3-D Visualization of a Gene Regulatory Network: Stochastic Search for Layouts

Transcription

1 3-D Viualization of a Gene Regulatory Network: Stochatic Search for Layout Naoki Hooyama Department of Electronic Engineering, Univerity of Tokyo, Japan hooyama@iba.k.u-tokyo.ac.jp Abtract- In recent year, bae equence have been increaingly uncrambled through attempt repreented by the human genome project. Accordingly, the etimation of the genetic network ha been accelerated. However, no definitive method ha become available for drawing a large effective graph. Thi paper propoe a method which allow for coping with an increae in the number of by laying out gene on plane of everal layer and then overlapping thee plane. Thi layout involve an optimization problem which require maximizing the fitne function. To demontrate the effectivene of our approach, we how ome graph uing actual data on 82 gene, 552 gene, and artificial data modeled from a cale-free network of 1,000 gene. We alo decribe how to lay out by mean of tochatic earche, e.g., tochatic hill-climbing and imulating annealing method. The experimental reult how the uperiority and uefulne of tochatic earche in comparion with the imple random earch. 1 Introduction Recently, the inference method of a gene regulatory network ha been rapidly developed. The target network ize ha become larger and larger with thi development. For example, we have to tackle a network of about 500 in typical tudie [Savoie03] [Aburatani03]. A a reult, the viualization technique of uch a large network i neceary o that the whole tructure can be graped at a glance. For the ake of viualization, it may be poible to arrange a network with 20 to 30 in a try-and-error manner. However, it i out of the quetion to cope with a network of about 500 manually. Therefore, the automatic generation of a clear layout i truly eential. There have been everal model propoed for gene regulatory network, e.g., boolean network [Akutu99], S-ytem [Sakamoto00], and bayeian network [Imoto02]. In mot of thee model, the cauality relationhip between gene i repreented by a directed arc. For the viualization of a network with hundred of, the pring model [Itoh00] and the fih-eye len model [gcope] are commonly ued. In the former model, in a network are arranged in a two-dimenional plane according to the pring dynamic. In the latter model, the emphai can be put upon a local area for the Hitohi Iba Department of Frontier Informatic, Univerity of Tokyo, Japan iba@iba.k.u-tokyo.ac.jp ake of focuing. However, there are ome limitation to thee model. For example, the arrangement of o many i difficult due to the pace capacity. In addition, although either the whole or a part of a network can be graped, it i hard to watch a local area while looking over the whole tructure. To olve thee difficultie, we propoe a new approach to viualizing a gene regulatory network in a threedimenional pace. Our model ha the following alient feature for the ake of effective viualization: 1. The arrangement of can arbitrarily be widened and the network can be viewed from a variety of direction. 2. The ignored area i diplayed emi-tranparently. A a reult of thi, the attention can be paid to ome local area without miing the whole image. 3. The overlapping of or arc i carefully avoided by uing the two-dimenional lattice-like arrangement in each layer. In order to derive a clear arrangement, we define a fitne function in term of the clarity and optimize the function by mean of tochatic earch method. We ue a random earch, a tochatic hill-climbing earch, and a imulated annealing earch. Their earch performance i empirically compared and dicued. Thi paper i organized a follow. The next ection give the definition of a tak and related work in thi tudy. After that, Section 3 explain our propoed technique. Section 4 preent an experimental reult with the viualization of three different network. Section 5 dicue reult of comparion and future reearche. Finally, a concluion i given. 2 Network Model and Viualization 2.1 Modeling a genetic network A number of model have been propoed to repreent the caual relation of gene. Thee model ue their own characteritic approache and handle different volume of information. For example, the S-Sytem [Savageau76] interpret a genetic network a a differential equation ytem and aume an invere problem in which the original parameter are etimated from the time erie of expreed data from the DNA micro-array. From the viewpoint of drawing the network, thi i roughly claified into 3 type of graph a hown in Table 1.

2 Table 1. 3 Type of target graph. Type Detail Correlation graph The information about the poitive / negative correlation between gene i (indirect decribed. Two related gene are graph) connected with an undirected arc. Caue-effect Decribing the relationhip caued by a graph gene acting upon another gene. (direct graph) Cauality i repreented by a directed arc, whoe direction how the caue Weighted graph (in the broad ene) and effect. Some qualitative meaning i attached to a graph within it arc. E.g., S-ytem or a Bayeian network. Thi paper addree the Caue-effect graph, conidering the nature of etimated data on the genetic network which can be etimated at preent. The given data i aumed to have the following characteritic: The data can include any arc which mutually ha a caual relation like A B. The data can have any cyclic tructure like A B C A. There can be tructural eparation like A B C and D E. 2.2 Layout policy A a viualization example conider the ame graph having 6 and 8 arc hown in different layout (ee Fig. 1). Figure 1: Eay ample of ueful layout. When the two graph are compared with each other, the tructure on the right i found eaier to undertand [Itoh00]. Table2: Ueful layout technique. Regulation Figure A Minimize the um of arc length. B Maintain at leat a given ditance between mutually adjacent. C Enure that no arc overlap with a different in any place other than the endpoint. 2.3 Spring model and molecular model The approach uing a dynamic model i known a a method of calculating, at a relatively high peed, a layout which meet thee requirement to ome extent (ee Fig. 2). In thi method, we firt apply a pring-like model to the arc. Thi allow for olving the problem with technique (Table 2.A) by coming cloer to the natural length and alo the technique (Table 2.B) governing the arc linking. In addition, we apply a repulive force model like intermolecular force, between. Thi allow for applying the technique (Table 2.B) for the which are not linked with an arc.. Figure 2: Spring Model & Molecular model The drawback of thi method include: (i) technique (Table 2.C), i.e., a overlapping with a different, which cannot be avoided at a high peed, and (ii) the expenive calculation cot which i required until the value are converged a the number of the increae. 2.4 Emphai of the part of the graph Conider a network with hundred of. It may be difficult to command a view of uch a large network. Even if we devie a orted diplacement to maintain the equal ditance between by uing a pring model, we cannot keep or arc from crowding. To olve thi difficulty we can ue an interactive ytem, in which the graph appearance i modified according to the uer' command. For thi purpoe, we diplay only the focued area and it urrounding relationhip. Thee technique are expected to be more effective for a network with a greater number of. Among them the fih-eye len method i very popular, by which only everal and their connection are enlarged and diplayed. GScope [gcope] i one uch application for the purpoe of diplaying the caual relationhip between gene. For example, it how Yeat bacteria' gene relationhip available from DNA microarray data, i.e., 417 protein and 564 caual relationhip between them. When you diplay all 417 in one creen at the ame time, you cannot read even a gene name, uch a "YGL127C". However, when you chooe the enlargement mode, then full name of everal gene are readable, which help undertand the relationhip between gene. In thi method, the following equation i ued for the ake of tranlating the ditance x from the view point to obtain the new ditance h(x): d + 1 h ( x) = (0 x 1), d + 1/ x h'( x = 0) d (1)...(2)

3 where d i a magnifying coefficient, i.e., the peripheral area around the view point i enlarged d+1 time (ee eq.(2)). Note that the angle from the view point i kept fixed for thi operation. In addition, if x i greater than 1, then no tranformation i carried out. To follow thi rule, all ditance are normalized at firt o that the ditance of the furthermot point from the view hould alway be maller than 1. When we ue the fih-eye len method, the central part i enlarged. On the other hand, the peripheral area become ditorted and reduced for the acrifice of thi enlargement. Fig. 3 how thi ditortion for the lattice. ditance between can never be decreaed beyond a certain level. In addition, thi approach minimize the frequency of vicinity between and arc and the overlapping between arc, which are laid out motly in mutually kew poition. Thu, all the condition in technique (A), (B), and (C) lited in Table 2 can be eaily atified without applying them. The internal layer can be viewed through a gap from an upper layer (Figure 5). A rotation allow you to change your viewpoint, e.g., to a lower layer. Thi offer the capability of complying with an increae in the number of that can be laid out in proportion to the number of layer. However, 2 to 4 layer are conidered deirable becaue the central tructure, in particular arc, will otherwie become difficult to view. Figure 3: Fih-eye len function (d=4). There are everal other method, uch a uing expanion/contraction [Shiozawa00] or diplaying only the urrounding of a elected [touch]. However, all the above method have ome difficultie with howing a large network, epecially in a two-dimenional diplay. 3 Propoing layout method 3.1 Baic idea Layout in a 3-dimenional pace i alo poible. Although a imple application will prevent you from interpreting the internal tructure from the outide becaue are denely located at the center (Fig. 4). Since a large number of cannot be laid out on a plane, improvement to a 3-dimenional preentation i deirable to draw a larger graph than a certain cale. Figure 5: Honeycomb graph view of network. Each of the three layer form a lattice uch that all are diplaced within a ditance of one from it center. The three lattice are overlapped and each hifted by 20 degree. Thi i to avoid overlapping from a particular angle. 3.2 Meaning of layer Thi time the and arc have been divided into 3 layer; pleae ee Table 3 and Figure 6. Table 3: Meaning of 3 layer. Color Behavior Top Green Only function a a controlling gene. Middle Red Function a both controlling and controlled gene. Bottom Blue Only function a a controlled gene. Figure 4: Sample 3D layout where the inner tructure cannot be een. In order to olve thee problem, a pace coniting of everal layer of plane i aumed with the layer laid out like a honeycomb. In thi approach, ince the coordinate which can be laid out like a honeycomb are previouly pecified, the Figure 6: Meaning of 3 layer. The red layer i equivalent to the backbone of the network that ha both an entrance and an exit, the green layer i equivalent to the prerequiite, and the blue layer i equivalent to the reult. Thi layout order make it poible to find at a glance the orientation of caual relation a green -> red -> blue and upper -> lower (Figure 7).

4 The core of the entire network i expreed by equation (5) for the um of core. A layout that maximize thi value i deirable. Figure 7: Diagram of layer. S = = 2 1 a b linked to a ( a, b)...(5) The direction of the red -> red arc will become difficult to find, but the arc will be colored in accordance with their orientation to enure that no orientational information i lot. Each layer conit of a regular hexagonal lattice, which i included within a circle of radiu one (ee Fig. 7). Let n be the number of in each layer and m be the number of arranged in each hexagonal egment. A you can ee from the figure, the larger the value m, the more are included in one layer. In order to derive the mot compact tructure, we derive the minimum value of m atifying the following inequalitie: 3 ( m 1) m + 1 < n 3m( m + 1) (3) Figure 8: All are included in a circle of radiu one. 3.3 Evaluation function The layout problem for a graph i conidered an optimization problem, which require that the cale determined to make the graph eaier to view be optimized acro the entire graph. In the pring model, the converging point at which the potential decreae in the phyical model howed a good core (local olution). In thi approach, the core for the pring-like model i referenced to calculate the core for each arc from the arc ditance a projected from the top, uing equation (4). Variable "a" and "b" indicate the connected to both end of a certain arc. Smaller core function how better evaluation reult. ( a, b) = (1...(4) ( a x 2 b ) + ( a x y b ) Baically, thi i a problem which require minimizing the ditance. The core remain unchanged after the upper limit of a certain ditance ha been exceeded, becaue no ignificant difference i ened when the viewed ditance i longer than a certain level; and, it i regarded a more important whether or not the network i compactly integrated beyond that limit. y 2 ) Random earch For the implet method, we firt ue a random earch. Node are randomly poitioned to derive the overall fitne value. Thi proce i repeated a certain number of time. We report the bet core after the repetition. Thi method cannot ue the earch hitory to improve the future earch. In addition, it may be very cotly to calculate the operation for all in the whole network, e.g., eq.(4). Thu, thi method i not deirable for thee reaon. 3.5 Stochatic hill-climbing earch It i conceivable to wap two poition o a to improve the whole fitne core gradually. Thi i called a hill-climbing earch. More preciely, we follow the proce decribed below: Step 1. Initially, diplace all randomly. Step 2. Chooe a lattice point p at random. Thi p may or may not have an incoming arc. Step 3. Chooe randomly another lattice point q in the ame layer with p. We aume that at leat one of thee p and q ha an incoming arc. Step 4. Derive the new core if p and q are wapped. If the core i increaed, then wap p and q. Step 5. If the earch i converged, then top. Ele, go back to Step 2. We calculate the equation (5) in order to derive the core value in Step 5. Note that only two are wapped and the ret of the network remain the ame, i.e., the core for the ret alo remain the ame. Thu, we aume that the core difference i given in eq.(6) when wapping p and q; and, if thi difference i greater than 0, then we accept the wapping. S = a linked to p a linked to p ( q, a) + ( p, a) a linked to q b linked to q ( p, a) ( q, a)...(6) Fig.9 how thi wapping proce. In the figure 9a, wapping p and q reult in a decreae in the total length of arc. Since the ret of the network remain the ame, thi wapping lead to an increae in the core value and i an accepted modification. On the other hand, in figure 9b, when wapping p and pace q, the core decreae. Thu, thi wapping i not regarded a a good modification and rejected.

5 We have choen the core function given by eq. (4) and (5), and evaluated the performance for the three method in ection 3.4, 3.5 and 3.6, i.e., random earch (rnd), tochatic hill-climbing earch (HC) and imulated annealing earch (SA). The number of tep and SA parameter are given in Table 5. Table 5: Parameter. # of tep SA parameter A 50,000 T=10, k=0.995 B 500,000 T=10, k= C 500,000 T=10, k= Figure 9: Swapping two. 3.6 Simulated-annealing earch The imple hill-climbing method often fall into a local optima. Thu, we ue another tochatic method, i.e., imulated-annealing earch. For thi purpoe, Step 4. in ection 3.5 i revied a follow: Step 4a. Derive the new core if p and q are wapped. Generate a random number r between 0 and 1. If S / T r e hold for the core difference S, then wap p and q. Step 4b. The temperature parameter i updated according to the following equation: T kt ( 0 < k < 1) Note that in the original Step 4 of ection 3.5, a imple direction to decreae the length i choen. By thi new method, we wap p and q if the core value increae with thi wapping. In addition, even if the core value decreae, the wapping i carried out with a certain probability. The probability i modified o that the higher the temperature T, the more often the wapping i accepted. The temperature i decreaing in a geometric progreion. In the end, the earch become equivalent to imple hill-climbing. 4 Experiment 4.1 Data et Three network are viualized by the propoed approach in Section 3 (Table 4). Table 4: Data Set. # of gene # of relation Source A [Akutu00] B [Savoie03][ Aburatani03 ] C (imulation) Data et A and B are real data inferred by a Boolean network mode, wherea data et C i generated by imulating a cale-free network [Albert00] extended to a directed graph. In general, a gene i conidered to be affected by 4 to 8 other gene. Thu, we et the average number of affector to be Reult of experiment. The averaged value of evaluation function are given in Table 6, and the fitne tranition in Figure 10. The data i baed on the average over 20 run. Table 6: Reult. rnd HC SA A B (1112) C (1156) Random earch i much lower than the other 2 method a it compute only the one which i related to two. Specifically in 6.B and 6.C, the peed difference i more than 100 time. Therefore, in thee experiment, we computed only tep and did not repeat it. From thi earch, we readily infered that good reult could not be achieved even if it wa allowed to continue. SA howed maller fitne increaer at the earlier tage of the earch, but it eventually reulted in a good performance. Epecially in the cae of 6.B and 6.C, where SA final fitne value i a little higher than HC. However, SA ued an additional operation, i.e., logarithmic calculation. In addition, the graph with SA and HC do not look o different. Thu, the uperiority of SA to HC remain to be een. We can oberve the relatively quick convergence in cae of data et (A). On the other hand, for the larger et (B) and (C), 100,000 tep are not enough to ettle down the earch, but even the major reviion ometime occurred at the final tage of the earch. The convergence time i exponentially increaed with the number of and arc, which how the ignificance of the efficient earch to olve thi computational burden. 4.3 Viualization Reult Figure 11a, 11b how the viualization of the whole image for data et (A). Figure 10a i a ide view, by which we can ee the whole tructure i divided into 3 layer. The regulatory direction i clearly hown upward and downward. Each i repreented by a cube, whoe volume i proportional to the number of it connection. i.e., the more a i connected to, the larger the cube i. Figure 10b i a top view, by which we can ee that there i a cluter in the lower left quadrant. In addition, we can oberve that the lower right quadrant i only connected to the upper right by one arc.

6 A Figure 11a: ide-view. Reult of 82 Gene. B Figure 11b: top-view. C Reult of 552 Gene. Reult of 1000 Gene. Figure 10: Reult. Graph Data et (B) and (C) are drawn in the ame way (Figure 13 and 14 repectively). In thee cae it i difficult to grap the whole tructure if all arc are drawn (Figure 12). Thu, only the focued and it adjacent area are drawn, by which we can ee that relatively neighboring are crowed within a certain ditance.

7 Figure 11: Viualization Reult of 82-gene. Figure12: 552-gene / all-arc. (a) 1 t tate. Step=0, Score=24.5 (b) 2 nd tate. Step=100, Score=38.0 (c) 3 rd tate. Step=500, Score=50.0 (d) 4 th tate. Step=2000, Score=57.7 Figure 15: Iteration of the HC method applied to data et (A). Figure 13: 552-gene / arc form notice gene and in middle layer. Figure 14: 1000-gene / arc from notice gene. 4.4 Number of tep and difference of appearance Figure 15a-d how the tranition of graph drawn for data et (A) when we ued the HC method. A can be een in the figure, a the arc length became maller and maller, the whole image look clearer and more ordered. 4.5 Interface A uer can manipulate the viualization proce with a moue (and a keyboard or joy pad, if neceary). For intance, he or he can look over the graph from a variety of direction. In addition, the following feature are provided for the ake of uer-friendly functionalitie: Tranlation and rotation in x and y direction Reizing, i.e., enlargement and reduction. If a uer chooe a certain, then the attention i focued on it. The focued and it adjacent with arc are alway diplayed. In the left corner, the number of connection to the focued i hown (ee Figure 16). Diplay only a whoe number of connection i greater than a threhold. In other cae, the i hown emi-tranparent. The fourth function i to clarify which work a a hub. Even the non-focued area i diplayed emitranparent, o that we will not mi the whole tructure. For example, with 10 or le connection are emitranparently diplayed in Figure 17. Thi ytem ha been implemented on a PentiumIII 1.0GHz, Mobility Radeon 16MB. The diplay peed i about 10~60fp. The oftware i available from the following URL:

8 Figure 16: Lit of which have relation with Gene MCD Optimization in layout For the future, two type of expanion will be conidered: one i to ue GA and GP for improved layout and the other i to further improve the reponivene o that the oftware can immediately calculate and viualize the layout when any arbitrary graph tructure for caual relation i given. A another improvement, there i the poibility that the preent layout on 3-layered plane might be changed to other poible layout uch a a pherical layout o that an eaier-to-view graph can be drawn while minimizing the overlapping of. 6 Concluion Figure 17: Semi-tranparent diplay. 5 Future work 5.1 Colliion detection Conider the network hown in Figure 10b. Thi i originally divided into 3 part. However, we can oberve only two of the three. The caue of thi problem i explained in Figure 18. Some may happen to be in the middle of an arc when they are connected within the middle layer. Note that thi doe not happen between in different layer. Figure 18: Node marked a x eem to be linked to an upper-left, but actually do not. In thi cae, it i very difficult to grap the graph characteritic. To avoid thi difficulty, we are currently working on an extenion of the fitne function. For intance, it might be ueful to add the penalty uch a if an arc penetrate through a, then the core i et to be zero. Thi paper decribed the conventional viualization method and identified their problem. A a olution to thee problem, we propoed a preentation technique which combine the 2-dimenional approach with the 3- dimenional one. The propoed interface ha been implemented in a current PC, whoe operating peed i atifactorily high. In addition, it ha been uccefully verified that the proper gene layout and eay-to-view drawing of a network of approximately 1,000 gene wa poible when the hill-climbing and the imulated annealing method were ued. Now that the diplay peed of 3D object ha become very high and almot the ame a 2D object, the viualization method with an appropriate interface will be developed further in the future. We believe that the viualization of a gene network will not be a final target, but commonly ued a a tool in bioinformatic. Acknowledgement Thi work wa partially upported by the Grant-in-Aid for Scientific Reearch on Priority Area (C), Genome Information Science (No ) from the Minitry of Education, Culture, Sport, Science and Technology in Japan. Bibliography [Aburatani03] S. ABURATANI et al., Dicovery of Novel Trancription Control Relationhip with Gene Regulatory Network Generated from Multiple-diruption Full Genome Expreion Librarie, DNA Reearch 10, 1-8 (2003) [Akutu99] T.Akutu et al, Identification of genetic network from a mall number of gene expreion pattern under the Boolean network model, Pacific Sympoium on Biocomputing 99, (1999). [Akutu00] Akutu, Miyano, and Kuhara, Algorithm for inferring qualitative model of biological network, The Pacific Sympoium on Biocomputing, 5: (2000). [Albert00] Re ka Albert et al, Error and attack tolerance of complex network, NATURE 406, (2000) [gcope] GScope,

9 [Imoto02] Imoto et al, Etimation of genetic network and functional tructure between gene by uing Bayeian network and nonparametric regreion, The Pacific Sympoium on Biocomputing, 7: (2002). [Itoh00] Takayuki Itoh et al, An Improvement of Forcedirected Graph Layout Method, Information Proceing Society of Japan, 2001-CG-103 (2001). [Sakamoto01] E,Sakamoto and H.Iba, Inferring a Sytem of Differential Equation for a Gene Regulatory Network by uing Genetic Programming, Genome Informatic 2001, (2001). [Savageau76] M.A.Savageau, Biochemical Sytem analyi: a tudy of function and deign in molecular biology, Addion Weley Reading (1976). [Savoie03] C.J. SAVOIE et al., Ue of Gene Network from Full Genome Microarray Librarie to Identify Functionally Relevant Drug-affected Gene and Gene Regulation Cacade, DNA Reearch 10, (2003). [Shiozawa00] Hidekazu SHIOZAWA et al, The Natto View: An Architecture for Interactive Information Viualization, Information Proceing Society of Japan Journal Abtract, Vol.38 No (2000). [touch] TouchGraph,