Evolutionary Dynamics  Exercises 5 -Solved

Problem 1: Pathways of carcinogenesis 
Consider three independent mutations {1,2,3}. Each mutation occurs after an exponentially distributed 
waiting time Ti ∼ exp(λi), i = 1,2,3. 
(a) What is the probability for the path P = 2 → 1 → 3? 

(b) Assume cancer arises if any two of the three genes are mutated. How many possible genotypes 
are there? How many pathways? Compute the expected waiting time until any two out of three 
genes are mutated. 

(c) Now consider d independent mutations. How many paths exist leading to the genotype {1,...,d} 
with all mutations present? If cancer already arises after any k mutations, how many different 
paths are there? 
 
Problem 2: Neutral Wright-Fisher process 
Consider the neutral Wright-Fisher process for a system of N cells of two different types {A,B}. Let 
X(t) denote the number of A-cells at time t. The process has the transition matrix 
Pi, j = Prob[X(t) = j | X(t −1) = i] =  N
j  
i
N 
 j  NN−i N−j 
, 
that is, X(t) | X(t −1) = i is binomially distributed with parameter p = i/N. 
(a) Compute the conditional expectation E[X(t) | X(0) = i]. 
(tutorial discussion) 
(b) Compute the conditional variance Var[X(t) | X(0) = i]. 

Hint: Show that 
Var[X(t) | X(0) = i] = V1 + (1−1/N)Var[X(t −1) | X(0) = i] 
where V1 = Var[X(1) | X(0) = i]. You can then use the expression above to derive the fifinal result 
(no explicit calculation needed for this last step). 
(c) Derive an approximation for Var[X(t) | X(0) = i] for large population size N. Compare the vari
ance of the Wright Fisher process to the variance of the Moran process, explain the difference(s). 

(d) Show that in the Wright-Fisher process, the heterozygosity Ht at time t satisfifies 
(1 point) 
E[Ht | X0 = i] = H0(i) 1− N1  t 
and hence decreases exponentially at rate 1/N. Compare this behaviour with the Moran model. 
1Note: Heterozygosity in this context is defifined as the probability that two individuals chosen at 
random from the population are of different types. 
(e) § Simulate the Wright-Fisher process. Compute empirical mean and variance and compare them 
with your analytical results. Use n = 100 simulations with population sizes of N ∈ {10,100}, 

Problem 3: Wave approximation 
Consider the wave approximation of the Wright-Fisher model for cancer progression. Here, the growth 
of a clone with j mutations is given by 
x˙j = sxj(j − hji ). 
For small times, the average fifitness sh ji = s∑j jxj can be considered constant. Use this throughout your 
calculations. 
(a) Find the analytic solution for the initial condition xj(0) = 1/N. 

(b) The rate at which an additional mutation occurs is given by udxj(t). Find the time τ when the 
cumulative probability exceeds 1/N. 

(c) Compute the waiting time until the next mutation for a mutation rate u = 10−7/cell generation, 
d = 90 genes and a fifitness advantage of s = 1.25% per mutation. Use that j − hji ≈ 
√ logN with 
N = 107 cells and assume a cell generation time of 1 day. 

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