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Publications by Eduardo D. Sontag in year 2006
Articles in journal or book chapters
  1. B. Dasgupta, P. Berman, and E.D. Sontag. Computational complexities of combinatorial problems with applications to reverse engineering of biological networks. In D. Liu and F-Y. Wan, editors, Advances in Computational Intelligence: Theory & Applications, pages 303-316. World Scientific, Hackensack, 2006. Keyword(s): systems biology, biochemical networks, gene and protein networks, reverse engineering, systems identification, theory of computing and complexity.


  2. 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. Ālvarez 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.


  3. W. Maass, P. Joshi, and E.D. Sontag. Principles of real-time computing with feedback applied to cortical microcircuit models. In Advances in Neural Information Processing Systems 18. MIT Press, Cambridge, 2006. Keyword(s): neural networks.


  4. 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.


  5. 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.


  6. M. Chaves, E.D. Sontag, and R. Albert. Methods of robustness analysis for Boolean models of gene control networks. IET Systems Biology, 153:154-167, 2006. [PDF] Keyword(s): systems biology, biochemical networks, boolean systems, gene and protein networks, hybrid systems.
    Abstract:
    As a discrete approach to genetic regulatory networks, Boolean models provide an essential qualitative description of the structure of interactions among genes and proteins. Boolean models generally assume only two possible states (expressed or not expressed) for each gene or protein in the network as well as a high level of synchronization among the various regulatory processes. In this paper, we discuss and compare two possible methods of adapting qualitative models to incorporate the continuous-time character of regulatory networks. The first method consists of introducing asynchronous updates in the Boolean model. In the second method, we adopt the approach introduced by L. Glass to obtain a set of piecewise linear differential equations which continuously describe the states of each gene or protein in the network. We apply both methods to a particular example: a Boolean model of the segment polarity gene network of Drosophila melanogaster. We analyze the dynamics of the model, and provide a theoretical characterization of the model's gene pattern prediction as a function of the timescales of the various processes.


  7. B. DasGupta, J.P. Hespanha, J. Riehl, and E.D. Sontag. Honey-pot constrained searching with local sensory information. Nonlinear Analysis, 65:1773-1793, 2006. [PDF] Keyword(s): search problems, algorithms, computational complexity.
    Abstract:
    This paper investigates the problem of searching for a hidden target in a bounded region of the plane by an autonomous robot which is only able to use limited local sensory information. It proposes an aggregation-based approach to solve this problem, in which the continuous search space is partitioned into a finite collection of regions on which we define a discrete search problem and a solution to the original problem is obtained through a refinement procedure that lifts the discrete path into a continuous one. The resulting solution is in general not optimal but one can construct bounds to gauge the cost penalty incurred. The discrete version is formalized and an optimization problem is stated as a `reward-collecting' bounded-length path problem. NP-completeness and efficient approximation algorithms for various cases of this problem are discussed.


  8. 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.


  9. 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.


  10. M. Malisoff, M. Krichman, and E.D. Sontag. Global stabilization for systems evolving on manifolds. Journal of Dynamical and Control Systems, 12:161-184, 2006. [PDF] Keyword(s): nonlinear stability, nonlinear control, feedback stabilization.
    Abstract:
    This paper shows that any globally asymptotically controllable system on any smooth manifold can be globally stabilized by a state feedback. Since discontinuous feedbacks are allowed, solutions are understood in the ``sample and hold'' sense introduced by Clarke-Ledyaev-Sontag-Subbotin (CLSS). This work generalizes the CLSS Theorem, which is the special case of our result for systems on Euclidean space. We apply our result to the input-to-state stabilization of systems on manifolds relative to actuator errors, under small observation noise.


  11. 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).


  12. 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.


  13. 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.


  14. 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.


  15. 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.


  16. N.A.W. van Riel and E.D. Sontag. Parameter estimation in models combining signal transduction and metabolic pathways: The dependent input approach. IET Systems Biology, 153:263-274, 2006. [PDF] Keyword(s): systems biology, biochemical networks, parameter identification.
    Abstract:
    Biological complexity and limited quantitative measurements impose severe challenges to standard engineering methodologies for systems identification. This paper presents an approach, justified by the theory of universal inputs for distinguishability, based on replacing unmodeled dynamics by fictitious `dependent inputs'. The approach is particularly useful in validation experiments, because it allows one to fit model parameters to experimental data generated by a reference (wild-type) organism and then testing this model on data generated by a variation (mutant), so long as the mutations only affect the unmodeled dynamics that produce the dependent inputs. As a case study, this paper addresses the pathways that control the nitrogen uptake fluxes in baker's yeast Saccharomyces cerevisiae enabling it to optimally respond to changes in nitrogen availability. Well-defined perturbation experiments were performed on cells growing in steady-state. Time-series data of extracellular and intracellular metabolites were obtained, as well as mRNA levels. A nonlinear model was proposed, and shown to be structurally identifiable given input/output data. The identified model correctly predicted the responses of different yeast strains and different perturbations.


Conference articles
  1. B. Andrews, P. Iglesias, and E.D. Sontag. Signal detection and approximate adaptation implies an approximate internal model. In Proc. IEEE Conf. Decision and Control, San Diego, Dec. 2006, pages 2364-2369, 2006. IEEE. [PDF] Keyword(s): biological adaptation, internal model principle.
    Abstract:
    This conference paper presented a version of an approximate internal model principle, for linear systems. A subsequent paper at the IFAC 2008 conference improved on this result by extending it to a class of nonlinear systems.


  2. 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.


  3. 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.


  4. 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.


  5. M. Chaves, E.D. Sontag, and R. Albert. Structure and timescale analysis in genetic regulatory networks. In Proc. IEEE Conf. Decision and Control, San Diego, Dec. 2006, pages 2358-2363, 2006. IEEE. [PDF] Keyword(s): genetic regulatory networks, Boolean systems, hybrid systems.
    Abstract:
    This work is concerned with the study of the robustness and fragility of gene regulation networks to variability in the timescales of the distinct biological processes involved. It explores and compares two methods: introducing asynchronous updates in a Boolean model, or integrating the Boolean rules in a continuous, piecewise linear model. As an example, the segment polarity network of the fruit fly is analyzed. A theoretical characterization is given of the model's ability to predict the correct development of the segmented embryo, in terms of the specific timescales of the various regulation interactions.


  6. 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.


  7. 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.



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