Functionality & Speciation in Boolean Networks
Jamie Luo (Warwick University Centre for Complexity Science)
Dr Matthew Turner (Warwick University Department of Physics and Centre for Complexity Science)
email: J.X.Luo@warwick.ac.uk
1. INTRODUCTION:
2. MODEL & METHODOLOGY:
Genetic regulatory networks (GRNs) form complex dynamical systems consisting of many interacting genes. We
investigate how the functional capacity of a GRN scales with the number of genes. We also explore the trade-off
between increasing functional complexity and innovation.
The use of Boolean Networks (BNs) to model GRNs was first introduced by Kauffman with a focus on how the
dynamics of Random BNs were affected by topology. Recent studies by A. Wagner et al [1] and others have focused
on the sub-class of Threshold BNs as a biologically relevant model for GRNs.
An N-size Threshold BN is a discrete time, discrete space Markovian dynamical system over the state space S={0,1}^N
and a matrix A ∈ {-1, 0, 1}^N . For any v(0) ∈ S , let h(t)=Av(t). Then the dynamics of the system are defined by:
Example N=2 Threshold BN:
Gene
1
Gene
1
ABSTRACTION
Gene
2
Gene
3
http://www.pnas.org/cgi/content-nw/full/104/31/12890/F2
5. DISCUSSION & FURTHER WORK:
Our approach has been to focus on the
importance of complete trajectories over the
start and end point approach adopted by
Wagner and others. We then use the length
T as a proxy measure for the complexity of a
function/trajectory .
By considering trajectories we garner new
insights into to the resulting metagraph
structure of trajectories. Furthermore we can
classify trajectories (eg. different θ) and have
demonstrated considerable variation in
attainability for different types of trajectory.
We intend to expand this analysis to
trajectories with different features (limit
cycles etc.) and to multiple trajectories
(multi-functionality.)
Validate the speciation hypothesis presented
in (4) using evolutionary algorithms inspired
by a trajectory focused approach.
4. THE METAGRAPH & SPECIATION:
Gene
2
For a fixed system size N we generate a sample trajectory through the state space { v(t) }t=0,1,...,T . A trajectory represents
a biological function for which we then find all matrices A (systems) which map v(t) to v(t+1) for all t=0,...,T-1. The
number of these matrices, call it |A| which satisfy a trajectory indicate how biologically attainable such a function is.
Results are presented below on how |A| scales with the length (complexity) T of a trajectory and the speed of the
trajectory θ.
3. ATTAINABLILITY OF DIFFERENT TRAJECTORY CLASSES:
Trajectories { v(t) }t=0,1,...,T are generated by sampling a v(0) ∈ S at random and then sampling each subsequent v(t) by
flipping each node of v(t-1) with a probability θ.
Example { v(t) }t=0,1,2 :
1
0
Many
Connected
Components
SPECIATION
One Large
Connected
Component
Increasing Complexity (Length T)
If A , B ∈ {-1, 0, 1}^N then we can define the distance
between these two matrices d(A,B) as the sum of the number
of entries which are different. If d(A,B)=1 then a single change
in one interaction (point mutation) can result in A becoming B
and vice versa. Any set {A} can then form a network which is
traversable by point mutations. As this is a graph of graphs we
shall refer to it as a metagraph (aka the neutral network).
Wagner et al [1] focused on finding those {A} which will map
from some v(0) to a fixed point v∞ eventually. They found the
resulting metagraph to largely comprise of one connected
component.
However in our approach by fixing the trajectory { v(t) }t=0,1,...,T
we find that as T increases it is highly probable that the
metagraph breaks up into isolated connected components. To
the left is plotted the mean log10 of the number of connected
components MC with increasing T for a thousand sample
trajectories. It is clear that there is typically more than one
connected component as T grows. Systems on these islands
can no longer reach other islands through point mutations,
suggesting that speciation may arise as systems attain more
complex functionality.
1
1
For a fixed speed θ all system sizes suffer an exponential
decrease in the mean number of matrices A which satisfy
the generated trajectories as they increase in length T.
Thus achieving linear increases in functional complexity
is exponentially costly as depicted in the plot on the right
for θ=0.1 .
This reduction in the number of solutions also leads to a
reduction in the mutational robustness of the matrices.
The mutational robustness of a matrix A is defined as the
number of matrices adjacent to it in its metagraph (4). So
more complex organisms are less robust to mutations.
As θ is varied the mean length T at which there are no
more matrices A which satisfy the trajectory varies as
depicted (right). Thus the speed at which a trajectory
traverses the state space affects its attainability.
Both the size of the state space |S|=2^N and the space
of matrices |A|=3^N2 grow at least exponentially with N.
However the mean length of attainable trajectories is
growing only approximately linearly with N.
References:
[1] S. Ciliberti, O. C. Martin and A. Wagner, (2007) ‘Innovation and robustness in complex regulatory gene networks.’
Proc. Natl. Acad. Sci. USA 104: 13591–13596.