The structure of infinitesimal homeostasis in input-output networks

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.

Design Principles for Perfect Adaptation in Biological Networks with Nonlinear Dynamics

Establishing a mapping between the emergent biological properties and the repository of network structures has been of great relevance in systems and synthetic biology. Adaptation is one such biological property of paramount importance that promotes regulation in the presence of environmental disturbances. This paper presents a nonlinear systems theory-driven framework to identify the design principles for perfect adaptation with respect to external disturbances of arbitrary magnitude. Based on the prior information about the network, we frame precise mathematical conditions for adaptation using nonlinear systems theory. We first deduce the mathematical conditions for perfect adaptation for constant input disturbances. Subsequently, we translate these conditions to specific necessary structural requirements for adaptation in networks of small size and then extend to argue that there exist only two classes of architectures for a network of any size that can provide local adaptation in the entire state space, namely, incoherent feed-forward (IFF) structure and negative feedback loop with buffer node (NFB). The additional positiveness constraints further narrow the admissible set of network structures. This also aids in establishing the global asymptotic stability for the steady state given a constant input disturbance. The proposed method does not assume any explicit knowledge of the underlying rate kinetics, barring some minimal assumptions. Finally, we also discuss the infeasibility of certain IFF networks in providing adaptation in the presence of downstream connections. Moreover, we propose a generic and novel algorithm based on non-linear systems theory to unravel the design principles for global adaptation. Detailed and extensive simulation studies corroborate the theoretical findings.

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.

Design Principles for Biological Adaptation: A Systems and Control-Theoretic Treatment

Establishing a mapping between (from and to) the functionality of interest and the underlying network structure (design principles) remains a crucial step toward understanding and design of bio-systems. Perfect adaptation is one such crucial functionality that enables every living organism to regulate its essential activities in the presence of external disturbances. Previous approaches to deducing the design principles for adaptation have either relied on computationally burdensome brute-force methods or rule-based design strategies detecting only a subset of all possible adaptive network structures. This chapter outlines a scalable and generalizable method inspired by systems theory that unravels an exhaustive set of adaptation-capable structures. We first use the well-known performance parameters to characterize perfect adaptation. These performance parameters are then mapped back to a few parameters (poles, zeros, gain) characteristic of the underlying dynamical system constituted by the rate equations. Therefore, the performance parameters evaluated for the scenario of perfect adaptation can be expressed as a set of precise mathematical conditions involving the system parameters. Finally, we use algebraic graph theory to translate these abstract mathematical conditions to certain structural requirements for adaptation. The proposed algorithm does not assume any particular dynamics and is applicable to networks of any size. Moreover, the results offer a significant advancement in the realm of understanding and designing complex biochemical networks.

Advancing physiology education by understanding the multiple dimensions of homeostasis

Homeostasis of the internal environment has been considered the central organizing concept of physiology. However, current definitions of it in textbooks and online teaching sources do not sufficiently reflect how homeostasis serves its central unifying role. Meanwhile, scientific understanding of the functions of the body's structures at multiple levels (molecular, cell, tissue, organ, organ system, and organism) has advanced significantly, but the understanding of homeostasis is still in the same place. In this article, the author describes some issues and insufficiencies in teaching about homeostasis in physiology education and proposes that homeostasis needs to be understood in terms of four dimensions rather than a simple definition: internal, functional organization; functional manifestation; mechanism; and effect or consequence. Each dimension has two subdimensions or sides. Throughout the elucidation of these dimensions and subdimensions, the original meaning of homeostasis is reinforced, what is lost in current understanding of homeostasis becomes clear, some insufficiencies mentioned above are supplemented, new insights into homeostasis develop, and how the four dimensions of homeostasis can be applied to physiology education is exampled. This new, comprehensive conceptualization advances the understanding of homeostasis and can facilitate teaching and learning about homeostasis and physiology.

On biological networks capable of robust adaptation in the presence of uncertainties: A linear systems-theoretic approach

Biological adaptation, the tendency of every living organism to regulate its essential activities in environmental fluctuations, is a well-studied functionality in systems and synthetic biology. In this work, we present a generic methodology inspired by systems theory to discover the design principles for robust adaptation, perfect and imperfect, in two different contexts: (1) in the presence of deterministic external and parametric disturbances and (2) in a stochastic setting. In all the cases, firstly, we translate the necessary qualitative conditions for adaptation to mathematical constraints using the language of systems theory, which we then map back as design requirements for the underlying networks. Thus, contrary to the existing approaches, the proposed methodologies provide an exhaustive set of admissible network structures without resorting to computationally burdensome brute-force techniques. Further, the proposed frameworks do not assume prior knowledge about the particular rate kinetics, thereby validating the conclusions for a large class of biological networks. In the deterministic setting, we show that unlike the incoherent feed-forward network structures (IFFLP or opposer modules), the modules containing negative feedback with buffer action (NFBLB or balancer modules) are robust to parametric fluctuations when a specific part of the network is assumed to remain unaffected. To this end, we propose a sufficient condition for imperfect adaptation and show that adding negative feedback in an IFFLP topology improves the robustness concerning parametric fluctuations. Further, we propose a stricter set of necessary conditions for imperfect adaptation. Turning to the stochastic scenario, we adopt a Wiener-Kolmogorov filter strategy to tune the parameters of a given network structure towards minimum output variance. We show that both NFBLB and IFFLP can be used as a reduced-order W-K filter. Further, we define the notion of nearest neighboring motifs to compare the output variances across different network structures. We argue that the NFBLB achieves adaptation at the cost of a variance higher than its nearest neighboring motifs whereas the IFFLP topology produces locally minimum variance while compared with its nearest neighboring motifs. We present numerical simulations to support the theoretical results. Overall, our results present a generic, systematic, and robust framework for advancing the understanding of complex biological networks.

Homeostasis and injectivity: a reaction network perspective

Homeostasis represents the idea that a feature may remain invariant despite changes in some external parameters. We establish a connection between homeostasis and injectivity for reaction network models. In particular, we show that a reaction network cannot exhibit homeostasis if a modified version of the network (which we call homeostasis-associated network) is injective. We provide examples of reaction networks which can or cannot exhibit homeostasis by analyzing the injectivity of their homeostasis-associated networks.

Universal structural requirements for maximal robust perfect adaptation in biomolecular networks

Adaptation is a running theme in biology. It allows a living system to survive and thrive in the face of unpredictable environments by maintaining key physiological variables at their desired levels through tight regulation. When one such variable is maintained at a certain value at the steady state despite perturbations to a single input, this property is called robust perfect adaptation (RPA). Here we address and solve the fundamental problem of maximal RPA (maxRPA), whereby, for a designated output variable, RPA is achieved with respect to perturbations in virtually all network parameters. In particular, we show that the maxRPA property imposes certain structural constraints on the network. We then prove that these constraints are fully characterized by simple linear algebraic stoichiometric conditions which differ between deterministic and stochastic descriptions of the dynamics. We use our results to derive a new internal model principle (IMP) for biomolecular maxRPA networks, akin to the celebrated IMP in control theory. We exemplify our results through several known biological examples of robustly adapting networks and construct examples of such networks with the aid of our linear algebraic characterization. Our results reveal the universal requirements for maxRPA in all biological systems, and establish a foundation for studying adaptation in general biomolecular networks, with important implications for both systems and synthetic biology.

Classification of infinitesimal homeostasis in four-node input-output networks

An input-output network has an input node [Formula: see text], an output node o, and regulatory nodes [Formula: see text]. Such a network is a core network if each [Formula: see text] is downstream from [Formula: see text] and upstream from o. Wang et al. (J Math Biol 82:62, 2021. https://doi.org/10.1007/s00285-021-01614-1 ) show that infinitesimal homeostasis can be classified in biochemical networks through infinitesimal homeostasis in core subnetworks. Golubitsky and Wang (J Math Biol 10:1-23, 2020) show that there are three types of 3-node core networks and three types of infinitesimal homeostasis in 3-node core networks. This paper uses the theory developed in Wang et al. (2021) to show that there are twenty types of 4-node core networks (Theorem 1.3) and seventeen types of infinitesimal homeostasis in 4-node core networks (Theorem 1.7). Biological contexts illustrate the classification theorems and show that the theory can be an aid when calculating homeostasis in specific biochemical networks.

A homeostasis criterion for limit cycle systems based on infinitesimal shape response curves

Homeostasis occurs in a control system when a quantity remains approximately constant as a parameter, representing an external perturbation, varies over some range. Golubitsky and Stewart (J Math Biol 74(1-2):387-407, 2017) developed a notion of infinitesimal homeostasis for equilibrium systems using singularity theory. Rhythmic physiological systems (breathing, locomotion, feeding) maintain homeostasis through control of large-amplitude limit cycles rather than equilibrium points. Here we take an initial step to study (infinitesimal) homeostasis for limit-cycle systems in terms of the average of a quantity taken around the limit cycle. We apply the "infinitesimal shape response curve" (iSRC) introduced by Wang et al. (SIAM J Appl Dyn Syst 82(7):1-43, 2021) to study infinitesimal homeostasis for limit-cycle systems in terms of the mean value of a quantity of interest, averaged around the limit cycle. Using the iSRC, which captures the linearized shape displacement of an oscillator upon a static perturbation, we provide a formula for the derivative of the averaged quantity with respect to the control parameter. Our expression allows one to identify homeostasis points for limit cycle systems in the averaging sense. We demonstrate in the Hodgkin-Huxley model and in a metabolic regulatory network model that the iSRC-based method provides an accurate representation of the sensitivity of averaged quantities.

Discovering adaptation-capable biological network structures using control-theoretic approaches

Constructing biological networks capable of performing specific biological functionalities has been of sustained interest in synthetic biology. Adaptation is one such ubiquitous functional property, which enables every living organism to sense a change in its surroundings and return to its operating condition prior to the disturbance. In this paper, we present a generic systems theory-driven method for designing adaptive protein networks. First, we translate the necessary qualitative conditions for adaptation to mathematical constraints using the language of systems theory, which we then map back as ‘design requirements’ for the underlying networks. We go on to prove that a protein network with different input–output nodes (proteins) needs to be at least of third-order in order to provide adaptation. Next, we show that the necessary design principles obtained for a three-node network in adaptation consist of negative feedback or a feed-forward realization. We argue that presence of a particular class of negative feedback or feed-forward realization is necessary for a network of any size to provide adaptation. Further, we claim that the necessary structural conditions derived in this work are the strictest among the ones hitherto existed in the literature. Finally, we prove that the capability of producing adaptation is retained for the admissible motifs even when the output node is connected with a downstream system in a feedback fashion. This result explains how complex biological networks achieve robustness while keeping the core motifs unchanged in the context of a particular functionality. We corroborate our theoretical results with detailed and thorough numerical simulations. Overall, our results present a generic, systematic and robust framework for designing various kinds of biological networks.

Biological systems display a remarkable diversity of functionalities, many of which can be conceived as the response of a large network composed of small interconnecting modules. Unravelling the connection pattern, i.e. design principles, behind important biological functionalities is one of the most challenging problems in systems biology. One such phenomenon is perfect adaptation, which merits special attention owing to its universal presence ranging from chemotaxis in bacterial cells to calcium homeostasis in mammalian cells. The present work focuses on finding the design principles for perfect adaptation in the presence of a stair-case type disturbance. To this end, the current work proposes a systems-theoretic approach to deduce precise mathematical (hence structural) conditions that comply with the key performance parameters for adaptation. The approach is agnostic to the particularities of the reaction kinetics, underlining the dominant role of the topological structure on the response of the network. Notably, the design principles obtained in this work serve as the most strict necessary structural conditions for a network of any size to provide perfect adaptation.