Robustness against Fluctuations and Mutation in Gene Regulatory Network
Macoto Kikuchi and Shintaro Nagata∗
Cybermedia Center, Osaka University, Japan. ∗ Department of Physics, Osaka University, Japan

Gene Regulatory Network (GRN)

Steady-State Response to Input
Fittest RGNs respond step-function-like (cooperatively) to input:
Emergence of bistability (two fixed points!).
1.0

0.8

0.8

0.6

0.6r(̄ σ)

1.0r(̄ σ)

The state of the cell is regulated by the degree of expressionof many genes, namely through quantities and balance ofmany proteins, adaptively to the environmental conditions.
The mutual regulations of genes form a complex network.

0.4

0.4

0.2

0.2

0.0

0.0
0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0σσ

Figure: F = 0.7 ∼ 0.71

Figure: The fittest ensemble

Dynamical Response using the Fixed-Point Switching
Purpose and Method∼ 60% of GRNs in the fittest ensemble can respond quickly to thenoisy input: Robustness against the input noise.

We investigate GRNs that respond cooperatively to the input
focusing their robustness in particular.
Robustness against the mutationRobustness against the input fluctuation
Robustness against the internal nose

1.0

N=162024
28
32

0.8

1.2

output1.0

0.8

P2r(t)

0.6



0.4



input

0.6

0.4

0.2

For that purpose, we produce the ensemble of GRNs withcooperative response.
We apply the multicanonical MC method for sampling GRNs
randomly instead of applying GA.

0.2
0.0

0.0

−0.2

0.5

0.6

0.7

F

0.8

0.9

1.0

0100

200

300

400

500

600t

Figure: Ratio of the GRNs having two Figure: Stable response to the noisy
input due to the fixed-point switching
fixed points

Model
Noise-Induced Sensitive Response (Effect of Internal Noise)

Directed random graph N nodes and K edges
Node: Gene
Edge: Regulatory relation

1.0

In some cases that the GRNs
cannot follow the rapid changeof input, the internal noise helps
them follow the input:∼ 70% of GRNs can follow thechange of the input in total

0.8

Self regulation and mutually-regulating pair are not considered.
r(t)We consider GRNs having 1 input gene and 1 output gene.

0.6

0.4

Discrete-Time Dynamics

0.2
w/o noisew noise0.0)

Sj (t + 1) = R σδj,1 + Σi Jij Si (t)tanh x + 1
R(x) =2500

1000

1500

2000

2500

3000

t0.8

Figure: Response to input with andwithout internal noise

0.6

R(x)

(01.0

0.4

0.2

Mutational Robustness

0.0
−2.0

−1.5

−1.0

−0.5

0.0

x

0.5

1.0

1.5

2.0

Si : Expression of ith gene (continuus variable of [−1, 1])
Jij : Interaction between ith and jth gene(±1)σ : Input signal from environment
R: Soft response function of each gene

Single-edge deletion.
Distribution of fitness after
the mutation splits into two
peaks for large fitness.

100000

80000

60000

Majority of edges are neutral
against mutation.
Only a small number ofedges are lethal.
FLarger GRNs are relatively
more rubust against mutation. Figure: Fitness distribution after themutation
40000

20000

Definition of the Fitness

00.0

0.2

0.4

0.6

0.8

1.0′

Sensitivity of gene i|S̄i [1] − S̄i [0]|S̄i [σ]: time average of the response of ith gene to the input σ
The node having the largest sensitivity is selected as theoutput gene.

Example of Robust GRN
inr : S of the output gene (response of the network)

Fitness F : sensitivity of the output gene.
This GRN has no lethaledge and robust against
both the input noise andthe internal noise.

Fitness Landscape
N=32 K=80
N=28 K=70
N=24 K=60
N=20 K=50
N=16 K=40

10−2P( F )

10−5
10−8

10−11
10−14
10−17
10−20

0.0

0.2

0.4

F0.6

0.8

1.0

Figure: Appearance probability offitness

Highly fitted GRNs are veryrare!
We investigate theuniversal properties of thefittest ensemble(F = 0.99 ∼ 1)out

ProposalThree robustnesses are characteristic properties accompanying to
the high fitness and realize (almost) irrespective to the pathway ofevolution.