Dynamical Properties of Protein Interaction
Networks in Yeast Nucleus
X. Jia, F.T. Li, and Q. Ouyang
Abstract
A dynamic model based on the protein-protein interaction networks in
yeast is introduced. By a defined evolution function, most states can finally
end their evolving processes with some fixstates. The number of initial states
associated with a given fixstate (Ns) differs from one fixstate to another，and
the preferred fixstates emerge with value of Ns much larger than the average.
The relation between Ns and the number of fixstates evolved from Ns initial
states follows power law. Moreover, two other kinds of networks are also
investigated, and the results indicate that a protein interaction network is more
like a scalefree network than a random one.
1. INTRODUCTION
Proteins play crucial roles in the execution of various biological functions. These
years, along with the development in biotechnology and bioscience, we are now able
to investigate living systems at a molecular level. In 1996, the first complete DNA
sequence of a eukaryotic genome that of the yeast Saccharomyces cerevisiae was
released in electronic form ① ② . From then on, as one of the most important
eukaryotic model organisms, yeast is widely used to study the basic cellular processes.
Recently, some systematic approaches for identifying protein-protein interaction
networks in yeast have been published③, these developments in post-genomic era
have expanded the protein's role, which was considered as an individual before, to an
element in a network of protein interaction. Many authors have reported their research
results about the topological properties of protein interaction networks in
yeast ④⑤⑥⑦ . Nevertheless, they all mainly concentrated on the features about
topology of the networks, such as some characteristics of connection.
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In this paper we take a step in
the direction of a study of the
evolving
process
of
a
protein-protein
interaction
network in yeast nucleus,
providing
a
statistical
characterization, and pointing out
that the possible states of protein
sequence in yeast are not all
equivalent. With this aim, we
introduce and investigate a
dynamic model to simulate the
evolution process for the states of
Fig.1 The protein-protein interaction networks in yeast
proteins. The network (see Fig.1)
nucleus, containing 123 proteins and 323 interactions. A red
is from database of Proteom.com
arrow represents a negative impact, whereas a green one
with 123 proteins and 323
represents the impact is positive.
interactions, which was kindly
supported by A. Snappen. We choose the simplest possible model, in which the
network is simplified into its
major part. The points are valued
as only 0 or 1 to represent
whether this protein is active. The
Subgroup
relations
between
two
coterminous nodes are classified
Subgroup
Main Group
into positive and negative ones.
By analyzing this simple model,
we show that the number of initial
Subgroup
states (Ns) associated with a given
state differs from one fixstate to
(a)
another, and Ns decays as a quasi
power law while the number of
fixstates evolved from Ns initial
states (S) increases, following Ns
~ S-γ.
2. CALCULATING MODEL
Let one node in network be
evaluated by 0 or 1, thus a
possible state of a network
containing n proteins can be
represented by a binary sequence
of n digit bit. As for the original
networks containing 123 nodes,
and 323 edges as well, there are
(b)
Fig.2 (a) The new network after simplifying the original
networks, which can be clearly separated into four parts as
shown above (b) the major part of the origin protein interaction
networks, based on which our simulation is performed.
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2123 possible states to be calculated, which is impossible to finish for our computer.
Therefore, we simplify this network as following steps:
We observe that there exists a kind of nodes which are affected by their adjacent
nodes but have no effect on others (such as the node GAL3 in Fig.1). The first
property of these nodes means that their values can be determined by their
neighboring nodes' values; and the second one shows the feasibility that in
calculations we can delete these nodes as well as the interactions between these nodes
and others. Based on this principle, we find that 42 nodes can be removed from Fig.1,
and then get a new network. Moreover, the operation we do above can be repeated
again and again, and finally we arrive at a much smaller network with only 80 nodes
(see Fig.2 (a)). Furthermore, in this network, we notice that all the 80 nodes can be
artificially separated into some subgroups, and the major one contains 58 nodes. In
this main group, we omit some nodes, which are only equipped with one outward
interaction (such as the node HAP3 in Fig.2 (a)), and consequently we get the major
part of the original network (see Fig.2 (b)) including merely 37 nodes, based on which
our simulation is performed.
The dynamic rule for the states' evolution of this network is defined as follows.
Consider at one time the network is standing on a certain state, at the next time value
of the ith node will update though the rule as
Ati1

1


0

 i
 At

n
when

j i
At j  0

j i
At j  0
j1
n
when
j1
n
when

j1
j i
(1)
At j  0
Where Ati represents the value of the ith node at time step t, and  ji is connection
strength of the link from the jth node to the ith node, following:
 j i
 1

  
 0

when positive interactio n
when negative interactio n
when no interactio n
(2)
Moreover, the sum in (1) is over all the n=37 nodes in the networks, even
self-feedback included.
3. CALCULATING RESULTS FOR YEAST NETWORK
Our simulation starts with initial states chosen randomly from all possible ones. At
each time step all the values of nodes will be updated together based upon the rule we
mentioned above. We found that, corresponding to different initial states, evolution
process will finally end with two different kinds of dynamic results. One possible
termination is a fixstate, i.e. a state that is an invariant under this evolving rule.
Moreover, it is observed that most initial states will terminate in this way, although
the final states they reach may be different. The other termination is limited cycle, i.e.
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the final state will change periodically among a certain number of states.
Our statistical work is mainly about such processes that end with a certain fixstate.
First, we notice that initial state will reach the fixstate fast as this yeast network is
concerned. Fig.3 (a) clearly demonstrates the distribution of total steps for different
initial states to evolve to their terminations. It shows that about 60 percent of all initial
states will take a typical number of
steps of 3, and all evolution
processes will finish within no more
than 6 steps.
Furthermore, we completely
enumerate the fixstates from all 237
possible initial states, showing that
1441172 invariable fixstates exist.
Then we randomly choose 2×107
initial states and let them evolve to
the finial fixstates according to the
dynamic rule. Finally, we obtain all
possible states associated with each
given fixstate, which means they
have that special fixstate as their
Fig.3 The distribution of evolution step for one random
unique fixstate. Thus the number of
initial state to reach its corresponding final fixstate in three
initial states (Ns) associated with a
kinds of networks (a) Protein interaction networks in yeast
given fixstate is quite a good
nucleus; (b) Scale-free networks; (c) Random networks.
representation of the attraction basin
of that fixstate.
It is remarkable that all fixstates differ between each other in terms of their
attraction basins. There are a few special fixstates that are the ones corresponded with
quite a number of initial states, and there are also some ''poor'' fixstates that can attract
only a few or even no initial states. For instance, as far as our data is concerned, the
average
number
of
states
associated
with
one
fixstate
Ns
is
N s =2×107/1441172≈14 , however, the most popular fixstate can attract 1297
different initial states of all 2×107 (Ns =1297 >> N s ), and there are also 650275
fixstates for which Ns=0. Moreover, the number of fixstates with a given Ns value
decreases almost monotonously as Ns increases, although with regular periodic small
fluctuation for some unknown reasons (Fig. 4 (a)). Linear fit implies that this
distribution satisfies the power law relation of Ns ~ S-γ, where γ=2.024. This
distribution differs markedly from Poisson distribution, which would be present if all
the fixstates are statistically equivalent.
4. COMPARING WITH SCALE-FREE AND RANDOM NETWORKS
In order to understand whether this quasi power law distribution and evolution
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property are unique for yeast networks, we have also investigated these dynamic
properties in two other most important networks today: scale-free and random
networks. We artificially produced several new networks of these kinds, keeping the
number of nodes the same as that of protein interaction network in yeast, as well as
the number of negative and positive interactions.
Fig.3 (b) (c) illustrates the differences in terms of evolution steps among them. It
is shown that, in the case of scale-free network, the number of time steps that one
certain initial state takes to reach its
corresponding fixstate is nearly the
same as that in yeast network, see Fig.3
(b). However, in random network, this
value is generally a little larger than
that in the former two kinds. For
example, some initial states even take 8
steps during the whole evolving process,
and the most probable evolution step is
4, both of which are somewhat more
than the corresponding values in
protein interaction networks.
There are also other differences
among these three kinds of networks.
Fig.4 (b) (c) demonstrates typical
characteristic of the relation between Ns
and the number of fixstates associated
Fig.4 The number of states with a given Ns in three
with a given Ns for the scale-free and
kinds of networks, (a) Protein interaction networks in
random networks. Compared with Fig.4
yeast nucleus; (b) Scale-free networks; (c) Random
(a), it is understandable that the
networks.
common feather among those three
distributions lies in the monotonously downward trend, and there also exist some
special states with Ns >> N s , which contribute to the long tail of the curve. However,
for the scale-free networks, this distribution does not satisfy the power law relation
any more; and for random networks, what differs from that distribution of protein
network consists in the fact that the data points in the long tail are separated for an
obvious reason of statistical fluctuation.
5. CONCLUSION AND DISCUSSION
In this paper we introduce a dynamic model and investigate the dynamic
properties of states in protein-protein interaction networks in yeast, and the two other
kinds of networks as well. Though the model is quite simple, it shows a number of
nontrivial behaviors. By performing the simulation of the evolution process based
upon such a simple model, we have shown that all possible fixstates in protein
networks are not equivalent, and some special fixstates seem more popular than others,
which suggests that there must be a principle that plays a crucial role in the selection
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of functional states for yeast. Besides, compared with scale-free and random networks,
our simulation implies that this special property of protein networks mainly results
from the unique topological structure. A protein interaction network is more like a
scale-free network, rather than a random one.
However, as mentioned in this letter, there is yet little known about the regular
periodic fluctuation in Fig.4 (a), moreover, at which aspect would a real protein
interaction network differs from other kinds of networks still requires more detailed
study.
6. ACKNOWLEDGEMENTS
The authors acknowledge the support of Tai-Zhao scholarship fund.
Reference

Electronic address: qi@mail.phy.pku.edu.cn
① Saccharomyces Genome Database (SGD) at http://genome-www.stanford.edu/Saccharomyces/;
Yeast Genome from MIPS (Martinsried Institute for Protein Sequences) at
http://speedy.mips.biochem.mpg.de/mips/yeast/; Yeast Protein Database (YPD) at
http://www.proteome.com/YPDhome.html
② A. Goffeau et al., Science, vol. 274, 546 (1996)
③ A. H. Y. Tong et al., Science, vol. 295, 321 (2002)
④ Benno Schwikowski, Peter Uetz, and Stanley Fields, Nature Biotechnology, vol. 18, 1257
(2000)
⑤ Tony R. Hazbun, and Stanley Fields, PNAS, vol. 98, 4227 (2001)
⑥ H. Jeong, S. P. Maso, A. L. Barabasi, and Z. N. Oltvai, Nature, vol. 411, 41 (2001)
⑦ Sergei Maslov, and Kim Sneppen, Science, vol. 296, 910 (2002)
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