arXiv subjects
Publications and source records attributed to Eduardo D. Sontag.
This is a short note with some (very elementary) remarks concerning realization theory and the choice of inputs for identifying systems. Its goal is to bring some of these facts to the attention of the systems biology community.
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.
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.
We show that strongly monotone systems of ordinary differential equations which have a certain translation-invariance property are so that all solutions converge to a unique equilibrium. The result may be seen as a dual of a well-known theorem of Mierczynski for systems that satisfy a conservation law. An application to a reaction of interest in biochemistry is provided as an illustration.
The fundamental property of strongly monotone systems, and strongly cooperative systems in particular, is the limit set dichotomy due to Hirsch: if x(0) < y(0), 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.
This note remarks that small-gain results for a negative feedback loop around a monotone system can be seen as consequences of results concerning an extended monotone system.
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.
Interactions between genes and gene products give rise to complex circuits that enable cells to process information and respond to external signals. Theoretical studies often describe these interactions using continuous, stochastic, or logical approaches. We propose a new modeling framework for gene regulatory networks, that combines the intuitive appeal of a qualitative description of gene states with a high flexibility in incorporating stochasticity in the duration of cellular processes. We apply our methods to the regulatory network of the segment polarity genes, thus gaining novel insights into the development of gene expression patterns. For example, we show that very short synthesis and decay times can perturb the wild type pattern. On the other hand, separation of timescales between pre- and posttranslational processes and a minimal prepattern ensure convergence to the wild type expression pattern regardless of fluctuations.
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.
In this note, we show how certain properties of Goldbeter's 1995 model for circadian oscillations can be proved mathematically, using techniques from the recently developed theory of monotone systems with inputs and outputs. The theory establishes global asymptotic stability, and in particular no oscillations, if the rate of transcription is somewhat smaller than that assumed by Goldbeter. This stability persists even under arbitrary delays in the feedback loop.
Weakly activated signaling cascades can be modeled as linear systems. The input-to-output transfer function and the internal gain of a linear system, provide natural measures for the propagation of the input signal down the cascade and for the characterization of the final outcome. The most efficient design of a cascade for generating sharp signals, is obtained by choosing all the off rates equal, and a ``universal'' finite optimal length.
Monotone systems constitute one of the most important classes of dynamical systems used in mathematical biology modeling. The objective of this paper is to extend the notion of monotonicity to systems with inputs and outputs, a necessary first step in trying to understand interconnections, especially including feedback loops, built up out of monotone components. Basic definitions and theorems are provided, as well as an application of a theorem regarding negative feedback loops to the study of a model of one of the cell's most important subsystems (MAPK cascades) .
This paper studies the emergence of multi-stability and hysteresis in those systems that arise, under positive feedback, starting from monotone systems with well-defined steady-state responses. Such feedback configurations appear routinely in several fields of application, and especially in biology. Characterizations of global stability behavior are stated in terms of easily checkable graphical conditions. An example of a signaling cascade under positive feedback is presented.
This note provides a theorem showing, under suitable technical assumptions, that if a system S adapts to a class of external signals U, in the sense of egulation against disturbances or tracking signals in U, then S must ecessarily contain a subsystem which is capable of generating all the signals in U. It is not assumed that regulation is robust, nor is there a prior requirement for the system to be partitioned into separate plant and controller components. Instead, one assumes that a ``signal detection'' property holds. The result was motivated by questions of adaptation in bacterial chemotaxis, but the general mathematical principle is of wide applicability.
We extend the setup in our previous paper to deal with the case in which more than one steady state may exist in feedback configurations. This provides a foundation for the analysis of multi-stability and hysteresis behaviour in high dimensional feedback systems.
The fundamental Filippov-Wazwski Relaxation Theorem states that the solution set of an initial value problem for a locally Lipschitz inclusion is dense in the solution set of the same initial value problem for the corresponding relaxation inclusion on compact intervals. In our recent work, a complementary result was provided for inclusions with finite dimensional state spaces which says that the approximation can be carried out over non-compact or infinite intervals provided one does not insist on the same initial values. This note extends the infinite-time relaxation theorem to the inclusions whose state spaces are Banach spaces. To illustrate the motivations for studying such approximation results, we briefly discuss a quick application of the result to output stability and uniform output stability properties.
Monotone systems constitute one of the most important classes of dynamical systems used in mathematical biology modeling. The objective of this paper is to extend the notion of monotonicity to systems with inputs and outputs, a necessary first step in trying to understand interconnections, especially including feedback loops, built up out of monotone components. Basic definitions and theorems are provided, as well as an application to the study of a model of one of the cell's most important subsystems.
Suppose that an equilibrium is asymptotically stable when external inputs vanish. Then, every bounded trajectory which corresponds to a control which approaches zero and which lies in the domain of attraction of the unforced system, must also converge to the equilibrium. This "well-known" but hard-to-cite fact is proved and slightly generalized here.