Physics Letters A 372 (2008) 6202–6206
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Physics Letters A
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Power-law distribution of gene expression fluctuations
J.C. Nacher
a
,∗
,T.Ochiai
b
a
Department of Complex Systems, Future University-Hakodate, 116-2 Kamedanakano-cho Hakodate, Hokkaido 041-8655, Japan
b
Faculty of Engineering, Toyama Prefectural University, 5180 Kurokawa Imizu-shi, Toyama 939-0398, Japan
article info abstract
Article history:
Received 7 July 2008
Received in revised form 7 August 2008
Accepted 11 August 2008
Availableonline15August2008
Communicated by V.M. Agranovich
Keywords:
Fluctuations
Gene expression
Scaling-law
Stochastic theory
Large-scale genomic technologies has opened new possibilities to infer gene regulatory networks from
time series data. Here, we investigate the relationship between the dynamic information of gene
expression in time series and the underlying network structure. First, our results show that the
distribution of gene expression fluctuations (i.e., standard deviation) follows a power-law. This finding
indicates that while most genes exhibit a relatively low variation in expression level, a few genes are
revealed as highly variable genes. Second, we propose a stochastic model that explains the emergence of
this power-law behavior. The model derives a relationship that connects the standard deviation (variance)
of each node to its degree. In particular, it allows us to identify a global property of the underlying genetic
regulatory network, such as the degree exponent, by only computing dynamic information. This result not
only offers an interesting link to explore the topology of real systems without knowing the real structure
but also supports earlier findings showing that gene networks may follow a scale-free distribution.
©
2008 Elsevier B.V. All rights reserved.
1. Introduction
Breathtaking advances in large-scale genomic technologies, such
as Gene Chips and Microarrays, have made it possible to monitor
mRNA levels (i.e., gene expression) for hundreds of thousands of
genes in parallel. Gene expression analyses have not only become
a widely used method for diagnosis and genetic disorder classi-
fication but also for observing the gene activities through several
periods of time. The existence of these genomic experiments has
opened new exciting avenues to infer gene regulatory networks
from time series analyses [1].
A number of mathematical models have been proposed to
deal with the problem of reverse engineering of genetic networks
[2–4], including powerful Boolean logic approaches [5,6],Bayesian
networks [7,8],graphtheory[9] and differential equations-based
models [10]. Most models aim to reconstruct gene regulatory
networks only based on transcriptional data. Then, the result-
ing network represents an effective regulatory interaction pattern
between genes [11]. Therefore, in these models many molecu-
lar elements associated to the complete transcriptional network
are typically omitted. In several works, correlations between ex-
pression levels of genes computed under a wide range of envi-
ronmental conditions have been routinely performed to construct
edges between genes [12,13,16]. Then, a complex regulatory net-
*
Corresponding author.
E-mail addresses: nacher@fun.ac.jp (J.C. Nacher), ochiai@pu-toyama.ac.jp
(T. Ochiai).
work emerges where each gene (node) is connected to other genes
by regulatory edges that either induce or repress their expression
level.
On the other hand, network analysis to biological and non-
biological systems has evolved rapidly and has contributed to un-
cover and characterize their topological complexity and hetero-
geneity [14–18]. Network science has made it possible to inves-
tigate systems composed of thousands elements and identify the
main mechanisms behind the emergence of these complex net-
worked structures. Power-laws have been found in many biological
and non-biological systems, and the degree exponents have been
classified [19]. While for most systems composed of basic nodes it
is possible to construct networks defined by known relationships
between nodes, this represents a complicated task in some partic-
ular systems where only dynamical data is available, such as, for
example, gene expression profiles and stock price data.
Here, we aim to predict global properties of the underlying
gene regulatory network by examining the fluctuations in gene ex-
pression dynamics. In this work, we use the term fluctuation for
the variations of gene expression. The variance of gene expression
for the active genes can be seen as a deterministic process with
noise where oscillatory patterns are clearly observed. This is partic-
ularly true when a reduced set of well-know genes is considered.
In many datasets, however, several thousands of genes can be in-
vestigated at the same time and only the molecular regulation of
a relatively small proportion of them is well-known. Furthermore,
molecular-level fluctuations in the reaction steps that are intrinsic
to the transcription process should also be considered as a source
of internal noise. In this work, in order to deal with the massive
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©
2008 Elsevier B.V. All rights reserved.
doi:10.1016/j.physleta.2008.08.023