Shanghai Center for Systems Biomedicine, State Key Laboratory of Oncogenes and Related Genes, Shanghai Jiao Tong University, 200240, Shanghai, China

Department of Mathematics, Xinyang Normal University, 464000, Xinyang, Henan, China

Department of Physics, Shanghai Jiao Tong University, 200240, Shanghai, China

Abstract

Background

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.

Methods

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.

Results

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.

Background

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

Biologists have suggested

Previous works mainly focused on the parameter regimes with lower or higher mutation rates. And models are represented as stochastic differential equation. In Ref.

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

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.

Methods

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

Here in one dimensional case, we consider one locus with two alleles (for example, _{1,0}, that is _{1,0 }_{0,0 }= 1 - _{0,1 }= 0, _{1,1 }_{0 }= 1 while that of individuals with allele _{1 }= 1 -

Eq.(1) describes the deterministic process that ignores random drift. Under the mutation-selection balance, the fixed points is

This means the population ultimately arrives at the state with allele frequency _{n,t}

The matrix of transition probabilities is **W **which is composed of elements _{nm}

Where **P**(

Here **v ^{T }**means the transpose of vector

Where **0** is a column vector with **v **is a column vector with _{m }= W_{0m}, where **w **is an _{nm }= W_{nm}**w**, this corresponding vector is the quasi-stationary probability density of allele

Eq.(8) has the solution

Because the leading large time behavior is determined by the eigenvalue of matrix **w**. From Perron-Frobenius theory this issue is transformed to solve the leading large eigenvalue of **w **and corresponding eigenvector. But

Where **I ^{T}**= (1

We can describe Eq.(8) as

where

Then the probability density is derived by

Where _{1 }is determined by _{1 }= 1-**v ^{T}q**.

Populations evolution is a natural and random process. We model the process as the discrete form. The boundary of the discrete model is determined by its transition matrix. Generally, The transition probabilities from generation

From the expression of transition probability, it can be seen that the transition probabilities are zero for any frequency ^{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

Diffusion approximation

Here we briefly outline the diffusion approximation from the discrete to continuous models. At generation

Where _{ij }_{0}. Let

Letting

In any time interval

and we can treat Eq.(21) as the following

that is

Because the function about the change of allele frequency in one generation is continuous and smooth enough, under the condition that

and according to the definition of

Among this

Adaptive landscape

Under the general diffusion approximation, frequency _{n,t }

with

With a prime denoting differentiation of a function with respect to its argument such as _{x}D

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

When the process lies at stationary state, the probability flux of the system is zero, and probability flux flows in

It has the form of Boltzmman-Gibbs distribution

We are interested in the dynamical property of adaptive landscape, so we treat Φ and Φ/∊ no difference in this respect, that is, for convenience we can take ∊ = 1 of

From the expression of adaptive landscape Φ(

The stationary distribution can be expressed as

Results

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.

According to general analysis of a dynamical system, letting

we get

We solved the Eq.(32) and found two fixed points. If we denote

They are

For two singular points

Here we address dynamical behavior by the positions of two real inequivalent fixed points _{1 }_{2 }first.

I) We find two different real fixed points in the regimes of

i) 1_{1 }_{2}

In the regimes of _{1 }_{2}. At the same time the singular point

ii) 1 = _{1 }_{2}

In the regions of _{1 }= 1, 1_{2}. The state with allele frequency

iii) 0 _{1 }_{2}

In the regimes of _{1 }_{2}. The fixed point _{1 }is unadaptive. There is only one unadaptive state with allele frequency _{1 }in the system, and two unadaptive states with allele frequency _{1}, populations tend to evolve to the adaptive state with allele frequency

iv) 0 _{1 }_{2 }

In the regimes of _{1 }_{2 }_{1 }is unadaptive while that with allele frequency _{2 }is adaptive. There are two adaptive states with allele frequency _{2 }and two unadaptive states with allele frequency _{1 }in the system. Populations evolve to which adaptive states dependent on the initial position.

v) 0 = _{1 }_{2}

In the regime of _{1 }_{2}. When selection rate

vi) _{1 }_{2 }

The case _{1 }_{2 }

II) Then we discuss the case of two equivalent real fixed points _{2 }_{1}.

In the regimes of

i) 1 _{1,2}

In the regimes of _{1,2}, and they are unadaptive. There is one adaptive state with allele frequency

ii) 1 = _{1,2}

At the two points of ((^{2}) and _{1,2 }= 1, and they are unadaptive. There is one adaptive state with allele frequency

iii) 0 _{1,2 }

In the regime of _{1,2 }

III) Finally we consider two imaginary fixed points _{1}_{2}

In the regime of

Especially Φ'(

Relation of fixed points and parameters for the system in all regimes

**Relation of fixed points and parameters for the system in all regimes**. The regime represented by **III **i) with parameters regions _{1}_{2}) and **II**) with parameters satisfying _{2 }**II**) occur in the intervals. The regimes denoted by I) with parameters satisfying _{2}_{5}) and _{2}_{5}) and _{1}_{5}) and _{1}_{5})

Irreversible mutation, selection and random drift balance

Concretely we divide mutation rates into three regimes. One is with mutation rates ^{2}, (2

Adaptive landscape against allele frequency

**Adaptive landscape against allele frequency x with mutation rates and selection rates in all regimes**.

Characterization of the single click time

We visualize the adaptive landscape, then one may wonder about how the population moves from one peak to another and how long it might be to move from one maximum to another. The process was first visualized by Wright in 1932. In addition, the problem of transition from metastable states is ubiquitous in almost all scientific areas. Most of previous works encounter finite potential barriers from the physical point of view. Interesting issue here is that we touch upon infinite potential barriers under the circumstance of well defined two adaptive states. Then we manifest the derivation of a single click time. The time of a click of the ratchet is recognized as the random time of loss of the fittest class

Adaptive landscape with two adaptive states

**Adaptive landscape with two adaptive states**.

After straightforward calculation, backward Fokker- Planck equation corresponding to Eq.(27) can be expressed with the property of time homogeneous in the following form

General single click time dependent on initial Dirac function satisfies

With

Above treatment is valid. Because populations evolution is according to Muller's ratchet, that is in the presence of deleterious mutation, without any recombination, but with selection and random drift. And the model in discrete manner demonstrates the transition probabilities are 0 from the boundary

here Φ(^{x}

Here the evolutionary process occurs when

and near

At the same time, if the central maximum of Φ(

From the expression of Eq.(44), the single click time is not sensitive to the assumption of Eq.(40). In the higher mutation rates regime, where

Here _{2}, and _{1}, parameters

Where α and β are the same as Eqs. (34) and (35). The approximated single click time varies with mutation rates in Figure _{1→ }increases with population size

The approximated single click time decreases with mutation rates increasing in the regime

**The approximated single click time decreases with mutation rates increasing in the regime μ ∊ (2N/4N(N - 1), 1) and **

For the lower mutation rates regime, where the potential barrier is infinite. The single click time can be estimated also, _{1 }that the population lies at the lowest potential.

From expression of Eq.(47), the single click time goes to infinity with mutation rates tends to zero in the parameters regimes of

Analogous to the derivation of _{1→0}, we can calculate

Compared with the singular point

Discussion

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

When allele frequency

that is, when

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

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.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

Shuyun Jiao carried the research and writing. Ping Ao oversaw the whole project, and participated in research and writing.

Acknowledgements

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.).

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