Fatihcan Atay (Leipzig), Tobias Galla (Manchester), Axel Hutt (Villers-lès-Nancy), Andre S. Ribeiro (Tampere), Martin-Luc Rosinberg (Paris), Raúl Toral (Palma de Mallorca), Jan Wehr (Tucson).

Time delays usually lead to oscillations in dynamical systems. We will introduce a simple time-delayed random walk which illustrates the mechanism of such a behavior. However, the occurence of oscillations may depend upon origins of time delays as it can be seen in some evolutionary games and gene expression models. We will briefly review models where time delays shift stationary states and those where stochastic stability of stationary states depends on time delays.

Fatihcan Atay (Leipzig): Consensus and Synchronization Problems in Networks with Time Delays. pdf

This talk will consider the network consensus problem as a special case of synchronization. Although the problem is a deterministic one, I will indicate some close relations to Markov processes, and specifically to random walks on graphs. I will then introduce time delays into the models and study their effects on network dynamics. I will consider both discrete and distributed delays, while distinguishing between signal propagation and signal processing delays.

Tobias Galla (Manchester): Stochastic processes with delays and their application to gene regulation and epidemics.

Many of the systems modelled in biology have memory, examples are translational or transcriptional delays in gene regulation, or recovery periods in the context of infectious diseases. At the same time such systems are often composed of a finite number of particles, subject to discrete reaction events. We here focus on the mathematical description of such non-Markovian systems with intrinsic stochasticity. Master equations cannot be formulated straightforwardly and it is not clear how to derive systematic Gaussian approximations. We demonstrate that progress can be made using a path-focused view, based on generating functionals. These do not describe the time-evolution of one-time probability distributions, instead they capture the probabilities of entire paths. We derive analytical expressions for Gaussian approximations for a wide class of delay systems, and apply these to two biological problems in which delay is relevant. One is the susceptible-infective-recovered model in epidemiology and the other a model of delayed autoinhibition in gene regulation. This allows us to characterise the phenomena arising from the combination of intrinsic noise and delayed dynamics.
T. Brett, T. Galla, Phys. Rev. Lett. 110, 250601 (2013) ; T. Galla, Phys. Rev. E 80, 021909 (2009)

Axel Hutt (Villers-lès-Nancy): Additive noise tunes the stability in nonlinear systems.

The talk motivates the study of additive noise by a numerical result on the stochastic Swift-Hohenberg equation revealing a shift of stability by additive noise. The corresponding analytical study based on stochastic center manifolds elucidates the underlying mechanism. A subsequent center manifold study of delayed scalar stochastic equations introduces the similar analysis in delayed systems revealing the same stabilizing and underlying mechanism.

Andre S. Ribeiro (Tampere): Delays as regulators of the dynamics of genetic circuits.

Time delays are a key component of the dynamics of genetic circuits. Relevantly, many of these delays have a kinetics which is, to a great extent, sequence dependent and, thus, evolvable. Here, we review recent findings on the role that time delays play in cellular processes. First, we address the effects of delays that take place in transcription initiation, along with the most recent measurements of the mean and variance of the duration of these events in live, individual cells. We further address the ability that regulatory molecules of transcription have to affect both mean and fluctuations of these delays. Next, we address the effects of sequence-dependent delays and how they may be used to regulate RNA and protein production kinetics. Also addressed is the delay in the intake of regulatory molecules and its effects on the cell to cell diversity in gene activity. Finally, we consider recent findings on the potential effects of the aforementioned delays on the dynamics of genetic circuits and, consequently, on cellular phenotypes and phenotypic diversity.

Martin-Luc Rosinberg (Paris): Entropy production and fluctuation theorems in stochastic systems with time delay. pdf

Stochastic thermodynamics has emerged over the last decade as a general theoretical framework that extends the central concepts of thermodynamics (work, heat, entropy) to the level of single stochastic trajectories [1]. It is aimed at studying the nonequilibrium behavior of small systems (colloidal particles, macromolecules, nanomechanical oscillators, etc...) subjected to large and measurable fluctuations. The so-called fluctuation theorems play a key role in this context, and the second law of thermodynamics for Markov systems is now understood as resulting from a universal identity that relates the entropy production to the probability of observing a trajectory and its time reversal. Recent developments, at the crossroad between statistical physics and information theory, have generalized this framework to stochastic systems operating under feedback control [2,3]. In this talk, I will present ongoing work [4] that focuses on Langevin systems submitted to a continuous, non-Markovian feedback control. The non-Markovian character results from a time lag between the input and output signals, which is a common feature in many biological and artificial systems. As a consequence, the feedback-controlled systems settle into a nonequilibrium steady state where entropy is permanently produced (and cooling or heating is achieved depending on the delay). We show that non-Markovianity gives rise to a specific contribution to the entropy production due to an apparent breaking of causality when time is reversed. This so-called "Jacobian effect" is illustrated by detailed analytical and numerical calculations for linear delay Langevin equations with additive Gaussian white noise. In particular, we show that a comprehensive path integral description of the steady-state behavior is available in the overdamped limit of the stochastic motion.
[1] See e.g. U. Seifert, Rep. Prog. Phys. 75, 126001 (2012) and references therein. [2] T. Sagawa and M. Ueda, Phys. Rev. E 85, 021104 (2012). [3] T. Munakata and M. L. Rosinberg, J. Stat. Mech. P05010 (2012); ibid P06014 ( 2013). [4] T. Munakata, M.L. Rosinberg, and G. Tarjus (in preparation).

Raúl Toral (Palma de Mallorca): Stochastic Description of Systems with Delay: Applications to Models of Protein Dynamics. pdf

The combined effects of delay and stochasticity are not completely understood, despite the fact that both effects appear simultaneously in a large variety of processes of relevance in many areas of science, such as physics, ecology or chemistry. From the mathematical point of view, stochastic processes including delay are difficult to analyze due to their non-Markovian character. Most of the previous approaches have focused on stochastic differential equations or random walks in discrete time. However the consideration of discrete variables and continuous time are the natural description of many systems such as gene regulations, chemical reactions, population dynamics or epidemic spreading, in which discreteness can be a major source of fluctuations. In this talk, I will introduce some simple, yet general, stochastic birth and death processes including delay and will discuss some of the inherent difficulties for their study. Most often, the delay time is taken to be a constant with zero fluctuations, a non very realistic assumption for the applications, since it is unusual to have a deterministic delay when the rest of the dynamics is stochastic. I will take this consideration into account by allowing the delay times to be random variables with arbitrary probability density functions. I will then present different approaches that have been developed for their analytical treatment, allowing us to derive effective Markovian models that incorporate most of the features of the delay. We apply the methodology to a protein-dynamics model that explicitly includes transcription and translation delays. In the case of delay in the degradation we rigorously derive the master equation for the processes and solve it exactly. We show that the equations for the mean values obtained differ from others intuitively proposed and that oscillatory behavior is not possible in this system. We discuss the calculation of correlation functions in stochastic systems with delay, stressing the differences with Markovian processes. The exact results allow us to clarify the interplay between stochasticity and delay.
Work in collaboration with Luis F. Lafuerza: -Role of delay in the stochastic creation process, Physical Review E84, 021128 (2011). - Exact solution of a stochastic protein dynamics model with delayed degradation, Physical Review E84, 051121 (2011). -Stochastic description of delayed systems, Philosophical Transactions A371, 20120458 (2013).

Janek Wehr (Tucson): Stochastic delay equations with colored noise --- theory and experiment.

In a recent experiment it was shown that a noisy electrical circuit exhibits an Ito-to-Stratonovich transition as some parameters are varied. I will explain this effect by studying a associated system of time-delayed stochastic differential equations. The results were obtained jointly with experimental physicists G. Pesce and G. Volpe, and with mathematics graduate students S. Hottovy and A. McDaniel. They were published in a recent issue (November 12) of Nature Communications.