It n

eait.

Then type i is at its maximum near t»U = -J— flog —+ T log ^LZ-^ . (7.22)

In particular, after time tn, the type xn will dominate.

Multidimensional competition dynamics. Equations (7.5-7.11) represent two parallel mutation cascades, that is, two sets of quasispecies equations, coupled via the common fitness term, (j). In order to use the

Fig. 7.3 Simulation of mutation cascades. In the picture above, we have ai = ai-i +2, for 1 < i < n and fcj = e. In the picture below, we have a, = Oj_i + 2(1 + and bi — e(l + Q), where and Q are some random numbers drawn from a uniform distribution between zero and one. For both pictures, n = 9, Ci = 0 for all i, e = 0.001 and ao = 1.

techniques developed above, let us write the equations for the mutational cascade in a simpler form,

by introducing the following obvious notations:

Zi ->• Si, z'i Mi, 0 <i<n, zn+1 ->• w, (7.26)

ai = Ri{l-us), a^Riil-Um), 0 < % < n - 1, (7.27)

an = Rn[l - us +au(l - es)], a'n = Rn[l - um + au(l - em)}, (7.28) bi = auRi-i(l-ea), b'^auRi-^l-ern), 1 < i < n, (7.29)

Ci = (1 - a)JRiu(l - es), c- = (1 - a)i?iU(l - em), 0 < i < n. (7.30)

In a matrix notation, equations (7.5-7.11) read:

where the fitness matrices fS:Tn are found from (7.23-7.25). The solution of the nonlinear system can be found by re-normalizing the solution of system (7.23-7.25), as before.

Fitness landscape. In order to analyze the dynamics of system (7.5-7.11), we have to make assumptions on the fitness landscape for the consecutive mutants (Figure 7.4).

(a) Apoptosis intact

(b) Apoptosis impaired

(a) Apoptosis intact

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