Introduction
The Coalescent
The Birth–Death Process
Our Work
Results and Discussion

Connections between the
Coalescent and Birth–Death
Sampling Processes
Spencer Enesa
University of Auckland
Department of Statistics

Project Presentation

S. Enesa

Connections between the Coalescent & BD Sampling Processes

Introduction
The Coalescent
The Birth–Death Process
Our Work
Results and Discussion

Outline
1 Introduction
2 The Coalescent

Standard coalescent
Variable population size
3 The Birth–Death Process

The birth–death-samplingm process
Connection to the coalescent
4 Our Work
5 Results and Discussion

Results
Discussion
Conclusion
S. Enesa

Connections between the Coalescent & BD Sampling Processes

Introduction
The Coalescent
The Birth–Death Process
Our Work
Results and Discussion

Standard coalescent
Variable population size

The Coalescent
The Wright–Fisher Model

S. Enesa

Connections between the Coalescent & BD Sampling Processes

Introduction
The Coalescent
The Birth–Death Process
Our Work
Results and Discussion

Standard coalescent
Variable population size

The Coalescent
The Wright–Fisher Model

Consider two lineages from a population of size N and trace their ancestry backward in time. The probability that they have not coalesced t generations ago is
P(T (2) > t) =

1−

1
N

t

i.e., the time for two lineages to coalesce is geometrically distributed. S. Enesa

Connections between the Coalescent & BD Sampling Processes

Introduction
The Coalescent
The Birth–Death Process
Our Work
Results and Discussion

Standard coalescent
Variable population size

The Coalescent
Kingman’s Approximation

Rescale time so that one unit of scaled time, τ , corresponds to N generations. Then,
1−

1
N

Nτ

→ eτ

as N → ∞.
We can ignore the event that more than two lineages coalesce simultaneously. S. Enesa

Connections between the Coalescent & BD Sampling Processes

Introduction
The Coalescent
The Birth–Death Process
Our Work
Results and Discussion

Standard coalescent
Variable population size

Variable population size

The coalescent with variable population size converges to the standard coalescent: if the changes in population size are deterministic or if the changes in population size are stochastic and the population sizes are independently and identically distributed.

We relax this assumption and allow the population size to vary according to a birth-death process.

S. Enesa

Connections between the Coalescent & BD Sampling Processes

Introduction
The Coalescent
The Birth–Death Process
Our Work
Results and Discussion

The birth–death-samplingm process
Connection to the coalescent

The Birth–Death Process

Begin with one individual. Each individual gives birth at rate λ and dies at rate µ.
We condition on having m individuals at the present.
We consider the special case where λ = µ, the critical branching process.

S. Enesa

Connections between the Coalescent & BD Sampling Processes

Introduction
The Coalescent
The Birth–Death Process
Our Work
Results and Discussion

The birth–death-samplingm process
Connection to the coalescent

The birth–death-samplingm process

Given a birth–death tree on m individuals, we sample n of these individuals uniformly at random and recover the process which leads to these n individuals.
When λ = µ, the expected time for n = 2 lineages to coalesce is m
E [TBDSM (2)] =
2λ

S. Enesa

Connections between the Coalescent & BD Sampling Processes

Introduction
The Coalescent
The Birth–Death Process
Our Work
Results and Discussion

The birth–death-samplingm process
Connection to the coalescent

Connection to the Coalescent

The expected time for two lineages to coalesce under the standard coalescent is 1 unit of scaled time (m generations).
The expected time for two lineages to coalesce under the standard coalescent is equivalent to that under the birth–death-samplingm process with rate λ = µ = m
2
However, their variances differ.

S. Enesa

Connections between the Coalescent & BD Sampling Processes

Introduction
The Coalescent
The Birth–Death Process
Our Work
Results and Discussion

The birth–death-samplingm process
Connection to the coalescent

Connection to the Coalescent
Density Plots

Figure: Density of the coalescent time of two individuals under the standard coalescent (black), under the birth–death-samplingm process with λ = µ = m/2 (blue).
S. Enesa

Connections between the Coalescent & BD Sampling Processes

Introduction
The Coalescent
The Birth–Death Process
Our Work
Results and Discussion

Our Work

In our model we extend the coalescent to allow the population to fluctuate according to a constant rate birth–death process:
The population size moves from state i to (i + 1) with birth rate λi = iλ and moves to state (i − 1) with death rate µi = iµ.
The process is reversible. This allows us to simulate the fluctuation in population size backwards in time, i.e., just like the coalescent.

S. Enesa

Connections between the Coalescent & BD Sampling Processes

Introduction
The Coalescent
The Birth–Death Process
Our Work
Results and Discussion

Our Work
Simulating the model with slow fluctuations

Figure: The population change (black), number of lineages (grey), for λ = µ = 0.001
S. Enesa

Connections between the Coalescent & BD Sampling Processes

Introduction
The Coalescent
The Birth–Death Process
Our Work
Results and Discussion

Our Work
Simulating the model with fast fluctuations

Figure: The population change (black), number of lineages (grey), for λ = µ = 0.1
S. Enesa

Connections between the Coalescent & BD Sampling Processes

Introduction
The Coalescent
The Birth–Death Process
Our Work
Results and Discussion

Our Work
Simulations

We use this simulation scheme to investigate the distribution of the time to coalescence for k = 2 individuals when the population size fluctuates according to a critical birth–death process (λ = µ).
We set the initial population size to be N = 100.

S. Enesa

Connections between the Coalescent & BD Sampling Processes

Introduction
The Coalescent
The Birth–Death Process
Our Work
Results and Discussion

Results
Discussion
Conclusion

Results
Table

λ
0.001
0.005
0.01
0.05
0.1
0.5
1
5
10

mean
50171.88
9946.71
5000.74
995.13
501.03
100.20
50.49
9.97
4.98

s.d.
50171.99
10015.17
5106.49
1089.55
584.97
153.52
89.00
22.73
12.85

N
2λ

50000
10000
5000
1000
500
100
50
10
5

Table: Simulation results.
S. Enesa

Connections between the Coalescent & BD Sampling Processes

Introduction
The Coalescent
The Birth–Death Process
Our Work
Results and Discussion

Results
Discussion
Conclusion

Results
Slow fluctuations

Figure: Simulation results for λ = 0.01.
S. Enesa

Connections between the Coalescent & BD Sampling Processes

Introduction
The Coalescent
The Birth–Death Process
Our Work
Results and Discussion

Results
Discussion
Conclusion

Results
Intermediate fluctuations

Figure: Simulation results for λ = 0.1.
S. Enesa

Connections between the Coalescent & BD Sampling Processes

Introduction
The Coalescent
The Birth–Death Process
Our Work
Results and Discussion

Results
Discussion
Conclusion

Results
Fast fluctuations

Figure: Simulation results for λ = 1.
S. Enesa

Connections between the Coalescent & BD Sampling Processes

Introduction
The Coalescent
The Birth–Death Process
Our Work
Results and Discussion

Results
Discussion
Conclusion

Results
Fast fluctuations

Figure: Simulation results for λ = 50.
S. Enesa

Connections between the Coalescent & BD Sampling Processes

Introduction
The Coalescent
The Birth–Death Process
Our Work
Results and Discussion

Results
Discussion
Conclusion

Discussion

For sufficiently small λ, the time is distributed as the standard coalescent. For sufficiently large λ, the time is distributed as the birth-death-samplingm process (??)
Hence, the effective population size only exists in expectation.

S. Enesa

Connections between the Coalescent & BD Sampling Processes

Introduction
The Coalescent
The Birth–Death Process
Our Work
Results and Discussion

Results
Discussion
Conclusion

Conclusion

We have shown that convergence to the standard coalescent is not always obtained when the population size fluctuates randomly.
Whether or not this happens depends on: whether the fluctuations are independent and the relative time scales at which coalescences and population fluctuations are operating.

S. Enesa

Connections between the Coalescent & BD Sampling Processes

Introduction
The Coalescent
The Birth–Death Process
Our Work
Results and Discussion

Results
Discussion
Conclusion

THE END

S. Enesa

Connections between the Coalescent & BD Sampling Processes

Introduction
The Coalescent
The Birth–Death Process
Our Work
Results and Discussion

Results
Discussion
Conclusion

Connection to the Coalescent
Time Scale

In the coalescent we measure time in units of N generations and in the birth-death model we measure time in calendar units. Using the fact that the expected times are equal for λ = µ = N , we can define 1 generation to correspond to
2
calendar time units.

1
2λ

This makes sense, if we consider that an individual dies when it gives birth to two new individuals (rate λ) or ‘actually’ dies
(with rate µ = λ).
Hence,

N
2λ

calendar time units correspond to N generations.
S. Enesa

Connections between the Coalescent & BD Sampling Processes

Introduction
The Coalescent
The Birth–Death Process
Our Work
Results and Discussion

Results
Discussion
Conclusion

Our Work
Simulating the model

3

Simulate a time to a population change, tN(i) from an exponential distribution with rate (λ + µ)N(i)
Run the standard coalescent with population size N(i) for
1
tN(i) time units. The pairwise rate of coalescence is
.
N(i)
If k = 1 stop, otherwise sample the next population size:

 N(i) + 1, with probability λ

 λ+µ N(i + 1) =

 N(i) − 1, with probability µ

λ+µ

4

Return to step 1 and set i = i + 1.

1

2

S. Enesa

Connections between the Coalescent & BD Sampling Processes