Robust Moment Closure Method for the Chemical Master Equation

The Chemical Master Equation (CME) is used to stochastically model biochemical reaction networks, under the Markovian assumption. The low-order statistical moments induced by the CME are often the key quantities that one is interested in. However, in most cases, the moments equation is not closed; in the sense that the first $n$ moments depend on the higher order moments, for any positive integer $n$. In this paper, we develop a moment closure technique in which the higher order moments are approximated by an affine function of the lower order moments. We refer to such functions as the affine Moment Closure Functions (MCF) and prove that they are optimal in the worst-case context, in which no a priori information on the probability distribution is available. Furthermore, we cast the problem of finding the optimal affine MCF as a linear program, which is tractable. We utilize the affine MCFs to derive a finite dimensional linear system that approximates the low-order moments. We quantify the approximation error in terms of the $$ induced norm of some linear system. Our results can be effectively used to approximate the low-order moments and characterize the noise properties of the biochemical network under study

Signaling architectures that transmit unidirectional information

Submitted for review.A signaling pathway transmits information from an upstream system to downstream systems, ideally unidirectionally. A key bottleneck to unidirectional transmission is retroactivity, which is the additional reaction flux that affects a system once its species interact with those of downstream systems. This raises the question of whether signaling pathways have developed specialized architectures that overcome retroactivity and transmit unidirectional signals. Here, we propose a general mathematical framework that provides an answer to this question. Using this framework, we analyze the ability of a variety of signaling architectures to transmit signals unidirectionally as key biological parameters are tuned. In particular, we find that single stage phosphorylation and phosphotransfer systems that transmit signals from a kinase show the following trade-off: either they impart a large retroactivity to their upstream system or they are significantly impacted by the retroactivity due to their downstream system. However, cascades of these architectures, which are highly represented in nature, can overcome this trade-off and thus enable unidirectional information transmission. By contrast, single and double phosphorylation cycles that transmit signals from a substrate impart a large retroactivity to their upstream system and are also unable to attenuate retroactivity due to their downstream system. Our findings identify signaling architectures that ensure unidirectional signal transmission and minimize crosstalk among multiple targets. Our results thus establish a way to decompose a signal transduction network into architectures that transmit information unidirectionally, while also providing a library of devices that can be used in synthetic biology to facilitate modular circuit design.NSF Expedition award number 1521925, NIGMS grant P50 GMO9879

Limitations and trade-offs in gene expression due to competition for shared cellular resources

Gene circuits share transcriptional and translational resources in the cell. The fact that these common resources are available only in limited amounts leads to unexpected couplings in protein expressions. As a result, our predictive ability of describing the behavior of gene circuits is limited. In this paper, we consider the simultaneous expression of proteins and describe the coupling among protein concentrations due to competition for RNA polymerase and ribosomes. In particular, we identify the limitations and trade-offs in gene expression by characterizing the attainable combinations of protein concentrations. We further present two application examples of our results: we show that even in the absence of regulatory linkages, genes can seemingly behave as repressors, and surprisingly, as activators to each other, purely due to the limited availability of shared cellular resources.United States. Air Force Office of Scientific Research (Grant FA9550-12-1-0129)National Institute of General Medical Sciences (U.S.) (Grant P50 GM098792

Effective interaction graphs arising from resource limitations in gene networks

Protein production in gene networks relies on the availability of resources necessary for transcription and translation, which are found in cells in limited amounts. As various genes in a network compete for a common pool of resources, a hidden layer of interactions among genes arises. Such interactions are neglected by standard Hill-function-based models. In this work, we develop a model with the same dimension as standard Hill-function-based models to account for the sharing of limited amounts of RNA polymerase and ribosomes in gene networks. We provide effective interaction graphs to capture the hidden interactions and find that the additional interactions can dramatically change network behavior. In particular, we demonstrate that, as a result of resource limitations, a cascade of activators can behave like an effective repressor or a biphasic system, and that a repression cascade can become bistable.United States. Air Force Office of Scientific Research (FA9550-12-1-0129)National Institute of General Medical Sciences (U.S.) (P50 GMO98792

Mitigation of ribosome competition through distributed sRNA feedback (extended version)

This paper is an extended version of a paper of the same title accepted to Proceedings of the 55th IEEE Conference on Decision and Control (2016).A current challenge in the robust engineering of synthetic gene networks is context dependence, the unintended interactions among genes and host factors. Ribosome competition is a specific form of context dependence, where all genes in the network compete for a limited pool of translational resources available for gene expression. Recently, theoretical and experimental studies have shown that ribosome competition creates a hidden layer of interactions among genes, which largely hinders our ability to predict design outcomes. In this work, we establish a control theoretic framework, where these hidden interactions become disturbance signals. We then propose a distributed feedback mechanism to achieve disturbance decoupling in the network. The feedback loop at each node consists of the protein product transcriptionally activating a small RNA (sRNA), which forms a translationally inactive complex with mRNA rapidly. We illustrate that with this feedback mechanism, protein production at each node is only dependent on its own transcription factor inputs, and almost independent of hidden interactions arising from ribosome competition.AFOSR grant FA9550-12-1-0129 and ONR grant N00014131007

Retroactivity Attenuation in Bio-Molecular Systems Based on Timescale Separation

As with several engineering systems, bio-molecular systems display impedance-like effects at interconnections, called retroactivity. In this paper, we propose a mechanism that exploits the natural timescale separation present in bio-molecular systems to attenuate retroactivity. Retroactivity enters the dynamics of a bio-molecular system as a state dependent disturbance multiplied by gains that can be very large. By virtue of the system structure, retroactivity can be arbitrarily attenuated by internal system gains even when these are much smaller than the gains multiplying retroactivity terms. This result is obtained by employing a suitable change of coordinates and a nested application of the singular perturbation theorem on the finite time interval. As an application example, we show that two modules extracted from natural signal transduction pathways have a remarkable capability of attenuating retroactivity, which is certainly desirable in any (engineered or natural) signal transmission system.United States. Air Force Office of Scientific Research (AFOSR Award FA9550-09-1-0211

Control for Safety Specifications of Systems With Imperfect Information on a Partial Order

In this paper, we consider the control problem for uncertain systems with imperfect information, in which an output of interest must be kept outside an undesired region (the bad set) in the output space. The state, input, output, and disturbance spaces are equipped with partial orders. The system dynamics are either input/output order preserving with output in R[superscript 2] or given by the parallel composition of input/output order preserving dynamics each with scalar output. We provide necessary and sufficient conditions under which an initial set of possible system states is safe, that is, the corresponding outputs are steerable away from the bad set with open loop controls. A closed loop control strategy is explicitly constructed, which guarantees that the current set of possible system states, as obtained from an estimator, generates outputs that never enter the bad set. The complexity of algorithms that check safety of an initial set of states and implement the control map is quadratic with the dimension of the state space. The algorithms are illustrated on two application examples: a ship maneuver to avoid an obstacle and safe navigation of an helicopter among buildings.National Science Foundation (U.S.) (CAREER Award CNS-0642719

Safety control of piece-wise continuous order preserving systems

This paper is concerned with safety control of systems with imperfect state information and disturbance input. Specifically, we consider the class of systems whose dynamics preserve a partial ordering. We provide necessary and sufficient conditions under which a given set of initial states is steerable away from a specified bad set. Moreover, a control strategy is provided that guarantees that the bad set is avoided. Such characterization is achieved for order preserving systems while for general systems only an approximated solution is achievable. A method for implementation of the control strategy is provided and the effectiveness of the proposed method is illustrated via a numerical example and employed for obstacle avoidance of a ship.National Science Foundation (U.S.) (NSF CAREER AWARD # CNS-0642719