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Spherical Cows Grazing in Flatland:
Constraints to Selection and Adaptation
Mark Blows
University of Queensland
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Bruce Walsh ([email protected])
University of Arizona
Geometry and Biology
Geometry
has a original
long andorthogonal
important variance
history in
biology
Fisher's (1918)
decomposition
D'Arcy Thompson (1917) On Growth and Form
Fisher's (1930) geometric model for the adaptation of
new mutations
Wright (1932)-Simpson (1944) concept of a phenotypic
adaptive topography
Lande-Arnold (1983) estimation of quadratic fitness
surfaces
When considering adaptation, the
appropriate geometry of the multivariate
phenotype (and the resulting vector of
breeding values) needs to be used,
otherwise we are left
with a misleading view of both selection
and adaptation.
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A “spherical cow” -- an overly-simplified representation
of a complex geometric structure
Geometric models for the adapativeness of new mutations
One of the first considerations
the role
of
Fisherof
(1930)
suggested
geometry in evolution is Fisher’s
work
on the
that the
number
of
probability that a new mutation
is adaptive
independent
traits(has
under
higher fitness than the wildtype
from
selection
haswhich
important
it is derived)
consequences for adaptation
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R. A. Fisher
Fisher used a fairly simple
geometric argument to make
this point
The (2-D) geometry behind Fisher’s model
d = distance between z and q
Optimal (highest)
Fitness value
in phenotypic
space
Phenotype
of mutant
r
q
d
Fitness contour
for wildtype
z
wildtype is here
New phenotypes for
a random mutation that
are a (random) distance
r from the wildtype
The probability the new mutation is adaptive is simply
the fraction of the arc of the circle inside of the
fitness contour of the starting phenotype. Function
of r, d, and n
Fisher asked if we have a mutation that randomly moves
a distance r from the current position, what is the chance
that an advantageous mutation (increased fitness) occurs.
If there are n traits under selection, Fisher showed that
this probability is given by
pf av
1
= p
2º
Z
1
exp(° y2 =2)dy = 1° erf (x)
x
where x = r
p
n
2d
Note that p decreases as x increases. Thus, increasing
n results in a lower chance of an adapative mutation
Prob(Adaptive mutation)
0.5
0.4
pf av
µ p ∂
r n
= 1 ° erf(x) = 1 ° erf
2d
0.3
0.2
0.1
0.0
0
0.5
1.0
1.5
2.0
r n1/2 / [2d]
2.5
3.0
Extension’s of Fisher’s model
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M. Kimura
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A. Orr
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S. Rice
Kimura and Orr offered an important extension of
Fisher’s model: Fisher simply consider the probability
that the mutation was favorable
The more relevant issue is the chance that the new
mutation is fixed. Favorable mutations might be rarer,
but have higher probability of fixation.
For example, as r -> 0, Prob(Favor) -> 0.5, but s -> 0,
and probability (fixation) -> neutral value (1/2N)
Orr showed that the optimal mutation size was x ~ 0.925,
or
d
r opt ' 1:85 ¢ p
n
Orr further showed that there is a considerable cost to
complexity (dimensions of selection n) with the rate of
adaptation (favorable mutation rate times fixation
probability) declining significantly faster that 1/n.
Thus, the constraint on dimensionality may be much
more severe than originally suggested by
Fisher.
Fisher’s model makes simplifying geometric assumptions
Phenotype
of mutant
r
q
d
Fitness contour
for wildtype
z
Two spherical cow
assumptions!
Equal (and spherical)
fitness contours
for all traits
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Equal (and spherical)
distribution of
mutational effects
Rice significantly relaxes the assumption of a
spherical fitness surface around a single optimal
value
The probability of adaptation on these surfaces
depends upon their ``effective curvature'', roughly
the harmonic mean of the individual curvatures.
Recalling that the harmonic mean is dominated by
small values, it follows that the probability of
adaptation is likewise dominated by those fitness
surfaces with low curvature (weak selection).
However, on such surfaces, s is small, and hence the
fixation probability small.
Multivariate Phenotypes and Selection Response
Now let’s move from the geometry of adaptive
mutations to the evolution of a vector of traits,
a multivariate phenotype
For univariate traits, the classic breeders’
equation R= h2 S relates the within-generation
change S in mean phenotype to the
between-generation change R (the response
to selection)
The Multivariate Breeders’ Equation
Lande (1979) extended the
univariate breeders’ equation
R = h2 S to the response for a
Vector R of traits
R = GP
° 1
S
Defining
R = Var(A)
the selection
Var-1(P) gradient
S
° 1
b by
Ø= P
S
yields the Lande Equation
Russ Lande
R = GØ
The selection gradient b
Robertson & Price showed that S = Cov(w,z), so
that the selection differential S is the covariance
between (relative) fitness and phenotypic value
Since S is the vector of covariances and P the
covariance matrix for z, it follows that Ø= P° 1S
is the vector of regression coefficients for
predicting fitness w given phenotypes zi, e.g.,
w = a+
Xn
i =1
Øi zi + ei
G, b, and selective constraints
A non-zero bi means that selection is acting
directly to change the mean of trait i.
The selection gradient b measures the direction
that selection is trying to move to population
mean to maximally improve fitness
Multiplying b by G results in a rotation (moving away
from the optimal direction) as well as a scaling (reducing
the response). Thus, G imposes constraints in the selection
response,
Thus G and b both describe something about the
geometry of selection
The vector b is the optimal direction to move to
maximally increase fitness
The covariance matrix G of breeding values
describes the space of potential constraints
on achieving this optimal response
Treating this multivariate problem as a series of
univariate responses is incredibly misleading
The problems working with a lowerdimensional projection from a
higher-dimensional space
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Edwin Abbott Abbott, writing as
A Square, 1884
The misleading univariate world of selection
For a single trait, we can express the breeders’
equation as R = Var(A)* b.
Consider two traits, z1 and z2, both heritable
and both under direct selection
Suppose b1 = 2, b2 =-1, Var(A1) = 10, Var(A2) = 40
One would thus expect that each trait would
respond to selection, with Ri = Var(Ai)* bi
What is the actual response?
Not enough information to tell --- need
Var(A1, A2).
µ
R = GØ=
10 0
0 40
∂ µ
2
°1
∂
µ
=
20
° 40
However, with a different covariance,
µ
R = GØ=
10 20
20 40
∂ µ
2
°1
∂
µ ∂
0
=
0
∂
The notion of multivariate constraints is not new
Dickerson (1955) -- genetic variation in all of the
components of a selection index, but no (additive)
variation in the index itself.
Lande also noted the possibility of constraints
There can be both phenotypic and genetic constraints
Singularity of P: Selection cannot independently act on
all components
Singularity of G: Certain combinations of traits show no
additive variance
If the covariance matrix is not singular, how can
we best quantify its constraints (if any)
One simple measure is the angle q between
the vectors of desired (b) and actual (R) responses
Recall that the angle between two vectors x and y
is simply given by
xT y
cos(q) =
jj x jj jj y jj
If the inner product of b and R is zero, q = 90o, and
there is an absolute constraint. If q = 0o, the
response and gradient point in exactly the same direction
(b is an eigenvector of G)
√
Trait 1
q
µ
Trait 2
Ø=
R
q = cos
° 1
b
2
°1
∂
T
R Ø
jj R jj jj Øjj
The plot is for the first
of our examples, where
G = µ 10
0
0
40
∂
Note here that q = 37o, even thought there is no
covariance between traits and hence this
reduces to two univariate responses.
The constraint arises because much more genetic
variation in trait 2 (the weaker-selected trait)
!
Constraints and Consequences
Thus, it is theoretically possible to have a very constrained
selection response, in the extreme none (G is a zero
eigenvalue and b is an associated eigenvector)
This is really an empirical question. At first blush, it
would seem incredibly unlikely that b “just happens” to
be near a zero eigenvector of G
However, selection tends to erode
away additive variation for a trait
under constant selection
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Drosophila serrata
Empirical study from Mark’s lab:
Cuticular hydrocarbons and mate choice in
Drosophila serrata
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Emma Hine
Stephen Chenoweth
Cuticular hydrocarbons
46
5,9-C25
Signal strength (pA)
2-Me-C28
42
5,9-C27
38
• D. serrata
34
9-C25
5,9-C24
2-Me-C26
5,9-C29
9-C26
2-Me-C30
30
8.0
8.5
9.0
9.5
Time (minutes)
10.0
10.5
For D. serrata, 8 cuticular hydrocarbons (CHC) were
found to be very predictive of mate choice.
Laboratory experiments measured both b for this
vector of 8 traits as well as the associated G
matrix.
While all CHC traits had significant heritabilities,
the covariance matrix was found to be ill-conditioned,
with the first two eigenvalues (g1, g2) accounting
for roughly 78% of the total genetic variation.
Computing the angles between each of these
two eigenvalues and b provides a measure of the
constraints in this system.
0 0:2321
B 0:132C
B 0:255C
B
C
B 0:536C
C
g1 = B
B 0:449C
B
C
B 0:363C
B
C
@0:430A
0:239
q(g1, b) = 81.5o
0
0:319 1
B 0:182 C
B 0:213 C
B
C
B ° 0:436C
C
g2 = B
B 0:642 C
B
C
B ° 0:362C
B
C
@° 0:014A
° 0:293
0 ° 0:0991
B ° 0:055C
B 0:133 C
B
C
B ° 0:186C
C
Ø= B
B ° 0:133C
B
C
B 0:779 C
B
C
@ 0:306 A
° 0:465
q(g2, b) = 99.7o
Thus much (at least 78%) of the usable genetic
variation is essentially orthogonal to the direction
b that selection is trying to move the population.
Evolution along “Genetic lines of least resistance”
Schluterconstant,
(1996) suggested
Assuming G remains (relatively)
can
that we can,
as feature
he observed
we relate population divergence
to any
of G?
that populations tend to
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given by the first principal
component of G (its leading
eigenvector)
Schluter called this evolution along “genetic lines of
least resistance”, noting that populations tend to diverge
in the direction of gmax, specifically the angle between the
vector of between-population divergence in means and
gmax was small.
Evolution along gmax
There are two ways to interpret Schluter’s observation.
(i) such lines constrain selection, with departures away from
such directions being difficult
(ii) such lines are also the directions on which maximal
genetic drift is expected to occur
Under a simple Brownian motion model of drift in the
vector of means is distributed as,
µ
∂
t
π(t) ª M V N π;
¢G
2N e
Maximal directions of change correspond to the leading
eigenvectors of G.
Looking at lines of least resistance in the Australian rainbow
fish (genus Melanotaenia )
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Megan Higgie
Katrina McGuigan
Two sibling species were measured, both of which have
populations differentially adapted to lake vs. stream
hydrodynamic environments
The vector of traits were morphological landmarks
associated with overall shape (and hence potential
performance in specific hydrodynamic environments)
Here, there was no b to estimate, rather the divergence
vector d between the mean vector for groups
(e.g., the two species, the two environments within a
species, etc.)
To test Schluter’s ideas, the angle between gmax and
different d’s we computed.
Divergence between species, as well as divergence among replicate
hydrodynamic populations within each species, followed Schluter's
results (small angular departures from the vector d of divergent means
and gmax).
However, hydrodynamic divergence between lake versus
stream populations within each species were along directions that were
quite removed from gmax (as well as the other eigenvectors of G
that described most of the genetic variation).
Thus, the between- and within-species divergence within the same
hydrodynamic environment are consistent with drift, while hydrodynamic
divergence within each species had to occur against a gradient of very
little genetic variation.
One cannot rule out that the adaptation to these environments
resulted in a depletion of genetic variation along these directions.
Indeed, this may indeed be the case.
Beyond gmax : Using Matrix Subspace
Projection to Measure Constraints
Schluter’s idea is to examine the angle between
the leading eigenvector of G and the vector
of divergence
More generally, one can construct a space
containing the first k eigenvalues, and examine
the angle between the projection of b onto
this space and b
This provides a measure on the constraints
imposed by a subset of the useable variation
An advantage of using a subspace projection is
that G is often ill-conditioned, in that
lmax / lmin is large.
In such cases (as well as others!) estimation of G
may result in estimates of eigenvalues that are
very close to zero or even negative.
Negative estimates arise due to sampling (Hill and
Thompson 1978), but values near zero may reflect
the true biology in that there is very little variation
in certain dimensions.
One can extract (estimate) a subspace of G that
accounts for the vast majority of useable genetic
variation by, for example, taking the leading k
eigenvectors.
It is often the case that G contains several
eigenvalues whose associated eigenvectors account
for almost no variation (i.e, lmax / tr(G) ~ 0) .
In such cases, most of the genetic variation
resides on a lower-dimensional subspace.
To do this, first construct the matrix A
of the first k eigenvalues
A = ( g1 ; g2 ; ¢¢¢; gk )
The projection matrix for this subspace is
given by
°
T
P r oj = A A A
¢°
1
AT
Thus, the projection of b into this subspace
is given by the vector
°
T
p = P r oj Ø = A A A
¢°
1
ATØ
Note that this is the generalization of the
projection of one vector onto another
The constraints imposed within this subspace
is given by the angle between p, the projection
of b into this space, and b.
For the Drosophia serrata CHC traits involved in
mate choice., the first two eigenvalues account
for roughly 80\% of the total variation in G.
The angle
q between b and the projection p of
b into the subspace of the genetic variance is 77.1o
Thus the direction of optimal response is 77o
away from the genetic variation described by this
subspace (which spans 78% of the total variance).
How typical is this amount of constraint?
The estimated G for these traits had 98% of the
total genetic variation
in at
the
five PCs in mate
Looked
9 first
CHC involved
(the first four had
95%inofDrosophila
the total variance).
choice
bunnanda
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The
angle
between b and its projection into this
Anna Van
Homrigh
5-dimensional subspace was 88.2o.
If the first four PCs were considered for the
subspace, the projection is even more constrained,
being 89.1o away for b.
When the entire space of G is considered,
the resulting angle between R and b is 67o
Evolution Under Constraints or
Evolution of Constraints?
G both constrains selection and also evolves under
selection. Over short time scales, if most alleles
have modest effects, G changes due to selection
generating linkage disequilibrium.
The within-generation change in G under
the infinitesimal model is
T
¢ G = ° G ØØ G = ° R R
T
Thus, the (within-generation) change in G
between traits i and j is
¢ Gij = ¢ æ(A i ; A j ) = ° Ri R j
The net result is that linkage disequilibrium
increases any initial constraints. A simple way to
see this is to consider selection on the index
I = S zi bi
Selection on this index (which is the predicted
fitness) results in decreased additive variance
in this composite trait (Bulmer 1971).
Thus, as pointed out by Shaw et al. (1995),
if one estimates G by first having several
generations of random mating in the laboratory
under little selection, existing linkage
disequilibrium decays, and the resulting
estimated G matrix may show less of a
constraint than the actual G operating in nature
(with its inherent linkage disequilibrium).
Why so much variation?
It is certainly not surprising that little usable
genetic variation may remain along a direction
of persistence directional selection.
What is surprising, however, is that considerable
genetic variation may exist along other directions.
The quandary is not why is there so little usable
variation but rather why is their so much?
Quantitative genetics is in the embarrassing
position as a field of having no models that
adequately explain one of its central observations
-- genetic variation (measured by single-trait
heritabilities) is common and typically in the
range of 0.2 to 0.4 for a wide variety of traits.
As Johnson and Barton (2005) point out, the
resolution of these issues likely resides in
more detailed considerations of pleiotropy,
wherein new mutations influence a number
of traits (back to Fisher’s model!)
Once again, it is likely we need to move to a
higher dimensional space to reasonably account
for observations based on a projection into
one dimension (i.e., standing heritability levels
for a trait).
The final consideration with pleiotropy
is not just the higher-dimensional fitness
surface for the vector of traits they influence
but also the distributional space of pleiotropic
mutations themselves.
The “deep” nature of G
Is the covariance structure G itself some
optimal configuration for certain sets of
highly-correlated traits?
Has there been selection on developmental
processes to facilitate morphological integration
(the various units of a complex trait functioning
smoothly together), which in turn would result
in constraints on the pattern of accessible
mutations under pleiotropy (Olson and Miller
1958, Lande 1980)?
Developmental systems are networks
First, they
areapparently
small-world graphs,
means of
that
Some
generalwhich
features
the mean path distance
between
any two nodes is short.
Biological
networks
The members live in a small world (Bacon, Erdos numbers)
The second feature that studied regulatory/
metabolic networks showed is that the degree
distribution (probability distribution that a node
is connected to k other others) follows a power
law
P(k) ~ k-g
Graphs with a power distribution of links are called
scale-free graphs.
Scale-free graphs show they very important feature
that they are fairly robust to perturbations. Most
randomly-chosen nodes can be removed with little
effect on the system.
Our spherical cow may in reality have a very
non-spherical distribution of new mutation
phenotypes around a current phenotype.
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Geometry of the fitness surface
and geometry of the mutational space
Raw material
Filter
Fuel
q
z
Effects of selection removing variation
(geometry of the fitness surface)
Residual variation: constraints and usable
evolutionary fuel (geometry of the subspace
of usable variation relative to direction of
selection
Stuart Barker
“For someone learning the trade of quantitative genetics
in the late 1980's, Stuart's work was like a beacon of
interest in a sea of allozymes; incisive reviews, classic
experimental designs (even with allozymes!), and
above all the innovative application of quantitative
genetics to important and interesting questions in
evolutionary biology.”
-- Mark Blows
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q
q
z
z