Lecture 10

Question 1

The coalescence model can be derived as —— from several ——.

Answer 1

a limiting distribution, population genetic models.

Question 2

What is the common assumption in the coalescent models ?

Answer 2

That the underlying population dynamics are deterministic, unlike birth death process

Question 3

What are Coalescent models often used for?

Answer 3

They are often used as the basis for phylodynamic inference of population size and dynamics.

Question 4

In the wright fisher process generations are —-.

Answer 4

discrete

Question 5

Each generation in the WF process consists of ——.

Answer 5

N individuals

Question 6

How do the individuals in the offspring population choose their parent?

Answer 6

uniformly at random from N parents.

Question 7

How are the number of offspring of a parent distributed?

Answer 7

Binomially

Question 8

If there are copies of a gene present in an individual, how can we account for this?

Answer 8

Ploidy can be taken into account by multiplying N by a factor which accounts for the number of copies of the gene present

Question 9

For a diploid organism, the number of copies of a gene in the population is —-?

Answer 9

2N

Question 10

Generations between internal nodes are related to —-.

Answer 10

population size

Question 11

When we sample a 2 individual phylogeny, what is the main question that we’re asking?

Answer 11

What is th probability of the individuals sharing a parent?

Question 12

P_coal for two individuals = ?

Answer 12

1*1/N

Question 13

What is the probability of coalescence in generation i - m?

Answer 13

p(m) = (1-p_coal)^(m-1) * p_coal = (1-1/N)^m-1*1/N. We keep trying for each generation—> no coalescence –>the first term, finally there is a coalesence –> 2nd term

Question 14

For large m, the probablity distiburiton becomes —-.

Answer 14

exponential

Question 15

If g is the calendar time of a generation, what is the calendar time span of m generations?

Answer 15

g*m

Question 16

What is the probablity density function for the coalesce time of two lineages?

Answer 16

1/gN * e^(-dt/gN)

Question 17

What is the average time to go back to find a common ancestor?

Answer 17

g*N

Question 18

In the large N limit, the time to coalescence is ?

Answer 18

exponentially distributed with mean gN

Question 19

How could we generalise p_coal for k samples?

Answer 19

k choose 2 *1/N

Question 20

Kingsman’s coalescent, uses —— time —— which produces —–.

Answer 20

continuous, Markov process, sampled time trees

Question 21

In KC, process occurs —– in time.

Answer 21

backwards

Question 22

In what sense is KC a Markov process?

Answer 22

if we start the tree simulation and move back in time, the only thing we need to know is the number of extant lineages and nothing else about the history (memorylessness aspect)

Question 23

How could KC be equivalent to WF?

Answer 23

It is equivalent to sampled trees produced by WF model when N is much larger than the number os samples

Question 24

Times between the coalesence events are drawn from —- distribution with rate parameter —-.

Answer 24

exponential, (k choose 2)*1/Ng

Question 25

Under the coalescent model the average time required for n lineages to coalescence into 1 is -----, which can be simplified to ---- for larger n. n is -----.

Answer 25

slide 17, 2Ng, number of leaves in the coalescent tree.

Question 26

What is the probability of a coalescence tree given Ng?

Answer 26

side 17

Question 27

What does each term represent in the coalescent tree probablity?

Answer 27

The nested term, what is the probablity that we go the entire duration without any common ancestor being found( k choose 2 being due to having k lineages, dt the duration in time , the other term is the probablity density that a particular part of lineage coalesce.

Question 28

Why would results from a coalesce model be different from a real tree?

Answer 28

1- the results could be biased since the real dynamics differ from WF dynamics 2- real population sizes are structured, where the WF population is assumed to be completely homogenous and that if we pick any individual, they have an equal chance of sharing a parent which is obv not true for many populations

Question 29

Inferred population size is referred to as the ----?

Answer 29

effective population size

Question 30

Inferred population size is the size of a ----- populations. does this share any similarity with the real population?

Answer 30

WF, yes, some statistical similarity, however care should be taken when drawing conclusion from the effective population sizes

Question 31

The coalescent distribution is derived as ----- of the WF process.

Answer 31

limit

Question 32

What other population process does WF appear as a limit as? How is the limit applied?

Answer 32

The canning model : generalisation of WF The Moran model : overlapping generations, fixed population size Stochastic logistic models :continuous time, population fluctuations

Question 33

What is the robustness of coalescent?

Answer 33

Coalescent distribution persists in the face of many departures from WF model is sometimes termed as robustness of coalescent.

Question 34

What are the 3 general assumption of coalescent?

Answer 34

1- samples are members of a population that is at demographic equilibrium. 2- Number of samples is small compared to the total population size 3- population is well mixed and samples are drawn uniformly at random.

Question 35

What does each assumption of coalescent justify?

Answer 35

demographic equilibrium : use of fixed or slowly varying population size small sample size compared to total population size: neglect of >2 lineages coalescing in the same generation well mixed: the coalescent rate between any pair of sample lineages being equal. population structure violates this assumption.

Question 36

Probability of a phylogenetic tree can be calculated via?

Answer 36

the rate of coalescence, so 1/ N(t), meaning the inverse of population as a function of time.

Question 37

When N(t) is larger we have a ---- coalescence rate and ----- branches.

Answer 37

slower, longer

Question 38

In paprmetric population dynamics inference rate of exponential distribution can/cant change through time, this is accounted for by having an ---- in the function.

Answer 38

can, integral

Question 39

Parametric population dynamic inference, leads to a likelihood that -----, therefore we can ---- different --- scenarios for a given ----.

Answer 39

Incorporates the growth rate term of population size N(t). compare and test, demographic, tree.

Question 40

How does non parametric population dynamics inference work?

Answer 40

Assume a population which has distinct constant values in each interval between coalescent events. we can obtain a separate ML estimate for each population size. Resulting population function estimate is the skyline plot.

Question 41

Can we develop coalescent distributions which approximate the probability density of sampled phylogenies generated by birth-death process? if yes, how?

Answer 41

yes. 1- Assume the ODE solution in slide 30 including the population size at present for linear birth death process is correct. Birth occurs at time t with overall rate BI(t). 2- Every brith is a potential coalesce between sampled lineages 3- probablity of choosing a sampled lineage pair is k choose 2 *I(t) choose two 4-we then approximate the coalesce rate 5- we can use the coalesce rate to compute an approximation to probablity of tree given the B,S,T.

Question 42

What does the quality of approximation depend on?

Answer 42

It depends on how well the birth death population dynamics are approximated by deterministic ODE solution. This approximation can perform very poorly when population size is small, as it is at the start of an epidemic.

Question 43

What are the parameters of birth death vs coalescent?

Answer 43

BD : transmission rate, removal rates, sampling rates/proportions C : effective population size (nOT THE ACTUAL POPULATION SIZE)

Question 44

BD models ------, while Coaslecent assumes -----.

Answer 44

sampling process (sampling times/locations are data), number of sampled lineages are small (k<

Question 45

What are the advantages and disadvantages of BD models?

Answer 45

+: 1- accounts for stochastic variability in population dynamics 2- generally easier interpretation of parameters 3- uses information about sampling -: 1- sensitive to un-modeled changes in sampling fractions 2- difficult to extend to complex population models

Question 46

What are the advantages and disadvantages of C models?

Answer 46

+: 1- Generally fast likelihood calculations 2- easy to extend to complex population dynamics 3- naturally account for incomplete sampling - : 1- sensitive to uncertainty in population dynamics at high sampling 2- sensitive to hidden population structure and nonrandom sampling

Question 47

Both BD and C models ----- relate a population's ---- to its -------, their difference is in their ----.

Answer 47

probabilistically, demography, phylogenetic history, parameterisation

Question 48

Under the WF modle, how many generation do we have to go back before we find the common ancestor of a pair of genes sampled from a haploid population of size N?

Answer 48

1/N

Question 49

Suppose you had a tree inferred using present day samples from a population that experience severe bottleneck in its recent past. How and why would this bottleneck likely affect our ability to infer ancestral population dynamics ?

Answer 49

Probability of coalescnce becomes very large at the bottle neck( lineages will be rapidly coalescing) and therefore languages coalesce and becomes very difficult to see past the bottleneck. so beyond this point it's like we dont have any coalesce anymore so we cant infer anything from a tree which is corresponding to just one tip.

Question 50

Imagine spreading a WF population across islands in an archipelago, so that movement between the islands is restricted but within each island the population is well mixed. Qualitatively, how would you expect this population structure to influence estimates of the effective population size?

Answer 50

if there is no movement between them whatsoever they all have 3 independent trees. if there is still some slow movement: within each island the tree and the timing between branching events is based on the population size, then within the islands, the coalesce rate is dependant on slow movement rate in the process so the length of the branches are going to be longer. see slide 3 of lecture 11 for the tree .