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 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 other physical phenomena is in that the former are rare phenomena made by evolution and exhibit robustness against mutation

Some questions How can we quantify the rareness of Life 1 phenomena What are the characteristics of the fitness 2 landscape Whether the mutational robustness is aquires 3 in the course of evolution or the high fitness inevitably produces robustness

We study a mathematical model of the gene regulatory networks To generate the ensemble of GRNs that 1 respond cooperatively (sensitively) to the input To investigate universal properties of GRNs in 2 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 of 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 offspring, 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 output gene

S i : Expression of i th gene (continuus variable of [ − 1 , 1]) J ij : Interaction between i th and j th gene J ij = ± 1 (activation or repression) σ : Input signal from outside Discrete-time dinamics S j ( t + 1) = R ( σδ j , 1 + Σ i J ij S i ( t )) R ( x ) = tanh x + 1 2 This type of model is frequently used for studying the evolution of GRN

1.0 0.8 0.6 R 0.4 0.2 0.0 −4 −2 0 2 4 S i 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 i th node against the input σ is defined by the time average of the expression τ + T S i [ σ ] ≡ 1 ¯ ∑ S i ( t ) T t = τ We take τ = T = 1000 Initial value: S i (0) = 0 . 5 (certainly irrelevant to the result)

Fitness Sensitivity of gene i d i = ¯ S i [1] − ¯ S i [0] The node having the largest d i is selected as the output gene Response of the network d MAX ≡ max { d i } We use d MAX 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 d MAX Investigate d MAX 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 d MAX as energy Uniform sampling with respect to d MAX Actually we divide d MAX 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 2 K / N = 5 , 6(fixed) Elementary process: select a node 1 cut one edge connected to that node 2 select disconnected node pair randomly and 3 connect them by J = 1 or − 1 Elementary process is rather arbitrary, because we do not follow the evolution dynamics

Result 1: Fitness landscape Probability Distribution of d MAX for 2 K / N = 5 N=32 K=80 10 −2 N=28 K=70 N=24 K=60 N=20 K=50 10 −5 N=16 K=40 P ( d MAX ) 10 −8 10 −11 10 −14 10 −17 10 −20 0.0 0.2 0.4 0.6 0.8 1.0 d MAX Probability distribution of d MAX for 2 K / N = 5

Probability Distribution of d MAX for 2 K / N = 6 N=32 K=96 10 −2 N=28 K=84 N=24 K=72 N=20 K=60 10 −5 N=16 K=48 P ( d MAX ) 10 −8 10 −11 10 −14 10 −17 10 −20 0.0 0.2 0.4 0.6 0.8 1.0 d MAX Probability distribution of d MAX for 2 K / 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 d MAX < 0 . 01 (non-functional) Threshold of rareness d ∗ at d MAX ≃ 0 . 2 More than 95% of GRNs are in d MAX < 0 . 2 GRNs are exponentially rare for d MAX > d ∗ More than exponentially rare for d MAX > 0 . 9 Appearance probability of GRNs in the fittest ensemble (0 . 99 ≤ d MAX ≤ 1) is 3 × 10 − 19 (for N = 32 , K = 5) GRNs in the fittest ensemble are rare

̄ result 2: cooperative response Response for d MAX ≃ 0.7( N = 32, K = 80) 1.0 0.8 0.6 S out ( σ ) 0.4 0.2 0.0 0.2 0.4 0.6 0.8 1.0 σ Steady state response of GRNs to the input (for d MAX ≃ 0 . 7, 20 samples)

̄ Response for d MAX ≃ 1 ( N = 32, K = 80) 1.0 0.8 S out ( σ ) 0.6 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 σ Steady state response of GRNs to the input (Fittest ensemble, 20 samples)

On the response GRNs having small d MAX 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 of 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

input 1.25 output 1.00 0.75 S out 0.50 0.25 0.00 −0.25 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 t Response to the noisy input

input 1.25 output 1.00 0.75 S out 0.50 0.25 0.00 −0.25 2960 2980 3000 3020 3040 t Transient of the response to the change of the input

input 1.25 output 1.00 0.75 S out 0.50 0.25 0.00 −0.25 5960 5980 6000 6020 6040 t 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 or that of the binding site* d MUT : d for the output node for the GRN after mutation (for the same output node)

N=32 K=80 d MAX = 0.99 0.40 0.9 0.8 0.35 0.7 0.6 0.30 0.5 P ( d MUT ) 0.25 0.20 0.15 0.10 0.05 0.00 0.0 0.2 0.4 0.6 0.8 1.0 d MUT Probability distribution of d MUT for all the possible mutations of all the samples

d MAX = 0.99 N=32 K=96 N=32 K=90 0.4 P ( d MUT ) 0.3 0.2 0.1 0.0 0.0 0.2 0.4 0.6 0.8 1.0 d MUT Probability distribution of d MUT ( K dependence for the fittest ensemble ( N = 32)

d MAX = 0.99 N=16 K=48 0.35 N=16 K=40 0.30 0.25 P ( d MUT ) 0.20 0.15 0.10 0.05 0.00 0.0 0.2 0.4 0.6 0.8 1.0 d MUT Probability distribution of d MUT ( K dependence for the fittest ensemble ( N = 16)