Coevolution of the mitotic and meiotic modes of eukaryotic cellular division

The genetic material of a eukaryotic cell (one whose nucleus and other organelles, including mitochondria, are enclosed within membranes) comprises both nuclear DNA (ncDNA) and mitochondrial DNA (mtDNA). These differ markedly in several aspects but nevertheless must encode proteins that are compatible with one another for the proper functioning of the organism. Here, we introduce a network model of the hypothetical coevolution of the two most common modes of cellular division for reproduction: by mitosis (supporting asexual reproduction) and by meiosis (supporting sexual reproduction). Our model is based on a random hypergraph, with two nodes for each possible genotype, each encompassing both ncDNA and mtDNA. One of the nodes is necessarily generated by mitosis occurring at a parent genotype, the other by meiosis occurring at two parent genotypes. A genotype's fitness depends on the compatibility of its ncDNA and mtDNA. The model has two probability parameters, $p$ and $r$, the former accounting for the diversification of ncDNA during meiosis, the latter for the diversification of mtDNA accompanying both meiosis and mitosis. Another parameter, $\lambda$, is used to regulate the relative rate at which mitosis- and meiosis-generated genotypes are produced. We have found that, even though $p$ and $r$ do affect the existence of evolutionary pathways in the network, the crucial parameter regulating the coexistence of the two modes of cellular division is $\lambda$. Depending on genotype size, $\lambda$ can be valued so that either mode of cellular division prevails. Our study is closely related to a recent hypothesis that brings mitochondria to the center stage and views the appearance of cellular division by meiosis, as opposed to division by mitosis, as an evolutionary strategy for boosting ncDNA diversification to keep up with that of mtDNA. Our results indicate that this may well have been the case, thus lending support to the first hypothesis in the field to take into account the role of such ubiquitous and essential organelles as mitochondria.

Figure 1

(a) Generation of genotype $i$ by mitosis from genotype $j$. Sequences $j_1$ and $j_2$ are copied to $i_1$ and $i_2$, respectively, while $i_{\mathrm{mt}}$ is a possibly mutated copy of $j_{\mathrm{mt}}$. (b) Generation of genotype $i$ by meiosis at parents $j$ and $k$. Sequence $i_1$ originates from recombination and mutation between $j_1$ and $j_2$, sequence $i_2$ from $k_1$ and $k_2$. Sequence $i_{\mathrm{mt}}$ is a possibly mutated copy of either $j_{\mathrm{mt}}$ or $k_{\mathrm{mt}}$.

Figure 2

A fragment of hypergraph $H$, showing some of its hyperedges and only those nodes that participate in them. The complete node sets $A$ and $B$ are identical (each contains nodes representing all genotypes), but a hyperedge leading to a node in $A$ is of a different nature than a hyperedge leading to a node in $B$ since nodes in set $A$ represent genotypes generated by meiosis while those in $B$ represent genotypes generated by mitosis. In this fragment, three hyperedges $S_1\to D_1,S_2\to D_2$, and $S_3\to D_3$ indicate genotype generation by meiosis, two from same-set parents, one from mixed parents. Two other hyperedges, $S_4\to D_4$ and $S_5\to D_5$, indicate genotype generation by mitosis, the former from a genotype in $A$, the latter from another genotype in $B$.

Figure 3

Evolution of the relative abundance of genotypes generated by meiosis ($x_A$) for $L=3$ (a)–(c) and $L=4$ (d)–(f). Parameter values omitted from each panel default to $p={10}^{-5},r=0.01$, and $\lambda =109.3$ (for $L=3$) or $\lambda =683$ (for $L=4$).

Figure 4

Probability density of the stationary-state relative abundance of genotype $i\in A\cup B$ for $L=4$ and $x_A\approx 0.04$ (i.e., mitosis dominates), with $p={10}^{-5},r=0.99$, and $\lambda =2.84\times {10}^3$. Each of panels (a)–(e) refers to genotypes having the same number ($k$) of loci at which the mtDNA sequence differs from both ncDNA sequences, therefore having the same fitness $f_i=2^{-k}$, as indicated. Panel (f) refers to all genotypes. All panels contain density spikes very near $x_i=0$ for both mitosis- and meiosis-generated genotypes. These are highlighted in the logarithmic-scale insets, with $x_i$ in the range ${10}^{-5}-{10}^{-2}$ and densities ranging as in the containing panels.

Figure 5

Probability density of the stationary-state relative abundance of genotype $i\in A\cup B$ for $L=4$ and $x_A\approx 0.5$ (i.e., mitosis and meiosis coexist at about the same proportion), with $p={10}^{-5},r=0.99$, and $\lambda =938$. Each of panels (a)–(e) refers to genotypes having the same number ($k$) of loci at which the mtDNA sequence differs from both ncDNA sequences, therefore having the same fitness $f_i=2^{-k}$, as indicated. Panel (f) refers to all genotypes.

Figure 6

Probability density of the stationary-state relative abundance of genotype $i\in A\cup B$ for $L=4$ and $x_A\approx 0.96$ (i.e., meiosis dominates), with $p={10}^{-5},r=0.99$, and $\lambda =27.7$. Each of panels (a)–(e) refers to genotypes having the same number ($k$) of loci at which the mtDNA sequence differs from both ncDNA sequences, therefore having the same fitness $f_i=2^{-k}$, as indicated. Panel (f) refers to all genotypes. Panels (a)–(c) contain density spikes very near $x_i=0$ for mitosis-generated genotypes.

Figure 7

Probability density of the stationary-state relative abundance of genotype $i\in A\cup B$ for $L=3$ (a)–(c) and $L=4$ (d)–(f), in both cases for $x_A\approx 0.96$. Probability $p$ is fixed at $p={10}^{-5}$; probability $r$ varies from $r=0.01$ (a), (d), to $r=0.9$ (b), (e), to $r=0.99$ (c), (f); the value of $\lambda$ for each panel is as follows: $\lambda =3.15$ (a), $\lambda =4.1$ (b), $\lambda =4.05$ (c), $\lambda =21.4$ (d), $\lambda =27.7$ (e), $\lambda =27.7$ (f). Panel (f) is identical to Fig. 6. Panels (a)–(c) contain density spikes very near $x_i=0$ for mitosis-generated genotypes.