Absorbing phenomena and escaping time for Muller's ratchet in adaptive landscape
© Jiao and Ao; licensee BioMed Central Ltd. 2012
Published: 16 July 2012
The accumulation of deleterious mutations of a population directly contributes to the fate as to how long the population would exist, a process often described as Muller's ratchet with the absorbing phenomenon. The key to understand this absorbing phenomenon is to characterize the decaying time of the fittest class of the population. Adaptive landscape introduced by Wright, a re-emerging powerful concept in systems biology, is used as a tool to describe biological processes. To our knowledge, the dynamical behaviors for Muller's ratchet over the full parameter regimes are not studied from the point of the adaptive landscape. And the characterization of the absorbing phenomenon is not yet quantitatively obtained without extraneous assumptions as well.
We describe how Muller's ratchet can be mapped to the classical Wright-Fisher process in both discrete and continuous manners. Furthermore, we construct the adaptive landscape for the system analytically from the general diffusion equation. The constructed adaptive landscape is independent of the existence and normalization of the stationary distribution. We derive the formula of the single click time in finite and infinite potential barrier for all parameters regimes by mean first passage time.
We describe the dynamical behavior of the population exposed to Muller's ratchet in all parameters regimes by adaptive landscape. The adaptive landscape has rich structures such as finite and infinite potential, real and imaginary fixed points. We give the formula about the single click time with finite and infinite potential. And we find the single click time increases with selection rates and population size increasing, decreases with mutation rates increasing. These results provide a new understanding of infinite potential. We analytically demonstrate the adaptive and unadaptive states for the whole parameters regimes. Interesting issues about the parameters regions with the imaginary fixed points is demonstrated. Most importantly, we find that the absorbing phenomenon is characterized by the adaptive landscape and the single click time without any extraneous assumptions. These results suggest a graphical and quantitative framework to study the absorbing phenomenon.
Muller's ratchet proposed in 1964 is that the genome of an asexual population accumulates deleterious mutations in an irreversible manner. It is a mechanism that has been suggested as an explanation for the evolution of sex . For asexually reproducing population, without recombination, chromosomes are directly passed down to offsprings. As a consequence, the deleterious mutations accumulate so that the fittest class loses. For sexually reproducing population, because of the existence of recombination between parental genomes, a parent carrying high mutational loads can have offspring with fewer deleterious mutations. The high cost of sexual reproduction is thus offset by the benefits of inhibiting the ratchet . Muller's ratchet has received growing attention recently. Most studies of Muller's ratchet are related to two issues. One is that without recombination, the genetic uniformity of the offspring leads to much lower genetic diversity, which is likely to make it more difficult to adapt . So its adaptiveness arouses concern. The other is that population lacking genetic repair should decay with time, due to successive loss of the fittest individuals [4, 5]. So the fixation probability arouses concern. In addition, Muller's ratchet is relevant to some replicators [6, 7], endosymbionts , and mitochondria . In order to assess the relevance of Muller's ratchet, it is necessary to determine the rate (or the time) for the accumulation of deleterious mutations . It is widely recognized that the rate of deleterious mutations being much higher than that of either reverse or beneficial mutations results in a serious threat to the survival of populations at the molecular level . Because models proposed must rest on the biological reality, which must be analyzed on their own without any injection of extraneous assumptions during the analysis . Overall, it has been a long interest to develop a suitable and quantitative theory for the ratchet mechanism and the incidental absorbing phenomenon.
Biologists have suggested [12, 10] that a quantitative framework is needed. The potential evolutionary importance of Muller's ratchet makes it desirable to carry out careful quantitative studies . And the incidental absorbing phenomenon is investigated quantitatively in broad literature. The simplest and earliest mathematical model is the pioneering work in Ref. . It described the same evolutionary process on the condition of deterministic mutation-selection balance according to the Wright-Fisher dynamics. And it indicated numerical evidence of relation between the total number of individuals and the average time between clicks of the ratchet, but it did not focus on the absorbing phenomenon. It treated the pioneering model as a diffusion approximation , and produced more accurate predictions over the relatively slow regime. It noted that the increasing importance of selection coefficients for the rate of the ratchet for increasing values of the total number of individuals. But it is represented as stochastic differential equations and did not get the predictions over all parameters regions. It employed simulation approaches to Muller's ratchet  and estimated how different between the distribution of mutations within a population and a Poisson distribution. But it did not emphasize the absorbing phenomenon. In Ref.  it obtained diffusion approximations for three different parameter regimes, depending on the speed of the ratchet. The model shed new light on . But it mainly focused on the property of the solution for these stochastic differential equations. In Ref.  it mapped Muller's ratchet to Wright-Fisher process, and got the prediction of the rate of accumulation of deleterious mutations when parameters lie in the fast and slow regimes of the operation of the ratchet. But it put the constraints of Dirac function on the boundary.
Previous works mainly focused on the parameter regimes with lower or higher mutation rates. And models are represented as stochastic differential equation. In Ref.  authors imagined the population evolved on an adaptive landscape, but they could not analytically construct it. It described discrete birth-death model and its corresponding diffusion manner by the adaptive landscape . But it did not discuss the absorbing phenomenon. The concept of adaptive landscape is proposed by Sewall Wright to build intuition for the complex biological phenomena . In the present article, inspired by [16, 10], we model Muller's ratchet as a Wright-Fisher process, analytically construct the adaptive landscape, where the non-normalizable stationary distribution occurs. Here the adaptive landscape is analytically quantified as a potential function from the physical point of view . We give the adaptive and unadaptive states for the whole parameters region by the adaptive landscape. We give the formula for the single click time of Muller's ratchet in the face of infinite and finite potential. In addition, we can handle the absorbing phenomenon without extraneous assumptions.
The key concept in constructing the adaptive landscape is of potential function as a scalar function. There is a long history of definition, interpretation, and generalization of the potential. Such potential has also been applied to biological systems in various ways. The usefulness of a potential reemerges in the current study of dynamics of gene regulatory networks , such as its application in genetic switch [20–23]. The role of potential is the same as that of adaptive landscape. In this article, we do not distinct them.
We now make the obvious advance to Muller's ratchet. We analytically construct the adaptive landscape. We demonstrate the position and adaptiveness of fixed points. This makes the dynamical behaviors of the population to be investigated. In addition, we give the area with imaginary fixed points. This makes the explaining for the imaginary fixed points biologically possible. Infinite potential barriers can be crossed over under some cases. We handle the absorbing phenomenon without any extraneous assumptions under the condition of diffusion approximation. Inversely, we demonstrate the power of the adaptive landscape.
Discrete model and absorbing boundary
We consider here in population genetics an important and widely applied mechanism- Muller's ratchet. It is the process by which the genomes of an asexual population accumulate deleterious mutations in an irreversible manner [24, 25]. It corresponds to the repeated irreversible loss of the fittest class of individuals because of the accumulation of the deleterious mutations, the effective absence of beneficial mutations, without any recombination, but with the random drift [25, 12]. Consider a population of haploid asexual individuals with discrete generations t = 0,1, 2,.... The common point in a generation is regarded as an adult stage, after all selection has occurred and immediately prior to reproduction. New mutations occur at reproduction and all mutations are assumed to deleteriously affect viability but have no effect on fertility. Supposed population size is fixed for each generation. The viability of a newly born individual is taken to be determined solely by the alleles they carry. This allows us to divide the population into different classes with different genotypes.
Where λ1 is determined by . But . In the end we get λ1 = 1-v T q.
From the expression of transition probability, it can be seen that the transition probabilities are zero for any frequency under the condition of parameter σ = 1. It means the population stays at its initial state. In addition, the transition probabilities from the boundary 0 to its next are (1,0,..., 0) T . This means boundary 0 can not output any probability flow to its next, it only absorbs probability from next. We call absorbing phenomenon occurring at the boundary 0.
Continuous model and adaptive landscape
Among this M(x) is the symbol for the change in allele frequency [26, 11] that occurs in one generation due to systematic force. The function V(x) is the variance in allele frequency after one generation of binomial sampling of N alleles .
The symmetric Eq.(27) has two advantages. On the one hand, the adaptive landscape is directly read out when the detailed balance is satisfied. On the other hand, the constructive method is dynamical, independent of existence and normalization of stationary distribution. We call f(x) directional transition rate, integrating the effects of M(x) and the derivative of V(x). Directional transition rate can give equilibrium states when it has the linear form.
It has the form of Boltzmman-Gibbs distribution , so the scalar function Φ(x) naturally acquires the meaning of potential energy . The value of Z determines the normalization of ρ(x, t = ∞) from the perspective of probability, and the finite value of Z manifests the normalization of ρ(x,t = ∞). The stationary distribution is not true in the face of infinite Z. It demonstrates absorbing phenomenon occurs at the boundary. Together with the flux at the boundary, the true stationary distribution could be got. The constant ∊ holds the same position as temperature of Boltzmman-Gibbs distribution in statistical mechanics. But it does not hold the nature of temperature in Boltzmman-Gibbs distribution.
From the expression of adaptive landscape Φ(x), we may find there are two singular points 0 and 1 of adaptive landscape, characterized by infinite value, infinity means adaptiveness or unadaptiveness of the system. Here the adaptive landscape is composed of three terms. The first term and the third term quantify the effect of the effect of irreversible mutations and selection respectively, the second term quantifies the effect of random drift.
Fixed points and their adaptiveness
To understand the mechanism of Muller's ratchet, a full characterization of dynamical process is a
prerequisite for obtaining more accurate decaying time. Here we study the dynamical behaviors by
investigating the position and adaptiveness of all fixed points. We further derive the parameter regions for all possible cases.
For two singular points x = 0,1, if x → 1, and σ ∊(μ, (2Nμ - 1)/(2Nμ - μ)), Φ(x) → -∞. So the population is unadaptive at x = 1. When x → 1, and σ ∊ ((2Nμ - 1/(2Nμ - μ), 1), Φ(x) → +∞. So the population is adaptive at x = 1. For x → 0, Φ(x) → +∞ in almost parameters regimes except σ = 1. So the population is always adaptive at x = 0. When σ = 1, the viability of the sub-fittest class is zero, so populations stay at the initial state, the corresponding minimum of adaptive landscape demonstrates the state with allele frequency x = 0.
We find two different real fixed points in the regimes of and σ ∊ (μ, 1); in the regimes of and except the regime of με[(2N-1)/4N(N-1)], and δ = (2Nμ-1)/(2Nμ-μ). We discuss the position between them and the boundary points 0, 1 and adaptiveness of them in the following.
1< x1 < x2
In the regimes of and σ ∊ (μ, (2N μ - 1)/(2Nμ - μ)); in the regimes of , (2N - 1)/4N(N - 1)) and σ ∊ , (2Nμ-1)/(2Nμ-μ)), the fixed points satisfy 1< x1 < x2. At the same time the singular point x = 1 is adaptive. There is one adaptive state with allele frequency x = 0 in the system. Populations tend to evolve to the adaptive state.
ii) 1 = x1 < x2
In the regions of μ ∊ (1/(2N - 1), (2N - 1)/4N(N - 1)) and σ = (2Nμ - 1)/(2Nμ - μ), the two fixed points satisfy x1 = 1, 1< x2. The state with allele frequency x = 1 is unadaptive. There is one adaptive state with allele frequency x = 0 in the system.
iii) 0 < x1 < 1< x2
In the regimes of μ ∊ (0,1/(2N - 1)) and σ ∊ (μ, 1); in the regimes of μ ∊ (1/(2N - 1), 1) and σ ∊ ((2Nμ - 1)/(2Nμ - μ), 1), the fixed points satisfy 0 < x1 < 1, 1< x2. The fixed point x1 is unadaptive. There is only one unadaptive state with allele frequency x = x1 in the system, and two unadaptive states with allele frequency x = 1 and x = 0 occur in the system. Populations tend to evolve to the adaptive states dependent on the position of the initial state. If the initial state with allele frequency is greater than x1, populations tend to evolve to the adaptive state with allele frequency x = 1.
iv) 0 < x1 < x2 < 1
0 = x1 < 1< x2
In the regime of μ ∊ (0,1) and σ = 1, the fixed points satisfy x1 = 0, 1< x2. When selection rate σ = 1, the process lies at the initial state because for this case, the viability of the sub-fittest class is zero.
vi) x1 < 0 or x2 < 0
The case x1 < 0 is impossible, and the case x2 < 0 is impossible.
II) Then we discuss the case of two equivalent real fixed points x2 = x1.
1 < x1,2
In the regimes of and , there are two same fixed points satisfy 1 < x1,2, and they are unadaptive. There is one adaptive state with allele frequency x = 0 in the process.
ii) 1 = x1,2
At the two points of ((2N - 1)/4N(N - 1),2N/(2N - 1)2) and , there are two same fixed points satisfy x1,2 = 1, and they are unadaptive. There is one adaptive state with allele frequency x = 0 in the process.
iii) 0 < x,1,2 < 1
In the regime of μ ∊ ((2N - 1)/4N(N - 1), (2N - 1)/(4N - 3)), ; in the regimes of μ ∊ ((2N - 1)/(4N - 3), 1) and , there are two same fixed points satisfy 0 < x1,2 < 1, and they are unadaptive. There is one adaptive state with allele frequency x = 0 in the process.
III) Finally we consider two imaginary fixed points |x1| < |x2|. Where the |.| denotes the length for an imaginary points.
In the regime of and , there are two imaginary fixed points in the system. There is only one adaptive state with allele frequency x = 0. Populations always evolve to the adaptive state.
Irreversible mutation, selection and random drift balance
Characterization of the single click time
here Φ(x) = ∫ x (f(x')/D(x'))dx'(∊ = 1).
From expression of Eq.(47), the single click time goes to infinity with mutation rates tends to zero in the parameters regimes of μ ∊ (0,1/(2N - 1)) and σ ∊ (μ, 1). From Figure 4, when parameters regions lie μ ∊ (1/(2N - 1), 1) and σ ∊ ((2Nμ - 1)/(2Nμ - μ), 1), the results of the single click time is not sensitive to the population size. Biologically if deleterious mutation accumulates, the viability of sub-fittest class increases, these results in the single click time longer.
Compared with the singular point x = 1, the difference between two singular points x = 0 and x = 1 is the mutation rates μ. This results in the power of (1-z) is not negative, so the single click time is finite from it. The infinity of the single click time from x = 0 comes from that the mutation from unfavored allele a to favored allele A is zero. This results in the second integral nonintegrable because of the negative power of z. Biologically because the absence of back mutation, the accumulation of deleterious mutation, once the population arrives at the state that almost individuals are with allele a, the population is absorbed the state and can not leave with high probability. Mathematically, because the second integral is singular for the singular point x = 0. And the integrated function is a fraction respect to argument x, but the highest power of denominator is smaller than that of numerator. That results in the power of x is 2N - 2. As a consequence the second integral is singular.
We analytically construct adaptive landscape. The constructive method is independent on the existence and normalization of stationary distribution. We demonstrate the position and adaptiveness of all fixed points for the whole parameters regimes under the condition of the diffusion approximation. An interesting thing is the imaginary fixed points occurring. We give the parameters regions of their occurrence. However, we have not found any study of Muller ratchet for the fixed points to give a complete description. In addition, we give the description of escape from infinite potential. However, intuitively infinite potential means the population lies at adaptive state. The transition from the adaptive state can not occur. Here we find that the escape from infinite potential can not occur when the boundary is absorbing. So we define the absorbing boundary by adaptive landscape and the single click time without any extraneous assumptions.
The model with discrete manner describes the nature of populations evolution. Here we give two special cases. One is that the population lies at that state with allele frequency x = 0, the other is that the population lies at the state with allele frequency x = 1. We compare the model with discrete and continuous manners to conclude the definition of boundary conditions without any extraneous assumptions. The model with continuous manner is derived under the condition of enough small space change δx in one generation, in another word, when population size N is much bigger. The model with continuous manner can correspond to the model with discrete manner.
that is, when μ > 0, the transition probabilities are not zero, the state with allele frequency x = 1 can input any flux to its next state. The single click time from the state is finite in the model with diffusion manner, however the potential at the state could be infinite, then we draw the conclusion this boundary is not absorbing. This is consistent with the biological understanding.
This article presents an approach to estimate the single click time of Muller's ratchet. Furthermore, it define the absorbing phenomenon by the single click time without any extraneous assumptions. Inspired by [16, 10], we connect Muller's ratchet to one locus Wright-Fisher model with asexual population including N haploid individuals. And our model is represented as a Fokker-Planck equation. We give a complete description for the position and adaptiveness of all fixed points in the whole parameters regimes. This is first done bases on diffusion approximation. The investigated elements is at the allele level. This is different from Ref. . Our method does not need the existence and normalization of the stationary distribution. Our constructive method is independent of the stationary distribution. Compared with the method based on diffusion approximation [15, 2], mathematically it is described as stochastic differential equations. Our method investigates the global dynamical property of the system, and reduces the complexity of calculating stochastic differential equations. In addition, the boundary condition of these stochastic differential equations is prescribed. Compared with Ref. , They added Dirac function to the boundary. But this is not appropriate for the adding non-differential Dirac function to stationary distribution, and stationary distribution should satisfy diffusion equation. However, the treatment is convenient for computing the stationary distribution. The stationary distribution of theirs is equivalent to our adaptive landscape. They had not given the shape of adaptive landscape when the mutation rate lies in the lower regime. We use the model defined in the interval (0, 1) to describe the absorbing boundary. We check the biological phenomenon by the model with both discrete and continuous manners. This is a new method to handle the boundary condition. We investigate the absorbing phenomenon by it without any extraneous assumptions.
To summarize, we have obtained two main sets of results in the present work. Most importantly, we find that the absorbing phenomenon is characterized by the adaptive landscape and the single click time without any extraneous assumptions. First, we demonstrate the adaptive landscape can be explicitly read out as a potential function from general diffusion equation. This not only allows computing the single click time of Muller's ratchet straightforward, but also characterizes the whole picture of the ratchet mechanism. The adaptive landscape has rich structures such as finite and infinite potential, real and imaginary fixed points. We analytically demonstrate the adaptive and unadaptive states for the whole parameters regimes. We find corresponding parameters regimes for different shapes of adaptive landscape. Second, we give the formula about the single click time with finite and infinite potential. And we find the single click time increases with selection rates and population size increasing, decreases with mutation rates increasing. These results give a new understanding of infinite potential and allow us a new way to handle the absorbing phenomenon. In this perspective our work may be a starting point for estimating the click time for Muller's ratchet in more general situations and for describing the boundary condition. Such demonstration suggests that adaptive landscape may be applicable to other levels of systems biology.
We would like to thank Yanbo Wang for drawing the figures, also thank Quan Liu for discussions and technical help, thank Song Xu for technical help. We thank Bo Yuan for some advice on writing and correcting some expression on language. This work was supported in part by the National 973 Projects No. 2010CB529200 (P.A.), and in part by No. 91029738 (P.A.) and No.Z-XT-003 (S.J.) and by the project of Xinyang Normal university No. 20100073 (S.J.).
This article has been published as part of BMC Systems Biology Volume 6 Supplement 1, 2012: Selected articles from The 5th IEEE International Conference on Systems Biology (ISB 2011). The full contents of the supplement are available online at http://www.biomedcentral.com/bmcsystbiol/supplements/6/S1.
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