Pulsed Feedback Defers Cellular Differentiation

Environmental signals induce diverse cellular differentiation programs. In certain systems, cells defer differentiation for extended time periods after the signal appears, proliferating through multiple rounds of cell division before committing to a new fate. How can cells set a deferral time much longer than the cell cycle? Here we study Bacillus subtilis cells that respond to sudden nutrient limitation with multiple rounds of growth and division before differentiating into spores. A well-characterized genetic circuit controls the concentration and phosphorylation of the master regulator Spo0A, which rises to a critical concentration to initiate sporulation. However, it remains unclear how this circuit enables cells to defer sporulation for multiple cell cycles. Using quantitative time-lapse fluorescence microscopy of Spo0A dynamics in individual cells, we observed pulses of Spo0A phosphorylation at a characteristic cell cycle phase. Pulse amplitudes grew systematically and cell-autonomously over multiple cell cycles leading up to sporulation. This pulse growth required a key positive feedback loop involving the sporulation kinases, without which the deferral of sporulation became ultrasensitive to kinase expression. Thus, deferral is controlled by a pulsed positive feedback loop in which kinase expression is activated by pulses of Spo0A phosphorylation. This pulsed positive feedback architecture provides a more robust mechanism for setting deferral times than constitutive kinase expression. Finally, using mathematical modeling, we show how pulsing and time delays together enable “polyphasic” positive feedback, in which different parts of a feedback loop are active at different times. Polyphasic feedback can enable more accurate tuning of long deferral times. Together, these results suggest that Bacillus subtilis uses a pulsed positive feedback loop to implement a “timer” that operates over timescales much longer than a cell cycle

Discrete time piecewise affine models of genetic regulatory networks

We introduce simple models of genetic regulatory networks and we proceed to the mathematical analysis of their dynamics. The models are discrete time dynamical systems generated by piecewise affine contracting mappings whose variables represent gene expression levels. When compared to other models of regulatory networks, these models have an additional parameter which is identified as quantifying interaction delays. In spite of their simplicity, their dynamics presents a rich variety of behaviours. This phenomenology is not limited to piecewise affine model but extends to smooth nonlinear discrete time models of regulatory networks. In a first step, our analysis concerns general properties of networks on arbitrary graphs (characterisation of the attractor, symbolic dynamics, Lyapunov stability, structural stability, symmetries, etc). In a second step, focus is made on simple circuits for which the attractor and its changes with parameters are described. In the negative circuit of 2 genes, a thorough study is presented which concern stable (quasi-)periodic oscillations governed by rotations on the unit circle -- with a rotation number depending continuously and monotonically on threshold parameters. These regular oscillations exist in negative circuits with arbitrary number of genes where they are most likely to be observed in genetic systems with non-negligible delay effects.Comment: 34 page

On delayed genetic regulatory networks with polytopic uncertainties: Robust stability analysis

Copyright [2008] IEEE. This material is posted here with permission of the IEEE. Such permission of the IEEE does not in any way imply IEEE endorsement of any of Brunel University's products or services. Internal or personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution must be obtained from the IEEE by writing to [email protected]. By choosing to view this document, you agree to all provisions of the copyright laws protecting it.In this paper, we investigate the robust asymptotic stability problem of genetic regulatory networks with time-varying delays and polytopic parameter uncertainties. Both cases of differentiable and nondifferentiable time-delays are considered, and the convex polytopic description is utilized to characterize the genetic network model uncertainties. By using a Lyapunov functional approach and linear matrix inequality (LMI) techniques, the stability criteria for the uncertain delayed genetic networks are established in the form of LMIs, which can be readily verified by using standard numerical software. An important feature of the results reported here is that all the stability conditions are dependent on the upper and lower bounds of the delays, which is made possible by using up-to-date techniques for achieving delay dependence. Another feature of the results lies in that a novel Lyapunov functional dependent on the uncertain parameters is utilized, which renders the results to be potentially less conservative than those obtained via a fixed Lyapunov functional for the entire uncertainty domain. A genetic network example is employed to illustrate the applicability and usefulness of the developed theoretical results

Robust H∞ feedback control for uncertain stochastic delayed genetic regulatory networks with additive and multiplicative noise

The official published version can found at the link below.Noises are ubiquitous in genetic regulatory networks (GRNs). Gene regulation is inherently a stochastic process because of intrinsic and extrinsic noises that cause kinetic parameter variations and basal rate disturbance. Time delays are usually inevitable due to different biochemical reactions in such GRNs. In this paper, a delayed stochastic model with additive and multiplicative noises is utilized to describe stochastic GRNs. A feedback gene controller design scheme is proposed to guarantee that the GRN is mean-square asymptotically stable with noise attenuation, where the structure of the controllers can be specified according to engineering requirements. By applying control theory and mathematical tools, the analytical solution to the control design problem is given, which helps to provide some insight into synthetic biology and systems biology. The control scheme is employed in a three-gene network to illustrate the applicability and usefulness of the design.This work was funded by Royal Society of the U.K.; Foundation for the Author of National Excellent Doctoral Dissertation of China. Grant Number: 2007E4; Heilongjiang Outstanding Youth Science Fund of China. Grant Number: JC200809; Fok Ying Tung Education Foundation. Grant Number: 111064; International Science and Technology Cooperation Project of China. Grant Number: 2009DFA32050; University of Science and Technology of China Graduate Innovative Foundation

Robust filtering for stochastic genetic regulatory networks with time-varying delay

This is the post print version of the article. The official published version can be obtained from the link - Copyright 2009 Elsevier LtdThis paper addresses the robust filtering problem for a class of linear genetic regulatory networks (GRNs) with stochastic disturbances, parameter uncertainties and time delays. The parameter uncertainties are assumed to reside in a polytopic region, the stochastic disturbance is state-dependent described by a scalar Brownian motion, and the time-varying delays enter into both the translation process and the feedback regulation process. We aim to estimate the true concentrations of mRNA and protein by designing a linear filter such that, for all admissible time delays, stochastic disturbances as well as polytopic uncertainties, the augmented state estimation dynamics is exponentially mean square stable with an expected decay rate. A delay-dependent linear matrix inequality (LMI) approach is first developed to derive sufficient conditions that guarantee the exponential stability of the augmented dynamics, and then the filter gains are parameterized in terms of the solution to a set of LMIs. Note that LMIs can be easily solved by using standard software packages. A simulation example is exploited in order to illustrate the effectiveness of the proposed design procedures.This work was supported in part by the Biotechnology and Biological Sciences Research Council (BBSRC) of the U.K. under Grants BB/C506264/1 and 100/EGM17735, an International Joint Project sponsored by the Royal Society of the U.K., the Research Grants Council of Hong Kong under Grant HKU 7031/06P, the National Natural Science Foundation of China under Grant 60804028, and the Alexander von Humboldt Foundation of Germany

Transcriptional delay stabilizes bistable gene networks

Transcriptional delay can significantly impact the dynamics of gene networks. Here we examine how such delay affects bistable systems. We investigate several stochastic models of bistable gene networks and find that increasing delay dramatically increases the mean residence times near stable states. To explain this, we introduce a non-Markovian, analytically tractable reduced model. The model shows that stabilization is the consequence of an increased number of failed transitions between stable states. Each of the bistable systems that we simulate behaves in this manner

An HIV feedback resistor: auto-regulatory circuit deactivator and noise buffer.

Animal viruses (e.g., lentiviruses and herpesviruses) use transcriptional positive feedback (i.e., transactivation) to regulate their gene expression. But positive-feedback circuits are inherently unstable when turned off, which presents a particular dilemma for latent viruses that lack transcriptional repressor motifs. Here we show that a dissipative feedback resistor, composed of enzymatic interconversion of the transactivator, converts transactivation circuits into excitable systems that generate transient pulses of expression, which decay to zero. We use HIV-1 as a model system and analyze single-cell expression kinetics to explore whether the HIV-1 transactivator of transcription (Tat) uses a resistor to shut off transactivation. The Tat feedback circuit was found to lack bi-stability and Tat self-cooperativity but exhibited a pulse of activity upon transactivation, all in agreement with the feedback resistor model. Guided by a mathematical model, biochemical and genetic perturbation of the suspected Tat feedback resistor altered the circuit's stability and reduced susceptibility to molecular noise, in agreement with model predictions. We propose that the feedback resistor is a necessary, but possibly not sufficient, condition for turning off noisy transactivation circuits lacking a repressor motif (e.g., HIV-1 Tat). Feedback resistors may be a paradigm for examining other auto-regulatory circuits and may inform upon how viral latency is established, maintained, and broken

Delay-Based Controller Design for Continuous-Time and Hybrid Applications

Motivated by the availability of different types of delays in embedded systems and biological circuits, the objective of this work is to study the benefits that delay can provide in simplifying the implementation of controllers for continuous-time systems. Given a continuous-time linear time-invariant (LTI) controller, we propose three methods to approximate this controller arbitrarily precisely by a simple controller composed of delay blocks, a few integrators and possibly a unity feedback. Different problems associated with the approximation procedures, such as finding the optimal number of delay blocks or studying the robustness of the designed controller with respect to delay values, are then investigated. We also study the design of an LTI continuous-time controller satisfying given control objectives whose delay-based implementation needs the least number of delay blocks. A direct application of this work is in the sampled-data control of a real-time embedded system, where the sampling frequency is relatively high and/or the output of the system is sampled irregularly. Based on our results on delay-based controller design, we propose a digital-control scheme that can implement every continuous-time stabilizing (LTI) controller. Unlike a typical sampled-data controller, the hybrid controller introduced here -— consisting of an ideal sampler, a digital controller, a number of modified second-order holds and possibly a unity feedback -— is robust to sampling jitter and can operate at arbitrarily high sampling frequencies without requiring expensive, high-precision computation