doi:10.1006/jtbi.2003.3190, available online at http://www.idealibrary.comon

## Modeling Genetic Switches with Positive Feedback Loops

### Tetsuya Kobayashi

n### w

### z

### , Luonan Ch eny

### and Kazuyuki Aihar aw

### u

wDepartment of Complexity Science and Engineering, Graduate School of Frontier Sciences, The University of Tokyo,Hongo 7-3-1,Bunkyo-Ku,Tokyo113-8656,JapanzJSPS Research Fellow.5-3-1

Kojimachi,Chiyoda-ku,Tokyo102-8471,JapanyDepartment of Electrical Engineering and Electronics,

Faculty of Engineering,Osaka Sangyo University,Nakagaito3-1-1,Daito,Osaka574–8530,Japan and

uCREST, JST, Kawaguchi,Saitama 332-0012, Japan

(Received on28 May2002, Accepted in revised form on 11October 2002)

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 investigatedlacI,

tetR, andcI genes for numerical simulation.

r2003 Elsevier Science Ltd. All rights reserved.

1. 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., 2000; Elowitz & Leibler, 2000; Becskei & Serrano, 2000; Becskei et al., 2001). Actually, based on the theoretical analy-sis, several simple genetic networks have been successfully constructed experimentally, e.g., genetic switches (Gardner et al., 2000; Isaacs

et al., 2002), repressilator (Elowitz & Leibler, 2000), and a single negative feedback loop network (Becskei & Settano, 2000). 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., 2001a,b; McMillen et al., 2002). Such simple models represent a ﬁrst step towards logical cellular control by monitoring and manipulating biolo-gical 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

n

Corresponding author. Fax: +81-3-5841-8594. E-mail address:[email protected] (T. Kobayashi).

applications (Hastyet al., 2001a; Agha-Moham-madi & Lotze, 2000; Imhof et al., 2000). 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., 2001; Weiss, 2001).

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, 2002a,b). Therefore, even for the designs of simple-structured switches and oscillators in the previous works, many important physiolo-gical 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 2002a; Smolenet al., 2000, 2001). Thus, one major obstacle to design multiple genetic networks with complicated dynamics is how to analyse high-dimensional nonlinear differen-tial equations, in particular, differendifferen-tial equa-tions with possibly uncertain delays, which generally have inﬁnite dimensions in a functional space (Hale & Lunel, 1993).

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, 1995). 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:

1. 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.

2. 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 func-tional differential equations (FDEs) to design synthetic genetic networks.

3. 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 qualita-tive experimental data, the proposed procedure can design biologically and experimentally fea-sible switches from simple abstract models and predict the dynamical behaviors of the synthe-sized genetic networks. This paper is organized as follows. In Section 2, we makes notations, deﬁnitions, and general assumptions for the mathematical model of genetic networks. In Section 3, we describe our main results including stability conditions and the reduction procedure. Several examples are demonstrated in Section 4, and the paper ends with concluding remarks in

Section 5. All proofs of theorems in this paper are given in Appendix A.

2. General Settings

In this paper, we model a genetic network by delayed differential equations or functional differential equations (FDEs). This model in-cludes many essential properties of the genetic networks in the previous works except stochas-ticity (Shea & Ackers,1985; Wolf & Eeckman, 1998; Drew, 2001; Smolen et al., 2000). 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 compo-nents (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 ¼fðx_{t}Þ DxðtÞ fðx_{t}Þ; ð1Þ

whereRþ _{is the set of nonnegative real numbers}

and xðtÞARþ

n

indicates the concentrations
of all components at time tAR: x_{t}ACþ

C_{ð½r}_{;}_{0}_{;}Rþn_{Þ} _{denotes} _{the} _{element} _{of}
Cð½r;0;RþnÞ whereCð½r;0;RþnÞ is the space
of continuous maps on½r;0 intoRþn_{:}_{In other}

words, xtACð½r;0;Rþ

n

Þ is deﬁned by xtðyÞ ¼

xðtþyÞ;r_{p}yp0; and its normis deﬁned by

jjxtjj ¼suprpyp0jxtðyÞj: When emphasizing the
dependence of a solution on an initial data
fACþ; we write xðt;fÞ or x_{t}ðfÞ for x_{t}: D¼

diagðd_{1};y;d_{n}Þis annndiagonal matrix withn
positive real diagonal components representing
the degradation rates of the chemical
compo-nents. fAC1ðCþ;Rþ

n

Þ:Cþ-Rþn _{indicates the}

synthesis rates of components, where

C1ðCþ_{;}RþnÞ means that the functions from Cþ

to Rþn _{are continuously differentiable. In }

addi-tion, we deﬁne N¼ f1;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, transloca-tions and modiﬁcation reactransloca-tions of proteins. In this paper, we make several assumptions as follows.

Assumption 2.1. The network described by eqn(1)

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 concentra-tion will eventually destroy the organism’s metabolism.

Assumption 2.2. The synthesis rates of all components of the network described by eqn (1) 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.2 in Appendix A).

Assumption 2.3. The synthesis rates of the net-work eqn (1) at t depend on the states of the network at a ﬁnite number of discrete past time instants.

Discrete time delays assumed here are simpli-ﬁed representations of actual time delays mainly caused by transcription, translation, transloca-tion, 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 che-mical components of the network.

Assumption 2.4. The synthesis rate of the i-th chemical component at t; namely, fiðxtÞ mono-tonously increases, or monomono-tonously decreases, or is unaffected if the concentration of only the

j-th chemical component at ttij is monoto-nously increased and the concentrations of the other chemical components are kept constant where i;jAN: In addition, this tendency is

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 monotoni-city, Assumption 2.4 is reasonable for most genetic networks. For instance, the following physiological reactions satisfy Assumption 2.4: transcription activation, transcription inhibition, translation, phosphorylation, enzymic reactions, normal chemical reactions, and translocation.

For the sake of simplicity, in Assumption 2.4, the synthesis rate fi of the i-th chemical component is assumed to directly depend on the concentration of the j-th chemical compo-nent at only one time point ttij; although there may exist multiple direct interactions from thej-th chemical component to thei-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 2.1 (Types of interactions). Suppose that the concentration of the j-th chemical component at ttij affects the synthesis rate of thei-th chemical component attwherei;jAN:

If the synthesis rate of the i-th chemical component at t; namely, fiðxtÞ monotonously increases (or decreases) as the concentration of the j-th chemical component at ttij mono-tonously increases, then the type of the interac-tion fromthe j-th chemical component to the

i-th chemical component is called positive (or negative), and we set sij ¼1 ðor1Þ: If the synthesis rate of thei-th chemical component att

is never affected by the change in the concentra-tion of thej-th chemical component, then we set

sij ¼0:

Thus, sij ¼1 ðor1Þ means that the j-th chemical component affects positively (or nega-tively) the i-th component with time delay tij: For instance, sij ¼1 for fi ¼xjðttijÞ=ð1þ

xjðttijÞÞ; and sij ¼ 1 for fi ¼1=ð1þxjðt

tijÞÞ: Examples of interactions are illustrated in Fig. 1. In addition, for simplicity, we assume that fi increases or decreases in the Michaelis– Menten or Hill’s manner as illustrated in Fig. 1. 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 (1). 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 2.2 (Interaction graph). An interac-tion graph,IGðfÞ;of the genetic network deﬁned by eqn (1) 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 sija0 withtijX0; that is, thej-th chemical component affects the synthesis rate of the i-th chemical component with time delaytij;the graph has an edge, eij; directed fromthe j-th node to the i-th node.

Figure 2 is an example of an interaction graph. In addition,irreducibilityof a graph is deﬁned as follows.

Deﬁnition 2.3 (Irreducibility). IGðfÞ is said to be irreducible only when there is at least one path pði;jÞ ¼ ðj¼p1 -ep2p1 p2 -ep3p2 ? -epipi1 pi¼iÞ; 1 2 3 4 5 2 4 6 8 10 sij = -1 sij = -1 sij = +1 xj f i sij = +1

Fig. 1. Examples of types of interactions. Solid and

dashed curves show positive and negative interactions, respectively.

fromthe j-th node to th i-th node for all

i;jAN ðiajÞ where p_{1};y;p_{i}AN and e_{p}

bpa is an

edge fromnodepa to node pb:

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 2.5. IGðfÞ of eqn (1) is irreducible. Actually, if a network can be divided into several irreducible sub-networks that are gener-ally 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 2.4 (Feedback loops and their types). If there is a path fromthe i-th node of an interaction graph to the samei-th node,pði;iÞ ¼ ði¼p1 -ep2p1 p2 -ep3p2 ? -epipi1 pi¼iÞ; then this path is said to be a feedback loop and furthermore be a self-feedback loop wheni is 2.

In addition, this feedback loop is said to be
positive (or negative) ifQi_{m}_{¼}1_{1} spmþ1pm¼1ðor1Þ:

Figure 2 is an example of an interaction graph with positive and negative feedback loops, e.g. this graph has one positive feedback loop 1 -þ 2 -þ 3 - 6 - 1; one negative feed-back loop 1 -þ 2 - 4 - 5 -þ 6 - 1;

and a positive self-feedback loop 5 -þ 5:

Finally, we make the following important assumption:

Assumption 2.6. The interaction graphIGðfÞof the model in eqn (1) has only positive feedback loops.

As shown in Fig. 2, a positive feedback loop can include negative interaction edges. We can see that for arbitrary two nodesi andj ofIGðfÞ;

all the paths fromthe i-th node to thej-th node have the same sign under Assumption 2.6 where the paths are allowed to include loops because of the irreducibility ofIGðfÞand the deﬁnitions of types of interactions and feedback loops. We can also see that Assumption 2.6 holds if for arbitrary two nodes i and j of IGðfÞ; 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 2.5 (Equilibria). The set of equilibria for eqn (1) is deﬁned asxtthat is constant for all

tX0:

Mathematically more rigorous representation of all the assumptions and the deﬁnitions is given in Appendix A.

3. 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 theoreti-cally. 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. e21 e32 e63 e65 e54 e16 τ16 τ21 τ32 τ63 τ65 τ42 τ54 e55 τ55 e42 1 2 3 4 5 6

Fig. 2. An example of an interaction graph with

feed-back loops. Signsþandon an edge indicates¼1 and

1;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 1st, 2nd, 4th, 5th, and 6th nodes, a positive feedback loop composed of the 1st, 2nd, 3rd and 6th nodes, and a positive self-feedback loop composed of the 5th node.

3.1. CONVERGENCE TO EQUILIBRIA

Theorem3.1 (Convergence to equilibria). If eqn

(1)satisfies Assumptions 2.1–2.6, then for almost
all initial conditions fACþ; x_{t}ðfÞ converges to

equilibria.

This theoremindicates that a genetic network with only positive feedback loops has no dyna-mical 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 Theorem3.1, if we design a synthetic genetic switch only with positive feed-back 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 complex-ity 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-equilibriumsolu-tions cannot be usually observed due to intracel-lular 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.

3.2. 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, Theorem3.1 does not provide any information on the stability of each equilibriumpoint, although it asserts that eqn (1) 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.

Theorem3.2. Let eqn(2)be the associated ODEs of eqn (1) obtained by ignoring all time delays of eqn(1):

’

xðtÞ ¼FðxðtÞÞ DxðtÞ FðxðtÞÞ; ð2Þ where we set tij ¼0 for all i;jAN; and F is f in

eqn(1)but without time delays. Then eqns(1)and

(2) have the identical equilibria. Moreover, if eqn

(1) satisfies all conditions of Theorem 3.1, then each corresponding equilibria of eqns (1) and (2)

have the identical stability.

Theorem3.2 means that if there exists an equilibriumpoint that is asymptotically stable (or unstable) for eqn (2), then it is also asymptotically stable (or unstable) for eqn (1), andvice versa. Based on this theorem, instead of the complicated FDEs of eqn (1), we can use the associated ODEs, eqn (2), 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 complex-ity of the problemand make a design problemof a large-scaled multi-gene network tractable.

3.3. A REDUCTION METHOD TO SIMPLIFY ANALYSIS OF GENETIC SWITCHES

Although Theorems 3.1–3.2 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.

Theorem3.3. Consider eqn(2)and its interaction graph IGðfÞ: Assume that the i-th node does not have any self-feedback loop, that is, an edge eii:

Then by removing x’i¼FiðxÞ dixi from eqn (2)

and by substituting xi¼FiðxÞ=di into remaining

equations, we obtain an n1 dimensional
differ-ential equations
’
x0 ¼F0ðx0Þ D0x0; ð3Þ
where
x0 ¼ ðx1;y;xi1;xiþ1;y;xnÞ;
F0¼ ðF_{1};y;F_{i}_{}_{1};F_{i}_{þ}_{1};y;F_{n}Þ;
D0¼diagðd_{1};y;d_{i}_{}_{1};d_{i}_{þ}_{1};y;d_{n}Þ:

The equilibria of eqn(3)correspond one to one to those of eqn(2),and their stabilities are the same. In addition, the Jacobian matrix of eqn (3) is irreducible.

Theorem3.3 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 self-feedback loop can be eliminated one by one according to Theorem3.3 as illustrated in Fig. 3. The reduction procedure is interpreted by the following operations on the interaction graph. First, we remove the target node. In Fig. 3(A) and (B), the target node is the 2nd 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. 3(A) and (B), there are two edges from the 1st and 5th nodes to the removed 2nd node and are three edges going out fromthe 2nd node to the 1st, 6th, and 7th nodes. Thus, new edges fromthe 1st node to the 1st, 6th, and 7th nodes and ones fromthe 5th node to the 1st, 6th, and 7th nodes are created. Here the edge fromthe 1st node to the 1st node is a positive self-feedback loop. In Fig. 3(A), because the sign of the edge fromthe 1st node to the 2nd node in the original graph is positive, the new edges fromthe 1st node to the 1st, 6th, and 7th nodes are positive,

positive, and negative, respectively. On the other hand, in Fig. 3(B), because the sign of the edge fromthe 1st node to the 2nd node is negative, the new edges fromthe 1st node to the 1st, 6th, and 7th nodes are positive, negative, and positive, respectively.

The low-dimensional ODEs ﬁnally obtained are easier to analyse than the original high-dimensional ODEs. In addition, Theorem 3.3 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 4 is a schematic diagram of this method shown by the interaction graphs IGðfÞ; where each node corresponds to a variable of eqn (1). By applying the proposed method, a node in

IGðfÞ 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 corre-sponding interaction graph with the number of nodes smaller than the original one of eqn (1). For instance, a four-node network is eventually reduced to a one-node network with two positive self-feedback loops in Fig. 4, which is a minimal model of the network. Figure 5 shows other examples. It should be noted that a reduced

(A) (B) 1 2 3 5 6 7 1 3 4 4 5 6 7 1 2 3 5 6 7 1 3 4 4 5 6 7

Fig. 3. A schematic diagram of the reduction procedure

described in Section 3.3. The signs attached to arrows indicate the types of interactions. (A) The case that the signs ofe12ande21are positive. (B) The case that the signs

interaction graph can have two edges fromthe

j-th node to thei-th node in the case of Fig. 5 (B).

fi as a function of xj increases or decreases sigmoidally in the case of Fig. 1 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),fi may not be sigmoidal. Ifi¼j;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). Becausefiof 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

fi of the latter is not necessarily sigmoidal as illustrated by the broken curve in Fig. 5(C), the 1 2 4 3 1 2 3 1 1 3

Fig. 4. A schematic diagram of the reduction method

proposed in Section 3.3. The original network with four components is reduced step by step to one with one component and two feedback loops. First, the 4th node is removed and the edgese43 ande14 are merged. Then, the

2nd and the 3rd nodes are removed in turn. Finally, we obtain a network with only the 1st node and two positive self-feedback loops. 1 2 3 4 5 2 4 6 8 10 xj f i (A) (C) (B) j k i j i j k i j i

Fig. 5. (A) A simple case of the reduction procedure where thek-th node is removed, and a new edge from thej-th node

to thei-th node is created. (B) Another more complicated case of the reduction procedure, where after removal of thek-th node, a new edge fromthej-th node to thei-th node is created, and thei-th node has two edges fromthej-th node. (C) A reducedfias a function ofxj:When the reduced interaction graph is composed of only thei-th node with only one self-feedback loop, thenfiis sigmoidal as depicted by a solid curve. When thei-th node in the reduced interaction graph has two self-feedback loops, thenfias a function ofxiis no longer simply sigmoidal, and its shape can be complex as depicted by a broken curve. A solid thin line represents the degradation of thei-th chemical component. The intersection points of the solid line andfiare equilibriumpoints of the system. The equilibriumpoints of one and two feedback loops are designated by closed circles and open circles, respectively.

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 repre-sent 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 equili-briumstates of the network.

3.4. A PROCEDURE TO DESIGN GENETIC SWITCHES Theorem3.3 instructs us how to reduce the dimensionality of a genetic network to simplify the analysis and the computation of the asso-ciated ODEs. However, when we design a genetic switch, it is convenient for us to start with a minimal model satisfying all require-ments, and then to extend the model to biologically plausible ones with higher dimen-sions. Namely, we reverse the procedure of Section 3.3 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.

Theorem3.4. Let a transformation from eqn (3)

to eqn (2)be

xi ¼Fiðx0Þ=di . x’i ¼FiðxÞ dixi: ð4Þ

Assume that eqns (2) and(3) satisfy Assumptions

2.1–2.6, and that the orbits of eqns (2) and (3)

have a compact closure in the state spaces. Then eqns (2) and (3) have the same equilibria whose stabilities are identical.

Based on Theorem3.4, we can design a genetic switch as follows:

1. 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.

2. Add components that satisfy the assump-tions required in Theorem3.4 one by one to the model in order to make the model more plausible and easier to implement experimen-tally. According to Theorem3.4, 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

1 2 1 2 4 3 1 2 4 5 6 3

Fig. 6. A schematic diagram of the designing procedure proposed in Section 3.4. The original simple network only with

two components is enlarged by adding components to obtain a biologically plausible network. First, the 3rd and the 4th nodes are added, and then the 5th and the 6th nodes are added.

of the reduction method in Section 3.3. 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 Theo-rems 3.1–3.4 guarantee that the systems with and without time delays have the identical equilibria and stabilities.

4. 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 Sec-tion 3.4, 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 corre-sponding 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, 2001, private comm.).

In this switch, we use lacI,tetR, andcIgenes, and PLtetO1;Ptrc2;andPRMpromoters. In fact,

lacI and tetR genes with PLtetO1 and Ptrc2 promoters were artiﬁcially engineered (Lutz & Bujard, 1997) and used to construct a two-state toggle switch (Gardneret al., 2000). On the other hand, the wild-type PRM promoter has three binding sites, i.e. OR1;OR2; and OR3 ( Ptashne, 1992). In our model, the binding siteOR3 of the PRM promoter is assumed to be artiﬁcially altered or mutated so that CI proteins cannot bind to OR3: Although there is no detailed experimental report on the mutated PRM; an

LacI TetR CI 1 2 1 2 3 (A) (C) (D) (B) LacI TetR CI mlacI mtetR1 mtetR2 mcI1 mcI1

Fig. 7. (A) A simple model with two components and

three feedback loops. Each node has a positive self-feedback loop, and their mutual interactions form a positive feedback loop. (B) An extension of the simple model in Fig. (A). The 3rd node is added in order to replace the positive self-feedback loop of the 1st node in Fig. (A) with mutual negative interactions between the 1st and the 3rd nodes. (C) A schematic diagram of a realization of the enlarged model in Fig. (B). LacI, CI, and TetR proteins are adopted to the 1st, the 2nd and the 3rd 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. (2000). The bold line indicates a self-feedback loop of cI2: (D) A further

extension of Fig. (C), which includes mRNAs.

PLtet O-1
*lacI*
LacI
Ptrc-2
PRM
*tetR1*
TetR
CI
*tetR2*
*cI2*
*cI1*

Fig. 8. An implementation of the model with two

components and three feedback loops in Fig. 7 with genes lacI,tetR, andcIand promoters PLtetO1;Ptrc2andPRM

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.ðtetR1;tetR2Þand ðcI1;cI2Þare the sametetR, andcIgenes but with different

abundance of experimental data on CI proteins and the wild-type PRM promoter allows us to predict their quantitative behaviors.

Due to the mutated OR3 site, protein CI is expected to activate only the expression of the genes located downstreamof the mutatedPRM; by binding toOR1 and OR2 sites of the mutated PRM:

As indicated in Fig. 8, after dimerization, the dimers of protein CI enhance promoterPRMand those of protein TetR repress promoter PLtetO1; respectively, whereas proteins LacI forms tetra-mers to inhibit promoter Ptrc2:

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 mtetR1 and mtetR2: 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:

dmlacI

dt ¼nPLtetO1fPLtetO1ðptetRðttPLtetO1ÞÞ dmlacImlacI;

dplacI

dt ¼smlacImlacIðttlacIÞ dplacIplacI;

dmtetR1 dt ¼nPtrc2 fPtrc2ðplacIðttPtrc2ÞÞ dmtetRmtetR1; dmtetR1 dt ¼nPRMfPRMðpcIðttPRMÞÞ dmtetRmtetR2; dptetR

dt ¼smtetR1mtetR1ðtttetRÞ

þ smtetR_{2}mtetR2ðtttetRÞ dptetRptetR;

dmcI1
dt ¼nPtrc2 fPtrc2ðplacIðttPtrc2ÞÞ dmcImcI1;
dmcI2
dt ¼nPRM fPRMðpcIðttPRMÞÞ dmcImcI2;
dpcI
dt ¼smcI1mcI1ðttcIÞ
þ smcI_{2}mcI2ðttcIÞ dpcIpcI;

where m and p indicate concentrations of mRNAs and proteins of genes assigned by subscripts.sand d indicate synthesis coefﬁcients of proteins, and degradation rates of mRNAs and proteins, respectively. nPLtetO1;nPtrc2; and nPRM 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 fs are monotone functions.

By applying the proposed procedure in Theorem3.3 and transforming the equations properly, we obtain the following two-dimen-sional differential equations, which preserve equilibria and their stabilities of the original network:

dptetR

dt ¼f1ðptetRÞ þetetRf2ðpcIÞ ptetR; ð5Þ

dpcI
dt ¼k½ecIf1ðptetRÞ þf2ðpcIÞ dpcI ; ð6Þ
where
f1ðptetRÞ ¼
smtetR_{1}nPtrc2
dmtetRdptetR
fPtrc2
smlacInPLtetO_{}_{1}fPLtetO1ðptetRÞ

dmlacIdplacI
;
f2ðpcIÞ ¼
smtetR_{1}nPRM
dmtetRdptetR
fPRMðpcIÞ;
etetR ¼
smtetR2
smtetR_{1}
;ecI ¼
smcI1
smcI_{2}
;
k ¼ dmtetR
smtetR_{1}
smcI2
dmcI
; and
d ¼ dpcI
kdptetR
:

Time t is implicitly normalized by the half-life time of protein TetR. Since fPtrc2ðplacIÞ and fPLtetO1ðptetRÞ are monotonously decreasing func-tions butfPRMðpcIÞ is a monotonously increasing function, both f1ðptetRÞ and f2ðpcIÞ are mono-tonously increasing functions, which satisfy the conditions of Theorems 3.1–3.4.

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., (2000)). According to the previous work, this sub-network can have two stable equilibria by inserting proper RBSs to tetR1 and lacI genes (Gardner et al., 2000) if isolated fromthe other components of the switch network. Therefore, if we set etetR small enough, namely, we choose a proper RBS fortetR2gene, 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 thecIgene with weaker one or by using a temperature-sensitive cI

mutant whose lifetime is much shorter than that of the wild-typecI, 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., 2002). 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 cI2 gene or by using a temperature-sensitivecIgene forcI2in Fig. 8. It should be also noted that the order of the genes in a polycistronically transcribed set can inﬂu-ence this condition because transcriptional efﬁ-ciency can be affected by such an order of the

genes. If we set ecI small enough by choosing a proper RBS forcI1;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., 2000) due to the sub-network with genestetR 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 placI;ptetR; and

pcI as placIðnMÞE84p0_{lacI};ptetRðnMÞE8p0_{tetR}; and

pcIðnMÞE8pcI: Then, we set

p0_{lacI}ðp0_{tetR}Þ ¼53nPLtetO1
1
1þp02
tetR
;
f1ðp0tetRÞ ¼5:3nPtrc_{}_{2}
1
1þp0
lacIðp0tetRÞ4
;
f2ðp0cIÞ ¼5:3nPRM
1þp02
cI þ22p0cI4
1þp02
cI þ2p0cI4
: ð7Þ

These functions are calculated using the para-meter values in Hasty et al. (2002) under the following assumptions: (1) CI dimers cannot bind to theOR3site; (2) the equilibriumconstant of TetR dimerization is the same as that of CI; (3) the equilibriumconstant of TetR dimers and PLtetO1 is the same as that of CI dimers and the

OR1site ofPRM;(4) the reaction rates of protein production fromPLtetO1 and Ptrc2 without TetR dimers and LacI tetramers are the same as that ofPRM without CI dimers; (5) the degrada-tion rates of TetR and LacI are 0:0025 s1that is set to be the same as that of Cro (Arkin et al., 1998). Figure 9 shows that eqns (5) and (6) have three stable equilibria, ðtetR;cIÞ ¼ ðOFF;OFFÞ;

ðOFF;ONÞ;ðON;ONÞ; and Theorems 3.1–3.4 guarantee that this designed switch converges to a stable equilibriumeven with uncertain delays, where etetR ¼0:2; ecI ¼0:2; k¼1; d¼ 25;nPLtetO1 ¼6;nPtrc2 ¼3;andnPRM ¼1:Notice that it is not difﬁcult to obtain three stable equilibria by settingetetR andecI 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 10-folds or more. Thus, in this sense,etetR¼0:2 and ecI ¼0:2 are reasonable, and this switch may

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. 10 where etetR¼0:2;ecI ¼0:1;

k¼1; d¼30; nPLtetO1 ¼6; nPtrc2 ¼5; and

nPRM ¼1:

Notice that the number of equilibria and their ON–OFF combinations can be controlled theo-retically by two experimentally regulable para-meters etetR and ecI 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, gA; gB; and gC rather thantetR,lacI, andcIonly ifgA andgBfunction as a toggle switch and gC forms a positive self-feedback loop with bistability or one equilibrium point at a low state. In addition, the plasmid

copy numbers also can be controllable para-meters. 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 mono-tone 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 ex-istence of non-equilibriumstates. This makes theoretical analysis and design of switches much easier and simpler. In addition, the fact that the

10 20 30 40 50 1 2 3 4 5 TetR CI (OFF,OFF) (OFF,ON) (ON,ON)

Fig. 9. Geometric structures of the null-clines of eqns (5) (broken curves) and (6) (solid curves) calculated by eqn (7)

withetetR¼0:1;ecI¼0:1;k¼1;andd¼1:The circles indicate stable equilibria. The switch has three equilibriumstates, namely,ðOFF;OFFÞ;ðOFF;ONÞ;andðON;ONÞ:Both the concentrations of proteins TetR and CI are normalized by their representative values (see the text).

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 particu-larly powerful when applied to eukaryotic

genetic networks. One reason is because time delays are ubiquitous and inﬂuential in eukar-yotes. 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 cer-tain 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 informa-tion on eventual effects of interacinforma-tions.

Finally, to demonstrate our theoretical results, we have designed realistic synthetic switches with three or four equilibriumstates using experi-mentally 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

10 20 30 40 50 1 2 3 4 5 TetR CI (OFF,OFF) (OFF,ON) (ON,ON) (ON,OFF)

Fig. 10. Geometric structures of eqns (5) (broken curves) and (6) (solid curves) calculated by eqn (7) withe_{tetR}¼0:1;

ecI¼0:01; k¼1; and d¼1: The circles indicate stable equilibria. The switch has four equilibriumstates, namely,

and the numerical simulation show that the proposed model can have three or four stable equilibriumstates only by properly adjusting two parameters, the RBSs oftetR2 and cI1:

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 inte-grated study of a theoretical model anda 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 oflacIand tetRgenes (Hastyet al., 2000; Isaacs

et al., 2002). Therefore, it is an important future problemto understand how stochastic effects inﬂuence the behaviors of the switch. In addi-tion, it is also necessary to investigate how time delays affect stochastic nature in genetic net-works 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 12208004.

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Appendix A

A.1. 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.1. The solution xtðfÞ of eqn (1)
is deﬁned for tX0 and jjx_{t}jjoN for all

tX0:

Assumption A.2. The synthesis ratesfðx_{t}Þ in eqn
(1) are bounded ifxt is bounded.

Assumption A.3. The synthesis ratesfðxtÞ in eqn (1) depend on xðtþsÞ; sA½r;0 at a ﬁnite

number of time instantss:

These assumptions are mathematical

repre-sentations of Assumptions 2.1–2.3 in

Section 2.

A.1.1. Properties of Solutions for eqn (1)

With Assumptions A.1–A.3, the following results hold for eqn (1):

1. Equation (1) generates a (local) semiﬂow F

onCþ _{by}

Fðt;fÞ ¼xtðfÞ; fACþ

fortX0 (Smith & Thieme, 1991).

2.Cþ_{is positively invariant for eqn (1), that is,}

for arbitrarytX0;

Fðt;Cþ_{Þ}CCþ

holds (Smith & Thieme, 1991).

3. Deﬁne the orbit of eqn (1) for the initial condition fACþ as

OþðfÞ ¼ fx_{t}ðfÞ:tX0g:

Then OþðfÞ is precompact in Cþ _{and the}

omega limit set, oðfÞ; is deﬁned by

oðfÞ ¼ \

sX0

fx_{t}ðfÞ:tXsg:

In addition, oðfÞ is non-empty, compact, con-nected, and invariant (Smith & Thieme, 1991).

Next, we give mathematical representations of Assumptions 2.4–2.6, which are generalized to allow multiple discrete time delays.

Assumption A.4. Suppose that the concentra-tions of the j-th chemical component at an lij number of different time instants, xjðt

t1

ijÞ;xjðtt2ijÞ;y;xjðttlijijÞ; affect the synthesis rate of the i-th chemical component at t:

Without loss of generality, we set

t1_{ij}ot2_{ij}o?otl_{ij}ij: Then, for f_{i}ðx_{t}Þ ¼f_{i}ðx_{t}Wx_{j}ðt
tk_{ij}Þ;xjðttkijÞÞ; qfiðxtWx_{j}ðttk_{ij}Þ;yÞ=qyX0 on

Cþ_{;} _{or} _{q}_{f}_{i}_{ðx}_{t}Wx_{j}ðttk_{ij}Þ;yÞ=qyp0 on Cþ holds

fori;jANandkAf1;y;l_{ij}g;wherex_{t}Wx_{j}ðttk_{ij}Þ

indicates all values of xt except xjðttkijÞ: This assumption is a more general mathematical representation of Assumption 2.4 in Section 2.

Deﬁnition A.1 (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:

Deﬁnefi as

fiðxtÞ ¼fiðxtWx_{j};x_{j}ðtt1_{ij}Þ;y;x_{j}ðttl_{ij}ijÞÞ;

where xtWx_{j} is x_{t} without x_{j}ðtt1_{ij}Þ;x_{j}ðtt2_{ij}Þ;

y;x_{j}ðttl_{ij}ijÞ:

Focusing on the dependency offionxjðttkijÞ;
we deﬁne sk_{ij}ðx_{t}Þ; the type of an interaction
between the i-th and the j-th components atxt;

as follows:
sk_{ij}ðx_{t}Þ ¼
þ1 :if @fið*;y;*;xjðtt
k
ijÞ;*;y;_{*}Þ
@xjðttk
ijÞ
j_{x}
t40;
1 :if @fið*;y;*;xjðtt
k
ijÞ;*;y;*Þ
@xjðttkijÞ
j_{x}
t
o0;
0 :if @fið*;y;*;xjðtt
k
ijÞ;*;y;_{*}Þ
@xjðttk
ijÞ
j_{x}
t ¼0:
8
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
:
ðA:1Þ
If sk

ijðxtÞ ¼1ðor1Þ; then the j-th chemical component is said to affect positively

(or negatively) the i-th component with time delay tk

ij at xt: Because of Assumption A.4,

sk_{ij}ðx_{t}ÞX0 ðorp0Þ for all x_{t}ACþ: In general,

sk

ijðxtÞ may not be equal to sk

0

ijðxtÞ for kak0: 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, sk

ijðxtÞsk

0

ijðxtÞX0 for

k;k0Af1;y;l_{ij}gand x_{t}ACþ: We sets_{ij} s1_{ij}:

In addition, we make the following assumption:

Assumption A.6. If sija0 and tlijij40 then

slij

ijðxtÞa0 for all xtACþ:

This assumption is biologically plausible.

Deﬁnition A.2 (Interaction graph). An interac-tion graph at xt; IGxtðfÞ; of a genetic network

deﬁned by eqn (1) is a directed graph whose
nodes represent the individual chemical
compo-nents of the genetic network and whose edges
represent the interactions between the nodes.
When sk_{ij}ðx_{t}Þa0 and tk_{ij}X0; that is, the j-th
chemical component affects the synthesis rate
of the i-th chemical component with time delay
tk

ij; the graph has an edge, ekijðxtÞ; directed from thej-th node to thei-th node.

In addition,irreducibilityof a graph is deﬁned as follows.

Deﬁnition A.3 (Irreducibility). IGxtðfÞ is said to

be irreducible when there is at least one path

pði;jÞ ¼ ðj ¼p1
-kp2p1
ep2p1ðxtÞ
p2
-kp3p2
ep3p2ðxtÞ
?
-kpipi1
epip_{i}_{}_{1}ðxtÞ
pi¼iÞ

fromthe j-th node to the i-th node for all

i;jANðiajÞ where p_{1};y;p_{i}AN and ek_{p}pbpa

bpaðxtÞ is

thekpbpa-th edge fromnodepa to nodepb:

Assumption A.7. IGxtðfÞof eqn (1) is irreducible

for all xtACþ:

Deﬁnition A.4 (Feedback loops and their types). If a path fromthe i-th node of an interaction graph, IGxtðfÞ; to the same i-th node,

pði;jÞ
¼ i¼p1
-kp2p1
ep2p1ðxtÞ
p2
-kp3p2
ep3p2ðxtÞ
?
-kpip_{i}_{}_{1}
epip_{i}_{}_{1}ðxtÞ
pi¼i
0
B
@
1
C
A
exists, then this path is said to be a feedback loop
and furthermore be a self-feedback loop when i
is 2. In addition, this feedback loop is said to be
positive (or negative) if Qi_{m}_{¼}1_{1} skpmþ1;pm

pmþ1pm ðxtÞ ¼

Qi1

m¼1 spmþ1pmðxtÞ ¼1 (or1).

Assumption A.8. The interaction graph IGxtðfÞ

of the model in eqn (1) has only positive feedback loops for all xtACþ:

A genetic network with only positive feedback loops may have both positive and negative interaction edges, which make analysis compli-cated. Then, we consider a coordinate transfor-mation to reduce the original genetic network with only positive feedback loops into a equiva-lent one with only positive interaction edges.

Choose a nodej arbitrarily. First, we setsj ¼ 1:IfsijðxtÞ ¼1ðor1Þfor somextACþ;then set

si ¼1ðor1Þ:

si 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. Usingsi;we deﬁne a transformationPdescribed

by a matrix as follows: P¼ s1 0 & 0 sn 0 B @ 1 C A

UsingP_{;}_{we deﬁne a coordinate transformation}

ofC¼Cð½t;0;RnÞ as

P_{ð}C_{Þ }C_{ð½}_{t}_{;}_{0}_{;}P_{ð}Rn_{ÞÞ}_{;}

and set C0_{¼}P_{ð}Cþn_{Þ}_{:} _{Then, we transform} _{f} _{to}
gAC1ðC0;R0Þ as follows:

g¼P3f3P;

where we use P1_{¼}P _{and set} R0 _{¼}P_{ð}Rþn_{Þ}_{:}

Using g; we deﬁne another FDEs on C0 _{as}

follows:

’

yðtÞ ¼gðy_{t}Þ DyðtÞ gðy_{t}Þ with yAC0: ðA:2Þ

Because @gi=@xjðttijÞ ¼sisj@fi=@xjðttijÞ and si and sj@fi=@xjðttijÞ have the same sign, @gi=@xjðttijÞX0: Thus, eqn (A.2) has only positive interaction edges, and each sij of

IGðgÞ is always 1 (Snoussi, 1998). Since P

merely reverses the directions of some axes of coordinates, eqn (A.2) has the same dynami-cal properties as eqn (1). Since all the theo-retical results proven for eqn (A.2) also hold for eqn (1), we prove the theorems only for eqn (A.2).

We also make some deﬁnitions that are used in the proofs of the theorems.

Deﬁnition A.5 (Equilibria). The set of equilibria for eqn (1) is deﬁned as

ef ¼ ffACþ:f¼xˆ

for somexARþ

n

satisfyingfðxÞ ¼ˆ 0g;

where ˆx is the natural inclusion from xARþ

n

to

ˆ

xACþ by ˆx_{t}ðyÞ ¼xðtpyp0Þ:

Deﬁnition A.6 (The set of convergent points).
The set of convergent points E_{f} _{of eqn (1) is}

deﬁned as

E_{f} _{¼ f}_{f}ACþ:oðfÞ ¼ fxgˆ for some ˆxAe_{f}g:

The set of convergent points E_{g} _{is similarly}

deﬁned for eqn (A.2), andE_{f} _{¼}P_{ð}E_{g}_{Þ} _{holds.}
Deﬁnition A.7 (Quasi-positive matrix). Annn

matrix Bis said to be quasi-positive when

BþlInX0 for all sufficiently largel or when all components of B except diagonal components are nonnegative, whereIn is ann

n identity matrix.

Note that a quasi-positive matrix is different from a positive semi-deﬁnite matrix.

Deﬁnition A.8 (Irreducible matrix). An nn

matrix A is irreducible when for every non-empty, proper subset N0 of N; there exists an

iAN0 and a jANWN0 such that ½A _{ij}a0:

A.1.2. Proofs of the Main Results

Deﬁnition A.9. (Stable point). fACþis a stable

point if for everye40 there existsd40 such that

jjFðt;fÞ Fðt;cÞjjoe;fortwhenevercACþ and

jjfcjjod:We deﬁneS_{f} _{to be the set of stable}

points of eqn (1). Note that iffAS_{f} then points

nearfhave limit sets nearoðfÞ:The set of stable
points of eqn (A.2),S_{g}_{;} _{is similarly deﬁned, and}
S_{g}¼PðS_{f}Þ and S_{f} ¼PðS_{g}Þ hold.

The following theorem(Theorem4.1 in (Smith (1995))) is crucial for proving Theorem3.1:

TheoremA.1 (Convergence to equilibria).
As-sume that eqn(A.2)is cooperative and irreducible
inC0 _{and the following condition}ðTÞ holds. Then
S_{g}CE_{g};andIntS_{g}is dense inC0 for the semiflow

generated by eqn (A.2).

(T)g maps bounded subsets ofC0 _{to bounded}

subsets of R0_{:} _{For each} _{f}AC0; yðt;fÞ is deﬁned

for tX0 and OþðfÞ is bounded. For each compact subset AC0 of C

0

; there exists a closed and bounded subset BC0 of C

0 _{such that}

oðfÞDBC0 for every fAAC0:

Proof of Theorem3.1. First, (T) holds, because of Assumptions A.1–A.3.

Second, we show that eqn (A.2) is cooperative.
Equation (A.2) is cooperative inC0 _{if}C0 _{is order}

convex and for all fAC0 and every c such that

cX0 andc_{i}ð0Þ ¼0 foriAN;

dgðfÞcX0;

where dgðfÞ is the derivative ofg at f:Because
of the deﬁnition ofP_{and Assumptions A.3, A.4}

and A.5, it is easy to see that eqn (A.2) is cooperative.

Finally, we show that eqn (A.2) is irreducible. Equation (A.2) is irreducible when for allytAC0

ðdgðytÞeˆ1;y;dgðytÞeˆnÞ;

is irreducible where ðe1;y;enÞ is an nn identity matrix, and when for all ytAC0 and all

jAN;there existsiAN such that

@giðytWy_{j}ðtt_{ij}lijÞ;y_{j}ðttl_{ij}ijÞ

yjðttlijijÞ

40:

Obviously, these are satisﬁed because of As-sumptions A.6 and A.7. &

Proof of Theorem3.2. Assume that eqn (A.2) satisﬁes all the conditions of TheoremA.1. We deﬁne the associated ODEs of eqn (A.2) obtained by ignoring all time delays as

’

yðtÞ ¼GðyðtÞÞ DyðtÞ GðyðtÞÞ; ðA:3Þ

where we set tij ¼0 for all i;jAN; and G is an

induced function of g without time delays. By applying Corollary 5.2 in Smith (1995) to eqns (A.2) and (A.3), it can be shown that eqns (A.2) and (A.3) have identical equilibria, and that each equilibriumof eqns (A.2) and (A.3) has identical stability. &

Proof of Theorem3.3. Assume that thei-th node
ofIGðfÞdeﬁned by eqn (2) does not have a
self-feedback loop, that is, an edge eii; then the i-th
node ofIGðgÞ also does not have a self-feedback
loop. By removing y’_{i}¼GiðyÞ diyi fromeqn
(A.3) and by substituting yi¼GiðyÞ=di into
remaining equations, we obtain ann1
dimen-sional differential equation

’
y0 ¼G0ðy0Þ D0y0; ðA:4Þ
where
y0¼ ðy1;y;yi1;yiþ1;y;ynÞ;
G0 ¼ ðG_{1};y;G_{i}_{}_{1};G_{i}_{þ}_{1};y;G_{n}Þ;
D0 ¼diagðd_{1};y;d_{i}_{}_{1};d_{i}_{þ}_{1};y;d_{n}Þ:
Because y¼P_{x} _{and} _{F} ¼P3G3P; the operation
to derive eqn (A.4) fromeqn (A.3) directly
corresponds to that to derive eqn (3) fromeqn
(2). Thus, results proven for eqns (A.3) and (A.4)
also hold for eqns (2) and (3). The irreducibility
of the reduced interaction graph is obvious
because of the operation to the interaction graph
shown in Fig. 3.

In this proof, we use the following property of a quasi-positive matrix.

For a quasi-positive matrix A; sðAÞ ¼

maxRelo0 is equivalent to the condition that there exists u40 such thatAuo0 where l runs over the eigenvalues ofAandsðAÞis the stability modulus ofA (Smith, 1986).

Since Gij @Gi=@yj ¼ Pl_{k}¼1 gkijX0 holds for
alll ¼1;2;y;l_{ij}wherei;jAN; iaj;the Jacobian

matrix of GðyÞ is quasi-positive. Moreover, the Jacobian matrix of GðyÞ is irreducible on R0

because of Assumption A.7. If a solution yðt;fÞ

of eqn (A.3) with an initial conditionfAR0has a

compact closure for tX0 then almost all
solu-tions starting at a point on R0 _{converge to}

equilibria (Smith, 1995).

The Jacobian matrix of GðyÞ at an
equili-briumis
A¼
G11 y ^ y
^ _{&} Gi1i
Gi1 y Gii1 Gii Giiþ1 y
Giþ11 Giþ1i
^ ^
0
B
B
B
B
B
B
@
1
C
C
C
C
C
C
A
D
¼JD: 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

that thei-th node does not have the edgeeii;that
is,Gii ¼0;
½Au _{i} ¼X
i1
k¼1
Gikukþ
Xn
k¼iþ1
Gikukdiui; ðA:6Þ

holds for arbitrary uARn: In addition, we

deﬁne J0 to be a matrix obtained by removing
thei-th row and the i-th column fromJ of eqn
(A.5) as
J0¼
G11 y G_{1}_{i}_{}_{1} G_{1}_{i}_{þ}_{1} y
^ _{&} ^ ^
Gi11 y Gi1i1 Gi1iþ1
Giþ11 y G_{i}_{þ}_{1}_{i}_{}_{1} G_{i}_{þ}_{1}_{i}_{þ}_{1}
^ _{&} ^ ^ _{&}
Gn1 y Gni1 Gniþ1 y
0
B
B
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
C
C
A
:
ðA:7Þ

Because of these deﬁnitions, for an arbitrary

n-dimensional real vector v¼ ðv_{1};y;v_{n}Þ and

ðn1Þ-dimensional real vector v0¼
ðv_{1};y;v_{i}_{}_{1};v_{i}_{þ}_{1};y;v_{n}Þ;

½Jv _{j} ¼ ½J0v0 _{j}þGjivi; ðA:8Þ

½Dv _{j} ¼ ½D0v0 _{j}; ðA:9Þ

hold forj ¼1;y;i1;iþ1;y;n:

Next, let us deﬁne as GkðyÞ ¼Gkðyi;y0Þ
where y0 ¼ ðy1;y;yi1;yiþ1;y;ynÞ and kai:
Notice that GiðyÞ ¼Giðy0Þ because the i-th
node of IGðgÞ has no self-feedback loop. By
substitutingyi¼Giðy0Þ=di into this equation, we
obtain
’
y_{k}¼G0_{k}ðy0Þ dkyk ¼Gk
1
di
Giðy0Þ;y0
dkyk;

for k¼1;y;i1;iþ1;y;n: The deriva-tives of this equation at an equilibriumpoint are Gkjþ 1 di GkiGijX0 if jak; Gkkþ 1 di GkiGikdk if j ¼k:

Thus, the Jacobian matrix of eqn (A.4) is

A0 ¼J0D0þ 1 di G1i ^ Gi1i Giþ1i ^ Gni 0 B B B B B B B B B @ 1 C C C C C C C C C A Gi1 ^ Gii1 Giiþ1 ^ Gin 0 B B B B B B B B B @ 1 C C C C C C C C C A t ;

wheretindicates the transpose. Apparently,A0is quasi-positive when Ais quasi-positive.

Assume that sðAÞo0; then there exists u40
such that Auo0: Let u0 be the ðn1Þ
-dimen-sional vector obtained by removing ui from u:
Note that u040: Then for alljAf1;y;i1;iþ
1;y;ng;
½A0u0 _{j} ¼ ½J0u0_{j} ½D0u0_{j}
þ Gji
di
Xi1
k¼1
Gikukþ
Xn
k¼iþ1
Gikuk
" #
¼ ½Au _{j}Gjiui
þ Gji
di
Xi1
k¼1
Gikukþ
Xn
k¼iþ1
Gikuk
" #
¼ ½Au _{j}
þ Gji
di
Xi1
k¼1
Gikukþ
Xn
k¼iþ1
Gikukdiui
" #
ðA:10Þ
¼ ½Au _{j}þGji
di
½Au _{i}; ðA:11Þ

where we use eqns (A.8), (A.9), and (A.6). According to eqn (A.11) with Auo0 and

GjiX0; we have A0u0o0: Thus, the theoremin Smith (1995) shows thatsðA0Þo0 ifsðAÞo0:

Assume thatsðA0Þo0;then there is anðn1Þ
-dimensional vector u0 _{¼ ðu}

1;y;u_{i}_{}_{1};u_{i}_{þ}_{1};y;u_{n}Þ
such that A0u0o0: If Gj of eqn (A.3) does not
depend on xi; that is, Gji ¼0; then ½Au jo0
because of eqn (A.10). Hence, we consider the
case with Gji40:Due toGikX0 and uk40;

Xi1 k¼1 Gikukþ Xn k¼iþ1 Gikuk40; ðA:12Þ

holds. Because of eqn (A.6) and di40; we
have ½Au _{i}o0 by setting ui40 sufﬁciently
arge. In addition, from ½A0u0_{j}o0 and
eqn (A.11), we have u such that ½Au _{j}_{o}0
and ½Au _{i}o0 for all j by setting ui
sufﬁ-ciently large, i.e. maxj ½A0u0joðGji=diÞ½Au io0:
Thus, we have u such that ½Au o0:

There-fore, sðAÞo0 if sðA0Þo0; which proves the theorem. &

Proof of Theorem3.4. By applying Theorem3.3 to the enlarged systemdeﬁned by the transfor-mation of eqn (4), we easily obtain the result in Theorem3.4. &