Systems biology of robustness and homeostatic mechanisms

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.

Artificial Homeostasis Systems Based on Feedback Reaction Networks: Design Principles and Future Promises

Feedback-controlled chemical reaction networks (FCRNs) are indispensable for various biological processes, such as cellular mechanisms, patterns, and signaling pathways. Through the intricate interplay of many feedback loops (FLs), FCRNs maintain a stable internal cellular environment. Currently, creating minimalistic synthetic cells is the long-term objective of systems chemistry, which is motivated by such natural integrity. The design, kinetic optimization, and analysis of FCRNs to exhibit functions akin to those of a cell still pose significant challenges. Indeed, reaching synthetic homeostasis is essential for engineering synthetic cell components. However, maintaining homeostasis in artificial systems against various agitations is a difficult task. Several biological events can provide us with guidelines for a conceptual understanding of homeostasis, which can be further applicable in designing artificial synthetic systems. In this regard, we organize our review with artificial homeostasis systems driven by FCRNs at different length scales, including homogeneous, compartmentalized, and soft material systems. First, we stretch a quick overview of FCRNs in different molecular and supramolecular systems, which are the essential toolbox for engineering different nonlinear functions and homeostatic systems. Moreover, the existing history of synthetic homeostasis in chemical and material systems and their advanced functions with self-correcting, and regulating properties are also emphasized.

A unified approach to dissecting biphasic responses in cell signaling

Biphasic responses are encountered at all levels in biological systems. At the cellular level, biphasic dose-responses are widely encountered in cell signaling and post-translational modification systems and represent safeguards against overactivation or overexpression of species. In this paper, we provide a unified theoretical synthesis of biphasic responses in cell signaling systems, by assessing signaling systems ranging from basic biochemical building blocks to canonical network structures to well-characterized exemplars on one hand, and examining different types of doses on the other. By using analytical and computational approaches applied to a range of systems across levels (described by broadly employed models), we reveal (i) design principles enabling the presence of biphasic responses, including in almost all instances, an explicit characterization of the parameter space (ii) structural factors which preclude the possibility of biphasic responses (iii) different combinations of the presence or absence of enzyme-biphasic and substrate-biphasic responses, representing safeguards against overactivation and overexpression, respectively (iv) the possibility of broadly robust biphasic responses (v) the complete alteration of signaling behavior in a network due to biphasic interactions between species (biphasic regulation) (vi) the propensity of different co-existing biphasic responses in the Erk signaling network. These results both individually and in totality have a number of important consequences for systems and synthetic biology.

A kinetic model of iron trafficking in growing Saccharomyces cerevisiae cells; applying mathematical methods to minimize the problem of sparse data and generate viable autoregulatory mechanisms

Iron is an essential transition metal for all eukaryotic cells, and its trafficking throughout the cell is highly regulated. However, the overall cellular mechanism of regulation is poorly understood despite knowing many of the molecular players involved. Here, an ordinary-differential-equations (ODE) based kinetic model of iron trafficking within a growing yeast cell was developed that included autoregulation. The 9-reaction 8-component in-silico cell model was solved under both steady-state and time-dependent dynamical conditions. The ODE for each component included a dilution term due to cell growth. Conserved rate relationships were obtained from the null space of the stoichiometric matrix, and the reduced-row-echelon-form was used to distinguish independent from dependent rates. Independent rates were determined from experimentally estimated component concentrations, cell growth rates, and the literature. Simple rate-law expressions were assumed, allowing rate-constants for each reaction to be estimated. Continuous Heaviside logistical functions were used to regulate rate-constants. These functions acted like valves, opening or closing depending on component "sensor" concentrations. Two cellular regulatory mechanisms were selected from 134,217,728 possibilities using a novel approach involving 6 mathematically-defined filters. Three cellular states were analyzed including healthy wild-type cells, iron-deficient wild-type cells, and a frataxin-deficient strain of cells characterizing the disease Friedreich's Ataxia. The model was stable toward limited perturbations, as determined by the eigenvalues of Jacobian matrices. Autoregulation allowed healthy cells to transition to the diseased state when triggered by a mutation in frataxin, and to the iron-deficient state when cells are placed in iron-deficient growth medium. The in-silico phenotypes observed during these transitions were similar to those observed experimentally. The model also predicted the observed effects of hypoxia on the diseased condition. A similar approach could be used to solve ODE-based kinetic models associated with other biochemical processes operating within growing cells.

Toward a universal measure of robustness across model organs and systems

The development of an individual must be capable of resisting the harmful effects of internal and external perturbations. This capacity, called robustness, can make the difference between normal variation and disease. Some systems and organs are more resilient in their capacity to correct the effects of internal disturbances such as mutations. Similarly, organs and organisms differ in their capacity to be resilient against external disturbances, such as changes in temperature. Furthermore, all developmental systems must be somewhat flexible to permit evolutionary change, and understanding robustness requires a comparative framework. Over the last decades, most research on developmental robustness has been focusing on specific model systems and organs. Hence, we lack tools that would allow cross-species and cross-organ comparisons. Here, we emphasize the need for a uniform framework to experimentally test and quantify robustness across study systems and suggest that the analysis of fluctuating asymmetry might be a powerful proxy to do so. Such a comparative framework will ultimately help to resolve why and how organs of the same and different species differ in their sensitivity to internal (e.g., mutations) and external (e.g., temperature) perturbations and at what level of biological organization buffering capacities exist and therefore create robustness of the developmental system.

Self-organized collective cell behaviors as design principles for synthetic developmental biology

Over the past two decades, molecular cell biology has graduated from a mostly analytic science to one with substantial synthetic capability. This success is built on a deep understanding of the structure and function of biomolecules and molecular mechanisms. For synthetic biology to achieve similar success at the scale of tissues and organs, an equally deep understanding of the principles of development is required. Here, we review some of the central concepts and recent progress in tissue patterning, morphogenesis and collective cell migration and discuss their value for synthetic developmental biology, emphasizing in particular the power of (guided) self-organization and the role of theoretical advances in making developmental insights applicable in synthesis.

Gene expression underlying parenting and being parented shows limited plasticity in response to different ambient temperatures

Flexible interactions between parents and offspring are essential for buffering families against variable, unpredictable, and challenging environmental conditions. In the subsocial carrion beetle, Nicrophorus orbicollis, mid-summer temperatures impose steep fitness costs on parents and offspring but do not elicit behavioural plasticity in parents. Here, we ask if plasticity of gene expression underpins this behavioural stability or facilitates independent compensation by larvae. To test this, we characterized gene expression of parents and offspring before and during active parenting under benign (20°C) and stressful (24°C) temperatures to identify genes of parents and offspring associated with thermal response, parenting/being parented, and gene expression plasticity associated with behavioural stability of parental care. The main effects of thermal and social condition each shaped patterns of gene expression in females, males, and larvae. In addition, we implicated 79 genes in females as "buffering" parental behaviour across environments. The majority of these underwent significant changes in expression in actively parenting mothers at the benign temperature, but not at the stressful temperature. Our results suggest that neither genetic programmes for parenting nor their effects on offspring gene expression are fundamentally different under stressful conditions, and that behavioural stability is associated primarily with the maintenance of existing genetic programmes rather than replacement or supplementation. Thus, while selection for compensatory gene expression could expand the range of thermal conditions parents will tolerate, without expanding the toolkit of genes involved selection is unlikely to lead to adaptive changes of function.

Energy Cost of Dynamical Stabilization: Stored versus Dissipated Energy

Dynamical stabilization processes (homeostasis) are ubiquitous in nature, but the needed energetic resources for their existence have not been studied systematically. Here, we undertake such a study using the famous model of Kapitza’s pendulum, which has attracted attention in the context of classical and quantum control. This model is generalized and rendered autonomous, and we show that friction and stored energy stabilize the upper (normally unstable) state of the pendulum. The upper state can be rendered asymptotically stable, yet it does not cost any constant dissipation of energy, and only a transient energy dissipation is needed. Asymptotic stability under a single perturbation does not imply stability with respect to multiple perturbations. For a range of pendulum–controller interactions, there is also a regime where constant energy dissipation is needed for stabilization. Several mechanisms are studied for the decay of dynamically stabilized states.

Principles of Endocrine Regulation: Reconciling Tensions Between Robustness in Performance and Adaptation to Change

Endocrine regulation in the hypothalamic-pituitary-thyroid (HPT) axis is orchestrated by physiological circuits which integrate multiple internal and external influences. Essentially, it provides either of the two responses to overt biological challenges: to defend the homeostatic range of a target hormone or adapt it to changing environmental conditions. Under certain conditions, such flexibility may exceed the capability of a simple feedback control loop, rather requiring more intricate networks of communication between the system's components. A new minimal mathematical model, in the form of a parametrized nonlinear dynamical system, is here formulated as a proof-of-concept to elucidate the principles of the HPT axis regulation. In particular, it allows uncovering mechanisms for the homeostasis of the key biologically active hormone free triiodothyronine (FT3). One mechanism supports the preservation of FT3 homeostasis, whilst the other is responsible for the adaptation of the homeostatic state to a new level. Together these allow optimum resilience in stressful situations. Preservation of FT3 homeostasis, despite changes in FT4 and TSH levels, is found to be an achievable system goal by joining elements of top-down and bottom-up regulation in a cascade of targeted feedforward and feedback loops. Simultaneously, the model accounts for the combination of properties regarded as essential to endocrine regulation, namely sensitivity, the anticipation of an adverse event, robustness, and adaptation. The model therefore offers fundamental theoretical insights into the effective system control of the HPT axis.

Senecavirus A Enhances Its Adaptive Evolution via Synonymous Codon Bias Evolution

Synonymous codon bias in the viral genome affects protein translation and gene expression, suggesting that the synonymous codon mutant plays an essential role in influencing virulence and evolution. However, how the recessive mutant form contributes to virus evolvability remains elusive. In this paper, we characterize how the Senecavirus A (SVA), a picornavirus, utilizes synonymous codon mutations to influence its evolution, resulting in the adaptive evolution of the virus to adverse environments. The phylogenetic tree and Median-joining (MJ)-Network of these SVA lineages worldwide were constructed to reveal SVA three-stage genetic development clusters. Furthermore, we analyzed the codon bias of the SVA genome of selected strains and found that SVA could increase the GC content of the third base of some amino acid synonymous codons to enhance the viral RNA adaptive evolution. Our results highlight the impact of recessive mutation of virus codon bias on the evolution of the SVA and uncover a previously underappreciated evolutionary strategy for SVA. They also underline the importance of understanding the genetic evolution of SVA and how SVA adapts to the adverse effects of external stress.

Towards organism-level systems biology by next-generation genetics and whole-organ cell profiling

The system-level identification and analysis of molecular and cellular networks in mammals can be accelerated by "next-generation" genetics, which is defined as genetics that can achieve desired genetic makeup in a single generation without any animal crossing. We recently established a highly efficient procedure for producing knock-out (KO) mice using the "Triple-CRISPR" method, which targets a single gene by triple gRNAs in the CRISPR/Cas9 system. This procedure achieved an almost perfect KO efficiency (96-100%). We also established a highly efficient procedure, the "ES-mouse" method, for producing knock-in (KI) mice within a single generation. In this method, ES cells were treated with three inhibitors to keep their potency and then injected into 8-cell-stage embryos. These procedures dramatically shortened the time required to produce KO or KI mice from years down to about 3 months. The produced KO and KI mice can also be systematically profiled at a single-cell resolution by the "whole-organ cell profiling," which was realized by tissue-clearing methods, such as CUBIC, and an advanced light-sheet microscopy. The review describes the establishment and application of these technologies above in analyzing the three states (NREM sleep, REM sleep, and awake) of mammalian brains. It also discusses the role of calcium and muscarinic receptors in these states as well as the current challenges and future opportunities in the next-generation mammalian genetics and whole-organ cell profiling for organism-level systems biology.

DOPA Homeostasis by Dopamine: A Control-Theoretic View

Dopamine (DA) is an important signal mediator in the brain as well as in the periphery. The term "dopamine homeostasis" occasionally found in the literature refers to the fact that abnormal DA levels can be associated with a variety of neuropsychiatric disorders. An analysis of the negative feedback inhibition of tyrosine hydroxylase (TH) by DA indicates, with support from the experimental data, that the TH-DA negative feedback loop has developed to exhibit 3,4-dihydroxyphenylalanine (DOPA) homeostasis by using DA as a derepression regulator. DA levels generally decline when DOPA is removed, for example, by increased oxidative stress. Robust DOPA regulation by DA further implies that maximum vesicular DA levels are established, which appear necessary for a reliable translation of neural activity into a corresponding chemical transmitter signal. An uncontrolled continuous rise (windup) in DA occurs when Levodopa treatment exceeds a critical dose. Increased oxidative stress leads to the successive breakdown of DOPA homeostasis and to a corresponding reduction in DA levels. To keep DOPA regulation robust, the vesicular DA loading requires close to zero-order kinetics combined with a sufficiently high compensatory flux provided by TH. The protection of DOPA and DA due to a channeling complex is discussed.

Retinoic Acid Fluctuation Activates an Uneven, Direction-Dependent Network-Wide Robustness Response in Early Embryogenesis

Robustness is a feature of regulatory pathways to ensure signal consistency in light of environmental changes or genetic polymorphisms. The retinoic acid (RA) pathway, is a central developmental and tissue homeostasis regulatory signal, strongly dependent on nutritional sources of retinoids and affected by environmental chemicals. This pathway is characterized by multiple proteins or enzymes capable of performing each step and their integration into a self-regulating network. We studied RA network robustness by transient physiological RA signaling disturbances followed by kinetic transcriptomic analysis of the recovery during embryogenesis. The RA metabolic network was identified as the main regulated module to achieve signaling robustness using an unbiased pattern analysis. We describe the network-wide responses to RA signal manipulation and found the feedback autoregulation to be sensitive to the direction of the RA perturbation: RA knockdown exhibited an upper response limit, whereas RA addition had a minimal feedback-activation threshold. Surprisingly, our robustness response analysis suggests that the RA metabolic network regulation exhibits a multi-objective optimization, known as Pareto optimization, characterized by trade-offs between competing functionalities. We observe that efficient robustness to increasing RA is accompanied by worsening robustness to reduced RA levels and vice versa. This direction-dependent trade-off in the network-wide feedback response, results in an uneven robustness capacity of the RA network during early embryogenesis, likely a significant contributor to the manifestation of developmental defects.

Uncovering the Role of Metabolism in Oomycete-Host Interactions Using Genome-Scale Metabolic Models

Metabolism is the set of biochemical reactions of an organism that enables it to assimilate nutrients from its environment and to generate building blocks for growth and proliferation. It forms a complex network that is intertwined with the many molecular and cellular processes that take place within cells. Systems biology aims to capture the complexity of cells, organisms, or communities by reconstructing models based on information gathered by high-throughput analyses (omics data) and prior knowledge. One type of model is a genome-scale metabolic model (GEM) that allows studying the distributions of metabolic fluxes, i.e., the "mass-flow" through the network of biochemical reactions. GEMs are nowadays widely applied and have been reconstructed for various microbial pathogens, either in a free-living state or in interaction with their hosts, with the aim to gain insight into mechanisms of pathogenicity. In this review, we first introduce the principles of systems biology and GEMs. We then describe how metabolic modeling can contribute to unraveling microbial pathogenesis and host-pathogen interactions, with a specific focus on oomycete plant pathogens and in particular Phytophthora infestans. Subsequently, we review achievements obtained so far and identify and discuss potential pitfalls of current models. Finally, we propose a workflow for reconstructing high-quality GEMs and elaborate on the resources needed to advance a system biology approach aimed at untangling the intimate interactions between plants and pathogens.

Resolving the Rules of Robustness and Resilience in Biology Across Scales

Why do some biological systems and communities persist while others fail? Robustness, a system's stability, and resilience, the ability to return to a stable state, are key concepts that span multiple disciplines within and outside the biological sciences. Discovering and applying common rules that govern the robustness and resilience of biological systems is a critical step toward creating solutions for species survival in the face of climate change, as well as the for the ever-increasing need for food, health, and energy for human populations. We propose that network theory provides a framework for universal scalable mathematical models to describe robustness and resilience and the relationship between them, and hypothesize that resilience at lower organization levels contribute to robust systems. Insightful models of biological systems can be generated by quantifying the mechanisms of redundancy, diversity, and connectivity of networks, from biochemical processes to ecosystems. These models provide pathways towards understanding how evolvability can both contribute to and result from robustness and resilience under dynamic conditions. We now have an abundance of data from model and non-model systems and the technological and computational advances for studying complex systems. Several conceptual and policy advances will allow the research community to elucidate the rules of robustness and resilience. Conceptually, a common language and data structure that can be applied across levels of biological organization needs to be developed. Policy advances such as cross-disciplinary funding mechanisms, access to affordable computational capacity, and the integration of network theory and computer science within the standard biological science curriculum will provide the needed research environments. This new understanding of biological systems will allow us to derive ever more useful forecasts of biological behaviors and revolutionize the engineering of biological systems that can survive changing environments or disease, navigate the deepest oceans, or sustain life throughout the solar system.

Testing Theoretical Minimal Genomes Using Whole-Cell Models

The minimal gene set for life has often been theorized, with at least ten produced for Mycoplasma genitalium (M. genitalium). Due to the difficulty of using M. genitalium in the lab, combined with its long replication time of 12-15 h, none of these theoretical minimal genomes have been tested, even with modern techniques. The publication of the M. genitalium whole-cell model provided the first opportunity to test them, simulating the genome edits in silico. We simulated minimal gene sets from the literature, finding that they produced in silico cells that did not divide. Using knowledge from previous research, we reintroduced specific essential and low essential genes in silico; enabling cellular division. This reinforces the need to identify species-specific low essential genes and their interactions. Any genome designs created using the currently incomplete and fragmented gene essentiality information will very likely require in vivo reintroductions to correct issues and produce dividing cells.

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).

Tumor-stroma biomechanical crosstalk: a perspective on the role of caveolin-1 in tumor progression

Tumor stiffening is a hallmark of malignancy that actively drives tumor progression and aggressiveness. Recent research has shed light onto several molecular underpinnings of this biomechanical process, which has a reciprocal crosstalk between tumor cells, stromal fibroblasts, and extracellular matrix remodeling at its core. This dynamic communication shapes the tumor microenvironment; significantly determines disease features including therapeutic resistance, relapse, or metastasis; and potentially holds the key for novel antitumor strategies. Caveolae and their components emerge as integrators of different aspects of cell function, mechanotransduction, and ECM-cell interaction. Here, we review our current knowledge on the several pivotal roles of the essential caveolar component caveolin-1 in this multidirectional biomechanical crosstalk and highlight standing questions in the field.

Intermediate progenitors support migration of neural stem cells into dentate gyrus outer neurogenic niches

The hippocampal dentate gyrus (DG) is a unique brain region maintaining neural stem cells (NCSs) and neurogenesis into adulthood. We used multiphoton imaging to visualize genetically defined progenitor subpopulations in live slices across key stages of mouse DG development, testing decades old static models of DG formation with molecular identification, genetic-lineage tracing, and mutant analyses. We found novel progenitor migrations, timings, dynamic cell-cell interactions, signaling activities, and routes underlie mosaic DG formation. Intermediate progenitors (IPs, Tbr2+) pioneered migrations, supporting and guiding later emigrating NSCs (Sox9+) through multiple transient zones prior to converging at the nascent outer adult niche in a dynamic settling process, generating all prenatal and postnatal granule neurons in defined spatiotemporal order. IPs (Dll1+) extensively targeted contacts to mitotic NSCs (Notch active), revealing a substrate for cell-cell contact support during migrations, a developmental feature maintained in adults. Mouse DG formation shares conserved features of human neocortical expansion.

Competitive Exclusion and Evolution: Convergence Almost Never Produces Ecologically Equivalent Species: (A Comment on McPeek, "Limiting Similarity? The Ecological Dynamics of Natural Selection among Resources and Consumers Caused by Both Apparent and Resource Competition")

In a recent modeling study ("Limiting Similarity? The Ecological Dynamics of Natural Selection among Resources and Consumers Caused by Both Apparent and Resource Competition") that appeared in the April 2019 issue of The American Naturalist, Mark A. McPeek argued that ecologically equivalent species may emerge via competition-induced trait convergence, in conflict with naive expectations based on the limiting similarity principle. Although the emphasis on the possibility of the convergence of competitors is very timely, here we show that the proposed mechanism will only lead to actual coexistence in the converged state for specially chosen fine-tuned parameter settings. It is therefore not a robust mechanism for the evolution of ecologically equivalent species. We conclude that invoking trait convergence as an explanation for the co-occurrence of seemingly fully equivalent species in nature would be premature.

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.