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The Structure, Function, and
Evolution of Biological Systems
Instructor: Van Savage
Spring 2010 Quarter
4/1/2010
Epistasis
Interactions among fitness effects for different alleles
  Cov(wx ,wy )  w xy  w x w y
Can now see covariance plays central role in all of
evolution. In fact, it is as central as fitness itself.

If no interaction, then the covariance is 0.
w xy  w x w y
This is know as additive
(or sometimes multiplicative).
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)
k1
Lethal number
of mutations
sk1
~e
This measure of epistasis and definition of ε is different than
in previous slide even though conceptually it is the same.

Modeling more than two mutations
wepi 1 sk
1
~ (1 s)
k1
sk1
~e
How do we interpret synergy and
antagonism?
Mutations to steps in sequence are antagonistic (green)
Mutations to steps in parallel can be synthetically lethal
if it knocks out a loop, which is extreme synergy (red), or
multiplicative (black).
Recent papers using models of epistasis:
Segre, DeLuna, Church, Kishony
Quantitative Epistatic Interactions
Perturbation X
Phenotype
(Growth Rate)
Perturbation Y
Synergy
Antagonism
Suppression
See also:
Boone, Science (2004)
Weissman, Cell (2006)
Boeke, Nature Genetics (2003); Cell (2006)
Giaever, Nature Genetics (2007)
Measures of epistasis
Since covariance is as fundamental as fitness, why not
define relative covariance instead of relative fitness. We
define it relative to tri-modally binned covariance that
itself varies, so relative to a shifting baseline.
Absolute covariance
  Cov(wx ,wy )  wxy  wx wy
Relative covariance
wxy  wx w y
˜


BinnedCov (wx ,w y ) w˜ xy  wx w y
Cov(wx ,wy )

Measures of epistasis—based onFBA
predictions in yeast
Sort of unimodal distribution goes to trimodal distribution
Opposite of Lenki et al. because synergy is enriched. Why?
Measures of epistasis—RNA viruses
A bit more continuous for real data. We will see more real
data later on.
Higher level epistasis—interactions
among functional groups rather than loci
Interactions are mostly monochromatic. No reason a priori
that this should be, except it signifies functional organization.
Cov(WModA,WModB )  WModA,ModB WModAWModB
Can we do reverse and cluster
monochromatically to find functional groups?
Construct network for all pairwise interactions,
Start with each gene in its own group.
Cluster by pairs if they interact with other genes in same way.
Require monochromaticity, each group must interact with all
other groups in same way
Within a group there is no requirement for monochromaticity
Make cluster sizes as large as possible
Do clustered groups correspond to
functionally annotated groups?
Statistical Test: N=total gene pairs, 1034
n=interacting pairs, 278
K=same annotation pairs, 104
K N  K  N 
 1  
/ 
i n  i  n 
i 0  
k1
PANN
Ways to choose Ways to choose
i annotated pairs n-i non-annotated
pairs
Ways to choose
n interacting pairs
Each term in sum is probability of choosing i annotated pairs
out of subset of n pairs relative to any subset of n pairs being
the interacting ones. Sum is probability of having k-1 or less.
Can we do reverse and cluster
monochromatically to find functional groups?
Cluster Movie
How clusterable are networks?
Is clustering unique?
If not, which instantiation is chosen?
Number of random networks
Monochromatic organization exists in the yeast metabolic
network, but is very unlikely in random networks
400
Random
300
200
100
0
Yeast
0
5
10
Number of non-monochromatic connections
Overlay of monochromatic and
functionally annotated groups?
A system-level view of modular genetic interactions in yeast metabolism
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Identifying Biological Functions in Higher Hierarchical Levels
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Future directions research using
digital organisms
1. I would use mixed selection criteria because I think
they are more biological:
a. Baseline allocation of CPU time is independent of
genome size
b. mathematical operations are rewarded with
additional CPU time
2. Run Prism algorithm on existing digital data. Is it
clusterable? What would make it clusterable?
Recent papers using models of epistasis:
Desai, Weissman, Feldman
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  
Evolution of Epistasis
Attempt to look at whether evolution favors additive,
synergy, or antagonism
Analyze evolutionary effects of different forms of epistasis
in asexuals and sexuals (with recombination) with mutations
Let one loci control degree of epistasis and itself be subject
to selection and evolution. Not necessarily a good model
because there could be mechanical, functional, and
mechanistic constraints not controlled by a single or any locus.
Claim there are no general patterns
Assume s>>Lμ and can ignore stochastic effects
Frequency distribution for number of
mutations
How do frequencies change in time?
wF  (1 L)F0
'
0
wF  wk Fk  (L  k)wk Fk  (L  1 k)wk1Fk1
'
k
 (1 (L

k)

)e

sk1
Fk  (L  1 k)e
Make approximations using Lμ<<s<<1


L

1 ˆ
ˆ
Fk ~   1 F0
 s  k!
k
s(k1)1
Fk1
Frequency distribution for number of
mutations for asexuals
Normalizing our frequency distribution gives
Fˆ0 ~
1
k
L
L  1
 s  k!1
k 0
And we also have

w  1 L
Note there is no dependence on ε, so no selection on epistasis
Additive case
Fˆ0 ~ e
When ε=0, notice that this is
Probability of having a mutation is


1 e


L
s
 
ˆ
Fk ~ e
k!
k
Frequency distribution is Poisson
L
s
Epistasis and recombination (sexual)
Proximity on genome increases recombination
M is locus that controls epistasis
Epistasis and recombination (sexual)
D  Cov(i, j)  Fˆ0 Fˆ2 (i, j)  Fˆ1(i) Fˆ1 ( j)
L  
ˆ
 F0  (2 1)
 s 
2
Disequilibirum and epistasis have opposite signs
Sign[Cov(i, j)]  Sign[Cov(w x ,w y )]
Full form of disequilibrium
wF  (1 L)Fˆ0  r( j  i)D(i, j)
'
0
i j
Same basic form as we already derived in class

2
 Lr 1  j  i L




2i
2
j


2
2
2
ˆ (i, j)    (2 1)Fˆ 1
D
Fˆ0 

(2  ( 1)  ( 1) 
0
s 
s
L
L

 s 2  2
Recombination decreases disequilibrium but does not change its sign.
Recombination-selection balance and mutation-selection balance
come in similar forms
Distribution of mutations
Synergistic population favors recombination because it breaks up
mutations to increase fitness. Opposite for antagonistic population
Time evolution of fitness for mixed
(synergy and antagonism) population
L  Lr
1 2 
w  w1  w 0 ~     R 1 1
 s   3
 2
2
R is recombination between epistasis locus and rest of loci.
Population 1 is antagonistic and population 0 is synergistic.
Comparison of analytical and
simulations
Does recombination favor synergy or
antagonism?
Average fitness is same. Antagonistic individuals will build up
mutations and recombination will shift those to synergistic individuals
and dramatically decrease their fitness.
Starting with a synergistic population,
recombination would be able to take
off and establish. Once established,
it would push the population towards
more antagonistic epistasis, which
might then help eliminate
recombination.