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Eduardo D. Sontag

Publications and source records attributed to Eduardo D. Sontag.

At least 73 records · Page 4Linked to original sources
arXiv2013

A remark about polynomials with specified local minima and no other critical points

The following observation must surely be "well-known", but it seems worth giving a simple and quite explicit proof. Take any finite subset X of Rn, n>1. Then, there is a polynomial function P:Rn -> R which has local minima on the set X, and has no other critical points. Applied to the negative gradient flow of P, this implies that there is a polynomial vector field with asymptotically stable equilibria on X and no other equilibria. Some trajectories of this vector field are not pre-compact; a complementary observation says that, again for arbitrary X, one can find a vector field with asymptotically stable equilibria on X, no other equilibria except saddles, and all omega-limit sets consisting of singletons.

math.DS↗
arXiv2012

The energy costs of biological insulators

Biochemical signaling pathways can be insulated from impedance and competition effects through enzymatic "futile cycles" which consume energy, typically in the form of ATP. We hypothesize that better insulation necessarily requires higher energy consumption, and provide evidence, through the computational analysis of a simplified physical model, to support this hypothesis.

q-bio.MN↗
arXiv2012

Logarithmic Lipschitz norms and diffusion-induced instability

This paper proves that contractive ordinary differential equation systems remain contractive when diffusion is added. Thus, diffusive instabilities, in the sense of the Turing phenomenon, cannot arise for such systems. An important biochemical system is shown to satisfy the required conditions.

eess.SY↗
arXiv2009

Global entrainment of transcriptional systems to periodic inputs

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 a fixed limit cycle. General results are proved, and the properties are verified in the specific case of some models of transcriptional systems. The basic mathematical results needed from contraction theory are proved in the paper, making it self-contained.

q-bio.QM↗
arXiv2009

An observation regarding systems which converge to steady states for all constant inputs, yet become chaotic with periodic inputs

This note provides a general construction, and gives a concrete example of, forced ordinary differential equation systems that have these two properties: (a) for each constant input u, all solutions converge to a steady state but (b) for some periodic inputs, the system has arbitrary (for example, "chaotic") behavior. An alternative example has the property that all solutions converge to the same state (independently of initial conditions as well as input, so long as it is constant).

math.DS↗
arXiv2009

Remarks on Feedforward Circuits, Adaptation, and Pulse Memory

This note studies feedforward circuits as models for perfect adaptation to step signals in biological systems. A global convergence theorem is proved in a general framework, which includes examples from the literature as particular cases. A notable aspect of these circuits is that they do not adapt to pulse signals, because they display a memory phenomenon. Estimates are given of the magnitude of this effect.

q-bio.QM↗
arXiv2008

Stabilizing and Destabilizing Effects of Embedding 3-node Subgraphs on State Space of Boolean Networks

We demonstrate the effects of embedding subgraphs using a Boolean network, which is one of the discrete dynamical models for transcriptional regulatory networks. After comparing the dynamical properties of network embedded seven different subgraphs including feedback and feedforward subgraphs, we found that complexity of the state space that increases with longer length of attractors and greater number of attractors is reduced for networks with more feedforward subgraphs. In addition, feedforward subgraphs can also provide higher mutual information with lower entropy in a temporal program of gene expression. Networks with other six subgraphs show opposite effects on dynamics of the networks, is roughly consistent with Thomas's conjecture. These results suggest that feedforward subgraphs are one of the favorable local structures in biological complex networks.

nlin.CG↗
arXiv2007

Shape, size and robustness: feasible regions in the parameter space of biochemical networks

The concept of robustness of regulatory networks has been closely related to the nature of the interactions among genes, and the capability of pattern maintenance or reproducibility. Defining this robustness property is a challenging task, but mathematical models have often associated it to the volume of the space of admissible parameters. Not only the volume of the space but also its topology and geometry contain information on essential aspects of the network, including feasible pathways, switching between two parallel pathways or distinct/disconnected active regions of parameters. A general method is presented here to characterize the space of admissible parameters, by writing it as a semi-algebraic set, and then theoretically analyzing its topology and geometry, as well as volume. This method provides a more objective and complete measure of the robustness of a developmental module. As an illustration, the segment polarity gene network is analyzed.

q-bio.MN↗
arXiv2007

Attractors in coherent systems of differential equations

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 coherent systems, defined by the property that no directed feedback loops are negative. Several new results for cooperative systems are obtained in the process.

math.DS↗
arXiv2007

A remark on the number of steady states in a multiple futile cycle

The multisite phosphorylation-dephosphorylation cycle is a motif repeatedly used in cell signaling. This motif itself can generate a variety of dynamic behaviors like bistability and ultrasensitivity without direct positive feedbacks. In this paper, we study the number of positive steady states of a general multisite phosphorylation-dephosphorylation cycle, and how the number of positive steady states varies by changing the biological parameters. We show analytically 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 (in particular, this implies that for n=2, including single levels of MAPK cascades, there are at most three 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.

q-bio.QM↗
arXiv2007

A passivity-based stability criterion for a class of interconnected systems and applications to biochemical reaction networks

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.

q-bio.QM↗
arXiv2007

Oscillations in I/O monotone systems under negative feedback

Oscillatory behavior is a key property of many biological systems. The Small-Gain Theorem (SGT) for input/output monotone systems provides a sufficient condition for global asymptotic stability of an equilibrium and hence its violation is a necessary condition for the existence of periodic solutions. One advantage of the use of the monotone SGT technique is its robustness with respect to all perturbations that preserve monotonicity and stability properties of a very low-dimensional (in many interesting examples, just one-dimensional) model reduction. This robustness makes the technique useful in the analysis of molecular biological models in which there is large uncertainty regarding the values of kinetic and other parameters. However, verifying the conditions needed in order to apply the SGT is not always easy. This paper provides an approach to the verification of the needed properties, and illustrates the approach through an application to a classical model of circadian oscillations, as a nontrivial ``case study,'' and also provides a theorem in the converse direction of predicting oscillations when the SGT conditions fail.

q-bio.QM↗
arXiv2006

Monotone and near-monotone network structure (part I)

This paper (parts I and II) provides an expository introduction to monotone and near-monotone dynamical systems associated to biochemical networks, those whose graphs are consistent or near-consistent. Many conclusions can be drawn from signed network structure, associated to purely stoichiometric information and ignoring fluxes. In particular, monotone systems respond in a predictable fashion to perturbations and have robust and ordered dynamical characteristics, making them reliable components of larger networks. Interconnections of monotone systems may be fruitfully analyzed using tools from control theory, by viewing larger systems as interconnections of monotone subsystems. This allows one to obtain precise bifurcation diagrams without appeal to explicit knowledge of fluxes or of kinetic constants and other parameters, using merely "input/output characteristics" (steady-state responses or DC gains). The procedure may be viewed as a "model reduction" approach in which monotone subsystems are viewed as essentially one-dimensional objects. The possibility of performing a decomposition into a small number of monotone components is closely tied to the question of how "near" a system is to being monotone. We argue that systems that are "near monotone" may be more biologically more desirable than systems that are far from being monotone. Indeed, there are indications that biological networks may be much closer to being monotone than random networks that have the same numbers of vertices and of positive and negative edges.

q-bio.MN↗
arXiv2006

Monotone and near-monotone network structure (part II)

This paper (parts I and II) provides an expository introduction to monotone and near-monotone dynamical systems associated to biochemical networks, those whose graphs are consistent or near-consistent. Many conclusions can be drawn from signed network structure, associated to purely stoichiometric information and ignoring fluxes. In particular, monotone systems respond in a predictable fashion to perturbations and have robust and ordered dynamical characteristics, making them reliable components of larger networks. Interconnections of monotone systems may be fruitfully analyzed using tools from control theory, by viewing larger systems as interconnections of monotone subsystems. This allows one to obtain precise bifurcation diagrams without appeal to explicit knowledge of fluxes or of kinetic constants and other parameters, using merely "input/output characteristics" (steady-state responses or DC gains). The procedure may be viewed as a "model reduction" approach in which monotone subsystems are viewed as essentially one-dimensional objects. The possibility of performing a decomposition into a small number of monotone components is closely tied to the question of how "near" a system is to being monotone. We argue that systems that are "near monotone" may be more biologically more desirable than systems that are far from being monotone. Indeed, there are indications that biological networks may be much closer to being monotone than random networks that have the same numbers of vertices and of positive and negative edges.

q-bio.MN↗
arXiv2006

Input Classes for Identification of Bilinear Systems

This paper asks what classes of input signals are sufficient in order to completely identify the input/output behavior of generic bilinear systems. The main results are that step inputs are not sufficient, nor are single pulses, but the family of all pulses (of a fixed amplitude but varying widths) do suffice for identification.

math.OC↗