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

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

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

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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.

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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).

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Example GRN p53 Mdm2 network: Example path through the state space: Mdm2p53

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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.

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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.

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Robustness Mutational Robustness (M d ) of a network is its metagraph degree. Noise Robustness (R n ) can be defined as the probability that a change in one genes 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.

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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),

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Example GRN Revisited p53 Mdm2 network: Example path through the state space: Mdm2p53

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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).

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Which Path to Take? Large number of paths for any given N. How to sample? Method 1 (speed ): Choose a [0 1]. Randomly sample an initial condition v(0) S. Then v i (t +1) = v i (t) with a probability 1- 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.

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

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Speed Kills? Mean path duration T end depends non-monotonically on .

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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 : r j {-1, 0, 1}} i be all the row solutions for gene i. Suppose v i (0) = 0 and v i (1) = 1, then h i (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 r j 1). Other cases are similarly accounted for (-1 = [-1,..., -1]).

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The Metagraph & Speciation

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

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

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T vs. (M d,R n ) Mutational Robustness and Noise Robustness are positively correlated but the strength of this correlation is T dependent.

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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. vv v(0) v(T) v(0)

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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, (M d,R n ) and the number of solutions all depend on path complexity. The breakup of the metagraph with increasing complexity is analogous to a speciation effect.

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

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Acknowledgements Matthew Turner Complexity DTC EPSRC Questions?