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.
Slide 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).
Slide 6
Example GRN p53 Mdm2 network: Example path through the state
space: Mdm2p53
Slide 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 . [1] Ciliberti S, Martin OC, Wagner A (2007) PLoS
Comput Biol 3(2): e15.
Slide 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]. [1] Ciliberti S,
Martin OC, Wagner A (2007) PLoS Comput Biol 3(2): e15.
Slide 9
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.
Slide 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]. [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),
Slide 11
Example GRN Revisited p53 Mdm2 network: Example path through
the state space: Mdm2p53
Slide 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).
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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.
Slide 14
Attainability of a Function Increasing duration T exponentially
constrains the topology.
Slide 15
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]).
Slide 17
The Metagraph & Speciation
Slide 18
Complexity to Speciation Increasing Complexity as measured by
duration T leads to a speciation effect. T = 1T > 1
Slide 19
Robustness Complexity Trade-off Mutational Robustness decreases
with increasing T.
Slide 20
T vs. (M d,R n ) Mutational Robustness and Noise Robustness are
positively correlated but the strength of this correlation is T
dependent.
Slide 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. vv v(0)
v(T) v(0)
Slide 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, (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.
Slide 23
Future Work & Design Multi-functionality. Paths with
Features. Genetic Sensors.
Slide 24
Acknowledgements Matthew Turner Complexity DTC EPSRC
Questions?