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Publications of Eduardo D. Sontag jointly with P. de Leenheer
Articles in journal or book chapters
  1. D. Angeli, P. de Leenheer, and E.D. Sontag. Persistence results for chemical reaction networks with time-dependent kinetics and no global conservation laws. SIAM Journal on Applied Mathematics, 71:128-146, 2011. [PDF] Keyword(s): biochemical networks, fluxes, Petri nets, persistence, biochemical networks with inputs.
    Abstract:
    New checkable criteria for persistence of chemical reaction networks are proposed, which extend and complement existing ones. The new results allow the consideration of reaction rates which are time-varying, thus incorporating the effects of external signals, and also relax the assumption of existence of global conservation laws, thus allowing for inflows (production) and outflows (degradation). For time-invariant networks parameter-dependent conditions for persistence of certain classes of networks are provided. As an illustration, two networks arising in the systems biology literature are analyzed, namely a hypoxia and an apoptosis network.


  2. 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.


  3. 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.


  4. D. Angeli, P. de Leenheer, and E.D. Sontag. Chemical networks with inflows and outflows: A positive linear differential inclusions approach. Biotechnology Progress, 25:632-642, 2009. [PDF] Keyword(s): biochemical networks, fluxes, differential inclusions, positive systems, Petri nets, persistence, switched systems.
    Abstract:
    Certain mass-action kinetics models of biochemical reaction networks, although described by nonlinear differential equations, may be partially viewed as state-dependent linear time-varying systems, which in turn may be modeled by convex compact valued positive linear differential inclusions. A result is provided on asymptotic stability of such inclusions, and applied to biochemical reaction networks with inflows and outflows. Included is also a characterization of exponential stability of general homogeneous switched systems


  5. D. Angeli, P. De Leenheer, and E.D. Sontag. A Petri net approach to persistence analysis in chemical reaction networks. 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 181-216. Springer-Verlag, Berlin, 2007. Note: See abstract for A Petri net approach to the study of persistence in chemical reaction networks.[PDF]


  6. 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.


  7. 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.


  8. 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.


  9. 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.


  10. 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.


  11. 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.


  12. 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.


  13. 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.


Conference articles
  1. D. Angeli, P. de Leenheer, and E.D. Sontag. On persistence of chemical reaction networks with time-dependent kinetics and no global conservation laws. In Proc. IEEE Conf. Decision and Control, Shanhai, Dec. 2009, pages 4559-4564, 2009. [PDF] Keyword(s): biochemical networks, fluxes, Petri nets, persistence, biochemical networks with inputs.
    Abstract:
    This is a very summarized version ofthe first part of the paper "Persistence results for chemical reaction networks with time-dependent kinetics and no global conservation laws".


  2. 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"


  3. 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.


  4. 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.


  5. 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.


  6. 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.



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Last modified: Thu Nov 23 10:40:56 2017
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