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Simultaneous Emergence of Cooperative Response and Mutational Robustness in Gene Regulatory Networks
Macoto Kikuchi and Shintaro Nagata
Osaka University, Japan
Simultaneous Emergence of Cooperative Response and Mutational - - PowerPoint PPT Presentation
Simultaneous Emergence of Cooperative Response and Mutational - - PowerPoint PPT Presentation
Simultaneous Emergence of Cooperative Response and Mutational Robustness in Gene Regulatory Networks Macoto Kikuchi and Shintaro Nagata Osaka University, Japan CCS2018 Motivation Living systems exhibit high fitness and robustnesses
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We study a simple model of the gene regulatory network
without using evolutionary simulations make an ensemble of GRNs with high fitness by Multi-canonical MC
To explore the universal properties of highly fitted GRNs. The robustnesses in particular.
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The gene regulatory network
The cell state is regulated by the expression levels of many genes adaptively to the environmental conditions. Gene expressions are regulated by the transcription factors (TF), which themselves are proteins produced from genes. Genes are mutually regulated through TF
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Gene regulation GRN
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Model
Directed random graph with N nodes and K edges
Node: Gene Edge: Regulatory relation
1 input gene and 1 output gene Average number of edges connected to a node is 2K/N = 5
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Discrete-time dinamics Si(t + 1) = R (σδi,1 + ΣjJijSj(t)) R(x) = tanh x + 1 2 Si : Expression of ith gene (continuus variable
- f [0, 1))
Jij : Interaction from jth to ith gene
Jij = ±1 (activation or repression) or 0 (no regulation)
σ : Input signal from outside
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−4 −2 2 4
Si
0.0 0.2 0.4 0.6 0.8 1.0
R
Response function
each gene respond moderately
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Effective response Consider the steady state Effective response of ith gene against the input σ ¯ Si[σ] ≡ 1 T
τ+T
∑
t=τ
Si(t) Fitness Sensitivity of gene i ∆i = | ¯ Si[1] − ¯ Si[0]| Fitness f ≡ max{∆i}
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Method
Rare event sampling by the Multicanonical ensemble Monte Carlo method regarding the fitness f as energy It ebables us to sample GRNs in a wide range
- f fitness randomly (in principle).
Wang-Landau method for determining the Monte Carlo weight
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Fitness Landscape
0.0 0.2 0.4 0.6 0.8 1.0
f
10−20 10−17 10−14 10−11 10−8 10−5 10−2
P(f)
N=32 K=80 N=28 K=70 N=24 K=60 N=20 K=50 N=16 K=40
Probability distribution of the fitness
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Steady-State Response
0.0 0.2 0.4 0.6 0.8 1.0
σ
0.2 0.4 0.6 0.8 1.0
̄ Sout(σ)
Response for f ≃ 0.7(N = 32, K = 80)
f ≃ 0.7
0.0 0.2 0.4 0.6 0.8 1.0
σ
0.0 0.2 0.4 0.6 0.8 1.0
̄ Sout(σ)
Response for f ≃ 1 (N = 32, K = 80)
f ≃ 1
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Emergence of fixed-point switching
For f ≃ 1, all the networks have two fixed points Emergence of the cooperative response to the input using the fixed point switching mechanism
A kind of inovation takes place inevitably for highly fitted GRNs.
Question Then, can they respond properly to the rapid change of input?
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Dynamical Response
Response to abruptly changing input
100 200 300 400 500 600
Time
0.0 0.2 0.4 0.6 0.8 1.0
Response
Some GRNs can follow,
100 200 300 400 500 600
Time
0.0 0.2 0.4 0.6 0.8 1.0
Response
Some cannot
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Effect of Internal Noise
Consider the number flucturations of TFs as the internal noise. Si(t) → Si(t) + ri ri: uniform random number in [−0.1, 0.1]
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100 200 300 400 500 600
Time
0.0 0.2 0.4 0.6 0.8 1.0
Response
without fluctuation with fluctuation
Robust against internal noise
100 200 300 400 500 600
Time
0.0 0.2 0.4 0.6 0.8 1.0
Response
without fluctuation with fluctuation
Noise-induced response
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w/o noise: ∼ 60% of GRNs can respond sensitively
They are robust against internal noise
w noise: ∼ 74% of GRNs can respond sensitively
Noise-Induced sensitive response
Internal noise makes GRNs to respond properly
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Robustness against mutation
0.0 0.2 0.4 0.6 0.8 1.0
f*
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40
P(f*)
N = 32, K = 80
f = 0.99 0.9 0.8 0.7 0.6 0.5
Fitness after mutation Consider single-edge deletion
A moderate mutation
All the possible cuts are tried
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0.0 0.2 0.4 0.6 0.8 1.0
f*
0.0 0.1 0.2 0.3 0.4
P(f*)
f = 0.99
N=32 K=96 N=32 K=90
For f ≃ 1 Majority of edges are neutral Small number of edges are lethal No intermediate edge
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20 40 60 80 100
n
0.00 0.02 0.04 0.06 0.08 0.10
P(n)
2K/N = 5
N=16 20 24 28 32
Number distribution of lethal edges Small number of lethal edges The peak is independent of N
Larger GRNs are relatively robust
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Summary
Result For the GRNs with high fitness, we found that the majority of the networks own the following robustnesses
1
Mutational Robustness
2
Robustness against internal noise
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Robustness against input noise (not shown)
Proposal These robustnesses are not the consequence of the evolution, but the characteristic properties accompanying to the high fitness irrespective to the pathway of evolution
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Rubustness ageinst input noise
1000 2000 3000 4000 5000 6000 7000 8000 9000
t
−0.25 0.00 0.25 0.50 0.75 1.00 1.25
Sout
input
- utput