Title: Gametic Phase Disequilibrium Multilocus Population Genetics
1Gametic Phase Disequilibrium (Multilocus
Population Genetics)
- Our Topics
- Two-locus gametic frequency matrices
- Definition of linkage disequilibrium (D)
- Decay of D to 0 with H-W assumptions
- Factors affecting D
2Gametic Phase Disequilibrium (Multilocus
Population Genetics)
- This is an introduction to the topic -- we will
revisit multilocus issues (several times) as we
enjoy our journey through the semester!
3Gametic Phase Disequilibrium (Multilocus
Population Genetics)
- Real organisms have 1 locus . . . so . . . we
should wonder if H-W extends to multiple loci,
i.e. - Are multilocus frequencies predictable from
single-locus frequencies? - Is an equilibrium reached in 1 generation?
4Two Loci, Each with Two Alleles Hypothetical
Gametic Frequency Matrix 1
pi freq(Ai) qi freq(Bi)
5Two Loci, Each with Two Alleles Hypothetical
Gametic Frequency Matrix 1
6Two Loci, Each with Two Alleles Hypothetical
Gametic Frequency Matrix 2
7Two Loci, Each with Two Alleles General Form of
the Gametic Frequency Matrix
Pij freq(AiBj)
8Two Loci, Each with Two Alleles General Form of
the Gametic Frequency Matrix
FACT p1 p2 1 FACT q1 q2 1 FACT
P11 P12 P21 P22 1
9Two Loci, Each with Two Alleles General Form of
the Gametic Frequency Matrix
The alleles of the two loci are statistically
independent if P11p1q1 and P12p1q2 and
P21p2q1 and P22p2q2 Such statistical
independence is called linkage equilibrium.
10Two Loci, Each with Two Alleles General Form of
the Gametic Frequency Matrix
If the alleles of the loci are statistically
dependent, we measure the departure from
independence by D P11 - p1q1 P11P22 -
P12P21 D is called the linkage disequilibrium
parameter. (Synonym gametic-phase
disequilibrium)
11D Computation in Our Hypothetical Gametic
Frequency Matrix 1
D P11 - p1q1 0.35 - (0.5)(0.7) 0 D
P11P22 - P12P21 (0.35)(0.15) - (0.15)(0.35) 0
12D Computation in Our Hypothetical Gametic
Frequency Matrix 2
D P11 - p1q1 0.20 - (0.5)(0.7) - 0.15 D
P11P22 - P12P21 (0.2)(0.0) - (0.3)(0.5) - 0.15
13D What Does It Mean?
- 1. If D 0 the two loci are statistically
independent - 2. If D ? 0 the magnitude of D is related to
strength of disequilibrium (statistical
non-randomness) - 3. If D ? 0 the sign of D indicates direction
of disequilibrium (i.e. which alleles occur
together in gametes more frequently than expected
by chance)
14The Fate of D
- What happens to D over time?
- (Recall single-locus H-W proportions are
reached in a single generation of random mating.) - Lets assume all H-W assumptions are met, except
that we now have two loci, each with two alleles.
15The Fate of D
- We need some definitions
- P11(0) Freq(A1B1) initially (generation 0)
- P11(1) Freq(A1B1) in generation 1
- P11(t) Freq(A1B1) in generation t
- D(t) D in generation t
- r recombination rate between A B
16The Fate of D
- We can express P11(1) as a function of P11(0), r,
and allele frequencies - P11(1) (1-r) P11(0) rp1q1
Chance of obtaining A1B1 by recombination
Chance of being A1B1 initially and not recombining
17The Fate of D
- Since P11(1) (1-r) P11(0) rp1q1,
- D(1) P11(1) - p1q1 (1-r) P11(0) rp1q1 -
p1q1 - (1-r)(P11(0) - p1q1)
- (1-r)D(0)
- PUNCH LINE This is not 0 (unless D(0) 0)!
18The Fate of D
- D(t) (1-r)1 D(t-1)
- (1-r)2 D(t-2)
- (1-r)3 D(t-3)
- (1-r)t D(0)
- Since 1/2 1-r 1
- D decays to 0 asymptotically as t increases
(unless r0) - If r is small, the rate of decay is SLOW
19The Fate of D
- Computer simulations of the decay of D will be
shown in class for various values of r
20Other Measures of Nonrandom Association Among Loci
- Notice that D is allele-frequency dependent
- D P11 - p1q1
- We would like a statistic that enables us to
compare disequilibrium among populations, even if
allele frequencies differ.
21Other Measures of Nonrandom Association among
Alleles
- D D / Dmax if D0
- D D / -Dmin if D
- Dmax minp1q2, p2q1
- Dmin max-p1q1, -p2q2
- -1
22Other Measures of Nonrandom Association among
Alleles
- Correlation Coefficient (?)
- ? D / (p1p2q1q2)1/2
- ?12 ?2n
- (n is gametes sampled for each locus, i.e. the
haploid sample size)
23What Is D in Real Populations?
- Factors that affect D
- 1. Physical linkage reduces rate of decay of D
- 2. Inbreeding reduces rate of decay of D
- 3. Drift can be strong for multiple loci
- 4. Population subdivision, migration, Wahlund
effect can cause D ? 0 - 5. Selection can increase D (e.g., imagine A1B1
A2B2 are favored but A1B2 A2B1 are
disfavored this is an example of epistatic
selection)
24What Is D in Real Populations?
- Predictions from Theory
- 1. D is likely to differ from 0 if
- r small
- inbreeding present
- N small
- Permanent nonzero D requires epistatic selection
- (and large permanent D requires tight linkage,
strong inbreeding, or strong epistasis)
25What Is D in Real Populations?
- Observations in Nature
- 1. Random mating populations (e.g. humans)
- D usually near 0 (except for tightly linked loci)
- 2. Inbreeding populations
- D ? 0 (almost always, for all pairs of loci)
26What Is D in Real Populations?
- Implications of D
- 1. Conservation biology
- 2. Breeding programs
- 3. DNA forensics
27The Power of Multilocus Analyses
- In outcrossers (such as humans), D is usually
near 0 . . . so . . . the frequency of a
multilocus genotype is approximately the product
of relevant single-locus genotype frequencies,
e.g. - If frequency of blue eyes 0.2
- And if frequency of red hair 0.4
- Then Frequency of blue eyes red hair
(0.2)(0.4) 0.08 - This exemplifies the product rule
28The Power of Multilocus Analyses
- TEAMS Use our overall class data to predict
the frequency of individuals with - LONG HALLUX (ha ha)
- AND
- BLOOD TYPE O
29The Power of Multilocus Analyses
- Consider DNA forensic loci (13 loci) . . .
- If 10 equally-frequent alleles exist at each
locus, each allele frequency is 0.1 and - Probability of any 13-locus heterozygote is
- 2(0.1)(0.1) 13 8.2 x 10-23
30THERE ARE MORE THAN 2 LOCI AND MORE THAN 2
ALLELES IN THE REAL WORLD!!!
- We have defined D, D, ? in terms of 2 loci, each
with 2 alleles - When there are more than 2 loci and/or more than
2 alleles . . . - 1. The math is more annoying!
- 2. In general, we find that we can ill afford to
ignore multi-locus associations
31YOUR TURN (TEAMS)!
From the above gametic matrix, please compute D