Functionality & Speciation in Boolean Networks

1. Functionality & Speciation in Boolean Networks Jamie Luo Warwick Complexity DTC Dr Matthew Turner Warwick Physics & Systems Biology

2. Gene Regulatory Networks http://www.cs.uiuc.edu/homes/sinhas/work.html

3. Gene Regulatory Networks http://www.pnas.org/cgi/content-nw/full/104/31/12890/F2

4. Why Study Boolean Networks? • How does the Topology influence the Dynamics? • Construct Predictive Models of Complex Biological Systems. • Network Inference. • How Dynamical Function Influences Topology? • Design and Shaping Intuition.

5. Threshold Dynamics • N-size (N genes) Threshold Boolean Network is a Markovian dynamical system over the state space S = {0,1}N. • Defined by an interaction matrix A ∈{-1, 0, 1}N . • For any v(t) ∈ S, let h(t) = Av(t).

6. Example GRN • p53 – Mdm2 network: • Example path through the state space: Mdm2 p53

7. Biological Functionality • Define a biological function or cell process. • Start – end point (v(0), v∞) definition of a function [1]. • Find all matrices A ∈{-1, 0, 1}N which attain this function. • Investigate the resulting space of matrices which map v(0) to the fixed point v∞. [1] Ciliberti S, Martin OC, Wagner A (2007) PLoS Comput Biol 3(2): e15.

8. Metagraph (Neutral Network) • For A , B ∈{-1, 0, 1}Ndefine a distance: • Metagraph where A and B are connected if d(A , B) = 1. • Start-end point (v(0), v∞) approach results in a single large connected component dominating the metagraph [1]. [1] Ciliberti S, Martin OC, Wagner A (2007) PLoS Comput Biol 3(2): e15.

9. Robustness • Mutational Robustness (Md) of a network is its metagraph degree. • Noise Robustness (Rn) can be defined as the probability that a change in one gene’s initial expression pattern in v(0) leaves the resulting steady state v∞ unchanged • Start-end point approach finds that Mutational Robustness and Noise Robustness are highly correlated. Furthermore Mutational robustness is found to have a broad distribution.

10. Intuition Shaping • Robustness is an evolvable property [1]. • The metagraph being connected and evolvability of robust networks may be a general organizational principle [1]. • Long-term innovation can only emerge in the presence of the robustness caused by a connected metagraph [2]. • Above conclusions rely on a largely connected metagraph. • Metagraph Islands [3]. [1] Ciliberti S, Martin OC, Wagner A (2007) PLoS Comput Biol 3(2): e15. [2] Ciliberti S, Martin OC, Wagner A (2007) PNAS vol. 104 no. 34 13591-13596 [3] G Boldhaus, K Klemm (2010), Regulatory networks and connected components of the neutral space. Eur. Phys. J. B (2010),

11. Example GRN Revisited • p53 – Mdm2 network: • Example path through the state space: Mdm2 p53

12. Redefining a Biological Function • Any start-end point function (v(0), v∞) encompasses the ensemble of all paths from v(0) to v∞. • Unrepresentative of many cellular processes (cell cycle, p53). • We propose using a path {v(t)}t=0,1,...,Tto define a function. • Crucially distinguish paths by duration T (complexity).

13. Which Path to Take? • Large number of paths for any given N. How to sample? • Method 1 (speed θ): Choose a θ ∈[0 1]. Randomly sample an initial condition v(0)∈S. Then vi(t +1) = vi(t) with a probability 1- θfor all t ≥ 0. • Method 2 (matrix sampling): Randomly sample an initial condition v(0)∈S. Then for each t ≥ 0 randomly sample a matrix A to map v(t) tov(t+1) and so on.

14. Attainability of a Function • Increasing duration T exponentially constrains the topology.

15. Speed Kills? • Mean path duration Tend depends non-monotonically on θ.

16. T=1 => Connected Metagraph • For any path {v(t)}t=0,1,...,Tof duration T = 1 the corresponding metagraph is connected. • Proof: Fix a path of the form {v(0), v(1)} Let {r : rj∈{-1, 0, 1}}i be all the row solutions for gene i. Suppose vi(0) = 0 and vi(1) = 1, then hi(0) >0. Therefore 1= [1 1 , . . . , 1] is always a valid row solution. Furthermore any other solution r can be mapped to 1 by point mutations (changing an entry to rj1). Other cases are similarly accounted for (-1= [-1 , . . . , -1]).

17. The Metagraph & Speciation

18. Complexity to Speciation • Increasing Complexity as measured by duration T leads to a speciation effect. T = 1 T > 1

19. Robustness Complexity Trade-off • Mutational Robustness decreases with increasing T.

20. T vs. ρ(Md,Rn) • Mutational Robustness and Noise Robustness are positively correlated but the strength of this correlation is T dependent.

21. Ensemble vs. Path • The start-end point definition of a biological function includes the ensemble of all paths from v(0) to the fixed point v∞. • Our definition isolates a single path. v(0) v∞ v(0) v(T)

22. Summary • A path definition of functionality leads to contrasting conclusions from the start – end point one. Conclusions based on the existence of a largely connected metagraph are not applicable under a functional path definition. • Metagraph connectivity, mutational robustness, ρ(Md,Rn) and the number of solutions all depend on path complexity. • The breakup of the metagraph with increasing complexity is analogous to a speciation effect.

23. Future Work & Design • Multi-functionality. • Paths with Features. • Genetic Sensors.

24. Acknowledgements • Matthew Turner • Complexity DTC • EPSRC • Questions?