of symbionts and hosts by considering the invasibility of their mutants in the equilibrium population of residents. Suppose that the populations of hosts and symbionts reach an equilibrium state (see the electronic supplementary material, S1 for the condition for the coexistence of hosts and symbionts), where we represent the densities at the equilibrium by symbols with hats: ${\hat{x}}_0$, ${\hat{x}}_1$, ${\hat{x}}_2$, $\hat{x}={\sum }_{i=0}^2{\hat{x}}_i$ and $\hat{y}$. The equilibrium population densities of residents in the following expressions are calculated numerically by equating the left-hand sides of equation (2.1) to zero (electronic supplementary material, figure S1).

We first ask whether mutant symbionts can invade the equilibrium population of resident symbionts. When mutants are rare, their population dynamics in the equilibrium population of the resident can be given approximately as follows:

$$\begin{array}{rl}& \frac{\mathrm{d}}{\mathrm{d}t}\left(\begin{array}{c}{x\mathrm{\prime }}_1\\ {x\mathrm{\prime }}_2\\ y\mathrm{\prime }\end{array}\right)=\hspace{0.17em}{J}_{\mathrm{mut}}\left(\begin{array}{c}{x\mathrm{\prime }}_1\\ {x\mathrm{\prime }}_2\\ y\mathrm{\prime }\end{array}\right),\\ J_{\mathrm{mut}}=& \left(\begin{array}{ccc}-(D\hat{x}+b\mathrm{\prime }+d)& B+2d& c{\hat{x}}_0\\ b\mathrm{\prime }& -\left(\frac{B}{2}+D\hat{x}+2b\mathrm{\prime }+2d\right)& 0\\ \hspace{0.17em}pD\hat{x}& \hspace{0.17em}p(2D\hat{x}+6b\mathrm{\prime })& b_0-d_0\hat{y}-c{\hat{x}}_0\end{array}\right),\end{array}$$ 2.2

where $x_i^{\mathrm{\prime }}$ and y′ are the densities of hosts that harbour i mutant symbionts and free-living mutant symbionts, respectively, and b^′ is the division rate of mutants (b is that of residents). By focusing on the singly infected state in the symbiont's life cycle, we can construct the next generation matrix by decomposing the matrix J_mut of the mutant dynamics (2.2) as J_mut = F − V, where F is the matrix made up only of the terms for the production of singly infected state and −V is the matrix for the other changes in the states:

$$\begin{array}{rl}F& =\left(\begin{array}{ccc}0& B+2d& c{\hat{x}}_0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right),\\ V& =\left(\begin{array}{ccc}D\hat{x}+b\mathrm{\prime }+d& 0& 0\\ -b\mathrm{\prime }& \frac{B}{2}+D\hat{x}+2b\mathrm{\prime }+2d& 0\\ -pD\hat{x}& -p(2D\hat{x}+6b\mathrm{\prime })& d_0\hat{y}+c{\hat{x}}_0-b_0\end{array}\right).\end{array}$$ 2.3

We define the next generation matrix as FV⁻¹ whose dominant eigenvalue represents the basic reproductive number ${\mathcal{R}}_0$ [18,19] of the mutant lineage in the resident population. Note that the basic reproductive number for a mutant defined here is different from the basic reproductive number in epidemiological models—the former assumes the equilibrium population of the resident genotype but the latter assumes a completely susceptible host population. Because only the first row of the next generation matrix FV⁻¹ has a non-zero element (electronic supplementary material, S2, equation (S2.2)), the (1, 1)-element, denoted by ${\mathcal{R}}_0(b\mathrm{\prime },\hspace{0.17em}b)$, gives the dominant eigenvalue, and thus it is the basic reproductive number:

$$\begin{array}{rl}{\mathcal{R}}_0(b\mathrm{\prime },b)=& \frac{b\mathrm{\prime }}{D\hat{x}+b\mathrm{\prime }+d}\frac{B+2d}{(B/2)+D\hat{x}+2b\mathrm{\prime }+2d}\\ & +\frac{\hspace{0.17em}pD\hat{x}}{D\hat{x}+b\mathrm{\prime }+d}\hspace{0.17em}\frac{c{\hat{x}}_0}{d_0\hat{y}+c{\hat{x}}_0-b_0}\\ & +\frac{b\mathrm{\prime }}{D\hat{x}+b\mathrm{\prime }+d}\frac{\hspace{0.17em}p(2D\hat{x}+6b\mathrm{\prime })}{(B/2)+D\hat{x}+2b\mathrm{\prime }+2d}\frac{c{\hat{x}}_0}{d_0\hat{y}+c{\hat{x}}_0-b_0}.\end{array}$$ 2.4

Note that reproduction in the free-living state is included in ${\mathcal{R}}_0$ by the term $c{\hat{x}}_0/(d_0\hat{y}+c{\hat{x}}_0-b_0)$ (electronic supplementary material, S2). In this way, we define the invasion fitness of the mutant ${\mathcal{R}}_0(b\mathrm{\prime },b)$ as the expected lifetime reproductive output of a mutant symbiont in a singly infected host, which is measured by the number of secondly produced host cells harbouring a single mutant symbiont. This is a kind of basic reproductive number for a mutant symbiont but is more precisely to be described as the type-reproduction number (TRN) for a singly infected state [20,21]. TRN and basic reproductive number give the same threshold condition for invasion.

The basic reproductive number (equation (2.4)) is made up of the sum of the three terms, each corresponding to different reproductive cycles of symbionts (figure 2): (i) a short cycle of symbiont reproduction through vertical transmission (the cycle between singly and doubly infected host cells); (ii) a shorter cycle of horizontal transmission (the cycle between free-living symbionts and singly infected host cells); and (iii) a longer cycle of horizontal transmission (the cycle starting from the transition of symbiont state from free-living to singly infected, then to a doubly infected state through symbiont reproduction within a host, ending up with the release of free-living symbionts from a doubly infected host). In particular, on cycle (i), reproduction of an endosymbiont and its host are coupled with each other, and they maintain their symbiotic relationship through the cycle. By contrast, in cycles (ii) and (iii), the reproduction of a symbiont is not coupled with that of the host—indeed, the symbiont escapes from its host through the death or burst of the host and re-associates with a new one. Thus, we regard the extent of reproductive coupling σ as the proportion of the contribution of cycle (i) in the total reproductive outcome ${\mathcal{R}}_0(b,b)$ of a symbiont as follows:

$$\begin{array}{rl}\sigma & =\frac{(b/D\hat{x}+b+d)\cdot (B+2d)/((B/2)+D\hat{x}+2b+2d)}{{\mathcal{R}}_0(b,b)}\\ & =\frac{b}{D\hat{x}+b+d}\frac{B+2d}{(B/2)+D\hat{x}+2b+2d},\end{array}$$ 2.5

where ${\mathcal{R}}_0(b,b)=1$ at an equilibrium population.

Figure 2.

Schematic diagram of life cycles derived from our model. (a) shows the life cycle of symbionts. Arrows represent the transitions and productions from the viewpoint of symbionts, and mathematical symbols denote these rates. From the viewpoint of lifetime reproduction, each rate is scaled by the average sojourn time of a symbiont in the state, which is the reciprocal of the rate at which a symbiont leaves the state: $1/(D\hat{x}+b+d)$ (a host holding one symbiont), $1/(B/2+D\hat{x}+2b+2d)$ (a host holding two symbionts), $1/(d_0\hat{y}+c{\hat{x}}_0-b_0)$ (free-living state; see the electronic supplementary material, S2 for detail). The reproductive outputs through sub-cycle (i), (ii) and (iii), respectively, correspond to the first, second and third term of the invasion fitness of a symbiont (equation (2.4)). (b) shows the life cycle of hosts. Sub-cycles (I), (II) and (III), respectively, correspond to the first, second and third term of the invasion fitness of a host (equation (2.6)). (Online version in colour.)

Since mutants can invade the population when their invasion fitness, ${\mathcal{R}}_0(b\mathrm{\prime },b)$, is larger than 1, the fitness gradient $g(b)\equiv \mathrm{\partial }{\mathcal{R}}_0/\mathrm{\partial }b\mathrm{\prime }{|}_{b\mathrm{\prime }=b}$ determines the direction of the evolution: if g(b) > 0, symbionts evolve to increase their division rate b; otherwise, if g(b) < 0, they decrease their division rate. The directional selection halts on an evolutionary singular point b* where g(b*) = 0. The singular point is convergence stable if dg/db|_b=b* < 0, and it is evolutionarily stable if ${\mathrm{\partial }}^2{\mathcal{R}}_0/\mathrm{\partial }{b}^{{\hspace{0.17em}}^{\mathrm{\prime }}2}{|}_{b\mathrm{\prime }=b=b^{\ast }}<0$ [17].

We next define the invasion fitness ${\mathcal{Q}}_0(d\mathrm{\prime },d)$ of a mutant host that adjusts their symbionts' mortality to rate d^′ in the equilibrium population of the resident host that adjusts the symbiont mortality to d, in a similar way as we defined the symbionts' invasion fitness (equation (2.4)). It can be described as follows (see the electronic supplementary material, S2 for derivation):

$$\begin{array}{rl}{\mathcal{Q}}_0(d\mathrm{\prime },d)& =\frac{b}{D\hat{x}+b+d\mathrm{\prime }}\frac{B+2d\mathrm{\prime }}{(B/2)+D\hat{x}+2b+2d\mathrm{\prime }}\\ & +\frac{B+d\mathrm{\prime }}{D\hat{x}+b+d\mathrm{\prime }}\frac{c\hat{y}}{D_0\hat{x}+c\hat{y}-B_0}\\ & +\frac{b}{D\hat{x}+b+d\mathrm{\prime }}\frac{(B/2)}{(B/2)+D\hat{x}+2b+2d\mathrm{\prime }}\frac{c\hat{y}}{D_0\hat{x}+c\hat{y}-B_0}.\end{array}$$ 2.6

Then, the fitness gradient is $G(d)\equiv \mathrm{\partial }{\mathcal{Q}}_0/\mathrm{\partial }d\mathrm{\prime }{|}_{d\mathrm{\prime }=d}$, and the evolutionary singular point is d* where G(d*) = 0, and the convergence and evolutionary stability, respectively, are determined by dG/dd|_d=d* and ${\mathrm{\partial }}^2{\mathcal{Q}}_0/\mathrm{\partial }{d}^{{\hspace{0.17em}}^{\mathrm{\prime }}2}{|}_{d\mathrm{\prime }=d=d^{\ast }}$.

According to Dieckmann & Law (1996) [22], the long-term coevolutionary dynamics of symbionts and hosts can be represented by using the invasion fitness as following:

$$\begin{array}{c}\dot{b}=\theta g(b),\\ \dot{d}=\mathit{\Theta }G(d),\end{array}$$ 2.7

where the dots indicate the time derivative in an evolutionary time scale, θ and Θ are parameters of symbionts and hosts that determine the speed of their evolution, which consist of the rate of mutation and the variance of its phenotypic effect.

3. Results

We will first examine the evolution of symbiont division rate in a host for a fixed generosity of hosts (equation (2.4)). We will next show the results for the evolution of the host's generosity for a fixed symbiont's division rate (equation (2.6)). Finally, we will show the consequence of the joint evolution of the symbiont's and host's traits (equation (2.7)).

(a). Evolution of self-limited cell division of symbionts

We found that the division rate of symbiont in a host evolves to a low level only when both symbionts and hosts enjoy large benefit from their symbiotic interaction (figure 3a). Conversely, if either symbiont's or host's symbiotic benefit is very low (the white region in figure 3a), the division rate of symbionts increases without limit (electronic supplementary material, S3). In particular, this is the case even if symbionts unilaterally enjoy large benefits.

Figure 3.

The effect of symbiotic benefits to hosts and symbionts on the evolution of symbionts. (a) shows the heat map for the evolved value of division rate of symbionts b*. A lighter colour in the panel indicates a higher value of the evolved division rate b*. In particular, the division rate of symbionts, b, evolves to infinity in the white region below the dashed line (electronic supplementary material, S3). In each axis, the death rates of the symbiotic state are varied, while those in the free-living state are kept constant. (b) shows the heat map for the extent of reproductive coupling, σ, when the division rate of symbionts is reached to the evolutionary endpoint shown in (a). A darker colour indicates that symbionts rely more heavily on vertical transmission through synchronized cell division. In the white region below, the dashed line (electronic supplementary material, S3), symbionts rely perfectly on horizontal transmission. Parameter values are B₀ = B = 1, D₀ = 0.5, b₀ = 3, d₀ = 2, c = 0.1 and p = 1. (Online version in colour.)

Note that, under large symbiotic benefits to both symbionts and hosts, the evolved division rate of symbionts, b*, can be lower than the division rate of free-living symbionts, b₀. More importantly, we find that it does not coincide with the division rate of hosts, B. Rather, the endogenous symbionts grow slower than the host in an evolutionary stable state. It is because, even if the endogenous symbionts have the same division rate as their host's, the chances are large that an earlier cell division of symbionts destroys the host by exceeding the threshold number. In fact, when the division rate of symbionts is equivalent to that of their host, the extent of reproductive coupling, σ, is less than 0.4—it approaches 1 only when the symbiont division rate is sufficiently smaller than that of the host (figure 3b).

These results are essentially unaffected by the change of the survival probability p of symbionts through the burst of their host cell (electronic supplementary material, S4). One may expect that symbionts should reduce their division rate if symbionts almost always die when their host dies (p ≈ 0), because bursting their host inevitably kills themselves. However, our result shows that this is not the case; even in such a situation, the division rate of symbionts can evolutionarily increase.

In summary, symbionts can evolve to limit their cell division for the synchronized cell division when the mutualistic interaction of hosts and symbionts brings large benefits to both of them. Interestingly, the result is not affected by varying the survival probability. The other parameters also have only marginal effects on the above results (electronic supplementary material, S5).

(b). Evolutionary maintenance of symbionts by generous hosts

We next study the opposite case where symbionts' trait does not evolve but hosts’ does. The evolution is described by equation (2.6). The endpoint of the evolutionary dynamics corresponds to the evolved host generosity (electronic supplementary material, S6), which shows that the host evolves to be generous (maintains their symbionts for a long time) only if symbionts divide slowly. In other words, if mutualistic symbionts rarely burst their host, the host generosity evolves. Conversely, if symbionts divide quickly and thus burst their host frequently, hosts become less generous and resist those virulent symbionts by killing them rapidly.

(c). Coevolutionary dynamics of hosts and symbionts

Finally, we study coevolution of symbionts and hosts. The coevolutionary dynamics of division rate of symbionts and generosity of hosts (equation (2.7)) are described by combining the evolutionary dynamics for each. Figure 4 shows typical phase planes of the coevolutionary dynamics. When the symbiotic benefit of hosts, D₀/D, is low, there are no coevolutionary equilibria for symbionts' and hosts’ traits (figure 4a). Symbionts evolve to increase their division rate indefinitely, while hosts retaliate such symbionts by evolutionarily decreasing their generosity towards symbionts, leading to an arms race between them. The symbionts are better off by dividing themselves as rapidly as possible and they come to rely exclusively on the horizontal transmission for their reproduction, as lytic parasites do. The hosts are better off by being intolerant as much as possible to symbionts and by eliminating them rapidly.

Figure 4.

The coevolutionary dynamics of division rate of symbionts and generosity of hosts. (a) and (b) are typical cases where the symbiotic benefit of hosts is low, log₁₀(D₀/D) = 0.68, and high (=1.0), respectively. Thick black lines are typical trajectories of the coevolution of symbiont-controlled trait b (cell division rate of endosymbiont) and host-controlled trait d₀/d (generosity). The solid one corresponds to the case where symbionts evolve 10 times slower than hosts (θ : Θ = 1 : 10), and the dashed one is the opposite case (θ : Θ = 10 : 1). Thin arrows and lines indicate the directions and nullclines of evolutionary dynamics, respectively. The dark green and dotted grey, respectively, correspond to symbionts and hosts. Note that the symbiont division rate b and host generosity d₀/d seems to be evolving towards infinity, but these should be best interpreted that they evolve to a physiological maximum. The other parameters are B₀ = B = 1, D₀ = 0.5, b₀ = 3, d₀ = 2, c = 0.1 and p = 1. (Online version in colour.)

On the other hand, when the symbiotic benefit of hosts is high, in addition to the aforementioned endpoint of parasitism (dashed line in figure 4b), another coevolutionary outcome arises, in which symbionts self-limit their division rate and are transmitted mainly vertically and hosts remain generous so that they keep their symbionts for a long time (solid line in figure 4b). When the latter coevolutionary outcome occurs, the interaction between hosts and symbionts is mutualistic and permanent and they are like hosts and organelles. Whether the coevolution results in the arms race or symbiogenesis depends on the initial condition of the coevolution and the speed of their evolution.

For example, suppose that the relationship between symbionts and hosts originates from a prey–predator relationship, where a host (a predator) is not generous and attempts to digest its symbionts (preys) rapidly and symbionts almost cannot divide within the host (high d and low b, corresponding to the left bottom region of figure 4b). If symbionts evolve much slower than hosts then we predict that the coevolution should lead to symbiogenesis (solid line in figure 4b). This condition means that the coevolutionary trajectory in figure 4b moves rightwards rapidly but upwards only slowly, until hosts and symbionts coevolve to vertically transmitted mutualism.

4. Discussion

The evolution of synchronized cell division is an essential step of the evolutionary transition towards symbiogenesis. In this study, we theoretically revealed the condition under which symbionts evolve to limit their own cell division for synchronization with hosts. Our study has shown that, even if symbionts can potentially increase their division rate unlimitedly, the self-limited cell division and thus synchronized cell division can evolve when the mutualistic interaction of hosts and symbionts brings large benefits to both of them. Moreover, its coevolution with host generosity, which is the tendency of a host to keep its symbionts alive, heads in the dichotomous direction depending on the initial state: an arms race between the symbionts that rapidly divide in their host and ungenerous hosts that try to kill them, or symbiogenesis, where symbionts limit their cell division and their host keeps them for a long time. In this way, our study indicates that symbiogenesis can proceed only after the mutualistic relationship is established, where symbionts are willing to reduce their division rate.

Our model suggests that a large symbiotic benefit for symbionts alone is not enough to bring the self-limited cell division—the benefit to hosts must also be large for it to evolve. There are two reasons why the benefit of hosts plays a critical role in the evolution of limited cell division of symbionts in our model. First, the large benefit of hosts can make it hard for a symbiont to find new free-living hosts, because symbiotic hosts dominate in the host population and thus the frequency of free-living hosts decreases. It will encourage symbionts to stay in their host without bursting it. Second, the prolonged life expectancy of a symbiotic host ensures that symbionts within the host enjoy the symbiosis for a long time. These two factors are also known as key factors for parasites to evolve towards heavier reliance on vertical transmission at the expense of reduced horizontal transmission [23–25], and thus our study shows that these factors can be implemented by large symbiotic benefits for hosts. Moreover, in our model, when symbionts benefit from the interaction, the evolved division rate of symbionts decreases along with the increase of symbiotic benefit for their host. The prediction might correspond to the experimental result that the average number of symbiotic Chlorella per host ciliate decreases when they become more beneficial for their host [26].

On the other hand, unless mutualistic benefits make vertical transmission more effective than a horizontal one, the evolved division rate can diverge infinitely in our model. It is because, in our model, the more rapidly symbionts divide and kill their hosts, the more efficient their horizontal transmission. It corresponds to the situation where infinitely large virulence evolves in pathogens when horizontal transmission rate increases linearly or more than linearly with virulence [27,28]. Moreover, these results are basically unaffected by the change of the survival probabilities p of released symbionts from burst or dead hosts. It is probably because the difficulty in surviving the host's death will be cancelled out by a large benefit of easily finding new hosts. Indeed, if p is small, almost all symbionts die with the host's death, indicating that the potential competitors for acquiring new free-living hosts decrease.

In addition to the condition of the evolution of self-limited division rate, our model also suggests that the coevolution of symbiont division rate and host generosity can proceed towards the arms race and symbiogenesis. A popular explanation for the evolutionary transition from eukaryotic algae to plastids is the ‘stuck-in-the-throat’ theory [5,29], which suggests that ‘a phagotrophic, non-photosynthetic eukaryote swallows but fails to digest a photosynthetic eukaryote’ ([29], p.36). Our result is, to our knowledge, the first to show that, as the theory suggests, prey–predator systems can proceed towards symbiogenesis. On the other hand, our result also shows that there is an alternative outcome, host–parasite arms race. This dichotomous outcome of the coevolution might correspond to the contrasting behaviours of two symbionts found in Paramecium ciliate: although both of them are ingested by Paramecium ciliate, the cell division of mutualistic algae Chlorella is synchronized with their hosts [8,9]; whereas, the cell division rate of parasitic bacterium Holospora is so fast that it disturbs host's cell division and kills the host [30]. Moreover, our model suggests that the difference in their benefit to the host or in evolutionary speed can explain the different evolutionary outcomes for Chlorella and Holospora. In particular, the rapid evolution of symbionts (prey) leads to the host–parasite arms race in our model, and it seems to be reasonable to assume that the bacterial parasites, Holospora, evolve much faster than the eukaryotic hosts, Paramecium, whereas the eukaryotic algae, Chlorella, does not.

Conventionally, theoretical studies have often hypothesized on the parasitic origin of mutualism and considered that the evolutionary transition from parasitism to mutualism is brought about by the evolution of exclusive vertical transmission [31–33]. However, our study suggests that such evolutionary transition is difficult in the symbiosis between unicellular hosts and their endosymbionts from the two following reasons. First, our theoretical result indicates the opposite order; that, because the evolved division rate can diverge infinitely unless mutualistic benefits for both hosts and symbionts are already present and large enough, mutualism is required in advance for the evolution of synchronized cell division and thus for the evolution of vertical transmission. Second, our theory predicts that host–parasite arms race and host-mutualist symbiogenesis are two opposite endpoints of the coevolution of the symbiont division rate and host generosity, and thus the evolutionary transitions between parasitism and mutualism are difficult.

In contrast to previous studies, our model suggests another pathway where mutualism evolves in advance of the evolution of synchronized cell division (exclusive vertical transmission) and then brings the evolution of synchronized cell division. Such transition has a close connection to the context of the evolution of an organelle as discussed above. However, if mutualism evolved in advance of the evolution of exclusive vertical transmission, how it evolved without vertical transmission has to be asked. One possibility that enhances the evolution of mutualism is ‘partner choice’, which is an active behavioural response to the quality or behaviour of partners to reward cooperative ones but not free-riding or less-cooperative ones [34]. In the example of the Paramecium–Chlorella relationship, it is experimentally shown that the ciliates can associate preferentially with photosynthesizing chlorella compared with ones whose photosynthesis is inhibited [35]. The preference by hosts might promote the evolution of nutrient supply by chlorella as a partner choice even if the vertical transmission is absent.

In order to discuss the evolutionary transition from partner choice-based mutualism to vertical transmission-based mutualism, several limitations of our model should be relaxed. Previous studies have shown that the evolutionary maintenance of mutualism by partner choice requires high variability in symbiont quality [36] (but see [37]), because such choosiness can no longer be favoured if symbionts have similar quality. However, our model assumed that a host is asexual and infected by only one strain of symbionts throughout its symbiotic period. The lack of multiple infection precludes the competition among coexisting symbionts in a host. Such assumption allowed us to concentrate on the trade-off between vertical and horizontal transmission through the division rate of symbionts, but eliminated the opportunity for intra-host competition between variable symbionts. To theoretically bridge the two types of mutualism discussed above, these effects should be incorporated into our model in the future.

Although it has been believed widely in the context of organelle evolution that a host controls the cell division of its symbionts, our study shows that mutualistic symbionts should have an incentive to reduce their own division rate. Therefore, the evolution of synchronized cell divisions between hosts and their symbionts has to be re-examined from the perspective of self-restriction of the symbiont's growth. In the latter evolutionary scenario, mutualism has to evolve in advance of the evolution of synchronized cell division and vertical transmission. In other words, the evolution of mutualism might not be led by vertical transmission as has been argued in the context of pathogen evolution. Rather, vertical transmission might be led by mutualism that evolved through partner choice or the like. However, there are only limited pieces of empirical and theoretical evidence that supports the pathway we proposed here. To reveal the evolutionary transitions towards symbiogenesis, possible pathways should be further investigated both empirically and theoretically.

Data accessibility

Additional data can be found in the electronic supplementary material.

Authors' contributions

Y.U., H.O. and A.S. designed the study and wrote the manuscript. All authors gave final approval for publication.

Competing interests

We have no competing interests.

Funding

This study was supported by JSPS KAKENHI grant number JP 15J10728 (to Y.U.).