On Monotonicity and Propagation of Order Properties

In this paper, a link between monotonicity of deterministic dynamical systems and propagation of order by Markov processes is established. The order propagation has received considerable attention in the literature, however, this notion is still not fully understood. The main contribution of this paper is a study of the order propagation in the deterministic setting, which potentially can provide new techniques for analysis in the stochastic one. We take a close look at the propagation of the so-called increasing and increasing convex orders. Infinitesimal characterisations of these orders are derived, which resemble the well-known Kamke conditions for monotonicity. It is shown that increasing order is equivalent to the standard monotonicity, while the class of systems propagating the increasing convex order is equivalent to the class of monotone systems with convex vector fields. The paper is concluded by deriving a novel result on order propagating diffusion processes and an application of this result to biological processes.Comment: Part of the paper is to appear in American Control Conference 201

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.Comment: Related work can be retrieved from second author's websit

Shaping Pulses to Control Bistable Biological Systems

In this paper we study how to shape temporal pulses to switch a bistable system between its stable steady states. Our motivation for pulse-based control comes from applications in synthetic biology, where it is generally difficult to implement real-time feedback control systems due to technical limitations in sensors and actuators. We show that for monotone bistable systems, the estimation of the set of all pulses that switch the system reduces to the computation of one non-increasing curve. We provide an efficient algorithm to compute this curve and illustrate the results with a genetic bistable system commonly used in synthetic biology. We also extend these results to models with parametric uncertainty and provide a number of examples and counterexamples that demonstrate the power and limitations of the current theory. In order to show the full potential of the framework, we consider the problem of inducing oscillations in a monotone biochemical system using a combination of temporal pulses and event-based control. Our results provide an insight into the dynamics of bistable systems under external inputs and open up numerous directions for future investigation.Comment: 14 pages, contains material from the paper in Proc Amer Control Conf 2015, (pp. 3138-3143) and "Shaping pulses to control bistable systems analysis, computation and counterexamples", which is due to appear in Automatic