Aivar Sootla

Shaping pulses to control bistable 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.

Properties of isostables and basins of attraction of monotone systems

In this paper, we investigate geometric properties of monotone systems by studying their isostables and basins of attraction. Isostables are boundaries of specific forward-invariant sets defined by the so-called Koopman operator, which provides a linear infinite-dimensional description of a nonlinear system. First, we study the spectral properties of the Koopman operator and the associated semigroup in the context of monotone systems. Our results generalize the celebrated Perron-Frobenius theorem to the nonlinear case and allow us to derive geometric properties of isostables and basins of attraction. Additionally, we show that under certain conditions we can characterize the bounds on the basins of attraction under parametric uncertainty in the vector field. We discuss computational approaches to estimate isostables and basins of attraction and illustrate the results on two and four state monotone systems.

Structured Projection-Based Model Reduction With Application to Stochastic Biochemical Networks

The chemical master equation (CME) is well known to provide the highest resolution models of a biochemical reaction network. Unfortunately, even simulating the CME can be a challenging task. For this reason, simpler approximations to the CME have been proposed. In this paper, we focus on one such model, the linear noise approximation (LNA). Specifically, we consider implications of a recently proposed LNA time-scale separation method. We show that the reduced-order LNA converges to the full-order model in the mean square sense. Using this as motivation, we derive a network structure-preserving reduction algorithm based on structured projections. We discuss when these structured projections exist and we present convex optimization algorithms that describe how such projections can be computed. The algorithms are then applied to a linearized stochastic LNA model of the yeast glycolysis pathway.

Shaping pulses to control bistable biological systems

In this paper, we present a framework for shaping pulses to control biological systems, and specifically systems in synthetic biology. By shaping we mean computing the magnitude and the length of a pulse, application of which results in reaching the desired control objective. Hence the control signals have only two parameters, which makes these signals amenable to wetlab implementations. We focus on the problem of switching between steady states in a bistable system. We show how to estimate the set of the switching pulses, if the trajectories of the controlled system can be bounded from above and below by the trajectories of monotone systems. We then generalise this result to systems with parametric uncertainty under some mild assumptions on the set of admissible parameters, thus providing some robustness guarantees. We illustrate the results on some example genetic circuits.

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. Order propagation has received a considerable attention in the literature, however, this notion is yet to be fully understood. The main contribution of this paper is a study of order propagation in the deterministic setting, which potentially can provide new techniques for analysis in the stochastic one. We take a close look at 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 propagation of the increasing order is equivalent to classical monotonicity, while the class of systems propagating the increasing convex order is equal 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.

An optimal control formulation of pulse-based control using Koopman operator

Abstract In many applications, and in systems/synthetic biology, in particular, it is desirable to solve the switching problem, i.e., to compute control policies that force the trajectory of a bistable system from one equilibrium (the initial point) to another equilibrium (the target point). It was recently shown that for monotone bistable systems, this problem admits easy-to-implement open-loop solutions in terms of temporal pulses (i.e., step functions of fixed length and fixed magnitude). In this paper, we develop this idea further and formulate a problem of convergence to an equilibrium from an arbitrary initial point. We show that the convergence problem can be solved using a static optimization problem in the case of monotone systems. Changing the initial point to an arbitrary state allows building closed-loop, event-based or open-loop policies for the switching/convergence problems. In our derivations, we exploit the Koopman operator, which offers a linear infinite-dimensional representation of an autonomous nonlinear system and powerful computational tools for their analysis. Our solutions to the switching/convergence problems can serve as building blocks for other control problems and can potentially be applied to non-monotone systems. We illustrate this argument on the problem of synchronizing cardiac cells by defibrillation.

Pulse-Based Control Using Koopman Operator Under Parametric Uncertainty

In applications, such as biomedicine and systems/synthetic biology, technical limitations in actuation complicate implementation of time-varying control signals. In order to alleviate some of these limitations, it may be desirable to derive simple control policies, such as step functions with fixed magnitude and length (or temporal pulses). In this technical note, we further develop a recently proposed pulse-based solution to the convergence problem, i.e., minimizing the convergence time to the target exponentially stable equilibrium, for monotone systems. In particular, we extend this solution to monotone systems with parametric uncertainty. Our solutions also provide worst case estimates on convergence times. Furthermore, we indicate how our tools can be used for a class of nonmonotone systems, and more importantly how these tools can be extended to other control problems. We illustrate our approach on switching under parametric uncertainty and regulation around a saddle point problems in a genetic toggle switch system.

On optimisation programmes with hidden convexity

The main result of this paper is a method for finding hidden convexity in some smooth nonconvex programmes. Specifically, the method is identifying an equivalent convex formulation of the underlying nonconvex programme. The main idea of our method is to view a constrained optimisation programme as a control system, where the dual variable plays a role of a control signal. Therefore the existence of a global stabilising feedback controller implies the existence of an equivalent convex optimisation programme. In detail, the case of programmes with linear constraints is considered, for which sufficient conditions for finding hidden convexity are derived. If these sufficient conditions are satisfied an equivalent convex formulation can be obtained by using control-theoretic tools without a considerable computational cost. The whole procedure can be seen as a generalisation of the augmented Lagrangian method. This observation allows to obtain a control-theoretic interpretation of the augmented Lagrangian method, and an extension to incorporate linear inequality constraints.