Design and statistical properties of robust functional networks:
A model study of biological signal transduction
Pablo Kaluza,
1
Mads Ipsen,
2
Martin Vingron,
3
and Alexander S. Mikhailov
1
1
Department of Physical Chemistry, Fritz Haber Institute of the Max Planck Society, Faradayweg 4-6, D-14195 Berlin, Germany
2
Department of Chemistry, H. C. Ørsted Institute, University of Copenhagen, Universitetsparken 5, Copenhagen Ø2100, Denmark
3
Department of Computational Molecular Biology, Max Planck Institute for Molecular Genetics, Ihnestrasse 63-73,
D-14195 Berlin, Germany
Received 10 March 2006; published 19 January 2007
A simple flow network model of biological signal transduction is investigated. Networks with prescribed
signal processing functions, robust against random node or link removals, are designed through an evolutionary
optimization process. Statistical properties of large ensembles of such networks, including their characteristic
motif distributions, are determined. Our analysis suggests that robustness against link removals plays the
principal role in the architecture of real signal transduction networks and developmental genetic transcription
networks.
DOI: 10.1103/PhysRevE.75.015101 PACS numbers: 89.75.Hc, 89.20.a, 89.75.Fb
Cells in a biological organism function in a stable, precise
way despite noise and destructive mutations 1. If the prin-
ciples of biological robustness were understood, they could
be further applied for the design of complex industrial pro-
duction and transportation systems, or for understanding so-
cial processes 2,3. The cells operate as dynamical networks
and their robustness is, to a large extent, determined by a
special network architecture. This has been demonstrated for
various biological functions, including chemotaxis 4, me-
tabolism 5, signal transduction 6, and the cell cycle 7. It
was shown that genetic networks with robust expression pat-
terns may spontaneously develop through biological evolu-
tion 8. The problems of robustness have also been dis-
cussed in an abstract context for large random networks,
aimed at developing optimal defense strategies for the Inter-
net and the WWW 9–11. The networks of a living cell are
not random. They are selected by biological evolution to
execute certain functions. In particular, a cell should activate
a fixed group of genes in response to each arriving stimulus.
These networks should maintain their prescribed, specific
functions, although possibly exposed to random damage or
parameter variations. The structure of such networks reflects
their functions. Is it possible, by adjustment of the network
structure, to develop systems with prescribed functions that
are, furthermore, robust against damage? How strongly
would the requirements of robustness against a particular
kind of damage affect their architecture?
In this paper, we study a toy flow model of biological
signal transduction. The network transports signals, applied
to input nodes, through a number of middle redistribution
nodes to a set of output nodes. In a cell, the analogy would
be to a particular set of genes that are turned on upon arrival
of a certain stimulus at the cell surface. This mapping be-
tween input stimulus and output gene activity is mediated
by a network of interactions among proteins in the cell.
These proteins are modeled as nodes in our networks, while
interactions between them are reflected in the existence of
links. Physically, a signal from the cell surface is passed on
through processes like protein phosphorylation or dephos-
phorylation, translocation, structural change, etc. In a gross
oversimplification of the real processes, we model this signal
transduction process by an abstract network flow. Proteins
undergo mutations which in turn can affect the links among
them or completely delete some nodes and introduce new
ones. Thus the network topology is subject to random local
changes.
By running an optimization process with structural muta-
tions and subsequent selection, we show that networks with
predefined output patterns can be constructed. Then, we ex-
tend the optimization criterion and design networks which,
while approximately retaining a fixed output pattern, become
robust against removal of randomly chosen links or nodes.
Statistical properties of robust functional networks, for an
ensemble of different optimization trajectories starting with
various initial conditions, are considered and distributions of
structural motifs in two kinds of networks, robust against
link or node removals, are then determined.
A considered network of size N = N
in
+ M + N
out
consists of
N
in
input nodes, M middle nodes, and N
out
output nodes. Its
architecture is specified by a directed graph of connections
between the nodes with the adjacency matrix A
ij
we have
A
ij
=1, if there is a link from node j to node i, and A
ij
=0
otherwise. An input node can be connected only with the
middle nodes, a middle node can be connected with other
middle nodes and with the output nodes see Fig. 1. Each
link j → i carries some signal flux u
ij
. The sum of all incom-
ing fluxes for any node is equal to the sum of all outgoing
fluxes. For any node, all outgoing fluxes are equal in inten-
sity and are obtained by splitting the total incoming signal
flux in equal parts between the outgoing connections. Thus
we have u
ik
=
l
A
lk
-1
j
A
kj
u
kj
for any node k. Introducing
the total fluxes x
i
=
j
A
ji
u
ji
passing through nodes i, this re-
distribution law can also be written as x
i
=
j
A
ij
x
j
k
A
kj
-1
for i =1,2,... N. External fluxes can be applied to the input
nodes and sinks are attached to the output nodes. An external
unit flux x
=1, applied to an input node =1,2,..., N
in
,
becomes distributed after passing through the network and
fractions x
= Q
of the applied flux reach different output
nodes =1,2,..., N
out
. The matrix Q with the elements Q
represents the output pattern of a given network. Note that
Q
= 1. The performance FG of a given network G is its
output pattern Q, i.e., FG = Q. The ideal performance of a
PHYSICAL REVIEW E 75, 015101R2007
RAPID COMMUNICATIONS
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