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Spherical Cows Grazing in Flatland: Constraints to Selection and Adaptation

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Spherical Cows Grazing in Flatland:

Constraints to Selection and Adaptation

Mark Blows

University of Queensland

Bruce Walsh (jbwalsh@u.arizona.edu)

University of Arizona

Geometry and Biology

Geometry has a long and important history in biology

Fisher's (1918) original orthogonal variance 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

A “spherical cow” -- an overly-simplified representation

of a complex geometric structure

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.

Geometric models for the adapativeness of new mutations

One of the first considerations of the role of

geometry in evolution is Fisher’s work on the

probability that a new mutation is adaptive (has

higher fitness than the wildtype from which

it is derived)

Fisher (1930) suggested

that the number of

independent traits under

selection has important

consequences for adaptation

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

Phenotype

of mutant

Optimal (highest)

Fitness value

in phenotypic

space

q

z

r

wildtype is here

d

New phenotypes for

a random mutation that

are a (random) distance

r from the wildtype

Fitness contour

for 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

r

n

x

=

2

d

Z

1

1

2

p

p

=

exp

(

°

y

=

2)

dy

=

1

°

er

f

(

x

)

f

av

2

º

where

x

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

Note that p decreases as x increases. Thus, increasing

n results in a lower chance of an adapative mutation

p

r

'

1

:

85

¢

opt

n

Orr showed that the optimal mutation size was x ~ 0.925,

or

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 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.

assumptions!

Phenotype

of mutant

q

z

r

Equal (and spherical)

fitness contours

for all traits

d

Equal (and spherical)

distribution of

mutational effects

Fitness contour

for wildtype

Fisher’s model makes simplifying geometric assumptions

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)

=

G

Ø

°

1

R

=

G

P

S

°

1

Ø

=

P

S

R = Var(A) Var-1(P) S

Defining the selection gradient

b by

Russ Lande

The Multivariate Breeders’ Equation

Lande (1979) extended the

univariate breeders’ equation

R = h2 S to the response for a

Vector R of traits

yields the Lande Equation

Since S is the vector of covariances and P the

covariance matrix for z, it follows that

°

1

Ø

=

P

S

is the vector of regression coefficients for

predicting fitness w given phenotypes zi, e.g.,

n

X

w

=

a

+

Ø

z

+

e

i

i

i

i

=1

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

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

Edwin Abbott Abbott, writing as

A Square, 1884

The problems working with a lower- dimensional projection from a higher-dimensional spaceThe 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

∂

µ

∂

µ

∂

10

0

2

20

R

=

G

Ø

=

=

0

40

°

1

°

40

µ

∂

µ

∂

µ

∂

10

20

2

0

R

=

G

Ø

=

=

20

40

°

1

0

What is the actual response?

Not enough information to tell --- need

Var(A1, A2).

However, with a different covariance,

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

x

y

cos

(

q

)

=

jj

x

jj

jj

y

jj

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

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)

of our examples, where

G =

√

!

T

R

Ø

°

1

q

=

cos

jj

R

jj

jj

Ø

jj

µ

∂

10

0

0

40

µ

∂

2

Ø

=

°

1

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

Stephen Chenoweth

Empirical study from Mark’s lab:

Cuticular hydrocarbons and mate choice in

Drosophila serrata

- D. serrata

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.

1

0

1

0

1

0

:

232

0

:

319

°

0

:

099

0

:

132

0

:

182

°

0

:

055

B

C

B

C

B

C

B

C

B

C

B

C

0

:

255

0

:

213

0

:

133

B

C

B

C

B

C

B

C

B

C

B

C

0

:

536

°

0

:

436

°

0

:

186

B

C

B

C

B

C

g

=

g

=

Ø

=

B

C

B

C

B

C

1

2

0

:

449

0

:

642

°

0

:

133

B

C

B

C

B

C

B

C

B

C

B

C

0

:

363

°

0

:

362

0

:

779

B

C

B

C

B

C

@

A

@

A

@

A

0

:

430

°

0

:

014

0

:

306

0

:

239

°

0

:

293

°

0

:

465

q(g1, b) = 81.5o

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.

Schluter (1996) suggested that we can, as he observed that populations tend to diverge along the direction given by the first principal component of G (its leading

eigenvector)

Evolution along “Genetic lines of least resistance”

Assuming G remains (relatively) constant, can

we relate population divergence to any feature of G?

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.

∂

t

π

(

t

)

ª

M

V

N

π

;

¢

G

2

N

e

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,

Maximal directions of change correspond to the leading

eigenvectors of G.

Looking at lines of least resistance in the Australian rainbow

fish (genus Melanotaenia )

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.

=

(

g

;

g

;

¢

¢

¢

;

g

)

1

2

k

°

¢

°

1

T

T

P

=

A

A

A

A

r

oj

°

¢

°

1

T

T

p

=

P

Ø

=

A

A

A

A

Ø

r

oj

To do this, first construct the matrix A

of the first k eigenvalues

The projection matrix for this subspace is

given by

Thus, the projection of b into this subspace

is given by the vector

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).

Looked at 9 CHC involved in mate

choice in Drosophila bunnanda

Anna Van Homrigh

How typical is this amount of constraint?

The estimated G for these traits had 98% of the

total genetic variation in the first five PCs

(the first four had 95% of the total variance).

The angle between b and its projection into this

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

T

¢

G

=

°

G

Ø

Ø

G

=

°

R

R

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

G

=

¢

æ

(

A

;

A

)

=

°

R

R

ij

i

j

i

j

Thus, the (within-generation) change in G

between traits i and j is

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 zibi

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).

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.

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)?

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

First, they are small-world graphs, which means that

the mean path distance between any two nodes is short.

The members live in a small world (Bacon, Erdos numbers)

Some apparently general features of

Biological networks

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.

q

Geometry of the fitness surface

and geometry of the mutational space

Raw material

Effects of selection removing variation

(geometry of the fitness surface)

Filter

Residual variation: constraints and usable

evolutionary fuel (geometry of the subspace

of usable variation relative to direction of

selection

Fuel

“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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