1. Continuous Coalescent Model
The continuous coalescent lends itself to generative models
Algorithm to construct a plausible genealogy for n genes
Note that this model runs backwards, it begins from the current population and posits ancestry, in contrast to a forward algorithm like those used in the first lecture
Start with k = n genes
Simulate the waiting time, , to the next event,
Choose a random pair (i, j) with 1 ≤ i < j ≤ k uniformly among the pairs
Merge I and J into one gene and decrease the sample size by one, k  k -1
Repeat from step 2 while k > 1
4/17/2017
Comp 790– Continuous-Time Coalescence

2. Comp 790– Continuous-Time Coalescence
In Python
A simulator in 12 lines
 T = [[i,0.0] for i in xrange(N)] # gene id, time of merge
k = N
t = 0.0
while k > 1:
t += expovariate(0.5*k*(k-1))
i = randint(0,k-1)
j = randint(0,k-1)
while i == j:
T[i] = [T[i], T[j], t]
T.pop(j)
k -= 1
4/17/2017
Comp 790– Continuous-Time Coalescence

3. Properties of a Coalescent Tree
The height, Hn, of the tree is the sum of time epochs, Tj, where there are j = n, n-1, n-2, … , 2, 1 ancestors.
As n ∞, E(Hn)  2,
and, if n=2, E(H2)=1.
Thus, the waiting time for n genes to find their common ancestor is less than twice the time for 2!
As n ∞, Var(Hn)  4(π2-9)/3,
and, if n=2, Var(H2)=1.
4/17/2017
Comp 790– Continuous-Time Coalescence

4. Comp 790– Continuous-Time Coalescence
Sampled Distribution
N =
4/17/2017
Comp 790– Continuous-Time Coalescence

5. Comp 790– Continuous-Time Coalescence
Example Trees
Observation: The contribution of T2, where the last two ancestors converge to a common root, is disproportionately large
4/17/2017
Comp 790– Continuous-Time Coalescence

6. Comp 790– Continuous-Time Coalescence
Total Branch Length
In contrast to Hn, the distribution of the total branch length Ln, has a simple form:
The mean of Ln is found by weighting the coalescent times by the number of active lineages
This sum does not converge for large n, but grows slowly. It fact, it is proportional to log(n)
4/17/2017
Comp 790– Continuous-Time Coalescence

7. Comp 790– Continuous-Time Coalescence
Shared History
E(Ln) can be used to get a sense of how much history genes share.
Genes would share the least history if they all arose from a common ancestor long ago and then propagated along distinct lineages.
If the mean time to the common ancestor is E(Hn) = 2(1 – 1/n), and we assume the split was a early as possible (thus minimizing the shared history), then the total branch length would be nE(Hn) = 2(n-1).
Comparing to E(Ln) as a fraction of this minimum shared-history case gives: …
4/17/2017
Comp 790– Continuous-Time Coalescence 7 7 7 7

8. Comp 790– Continuous-Time Coalescence
Plot of Shared History
Even for small n, samples, on average, share considerable history
share(5) = 48%
share(10) = 69%
share(20) = 81%
Sharing is the fraction of a genealogy that an average gene shares with two or more other extant genes
4/17/2017
Comp 790– Continuous-Time Coalescence

9. Variance of Total Branch Length
The variance in the total branch length is: which converges to 2π2/3 ≈ as n ∞.
This implies that for large n, Ln is narrowly centered around E(Ln). Likewise, sharing is also relatively consistent.
4/17/2017
Comp 790– Continuous-Time Coalescence

10. Implications on Sampling Paths
Sampling multiple paths from extant genes along their ancestors is less effective than one might think.
Most long branches are covered by relatively few samples
Not surprising since the E(H40) = 1.95 and E(H10) = 1.8 (a 4x increase in samples increases height by less than 10%).
4/17/2017
Comp 790– Continuous-Time Coalescence

11. Effective Population Size
Real populations are not likely to satisfy the Wright-Fisher model.
In particular, most real populations show some sort of reproductive structure, either due to geography or societal constraints
Also likely that the number of descendents is a generation depends on many factors (health, disease, etc.), as opposed to the implicit Poisson model
Total population size is not fixed, but changes over time
4/17/2017
Comp 790– Continuous-Time Coalescence

12. Comp 790– Continuous-Time Coalescence
Sanity Check
When the Wright-Fisher model, or the basic coalescent, is used to model a real population, the size of the population (2N) cannot be taken literally.
For example, many human genes have a MRCA less than 200,000 years ago. If we consider one generation per 20 years then N should be less than 200,000/(4*20) = 2500, which is too small (recall the maximum tree height for the entire population is 2. and 2(2 generation_time) = 4*20)
4/17/2017
Comp 790– Continuous-Time Coalescence