Function and Dynamic constrains on the topology of biological networks

1. Function and Dynamic constrains on the topology of biological networks Qi Ouyang Center for Theoretical Biology and School of Physics, Peking University, Beijing, China PICB, Shanghai, April. 30, 2009

2. Our interest: Understanding the behavior of biological networks • Topological properties; • Dynamic properties; • Biological functions; • Their relations. topology dynamics function

3. Previous work: topology dynamics Budding Yeast cell cycle network The states of proteins are defined as: 0 - inactive, 1 - active. The interaction rules: The protein states in the next logic time step are determined by the protein states in the present step. 11Nodes; 16 Green lines; 13 Red lines; 5 Yellow lines for positive interactions (green arrows); for negative interactions (red arrows), self-degradation (yellow lines).

4. Results Signal: Cln3 from 0 to 1.

5. Fixed point of the dynamics 2048 initial states Question: the distribution of attractor size of the network. 1764 of 2048 initial states (86%) evolve to G1 states. Making the G1 state the only global attractor.

6. Biological trajectory in phase space • The trajectory from excited G1 state to rest G1 state consists a globally stable manifold, which corresponds to biological pathway. PNAS (2004) ; J. Theo. Biol. (2006); Physica D (2006)

7. Some reverse questions: • Given a function of the control system, what kinds of biological networks can perform this function? • How many of them can do it robustly (dynamically and structural stable)? • Are there any structures of the biologically functional networks? Molecular Systems Biology 2, (12 Dec. 2006)

8. Cell-cell signaling in pattern formation Gene cascade of segmentation

9. Function of the SP network Setting up sharp and stable boundaries between parasegments Segmentation polarity network From von Dassow, et al., J Exp Zool. 2002 Oct 15;294(3):179-215.

10. The constrain of dynamics —Structure stability Biological network are robustly designed G. Dassow, E. Meir, E.M. Munro & G.M. Odell, NATURE406, 188 (2000) Segmentation polarity network • Randomly sample the parameter space • Hit rate (Q) = fraction of parameter space that can perform the biological function • Hit rate: 0.5% ~ 90%48 • A robust functional module! 25%=50%2

11. Enumerate all 2-node networks … … … … A B A B A B 4x2=8 edges 3 possibilities per edge 38=6561 networks E E W W

12. Model of regulation n,k A B t n/4k B A A1 B A A k A2 Define then

13. An example A B A B A B … … Q=fraction of parameter space that can perform the function

14. Distribution of Q values What are these 45 networks?

15. Skeletons and families Essential Neutral Bad Very bad Three and half topological features: Positive loop on E Positive loop on W Mutual intercellular activation of E and W Mutual repression if extracellular loop

16. Functional modules Bistability Sharp boundaries Bistability

17. 3-node networks 3x6=18 edges 318=387,420,489 networks E E Only two extracellular signaling 315=14,348,907 S S W W

18. ? Distribution of Q values

19. Modules for 3-node networks

20. 108 possible combinations

21. 44 combinations form the skeletons for all robust networks (Q>0.1) Q=0.58 Q=0.66 Q=0.63 Q=0.59 Q=0.50 Q=0.66 Q=0.63 Q=0.29 Q=0.34 Q=0.26 Q=0.48

22. Two candidates for bionetwork ? Derek Lessing and Roel Nusse, (1998) Development 125, 1469-1476 Marita Buescher, et al. (2004) Current Biology, 14, 1694-1702 Hsiu-Hsiang Lee and Manfred Frasch, Development 127, 5497-5508 (2000) ?

23. W E W E       W W E E W W W W E zw3 mutant, or ectopic expression of Wg W E E E E W W E E Wild type patched mutant Mutant tests for the two candidates

24. Summary • Functionality of the system drastically limits network topology; • Using different initial conditions can further limits the network topology; • Robustness constrains on the network further reduce the network variability; • Modularity provides combinatorial variability; • In order to narrow down the topology variability, one need to find other constrains (Modularity? Evolutionary plausible? …)