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Publications about 'multistability'
Articles in journal or book chapters
  1. M. A. Al-Radhawi, D. Del Vecchio, and E. D. Sontag. Multi-modality in gene regulatory networks with slow gene binding. 2017. Note: Submitted. Preprint in arXiv:1705.02330, May 2017 rev Nov 2017. [PDF] Keyword(s): multistability, gene networks, Markov Chains, Master Equation, cancer heterogeneity, phenotypic variation, nonlinear systems, stochastic models, epigenetics.
    Abstract:
    In biological processes such as embryonic development, hematopoietic cell differentiation, and the arising of tumor heterogeneity and consequent resistance to therapy, mechanisms of gene activation and deactivation may play a role in the emergence of phenotypically heterogeneous yet genetically identical (clonal) cellular populations. Mathematically, the variability in phenotypes in the absence of genetic variation can be modeled through the existence of multiple metastable attractors in nonlinear systems subject with stochastic switching, each one of them associated to an alternative epigenetic state. An important theoretical and practical question is that of estimating the number and location of these states, as well as their relative probabilities of occurrence. This paper focuses on a rigorous analytic characterization of multiple modes under slow promoter kinetics, which is a feature of epigenetic regulation. It characterizes the stationary distributions of Chemical Master Equations for gene regulatory networks as a mixture of Poisson distributions. As illustrations, the theory is used to tease out the role of cooperative binding in stochastic models in comparison to deterministic models, and applications are given to various model systems, such as toggle switches in isolation or in communicating populations and a trans-differentiation network.


  2. 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.


  3. L. Wang and E.D. Sontag. On the number of steady states in a multiple futile cycle. Journal of Mathematical Biology, 57:29-52, 2008. [PDF] Keyword(s): singular perturbations, futile cycles, MAPK cascades, systems biology, biochemical networks, multistability.
    Abstract:
    This note studies the number of positive steady states in biomolecular reactions consisting of activation/deactivation futile cycles, such as those arising from phosphorylations and dephosphorylations at each level of a MAPK cascade. It is shown that: (1) for some parameter ranges, there are at least n+1 (if n is even) or n (if n is odd) steady states; (2) there never are more than 2n-1 steady states (so, for n=2, there are no more than 3 steady states); (3) for parameters near the standard Michaelis-Menten quasi-steady state conditions, there are at most n+1 steady states; and (4) for parameters far from the standard Michaelis-Menten quasi-steady state conditions, there is at most one steady state.


  4. 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.


  5. 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.


  6. 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.


  7. 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.


Conference articles
  1. N.S. Kumar, M. A. Al-Radhawi, D. Del Vecchio, and E. D. Sontag. Stochasticity is necessary for multiple attractors in a class of differentiation networks. In 2017 American Control Conference (ACC), pages submitted, 2017. Keyword(s): systems biology, genetic regulatory, multistability, gene networks.
    Abstract:
    Deterministic models remain the most common option for modeling gene regulatory networks even when the underlying assumptions of high copy numbers and fast promoter kinetics are unsatisfied. Here, we analyze a widely studied differentiation network motif known as the PU.1-GATA-1 circuit and we show that an ODE model of the biomolecular reactions consistent with known biology is incapable of exhibiting multistability, a defining behaviour for such a network. Thus, we consider the chemical master equation model of the same biomolecular reactions and using results recently developed by the authors, we analytically construct the stationary distribution. We show that this distribution is indeed capable of admitting a multitude of modes. We illustrate the results with a numerical example.


  2. 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.


  3. 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.


  4. 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.


  5. 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.



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