Gametic Phase Disequilibrium Multilocus Population Genetics

1
Gametic 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

2
Gametic 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!

3
Gametic 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?

4
Two Loci, Each with Two Alleles Hypothetical
Gametic Frequency Matrix 1
pi freq(Ai) qi freq(Bi)
5
Two Loci, Each with Two Alleles Hypothetical
Gametic Frequency Matrix 1
6
Two Loci, Each with Two Alleles Hypothetical
Gametic Frequency Matrix 2
7
Two Loci, Each with Two Alleles General Form of
the Gametic Frequency Matrix
Pij freq(AiBj)
8
Two 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
9
Two 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.
10
Two 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)
11
D 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
12
D 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
13
D 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)

14
The 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.

15
The 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

16
The 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
17
The 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)!

18
The 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

19
The Fate of D

• Computer simulations of the decay of D will be
  shown in class for various values of r

20
Other 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.

21
Other Measures of Nonrandom Association among
Alleles

• D D / Dmax if D0
• D D / -Dmin if D
• Dmax minp1q2, p2q1
• Dmin max-p1q1, -p2q2
• -1

22
Other 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)

23
What 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)

24
What 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)

25
What 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)

26
What Is D in Real Populations?

• Implications of D
• 1. Conservation biology
• 2. Breeding programs
• 3. DNA forensics

27
The 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

28
The 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

29
The 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

30
THERE 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

31
YOUR TURN (TEAMS)!
From the above gametic matrix, please compute D