The Structure, Function, and Evolution of Biological Systems

Instructor: Van SavageSpring 2010 Quarter

4/1/2010

Crash Course in Evolutionary Theory

What is fitness and what does it describe?

Ability of an entity to survive and propagate forwardin time. It is inherently a dynamic (time evolving property). Can assign fitness to

1. Individuals2. Genes3. Phenotypes4. Behaviors5. Strategies (economic, cultural, games, etc)6. Tumor cells and tumor treatment 7. Antibiotic resistance8. Language

Evolution of allele frequency and Wright’s equations

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Δpt = p(1− p)2

d lnwdp

= p(1− p)2

d(rG)dp

Conclusions1. Increases in direction of slope of fitness function2. Allele frequency climbs peak until maximal fitness

and this derivative or slope is zero3. Peak occurs when marginal fitness for A1 and A2

are equal, implying relative fitness of heterozygote4. Prefactor is actually a variance, so strength of

selection depends on variance. No variance implies no selection.

How do we maintain variance?

Mutation and migration

What is typical effect of a mutation?

Wild Type fitness=1 (relative fitness)Hetero. Mutant fitness=1-hsDeleterious double mutant=1-s

Genetic Load=

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1− w = sp2 + 2phs(1− p)

Mutation-selection balance

Given a forward mutation rate, μ, and backward mutationrate, ν

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ˆ p ~ μhs

Special case that h=0, we have

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ˆ p ~ μs

and Genetic Load

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~ sˆ p 2 = μ

A1 A2

μ(1-p)

νp

How good are these approximations?

Other important factors

1. Density dependence

2. Multiple alleles (more then two)

3. Multiple Loci (more than one)

4. Fertility selection is pair specific

Do better for finite-size populations with conditional probabilities

Fundamental formula in statistics is

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P(A1 I A2) = P(A1)P(A2 | A1)

Note that P(A1)=p and we define

So the marginal fitness is

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w1* ~ γ11

pw11 + γ12

pw12

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γij = P(Ai I A j )

Do better for finite-size populations with conditional probabilities

Definition of average fitness is now

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w = γ ijwijj

∑i

∑

Measure, gij, is the proportion of A1 alleles within a genotype, so mean value of g is p

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Cov(w,g) = γ ij (gij − g)(wij − w)j

∑i

∑

Special case of Price’s Theorem

We will learn full version in much greater detail soon.

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Δp = Δg = Cov(w,g)w

Additive Genetic Variance

From statistics

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Cov(w,g) = βwgVar (g) = βwgp(1− p)

2

Least-squares regression of w on gKnown as additive genetic variance andused by breeders

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VA = Var (p)β wg2

Variance in fitness is square of deviations in fitness, s

Special case of Fisher’s Fundamental Theorem of Natural selection

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Δw = VA

wThis term captures selection favoring the most fit.Need variance for selection to act. Small values of fitness lead to rapid changes to increase it. Large valuelead to small changes because we are near the peak.Fitness is always increasingMore general form of Theorem is

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Δw = VA

w+ E (δw)

Extra term captures effects of density dependence. Also,need to account for fluctuating environments

Additional effects for more than two loci

1. Recombination—breaking, rejoining, and rearranging of genetic material. Major extra source of variation.

2. Epistasis—interactions between loci (i.e., non-independence). Fitness effects of alleles affect each other in non-additive way.

RecombinationWhy do we need two loci for re-arrangements to matter?

A1

A2

A2

A1

up versus down makes no difference in our model

A1B1

A2B2

up and down are now differentiatedby the B alleles

A2B1

A1B2

A1B1

A2B1

Does this re-arrangement makea difference?

A2B1

A1B1

RecombinationNow need four frequencies for each possible pairing ofA and B alleles?

A1B1Freq of =x11

A2B1

A1B2

A2B2

Freq of =x21

Freq of =x12

Freq of =x22

A1Freq of =p1=x11+x12

A2Freq of =p2=x21+x22

Freq of Bi=qi=

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x iji

∑

Freq of Ai=pi=

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x ijj

∑

RecombinationFor which genotypes with will recombination have an effect A1B1?Take all possible genotypes with an A1 or B1

A1B1

A2B1

A1B1

A1B1

A1B2

A2B1

A1B1

A2B2

A1B1

A1B2

A1B2

A2B1

A1B1

A2B2

r

1-r

Recombination

Can understand all of this again in terms of covariance.Covariance of A and B implies effect of recombination.Zero covariance implies no recombination

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Cov(A,B) = E(AB) − E(A)E(B) = x11 − p1q1 = D

D is the measure of gametic disequilibrium and timeevolution can be expressed in terms of this and therecombination rate

x’ij=xij+(-1)i+jrD

D’=D(1-r)

Recombination with selection

Must assign fitness and then use formulas and do algebrasimilar to what we have been doing.

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Δx ij = 1w

[Cov(w,gij ) ± rDw1122 ]

Additional term captures effects of recombinationand whether it slows or speeds up evolution. “-” if i=j and “+” is I does not equal j

Epistasis

Interactions among fitness effects for different alleles

If no interaction, then the covariance is 0.

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Cov(wx,wy ) = w xy − w x w y

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w xy = w x w y

This is know as additive (or sometimes multiplicative.

Additive

Choose relative fitness so that the wild type fitness is 1,and look at exponential (continuous) versions

Still assuming a mutation is deleterious, we lookat combined effects of two mutations

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wWT =1= e0

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w x =1− sx ~ e−sx

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w y =1− sy ~ e−syand

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w x w y = e−sx e−sy = e−(sx +sy ) ~ 1− (sx + sy )

Non-Additive

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w xy ≠ w x w y

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w xy < w x w y

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w xy > w x w y

Synergistic (negative epistasis)

Antagonistic (positive epistasis)

What is the distribution of these effects?What fraction of mutation pairs are antagonistic?What fraction of mutation pairs are synergistic?

Graphical representation

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φ=α +βφ,gAgA + β w,gB

gB + βφ,gA gBgA gB

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φ=α +βφ,gAgA + β w,gB

gB

Modeling more than two mutations

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w = e−ks ~ 1− s( )k ~ 1− ks =1− k

kL

If all mutations have the same deleterious effect, andk mutations are lethal, then

How can we modify this for epistasis?

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wepi =1− sk1+ε ~ (1− s)k1+ε

~ e−sk1+ε

What about these forms for epistasis?

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wepi = (1− ks)1+ε

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wepi =1− kkL

⎛ ⎝ ⎜

⎞ ⎠ ⎟

1+ε

or

Lethal numberof mutations

Next class we will move onto interactions between loci and genes and possible touch on drift and coalescence.

Some material is in Chapter 2 of Sean Rice’s book, but youdon’t need to know more beyond what was covered in class

Read papers for next week on distribution of epistatic interactions, modeling epistasis, the evolution of sex, and the evolution of antibiotic resistance.