1. Functionality & Speciation in Boolean Networks
Jamie Luo
Warwick Complexity DTC
Dr Matthew Turner
Warwick Physics & Systems Biology
Model Genetic Regulatory Networks using Boolean Networks.
Motivation + Approach + Model
Previous Work – A. Wagner and others – results + functionality definition
Results – based on reformulated definition of functionality (insilico)
Discuss Outlook

2. Gene Regulatory Networks

3. Gene Regulatory Networks
Not our Afghanistan Strategy.
Non-linear – lots of feedback noise.
Angiogenic 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), protein–protein 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.

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.
Why use BNs to model GRNs ? - GRNs have many components, non-linear, abstraction of BNs allows for qualitative insights
Classical RBNs Kauffman in his 1963 book ‘The Origins of Order’ RBNs – Markovian system over {0 1}^N – Finite space and -> attractor
Q: 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 phenotypes
Boolean Networks as a basis for Network Inference Models
Inverse Question
Andreas Wagner et al (Klemm)
Intuition in Biology is Tricky – Scale, Experimental Limitations (observations & questions one can ask)

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).
Genes are on/off.
Matrix – edges - +ve/-ve – up/down regulating
Non-linearity
Abstract, simple, reflection of GRNs.
Stripped out more complex downstream interactions, all produced proteins are TFs or are discarded.

6. Example GRN p53 – Mdm2 network: Example path through the state space:
p53 (also known as protein 53 or tumor protein 53), is a tumor suppressor protein that in humans is encoded by the TP53 gene
Explain matrix + graph
Actual 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.

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∞.
Defn proposed by Andreas Wagner – Phenotype – Genotype mapping.
Finding A’s complete enumeration + sampling or monte carlo
[1] Ciliberti S, Martin OC, Wagner A (2007) PLoS Comput Biol 3(2): e15.

8. 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].
d = the number of different entries
d(A,B)=1 – point mutations
Connectedness – point mutations – neutral – allow any network to find another for most accessible genotypes
Core 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?
[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.
Point Mutations.
Four defns of noise robustness all highly correlated.
Spearman’s s = 0.7, p<10^-15 10^3 sampled matrices.
Add Figure to supplements.

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].
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 path. Li – identified a main path through the state space = cell cycle (T=12/13).
Metagraph was found for this space and it was not connected – comprised O(10^8) components of size 10^24 – 10^26
GB – hypothesised that increasing number of constraints may be responsible for Metagraph disconnectedness.
22mins
[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
[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
How suitable is the start-end defn.
Consider the example path.
p53 (also known as protein 53 or tumor protein 53), is a tumor suppressor protein that in humans is encoded by the TP53 gene
Explain matrix + graph
Actual 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.

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,...,T to define a function.
Crucially distinguish paths by duration T (complexity).
Duration analogous to biological complexity of the function.

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- 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.
(2^N)!/(2^N-k)!
Method 1 – analogous to setting a speed or binomial distribution for the number of flips for any time step (more true sampling method)
Method 2 – natural method defined by the space and is useful for generating longer paths.

14. Attainability of a Function
Increasing duration T exponentially constrains the topology.
Method 2 – also true for method 1
Number of Matrices – proxy measurement for how biologically realisable a function is.
Not an unexpected result as any loss in the number of rows leads to an exponential loss

15. Speed Kills? Mean path duration Tend depends non-monotonically on θ.
T_end exists because one eventually makes a move which has no solutions.
Number of flips is not the only factor in determining whether a solution is attainable.
Refractory period positively correlated with T_end.
N= (1000trajs) N=9, 10 (100trajs)
O(N) or O(N^2) increase in T_end with N.
Return to the Metagraph

16. 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]).
Also applies to {v(0), v(1), v(1)}

17. The 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 effect

18. Complexity to Speciation
Increasing Complexity as measured by duration T leads to a speciation effect.
T > 1
T = 1
More complex organisms speciate.
Opposite result implied by Wagner’s work.
Where does this leave robustness?
Is robustness evolvable?
35 mins

19. Robustness Complexity Trade-off
Mutational 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.

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

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∞
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.
v(T)
v(0)

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.
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.

23. Future Work & Design Multi-functionality. Paths with Features.
Genetic Sensors.
Parallels with bi-functionality in Wagner’s 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 network
Genetic sensors – sensors built from the same components as the underlying dynamical system

24. Acknowledgements Matthew Turner Complexity DTC EPSRC Questions?
Thank You all for listening