Considerations for using integral feedback control to construct a perfectly adapting synthetic gene network

It has long been known to control theorists and engineers that integral feedback control leads to, and is necessary for, “perfect” adaptation to step input perturbations in most systems. Consequently, implementation of this robust control strategy in a synthetic gene network is an attractive prospect. However, the nature of genetic regulatory networks (density-dependent kinetics and molecular signals that easily reach saturation) implies that the design and construction of such a device is not straightforward. In this study, we propose a generic two-promoter genetic regulatory network for the purpose of exhibiting perfect adaptation; our treatment highlights the challenges inherent in the implementation of a genetic integral controller. We also present a numerical case study for a specific realization of this two-promoter network, “constructed” using commonly available parts from the bacterium Escherichia coli. We illustrate the possibility of optimizing this network's transient response via analogy to a linear, free-damped harmonic oscillator. Finally, we discuss extensions of this two-promoter network to a proportional-integral controller and to a three-promoter network capable of perfect adaptation under conditions where first-order protein removal effects would otherwise disrupt the adaptation.     

Coupled positive and negative feedback circuits form an essential building block of cellular signaling pathways

Cellular circuits have positive and negative feedback loops that allow them to respond properly to noisy external stimuli. It is intriguing that such feedback loops exist in many cases in a particular form of coupled positive and negative feedback loops with different time delays. As a result of our mathematical simulations and investigations into various experimental evidences, we found that such coupled feedback circuits can rapidly turn on a reaction to a proper stimulus, robustly maintain its status, and immediately turn off the reaction when the stimulus disappears. In other words, coupled feedback loops enable cellular systems to produce perfect responses to noisy stimuli with respect to signal duration and amplitude. This suggests that coupled positive and negative feedback loops form essential signal transduction motifs in cellular signaling systems.     

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.     

Dynamical compensation in physiological circuits

Biological systems can maintain constant steady‐state output despite variation in biochemical parameters, a property known as exact adaptation. Exact adaptation is achieved using integral feedback, an engineering strategy that ensures that the output of a system robustly tracks its desired value. However, it is unclear how physiological circuits also keep their output dynamics precise—including the amplitude and response time to a changing input. Such robustness is crucial for endocrine and neuronal homeostatic circuits because they need to provide a precise dynamic response in the face of wide variation in the physiological parameters of their target tissues; how such circuits compensate their dynamics for unavoidable natural fluctuations in parameters is unknown. Here, we present a design principle that provides the desired robustness, which we call dynamical compensation (DC). We present a class of circuits that show DC by means of a nonlinear feedback loop in which the regulated variable controls the functional mass of the controlling endocrine or neuronal tissue. This mechanism applies to the control of blood glucose by insulin and explains several experimental observations on insulin resistance. We provide evidence that this mechanism may also explain compensation and organ size control in other physiological circuits.     

The Control of the Controller: Molecular Mechanisms for Robust Perfect Adaptation and Temperature Compensation

Organisms have the property to adapt to a changing environment and keep certain components within a cell regulated at the same level (homeostasis). “Perfect adaptation” describes an organism's response to an external stepwise perturbation by regulating some of its variables/components precisely to their original preperturbation values. Numerous examples of perfect adaptation/homeostasis have been found, as for example, in bacterial chemotaxis, photoreceptor responses, MAP kinase activities, or in metal-ion homeostasis. Two concepts have evolved to explain how perfect adaptation may be understood: In one approach (robust perfect adaptation), the adaptation is a network property, which is mostly, but not entirely, independent of rate constant values; in the other approach (nonrobust perfect adaptation), a fine-tuning of rate constant values is needed. Here we identify two classes of robust molecular homeostatic mechanisms, which compensate for environmental variations in a controlled variable's inflow or outflow fluxes, and allow for the presence of robust temperature compensation. These two classes of homeostatic mechanisms arise due to the fact that concentrations must have positive values. We show that the concept of integral control (or integral feedback), which leads to robust homeostasis, is associated with a control species that has to work under zero-order flux conditions and does not necessarily require the presence of a physico-chemical feedback structure. There are interesting links between the two identified classes of homeostatic mechanisms and molecular mechanisms found in mammalian iron and calcium homeostasis, indicating that homeostatic mechanisms may underlie similar molecular control structures.     

Computational aspects of feedback in neural circuits

It has previously been shown that generic cortical microcircuit models can perform complex real-time computations on continuous input streams, provided that these computations can be carried out with a rapidly fading memory. We investigate the computational capability of such circuits in the more realistic case where not only readout neurons, but in addition a few neurons within the circuit, have been trained for specific tasks. This is essentially equivalent to the case where the output of trained readout neurons is fed back into the circuit. We show that this new model overcomes the limitation of a rapidly fading memory. In fact, we prove that in the idealized case without noise it can carry out any conceivable digital or analog computation on time-varying inputs. But even with noise, the resulting computational model can perform a large class of biologically relevant real-time computations that require a nonfading memory. We demonstrate these computational implications of feedback both theoretically, and through computer simulations of detailed cortical microcircuit models that are subject to noise and have complex inherent dynamics. We show that the application of simple learning procedures (such as linear regression or perceptron learning) to a few neurons enables such circuits to represent time over behaviorally relevant long time spans, to integrate evidence from incoming spike trains over longer periods of time, and to process new information contained in such spike trains in diverse ways according to the current internal state of the circuit. In particular we show that such generic cortical microcircuits with feedback provide a new model for working memory that is consistent with a large set of biological constraints. Although this article examines primarily the computational role of feedback in circuits of neurons, the mathematical principles on which its analysis is based apply to a variety of dynamical systems. Hence they may also throw new light on the computational role of feedback in other complex biological dynamical systems, such as, for example, genetic regulatory networks.     

Modeling genetic switches with positive feedback loops

In this paper, we develop a new methodology to design synthetic genetic switch networks with multiple genes and time delays, by using monotone dynamical systems. We show that the networks with only positive feedback loops have no stable oscillation but stable equilibria whose stability is independent of the time delays. In other words, such systems have ideal properties for switch networks and can be designed without consideration of time delays, because the systems can be reduced from functional spaces to Euclidian spaces. Therefore, we can ensure that the designed switches function correctly even with uncertain delays. We first prove the basic properties of the genetic networks composed of only positive feedback loops, and then propose a procedure to design the switches, which drastically simplifies analysis of the switches and makes theoretical analysis and design tractable even for large-scaled systems. Finally, to demonstrate our theoretical results, we show biologically plausible examples by designing a synthetic genetic switch with experimentally well investigated lacI, tetR, and cI genes for numerical simulation.     

Perfect and near-perfect adaptation in a model of bacterial chemotaxis

The signaling apparatus mediating bacterial chemotaxis can adapt to a wide range of persistent external stimuli. In many cases, the bacterial activity returns to its prestimulus level exactly, and this perfect adaptability is robust against variations in various chemotaxis protein concentrations. We model the bacterial chemotaxis signaling pathway, from ligand binding to CheY phosphorylation. By solving the steady-state equations of the model analytically, we derive a full set of conditions for the system to achieve perfect adaptation. The conditions related to the phosphorylation part of the pathway are discovered for the first time, while other conditions are generalizations of the ones found in previous works. Sensitivity of the perfect adaptation is evaluated by perturbing these conditions. We find that, even in the absence of some of the perfect adaptation conditions, adaptation can be achieved with near-perfect precision as a result of the separation of scales in both chemotaxis protein concentrations and reaction rates, or specific properties of the receptor distribution in different methylation states. Since near-perfect adaptation can be found in much larger regions of the parameter space than that defined by the perfect adaptation conditions, their existence is essential to understand robustness in bacterial chemotaxis.     

Synthetic circuit of inositol phosphorylceramide synthase in Leishmania: a chemical biology approach

Building circuits and studying their behavior in cells is a major goal of systems and synthetic biology. Synthetic biology enables the precise control of cellular states for systems studies, the discovery of novel parts, control strategies, and interactions for the design of robust synthetic systems. To the best of our knowledge, there are no literature reports for the synthetic circuit construction for protozoan parasites. This paper describes the construction of genetic circuit for the targeted enzyme inositol phosphorylceramide synthase belonging to the protozoan parasite Leishmania. To explore the dynamic nature of the circuit designed, simulation was done followed by circuit validation by qualitative and quantitative approaches. The genetic circuit designed for inositol phosphorylceramide synthase (Biomodels Database—MODEL1208030000) shows responsiveness, oscillatory and bistable behavior, together with intrinsic robustness.     

The modularity of biological regulatory networks

A useful approach to complex regulatory networks consists of modeling their elements and interactions by Boolean equations. In this context, feedback circuits (i.e. circular sequences of interactions) have been shown to play key dynamical roles: whereas positive circuits are able to generate multistationarity, negative circuits may generate oscillatory behavior. In this paper, we principally focus on the case of gene networks. These are represented by fully connected Boolean networks where each element interacts with all elements including itself. Flexibility in network design is introduced by the use of Boolean parameters, one associated with each interaction or group of interactions affecting a given element. Within this formalism, a feedback circuit will generate its typical dynamical behavior (i.e. multistationarity or oscillations) only for appropriate values of some of the logical parameters. Whenever it does, we say that the circuit is 'functional'. More interestingly, this formalism allows the computation of the constraints on the logical parameters to have any feedback circuit functional in a network. Using this methodology, we found that the fraction of the total number of consistent combinations of parameter values that make a circuit functional decreases geometrically with the circuit length. From a biological point of view, this suggests that regulatory networks could be decomposed into small and relatively independent feedback circuits or 'regulatory modules'.     

Positive feedback circuits and memory

The concept of regulatory feedback circuit refers to oriented cyclic interactions between elements of a system. There are two classes of circuits, positive and negative, whose properties are in striking contrast. Positive circuits are a prerequisite for the occurrence of multiple steady states (multistationarity), and hence, they are involved in all processes showing hysteresis or memory. Endogenous or exogenous perturbations can lead the system to exhibit or to evoke one particular stable regime. The role of positive circuits in cell differentiation and in immunology is well documented. Negative circuits are involved in homeostatic regulation, with or without oscillations. The aim of this paper is to show: a) that positive circuits account for many features of memory stricto sensu (i.e., neural memory and mnesic evocation) as well as largo sensu (e.g. differentiation or immunological memory); and b) that simple combinations of positive and negative circuits provide powerful regulatory modules, which can also be associated in batteries. These entities have vast dynamical possibilities in the field of neurobiology, as well as in the fields of differentiation and immunology. Here we consider a universal minimal regulatory module, for which we suggest to adopt the term 'logical regulon', which can be considered as an atom of Jacob's integron. It comprises a positive and a negative circuit in its interaction matrix, and we recall the main results related to the simultaneous presence of these circuits. Finally, we give three applications of this type of interaction matrix. The first two deal with the coexistence of multiple stable steady states and periodicity in differentiation and in an immunological system showing hysteretic properties. The third deals with the dual problems of synchronization and desynchronization of a neural model for hippocampus memory evocation processes.     

Optimizing interneuron circuits for compartment-specific feedback inhibition

Cortical circuits process information by rich recurrent interactions between excitatory neurons and inhibitory interneurons. One of the prime functions of interneurons is to stabilize the circuit by feedback inhibition, but the level of specificity on which inhibitory feedback operates is not fully resolved. We hypothesized that inhibitory circuits could enable separate feedback control loops for different synaptic input streams, by means of specific feedback inhibition to different neuronal compartments. To investigate this hypothesis, we adopted an optimization approach. Leveraging recent advances in training spiking network models, we optimized the connectivity and short-term plasticity of interneuron circuits for compartment-specific feedback inhibition onto pyramidal neurons. Over the course of the optimization, the interneurons diversified into two classes that resembled parvalbumin (PV) and somatostatin (SST) expressing interneurons. Using simulations and mathematical analyses, we show that the resulting circuit can be understood as a neural decoder that inverts the nonlinear biophysical computations performed within the pyramidal cells. Our model provides a proof of concept for studying structure-function relations in cortical circuits by a combination of gradient-based optimization and biologically plausible phenomenological models.     

Designing and engineering evolutionary robust genetic circuits

Background  

One problem with engineered genetic circuits in synthetic microbes is their stability over evolutionary time in the absence of selective pressure. Since design of a selective environment for maintaining function of a circuit will be unique to every circuit, general design principles are needed for engineering evolutionary robust circuits that permit the long-term study or applied use of synthetic circuits.     

The Biological Property of Synthetic Evolved Digital Circuits with ESD Immunity – Redundancy or Degeneracy?

In the ongoing evolutionary process, biological systems have displayed a fundamental and remarkable property of robustness, i.e., the property allows the system to maintain its functions despite external and internal perturbations. Redundancy and degeneracy are thought to be the underlying structural mechanisms of biological robustness. Inspired by this, we explored the proximate cause of the immunity of the synthetic evolved digital circuits to ESD interference and discussed the biological characteristics behind the evolutionary circuits. First, we proposed an evolutionary method for intrinsic immune circuit design. The circuits' immunity was evaluated using the functional fault models based on probability distributions. Then, several benchmark circuits, including ADDER, MAJORITY, and C17, were evolved for high intrinsic immunity. Finally, using the quantitative definitions based on information theory, we measured the topological characteristics of redundancy and degeneracy in the evolved circuits and compared their contributions to the immunity. The results show that redundant elements are necessary for the ESD immune circuit design, whereas degeneracy is the key to making use of the redundancy robustly and efficiently.