Functionality & Speciation in Boolean Networks Jamie Luo Warwick Complexity DTC Dr Matthew Turner Warwick Physics & Systems Biology Gene Regulatory Networks http://www.cs.uiuc.edu/homes/sinhas/work.html Gene Regulatory Networks http://www.pnas.org/cgi/content-nw/full/104/31/12890/F2 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. 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). Example GRN p53 – Mdm2 network: Mdm2 p53 Example path through the state space: 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. Metagraph (Neutral Network) For A , B ∈ {-1, 0, 1}N define 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. 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. 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), Example GRN Revisited p53 – Mdm2 network: Mdm2 p53 Example path through the state space: 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,...,T to define a function. Crucially distinguish paths by duration T (complexity). 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- θ for 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) to v(t+1) and so on. Attainability of a Function Increasing duration T exponentially constrains the topology. 50 Mean |A| for Paths of length T 10 N = 10 N=9 N=8 N=7 N=6 N=5 40 10 30 10 20 10 10 10 0 10 0 10 20 30 T 40 50 60 Speed Kills? Mean path duration Tend depends non-monotonically on θ. 12 N= N= N= N= N= N= 11 Mean T end 10 9 5 6 7 8 9 10 8 7 6 5 4 0 0.1 0.2 0.3 0.4 0.5 Speed 0.6 0.7 0.8 0.9 1 T=1 => Connected Metagraph For any path {v(t)}t=0,1,...,T of 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 rj 1). Other cases are similarly accounted for (-1 = [-1 , . . . , -1]). The Metagraph & Speciation N=5 N=6 N=7 2 1.5 1 10 Mean log (# connected components) 2.5 0.5 0 0 5 10 15 T 20 25 30 Complexity to Speciation Increasing Complexity as measured by duration T leads to a speciation effect. T> =1 Robustness Complexity Trade-off Mutational Robustness decreases with increasing T. = 0.1 80 Mean Metagraph Degree 70 N=5 N=6 N=7 60 50 40 30 20 10 0 0 5 10 T 15 T vs. ρ(Md,Rn) Mutational Robustness and Noise Robustness are positively correlated but the strength of this correlation is T dependent. = 0.05 N=6 0.6 d n <p( M , R )> 0.8 0.4 0.2 0 -0.2 0 2 4 6 8 T 10 12 14 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∞. v(0) v∞ Our definition isolates a single path. v(0) v(T) 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. Future Work & Design Multi-functionality. Paths with Features. Genetic Sensors. Acknowledgements Matthew Turner Complexity DTC EPSRC Questions?