Functionality & Speciation in Boolean Networks

Jamie Luo Warwick Complexity DTC

Dr Matthew Turner Warwick Physics & Systems BiologyFunctionality & Speciation in Boolean NetworksModel Genetic Regulatory Networks using Boolean Networks.Motivation + Approach + ModelPrevious Work A. Wagner and others results + functionality definitionResults based on reformulated definition of functionality (insilico)Discuss Outlook1

http://www.cs.uiuc.edu/homes/sinhas/work.htmlGene Regulatory Networks2Gene Regulatory Networks

http://www.pnas.org/cgi/content-nw/full/104/31/12890/F2Not our Afghanistan Strategy.Non-linear lots of feedback noise.

http://www.pnas.org/cgi/content-nw/full/104/31/12890/F2Angiogenic signalling network. A gene regulatory network constructed from inversely regulated proangiogenic genes. All presented genes are down-regulated after endostatin although up-regulated after VEGF/bFGF treatment (except APC gene; arrow demonstrates opposite regulation). The direction of gene regulation and the high degree of cooperative networking between the selected genes point to a switchable angiogenic network. The concerted up-regulation of the network genes indicates the proangiogenic state (On). Highlighted are gene interactions based on promotor-binding site (green connection lines), protein modification (yellow connection lines), proteinprotein binding (violet connection lines), gene expression (blue connection lines), and gene regulation (black connection lines). Two signalling pathways, STAT3 (yellow circles) and PPAR/-catenin (red shadows), are highlighted and demonstrate the interconnectedness of the pathways within the angiogenic network.

3Why 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.Why use BNs to model GRNs ? - GRNs have many components, non-linear, abstraction of BNs allows for qualitative insightsClassical RBNs Kauffman in his 1963 book The Origins of Order RBNs Markovian system over {0 1}^N Finite space and -> attractorQ: how does topology influence the distribution of attractors etc.Q: Degree or number of edges influence dynamics all update rules. Derrida results (analytical) k=2 is CRITICAL.People still do this work simulation based.Fangting Li (Beijing) et al 2004 yeast cell cycle; Expanded - Hao Ge stochastic variation; Stefan Bornholdt built similar models for fission yeast; Konstantin Klemm (Gunnar Boldhaus)Drosophila (model variant) dynamics of early developmental genes & in predicting mutant phenotypesBoolean Networks as a basis for Network Inference ModelsInverse QuestionAndreas Wagner et al (Klemm)Intuition in Biology is Tricky Scale, Experimental Limitations (observations & questions one can ask)4Threshold DynamicsN-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).

Genes are on/off.Matrix edges - +ve/-ve up/down regulatingNon-linearityAbstract, simple, reflection of GRNs.Stripped out more complex downstream interactions, all produced proteins are TFs or are discarded.5Example GRNp53 Mdm2 network:

Example path through the state space:

Mdm2p53

p53 (also known as protein 53 or tumor protein 53), is a tumor suppressor protein that in humans is encoded by the TP53 geneExplain matrix + graphActual p53 pathway / GRN is more complex but this is a minimalistic model for it.(0,0) is an attracting state always.

Hao Ge stochastic version of the above.6Biological FunctionalityDefine 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.Defn proposed by Andreas Wagner Phenotype Genotype mapping.Finding As complete enumeration + sampling or monte carlo7Metagraph (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.

d = the number of different entriesd(A,B)=1 point mutationsConnectedness point mutations neutral allow any network to find another for most accessible genotypesCore insilico result from which a variety of conclusions are built upon.Used to conclude that robustness is an evolvable quality (through phenotype neutral evolutions)What does one mean by robustness?8RobustnessMutational Robustness (Md) of a network is its metagraph degree.

Noise Robustness (Rn) 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.

Point Mutations.Four defns of noise robustness all highly correlated.Spearmans s = 0.7, p Connected MetagraphFor 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]).Also applies to {v(0), v(1), v(1)}

16The Metagraph & Speciation

Method 2 (matrix) generates longer trajectories (T inconsistent)1000 for N=5 +6, 100 for N=7 trajs.Mc number of metagraph components.Mc can vary over orders of magnitudes log10. But size of components is comparable [Figure for that].Speciation effect.Reconcile these two results from different functional definitions? Ensemble versus path effect17Complexity to SpeciationIncreasing Complexity as measured by duration T leads to a speciation effect.T = 1T > 1More complex organisms speciate.

Opposite result implied by Wagners work.

Where does this leave robustness?

Is robustness evolvable?

35 mins18Robustness Complexity Trade-offMutational Robustness decreases with increasing T.

Mutational Robustness is inherently constrained by the path complexity/duration.Mean Variance in robustness is not much more than 10.19T vs. (Md,Rn)Mutational Robustness and Noise Robustness are positively correlated but the strength of this correlation is T dependent.

Path recovery definition of noise.20Ensemble vs. PathThe 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(0)v(T)v(0)Metagraph:We know the T=1 path is connected backbone for other connections.Also shorter paths have exponentially more solutions and are also much more probabilistically likely to be attainable than longer ones.Mutational Robustness:Sample over entire ensemble then a broad distribution of robustness is likely.T vs p(Md,Rn):Not so sure.21SummaryA 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, (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.

Robustness is an evolvable property [REF]. The metagraph being connected and evolvability of robust networks may be a general organizational principle [REF].Long-term innovation can only emerge in the presence of the robustness caused by a connected metagraph [REF].Conclusions rely on a largely connected metagraph.Metagraph Islands [REF].

Metagraph is connected by gradual mutations.General organizational principle applies to RNA and protein structures.Innovation phenotype change.Need to travel long distances to access all phenotypes and this can only be done on a large metagraph. Large Diameter of Neutral Networks is crucial.Gunnar Boldhaus work on the Yeast Cell Cycle trajectory.22Future Work & DesignMulti-functionality.

Paths with Features.

Genetic Sensors.Parallels with bi-functionality in Wagners work.Path duration is a better variable.Our results are consistent with sum of path lengths.L.c.s, stars etc...

Design : Wagner design of a robust gene networkGenetic sensors sensors built from the same components as the underlying dynamical system23AcknowledgementsMatthew Turner

Complexity DTC

EPSRC

Questions?

Thank You all for listening24