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?