# Presentation: 801761

## Estimating Human Recombination Through Computational Methods

- Bert Gold
- April 24, 2008

## Precis

- Recombination is a frequent and obligatory process during human meiosis.
- Mutation is frequent enough to be a confounding factor in computational simulations of recombination.
- Population history reconstruction through the construction of coalescent trees provides a way to view recombination history as well as past population dynamics.
- Numerous population genetic models of human recombination have been conceived in the past ten years, and nearly all of them have some heuristic value.
- We are focusing on two of these models (with options on a third).
- For actual results see: _[http://theta2.ncifcrf.gov](http://theta2.ncifcrf.gov/)_

## Meiosis in the Human Male

- 2

## Cytogenetics of Human Male Recombination

## Products of Human Male Recombination After MII

## Meiotic Movie Diversion

## Observation of Human Recombination

## Phenomenology of Human Recombination

## Recombination Movie Diversion

## Recombination Mapping by RFLP or STRs

## Mutation Facts

- ~1014 cells in an adult human being
- Haploid # = c = 3 x 109 bp
- Estimated mutation rate =          nucleotides
- This reduces to ~ 1 mutation per 10,000 genes per generation, or about 3 mutations per haploid gene complement.
- Therefore, every cell contains ~ 120 errors and each person about 6 mutations in genes in every cell.
- Single human sperm ejaculate ~ 1.8 x 106 sperm
- Probably ~ 90% of mutations arise in males during spermatogenesis
- Each of us has probably had a lesion in each of our genes in some cell or another.

## Coalescence

- In a population of 1000 people who choose their mates at random, 10 generations are normally enough to guarantee that any two people have some ancestor in common.
- And, 18 generations normally guarantee that any two people have all their ancestors in common.

## Knowing  allows back calculation of Ne

- Because:

-  = 4 Ne

- per generation
- mutation rate

- population mutation
- rate per generation

- But the mutation rate varies by locus!!!

## From Andrew Clark’s Mini-Review

## Methods for Calculating and Estimating Human Recombination

- Observations in Families
- STR markers coupled with segregation analysis
- Marshfield, Genethon, Decode maps
- Sperm Typing
- Known somatic haplotypes provide basis for mapping sperm events.
- Statistical Methods based upon haplotypes
- Moment Estimators
- Full Likelihood Methods (Ewens Sampling)
- Composite Likelihood Methods (Watterson’s Estimate)
- Approximate Conditionals Methods (new  distributions)

## How Well Do Hot Spots and LD Correlate?

## An ARG (Ancestral Recombination Graph)

**Notes:**
Statistical inferences of evolutionary processes often centre on a description of the genealogy underlying a population or sample. The coalescent is a stochastic process that generates such genealogies. In the legend, 0 and 1 denote ancestral and derived alleles, respectively. a | The genealogy of a single hypothetical locus is represented by a single bifurcating tree. A mutation event of 0right arrow1 (arrow) gives rise to a derived allele. b | The genealogy of a second locus (red) that is physically close to the locus depicted in part a is shown; its genealogy is partially correlated with the original (blue) genealogy (these are known as MARGINAL GENEALOGIES). If mutations occur along the two lineages (indicated by the solid arrows) then the recombination event will be detected in the resulting two-locus gametes, because, as shown here, all four possible gametes (0,0; 0,1; 1,0; 1,1) are observed in the sample. It should be noted that there are two lineages along the red genealogy for which a mutation event can cause the recombination event to be detected (red solid and dashed arrows). c | In these two genealogies the recombination event cannot be detected from the resulting data, no matter which lineages mutations occur on. This is because there is no combination of lineages among the two marginal genealogies along which mutations will give rise to all four possible two-locus gametes. For this reason smaller samples are less informative about recombination than larger samples.

## Another ARG

**Notes:**
An ancestral recombination graph of the geneologies at three linked loci for a sample size of 4:  The ARG has been drawn back to the oldest most recent common ancestor (OMRCA).  Mutations are depicted by circles.  Both the types of loci at the top of the graph and the types of mutations have been specified.  This uniquely determines the type of the sample.

## Statistical Methods for Estimating Recombination

- Moment Estimators
- Full Likelihood
- Composite Likelihood
- Product of Approximate Conditionals

## The Population Recombination Rate

-  = 4Ner

- per generation
- recombination rate

- effective population size

- population recombination
- rate per generation

- a function of
- demographic history

- a function of
- local factors, DNA structure

## World Human Population Growth

## Ne = 104?

## Full likelihood methods

- Demographic history given
- the coalescent model and its
- parameters

- per generation
- recombination rate

- per generation
- mutation rate

## Fearnhead and Donnelly Method

- If r and  denote the probability of recombination and mutation
- in a region of interest, and     = 4Nr and  = 4N, where
- N is the effective population size, then, no problem estimating
- that observed data is some function of  and .
- Counting recombination and mutation events and dividing
- these over the time over which they’ve occurred would give
- an adequate history.  But we can’t do this.
- So, what we end up with is counting an average (or integral)
- Over the (uncountable) number of possible histories consistent
- With the data.  Monte Carlo methods are needed because it would
- Take 1050 simulations to find one history consistent with the data.

## Fearnhead and Donnelly continued

- Therefore, use importance sampling (generate genealogical histories consistent with the data) to distribute a weight of possible population history reconstruction scenarios over a “likelihood surface”.
- Likelihood surfaces plot  and  versus likelihoods that that population history reconstruction is consistent with the data.

## Likelihood Surface

**Notes:**
Figure 2 from Fearnhead et al., 2004; The likelihood surfaces in the scaled mutation and recombination rates theta=4Neu and rho=4Ner where u and r are the mutation and recombination rates per generation across the relevant region, and Ne is the effective population size.  (a) Region I, The Gambia; (b) Region I, United Kingdom; © region IV, The Gambia; (d) Region IV, United Kingdom.

## Standard Coalescent Model 

- Wright Fisher
- Neutrally evolving (sometimes called “neutral coalescent”)
- Constant-sized
- Panmictic population
- Infinite Sites - the infinite sites model assumes no repeat mutation.

## Ewens’ Sampling Formula

- Where a1, a2 etc. are singly sampled alleles,
- in n gametes and j provides the number of draws.

## Recursive Methods

- Possible ancestral states are inferred.
- The observed product is then removed from the dataset.
- Then a possible ancestral state is again inferred.
- This method, ‘importance sampling’ permits a description of the likelihood surface.

## Markov Chain Monte Carlo Methods (MCMC)

- These are applicable to full likelihood, composite likelihood, or approximate likelihood methods
- ‘Stochastic’ because they use an initial guess, within bounds, and then make subsequent adjustments
- Changes that are more probable are accepted, steps less likely rejected, with probabilities proportional to how likely they are to have occurred under the assumed model

## Pairwise (Composite) Likelihoods

- 1

- 1

- 1

- 1

- 1

- 2

- 2

- 2

- 2

- 2

- 2

- 2

- Definite recombination,
- but where did it occur?

- CL = prob. r + prob  = near 1 x next trio

-  A major advantage is the development of a lookup table
- for all pairwise possibilities

-  Another advantage is that no haplotypes are needed!

-  Finally, you can leave out alleles below a threshold frequency
- and consider only a few SNPs at a time.

## Watterson’s Estimate(of mismatch distribution)

- number of segregating sites
- in a sample of i genes

- Where  = 4Ne

- Where  is the
- mutation rate
- per gene (not
- site).

## Watterson’s continued

- In the special case where there are no differences
- between sites i = 0 and the equation simplifies to:

## Recombination Estimate Methods Continuum

- Full Likelihood

- Composite, Approximate Likelihood

- Moment Estimators

- Computationally intensive,
- Extensive information inputs
- So far ‘impractical’for large
- Genomic regions or populations

- Computationally practical,
- Leaves out either minor alleles under
- Some threshold, doesn’t employ
- Haplotypes (composite, pairwise), or
- Or considers only a small number of
- Markers at a time.
- Amenable to computation through the
- Use of lookup tables (CL), or sampling
- Algorithms (AL).

- Computationally rapid,
- Uses LD to estimate
- Recombination directly.
- Subject to perturbations
- And inconsistencies.

## Product of Approximate Conditionals Method

- Or by approximation:

- Redefining:

## Rudiments of the    A Distribution

- 1. Make a distribution of values of c(x) where
-      represents the background rate.
- 2. one where you express      in terms of a, b, 
- and     , which is where  is greater than one,
- and one otherwise.

## Variables that make up the     distributions

- = background crossover rate

- N

- = effective population size

- j

- = multiplier controlling for
- between marker crossover rate variation
- (aka crossover rate between a and b)

- 

- = magnitude of recombination hotspot =

- S

- = number of SNPs

- a

- b

- = left hand side of the region under consideration

- = right hand side of the region under consideration

- = 4N

## Contribution to local rate

- x

- 

- 

- or

- 10x variation, p ~ 95%

## Comparing Estimators

- Bias (PACB corrects for bias, PACA does not)
- Variance
- Statistical Efficiency
- Robustness to deviation from the assumed model
- Computational efficiency
- Consistency

- Attendant Questions

- Do we expect the estimator to obtain the correct value?
- How much variability do we expect in estimates for a given value of the underlying recombination rate?
- How much of the information contained in the data does each estimator use?
- How wrong will the estimate be if the assumptions are incorrect?
- How fast can the estimate be computed?
- Would the estimator be perfect if we had unlimited amounts of data?

## Four Questions to be Answered

- How much recombination occurs in one region compared with another?
- Is there variation in recombination rate within the region being analyzed?
- Is the effective population size of the group being examined, greater than, or smaller than that of another population?
- How good is the neutral coalescent with recombination at describing patterns of variation?
- (that is, is there evidence for natural selection or complex demographic processes?)

## For datasets with a small number of SNPs (≤ 12) all methods do very poorly and    CL behaves much better than either     PAC-A or            PAC-B, especially for datasets that seem to support very small values of recombination rates.

- A practical consideration

## A “practical” test on simulated data

## Slide 43

## Summary

- Recombination is a human obligation.
- Mutation can be thought of as “imperfect” recombination and confounds population history reconstructions of recombination.
-      (examples:  Slip-strand mispairing, gene conversion, retrotransposition)
- Full Likelihood coalescent modeling provides a way to simulate recombination in small stretches of human genome.
- Composite Likelihood and Approximate Conditionals methods use simplifications of Full Likelihoods to provide useful approximations of recombination histories throughout the genome.
- There is heuristic and didactic value to all these models, although their accuracy may be doubted.
- Simulation is valuable.  We have focused on two of these models (with options on a third).
- For actual results see: _[http://theta2.ncifcrf.gov](http://theta2.ncifcrf.gov/)_

## ACKNOWLEDGEMENT

- Mary Carrington
- Mike Cullen
- Judith Hall (UBC)
- Alec Jeffreys (Leicester)
- David Ledbetter (Emory)
- Na (Michael) Li (U. Minnesota)
- Stefan Stefanov
- Kate McGee
- Eckart Bindewald
- Matthew Stephens (U. Washington)

## The END

- (supplementary slides follow)

## Female Meiosis is Different I

## Female Meiosis is Different II

## Female Meiosis Completes After Fertilization