Cells of both types have a probability to die proportional to their abundance, i.e. the probability that a cell of type "A" (or "B") dies is given respectively by p _ a p _ b

We will refer to an event consisting of one replication and one cell death by an elementary event.

The resulting population dynamics is a Markov process with states b = 0,1,..., N, and time steps of length l/N. The probability that an elementary event results in an increase of the number of cells of type "B", is equal to P+bP-a, and the probability that the number of cells of type "B" decreases is equal to P-bP+a• If Pj is the probability to go to state b = j from state b = i, then the transition matrix is given by u(N-i)+ri N-i A il

0 otherwise, where 0 < i,j < N, and we introduced the notation

The corresponding Markov process is a biased random walk with one absorbing state, b = N. Let us set the initial condition to be b = 0 (all cells are of type "A" ) and study the dynamics of absorption into the state b = N.

Notation for the time-variable. In this chapter, we will adopt the upper case variable, T, for measuring the time in terms of elementary events. The lower-case notation is reserved for time measured in terms of generations. For constant population processes, we have the simple relation, t = T/N.

3.1.2 Analysis of a one-hit process

Diffusion approximation. Let us denote the probability to be in state a = i at time T as <fi(T). Using the transition matrix for two types, (3.2), we can write down the Kolmogorov forward equation for ip\

[r(N ~{i + 1)) + (i + l)u](i + 1) [r(N - i) + iu](N - i)

It is convenient to introduce the variable rj = i/N. Taking the continuous limit and expanding into the Taylor series up to the second order, we obtain the following partial differential equation for ip(t],T):


M .. y(l - r)(l - r,) - u y_ 1 ??[(! — T))(l + r) — u(l — 2r])} ^ r/(r — 1) — r ' 2 r)(r - 1) - r

When r = 1 + s/N with s <C N, we have the following equation:

This equation is studied in [Kimura (1994)]. In the case s<l the principal term in the expression for (p(r},T) is proportional to e~tJ,°T, where

This sets the typical time-scale of the process.

We can also study the case 1 <C |s| <C N. In that limit, for s > 0, the region of interest is rj <C 1 (remember that rj = 0 corresponds to all the "B" states). Thus the equation simplifies to

This equation could be solved in terms of Laguerre polynomials, in general:

The Laguerre polynomials, L"(x), satisfy the differential equation

Note that the leading transient gives ¡jlq = ^ = |1 — r\ in this limit. One could similarly treat the case of s < 0, |s| <C 1. In general, [Iq = jff(s) where f(s) = 1 + 0(s2) for small s, but /(s) » |s| for large s.

Absorption time. The method presented above provides a lot of information about the process. However, we can address some interesting questions without such a detailed description. For example, if we are only interested in the time it takes for a mutant of type "B" to appear and invade the population, we can do this directly, by looking at the absorption time for the Markov process. If we denote the number of elementary events until absorption starting from state i as Tj, we have


The absorption time is then given by To- Solving system (3.4) directly is cumbersome, so we will use some approximations.

There are two processes that go on in the system: mutation and selection. If the characteristic time scales of the two processes are vastly different, our task of finding the absorption time simplifies greatly. Let us assume that u is very small, so that once a mutant of type "B" is produced, it typically has time to get fixated or die out before a new mutation occurs. In other words, once a mutant is produced, it is safe to assume that during its life-time no other mutations occur. In this case of rare mutations, the inverse time to absorption is roughly u times the probability to get absorbed in the state b = N from the state b = 1 assuming u = 0.

For u — 0, the system has two absorbing states, b — 0 and b = N. Let us denote the probability to get absorbed in b = N starting from the state b = i as 7T». Then we have approximately

tabs where the quantity 7Ti is given by the system:



note that we set u = 0 in the expression for P. System (3.6) can be rewritten as

1 < i < N- 1, where we canceled the common multiplier in terms of the matrix I — P in the same row. The boundary conditions are

(r + 1)71*1 — r"7r2 = 0, -7rjv-2 + (r + 1)ît jv-i = r.

We can look for a solution in the form 7r* = a1. The quadratic equation for a gives the roots a = l/r and a = 1. Substituting 7^ = Ar~l + B into the boundary conditions we obtain the solution, rN-i(l_ri) ^ = \ — rN ■ (3-7)

Let us reserve the notation p for the quantity iri:

The same result is obtained if we solve system (3.4) explicitly and then take the first term in the Taylor expansion of To in u.

In order for approximation (3.5) to be valid, we need to make sure that the time-scale related to mutation ((jVu)-1) is much longer than the time-scale of the fixation/extinction processes. Only the fraction p of all mutants will successfully reach fixation, whereas the rest will be quickly driven to extinction. In order for each mutant lineage to be treated independently, we need to require that the time it takes to produce a successful mutant, (pNu)~l, is much larger than the typical time-scale of fixation, Pq1. The value /i0 1 is calculated above. We have the general expression,

In the case of neutral mutations, p = 1/N, po = 1 /N and we arrive at the intuitive condition, if|l-r|«^. (3.10)

In the case where the mutation is positively or negatively selected, we have

The approximation of "almost absorbing" states. We will call a state of the system homogeneous, or pure, if all the N cells are of the same type. In the two-species model, these are the states 6 = 0 and b = N. States containing more than one type of cells (1 < b < N) will be referred to as heterogeneous, or mixed states.

Since the mutation rate is very low relative to the absorption processes in the system (conditions (3.10-3.12)), the probability of finding the system in a heterogeneous state is very low. More precisely, the probability of finding the state with b cells of species "B" is of the order u for 1 < b < N. The system spends most of the time in the states 6 = 0 and b = N. This allows us to make a further approximation of "almost absorbing" states.

Let us use the capital letters A and B for the probability to find the system in the state 6 = 0 and 6 = N respectively. Strictly speaking, the state b = 0 is not absorbing, but it is long-lived. We have approximately, A + B = 1. Let us define the following "coarse-grained", continuous time stochastic process: the system jumps between two states, A = 0 and ^4=1, with the following probabilities:

P(A = 0 ,t + At\A = 1 ,t) = up At, P(A = 0 ,t + At\A = 0, t) = 1, P(A = 1 ,t + At\A = 1,t) = 1 - upAt, P{A = 1 ,t + At\A = 0,t) = 0.

The Kolmogorov forward equations for this simple system can be written down, which describe the dynamics of the two-species model, (3.1):

where A is the probability to find the entire system in state "A", B is the probability to find the entire system in state "B" and p is given by equation (3.7). Equations (3.13-3.14) lead to the solution A(t) = exp(-uNpt) and B = 1 - exp(-uNpt).

A short-hand notation for coarse-grained differential equations (3.133.14) is as follows:

««(r^JV)-1, ifrcl, — <C|l-r|«l, (3.11) uCr/N, if r > 1, -j-« |l-r| «1. (3.12)

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