## Y

and obtain a Riccati equation for Y:

The solution can be easily obtained:

&+l + ^62e(b2-bl)i V ~ ~L(1 - ui)(l + Ae^-bi)*)'

where

= h + L(1 -m) b2 + L{l-uiy and b\ > 62 are roots of the quadratic equation, b2 + (L + D)b + L(1 - m)D = 0.

This is similar to equation (3.21). The difference is that now, we allow for expansion and contraction of the population by using different values for L and D.

3.3.3 Three types of dynamics

Again, there are three limits in this problem.

Slow expansion or contraction. In the limit where \L — D\ 2-JLDui, we have

61 = -L{ 1 - v^I), b2 = -L( 1 + V^T), and the behavior of y is as follows:

• for small t, that is when Ly/ult <C 1, we have y = 1 — Luit, whereas

• for larger t such that L^Juit > 1, y(t) —> yac, where

Shrinking population. If D > L and D - L » 2\fLDu[, we have

LDux LDu\

and the solution has the following shape:

y(t) = 1 — uiLt for (D — L)t <C 1, as before, and for t such that (D — L)t > 1, y(t) tends to a constant,

Fast expansion. In the opposite case where L — D 2^/LDu\, we have

and the solution has the following shape:

• y(t) = 1 - UiLt for (L — D)t <C 1, as before, and

• for t such that (L — D)t > 1, y(t) saturates at a different value,

3.3.4 Probability to create a mutant of type "C"

- For slow expansion/contraction, initially (L^Ju[T <C 1) we have a linear growth, Pi(t) « Lu\t, and then the probability saturates at

- For a shrinking population, initially ((D—L)t <C 1) we have Pi (t) ~ Luit, and then it saturates at Pi = Lu\/{D — L).

- For an expanding population, we have initially (when (L—D)t 1) Pi(t) « Lu\t, and then Pi = 1 - D/L.

The function Pi(t) is presented in Figure 3.2, the upper line.

3.3.5 A two-hit process

Next, we can solve the equation for x, equation (3.39), in order to obtain

Fig. 3.2 The functions -Pi(i) and P2(t) for an expanding population, corresponding to one-hit and two-hit processes. The function Pi(t) denotes the probability to create at least one "dangerous" mutant by time t, in an ¿-hit process. The parameter values are L = 3, D = l,u = ui=5x 10"4.

Fig. 3.2 The functions -Pi(i) and P2(t) for an expanding population, corresponding to one-hit and two-hit processes. The function Pi(t) denotes the probability to create at least one "dangerous" mutant by time t, in an ¿-hit process. The parameter values are L = 3, D = l,u = ui=5x 10"4.

the probability that a mutant of type "C" has been created starting from one cell of type "A". Using the same change of variables,

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