Ioannis Lestas

Fundamental limits on the suppression of molecular fluctuations

Negative feedback is common in biological processes and can increase a system’s stability to internal and external perturbations. But at the molecular level, control loops always involve signalling steps with finite rates for random births and deaths of individual molecules. Here we show, by developing mathematical tools that merge control and information theory with physical chemistry, that seemingly mild constraints on these rates place severe limits on the ability to suppress molecular fluctuations. Specifically, the minimum standard deviation in abundances decreases with the quartic root of the number of signalling events, making it extremely expensive to increase accuracy. Our results are formulated in terms of experimental observables, and existing data show that cells use brute force when noise suppression is essential; for example, regulatory genes are transcribed tens of thousands of times per cell cycle. The theory challenges conventional beliefs about biochemical accuracy and presents an approach to the rigorous analysis of poorly characterized biological systems.

Scalable Decentralized Robust Stability Certificates for Networks of Interconnected Heterogeneous Dynamical Systems

We derive scalable decentralized conditions that can guarantee robust stability for networks of linearly interconnected, stable, linear time invariant dynamical systems. Unlike previous results of this kind, we allow for heterogeneous dynamics on arbitrary interconnection topologies (i.e., linear systems on arbitrary underlying graphs). Robust stability of the entire network is guaranteed by satisfying local rules that involve only an agent and its neighboring dynamics; each new agent will introduce only one additional condition of this kind and hence the stability certificates scale with the network size. An application of this theory is given, where robustness analysis of Internet congestion control protocols is carried out in the general case of dynamics at both users and resources without any global bounds on the dynamics

Noise in Gene Regulatory Networks

Life processes in single cells and at the molecular level are inherently stochastic. Quantifying the noise is, however, far from trivial, as a major contribution comes from intrinsic fluctuations, arising from the randomness in the times between discrete jumps. It is shown in this paper how a noise-filtering setup with an operator theoretic interpretation can be relevant for analyzing the intrinsic stochasticity in jump processes described by master equations. Such interpretation naturally exists in linear noise approximations, but it also provides an exact description of the jump process when the transition rates are linear. As an important example, it is shown in this paper how, by addressing the proximity of the underlying dynamics in an appropriate topology, a sequence of coupled birth-death processes, which can be relevant in gene expression, tends to a pure delay; this implies important limitations in noise suppression capabilities. Despite the exactness, in a linear regime, of the analysis of noise in conjunction with the network dynamics, we emphasize in this paper the importance of also analyzing dynamic behavior when transition rates are highly nonlinear; otherwise, steady-state solutions can be misinterpreted. The examples are taken from systems with macroscopic models leading to bistability. It is discussed that bistability in the deterministic mass action kinetics and bimodality in the steady-state solution of the master equation neither always imply one another nor do they necessarily lead to efficient switching behaviours: the underlying dynamics need to be taken into account. Finally, we explore some of these issues in relation to a model of the lac operation.

Heterogeneity and scalability in group agreement protocols: Beyond small gain and passivity approaches

It is shown in this paper how by introducing interconnection symmetries as in bidirectional communication schemes, stability of consensus protocols on arbitrary topologies can be guaranteed in a decentralized and scalable way, despite the presence of higher order heterogeneous dynamics. The analysis is centred round the notion of an S-hull and other related convexifications in the complex plane, that can lead to decentralized certificates by means of their dual interpretation. In the case of linear dynamics the certificates derived include, as special cases, passivity and dissipativity approaches, though a wider class of dynamics is allowed by employing convexification arguments that exploit the interconnection structure. Examples are given of networks comprised of agents with heterogeneous higher order dynamics and also with non-identical input and communication delays. Special cases of the results recover small gain related delay independent stability, but can also lead to less conservative delay dependent conditions that are fully decentralized and can be necessary and sufficient. Time domain interpretations are finally discussed for certain formulations of the problem.

Brief paper: Scalable robust stability for nonsymmetric heterogeneous networks

Scalable decentralized stability certificates in networks i.e. decentralized stability guarantees for an arbitrary interconnection of heterogeneous dynamical systems, are often based on certain symmetry assumptions in the way the systems are interconnected. Such structure simplifies the mathematical analysis significantly, nevertheless a potential pitfall needs to be addressed: the stability proof might offer no robustness guarantees to deviations from protocol symmetry. This is, for example, the case for Internet congestion control stability results for arbitrary networks i.e. the stability proofs break down with an arbitrarily small deviation from protocol symmetry. We propose in the paper conditions that can guarantee scalable robust stability in a nonsymmetric interconnection setting for a class of networks that includes Internet congestion control models and consensus protocols. The certificates derived are decentralized and scale with the degree of nonsymmetry.

Scalability in heterogeneous vehicle platoons

It is known that vehicle platoons exhibit string instability when each vehicle tries to maintain a fixed distance from its predecessor. This can be avoided if sufficiently strong coupling with the leader is employed. If instead each vehicle tracks the average distance form its neighbours, the interconnection can still be ill-conditioned in the sense that the response to disturbances is not uniformly bounded with the size of the platoon. We show in the paper that in a symmetric bidirectional scheme, arbitrarily weak coupling with the leader can make the platoon scalable. In addition, we show that despite the additional feedback in a bidirectional control law, the symmetry of the information flow enables the derivation of local conditions which, if satisfied, guarantee that an arbitrarily long heterogeneous interconnection is robustly stable.

SCALABLE ROBUSTNESS FOR CONSENSUS PROTOCOLS WITH HETEROGENEOUS DYNAMICS

Abstract It is shown in the paper how robustness can be guaranteed for consensus protocols with heterogeneous dynamics in a scalable and decentralized way i.e. by each agent satisfying a test that does not require knowledge of the entire network. Random graph examples illustrate that the proposed certificates are not conservative for classes of large scale networks, despite the heterogeneity of the dynamics, which is a distinctive feature of this work. The conditions hold for symmetric protocols and more conservative stability conditions are given for general nonsymmetric interconnections. Nonlinear extensions in an IQC framework are finally discussed.

On the stability of the Foschini-Miljanic algorithm with time-delays

Many of the distributed power control algorithms for wireless networks in the literature ignore the fact that while the algorithms necessitate communication among users, propagation delays exist in the network. This problem is of vital importance, since propagation delays are omnipresent in wireless networks. The Foschini-Miljanic algorithm is provably stable if there are no time-delays in the execution of the algorithm. However, since the interference measurements are fed back to the transmitter by its corresponding receiver, time-delays are inevitably introduced into the system. This work presents a more realistic version of the well known Foschini-Miljanic algorithm for distributed power control since it considers the time-delays introduced to the system due to propagation delays. In both the continuous and discrete time cases we prove global stability of the system in the presence of propagation delays.

The S-hull approach to consensus

It is shown in the paper how by introducing interconnection symmetries by use of bidirectional communication, stability of consensus protocols on arbitrary topologies can be guaranteed in a decentralized and scalable way, despite the presence of higher order heterogeneous dynamics. The analysis is centred round the notion of an S-hull a relaxed convex hull in the complex plane. In the case of linear dynamics the certificates derived include, as special cases, passivity and dissipativity approaches, though a wider class of dynamics is allowed by employing convexification arguments. As a corollary, it is shown that in the presence of heterogeneous delays in the relative information (in which case stability is delay dependent) a simple scaling can guaranty stability on an arbitrary topology. The condition becomes necessary and sufficient when the delays are homogeneous. Time domain interpretations are finally discussed for certain formulations of the problem.

On the convergence to saddle points of concave-convex functions, the gradient method and emergence of oscillations

It is known that for a strictly concave-convex function, the gradient method introduced by Arrow and Hurwicz [1], has guaranteed global convergence to its saddle point. Nevertheless, there are classes of problems where the function considered is not strictly concave-convex, in which case convergence to a saddle point is not guaranteed. In the paper we provide a characterization of the asymptotic behaviour of the gradient method, in the general case where this is applied to a general concave-convex function. We prove that for any initial conditions the gradient method is guaranteed to converge to a trajectory described by an explicit linear ODE. We further show that this result has a natural extension to subgradient methods, where the dynamics are constrained in a prescribed convex set. The results are used to provide simple characterizations of the limiting solutions for special classes of optimization problems, and modifications of the problem so as to avoid oscillations are also discussed.