Structural identifiability of biomolecular controller motifs with and without flow measurements as model output.

Controller motifs are simple biomolecular reaction networks with negative feedback. They can explain how regulatory function is achieved and are often used as building blocks in mathematical models of biological systems. In this paper we perform an extensive investigation into structural identifiability of controller motifs, specifically the so-called basic and antithetic controller motifs. Structural identifiability analysis is a useful tool in the creation and evaluation of mathematical models: it can be used to ensure that model parameters can be determined uniquely and to examine which measurements are necessary for this purpose. This is especially useful for biological models where parameter estimation can be difficult due to limited availability of measureable outputs. Our aim with this work is to investigate how structural identifiability is affected by controller motif complexity and choice of measurements. To increase the number of potential outputs we propose two methods for including flow measurements and show how this affects structural identifiability in combination with, or in the absence of, concentration measurements. In our investigation, we analyze 128 different controller motif structures using a combination of flow and/or concentration measurements, giving a total of 3648 instances. Among all instances, 34% of the measurement combinations provided structural identifiability. Our main findings for the controller motifs include: i) a single measurement is insufficient for structural identifiability, ii) measurements related to different chemical species are necessary for structural identifiability. Applying these findings result in a reduced subset of 1568 instances, where 80% are structurally identifiable, and more complex/interconnected motifs appear easier to structurally identify. The model structures we have investigated are commonly used in models of biological systems, and our results demonstrate how different model structures and measurement combinations affect structural identifiability of controller motifs.

Homeostatic controllers compensating for growth and perturbations.

Cells and organisms have developed homeostatic mechanisms which protect them against a changing environment. How growth and homeostasis interact is still not well understood, but of increasing interest to the molecular and synthetic biology community to recognize and design control circuits which can oppose the diluting effects of cell growth. In this paper we describe the performance of selected negative feedback controllers in response to different applied growth laws and time dependent outflow perturbations of a controlled variable. The approach taken here is based on deterministic mass action kinetics assuming that cell content is instantaneously mixed. All controllers behave ideal in the sense that they for step-wise perturbations in volume and a controlled compound A are able to drive A precisely back to the controllers' theoretical set-points. The applied growth kinetics reflect experimentally observed growth laws, which range from surface to volume ratio growth to linear and exponential growth. Our results show that the kinetic implementation of integral control and the structure of the negative feedback loop are two properties which affect controller performance. Best performance is observed for controllers based on derepression kinetics and controllers with an autocatalytic implementation of integral control. Both are able to defend exponential growth and perturbations, although the autocatalytic controller shows an offset from its theoretical set-point. Controllers with activating signaling using zero-order or bimolecular (antithetic) kinetics for integral control behave very similar but less well. Their performance can be improved by implementing negative feedback structures having repression/derepression steps or by increasing controller aggressiveness. Our results provide a guide what type of feedback structures and integral control kinetics are suitable to oppose the dilution effects by different growth laws and time dependent perturbations on a deterministic level.

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.

Performance of Homeostatic Controller Motifs Dealing with Perturbations of Rapid Growth and Depletion.

An essential property of life is that cells and organisms have the ability to protect themselves against external disturbances/attacks by using homeostatic mechanisms. These defending mechanisms are based on negative feedback regulation and often contain additional features, such as integral control, where the integrated error between a controlled variable and its set-point is used to achieve homeostasis. Although the concept of integral control has its origin in industrial processes, recent findings suggest that biological systems are also capable of showing integral control. We recently described a basic set of negative feedback structures (controller motifs) where robust homeostasis is achieved against different but constant perturbations. As many perturbations in biology, such as infections, increase rapidly over time, we investigated how the different controller motifs equipped with different implementations of integral control perform in relation to rapidly changing perturbations, including exponential and hyperbolic changes. The findings show that the construction of an optimum biochemical controller design for time-dependent perturbations requires a certain match between the structure of the negative feedback loop, its signaling kinetics, and the kinetics of how integral control is implemented within the negative feedback loop.

Modeling the diversion of primary carbon flux into secondary metabolism under variable nitrate and light/dark conditions.

In plants, the partitioning of carbon resources between growth and defense is detrimental for their development. From a metabolic viewpoint, growth is mainly related to primary metabolism including protein, amino acid and lipid synthesis, whereas defense is based notably on the biosynthesis of a myriad of secondary metabolites. Environmental factors, such as nitrate fertilization, impact the partitioning of carbon resources between growth and defense. Indeed, experimental data showed that a shortage in the nitrate fertilization resulted in a reduction of the plant growth, whereas some secondary metabolites involved in plant defense, such as phenolic compounds, accumulated. Interestingly, sucrose, a key molecule involved in the transport and partitioning of carbon resources, appeared to be under homeostatic control. Based on the inflow/outflow properties of sucrose homeostatic regulation we propose a global model on how the diversion of the primary carbon flux into the secondary phenolic pathways occurs at low nitrate concentrations. The model can account for the accumulation of starch during the light phase and the sucrose remobilization by starch degradation during the night. Day-length sensing mechanisms for variable light-dark regimes are discussed, showing that growth is proportional to the length of the light phase. The model can describe the complete starch consumption during the night for plants adapted to a certain light/dark regime when grown on sufficient nitrate and can account for an increased accumulation of starch observed under nitrate limitation.

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.

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.

Transepithelial glucose transport and Na+/K+ homeostasis in enterocytes: an integrative model.

The uptake of glucose and the nutrient coupled transcellular sodium traffic across epithelial cells in the small intestine has been an ongoing topic in physiological research for over half a century. Driving the uptake of nutrients like glucose, enterocytes must have regulatory mechanisms that respond to the considerable changes in the inflow of sodium during absorption. The Na-K-ATPase membrane protein plays a major role in this regulation. We propose the hypothesis that the amount of active Na-K-ATPase in enterocytes is directly regulated by the concentration of intracellular Na(+) and that this regulation together with a regulation of basolateral K permeability by intracellular ATP gives the enterocyte the ability to maintain ionic Na(+)/K(+) homeostasis. To explore these regulatory mechanisms, we present a mathematical model of the sodium coupled uptake of glucose in epithelial enterocytes. Our model integrates knowledge about individual transporter proteins including apical SGLT1, basolateral Na-K-ATPase, and GLUT2, together with diffusion and membrane potentials. The intracellular concentrations of glucose, sodium, potassium, and chloride are modeled by nonlinear differential equations, and molecular flows are calculated based on experimental kinetic data from the literature, including substrate saturation, product inhibition, and modulation by membrane potential. Simulation results of the model without the addition of regulatory mechanisms fit well with published short-term observations, including cell depolarization and increased concentration of intracellular glucose and sodium during increased concentration of luminal glucose/sodium. Adding regulatory mechanisms for regulation of Na-K-ATPase and K permeability to the model show that our hypothesis predicts observed long-term ionic homeostasis.

Integrating fluctuating nitrate uptake and assimilation to robust homeostasis.

Nitrate is an important nitrogen source used by plants. Despite of the considerable variation in the amount of soil nitrate, plants keep cytosolic nitrate at a homeostatic controlled level. Here we describe a set of homeostatic controller motifs and their interaction that can maintain robust cytosolic nitrate homeostasis at fluctuating external nitrate concentrations and nitrate assimilation levels. The controller motifs are divided into two functional classes termed as inflow and outflow controllers. In the presence of high amounts of environmental nitrate, the function of outflow controllers is associated to efflux mechanisms removing excess of nitrate from the cytosol that is taken up by low-affinity transporter systems (LATS). Inflow controllers on the other hand maintain homeostasis in the presence of a high demand of nitrate by the cell relative to the amount of available environmental nitrate. This is achieved by either remobilizing nitrate from a vacuolar store, or by taking up nitrate by means of high-affinity transporter systems (HATS). By combining inflow and outflow controllers we demonstrate how nitrate uptake, assimilation, storage and efflux are integrated to a regulatory network that maintains cytosolic nitrate homeostasis at changing environmental conditions.

On the relationship between sensitivity coefficients and transfer functions of reaction kinetic networks.

Metabolic control analysis (MCA) and biochemical systems theory (BST) have become established methods when analyzing the behavior/kinetics of biochemical reaction systems. While the usage of MCA and BST involves the determination of sensitivities, e.g., steady state control coefficients (CCs), typically between reaction rates and concentrations/fluxes, transfer functions (TFs) from control engineering allow to analyze the connectivity between arbitrary input signals (e.g., rate constants or temperature) and arbitrary output signals (e.g., concentrations or fluxes) in the complex-valued s- or frequency domain. As CCs generally do not provide information about the connectivity between input and output signals, we became interested in the question of how CCs and TFs, or more generally, how arbitrary sensitivity coefficients (SCs) and TFs are related to each other. In this work, we describe a general relationship between SCs and their corresponding TFs from a general kinetic (state space) approach and show that the state space approach can describe the SC-TF relationship by a single equation. During our work, we became aware of an alternative method which relates CCs and TFs by using a stoichiometric network approach. In this work, we describe a procedure to identify conditions to determine whether a receptor-mediated input to a reaction kinetic network can show robust (perturbation independent) or nonrobust (balanced or perturbation dependent) adaptive or homeostatic behavior in an output. Compared to the stoichiometric network approach, the here described method allows for dealing with arbitrary (including empirically identified) kinetic expressions.

Studying adaptation and homeostatic behaviors of kinetic networks by using MATLAB.

Organisms have the ability to counteract environmental perturbations and keep certain components within a cell homeostatically regulated. Closely related to homeostasis is the behavior of perfect adaptation where an organism responds to a step-wise perturbation by regulating some of its components, after a transient period, to their original pre-perturbation values. A particular interesting type of model relates to the so-called robust behavior where the homeostatic or perfect adaptation property is independent of the magnitude of the applied step-wise perturbation. It has been shown that this type of behavior is related to the control-theoretic concept of integral feedback (or integral control). Using downloadable MATLAB examples, we demonstrate how robust perfect adaptation sites can be identified in reaction kinetic networks by linearizing the system, applying the Laplace transform and inspecting the transfer function. We also show how the homeostatic set point in perfect adaptation is related to the presence of zero-order fluxes.

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.

Predicting perfect adaptation motifs in reaction kinetic networks.

Adaptation and compensation mechanisms are important to keep organisms fit in a changing environment. "Perfect adaptation" describes an organism's response to an external stepwise perturbation by resetting some of its variables precisely to their original preperturbation values. Examples of perfect adaptation are found in bacterial chemotaxis, photoreceptor responses, or MAP kinase activities. Two concepts have evolved for how perfect adaptation may be understood. In one approach, so-called "robust perfect adaptation", the adaptation is a network property (due to integral feedback control), which is independent of rate constant values. In the other approach, which we have termed "nonrobust perfect adaptation", a fine-tuning of rate constant values is needed to show perfect adaptation. Although integral feedback describes robust perfect adaptation in general terms, it does not directly show where in a network perfect adaptation may be observed. Using control theoretic methods, we are able to predict robust perfect adaptation sites within reaction kinetic networks and show that a prerequisite for robust perfect adaptation is that the network is open and irreversible. We applied the method on various reaction schemes and found that new (robust) perfect adaptation motifs emerge when considering suggested models of bacterial and eukaryotic chemotaxis.