1. Steady-state Analysis of Gene Regulatory Networks via G-networks Intelligent Systems & Networks Group Dept. Electrical and Electronic Engineering Haseong Kim, Erol Gelenbe

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

3. 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

4. Introduction Queuing Networks G-networks 4-gene Network Yeast Cell Cycle A Simple Queuing System Queue Server Queuing system Customer : Input rate  : Service rate q : Utilization rate (Steady-state probability that a server is busy) Queuing Networks

5. Introduction Queuing Networks G-networks 4-gene Network Yeast Cell Cycle A Jackson Network (The Simplest Queuing Network) Let k i be the length of ith queue. P(K 1 =k 1, K 2 =k 2, K 3 =k 3, K 4 =k 4 ) =P(K 1 =k 1 )P(K 2 =k 2 )P(K 3 =k 3 )P(K 4 =k 4 ) where P(K i =k i )=q i k i (1-q i ) James R. Jackson, 1963 Queuing Networks

6. 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

7. Introduction Queuing Networks G-networks 4-gene Network Yeast Cell Cycle G-networks for GRNs A. Arazi et. al., 2004 E. Gelenbe, 2007 G-networks

8. Introduction Queuing Networks G-networks 4-gene Network Yeast Cell Cycle E. Gelenbe, 2007 G-networks

9. Introduction Queuing Networks G-networks 4-gene Network Yeast Cell Cycle The Solution of the G-networks E. Gelenbe, 2007 G-networks

10. Introduction Queuing Networks G-networks 4-gene Network Yeast Cell Cycle Parameter Estimation r i = number of outdegrees of gene i  i = mRNA degradation rate of gene i G-networks P + (i,j) =P - (i,j) =Q(i,j,l) =Q(j,i,l) =1

11. Introduction Queuing Networks G-networks 4-gene Network Yeast Cell Cycle Parameter Estimation Boundary of total input rate  i Initial transcription rate without any external effects Positive Inputs from other genes are zero and queues fully work G-networks

12. Introduction Queuing Networks G-networks 4-gene Network Yeast Cell Cycle Parameter Estimation Compute q iu by solving the following equation numerically Select  i * and q i * which are maximizing the L iu G-networks

13. Introduction Queuing Networks G-networks 4-gene Network Yeast Cell Cycle Stochastic Gene Expression Model 4-gene Networks H. McAdams and A. Arkin, 1997 J. Paulsson, 2005 A. Riberio et al., 2006

14. Introduction Queuing Networks G-networks 4-gene Network Yeast Cell Cycle 4-Gene Network Example Gillespie Algorithm (D. Gillespie, 1977) Generalized Gillespie Algorithm (D. Bratsun, 2005) 4-gene Networks

15. Introduction Queuing Networks G-networks 4-gene Network Yeast Cell Cycle Parameters of the Stochastic Gene Expression Model ParametersValuesReferences Transcription initiation 2 0.0025sec -1 Golding, et al., 2005; Thattai and van Oudenaarden, 2001 Translation initiation 3 0.0612sec -1 Paulsson, 2005; Thattai and van Oudenaarden, 2001 mRNA degradation 2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 associationk a1 0.1Buchler, et al., 2005 Dimer dissociationk d1 0.01Buchler, et al., 2005 DNA-protein associationk a2 0.189Goeddel et al., 1977 DNA-protein dissociationk d2 0.0157Goeddel et al., 1977 Burst sizeb10Paulsson, 2005; Thattai and van Oudenaarden, 2001 Accumulation time of proteins tt 0.1Bratsun 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

16. 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 G A = 0.0012 sec -1 is a half of the normal transcription rate 0.0025 sec -1 4-gene Networks

17. Introduction Queuing Networks G-networks 4-gene Network Yeast Cell Cycle Normal Abnormal 4-gene Networks

18. Introduction Queuing Networks G-networks 4-gene Network Yeast Cell Cycle Simulation Results 20 datasets each of which have randomly selected 50 samples Compute steady-state probabilities and p-values of t-test 4-gene Networks

19. Introduction Queuing Networks G-networks 4-gene Network Yeast Cell Cycle Yeast Cell Cycle Wittenberg C. 2005 Bahler J. 2005 Bloom J. 2007 Yeast Cell Cycle

20. Introduction Queuing Networks G-networks 4-gene Network Yeast Cell Cycle Reconstructed Cell Cycle GRN Yeast Cell Cycle

21. 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

22. Introduction Queuing Networks G-networks 4-gene Network Yeast Cell Cycle 13 Genes Expression Profiles Yeast Cell Cycle

23. Introduction Queuing Networks G-networks 4-gene Network Yeast Cell Cycle Steady-State Probabilities StateCellsCLN3WHI5SWI4MBP1CLB2YOX1YHP1HCM1FKH2NDD1SWI5ACE2SIC1 S1 WT 0.8800.8130.8290.8390.7840.990.8030.8430.8550.8360.7990.99 CM 0.8780.8140.8180.8480.770.990.8020.8420.8640.8390.7870.99 S2 WT 0.8820.845 0.8400.8470.990.8500.8700.863 0.8250.99 CM 0.8760.8370.8460.8470.7690.990.8530.8730.8650.8610.8070.99 S3 WT 0.8900.8400.8260.8460.8860.990.8440.8550.8630.8540.8710.99 CM 0.880.8460.820.8490.7510.990.863 0.8690.870.840.99 S4 WT 0.8900.8410.8370.8450.8660.990.8390.870.8620.8530.8570.99 CM 0.8790.8350.8210.8490.7570.990.864 0.8590.8630.8450.99 S5 WT 0.8910.8500.8370.8460.8770.990.8390.8690.8620.8560.8650.99 CM 0.8690.830.8230.8420.7560.990.862 0.8570.8610.8450.99

24. 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)