spherical cows grazing in flatland: constraints to selection and adaptation bruce walsh...

Spherical Cows Grazing in Flatland: Constraints to Selection and Adaptation

Bruce Walsh ([email protected])University of Arizona

Mark Blows University of Queensland

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Geometry and 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

Geometry has a long and important history in biology



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

leftwith a misleading view of both

selection and adaptation.

Geometric models for the adapativeness of new mutations



R. A. Fisher

Fisher (1930) suggestedthat the number of independent traits underselection has importantconsequences for adaptation

Fisher used a fairly simplegeometric argument to makethis point

One of the first considerations of the role ofgeometry in evolution is Fisher’s work on theprobability that a new mutation is adaptive (hashigher fitness than the wildtype from whichit is derived)

z

Fitness contourfor wildtype

Phenotypeof mutant

r

d

The (2-D) geometry behind Fisher’s model

wildtype is here

Optimal (highest) Fitness value in phenotypic space

New phenotypes fora random mutation thatare a (random) distance

r from the wildtype

The probability the new mutation is adaptive is simplythe fraction of the arc of the circle inside of the

fitness contour of the starting phenotype. Functionof r, d, and n

d = distance between z and

Fisher asked if we have a mutation that randomly moves a distance r from the current position, what is the chancethat an advantageous mutation (increased fitness) occurs.

pf av =1

p2º

Z 1

xexp(°y2=2)dy = 1°erf (x)

If there are n traits under selection, Fisher showed that this probability is given by

x =r

pn

2dwhere

Note that p decreases as x increases. Thus, increasingn results in a lower chance of an adapative mutation

0.5

0.4

0.3

0.2

0.1

0.0

Pro

b(A

dap

tive m

uta

tion)

0 0.5 1.0 1.5 2.0 2.5 3.0

pf av = 1 ° erf(x) = 1 ° erfµ

rp

n2d

∂

r n1/2 / [2d]





M. Kimura A. Orr

Extension’s of Fisher’s model



S. Rice

Kimura and Orr offered an important extension ofFisher’s model: Fisher simply consider the probabilitythat the mutation was favorable

The more relevant issue is the chance that the newmutation 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

ropt ' 1:85¢dp

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 fixationprobability) declining significantly faster that 1/n.

Thus, the constraint on dimensionality may be much more severe than originally suggested by

Fisher.

z

Fitness contourfor wildtype

Phenotypeof mutant

r

d



Two spherical cowassumptions!

Equal (and spherical) fitness contours

for all traits

Equal (and spherical) distribution of

mutational effects

Fisher’s model makes simplifying geometric assumptions

Rice significantly relaxes the assumption of aspherical fitness surface around a single optimalvalue

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 adaptivemutations 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)

Russ Lande

The Multivariate Breeders’ Equation

R = GP °1S

Lande (1979) extended theunivariate breeders’ equationR = h2 S to the response for aVector R of traits

R = Var(A) Var-1(P) SDefining the selection gradient by Ø = P °1S

yields the Lande Equation

R = GØ

w = a+nX

i=1

Øizi +ei

The selection gradient

Robertson & Price showed that S = Cov(w,z), sothat the selection differential S is the covariancebetween (relative) fitness and phenotypic value

Ø= P °1SSince S is the vector of covariances and P thecovariance matrix for z, it follows that

is the vector of regression coefficients for predicting fitness w given phenotypes zi, e.g.,

G, , and selective constraints

The selection gradient measures the direction that selection is trying to move to population mean to maximally improve fitness

A non-zero i means that selection is acting directly to change the mean of trait i.

Multiplying by G results in a rotation (moving awayfrom the optimal direction) as well as a scaling (reducingthe response). Thus, G imposes constraints in the selectionresponse,

Thus G and both describe something about thegeometry of selection

The vector is the optimal direction to move tomaximally increase fitness

The covariance matrix G of breeding valuesdescribes the space of potential constraintson achieving this optimal response

Treating this multivariate problem as a series ofunivariate responses is incredibly misleading





Edwin Abbott Abbott, writing asA Square, 1884

The problems working with a lower- dimensional projection

from a higher-dimensional space

The misleading univariate world of selection

For a single trait, we can express the breeders’equation as R = Var(A)* .

Consider two traits, z1 and z2, both heritableand both under direct selection

Suppose 1 = 2, 2 =-1, Var(A1) = 10, Var(A2) = 40

One would thus expect that each trait wouldrespond to selection, with Ri = Var(Ai)* i

R = GØ =µ

10 00 40

∂ µ2

°1

∂=

µ20

°40

∂

R = GØ =µ

10 2020 40

∂ µ2

°1

∂=

µ00

∂

What is the actual response?

Not enough information to tell --- needVar(A1, A2).

However, with a different covariance,

Dickerson (1955) -- genetic variation in all of the components of a selection index, but no (additive) variation in the index itself.

Singularity of G: Certain combinations of traits show no additive variance

The notion of multivariate constraints is not new

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

cos() =xT y

jjxjj jjyjj

One simple measure is the angle betweenthe vectors of desired () and actual (R) responses

If the covariance matrix is not singular, how canwe best quantify its constraints (if any)

Recall that the angle between two vectors x and yis simply given by

If the inner product of and R is zero, = 90o, andthere is an absolute constraint. If = 0o, theresponse and gradient point in exactly the same direction( is an eigenvector of G)

R

Trait1

Trait2

=cos°1

√RTØ

jjRjj jjØjj

!

The plot is for the firstof our examples, where

G = µ10 00 40

∂

Note here that = 37o, even thought there is nocovariance between traits and hence thisreduces to two univariate responses.

Ø =µ

2°1

∂

The constraint arises because much more geneticvariation in trait 2 (the weaker-selected trait)

Constraints and Consequences

Thus, it is theoretically possible to have a very constrainedselection response, in the extreme none (G is a zeroeigenvalue and is an associated eigenvector)

This is really an empirical question. At first blush, itwould seem incredibly unlikely that “just happens” to be near a zero eigenvector of G

However, selection tends to erodeaway additive variation for a traitunder constant selection



Drosophila serrata



Stephen ChenowethEmma Hine

Empirical study from Mark’s lab: Cuticular hydrocarbons and mate choice inDrosophila serrata

Cuticular hydrocarbons

Time (minutes)8.0 8.5 9.0 9.5 10.0 10.5

30

34

38

42

46

Signal strength (pA)

5,9-C24

5,9-C25

9-C259-C26

2-Me -C26

5,9 -C27

5,9 -C29

2-Me -C28

2-Me -C30

Time (minutes)8.0 8.5 9.0 9.5 10.0 10.5

30

34

38

42

46

Signal strength (pA)

5,9-C24

5,9-C25

9-C259-C26

2-Me -C26

5,9 -C27

5,9 -C29

2-Me -C28

2-Me -C30

• D. serrata

For D. serrata, 8 cuticular hydrocarbons (CHC) werefound to be very predictive of mate choice.

Laboratory experiments measured both for thisvector of 8 traits as well as the associated Gmatrix.

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 thesetwo eigenvalues and provides a measure of theconstraints in this system.

g1 =

0

BBBBBBBBB@

0:2320:1320:2550:5360:4490:3630:4300:239

1

CCCCCCCCCA

g2 =

0

BBBBBBBBB@

0:3190:1820:213

°0:4360:642

°0:362°0:014°0:293

1

CCCCCCCCCA

Ø =

0

BBBBBBBBB@

°0:099°0:0550:133

°0:186°0:1330:7790:306

°0:465

1

CCCCCCCCCA

(g1, ) = 81.5o (g2, ) = 99.7o

Thus much (at least 78%) of the usable genetic variation is essentially orthogonal to the direction that selection is trying to move the population.

Evolution along “Genetic lines of least resistance”



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)

Assuming G remains (relatively) constant, can we relate population divergence to any feature of G?

Schluter called this evolution along “genetic lines ofleast resistance”, noting that populations tend to diverge in the direction of gmax, specifically the angle between thevector of between-population divergence in means and gmax was small.

π(t) ª MV Nµ

π;t

2Ne¢G

∂

There are two ways to interpret Schluter’s observation.

Evolution along gmax

(ii) such lines are also the directions on which maximal genetic drift is expected to occur

(i) such lines constrain selection, with departures away from such directions being difficult

Under a simple Brownian motion model of drift in thevector of means is distributed as,

Maximal directions of change correspond to the leadingeigenvectors of G.



Looking at lines of least resistance in the Australian rainbowfish (genus Melanotaenia )





Katrina McGuigan

Megan Higgie

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 potentialperformance in specific hydrodynamic environments)

Here, there was no to estimate, rather the divergencevector 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 anddifferent 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 environmentsresulted 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 betweenthe leading eigenvector of G and the vectorof divergence

More generally, one can construct a spacecontaining the first k eigenvalues, and examinethe angle between the projection of ontothis space and

This provides a measure on the constraintsimposed by a subset of the useable variation

An advantage of using a subspace projection is that G is often ill-conditioned, in that max / min 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 geneticvariation by, for example, taking the leading keigenvectors.

In such cases, most of the genetic variation resides on a lower-dimensional subspace.

It is often the case that G contains several eigenvalues whose associated eigenvectors accountfor almost no variation (i.e, max / tr(G) ~ 0) .

P roj = A°AT A

¢°1AT

p = P rojØ = A°AT A

¢°1ATØ

A = (g1; g2; ¢¢¢;gk )

To do this, first construct the matrix Aof the first k eigenvalues

The projection matrix for this subspace isgiven by

Thus, the projection of into this subspaceis given by the vector

Note that this is the generalization of the projection of one vector onto another

The constraints imposed within this subspaceis given by the angle between p, the projectionof into this space, and .

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 between and the projection p of

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?



Anna Van Homrigh

Looked at 9 CHC involved in mate choice in Drosophila bunnanda

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 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 .

When the entire space of G is considered,the resulting angle between R and is 67o

Evolution Under Constraints or Evolution of Constraints?

G both constrains selection and also evolves under selection. Over short time scales, if most alleleshave modest effects, G changes due to selection generating linkage disequilibrium.

¢ G = ° GØØT G = ° R RT

The within-generation change in G under the infinitesimal model is

¢G ij = ¢æ(A i; A j ) = ° R iR 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 = zi i

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.

Why so much variation?

What is surprising, however, is that considerablegenetic 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 ofhighly-correlated traits?

The “deep” nature of G

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 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)

The second feature that studied regulatory/metabolic networks showed is that the degreedistribution (probability distribution that a node is connected to k other others) follows a powerlaw

P(k) ~ k-

Graphs with a power distribution of links are called scale-free graphs.

Scale-free graphs show they very important featurethat they are fairly robust to perturbations. Mostrandomly-chosen nodes can be removed with littleeffect on the system.

Some apparently general features ofBiological networks

Our spherical cow may in reality have a very non-spherical distribution of new mutation phenotypes around a current phenotype.





z

Geometry of the fitness surfaceand geometry of the mutational space

Effects of selection removing variation(geometry of the fitness surface)

Residual variation: constraints and usableevolutionary fuel (geometry of the subspaceof usable variation relative to direction of selection

Raw material

Filter

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

Stuart Barker

spherical cows grazing in flatland: constraints to selection and adaptation bruce walsh...

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