Publications by Eduardo D. Sontag in year 2014 |
Articles in journal or book chapters |
This paper gives conditions that guarantee spatial uniformity of the solutions of reaction-diffusion partial differential equations, stated in terms of the Jacobian matrix and Neumann eigenvalues of elliptic operators on the given spatial domain, and similar conditions for diffusively-coupled networks of ordinary differential equations. Also derived are numerical tests making use of linear matrix inequalities that are useful in certifying these conditions. |
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. |
This paper introduces a small-gain result for interconnected orthant-monotone systems for which no matching condition is required between the partial orders in input and output spaces. Previous results assumed that the partial orders adopted would be induced by positivity cones in input and output spaces and that such positivity cones should fulfill a compatibility rule: namely either be coincident or be opposite. Those two configurations correspond to positive feedback or negative feedback cases. We relax those results by allowing arbitrary orthant orders. |
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. |
This paper describes a potential pitfall of perturbation-based approaches to network inference It is shows experimentally, and then explained mathematically, how even in the simplest signaling systems, perturbation methods may lead to paradoxical conclusions: for any given pair of two components X and Y, and depending upon the specific intervention on Y, either an activation or a repression of X could be inferred. The experiments are performed in an in vitro minimal system, thus isolating the effect and showing that it cannot be explained by feedbacks due to unknown intermediates; this system utilizes proteins from a pathway in mammalian (and other eukaryotic) cells that play a central role in proliferation, gene expression, differentiation, mitosis, cell survival, and apoptosis and is a perturbation target of contemporary therapies for various types of cancers. The results show that the simplistic view of intracellular signaling networks being made up of activation and repression links is seriously misleading, and call for a fundamental rethinking of signaling network analysis and inference methods. |
A transcriptional system is built based on a 'resource allocator' that sets a core RNAP concentration, which is then shared by multiple sigma fragments, which provide specificity. Adjusting the concentration of the core sets the maximum transcriptional capacity available to a synthetic system. |
This paper studies the direction of change of steady states to parameter perturbations in chemical reaction networks, and, in particular, to changes in conserved quantities. Theoretical considerations lead to the formulation of a computational procedure that provides a set of possible signs of such sensitivities. The procedure is purely algebraic and combinatorial, only using information on stoichiometry, and is independent of the values of kinetic constants. Two examples of important intracellular signal transduction models are worked out as an illustration. In these examples, the set of signs found is minimal, but there is no general guarantee that the set found will always be minimal in other examples. The paper also briefly discusses the relationship of the sign problem to the question of uniqueness of steady states in stoichiometry classes. |
Conference articles |
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. |
In this paper, we sketch recent results for synchronization in a network of identical ODE models which are diffusively interconnected. In particular, 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. |
This conference paper (a) summarizes material from "A fundamental limitation to fold-change detection by biological systems with multiple time scales" (IET Systems Biology 2014) and presents additional remarks regarding (b) expansion techniques to compute FCD error and (c) stochastic adaptation and FCD |
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. |
We introduce three forms of generalized contraction~(GC). Roughly speaking, these are motivated by allowing contraction to take place after small transients in time and/or amplitude. Indeed, contraction is usually used to prove asymptotic properties, like convergence to an attractor or entrainment to a periodic excitation, and allowing initial transients does not affect this asymptotic behavior. We provide sufficient conditions for GC, and demonstrate their usefulness using examples of systems that are not contractive, with respect to any norm, yet are~GC. |
Internal reports |
The connection between optimal biological function and energy use, measured for example by the rate of metabolite consumption, is a current topic of interest in the systems biology literature which has been explored in several different contexts. In [J. P. Barton and E. D. Sontag, Biophys. J. 104, 6 (2013)], we related the metabolic cost of enzymatic futile cycles with their capacity to act as insulators which facilitate modular interconnections in biochemical networks. There we analyzed a simple model system in which a signal molecule regulates the transcription of one or more target proteins by interacting with their promoters. In this note, we consider the case of a protein with an active and an inactive form, and whose activation is controlled by the signal molecule. As in the original case, higher rates of energy consumption are required for better insulator performance. |
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