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Publications about 'dynamical systems'
Articles in journal or book chapters
  1. M. Margaliot, E.D. Sontag, and T. Tuller. Contraction after small transients. Automatica, 67:178-184, 2016. [PDF] Keyword(s): entrainment, nonlinear systems, stability, contractions, contractive systems.
    Abstract:
    Contraction theory is a powerful tool for proving asymptotic properties of nonlinear dynamical systems including convergence to an attractor and entrainment to a periodic excitation. We introduce three new forms of generalized contraction (GC) that are motivated by allowing contraction to take place after small transients in time and/or amplitude. These forms of GC are useful for several reasons. First, allowing small transients does not destroy the asymptotic properties provided by standard contraction. Second, in some cases as we change the parameters in a contractive system it becomes a GC just before it looses contractivity. In this respect, GC is the analogue of marginal stability in Lyapunov stability theory. We provide checkable sufficient conditions for GC, and demonstrate their usefulness using several models from systems biology that are not contractive, with respect to any norm, yet are GC.


  2. 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.


  3. 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.


  4. Z. Aminzare and E.D. Sontag. Synchronization of diffusively-connected nonlinear systems: results based on contractions with respect to general norms. IEEE Transactions on Network Science and Engineering, 1(2):91-106, 2014. [PDF] Keyword(s): matrix measures, logarithmic norms, synchronization, consensus, contractions, contractive systems.
    Abstract:
    Contraction theory provides an elegant way to analyze the behavior of certain nonlinear dynamical systems. In this paper, we discuss the application of contraction to synchronization of diffusively interconnected components described by nonlinear differential equations. We provide estimates of convergence of the difference in states between components, in the cases of line, complete, and star graphs, and Cartesian products of such graphs. We base our approach on contraction theory, using matrix measures derived from norms that are not induced by inner products. Such norms are the most appropriate in many applications, but proofs cannot rely upon Lyapunov-like linear matrix inequalities, and different techniques, such as the use of the Perron-Frobenious Theorem in the cases of L1 or L-infinity norms, must be introduced.


  5. M. Margaliot, E.D. Sontag, and T. Tuller. Entrainment to periodic initiation and transition rates in a computational model for gene translation. PLoS ONE, 9(5):e96039, 2014. [WWW] [PDF] [doi:10.1371/journal.pone.0096039] Keyword(s): ribosomes, entrainment, nonlinear systems, stability, contractions, contractive systems.
    Abstract:
    A recent biological study has demonstrated that the gene expression pattern entrains to a periodically varying abundance of tRNA molecules. This motivates developing mathematical tools for analyzing entrainment of translation elongation to intra-cellular signals such as tRNAs levels and other factors affecting translation. We consider a recent deterministic mathematical model for translation called the Ribosome Flow Model (RFM). We analyze this model under the assumption that the elongation rate of the tRNA genes and/or the initiation rate are periodic functions with a common period T. We show that the protein synthesis pattern indeed converges to a unique periodic trajectory with period T. The analysis is based on introducing a novel property of dynamical systems, called contraction after a short transient (CAST), that may be of independent interest. We provide a sufficient condition for CAST and use it to prove that the RFM is CAST, and that this implies entrainment. Our results support the conjecture that periodic oscillations in tRNA levels and other factors related to the translation process can induce periodic oscillations in protein levels, and suggest a new approach for engineering genes to obtain a desired, periodic, synthesis rate.


  6. 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.


  7. G. Russo, M. di Bernardo, and E.D. Sontag. Global entrainment of transcriptional systems to periodic inputs. PLoS Computational Biology, 6:e1000739, 2010. [PDF] Keyword(s): contractive systems, contractions, systems biology, biochemical networks, gene and protein networks.
    Abstract:
    This paper addresses the problem of giving conditions for transcriptional systems to be globally entrained to external periodic inputs. By using contraction theory, a powerful tool from dynamical systems theory, it is shown that certain systems driven by external periodic signals have the property that all solutions converge to fixed limit cycles. General results are proved, and the properties are verified in the specific case of some models of transcriptional systems.


  8. 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.


  9. 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.


  10. M. Arcak and E.D. Sontag. A passivity-based stability criterion for a class of interconnected systems and applications to biochemical reaction networks. Mathematical Biosciences and Engineering, 5:1-19, 2008. Note: Also, preprint: arxiv0705.3188v1 [q-bio], May 2007. [PDF] Keyword(s): systems biology, biochemical networks, cyclic feedback systems, secant condition, nonlinear stability, dynamical systems.
    Abstract:
    This paper presents a stability test for a class of interconnected nonlinear systems motivated by biochemical reaction networks. One of the main results determines global asymptotic stability of the network from the diagonal stability of a "dissipativity matrix" which incorporates information about the passivity properties of the subsystems, the interconnection structure of the network, and the signs of the interconnection terms. This stability test encompasses the "secant criterion" for cyclic networks presented in our previous paper, and extends it to a general interconnection structure represented by a graph. A second main result allows one to accommodate state products. This extension makes the new stability criterion applicable to a broader class of models, even in the case of cyclic systems. The new stability test is illustrated on a mitogen activated protein kinase (MAPK) cascade model, and on a branched interconnection structure motivated by metabolic networks. Finally, another result addresses the robustness of stability in the presence of diffusion terms in a compartmental system made out of identical systems.


  11. 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.


  12. 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''.


  13. D. Angeli, P. de Leenheer, and E.D. Sontag. A Petri net approach to the study of persistence in chemical reaction networks. Mathematical Biosciences, 210:598-618, 2007. Note: Please look at the paper ``A Petri net approach to persistence analysis in chemical reaction networks'' for additional results, not included in the journal paper due to lack of space. See also the preprint: arXiv q-bio.MN/068019v2, 10 Aug 2006. [PDF] Keyword(s): Petri nets, systems biology, biochemical networks, nonlinear stability, dynamical systems, futile cycles.
    Abstract:
    Persistency is the property, for differential equations in Rn, that solutions starting in the positive orthant do not approach the boundary. For chemical reactions and population models, this translates into the non-extinction property: provided that every species is present at the start of the reaction, no species will tend to be eliminated in the course of the reaction. This paper provides checkable conditions for persistence of chemical species in reaction networks, using concepts and tools from Petri net theory, and verifies these conditions on various systems which arise in the modeling of cell signaling pathways.


  14. 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.


  15. 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.


  16. 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.


  17. 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.


  18. M. Arcak and E.D. Sontag. Diagonal stability of a class of cyclic systems and its connection with the secant criterion. Automatica, 42:1531-1537, 2006. [PDF] Keyword(s): passive systems, systems biology, biochemical networks, cyclic feedback systems, secant condition, nonlinear stability, dynamical systems.
    Abstract:
    This paper considers a class of systems with a cyclic structure that arises, among other examples, in dynamic models for certain biochemical reactions. We first show that a criterion for local stability, derived earlier in the literature, is in fact a necessary and sufficient condition for diagonal stability of the corresponding class of matrices. We then revisit a recent generalization of this criterion to output strictly passive systems, and recover the same stability condition using our diagonal stability result as a tool for constructing a Lyapunov function. Using this procedure for Lyapunov construction we exhibit classes of cyclic systems with sector nonlinearities and characterize their global stability properties.


  19. M. Chaves and E.D. Sontag. Exact computation of amplification for a class of nonlinear systems arising from cellular signaling pathways. Automatica, 42:1987-1992, 2006. [PDF] Keyword(s): systems biology, biochemical networks, nonlinear stability, dynamical systems.
    Abstract:
    A commonly employed measure of the signal amplification properties of an input/output system is its induced L2 norm, sometimes also known as H-infinity gain. In general, however, it is extremely difficult to compute the numerical value for this norm, or even to check that it is finite, unless the system being studied is linear. This paper describes a class of systems for which it is possible to reduce this computation to that of finding the norm of an associated linear system. In contrast to linearization approaches, a precise value, not an estimate, is obtained for the full nonlinear model. The class of systems that we study arose from the modeling of certain biological intracellular signaling cascades, but the results should be of wider applicability.


  20. 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.


  21. 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.


  22. E.P. Ryan and E.D. Sontag. Well-defined steady-state response does not imply CICS. Systems and Control Letters, 55:707-710, 2006. [PDF] [doi:10.1016/j.sysconle.2006.02.001] Keyword(s): nonlinear stability, dynamical systems.
    Abstract:
    Systems for which each constant input gives rise to a unique globally attracting equilibrium are considered. A counterexample is provided to show that inputs which are only asymptotically constant may not result in states converging to equilibria (failure of the converging-input converging state, or ``CICS'' property).


  23. E.D. Sontag. Passivity gains and the ``secant condition'' for stability. Systems Control Lett., 55(3):177-183, 2006. [PDF] Keyword(s): cyclic feedback systems, systems biology, biochemical networks, nonlinear stability, dynamical systems, passive systems, secant condition, biochemical networks.
    Abstract:
    A generalization of the classical secant condition for the stability of cascades of scalar linear systems is provided for passive systems. The key is the introduction of a quantity that combines gain and phase information for each system in the cascade. For linear one-dimensional systems, the known result is recovered exactly.


  24. 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.


  25. 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.


  26. 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.


  27. 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.


  28. 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.


  29. 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.


  30. 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.


  31. 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.


  32. 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.


  33. 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.


  34. M. Chaves, R.J. Dinerstein, and E.D. Sontag. Optimal length and signal amplification in weakly activated signal transduction cascades. J. Physical Chemistry, 108:15311-15320, 2004. [PDF] Keyword(s): systems biology, biochemical networks, dynamical systems.
    Abstract:
    Weakly activated signaling cascades can be modeled as linear systems. The input-to-output transfer function and the internal gain of a linear system, provide natural measures for the propagation of the input signal down the cascade and for the characterization of the final outcome. The most efficient design of a cascade for generating sharp signals, is obtained by choosing all the off rates equal, and a "universal" finite optimal length.


  35. M. Chaves, E.D. Sontag, and R. J. Dinerstein. Steady-states of receptor-ligand dynamics: A theoretical framework. J. Theoret. Biol., 227(3):413-428, 2004. [PDF] Keyword(s): zero-deficiency networks, systems biology, biochemical networks, receptor-ligand models, dynamical systems.
    Abstract:
    This paper studies aspects of the dynamics of a conventional mechanism of ligand-receptor interactions, with a focus on the stability and location of steady-states. A theoretical framework is developed, and, as an application, a minimal parametrization is provided for models for two- or multi-state receptor interaction with ligand. In addition, an "affinity quotient" is introduced, which allows an elegant classification of ligands into agonists, neutral agonists, and inverse agonists.


  36. 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.


  37. 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.


  38. 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.


  39. 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.


  40. 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.


  41. 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.


  42. M. Chaves and E.D. Sontag. State-Estimators for chemical reaction networks of Feinberg-Horn-Jackson zero deficiency type. European J. Control, 8:343-359, 2002. [PDF] Keyword(s): observability, zero-deficiency networks, systems biology, biochemical networks, observers, nonlinear stability, dynamical systems.
    Abstract:
    This paper provides a necessary and sufficient condition for detectability, and an explicit construction of observers when this condition is satisfied, for chemical reaction networks of the Feinberg-Horn-Jackson zero deficiency type.


  43. E.D. Sontag. Correction to: ``Structure and stability of certain chemical networks and applications to the kinetic proofreading model of T-cell receptor signal transduction'' [IEEE Trans. Automat. Control 46 (2001), no. 7, 1028--1047; MR1842137 (2002e:92006)]. IEEE Trans. Automat. Control, 47(4):705, 2002. [PDF] Keyword(s): zero-deficiency networks, systems biology, biochemical networks, nonlinear stability, dynamical systems.
    Abstract:
    errata for Structure and stability of certain chemical networks and applications to the kinetic proofreading model of T-cell receptor signal transduction


  44. W. Desch, H. Logemann, E. P. Ryan, and E.D. Sontag. Meagre functions and asymptotic behaviour of dynamical systems. Nonlinear Anal., 44(8, Ser. A: Theory Methods):1087-1109, 2001. [PDF] [doi:http://dx.doi.org/10.1016/S0362-546X(99)00323-5] Keyword(s): invariance principle.
    Abstract:
    A measurable function x from a subset J of R into a metric space X is said to be C-meagre if C is non-empty subset of X and, for every closed subset K of X disjoint from C, the preimage of K under x has finite Lebesgue measure. This concept of meagreness, applied to trajectories, is shown to provide a unifying framework which facilitates a variety of characterizations, extensions or generalizations of diverse facts pertaining to asymptotic behaviour of dynamical systems.


  45. E.D. Sontag. Structure and stability of certain chemical networks and applications to the kinetic proofreading model of T-cell receptor signal transduction. IEEE Trans. Automat. Control, 46(7):1028-1047, 2001. [PDF] Keyword(s): zero-deficiency networks, systems biology, biochemical networks, nonlinear stability, dynamical systems.
    Abstract:
    This paper deals with the theory of structure, stability, robustness, and stabilization for an appealing class of nonlinear systems which arises in the analysis of chemical networks. The results given here extend, but are also heavily based upon, certain previous work by Feinberg, Horn, and Jackson, of which a self-contained and streamlined exposition is included. The theoretical conclusions are illustrated through an application to the kinetic proofreading model proposed by McKeithan for T-cell receptor signal transduction.


  46. D. Angeli and E.D. Sontag. Forward completeness, unboundedness observability, and their Lyapunov characterizations. Systems Control Lett., 38(4-5):209-217, 1999. [PDF] Keyword(s): observability, input to state stability, dynamical systems.
    Abstract:
    A finite-dimensional continuous-time system is forward complete if solutions exist globally, for positive time. This paper shows that forward completeness can be characterized in a necessary and sufficient manner by means of smooth scalar growth inequalities. Moreover, a version of this fact is also proved for systems with inputs, and a generalization is also provided for systems with outputs and a notion (unboundedness observability) of relative completeness. We apply these results to obtain a bound on reachable states in terms of energy-like estimates of inputs.


  47. F. Albertini and E.D. Sontag. Discrete-time transitivity and accessibility: analytic systems. SIAM J. Control Optim., 31(6):1599-1622, 1993. [PDF] [doi:http://dx.doi.org/10.1137/0331075]
    Abstract:
    A basic open question for discrete-time nonlinear systems is that of determining when, in analogy with the classical continuous-time "positive form of Chow's Lemma", accessibility follows from transitivity of a natural group action. This paper studies the problem, and establishes the desired implication for analytic systems in several cases: (i) compact state space, (ii) under a Poisson stability condition, and (iii) in a generic sense. In addition, the paper studies accessibility properties of the "control sets" recently introduced in the context of dynamical systems studies. Finally, various examples and counterexamples are provided relating the various Lie algebras introduced in past work.


  48. M. L. J. Hautus and E.D. Sontag. An approach to detectability and observers. In Algebraic and geometric methods in linear systems theory (AMS-NASA-NATO Summer Sem., Harvard Univ., Cambridge, Mass., 1979), volume 18 of Lectures in Appl. Math., pages 99-135. Amer. Math. Soc., Providence, R.I., 1980. [PDF] Keyword(s): observability.
    Abstract:
    This paper proposes an approach to the problem of establishing the existence of observers for deterministic dynamical systems. This approach differs from the standard one based on Luenberger observers in that the observation error is not required to be Markovian given the past input and output data. A general abstract result is given, which special- izes to new results for parametrized families of linear systems, delay systems and other classes of systems. Related problems of feedback control and regulation are also studied.


  49. Y. Rouchaleau and E.D. Sontag. On the existence of minimal realizations of linear dynamical systems over Noetherian integral domains. J. Comput. System Sci., 18(1):65-75, 1979. [PDF] Keyword(s): systems over rings.
    Abstract:
    This paper studies the problem of obtaining minimal realizations of linear input/output maps defined over rings. In particular, it is shown that, contrary to the case of systems over fields, it is in general impossible to obtain realizations whose dimiension equals the rank of the Hankel matrix. A characterization is given of those (Noetherian) rings over which realizations of such dimensions can he always obtained, and the result is applied to delay-differential systems.


  50. E.D. Sontag. On linear systems and noncommutative rings. Math. Systems Theory, 9(4):327-344, 1975. [PDF] Keyword(s): systems over rings.
    Abstract:
    This paper studies some problems appearing in the extension of the theory of linear dynamical systems to the case in which parameters are taken from noncommutative rings. Purely algebraic statements of some of the problems are also obtained. Through systems defined by operator rings, the theory of linear systems over rings may be applied to other areas of automata and control theory; several such applications are outlined.


Conference articles
  1. A. O. Hamadeh, E.D. Sontag, and D. Del Vecchio. A contraction approach to output tracking via high-gain feedback. In Proc. IEEE Conf. Decision and Control, Dec. 2015, pages 7689-7694, 2015. [PDF]
    Abstract:
    This paper adopts a contraction approach to the analysis of the tracking properties of dynamical systems under high gain feedback when subject to inputs with bounded derivatives. It is shown that if the tracking error dynamics are contracting, then the system is input to output stable with respect to the input signal derivatives and the output tracking error. As an application, it iss hown that the negative feedback connection of plants composed of two strictly positive real LTI subsystems in cascade can follow external inputs with tracking errors that can be made arbitrarily small by applying a sufficiently large feedback gain. We utilize this result to design a biomolecular feedback for a synthetic genetic sensor to make it robust to variations in the availability of a cellular resource required for protein production.


  2. Z. Aminzare and E.D. Sontag. Contraction methods for nonlinear systems: A brief introduction and some open problems. In Proc. IEEE Conf. Decision and Control, Los Angeles, Dec. 2014, pages 3835-3847, 2014. [PDF] Keyword(s): contractions, contractive systems, stability, reaction-diffusion PDE's, synchronization, contractive systems, stability.
    Abstract:
    Contraction theory provides an elegant way to analyze the behaviors of certain nonlinear dynamical systems. Under sometimes easy to check hypotheses, systems can be shown to have the incremental stability property that trajectories converge to each other. The present paper provides a self-contained introduction to some of the basic concepts and results in contraction theory, discusses applications to synchronization and to reaction-diffusion partial differential equations, and poses several open questions.


  3. M. Skataric and E.D. Sontag. Remarks on model-based estimation of nonhomogeneous Poisson processes and applications to biological systems. In Proc. European Control Conference, Strasbourg, France, June 2014, pages 2052-2057, 2014. [PDF] Keyword(s): systems biology, random dynamical systems.
    Abstract:
    This paper studies model-based estimation methods of a rate of a nonhomogeneous Poisson processes that describes events arising from modeling biological phenomena in which discrete events are measured. We describe an approach based on observers and Kalman filters as well as preliminary simulation results, and compare these to other methods (not model-based) in the literature. The problem is motivated by the question of identification of internal states from neural spikes and bacterial tumbling behavior.


  4. 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.


  5. D. Angeli, P. de Leenheer, and E.D. Sontag. Petri nets tools for the analysis of persistence in chemical networks. In Proc. 7th IFAC Symposium on Nonlinear Control Systems (NOLCOS 2007), Pretoria, South Africa, 22-24 August, 2007, 2007. Keyword(s): Petri nets, systems biology, biochemical networks, nonlinear stability, dynamical systems, futile cycles.


  6. M. Arcak and E.D. Sontag. A passivity-based stability criterion for a class of interconnected systems and applications to biochemical reaction networks. In Proc. IEEE Conf. Decision and Control, New Orleans, Dec. 2007, pages 4477-4482, 2007. Note: Conference version of journal paper with same title. Keyword(s): systems biology, biochemical networks, cyclic feedback systems, secant condition, nonlinear stability, dynamical 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. M. Arcak and E.D. Sontag. Connections between diagonal stability and the secant condition for cyclic systems. In Proc. American Control Conference, Minneapolis, June 2006, pages 1493-1498, 2006. Keyword(s): systems biology, biochemical networks, cyclic feedback systems, secant condition, nonlinear stability, dynamical systems.


  10. 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.


  11. 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.


  12. 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.


  13. E.D. Sontag. A notion of passivity gain and a generalization of the `secant condition' for stability. In Proc. IEEE Conf. Decision and Control, Seville, Dec. 2005, IEEE Publications, pages 5645-5649, 2005. Keyword(s): nonlinear stability, dynamical systems.


  14. E.D. Sontag and M. Chaves. Computation of amplification for systems arising from cellular signaling pathways. In Proc. 16th IFAC World Congress, Prague, July 2005, 2005. Keyword(s): systems biology, biochemical networks, dynamical systems.


  15. 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.


  16. 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.


  17. M. Chaves, E.D. Sontag, and R.J. Dinerstein. Gains and optimal design in signaling pathways. In Proc. IEEE Conf. Decision and Control, Paradise Island, Bahamas, Dec. 2004, IEEE Publications, pages 596-601, 2004. Keyword(s): systems biology, biochemical networks, dynamical systems.


  18. 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.


  19. 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.


  20. M. Chaves and E.D. Sontag. An alternative observer for zero deficiency chemical networks. In Proc. Nonlinear Control System Design Symposium, St. Petersburg, July 2001, pages 575-578, 2001. Keyword(s): observability, observers, zero-deficiency networks, systems biology, biochemical networks, nonlinear stability, dynamical systems.


  21. M. Chaves and E.D. Sontag. Observers for certain chemical reaction networks. In Proc. 2001 European Control Conf., Sep. 2001, pages 3715-3720, 2001. Keyword(s): zero-deficiency networks, systems biology, biochemical networks, nonlinear stability, dynamical systems, observability, observers.


  22. E.D. Sontag. Spaces of observables in nonlinear control. In Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), Basel, pages 1532-1545, 1995. Birkhäuser. [PDF] Keyword(s): observability, dynamical systems.
    Abstract:
    Invited talk at the 1994 ICM. Paper deals with the notion of observables for nonlinear systems, and their role in realization theory, minimality, and several control and path planning questions.


  23. F. Albertini and E.D. Sontag. Some connections between chaotic dynamical systems and control systems. In Proc. European Control Conf. , Vol 1, Grenoble, July 1991, pages 58-163, 1991. [PDF] Keyword(s): chaotic systems, controllability.
    Abstract:
    This paper shows how to extend recent results of Colonius and Kliemann, regarding connections between chaos and controllability, from continuous to discrete time. The extension is nontrivial because the results all rely on basic properties of the accessibility Lie algebra which fail to hold in discrete time. Thus, this paper first develops further results in nonlinear accessibility, and then shows how a theorem can be proved, which while analogous to the one given in the work by Colonius and Klieman, also exhibits some important differences. A counterexample is used to show that the theorem given in continuous time cannot be generalized in a straightforward manner.


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.


  2. F. Albertini and E.D. Sontag. Some connections between chaotic dynamical systems and control systems. Technical report SYCON-90-13, Rutgers Center for Systems and Control, 1990.



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