Exploring the Fitness Landscape and Emergence of Mutational Robustness
in Gene Regulatory Network
Macoto Kikuchi and Shintaro Nagata∗
Cybermedia Center, Osaka University, Japan. ∗ Department of Physics, Osaka University, Japan

Motivation

Fitness LandscapeSignificant difference of Life phenomena from other physicalphenomena is in that the former are rare phenomena made byevolution and exhibit robustness against mutation.

Highly fitted GRNs are very rare!
Probability Distribution of dMAX for 2K/N = 5
N=32 K=80
N=28 K=70
N=24 K=60
N=20 K=50
N=16 K=40

10−2

Gene Regulatory Network (GRN)

10−810−11

10−510−810−1110−1410−1410−1710−1710−20

N=32 K=96
N=28 K=84
N=24 K=72
N=20 K=60
N=16 K=48

10−2P(dMAX)

What are the characteristics of the fitness landscape?
2 Is mutational robustness aquired in the course of evolution or
the high fitness inevitably produces robustness?
1P(dMAX)

10−5

Two Questions

Probability Distribution of dMAX for 2K/N = 6

0.0

0.2

0.4

dMAX

0.6

0.8

10−20

1.0

0.0

0.2

0.4

dMAX

0.6

0.8

1.0

Response to Input
Fittest RGNs respond step-function-like (cooperatively) to input:
Emergence of bistability
Response for dMAX ≃ 0.7(N = 32, K = 80)

1.0

Response for dMAX ≃ 1 (N = 32, K = 80)
1.0
0.8

0.6

0.6

̄ (σ)
Sout

0.8

̄ (σ)
Sout

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.
Genes are mutually regulated through the transcription factors.
The mutual regulations of genes form a complex network.

0.4

0.4

0.2

0.2

0.0
0.0

0.2

0.4

σ0.6

0.81.00.0

0.2

0.4

0.6

σ0.8

1.0

Dynamical Response to Noisy Input
input

Mutational Robustness
Consider the single-edge deletion.
Distribution of fitness afterthe mutation splits into twopeaks for large fitness(> 0.8).
Majority of edges are neutral
against mutation.
Only a small number ofedges are lethal.

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

Sj (t + 1) = R σδj,1 + Σi Jij Si (t)tanh x + 1
R(x) =2

Definition of the Fitness
Sensitivity of gene idi = S̄i [1] − S̄i [0]S̄i [σ]: time average of the response of ith gene to the input σ
The node having the largest di is selected as the output gene.
Fitness (Response of the network)
dMAX ≡ max{di }

0.00

−0.25

1000

2000

3000

4000

t

5000

6000

7000

8000

9000

N=32 K=80
dMAX = 0.99
0.9
0.8
0.7
0.6
0.5

0.40
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00

0.0

0.2

0.4

dMUT

0.6

0.8

1.0

The peak of the number distribution of the lethal edges is
independent of N:Larger GRNs are relatively robust.
Number distribution of lethal edges for 2K/N=5
N=16202428
32

0.10
0.08

Number distribution of lethal edges for 2K/N=60.08

N=16202428
32

0.07
0.06
0.05

0.06

0.04
0.03

0.04

0.02

0.02

Si : Expression of ith gene (continuus variable of [−1, 1])
Jij : Interaction between ith and jth gene(±1)σ : Input signal from outside

0.25

P(n)

Discrete-Time Dynamics

0.50

Probability Distribution of the Lethal Edges

P(n)We deal with GRNs having 1 input gene and 1 output gene.

0.75

0ModelSelf regulation and mutually-regulating pair are not considered (although
they exist in real GRNs).

1.00

Sout

We investigate GRNs that respond cooperatively to the input
focusing their robustness in particular.
Robustness against the mutation
Robustness against the input fluctuation
For that purpose, we produce the ensemble of GRNs withcooperative response.
We do not apply GA: We would like to explore properties
independent of the evolutionaly path.
We apply the multicanonical MC method instead for sampling
GRNs randomly.

outputP(dMUT)

Purpose and Method∼ 60% of GRNs in the fittest ensemble can respond quickly tothe noisy input: Robustness against the input
noise

1.25

0.01

0.00

0.00
02040

n60

80

100

02040

n60

80

100

Summary
GRNs in the fittest ensemble exhibit the following properties:
1 Cooperative response using the bistability.
2 Majority of GRNs respond stably to the noisy input.
3 Robust against mutation.
Proposal
Two robustnesses are characteristic properties accompanying to
the high fitness and realize irrespective to the pathway of evolution.