Muller’s ratchet in symbiont populations

Abstract

Muller’s ratchet, the inevitable accumulation of deleterious mutations in asexual populations, has been proposed as a major factor in genome degradation of obligate symbiont organisms. Essentially, if left unchecked the ratchet will with certainty cause extinction due to the ever increasing mutational load. This paper examines the evolutionary fate of insect symbionts, using mathematical modelling to simulate the accumulation of deleterious mutations. We investigate the effects of a hierarchical two level population structure. Since each host contains its own subpopulation of symbionts, there will be a large number of small symbiont populations linked indirectly via selection on the host level. We show that although the separate subpopulations will accumulate deleterious mutations quickly, the symbiont population as a whole will be protected from extinction by selection acting on the hosts. As a consequence, the extent of genome degradation observed in present day symbionts is more likely to represent loss of functions that were (near-) neutral to the host, rather than a snap shot of a decline towards complete genetic collapse.

Appendix

Deterministic growth and accumulation of mutations

Assume that a population grows during time T from M ₀ individuals. Unmutated individuals grow with rate 1, individuals that carry k mutations grow with rate (1 + s)^k. Mutations appear with probability μ in each replication. Each mutation increases the growth rate by a factor 1 + s. The deterministic growth equations for the number of individuals, n _k (t), that carry k mutations at time t follow

$${\frac{{{\text{d}}n_{k} }} {{{\text{d}}t}}\; = \;{\left( {{\text{1 }} + s} \right)}^{k} {\left( {{\text{1 }} - \mu } \right)}n_{k} + \mu {\left( {{\text{1 }} + s} \right)}^{{k - {\text{1}}}} n_{{k - {\text{1}}}} }$$

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This is valid for k ≥ 0 if ${n_{{ - {\text{1}}}} \equiv {\text{0 }}{\text{.}}}$ Introducing new variables

$${f_{k} {\left( t \right)}\; = \;{\text{e}}^{{ - {\left( {{\text{1 }} - \mu } \right)}{\left( {{\text{1 }} + s} \right)}^{k} \cdot t}} n_{k} {\left( t \right)}} $$

(6)

the growth equations simplify to

$$ {\frac{{{\text{d}}f_{k} }} {{{\text{d}}t}}\; = \;\mu {\left( {{\text{1 }} + s} \right)}^{{k - {\text{1}}}} {\text{e}}^{{ - {\left( {{\text{1 }} - \mu } \right)}{\left( {{\text{1 }} + s} \right)}^{{k - {\text{1}}}} \cdot st}} f_{{k - {\text{1}}}} {\left( t \right)}}. $$

(7)

For k = 0, the solution is f ₀(t) = f ₀(0) which gives ${n_{{\text{0}}} {\left( t \right)}\; = \;{\text{e}}^{{{\left( {{\text{1 }} - \mu } \right)} \cdot t}} \cdot n_{{\text{0}}} {\left( {{\text{0 }}} \right)}{\text{.}}}$ Inserting f ₀(t) = n ₀(0) into the growth equation for k = 1, we can solve for f ₁(t) giving

$$ {n_{{\text{1}}} {\left( t \right)} = {\text{e}}^{{{\left( {1 - \mu } \right)}{\left( {1 + s} \right)} \cdot t}} {\left\{ {n^{{{\text{0 }}}}_{{\text{1}}} + n^{{0 }}_{0} \frac{\mu } {{s{\left( {{\text{1 }} - \mu } \right)}}}{\left[ {{\text{1 }} - {\text{e}}^{{ - {\left( {{\text{1 }} - \mu } \right)} \cdot st}} } \right]}} \right\}}}, $$

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where n ⁰₁  = n ₁(0) etc. Continuing the recursion to k = 2 gives

$$ \eqalign{{} n_{{\text{2}}} {\left( t \right)}\, = \,{\text{e}}^{{{\left( {1 - \mu } \right)}{\left( {1 + s}\right)}^{{2 }} \cdot t}} \left\{ {n^{\text{0 }}_{{\text{2}}} +n^{{0 }}_{1} \frac{\mu } {{s{\left( {{\text{1 }} - \mu }\right)}}}{\left[ {{\text{1 }} - {\text{e}}^{{ - {\left( {{\text{1}} - \mu } \right)}{\left( {{\text{1 }} + s} \right)} \cdot st}} }\right]}} \right. \cr + n^{{0 }}_{0} {\left({\frac{\mu } {{s{\left( {{\text{1 }} - \mu } \right)}}}}\right)}^{{{\text{2 }}}} \left[ \frac{{{\text{1 }}}} {{{\text{2 }}+ s}} - {\text{e}}^{{ - {\left( {{\text{1 }} - \mu }\right)}{\left( {{\text{1 }} + s} \right)} \cdot st}}\right.\cr +\left.\left.\frac{{{\text{1 }} + s}} {{{\text{2 }} + s}}{\text{e}}^{{ -{\left( {{\text{1 }} - \mu } \right)}{\left( {{\text{2 }} + s}\right)} \cdot st}} \right] \right\}.}$$

(9a)

In principle, the recursion could continue for all values of k with increasing complexity. If for simplicity it is assumed that at most two mutations can be acquired during one growth period, we get

$$\begin{array}{*{20}c} n_{k} \left( t \right) = {\text{e}}^{{\left({1 - \mu } \right)}{\left( {1 + s} \right)}^{k} \cdot t} \left\{n^{0}_{k} + n^{0}_{k - 1} \frac{\mu }{s{\left({{\text{1 }} - \mu } \right)}}\left[ {{\text{1 }} - {\text{e}}^{{- {\left( {{\text{1 }} - \mu } \right)}{\left( {\text{1} + s}\right)}^{{k - {\text{1}}}} \cdot st}} } \right] \right.\cr\left. + n^{0 }_{k - 2} \left( \frac{\mu } {s{\left( {\text{1 } -\mu } \right)}} \right)^{\text{2 } }\left[ \frac{{\text{1 }}}{\text{2 }} + s - {\text{e}}^{{- {\left( {{\text{1 }} - \mu } \right)}{\left( {\text{1} + s}\right)}^{{k - {\text{1}}}} \cdot st}} \right. \right. \cr \left.\left. + \frac{{{\text{1 }} + s}}{{{\text{2 }} +s}}{\text{e}}^{{ - {\left( {{\text{1 }} - \mu } \right)}{\left({{\text{1 }} + s} \right)}^{{k - {\text{2}}}} {\left( {{\text{2 }}+ s} \right)} \cdot st}}\right]\right\}.\end{array} $$

(9b)

In the application to symbiont growth in the main text, the parameter s should be replaced by S _S. Here n _k (t) is the number of symbionts with k mutations at time t, and t is the number of generations needed to get to M _T from M ₀, which is calculated as:

$$ {t = \frac{{{\text{ln}}{\left( {M_{T} /M_{{\text{0}}} } \right)}}} {{{\text{ln}}{\left( {{\text{2 }}} \right)}}}}. $$

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