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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., Competing Interests: The authors have declared that no competing interests exist., (Copyright: © 2023 Haus et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.)

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., (© 2017 The Authors. Physiological Reports published by Wiley Periodicals, Inc. on behalf of The Physiological Society and the American Physiological Society.)

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., (Copyright © 2016 Elsevier Ltd. All rights reserved.)

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.

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., (Copyright © 2014 the American Physiological Society.)

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., (© 2011 American Chemical Society)

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.