1. Introduction
Bacterial speciation differs from that in animals or plants, where the natural limitations on breeding exist, due to the lack of sexual reproduction in prokaryotes. Nonetheless, bacteria are capable of obtaining genetic information from sources other than their maternal cells.
Some species can acquire DNA directly from the environment. This process is called natural transformation. Approximately 1% of bacterial species are known to have this ability, that is, are competent (Jonas et al., 2001; Thomas and Nielsen, 2005). Many of these species are not permanently competent, their ability to uptake DNA being induced by many factors such as stress and starvation.
Other mechanisms for horizontal DNA transfer are conjugation and transduction. Noncompetent species, such as Escherichia coli, acquire DNA from other bacteria through conjugative plasmids (conjugation) or phages (transduction) (Weinbauer and Rassoulzadegan, 2004; Arutyunov and Frost, 2013; Dixit et al., 2015).
Following uptake, DNA can be used by a cell as food or integrated in the genome by homologous recombination. As it has been demonstrated in vitro, the probability of successful homologous recombination depends, first, on the similarity of the recombining segments and, second, on their length (Shen and Huang, 1986; Vulić et al., 1997; Majewski and Cohan, 1999).
Homologous recombination plays a major role in shaping bacterial species (Chan et al., 2011; Yahara et al., 2012). The process of homologous recombination is believed to be more intensive within bacterial species than between them due to higher similarity of genomes and common environment (Skippington and Ragan, 2012). Thus, bacterial species should be homogeneous, but, in fact, they often form stable subspecies or phylogenetic groups (Guttman and Dykhuizen, 1994; Chaudhuri and Henderson, 2012), which may be considered as the early stage of the bacterial speciation.
The emergence of clusters of genomes as a result of niche specialization, geographical isolation, or selective pressure is possible (Koeppel et al., 2013; Polz et al., 2013; Cheng et al., 2015), but it is not obvious whether clusters may emerge in neutral models with only the mutation and homologous recombination processes.
Previous studies generated no consensus on the emergence of stable clusters of genomes in neutral models. Falush et al. (2006) have shown that stable isolated clusters emerge in the neutral model with appropriate values of the mutation rate to the recombination rate ratio and other parameters of simulation. More general simulations showed that the emergence of clusters is likely in the absence or low rate of homologous recombination, where the clonal populations form clusters, whereas the high rate of homologous recombination acts like a cohesive force (Fraser et al., 2007).
Furthermore, it has been analytically shown that distinct populations may be maintained by the mutation and homologous recombination processes without other factors (Doroghazi and Buckley, 2011). However, in this study, the distance between two populations was defined as the mean distance between all pairs of genomes, so if two similar populations with high variance formed one cluster, they still had nonzero distance between them.
An experimental study on dependence of recombination rate on sequence similarity in vivo (Bao et al., 2014) demonstrated that if the recombination rate fell as sequence divergence increased, no clear-cut genomic boundaries between species emerged. On the contrary, such boundaries are observed (Tang et al., 2013), and the process of uptake exogenous DNA in vivo differs significantly from that in vitro. Understanding of the bacterial population behavior in the neutral model entails understanding of bacterial subspecies isolation and reduction of homologous recombination between them (Ellegaard et al., 2013).
In this study, we consider the possibility of phylogroup emergence in the neutral model due to solely mutations and recombination. In Lyubich and Yu (1971) and Lyubich (1992), this situation was analyzed for a diploid population. The convergence to equilibrium was proved, but dependence of the recombination rate on sequence similarity was not considered. This property of homologous recombination is essential in all studies on bacterial speciation in the neutral model. Special models of the recombination process were studied in Baake and Baake (2003) and Baake (2011a, 2011b) using explicit formulas. We develop the qualitative theory of such processes based on the method of Lyapunov functions.
We define a bacterial population as a set of genomes that continuously exchange genetic information through homologous recombination. For simplicity, we assume that the genomes can be aligned throughout their entire length, so that coordinates in a genome completely define the homologous region in another genome. Below, after giving formal definitions, we write a differential equation that describes a population under mutation and recombination processes in terms of probability measures on the space of genomes and examine its fixed points. The equation describes the behavior of a population in the infinite size limit. For the finite size, there is no closed system of equations for the average fractions of different genomes in the population. Our main tool, the monotonicity of the entropy, was used in other situations in Kun and Lyubich (1980) and by L. Boltzmann in statistical physics. The monotonicity of the relative entropy was studied for some equations of chemical kinetics (Sontag, 2001; Gunawardena, 2003; Batishcheva and Vedenyapin, 2005). We used it to study the recombination processes. A nontrivial behavior of the equation solutions would correspond to a complex population structure that hypothetically could emerge in this model.
2. Results
Let K be a finite alphabet (a set of nucleotides) and let a genome x be a word of length
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$$n$$
\end{document}
over it. We consider two transformations of a genome:
(1) Mutation, when one letter changes to another
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$${x_i} \to {y_i}$$
\end{document}
,
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$$i \in \Lambda = \{ 1 , \ldots , n \} $$
\end{document}
, the mutation matrix is supposed to be irreducible, that is, it is possible to get any letter from any other by several mutations.
(2) Homologous recombination, when a substring
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$${x_I}$$
\end{document}
changes with a certain probability to substring
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$${y_I}$$
\end{document}
with the same coordinates from another genome
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$$y$$
\end{document}
. Here
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$$I$$
\end{document}
is any subset of
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$$\Lambda$$
\end{document}
,
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$$I \subset \Lambda$$
\end{document}
(hence, this definition is more general than in biology, where
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$$I$$
\end{document}
should be an interval in
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$$\Lambda$$
\end{document}
).
The fundamental difference between these two transformations is that mutations occur in a genome independently of other genomes. Formally, for any position
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$$i$$
\end{document}
in a genome, for any nucleotides
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$$a , b \in K , a \ne b$$
\end{document}
, there exists a probability of transition
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$$a \to b$$
\end{document}
, denoted by
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$${ \alpha _i} ( a , b )$$
\end{document}
. This means that for a small period of time
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$$dt$$
\end{document}
, the probability of mutation of nucleotide
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$$a$$
\end{document}
to nucleotide
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$$b$$
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approximately equals
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$${ \alpha _i} ( a , b ) dt$$
\end{document}
.
Homologous recombination results from interaction of genomes in the space of genomes
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$$X$$
\end{document}
. The recombination probability depends on the distribution of genomes in the space
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$$X$$
\end{document}
and a function
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$$\varphi ( {x_I} , {y_I} )$$
\end{document}
, which defines similarity between genomes
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$$x$$
\end{document}
and
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$$y$$
\end{document}
on substring
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$$I$$
\end{document}
. This function is symmetric and non-negative. The distribution of genomes in
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$$X$$
\end{document}
is characterized by the probability distribution
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$${ \mu _ \Lambda } ( x )$$
\end{document}
. Thus, the probability
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$$P_ \mu ^{ ( I ) } ( x \to y )$$
\end{document}
of substitution of a substring
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$${x_I}$$
\end{document}
in genome
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$$x$$
\end{document}
to substring
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$${y_I}$$
\end{document}
from genome
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$$y$$
\end{document}
equals
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$$\varkappa \varphi ( {x_I} , {y_I} ) { \mu _I} ( {y_I} ) dt$$
\end{document}
up to terms of order
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$${ ( dt ) ^2}$$
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,
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$$\varkappa$$
\end{document}
is a constant and
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$${ \mu _I} ( {y_I} )$$
\end{document}
is the marginal distribution, that is, the probability distribution of substring
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$${y_I}$$
\end{document}
.
Importantly, the probability of recombination on a substring
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$$I$$
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in a genome depends on the probability distribution of all genomes in
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$$X$$
\end{document}
. Such processes are called continuous-time nonlinear Markov processes in the sense of McKean (1996) (i.e., Markov processes whose generator depends on a measure). The dependence of the probability distribution
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$$\mu ( x )$$
\end{document}
on time is described by a nonlinear differential equation
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\begin{align*}
\frac {d{ \mu _ \Lambda } ( {x_ \Lambda } ) } {dt} =& \sum
\limits_i \sum \limits_{{y_i}} \left( {{ \alpha _i} ( {y_i} ,
{x_i} ) { \mu _ \Lambda } ( {x_{ \Lambda \backslash i}} , {y_i} )
- { \alpha _i} ( {x_i} , {y_i} ) { \mu _ \Lambda } ( {x_ \Lambda }
) } \right) \\
\qquad\qquad\qquad\qquad\qquad\quad &{+ \varkappa \sum \limits_I
\sum \limits_{{y_I}} \left( { \varphi ( {y_I} , {x_I} ) { \mu _I}
( {x_I} ) { \mu_\Lambda } ( {x_{ \Lambda \backslash I}} , {y_I} )
- \varphi ( {x_I} , {y_I} ) { \mu _I} ( {y_I} ) { \mu _ \Lambda }
( {x_ \Lambda } ) } \right), } \tag{1}
\end{align*}
\end{document}
(unlike the linear Kolmogorov forward equation for usual Markov processes). The right-hand side of this equation is the sum of the following terms:
(1) Linear terms for mutations.
(2) Nonlinear terms for substrings
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$$I \subset \Lambda$$
\end{document}
, where recombination is possible.
As proved by Kurtz (1970) and Ethier and Kurtz (1986), this equation is exact in the infinite size population limit. Such deterministic description of recombination processes was used also in a study by Lyubich, 1992; Buerger, 2000.
In this study, we prove that if only mutation and recombination processes are considered, and the similarity function
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$$\varphi ( {x_I} , {y_I} )$$
\end{document}
is symmetric, then for all values of other parameters, such as the ratio of the intensity of mutation and recombination events, or an initial distribution of genomes, there is a unique fixed point. This fixed point, as we show below, is the stationary distribution
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$${q_ \Lambda }$$
\end{document}
for the pure mutation process (the process without recombination).
Theorem. Equation (1) has a unique fixed point
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$${q_ \Lambda }$$
\end{document}
and all trajectories of (1)
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$${ \mu _ \Lambda } ( t ) \to {q_ \Lambda }$$
\end{document}
as
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$$t \to \infty$$
\end{document}
.
Note. From the convergence of trajectories it follows that for a population consisting of
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$$N$$
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individual bacteria (in the stationary state), the fraction
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$${f_N} ( x )$$
\end{document}
of bacteria having genome
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$$x$$
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converges in probability to
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$${q_ \Lambda } ( x )$$
\end{document}
when
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$$N$$
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tends to infinity, see Liggett (2005, chapter 1). It follows also that
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$$E{f_N} ( x ) \to {q_ \Lambda } ( x )$$
\end{document}
when
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$$N \to \infty$$
\end{document}
.
We have no detailed information about the dependencies between genomes for finite
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$$N$$
\end{document}
. However, in the limit of infinite population size, genomes sampled from the population are independent. The asymptotic independence also follows from the convergence of fractions
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$${f_N} ( x )$$
\end{document}
(Pirogov and Petrova, 2014).
To prove the Theorem, we use the Lyapunov method. The Lyapunov function is the Kullback–Leibler divergence (relative entropy) of
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$${ \mu _ \Lambda }$$
\end{document}
with respect to
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$${q_ \Lambda }$$
\end{document}
.
Consider the mutation and recombination processes separately. As mentioned above, if
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$$dt$$
\end{document}
is small, the recombination process on the substring
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$$I$$
\end{document}
can be described as a nonlinear discrete time Markov chain on the space
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$$X$$
\end{document}
with transition probabilities
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\begin{align*}
P_ \mu ^{ ( I ) } ( x \to y ) = \varkappa \delta ( {x_{ \Lambda
\backslash I}} , {y_{ \Lambda \backslash I}} ) \varphi ( {x_I} ,
{y_I} ) { \mu _I} ( {y_I} ) dt \tag{2}
\end{align*}
\end{document}
for
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$$y \ne x$$
\end{document}
, and
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$$P_ \mu ^{ ( I ) } ( x \to x ) = 1 - \sum \nolimits_{y \ne x} P_ \mu ^{ ( I ) } ( x \to y )$$
\end{document}
. Here
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$$\delta$$
\end{document}
is the Kronecker delta. It means that the genome x changes with the rate depending on all genomes in the population.
Obviously, for this Markov chain, the probability distribution
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\begin{align*}
{ \hat \mu _ \Lambda } ( {x_ \Lambda } ) = { \mu _{ \Lambda \backslash I}} ( {x_{ \Lambda \backslash I}} ) { \mu _I} ( {x_I} )
\end{align*}
\end{document}
is an invariant measure (here it is important that the similarity function
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$$\varphi ( {x_I} , {y_I} ) = \varphi ( {y_I} , {x_I} )$$
\end{document}
is symmetric). Moreover, any measure
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$${ \nu _ \Lambda } ( x )$$
\end{document}
on the space
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$$X$$
\end{document}
with marginal distributions
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$${ \mu _{ \Lambda \backslash I}} ( {x_{ \Lambda \backslash I}} )$$
\end{document}
and
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$${ \mu _I} ( {x_I} )$$
\end{document}
turns to a measure
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$${ \nu _ \Lambda }P_ \mu ^{ ( I ) } ( x ) = \sum \nolimits_{y \in X} { \nu _ \Lambda } ( y ) P_ \mu ^{ ( I ) } ( y \to x )$$
\end{document}
having the same marginal distributions. So for the given measure
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$${ \mu _ \Lambda }$$
\end{document}
, the matrix
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$$P_ \mu ^{ ( I ) }$$
\end{document}
is the transition matrix of the usual (linear) Markov chain with the invariant measure
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$${ \hat \mu _ \Lambda }$$
\end{document}
.
We now use an inequality for finite Markov chains, although it is more general in Yosida (1940, 1965).
Lemma. Let
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$$P$$
\end{document}
be a stochastic matrix, that is, matrix
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$${P_{xy}}$$
\end{document}
such that
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$${P_{xy}} \ge 0$$
\end{document}
and
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$$\sum \nolimits_y {P_{xy}} = 1$$
\end{document}
, and let
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$$\hat \mu$$
\end{document}
be an invariant probability measure,
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$$\hat \mu = \hat \mu P$$
\end{document}
. Suppose
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$$\hat \mu ( x ) > 0$$
\end{document}
for any
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$$x$$
\end{document}
. Then, for any probability measure
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$$\mu$$
\end{document}
,
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\begin{align*}
\mathop \sum \limits_x \left( { \ln { \frac { \left( { \mu P } \right) ( x ) } { \hat \mu ( x ) } } } \right) ( \mu P ) ( x ) \le \mathop \sum \limits_x \left( { \ln { \frac { \mu ( x ) } { \hat \mu ( x ) } } } \right) \mu ( x ) \tag { 3 }
\end{align*}
\end{document}
(Here as always
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$$0 \ln 0 = 0$$
\end{document}
).
In our case,
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$${ \hat \mu _ \Lambda } ( {x_ \Lambda } ) = { \mu _{ \Lambda \backslash I}} ( {x_{ \Lambda \backslash I}} ) { \mu _I} ( {x_I} )$$
\end{document}
, so
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$$\ln \hat \mu = \ln { \mu _I} ( {x_I} ) + \ln { \mu _{ \Lambda \backslash I}} ( {x_{ \Lambda \backslash I}} )$$
\end{document}
is a sum of functions depending only on
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$${x_I}$$
\end{document}
and
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$${x_{ \Lambda \backslash I}}$$
\end{document}
, respectively. Since
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$$P_ \mu ^{ ( I ) }$$
\end{document}
, acting on the measure
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$${ \mu _ \Lambda }$$
\end{document}
, retains marginal distributions of
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$${x_I}$$
\end{document}
and
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$${x_{ \Lambda \backslash I}}$$
\end{document}
, it follows that
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\begin{align*}
\mathop \sum \limits_x \left( { \ln {{ \hat \mu }_ \Lambda } ( x ) } \right) ( { \mu _ \Lambda }P_ \mu ^{ ( I ) } ) ( x ) = \mathop \sum \limits_x \left( { \ln {{ \hat \mu }_ \Lambda } ( x ) } \right) { \mu _ \Lambda } ( x ) \tag{4}
\end{align*}
\end{document}
Finally, the Lemma yields the entropic inequality
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\begin{align*}
\mathop \sum \limits_x \left( { \ln \left( {{ \mu _ \Lambda }P_ \mu ^{ ( I ) }} \right) ( x ) } \right) \left( {{ \mu _ \Lambda }P_ \mu ^{ ( I ) }} \right) ( x ) \le \mathop \sum \limits_x \left( { \ln { \mu _ \Lambda } ( x ) } \right) { \mu _ \Lambda } ( x ) \tag{5}
\end{align*}
\end{document}
Now consider mutations. It is supposed that transition intensities
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$${ \alpha _i} ( a , b )$$
\end{document}
define a connected continuous-time Markov chain on alphabet
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$$K$$
\end{document}
, so it is possible to pass from any
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$$a$$
\end{document}
to any
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$$b$$
\end{document}
in several steps. By definition,
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$${ \alpha _i} ( a , a ) = - \sum \nolimits_{b \ne a} { \alpha _i} ( a , b )$$
\end{document}
. Matrix
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$${A_i} = \left( {{ \alpha _i} ( a , b ) , a , b \in K} \right)$$
\end{document}
is called the infinitesimal matrix of a time-continuous Markov chain. It is well known that for such chain, there exists a unique invariant distribution
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$${q_i} ( a ) , a \in K$$
\end{document}
and
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$${q_i} ( a ) > 0$$
\end{document}
. In terms of matrix
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$${A_i}$$
\end{document}
, this means that
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$${q_i}{A_i} = 0$$
\end{document}
(by definition
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$$\left( {{q_i}{A_i}} \right) ( x ) = \sum \nolimits_y {q_i} ( y ) { \alpha _i} ( y , x )$$
\end{document}
).
To describe mutations in any arbitrary position in the genome, consider the following continuous-time Markov chain. Let
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$${A_ \Lambda } = ( {a_ \Lambda } ( x , y ) , x , y \in X )$$
\end{document}
be the infinitesimal matrix,
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$${a_ \Lambda } ( x , y ) = \sum \nolimits_i \delta \left( {{x_{ \Lambda \backslash i}} , {y_{ \Lambda \backslash i}}} \right) { \alpha _i} ( {x_i} , {y_i} )$$
\end{document}
. The invariant distribution of the chain, defined by matrix
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$${A_ \Lambda }$$
\end{document}
, is
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\begin{align*}
{q_ \Lambda } ( {x_ \Lambda } ) = \prod \limits_i {q_i} ( {x_i} )
\end{align*}
\end{document}
Obviously, this chain is connected on the space
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$$X$$
\end{document}
.
Finally, we use a general statement about the entropy monotonicity that is well known from the folklore and from results of Batishcheva and Vedenyapin (2005) as a special case.
Lemma. Let
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$${ \alpha _{xy}}$$
\end{document}
be the transition intensities of a connected finite continuous time Markov chain and let
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$${q_x}$$
\end{document}
be its stationary distribution. Then, the relative entropy
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$$D ( p \vert q ) = \sum \nolimits_x p ( x ) \ln { \frac { p ( x ) } { q ( x ) } } $$
\end{document}
is strictly decreasing and, furthermore, its derivative is strictly negative along the trajectory of the Kolmogorov forward equation
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$$\dot p = pA$$
\end{document}
, where
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$$A$$
\end{document}
is the infinitesimal matrix of the considered Markov chain.
Proof. (for the reader's convenience).
Let
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$$p ( t )$$
\end{document}
be the solution of the Kolmogorov forward equation and denote
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$$ { \frac { { p_x } } { { q_x } } } $$
\end{document}
by
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$${f_x}$$
\end{document}
, then the derivative
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$$ \frac { d } { { dt } } D ( p ( t ) \vert q )$$
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can be written as
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\begin{align*}
{ \frac { dD } { dt } } = - \mathop \sum \limits_ { x , y } \left( { { \frac { { f_x } } { { f_y } } } \ln { \frac { { f_x } } { { f_y } } } - { \frac { { f_x } } { { f_y } } } + 1 } \right) { q_x } { \alpha _ { xy } } { f_y }
\end{align*}
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Obviously, after removing parentheses, the two last terms in this formula cancel out, but they are needed to prove monotonicity. The expressions in parentheses are non-negative and, as the Markov chain is connected, they can be simultaneously equal to
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$$0$$
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only if
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$${f_x} = {f_y}$$
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for all
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$$x , y$$
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, i.e., if the distributions
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$$p$$
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and
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$$q$$
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are the same. ■
We now collect the properties of the mutation and homologous-recombination processes described above.
(1) For the recombination process on substring
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$$I$$
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\begin{align*}
H ( { \mu _ \Lambda } ) = \mathop \sum \limits_x \left( { \ln { \mu _ \Lambda } ( x ) } \right) { \mu _ \Lambda } ( x )
\end{align*}
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monotonically (maybe, nonstrictly) decreases, so its time derivative is nonpositive.
(2) For the same process, the value
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$$\sum \nolimits_x ( \ln {q_ \Lambda } ) { \mu _ \Lambda } ( x )$$
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does not change, because this logarithm is the sum of functions of
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$${x_I}$$
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and
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$${x_{ \Lambda \backslash I}}$$
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, and as shown above, the means of such functions remain constant.
(3) Hence, the relative entropy.
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\begin{align*}
D ( { \mu _ \Lambda } \vert { q_ \Lambda } ) = \mathop \sum \limits_x \left( { \ln { \frac { { \mu _ \Lambda } ( x ) } { { q_ \Lambda } ( x ) } } } \right) { \mu _ \Lambda } ( x )
\end{align*}
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also has a nonpositive derivative.
(4) For the mutation process, the relative entropy
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$$D ( { \mu _ \Lambda } \vert {q_ \Lambda } )$$
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has a strictly negative derivative.
The right-hand side of Equation (1) consists of the terms for the recombination process on all substrings
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$$I$$
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, and for the mutation process. Since the relative entropy has a nonpositive derivative by equations for the recombination process and a strictly negative derivative for the mutation process, its derivative by Equation (1) is strictly negative, if
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$${ \mu _ \Lambda } \ne {q_ \Lambda }$$
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. This means that the relative entropy strictly decreases along the trajectory of Equation (1) and this equation has a unique fixed point
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$${q_ \Lambda }$$
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. As noted above, fixed points of Equation (1) correspond to different population structures. A unique fixed point
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$${q_ \Lambda }$$
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depends only on the infinitesimal matrix for the mutation process, so it gives us a population without a nontrivial structure; if
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$${q_i}$$
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does not depend on
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$$i$$
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, then the probability of a genome depends only on its nucleotide composition. Note that if the similarity function
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$$\varphi ( {x_I} , {y_I} )$$
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and the constant
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$$\varkappa$$
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depend on time, it would not affect the aforementioned calculations.
