The Structure, Function, and
Evolution of Biological Systems
Instructor: Van Savage
Spring 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 forward
in time. It is inherently a dynamic (time evolving
property). Can assign fitness to
1.
2.
3.
4.
5.
6.
7.
8.
Individuals
Genes
Phenotypes
Behaviors
Strategies (economic, cultural, games, etc)
Tumor cells and tumor treatment
Antibiotic resistance
Language
Evolution of allele frequency and
Wright’s equations
p(1 p) d ln w p(1 p) d(rG)
pt 

2
dp
2
dp
Conclusions
1. Increases in direction of slope of fitness function
2. Allele frequency climbs peak until maximal fitness
and this derivative or slope is zero
3. Peak occurs when marginal fitness for A1 and A2
are equal, implying relative fitness of heterozygote
4. 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-hs
Deleterious double mutant=1-s
Genetic Load= 1 w  sp 2  2 phs(1 p)
Mutation-selection balance
μ(1-p)
A1
A2
νp
Given a forward mutation rate, μ, and backward mutation
rate, ν
pˆ ~

hs
Special case that h=0, we have
pˆ ~

s
and Genetic Load ~ spˆ 2  
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
P(A1 I A2 )  P(A1)P(A2 | A1)
Note that P(A1)=p and we define  ij  P(Ai I A j )

So the marginal fitness is
*
1
w ~

11
p
w11 
12
p
w12
Do better for finite-size populations
with conditional probabilities
Definition of average fitness is now
w   ij wij
i
j
Measure, gij, is the proportion of A1 alleles within a
genotype, so mean value of g is p

Cov(w,g)   ij (gij  g)(wij  w)
i
j
Special case of Price’s Theorem
We will learn full version in much greater detail soon.
Cov(w,g)
p  g 
w


Additive Genetic Variance
From statistics
Cov(w,g)  wgVar(g)  wg
p(1 p)
2
Least-squares regression of w on g
Known as additive genetic variance and
used by breeders
Variance in fitness is square of deviations in fitness, s
2
VA  Var( p) wg
Special case of Fisher’s Fundamental
Theorem of Natural selection
VA
w 
w
This 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 value
lead to small
 changes because we are near the peak.
Fitness is always increasing
More general form of Theorem is
VA
w 
 E(w)
w
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.
Recombination
Why do we need two loci for re-arrangements to matter?
A1
A2
A2
A1
up versus down makes no
difference in our model
A1 B 1
A2 B 2
A2 B 1
A1 B 2
up and down are now differentiated
by the B alleles
A1 B 1
A2 B 1
A2 B 1
A1 B 1
Does this re-arrangement make
a difference?
Recombination
Now need four frequencies for each possible pairing of
A and B alleles?
Freq of
A1 B 1
=x11
Freq of
A2 B 1
=x21
Freq of A1=p1=x11+x12
Freq of A2=p2=x21+x22
x
Freq of Ai=pi=
Freq of
A1 B 2
=x12
j
x
Freq of Bi=qi=
i
Freq of A2B2
=x22
ij

ij
Recombination
For which genotypes with will recombination have
an effect A1B1?
Take all possible genotypes with an A1 or B1
A1 B 1
A1 B 2
A1 B 1
A2 B 1
A1 B 2
A2 B 1
A1 B 1
A1 B 1
r
1-r
A1 B 1
A2 B 2
A1 B 1
A2 B 2
A1 B 2
A2 B 1
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
Cov(A,B)  E(AB)  E(A)E(B)  x11  p1q1  D
D is the measure of gametic disequilibrium and time
evolution can be expressed in terms of this and the
recombination rate
x’ij=xij+(-1)i+jrD
D’=D(1-r)

Recombination with selection
Must assign fitness and then use formulas and do algebra
similar to what we have been doing.
1
x ij  [Cov(w,gij )  rDw1122]
w
Additional term captures effects of recombination
and 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
Cov(wx ,wy )  w xy  w x w y
If no interaction, then the covariance is 0.
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
wWT 1 e
0
Still assuming a mutation is deleterious, we look
at combined effects of two mutations
w x  1
s
~
e
x

sx
sx sy
wx wy  e e
w y  1 sy ~ e
and
e
(sx sy )
~ 1 (sx  sy )
sy
Non-Additive
w xy  w x w y
w xy  w x w y

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
     ,g gA   w,g gB
A
B

     ,g gA   w,g gB   ,g
A
B
A gB
gA gB
Modeling more than two mutations
If all mutations have the same deleterious effect, and
k mutations are lethal, then
ks
we
k
~ 1 s ~ 1 ks  1
kL
k
How can we modify this for epistasis?
wepi 1 sk
1
~ (1 s)
k 1
wepi  (1 ks)
or
sk1
~e
What about these forms for epistasis?
1
Lethal number
of mutations
1
 k 
w epi  1  
kL 
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 you
don’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.