Monotone and nearmonotone biochemical networks
 1.5k Downloads
 97 Citations
Abstract
Monotone subsystems have appealing properties as components of larger networks, since they exhibit robust dynamical stability and predictability of responses to perturbations. This suggests that natural biological systems may have evolved to be, if not monotone, at least close to monotone in the sense of being decomposable into a “small” number of monotone components, In addition, recent research has shown that much insight can be attained from decomposing networks into monotone subsystems and the analysis of the resulting interconnections using tools from control theory. This paper provides an expository introduction to monotone systems and their interconnections, describing the basic concepts and some of the main mathematical results in a largely informal fashion.
Keywords
Biochemical networks Dynamical systems Monotone systemsIntroduction
The field of systems molecular biology is largely concerned with the study of biochemical networks consisting of proteins, RNA, DNA, metabolites, and other molecules. These networks participate in control and signaling in development, regulation, and metabolism, by processing environmental signals, sequencing internal events such as gene expression, and producing appropriate cellular responses. It is of great interest to be able to infer dynamical properties of a biochemical network through the analysis of wellcharacterized subsystems and their interconnections. This paper discusses recent work which makes use of both topology (graph structure) and sign information in order to deduce such properties.

Interesting and nontrivial conclusions can be drawn from (signed) network structure alone. This structure is associated to purely stoichiometric information about the system and ignores fluxes. Consistency, or close to consistency, is an important property in this regard.

Interpreted as dynamical systems, consistent networks define monotone systems, which have highly predictable and ordered behavior.

It is often useful to analyze larger systems by viewing them as interconnections of a small number of monotone subsystems. This allows one to obtain precise bifurcation diagrams without appeal to explicit knowledge of fluxes or of kinetic constants and other parameters, using merely “input/output characteristics” (steadystate responses or DC gains). The procedure may be viewed as a “model reduction” approach in which monotone subsystems are viewed as essentially onedimensional objects.

The possibility of performing a decomposition into a small number of monotone components is closely tied to the question of how “near” a system is to being monotone.

We argue that systems that are “near monotone” are biologically more desirable than systems that are far from being monotone.

There are indications that biological networks may be much closer to being monotone than random networks that have the same numbers of vertices and of positive and negative edges.
The need for robust structures and robust analysis tools
In contrast to many areas of applied mathematics and engineering, the study of dynamics in cell biology should take into account the often huge degree of uncertainty inherent in models of cellular biochemical networks, which arises from environmental fluctuations or from variability among cells of the same type. From a mathematical analysis perspective, this uncertainty translates into the difficulty of measuring the relevant model parameters such as kinetic constants or cooperativity indices, and hence the impossibility of obtaining a precise model.
Consistent graphs, monotone systems, and nearmonotonicity
We now introduce the basic notions of monotonicity and consistency. The present section deals exclusively with graphtheoretic information, which is derived from stoichiometric constraints. Complementary to this analysis, bifurcation phenomena can be sometimes analyzed using a combination of these graphical techniques together with information on steadystate gains; that subject is discussed in section “I/O monotone systems.” In order to preserve readability, the discussion in this section is informal, and not all mathematical technicalities are explained; references are given that will allow the reader to fillin the missing details, and also section “I/O monotone systems” has more rigorous mathematical statements, presented in the more general context of systems with external inputs and outputs.
One may also study more complicated descriptions of dynamics that those given by ordinary differential and difference equations; many of the results that we discuss here have close analogs that apply to more general classes of (deterministic) dynamical systems, including reaction–diffusion partial differential equations, which are used for spacedependent problems with slow diffusion and no mixing, delaydifferential systems, which help model delays due to transport and other cellular phenomena in which concentrations of one species only affect others after a time interval, and integrodifferential equations (Smith 1995; Hirsch and Smith 2005; Sontag 2004; Enciso et al. 2006). In a different direction, one may consider systems with external inputs and outputs (Angeli and Sontag 2003).
The graph associated to a system
There are at least two types of graphs that can be naturally associated to a given biochemical network. One type, sometimes called the speciesreaction graph, is a bipartite graph with nodes for reactions (fluxes) and species, which leads to useful analysis techniques based on Petri net theory and graph theory (Feinberg 1991; Reddy et al. 1993; ZevedeiOancea and Schuster 2003; Craciun and Feinberg 2005; Craciun and Feinberg 2006; Angeli and Sontag 2007; Angeli et al. 2006, 2007). We will not discuss speciesreaction graphs here. A second type of graph, which we will discuss, is the species graph G. It has n nodes (or “vertices”), which we denote by \(v_1,\ldots ,v_n,\) one node for each species. No edge is drawn from node v _{ j } to node v _{ i } if the partial derivative \({\partial f_i}/{\partial x_j}(x)\) vanishes identically, meaning that there is no direct effect of the jth species upon the ith species. If this derivative is not identically zero, then there are three possibilities: (1) it is ≥0 for all x, (2) it is ≤0 for all x, or (3) it changes sign depending on the particular entries of the concentration vector x. In the first case (activation), we draw an edge labeled +, +1, or just an arrow →. In the second case (repression or inhibition), we draw an edge labeled −, −1, or use the symbol \(\dashv.\) In the third case, when the sign is ambiguous, we draw both an activating and an inhibiting edge from node v _{ j } to node v _{ i }. The graph G is an example of a signed graph (Zaslavsky 1998), meaning that its edges are labeled by signs.
For continuoustime systems, no selfedges (edges from a node v _{ i } to itself) are included in the graph G, whatever the sign of the diagonal entry \(\partial f_i/\partial x_i\) of the Jacobian. For discretetime systems, on the other hand, selfedges are included (we later discuss the reason for these different definitions for differential and difference equations).
Although adding new edges as explained above is a purely formal construction with graphs, it may be explained biologically as follows. Often, ambiguous signs in Jacobians reflect heterogeneous mechanisms. For example, take the case where protein A enhances the transcription rate of gene B if present at high concentrations, but represses B if its concentration is lower than some threshold. Further study of the chemical mechanism might well reveal the existence of, for example, a homodimer that is responsible for this ambiguous effect. Mathematically, the rate of transcription of B might be given algebraically by the formula \(k_2a^2k_1a,\) where a denotes the concentration of A. Introducing a new species C to represent the homodimer, we may rewrite this rate as \(k_2ck_1a,\) where c is the concentration of C, plus an new equation like \(dc/dt = k_3a^2k_4c\) representing the formation of the dimer and its degradation. This is exactly the situation in Fig. 3.
Spin assignments and consistency
We will say that Σ is a consistent spin assignment for the graph G (or simply that G is consistent) if every edge of G is consistent with Σ. In other words, for any pair of vertices v _{ i } and v _{ j }, if there is a positive edge from node v _{ j } to node v _{ i }, then v _{ j } and v _{ i } must have the same spin, and if there is a negative edge connecting v _{ j } to v _{ i }, then v _{ j } and v _{ i } must have opposite spins. (If there is no edge from v _{ j } to v _{ i }, this requirement imposes no restriction on their spins.)
The balancing property, in turn, can be checked with a fast dynamic programminglike algorithm. For connected graphs, there can be at most two consistent assignments, each of which is the reverse (flip every spin) of the other.
Monotone systems
A dynamical system is said to be monotone if there exists at least one consistent spin assignment for its associated graph G. Monotone systems (Smith 1995; Hirsch 1983, 1985) were introduced by Hirsch, and constitute a class of dynamical systems for which a rich theory exists. (To be precise, we have only defined the subclass of systems that are monotone with respect to some orthant order. The notion of monotonicity can be defined with respect to more general orders.)
Consistent response to perturbations
In contrast, consider next the graph in Fig. 4c, where the edge from 1 to 2 is now positive. There are two paths from node 1 to node 4, one of which (through 3) is positive and the other of which (through 2) is negative. Equivalently, the undirected loop 1,3,4,2,1 (“undirected” because the last two edges are transversed backward) has a net negative parity. Therefore, the loop test for consistency fails, so that there is no possible consistent spinassignment for this graph, and therefore the corresponding dynamical system is not monotone. Reflecting this fact, the net effect of an increase in node 1 is ambiguous. It is impossible to conclude from the graphical information alone whether node 4 will be repressed (because of the path through 2) or activated (because of the path through 3). There is no way to resolve this ambiguity unless equations and precise parameter values are assigned to the arrows.
The uncertainty associated to a graph like the one in Fig. 4c might be undesirable in natural systems. Cells of the same type differ in concentrations of ATP, enzymes, and other chemicals, and this affects the values of model parameters, so two cells of the same type may well react differently to the same “stimulus” (increase in concentration of chemical 1). While such epigenetic diversity is sometimes desirable, it makes behavior less predictable and robust. From an evolutionary viewpoint, a “change in wiring” such as replacing the negative edge from 1 to 2 by a positive one (or, instead, perhaps introducing an additional inconsistent edge) could lead to unpredictable effects, and so the fitness of such a mutation may be harder to evaluate. In a monotone system, in contrast, a stimulus applied to a component is propagated in an unambiguous manner throughout the circuit, promoting a predictably consistent increase or consistent decrease in the concentrations of all other components.
Similarly, consistency also applies to feedback loops. For example, consider the graph shown in Fig. 4d. The negative feedback given by the inconsistent path 1,3,4,2,1 means that the instantaneous effect of an upperturbation of node 1 feeds back into a negative effect on node 1, while a downperturbation feeds back as a positive effect. In other words, the feedback loop acts against the perturbation.
Of course, negative feedback as well as inconsistent feedforward circuits are important components of biomolecular networks, playing a major role in homeostasis and in signal detection. The point being made here is that inconsistent networks may require a more delicate tuning in order to perform their functions.
In rigorous mathematical terms, this predictability property can be formulated as Kamke’s Theorem. Suppose that \(\Sigma =\{\sigma_i,i=1,\ldots ,n\}\) is a consistent spin assignment for the system graph G. Let x(t) be any solution of \(dx/dt=f(x).\) We wish to study how the solution z(t) arising from a perturbed initial condition \(z(0)=x(0)+\Delta \) compares to the solution x(t). Specifically, suppose that a positive perturbation is performed at time t = 0 on the ith coordinate, for some index \(i\in \{1,\ldots ,n\}\): \(z_i(0) > x_i(0)\) and \(z_j(0)=x_j(0)\) for all \(j\neq i.\) For concreteness, let us assume that the perturbed node i has been labeled by \(\sigma_i=+1.\) Then, Kamke’s Theorem says the following: for each node that has the same parity (i.e., each index j such that \(\sigma_j=+1\)), and for every future time t, \(z_j(t)\ge x_j(t).\) Similarly, for each node with opposite parity (\(\sigma_j=1\)), and for every time t, \(z_j(t)\le x_j(t).\) (Moreover, one or more of these inequalities must be strict.) This is the precise sense in which an upperturbation of the species represented by node v _{ i } unambiguously propagates into up or downbehavior of all the other species. See Smith (1995) for a proof, and see Angeli and Sontag (2003) for generalizations to systems with external input and output channels.
For difference equations (discrete time systems), once that selfloops have been included in the graph G and the definition of consistency, Kamke’s theorem also holds; in this case the proof is easy, by induction on time steps.
Removing the smallest number of edges so as to achieve consistency
Let us call the consistency deficit (CD) of a graph G the smallest possible number of edges that should be removed from G in order that there remains a consistent graph, and, correspondingly, a monotone system.
After deleting the diagonal, a consistent spin assignment Σ is: \(\sigma_1=\sigma_3=1\) and \(\sigma_2=\sigma_4=1,\) see Fig. 6b. (Another assignment is the one with all spins reversed: \(\sigma_1=\sigma_2=1\) and \(\sigma_3=\sigma_4=1\).) If we now bring back the deleted edge, we see that in the original graph only the one edge from node 1 to node 4 is inconsistent for the spin assignment Σ (Fig. 6c).
This example illustrates a general fact: minimizing the number of edges that must be removed so that there remains a consistent graph is equivalent to finding a spin assignment Σ for which the number of inconsistent edges (those for which \(J_{ij}\sigma_i\sigma_j=1\)) is minimized.
A very special case is when the graph has all of its edges labeled negative, that is, \(J_{ij}=1\) for all i,j. Stated in the language of partitions, the CD problem amounts to searching for a partition such that \(n_1+n_{1}\) is minimized (as there are no positive edges, p = 0). Moreover, since there are no positive edges, \(n_1+n_{1}\) is actually the total number of edges between any two nodes in A _{1} or in A _{−1}. Thus, \(N(n_1+n_{1})\) is the number of remaining edges, that is, the number of edges between nodes in A _{1} and A _{−1}. Therefore, minimizing \(n_1+n_{1}\) is the same as maximizing \(N(n_1+n_{1}).\) This is precisely the standard “MAXCUT” problem in computer science.
The enlarged graph has only negative edges, and it is easy to see that the minimal number of edges that have to be removed in order to achieve consistency is the same as the number of edges that would have had to be removed in the original graph. Unfortunately, the MAXCUT problem is NPhard. However, the paper (DasGupta et al. 2007) gave an approximation polynomialtime algorithm for the CD problem, guaranteed to solve the problem to within 87.9% of the optimum value, as an adaptation of the semidefinite programming relaxation approach to MAXCUT based on Goemans and Williamson’s work (1995). (Is not enough to simply apply the MAXCUT algorithm to the enlarged graph obtained by the above trick, because the approximation bound is degraded by the additional edges, so the construction takes some care.) The recent paper (Hüffner et al. 2007) substantially improved upon the approach in DasGupta et al. (2007), resulting in a very efficient algorithm.
Relation to Ising spinglass models
Nearmonotone systems may be “practically” monotone
Obviously, there is no reason for large biochemical networks to be consistent, and they are not. However, when the number of inconsistencies in a biological interaction graph is small, it may well be the case that the network is in fact consistent in a practical sense. For example, a gene regulatory network represents all potential effects among genes. These effects are often mediated by proteins which themselves need to be activated in order to perform their function, and this activation will, in turn, be contingent on the “environmental” context: extracellular ligands, additional genes being expressed which may depend on cell type or developmental stage, and so forth. Thus, depending on the context, different subgraphs of the original graph describe the system, and these graphs may be individually consistent even if the entire graph, the union of all these subgraphs, is not. As an illustration, take the system in Fig. 4c. Suppose that under environmental conditions A, the edge from 1 to 2 is not present, and under nonoverlapping conditions B, the edge from 1 to 3 is not be present. Then, under either conditions, A or B, the graph is consistent, even though, formally speaking, the entire network is not consistent.
The closer to consistent, the more likely that this phenomenon may occur.
Some evidence suggesting nearmonotonicity of natural networks
Since consistency in biological networks may be desirable, one might conjecture that natural biological networks tend to be consistent. As a way to test this hypothesis, the CD algorithm from DasGupta et al. (2007) was run on the yeast Saccharomyces cerevisiae gene regulatory network from Milo et al. (2002), downloaded from http://www.weizmann.ac.il/mcb/UriAlon/Papers/networkMotifs/yeastData.mat (Milo et al. (2002) used the YPD database (Costanzo et al. 2001). Nodes represent genes, and edges are directed from transcription factors, or protein complexes of transcription factors, into the genes regulated by them.) This network has 690 nodes and 1,082 edges, of which 221 are negative and 861 are positive (we labeled the one “neutral” edge as positive; the conclusions do not change substantially if we label it negative instead, or if we delete this one edge). The approximation algorithm from DasGupta et al. (2007) estimated the CD at 43, and the exact algorithm from Hüffner et al. (2007) later improved this estimate to a precise value CD = 41. In other words, deleting a mere 4% of edges makes the network consistent. Also remarkable is the following fact. The original graph has 11 components: a large one of size 664, one of size 5, three of size 3, and six of size 2. All of these components remain connected after edge deletion. The deleted edges are all from the largest component, and they are incident on a total of 65 nodes in this component.
To better appreciate if a small CD might happen by chance, the algorithm was also run on random graphs having 690 nodes and 1082 edges (chosen uniformly), of which 221 edges (chosen uniformly) are negative. It was found that, for such random graphs, about 12.6% (136.6 ± 5) of edges have to be removed in order to achieve consistency. (To analyze the scaling of this estimate, we generated random graphs with N nodes and 1.57N edges of which 0.32N are negative. We found that for N > 10, approximately N/5 nodes must be removed, thus confirming the result for N = 690.) Thus, the CD of the biological network is roughly 15 standard deviations away from the mean for random graphs. Both topology (i.e., the underlying graph) and actual signs of edges contribute to this nearconsistency of the yeast network. To justify this assertion, the following numerical experiment was performed. We randomly changed the signs of 50 positive and 50 negative edges, thus obtaining a network that has the same number of positive and negative edges, and the same underlying graph, as the original yeast network, but with 100 edges, picked randomly, having different signs. Now, one needs 8.2% (88.3 ± 7.1) deletions, an amount inbetween that obtained for the original yeast network and the one obtained for random graphs. Changing more signs, 100 positives and 100 negatives, leads to a less consistent network, with 115.4 ± 4.0 required deletions, or about 10.7% of the original edges, although still not as many as for a random network.
Decomposing systems into monotone components
For example, let us take the graph shown in Fig. 6a. The procedure of dropping the diagonal edge and seeing it instead as an external feedback loop can be modeled as follows. The original differential equation \({dx_1}/{dt}=f_1(x_1,x_2,x_3,x_4)\) is replaced by the equation \({dx_1}/{dt}=f_1(x_1,x_2,x_3,u),\) where the symbol u, which represents an external input signal, is inserted instead of the state variable x _{4}. The consistent system in Fig. 8 includes the remaining four edges, and the “negative” feedback (negative in the sense that it is inconsistent with the rest of the system) is the connection from x _{4}, seen as an “output” variable, back into the input channel represented by u. The closedloop system obtained by using this feedback is the original system, now viewed as a negative feedback around the consistent system in Fig. 6b.
Generally speaking, the decomposition techniques in Angeli and Sontag (2003, 2004a), Angeli et al. (2004a, b), Sontag (2004, 2005), Enciso et al. (2006), de Leenheer et al. (2005), Enciso and Sontag (2005b, 2006), De Leenheer and Malisoff (2006), Gedeon and Sontag (2007) are most useful if the feedback loop involves few variables. This is equivalent to asking that the graph G associated to the system be close to consistent, in the sense of the CD of G being small. This view of systems as monotone systems—which have strong stability properties, as discussed next, with negativefeedback regulatory loops around them is very appealing from a control engineering perspective as well.
Dynamical behavior of monotone systems
Continuoustime monotone systems have convergent behavior. For example, they cannot admit any possible stable oscillations (Hirsch and Smith 2005; Hadeler and Glas 1983; Hirsch 1984). When there is only one steadystate, a theorem of Dancer (1998) shows—under mild assumptions regarding possible constraints on the values of the variables, which are often satisfied, and boundedness of solutions, which usually follows from conservation laws—that every solution converges to this unique steadystate (monostability). When, instead, there are multiple steadystates, the Hirsch Generic Convergence Theorem (Smith 1995; Hirsch and Smith 2005; Hirsch 1983, 1985) is the fundamental result. A strongly monotone system is one for which the an initial perturbation \(z_i(0) > x_i(0)\) on the concentration of any species propagates as a strict up or down perturbation: \(z_j(t) > x_j(t)\) for all t > 0 and all indices j for which \(\sigma_j=\sigma_i,\) and \(z_j(t) < x_j(t)\) for all t > 0 and all j for which \(\sigma_j=\sigma_i.\) Observe that this requirement is stronger (hence the terminology) than merely weak inequalities: \(z_j(t)\ge x_j(t)\) or \(z_j(t)\le x_j(t)\), respectively as in Kamke’s Theorem. A sufficient condition for strong monotonicity is that the Jacobian matrices must be irreducible for all x, which basically amounts to asking that the graph G must be strongly connected and that every nonidentically zero Jacobian entry be everywhere nonzero. Even though they may have arbitrarily large dimensionality, monotone systems behave in many ways like onedimensional systems: Hirsch’s Theorem asserts that generic bounded solutions of strongly monotone differential equation systems must converge to the set of steadystates. (“Generic” means “every solution except for a measurezero set of initial conditions.”) In particular, no “chaotic” or other “strange” dynamics can occur. For discretetime strongly monotone systems, generically also stable oscillations are allowed besides convergence to equilibria, but no more complicated behavior.
The ordered behavior of monotone systems is robust with respect to spatial localization effects as well as signaling delays (such as those arising from transport, transcription, or translation). Moreover, their stability character does not change much if some inconsistent connections are inserted, but only provided that these added connections are weak (“small gain theorem”) or that they operate at a comparatively fast time scale (Wang and Sontag 2006a).
For general, nonmonotone systems, on the other hand, no dynamical behavior, including chaos, can be mathematically ruled out. This is in spite of the fact that some features of nonmonotone systems are commonly regarded as having a stabilizing effect. For example, negative feedback loops confer robustness with regard to certain types of structural as well as external perturbations (Doyle et al. 1990; Sepulchre et al. 1997; Sontag 1999; Khalil 2002). However, and perhaps paradoxically, the behavior of nonmonotone systems may also be very fragile: for instance, they can be destabilized by delays in negative feedback paths. Nonetheless, we conjecture that systems that are close to monotone must be betterbehaved, generically, than those that are far from monotone. Preliminary evidence (unpublished) for this has been obtained from the analysis of random Boolean networks, at least for discrete analogs of the continuous system, but the work is not yet definitive.
Directed cycles
Intuition suggests that somewhat less than monotonicity should suffice for guaranteeing that no chaotic behavior may arise, or even that no stable limit cycles exist. Indeed, monotonicity amounts to requiring that no undirected negativeparity cycles be present in the graph, but a weaker condition, that no directed negative parity cycles exist, should be sufficient to insure these properties. For a strongly connected graph, the property that no directed negative cycles exist is equivalent to the property that no undirected negative cycles exist, because the same proof as given earlier, but applied to directed paths, insures that a consistent spin assignment exists (and hence there cannot be any undirected negative cycles). However, for nonstrongly connected graphs, the properties are not the same. On the other hand, every graph can be decomposed as a cascade of graphs that are strongly connected. This means (aside from some technicalities having to do with Jacobian entries being not identically zero but vanishing on large sets) that systems having no directed negative cycles can be written as a cascade of strongly monotone systems. Therefore, it is natural to conjecture that such cascades have nice dynamical properties. Indeed, under appropriate technical conditions for the systems in the cascade, one may recursively prove convergence to equilibria in each component, appealing to the theory of asymptotically autonomous systems (Thieme 1992) and thus one may conclude global convergence of the entire system (Hirsch 1989; Smith 1991). For example, a cascade of the form \(dx/dt=f(x),\) \(dy/dt=g(x,y)\) where the x system is monotone and where the system \(dy/dt=g(x_0,y)\) is monotone for each fixed x _{0}, cannot have any attractive periodic orbits (except equilibria). This is because the projection of such an orbit on the first system must be a point x _{0}, and hence the orbit must have the form \((x_0,y(t)).\) Therefore, it is an attractive periodic orbit of \(dy/dt=g(x_0,y),\) and by monotonicity of this latter system we conclude that \(y(t)\equiv \) a constant as well. The argument generalizes to any cascade, by an inductive argument. Also, chaotic attractors cannot exist (D. Angeli et al. in preparation).
The condition of having no directed negative cycles is the weakest one that can be given strictly on the basis of the graph G, because for any graph G with a negative feedback loop there is a system with graph G which admits stable periodic orbits. (First find a limit cycle for the loop, and then use a small perturbation to define a system with nonzero entries as needed, which will still have a limit cycle.)
Positive feedback and stability
The strong global convergence properties of monotone systems mentioned above would seemingly contradict the fact that positive feedback, which tends to increase the direction of perturbations, is allowed in monotone systems, but negative feedback, which tends to stabilize systems, is not. One explanation for this apparent paradox is that the main theorems in monotone systems theory only guarantee that bounded solutions converge, but they do not make any assertions about unbounded solutions. For example, the system \(dx/dt=x + x^2\) has the property that every solution starting at an x(0) > 1 is unbounded, diverging to + ∞, a fact which does not contradict its monotonicity (every onedimensional system is monotone). This is not as important a restriction as it may seem, because for biochemical systems it is often the case that all trajectories must remain bounded, due to conservation of mass and other constraints. A second explanation is that negative selfloops are not ruled out in monotone systems, and such loops, which represent degradation or decay diagonal terms, help insure stability.
Intuition on why negative selfloops do not affect monotonicity
In the definition of the graph associated to a continuoustime system, selfloops (diagonal terms in the Jacobian of the vector field f) were ignored. The theory (Kamke’s condition) does not require selfloop information in order to guarantee monotonicity. Intuitively, the reason for this is that a larger initial value for a variable x _{ i } implies a larger value for this variable, at least for short enough time periods, independently of the sign of the partial derivative \(df_i/dx_i\) (continuity of flow with respect to initial conditions). For example, consider a degradation equation \(dp/dt = p,\) for the concentration p(t) of a protein P. At any time t, we have that \(p(t)=e^{t}p(0),\) where p(0) is the initial concentration. The concentration p(t) is positively proportional to p(0), even though the partial derivative \({\partial(p)}/{\partial p}=1\) is negative. Note that, in contrast, for a difference equation, a jump may occur: for instance the iteration \(p(t+1)=p(t)\) has the property that the order of two elements is reversed at each time step. Thus, for difference equations, diagonal terms matter.
Multiple time scale analysis may make systems monotone
A system may fail to be monotone due to the effect of negative regulatory loops that operate at a faster time scale than monotone subsystems. In such a case, sometimes an approximate but monotone model may be obtained, by collapsing negative loops into selfloops. Mathematically: a nonmonotone system might be a singular perturbation of a monotone system. A trivial linear example that illustrates this point is \(dx/dt=xy,\) \(\varepsilon dy/dt=y+x,\) with \(\varepsilon > 0.\) This system is not monotone (with respect to any orthant cone). On the other hand, for \(\varepsilon \ll1,\) the fast variable y tracks x, so the slow dynamics is wellapproximated by \(dx/dt=2x\) (monotone, since every scalar system is). More generally, one may consider \(dx/dt=f(x,y),\) \(\varepsilon dy/dt=g(x,y)\) such that the fast system \(dy/dt=g(x,y)\) has a unique globally asymptotically stable steadystate y = h(x) for each x (and possibly a mild input to state stability requirement, as with the special case \(\varepsilon dy/dt=y+h(x)\)), and the slow system \(dx/dt=f(x,h(x))\) is (strongly) monotone. Then one may expect that the original system inherits global convergence properties, at least for all \(\varepsilon > 0\) small enough. The paper (Wang and Sontag 2006b) employs tools from geometric invariant manifold theory (Fenichel 1979; Jones 1994), taking advantage of the existence of a manifold \(M_\varepsilon \) invariant for the dynamics, which attracts all nearenough solutions, and with an asymptotic phase property. The system restricted to the invariant manifold \(M_\varepsilon \) is a regular perturbation of the fast (\(\varepsilon =0\)) system, and hence inherits strong monotonicity properties. So, solutions in the manifold will be generally wellbehaved, and asymptotic phase implies that solutions track solutions in \(M_\varepsilon,\) and hence also converge to equilibria if solutions on \(M_\varepsilon\) do. However, the technical details are delicate, because strong monotonicity only guarantees generic convergence, and one must show that the generic tracking solutions start from the “good” set of initial conditions, for generic solutions of the large system.
Discretetime systems
As discussed, for autonomous differential equations monotonicity implies that stable periodic behaviors will not be observed, and moreover, under certain technical assumptions, all trajectories must converge to steadystates. This is not exactly true for difference equation models, but a variant does hold: for discretetime monotone systems, trajectories must converge to either steadystates or periodic orbits. In general, even the simplest difference equations may exhibit arbitrarily complicated (chaotic) behavior, as shown by the logistic iteration in one dimension \(x(t+1)=kx(t)(1x(t))\) for appropriate values of the parameter k (Devaney 1989). However, for monotone difference equations, a close analog of Hirsch’s Generic Convergence Theorem is known. Specifically, suppose that the equations are pointdissipative, meaning that all solutions converge to a bounded set (Hale 1988), and that the system is strongly monotone, in the sense that the Jacobian matrix \((\partial f_i / \partial x_j)\) is irreducible at all states. Then, a result of Tereščák and coworkers (Poláčik and Tereščák 1992; Poláčik and Tereščák 1993; Hess and Poláčik 1993; Tereščák 1996) shows that there is a positive integer m such that generic solutions (in an appropriate sense of genericity) converge to periodic orbits with period at most m. Results also exist under less than strong monotonicity, just as in the continuous case, for example when steadystates are unique (Dancer 1998).
Oscillatory behaviors
Stable periodic behaviors are ruledout in autonomous monotone continuoustime systems. However, stable periodic orbits may arise through various external mechanisms. Three examples are (1) inhibitory negative feedback from some species into others in a monotone monostable system, (2) the generation of relaxation oscillations from a hysteresis parametric behavior by negative feedback on parameters by species in a monotone system, and (3) entrainment of external periodic signals. These general mechanisms are classical and wellunderstood for simple, one or twodimensional, dynamics, and they may be generalized to the case where the underlying system is higherdimensional but monotone.
Embeddings in monotone systems
Discrete systems
We remark that one may also study difference equations for which the state components are only allowed to take values out of a finite set. For example, in Boolean models of biological networks, each variable x _{ i }(t) can only attain two values (0/1 or “on/off”). These values represent whether the ith gene is being expressed, or the concentration of the ith protein is above certain threshold, at time t. When detailed information on kinetic rates of protein–DNA or protein–protein interactions is lacking, and especially if regulatory relationships are strongly sigmoidal, such models are useful in theoretical analysis, because they serve to focus attention on the basic dynamical characteristics while ignoring specifics of reaction mechanisms (Kauffman 1969a, b; Kauffman and Glass 1973; Albert and Othmer 2003; Chaves et al. 2005).
For difference equations over finite sets, such as Boolean systems, it is quite clear that all trajectories must either settle into equilibria or to periodic orbits, whether the system is monotone or not. However, cycles in discrete systems may be arbitrarily long and these might be seen as “chaotic” motions. Monotone systems, while also settling into steadystates or periodic orbits, have generally shorter cycles. This is because periodic orbits must be antichains, i.e.no two different states can be compared; see Smith (1995) and Gilbert (1954). For example, consider a discretetime system in which species concentrations are quantized to the k values \(\{0,\ldots ,{k1}\};\) we interpret monotonicity with respect to the partial order: \((a_1,\ldots ,a_n)\le (b_1,\ldots ,b_n)\) if every coordinate \(a_i\le b_i.\) For nonmonotone systems, orbits can have as many as k ^{ n } states. On the other hand, monotone systems cannot have orbits of size more than the width (size of largest antichains) of \(P=\{0,\ldots ,{k1}\}^n,\) which can be interpreted as the set of multisubsets of an nelement set, or equivalently as the set of divisors of a number of the form \((p_1p_2\ldots p_n)^{k1}\) where the p _{ i }’s are distinct primes. The width of P is the number of possible vectors \((i_1,\ldots ,i_n)\) such that \(\sum i_j =\lfloor{kn/2}\rfloor\) and each \(i_j\in \{0,\ldots ,{k1}\}.\) This is a generalization of Sperner’s Theorem; see Anderson (2002). For example, for n = 2, periodic orbits in a monotone system evolving on \(\{0,\ldots ,{k1}\}^2\) cannot have length larger than k, while nonmonotone systems on \(\{0,\ldots ,{k1}\}^2\) can have a periodic orbit of period k ^{2}. As another example, arbitrary Boolean systems (i.e., the state space is \(\{0,1\}^n\)) can have orbits of period up to 2^{ n }, but monotone systems cannot have orbits of size larger than \({n \choose \lfloor{n/2}\rfloor}\approx 2^n \sqrt{2 / (n \pi)}.\) These are all classical facts in Boolean circuit design (Gilbert 1954). It is worth pointing out that any antichain P _{0} can be seen as a periodic orbit of a monotone system. This is proved as follows: we enumerate the elements of P _{0} as \(x_1,\ldots ,x_\ell,\) and define \(f(x_i)=x_{i1}\) for all i modulo ℓ. Then, f can be extended to all elements of the state space by defining \(f(x)=(0,\ldots ,0)\) for every x which has the property that \(x < x_i\) for some \(x_i\in P_0\) and \(f(x)=({k1},\ldots,{k1})\) for every x which is not \(\le x_i\) for any \(x_i\in P_0.\) It is easy to see that this is a monotone map (Gilbert 1954; Aracena et al. 2004).
I/O monotone systems
We next describe recent work on monotone input/output systems (“MIOS” from now on). Monotone i/o systems originated in the analysis of mitogenactivated protein kinase cascades and other cell signaling networks, but later proved useful in the study of a broad variety of other biological models. Their surprising breath of applicability notwithstanding, of course MIOS constitute a restricted class of models, especially when seen in the context of large biochemical networks. Indeed, the original motivation for introducing MIOS, in the 2003 paper (Angeli and Sontag 2003), was to study an existing nonmonotone model of negative feedback in MAPK cascades. The key breakthrough was the realization that this example, and, as it turned out, many others, can be profitably studied by decompositions into MIOS. In other words, a nonmonotone system is viewed as an interconnection of monotone subsystems. Based on the architecture of the interconnections between the subsystems (“network structure”), one deduces properties of the original, nonmonotone, system. (Later work, starting with Angeli and Sontag (2004a), showed that even monotone systems can be usefully studied through this decompositionbased approach.)
Given three partial orders on X,U,Y (we use the same symbol \(\prec \) for all three orders), a monotone I/O system (MIOS), with respect to these partial orders, is a system (1000) such that h is a monotone map (it preserves order) and: for all initial states \(x_1,x_2\) for all inputs \(u_1,u_2,\) the following property holds: if \(x_1\preceq x_2\) and \(u_1\preceq u_2\) (meaning that \(u_1(t)\preceq u_2(t)\) for all t ≥ 0), then \(\varphi(t,x_1,u)\preceq\varphi(t,x_2,u_2)\) for all t > 0. Here we consider partial orders induced by closed proper cones \(K\subseteq{\mathbb{R}}^\ell,\) in the sense that \(x\preceq y\) iff \(yx\in K.\) The cones K are assumed to have a nonempty interior and are pointed, i.e. \(K\bigcap K=\{0\}.\) A strongly monotone system is one which satisfies the following stronger property: if \(x_1\preceq x_2,\) \(x_1\neq x_2,\) and \(u_1\preceq u_2,\) then the strict inequality \(\varphi(t,x_1,u)\prec\!\prec \varphi(t,x_2,u_2)\) holds for all t > 0, where \(x\prec\!\prec y\) means that y−x is in the interior of the cone K.
The most interesting particular case is that in which K is an orthant cone in \({\mathbb{R}}^n,\) i.e. a set \(S_\varepsilon \) of the form \(\{x\in {\mathbb{R}}^n  \varepsilon_i x_i\ge 0\},\) where \(\varepsilon_i=\pm 1\) for each i.
When there are no inputs nor outputs, the definition of monotone systems reduces to the classical one of monotone dynamical systems studied by Hirsch, Smith, and Others (1995). This is what we discussed earlier, for the case of orthant cones. When there are no inputs, strongly monotone classical systems have especially nice dynamics. Not only is chaotic or other irregular behavior ruled out, but, in fact, almost all bounded trajectories converge to the set of steady states (Hirsch’s generic convergence theorem (see Hirsch (1983, 1985)).
A useful test for monotonicity with respect to orthant cones, which generalizes Kamke’s condition to the i/o case, is as follows. Let us assume that all the partial derivatives \(\frac{\partial f_i}{\partial x_j}(x,u)\) for \(i\neq j,\) \(\frac{\partial f_i}{\partial u_j}(x,u)\) for all i,j, and \(\frac{\partial h_i}{\partial x_j}(x)\) for all i,j (subscripts indicate components) do not change sign, i.e., they are either always ≥0 or always ≤0. We also assume that X is convex (much less is needed.) We then associate a directed graph G to the given MIOS, with n + m + p nodes, and edges labeled “+” or “−” (or ±1), whose labels are determined by the signs of the appropriate partial derivatives (ignoring diagonal elements of \(\partial f/\partial x\)). One may define in an obvious manner undirected loops in G, and the parity of a loop is defined by multiplication of signs along the loop. (See e.g. Angeli and Sontag 2004a, b for more details.) Then, it is easy to show that a system is monotone with respect to some orthant cones in X,U,Y if and only if there are no negative loops in G. A sufficient condition for strong monotonicity is that, in addition to monotonicity, the partial Jacobians of f with respect to x should be everywhere irreducible. (“Almosteverywhere” often suffices; see Smith (1995), Hirsch and Smith 2005). See these references also for extensions to nonorthant cones in the case of no inputs and outputs, based on work of Schneider and Vidyasagar, Volkmann, and others (Schneider and Vidyasagar 1970; Volkmann 1972; Walcher 2001; Walter 1970).
In inhibitory feedback, a chemical species x _{ j } typically affects the rate of formation of another species x _{ i } through a term like \(h(x_j)={V}/({K+x_j}).\) The decreasing function h(x _{ j }) can be seen as the output of an antimonotone system, i.e. a system which satisfies the conditions for monotonicity, except that the output map reverses order: \(x_1\preceq x_2 \Rightarrow h(x_2)\preceq h(x_1).\)
An interconnection of monotone subsystems, that is to say, an entire system made up of monotone components, may or may not be monotone: “positive feedback” (in a sense that can be made precise) preserves monotonicity, while “negative feedback” destroys it. Thus, oscillators such as circadian rhythm generators require negative feedback loops in order for periodic orbits to arise, and hence are not themselves monotone systems, although they can be decomposed into monotone subsystems (cf. Angeli and Sontag 2004c). A rich theory is beginning to arise, characterizing the behavior of nonmonotone interconnections. For example, Angeli and Sontag (2003) shows how to preserve convergence to equilibria; see also the followup papers (Enciso et al. 2006; Angeli et al. 2004b; de Leenheer et al. 2005; Enciso and Sontag 2006; Gedeon and Sontag 2007). Even for monotone interconnections, the decomposition approach is very useful, as it permits locating and characterizing the stability of steadystates based upon input/output behaviors of components, as described in Angeli and Sontag (2004a); see also the followup papers Angeli et al. (2004a), Enciso and Sontag (2005b), De Leenheer and Malisoff (2006).
Moreover, a key point brought up in Angeli and Sontag (2003, 2004a), Sontag (2004, 2005) is that new techniques for monotone systems in many situations allow one to characterize the behavior of an entire system, based upon the “qualitative” knowledge represented by general network topology and the inhibitory or activating character of interconnections, combined with only a relatively small amount of quantitative data. The latter data may consist of steadystate responses of components (doseresponse curves and so forth), and there is no need to know the precise form of dynamics or parameters such as kinetic constants in order to obtain global stability conclusions and study global bifurcation behavior. We now discuss these issues, first for positive and then for negative feedback loops.
Positive feedback and possible multistability
We first discuss how multistability in cell signaling networks may arise from positive feedback loops. The general framework is that in which two input/output systems, each of which is monostable in isolation, can combine to produce a multistable closedloop behavior when interconnected in closedloop. Schematically, we consider two systems, one of which processes an input signal u and produces an output y, and a second one which processes the signal y to produce u.
Stepinput steadystate responses (characteristics) of openloop systems
Characteristics (dose–response curves, activity plots, steadystate expression of a gene in response to an external ligand, etc.) are frequently available from experimental data, especially in molecular biology and pharmacology, for instance in the modeling of receptor–ligand interactions (Chaves et al. 2004).
The results to be described are also valid under weaker definitions of characteristics, such as not requiring GAS properties, or allowing setvalued characteristics (Angeli and Sontag 2003; Angeli et al. 2004b; de Leenheer et al. 2005; De Leenheer and Malisoff 2006; Enciso and Sontag 2005a, 2006; G.A. Enciso and E.D. Sontag, in preparation).
It is worth pointing out that, if a system is monotone, then the stability property in the definition of characteristic is often automatically satisfied, provided that uniqueness of steadystates holds. More precisely, if one knows that (a) trajectories are bounded, and (b) the state space X has the property that least upper bounds and greatest lower bounds exist for any two elements of X (for example, if the state space is a “cube” with respect to the order cone K), then just knowing that K(u _{0}) has only one point is enough to conclude that K(u _{0}) is in fact a GAS state for \(dx/dt=f(x,u_0)\) (Dancer 1998; Jiang 1994).
Hyperbolic and sigmoidal characteristics
Before reviewing theorems about feedback interconnections of MIOS systems, we discusse a very simple example which does not require any theory. Often, models of systems representing signaling and other molecular biology networks have a hyperbolic or a sigmoidal steadystate response.
It is believed that sigmoidal responses in signaling pathways are used in those situations in which binary decisions must be taken, such as when a cell must “decide” whether a gene should be transcribed or not, depending on the value of an extracellular signal (Novic and Weiner 1957; Ptashne 1992; Thomas and Kaufman 2001; Sha et al. 2003; Pomerening et al. 2003; Ferrell and Xiong 2001; Lisman 1985; Laurent and Kellershohn 1999; Gardner et al. 2000; Ferrell and Machleder 1998; Bagowski and Ferrell Jr. 2001; Bhalla et al. 2002; Cross et al. 2002; Becskei et al. 2001; Bagowski et al. 2003). Sigmoidal responses with large r > 1 (“ultrasensitive responses”) can be obtained by cascading simple enzymatic reactions provided that each reaction in the cascade has a Hill coefficient r > 1 (Ferrell Jr. 1996). (Basically, this statement amounts to the chain rule for derivatives.)
Creating bistability from sigmoidal responses
The simplest way to create bistability from a sigmoidal response is through positive feedback. We illustrate this procedure using the example just discussed. Schematically, we start with the “open loop” system that produces the protein P, with its concentration y(t) = p(t) considered as an output and the concentation u(t) of U seen as an input. We then “close the loop” by introducing a second system, one that simply produces U from P in such a manner that the concentration of U is proportional to that of P, as in Fig. 11. We ignore, for the purpose of this expository example, the details of the mechanism that implements the autocatalytic process in which U is produced from P. The mechanism might involve several intermediate proteins as well as time delays. For simplicity, we assume that there results an instantaneous change in the concentration of U proportional to the concentration of P. (One of the tools to be discussed, the theory of monotone input/output systems, provides conditions that explain when this simplification is justified.)
We plot in Fig. 14 both the formation rate \(({V_{max}p^r})/({k_m^r+p^r})\) together with the degradation/dilution rate kp, in cases where r = 1 (left) or r > 1 (right). We assume, in the sigmoidal case, that the slope of the degradation curve is so that three intersections result, as shown in the plots. (For different k’s, the line will have different slopes, and anywhere from one to three intersections are possible.)
Positive feedback and multistability in monotone I/O systems
The elementary and intuitive proof of bistability for the simple production/degradation system with sigmoidal characteristics just discussed can be generalized to a feedback interconnections of individually monostable systems, applying even if the \(u\ \mapsto\ y\) and \(y\ \mapsto\ u\) systems in Fig. 11 are far more complicated than the onedimensional system \(dy/dt=\frac{V_{max}u^r}{k_m^r+u^r} kp\) and the memoryless system u = λy, respectively.
For expository reasons (see Enciso and Sontag (2005b) for a generalization to highdimensional inputs and outputs), we assume as in Angeli and Sontag (2004a) that the inputs and outputs of both systems are scalar: \(m_1=m_2=p_1=p_2=1.\)
Of course, the systems may be more complicated than \(du/dt=yg^{1}(u)\) and y = k(u), so that the above paragraph does not constitute a proof. Nonetheless, a theorem to be explained below provides conditions insuring the validity of this argument. Before explaining the generalization, however, we provide a cautionary note, with the purpose of showing that intuition may sometimes fail.
A cautionary counterexample
A general theorem
We remark that the theorems remain true even if arbitrary delays are allowed in the feedback loop and/or if spacedependent models are considered and diffusion is allowed (see Sontag (2005) for a discussion). A new approach (Angeli 2006), based not on monotone theory but on a notion of “counterclockwise dynamics,” extends in a different direction the range of applicability of this methodology.
We wish to emphasize the potential practical relevance of this result (and others such as Angeli (2006)). The equations describing each of the systems are often poorly, or not at all, known. But, as long as we can assume that each subsystem is monotone and monostable, we can use the information from the planar plots in Fig. 15b to understand the global dynamics of the closedloop system, no matter how large the number of state variables. It is often said that the field of molecular systems biology is characterized by a datarich/datapoor paradox: while on the one hand a huge amount of qualitative network (schematic modeling) knowledge is available for signaling, metabolic, and gene regulatory networks, on the other hand little of this knowledge is quantitative, at least at the level of precision demanded by most mathematical tools of analysis. On the other hand, input/output steadystate data (from a signal such as a ligand, to a reporter variable such as the expression of a gene monitored by GFP, or the activity of a protein measured by a Western blot) is frequently available. The problem of exploiting qualitative knowledge, and effectively integrating relatively sparse quantitative data, is among the most challenging issues confronting systems biology. The MIOS approach provides one way to combine these two types of data, hence addressing the “datarich/datapoor” issue (Sontag 2004, 2005). When applicable, MIOS analysis allows one to combine the numerical information provided by the shape of the graphs of characteristics with the qualitative information given by (signed) network topology in order to predict global bifurcation behavior. This information is often easier to obtain from experimental data, at least in interpolated form, than kinetic constants (of which there may be a very large number). An analysis based on characteristics, when it can be done, is “robust” with respect to uncertainty in internal parameters of the system, and serves as a “qualitativequantitative approach” to systems biology (Sontag 2005). In addition, characteristics are also a very powerful tool for the purely mathematical analysis of existing models. Monotone systems with welldefined characteristics constitute a very wellbehaved set of building blocks for arbitrary systems, as illustrated by the fact that cascades of such systems inherit the same properties (monotone, monostable response) and by the feedback theorems reviewed here, originally presented in the works (Angeli and Sontag 2004a; Angeli and Sontag 2003).
More discussion through an example: MAPK cascades
We make here the simplest assumptions about the dynamics, amounting basically to a quasisteadystate approximation of enzyme kinetics. (For related results using more realistic, massaction, models, see Angeli and Sontag (2007) and Angeli et al. (2006, 2007).) For example, take the reaction shown in the square in Fig. 19a. As y _{3} (MAPKKPP) facilitates the conversion of z _{1} into z _{2} (MAPK to MAPKP), the rate of change dz _{2}/dt should include a term \(\alpha (z_1,y_3)\) (and \(dz_1/dt\) has a term \(\alpha (z_1,y_3)\)) for some (otherwise unknown) function α such that \(\alpha (0,y_3)=0\) and \(\frac{\partial \alpha}{\partial z_1} > 0,\) \(\frac{\partial \alpha}{\partial y_3} > 0\) when \(z_1 > 0.\) (Nothing happens if there is no substrate, but more enzyme or more substrate results in a faster reaction.) There will also be a term \(+\beta (z_2)\) to reflect the phosphatase action. Similarly for the other species. The system as given would be represented by a set of seven ordinary differential equations (or reaction–diffusion PDE’s, if spatial localization is of interest, or delaydifferential equations, if appropriate).
This system is not monotone (at least with respect to any orthant cone), as is easy to verify graphically. However, as with many other examples of biochemical networks, the system is “monotone in disguise”, so to speak, in the sense that a judicious change of variables allows one to apply MIOS tools. (Far more subtle forms of this argument are key to applications to signaling cascades. A substantial research effort, not reviewed here because of lack of space, addresses the search for graphtheoretic conditions that allow one to find such “monotone systems in disguise”; see Sontag (2004, 2005) and Angeli et al. (2006) for references).
Positive and negative feedback loops around MAPK cascades have been a topic of interest in the biological literature. For example, see Ferrell and Machleder (1998) and Bhalla et al. (2002) for positive feedback and Kholodenko (2000) and Shvartsman et al. (2000) for negative feedback. Since we know that the system is monotone and has a characteristic, MIOS theory as described here can indeed be applied to the example. We study next the effect of a positive feedback u = g y obtained by “feeding back” into the input a scalar multiple g of the output. (This is a somewhat unrealistic model of feedback, since feedbacks act for example by enhancing the activity of a kinase. We pick it merely for illustration of the techniques.)
With these choices, the steadystate step response is the sigmoidal curve shown in Fig. 19c, where y is the output z _{3}. We plotted in the same figure the inverse g ^{−1} of the characteristic of the feedback system, in this case just the linear mapping y = (1/g)u, for three typical “feedback gains” (g = 1/0.98,1/2.1,1/6).
This resulting complete bifurcation diagram showing points of saddlenode bifurcation can be also completely determined just from the characteristic, with no need to know the equations of the system. Relaxation oscillations may be expected under such circumstances if a second, slower, feedback loop is used to negatively adapt the gain as a function of the output. Reasons of space preclude describing a very general theorem, which shows that indeed, relaxation oscillations can be guaranteed in this fashion: see Gedeon and Sontag (2007) for technical details, and Sontag (2005) for a more informal discussion. Fig. 22b shows a simulation confirming the theoretical prediction (details in Sontag (2005) and Gedeon and Sontag (2007)).
Negative feedback and possible oscillations
Then, if solutions of the closedloop system are bounded and if this iteration has a globally attractive fixed point \({\bar u},\) as shown in Fig. 23b, then the feedback system has a globally attracting steadystate. (An equivalent condition, see Enciso and Sontag (2006), is that the iteration have no nontrivial periodtwo orbits.) We call this result a small gain theorem (“SGT”), because of its analogy to concepts in control theory.
It is easy to see that arbitrary delays may be allowed in the feedback loop. In other words, the feedback could be of the form \(u(t) = y(th),\) and such delays (even with h = h(t) time varying or even statedependent, as long as \(th(t)\rightarrow \infty \) as \(t\rightarrow \infty \)) do not destroy global stability of the closed loop. In Enciso et al. (2006), we have now shown also that diffusion does not destroy global stability either. In other words, a reaction–diffusion system (Neumann boundary conditions) whose reaction can be modeled in the shown feedback configuration, has the property that all solutions converge to a (unique) uniform in space solution. This is not immediately obvious, since standard parabolic comparison theorems do not immediately apply to the feedback system, which is not monotone.
Example: MAPK cascade with negative feedback
Example: testosterone model
Example: Lac operon
Based on results on rational difference equations from Kulenovic and Ladas (2002), one concludes that there are no nontrivial 2periodic orbits, provided that \(\rho < (\sqrt{K}+1)/(K1),\) for arbitrary \(b_1,b_2,b_3,r,S.\) Hence, by the theorem, there is a unique steadystate of the original system, which is GAS, even when arbitrary delays are present.
These and other conditions are analyzed in Enciso and Sontag (2006), where it is also shown that the results from Mahaffy and Savev (1999) are recovered as a special case. Among other advantages of this approach, besides generalizing the result and giving a conceptually simple proof, we have (because of Enciso et al. (2006)) the additional conclusion that also for the corresponding reaction–diffusion system, in which localization is taken account of, the same globally stable behavior can be guaranteed.
Example: Circadian oscillator
A counterexample
Note that, for each constant input \(u\equiv u_0,\) the solution of the system converges to (0, u _{0}/2), and therefore the output converges to u _{0}, so indeed the characteristic k is the identity. We only need to modify the feedback law in order to make solutions of the closedloop globally bounded. For the feedback law we pick \(g(x) = 0.5 \hbox{sat}(y),\) where \(\hbox{sat}(\cdot):=\hbox{sign}(\cdot)\hbox{min}\{1,\cdot\}\) is a saturation function. The only equilibrium of the closedloop system is at (0,0).
With an arbitrary initial condition u _{0}, we have that \(u_1=(1/2)\hbox{sat}(u_0),\) so that \(u_1\le 1/2.\) Thus \(u_k=(1/2)u_{k1}\) for all k ≥ 2, and indeed \(u_k\rightarrow 0\) so global convergence of the iteration holds.
Conclusions
There is a clear need in systems biology to study robust structures and to develop robust analysis tools. The theory of monotone systems provides one such tool. Interesting and nontrivial conclusions can be drawn from (signed) network structure alone, which is associated to purely stoichiometric information about the system, and ignores fluxes.
Associating a graph to a given system, we may define spin assignments and consistency, a notion that may be interpreted also as nonfrustration of Ising spinglass models. Every species in a monotone system (one whose graph is consistent) responds with a consistent sign to perturbations at every other species. This property would appear to be desirable in biological networks, and, indeed, there is some evidence suggesting the nearmonotonicity of some natural networks. Moreover, “near”monotone systems might be “practically” monotone, in the sense of being monotone under disjoint environmental conditions.
Dynamical behavior of monotone systems is ordered and “nonchaotic.” Systems close to monotone may be decomposed into a small number of monotone subsystems, and such decompositions may be usefully employed to study nonmonotone dynamics as well as to help detect bifurcations even in monotone systems, based only upon sparsenumerical data, resulting in a sometimes useful modelreduction approach.
Notes
Acknowledgments
Much of the author’s work on I/O monotone systems was done in collaboration with David Angeli, as well as Patrick de Leenheer, German Enciso, Bhas kar Dasgupta, and Hal Smith. The author also wishes to thank Moe Hirsch, Reka Albert, Tom Knight, Avi Maayan, Alex van Oudenaarden, and many others, for useful comments and suggestions regarding the material discussed here.
References
 Albert R, Othmer HG (2003) The topology of the regulatory interactions predicts the expression pattern of the drosophila segment polarity genes. J Theoret Biol 223:1–18Google Scholar
 Allwright DJ (1977) A global stability criterion for simple control loops. J Math Biol 4:363–373Google Scholar
 Anderson I (2002) Combinatorics of finite sets. Dover Publications, Mineola, NYGoogle Scholar
 Angeli D (2006) Systems with counterclockwise input–output dynamics. IEEE Trans Automatic Control 51:1130–1143Google Scholar
 Angeli D, Sontag ED (2003) Monotone control systems. IEEE Trans Automat Control 48(10):1684–1698. Errata are here: http://www.math.rutgers.edu/(tilde)sontag/FTPDIR/angelisontagmonotoneTAC0 3typos.txtGoogle Scholar
 Angeli D, Sontag ED (2004a) Multistability in monotone input/output systems. Syst Control Lett 51(3–4):185–202Google Scholar
 Angeli D, Sontag ED (2004b) Interconnections of monotone systems with steadystate characteristics. In: Optimal control, stabilization and nonsmooth analysis, vol 301 of Lecture Notes in Control and Information Science. Springer, Berlin, pp 135–154Google Scholar
 Angeli D, Sontag ED (2004c) An analysis of a circadian model using the smallgain approach to monotone systems. In Proc. IEEE Conf. Decision and Control, Paradise Island, Bahamas, Dec. IEEE Publications, pp 575–578Google Scholar
 Angeli D, Sontag ED (2007) Analysis of a circadian model using the smallgain approach to monotone systems. In: IEEE transactions automation and control, Special Issue on Systems Biology, January 2007 (submitted). Preprint version in arXiv qbio.QM/0701018, 14 Jan 2007Google Scholar
 Angeli D, Sontag ED (2007) Translationinvariant monotone systems, and a global convergence result for enzymatic futile cycles. Nonlinear Anal Ser B: Real World Appl (to appear)Google Scholar
 Angeli D, Ferrell JE, Sontag ED (2004a) Detection of multistability, bifurcations, and hysteresis in a large class of biological positivefeedback systems. Proc Natl Acad Sci USA 101(7):1822–1827, February 2004. A revision of Suppl. Fig.7(b) is here:http://www.math.rutgers.edu/(tilde)sontag/FTPDIR/nullclinesfgREV.jpg; and typos can be found here: http://www.math.rutgers.edu/(tilde)sontag/FTPDIR/angeliferrellsontagpnas0 4errata.txt.Google Scholar
 Angeli D, de Leenheer P, Sontag ED (2004b) A smallgain theorem for almost global convergence of monotone systems. Syst Control Lett 52(5):407–414Google Scholar
 Angeli D, de Leenheer P, Sontag ED (2006) On the structural monotonicity of chemical reaction networks. In: Proc. IEEE Conf. Decision and Control, San Diego, Dec. 2006, p WeA01.2. IEEE, 2006Google Scholar
 Angeli D, de Leenheer P, Sontag ED (2007) A Petri net approach to the study of persistence in chemical reaction networks. Math Biosci (to appear). Also arXiv qbio.MN/068019v2, 10 Aug 2006Google Scholar
 Aracena J, Demongeot J, Goles E (2004) On limit cycles of monotone functions with symmetric connection graph. Theoret Comput Sci 322(2):237–244Google Scholar
 Asthagiri AR, Lauffenburger DA (2001) A computational study of feedback effects on signal dynamics in a mitogenactivated protein kinase (MAPK) pathway model. Biotechnol Prog 17:227–239PubMedGoogle Scholar
 Bagowski CP, Ferrell JE Jr (2001) Bistability in the JNK cascade. Curr Biol 11:1176–1182PubMedGoogle Scholar
 Bagowski CP, Besser J, Frey CR, Ferrell JE Jr (2003) The JNK cascade as a biochemical switch in mammalian cells: ultrasensitive and allornone responses. Curr Biol 13:315–320PubMedGoogle Scholar
 Barahona F (1982) On the computational complexity of Ising spin glass models. J Phys A Math Gen 15:3241–3253Google Scholar
 Becskei A, Seraphin B, Serrano L (2001) Positive feedback in eukaryotic gene networks: cell differentiation by graded to binary response conversion. EMBO J 20:2528–2535PubMedGoogle Scholar
 Bhalla US, Ram PT, Iyengar R (2002) Map kinase phosphatase as a locus of flexibility in a mitogenactivated protein kinase signaling network. Science 297:1018–1023PubMedGoogle Scholar
 Cartwright M, Husain MA (1986) A model for the control of testosterone secretion. J Theoret Biol 123:239–250Google Scholar
 Chaves M, Sontag ED, Dinerstein RJ (2004) Steadystates of receptorligand dynamics: a theoretical framework. J Theoret Biol 227(3):413–428Google Scholar
 Chaves M, Albert R, Sontag ED (2005) Robustness and fragility of Boolean models for genetic regulatory networks. J Theoret Biol 235(3):431–449Google Scholar
 Cinquin O, Demongeot J (2002) Positive and negative feedback: striking a balance between necessary antagonists. J Theoret Biol 216:229–241Google Scholar
 Clarke BL (1980) Stability of complex reaction networks. In: Prigogine I, Rice SA (eds) Advances in chemical physics. John Wiley, New York, pp 1–215Google Scholar
 Clive Maxfield (2006) How to invert three signals with only two not gates (and *no* xor gates). Technical report, http://www.mobilehandsetdesignline.comGoogle Scholar
 Costanzo MC, Crawford ME, Hirschman JE, Kranz JE, Olsen P, Robertson LS, Skrzypek MS, Braun BR, Hopkins KL, Kondu P, Lengieza C, LewSmith JE, Tillberg M, Garrels JI (2001) YPDTM, PombePDTM and WormPDTM: model organism volumes of the BioKnowledgeTM Library, an integrated resource for protein information. Nucl Acids Res 29(1):75–79PubMedGoogle Scholar
 Craciun G, Feinberg M (2005) Multiple equilibria in complex chemical reaction networks: I. The injectivity property. SIAM J Appl Mathematics 65:1526–1546Google Scholar
 Craciun G, Feinberg M (2006) Multiple equilibria in complex chemical reaction networks: II. the speciesreactions graph. SIAM J Appl Math 66:1321–1338Google Scholar
 Cross FR, Archambault V, Miller M, Klovstad M (2002) Testing a mathematical model of the yeast cell cycle. Mol Biol Cell 13:52–70PubMedGoogle Scholar
 Dancer EN (1998) Some remarks on a boundedness assumption for monotone dynamical systems. Proc AMS 126:801–807Google Scholar
 DasGupta B, Enciso GA, Sontag ED, Zhang Y (2007) Algorithmic and complexity aspects of decompositions of biological networks into monotone subsystems. BioSystems (to appear)Google Scholar
 De Leenheer P, Malisoff M (2006) A smallgain theorem for monotone systems with multivalued inputstate characteristics. IEEE Trans Automat Control 51:287–292Google Scholar
 De Simone C, Diehl M, Junger M, Mutzel P, Reinelt G, Rinaldi G (1995) Exact ground states of Ising spin glasses: new experimental results with a branch and cut algorithm. J Stat Phys 80:487–496Google Scholar
 DeAngelis DL, Post WM, Travis CC (1986) Positive feedback in natural systems. SpringerVerlag, New YorkGoogle Scholar
 Devaney R (1989) An introduction to chaotic dynamical systems, 2nd edn. AddisonWesley, Redwood CityGoogle Scholar
 Doyle JC, Francis B, Tannenbaum A (1990) Feedback control theory. MacMillan Publishing Co.Google Scholar
 EdelsteinKeshet L (2005) Mathematical models in biology. SIAM, PhiladelphiaGoogle Scholar
 Enciso GA, Sontag ED (2004) On the stability of a model of testosterone dynamics. J Math Biol 49(6):627–634PubMedGoogle Scholar
 Enciso GA, Sontag ED (2005a) A remark on multistability for monotone systems ii. In: Proc. IEEE conf. decision and control, Seville, Dec. 2005, IEEE Publications, pp 2957–2962Google Scholar
 Enciso GA, Sontag ED (2005) Monotone systems under positive feedback: multistability and a reduction theorem. Syst Control Lett 54(2):159–168Google Scholar
 Enciso GA, Sontag ED (2006) Global attractivity, I/O monotone smallgain theorems, and biological delay systems. Discrete Contin Dyn Syst 14(3):549–578Google Scholar
 Enciso GA, Smith HL, Sontag ED (2006) Nonmonotone systems decomposable into monotone systems with negative feedback. J Differ Eqs 224:205–227Google Scholar
 Feinberg M (1987) Chemical reaction network structure and the stability of complex isothermal reactors – i. the deficiency zero and deficiency one theorems. Chem Eng Sci 42:2229–2268Google Scholar
 Feinberg M (1991) Some recent results in chemical reaction network theory. In: Aris R, Aronson DG, Swinney HL (eds) Patterns and dynamics in reactive media, IMA, Vol Math Appl 37. Springer, Berlin, p 4370Google Scholar
 Feinberg M (1995) The existence and uniqueness of steady states for a class of chemical reaction networks. Arch Rational Mech Anal 132:311–370Google Scholar
 Feinberg M, Horn FJM (1974) Dynamics of open chemical systems and algebraic structure of underlying reaction network. Chem Eng Sci 29:775–787Google Scholar
 Fenichel N (1979) Geometric singular perturbation theory for ordinary differential equations. J Differ Eqs 31:53–98Google Scholar
 Ferrell JE Jr (1996) Tripping the switch fantastic: how a protein kinase cascade can convert graded inputs into switchlike outputs. Trends Biochem Sci 21:460–466PubMedGoogle Scholar
 Ferrell JE Jr, Machleder EM (1998) The biochemical basis of an allornone cell fate switch in xenopus oocytes. Science 280:895–898PubMedGoogle Scholar
 Ferrell JE Jr, Xiong W (2001) Bistability in cell signaling: how to make continuous processes discontinuous, and reversible processes irreversible. Chaos 11:227–236PubMedGoogle Scholar
 Gardner TS, Cantor CR, Collins JJ (2000) Construction of a genetic toggle switch in escherichia coli. Nature 403:339–342PubMedGoogle Scholar
 Gedeon T (1998) Cyclic feedback systems. Mem Am Math Soc 134:1–73Google Scholar
 Gedeon T, Sontag ED (2007) Oscillations in multistable monotone systems with slowly varying feedback. J Differ Eqs (to appear)Google Scholar
 Gilbert EN (1954) Lattice theoretic properties of frontal switching functions. J Math Phys 33:57–67Google Scholar
 Goemans M, Williamson D (1995) Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J ACM 42:1115–1145Google Scholar
 Goldbeter A (1995) A model for circadian oscillations in the drosophila period protein (per). Proc Roy Soc Lond B 261:319–324Google Scholar
 Goldbeter A (1996) Biochemical oscillations and cellular rhythms. Cambridge University Press, CambridgeGoogle Scholar
 Gouze JL (1988) A criterion of global convergence to equilibrium for differential systems. application to LotkaVolterra systems. Technical Report RR0894, INRIAGoogle Scholar
 Gouze JL (1998) Positive and negative circuits in dynamical systems. J Biol Syst 6:11–15Google Scholar
 Gouze JL, Hadeler KP (1994) Order intervals and monotone flow. Nonlinear World 1:23–34Google Scholar
 Hadeler K, Glas D (1983) Quasimonotone systems and convergence to equilibrium in a population genetics model. J Math Anal Appl 95:297–303Google Scholar
 Hale JK (1988) Asymptotic behavior of dissipative systems. Amer Math Soc, ProvidenceGoogle Scholar
 Harary F (1953) On the notion of balance of a signed graph. Michigan Math J 2:143–146Google Scholar
 Hastings S, Tyson J, Webester D (1977) Existence of periodic solutions for negative feedback cellular control systems. J Differ Eqs 25:39–64Google Scholar
 Hess P, Poláčik P (1993) Boundedness of prime periods of stable cycles and convergence to fixed points in discrete monotone dynamical systems. SIAM J Math Anal 24:1312–1330Google Scholar
 Hirsch M (1983) Differential equations and convergence almost everywhere in strongly monotone flows. Contemp Math 17:267–285Google Scholar
 Hirsch MW (1984) The dynamical systems approach to differential equations. Bull AMS 11:1–64Google Scholar
 Hirsch M (1985) Systems of differential equations that are competitive or cooperative ii: convergence almost everywhere. SIAM J Math Anal 16:423–439Google Scholar
 Hirsch M (1989) Convergent activation dynamics in continuoustime networks. Neural Networks 2:331–349Google Scholar
 Hirsch M, Smith HL (2005) Monotone dynamical systems. In: Handbook of differential equations, ordinary differential equations (second volume). Elsevier, AmsterdamGoogle Scholar
 Horn FJM (1974) The dynamics of open reaction systems. In: Mathematical aspects of chemical and biochemical problems and quantum chemistry. Proc. SIAMAMS Sympos Appl Math New York, 1974, pp 125–137. Amer Math Soc, Providence, 1974. SIAMAMS Proceedings, vol. VIIIGoogle Scholar
 Horn FJM, Jackson R (1972) General mass action kinetics. Arch Rational Mech Anal 49:81–116Google Scholar
 Huang CYF, Ferrell JE Jr (1996) Ultrasensitivity in the mitogenactivated protein kinase cascade. Proc Natl Acad Sci USA 93:10078–10083PubMedGoogle Scholar
 Hüffner F, Betzler N, Niedermeier R (2007) Optimal edge deletions for signed graph balancing. In: Proceedings of the 6th workshop on experimental algorithms (WEA07), June 6–8, 2007, Rome. SpringerVerlagGoogle Scholar
 Istrail S (2000) Statistical mechanics, threedimensionality and NPcompleteness: I. Universality of intractability of the partition functions of the Ising model across nonplanar lattices. In: Proceedings of the 32nd ACM symposium on the theory of computing (STOC00). ACM Press, pp 87–96Google Scholar
 Jiang JF (1994) On the global stability of cooperative systems. Bull London Math Soc 6:455–458Google Scholar
 Jones CKRT (1994) Geometric singular perturbation theory. In: Dynamical systems (Montecatini Terme), Lect Notes in Math, vol 1609. SpringerVerlag, BerlinGoogle Scholar
 Kauffman SA (1969a) Metabolic stability and epigenesis in randomly constructed genetic nets. J Theoret Biol 22:437–467Google Scholar
 Kauffman SA (1969b) Homeostasis and differentiation in random genetic control networks. Nature 224:177–178PubMedGoogle Scholar
 Kauffman SA, Glass K (1973) The logical analysis of continuous, nonlinear biochemical control networks. J Theoret Biol 39:103–129Google Scholar
 Keener JP, Sneyd J (1998) Mathematical physiology. SpringerVerlag, New YorkGoogle Scholar
 Khalil HK (2002) Nonlinear systems, 3rd edn. Prentice Hall, Upper Saddle River, NJGoogle Scholar
 Kholodenko BN (2000) Negative feedback and ultrasensitivity can bring about oscillations in the mitogenactivated protein kinase cascades. Eur J Biochem 267:1583–1588PubMedGoogle Scholar
 Kulenovic MRS, Ladas G (2002) Dynamics of second order rational difference equations. Chapman & Hall/CRC, New YorkGoogle Scholar
 Laurent M, Kellershohn N (1999) Multistability: a major means of differentiation and evolution in biological systems. Trends Biochem Sci 24:418–422PubMedGoogle Scholar
 de Leenheer P, Angeli D, Sontag ED (2005) On predator–prey systems and smallgain theorems. Math Biosci Eng 2(1):25–42Google Scholar
 de Leenheer P, Angeli D, Sontag ED (2007) Monotone chemical reaction networks. J Math Chem (to appear)Google Scholar
 Lewis J, Slack JM, Wolpert L (1977) Thresholds in development. J Theoret Biol 65:579–590Google Scholar
 Lisman JE (1985) A mechanism for memory storage insensitive to molecular turnover: a bistable autophosphorylating kinase. Proc Natl Acad Sci USA 82:3055–3057PubMedGoogle Scholar
 Mahaffy J, Savev ES (1999) Stability analysis for a mathematical model of the lac operon. Quart Appl Math LVII:37–53Google Scholar
 MalletParet J, Smith HL (1990) The PoincaréBendixson theorem for monotone cyclic feedback systems. J Dyn Differ Eqs 2:367–421Google Scholar
 Mangan S, Alon U (2003) Structure and function of the feedforward loop network motif. Proc Natl Acad Sci USA 110:11980–11985Google Scholar
 Mangan S, Zaslaver A, Alon U (2003) The coherent feedforward loop serves as a signsensitive delay element in transcription networks. J Mol Biol 334:197–204PubMedGoogle Scholar
 Milo R, ShenOrr S, Itzkovitz S, Kashtan N, Chklovskii D, Alon U (2002) Network motifs: simple building blocks of complex networks. Science 298:824–827PubMedGoogle Scholar
 Minsky ML (1967) Computation: finite and infinite machines. PrenticeHall, Englewood Cliffs, NJGoogle Scholar
 Monod J, Jacob F (1961) Teleonomic mechanisms in cellular metabolism, growth and differentiation. Cold Spring Harb Symp Quant Biol 26:389–401PubMedGoogle Scholar
 Murray JD (2002) Mathematical biology, I, II: an introduction. SpringerVerlag, New YorkGoogle Scholar
 Novic A, Weiner M (1957) Enzyme induction as an allornone phenomenon. Proc Natl Acad Sci USA 43:553–566Google Scholar
 Othmer HG (1976) The qualitative dynamics of a class of biochemical control circuits. J Math Biol 3:53–78PubMedGoogle Scholar
 Plahte E, Mestl T, Omholt WS (1995) Feedback circuits, stability and multistationarity in dynamical systems. J Biol Syst 3:409–413Google Scholar
 Poláčik P, Tereščák I (1992) Convergence to cycles as a typical asymptotic behavior in smooth strongly monotone discretetime dynamical systems. Arch Rational Mech Anal 116:339–360Google Scholar
 Poláčik P, Tereščák I (1993) Exponential separation and invariant bundles for maps in ordered banach spaces with applications to parabolic equations. J Dyn Differ Eqs 5:279–303Google Scholar
 Pomerening JR, Sontag ED, Ferrell JE (2003) Building a cell cycle oscillator: hysteresis and bistability in the activation of cdc2. Nat Cell Biol 5(4):346–351, April 2003. Supplementary materials 2–4 are here: http://www.math.rutgers.edu/(tilde)sontag/FTPDIR/pomereningsontagferrella dditional.pdfGoogle Scholar
 Ptashne M (1992) A Genetic switch: phage λ and higher organisms. Cell Press and Blackwell Scientific Publications, Cambridge MAGoogle Scholar
 Rapp PE (1975) A theoretical investigation of a large class of biochemical oscillations. Math Biosci 25:165–188Google Scholar
 Reddy VN, Mavrovouniotis ML, Liebman MN (1993) Petri net representations in metabolic pathways. Proc Int Conf Intell Syst Mol Biol 1:328–336PubMedGoogle Scholar
 Remy E, Mosse B, Chaouiya C, Thieffry D (2003) A description of dynamical graphs associated to elementary regulatory circuits. Bioinformatics 19(Suppl 2):ii172–ii178PubMedGoogle Scholar
 Schneider H, Vidyasagar M (1970) Crosspositive matrices. SIAM J Numer Anal 7:508–519Google Scholar
 Segel LA (1984) Modeling dynamic phenomena in molecular and cellular biology. Cambridge University Press, CambridgeGoogle Scholar
 Sepulchre R, Jankovic M, Kokotović PV (1997) Constructive nonlinear control. SpringerVerlag, LondonGoogle Scholar
 Sha W, Moore J, Chen K, Lassaletta AD, Yi CS, Tyson JJ, Sible JC (2003) Hysteresis drives cellcycle transitions in xenopus laevis egg extracts. Proc Natl Acad Sci USA 100:975–980PubMedGoogle Scholar
 Shvartsman SY, Wiley HS, Lauffenburger DA (2000) Autocrine loop as a module for bidirectional and contextdependent cell signaling. Technical report, MIT Chemical Engineering DepartmentGoogle Scholar
 Smale S (1976) On the differential equations of species in competition. J Math Biol 3:5–7PubMedGoogle Scholar
 Smillie J (1984) Competitive and cooperative tridiagonal systems of differential equations. SIAM J Math Anal 15:530–534Google Scholar
 Smith HL (1987) Oscillations and multiple steady states in a cyclic gene model with repression. J Math Biol 25:169–190PubMedGoogle Scholar
 Smith HL (1991) Convergent and oscillatory activation dynamics for cascades of neural nets with nearest neighbor competitive or cooperative interactions. Neural Networks 4:41–46Google Scholar
 Smith H (1995) Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems, Mathematical Surveys and Monographs, vol 41, AMS, Providence, RIGoogle Scholar
 Snoussi EH (1998) Necessary conditions for multistationarity and stable periodicity. J Biol Syst 6:3–9Google Scholar
 Sontag ED (1998) Mathematical control theory. Deterministic finitedimensional systems, vol 6 of texts in applied mathematics, 2nd edn. SpringerVerlag, New YorkGoogle Scholar
 Sontag ED (1999) Stability and stabilization: discontinuities and the effect of disturbances. In: Nonlinear analysis, differential equations and control, Montreal, QC, 1998, vol 528 of NATO Sci Ser C Math Phys Sci. Kluwer Acad Publ, Dordrecht, pp 551–598Google Scholar
 Sontag ED (2004) Some new directions in control theory inspired by systems biology. IEE Proc Syst Biol 1:9–18Google Scholar
 Sontag ED (2005) Molecular systems biology and control. Eur J Control 11(4–5):396–435Google Scholar
 Tereščák I (1996) Dynamics of c ^{1} smooth strongly monotone discretetime dynamical system. Technical report, Comenius University, BratislavaGoogle Scholar
 Thieme HR (1992) Convergence results and a PoincaréBendixson trichotomy for asymptotically autonomous differential equations. J Math Biol 30:755–763Google Scholar
 Thomas R (1981) On the relation between the logical structure of systems and their ability to generate multiple steady states or sustained oscillations. Springer Ser Synerget 9:180–193Google Scholar
 Thomas R, D’ari R (1990) Biological feedback. CRC Press, Boca RatonGoogle Scholar
 Thomas R, Kaufman M (2001) Multistationarity, the basis of cell differentiation and memory. i. structural conditions of multistationarity and other nontrivial behavior. Chaos 11:170–179PubMedGoogle Scholar
 Thron CD (1991) The secant condition for instability in biochemical feedbackcontrol.1. The role of cooperativity and saturability. Bull Math Biol 53:383–401Google Scholar
 Tyson J, Othmer HG (1978) The dynamics of feedback control circuits in biochemical pathways. Prog Theoret Biol 5:1–60Google Scholar
 Tyson JJ, Chen K, Novak B (2003) Sniffers, buzzers, toggles, and blinkers: dynamics of regulatory and signaling pathways in the cell. Curr Opin Cell Biol 15:221–231PubMedGoogle Scholar
 Volkmann P (1972) Gewohnliche differentialungleichungen mit quasimonoton wachsenden funktionen in topologischen vektorraumen. Math Z 127:157–164Google Scholar
 Walcher S (2001) On cooperative systems with respect to arbitrary orderings. J Math Anal Appl 263:543–554Google Scholar
 Walter W (1970) Differential and Integral Inequalities. SpringerVerlag, BerlinGoogle Scholar
 Wang L, Sontag ED (2006a) Almost global convergence in singular perturbations of strongly monotone systems. In: Positive Systems. SpringerVerlag, Berlin, pp 415–422. Lecture Notes in Control and Information Sciences, vol 341, Proceedings of the second multidisciplinary international symposium on positive systems: theory and applications (POSTA 06) Grenoble, FranceGoogle Scholar
 Wang L, Sontag ED (2006b) A remark on singular perturbations of strongly monotone systems. In: Proc. IEEE conf. decision and control, San Diego, Dec. 2006, p WeB10.5. IEEEGoogle Scholar
 Widmann C, Spencer G, Jarpe MB, Johnson GL (1999) Mitogenactivated protein kinase: conservation of a threekinase module from yeast to human. Physiol Rev 79:143–180PubMedGoogle Scholar
 Zaslavsky T (1998) Bibliography of signed and gain graphs. Electron J Combin DS8Google Scholar
 ZevedeiOancea I, Schuster S (2003) Topological analysis of metabolic networks based on petri net theory. In Silico Biol 3:0029Google Scholar