Modeling Genetic Switches with Positive Feedback Loops

Transcription

1 J. theor. Biol. (00), doi:0.006/jtbi.00.90, available online at Modeling Genetic Switches with Positive Feedback Loops Tetsuya Kobayashi n wz, LuonanCheny and Kazuyuki Aiharawu wdepartment of Complexity Science and Engineering, Graduate School of Frontier Sciences, The University of Tokyo, Hongo 7--, Bunkyo-Ku, Tokyo -8656, Japan zjsps Research Fellow. 5-- Kojimachi, Chiyoda-ku, Tokyo 0-87, Japan ydepartment of Electrical Engineering and Electronics, Faculty of Engineering, Osaka Sangyo University, Nakagaito --, Daito, Osaka , Japan and ucrest, JST, Kawaguchi, Saitama -00, Japan (Received on 8 May 00, Accepted in revised form on October 00) In this paper, we develop a new methodology to design synthetic genetic switch networks with multiple genes and time delays, by using monotone dynamical systems. We show that the networks with only positive feedback loops have no stable oscillation but stable equilibria whose stability is independent of the time delays. In other words, such systems have ideal properties for switch networks and can be designed without consideration of time delays, because the systems can be reduced from functional spaces to Euclidian spaces. Therefore, we can ensure that the designed switches function correctly even with uncertain delays. We first prove the basic properties of the genetic networks composed of only positive feedback loops, and then propose a procedure to design the switches, which drastically simplifies analysis of the switches and makes theoretical analysis and design tractable even for large-scaled systems. Finally, to demonstrate our theoretical results, we show biologically plausible examples by designing a synthetic genetic switch with experimentally well investigated laci, tetr, and ci genes for numerical simulation. r 00 Elsevier Science Ltd. All rights reserved.. Introduction Recent development in genetic engineering has made the design and the implementation of synthesized genetic networks realistic from both theoretical and experimental viewpoints, in particular for yeast and bacteria such as E. coli (Gardner et al., 000; Elowitz & Leibler, 000; Becskei & Serrano, 000; Becskei et al., 00). Actually, based on the theoretical analysis, several simple genetic networks have been successfully constructed experimentally, e.g., genetic switches (Gardner et al., 000; Isaacs n Corresponding author. Fax: address: tetsuya@sat.t.u-tokyo.ac.jp (T. Kobayashi). et al., 00), repressilator (Elowitz & Leibler, 000), and a single negative feedback loop network (Becskei & Settano, 000). The data in these experiments well agree with the predictions of mathematical models; this implies that mathematical models can be a powerful tool for designing synthesized genetic networks, especially when designing complicated networks with multiple genes (Hasty et al., 00a,b; McMillen et al., 00). Such simple models represent a first step towards logical cellular control by monitoring and manipulating biological processes at the DNA level, and not only can be used as building modules to synthesize artificial biological systems, but also have great potential for biotechnological and therapeutic 00-59/0/$5.00 r 00 Elsevier Science Ltd. All rights reserved.

2 80 T. KOBAYASHI ET AL. applications (Hasty et al., 00a; Agha-Mohammadi & Lotze, 000; Imhof et al., 000). In addition to simple switching with on and off jumps, genetic switches can also be used to construct complicated logical circuits with high computational ability such as biological AND, OR, XOR gates, bio-memories, and even bio-computers (Simpson et al., 00; Weiss, 00). However, the more complicated a synthetic network becomes, the more difficult it is to design and analyse the behaviors of the system, which are usually described by high-dimensional nonlinear differential equations. Moreover, many parameters like time delays in biological systems are uncertain, e.g. due to stochastic influence, and their values are not available due to the lack of accurate measurements (Chen & Aihara, 00a,b). Therefore, even for the designs of simple-structured switches and oscillators in the previous works, many important physiological factors such as translation processes and time delays are simply ignored or abbreviated, in order to reduce dimensionality and complexity of the systems. It is well known, however, that such factors may play important roles in dynamics of genetic networks, and theoretical models without consideration of these factors may even provide wrong predictions (Chen & Aihara 00a; Smolen et al., 000, 00). Thus, one major obstacle to design multiple genetic networks with complicated dynamics is how to analyse high-dimensional nonlinear differential equations, in particular, differential equations with possibly uncertain delays, which generally have infinite dimensions in a functional space (Hale & Lunel, 99). In this paper, we develop a new methodology to design genetic switch networks with multiple genes and time delays, by using monotone dynamical systems (Smith, 995). As indicated in this paper, the networks with only positive feedback loops have no stable oscillation but equilibria whose stability is independent of the time delays. In other words, such systems have ideal properties for switch networks and can be designed without consideration of time delays, because the systems can be reduced from functional spaces to Euclidian spaces. Therefore, we can ensure that the designed switches function correctly even with many uncertain delays. In this paper, we first show that genetic networks with only positive feedback loops have the following desirable properties as genetic switches:. It is guaranteed that a genetic network with only positive feedback loops converges to stable equilibriumstates for almost all initial states. In other words, there are neither a stable oscillation nor other non-equilibriumattractors.. The systems with and without delays have the identical equilibria with the same stability. This property means that we can use ordinary differential equations rather (ODEs) than functional differential equations (FDEs) to design synthetic genetic networks.. The dimensions of the model can be further reduced by changing some ordinary differential equations into algebraic equations, keeping the equilibriumpoints and their stabilities invariant. This property makes analysis of a large-scaled systemtractable. Owing to such properties, the designed switches are robust to time delay variations, and guaranteed to converge to stable equilibria. Then we propose a simple procedure to design synthesized genetic switches, which drastically simplifies analysis of the switches and makes theoretical analysis and design tractable even for large-scaled systems. Finally, to demonstrate our theoretical results, we give biologically plausible examples by designing synthesized genetic switches with experimentally well investigated laci,tetr, and ci genes for numerical simulation. As indicated in the paper, by using quantitative and qualitative experimental data, the proposed procedure can design biologically and experimentally feasible switches from simple abstract models and predict the dynamical behaviors of the synthesized genetic networks. This paper is organized as follows. In Section, we makes notations, definitions, and general assumptions for the mathematical model of genetic networks. In Section, we describe our main results including stability conditions and the reduction procedure. Several examples are demonstrated in Section, and the paper ends with concluding remarks in

3 MODELING GENETIC SWITCHES 8 Section 5. All proofs of theorems in this paper are given in Appendix A.. General Settings In this paper, we model a genetic network by delayed differential equations or functional differential equations (FDEs). This model includes many essential properties of the genetic networks in the previous works except stochasticity (Shea & Ackers,985; Wolf & Eeckman, 998; Drew, 00; Smolen et al., 000). Note that many previous works on theoretical models of genetic networks ignore time delays in spite of their importance. Next, we describe necessary definitions as well as assumptions. Assume that there are n chemical components (i.e. proteins, mrnas, modified proteins, and proteins at different locations in a cell) in the network. Then the network can be described as xðtþ ¼f ðx t Þ DxðtÞ fðx t Þ; ðþ where R þ is the set of nonnegative real numbers and xðtþar þn indicates the concentrations of all components at time tar: x t AC þ Cð½ r; 0Š; R þn Þ denotes the element of Cð½ r; 0Š; R þn Þ where Cð½ r; 0Š; R þn Þ is the space of continuous maps on ½ r; 0Š into R þn : In other words, x t ACð½ r; 0Š; R þn Þ is defined by x t ðyþ ¼ xðt þ yþ; rpyp0; and its normis defined by jjx t jj ¼ sup rpyp0 jx t ðyþj: When emphasizing the dependence of a solution on an initial data fac þ ; we write xðt; fþ or x t ðfþ for x t : D ¼ diagðd ; y; d n Þ is an n n diagonal matrix with n positive real diagonal components representing the degradation rates of the chemical components. f AC ðc þ ; R þn Þ : C þ -R þn indicates the synthesis rates of components, where C ðc þ ; R þn Þ means that the functions from C þ to R þn are continuously differentiable. In addition, we define N ¼f; y; ng: Note that this model can describe not only synthesis and degradation reactions of the components but also a variety of other chemical reactions, if required, such as enzymic reactions, translocations and modification reactions of proteins. In this paper, we make several assumptions as follows. Assumption.. The network described by eqn () does not produce an infinite amount of chemical components. This assumption is clearly reasonable for all biological systems because the amount of a chemical component cannot be increased to infinity and too great a chemical concentration will eventually destroy the organism s metabolism. Assumption.. The synthesis rates of all components of the network described by eqn () are finite. This assumption of the bounded synthesis rates is naturally valid because the synthesis rates of chemicals, that is, f generally have a saturation property (see Assumption A. in Appendix A). Assumption.. The synthesis rates of the network eqn () at t depend on the states of the network at a finite number of discrete past time instants. Discrete time delays assumed here are simplified representations of actual time delays mainly caused by transcription, translation, translocation, and diffusion processes. These time delays may generally take not discrete but continuous distributed values, affected by the concentrations of chemical components at continuous time points or degradations. Since the regulation of genetic networks via changes in time delays is not a major topic of this paper, all delays are assumed to be discrete fixed values for the sake of simplicity. Next, we mathematically define the types of an interaction, positive and negative, between chemical components of the network. Assumption.. The synthesis rate of the i-th chemical component at t; namely, f i ðx t Þ monotonously increases, or monotonously decreases, or is unaffected if the concentration of only the j-th chemical component at t t ij is monotonously increased and the concentrations of the other chemical components are kept constant where i; jan: In addition, this tendency is

4 8 T. KOBAYASHI ET AL. independent of the concentrations of the other chemical components. The basic mechanism of chemical reactions is stochastic collisions of chemical components (Law of Mass Action), and the probability that a collision occurs monotonously increases with the concentrations of chemical components involved in the reaction. Since the speed of a chemical reaction often inherits this monotonicity, Assumption. is reasonable for most genetic networks. For instance, the following physiological reactions satisfy Assumption.: transcription activation, transcription inhibition, translation, phosphorylation, enzymic reactions, normal chemical reactions, and translocation. For the sake of simplicity, in Assumption., the synthesis rate f i of the i-th chemical component is assumed to directly depend on the concentration of the j-th chemical component at only one time point t t ij ; although there may exist multiple direct interactions from the j-th chemical component to the i-th chemical component with different time delays, which are discussed in Appendix A in details. Based on these assumptions, we define the types of interactions as follows: Definition. (Types of interactions). Suppose that the concentration of the j-th chemical component at t t ij affects the synthesis rate of the i-th chemical component at t where i; jan: If the synthesis rate of the i-th chemical component at t; namely, f i ðx t Þ monotonously increases (or decreases) as the concentration of the j-th chemical component at t t ij monotonously increases, then the type of the interaction fromthe j-th chemical component to the i-th chemical component is called positive (or negative), and we set s ij ¼ ðor Þ: If the synthesis rate of the i-th chemical component at t is never affected by the change in the concentration of the j-th chemical component, then we set s ij ¼ 0: Thus, s ij ¼ ðor Þ means that the j-th chemical component affects positively (or negatively) the i-th component with time delay t ij : For instance, s ij ¼ for f i ¼ x j ðt t ij Þ=ð þ x j ðt t ij ÞÞ; and s ij ¼ for f i ¼ =ð þ x j ðt t ij ÞÞ: Examples of interactions are illustrated in Fig.. In addition, for simplicity, we assume that f i increases or decreases in the Michaelis Menten or Hill s manner as illustrated in Fig.. A more rigorous and general representation of this assumption is given in Appendix A. Next, we define an interaction graph of the model in eqn (). The concept of the interaction graph not only enables us to understand the relations between the components intuitively but also provides a graphical interpretation of the theoretical results in this paper. Definition. (Interaction graph). An interaction graph, IGð f Þ; of the genetic network defined by eqn () is a directed graph whose nodes represent the individual chemical components of the genetic network and whose edges represent the interactions between the nodes. When s ij a0 with t ij X0; that is, the j-th chemical component affects the synthesis rate of the i-th chemical component with time delay t ij ; the graph has an edge, e ij ; directed fromthe j-th node to the i-th node. Figure is an example of an interaction graph. In addition, irreducibility of a graph is defined as follows. Definition. (Irreducibility). IGð f Þ is said to be irreducible only when there is at least one path fi pði; jþ ¼ðj ¼ p ēpp p ēpp? - epipi sij = - sij = - sij = + p i ¼ iþ; 5 xj sij = + Fig.. Examples of types of interactions. Solid and dashed curves show positive and negative interactions, respectively.

5 MODELING GENETIC SWITCHES 8 τ e In addition, this feedback loop is said to be positive (or negative) if Q i m¼ s p m þp m ¼ ðor Þ: e τ e6 τ6 e τ e6 τ6 e5 τ5 5 τ65 e65 e55 τ55 Figure is an example of an interaction graph with positive and negative feedback loops, e.g. this graph has one positive feedback loop - þ - þ ; one negative feedback loop - þ þ 6 - ; and a positive self-feedback loop 5 - þ 5: Finally, we make the following important assumption: 6 Fig.. An example of an interaction graph with feedback loops. Signs þ and on an edge indicate s ¼ and ; respectively. A feedback loop designated by solid curve (or dashed curve) is a positive (or negative) feedback loop. In this graph, there is a negative feedback loop composed of the st, nd, th, 5th, and 6th nodes, a positive feedback loop composed of the st, nd, rd and 6th nodes, and a positive self-feedback loop composed of the 5th node. fromthe j-th node to th i-th node for all i; jan ðiajþ where p ; y; p i AN and e pb p a is an edge fromnode p a to node p b : When the interaction graph of a genetic network is irreducible, the network cannot be divided into two or more sub-networks, for which a sub-network is not affected by the other ones. For the sake of simplicity, we have the following assumption: Assumption.5. IGð f Þ of eqn () is irreducible. Actually, if a network can be divided into several irreducible sub-networks that are generally easy to analyse in contrast to the whole network, we can only examine each individual sub-network by applying our method to each irreducible sub-network. Next, we define the types of feedback loops, which are qualitative characteristics of genetic networks. Definition. (Feedback loops and their types). If there is a path fromthe i-th node of an interaction graph to the same i-th node, pði; iþ ¼ ði ¼ p - epp p - epp? - epipi p i ¼ iþ; then this path is said to be a feedback loop and furthermore be a self-feedback loop when i is. Assumption.6. The interaction graph IGð f Þ of the model in eqn () has only positive feedback loops. As shown in Fig., a positive feedback loop can include negative interaction edges. We can see that for arbitrary two nodes i and j of IGð f Þ; all the paths fromthe i-th node to the j-th node have the same sign under Assumption.6 where the paths are allowed to include loops because of the irreducibility of IGð f Þ and the definitions of types of interactions and feedback loops. We can also see that Assumption.6 holds if for arbitrary two nodes i and j of IGð f Þ; all the paths fromthe i-th node to the j-th node have the same sign. Finally, we define an equilibriumpoint that is used in our main theorems. Definition.5 (Equilibria). The set of equilibria for eqn () is defined as x t that is constant for all tx0: Mathematically more rigorous representation of all the assumptions and the definitions is given in Appendix A.. Main Results In this section, we derive three main theorems, which show that a genetic network with only positive feedback loops has advantages as a genetic switch and is easy to analyse theoretically. Furthermore, a procedure to design a synthesized genetic switch is proposed based on these theorems. All proofs for the theorems as well as several generalized assumptions are given in Appendix A.

6 8 T. KOBAYASHI ET AL... CONVERGENCE TO EQUILIBRIA Theorem. (Convergence to equilibria). If eqn () satisfies Assumptions..6, then for almost all initial conditions fac þ ; x t ðfþ converges to equilibria. This theoremindicates that a genetic network with only positive feedback loops has no dynamical attractors. When we design a genetic switch, it is important to ensure that the designed switch does not show any dynamical behaviors except asymptotic convergence to stable equilibria. However, it is generally not easy to guarantee such stable behavior even for a small genetic network with a few components and without any time delays, due to the nonlinearity of the system. As indicated in Theorem., if we design a synthetic genetic switch only with positive feedback loops then the systemis guaranteed to converge to an stable equilibriumpoint in spite of nonlinearity, sizes, and delays of the network. Such property significantly reduces the complexity of designing and analysing genetic switches. It should be noted that this theoremdoes not exclude the existence of unstable non-equilibrium solutions such as an unstable periodic orbit. However, such unstable non-equilibriumsolutions cannot be usually observed due to intracellular noise. Thus, in this sense, the theorem asserts that a genetic network composed of only positive feedback loops inevitably converges to stable equilibria. In addition, it is worth noting that this theoremcan be extended not only for networks with multiple time delays (see Appendix A) but also for some networks with non-positive feedback loops... STABILITY OF EQUILIBRIA The stability of equilibria is one of the most important factors for design of a genetic switch because the switch is required to stay at stable equilibria robustly to perturbations. However, Theorem. does not provide any information on the stability of each equilibriumpoint, although it asserts that eqn () does not have any attractors except equilibria. In general, it is much more difficult to determine the stabilities of equilibria in FDEs than those of ODEs due to the transcendental characteristic equations of FDEs. To overcome this problem, we derive the second theorem, which shows that we can use ODEs to equivalently analyse the stability in FDEs at equilibria. In other words, the stabilities of equilibria for both ODEs and FDEs are actually invariant, and we can even ignore the time delays. Theorem.. Let eqn () be the associated ODEs of eqn () obtained by ignoring all time delays of eqn (): xðtþ ¼FðxðtÞÞ DxðtÞ FðxðtÞÞ; ðþ where we set t ij ¼ 0 for all i; jan; and F is f in eqn () but without time delays. Then eqns () and () have the identical equilibria. Moreover, if eqn () satisfies all conditions of Theorem., then each corresponding equilibria of eqns () and () have the identical stability. Theorem. means that if there exists an equilibriumpoint that is asymptotically stable (or unstable) for eqn (), then it is also asymptotically stable (or unstable) for eqn (), and vice versa. Based on this theorem, instead of the complicated FDEs of eqn (), we can use the associated ODEs, eqn (), to design and analyse a genetic switch network only with positive feedback loops. By using the ODEs instead of the FDEs, we can significantly reduce complexity of the problemand make a design problemof a large-scaled multi-gene network tractable... A REDUCTION METHOD TO SIMPLIFY ANALYSIS OF GENETIC SWITCHES Although Theorems.. allow us to design or examine equilibria and their stabilities by the much simpler associated ODEs, it is still difficult to analyse nonlinear ODEs especially with high dimensions. To cope with this problem, we propose a reduction method to further simplify the ODEs to lower-dimensional ODEs but with the same equilibria and stabilities as the original system. Theorem.. Consider eqn () and its interaction graph IGð f Þ: Assume that the i-th node does not have any self-feedback loop, that is, an edge e ii :

7 MODELING GENETIC SWITCHES 85 Then by removing x i ¼ F i ðxþ d i x i from eqn () and by substituting x i ¼ F i ðxþ=d i into remaining equations, we obtain an n dimensional differential equations x 0 ¼ F 0 ðx 0 Þ D 0 x 0 ; ðþ where x 0 ¼ðx ; y; x i ; x iþ ; y; x n Þ; (A) F 0 ¼ðF ; y; F i ; F iþ ; y; F n Þ; D 0 ¼ diagðd ; y; d i ; d iþ ; y; d n Þ: The equilibria of eqn () correspond one to one to those of eqn (), and their stabilities are the same. In addition, the Jacobian matrix of eqn () is irreducible. Theorem. shows a procedure to reduce the dimension of a genetic network. By using this theorem, the associated ODEs can be reduced to a lower dimensional network step by step until all the remaining nodes of the interaction graph of the reduced network have self-feedback edges. In other words, all nodes without any selffeedback loop can be eliminated one by one according to Theorem. as illustrated in Fig.. The reduction procedure is interpreted by the following operations on the interaction graph. First, we remove the target node. In Fig. (A) and (B), the target node is the nd node. Then for all nodes fromwhich an edge goes out to the target node, we create new edges fromthe nodes to all nodes to which an edge goes in fromthe target node. The sign of each new edge is the same as that of the path from the start node of the new edge to the end node of the new edge through the target node in the original graph. In Fig. (A) and (B), there are two edges from the st and 5th nodes to the removed nd node and are three edges going out fromthe nd node to the st, 6th, and 7th nodes. Thus, new edges fromthe st node to the st, 6th, and 7th nodes and ones fromthe 5th node to the st, 6th, and 7th nodes are created. Here the edge fromthe st node to the st node is a positive self-feedback loop. In Fig. (A), because the sign of the edge fromthe st node to the nd node in the original graph is positive, the new edges fromthe st node to the st, 6th, and 7th nodes are positive, (B) Fig.. A schematic diagram of the reduction procedure described in Section.. The signs attached to arrows indicate the types of interactions. (A) The case that the signs of e and e are positive. (B) The case that the signs are negative. positive, and negative, respectively. On the other hand, in Fig. (B), because the sign of the edge fromthe st node to the nd node is negative, the new edges fromthe st node to the st, 6th, and 7th nodes are positive, negative, and positive, respectively. The low-dimensional ODEs finally obtained are easier to analyse than the original highdimensional ODEs. In addition, Theorem. indicates that a genetic network can be reduced to a minimal model in terms of the number of nodes with a dimension at least as low as the number of loops that the original network has. Figure is a schematic diagram of this method shown by the interaction graphs IGð f Þ; where each node corresponds to a variable of eqn (). By applying the proposed method, a node in IGð f Þ without any self-feedback loop can be eliminated, and the edges coming into and going out fromthis node are merged. Then, we finally obtain lower dimensional ODEs and the corresponding interaction graph with the number of nodes smaller than the original one of eqn (). For instance, a four-node network is eventually reduced to a one-node network with two positive self-feedback loops in Fig., which is a minimal model of the network. Figure 5 shows other examples. It should be noted that a reduced 5 6 7

8 86 T. KOBAYASHI ET AL. Fig.. A schematic diagram of the reduction method proposed in Section.. The original network with four components is reduced step by step to one with one component and two feedback loops. First, the th node is removed and the edges e and e are merged. Then, the nd and the rd nodes are removed in turn. Finally, we obtain a network with only the st node and two positive self-feedback loops. interaction graph can have two edges fromthe j-th node to the i-th node in the case of Fig. 5 (B). f i as a function of x j increases or decreases sigmoidally in the case of Fig. if the reduced graph has only one edge between two nodes as illustrated in Fig. 5(A). On the other hand, if the reduced graph has two or more edges from the j-th node to the i-th node as illustrated in Fig. 5(B), f i may not be sigmoidal. If i ¼ j; then the reduced graph has one node with one positive self-feedback loop in the case of Fig. 5(A), and the reduced graph has one node with two positive self-feedback loops in the case of Fig. 5(B). Because f i of the former is sigmoidal as illustrated by the thick solid curve in Fig. 5(C), such a systemhas at most three equilibriumstates. On the other hand, because f i of the latter is not necessarily sigmoidal as illustrated by the broken curve in Fig. 5(C), the j k i j k i j i j i (A) (B) 0 8 fi 6 (C) 5 xj Fig. 5. (A) A simple case of the reduction procedure where the k-th node is removed, and a new edge from the j-th node to the i-th node is created. (B) Another more complicated case of the reduction procedure, where after removal of the k-th node, a new edge fromthe j-th node to the i-th node is created, and the i-th node has two edges fromthe j-th node. (C) A reduced f i as a function of x j : When the reduced interaction graph is composed of only the i-th node with only one selffeedback loop, then f i is sigmoidal as depicted by a solid curve. When the i-th node in the reduced interaction graph has two self-feedback loops, then f i as a function of x i is no longer simply sigmoidal, and its shape can be complex as depicted by a broken curve. A solid thin line represents the degradation of the i-th chemical component. The intersection points of the solid line and f i are equilibriumpoints of the system. The equilibriumpoints of one and two feedback loops are designated by closed circles and open circles, respectively.

9 MODELING GENETIC SWITCHES 87 system may have three or more equilibrium states. This implies that the number of feedback loops of a reduced network influences the number of equilibrium states of the network. In the case where a reduced network has two or more edges between two nodes, the similar result holds. It is not easy to figure out intuitively how many equilibrium states a network has just from observation of differential equations that represent the dynamics of the network. Thus, the interaction graph has advantages not only in giving a graphical interpretation of the reduction method but also in allowing us to extract intuitively information on the number of equilibriumstates of the network... A PROCEDURE TO DESIGN GENETIC SWITCHES Theorem. instructs us how to reduce the dimensionality of a genetic network to simplify the analysis and the computation of the associated ODEs. However, when we design a genetic switch, it is convenient for us to start with a minimal model satisfying all requirements, and then to extend the model to biologically plausible ones with higher dimensions. Namely, we reverse the procedure of Section. by increasing the dimensionality of the network in this section. The following theoremshows how to extend a genetic switch model while preserving equilibria and their stabilities that are the important properties of a genetic switch. Theorem.. Let a transformation from eqn () to eqn () be x i ¼ F i ðx 0 Þ=d i. x i ¼ F i ðxþ d i x i : ðþ Assume that eqns () and () satisfy Assumptions..6, and that the orbits of eqns () and () have a compact closure in the state spaces. Then eqns () and () have the same equilibria whose stabilities are identical. Based on Theorem., we can design a genetic switch as follows:. Design a switch with the simplest or minimal model satisfying the requirements for configuration, equilibria, and their stabilities, even if such a model may not be plausible from a biological viewpoint.. Add components that satisfy the assumptions required in Theorem. one by one to the model in order to make the model more plausible and easier to implement experimentally. According to Theorem., the enlarged model preserves the static properties of the systemin terms of equilibria and their stabilities. This procedure is schematically described in Fig. 6, and can be viewed as a reverse procedure 6 Fig. 6. A schematic diagram of the designing procedure proposed in Section.. The original simple network only with two components is enlarged by adding components to obtain a biologically plausible network. First, the rd and the th nodes are added, and then the 5th and the 6th nodes are added. 5

10 88 T. KOBAYASHI ET AL. of the reduction method in Section.. In Fig. 6, starting with an abstract model of a genetic switch and the corresponding interaction graph, we obtain a biologically plausible enlarged model by adding components and edges to the interaction graph. Note that we do not need to add time delays into the model because Theorems.. guarantee that the systems with and without time delays have the identical equilibria and stabilities.. Implementation and Numerical Simulation In this section, we demonstrate our theoretical results by designing a genetic switch with three or four stable equilibria. First, we start with an abstract genetic switch with two components, as shown in Fig. 7(A) where all feedback loops are positive. Simple algebraic analysis shows that this network can have three or four stable equilibria. By applying the procedure proposed in Section., we extend the abstract two-node network to a realistic three-node network as illustrated in Fig. 7(B). We adopt three different proteins to the three nodes of this network as in Fig. 7(C). It should be noted that the nodes of Fig. 7(C) are not genes but the concentrations of proteins because the nodes of Fig. 7(B) represent the variables of the FDEs that determine the dynamics of the network. Furthermore, we extend Fig. 7(C) by incorporating the corresponding mrnas as in Fig. 7(D). An example of a real implementation of Fig. 7(D) is shown in Fig. 8 where three pairs of the genes and the promoters are adopted (Araki, 00, private comm.). In this switch, we use laci, tetr, and ci genes, and P LtetO ; Ptrc ; and P RM promoters. In fact, laci and tetr genes with P LtetO and Ptrc promoters were artificially engineered (Lutz & Bujard, 997) and used to construct a two-state toggle switch (Gardner et al., 000). On the other hand, the wild-type P RM promoter has three binding sites, i.e. O R ; O R ; and O R ( Ptashne, 99). In our model, the binding site O R of the P RM promoter is assumed to be artificially altered or mutated so that CI proteins cannot bind to O R : Although there is no detailed experimental report on the mutated P RM ; an (A) LacI TetR CI (C) (D) mlaci (B) LacI TetR CI mtetr mci mtetr mci Fig. 7. (A) A simple model with two components and three feedback loops. Each node has a positive selffeedback loop, and their mutual interactions form a positive feedback loop. (B) An extension of the simple model in Fig. (A). The rd node is added in order to replace the positive self-feedback loop of the st node in Fig. (A) with mutual negative interactions between the st and the rd nodes. (C) A schematic diagram of a realization of the enlarged model in Fig. (B). LacI, CI, and TetR proteins are adopted to the st, the nd and the rd nodes in Fig. (B), respectively. The broken line indicates the feedback loop with proteins LacI and TetR, which is identical to a toggle switch examined in Gardner et al. (000). The bold line indicates a self-feedback loop of ci : (D) A further extension of Fig. (C), which includes mrnas. LacI laci Ptrc- PLtet O- tetr TetR ci CI PRM tetr Fig. 8. An implementation of the model with two components and three feedback loops in Fig. 7 with genes laci, tetr, and ci and promoters P LtetO ; Ptrc and P RM where the mrnas of laci, tetr, and ci are omitted for simplicity. The signs indicate the types of interactions between the proteins LacI, TetR, and CI. ðtetr ; tetr Þ and ðci ; ci Þ are the same tetr, and ci genes but with different promoters and RBSs. ci

11 MODELING GENETIC SWITCHES 89 abundance of experimental data on CI proteins and the wild-type P RM promoter allows us to predict their quantitative behaviors. Due to the mutated O R site, protein CI is expected to activate only the expression of the genes located downstreamof the mutated P RM ; by binding to O R and O R sites of the mutated P RM : As indicated in Fig. 8, after dimerization, the dimers of protein CI enhance promoter P RM and those of protein TetR repress promoter P LtetO ; respectively, whereas proteins LacI forms tetramers to inhibit promoter Ptrc : Adopting operons in Fig. 8 has an additional advantage, i.e. we do not need to construct OR and AND gates to connect the genes in the network of Fig. 7(C). To formulate the mathematical equations, we distinguish the concentrations of mrnas of tetr and ci genes with different ribosome binding sites (RBSs) or promoters by subscripts such as m tetr and m tetr : However, we do not distinguish proteins TetR and CI translated fromthese mrnas because the difference for the mrnas is only their RBSs. Then, the FDEs for this network can be described as follows: dm laci dt dp laci dt dm tetr dt ¼ n PLtetO f PLtetO ð p tetr ðt t PLtetO ÞÞ d mlaci m laci ; ¼ s mlaci m laci ðt t laci Þ d placi p laci ; ¼ n Ptrc f Ptrc ð p laci ðt t Ptrc ÞÞ d mtetr m tetr ; dm ci dt dp ci dt ¼ n PRM f PRM ðp ci ðt t PRM ÞÞ d mci m ci ; ¼ s mci m ci ðt t ci Þ þ s mci m ci ðt t ci Þ d pci p ci ; where m and p indicate concentrations of mrnas and proteins of genes assigned by subscripts. s and d indicate synthesis coefficients of proteins, and degradation rates of mrnas and proteins, respectively. n PLtetO ; n Ptrc ; and n PRM are the plasmid copy numbers. f s indicate synthesis rates of mrnas enhanced or repressed by the corresponded proteins and promoters. ts represent time delays due to transcriptions, translations, and translocations. All f s are monotone functions. By applying the proposed procedure in Theorem. and transforming the equations properly, we obtain the following two-dimensional differential equations, which preserve equilibria and their stabilities of the original network: dp tetr dt dp ci dt where ¼ f ð p tetr Þþe tetr f ð p ci Þ p tetr ; ¼ k½e ci f ð p tetr Þþf ð p ci Þ dp ci Š; f ðp tetr Þ¼ s m tetr n Ptrc f Ptrc d mtetr d ptetr s mlaci n PLtetO f PLtetO ð p tetr Þ ; d mlaci d placi ð5þ ð6þ dm tetr dt ¼ n PRM f PRM ð p ci ðt t PRM ÞÞ d mtetr m tetr ; f ðp ci Þ¼ s m tetr n PRM d mtetr d ptetr f PRM ð p ci Þ; dp tetr dt ¼ s mtetr m tetr ðt t tetr Þ þ s mtetr m tetr ðt t tetr Þ d ptetr p tetr ; e tetr ¼ s m tetr s mtetr ; e ci ¼ s m ci s mci ; k ¼ d m tetr s mtetr s mci d mci ; and dm ci dt ¼ n Ptrc f Ptrc ð p laci ðt t Ptrc ÞÞ d mci m ci ; d ¼ d p ci kd ptetr :

12 90 T. KOBAYASHI ET AL. Time t is implicitly normalized by the half-life time of protein TetR. Since f Ptrc ðp laci Þ and f PLtetO ðp tetr Þ are monotonously decreasing functions but f PRM ðp ci Þ is a monotonously increasing function, both f ðp tetr Þ and f ðp ci Þ are monotonously increasing functions, which satisfy the conditions of Theorems... The necessary condition for the genetic switch to have three or four stable equilibria is that each sub-network denoted by the dashed line and the bold line in Fig. 7(C) has one stable equilibrium at a low state or two stable equilibria, which means that the null-clines of eqns (5) and (6) can be bimodal. The sub-network denoted by the broken line is identical to a toggle switch in (Gardner et al., (000)). According to the previous work, this sub-network can have two stable equilibria by inserting proper RBSs to tetr and laci genes (Gardner et al., 000) if isolated fromthe other components of the switch network. Therefore, if we set e tetr small enough, namely, we choose a proper RBS for tetr gene, the null-cline of eqn (5) can be bimodal. On the other hand, the sub-network indicated by the bold line in Fig. 7(C) has only one stable equilibriumat a high expression level because the activation of the protein CI is too strong. In other words, this sub-network does not satisfy the condition of only one stable equilibriumat a low state or two stable equilibria. However, by changing the RBS of the ci gene with weaker one or by using a temperature-sensitive ci mutant whose lifetime is much shorter than that of the wild-type ci, the expression level of CI can be reduced, and the sub-network indicated by the bold line in Fig. 7(C) can have one stable equilibriumat a low expression level (Isaacs et al., 00). Hence, although the sub-network indicated by bold line in Fig. 7(C) has only one stable equilibriumpoint at a high expression level without any repressive factors, we could realize one stable equilibriumat a low expression level or two stable equilibria by adding such a proper RBS to the ci gene or by using a temperature-sensitive ci gene for ci in Fig. 8. It should be also noted that the order of the genes in a polycistronically transcribed set can influence this condition because transcriptional efficiency can be affected by such an order of the genes. If we set e ci small enough by choosing a proper RBS for ci ; the null-cline for eqn (6) can be bimodal. Note that the switch of Fig. 8 can be viewed as an extension of the previous two-state toggle switch (Gardner et al., 000) due to the subnetwork with genes tetr and laci. To demonstrate qualitative analysis of the genetic switch, we numerically analyse the model of Fig. 8 and calculate null-clines of eqns (5) and (6). First, we normalize p laci ; p tetr ; and p ci as p laci ðnmþe8p 0 laci ; p tetrðnmþe8p 0 tetr ; and p ci ðnmþe8p ci : Then, we set p 0 laci ð p0 tetr Þ¼5n P LtetO þ p 0 ; tetr f ð p 0 tetr Þ¼5:n P trc þ p 0 laci ðp0 tetr Þ; f ð p 0 ci Þ¼5:n þ p 0 ci þ p0 ci P RM þ p 0 ci þ : ð7þ p0 ci These functions are calculated using the parameter values in Hasty et al. (00) under the following assumptions: () CI dimers cannot bind to the O R site; () the equilibriumconstant of TetR dimerization is the same as that of CI; () the equilibriumconstant of TetR dimers and P LtetO is the same as that of CI dimers and the O R site of P RM ; () the reaction rates of protein production fromp LtetO and Ptrc without TetR dimers and LacI tetramers are the same as that of P RM without CI dimers; (5) the degradation rates of TetR and LacI are 0:005 s that is set to be the same as that of Cro (Arkin et al., 998). Figure 9 shows that eqns (5) and (6) have three stable equilibria, ðtetr; ciþ ¼ðOFF; OFF Þ; ðoff; ONÞ; ðon; ONÞ; and Theorems.. guarantee that this designed switch converges to a stable equilibriumeven with uncertain delays, where e tetr ¼ 0:; e ci ¼ 0:; k ¼ ; d ¼ 5; n PLtetO ¼ 6; n Ptrc ¼ ; and n PRM ¼ : Notice that it is not difficult to obtain three stable equilibria by setting e tetr and e ci small if the two sub-networks have two stable equilibria or one stable equilibriumat a low expression level. In addition, by inserting a proper RBS to a gene, the translation of the gene is enhanced about 0- folds or more. Thus, in this sense, e tetr ¼ 0: and e ci ¼ 0: are reasonable, and this switch may

13 MODELING GENETIC SWITCHES 9 5 CI (OFF,ON) (ON,ON) (OFF,OFF) TetR Fig. 9. Geometric structures of the null-clines of eqns (5) (broken curves) and (6) (solid curves) calculated by eqn (7) with e tetr ¼ 0:; e ci ¼ 0:; k ¼ ; and d ¼ : The circles indicate stable equilibria. The switch has three equilibriumstates, namely, ðoff; OFFÞ; ðoff; ONÞ; and ðon; ONÞ: Both the concentrations of proteins TetR and CI are normalized by their representative values (see the text). be implemented experimentally without much difficulty. In addition, by setting other parameter values, we also have four stable equilibria, ðtetr; ciþ ¼ ðoff; OFFÞ; ðon; OFFÞ; ðoff; ONÞ; ðon; ONÞ as shown in Fig. 0 where e tetr ¼ 0:; e ci ¼ 0:; k ¼ ; d ¼ 0; n PLtetO ¼ 6; n Ptrc ¼ 5; and n PRM ¼ : Notice that the number of equilibria and their ON OFF combinations can be controlled theoretically by two experimentally regulable parameters e tetr and e ci provided that both sub-networks indicated by the dashed and bold lines in Fig. 7(C) have one stable equilibriumat a low state or two stable equilibria; this implies that the genetic switch proposed in this section can be implemented independently of its components. In other words, we can implement such a switch using other three genes, g A ; g B ; and g C rather than tetr, laci, and ci only if g A and g B function as a toggle switch and g C forms a positive selffeedback loop with bistability or one equilibrium point at a low state. In addition, the plasmid copy numbers also can be controllable parameters. This flexibility in implementation is a significant advantage to design genetic networks with a simple abstract model and to extend it to a biologically plausible one in particular when constructing large-scaled genetic switch networks. 5. Conclusion In this paper, we first have proven that genetic networks only with positive feedback loops have desirable properties as switches by using monotone dynamical theory. That is, networks only with positive feedback loops have no stable oscillation but stable equilibria whose stabilities are independent of time delays. Due to such properties, a genetic switch only with positive feedback loops can be analysed or designed without consideration of time delays and existence of non-equilibriumstates. This makes theoretical analysis and design of switches much easier and simpler. In addition, the fact that the

14 9 T. KOBAYASHI ET AL. 5 CI (OFF,ON) (ON,ON) (OFF,OFF) (ON,OFF) Fig. 0. Geometric structures of eqns (5) (broken curves) and (6) (solid curves) calculated by eqn (7) with e tetr ¼ 0:; e ci ¼ 0:0; k ¼ ; and d ¼ : The circles indicate stable equilibria. The switch has four equilibriumstates, namely, ðoff; OFFÞ; ðon; OFFÞ; ðoff; ONÞ; and ðon; ONÞ: TetR results are independent of time delays also ensures that the designed switch is robust to delay variations. After proving the basic properties of the genetic networks only with positive feedback loops, we next have developed a new procedure to design genetic switch networks with multiple genes and time delays, based on the theorems proven in this paper. The proposed method designs a genetic switch, starting with a simple minimal model that is easy to analyse, and then extending the model to a biologically plausible one with the same equilibria and stabilities as the original one. Thus, the procedure drastically simplifies the process of designing a genetic switch and makes theoretical analysis and design tractable even for large-scaled systems. It may be worth noting that these theorems and the procedure also provide a theoretical background for many reduction methods, which are applied in the previous works. Moreover, the proposed method is particularly powerful when applied to eukaryotic genetic networks. One reason is because time delays are ubiquitous and influential in eukaryotes. Another is because the method allows us to model networks even if we do not have full information on the pathway of interactions. For example, we can still construct a genetic switch, if we only know that some transcriptional factors eventually activate transcription of certain genes but lack detailed information on how the transcriptional factors are translocated in the cytoplasmand the nucleus. Therefore, our reduction methods ensure that we can model a network provided that we have partial information on eventual effects of interactions. Finally, to demonstrate our theoretical results, we have designed realistic synthetic switches with three or four equilibriumstates using experimentally well investigated laci, tetr, and ci genes and their corresponding promoters, for numerical simulation. By virtue of the proposed procedure, the design of these switches is reduced to analysis of the two-dimensional ODEs. Furthermore, the theoretical analysis

15 MODELING GENETIC SWITCHES 9 and the numerical simulation show that the proposed model can have three or four stable equilibriumstates only by properly adjusting two parameters, the RBSs of tetr and ci : In this paper, we have examined genetic switches without consideration of stochastic noise, which is also an important factor in the dynamics of genetic networks. A recent integrated study of a theoretical model and a de novo synthesized genetic network shows that the ci positive feedback loop is easily subject to such a stochastic effect than the toggle switch composed of laci and tetr genes (Hasty et al., 000; Isaacs et al., 00). Therefore, it is an important future problemto understand how stochastic effects influence the behaviors of the switch. In addition, it is also necessary to investigate how time delays affect stochastic nature in genetic networks via both theoretical and experimental approaches. We express special thanks to Dr M. Araki for modeling of the three-gene network, and thanks Drs J. J. Collins, N. Kopell, J. Distefano, F. Arnold, Y. Yokobayashi, N. Ichinose, and Mr Y. Morishita for their helpful discussion. This research was supported by JSPS Research Fellowships for Young Scientist and by the Scientific Research fromthe Ministry of Education, Science and Culture, Japan, under Grant REFERENCES Agha-Mohammadi, S.& Lotze, M. T. (000). Regulatable systems: applications in gene therapy and replicating viruses. J.Clin. Invest. 05, Arkin, A., Ross, J.& McAdams, H. H. (998). Stochastic kinetic analysis of developmental pathway bifurcation in phage l-infected Escherichia coli cells. Genetics 9, Becskei, A., S!eraphin, B.&Serrano, L. (00). Positive feedback in eukaryotic gene networks: cell differentiation by graded to binary response conversion. EMBO J. 0, Becskei, A.& Serrano, L. (000). Engineering stability in gene networks by autoregulation. Nature 05, Chen, L. & Aihara, K. (00a). Stability of genetic regulatory networks with time delay. IEEE Trans. Circuits Syst. I 9, Chen, L. & Aihara, K. (00b). A model of periodic oscillation for genetic regulatory systems. IEEE Trans. Circuits Syst. I 9, 9 6. Drew, D. A. (00). A mathematical model for prokaryotic protein synthesis. Bull. Math. Biol. 6, 9 5. Elowitz, M. B. & Leibler, S. (000). A synthetic oscillatory network of transcriptional regulators. Nature 0, 5 8. Gardner, T. S., Cantor, C.R.&Collins, J. J. (000). Construction of a genetic toggle switch in Escherichia coli. Nature 0, 9. Hale, J. K. & Lunel, S. M. V. (99). Introduction to Functional Differential Equations. Berlin: Springer- Verlag. Hasty, J., Dolnik, M., Fer,V.R.&Collins, J. J. (00). Synthetic gene network for entraining and amplifying cellular oscillations. Phys. Rev. Lett. 88,. Hasty, J., Isaacs, F., Dolnik, M., McMillen, D. & Collins, J. J. (00a). Designer gene networks: towards fundamental cellular control. Chaos, Hasty, J., McMillen, D., Isaacs, F.&Collins, J.J. (00b). Computational studies of gene regulatory networks: in numero molecular biology. Nat. Rev. Genet., Hasty, J., Pradines, J., Dolnik, M.&Collins, J.J. (000). Noise-based switches and amplifiers for gene expression. Biophysics 97, Imhof, M. O., Chatellard, P.&Metmod, N. (000). A regulatory network for the efficient control of transgene expression. J. Gene Med., Isaacs, F. J., Hasty, J., Cantor, C.R.&Collins, J.J. (00). Autocatalytic gene expression: in numero quantitative predictions, in vivo stochastic dynamics. Preprint. Lutz, R. & Bujard, H. (997). Independent and tight regulation of transcriptional units in Escherichia coli via the LacR/O, the TetR/O and AraC=I I regulatory elements. Nucleic Acid Res. 5, 0 0. McMillen, D., Kopell, N., Hasty, J.&Collins, J.J. (00). Synchronizing genetic relaxation oscillators by intercell signalling. Proc. Natl. Acad. Sci. U.S.A. 99, Ptashne, M. (99). A Genetic Switch, nd edn. Cambridge, Cell and BSI. Shea,M.A.&Ackers, G. K.(985). The o r control system of bacteriophage lamda a physical chemical model for gene regulation. J. Mol. Biol. 8,. Simpson, M. L., Sayler, G. S., Fleming, J. T. & Applegate, B. (00). Whole-cell biocomputing. Trends Biotechnol. 9, 7. Smith, H. (986). Competing subcommunities of mutualists and a generalized Kamke theorem. SIAM J. Appl. Math. 6, Smith, H. (995). Monotone Dynamical Systems, Vol.. American Mathematical Society: Providence, RI. Smith, H.& Thieme, H. (99). Strongly order preserving semiflows generated by functional differential equations. J. Differential Equations, 9, 6. Smolen, P., Baxter, D. A. & Byrne, J. H. (000). Mathematical modelling of gene networks. Neuron 6, Smolen, P., Baxter, D. A. & Byrne, J. H. (00). Modelling circadian oscillations with interlocking positive and negative feedback loops. J. Neurosci., Snoussi, E. H. (998). Necessary conditions for multistationarity and stable periodicity. J. Biol. Syst. 6, 9.

16 9 T. KOBAYASHI ET AL. Weiss, R. (00). Cellular computation and communications using engineered genetic regulatory networks. Ph.D. Thesis, Massachusetts Institute of Technology. Wolf, D.M.&Eeckman, F. H. (998). On the relationship between genomic regulatory element organization and gene regulatory dynamics. J. theor. Biol. 95, Define the orbit of eqn () for the initial condition fac þ as O þ ðfþ ¼fx t ðfþ : tx0g: Appendix A Then O þ ðfþ is precompact in C þ omega limit set, oðfþ; is defined by and the A.. PROOFS OF THEOREMS In this appendix, we give detail proofs for the main results. All the assumptions and the definitions are mathematically represented here, and some of them are generalized to allow multiple time delays. We prove some theoretical results under these generalized conditions with multiple time delays. Assumption A.. The solution x t ðfþ of eqn () is defined for tx0 and jjx t jjon for all tx0: Assumption A.. The synthesis rates f ðx t Þ in eqn () are bounded if x t is bounded. Assumption A.. The synthesis rates f ðx t Þ in eqn () depend on xðt þ sþ; sa½ r; 0Š at a finite number of time instants s: These assumptions are mathematical representations of Assumptions.. in Section. A... Properties of Solutions for eqn () With Assumptions A. A., the following results hold for eqn ():. Equation () generates a (local) semiflow F on C þ by Fðt; fþ ¼x t ðfþ; fac þ for tx0 (Smith & Thieme, 99).. C þ is positively invariant for eqn (), that is, for arbitrary tx0; Fðt; C þ ÞCC þ holds (Smith & Thieme, 99). oðfþ ¼ \ sx0 fx t ðfþ : txsg: In addition, oðfþ is non-empty, compact, connected, and invariant (Smith & Thieme, 99). Next, we give mathematical representations of Assumptions..6, which are generalized to allow multiple discrete time delays. Assumption A.. Suppose that the concentrations of the j-th chemical component at an l ij number of different time instants, x j ðt t ij Þ; x jðt t ij Þ; y; x jðt t l ij ij Þ; affect the synthesis rate of the i-th chemical component at t: Without loss of generality, we set t ij ot ij o?otl ij ij : Then, for f iðx t Þ¼f i ðx t Wx j ðt t k ij Þ; x jðt t k ij ÞÞ; qf iðx t Wx j ðt t k ijþ; yþ=qy X0 on C þ ; or qf i ðx t Wx j ðt t k ij Þ; yþ=qyp0 oncþ holds for i; jan and kaf; y; l ij g; where x t Wx j ðt t k ij Þ indicates all values of x t except x j ðt t k ij Þ: This assumption is a more general mathematical representation of Assumption. in Section. Definition A. (Types of interactions). Suppose that the concentration of the j-th chemical component affects the synthesis rate of the i-th chemical component where iaj: Define f i as f i ðx t Þ¼f i ðx t Wx j ; x j ðt t ij Þ; y; x jðt t l ij ij ÞÞ; where x t Wx j is x t without x j ðt t ij Þ; x jðt t ij Þ; y; x j ðt t l ij ij Þ: Focusing on the dependency of f i on x j ðt t k ij Þ; we define s k ij ðx tþ; the type of an interaction between the i-th and the j-th components at x t ;

17 MODELING GENETIC SWITCHES 95 as follows: 8 þ >< s k ij ðx tþ¼ >: : ið * ;y; * ;x j ðt t k ij Þ; * ;y; * j ðt t k ij Þ j xt 0; : ið * ;y; * ;x j ðt t k ij Þ; * ;y; * Þ j ðt t k ij Þ xt o0; 0 : ið * ;y; * ;x j ðt t k ij Þ; * ;y; * j ðt t k ij Þ j xt ¼ 0: ða:þ If s k ij ðx tþ¼ ðor Þ; then the j-th chemical component is said to affect positively (or negatively) the i-th component with time delay t k ij at x t : Because of Assumption A., s k ij ðx tþx0 ðor p0þ for all x t AC þ : In general, s k ij ðx tþ may not be equal to s k0 ij ðx tþ for kak 0 : However, since this situation is not so common for genetic networks, we assume the following for the sake of simplicity: Assumption A.5. The j-th chemical component affects the i-th component either positively or negatively, namely, s k ij ðx tþs k0 ij ðx tþx0 for k; k 0 Af; y; l ij g and x t AC þ : We set s ij s ij : In addition, we make the following assumption: Assumption A.6. If s ij a0 and t l ij s l ij ij ðx tþa0 for all x t AC þ : ij 0 This assumption is biologically plausible. then Definition A. (Interaction graph). An interaction graph at x t ; IG xt ðf Þ; of a genetic network defined by eqn () is a directed graph whose nodes represent the individual chemical components of the genetic network and whose edges represent the interactions between the nodes. When s k ij ðx tþa0 and t k ijx0; that is, the j-th chemical component affects the synthesis rate of the i-th chemical component with time delay t k ij ; the graph has an edge, ek ij ðx tþ; directed from the j-th node to the i-th node. In addition, irreducibility of a graph is defined as follows. Definition A. (Irreducibility). IG xt ð f Þ is said to be irreducible when there is at least one path pði; jþ ¼ðj ¼ p - kpp e pp ðx t Þ? - p i ¼ iþ kpipi e ðx t Þ pipi p - kpp e pp ðx t Þ fromthe j-th node to the i-th node for all i; janðiajþ where p ; y; p i AN and e k p b pa p b p a ðx t Þ is the k pb p a -th edge fromnode p a to node p b : Assumption A.7. IG xt ð f Þ of eqn () is irreducible for all x t AC þ : Definition A. (Feedback loops and their types). If a path fromthe i-th node of an interaction graph, IG xt ð f Þ; to the same i-th node, pði; jþ 0 B i ¼ p - kpp e pp ðx t Þ p - kpp e pp ðx t Þ? - kpipi e pipi ðx t Þ C p i ¼ ia exists, then this path is said to be a feedback loop and furthermore be a self-feedback loop when i is. In addition, this feedback loop is said to be positive Q (or negative) if i m¼ s p mþ p m ðx t Þ¼ (or ). Q i m¼ sk p mþ ;pm p mþ p m ðx t Þ¼ Assumption A.8. The interaction graph IG xt ð f Þ of the model in eqn () has only positive feedback loops for all x t AC þ : A genetic network with only positive feedback loops may have both positive and negative interaction edges, which make analysis complicated. Then, we consider a coordinate transformation to reduce the original genetic network with only positive feedback loops into a equivalent one with only positive interaction edges. Choose a node j arbitrarily. First, we set s j ¼ : If s ij ðx t Þ¼ ðor Þ for some x t AC þ ; then set s i ¼ ðor Þ: s i is well defined because all paths fromthe j-th node to the i-th node have the same sign even if the path is allowed to include loops. Using s i ; we define a transformation P described

18 96 T. KOBAYASHI ET AL. by a matrix as follows: 0 s 0 B C P & A 0 s n Using P; we define a coordinate transformation of C ¼ Cð½ t; 0Š; R n Þ as PðCÞ Cð½ t; 0Š; PðR n ÞÞ; and set C 0 ¼ PðC þn Þ: Then, we transform f to gac ðc 0 ; R 0 Þ as follows: g ¼ Pf P; where we use P ¼ P and set R 0 ¼ PðR þn Þ: Using g; we define another FDEs on C 0 as follows: yðtþ ¼gðy t Þ DyðtÞ gðy t Þ with yac 0 : ða:þ i =@x j ðt t ij Þ¼s i s i =@x j ðt t ij Þ and s i and s i =@x j ðt t ij Þ have the same i =@x j ðt t ij ÞX0: Thus, eqn (A.) has only positive interaction edges, and each s ij of IGðgÞ is always (Snoussi, 998). Since P merely reverses the directions of some axes of coordinates, eqn (A.) has the same dynamical properties as eqn (). Since all the theoretical results proven for eqn (A.) also hold for eqn (), we prove the theorems only for eqn (A.). We also make some definitions that are used in the proofs of the theorems. Definition A.5 (Equilibria). The set of equilibria for eqn () is defined as e f ¼ffAC þ : f ¼ xˆ for some xar þn satisfying fðxþ ˆ ¼0g; where xˆ is the natural inclusion from xar þn to xac ˆ þ by xˆ t ðyþ ¼xð tpyp0þ: Definition A.6 (The set of convergent points). The set of convergent points E f of eqn () is defined as E f ¼ffAC þ : oðfþ ¼fˆ xg for some xae ˆ f g: The set of convergent points E g is similarly defined for eqn (A.), and E f ¼ PðE g Þ holds. Definition A.7 (Quasi-positive matrix). An n n matrix B is said to be quasi-positive when B þ li n X0 for all sufficiently large l or when all components of B except diagonal components are nonnegative, where I n is an n n identity matrix. Note that a quasi-positive matrix is different from a positive semi-definite matrix. Definition A.8 (Irreducible matrix). An n n matrix A is irreducible when for every nonempty, proper subset N 0 of N; there exists an ian 0 and a janwn 0 such that ½AŠ ij a0: A... Proofs of the Main Results Definition A.9. (Stable point). fac þ is a stable point if for every e0 there exists d0 such that jjfðt; fþ Fðt; cþjjoe; for t whenever cac þ and jjf cjjod: We define S f to be the set of stable points of eqn (). Note that if fas f then points near f have limit sets near oðfþ: The set of stable points of eqn (A.), S g ; is similarly defined, and S g ¼ PðS f Þ and S f ¼ PðS g Þ hold. The following theorem(theorem. in (Smith (995))) is crucial for proving Theorem.: TheoremA. (Convergence to equilibria). Assume that eqn (A.) is cooperative and irreducible in C 0 and the following condition ðtþ holds. Then S g CE g ; and Int S g is dense in C 0 for the semiflow generated by eqn (A.). (T) g maps bounded subsets of C 0 to bounded subsets of R 0 : For each fac 0 ; yðt; fþ is defined for tx0 and O þ ðfþ is bounded. For each compact subset A C 0 of C 0 ; there exists a closed and bounded subset B C 0 of C 0 such that oðfþdb C 0 for every faa C 0: Proof of Theorem.. First, (T) holds, because of Assumptions A. A..

19 MODELING GENETIC SWITCHES 97 Second, we show that eqn (A.) is cooperative. Equation (A.) is cooperative in C 0 if C 0 is order convex and for all fac 0 and every c such that cx0 and c i ð0þ ¼0 for ian; dgðfþcx0; where y 0 ¼ðy ; y; y i ; y iþ ; y; y n Þ; G 0 ¼ðG ; y; G i ; G iþ ; y; G n Þ; where dgðfþ is the derivative of g at f: Because of the definition of P and Assumptions A., A. and A.5, it is easy to see that eqn (A.) is cooperative. Finally, we show that eqn (A.) is irreducible. Equation (A.) is irreducible when for all y t AC 0 ðdgðy t Þeˆ ; y; dgðy t Þeˆ n Þ; is irreducible where ðe ; y; e n Þ is an n n identity matrix, and when for all y t AC 0 and all jan; there exists ian such i ðy t Wy j ðt t l ij ij Þ; y jðt t l ij ij Þ y j ðt t l ij ij Þ 0: Obviously, these are satisfied because of Assumptions A.6 and A.7. & Proof of Theorem.. Assume that eqn (A.) satisfies all the conditions of TheoremA.. We define the associated ODEs of eqn (A.) obtained by ignoring all time delays as yðtþ ¼GðyðtÞÞ DyðtÞ GðyðtÞÞ; ða:þ where we set t ij ¼ 0 for all i; jan; and G is an induced function of g without time delays. By applying Corollary 5. in Smith (995) to eqns (A.) and (A.), it can be shown that eqns (A.) and (A.) have identical equilibria, and that each equilibriumof eqns (A.) and (A.) has identical stability. & Proof of Theorem.. Assume that the i-th node of IGðf Þ defined by eqn () does not have a selffeedback loop, that is, an edge e ii ; then the i-th node of IGðgÞ also does not have a self-feedback loop. By removing y i ¼ G i ðyþ d i y i fromeqn (A.) and by substituting y i ¼ G i ðyþ=d i into remaining equations, we obtain an n dimensional differential equation y 0 ¼ G 0 ðy 0 Þ D 0 y 0 ; ða:þ D 0 ¼ diagðd ; y; d i ; d iþ ; y; d n Þ: Because y ¼ Px and F ¼ PGP; the operation to derive eqn (A.) fromeqn (A.) directly corresponds to that to derive eqn () fromeqn (). Thus, results proven for eqns (A.) and (A.) also hold for eqns () and (). The irreducibility of the reduced interaction graph is obvious because of the operation to the interaction graph shown in Fig.. In this proof, we use the following property of a quasi-positive matrix. For a quasi-positive matrix A; sðaþ ¼ max Relo0 is equivalent to the condition that there exists u0 such that Auo0 where l runs over the eigenvalues of A and sðaþ is the stability modulus of A (Smith, 986). Since G i =@y j ¼ P l k¼ gk ijx0 holds for all l ¼ ; ; y; l ij where i; jan; iaj; the Jacobian matrix of GðyÞ is quasi-positive. Moreover, the Jacobian matrix of GðyÞ is irreducible on R 0 because of Assumption A.7. If a solution yðt; fþ of eqn (A.) with an initial condition far 0 has a compact closure for tx0 then almost all solutions starting at a point on R 0 converge to equilibria (Smith, 995). The Jacobian matrix of GðyÞ at an equilibriumis 0 G y ^ y ^ & G i i A ¼ G i y G ii G ii G iiþ y D B A G iþ ^ ¼ J D: G iþi ^ ða:5þ We do not explicitly indicate the equilibrium point at which A and J are calculated for readability. Note that because of the assumption

20 98 T. KOBAYASHI ET AL. that the i-th node does not have the edge e ii ; that is, G ii ¼ 0; ½AuŠ i ¼ Xi k¼ G ik u k þ Xn k¼iþ G ik u k d i u i ; ða:6þ holds for arbitrary uar n : In addition, we define J 0 to be a matrix obtained by removing the i-th row and the i-th column from J of eqn (A.5) as 0 G y G i G iþ y ^ & ^ ^ J 0 G i y G i i G i iþ ¼ : G iþ y G iþi G iþiþ ^ & ^ ^ & C A G n y G ni G niþ y ða:7þ Because of these definitions, for an arbitrary n-dimensional real vector v ¼ðv ; y; v n Þ and ðn Þ-dimensional real vector v 0 ¼ ðv ; y; v i ; v iþ ; y; v n Þ; ½JvŠ j ¼½J 0 v 0 Š j þ G ji v i ; ½DvŠ j ¼½D 0 v 0 Š j ; ða:8þ ða:9þ hold for j ¼ ; y; i ; i þ ; y; n: Next, let us define as G k ðyþ ¼G k ðy i ; y 0 Þ where y 0 ¼ðy ; y; y i ; y iþ ; y; y n Þ and kai: Notice that G i ðyþ ¼G i ðy 0 Þ because the i-th node of IGðgÞ has no self-feedback loop. By substituting y i ¼ G i ðy 0 Þ=d i into this equation, we obtain y k ¼ Gk 0 ðy0 Þ d k y k ¼ G k G i ðy 0 Þ; y 0 d k y k ; d i for k ¼ ; y; i ; i þ ; y; n: The derivatives of this equation at an equilibriumpoint are G kj þ d i G ki G ij X0 if jak; G kk þ d i G ki G ik d k if j ¼ k: Thus, the Jacobian matrix of eqn (A.) is 0 0 t G i G i ^ ^ A 0 ¼ J 0 D 0 þ G i i G ii ; d i G iþi G iiþ ^ CB A@ ^ C A G ni G in where t indicates the transpose. Apparently, A 0 is quasi-positive when A is quasi-positive. Assume that sðaþo0; then there exists u0 such that Auo0: Let u 0 be the ðn Þ-dimensional vector obtained by removing u i from u: Note that u 0 0: Then for all jaf; y; i ; i þ ; y; ng; ½A 0 u 0 Š j ¼½J 0 u 0 Š j ½D 0 u 0 Š j " # þ G X ji i G ik u k þ Xn G ik u k d i k¼ k¼iþ ¼½AuŠ j G ji u i " # þ G X ji i G ik u k þ Xn G ik u k d i ¼½AuŠ j þ G ji d i k¼ k¼ k¼iþ " # X i G ik u k þ Xn G ik u k d i u i k¼iþ ¼½AuŠ j þ G ji d i ½AuŠ i ; ða:0þ ða:þ where we use eqns (A.8), (A.9), and (A.6). According to eqn (A.) with Auo0 and G ji X0; we have A 0 u 0 o0: Thus, the theoremin Smith (995) shows that sða 0 Þo0 ifsðaþo0: Assume that sða 0 Þo0; then there is an ðn Þdimensional vector u 0 ¼ðu ; y; u i ; u iþ ; y; u n Þ such that A 0 u 0 o0: If G j of eqn (A.) does not depend on x i ; that is, G ji ¼ 0; then ½AuŠ j o0 because of eqn (A.0). Hence, we consider the case with G ji 0: Due to G ik X0 and u k 0; X i k¼ G ik u k þ Xn k¼iþ G ik u k 0; ða:þ