## N

We will use this notation later to describe the Kolmogorov forward equation of more complex mutation-selection networks.

### 3.2 A two-hit model

Now we will consider a two-hit model. In this section we will restrict ourselves to the Moran process (a constant population process). Later on, we will also discuss models with a changing population size.

### 3.2.1 Process description

We suppose that there are three types of cells: type "A", type "B" and type "C", and the mutation-selection network that governs the dynamics is as follows:

The reproductive rates are respectively 1, r and r\. As before, the reproductive rates must be interpreted as relative probabilities to be chosen for reproduction, rather than parameters defining the time-scale. We assume that 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.

This model describes several biologically relevant situations. For instance, it may be directly applied for the two-hit hypothesis, that is, the process of the inactivation of a tumor suppressor gene. In the simplest case, the inactivation of the first allele of a tumor suppressor gene (TSP) does not lead to a phenotypic change, which corresponds to the value r = 1. This rigid definition can be relaxed to allow for certain gene dosage effects. For instance, the loss of one copy can lead to a certain change in the phe-notype, and the loss of both copies will increase this effect. In this case, we could have 1 < r < n. Finally, the case r < 1, n > 1 means that the intermediate cell has a disadvantage compared to wild type cells. For example, this may correspond to the situation where the inactivation of the first allele is achieved by a large scale genomic alteration, such as a loss-of-heterozygocity event where many genes have been lost. This would lead to the intermediate product having a disadvantage compared to the wild type cells. Losing the remaining allele of the tumor suppressor gene will give the cell a growth advantage which may override the fitness loss of the previous event, resulting in n > 1.

In general, the two-hit model described above refers to any two consecutive mutations, such that the first one may be positively or negatively selected (or neutral), and the second one confers a selective advantage to the cell.

Let us specify the states of the system by the variables a, b and c, which correspond to the number of cells of species "A", "B" and "C", respectively. They satisfy the constraint a + b + c = N. We can characterize a state as a vector (b,c). In this notation, the state we start with is (0,0), which is all "A". The final state, which is the state of interest, is (0, N), or all "C". The question we will study is again, the time of absorption in the state c = N.

We are interested in the case where the type "C" has a large selective advantage, i.e. r\ (l,r), so that once there is one cell of type "C", this type will invade instantaneously with probability one. Under this assumption we can use a trick which allows us to view the dynamics as a one-dimensional process. Namely, let us consider the following reduced Markov process with the independent stochastic variable b: the states b = i with 0 < i < N correspond to a = N — i, b = i, c = 0, and the state b = N + 1 contains all states with c > 1. The state b = N + 1 is absorbing, because we assume that once a mutant of type "C" appears, then cells "C" invade, so the system cannot go back to a state with c~ 0. The transition probabilities are given by for 0 < i < N, PN+hN+1 = 1 and PN+i,j = 0 for all j ± N + 1. In some special cases, the absorption time can be found from equation (3.4), however, a direct solution is not possible in the general case, and we will use some approximations.

### 3.2.2 Two ways to acquire the second hit

Let us start from the all "A" state. If we are in the regime of homogeneous states, conditions (3.10-3.12), we can consider the lineages of each mutant of type "B" separately. Once a mutant of type "B" is created, it can either go extinct, or get fixated. A mutant of type "C" can be created before or

- Pi,N+1, j = i, otherwise, after type "B" reaches fixation. This gives rise to two possible scenarios [Komarova et al. (2003)].

We will reserve the name genuine two-step process for a sequence of steps where starting from (0,0), after some time the system finds itself in the state (N, 0) and then gets absorbed in the state (0, N). In other words, starting from the all "A" state, the system gets to the state where the entire population consists of cells of type "B" and finally reaches fixation in the all "C" state.

We will use the term tunneling for such processes where the system goes from (0,0) to (0,N) without ever visiting state (N,0). This means that from the all "A" state the system gets absorbed in the all "C" state, skipping the intermediate fixation of type "B".

It turns out that the computation of the waiting time for a mutant of type "C" to appear will be different in the two regimes. We start from looking at the tunneling regime and then talk about a genuine two-step process.

### 3.2.3 The regime of tunneling

The hazard function. We would like to calculate the probability, P{t), that by the time t, at least one cell of type "C" has been produced, starting from all cells in the state "A" at time t = 0. This can be calculated by using the so-called hazard function, h(t), which is defined as the probability to create a mutant of type "C" in the next interval At, given that it has not been produced so far. We have

Let <Pj k{t) be the probability that at time t, we have b = j and c = k. It is convenient to introduce the probability generating function,

The hazard function can be expressed in terms of ^ as follows:

A trivial calculation shows that the probability P(t) can be expressed in terms of the function $ as follows:

The meaning of function "¡/(l, 0; i) is the probability that by time t, no cells of type "C" have been created.

Doubly stochastic process in a constant population. Let us calculate the function Vl>(y, z\ t). The initial condition of this problem is that all cells are normal (type "A"). In what follows we will assume that the time of interest is sufficiently short so that most cells remain type "A". In the Moran process, this means that a « N, and b, c <C Af. This assumption simplifies the problem. Following Moolgavkar, we will consider a filtered (or doubly-stochastic) Poisson process, where cells of type "B" are produced by mutations at the rate Nu (if we measure the time in terms of generations). Each cell of type "B" can produce a lineage (a clone). These lineages are independent of each other, so that the numbers of offspring of each "initial" cell are independent identically distributed random variables. Note that the assumption of the independence of the lineages breaks down as soon as a mutant of type "B" gets fixated. This happens with the probability p, see equation (3.8).

Assuming that fixation does not take place (the tunneling regime), we can write down the probability distribution, 0,fc> which is the probability to have j cells of type "B" and k cells of type "C" starting from one cell of type "B" and no further "A"—>"B" mutations. The corresponding probability generating function is given by

The function R(t) = 1 - <&{y,z\t), which we call the tunneling rate, can be calculated directly. We start by writing down the Kolmogorov forward equation for 0,fc:

Ö,fe = Cj-i,k(j ~ l)r(I - ui) + Cj+i,fcC? + 1) + Cj,k-ijrui - 0,fc(r + 1).

According to Parzen [Parzen (1962)], we have

Here we use matrix (3.16) with u = 0, measure time in terms of generations and assume that a ~ N. Rewriting this for <&(y, z; t) we obtain z;t) = (y2r( 1 - Ul) + yzru1 - (r + 1 )y + l)

We want to find i(l,0;t). Setting z = 0, we can write the equation for characteristics, which is a Riccati equation,

Following the standard method, we change the variables, y = ~2(i_Ml) f > and obtain a second order linear equation for z(t). Using the initial condition 0; 0) = y, we can write down the solution:

> and W > b2 are the roots of the quadratic equation, b2+ {r + l)b + r(l-Ul) = 0. (3.21)

Behavior of the tunneling rate. There are three important limits:

(i) Disadvantageous intermediate step, r < 1, |1 — r\ y/u±. Then

(ii) Neutral intermediate step, |1 — r\ -C Then b\ = — 1 + y/ui, b2 = -1 - y/ui-

(iii) Advantageous intermediate step, r > 1, |1 — r\ 2^/mJ. Then

Let us plot the tunneling rate, R(t) = 1 — $(1,0; i), equation (3.20), as a function of time, see Figure 3.1. It starts at zero at t = 0, grows monotonically and reaches a saturation. We can identify three distinct regimes in the behavior of the function $(1,0; i). For very short times, where max-f^uT, |1 — r\}t <C 1, we have the linear regime, where the function R(t) grows linearly with time; we have

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