Modeling Genetic Network: Boolean Network

1. Modeling Genetic Network: Boolean Network Yongyeol Ahn 2004.08.18. KAIST

2. Genetic Network • Genes interact with each other via proteins, RNAs and themselves.

3. Main Objectives • To infer genetic network from biological data • To explain and predict the behaviors of genetic regulatory network

4. Modeling Genetic Networks • Statistics rules! • Bayesian network Hmm.. We need some dynamics. • Let’s be realistic!  Differential equation approach • Simple is the best! Boolean network

5. Bayesian Network : Information Theory • Shannon entropy: • Joint probability: Pr(x,y) • Conditional probability: Pr(y|x) = Pr(x,y)/Pr(x) • Mutual information:

6. Bayesian Network • Find a directed acyclic graph which shows the relationships of nodes well. Xi : Expression level

7. C 0.2 1 1 2 0.5 A + R 10 5 50 50 50 0.01 500 A A 1 1 A A 50 Gene A 100 Gene R Differential Equation Approach ‘The Clock’

8. R mRNAs Expressed genes A C R A Differential Equation Approach C R

9. Why Boolean network? • It tells about the dynamics (vs. bayesian network) • ‘Gene switch’ : There are attempts to make a ‘genetic computer’ using genetic ‘logic gates’.  Binary state approximation is fine. • In many cases, the exact timing may not be important. • Simple, general, easy to implement, …

10. The Boolean Network • Nodes, Directed links • Synchronous dynamics • Binary states: ‘on’ or ‘off’ • A node’s state is determined by states of other nodes which have a link to the node(by assigned boolean functions).

11. 2 • 0 0 1 • 0 1 0 • 0 1 • 1 1 1 Example Node 2 Node 0 0 • 0 1 • 0 1 • 0 1 2 • 0 0 0 • 0 1 • 0 0 1 • 0 0 1 2 t

12. Boolean Network Variations • Multi-Valued model • Different updating scheme (asynchronous, …) • Probabilistic model

13. Classification of boolean networks (Gershenson2004)

14. Tools • DDLab • http://www.ddlab.com • RBNLab • http://rbn.sourceforge.net • BN/PBN toolbox: • http://www2.mdanderson.org/app/ilya/PBN/PBN.htm

15. (Classical) Random Boolean Network • Parameter: N, K, p • N: number of nodes • K: average in-degree • p: probability of ‘1’ in each boolean function • Large ensemble ,state space(2^N) • So big! • Very high standard deviations

16. Phase Transition • Stable (K<=2) • Critical • Chaotic (K>=3) • Visualization method • Active nodes: ‘green’ • Frozen nodes: ‘red’

17. Phase Transition • Islands • Chaotic: green sea percolate & red islands • Stable: frozen red sea & green islands • Robustness • Chaotic: damage spreads • Stable: robust • Convergence and divergence of traj. (Lyapunov exponent) • Chaotic: similar states tend to diverge • Stable: tend to converge

18. Loops  Trees • For active dynamics, network needs Loops. • Loops activate other parts (trees). • Active wave propagates from loops to trees.

19. G – Density • G-density : Garden of Eden states density • Ordered: very high G-density, high in-degree frequency • Critical: power-law in-degree distribution • Chaotic: lower G-density,

20. Analytical Result of Phase Transition • Derrida’s annealed approx. : Assuming connections and boolean functions are randomly reshuffled at each time step. • Define overlap = 1 – Normalized Hamming distance between two states • What will happen at tinf ?

21. Analytical Result of Phase Transition For a network with in-degree k, Transforming with Hamming distance and consider bias

22. Derrida Curves K=5 Critical connectivity K=2

23. Phase diagram (In practice, the size of the network can play a role in the phase transitions)

24. Topology of boolean network • In reality, genetic networks have very inhomogeneous degree distribution • Using Derrida’s annealed approximation, the phase diagram for scale-free network can be obtained.

25. Derrida’s Annealed Approx. For Power-law Degree Distribution • By the assumption, x(t) obeys the equation Where,

26. Contd.

27. Contd.

28. Topology of boolean network: Scale-free boolean network

29. Attractors

30. The Number of Attractors • The number of attractors grows faster than any power law with system size. (Samuelsson2003)

31. The Length of Attractor • For K=1, root(N/2) • At critical phase, it is long believed that the length proportional to root N ( Kauffman argued that this is related to the number of cell types ) • But it is linear

32. Applications • Reverse Engineering • Morphogenesis model • Segment polarity development • Yeast transcriptional network

33. Reverse engineering: REVEAL • REVerse Engineering Algorithm • It finds a minimal solution for a boolean network given any set of time-series. • Use entropy, mutual information

34. Neutral Mutation and Punctuated Equilibrium (Bornholdt1998) • The model evolves under robustness principle (look for silent mutations) • Threshold networks (restricted set of the boolean networks) Weight = ± 1, 0

35. Evolutionary Rule • Create a daughter network by ‘adding’, ‘removing’, ‘adding and removing a weight in the coupling matrix’ at random. (each p = 1/3) • With a random initial state, if mother & daughter reach the same attractor, replace the mother with the daughter. In other case, keep the mother network.

36. Punctuated Equilibrium • The evolution shows punctuated network connectivity (lifetime ~ 1/t^2) • Evolved networks have much shorter attractors, large frozen components

37. Model for Morphogenesis(Sole2003) • Modeling an organism with one dimensional cell array. • Each cells have the same set of genes and hormones. • Genes interact within the cell. • Hormones communicate with neighboring cells. • Threshold model.

38. Morphogenesis Model

39. Development

40. Adaptive Walks • ‘Toward more complex organism’ • Complexity measure: the number of cell types • Rule • Evolve many organism in parallel • Addition, Removal, randomization of link, link’s weight (each p=1/3) • Check the complexity

41. Logarithmic Increase of the Number of Patterns • Consistent with Kauffman’s ‘rugged landscape’ explanation of Cambrian Explosion

42. Segment Polarity Network in Fly (Albert2003) • The genetic network of Fly development • This network is simulated with ODE(Dassow, 2002) and has shown a good result.

43. Boolean Network Construction Rules • The effect of transcriptional activators and inhibitors is never additive, but rather, inhibitors are dominant. • Transcription and translation are ON/OFF functions of the state • If transcription/translation is ON, mRNAs/proteins are synthesized in one time step • mRNAs decay in one time step if not transcribed • Transcription factors and proteins undergoing post-translational modification decay in one time step if their mRNA is not present.

44. Constructed Boolean Network

45. Results • Stable state is same as the real fly. • Essential gene deletion results also agree with real data. • There are six distinct steady states in the model. Three of these are well-known experimentally

46. Yeast transcriptional network(Kauffman2003) • For a given network structure, they generate boolean network ensembles with nested canalyzing functions. • The ensemble of networks are very stable.

47. Canalyzing Function • Canalyzing boolean function has at least one input which the output value is fixed. • In most cases, genetic networks consist of canalyzing functions.

48. Yeast Transcriptional Network • Find topological transition point (to determine the confidence p value) • Remove genes that have no output to other genes

49. Yeast Transcriptional Network • For a given network structure.. • Functions with null hypothesis • Functions based on literature (canalyzing)

50. Nested Canalyzing Function • Inputs im, outputs Om, in degree k • Assume i1 is canalyzing input, then we can define a new rule without i1 (with indegree k-1) • In most cases, this new rule is also canalyzing.