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 Articles in journal or book chapters
1. M. Margaliot and E.D. Sontag. Revisiting totally positive differential systems: A tutorial and new results. Automatica, 101:1-14, 2019. [PDF] Keyword(s): tridiagonal systems, cooperative systems, monotone systems.
Abstract:
 A matrix is totally nonnegative (resp., totally positive) if all its minors are nonnegative (resp., positive). This paper draws connections between B. Schwarz's 1970 work on TN and TP matrices to Smillie's 1984 and Smith's 1991 work on stability of nonlinear tridiagonal cooperative systems, simplifying proofs in the later paper and suggesting new research questions.

2. S. J. Rahi, J. Larsch, K. Pecani, N. Mansouri, A. Y. Katsov, K. Tsaneva-Atanasova, E. D. Sontag, and F. R. Cross. Oscillatory stimuli differentiate adapting circuit topologies. Nature Methods, 14:1010-1016, 2017. [PDF] Keyword(s): biochemical networks, periodic behaviors, monotone systems, entrainment, oscillations, incoherent feedforward loop, feedforward, IFFL.
Abstract:
 Elucidating the structure of biological intracellular networks from experimental data remains a major challenge. This paper studies two types of response signatures'' to identify specific circuit motifs, from the observed response to periodic inputs. In particular, the objective is to distinguish negative feedback loops (NFLs) from incoherent feedforward loops (IFFLs), which are two types of circuits capable of producing exact adaptation. The theory of monotone systems with inputs is used to show that period skipping'' (non-harmonic responses) is ruled out in IFFL's, and a notion called refractory period stabilization'' is also analyzed. The approach is then applied to identify a circuit dominating cell cycle timing in yeast, and to uncover a calcium-mediated NFL circuit in \emph{C.elegans} olfactory sensory neurons.

3. J.A. Ascensao, P. Datta, B. Hancioglu, E.D. Sontag, M.L. Gennaro, and O.A. Igoshin. Non-monotonic response dynamics of glyoxylate shunt genes in Mycobacterium tuberculosis. PLoS Computational Biology, 12:e1004741, 2016. [PDF]
Abstract:
 Understanding how dynamical responses of biological networks are constrained by underlying network topology is one of the fundamental goals of systems biology. Here we employ monotone systems theory to formulate a theorem stating necessary conditions for non-monotonic time-response of a biochemical network to a monotonic stimulus. We apply this theorem to analyze the non-monotonic dynamics of the sigmaB-regulated glyoxylate shunt gene expression in Mycobacterium tuberculosis cells exposed to hypoxia. We first demonstrate that the known network structure is inconsistent with observed dynamics. To resolve this inconsistency we employ the formulated theorem, modeling simulations and optimization along with follow-up dynamic experimental measurements. We show a requirement for post-translational modulation of sigmaB activity in order to reconcile the network dynamics with its topology. The results of this analysis make testable experimental predictions and demonstrate wider applicability of the developed methodology to a wide class of biological systems.

4. E.V. Nikolaev and E.D. Sontag. Quorum-sensing synchronization of synthetic toggle switches: A design based on monotone dynamical systems theory. PLoS Computational Biology, 12:e1004881, 2016. [PDF] Keyword(s): quorum sensing, toggle switches, monotone systems.
Abstract:
 Synthetic constructs in biotechnology, bio-computing, and proposed gene therapy interventions are often based on plasmids or transfected circuits which implement some form of on-off (toggle or flip-flop) switch. For example, the expression of a protein used for therapeutic purposes might be triggered by the recognition of a specific combination of inducers (e.g., antigens), and memory of this event should be maintained across a cell population until a specific stimulus commands a coordinated shut-off. The robustness of such a design is hampered by molecular (intrinsic) or environmental (extrinsic) noise, which may lead to spontaneous changes of state in a subset of the population and is reflected in the bimodality of protein expression, as measured for example using flow cytometry. In this context, a majority-vote correction circuit, which brings deviant cells back into the required state, is highly desirable. To address this concrete challenge, we have developed a new theoretical design for quorum-sensing (QS) synthetic toggles. QS provides a way for cells to broadcast their states to the population as a whole so as to facilitate consensus. Our design is endowed with strong theoretical guarantees, based on monotone dynamical systems theory, of global stability and no oscillations, and which leads to robust consensus states.

5. M. Marcondes de Freitas and E.D. Sontag. A small-gain theorem for random dynamical systems with inputs and outputs. SIAM J. Control and Optimization, 53:2657-2695, 2015. [PDF] Keyword(s): random dynamical systems, monotone systems, small-gain theorem, stochastic systems.
Abstract:
 A formalism for the study of random dynamical systems with inputs and outputs (RDSIO) is introduced. An axiomatic framework and basic properties of RDSIO are developed, and a theorem is shown that guarantees the stability of interconnected systems.

6. D. Angeli, G.A. Enciso, and E.D. Sontag. A small-gain result for orthant-monotone systems under mixed feedback. Systems and Control Letters, 68:9-19, 2014. [PDF] Keyword(s): small-gain theorem, monotone systems.
Abstract:
 This paper introduces a small-gain result for interconnected orthant-monotone systems for which no matching condition is required between the partial orders in input and output spaces. Previous results assumed that the partial orders adopted would be induced by positivity cones in input and output spaces and that such positivity cones should fulfill a compatibility rule: namely either be coincident or be opposite. Those two configurations correspond to positive feedback or negative feedback cases. We relax those results by allowing arbitrary orthant orders.

7. D. Angeli and E.D. Sontag. Behavior of responses of monotone and sign-definite systems. In K. Hüper and Jochen Trumpf, editors, Mathematical System Theory - Festschrift in Honor of Uwe Helmke on the Occasion of his Sixtieth Birthday, pages 51-64. CreateSpace, 2013. [PDF] Keyword(s): monotone systems, reverse engineering, systems biology.
Abstract:
 This paper study systems with sign-definite interactions between variables, providing a sufficient condition to characterize the possible transitions between intervals of increasing and decreasing behavior. It also provides a discussion illustrating how our approach can help identify interactions in models, using information from time series of observations.

8. M. Marcondes de Freitas and E.D. Sontag. Random dynamical systems with inputs. In C. Pötzsche and P. Kloeden, editors, Nonautonomous Dynamical Systems in the Life Sciences, Lecture Notes in Mathematics vol. 2102, pages 41-87. Springer-Verlag, 2013. [PDF] Keyword(s): random dynamical systems, monotone systems.
Abstract:
 This work introduces a notion of random dynamical systems with inputs, providing several basic definitions and results on equilibria and convergence. It also presents a "converging input to converging state" result, a concept that plays a key role in the analysis of stability of feedback interconnections, for monotone systems.

9. G. Craciun, C. Pantea, and E.D. Sontag. Graph-theoretic analysis of multistability and monotonicity for biochemical reaction networks. In H. Koeppl, G. Setti, M. di Bernardo, and D. Densmore, editors, Design and Analysis of Biomolecular Circuits, pages 63-72. Springer-Verlag, 2011. [PDF] Keyword(s): biochemical networks, monotone systems.
Abstract:
 This is a short expository article describing how the species-reaction graph (SR graph) can be used to analyze both multistability and monotonicity of biochemical networks.

10. D. Angeli, P. de Leenheer, and E.D. Sontag. Graph-theoretic characterizations of monotonicity of chemical networks in reaction coordinates. J. Mathematical Biology, 61:581-616, 2010. [PDF] Keyword(s): biochemical networks, fluxes, monotone systems, reaction cordinates, Petri nets, persistence, futile cycles.
Abstract:
 This paper derives new results for certain classes of chemical reaction networks, linking structural to dynamical properties. In particular, it investigates their monotonicity and convergence without making assumptions on the form of the kinetics (e.g., mass-action) of the dynamical equations involved, and relying only on stoichiometric constraints. The key idea is to find an alternative representation under which the resulting system is monotone. As a simple example, the paper shows that a phosphorylation/dephosphorylation process, which is involved in many signaling cascades, has a global stability property.

11. L. Wang, P. de Leenheer, and E.D. Sontag. Conditions for global stability of monotone tridiagonal systems with negative feedback. Systems and Control Letters, 59:138-130, 2010. [PDF] Keyword(s): systems biology, monotone systems, tridiagonal systems, global stability.
Abstract:
 This paper studies monotone tridiagonal systems with negative feedback. These systems possess the Poincar{\'e}-Bendixson property, which implies that, if orbits are bounded, if there is a unique steady state and this unique equilibrium is asymptotically stable, and if one can rule out periodic orbits, then the steady state is globally asymptotically stable. Different approaches are discussed to rule out period orbits. One is based on direct linearization, while the other uses the theory of second additive compound matrices. Among the examples that will illustrate our main theoretical results is the classical Goldbeter model of the circadian rhythm.

12. D. Angeli, M.W. Hirsch, and E.D. Sontag. Attractors in coherent systems of differential equations. J. of Differential Equations, 246:3058-3076, 2009. [PDF] Keyword(s): monotone systems, positive feedback systems.
Abstract:
 Attractors of cooperative dynamical systems are particularly simple; for example, a nontrivial periodic orbit cannot be an attractor. This paper provides characterizations of attractors for the wider class of systems defined by the property that all directed feedback loops are positive. Several new results for cooperative systems are obtained in the process.

13. D. Angeli and E.D. Sontag. Oscillations in I/O monotone systems. IEEE Transactions on Circuits and Systems, Special Issue on Systems Biology, 55:166-176, 2008. Note: Preprint version in arXiv q-bio.QM/0701018, 14 Jan 2007. [PDF] Keyword(s): monotone systems, hopf bifurcations, circadian rhythms, tridiagonal systems, nonlinear dynamics, systems biology, biochemical networks, oscillations, periodic behavior.
Abstract:
 In this note, we show how certain properties of Goldbeter's 1995 model for circadian oscillations can be proved mathematically, using techniques from the recently developed theory of monotone systems with inputs and outputs. The theory establishes global asymptotic stability, and in particular no oscillations, if the rate of transcription is somewhat smaller than that assumed by Goldbeter, based on the application of a tight small gain condition. This stability persists even under arbitrary delays in the feedback loop. On the other hand, when the condition is violated a Poincare'-Bendixson result allows to conclude existence of oscillations, for sufficiently high delays.

14. D. Angeli and E.D. Sontag. Translation-invariant monotone systems, and a global convergence result for enzymatic futile cycles. Nonlinear Analysis Series B: Real World Applications, 9:128-140, 2008. [PDF] [doi:10.1016/j.nonrwa.2006.09.006] Keyword(s): systems biology, biochemical networks, nonlinear stability, dynamical systems, monotone systems.
Abstract:
 Strongly monotone systems of ordinary differential equations which have a certain translation-invariance property are shown to have the property that all projected solutions converge to a unique equilibrium. This result may be seen as a dual of a well-known theorem of Mierczynski for systems that satisfy a conservation law. As an application, it is shown that enzymatic futile cycles have a global convergence property.

15. G.A. Enciso and E.D. Sontag. Monotone bifurcation graphs. Journal of Biological Dynamics, 2:121-139, 2008. [PDF]
Abstract:
 This paper generalizes the approach to bistability based on the existence of characteristics for open-loop monotone systems to the case when characteristics do not exist. A set-valued version is provided, instead.

16. A. Maayan, R. Iyengar, and E.D. Sontag. Intracellular Regulatory Networks are close to Monotone Systems. IET Systems Biology, 2:103-112, 2008. [PDF]
Abstract:
 We find that three intracellular regulatory networks contain far more positive "sign-consistent" feedback and feed-forward loops than negative loops. Negative inconsistent loops can be more easily removed from real regulatory network topologies compared to removing negative loops from shuffled networks. The abundance of positive feed-forward loops and feedback loops in real networks emerges from the presence of hubs that are enriched with either negative or positive links, and from the non-uniform connectivity distribution. Boolean dynamics applied to the signaling network further support the stability of its topology. These observations suggest that the "close-to-monotone" structure of intracellular regulatory networks may contribute to the dynamical stability observed in cellular behavior.

17. E.D. Sontag, A. Veliz-Cuba, R. Laubenbacher, and A.S. Jarrah. The effect of negative feedback loops on the dynamics of Boolean networks. Biophysical Journal, 95:518-526, 2008. [PDF] Keyword(s): monotone systems, positive feedback systems, Boolean networks, limit cycles.
Abstract:
 Feedback loops play an important role in determining the dynamics of biological networks. In order to study the role of negative feedback loops, this paper introduces the notion of "distance to positive feedback (PF-distance)" which in essence captures the number of "independent" negative feedback loops in the network, a property inherent in the network topology. Through a computational study using Boolean networks it is shown that PF-distance has a strong influence on network dynamics and correlates very well with the number and length of limit cycles in the phase space of the network. To be precise, it is shown that, as the number of independent negative feedback loops increases, the number (length) of limit cycles tends to decrease (increase). These conclusions are consistent with the fact that certain natural biological networks exhibit generally regular behavior and have fewer negative feedback loops than randomized networks with the same numbers of nodes and connectivity.

18. L. Wang and E.D. Sontag. Singularly perturbed monotone systems and an application to double phosphorylation cycles. J. Nonlinear Science, 18:527-550, 2008. [PDF] Keyword(s): singular perturbations, futile cycles, MAPK cascades, systems biology, biochemical networks, nonlinear stability, nonlinear dynamics, multistability, monotone systems.
Abstract:
 The theory of monotone dynamical systems has been found very useful in the modeling of some gene, protein, and signaling networks. In monotone systems, every net feedback loop is positive. On the other hand, negative feedback loops are important features of many systems, since they are required for adaptation and precision. This paper shows that, provided that these negative loops act at a comparatively fast time scale, the main dynamical property of (strongly) monotone systems, convergence to steady states, is still valid. An application is worked out to a double-phosphorylation "futile cycle" motif which plays a central role in eukaryotic cell signaling.

19. E.D. Sontag. Monotone and near-monotone systems. In I. Queinnec, S. Tarbouriech, G. Garcia, and S-I. Niculescu, editors, Biology and Control Theory: Current Challenges (Lecture Notes in Control and Information Sciences Volume 357), pages 79-122. Springer-Verlag, Berlin, 2007. Note: Conference version of Monotone and near-monotone biochemical networks,'' basically the same paper.Keyword(s): systems biology, biochemical networks, monotone systems, Ising spin models, nonlinear stability, dynamical systems, consistent graphs, gene networks.
Abstract:
 See abstract and pdf for Monotone and near-monotone biochemical networks''.

20. B. DasGupta, G.A. Enciso, E.D. Sontag, and Y. Zhang. Algorithmic and complexity aspects of decompositions of biological networks into monotone subsystems. BioSystems, 90:161-178, 2007. [PDF] [doi:http://dx.doi.org/10.1016/j.biosystems.2006.08.001] Keyword(s): monotone systems, systems biology, biochemical networks.
Abstract:
 A useful approach to the mathematical analysis of large-scale biological networks is based upon their decompositions into monotone dynamical systems. This paper deals with two computational problems associated to finding decompositions which are optimal in an appropriate sense. In graph-theoretic language, the problems can be recast in terms of maximal sign-consistent subgraphs. The theoretical results include polynomial-time approximation algorithms as well as constant-ratio inapproximability results. One of the algorithms, which has a worst-case guarantee of 87.9% from optimality, is based on the semidefinite programming relaxation approach of Goemans-Williamson. The algorithm was implemented and tested on a Drosophila segmentation network and an Epidermal Growth Factor Receptor pathway model.

21. T. Gedeon and E.D. Sontag. Oscillations in multi-stable monotone systems with slowly varying feedback. J. of Differential Equations, 239:273-295, 2007. [PDF] Keyword(s): systems biology, biochemical networks, nonlinear stability, dynamical systems, monotone systems.
Abstract:
 This paper gives a theorem showing that a slow feedback adaptation, acting entirely analogously to the role of negative feedback for ordinary relaxation oscillations, leads to periodic orbits for bistable monotone systems. The proof is based upon a combination of i/o monotone systems theory and Conley Index theory.

22. E.D. Sontag. Monotone and near-monotone biochemical networks. Systems and Synthetic Biology, 1:59-87, 2007. [PDF] [doi:10.1007/s11693-007-9005-9] Keyword(s): systems biology, biochemical networks, monotone systems, Ising spin models, nonlinear stability, dynamical systems, consistent graphs, gene networks.
Abstract:
 This paper provides an expository introduction to monotone and near-monotone biochemical network structures. Monotone systems respond in a predictable fashion to perturbations, and have very robust dynamical characteristics. This makes them reliable components of more complex networks, and suggests that natural biological systems may have evolved to be, if not monotone, at least close to monotone. In addition, interconnections of monotone systems may be fruitfully analyzed using tools from control theory.

23. P. de Leenheer, D. Angeli, and E.D. Sontag. Monotone chemical reaction networks. J. Math Chemistry, 41:295-314, 2007. [PDF] [doi:10.1007/s10910-006-9075-z] Keyword(s): systems biology, biochemical networks, nonlinear stability, dynamical systems, monotone systems.
Abstract:
 We analyze certain chemical reaction networks and show that every solution converges to some steady state. The reaction kinetics are assumed to be monotone but otherwise arbitrary. When diffusion effects are taken into account, the conclusions remain unchanged. The main tools used in our analysis come from the theory of monotone dynamical systems. We review some of the features of this theory and provide a self-contained proof of a particular attractivity result which is used in proving our main result.

24. B. Dasgupta, G.A. Enciso, E.D. Sontag, and Y. Zhang. Algorithmic and complexity results for decompositions of biological networks into monotone subsystems. In C. Àlvarez and M. Serna, editors, Lecture Notes in Computer Science: Experimental Algorithms: 5th International Workshop, WEA 2006, pages 253-264. Springer-Verlag, 2006. Note: (Cala Galdana, Menorca, Spain, May 24-27, 2006). Keyword(s): systems biology, biochemical networks, monotone systems, theory of computing and complexity.

25. G.A. Enciso, H.L. Smith, and E.D. Sontag. Non-monotone systems decomposable into monotone systems with negative feedback. J. of Differential Equations, 224:205-227, 2006. [PDF] Keyword(s): nonlinear stability, dynamical systems, monotone systems.
Abstract:
 Motivated by the theory of monotone i/o systems, this paper shows that certain finite and infinite dimensional semi-dynamical systems with negative feedback can be decomposed into a monotone open loop system with inputs and a decreasing output function. The original system is reconstituted by plugging the output into the input. By embedding the system into a larger symmetric monotone system, this paper obtains finer information on the asymptotic behavior of solutions, including existence of positively invariant sets and global convergence. An important new result is the extension of the "small gain theorem" of monotone i/o theory to reaction-diffusion partial differential equations: adding diffusion preserves the global attraction of the ODE equilibrium.

26. G.A. Enciso and E.D. Sontag. Global attractivity, I/O monotone small-gain theorems, and biological delay systems. Discrete Contin. Dyn. Syst., 14(3):549-578, 2006. [PDF] Keyword(s): systems biology, biochemical networks, nonlinear stability, dynamical systems, monotone systems.
Abstract:
 This paper further develops a method, originally introduced in a paper by Angeli and Sontag, for proving global attractivity of steady states in certain classes of dynamical systems. In this aproach, one views the given system as a negative feedback loop of a monotone controlled system. An auxiliary discrete system, whose global attractivity implies that of the original system, plays a key role in the theory, which is presented in a general Banach space setting. Applications are given to delay systems, as well as to systems with multiple inputs and outputs, and the question of expressing a given system in the required negative feedback form is addressed.

27. E.D. Sontag and Y. Wang. A cooperative system which does not satisfy the limit set dichotomy. J. of Differential Equations, 224:373-384, 2006. [PDF] Keyword(s): dynamical systems, monotone systems.
Abstract:
 The fundamental property of strongly monotone systems, and strongly cooperative systems in particular, is the limit set dichotomy due to Hirsch: if x < y, then either Omega(x) < Omega (y), or Omega(x) = Omega(y) and both sets consist of equilibria. We provide here a counterexample showing that this property need not hold for (non-strongly) cooperative systems.

28. P. de Leenheer, D. Angeli, and E.D. Sontag. Crowding effects promote coexistence in the chemostat. Journal of Mathematical Analysis and Applications, 319:48-60, 2006. [PDF] Keyword(s): systems biology, biochemical networks, nonlinear stability, dynamical systems, monotone systems.
Abstract:
 We provide an almost-global stability result for a particular chemostat model, in which crowding effects are taken into consideration. The model can be rewritten as a negative feedback interconnection of two monotone i/o systems with well-defined characteristics, which allows the use of a small-gain theorem for feedback interconnections of monotone systems. This leads to a sufficient condition for almost-global stability, and we show that coexistence occurs in this model if the crowding effects are large enough.

29. P. de Leenheer, S.A. Levin, E.D. Sontag, and C.A. Klausmeier. Global stability in a chemostat with multiple nutrients. J. Mathematical Biology, 52:419-438, 2006. [PDF] Keyword(s): systems biology, biochemical networks, nonlinear stability, dynamical systems, monotone systems.
Abstract:
 We study a single species in a chemostat, limited by two nutrients, and separate nutrient uptake from growth. For a broad class of uptake and growth functions it is proved that a nontrivial equilibrium may exist. Moreover, if it exists it is unique and globally stable, generalizing a previous result by Legovic and Cruzado.

30. G.A. Enciso and E.D. Sontag. Monotone systems under positive feedback: multistability and a reduction theorem. Systems Control Lett., 54(2):159-168, 2005. [PDF] Keyword(s): multistability, systems biology, biochemical networks, nonlinear stability, dynamical systems, monotone systems.
Abstract:
 For feedback loops involving single input, single output monotone systems with well-defined I/O characteristics, a previous paper provided an approach to determining the location and stability of steady states. A result on global convergence for multistable systems followed as a consequence of the technique. The present paper extends the approach to multiple inputs and outputs. A key idea is the introduction of a reduced system which preserves local stability properties. New results characterizing strong monotonicity of feedback loops involving cascades are also presented.

31. E.D. Sontag. Molecular systems biology and control. Eur. J. Control, 11(4-5):396-435, 2005. [PDF] Keyword(s): cell biology, systems biology, biochemical networks, nonlinear stability, dynamical systems, monotone systems, molecular biology, systems biology, cellular signaling.
Abstract:
 This paper, prepared for a tutorial at the 2005 IEEE Conference on Decision and Control, presents an introduction to molecular systems biology and some associated problems in control theory. It provides an introduction to basic biological concepts, describes several questions in dynamics and control that arise in the field, and argues that new theoretical problems arise naturally in this context. A final section focuses on the combined use of graph-theoretic, qualitative knowledge about monotone building-blocks and steady-state step responses for components.

32. P. de Leenheer, D. Angeli, and E.D. Sontag. On predator-prey systems and small-gain theorems. Math. Biosci. Eng., 2(1):25-42, 2005. [PDF] Keyword(s): systems biology, biochemical networks, nonlinear stability, dynamical systems, monotone systems.
Abstract:
 This paper deals with an almost global attractivity result for Lotka-Volterra systems with predator-prey interactions. These systems can be written as (negative) feedback systems. The subsystems of the feedback loop are monotone control systems, possessing particular input-output properties. We use a small-gain theorem, adapted to a context of systems with multiple equilibrium points to obtain the desired almost global attractivity result. It provides sufficient conditions to rule out oscillatory or more complicated behavior which is often observed in predator-prey systems.

33. D. Angeli and E.D. Sontag. Interconnections of monotone systems with steady-state characteristics. In Optimal control, stabilization and nonsmooth analysis, volume 301 of Lecture Notes in Control and Inform. Sci., pages 135-154. Springer, Berlin, 2004. [PDF] Keyword(s): systems biology, biochemical networks, nonlinear stability, dynamical systems, monotone systems.
Abstract:
 One of the key ideas in control theory is that of viewing a complex dynamical system as an interconnection of simpler subsystems, thus deriving conclusions regarding the complete system from properties of its building blocks. Following this paradigm, and motivated by questions in molecular biology modeling, the authors have recently developed an approach based on components which are monotone systems with respect to partial orders in state and signal spaces. This paper presents a brief exposition of recent results, with an emphasis on small gain theorems for negative feedback, and the emergence of multi-stability and associated hysteresis effects under positive feedback.

34. D. Angeli, J. E. Ferrell, and E.D. Sontag. Detection of multistability, bifurcations, and hysteresis in a large class of biological positive-feedback systems.. Proc Natl Acad Sci USA, 101(7):1822-1827, 2004. Note: A revision of Suppl. Fig. 7(b) is here: http://www.math.rutgers.edu/(tilde)sontag/FTPDIR/nullclines-f-g-REV.jpg; and typos can be found here: http://www.math.rutgers.edu/(tilde)sontag/FTPDIR/angeli-ferrell-sontag-pnas04-errata.txt. [WWW] [PDF] [doi:10.1073/pnas.0308265100] Keyword(s): multistability, systems biology, biochemical networks, nonlinear stability, dynamical systems, monotone systems.
Abstract:
 Multistability is an important recurring theme in cell signaling, of particular relevance to biological systems that switch between discrete states, generate oscillatory responses, or "remember" transitory stimuli. Standard mathematical methods allow the detection of bistability in some very simple feedback systems (systems with one or two proteins or genes that either activate each other or inhibit each other), but realistic depictions of signal transduction networks are invariably much more complex than this. Here we show that for a class of feedback systems of arbitrary order, the stability properties of the system can be deduced mathematically from how the system behaves when feedback is blocked. Provided that this "open loop," feedback-blocked system is monotone and possesses a sigmoidal characteristic, the system is guaranteed to be bistable for some range of feedback strengths. We present a simple graphical method for deducing the stability behavior and bifurcation diagrams for such systems, and illustrate the method with two examples taken from recent experimental studies of bistable systems: a two-variable Cdc2/Wee1 system and a more complicated five-variable MAPK cascade.

35. D. Angeli and E.D. Sontag. Multi-stability in monotone input/output systems. Systems Control Lett., 51(3-4):185-202, 2004. [PDF] Keyword(s): multistability, systems biology, biochemical networks, nonlinear stability, dynamical systems, monotone systems.
Abstract:
 This paper studies the emergence of multi-stability and hysteresis in those systems that arise, under positive feedback, from monotone systems with well-defined steady-state responses. Such feedback configurations appear routinely in several fields of application, and especially in biology. The results are stated in terms of directly checkable conditions which do not involve explicit knowledge of basins of attractions of each equilibria.

36. D. Angeli, P. de Leenheer, and E.D. Sontag. A small-gain theorem for almost global convergence of monotone systems. Systems Control Lett., 52(5):407-414, 2004. [PDF] Keyword(s): systems biology, biochemical networks, nonlinear stability, dynamical systems, monotone systems.
Abstract:
 A small-gain theorem is presented for almost global stability of monotone control systems which are open-loop almost globally stable, when constant inputs are applied. The theorem assumes "negative feedback" interconnections. This typically destroys the monotonicity of the original flow and potentially destabilizes the resulting closed-loop system.

37. G.A. Enciso and E.D. Sontag. On the stability of a model of testosterone dynamics. J. Math. Biol., 49(6):627-634, 2004. [PDF] Keyword(s): systems biology, biochemical networks, nonlinear stability, dynamical systems, monotone systems.
Abstract:
 We prove the global asymptotic stability of a well-known delayed negative-feedback model of testosterone dynamics, which has been proposed as a model of oscillatory behavior. We establish stability (and hence the impossibility of oscillations) even in the presence of delays of arbitrary length.

38. E.D. Sontag. Some new directions in control theory inspired by systems biology. IET Systems Biology, 1:9-18, 2004. [PDF] Keyword(s): systems biology, biochemical networks, nonlinear stability, dynamical systems, monotone systems, cellular signaling.
Abstract:
 This paper, addressed primarily to engineers and mathematicians with an interest in control theory, argues that entirely new theoretical problems arise naturally when addressing questions in the field of systems biology. Examples from the author's recent work are used to illustrate this point.

39. P. de Leenheer, D. Angeli, and E.D. Sontag. A feedback perspective for chemostat models with crowding effects. In Positive systems (Rome, 2003), volume 294 of Lecture Notes in Control and Inform. Sci., pages 167-174. Springer, Berlin, 2003. Keyword(s): systems biology, biochemical networks, nonlinear stability, dynamical systems, monotone systems.

40. P. de Leenheer, D. Angeli, and E.D. Sontag. Small-gain theorems for predator-prey systems. In Positive systems (Rome, 2003), volume 294 of Lecture Notes in Control and Inform. Sci., pages 191-198. Springer, Berlin, 2003. Keyword(s): systems biology, biochemical networks, nonlinear stability, dynamical systems, monotone systems.

41. D. Angeli and E.D. Sontag. Monotone control systems. IEEE Trans. Automat. Control, 48(10):1684-1698, 2003. Note: Errata are here: http://www.math.rutgers.edu/(tilde)sontag/FTPDIR/angeli-sontag-monotone-TAC03-typos.txt. [PDF] Keyword(s): systems biology, biochemical networks, nonlinear stability, dynamical systems, monotone systems.
Abstract:
 Monotone systems constitute one of the most important classes of dynamical systems used in mathematical biology modeling. The objective of this paper is to extend the notion of monotonicity to systems with inputs and outputs, a necessary first step in trying to understand interconnections, especially including feedback loops, built up out of monotone components. Basic definitions and theorems are provided, as well as an application to the study of a model of one of the cell's most important subsystems.

42. J. R. Pomerening, E.D. Sontag, and J. E. Ferrell. Building a cell cycle oscillator: hysteresis and bistability in the activation of Cdc2. Nature Cell Biology, 5(4):346-351, 2003. Note: Supplementary materials 2-4 are here: http://www.math.rutgers.edu/(tilde)sontag/FTPDIR/pomerening-sontag-ferrell-additional.pdf. [WWW] [PDF] [doi:10.1038/ncb954] Keyword(s): systems biology, biochemical networks, oscillations, nonlinear stability, dynamical systems, monotone systems.
Abstract:
 In the early embryonic cell cycle, Cdc2-cyclin B functions like an autonomous oscillator, at whose core is a negative feedback loop: cyclins accumulate and produce active mitotic Cdc2-cyclin B Cdc2 activates the anaphase-promoting complex (APC); the APC then promotes cyclin degradation and resets Cdc2 to its inactive, interphase state. Cdc2 regulation also involves positive feedback4, with active Cdc2-cyclin B stimulating its activator Cdc25 and inactivating its inhibitors Wee1 and Myt1. Under the correct circumstances, these positive feedback loops could function as a bistable trigger for mitosis, and oscillators with bistable triggers may be particularly relevant to biological applications such as cell cycle regulation. This paper examined whether Cdc2 activation is bistable, confirming that the response of Cdc2 to non-degradable cyclin B is temporally abrupt and switchlike, as would be expected if Cdc2 activation were bistable. It is also shown that Cdc2 activation exhibits hysteresis, a property of bistable systems with particular relevance to biochemical oscillators. These findings help establish the basic systems-level logic of the mitotic oscillator.

 Conference articles
1. M. Margaliot and E.D. Sontag. Analysis of nonlinear tridiagonal cooperative systems using totally positive linear differential systems. In Proc. 2018 IEEE Conf. Decision and Control, pages 3104-3109, 2018. [PDF] Keyword(s): tridiagonal systems, cooperative systems, monotone systems.
Abstract:
 This is a conference version of "Revisiting totally positive differential systems: A tutorial and new results".

2. M. Marcondes de Freitas and E.D. Sontag. A class of random control systems: Monotonicity and the convergent-input convergent-state property. In Proc. American Control Conference, pages 4564-4569, 2013. [PDF] Keyword(s): random dynamical systems, monotone systems.

3. D. Angeli and E.D. Sontag. Remarks on the invalidation of biological models using monotone systems theory. In Proc. IEEE Conf. Decision and Control, Maui, Dec. 2012, 2012. Note: Paper TuC09.3.[PDF]
Abstract:
 This paper presents techniques for finding out what type of solutions are compatible with a given sign pattern of interactions between state/input variables once the input behaviour is also known. By type'' of solutions we essentially refer to the sequence of upwards or downwards segments that variables can exhibit (essentially sign-patterns of variables derivatives) once input profiles are also specified. A concrete experimental example of how such techniques can invalidate models is also provided.

4. D. Angeli and E.D. Sontag. A small-gain result for orthant-monotone systems in feedback: the non sign-definite case. In Proc. IEEE Conf. Decision and Control, Orlando, Dec. 2011, pages WeC09.1, 2011. Keyword(s): small-gain theorem, monotone systems.
Abstract:
 This note introduces a small-gain result for interconnected MIMO orthant-monotone systems for which no matching condition is required between the partial orders in input and output spaces of the considered subsystems. Previous results assumed that the partial orders adopted would be induced by positivity cones in input and output spaces and that such positivity cones should fulfill a compatibility rule: namely either be coincident or be opposite. Those two configurations corresponded to positive-feedback or negative feedback cases. We relax those results by allowing arbitrary orthant orders.

5. L. Wang, P. de Leenheer, and E.D. Sontag. Global stability for monotone tridiagonal systems with negative feedback. In Proc. IEEE Conf. Decision and Control, Cancun, Dec. 2008, pages 4091-4096, 2008. Keyword(s): systems biology, monotone systems, tridiagonal systems, global stability.
Abstract:
 Conference version of paper "Conditions for global stability of monotone tridiagonal systems with negative feedback"

6. L. Wang and E.D. Sontag. Further results on singularly perturbed monotone systems, with an application to double phosphorylation cycles. In Proc. IEEE Conf. Decision and Control, New Orleans, Dec. 2007, pages 627-632, 2007. Note: Conference version of Singularly perturbed monotone systems and an application to double phosphorylation cycles.Keyword(s): singular perturbations, futile cycles, MAPK cascades, systems biology, biochemical networks, nonlinear stability, nonlinear dynamics, multistability, monotone systems.

7. D. Angeli and E.D. Sontag. A note on monotone systems with positive translation invariance. In Control and Automation, 2006. MED '06. 14th Mediterranean Conference on, 28-30 June 2006, pages 1-6, 2006. IEEE. Note: Available from ieeexplore.ieee.org. [PDF] [doi:10.1109/MED.2006.3287822B2B2B2B2B2B] Keyword(s): systems biology, biochemical networks, nonlinear stability, dynamical systems, monotone systems.
Abstract:
 Strongly monotone systems of ordinary differential equations which have a certain translation-invariance property are shown to have the property that all projected solutions converge to a unique equilibrium. This result may be seen as a dual of a well-known theorem of Mierczynski for systems that satisfy a conservation law. As an application, it is shown that enzymatic futile cycles have a global convergence property.

8. D. Angeli, P. de Leenheer, and E.D. Sontag. On the structural monotonicity of chemical reaction networks. In Proc. IEEE Conf. Decision and Control, San Diego, Dec. 2006, pages 7-12, 2006. IEEE. [PDF] Keyword(s): monotone systems, systems biology, biochemical networks, nonlinear stability, dynamical systems.
Abstract:
 This paper derives new results for certain classes of chemical reaction networks, linking structural to dynamical properties. In particular, it investigates their monotonicity and convergence without making assumptions on the structure (e.g., mass-action kinetics) of the dynamical equations involved, and relying only on stoichiometric constraints. The key idea is to find a suitable set of coordinates under which the resulting system is cooperative. As a simple example, the paper shows that a phosphorylation/dephosphorylation process, which is involved in many signaling cascades, has a global stability property.

9. L. Wang and E.D. Sontag. A remark on singular perturbations of strongly monotone systems. In Proc. IEEE Conf. Decision and Control, San Diego, Dec. 2006, pages 989-994, 2006. IEEE. [PDF] Keyword(s): systems biology, biochemical networks, nonlinear stability, dynamical systems, singular perturbations, monotone systems.
Abstract:
 This paper deals with global convergence to equilibria, and in particular Hirsch's generic convergence theorem for strongly monotone systems, for singular perturbations of monotone systems.

10. L. Wang and E.D. Sontag. Almost global convergence in singular perturbations of strongly monotone systems. In C. Commault and N. Marchand, editors, Positive Systems, pages 415-422, 2006. Springer-Verlag, Berlin/Heidelberg. Note: (Lecture Notes in Control and Information Sciences Volume 341, Proceedings of the second Multidisciplinary International Symposium on Positive Systems: Theory and Applications (POSTA 06) Grenoble, France). [PDF] [doi:10.1007/3-540-34774-7] Keyword(s): systems biology, biochemical networks, nonlinear stability, dynamical systems, singular perturbations, monotone systems.
Abstract:
 This paper deals with global convergence to equilibria, and in particular Hirsch's generic convergence theorem for strongly monotone systems, for singular perturbations of monotone systems.

11. G.A. Enciso and E.D. Sontag. A remark on multistability for monotone systems II. In Proc. IEEE Conf. Decision and Control, Seville, Dec. 2005, IEEE Publications, pages 2957-2962, 2005. Keyword(s): multistability, systems biology, biochemical networks, nonlinear stability, dynamical systems, monotone systems.

12. D. Angeli and E.D. Sontag. An analysis of a circadian model using the small-gain approach to monotone systems. In Proc. IEEE Conf. Decision and Control, Paradise Island, Bahamas, Dec. 2004, IEEE Publications, pages 575-578, 2004. [PDF] Keyword(s): circadian rhythms, tridiagonal systems, nonlinear dynamics, systems biology, biochemical networks, oscillations, periodic behavior, monotone systems.
Abstract:
 We show how certain properties of Goldbeter's original 1995 model for circadian oscillations can be proved mathematically. We establish global asymptotic stability, and in particular no oscillations, if the rate of transcription is somewhat smaller than that assumed by Goldbeter, but, on the other hand, this stability persists even under arbitrary delays in the feedback loop. We are mainly interested in illustrating certain mathematical techniques, including the use of theorems concerning tridiagonal cooperative systems and the recently developed theory of monotone systems with inputs and outputs.

13. D. Angeli, P. de Leenheer, and E.D. Sontag. A tutorial on monotone systems- with an application to chemical reaction networks. In Proc. 16th Int. Symp. Mathematical Theory of Networks and Systems (MTNS 2004), CD-ROM, WP9.1, Katholieke Universiteit Leuven, 2004. [PDF] Keyword(s): systems biology, biochemical networks, nonlinear stability, dynamical systems, monotone systems.
Abstract:
 Monotone systems are dynamical systems for which the flow preserves a partial order. Some applications will be briefly reviewed in this paper. Much of the appeal of the class of monotone systems stems from the fact that roughly, most solutions converge to the set of equilibria. However, this usually requires a stronger monotonicity property which is not always satisfied or easy to check in applications. Following work of J.F. Jiang, we show that monotonicity is enough to conclude global attractivity if there is a unique equilibrium and if the state space satisfies a particular condition. The proof given here is self-contained and does not require the use of any of the results from the theory of monotone systems. We will illustrate it on a class of chemical reaction networks with monotone, but otherwise arbitrary, reaction kinetics.

14. D. Angeli, P. de Leenheer, and E.D. Sontag. Remarks on monotonicity and convergence in chemical reaction networks. In Proc. IEEE Conf. Decision and Control, Paradise Island, Bahamas, Dec. 2004, IEEE Publications, pages 243-248, 2004. Keyword(s): systems biology, biochemical networks, nonlinear stability, dynamical systems, monotone systems.

15. G.A. Enciso and E.D. Sontag. A remark on multistability for monotone systems. In Proc. IEEE Conf. Decision and Control, Paradise Island, Bahamas, Dec. 2004, IEEE Publications, pages 249-254, 2004. Keyword(s): multistability, systems biology, biochemical networks, nonlinear stability, dynamical systems, monotone systems.

16. D. Angeli and E.D. Sontag. A note on multistability and monotone I/O systems. In Proc. IEEE Conf. Decision and Control, Maui, Dec. 2003, IEEE Publications, 2003, pages 67-72, 2003. Keyword(s): systems biology, biochemical networks, nonlinear stability, dynamical systems, monotone systems.

 Internal reports
1. M. Marcondes de Freitas and E.D. Sontag. Remarks on random dynamical systems with inputs and outputs and a small-gain theorem for monotone RDS. Technical report, http://arxiv.org/abs/1207.1690, July 2012. Keyword(s): random dynamical systems, monotone systems.

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