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Hardy-Weinberg Equilibrium
No Selection 
No Mutation 
No Exchange of Genes (today)
Infinite (very large) Population Size (Monday)
Random Mating (after midterm)
Gene flow
dispersal = movement of individuals between popns
(necessary but not sufficient for gene flow)
gene flow
individuals leave their natal population
reach new suitable habitat
successfully reproduce
infer dispersal from studies of movement
infer gene flow from allele frequency patterns
model this as genetic exchange among demes
deme = subpopulation that is genetically
connected to other subpopulations
effects of gene flow:
1) introduce new alleles into a population
2) eliminate genetic differences among
populations (reduce among-population
genetic variance)
3) reduce the probability of fixation of
neutral alleles by genetic drift
4) may retard adaptation to local conditions
via natural selection
neutral alleles, two populations
m
let
and
m
m = fraction of immigrants
1-m = fraction of natives
qA = f(A2) in popn A
qB = f(A2) in popn B
if qA == qB ??
Single generation recursion:
qA’
= (1-m)qA + mqB
= qA - mqA + mqB
= qA - m(qA – qB)
qA’ = qA-m(qA – qB)
DqA
= qA’ - qA
= qA - m(qA – qB) - qA
= -m(qA – qB)
at equilibrium, DqA = 0
0 = -m(qA – qB)
qA = qB
gene flow homogenizes
allele frequencies
rate of convergence
determined by m
neutral alleles, many populations
qi = f(A2) in popn i,
q = f(A2) in all other popns
qi’ = (1-m)qi + mq
= qi - m(qi –q)
Dqi = qi’ - qi
= qi - m(qi –q) - qi
= -m(qi –q)
v
at equilibrium, qi = q
i
m = fraction immigrants
1-m = fraction natives
measuring gene flow in natural populations
models: gene flow equalizes frequency of neutral
alleles among populations, independent of their
frequency
alleles that are moderately common should be
present in all demes at ~same frequency
only rare alleles should be restricted to one
or a few demes
conditional average frequency -- mean frequency
of an allele (when it is present) as a function
of its distribution
Average frequency of allele
*
m=
m=
m=
m=
m=
0.001
0.005
0.01
0.05
0.1
for all, m = 10-4
d = 10
N = 25
Number of demes where an allele is found
gene flow and selection
deme i
wij
A1A1 A1A2 A2A2
1
1-s
1-2s
i
if selection is weak,
Dq ~
~
-sqi(1-qi)
w
-sqi(1-qi)
if deme i is now connected to a set of populations
where A2 is not deleterious, what happens??
selection will decrease f(A2), but gene flow will
increase f(A2)
qi decreases via selection
qi increases via gene flow
-sqi(1 - qi)
m(qi – q)
Dq = -sqi(1 - qi) + m(qi – q)
at equilibrium Dq = 0,
&
(m+s) + [(m+s)2 – 4smq]
qi =
2s
three biological outcomes:
m>>s
gene flow replaces A2 faster than
selection removes it qi ~ q
m<<s
selection eliminates A2 faster than
gene flow replaces it qi ~ 0
m~s
gene flow maintains A2 at a frequency
higher than under selection alone,
but its frequency in deme i does not
converge on the other demes
qi ~ q(m/s)
v
v
v
&
(m+s) + [(m+s)2 – 4smq]
qi =
2s
Dq = -sqi(1 - qi) + m(qi – q)
interaction of selection and gene flow
-- evolution of metal tolerance in plants
soil near mines contaminated by tailings or seepage
copper, lead, zinc
low in nitrogen, phosphorus, potassium
adaptations -- metal not taken up
metal taken up but sequestered
metal required
degree of adaptation measured by a tolerance index (TI)
root growth in metal
TI =
root growth in control
trade-off: strong advantage on contaminated soil,
but overgrown on clean soil
w =
Agrostis
Anthoxanthum
Plantago
Rumex
Xyield tolerant
Xyield susceptible
wij
contaminated
soil
metal-free
soil
tolerant
1
0.6 – 0.7
0.16 - 0.32
0.001 - 0.03
0.03 - 0.28
0.23 - 0.27
susceptible
0.01 – 0.05
1
1
gene flow via wind pollination
tolerant
favored
wtol
susceptible
favored
susceptible
favored
0
mine
adults
seeds
dispersal is necessary, but not sufficient, for gene flow
gene flow reduces among population genetic variance
gene flow can maintain a deleterious allele (prevent
adaptation to local conditions
the degree of gene flow can be inferred from the
distribution of neutral alleles across a set
of populations