Steady-state Analysis of Gene Regulatory Networks via G-networks

Introduction Queuing Networks G-networks Parameter Estimation Simulation Study Yeast Cell Cycle Networks Discussion

Intelligent Systems & Networks GroupDept. Electrical and Electronic Engineering

Haseong Kim, Erol Gelenbe

Introduction Queuing Networks G-networks 4-gene Network Yeast Cell Cycle

Introductiono Fundamental challenges of systems biology

Modeling regulatory interactions of genes by using mathematical & statistical methods

Exploring the dynamics of the gene regulatory networks (GRNs) by analyzing their long-run (steady-state) behaviors

Introduction

Introduction Queuing Networks G-networks 4-gene Network Yeast Cell Cycle

Objective

Infer the steady-state probabilities of genes in GRNs

G-network Theory

Gene Regulatory Network Structures

Microarray Gene Expression

Introduction

Introduction Queuing Networks G-networks 4-gene Network Yeast Cell Cycle

A Simple Queuing System

Queue

Server

Queuing system

Customer

l: Input ratem : Service rateq : Utilization rate (Steady-state probability that a server is busy)

Queuing Networks

Introduction Queuing Networks G-networks 4-gene Network Yeast Cell Cycle

A Jackson Network (The Simplest Queuing Network)

Let ki be the length of ith queue.

P(K1=k1, K2=k2, K3=k3, K4=k4)

=P(K1=k1)P(K2=k2)P(K3=k3)P(K4=k4)

where P(Ki=ki)=qiki(1-qi)

James R. Jackson, 1963

Queuing Networks

Introduction Queuing Networks G-networks 4-gene Network Yeast Cell Cycle

G-Networks• G-networks have positive, negative customers and signals

E. Gelenbe, 1991, 1993

G-networks

Introduction Queuing Networks G-networks 4-gene Network Yeast Cell Cycle

G-networks for GRNs

A. Arazi et. al., 2004E. Gelenbe, 2007

G-networks

Introduction Queuing Networks G-networks 4-gene Network Yeast Cell Cycle

)0(),())(1(

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E. Gelenbe, 2007

G-networks

Introduction Queuing Networks G-networks 4-gene Network Yeast Cell Cycle

The Solution of the G-networks

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E. Gelenbe, 2007

G-networks

Introduction Queuing Networks G-networks 4-gene Network Yeast Cell Cycle

Parameter Estimation

)()(

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where

ri = number of outdegrees of gene i mi = mRNA degradation rate of gene i

G-networks

P+(i,j) =P-(i,j) =Q(i,j,l) =Q(j,i,l) =1

Introduction Queuing Networks G-networks 4-gene Network Yeast Cell Cycle

Parameter Estimation

Boundary of total input rate i

)())(( jijiiiii qfqfrq m

0025.0loweri

1 where))(( jijiiii

upperi qqqfrq m

Initial transcription rate without any external effects

Positive Inputs from other genes are zero and queues fully work

G-networks

Introduction Queuing Networks G-networks 4-gene Network Yeast Cell Cycle

Parameter Estimation

)(logmaxarg

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Compute qiu by solving the following equation numerically

Select i* and qi* which are maximizing the Liu

G-networks

Introduction Queuing Networks G-networks 4-gene Network Yeast Cell Cycle

Stochastic Gene Expression Model4-gene Networks

H. McAdams and A. Arkin, 1997J. Paulsson, 2005A. Riberio et al., 2006

Introduction Queuing Networks G-networks 4-gene Network Yeast Cell Cycle

4-Gene Network Example

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Gillespie Algorithm (D. Gillespie, 1977)Generalized Gillespie Algorithm (D. Bratsun, 2005)

4-gene Networks

Introduction Queuing Networks G-networks 4-gene Network Yeast Cell Cycle

Parameters of the Stochastic Gene Expression Model

Parameters Values References

Transcription initiation l2 0.0025sec-1 Golding, et al., 2005; Thattai and van Oudenaarden, 2001

Translation initiation l3 0.0612sec-1 Paulsson, 2005; Thattai and van Oudenaarden, 2001

mRNA degradation 2 0.00578sec-1 Thattai and van Oudenaarden, 2001

Monomer degradation 3,mono 0.00077sec-1 Thattai and van Oudenaarden, 2001; Buchler, et al., 2005

Dimer degradation 3,dimer 0.00057sec-1 Thattai and van Oudenaarden, 2001; Buchler, et al., 2005

Dimer association ka1 0.1 Buchler, et al., 2005

Dimer dissociation kd1 0.01 Buchler, et al., 2005

DNA-protein association ka2 0.189 Goeddel et al., 1977

DNA-protein dissociation kd2 0.0157 Goeddel et al., 1977

Burst size b 10 Paulsson, 2005; Thattai and van Oudenaarden, 2001

Accumulation time of pro-teins

t 0.1 Bratsun et. al., 2005

Gene ON and OFF rate is set to zero. Cell growth rate and the cell volume is fixed.

4-gene Networks

Table 1

Introduction Queuing Networks G-networks 4-gene Network Yeast Cell Cycle

Data Generation• Two sets of data

– Normal vs. Abnormal – The normal set is obtained by using the parameters in Table 1– The abnormal set is the same as the normal set except the transcription

rate of GA = 0.0012 sec-1 is a half of the normal transcription rate 0.0025 sec-1

4-gene Networks

Introduction Queuing Networks G-networks 4-gene Network Yeast Cell Cycle

Normal

Abnormal

4-gene Networks

Introduction Queuing Networks G-networks 4-gene Network Yeast Cell Cycle

Simulation Results

20 datasets each of which have randomly selected 50 samplesCompute steady-state probabilities and p-values of t-test

4-gene Networks

Introduction Queuing Networks G-networks 4-gene Network Yeast Cell Cycle

Yeast Cell Cycle

Wittenberg C. 2005Bahler J. 2005Bloom J. 2007

Yeast Cell Cycle

Introduction Queuing Networks G-networks 4-gene Network Yeast Cell Cycle

Reconstructed Cell Cycle GRN

Yeast Cell Cycle

Introduction Queuing Networks G-networks 4-gene Network Yeast Cell Cycle

Expression Data

• D. Olando et. al., 2008• Yeast 2.0 oligonucleotide array• To determine which transcription factors contribute to

CDKs and to global regulation of the cell cycle transitions

• Two types of groups– Wide-type (WT) (30 time points)– Cyclin-mutant (CM) (30 time points)

Yeast Cell Cycle

Introduction Queuing Networks G-networks 4-gene Network Yeast Cell Cycle

13 Genes Expression Profiles

Yeast Cell Cycle

Introduction Queuing Networks G-networks 4-gene Network Yeast Cell Cycle

Steady-State Probabilities

State Cells CLN3 WHI5 SWI4 MBP1 CLB2 YOX1 YHP1 HCM1 FKH2 NDD1 SWI5 ACE2 SIC1

S1WT 0.880 0.813 0.829 0.839 0.784 0.99 0.803 0.843 0.855 0.836 0.799 0.99 0.99

CM 0.878 0.814 0.818 0.848 0.77 0.99 0.802 0.842 0.864 0.839 0.787 0.99 0.99

S2WT 0.882 0.845 0.845 0.840 0.847 0.99 0.850 0.870 0.863 0.863 0.825 0.99 0.99

CM 0.876 0.837 0.846 0.847 0.769 0.99 0.853 0.873 0.865 0.861 0.807 0.99 0.99

S3WT 0.890 0.840 0.826 0.846 0.886 0.99 0.844 0.855 0.863 0.854 0.871 0.99 0.99

CM 0.88 0.846 0.82 0.849 0.751 0.99 0.863 0.863 0.869 0.87 0.84 0.99 0.99

S4WT 0.890 0.841 0.837 0.845 0.866 0.99 0.839 0.87 0.862 0.853 0.857 0.99 0.99

CM 0.879 0.835 0.821 0.849 0.757 0.99 0.864 0.864 0.859 0.863 0.845 0.99 0.99

S5WT 0.891 0.850 0.837 0.846 0.877 0.99 0.839 0.869 0.862 0.856 0.865 0.99 0.99

CM 0.869 0.83 0.823 0.842 0.756 0.99 0.862 0.862 0.857 0.861 0.845 0.99 0.99

Introduction Queuing Networks G-networks 4-gene Network Yeast Cell Cycle

Conclusions & Discussions

• Analyze the steady-state of GRNs by using G-networks– In simulation study, our model provides more reliable measure then the t-

statistics.– Our G-networks are applied to the yeast cell cycle data

• The structure is too simple to draw the same conclusion with the original paper of the experiment data.

• More complex and large-scale networks are required

• Future works– Improve G-network model by providing proper probabilities P+(j,i), P-(j,i),

Q(i,j,l) with ensemble base GRN estimation method (H. Kim et al, 2009)– Steady-state analysis for both transcriptional and post-transcriptional

networks (E. Gelenbe., 2008)