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A Boolean Network Simulation of Heterosis

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Peter Emmrich

on 27 August 2013

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Transcript of A Boolean Network Simulation of Heterosis

fitness improves over many generations
A Gene Regulatory Network Simulation of Heterosis
Peter Emmrich, Hannah Roberts
and Vera Pancaldi
Heterosis in Maize
Mutation
Recombination
Selection
Diploids
Boolean Gene Regulatory Networks
ON
or
OFF
Two possible
states:
Gene 1
Gene 2
Gene 3
Gene 3 is turned
ON

in the following step if

Gene 1 is
ON
and Gene 2 is
OFF
G3 = G1 & ! G2
Parent1: Mo17
Parent 2: B73
+
high yielding hybrid
Under
each environment
select for
one module
to be turned
ON,
and others to be turned
OFF
Module 1
Module 2
E1
E2
E3
G1
G3
G2
G5
G4
G6
G8
G13
G7
G10
G9
G11
G12
G15
G14
G16
define
environments
as states of input nodes
Environment 1:
E1 =
ON
E2 =
ON
E3 =
OFF
Environment 2:
E1 =
OFF
E2 =
ON
E3 =
ON
haploid networks are allowed to mutate
1) remove output
G1
G2
G2'
2) remove input
G1
G2
G1'
3) add edge
G1
G2
G3
+
+
+
+
4) duplicate node
G1
G2
G1'
+
+
+
+
+
+
+
+
5) remove node
G1
G2
G3
6) do nothing
G1
G2
G3
G1
Parent 1
Parent 2
Diploid
G2
G3
G4
G1
G2'
G3
G5
G1
G2
G3
G4
G5
G2'
+
huge improvement over inbred lines
Breeding Scheme
E1
E2
E3
G1
G3
G2
G5
G4
G6
G8
G13
G7
G10
G9
G11
G12
G15
G14
G16
original network with realistic properties
all genes are homozygous
phenotype ≈ 0.0
duplicate to form
original population
Evolve!
(hundreds of cycles)
homozygotic
(from both parents)
from parent 2
from parent 1
G2
G3
G1
G2'
G3
G5
G4
G1
G1'
-
-
+
G2
G3
G1
G2'
G3
G5
G4
G1
G1'
-
+
-
G2'
G3
G4
G1
G1'
-
start with two haploids
randomly pick alleles from either parent
many different recom-binations are possible
Parent 1
Parent 2
Parent 1
Parent 2
new haploid
to select networks from the population, we need to measure the
fitness
of each network...
Evolve!
Evolve!
Evolve!
Evolve!
duplicate population
evolve separately
populations diverge over time
fitness is defined by the
phenotype
in response to the
environment
...
under
each environment
, the fitness is the average state

(between
0
and
1
) of the
corresponding module
minus the average of
all other modules
...
the
network fitness
is the average of the fitnesses for each environment
We use a
Metropolis-Hastings
algorithm for selection...
Better individuals are more likely to survive,
but occasionally worse ones get passed on as well...
This helps against getting trapped in a local maximum
p = e
accept
ß*(F - F )
new
old
p
accept
ß
a constant
probability of surviving
F
phenotype of network
phenotype of parents
F
old
new
Adaptation
Evolution!
then often progresses in jumps
starts off very fast, but soon reaches equilibrium,
populations can remain
in equilibrium for long periods of time
Thus a naive (unadapted) network has a
fitness ≈ 0.0
the nodes of the initial network are just as likely to be
ON
or
OFF
at any given point
Hybrid Vigour
Mutation rates
low rates
change one edge: 2%
change one node: 0.02%
standard
change one edge: 10%
change one node: 0.1%
add/kill nodes
change one edge: 10%
change one node: 1%
fitness
generation
Theories of Heterosis
Dominance
F = F > F
A
Aa
a
A
a
A
= >
A
Overdominance
F < F > F
A
AA'
A'
< >
A'
A
Under
dominance
F > F < F
A
AA'
A'
> <
This leads to hybrid
dis
advantage, the opposite of heterosis
Epistasis
Dominance
F < F > F
Ab
AbaB
aB
Pseudo-Over-
a
a
A
A'
A
A'
A'
A
A
A'
A
A'
< >
a
A
B
b
a
A
B
b
a
A
B
b
but really this is just
dominance!
requires gene linkage (not considered here)
A'
A
B
B'
A'
B
>
F > F
A'BAB'
A'B
positive genetic interaction between alleles of different genes
Don't need
hetero-zygotes
for this...
but highly heterozygotic indviduals have a greater number of
different alleles
...
so there is
more potential
for epistasis in hybrids
interaction can also be negative!
and select modules with
high connectivity

as outputs
generations
split populations
populations
start diverging
initial adjustment
fitness and diversity of the population increase
repeat as populations diverge
In theory, hybrids should better than the inbreds
Make hybrids and measure fitness
until they are so far apart they become incompatible
F > F
H
I
F < F
H
I
Heterosis
= Hybrid vigour
offspring of
crosses
between
genetically distinct
parents often show very favourable properties
they often
outperform their parents
in
yield, stress resistance, growth speed etc.
Parent 2
Parent 1
Hybrid
performance
Today vast majority of maize crop is hybrids
But heterosis remains poorly understood...
Is it due to individual loci or is it a general effect?
How important is epigenetics?
How can it be predicted?
What genetic mechanisms are responsible?
We use abstract
boolean networks
to study heterosis
Biological
Gene Regulatory Networks
scale-
free
small-
world
Wildenhain and Crampen 2006
hierar-
chical
Our algorithm generates networks with these properties.
One such network is copied to make a population...
If there are several alleles of one gene, there are several nodes
In most cases,
hybrids perform better
than their parent populations after just a few generations of separation
But after a while, hybrid fitness
collapses
The populations have become
incompatible
the number of generations until the collapse varies
Future questions for this model
what if hybrids are crossed again?
How important are gene linkage and epistasis?
Summary
Our model simulates the evolution of gene regulatory networks
it predicts increases in fitness for hybrids
overdominance appears to be the most important mechanism
Department of Plant Sciences, University of Cambridge
Major Mechanisms
Department of Plant Sciences
Special Thanks to:
Vera Pancaldi
Hannah Roberts
and the bioinformatics group of the Baulcombe lab
Network sizes
The algorithm can handle any size, but soon becomes computationally intensive
most networks were between 20 and 200 nodes
larger networks evolve much slower
inbred populations
hybrids
fitness
generation after split
count per network
generation after split
moment of fitness collapse
Underdominance
Overdominance
Dominance
Can heterosis potential be predicted from the population before the split?
power-law degree distribution emerges during evolution
networks are becoming more
scale-free
A network which completely turns on the correct modules under each environment would have a
fitness = 1
G1
Parent 1
Parent 2
Diploid
G2
G3
G4
G1
G2'
G3'
G5
G1
G2
G3
G4
G5
G2'
+
homozygotic
(from both parents)
from parent 2
from parent 1
G4
E1
Parent 1
Parent 2
Diploid
+
E2
G1
G2'
G4
E1
E2
G3'
G1
G2
G3
G4
E1
E2
G3'
G2'
Full transcript