Evolving Small GRNs with a Top-Down Approach
Javier Garcia-Bernardo* and Margaret J. Eppstein
)
Department of Computer Science and Vermont Complex Systems Center
University of Vermont
*jgarciab@uvm.edu
Abstract
Hierarchical Evolution of GRNs
Being able to design genetic regulatory networks (GRNs) to achieve a desired
cellular function is one of the main goals of synthetic biology1. While several
researchers have used evolutionary algorithms to evolve network motifs or
synthetic networks with a desired behavior2,3, previous studies have rarely
evolved minimal or near-minimal GRNs. Even when methods were explicitly
designed to try to prune excess parts4,5, the number of interactions in the
evolving GRNs increased over generations.
In this paper, we use a top-down approach, where we start with relatively
dense GRNs and use differential evolution6 (DE) to evolve interaction
coefficients. When the target dynamical behavior is found embedded in a
dense GRN, we narrow the focus of the search and attempt to remove excess
interactions. Three methods were tested to encourage parsimonious
networks.
We first validated the method by evolving GRNs for two known circuits:
(a) Bistable, a simple toggle switch, and
(b) Oscillator, an oscillatory circuit.
We then included those known GRNs as non-evolvable subnetworks in the
subsequent evolution of two more complex, modular GRNs:
(c) ConditionalOscillator, in which we evolved indirect interactions
between non-evolvable subnetworks for a toggle switch and an
oscillatory circuit, and
(d) DualOscillator, which combined two mutually exclusive
oscillators; to make this problem more difficult we only included
one copy of a non-evolvable oscillatory circuit.
We first show that the method can quickly rediscover known small GRNs for a
toggle switch and an oscillatory circuit. Next we include these known small
GRNs as non-evolvable subnetworks in the subsequent evolution of more
complex, modular GRNs. By incorporating aggressive pruning after the
desired behavior was found, along with a dynamic penalty term, the DE was
able to evolve minimal, or nearly minimal, GRNs in all 4 test problems.
Small GRNs evolved by the ForcedReduction method.
Results (out of 25 repetitions each)
ODE System Model of GRNs
p1
dmi
i
 mi 
 0
n1,i
n2,i
nG ,i
dt
1  K1,i p1  K 2,i p2  ...  KG ,i pG
Number of Interactions
p2
(a)
regulation
dpi
  i  pi  mi 
dt
translation
transcription
Protein pi
• For each pair of genes i and j, the coefficients Ki,j and nj,i determine the
strength of the positive or negative regulatory interaction of the protein
coded for by gene j on the expression of gene i.
• Positive nj,i indicates a negative regulation of the gene i by the protein j,
and vice versa; nj,i = 0 means no interaction exists.
Encouraging Evolution of Minimal GRNs
• DE was used to evolve the parameters Ki,j and nj,i of up to 6 genes (up to 84
decision variables).
• Once a desired behavior was found, the DE strategy was shifted from
DE/rand/1/bin to DE/rand-to-best/1/bin, to focus the search.
• DE was combined with three approaches designed to encourage
parsimonious networks, as follows:
Dynamic penalty for #
and strength of
interactions:
Method
gen
Penalty 
n j ,i

C i j
ForcedReduction
yes
NoPenalty
no
WithPenalty
yes
Discard weak
interactions:
if |ni,j| < 0.5,
set |ni,j|= 0.
Aggressive
Pruning
(see below)
no
yes
yes
yes
no
no
Aggressive Pruning
Initially
dense
networks
Remove 1 interaction if fitness
not degraded > 10%
for up to 5
interactions
removed each
generation
ForcedReduction
NoPenalty
Penalty
20
10
The ForcedReduction method is able to relatively quickly evolve minimal or
near-minimal genetic networks 0with the desired behaviors. The other two
0 the1000
2000
3000 4000
6000
methods also succeed in evolving
desired
behaviors,
but the5000
resulting
Number of Evaluations
GRNs have many excess interactions.
(b)
40
In ForcedReduction, the
NoPenalty
number of interactions
Penalty
30
grows until the desired
behavior is found, then
Representative run for the evolution
of the DualOscillatory network.
excess interactions are
20
removed. Despite more
fitness evaluations per
10
generation, this method
ForcedReduction
converges with fewer total
0
0
1000 2000 3000 4000 5000 6000
fitness evaluations.
Number of Evaluations
(c)
Summary
40
30
In summary, we present a top-down, evolutionary approach that can be
recursively applied to hierarchically evolve increasingly complex, modular
20
GRNs with few, if any, excess interactions.
10
While our primary motivation is for designing GRNs for synthetic biology,
the algorithm introduced in this
work could be used for the inference of
0
2000 3000
5000 6000
naturally-occurring GRNs from0 gene1000
expression
data. 4000
Additionally,
our
Number oftopologies
Evaluations for Artificial
approach could be used for evolving parsimonious
Neural Networks.
References
Final
networks
very sparse
In aggressive pruning, at the end of each DE generation subsequent to the
strategy shift, for each solution that exhibited the desired dynamics, we
iterated through each Hill coefficient nj,i, testing raw fitness (before penalty)
with nj,i = 0. If fitness degraded by 10% or less, we kept nj,i = 0. This process
was continued until we had replaced five coefficients with zero or we ran out
of Hill coefficients to try.
30
Number of Interactions
mRNA mi
Number of Interactions
DNA gene i bound with
regulatory proteins pj
40
1 A.S.
Khalil and J.J. Collins, “Synthetic Biology: applications come of age”, Nature Reviews Genetics,
5:367-379, 2010.
2 P. François and V. Hakim, “Design of genetic networks with specified functions by evolution in silico”,
PNAS, 101:580–585, 2004.
3 N. Noman, L. Palafox, and H. Iba, “Evolving Genetic Networks for Synthetic Biology”, New Gener.
Comput., vol. 31, no. 2, pp. 71–88, 2013.
4 B. Drennan and R. Beer, “Evolution of repressilators using a biologically-motivated model of gene
expression”, ALIFE X, pp. 22–27, 2006.
5 M. Dorp, B. Lannoo, and E. Carlon, “Evolutionary Generation of Small Oscillating Genetic Networks”, in
Adaptive and Natural Computing Algorithms, pp. 120–129, 2013.
6 R. Storn and K. Price, “Differential Evolution – A Simple and Efficient Heuristic for global Optimization
over Continuous Spaces”, J. Glob. Optim., 11:341–359, 1997.