Cross talking of network motifs in gene regulation that generates temporal pulses and spatial stripes

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1 Cross talking of network motifs in gene regulation that Oxford, GTC Genes ? 0 Original S 2005 Ishihara Blackwell talking to U Article Cells Publishing, et al. of network Ltd. Ltd motifs generates temporal pulses and spatial stripes Shuji Ishihara, oichi Fujimoto,2 and Tatsuo Shibata 3, * Department of Pure and Applied Sciences, University of Tokyo, omaba, Meguro-ku, Tokyo , Japan 2 ERATO, Complex Systems Biology Project, JST 3 Department of Mathematical and Life Sciences, University of iroshima, -3-, agamiyama, igashi-iroshima, , Japan Gene regulatory networks contain several substructures called network motifs, which frequently exist throughout the networks. One of such motifs found in Escherichia coli, Saccharomyces cerevisiae, and Drosophila melanogaster is the feed-forward loop, in which an effector regulates its target by a direct regulatory interaction and an indirect interaction mediated by another gene product. ere, we theoretically analyze the behavior of networks that contain feed-forward loops cross talking to each other. In response to levels of the effecter, such networks can generate multiple rise-and-fall temporal expression profiles and spatial stripes, which are typically observed in developmental processes. The mechanism to generate these responses reveals the way of inferring the regulatory pathways from experimental results. Our database study of gene regulatory networks indicates that most feed-forward loops actually cross talk. We discuss how the feed-forward loops and their cross talks can play important roles in morphogenesis. Introduction As more information about genetic and biochemical interactions in cells becomes available, it provides us the overall structure of molecular interactions. When molecular interactions involved in a gene regulation are represented by arrows, and gene products by nodes, the structure of the gene regulation is described by a network. In this abstraction, many different regulatory mechanisms are reduced to a common representation. The question is how to understand and predict the functioning of a system from the structure of its molecular interaction networks. From this viewpoint, Shen-Orr and colleagues recently found several types of small-scale structures that frequently appear in many regulatory networks (Shen-Orr et al. 2002), and named network motifs composed of several nodes and arrows. One of the most prominent network motifs is the feed-forward loop (Shen-Orr et al. 2002), which consists of three gene products and three transcriptional interactions, as shown as X, Y, and Z, and, 2, and 3, respectively in Fig. A,B. Transcription factor X regulates both Y and Z, and Y regulates Z. Thus, X Communicated by: Nobuo Shimamoto *Correspondence: shibata@hiroshima-u.ac.jp modulates Z expression by two pathways, a direct pathway (X Z), and an indirect pathway mediated by Y (X Y Z). The effect of a transcriptional interaction is either positive (activation) or negative (repression). Therefore, the overall effect of the indirect pathway mediated by Y on Z depends on the regulatory effects of interactions and 3. The effects of X and Y on Z are integrated at the promoter region of the gene that encodes Z. The dependence of the level of Z on the concentrations of X and Y is described by a cis-regulatory function of Z (Buchler et al. 2003; Setty et al. 2003). In this approach, before studying behaviors of a particular network containing a huge number of genes and interactions, functions of the motifs are studied independent of particular contexts. So far, several functions of a single feed-forward loop have been studied theoretically and experimentally (Shen-Orr et al. 2002; Mangan & Alon 2003; Mangan et al. 2003; Basu et al. 2004). It has also been discussed that these functions play roles in the transcriptional programming in the processes of skeletal myogenesis (Penn et al. 2004), flagella biosynthesis in Escherichia coli (alir & Alon 2004), and sporulation in Bacillus subtilis (Eichenberger et al. 2004). The network motifs do not often represent independent units that are functionally isolated from the rest of the DOI: 0./j x Blackwell Publishing Limited Genes to Cells (2005) 0,

2 S Ishihara et al. are responsible for generating temporally multiple riseand-fall expression pattern and spatially multiple stripes. It is then shown that the cross talks are found in the regulatory networks of E. coli, Saccharomyces cerevisiae, Strongylocentrotus purpuratus, and Drosophila melanogaster. Results Figure The feed-forward loop network motif. (A) Transcription factor X regulates both gene products Y and Z, and Y regulates the expression of gene product Z. Each transcriptional interaction, 2, and 3 is positive control (activation) or negative control (repression). Interactions 2 and 3 are integrated in the promoter region of the gene encoding Z (see Procedures). (B) The network representation of the feed-forward loop. network. Several network motifs interact or cross talk with each other to form a complex structure. The functioning of the network may be cooperatively generated by the cross talks of motifs as a kind of collective behavior. In this respect, the network motifs can be considered as building blocks of a network. So far, a few reports have appeared on how network motifs combine to form a larger structure (Dobrin et al. 2004; ashtan et al. 2004; Yeger-Lotem et al. 2004). Their attentions are mainly on the topological aspects of the network. But more attention should be paid on functional aspects that cross talks between motifs give rise to (ashtan et al. 2004). In this paper, we focused our attention to this most prominent network motif, the feed-forward loop. At first, the stimulus response profile of single feed-forward loop is analyzed. Secondly, we show that cross talking of feed-forward loops can give rise to a new feature of response that a single network motif alone cannot produce. These cross talks between the feed-forward loops Bell-shaped response of single feed-forward loop We consider the stimulus response profile of single feed-forward loop as the dependence of Z expression on the level of transcription factor X. The feed-forward loops are classified into two types. When both direct and indirect pathways from X to Z are either positive or negative (same effects or synergistic), it is called coherent feedforward loop; when the effects of both pathways are both positive and negative (opposing effects or antagonistic), it is called incoherent feed-forward loop (Shen-Orr et al. 2002). In a coherent feed-forward loop, the level of Z monotonically increases or decreases as the level of X increases, whereas in an incoherent feed-forward loop, it shows nonmonotonic and bell-shaped (convex or concave) responses (Fig. 2). In the example shown in Fig. 2A, an incoherent feedforward loop is composed of positive control (activation) for interactions and 3, and negative control (repression) for interaction 2. The expression of Y is activated by X, otherwise the level of Y is maintained to be negligible. The contribution of interactions and 3 mediated by Y to the level of Z is displayed by the blue broken curve, and the contribution of interaction 2 as the red curve in Fig. 2A. For the cis-regulatory function of Z, consider the function shown in Fig. 2B, where the level of Z is elevated when X exists at a low concentration and Y at a high concentration. Because Y is controlled by X, the level of Z changes along the red curve in Fig. 2B, and the overall dependence of the level of Z on X exhibits a bellshaped (convex) profile as shown in Fig. 2A. The expression profile of Z can depend on the cisregulation function of Z. Let us consider a case when the level of Z is activated by either X or Y (Fig. 2D). If interaction is negative and interactions 2 and 3 are positive (Fig. 2C), the net effect of indirect pathway is negative as displayed in the red curve, and the direct regulatory one in the blue curve. Because Y is negatively controlled by X, the level of Z changes along the red curve in Fig. 2D. Therefore, the level of Z again exhibits a bell-shaped (concave) profile (Fig. 2C). As shown in these examples, the bell-shaped (convex or concave) profile with a single extreme point is a general and robust characteristic of incoherent feed-forward loops. The condition to obtain 026 Genes to Cells (2005) 0, Blackwell Publishing Limited

3 Cross talking of network motifs Figure 2 The bell-shaped (convex and concave) response of a single incoherent feed-forward loop. (A), (C) The bell-shaped (convex and concave) response profiles of the networks, respectively. The level of Z is plotted as a function of the concentration of X (solid curve). The regulatory activities of the direct pathway (red curve), and the indirect regulatory pathway mediated by Y (blue curve) are plotted as function of X. In this paper, all axes are displayed as relative scale with arbitrary unit, which does not mean the absolute level. For the mathematical expressions, see Procedures section. ( B), (D) The activities of cis-regulatory function used in the networks shown in (A) and (C) plotted as a function of the levels of X and Y. The concentration of Z changes along the red curve when Y is controlled by X in the respective feed-forward loop. a bell-shaped response is that an activation threshold should be smaller (larger) than a repression threshold if the level of Z is not activated (is activated) when X product is absent (see Appendix for mathematical condition to obtain bell-shaped responses). Cross talk of feed-forward loops generates a response with multiple peaks When two network motifs share the same gene product (node), the two motifs are considered to be interacting or cross talking. Between two feed-forward loops, 2 types of interactions are possible (not shown). Among them, we found that five networks (Fig. 3A,C,E) exhibit stimulus response profiles with target gene expressions that are activated within two intervals of stimulus concentration, whereas the rest cannot show such a response. The network shown in Fig. 3A consists of two feed-forward loops connected in series. The output of the upstream feed-forward loop, Z, is the input for the downstream. In the upstream feed-forward loop, the level of Z exhibits a bell-shaped profile against X (the dotted curve in Fig. 3B) as shown in Fig. 2A. Therefore, the profile of Z 2 has one peak when the level of Z has an increasing profile from zero to maximum, and then again another peak appears when it has a decreasing profile. As a result, the overall dependence of the level of Z 2 on the level of X exhibits a double-peak profile (Fig. 3B solid line). This doubling in the number of peaks is a characteristic of incoherent feed-forward loops connected in series. When n is the number of feed-forward loops connected in series, the number of peaks with activation can increase at most as 2 n. The second example shown in Fig. 3C is also a network made up of two feed-forward loops. In this case, Blackwell Publishing Limited Genes to Cells (2005) 0,

4 S Ishihara et al. Figure 3 Networks made up of several feed-forward loops and their response with multiple peaks. (A) A network of two incoherent feed-forward loops connected in series, made up of five gene products X, Y, Z, Y 2, and Z 2 and six interactions. (B) The response of Z 2 with double peaks (solid line) and the bell-shaped response of Z (dotted line) plotted as functions of X concentration. (C) A network of two incoherent feed-forward loops connected in parallel, made up of four gene products X, Y, Z, and Z 2, and five interactions. The concentration profile of Z is plotted. (D) The response of Z 2 with multiple peaks and the bell-shaped response of Z. ( E) Three networks made up of four gene products and five interactions can also exhibit responses with multiple peaks. Note that the different combinations of activation and repression for the interactions are possible to obtain the same type of response. two products, Y and Z, and the pathway between them are shared by both feed-forward loops. This network also shows a response with multiple peaks. For the cis-regulatory input functions of the genes encoding Z and Z 2, we adopt the same type of function as is shown in Fig. 2C. In this network, X modulates the Z 2 expression along three pathways, the two positive pathways on Z 2 (X Y Z 2, and X Z Z 2 ), and the other negative pathway (X Y Z Z 2 ). Each pathway introduces a threshold of activation or repression of Z 2 in the concentration of X. ere, the threshold is defined as a concentration of X at which the concentration of Z 2 is half of an extreme value. These thresholds determine the two intervals where the Z 2 expression is elevated (Fig. 3D). The networks shown in Fig. 3E are the same type as the network shown in Fig. 3C, in which two products and one pathway are shared by both the feed-forward loops. In these networks, the Z expression can be elevated also within two intervals in the concentration of X. When a feed-forward loop is added to a network in parallel, a new threshold of activation or repression of Z is introduced in the concentration of X so that the response of Z to the level of X exhibits a profile that is more complex. A stimulus response profile with a single or with multiple peaks is possible, if two or more thresholds of activation and repression exist in the stimulus concentration. Each threshold corresponds to one direct or indirect pathway 028 Genes to Cells (2005) 0, Blackwell Publishing Limited

5 Cross talking of network motifs from a regulatory gene to a target gene. ere, for simplicity we consider the case when a single pathway determines a single threshold value (a more complicated case, such a pathway which introduces several thresholds, may be possible when a cis-regulatory region is made up of two or more modules). The numbers of direct and indirect pathways from the most upstream regulatory factor to the most downstream target gene in a single feed-forward loop (Fig. 2), the series type (Fig. 3A), and the parallel type networks (Fig. 3C,E) are two, four, and three, respectively. Thus, these networks have two, four, and three thresholds, respectively, which determine intervals where the level of target gene expression is elevated. Note that the examples shown here are not special kinds of networks that exhibit responses with multiple peaks. The multiple-peaks response is a robust property with respect to parameter variations, assuming the thresholds of activation and repression are ordered appropriately. For instance, in the case of Fig. 3D, the repression threshold must be set between the two activation thresholds (see Appendix for mathematical condition to obtain responses with a single and multiple peaks bounded by sharp thresholds). Inferring regulatory pathways from multiple-peak responses Based on this correspondence between a pathway and a threshold, one can infer regulatory pathways from experimental data. Once a response with multiple-peak profile has been obtained, one can estimate the minimum number of pathways of activation and repression between the two components, which is the number of thresholds in the stimulus response profile. For instance, if a database about interactions between genes and components is available, one may infer the pathways that are responsible for the responses, and may indicate that there should be unknown components if the number of known pathways is smaller than the expected value. It is also suggested that if a response with multiple peaks performs a function, it is expected that many feed-forward loops are involved in the network. Feed-forward loops work as concentration detectors In a network composed of one or more feed-forward loops, when a stimulus concentration is lying within an interval between thresholds of activation and repression, the target gene expression is activated; otherwise, the gene expression is repressed. Thus, the network can work as a kind of concentration detector. Such a concentration detection mechanism can perform various kinds of temporal and spatial information processing functions. One biological function is temporal pulse generation (Basu et al. 2004). When the signal to the network increases with time, the target gene expression is transiently elevated when the signal concentration is lying between the thresholds. As an example, we studied temporal response of a network that consists of three feedforward loops connected in a series (Fig. 4A). Consider the case where the level of X is 0 at time t = 0, and then X increases exponentially with time until it reaches a saturating concentration X = X f. In Fig. 4B, the time course of the levels of X, Z, Z 2, and Z 3 is plotted (see Methods for the equations describing this temporal evolution of expression levels). As the level of X increases with time, genes Z, Z 2, and Z 3 exhibit one, two, and four temporal pulses, respectively. Another function of the concentration detection mechanism can be a spatial pattern formation. When a signal component is distributed spatially over cells and forms a gradient, it can serve as a morphogen, which provides positional information (Wolpert 969). The network of incoherent feed-forward loops in each cell can detect the positional information by responding to the morphogen concentration. As a result, the target gene is expressed within particular spatial regions. For example, we consider spatial response of the network shown in Fig. 4A. Suppose that transcription factor X forms an exponential gradient (Fig. 4C). Such gradient is obtained if X is expressed only at the position, l = 0, and X effectively diffuses in space and is depleted at a constant rate. Because the response profiles of Z, Z 2, and Z 3 exhibit one, two, and four peaks respectively, as the concentration of X increases, the products of Z, Z 2, and Z 3 spatially form single, double, and quadruple stripes, respectively, as shown in Fig. 4C. In this way, a network of feed-forward loops can generate spatial stripes from a gradient of a stimulus. If a temporal rise-and-fall profile or spatial stripe pattern is obtained experimentally, a regulatory factor may exist in the upstream components, which shows temporally or spatially a monotonic increasing profile of the concentration. For instance, in Fig. 4B,C, the upstream gene X exhibits a monotonic increasing profile, whereas Z, Z 2, and Z 3 exhibit temporal rise-and-fall profiles and spatial stripes patterns, respectively. From these temporal and spatial profiles, the stimulus response profiles can be reconstructed by plotting the concentrations of Z, Z 2, and Z 3 against the level of X for each time point, and each location, respectively. From this reconstruction of the stimulus response profile, the network responsible for generating these patterns can be inferred, if it is not known completely. Blackwell Publishing Limited Genes to Cells (2005) 0,

6 S Ishihara et al. Figure 4 Spatial stripe formation by the incoherent feed-forward loops. (A) Left: The network consisting of three feedforward loops connected in series. Right: The dependence of the expression levels of genes Z, Z 2, and Z 3 plotted as a function of the level of X. (B) The temporal expression profile plotted as a function of time. Initially, the concentration of X is zero, and then increases exponentially until it reaches a saturating value. The expression profiles of Z, Z 2, and Z 3 show complex rise-and-fall patterns temporally. (C) The stripe patterns formed by the network plotted as a function of position, l. The concentration of X exhibits an exponential gradient and the expression profiles of Z, Z 2, and Z 3 form single, double, and quadruple stripes, respectively. Table Statistics on the presence of feed-forward loops (FFLs) in the gene-regulatory networks of E. coli, S. cerevisiae, S. purpuratus, and D. melanogaster (see Procedures) Appearance of FFL in real networks (appearance in randomized network (mean ± s.d.)) Appearance of incoherent FFLs (FFLs that can be distinguished between coherent and incoherent) The frequency of FFLs that share genes and regulations with other FFLs E. coli 42 (8. ± 3.2) 6 (36) 0.20 S. cerevisiae 27 (40.0 ± 7.4) 6 (27) 0.08 S. purpuratus 20 (5.3 ± 3.0) 5 (20) 0.35 D. melanogaster 57 (03.7 ±.) 43 (83) 0.37 Frequency of cross talks and possibility of multiple-peak responses in transcriptional regulatory networks The question now arises whether feed-forward loops form cross talks and the series and parallel types of interactions actually exist in real transcriptional regulatory networks. To investigate this, we performed a database analysis of the regulatory networks in E. coli, S. cerevisiae, S.purpuratus and D. melanogaster. In these four species, the number of feed-forward loops is much larger than the expected values of randomized networks (Table ), indicating that the loop is indeed considered as a network motif. ere, for the calculation of the expected value, see Procedures. The feed-forward loops found in the data sets of E. coli and 030 Genes to Cells (2005) 0, Blackwell Publishing Limited

7 Cross talking of network motifs Figure 5 The networks of feed-forward loops (FFLs) in E. coli (A) and D. melanogaster (B). The feed-forward loops found in the database of the species. The single-role genes X (red), Y (yellow), and Z (blue), and the genes with several different roles (white) are indicated. The interactions are indicated as arrows with positive control (blue), negative control (red), and unknown (black). D. melanogaster, shown in Fig. 5, are interacting and cross talking (Dobrin et al. 2004, ashtan et al. 2004). The number of feed-forward loops that are separated from the others is only one in E. coli, and none in S. cerevisiae, S. purpuratus, and D. melanogaster. We enumerated the frequency with which two loops share genes in the four species. The frequency is defined as the ratio between the number of pairs that share genes and the number of possible combination of any two loops. The frequency tends to increase with the genome size (Table ). While most feed-forward loops are cross talking in these four species, the structural characteristics of the networks in E. coli and D. melanogaster seem to be quite different (see Fig. 5). To see this difference, we study the distribution of gene products playing each role of X, Y, and Z in feed-forward loops. It has been reported that in E. coli, a small number of X regulates a large number of Y and Z (red, yellow, and blue bars in Figs. 6A and 5A) (alir & Alon 2004; ashtan et al. 2004). In most cases, products Y and Z are not shared between two feed-forward loops. In S. cerevisiae, S. purpuratus, and D. melanogaster, such tendency is not found. In particular, in D. melanogaster, the loops form more complex structure (see Fig. 5B). The cross talk can cause a product to play different roles of X, Y, and Z in different feed-forward loops. Such a product is typically found in the series and parallel networks of the loops. For example, in the network shown in Fig. 3A, Z plays both roles X and Z, and in the network in Fig. 3C, Y and Z play both roles X and Y, and Y and Z, respectively. Because 43% of the gene products in the loops play different roles in D. melanogaster, the tendency to form these types of network is most distinct. owever, in E. coli, only 5% of the genes are those with different roles. In Fig. 6B, the numbers of the series and the parallel type networks are shown (Fig. 6B). No seriestype network is found in E. coli. In S. cerevisiae, and D. melanogaster, both series and parallel types are found. We enumerated the frequency of the five types of interactions shown in Fig. 3A,C,E among the 2 types of possible interactions between two loops. Whereas in E. coli, the frequency is approximately 3%, in S. cerevisiae and D. melanogaster, the frequencies are 2% and 6%, respectively. Because these five types can exhibit a response with multiple peaks and the rest cannot, such a response can be found more possibly in S. cerevisiae and D. melanogaster than in E. coli. Discussion To develop a concept that network motifs are building blocks of a network, we showed how interactions and cross talks of feed-forward loops, which is the most prominent network motif, give rise to a new feature in stimulus response profile. ere, we consider that elements of a network are the motifs rather than individual gene products, which are then elements of the motifs. A qualitative prediction of the function of a network could Blackwell Publishing Limited Genes to Cells (2005) 0,

8 S Ishihara et al. Figure 6 Statistics of interactions between feed-forward loops (FFLs). (A) The numbers of genes playing each of the roles of X (red), Y (yellow), and Z (blue). The numbers of genes playing both X and Y (orange), Y and Z (green), Z and X (dark blue), and X, Y, and Z (purple). The fraction of genes playing several different roles (red line). (B) The numbers and frequencies of serial feed-forward loop networks (green) and parallel FFL networks (orange). be easier by seeing a network in such a way than considering the network as just a set of nodes and arrows. Note that in this paper we have limited our targets within transcriptional regulations. owever, the consideration can be extended into other regulations involved in translation, post-translation, signal transduction, and so on without much complication. play roles in the species. In the early developmental stage of D. melanogaster, the maternal factors have formed gradients in the embryo and some of the downstream genes are going to exhibit stripe expression patterns. Our results suggest that the feed-forward loops may contribute to the pattern formation during early development. The difference in the structure of networks may be related to their roles Both in E. coli and in D. melanogaster, most feed-forward loops share genes and regulations. The structures of networks are, however, different between these species, which could relate to the difference in the function of the networks. In E. coli, the feed-forward loops may play roles for quick and coordinated responses against environmental variations (alir & Alon 2004). In contrast, because there are many networks of the series type and the parallel type in the regulatory network of D. melanogaster (Fig. 6B), the responses with multiple peaks may Conditions of a response with multiple peaks It is not obvious whether multiple thresholds can be set in a stimulus concentration of a target gene and multiple peaks are generated in vivo temporally or spatially. From a theoretical point of view, the following conditions can be pointed out to have multiple peaks. First, the activation and repression thresholds, each of which is determined by a direct or indirect pathway, should appear appropriately on the concentration coordinate, i.e. as the stimulus concentration increases, for instance, in the profile shown in Fig. 3D, first activation threshold appears, then repression, and finally activation. Otherwise, two activation 032 Genes to Cells (2005) 0, Blackwell Publishing Limited

9 Cross talking of network motifs intervals are merged into a single interval. To obtain sufficiently sharp thresholds, successive activation and repression thresholds are sufficiently deviated and the ill coefficients around the thresholds should be large. Finally, the stimulus concentration changes over these thresholds under physiological condition. Implication of feed-forward loops in pattern formation Several examples are now known, where gene expression depends on morphogen concentration, and in a morphogen gradient, activation of the gene is induced within a spatial interval that is set by response thresholds of morphogen concentration (for review, see Gurdon & Bourillot 200). For instance, in Xenopus blastula cells, the gene, Xbrachyury (Xbra) responses to a morphogen activin in a concentration range bounded by thresholds (Green et al. 992, 994; Gurdon et al. 994). It was also indicated that the gene, Xbra might respond to two different concentration ranges of activin (Green et al. 992, 994). In these examples, incoherent feed-forward loops and their interactions may play roles to generate these patterns. In the case of D. melanogaster, it is not completely clear if multiple stripes of a gene product are determined by a multiple-thresholds mechanism that we presented in this paper. ere, we want to propose that a parallel-type interaction of two feed-forward loops may play a role in generating the two stripes of gene giant (gt) by interpreting the anterior to posterior gradient of unchback (b) protein formed in the early stage (Driever & Nüsslein-Volhard 989). In the first feed-forward loop, b represses both knirps (kni) and rüppel (r) directly, and the product of kni represses r ( Jäckle et al. 986; och et al. 992; Wu et al. 200; Jaeger et al. 2004a,b). As a result, r forms a stripe, and the borders are determined by the two pathways from hb to r (hb r, and hb kni r). Because r represses gt, the two pathways (hb r gt and hb kni r gt) determine the two borders of gt domains; the posterior border of anterior domain and the anterior border of posterior domain (Eldon & Pirrotta 99; raut & Levine 99a). As far as we know, there is no report concerning the factor that determines the anterior border of anterior gt domain. One candidate is hb, because it actually represses gt in the posterior region (Eldon & Pirrotta 99; raut & Levine 99a). In this case, hb, kni, r, and gt construct one of the parallel type networks shown in Fig. 3E (but each direct pathway is negative), and the three direct and indirect pathways from hb to gt establish the three borders of gt domains. Another candidate is tailless (tll), which is activated by Bicoid (Bcd) in the anterior tip region (Pignoni et al. 992; Liaw & Lengyel 993), and known as a repressor of gt in the posterior region of the embryo (Eldon & Pirrotta 99; raut & Levine 99a). In this case, the network that determines the borders of gt domains may contain other genes and pathways including the activation of hb by Bcd. Most recently, in order to test the idea that a network can generate a stripe pattern by applying a concentration gradient, a synthetic experiment has been performed in vitro by engineering a gene circuit. Isalan et al. (2005) designed a network that consists of three gene products, A, B, and C, whose expression is controlled by transcription with T7 and SP6 RNA polymerases. From our viewpoint, one of the networks they made is essentially the same as shown in Fig. 3C. Among these five components, four, A, B, and C and T7 polymerase, form two feedforward loops connected in parallel that are responsible for the stripe-pattern formation. They expected that the concentration of T7 polymerase forms a spatial gradient. The network interprets the gradient, forming the stripe pattern. They also introduced cross repressions between genes. Although the increase of repression interaction lowered the overall protein production, they reported that the thresholds became sharper. Cross repressions between genes are actually observed in the gene regulatory network of D. melanogaster. For instance, the factor gt, which is inhibited by r (see previous discussion), also represses r (raut & Levine 99b; Jaeger et al. 2004a,b). Such cross repression is contained in two types of network motifs made up of three genes, X, Y, and Z. In addition to the three interactions contained in a feed-forward loop, one contains an interaction from Y to X, and the other one contains an interaction from Z to Y, i.e. a feed-forward loop with cross-repression between Y and X, or Z to Y (Milo et al. 2004). A cross-repression between two genes sometimes gives rise to bistable genetic states as exemplified by genetic toggle switch (Gardner et al. 2000). In the case of the two motifs, however, as the level of X increases, bistable states are found only in a narrow concentration range, and the level of Z exhibits a bell-shaped profile with sharp thresholds against X concentration. Thus, we consider that a cross-repression introduced in a feedforward loop is one of the ways to increase the ill coefficient around a threshold and the threshold mechanism to generate a stripe pattern also works in these motifs (S.I. et al., manuscript in preparation). In the case of the stripe formation of gt, as r domain forms earlier than gt domain (Wu et al. 200), r determines the domain of gt at first and then cross repression makes the boundary sharp. Blackwell Publishing Limited Genes to Cells (2005) 0,

10 S Ishihara et al. Theoretical studies of morphogenesis Wolpert (969) proposed the model, in which the gradient of a morphogen conveys positional information and the cells interpret the concentration of morphogen to activate specific developmental programs. The subsequent question is how discrete thresholds are set within target cells in response to the gradient of morphogen. One possibility is to employ the concentration detection mechanism of feed-forward loops. The theoretical study of morphogenesis was started by the pioneering work of Turing (953). e proposed a reaction-diffusion mechanism that consists of two components, activator and inhibitor. These components form stripes and spot patterns spontaneously without any preceding gradient (ondo & Asai 995; Meinhardt & Gierer 2000). A uniform distribution at an initial stage is spontaneously destabilized and then a non-uniform distribution is evolved. One characteristic of this mechanism is that stripes have the intrinsic width. Therefore, the number of stripes is linearly proportional to the length of the system with stripes. In contrast, according to the concentration detection mechanism of feedforward loops, a gradient of morphogen is indispensable. The formation of a pattern is a consequence of a response to a morphogen concentration at every location. Therefore, the number of stripes is maintained against the variations in the length of the system. Outlook There are several kinds of network motifs such as motifs with cross repression as well as feed-forward loop (Lee et al. 2002). Why do the regulatory networks contain particular structural elements so frequently? The reason may be related to their biological roles. In addition to the functions of individual network motifs, the interaction between them could be important for the motifs to play the roles. It would be interesting to explore what kinds of temporal and spatial signal processing abilities are possible by networks composed of several types of motifs with their interactions. Procedures Equations for activation and repression regulations of gene expression When gene Y is regulated by transcription factor X, the steady state level of Y expression is described by a function of the concentration of X. Let X and Y be the respective levels of X and Y expression, the level of Y expression is written by Y = f (X), where f( X) is a regulatory function (McAdams & Arkin 998; Smolen et al. 2000; Mangan & Alon 2003). The time evolution of the level of Y is written by an ordinary differential equation: dy dt = γ( Y f( X)) where γ is the depletion or dilution rate of the product Y. The essential characteristic of the regulatory function f is described by the ill equation: for positive control (activation): X AX ( ) = V () X + and for negative control (repression): RX ( ) = V (2) X + where is the Michaelis constant, is the ill coefficient, and V is the maximum expression level. Cis-regulatory input functions The level of Z is determined by both X and Y, Z = G(X,Y), where G(X,Y) is a cis-regulatory input function. Then, the time evolution of the level of Z is written by an ordinary differential equation: dz dt = γ( Z G( XY, )) where γ is the depletion or dilution rate of the product Y. The function, G can show complex dependence on the concentration X and Y (Buchler et al. 2003; Setty et al. 2003). ere, we do not aim to present a model that accounts for the detailed biochemistry of particular cis-regulations, but a toy model that captures the essential characteristics. In most cases, the function is essentially described by a combination of the regulatory functions, A(X) and R(X). In this paper, we adopted two types of cis-regulatory input function as examples. For the first case, Y activates the Z expression, and the activation by Y is inhibited by X. The respective cis-regulatory function is written as the product of activation and repression regulatory functions: G(X,Y ) = R(X ) A(Y ) where R(X) is the repression regulation by X, and A(Y) is the activation regulation by Y. The parameters,, and V are suitably chosen for each binding of X and Y. The Michaelis constant for the repression by X should be larger than for the activation by Y; otherwise, the Z expression cannot be activated by Y. As is displayed in Fig. 2B, the Z expression is activated when the concentrations of Y are high and that of X are low enough. Such a cisregulatory input function is found in one of the best characterized regulatory region lac operon of E. coli, which is regulated by the activator CAP, and the repressor LacI. The lac operon is activated only when LacI repressor is dissociated and activator CAP binds to the activator site. 034 Genes to Cells (2005) 0, Blackwell Publishing Limited

11 Cross talking of network motifs For the second case, either X and Y can activate Z expression. The cis-regulatory input function G(X,Y) can be written as G(X,Y ) = A(X + Y ) which corresponds to the case that X and Y exclusively (competitively) bind to the same activation binding site. As is displayed in Fig. 2D, the Z expression is activated when either of the concentrations of X and Y are high. Such a cis-regulatory input function is found in the f lil promoter of E. coli, which is regulated by FliA and FlhDC (alir & Alon 2004). Equations of feed-forward loops and parameters Using the previously mentioned regulatory function A(X) and R(X) and cis-regulatory functions G(X,Y), the expression level of Z of a feed-forward loop can be written as a function of the X concentration X Z = G(X, f (X )) where the function of f(x) is either A(X) or R(X). For the incoherent feed-forward loop shown in Fig. 2A, the level of Z expression is written by Z = R 2 (X )A 3 (A (X )) (3) where A, R 2, and A 3 are the regulation functions for interactions, 2, and 3 respectively. The parameters we used in Fig. 2A are given as V = 0.0, = 0.0, =.0, V 2 =.0, 2 = 2.0, 2 = 2.0, V 3 =.0, 3 = 0., and 3 = 2.0. ere, V i, i, and i are the parameters of the function, A i (X) or R i (X ). For the feedforward loop shown in Fig. 2C, the level of Z expression is written by Z = A Z (X + R (X)). The parameters used in Fig. 2C are given as V =.0, = 0.002, and =.0 for R (X), and V Z =.0, Z = 0.2, and Z = 2.0 for A Z (X). The extension of this expression to networks of several loops can be carried out by using the regulatory functions, A(X) and R(X), and pertinent cis-regulatory functions G(X,Y ). For instance, the network of two loops shown in Fig. 3A, the level of Z and Z 2 is given by Z = R ( X) A ( A ( X)) ZX ZY YX Z = R ( Z ) A ( A ( Z )) 2 ZZ 2 ZY 2 2 YZ 2 V where A YX and R YX are the regulatory function for the regulation on gene Y by X. The parameters used in Fig. 3A are given as V Y X = 5.0, =.0, = 2.0, =.0, = 2.0, YX V YX ZX ZX V ZX = 3.0, = 30.0, = 2.0, = 2.0, = 4.0, ZY ZY V ZY YZ 2 YZ = 0.5, = 2.0, =.0, = 5.0, = 3.0, 2 YZ 2 ZZ YZ YZ =.0, 2 = 2.0, and 2 2 = 2.0. ere each parameter V ZY 2 2 YY 2 2 YY 2 2 corresponds to the function with the same subscript. For the network of two loops shown in Fig. 3C, the level of Z 2 expression is given by Z A ( A ( X)) R ( A ( A ( X)) R ( X)) = ZY Y X ZZ ZY Y X Z X where A YX and R YX are the regulatory function for the regulation on gene Y by X. In Fig. 3C, the parameters for the functions are given as V YX = 50.0, YX = 5.0, YX = 2.0, V Z X =.0, V Z X = 0.0, = 4.0, =.0, = 5.0, = 2.0, ZX ZY ZY V ZY V ZZ =.0, = 0.2, = 2.0, =.0, = 0., ZZ ZZ ZY Z Y and ZY = 4.0, where each parameter corresponds to the function 2 with the same subscript. Temporal and spatial responses of feed-forward loops In order to study temporal and spatial responses of the network shown in Fig. 4A, we use the following equations and parameters values. The temporal response of the network is described by the following ordinary differential equations. For transcription factor X, the concentration of X at time t = 0, X(t) = 0, and the temporal evolution is described by dx dt = γ ( X X ) X f where γ X is the depletion rate and X f is the saturating concentration. The feed-forward loops are described by a set of equations written by Z0 = X dyi = γy( Yi Ai( Z i i )), dt ( i = 23,, ) dzi = γz( Zi Gi( Zi Y i, i)) dt γ Yi γ Zi where and are the depletion rate of Y i and Z i, respectively, A i (Z i ) is the activation regulatory function for gene Y i, and G i (Z i,y i ) is the cis-regulatory function of gene Z i. For the cisregulatory function, we adopt the function shown in Fig. 2B, i.e. G i (Z i,y i ) = R i (Z i )A i (Y i ). For Fig. 4B, the expression level of X develops with the parameters, γ X =.0 0 3, X f = 0.5. The parameters for all Y i (i =, 2, 3) are given as V Yi =.0 and Z Y = 0.08 in the function A i (Z i ) and γ for the i i Yi = 0. depletion rate. The parameters for all Z i (i =, 2, 3) are given as V Z i = 2.0, Z Z = 0. and = 0. in the function G i (Z i,y i ), i i YZ i i and γ Z i =.0 for the depletion rate. The spatial expression profile shown in Fig. 4C is the solution of the following partial differential equations. For X, the translation of X takes place only at the localized area in the anterior pole (0 < l < l 0 ) at the rate of S X. Then the spatiotemporal expression profile follows the equation given by X t = γ X + D X X 2 X l 2 Blackwell Publishing Limited Genes to Cells (2005) 0,

12 S Ishihara et al. where the first term describes the depletion process, and the second term is the diffusion process. Furthermore, the term S X is added between the range 0 < l < l 0, representing the translation of X. The steady state solution of this equation exhibits the exponential profile shown in Fig. 4C. The expression profile of the feed-forward loops is described by a set of equations given by Z0 = X 2 Yi Y A Z D Y i = γy( i Y( i i i )) + Y, i 2 t l ( i = 23,, ) 2 Zi Zi = γz ( Zi GY( Zi Yi D i i, )) + Zi 2 t l where the first terms on the right hand sides are the same as the case of temporal response, and the second term describes the diffusion process. For Fig. 4C, we used the same parameters as in Fig. 4B for γ X (= γ Z 0 ), γ Yi, γ Zi, Z Y,,,, and i i Z Z i i YZ i V Yi V Z i (i =, 2, 3). All the diffusion coefficients are set as D X = D Yi = D Z = 0.0 (i =, 2, 3). The system size is set as i L = 00.0, and the translation of X takes place in the region 0 < l <.0 with rate S X =.0. Transcription network database For the bacterium, E. coli, we used the database released by Shen- Orr and collaborators (2002) ( UriAlon), consisting of operons regulated by transcription factors. The data set contains 423 genes and 578 regulations. For the yeast, S. cerevisiae, we used the literature-based database YPD (yeast protein database) containing directed positive or negative interactions (Costanzo et al. 200) ( YPDhome.html). The data set contains 807 genes and 862 regulations. For the fruit fly, D. melanogaster, we used the literaturebased database GeNet (Selov et al. 998) ( Institute/gorb_Department/inbios/genet/genet.htm), containing the transcription interactions involved in early development. Because the original data contains many ambiguous interactions, we chose regulatory paths indicated as direct or possibly direct in the original data. The data set contains 22 genes and 309 regulations. We also analyzed the transcription interactions of the sea urchin S. purpuratus, which govern the endomesoderm specification (Oliveri & Davidson 2004) ( mirsky/), although the size of the data set may still not be large enough to determine the large-scale distribution of network motifs. The data set contains 43 genes and 87 regulations. In order to study the abundance of feed-forward loops, we compared the number of feed-forward loops that appeared and the expected number of the loops in a set of randomized network. The set of networks is generated by randomizing the connections in a real network without changing the numbers of incoming and outgoing arrows of each node as well as the numbers of nodes and arrows. Then, the expected value is obtained by averaging the number of the feed-forward loops that appeared in each network over the set. If the number of the loops is much larger than the expected number, we consider it as a network motif (Milo et al. 2002; Shen-Orr et al. 2002). To determine the number of incoherent feed-forward loops, the regulations that are not specified as activation or repression were excluded. Then, the numbers of feed-forward loops and incoherent feed-forward loops were counted again. The frequency of feed-forward loops that share genes and interactions with other feed-forward loops was calculated as the ratio between the number of such feed-forward loops and the total number of combinations between two loops. Appendix ere, we briefly summarize the mathematical conditions in order to obtain a bell-shaped profile for an incoherent feed-forward loop. As an example, we consider the incoherent feed-forward loop shown in Fig. 2A. The interval of Z to be activated in the X concentration is characterized by the two thresholds: activation and repression. The activation threshold is determined by the overall effect of indirect pathway mediated by Y. When the concentration of Y reaches the Michaelis constant YZ of interaction 3, the strength of activation of Z by Y reaches half of the maximum value. The activation threshold is given by the concentration of X at which the concentration of Y reaches YZ. As shown in Equation in the Procedures section, the expression level of Y is given by Y = VY XY XY X + X XY XY where V Y is the maximum expression level of Y, and XY is the Michaelis constant, and XY is the ill coefficient of interaction. When the concentration of Y reaches YZ, the concentration X is given by XYZ : XYZ = XY YZ V Y YZ This expression gives the activation threshold of Z expression. In order for Z to be activated, YZ must be smaller than V Y ; otherwise, the activation regulation by X does not function. The indirect negative control of Z by X mediated by Y is effectively described by the activation regulatory function A(X), shown in Equation, in which XYZ gives the Michaelis constant and XYZ = XY YZ is the ill coefficient in the function. The repression threshold is determined by the Michaelis constant XZ of the direct interaction 2. XY 036 Genes to Cells (2005) 0, Blackwell Publishing Limited

13 Cross talking of network motifs When the concentration of X is XZ, the strength of repression by X reaches half of the maximum value. In the feed-forward loops, as the level of X increases, Z is activated approximately at X = XYZ and repressed at X = XZ, and the activation interval of Z is described by these two concentrations. Therefore, the condition XYZ < XZ must be satisfied for Z to be activated and thus to exhibit a bell-shaped expression profile. The sensitivity of Z to the change in the concentration of X is characterized by a ill coefficient. For a bellshaped profile, two ill coefficients are defined: one is for the intervals indicating increasing profile, and the other one is for the intervals showing decreasing profile. These coefficients are approximately given by XYZ for increasing profile (activation) and XZ for decreasing profile (repression). The maximum expression level of Z should be sufficiently strong; otherwise, the bell-shaped profile does not function. Suppose that the indirect pathway is approximately and effectively characterized by a ill equation with Michaelis constant XYZ and ill coefficient XYZ. Then, the expression level of Z, given by Equation 3, is approximately written by Z = VZ XZ XZ XZ XZ X XYZ + X + X XZ XZY XYZ XYZ We further suppose that the ill coefficients, XYZ and XZ are the same value,. Then, the maximum value of Z expression is given by Z max = V Z / XZ XYZ XZ/ XYZ + Thus, under the condition XYZ < XZ, the maximum value Z max increases with and XZ / XYZ. Even when XYZ and XZ take different values, the dependence of Z max on the parameters is not qualitatively different. This consideration is extended to the profile with multiple peaks. As an example, we study the network shown in Fig. 3D. There are three thresholds,, 2, and 3, each of which corresponds to a particular pathway from X to Z 2 in Fig. 3C. Suppose that and 3 are activation thresholds ( < 3 ), and 2 is the repression threshold. To have the two distinct intervals where the level of Z is elevated as shown in Fig. 3D, the condition < 2 < 3 should be satisfied. Around the concentration of each threshold, a ill coefficient can be defined. A profile with sharp peaks is obtained when the ill coefficients are sufficiently large. 2 Acknowledgements We are grateful to U. Alon,. aneko, Y. Maeda, A. S. Mikhailov, M. Sano, Sneppen, and T. Yamamoto for discussions. We also thank N. Shimamoto for discussions and critical reading of this manuscript. TS acknowledges the support by the grant-in-aid for young scientists from the ministry of education, culture, sports, science, and technology, Japan. References Basu, S., Mehreja, R., Thiberge, S., Chen, M.T. & Weiss, R. (2004) Spatiotemporal control of gene expression with pulsegenerating networks. Proc. Natl. Acad. Sci. USA 0, Buchler, N.E., Gerland, U. & wa, T. (2003) On schemes of combinatorial transcription logic. Proc. Natl. Acad. Sci. USA 00, Costanzo, M.C., Crawford, M.E., irschman, J.E., et al. (200) YPD, PombePD and WormPD: model organism volumes of the Bionowledge Library, an integrated resource for protein information. Nucleic Acids Res. 29, Dobrin, R., Beg, Q.., Barabasi, A.L. & Oltvai, Z.N. (2004) Aggregation of topological motifs in the Escherichia coli transcriptional regulatory network. BMC Bioinformatics 5, 0. Driever, W. & Nüsslein-Volhard, C. (989) The bicoid protein is positive regulator of hunchback transcription in the early Drosophila embryo. Nature 337, Eichenberger, P., Fujita, M., Jensen, S.T., et al. (2004) The program of gene transcription for a single differentiating cell type during sporulation in Bacillus subtilis. Plos Biol. 2, e328. Eldon, E.D. & Pirrotta, V. (99) Interactions of the Drosophila gap gene giant with maternal and zygotic pattern-forming genes. Development, Gardner, T.S., Cantor, C.R. & Collins, J.J. (2000) Construction of a genetic toggle switch in Escherichia coli. Nature 403, Green, J.B., New,.V. & Smith, J.C. (992) Responses of embryonic Xenopus cells to activin and FGF are separated by multiple dose thresholds and correspond to distinct axes of the mesoderm. Cell 7, Green, J.B., Smith, J.C. & Gerhart, J.C. (994) Slow emergence of a multithreshold response to activin requires cell-contactdependent sharpening but not prepattern. Development 20, Gurdon, J.B. & Bourillot, P.Y. (200) Morphogen gradient interpretation. Nature 43, Gurdon, J.B., arger, P., Mitchell, A. & Lemaire, P. (994) Activin signalling and response to a morphogen gradient. Nature 37, och, M., Gerwin, N., Taubert,. & Jäckle,. (992) Competition for overlapping sites in the regulatory region of the Drosophila gene rüppel. Science 256, Isalan, M., Lemerle, C. & Serrano, L. (2005) Engineering gene networks to emulate Drosophila embryonic pattern formation. Plos Biol. 3, e64. Blackwell Publishing Limited Genes to Cells (2005) 0,