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time

Fig. 3.1 The tunneling rate, R(t) = 1 — $(1,0; i), equation (3.20), vs time, is represented by thick lines. There are three cases, (a) the intermediate mutant is disadvantageous, (b) it is neutral, and (c) it is advantageous. The linear regime and the regime of saturation are marked. Note that in (c), we have an intermediate, exponential growth regime. The parameter values are as follows: (a) u = 10-7, r = 0.99, (b) u = 10~7, r = 0.999999, (c) u = 10~6, r = 1.01.

Next, during the times where l/(r — 1) < t < | logui|/(r — 1), we have the intermediate regime where the function R(t) grows faster than linear. If we assume that the intermediate mutant is advantageous, such that \r—1| u\, then the expression for R(t) can be simplified to give

As time increases, the function $(1,0; t) quickly reaches saturation, that is, a steady-state, $(1,0; t) = —bi/[r( 1 - ui)]. In the case where the intermediate step is disadvantageous, r < 1, |1 - r\ » y/ui, the saturated value is m = fr;-

where R is the "tunneling rate". In the case where the intermediate step is neutral, |1 - r| C 1Ju[, the saturated value is

In the case where the intermediate step is advantageous, r > 1, |1 — r\ -C 2^/rwi, the saturated value is

Probability of double mutations for disadvantageous, neutral and advantageous intermediate mutants. Now we can use the detailed information about the behavior of the function $(1,0; t) to evaluate the integral in (3.19) in different limiting cases. Let us first examine the case where the intermediate mutant is disadvantageous. Roughly speaking, the behavior of the function 1 — $(l,0;i) changes from linear to constant at t = tc, where tc = 1/|1 - r\. At this value of time, the integral /Qic( 1 — $(1,0; t')) dt ~ 1. In order to estimate the expression in (3.19), we need to know which of the regimes of growth makes the largest contribution. This is the same as to determine whether the exponent in (3.19) is negligible at the critical time, t = tc. It is easy to see that if uN -c 1, then there is a large contribution from the regime of saturation. On the other hand, for uN 1, it is only the linear regime that contributes. Therefore we have the following answer:

Next, we consider the case of a neutral intermediate mutant. The point of regime change is tc = l/^/ul- Again, if uN <C 1, then there is a large contribution from the regime of saturation. On the other hand, for uN » 1, it is only the linear regime that contributes, and we have the result:

Finally, we consider the case of advantageous mutants. There are

three regimes. The two critical times where the regimes change are

The value of the function 7Z(t) = fg( 1 — $(1, 0; t')) dt can be estimated at these points by using the expression for the intermediate regime; we have vu \ "i(e~2) <r>u ^ 1 + log Ml

Therefore, we have three regimes depending on the value of N: in the case where N <C 1 /u, the saturated regime contributes the most, for \/u <C N -C l/(uui), we have the contribution from the intermediate regime, and for N l/(uui), we have the contribution from the linear regime only. In summary,

(l-exp(-Nu^t), 7V<l/u, 1 - exp (-Nuu!l(t)), 1/u CiV<C l/(w«i), (3.22)

l-exp(-^f^), JV»l/(uui), with I(t) = -pzj rli"1 ~ t) f°r the intermediate regime.

Applicability of the method. The method assumes independence of the lineages of the intermediate mutant. Thus for the method to work, the probability of fixation of intermediate mutants must be small compared to the probability of "tunneling". Therefore, the applicability is defined by the inequality, p(r) < lim R, t—>00

where the right hand side is the saturated value of 1 - $(1,0; i). This condition can be written as

where for disadvantageous intermediate mutants, we have:

logr

For neutral intermediate mutants, we have

For advantageous mutants, we have log

log r

Summary: tunneling rates. Having a high mutation rate, u\ will increase the probability of tunneling. Also, in the case of a large population size, N, the fixation of type "B" becomes less probable thus making tunneling a more likely scenario. Finally, if type "B" is greatly disadvantageous, we also expect the system to tunnel from "A" to "B".

It is interesting that tunneling can be interpreted as making a two-hit process behaves effectively as a one-step process. Let us concentrate on the case where

In the general case, we have the following diagram:

This corresponds to the differential equations,

This is similar to the one-hit model, which we considered in the previous sections, see equations (3.13-3.14). The tunneling rate, Ra^c, is different depending on whether the intermediate mutant, "B", is positively or negatively selected. We have three cases:

• Type "B" negatively selected. If r < 1 and 1 — r | yfu{, then we have tunneling from "A" to "C" with the rate

• Type "B" neutral. If II — rj <C sju\, then we have tunneling from "A" to "C" with the rate

• Type "B" positively selected. If r > 1 and |1 — r\ -C y/u\, then we have tunneling from "A" to "C" with the rate

3.2.4 Genuine two-step processes

If the number of cells in the population, N, is sufficiently small, then the assumption of the previous section (3.23) breaks down, and tunneling does not happen. In this case, the dynamics can be represented by the diagram

with

Here we assumed that IZy^N ~ 1- The corresponding differential equations are

3.2.5 Summary of the two-hit model with a constant population

To put all the results together, we will describe the dynamics of the acquisition of a double mutant, as a function of time. Depending on the population size, the behavior is quite different. For small populations, where

This comes from solving equations (3.32-3.34) and setting P(t) = C(t). The function p depends on the fitness of the intermediate mutant, p — jBtfpr-The value Ntun is also defined by r, see formulas (3.24), (3.25) and (3.26). For intermediate values on N, such that

Ntun <N < 1/U, we have the following behavior:

P(t) = l~e~RA-ct, which comes from the solution of equations (3.27-3.28). The rate of tunneling, Ra^c again depends on the fitness of the intermediate mutant, see formulas (3.29), (3.30) and (3.31).

Finally, for very large population sizes, such that

For advantageous mutants we have another intermediate regime, which comes for 1/u < N < 1 /(uui). This is given in equation (3.22).

3.3 Modeling non-constant populations 3.3.1 Description of the model

Consider the process described by the following mutation-selection network:

out out

Here the reproductive rates of types "A" and "B" are the same and equal to L, and the death rates are D. Type "A" can mutate into type "B" with probability u, and type "B" can mutate to type "C" with probability u\. There are no other mutation processes in the system. We assume that « « Hi, and that the dynamics follow a Poisson process, where in time interval Ai, the following events can occur:

• With probability 1/(1 — u)At a cell of type "A" reproduces, creating an identical copy of itself,

• With probability Lu a cell of type "A" reproduces with a mutation, creating a cell of type "B",

• With probability L(l—ui)At a cell of type "B" reproduces, creating an identical copy of itself,

• With probability Lu\ a cell of type "B" reproduces with a mutation, creating a cell of type "C",

• With probability DAt a cell of type "A" dies,

• With probability DAt a cell of type "B" dies.

We start with one cell of type "A", and follow the process until the first cell of type "C" has been created. As before, we would like to calculate the probability, P (t), that one cell of type "C" has been created as a function of time. Before, we used the approximation of a doubly-stochastic process, see equation (3.19). A similar approach can be used to describe expanding populations, as long as we can assume that the expansion process is nearly deterministic. On the other hand, if we want to take account of the stochasticity in the colony growth, we need to perform a more general calculation, which is described below. An example of a system where such a calculation is necessary is a colony which grows from very low numbers, such that at the beginning, stochastic effects define the growth (or death) of the cell population.

Let us consider the probability £i,j,fc(t) that at time t, we have a = i, b = j and c = k. We have the Kolmogorov forward equation,

èi,j,k = Çi-i,j,kL{i - 1)(1 - u) + £i+i,j,kD{i + 1)

+ &,j-i,k[L(j - 1)(1 - mi) + Liu] + &j+i,kD(j + 1) - £i,j,k-iLjui - &d,k(L + L>)(i+ j). (3.36)

Note that here we do not consider the dynamics of the double-mutants: once produced, they remain in the colony. Birth-death processes of double-mutants can be incorporated leading to a slightly more complicated system. Let us define the generating function

The quantity \P(l,l,0;i) has the meaning of the probability that at time t, no cells of type "C" have been created. The quantity in question, the probability that at least one cell of type "C" has been created by time t, is given by

The subscript "2" refers to the number of hits (from "A" to "B" and from "B" to "C"). The function ty(x,y, z;t) satisfies the following equation:

+ Qjjly L( 1 - Ui) + £> + zyLux - (L + D)y\. (3.38)

The equations for characteristics are:

(note that the last equation is trivial because we suppress the dynamics of double-mutants; if we include their dynamics, the equation for 2 would reflect that). We want to obtain the expression for "f (1,1,0; t), thus we can set the initial conditions x(0) = 1, y(0) = 1, *(0) - 0.

We obtain immediately from equation (3.41) that z = 0.

3.3.2 A one-hit process

First, we consider a simplified model with only one hit. Note that the function

with the initial condition as above, has the meaning of the probability that a cell of type "C" has been created starting with one cell of type "B". The corresponding diagram is this:

The equation for y(t), (3.40), can then be solved exactly with z = 0. We set

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