Publications by Eduardo D. Sontag in year 2013 |
Articles in journal or book chapters |
This paper study systems with sign-definite interactions between variables, providing a sufficient condition to characterize the possible transitions between intervals of increasing and decreasing behavior. It also provides a discussion illustrating how our approach can help identify interactions in models, using information from time series of observations. |
This work introduces a notion of random dynamical systems with inputs, providing several basic definitions and results on equilibria and convergence. It also presents a "converging input to converging state" result, a concept that plays a key role in the analysis of stability of feedback interconnections, for monotone systems. |
This paper proves that ordinary differential equation systems that are contractive with respect to $L^p$ norms remain so when diffusion is added. Thus, diffusive instabilities, in the sense of the Turing phenomenon, cannot arise for such systems, and in fact any two solutions converge exponentially to each other. The key tools are semi-inner products and logarithmic Lipschitz constants in Banach spaces. An example from biochemistry is discussed, which shows the necessity of considering non-Hilbert spaces. An analogous result for graph-defined interconnections of systems defined by ordinary differential equations is given as well. |
Complex networks of biochemical reactions, such as intracellular protein signaling pathways and genetic networks, are often conceptualized in terms of ``modules,'' semi-independent collections of components that perform a well-defined function and which may be incorporated in multiple pathways. However, due to sequestration of molecular messengers during interactions and other effects, collectively referred to as retroactivity, real biochemical systems do not exhibit perfect modularity. Biochemical signaling pathways can be insulated from impedance and competition effects, which inhibit modularity, through enzymatic ``futile cycles'' which consume energy, typically in the form of ATP. We hypothesize that better insulation necessarily requires higher energy consumption. We test this hypothesis through a combined theoretical and computational analysis of a simplified physical model of covalent cycles, using two innovative measures of insulation, as well as a new way to characterize optimal insulation through the balancing of these two measures in a Pareto sense. Our results indicate that indeed better insulation requires more energy. While insulation may facilitate evolution by enabling a modular ``plug and play'' interconnection architecture, allowing for the creation of new behaviors by adding targets to existing pathways, our work suggests that this potential benefit must be balanced against the metabolic costs of insulation necessarily incurred in not affecting the behavior of existing processes. |
The chemotaxis pathway of the bacterium Rhodobacter sphaeroides has many similarities to that of Escherichia coli. It exhibits robust adaptation and has several homologues of the latter's chemotaxis proteins. Recent theoretical results have correctly predicted that, in response to a scaling of its ligand input signal, Escherichia coli exhibits the same output behavior, a property known as fold-change detection (FCD). In light of recent experimental results suggesting that R. sphaeroides may also show FCD, we present theoretical assumptions on the R. sphaeroides chemosensory dynamics that can be shown to yield FCD behavior. Furthermore, it is shown that these assumptions make FCD a property of this system that is robust to structural and parametric variations in the chemotaxis pathway, in agreement with experimental results. We construct and examine models of the full chemotaxis pathway that satisfy these assumptions and reproduce experimental time-series data from earlier studies. We then propose experiments in which models satisfying our theoretical assumptions predict robust FCD behavior where earlier models do not. In this way, we illustrate how transient dynamic phenotypes such as FCD can be used for the purposes of discriminating between models that reproduce the same experimental time-series data. |
This work introduces an experimental platform customized for the development and verification of reverse engineering and pathway characterization algorithms in mammalian cells. Specifically, we stably integrate a synthetic gene network in human kidney cells and use it as a benchmark for validating reverse engineering methodologies. The network, which is orthogonal to endogenous cellular signaling, contains a small set of regulatory interactions that can be used to quantify the reconstruction performance. By performing successive perturbations to each modular component of the network and comparing protein and RNA measurements, we study the conditions under which we can reliably reconstruct the causal relationships of the integrated synthetic network. |
This paper shows that metastatic breast cancer cells cooperatively invade a 3D collagen matrix while following a glucose gradient. The front cell leadership is dynamic, and invading cells act in a cooperative manner by exchanging leaders in the invading front. |
This paper studies networks of components, and shows that a contraction property on the interconnection matrix, coupled with contractivity of the individual component subsystems, suffices to insure contractivity of the overall system. |
Using synthetic circuits stably integrated in human kidney cells, we study the effect of negative feedback regulation on cell-wide (extrinsic) and gene-specific (intrinsic) sources of uncertainty. We develop a theoretical approach to extract the two noise components from experiments and show that negative feedback reduces extrinsic noise while marginally increasing intrinsic noise, resulting to significant total noise reduction. We compare the results to simple negative regulation, where a constitutively transcribed transcription factor represses a reporter protein. We observe that the control architecture also reduces the extrinsic noise but results in substantially higher intrinsic fluctuations. We conclude that negative feedback is the most efficient way to mitigate the effects of extrinsic fluctuations by a sole regulatory wiring. |
Conference articles |
Recent experimental work has shown that transient E. coli chemotactic response is unchanged by a scaling of its ligand input signal (fold change detection, or FCD), and this is in agreement with earlier mathematical predictions. However, this prediction was based on certain particular assumptions on the structure of the chemotaxis pathway. In this work, we begin by showing that behavior similar to FCD can be obtained under weaker conditions on the system structure. Namely, we show that under relaxed conditions, a scaling of the chemotaxis system's inputs leads to a time scaling of the output response. We propose that this may be a contributing factor to the robustness of the experimentally observed FCD. We further show that FCD is a special case of this time scaling behavior for which the time scaling factor is unity. We then proceed to extend the conditions for output time scaling to more general adapting systems, and demonstrate this time scaling behavior on a published model of the chemotaxis pathway of the bacterium Rhodobacter sphaeroides. This work therefore provides examples of how robust biological behavior can arise from simple yet realistic conditions on the underlying system structure. |
We present conditions that guarantee spatial uniformity in diffusively-coupled systems. Diffusive coupling is a ubiquitous form of local interaction, arising in diverse areas including multiagent coordination and pattern formation in biochemical networks. The conditions we derive make use of the Jacobian matrix and Neumann eigenvalues of elliptic operators, and generalize and unify existing theory about asymptotic convergence of trajectories of reaction-diffusion partial differential equations as well as compartmental ordinary differential equations. We present numerical tests making use of linear matrix inequalities that may be used to certify these conditions. We discuss an example pertaining to electromechanical oscillators. The paper's main contributions are unified verifiable relaxed conditions that guarantee synchrony. |
Internal reports |
This note works out an advection-diffusion approximation to the density of a population of E. coli bacteria undergoing chemotaxis in a one-dimensional space. Simulations show the high quality of predictions under a shallow-gradient regime. |
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. |
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