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Simultaneous Emergence of Cooperative Response and Mutational Robustness in Gene Regulatory Networks

Macoto Kikuchi and Shintaro Nagata

Cybermedia Center, Osaka University

Outline

Setup of the problem Gene expression and gene regulatory networks Model Method: rare event sampling Results

Fitness landscape Robustness

Summary and outlook

Setup of the problem

Premise Significant difference of Life phenomena from

• ther physical phenomena is in that the former

are rare phenomena made by evolution and exhibit robustness against mutation

Some questions

1

How can we quantify the rareness of Life phenomena

2

What are the characteristics of the fitness landscape

3

Whether the mutational robustness is aquires in the course of evolution or the high fitness inevitably produces robustness

We study a mathematical model of the gene regulatory networks

1

To generate the ensemble of GRNs that respond cooperatively (sensitively) to the input

2

To investigate universal properties of GRNs in the ensemb.e

We focus on robustness in particular

What is the gene expression

DNA contains many genes

A gene carries information for producing a protein

The RNA polymerase read the gene and produce a mRNA (messenger RNA) which carries the equivalent information as the gene. The ribosome (one of the inner-cell organs) produces a protein according to the information given from mRNA.

Gene expression

What is the gene regulatory network

The state of the cell is regulated by the degree

• f expression of many genes, namely through

quantities and balance of many proteins, adaptively to the environmental conditions. Expression of a gene is regulated by the appropriate transcription factor (TF), which itself is a protein produced by another gene. Genes are mutually regulated through TF

The mutual regulations of genes form a complex network: GRN

Gene regulation

Gene Regulatory Network

Background

GRNs are basic mechanism for regulating the cell state GRNs respond sensitively to change of the environmental conditions

Cooperative response

Responses of GRNs are robust against many types of fluctuations

Number fluctuation of molecules, Thermal fluctuation

GRNs are robust against mutation

Purpose and Method

We investigated the gene regulatory networks (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 produced the ensemble of GRNs with cooperative response.

We did not apply the genetic algorithm because it cannot sample GRNs randomly. We applied the rare-event sampling using the multicanonical MC method instead.

Popular approach to study evoution Genetic algorighm (GA)

It mimics the real Darwinian evolution through the

• ffspring, muation and crossing processes

Why we employ a different approach Highly evolved GRNs generated by GA may reflect the pathway of evolution We would like to study universal properties of highly fitted GRNs that are independent of the evolutional history

Model

Directed random graph N nodes and K edges

Node: Gend Edge: Regulatory relation

Detailed process of gene expression is ignored (longer time scale) Self regulation and mutually-regulating pair are not included (although they exist in real GRNs We consider GRNs having 1 input gene and 1

• utput gene

Si : Expression of ith gene (continuus variable

• f [−1, 1])

Jij : Interaction between ith and jth gene

Jij = ±1 (activation or repression)

σ : Input signal from outside Discrete-time dinamics Sj(t + 1) = R (σδj,1 + ΣiJijSi(t)) R(x) = tanh x + 1 2 This type of model is frequently used for studying the evolution of GRN

−4 −2 2 4

Si

0.0 0.2 0.4 0.6 0.8 1.0

R

Response function: same for all the genes

Input and output nodes Input: Randomly chosen from the nodes having paths to all the other nodes Output: The most sensitive gene among the nodes having paths from all the other nodes (determined only after the dynamics is run)

Effective response Consider in the steady state Effective response of ith node against the input σ is defined by the time average of the expression ¯ Si[σ] ≡ 1 T

τ+T

∑

t=τ

Si(t)

We take τ = T = 1000 Initial value: Si(0) = 0.5 (certainly irrelevant to the result)

Fitness Sensitivity of gene i di = ¯ Si[1] − ¯ Si[0] The node having the largest di is selected as the output gene Response of the network dMAX ≡ max{di}

We use dMAX as the fitness of evolution

Premise Evolution can produce GRNs with high fitness What we want to do Make the ensemble of GRNs for each value of dMAX Investigate dMAX dependence of characters of GRN

Characters of the fittest ensemble in particular

Use of equilibrium statistical mechanics approach

method Rare event sampling by the multicanonical Monte Carlo method regarding the fitness dMAX as energy Uniform sampling with respect to dMAX Actually we divide dMAX into 100 bins and perform the entropic sampling

microcanonical ensemble in each bin

Weight for the entropic sampling is determined by Wang-Landau method

Detail N = 16 − 32 2K/N = 5, 6(fixed) Elementary process:

1

select a node

2

cut one edge connected to that node

3

select disconnected node pair randomly and connect them by J = 1 or −1

Elementary process is rather arbitrary, because we do not follow the evolution dynamics

Result 1: Fitness landscape

0.0 0.2 0.4 0.6 0.8 1.0

dMAX

10−20 10−17 10−14 10−11 10−8 10−5 10−2

P(dMAX)

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

Probability distribution of dMAX for 2K/N = 5

0.0 0.2 0.4 0.6 0.8 1.0

dMAX

10−20 10−17 10−14 10−11 10−8 10−5 10−2

P(dMAX)

Probability Distribution of dMAX for 2K/N = 6

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

Probability distribution of dMAX for 2K/N = 6

On the probability distribution It can be regarded as the fitness landscape in the same meaning as the energy landscape More than 50% of GRNs are in dMAX < 0.01 (non-functional) Threshold of rareness d∗ at dMAX ≃ 0.2

More than 95% of GRNs are in dMAX < 0.2

GRNs are exponentially rare for dMAX > d∗

More than exponentially rare for dMAX > 0.9 Appearance probability of GRNs in the fittest ensemble (0.99 ≤ dMAX ≤ 1) is 3 × 10−19 (for N = 32, K = 5)

GRNs in the fittest ensemble are rare

result 2: cooperative 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 dMAX ≃ 0.7(N = 32, K = 80)

Steady state response of GRNs to the input (for dMAX ≃ 0.7, 20 samples)

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 dMAX ≃ 1 (N = 32, K = 80)

Steady state response of GRNs to the input (Fittest ensemble, 20 samples)

On the response GRNs having small dMAX respond smoothly to the input signal GRNs belonging to the fittest ensemble respond step-function-like to the input (cooperative response)

Implies that the emergence of the fixed point switching mechanism of the dynamical system

Question If the GRNs uses the fixed point switching, can they respond quickly to the dynamical change

• f the input? (Dynamical histeresis?)

Answer: For N = 32, K = 80, 61% GRNs can respond quickly. This is a sufficiently high probability

Result 3: Robustness against input noise

Study the response to the noisy input signal of the quickly responding GRNs in the fittest ensemble

We consider the finiteness of the input molecules as the source of the noise Uncorrelated Gaussian noise Allow the negative input for simplicity

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

Response to the noisy input

2960 2980 3000 3020 3040

t

−0.25 0.00 0.25 0.50 0.75 1.00 1.25

Sout

input

• utput

Transient of the response to the change of the input

5960 5980 6000 6020 6040

t

−0.25 0.00 0.25 0.50 0.75 1.00 1.25

Sout

input

• utput

Transient of the response to the change of the input

On the robustnes against input noise Stable response to the fluctuation of input

Filtering the noise thanks to the bistability due to the fixed point switching

Quick following to the change of input

transient of ∼ 10 steps

Result 4: Robustness against mutation

Consider a mutation of single-edge deletion

A moderate mutation Supposing the mutation of the transcription factor

• r that of the binding site*dMUT: d for the output

node for the GRN after mutation (for the same

• utput node)

0.0 0.2 0.4 0.6 0.8 1.0

dMUT

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

P(dMUT)

N=32 K=80

dMAX = 0.99 0.9 0.8 0.7 0.6 0.5

Probability distribution of dMUT for all the possible mutations of all the samples

0.0 0.2 0.4 0.6 0.8 1.0

dMUT

0.0 0.1 0.2 0.3 0.4

P(dMUT)

dMAX = 0.99

N=32 K=96 N=32 K=90

Probability distribution of dMUT (K dependence for the fittest ensemble (N = 32)

0.0 0.2 0.4 0.6 0.8 1.0

dMUT

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

P(dMUT)

dMAX = 0.99

N=16 K=48 N=16 K=40

Probability distribution of dMUT (K dependence for the fittest ensemble (N = 16)

On the effects of mutation Single peak distribution for GRNs of small

• dMAX. Majority of the edges are dMUT ≃ dMAX

(neutral mutation) Distribution splits into two peaks for GRNs of large dMAX (> 0.8)

Majority of edges are neutral for mutation Small number of edges are dMUT ≃ 0 (lethal mutation) Almost no intermediate edge

20 40 60 80 100

n

0.00 0.02 0.04 0.06 0.08 0.10

P(n)

Number distribution of lethal edges for 2K/N=5

N=16 20 24 28 32

Probability distribution of the lethal edges for the fittest ensemble (for 2N/K = 5)

20 40 60 80 100

n

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

P(n)

Number distribution of lethal edges for 2K/N=6

N=16 20 24 28 32

Probability distribution of the lethal edges for the fittest ensemble (for 2N/K = 6)

On the robustness against mutation Small number of lethal edges

For N = 32, K = 80、86% edges are n ≤ 20 Robustness agains mutation is a characteristic property of the fittest ensemble

The peak of the number distribution of the lethal edges is independent of N

Larger GRNs are relatively robust

There are a few GRNs having no lethal edge

All the edges cooperatively respond

Summary : robulsness and evolution

GRNs in the fittest ensemble exhibit the following properties

1

Cooperative response using the fixed point switching of the dynamical system

2

Majority of GRNs respond stably to the noisy input

3

Robust against mutation

4

Larger GRNs are relatively robust

Proposal Two robustnesses are characteristic properties accompanying to the high fitness and realize irrespective to the pathway of evolution

Remaining problems

Model dependence

Or reallity

Relationship between the fitness landscape and the evolutionally process