Download Evolutionary quantitative genetics and one

“Coarse” Notes
Population Genetics
Evolutionary quantitative genetics and one-locus population genetics
READING: Hedrick pp. 57–63, 587–596
• Most evolutionary problems involve questions about phenotypic means
• Goal: determine how selection causes evolutionary change in the mean of a quantitative
character.
• Aside:
Quantitative vs. Qualitative Traits
• Back to our main story.... selection on a trait such as body size
– body size of individual is denoted z
– mean body size in a population: z
– Suppose selection alters allele frequencies at loci that affect z
– Consider the effect on body size of one specific locus, i
– In general, to determine how much z changes, one needs to determine:
1) how selection changes p (i ) at all loci affecting z
2) how changes in p (i ) combine to change z .
body
size
z
Δz
( i)
( i)
Δp
(i )
0
p
(i )
– We'll focus on the effects of one of the loci that affects z
• The phenotype of an individual depends on:
1) its genotype at locus i
2) its genotype at other loci that affect the trait
3) its environmental experience
VI-1
1
“Coarse” Notes
Population Genetics
• lump 2 & 3 into a single factor, c
• Can think of an individual's phenotype as z = a jk + c
a22
z =c+
a12
a 11
A1 A1
A1 A2
A2 A2
• The average phenotype can similarly be broken down into the average contribution of
locus i and the average over other loci and environmental effects: z = a + c
• Consider the effect of a change in p (i ) on z (holding all other loci constant)
!z ( i)
(drop i)
(
) ( )
=
z p( i) + !p( i) " z p( i)
≈
dz
(i )
(i ) !p
dp
≈
%
"
d ln w '
dz $ 1
(term in brackets is just our old friend)
2 p(1 ! p)
24
3 dp '
dp $ 14
variance
123
#
selection &
≈
dz " 1
dz d ln w %
(using the "chain rule")
2 p(1 ! p)
dp $#
dp dz '&
≈
1
2
2
or !z
( i)
! dz $ d ln w
pq#
" dp % dz
(form emphasizes the dependency of fitness w on
phenotype z)
– What is
dz
?
dp
VI-2
“Coarse” Notes
Population Genetics
dz
d
(a + c )
=
dp dp
d 2
=
p a1 1 + 2pqa1 2 + q 2 a2 2 + c ) (assuming H- W)
(
dp
= 2pa1 1 + 2qa1 2 ! 2 pa1 2 ! 2qa2 2 + 0
– So, putting all the pieces together:
!z ( i) " 12 # pq # 4( pa1 1 + qa1 2 $ pa1 2 $ qa2 2)
2
d ln w
dz
or
!z ( i) " G( i) #
(i )
where G = 2pq[( pa1 1 + qa1 2 ) ! ( pa1 2 + qa2 2 )]
2
and ! =
d ln w
.
dz
– What is ! ?
• Called the selection gradient
• It is a measure of the strength of selection acting on a trait:
(i )
– What is G ?
• It is an extremely important quantity called the additive genetic variance
contributed by locus i
• Why "additive"?
– Dominance Variance
• Consider the total amount of variation in z due to variation at locus i:
( )
var z( i) = var ( ajk + c ) = var ( ajk ) since we have fixed c.
( )
– So, var z( i) =
(assuming H-W equilibrium)
=
var ( ajk )
(p a
2 2
11
[
]
E( a jk ) ! E( a jk )
2
=
2
+ 2 pqa1 2 + q 2 a2 2) ! ( p2 a1 1 + 2pqa1 2 + q 2 a2 2)
2
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2
2
“Coarse” Notes
Population Genetics
(after a lot of tedious algebra)
=
=
G
(i )
a + a $)
&
"
+ '2 pq a1 2 ! 1 1 2 2 *
#
2 %+
(
2
G( i ) + D(i )
2
a + a $)
&
"
where D = '2 pq a1 2 ! 1 1 2 2 * .
#
2 %+
(
(i )
(i )
• What is D ?
– It's called the dominance variance contributed by locus i.
(i )
• Note: D is never negative and has units of z2
– Why "dominance"?
(i )
(i )
– Relationship between G and D
• Case 1: No Dominance
• Case 2: Symmetric Overdominance
– What about other loci that contribute to the traits?
• Generally difficult (recall complications of 2-locus population genetics) to
understand;
• Assuming linkage equilibrium for all loci contributing to a trait:
!z = !z (1) + !z ( 2 ) + !z( 3 ) +L
[
]
= G(1 ) + G (2 ) + G (3 ) +L "
" n (i ) %
= $! G '( or
# i =1
&
!z = G "
VI-4
(recall that ! reflects how a change in
mean
phenotype
changes
mean
population fitness: it doesn't care what
the cause of the change in phenotype is.)
“Coarse” Notes
Population Genetics
– This is the central equation of quantitative genetics
– Note its similarity to the equation for selection at one locus.
– Note that it's written in several other ways
• Important Message: It's not the total phenotypic variance nor even the total
genotypic variance that determines how fast a population mean evolve in
response to selection
n
– Total genotypic variance = G + D (where D = D( 1) + D( 2 ) +L= ! D( i) )
i =1
• In general, there's between-locus genetic "interaction" variance as well.
– Total phenotypic variance = G + D + E (also written V P = V A + V D + V E )
• E is the variance that's due to environmental effects.
• A few quibbles with “heritability”
Selection on a single character ( !z = G " ): Other ways of determining G
• We now have an algebraic definition of G in terms of a jk ' s and p's at all loci
• There are, however, other ways to get G
– We'll skip the tedious algebra (see Falconer), but the methods are important since they
show how G can actually be measured!
(most of Falconer's book concerns estimating G in agricultural or laboratory settings.
See, e.g., Simms & Rausher for methods of measuring G in natural populations)
(1) Breeding Values
– Measure the type of offspring an individual tends to produce in a particular reference
population
• breeding values were originally used by plant and animal breeders for assessing the
value of an individuals for mating.
– Definition: g = z + 2(zo ! z )
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“Coarse” Notes
Population Genetics
• note that an individual's own phenotype doesn't enter in this - except through z
• zo = average value of the trait among the individual's offspring when she/he is
mated to a large # of randomly chosen individuals
– alternatively, zo may be thought of as the expected phenotype the individual's
offspring were it to mate with a randomly chosen individual in the population.
• Why the 2?
– Only 1/2 of an offspring phenotype is attributable to that individual. The other
1/2 comes from its mate.
• Important note: the breeding value of an individual refers to a particular population
as well as the individual.
– That is, the same individual could have a different breeding value were it moved
to a different population.
• Uses of breeding value
– Suppose a farmer (or nature) chooses a set of individuals to reproduce
– The mean value of the offspring will equal the average breeding value of the
selected individuals
• Breeding values can be estimated
– In principle, one could
1) mate an individual to many others
2) raise and measure the offspring
3) estimate the breeding value using the average offspring phenotype
• This is actually done for important breeding stock, dairy cattle, race
horses, etc.
– What about variation in breeding values?
• It's obviously related to a population's genetic variation
• can actually compute it under our earlier assumptions
• It turns out that
var ( g) = G
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“Coarse” Notes
Population Genetics
– Significance
• Statistically, can think of the phenotype of an individual as z = g + d + e
– g and d are inherited from its parents, but the individual will only pass on g
– By the way g and d are defined, d is acquired by an offspring independently of g
• These results suggest a way to measure G without ever knowing the p's and aij's!
– could experimentally determine breeding values of individuals in a population,
then compute their variance
– this is, essentially, what a "half-sib" breeding design does
– Drawback: this method is rather tedious, expensive, and labor intensive
• Turns out there's an easier way...
(2) Parent-offspring regression
– The response to selection, "z , depends on the relationship between parents and
offspring diagrammed here.
– "z depends on
!
!
• how strongly selection acts on the mean (s)
• the orientation of the cloud of points
– note that the orientation depends on inheritance
• e.g., if there's no genetic variance, the regression line would be flat.
– This orientation is related to genetics. It can be show (see H&C or Falconer)
that, under our previous assumptions:
G = 2cov ( zo ,zp )
– This is measurable (and, once again, doesn't require estimating the p's and aij's).
– It's one of the major methods for estimating G
Selection on a single character ( !z = G " ): The selection gradient revisited
• We'll now consider two definitions of the selection gradient
VI-7
“Coarse” Notes
Population Genetics
(1) ! = change in the logarithm of a population's mean fitness that would result if the
population's mean phenotype was increased by one "unit"
• Mathematically, ln[ w (z + ! )] " ln[ w (z )] #
i.e., ! =
d ln w ln w ( z + # ) $ ln w (z )
"
#
dz
d ln w
! = $! ,
dz
• ! measures steepness of mean population fitness function
– this is the relevant measure of the strength of selection acting on the mean.
• The units of ! are 1/char.
• The relationship between w (z) , w , and ! :
– w (z) : individual fitness function
• describes the fitness of an individual as a function of it's phenotype
– w : population mean fitness
• it's the average fitness of individuals in the population
• i.e., it's the average of w (z) with respect to the distribution of phenotypes z
in the population
– The relationship between w (z) and w :
• Suppose that ecologists/evolutionary biologists determine the following
fitness function
a) If P = 0 (i.e., there is no phenotypic variation => all individuals are
identical) then, w = w( z) = w(z ) since z = z .
– Note: w = w ( z ) :
b) If there's lots of phenotypic variation (P >> 0):
• In general, w is, roughly, a smooth version of w (z) .
• "weak selection":
– ! is approximately independent of P
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“Coarse” Notes
Population Genetics
• or, w (z) changes little over the distribution of z
– refers to stabilizing or disruptive component of selection
– note that the directional component of selection may still be strong
(2) ! = regression of relative fitness onto the value of the character
• Mathematically, ! =
cov ( w, z)
var (z )
– this indicates how ! can be measured!
Effects of selection within a generation: the "selection differential"
• The selection differential measures the within-generation impact of selection on a
population's mean phenotype.
*
• Mathematically, the selection differential is defined as s = z ! z
• Another mathematically equivalent definition (due to Robertson & Price): s = cov (w, z)
• The relationship between the selection gradient and selection differential:
s = P!
!=s P
– Note: for a single trait, there seems little difference between s and ! . However, when
the simultaneous evolution of several quantitative traits is being considered, the
selection gradient have many advantages over the selection differential (both
mathematically and in biological interpretation).
• Traditional form for the response to selection:
!z = G " = G
s G
G
2
2
is the "narrow-sense" heritability of z.
= s = h s where h =
P P
P
– Comments on heritability:
• There are two types: "broad-sense" and "narrow-sense"
– Broad-sense heritability is the fraction of phenotypic variability that is due to all
forms genetic variation = (G + D + "epistatic variance")/P.
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“Coarse” Notes
Population Genetics
2
– Narrow-sense heritability ( h ) is the fraction of phenotypic variability due to
additive-genetic variance.
2
• h and s are relatively uninformative regarding rates of evolution (compared to G
and ! ):
– without changing G and ! (and hence the rate of evolution !z = G " ), one can
2
↑ s simply by ↑ P (=> h ↓).
• Breeders like s because it is easier to control.
Evolutionary Equilibrium under selection: !z = G "
• !z = 0 for two possible reasons:
(1) ! = 0
(2) G = 0.
– Possibility (1) corresponds to a balance of ecological forces
• ˆz (the equilibrium mean of the population) is either a maximum of minimum of w
– the former is stable, the latter is unstable
• If selection is sufficiently weak, z will evolve in such a way that mean fitness will
↑.
– Note: fitness refers to overall fitness, not to antagonistic components of
selection that may occur within a generation
e.g., Darwin's finches: Price found that small size is favored in juveniles
(possibly due to energy conservation) but large size is favored in adults
(easier to crack seeds).
– Possibility (2) corresponds to a lack of evolutionary "material"
• This situation correponds to what are commonly called "developmental" or
"genetic" constraints.
• additive variance can reappear in several ways
– mutation & migration
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“Coarse” Notes
Population Genetics
– recombination between loci
• linkage disequilibrium affects the amount of variance expressed.
e.g., a population with only (+ -) (- +) gametes has no phenotypic variance
despite having copious amounts of genetic variance.
– recombination can create additive genetic variance
– recombination can destroy variation as well (e.g., (+ +) (- -) population.)
VI-11