Physical constraints on biological integral control design for homeostasis and sensory adaptation

Homeostasis in input-output networks: Structure, Classification and Applications

Homeostasis is concerned with regulatory mechanisms, present in biological systems, where some specific variable is kept close to a set value as some external disturbance affects the system. Many biological systems, from gene networks to signaling pathways to whole tissue/organism physiology, exhibit homeostatic mechanisms. In all these cases there are homeostatic regions where the variable is relatively to insensitive external stimulus, flanked by regions where it is sensitive. Mathematically, the notion of homeostasis can be formalized in terms of an input-output function that maps the parameter representing the external disturbance to the output variable that must be kept within a fairly narrow range. This observation inspired the introduction of the notion of infinitesimal homeostasis, namely, the derivative of the input-output function is zero at an isolated point. This point of view allows for the application of methods from singularity theory to characterize infinitesimal homeostasis points (i.e. critical points of the input-output function). In this paper we review the infinitesimal approach to the study of homeostasis in input-output networks. An input-output network is a network with two distinguished nodes 'input' and 'output', and the dynamics of the network determines the corresponding input-output function of the system. This class of dynamical systems provides an appropriate framework to study homeostasis and several important biological systems can be formulated in this context. Moreover, this approach, coupled to graph-theoretic ideas from combinatorial matrix theory, provides a systematic way for classifying different types of homeostasis (homeostatic mechanisms) in input-output networks, in terms of the network topology. In turn, this leads to new mathematical concepts, such as, homeostasis subnetworks, homeostasis patterns, homeostasis mode interaction. We illustrate the usefulness of this theory with several biological examples: biochemical networks, chemical reaction networks (CRN), gene regulatory networks (GRN), Intracellular metal ion regulation and so on.

The Duality of Astrocyte Neuromodulation: Astrocytes Sense Neuromodulators and Are Neuromodulators

Neuromodulation encompasses different processes that regulate neuronal and network function. Classical neuromodulators originating from long-range nuclei, such as acetylcholine, norepinephrine, or dopamine, act with a slower time course and wider spatial range than fast synaptic transmission and action potential firing. Accumulating evidence in vivo indicates that astrocytes, which are known to actively participate in synaptic function at tripartite synapses, are also involved in neuromodulatory processes. The present article reviews recent findings obtained in vivo indicating that astrocytes express receptors for neuromodulators that elevate their internal calcium and stimulate the release of gliotransmitters, which regulate synaptic and network function, and hence mediate, at least partially, the effects of neuromodulators. In addition, we propose that astrocytes act in local support of neuromodulators by spatially and temporally integrating neuronal and neuromodulatory signals to regulate neural network function. The presence of astrocyte-neuron hysteresis loops suggests astrocyte-neuron interaction at tripartite synapses scales up to astrocyte-neuronal networks that modulate neural network function. We finally propose that astrocytes sense the environmental conditions, including neuromodulators and network function states, and provide homeostatic control that maximizes the dynamic range of neural network activity. In summary, we propose that astrocytes are critical in mediating the effects of neuromodulators, and they also act as neuromodulators to provide neural network homeostasis thus optimizing information processing in the brain. Hence, astrocytes sense ongoing neuronal activity along with neuromodulators and, acting as neuromodulators, inform the neurons about the state of the internal system and the external world.

Background compensation revisited: Conserved phase response curves in frequency controlled homeostats with coherent feedback

Background compensation is the ability of a controlled variable to respond to an applied perturbation in an unchanged manner and independent of different but constant background signals which act in parallel to the perturbation. Background compensation occurs by 'coherent feedback' mechanisms where additional control variables feed directly back to the controlled variable. This paper extends a previous study on background compensation to include phase responses in frequency controlled coherent feedback oscillators. While the frequency resetting amplitude in coherent feedback oscillators is found to be dependent on the inflow/outflow perturbation of the controlled variable and thereby become phase dependent, the frequency resetting itself and the corresponding phase response curves are found to be background compensated. It is speculated that this type of background compensation may be an additional way how ambient noise can be 'ignored' by organisms.

Homeostasis in networks with multiple inputs

Homeostasis, also known as adaptation, refers to the ability of a system to counteract persistent external disturbances and tightly control the output of a key observable. Existing studies on homeostasis in network dynamics have mainly focused on 'perfect adaptation' in deterministic single-input single-output networks where the disturbances are scalar and affect the network dynamics via a pre-specified input node. In this paper we provide a full classification of all possible network topologies capable of generating infinitesimal homeostasis in arbitrarily large and complex multiple inputs networks. Working in the framework of 'infinitesimal homeostasis' allows us to make no assumption about how the components are interconnected and the functional form of the associated differential equations, apart from being compatible with the network architecture. Remarkably, we show that there are just three distinct 'mechanisms' that generate infinitesimal homeostasis. Each of these three mechanisms generates a rich class of well-defined network topologies-called homeostasis subnetworks. More importantly, we show that these classes of homeostasis subnetworks provides a topological basis for the classification of 'homeostasis types': the full set of all possible multiple inputs networks can be uniquely decomposed into these special homeostasis subnetworks. We illustrate our results with some simple abstract examples and a biologically realistic model for the co-regulation of calcium ( Ca ) and phosphate ( PO 4 ) in the rat. Furthermore, we identify a new phenomenon that occurs in the multiple input setting, that we call homeostasis mode interaction, in analogy with the well-known characteristic of multiparameter bifurcation theory.

Noise properties of adaptation-conferring biochemical control modules

A key goal of synthetic biology is to develop functional biochemical modules with network-independent properties. Antithetic integral feedback (AIF) is a recently developed control module in which two control species perfectly annihilate each other's biological activity. The AIF module confers robust perfect adaptation to the steady-state average level of a controlled intracellular component when subjected to sustained perturbations. Recent work has suggested that such robustness comes at the unavoidable price of increased stochastic fluctuations around average levels. We present theoretical results that support and quantify this trade-off for the commonly analyzed AIF variant in the idealized limit with perfect annihilation. However, we also show that this trade-off is a singular limit of the control module: Even minute deviations from perfect adaptation allow systems to achieve effective noise suppression as long as cells can pay the corresponding energetic cost. We further show that a variant of the AIF control module can achieve significant noise suppression even in the idealized limit with perfect adaptation. This atypical configuration may thus be preferable in synthetic biology applications.

Self-complexification through integral feedback in eusocial paper wasps of various levels of sociality

We investigate how simple physical interactions can generate remarkable diversity in the life history of social agents using data of social wasps, yielding complex scalable task partitioning. We built and analyzed a computational model to investigate how diverse task allocation patterns found in nature can emerge from the same behavioral blueprint. Self-organizing mechanisms of interwoven behavioral feedback loops, task-dependent time delays and simple material flows between interacting individuals yield an emergent homeostatic self-regulation while keeping the global colony performance scalable. Task allocation mechanisms based on implicitly honest signaling via material flows are not only very robust but are also highly evolvable due to their simplicity and reliability. We find that task partitioning has evolved to be scalable and adaptable to life history traits, such as expected colony size or temporal bottlenecks in the available workforce or materials. By tuning solely the total number of agents and a social connectivity-related parameter in the model, our simulations yield the whole range of emergent patterns in task allocation and task fidelity akin to observed field data. Our model suggests that the material exchange ("common stomach mechanism") found in many paper wasps provides a common functional "core" across these genera, which not only provides self-regulation of the colony, but also provides a scalable mechanism allowing natural selection to yield complex social integration in larger colonies over the course of their evolutionary trajectory.

Coherent feedback leads to robust background compensation in oscillatory and non-oscillatory homeostats

When in a reaction kinetic integral controller a step perturbation is applied besides a constant background, the concentration of a controlled variable (described as A) will generally respond with decreased response amplitudes ΔA as backgrounds increase. The controller variable E will at the same time provide the necessary compensatory flux to move A back to its set-point. A typical example of decreased response amplitudes at increased backgrounds is found in retinal light adaptation. Due to remarks in the literature that retinal light adaptation would also involve a compensation of backgrounds we became interested in conditions how background compensation could occur. In this paper we describe novel findings how background influences can be robustly eliminated. When such a background compensation is active, oscillatory controllers will respond to a defined perturbation with always the same (damped or undamped) frequency profile, or in the non-oscillatory case, with the same response amplitude ΔA, irrespective of the background level. To achieve background compensation we found that two conditions need to apply: (i) an additional set of integral controllers (here described as I1 and I2) have to be employed to keep the manipulated variable E at a defined set-point, and (ii), I1 and I2 need to feed back to the A-E signaling axis directly through the controlled variable A. In analogy to a similar feedback applied in quantum control theory, we term these feedback conditions as 'coherent feedback'. When analyzing retinal light adaptations in more detail, we find no evidence of the presence of background compensation mechanisms. Although robust background compensation, as described theoretically here, appears to be an interesting regulatory property, relevant biological or biochemical examples still need to be identified.

Protein-protein complexes can undermine ultrasensitivity-dependent biological adaptation

Robust perfect adaptation (RPA) is a ubiquitously observed signalling response across all scales of biological organization. A major class of network architectures that drive RPA in complex networks is the Opposer module-a feedback-regulated network into which specialized integral-computing 'opposer node(s)' are embedded. Although ultrasensitivity-generating chemical reactions have long been considered a possible mechanism for such adaptation-conferring opposer nodes, this hypothesis has relied on simplified Michaelian models, which neglect the presence of protein-protein complexes. Here we develop complex-complete models of interlinked covalent-modification cycles with embedded ultrasensitivity, explicitly capturing all molecular interactions and protein complexes. Strikingly, we demonstrate that the presence of protein-protein complexes thwarts the network's capacity for RPA in any 'free' active protein form, conferring RPA capacity instead on the concentration of a larger protein pool consisting of two distinct forms of a single protein. We further show that the presence of enzyme-substrate complexes, even at comparatively low concentrations, play a crucial and previously unrecognized role in controlling the RPA response-significantly reducing the range of network inputs for which RPA can obtain, and imposing greater parametric requirements on the RPA response. These surprising results raise fundamental new questions as to the biochemical requirements for adaptation-conferring Opposer modules within complex cellular networks.

Homeostasis at different backgrounds: The roles of overlayed feedback structures in vertebrate photoadaptation

We have studied the resetting behavior of eight basic integral controller motifs with respect to different but constant backgrounds. We found that the controllers split symmetrically into two classes: one class, based on derepression of the compensatory flux, leads to more rapid resetting kinetics as backgrounds increase. The other class, which directly activates the compensatory flux, shows a slowing down in the resetting at increased backgrounds. We found a striking analogy between the resetting kinetics of vertebrate photoreceptors and controllers based on derepression, i.e. vertebrate rod or cone cells show decreased sensitivities and accelerated response kinetics as background illuminations increase. The central molecular model of vertebrate photoadaptation consists of an overlay of three negative feedback loops with cytosolic calcium ([Formula: see text]), cyclic guanosine monophosphate (cGMP) and cyclic nucleotide-gated (CNG) channels as components. While in one of the feedback loops the extrusion of [Formula: see text] by potassium-dependent sodium-calcium exchangers (NCKX) can lead to integral control with cGMP as the controlled variable, the expected robust perfect adaptation of cGMP is lost, because of the two other feedback loops. They avoid that [Formula: see text] levels become too high and toxic. Looking at psychophysical laws, we found that in all of the above mentioned basic controllers Weber's law is followed when a "just noticeable difference" (threshold) of 1% of the controlled variable's set-point was considered. Applying comparable threshold pulses or steps to the photoadaptation model we find, in agreement with experimental results, that Weber's law is followed for relatively high backgrounds, while Stephens' power law gives a better description when backgrounds are low. Limitations of our photoadaption model, in particular with respect to potassium/sodium homeostasis, are discussed. Finally, we discuss possible implication of background perturbations in biological controllers when compensatory fluxes are based on activation.

Kinetics and mechanisms of catalyzed dual-E (antithetic) controllers

Homeostasis plays a central role in our understanding how cells and organisms are able to oppose environmental disturbances and thereby maintain an internal stability. During the last two decades there has been an increased interest in using control engineering methods, especially integral control, in the analysis and design of homeostatic networks. Several reaction kinetic mechanisms have been discovered which lead to integral control. In two of them integral control is achieved, either by the removal of a single control species E by zero-order kinetics (“single-E controllers”), or by the removal of two control species by second-order kinetics (“antithetic or dual-E control”). In this paper we show results when the control species E₁ and E₂ in antithetic control are removed enzymatically by ping-pong or ternary-complex mechanisms. Our findings show that enzyme-catalyzed dual-E controllers can work in two control modes. In one mode, one of the two control species is active, but requires zero-order kinetics in its removal. In the other mode, both controller species are active and both are removed enzymatically. Conditions for the two control modes are put forward and biochemical examples with the structure of enzyme-catalyzed dual-E controllers are discussed.

Astrocyte-neuronal network interplay is disrupted in Alzheimer's disease mice

Alzheimer's disease (AD) is associated with senile plaques of beta-amyloid (Aβ) that affect the function of neurons and astrocytes. Brain activity results from the coordinated function of neurons and astrocytes in astroglial-neuronal networks. However, the effects of Aβ on astroglial and neuronal network function remains unknown. Simultaneously monitoring astrocyte calcium and electric neuronal activities, we quantified the impact of Aβ on sensory-evoked cortical activity in a mouse model of AD. At rest, cortical astrocytes displayed spontaneous hyperactivity that was related to Aβ density. Sensory-evoked astrocyte responsiveness was diminished in AD mice, depending on the density and distance of Aβ, and the responses showed altered calcium dynamics. Hence, astrocytes were spontaneously hyperactive but hypo-responsive to sensory stimulation. Finally, AD mice showed sensory-evoked electrical cortical hyperresponsiveness associated with altered astrocyte-neuronal network interplay. Our findings suggest dysfunction of astrocyte networks in AD mice may dysregulate cortical electrical activity and contribute to cognitive decline.

Frequency-preference response in covalent modification cycles under substrate sequestration conditions

Covalent modification cycles (CMCs) are basic units of signaling systems and their properties are well understood. However, their behavior has been mostly characterized in situations where the substrate is in excess over the modifying enzymes. Experimental data on protein abundance suggest that the enzymes and their target proteins are present in comparable concentrations, leading to substrate sequestration by the enzymes. In this enzyme-in-excess regime, CMCs have been shown to exhibit signal termination, the ability of the product to return to a stationary value lower than its peak in response to constant stimulation, while this stimulation is still active, with possible implications for the ability of systems to adapt to environmental inputs. We characterize the conditions leading to signal termination in CMCs in the enzyme-in-excess regime. We also demonstrate that this behavior leads to a preferred frequency response (band-pass filters) when the cycle is subjected to periodic stimulation, whereas the literature reports that CMCs investigated so far behave as low-pass filters. We characterize the relationship between signal termination and the preferred frequency response to periodic inputs and we explore the dynamic mechanism underlying these phenomena. Finally, we describe how the behavior of CMCs is reflected in similar types of responses in the cascades of which they are part. Evidence of protein abundance in vivo shows that enzymes and substrates are present in comparable concentrations, thus suggesting that signal termination and frequency-preference response to periodic inputs are also important dynamic features of cell signaling systems, which have been overlooked.

Robust adaptation of PKCζ-IRS1 insulin signaling pathways through integral feedback control

Insulin signaling pathways in muscle tissue play a major role in maintaining glucose homeostasis. Dysregulation in these pathways results in the onset of serious metabolic disorders like type 2 diabetes. Robustness is an essential characteristic of insulin signaling pathways that ensures reliable signal transduction in the presence of perturbations as a result of several feedback mechanisms. Integral control, according to control engineering, provides reliable setpoint tracking and disturbance rejection. The presence of negative feedback and integrating process is crucial for biological processes to achieve integral control. The existence of an integral controller leads to the rejection of perturbations which resulted in the robust regulation of biochemical entities within acceptable levels. In the presentin silicoresearch work, the presence of integral control in the protein kinase Cζ- insulin receptor substrate-1 (PKCζ-IRS1) pathway is identified, verified mathematically and model is simulated in Cell Designer. The data is exported to Minitab software and robustness analysis is carried out statistically using the Mann-Whitney test. The p-value of the results obtained with given parameters perturbed by ±1% is greater than the significance level of 0.05 (0.2132 for 1% error in k7(rate constant of IRS1 phosphorylation), 0.2096 for -1% error in k7, 0.9037 for both ±1% error in insulin and 0.9037 for ±1% error in k1(association rate constant of the first molecule of insulin to bind the insulin receptor), indicated that our hypothesis is proved The results satisfactorily indicate that even when perturbations are present, glucose homeostasis in muscle tissue is robust due to the presence of integral regulation in the PKCζ-IRS1 insulin signaling pathways. In this paper, we have analysed the findings from the framework of robust control theory, which has allowed us to examine that how PKCζ-IRS1 insulin signaling pathways produces desired output in presence of perturbations.

Perfect adaptation in biology

A distinctive feature of many biological systems is their ability to adapt to persistent stimuli or disturbances that would otherwise drive them away from a desirable steady state. The resulting stasis enables organisms to function reliably while being subjected to very different external environments. This perspective concerns a stringent type of biological adaptation, robust perfect adaptation (RPA), that is resilient to certain network and parameter perturbations. As in engineered control systems, RPA requires that the regulating network satisfy certain structural constraints that cannot be avoided. We elucidate these ideas using biological examples from systems and synthetic biology. We then argue that understanding the structural constraints underlying RPA allows us to look past implementation details and offers a compelling means to unravel regulatory biological complexity.

The structure of infinitesimal homeostasis in input-output networks

Homeostasis refers to a phenomenon whereby the output [Formula: see text] of a system is approximately constant on variation of an input [Formula: see text]. Homeostasis occurs frequently in biochemical networks and in other networks of interacting elements where mathematical models are based on differential equations associated to the network. These networks can be abstracted as digraphs [Formula: see text] with a distinguished input node [Formula: see text], a different distinguished output node o, and a number of regulatory nodes [Formula: see text]. In these models the input-output map [Formula: see text] is defined by a stable equilibrium [Formula: see text] at [Formula: see text]. Stability implies that there is a stable equilibrium [Formula: see text] for each [Formula: see text] near [Formula: see text] and infinitesimal homeostasis occurs at [Formula: see text] when [Formula: see text]. We show that there is an [Formula: see text] homeostasis matrix [Formula: see text] for which [Formula: see text] if and only if [Formula: see text]. We note that the entries in H are linearized couplings and [Formula: see text] is a homogeneous polynomial of degree [Formula: see text] in these entries. We use combinatorial matrix theory to factor the polynomial [Formula: see text] and thereby determine a menu of different types of possible homeostasis associated with each digraph [Formula: see text]. Specifically, we prove that each factor corresponds to a subnetwork of [Formula: see text]. The factors divide into two combinatorially defined classes: structural and appendage. Structural factors correspond to feedforward motifs and appendage factors correspond to feedback motifs. Finally, we discover an algorithm for determining the homeostasis subnetwork motif corresponding to each factor of [Formula: see text] without performing numerical simulations on model equations. The algorithm allows us to classify low degree factors of [Formula: see text]. There are two types of degree 1 homeostasis (negative feedback loops and kinetic or Haldane motifs) and there are two types of degree 2 homeostasis (feedforward loops and a degree two appendage motif).

An amplified derepression controller with multisite inhibition and positive feedback

How organisms are able to maintain robust homeostasis has in recent years received increased attention by the use of combined control engineering and kinetic concepts, which led to the discovery of robust controller motifs. While these motifs employ kinetic conditions showing integral feedback and homeostasis for step-wise perturbations, the motifs’ performance differ significantly when exposing them to time dependent perturbations. One type of controller motifs which are able to handle exponentially and even hyperbolically growing perturbations are based on derepression. In these controllers the compensatory reaction, which neutralizes the perturbation, is derepressed, i.e. its reaction rate is increased by the decrease of an inhibitor acting on the compensatory flux. While controllers in this category can deal well with different time-dependent perturbations they have the disadvantage that they break down once the concentration of the regulatory inhibitor becomes too low and the compensatory flux has gained its maximum value. We wondered whether it would be possible to bypass this restriction, while still keeping the advantages of derepression kinetics. In this paper we show how the inclusion of multisite inhibition and the presence of positive feedback loops lead to an amplified controller which is still based on derepression kinetics but without showing the breakdown due to low inhibitor concentrations. By searching for the amplified feedback motif in natural systems, we found it as a part of the plant circadian clock where it is highly interlocked with other feedback loops.

Stochastic models for single-cell data: Current challenges and the way forward

Although the quantity and quality of single-cell data have progressed rapidly, making quantitative predictions with single-cell stochastic models remains challenging. The stochastic nature of cellular processes leads to at least three challenges in building models with single-cell data: (a) because variability in single-cell data can be attributed to multiple different sources, it is difficult to rule out conflicting mechanistic models that explain the same data equally well; (b) the distinction between interesting biological variability and experimental variability is sometimes ambiguous; (c) the nonstandard distributions of single-cell data can lead to violations of the assumption of symmetric errors in least-squares fitting. In this review, we first discuss recent studies that overcome some of the challenges or set up a promising direction and then introduce some powerful statistical approaches utilized in these studies. We conclude that applying and developing statistical approaches could lead to further progress in building stochastic models for single-cell data.

Guidelines for designing the antithetic feedback motif

Integral feedback control is commonly used in mechanical and electrical systems to achieve zero steady-state error following an external disturbance. Equivalently, in biological systems, a property known as robust perfect adaptation guarantees robustness to environmental perturbations and return to the pre-disturbance state. Previously, Briat et al proposed a biomolecular design for integral feedback control (robust perfect adaptation) called the antithetic feedback motif. The antithetic feedback controller uses the sequestration binding reaction of two biochemical species to record the integral of the error between the current and the desired output of the network it controls. The antithetic feedback motif has been successfully built using synthetic components in vivo in Escherichia coli and Saccharomyces cerevisiae cells. However, these previous synthetic implementations of antithetic feedback have not produced perfect integral feedback control due to the degradation and dilution of the two controller species. Furthermore, previous theoretical results have cautioned that integral control can only be achieved under stability conditions that not all antithetic feedback motifs necessarily fulfill. In this paper, we study how to design antithetic feedback motifs that simultaneously achieve good stability and small steady-state error properties, even as the controller species are degraded and diluted. We provide simple tuning guidelines to achieve flexible and practical synthetic biological implementations of antithetic feedback control. We use several tools and metrics from control theory to design antithetic feedback networks, paving the path for the systematic design of synthetic biological controllers.

Frequency switching between oscillatory homeostats and the regulation of p53

Homeostasis is an essential concept to understand the stability of organisms and their adaptive behaviors when coping with external and internal assaults. Many hormones that take part in homeostatic control come in antagonistic pairs, such as glucagon and insulin reflecting the inflow and outflow compensatory mechanisms to control a certain internal variable, such as blood sugar levels. By including negative feedback loops homeostatic controllers can exhibit oscillations with characteristic frequencies. In this paper we demonstrate the associated frequency changes in homeostatic systems when individual controllers -in a set of interlocked feedback loops- gain control in response to environmental changes. Taking p53 as an example, we show how Per2, ATM and Mdm2 feedback loops -interlocked with p53- gain individual control in dependence to the level of DNA damage, and how each of these controllers provide certain functionalities in their regulation of p53. In unstressed cells, the circadian regulator Per2 ensures a basic p53 level to allow its rapid up-regulation in case of DNA damage. When DNA damage occurs the ATM controller increases the level of p53 and defends it towards uncontrolled degradation, which despite DNA damage, would drive p53 to lower values and p53 dysfunction. Mdm2 on its side keeps p53 at a high but sub-apoptotic level to avoid premature apoptosis. However, with on-going DNA damage the Mdm2 set-point is increased by HSP90 and other p53 stabilizers leading finally to apoptosis. An emergent aspect of p53 upregulation during cell stress is the coordinated inhibition of ubiquitin-independent and ubiquitin-dependent degradation reactions. Whether oscillations serve a function or are merely a by-product of the controllers are discussed in view of the finding that homeostatic control of p53, as indicated above, does in principle not require oscillatory homeostats.

Infinitesimal homeostasis in three-node input-output networks

Homeostasis occurs in a system where an output variable is approximately constant on an interval on variation of an input variable [Formula: see text]. Homeostasis plays an important role in the regulation of biological systems, cf.  Ferrell (Cell Syst 2:62-67, 2016), Tang and McMillen (J Theor Biol 408:274-289, 2016), Nijhout et al. (BMC Biol 13:79, 2015), and Nijhout et al. (Wiley Interdiscip Rev Syst Biol Med 11:e1440, 2018). A method for finding homeostasis in mathematical models is given in the control theory literature as points where the derivative of the output variable with respect to [Formula: see text] is identically zero. Such points are called perfect homeostasis or perfect adaptation. Alternatively, Golubitsky and Stewart (J Math Biol 74:387-407, 2017) use an infinitesimal notion of homeostasis (namely, the derivative of the input-output function is zero at an isolated point) to introduce singularity theory into the study of homeostasis. Reed et al. (Bull Math Biol 79(9):1-24, 2017) give two examples of infinitesimal homeostasis in three-node chemical reaction systems: feedforward excitation and substrate inhibition. In this paper we show that there are 13 different three-node networks leading to 78 three-node input-output network configurations, under the assumption that there is one input node, one output node, and they are distinct. The different configurations are based on which node is the input node and which node is the output node. We show nonetheless that there are only three basic mechanisms for three-node input-output networks that lead to infinitesimal homeostasis and we call them structural homeostasis, Haldane homeostasis, and null-degradation homeostasis. Substantial parts of this classification are given in Ma et al. (Cell 138:760-773, 2009) and Ferrell (2016) among others. Our contributions include giving a complete classification using general admissible systems (Golubitsky and Stewart in Bull Am Math Soc 43:305-364, 2006) rather than specific biochemical models, relating the types of infinitesimal homeostasis to the graph theoretic existence of simple paths, and providing the basis to use singularity theory to study higher codimension homeostasis singularities such as the chair singularities introduced in Nijhout and Reed (Integr Comp Biol 54(2):264-275, 2014. https://doi.org/10.1093/icb/icu010) and Nijhout et al. (Math Biosci 257:104-110, 2014). See Golubitsky and Stewart (2017). The first two of these mechanisms are illustrated by feedforward excitation and substrate inhibition. Structural homeostasis occurs only when the network has a feedforward loop as a subnetwork; that is, when there are two distinct simple paths connecting the input node to the output node. Moreover, when the network is just the feedforward loop motif itself, one of the paths must be excitatory and one inhibitory to support infinitesimal homeostasis. Haldane homeostasis occurs when there is a single simple path from the input node to the output node and then only when one of the couplings along this path has strength 0. Null-degradation homeostasis is illustrated by a biochemical example from Ma et al. (2009); this kind of homeostasis can occur only when the degradation constant of the third node is 0. The paper ends with an analysis of Haldane homeostasis infinitesimal chair singularities.

Low-Burden Biological Feedback Controllers for Near-Perfect Adaptation

The robustness and reliability of synthetic biological systems can be substantially improved by the introduction of feedback control architectures that parallel those employed in traditional engineering disciplines. One common control goal is adaptation (or disturbance rejection), which refers to a system's ability to maintain a constant output despite variation in some of its constituent processes (as frequently occurs in noisy cellular environments) or external perturbations. In this paper, we propose and analyze a control architecture that employs integrase and excisionase proteins to invert regions of DNA and an mRNA-mRNA annihilation reaction. Combined, these components approximate the functionality of a switching controller (as employed in classical control engineering) with three distinct operational modes. We demonstrate that this system is capable of near-perfect adaptation to variation in rates of both transcription and translation and can also operate without excessive consumption of cellular resources. The system's steady-state behavior is analyzed, and limits on its operating range are derived. Deterministic simulations of its dynamics are presented and are then extended to the stochastic case, which treats biochemical reactions as discrete events.

Homeostasis in a feed forward loop gene regulatory motif

The internal state of a cell is affected by inputs from the extra-cellular environment such as external temperature. If some output, such as the concentration of a target protein, remains approximately constant as inputs vary, the system exhibits homeostasis. Special sub-networks called motifs are unusually common in gene regulatory networks (GRNs), suggesting that they may have a significant biological function. Potentially, one such function is homeostasis. In support of this hypothesis, we show that the feed-forward loop GRN produces homeostasis. Here the inputs are subsumed into a single parameter that affects only the first node in the motif, and the output is the concentration of a target protein. The analysis uses the notion of infinitesimal homeostasis, which occurs when the input-output map has a critical point (zero derivative). In model equations such points can be located using implicit differentiation. If the second derivative of the input-output map also vanishes, the critical point is a chair: the output rises roughly linearly, then flattens out (the homeostasis region or plateau), and then starts to rise again. Chair points are a common cause of homeostasis. In more complicated equations or networks, numerical exploration would have to augment analysis. Thus, in terms of finding chairs, this paper presents a proof of concept. We apply this method to a standard family of differential equations modeling the feed-forward loop GRN, and deduce that chair points occur. This function determines the production of a particular mRNA and the resulting chair points are found analytically. The same method can potentially be used to find homeostasis regions in other GRNs. In the discussion and conclusion section, we also discuss why homeostasis in the motif may persist even when the rest of the network is taken into account.

Realizing 'integral control' in living cells: how to overcome leaky integration due to dilution?

A major problem in the design of synthetic genetic circuits is robustness to perturbations and uncertainty. Because of this, there have been significant efforts in recent years in finding approaches to implement integral control in genetic circuits. Integral controllers have the unique ability to make the output of a process adapt perfectly to disturbances. However, implementing an integral controller is challenging in living cells. This is because a key aspect of any integral controller is a 'memory' element that stores the accumulation (integral) of the error between the output and its desired set-point. The ability to realize such a memory element in living cells is fundamentally challenged by the fact that all biomolecules dilute as cells grow, resulting in a 'leaky' memory that gradually fades away. As a consequence, the adaptation property is lost. Here, we propose a general principle for designing integral controllers such that the performance is practically unaffected by dilution. In particular, we mathematically prove that if the reactions implementing the integral controller are all much faster than dilution, then the adaptation error due to integration leakiness becomes negligible. We exemplify this design principle with two synthetic genetic circuits aimed at reaching adaptation of gene expression to fluctuations in cellular resources. Our results provide concrete guidance on the biomolecular processes that are most appropriate for implementing integral controllers in living cells.

Adaptive information processing of network modules to dynamic and spatial stimuli

BACKGROUND

Adaptation and homeostasis are basic features of information processing in cells and seen in a broad range of contexts. Much of the current understanding of adaptation in network modules/motifs is based on their response to simple stimuli. Recently, there have also been studies of adaptation in dynamic stimuli. However a broader synthesis of how different circuits of adaptation function, and which circuits enable a broader adaptive behaviour in classes of more complex and spatial stimuli is largely missing.

RESULTS

We study the response of a variety of adaptive circuits to time-varying stimuli such as ramps, periodic stimuli and static and dynamic spatial stimuli. We find that a variety of responses can be seen in ramp stimuli, making this a basis for discriminating between even similar circuits. We also find that a number of circuits adapt exactly to ramp stimuli, and dissect these circuits to pinpoint what characteristics (architecture, feedback, biochemical aspects, information processing ingredients) allow for this. These circuits include incoherent feedforward motifs, inflow-outflow motifs and transcritical circuits. We find that changes in location in such circuits where a signal acts can result in non-adaptive behaviour in ramps, even though the location was associated with exact adaptation in step stimuli. We also demonstrate that certain augmentations of basic inflow-outflow motifs can alter the behaviour of the circuit from exact adaptation to non-adaptive behaviour. When subject to periodic stimuli, some circuits (inflow-outflow motifs and transcritical circuits) are able to maintain an average output independent of the characteristics of the input. We build on this to examine the response of adaptive circuits to static and dynamic spatial stimuli. We demonstrate how certain circuits can exhibit a graded response in spatial static stimuli with an exact maintenance of the spatial mean-value. Distinct features which emerge from the consideration of dynamic spatial stimuli are also discussed. Finally, we also build on these results to show how different circuits which show any combination of presence or absence of exact adaptation in ramps, exact mainenance of time average output in periodic stimuli and exact maintenance of spatial average of output in static spatial stimuli may be realized.

CONCLUSIONS

By studying a range of network circuits/motifs on one hand and a range of stimuli on the other, we isolate characteristics of these circuits (structural) which enable different degrees of exact adaptive and homeostatic behaviour in such stimuli, how they may be combined, and also identify cases associated with non-homeostatic behaviour. We also reveal constraints associated with locations where signals may act to enable homeostatic behaviour and constraints associated with augmentations of circuits. This consideration of multiple experimentally/naturally relevant stimuli along with circuits of adaptation of relevance in natural and engineered biology, provides a platform for deepening our understanding of adaptive and homeostatic behaviour in natural systems, bridging the gap between models of adaptation and experiments and in engineering homeostatic synthetic circuits.

Integral feedback control is at the core of task allocation and resilience of insect societies

Homeostatic self-regulation is a fundamental aspect of open dissipative systems. Integral feedback has been found to be important for homeostatic control on both the cellular and molecular levels of biological organization and in engineered systems. Analyzing the task allocation mechanisms of three insect societies, we identified a model of integral control residing at colony level. We characterized a general functional core mechanism, called the "common stomach," where a crucial shared substance for colony function self-regulates its own quantity via reallocating the colony's workforce, which collects and uses this substance. The central component in a redundant feedback network is the saturation level of this substance in the colony. An interaction network of positive and negative feedback loops ensures the homeostatic state of this substance and the workforce involved in processing this substance. Extensive sensitivity and stability analyses of the core model revealed that the system is very resilient against perturbations and compensates for specific types of stress that real colonies face in their ecosystems. The core regulation system is highly scalable, and due to its buffer function, it can filter noise and find a new equilibrium quickly after environmental (supply) or colony-state (demand) changes. The common stomach regulation system is an example of convergent evolution among the three different societies, and we predict that similar integral control regulation mechanisms have evolved frequently within natural complex systems.

A quasi-integral controller for adaptation of genetic modules to variable ribosome demand

The behavior of genetic circuits is often poorly predictable. A gene’s expression level is not only determined by the intended regulators, but also affected by changes in ribosome availability imparted by expression of other genes. Here we design a quasi-integral biomolecular feedback controller that enables the expression level of any gene of interest (GOI) to adapt to changes in available ribosomes. The feedback is implemented through a synthetic small RNA (sRNA) that silences the GOI’s mRNA, and uses orthogonal extracytoplasmic function (ECF) sigma factor to sense the GOI’s translation and to actuate sRNA transcription. Without the controller, the expression level of the GOI is reduced by 50% when a resource competitor is activated. With the controller, by contrast, gene expression level is practically unaffected by the competitor. This feedback controller allows adaptation of genetic modules to variable ribosome demand and thus aids modular construction of complicated circuits.

Competition for shared cellular resources often renders genetic circuits poorly predictable. Here the authors design a biomolecular quasi-integral controller that allows gene expression to adapt to variable demand in translation resources.

A Framework for Engineering Stress Resilient Plants Using Genetic Feedback Control and Regulatory Network Rewiring

Crop disease leads to significant waste worldwide, both pre- and postharvest, with subsequent economic and sustainability consequences. Disease outcome is determined both by the plants' response to the pathogen and by the ability of the pathogen to suppress defense responses and manipulate the plant to enhance colonization. The defense response of a plant is characterized by significant transcriptional reprogramming mediated by underlying gene regulatory networks, and components of these networks are often targeted by attacking pathogens. Here, using gene expression data from Botrytis cinerea-infected Arabidopsis plants, we develop a systematic approach for mitigating the effects of pathogen-induced network perturbations, using the tools of synthetic biology. We employ network inference and system identification techniques to build an accurate model of an Arabidopsis defense subnetwork that contains key genes determining susceptibility of the plant to the pathogen attack. Once validated against time-series data, we use this model to design and test perturbation mitigation strategies based on the use of genetic feedback control. We show how a synthetic feedback controller can be designed to attenuate the effect of external perturbations on the transcription factor CHE in our subnetwork. We investigate and compare two approaches for implementing such a controller biologically-direct implementation of the genetic feedback controller, and rewiring the regulatory regions of multiple genes-to achieve the network motif required to implement the controller. Our results highlight the potential of combining feedback control theory with synthetic biology for engineering plants with enhanced resilience to environmental stress.

Variable setpoint as a relaxing component in physiological control

Setpoints in physiology have been a puzzle for decades, and especially the notion of fixed or variable setpoints have received much attention. In this paper, we show how previously presented homeostatic controller motifs, extended with saturable signaling kinetics, can be described as variable setpoint controllers. The benefit of a variable setpoint controller is that an observed change in the concentration of the regulated biochemical species (the controlled variable) is fully characterized, and is not considered a deviation from a fixed setpoint. The variation in this biochemical species originate from variation in the disturbances (the perturbation), and thereby in the biochemical species representing the controller (the manipulated variable). Thus, we define an operational space which is spanned out by the combined high and low levels of the variations in (1) the controlled variable, (2) the manipulated variable, and (3) the perturbation. From this operational space, we investigate whether and how it imposes constraints on the different motif parameters, in order for the motif to represent a mathematical model of the regulatory system. Further analysis of the controller's ability to compensate for disturbances reveals that a variable setpoint represents a relaxing component for the controller, in that the necessary control action is reduced compared to that of a fixed setpoint controller. Such a relaxing component might serve as an important property from an evolutionary point of view. Finally, we illustrate the principles using the renal sodium and aldosterone regulatory system, where we model the variation in plasma sodium as a function of salt intake. We show that the experimentally observed variations in plasma sodium can be interpreted as a variable setpoint regulatory system.

Realizing 'integral control' in living cells: how to overcome leaky integration due to dilution?

A major problem in the design of synthetic genetic circuits is robustness to perturbations and uncertainty. Because of this, there have been significant efforts in recent years in finding approaches to implement integral control in genetic circuits. Integral controllers have the unique ability to make the output of a process adapt perfectly to disturbances. However, implementing an integral controller is challenging in living cells. This is because a key aspect of any integral controller is a 'memory' element that stores the accumulation (integral) of the error between the output and its desired set-point. The ability to realize such a memory element in living cells is fundamentally challenged by the fact that all biomolecules dilute as cells grow, resulting in a 'leaky' memory that gradually fades away. As a consequence, the adaptation property is lost. Here, we propose a general principle for designing integral controllers such that the performance is practically unaffected by dilution. In particular, we mathematically prove that if the reactions implementing the integral controller are all much faster than dilution, then the adaptation error due to integration leakiness becomes negligible. We exemplify this design principle with two synthetic genetic circuits aimed at reaching adaptation of gene expression to fluctuations in cellular resources. Our results provide concrete guidance on the biomolecular processes that are most appropriate for implementing integral controllers in living cells.

The Interplay between Feedback and Buffering in Cellular Homeostasis

Buffering, the use of reservoirs of molecules to maintain concentrations of key molecular species, and negative feedback are the primary known mechanisms for robust homeostatic regulation. To our knowledge, however, the fundamental principles behind their combined effect have not been elucidated. Here, we study the interplay between buffering and negative feedback in the context of cellular homeostasis. We show that negative feedback counteracts slow-changing disturbances, whereas buffering counteracts fast-changing disturbances. Furthermore, feedback and buffering have limitations that create trade-offs for regulation: instability in the case of feedback and molecular noise in the case of buffering. However, because buffering stabilizes feedback and feedback attenuates noise from slower-acting buffering, their combined effect on homeostasis can be synergistic. These effects can be explained within a traditional control theory framework and are consistent with experimental observations of both ATP homeostasis and pH regulation in vivo. These principles are critical for studying robustness and homeostasis in biology and biotechnology.

Towards combinatorial transcriptional engineering

The modular nature of the transcriptional unit makes it possible to design robust modules with predictable input-output characteristics using a ‘parts- off a shelf’ approach. Customized regulatory circuits composed of multiple such transcriptional units have immense scope for application in diverse fields of basic and applied research. Synthetic transcriptional engineering seeks to construct such genetic cascades. Here, we discuss the three principle strands of transcriptional engineering: promoter and transcriptional factor engineering, and programming inducibilty into synthetic modules. In this context, we review the scope and limitations of some recent technologies that seek to achieve these ends. Our discussion emphasizes a requirement for rational combinatorial engineering principles and the promise this approach holds for the future development of this field.

Design of a Synthetic Integral Feedback Circuit: Dynamic Analysis and DNA Implementation

The design and implementation of regulation motifs ensuring robust perfect adaptation are challenging problems in synthetic biology. Indeed, the design of high-yield robust metabolic pathways producing, for instance, drug precursors and biofuels, could be easily imagined to rely on such a control strategy in order to optimize production levels and reduce production costs, despite the presence of environmental disturbance and model uncertainty. We propose here a motif that ensures tracking and robust perfect adaptation for the controlled reaction network through integral feedback. Its metabolic load on the host is fully tunable and can be made arbitrarily close to the constitutive limit, the universal minimal metabolic load of all possible controllers. A DNA implementation of the controller network is finally provided. Computer simulations using realistic parameters demonstrate the good agreement between the DNA implementation and the ideal controller dynamics.

Ultrasensitive Negative Feedback Control: A Natural Approach for the Design of Synthetic Controllers

Many of the most important potential applications of Synthetic Biology will require the ability to design and implement high performance feedback control systems that can accurately regulate the dynamics of multiple molecular species within the cell. Here, we argue that the use of design strategies based on combining ultrasensitive response dynamics with negative feedback represents a natural approach to this problem that fully exploits the strongly nonlinear nature of cellular information processing. We propose that such feedback mechanisms can explain the adaptive responses observed in one of the most widely studied biomolecular feedback systems—the yeast osmoregulatory response network. Based on our analysis of such system, we identify strong links with a well-known branch of mathematical systems theory from the field of Control Engineering, known as Sliding Mode Control. These insights allow us to develop design guidelines that can inform the construction of feedback controllers for synthetic biological systems.

Control theory meets synthetic biology

The past several years have witnessed an increased presence of control theoretic concepts in synthetic biology. This review presents an organized summary of how these control design concepts have been applied to tackle a variety of problems faced when building synthetic biomolecular circuits in living cells. In particular, we describe success stories that demonstrate how simple or more elaborate control design methods can be used to make the behaviour of synthetic genetic circuits within a single cell or across a cell population more reliable, predictable and robust to perturbations. The description especially highlights technical challenges that uniquely arise from the need to implement control designs within a new hardware setting, along with implemented or proposed solutions. Some engineering solutions employing complex feedback control schemes are also described, which, however, still require a deeper theoretical analysis of stability, performance and robustness properties. Overall, this paper should help synthetic biologists become familiar with feedback control concepts as they can be used in their application area. At the same time, it should provide some domain knowledge to control theorists who wish to enter the rising and exciting field of synthetic biology.

Synthetic biology and regulatory networks: where metabolic systems biology meets control engineering

Metabolic pathways can be engineered to maximize the synthesis of various products of interest. With the advent of computational systems biology, this endeavour is usually carried out through in silico theoretical studies with the aim to guide and complement further in vitro and in vivo experimental efforts. Clearly, what counts is the result in vivo, not only in terms of maximal productivity but also robustness against environmental perturbations. Engineering an organism towards an increased production flux, however, often compromises that robustness. In this contribution, we review and investigate how various analytical approaches used in metabolic engineering and synthetic biology are related to concepts developed by systems and control engineering. While trade-offs between production optimality and cellular robustness have already been studied diagnostically and statically, the dynamics also matter. Integration of the dynamic design aspects of control engineering with the more diagnostic aspects of metabolic, hierarchical control and regulation analysis is leading to the new, conceptual and operational framework required for the design of robust and productive dynamic pathways.

Cell-matrix interaction during strain-dependent remodelling of simulated collagen networks

The importance of tissue remodelling is widely accepted, but the mechanism by which the remodelling process occurs remains poorly understood. At the tissue scale, the concept of tensional homeostasis, in which there exists a target stress for a cell and remodelling functions to move the cell stress towards that target, is an important foundation for much theoretical work. We present here a theoretical model of a cell in parallel with a network to study what factors of the remodelling process help the cell move towards mechanical stability. The cell-network system was deformed and kept at constant stress. Remodelling was modelled by simulating strain-dependent degradation of collagen fibres and four different cases of collagen addition. The model did not lead to complete tensional homeostasis in the range of conditions studied, but it showed how different expressions for deposition and removal of collagen in a fibre network can interact to modulate the cell's ability to shield itself from an imposed stress by remodelling the surroundings. This study also showed how delicate the balance between deposition and removal rates is and how sensitive the remodelling process is to small changes in the remodelling rules.

The Organization of Controller Motifs Leading to Robust Plant Iron Homeostasis

Iron is an essential element needed by all organisms for growth and development. Because iron becomes toxic at higher concentrations iron is under homeostatic control. Plants face also the problem that iron in the soil is tightly bound to oxygen and difficult to access. Plants have therefore developed special mechanisms for iron uptake and regulation. During the last years key components of plant iron regulation have been identified. How these components integrate and maintain robust iron homeostasis is presently not well understood. Here we use a computational approach to identify mechanisms for robust iron homeostasis in non-graminaceous plants. In comparison with experimental results certain control arrangements can be eliminated, among them that iron homeostasis is solely based on an iron-dependent degradation of the transporter IRT1. Recent IRT1 overexpression experiments suggested that IRT1-degradation is iron-independent. This suggestion appears to be misleading. We show that iron signaling pathways under IRT1 overexpression conditions become saturated, leading to a breakdown in iron regulation and to the observed iron-independent degradation of IRT1. A model, which complies with experimental data places the regulation of cytosolic iron at the transcript level of the transcription factor FIT. Including the experimental observation that FIT induces inhibition of IRT1 turnover we found a significant improvement in the system’s response time, suggesting a functional role for the FIT-mediated inhibition of IRT1 degradation. By combining iron uptake with storage and remobilization mechanisms a model is obtained which in a concerted manner integrates iron uptake, storage and remobilization. In agreement with experiments the model does not store iron during its high-affinity uptake. As an iron biofortification approach we discuss the possibility how iron can be accumulated even during high-affinity uptake.

Synthetic feedback control using an RNAi-based gene-regulatory device

Background

Homeostasis within mammalian cells is achieved through complex molecular networks that can respond to changes within the cell or the environment and regulate the expression of the appropriate genes in response. The development of biological components that can respond to changes in the cellular environment and interface with endogenous molecules would enable more sophisticated genetic circuits and greatly advance our cellular engineering capabilities.

Results

Here we describe a platform that combines a ligand-responsive ribozyme switch and synthetic miRNA regulators to create an OFF genetic control device based on RNA interference (RNAi). We developed a mathematical model to highlight important design parameters in programming the quantitative performance of RNAi-based OFF control devices. By modifying the ribozyme switch integrated into the system, we demonstrated RNAi-based OFF control devices that respond to small molecule and protein ligands, including the oncogenic protein E2F1. We utilized the OFF control device platform to build a negative feedback control system that acts as a proportional controller and maintains target intracellular protein levels in response to increases in transcription rate.

Conclusions

Our work describes a novel genetic device that increases the level of silencing from a miRNA in the presence of a ligand of interest, effectively creating an RNAi-based OFF control device. The OFF switch platform has the flexibility to be used to respond to both small molecule and protein ligands. Finally, the RNAi-based OFF switch can be used to implement a negative feedback control system, which maintains target protein levels around a set point level. The described RNAi-based OFF control device presents a powerful tool that will enable researchers to engineer homeostasis in mammalian cells.

Electronic supplementary material

The online version of this article (doi:10.1186/s13036-015-0002-3) contains supplementary material, which is available to authorized users.

Molecular signaling network motifs provide a mechanistic basis for cellular threshold responses

BACKGROUND

Increasingly, there is a move toward using in vitro toxicity testing to assess human health risk due to chemical exposure. As with in vivo toxicity testing, an important question for in vitro results is whether there are thresholds for adverse cellular responses. Empirical evaluations may show consistency with thresholds, but the main evidence has to come from mechanistic considerations.

OBJECTIVES

Cellular response behaviors depend on the molecular pathway and circuitry in the cell and the manner in which chemicals perturb these circuits. Understanding circuit structures that are inherently capable of resisting small perturbations and producing threshold responses is an important step towards mechanistically interpreting in vitro testing data.

METHODS

Here we have examined dose-response characteristics for several biochemical network motifs. These network motifs are basic building blocks of molecular circuits underpinning a variety of cellular functions, including adaptation, homeostasis, proliferation, differentiation, and apoptosis. For each motif, we present biological examples and models to illustrate how thresholds arise from specific network structures.

DISCUSSION AND CONCLUSION

Integral feedback, feedforward, and transcritical bifurcation motifs can generate thresholds. Other motifs (e.g., proportional feedback and ultrasensitivity)produce responses where the slope in the low-dose region is small and stays close to the baseline. Feedforward control may lead to nonmonotonic or hormetic responses. We conclude that network motifs provide a basis for understanding thresholds for cellular responses. Computational pathway modeling of these motifs and their combinations occurring in molecular signaling networks will be a key element in new risk assessment approaches based on in vitro cellular assays.

Robust concentration and frequency control in oscillatory homeostats

Homeostatic and adaptive control mechanisms are essential for keeping organisms structurally and functionally stable. Integral feedback is a control theoretic concept which has long been known to keep a controlled variable A robustly (i.e. perturbation-independent) at a given set-point A(set) by feeding the integrated error back into the process that generates A. The classical concept of homeostasis as robust regulation within narrow limits is often considered as unsatisfactory and even incompatible with many biological systems which show sustained oscillations, such as circadian rhythms and oscillatory calcium signaling. Nevertheless, there are many similarities between the biological processes which participate in oscillatory mechanisms and classical homeostatic (non-oscillatory) mechanisms. We have investigated whether biological oscillators can show robust homeostatic and adaptive behaviors, and this paper is an attempt to extend the homeostatic concept to include oscillatory conditions. Based on our previously published kinetic conditions on how to generate biochemical models with robust homeostasis we found two properties, which appear to be of general interest concerning oscillatory and homeostatic controlled biological systems. The first one is the ability of these oscillators ("oscillatory homeostats") to keep the average level of a controlled variable at a defined set-point by involving compensatory changes in frequency and/or amplitude. The second property is the ability to keep the period/frequency of the oscillator tuned within a certain well-defined range. In this paper we highlight mechanisms that lead to these two properties. The biological applications of these findings are discussed using three examples, the homeostatic aspects during oscillatory calcium and p53 signaling, and the involvement of circadian rhythms in homeostatic regulation.

Retropath: automated pipeline for embedded metabolic circuits

Metabolic circuits are a promising alternative to other conventional genetic circuits as modular parts implementing functionalities required for synthetic biology applications. To date, metabolic design has been mainly focused on production circuits. Emergent applications such as smart therapeutics, however, require circuits that enable sensing and regulation. Here, we present RetroPath, an automated pipeline for embedded metabolic circuits that explores the circuit design space from a given set of specifications and selects the best circuits to implement based on desired constraints. Synthetic biology circuits embedded in a chassis organism that are capable of controlling the production, processing, sensing, and the release of specific molecules were enumerated in the metabolic space through a standard procedure. In that way, design and implementation of applications such as therapeutic circuits that autonomously diagnose and treat disease, are enabled, and their optimization is streamlined.

Design and characterization of a dual-mode promoter with activation and repression capability for tuning gene expression in yeast

Modularity in controlling gene expression artificially is becoming an essential aspect of synthetic biology. Artificial transcriptional control of gene expression is one of the most well-developed methods for the design of novel synthetic regulatory networks. Such networks are intended to help understand natural cellular phenomena and to enable new biotechnological applications. Promoter sequence manipulation with cis-regulatory elements is a key approach to control gene expression transcriptionally. Here, we have designed a promoter that can be both activated and repressed, as a contribution to the library of synthetic biological 'parts'. Starting with the minimal cytochrome C (minCYC) promoter in yeast, we incorporated five steroid hormone responsive elements (SHREs) and one lac operator site, respectively, upstream and downstream of the TATA box. This allows activation through the testosterone-responsive androgen receptor, and repression through the LacI repressor. Exposure to varying concentrations of testosterone (to vary activation) and IPTG (to vary repression) demonstrated the ability to tune the promoter's output curve over a wide range. By integrating activating and repressing signals, the promoter permits a useful form of signal integration, and we are optimistic that it will serve as a component in future regulatory networks, including feedback controllers.

Tuning response curves for synthetic biology

Synthetic biology may be viewed as an effort to establish, formalize, and develop an engineering discipline in the context of biological systems. The ability to tune the properties of individual components is central to the process of system design in all fields of engineering, and synthetic biology is no exception. A large and growing number of approaches have been developed for tuning the responses of cellular systems, and here we address specifically the issue of tuning the rate of response of a system: given a system where an input affects the rate of change of an output, how can the shape of the response curve be altered experimentally? This affects a system's dynamics as well as its steady-state properties, both of which are critical in the design of systems in synthetic biology, particularly those with multiple components. We begin by reviewing a mathematical formulation that captures a broad class of biological response curves and use this to define a standard set of varieties of tuning: vertical shifting, horizontal scaling, and the like. We then survey the experimental literature, classifying the results into our defined categories, and organizing them by regulatory level: transcriptional, post-transcriptional, and post-translational.