Layout Search of a Gene Regulatory Network for 3-D Visualization

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1 104 Genome Informatics 14: (2003) Layout Search of a Gene Regulatory Network for 3-D Visualization Naoki Hosoyama 1 Noman Nasimul 2 Hitoshi Iba 2 hosoyama@iba.k.u-tokoy.ac.jp noman@iba.k.u-tokyo.ac.jp iba@iba.k.u-tokyo.ac.jp 1 Department of Electronics Engineering, University of Tokyo, Japan 2 Department of Frontier Informatics, University of Tokyo, Japan Abstract In recent years, base sequences have been increasingly unscrambled through attempts represented by the human genome project. Accordingly, the estimation of the genetic network has been accelerated. However, no definitive method has become available for drawing a large effective graph. This paper proposes a method which allows for coping with an increase in the number of nodes by laying out genes on planes of several layers and then overlapping these planes. This layout involves an optimization problem which requires maximizing the fitness function. To demonstrate the effectiveness of our approach, we show some graphs using actual data on 82 genes and 552 genes. We also describe how to lay out nodes by means of stochastic searches, e.g., stochastic hill-climbing and incremental methods. The experimental results show the superiority and usefulness of two search methods in comparison with the simple random search. Keywords: gene regulatory network, visualization 1 Introduction Recently, the inference method of a gene regulatory network has been rapidly developed. The target network size has become larger and larger with this development. For example, we have to tackle a network of about 500 nodes in typical studies [1, 9]. As a result, the visualization technique of such a large network is necessary so that the whole structure can be grasped at a glance. For the sake of visualization, it may be possible to arrange a network with 20 to 30 nodes in a try-and-error manner. However, it is out of the question to cope with a network of about 500 nodes manually. Therefore, the automatic generation of a clear layout is truly essential. There have been several models proposed for gene regulatory networks, e.g., Boolean network [3], S-system [7], and Bayesian network [5]. In most of these models, the causality relationship between genes is represented by a directed arc. For the visualization of a network with hundreds of nodes, the spring model [6] and the fish-eye lens model [10] are commonly used. In the former model, nodes in a network are arranged in a two-dimensional plane according to the spring dynamics. In the latter model, the emphasis can be put upon a local area for the sake of focusing. However, there are some limitations to these models. For example, the arrangement of so many nodes is difficult due to the space capacity. In addition, although either the whole or a part of a network can be grasped, it is hard to watch a local area while looking over the whole structure. To solve these difficulties, we have proposed a new approach to visualizing a gene regulatory network in a three-dimensional space [4]. Our model has the following salient features for the sake of effective visualization: 1. From biological observation, we can assume that one gene is usually affected by other four to eight genes. In this sense, our model can grasp the characteristics of a gene network, in which each node has between 1 and 8 relations.

2 Layout Search of a Gene Regulatory Network The arrangement of nodes can arbitrarily be widened and the network can be viewed from a variety of directions. 3. The ignored area is displayed semi-transparently. As a result of this, the attention can be paid to some local area without missing the whole image. 4. The overlapping of nodes or arcs is carefully avoided by using the two-dimensional lattice-like arrangement in each layer. In our previous paper, for the sake of deriving a clear arrangement, we have defined a fitness function in terms of the clarity and optimized the function by means of stochastic search methods, i.e., stochastic hill-climbing method and simulated annealing method. In this paper, we introduce a more efficient search method, i.e., an incremental search, and utilize a more elaborate fitness function for the purpose of clearer lay-out. Their search performance is empirically compared and discussed. This paper is organized as follows. The next section gives the definition of a task and related works in this study. After that, Section 3 explains our proposed technique. Section 4 presents an experimental result with the visualization of three different networks. Section 5 discusses results of comparison and future researches. Finally, a conclusion is given. 2 Network Models and Visualization 2.1 Modeling a Genetic Network A number of models have been proposed to represent the causal relations of genes. These models use their own characteristic approaches and handle different volumes of information. For example, the S-System [8] interprets a genetic network as a differential equation system and assumes an inverse problem in which the original parameters are estimated from the time series of expressed data from the DNA micro-array. From the viewpoint of drawing the network, this is roughly classified into 3 types of graphs as shown in Table 1. Table 1: 3 Types of target graphs. Type Correlation graph (undirected graph) Cause-effect graph (direct graph) Weighted graph (in the broad sense) Detail The information about the positive / negative correlation between genes is described. Two related genes are connected with an undirected arc. Describing the relationship caused by a gene acting upon another gene. Causality is represented by a directed arc, whose direction shows the cause and effect. Some qualitative meaning is attached to a graph within its arcs. E.g., S-system or a Bayesian network. This paper addresses the Cause-effect graph, considering the nature of estimated data on the genetic network which can be estimated at present. The given data is assumed to have the following characteristics: The data can include any arc which mutually has a causal relation like A B. The data can have any cyclic structure like A B C A. There can be structural separation like A B C and D E. 2.2 Layout Policy As a visualization example consider the same graph having 6 nodes and 8 arcs shown in different layouts (see Fig. 1).

3 106 Hosoyama et al. Figure 1: Easy sample of useful layout. Figure 2: Spring Model Molecular & model. When the two graphs are compared with each other, the structure on the right is found easier to understand. To make useful layout, following layout techniques are known (see Table 2) [6]. The approach using a dynamic model is known as a method of calculating, at a relatively high speed, a layout which meets these requirements to some extent (see Fig. 2). In this method, we first apply a spring-like model to the arcs. This allows for solving the problem with technique A by coming closer to the natural length and also the technique governing the arcs linking nodes. In addition, we apply a repulsive force model like intermolecular force, between nodes. This allows for applying the technique B for the nodes which are not linked with an arc. The drawbacks of this method include: (i) technique C, i.e., a node overlapping with a different node, which cannot be avoided at a high speed, and (ii) the expensive calculation cost which is required until the values are converged as the number of the nodes increases. Table 2: Useful node layout techniques. Technique Regulation Figure A Minimize the sum of arc lengths. B Maintain at least a given distance between mutually adjacent nodes. C Ensure that no arc overlaps with a different node in any place other than the endpoints. 2.3 Emphasis of the Part of the Graph Consider a network with hundreds of nodes. It may be difficult to command a view of such a large network. Even if we devise a sorted displacement to maintain the equal distance between nodes by using a spring model, we cannot keep nodes or arcs from crowding. To solve this difficulty we can use an interactive system, in which the graph appearance is modified according to the user s command. For this purpose, we display only the focused area and its surrounding relationships. These techniques are expected to be more effective for a network with a greater number of nodes. 3 Proposed Layout Method 3.1 Basic Idea Layout in a 3-dimensional space is also possible. Though with a simple application, you cannot see internal structure from the outside (Fig. 3). Since a large number of nodes cannot be laid out on a plane, improvement to a 3-dimensional presentation is desirable to draw a larger graph than a certain scale.

4 Layout Search of a Gene Regulatory Network 107 Figure 3: Sample 3D layout where the inner structure cannot be seen. Figure 4: Honeycomb graph view of network. Each of the three layers forms a lattice such that all nodes are displaced within a distance of one from its center. The three lattices are overlapped and each shifted by 20 degrees (Fig. 4). This is to avoid overlapping nodes from a particular angle. 3.2 Meaning of Layers This time the nodes and arcs have been divided into 3 layers; please see Table 3 and Figure 5. Table 3: Meaning of 3 layers. Color Behavior Top Green Only functions as a controlling gene. Middle Red Functions as both controlling and controlled genes. Bottom Blue Only functions as a controlled gene. Figure 5: Meaning of 3 layers. Figure 6: Diagram of layers. The direction of the red -> red arc will become difficult to find, but the arcs will be colored in accordance with their orientations to ensure that no orientational information is lost. Each layer consists of a regular hexagonal lattice, which is included within a circle of radius one (see Fig. 6, 7). Let n be the number of nodes in each layer and m be the number of nodes arranged in each hexagonal segment. As you can see from the figure, the larger the value m, the more nodes are included in one layer. In order to derive the most compact structure, we derive the minimum value of m satisfying the following inequalities: 3(m 1)m + 1 < n 3m(m + 1) + 1 (3) 3.3 Evaluation Function The layout problem for a graph is considered an optimization problem, which requires that the scale determined to make the graph easier to view be optimized across the entire graph. In the spring model, the converging point at which the potential decreases in the physical model showed a good score (local solution). In this approach, the score for the spring-like model is referenced to calculate the score for each arc from the arc distance as projected from the top, using equation (4). Variables a and b indicate the nodes connected to both ends of a certain arc. Smaller score functions show better evaluation results.

5 108 Hosoyama et al. Figure 7: All nodes are included in a circle of radius one. This score function is different from the one used in our previous paper [4] and improved in the sense that it results in a more incomprehensible layout. This is because the new fitness function has removed the previous defect that the score gets larger for a distance longer than a certain threshold [4]. s node (a, b) = ( 1 (a x b x ) 2 + (a y b y ) 2 ) 2 ((a x b x ) 2 + (a y b y ) 2 1) 0 ((a x b x ) 2 + (a y b y ) 2 > 1) (4) Basically, this is a problem which requires minimizing the distance. The score remains unchanged after the upper limit of a certain distance has been exceeded, because no significant difference is sensed when the viewed distance is longer than a certain level; and, it is regarded as more important whether or not the network is compactly integrated beyond that limit. The score of the entire network is expressed by equation (5) for the sum of scores. A layout that maximizes this value is desirable. S = s node = 1 s node (a, b) (5) 2 a b linked to a 3.4 Random Search For the simplest method, we first use a random search. Nodes are randomly positioned to derive the overall fitness value. This process is repeated a certain number of times. We report the best score after the repetition. This method cannot use the search history to improve the future search. In addition, it may be very costly to calculate the operation for all nodes in the whole network, e.g., eq.(4). Thus, this method is not desirable for these reasons. 3.5 Stochastic Hill-Climbing Search It is conceivable to swap two nodes positions so as to improve the whole fitness score gradually. This is called a hill-climbing search. More precisely, we follow the process described below: Step1. Initially, displace all nodes randomly. Step2. Choose a lattice point p at random. This p may or may not have an incoming arc. Step3. Choose randomly another lattice point q in the same layer with p. We assume that at least one of these p and q has an incoming arc. Step4. Derive the new score if p and q are swapped. If the score is increased, then swap p and q. Step5. If the search is converged, then stop. Else, go back to Step 2. We calculate the equation (5) in order to derive the score value in Step 5. Note that only two nodes are swapped and the rest of the network remains the same, i.e., the score for the rest also remains the same. Thus, we assume that the score difference is given in eq.(6) when swapping nodes p and q; and, if this difference is greater than 0, then we accept the swapping. S = s node (q, a) + s node (p, a) a linked to p a linked to p s node (p, a) + a linked to q b linked to q s node (q, a) (6)

6 Layout Search of a Gene Regulatory Network 109 Fig. 8 shows this swapping process. In the figure 8a, swapping nodes p and q results in a decrease in the total length of arcs. Since the rest of the network remains the same, this swapping leads to an increase in the score value and is an accepted modification. On the other hand, in figure 8b, when swapping node p and space q, the score decreases. Thus, this swapping is not regarded as a good modification and rejected.!" #!"%$ # Figure 8: Swapping two nodes. Figure 9: incremental search. 3.6 Incremental Search Following the previous studies on graph layout techniques [6], we establish an incremental search method for nodes arrangement. In this search method, all nodes are not positioned initially, but they are incrementally added during the search (see Fig. 9). More precisely, we follow the process described below: Step1. Sort all nodes in a descendant order according to their numbers of connections. Make the sorted list. Step2. Choose one node, i.e., a, from the list. Node a is as a candidate which is put in an appropriate place additionally. Remove node a from the list. Step3. Search for a node b which belongs to the same layer with a, such that the score becomes the greatest when a and b are swapped with each other. Step4. If the score obtained by swapping is greater than the current best score, then swap a and b. And apply the above Step3 recursively to all nodes connected to a and b. Step5. If the search is converged, then stop. Else, go back to Step 2. By using this incremental method, the computational cost is relatively reduced in comparison with the other methods, by which all nodes are placed initially and some of them are swapped. This is because fewer nodes are compared at one time during the search. Thus, we can expect that this will work more efficiently. However, once nodes are swapped, the computational time will increase exponentially. In addition, the search time and solution quality depend on the ordering of the addition. We use a descendant order for the heuristics. 4 Experiment 4.1 Data Sets Two networks are visualized by the proposed approach in Section 3 (Table 4). Data sets A and B are real data inferred by a Table 4: Data sets. Boolean network mode. # of genes # of relations Source We have chosen the score functions given by eqs. A [2] (4) and (5), and evaluated the performances for the B [1, 9] three methods in Section 3.4, 3.5 and 3.6, i.e., random search (rnd), stochastic hill-climbing search (HC) and incremental search.

7 76 % 110 Hosoyama et al. 4.2 Results of Experiments The averaged values of evaluation functions are given in Table 5, and the fitness transition in Figure 10. The data is based on the average over 10 runs. Table 5: Results. Rnd (Time [s]) HC (Time [s]) Inc (Time [s]) A 159.4±1.1 (0:00:29) 256.0±1.0 (0:00:07) 257.5±1.8 (0:00:02) B 10.6±0.2 (0:11:02) 19.3±0.3 (0:07:38) 19.9±0.2 (1:21:41) Random search is much worse than the other 2 methods as it computes only the one which is related to two nodes. Incremental search gave better performance than the other two methods in terms of the obtained scores. However, the longer computational time has been required for the larger network lay-out. ) % % "! # $! 345 ( & % ( % % ' & % ' % % & % & & ( 89: & & ( 8 ;"< = % & % ' ' & % % % ( %*% +, -./ 0. 1*2 Figure 10a: 82 genes. Figure 10b: 552 genes. Figure 10: Results. We can observe the relatively quick convergence in case of data set (A). On the other hand, for the larger set (B), 100,000 steps are not enough to settle down the search, but even the major revision sometimes occurred at the final stage of the search. The convergence time is exponentially increased with the number of nodes and arcs, which shows the significance of the efficient search to solve this computational burden. 4.3 Visualization Result Figure 11a: Side-view. Figure 11b: Top-view. Figure 11: Visualization results of 82-genes. Figures 11a, 11b show the visualization of the whole image for data set (A). Figure 11a is a side view, by which we can see the whole structure is divided into 3 layers. The regulatory direction is clearly shown upward and downward. Each node is represented by a cube, whose volume is proportional to the number of its connections. i.e., the more nodes a node is connected to, the larger the

8 Layout Search of a Gene Regulatory Network 111 cube is. Figure 11b is a top view, by which we can see that there is a cluster in the lower left quadrant. In addition, we can observe that the lower right quadrant is only connected to the upper right by one arc. Figure 13 shows data set (B). In this case it is difficult to grasp the whole structure if all arcs are drawn (Figure 12). Thus, only the focused node and its adjacent area are drawn, by which we can see that relatively neighboring nodes are crowed within a certain distance. Figure 12: 552-genes / all-arcs. Figure 13: 552-genes / arcs form notice gene and in middle layer. 4.4 Number of Steps and Differences of Appearances Figures 14a-d show the transition of graphs drawn for data set (A) when we used the HC method. As can be seen in the figure, as the arc length becomes smaller and smaller, the whole image looks clearer and more ordered. (a) 1st state. (b) 2nd state. (c) 3rd state. (d) 4th state. Figure 14: Iterations of the HC method applied to data set (A). 4.5 Interface A user can manipulate the visualization process with a mouse (and a keyboard or joy pad, if necessary). For instance, he or she can look over the graph from a variety of directions. In addition, the following features are provided for the sake of user-friendly functionalities: Translation and rotation in x and y directions. Resizing, i.e., enlargement and reduction. The focused node and its adjacent nodes with arcs are always displayed. In the left corner, the number of connections to the focused node is shown (see Figure 15).

9 112 Hosoyama et al. Display only a node whose number of connections is greater than a threshold. In other cases, the node is shown semi-transparent. The fourth function is to clarify which node works as a hub. Even the non-focused area is displayed semi-transparent, so that we will not miss the whole structure. For example, nodes with 10 or less connections are semi-transparently displayed in Figure 16. This system has been implemented on a PentiumIII 1.0GHz, Mobility Radeon 16MB. The display speed is about 10 60fps. The software is available from the following URL: giw14.zip Figure 15: List of nodes which have relation with Gene MCD1. 5 Future Works Figure 16: Semi-transparent display. 5.1 Collision Detection Consider the network shown in Figure 8b. This is originally divided into 3 parts. However, we can observe only two of the three. The cause of this problem is explained in Figure 17. Some nodes may happen to be in the middle of an arc when they are connected within the middle layer. Note that this does not happen between nodes in different layers. Figure 17: Nodes marked as x seem to be linked to an upper-left node, but actually do not. In this case, it is very difficult to grasp the graph characteristics. To avoid this difficulty, we are currently working on an extension of the fitness function. For instance, it might be useful to add the penalty such as if an arc penetrates through a node, then the score is set to be zero. 5.2 Optimization in Layout For the future, two types of expansion will be considered: one is to use GA and GP for improved layouts and the other is to further improve the responsiveness so that the software can immediately calculate and visualize the layout when any arbitrary graph structure for causal relations is given. As another improvement, there is the possibility that the present layout on 3-layered planes might be changed to other possible layouts such as a spherical layout so that an easier-to-view graph can be drawn while minimizing the overlapping of nodes.

10 Layout Search of a Gene Regulatory Network Conclusions This paper described the conventional visualization methods and identified their problems. As a solution to these problems, we proposed a presentation technique which combines the 2-dimensional approach with the 3-dimensional one. The proposed interface has been implemented in a current PC, whose operating speed is satisfactorily high. In addition, it has been successfully verified that the proper gene layout and easy-to-view drawing of a network of approximately 500 genes was possible when the hill-climbing method was used. Now that the display speed of 3D objects has become very high and almost the same as 2D objects, the visualization method with an appropriate interface will be developed further in the future. We believe that the visualization of a gene network will not be a final target, but commonly used as a tool in bioinformatics. Acknowledgments This work was partially supported by the Grants-in-Aid for Scientific Research on Priority Areas (C), Genome Information Sciences (No ) from the Ministry of Education, Culture, Sports, Science and Technology in Japan. References [1] Aburatani, S., et al., Discovery of novel transcription control relationships with gene regulatory networks generated from multiple-disruption full genome expression libraries, DNA Research, 10:1 8, [2] Akutsu, T., Miyano, S., and Kuhara, S., Algorithms for inferring qualitative models of biological networks, The Pacific Symposium on Biocomputing, 5: , [3] Akutsu, T., Miyano, S., and Kuhara, S., Identification of genetic networks from a small number of gene expression patterns under the Boolean network model, Pacific Symposium on Biocomputing 99, 17 28, [4] Hosoyama, N. and Iba, H., 3-D Visualization of a gene regulatory network: stochastic search for layouts, Proc. of IEEE Conference on Electronic Commerce, [5] Imoto, S., Goto, T., and Miyano, S., Estimation of genetic networks and functional structures between genes by using Bayesian networks and nonparametric regression, Pacific Symposium on Biocomputing, 7: , [6] Itoh, T., Inoue, K., Doi, J., Kajinaga, Y., and Ikehata, Y., An Improvement of force-directed graph layout method, Information Processing Society of Japan, CG103:2, [7] Sakamoto, E. and Iba, H., Inferring a system of differential equations for a gene regulatory network by using genetic programming, Genome Informatics, 12: , [8] Savageau, M.A., Biochemical Systems analysis: a study of function and design in molecular biology, Addison Wesley Reading, [9] Savoie, C.J., et al., Use of gene networks from full genome microarray libraries to identify functionally relevant drug-affected genes and gene regulation cascades, DNA Research, 10:19 25, [10]