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. -1- 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. -2- 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 Ati1 1 0 i At n when j i At j 0 j i At j 0 j1 n when j1 n when j1 j i (1) At j 0 Where Ati represents the value of the ith node at time step t, and ji 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. -3- 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 -4- 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 -5- 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. 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