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.

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.

Next Steps in Integrative Biology: Mapping Interactive Processes Across Levels of Biological Organization

Emergent biological processes result from complex interactions within and across levels of biological organization, ranging from molecular to environmental dynamics. Powerful theories, database tools, and modeling methods have been designed to characterize network connections within levels, such as those among genes, proteins, biochemicals, cells, organisms and species. Here, we propose that developing integrative models of organismal function in complex environments can be facilitated by taking advantage of these methods to identify key nodes of communication across levels of organization. Mapping key drivers or connections among levels of organization will provide data and leverage to model potential rule-sets by which organisms respond and adjust to perturbations at any level of biological organization.

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).

Genetic assimilation and accommodation: Models and mechanisms

Genetic assimilation and genetic accommodation are mechanisms by which novel phenotypes are produced and become established in a population. Novel characters may be fixed and canalized so they are insensitive to environmental variation, or can be plastic and adaptively responsive to environmental variation. In this review we explore the various theories that have been proposed to explain the developmental origin and evolution of novel phenotypes and the mechanisms by which canalization and phenotypic plasticity evolve. These theories and models range from conceptual to mathematical and have taken different views of how genes and environment contribute to the development and evolution of the properties of phenotypes. We will argue that a deeper and more nuanced understanding of genetic accommodation requires a recognition that phenotypes are not static entities but are dynamic system properties with no fixed deterministic relationship between genotype and phenotype. We suggest a mechanistic systems-view of development that allows one to incorporate both genes and environment in a common model, and that enables both quantitative analysis and visualization of the evolution of canalization and phenotypic plasticity.

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.

Rat liver folate metabolism can provide an independent functioning of associated metabolic pathways

Folate metabolism in mammalian cells is essential for multiple vital processes, including purine and pyrimidine synthesis, histidine catabolism, methionine recycling, and utilization of formic acid. It remains unknown, however, whether these processes affect each other via folate metabolism or can function independently based on cellular needs. We addressed this question using a quantitative mathematical model of folate metabolism in rat liver cytoplasm. Variation in the rates of metabolic processes associated with folate metabolism (i.e., purine and pyrimidine synthesis, histidine catabolism, and influxes of formate and methionine) in the model revealed that folate metabolism is organized in a striking manner that enables activation or inhibition of each individual process independently of the metabolic fluxes in others. In mechanistic terms, this independence is based on the high activities of a group of enzymes involved in folate metabolism, which efficiently maintain close-to-equilibrium ratios between substrates and products of enzymatic reactions.

Coordinating Development: How Do Animals Integrate Plastic and Robust Developmental Processes?

Our developmental environment significantly affects myriad aspects of our biology, including key life history traits, morphology, physiology, and our susceptibility to disease. This environmentally-induced variation in phenotype is known as plasticity. In many cases, plasticity results from alterations in the rate of synthesis of important developmental hormones. However, while developmental processes like organ growth are sensitive to environmental conditions, others like patterning - the process that generates distinct cell identities - remain robust to perturbation. This is particularly surprising given that the same hormones that regulate organ growth also regulate organ patterning. In this review, we revisit the current approaches that address how organs coordinate their growth and pattern, and outline our hypotheses for understanding how organs achieve correct pattern across a range of sizes.

Nutrition, the visceral immune system, and the evolutionary origins of pathogenic obesity

The global obesity epidemic is the subject of an immense, diversely specialized research effort. An evolutionary analysis reveals connections among disparate findings, starting with two well-documented facts: Obesity-associated illnesses (e.g., type-2 diabetes and cardiovascular disease), are especially common in: (i) adults with abdominal obesity, especially enlargement of visceral adipose tissue (VAT), a tissue with important immune functions; and (ii) individuals with poor fetal nutrition whose nutritional input increases later in life. I hypothesize that selection favored the evolution of increased lifelong investment in VAT in individuals likely to suffer lifelong malnutrition because of its importance in fighting intraabdominal infections. Then, when increased nutrition violates the adaptive fetal prediction of lifelong nutritional deficit, preferential VAT investment could contribute to abdominal obesity and chronic inflammatory disease. VAT prioritization may help explain several patterns of nutrition-related disease: the paradoxical increase of chronic disease with increased food availability in recently urbanized and migrant populations; correlations between poor fetal nutrition, improved childhood (catch-up) growth, and adult metabolic syndrome; and survival differences between children with marasmus and kwashiorkor malnutrition. Fats and sugars can aggravate chronic inflammation via effects on intestinal bacteria regulating gut permeability to visceral pathogens. The extremes in a nutrition-sensitive trade-off between visceral (immune-function) vs. subcutaneous (body shape) adiposity may have been favored by selection in highly stratified premedicine societies. Altered adipose allocation in populations with long histories of social stratification and malnutrition may be the result of genetic accommodation of developmental responses to poor maternal/fetal conditions, increasing their vulnerability to inflammatory disease.

Sex differences in hepatic one-carbon metabolism

Background

There are large differences between men and women of child-bearing age in the expression level of 5 key enzymes in one-carbon metabolism almost certainly caused by the sex hormones. These male-female differences in one-carbon metabolism are greatly accentuated during pregnancy. Thus, understanding the origin and consequences of sex differences in one-carbon metabolism is important for precision medicine.

Results

We have created a mathematical model of hepatic one-carbon metabolism based on the underlying physiology and biochemistry. We use the model to investigate the consequences of sex differences in gene expression. We give a mechanistic understanding of observed concentration differences in one-carbon metabolism and explain why women have lower S-andenosylmethionine, lower homocysteine, and higher choline and betaine. We give a new explanation of the well known phenomenon that folate supplementation lowers homocysteine and we show how to use the model to investigate the effects of vitamin deficiencies, gene polymorphisms, and nutrient input changes.

Conclusions

Our model of hepatic one-carbon metabolism is a useful platform for investigating the mechanistic reasons that underlie known associations between metabolites. In particular, we explain how gene expression differences lead to metabolic differences between males and females.

Electronic supplementary material

The online version of this article (doi:10.1186/s12918-018-0621-7) contains supplementary material, which is available to authorized users.

Achilles and the tortoise: Some caveats to mathematical modeling in biology

Mathematical modeling has recently become a much-lauded enterprise, and many funding agencies seek to prioritize this endeavor. However, there are certain dangers associated with mathematical modeling, and knowledge of these pitfalls should also be part of a biologist's training in this set of techniques. (1) Mathematical models are limited by known science; (2) Mathematical models can tell what can happen, but not what did happen; (3) A model does not have to conform to reality, even if it is logically consistent; (4) Models abstract from reality, and sometimes what they eliminate is critically important; (5) Mathematics can present a Platonic ideal to which biologically organized matter strives, rather than a trial-and-error bumbling through evolutionary processes. This "Unity of Science" approach, which sees biology as the lowest physical science and mathematics as the highest science, is part of a Western belief system, often called the Great Chain of Being (or Scala Natura), that sees knowledge emerge as one passes from biology to chemistry to physics to mathematics, in an ascending progression of reason being purification from matter. This is also an informal model for the emergence of new life. There are now other informal models for integrating development and evolution, but each has its limitations.

Analysis of Homeostatic Mechanisms in Biochemical Networks

Cell metabolism is an extremely complicated dynamical system that maintains important cellular functions despite large changes in inputs. This "homeostasis" does not mean that the dynamical system is rigid and fixed. Typically, large changes in external variables cause large changes in some internal variables so that, through various regulatory mechanisms, certain other internal variables (concentrations or velocities) remain approximately constant over a finite range of inputs. Outside that range, the mechanisms cease to function and concentrations change rapidly with changes in inputs. In this paper we analyze four different common biochemical homeostatic mechanisms: feedforward excitation, feedback inhibition, kinetic homeostasis, and parallel inhibition. We show that all four mechanisms can occur in a single biological network, using folate and methionine metabolism as an example. Golubitsky and Stewart have proposed a method to find homeostatic nodes in networks. We show that their method works for two of these mechanisms but not the other two. We discuss the many interesting mathematical and biological questions that emerge from this analysis, and we explain why understanding homeostatic control is crucial for precision medicine.

A Promising Approach to Integrally Evaluate the Disease Outcome of Cerebral Ischemic Rats Based on Multiple-Biomarker Crosstalk

Purpose

The study was designed to evaluate the disease outcome based on multiple biomarkers related to cerebral ischemia.

Methods

Rats were randomly divided into sham, permanent middle cerebral artery occlusion, and edaravone-treated groups. Cerebral ischemia was induced by permanent middle cerebral artery occlusion surgery in rats. To form a simplified crosstalk network, the related multiple biomarkers were chosen as S100β, HIF-1α, IL-1β, PGI₂, TXA₂, and GSH-Px. The levels or activities of these biomarkers in plasma were detected before and after ischemia. Concurrently, neurological deficit scores and cerebral infarct volumes were assessed. Based on a mathematic model, network balance maps and three integral disruption parameters (k, φ, and u) of the simplified crosstalk network were achieved.

Results

The levels or activities of the related biomarkers and neurological deficit scores were significantly impacted by cerebral ischemia. The balance maps intuitively displayed the network disruption, and the integral disruption parameters quantitatively depicted the disruption state of the simplified network after cerebral ischemia. The integral disruption parameter u values correlated significantly with neurological deficit scores and infarct volumes.

Conclusion

Our results indicate that the approach based on crosstalk network may provide a new promising way to integrally evaluate the outcome of cerebral ischemia.

Robustness, flexibility, and sensitivity in a multifunctional motor control model

Motor systems must adapt to perturbations and changing conditions both within and outside the body. We refer to the ability of a system to maintain performance despite perturbations as "robustness," and the ability of a system to deploy alternative strategies that improve fitness as "flexibility." Different classes of pattern-generating circuits yield dynamics with differential sensitivities to perturbations and parameter variation. Depending on the task and the type of perturbation, high sensitivity can either facilitate or hinder robustness and flexibility. Here we explore the role of multiple coexisting oscillatory modes and sensory feedback in allowing multiphasic motor pattern generation to be both robust and flexible. As a concrete example, we focus on a nominal neuromechanical model of triphasic motor patterns in the feeding apparatus of the marine mollusk Aplysia californica. We find that the model can operate within two distinct oscillatory modes and that the system exhibits bistability between the two. In the "heteroclinic mode," higher sensitivity makes the system more robust to changing mechanical loads, but less robust to internal parameter variations. In the "limit cycle mode," lower sensitivity makes the system more robust to changes in internal parameter values, but less robust to changes in mechanical load. Finally, we show that overall performance on a variable feeding task is improved when the system can flexibly transition between oscillatory modes in response to the changing demands of the task. Thus, our results suggest that the interplay of sensory feedback and multiple oscillatory modes can allow motor systems to be both robust and flexible in a variable environment.

Homeostasis, singularities, and networks

Homeostasis occurs in a biological or chemical system when some output variable remains approximately constant as an input parameter [Formula: see text] varies over some interval. We discuss two main aspects of homeostasis, both related to the effect of coordinate changes on the input-output map. The first is a reformulation of homeostasis in the context of singularity theory, achieved by replacing 'approximately constant over an interval' by 'zero derivative of the output with respect to the input at a point'. Unfolding theory then classifies all small perturbations of the input-output function. In particular, the 'chair' singularity, which is especially important in applications, is discussed in detail. Its normal form and universal unfolding [Formula: see text] is derived and the region of approximate homeostasis is deduced. The results are motivated by data on thermoregulation in two species of opossum and the spiny rat. We give a formula for finding chair points in mathematical models by implicit differentiation and apply it to a model of lateral inhibition. The second asks when homeostasis is invariant under appropriate coordinate changes. This is false in general, but for network dynamics there is a natural class of coordinate changes: those that preserve the network structure. We characterize those nodes of a given network for which homeostasis is invariant under such changes. This characterization is determined combinatorially by the network topology.

Evaluating Alzheimer's Disease Progression by Modeling Crosstalk Network Disruption

Aβ, tau, and P-tau have been widely accepted as reliable markers for Alzheimer's disease (AD). The crosstalk between these markers forms a complex network. AD may induce the integral variation and disruption of the network. The aim of this study was to develop a novel mathematic model based on a simplified crosstalk network to evaluate the disease progression of AD. The integral variation of the network is measured by three integral disruption parameters. The robustness of network is evaluated by network disruption probability. Presented results show that network disruption probability has a good linear relationship with Mini Mental State Examination (MMSE). The proposed model combined with Support vector machine (SVM) achieves a relative high 10-fold cross-validated performance in classification of AD vs. normal and mild cognitive impairment (MCI) vs. normal (95% accuracy, 95% sensitivity, 95% specificity for AD vs. normal; 90% accuracy, 94% sensitivity, 83% specificity for MCI vs. normal). This research evaluates the progression of AD and facilitates AD early diagnosis.

Using mathematical models to understand metabolism, genes, and disease

Mathematical models are a useful tool for investigating a large number of questions in metabolism, genetics, and gene–environment interactions. A model based on the underlying biology and biochemistry is a platform for in silico biological experimentation that can reveal the causal chain of events that connect variation in one quantity to variation in another. We discuss how we construct such models, how we have used them to investigate homeostatic mechanisms, gene–environment interactions, and genotype–phenotype mapping, and how they can be used in precision and personalized medicine.

Feedback control as a framework for understanding tradeoffs in biology

Control theory arose from a need to control synthetic systems. From regulating steam engines to tuning radios to devices capable of autonomous movement, it provided a formal mathematical basis for understanding the role of feedback in the stability (or change) of dynamical systems. It provides a framework for understanding any system with regulation via feedback, including biological ones such as regulatory gene networks, cellular metabolic systems, sensorimotor dynamics of moving animals, and even ecological or evolutionary dynamics of organisms and populations. Here, we focus on four case studies of the sensorimotor dynamics of animals, each of which involves the application of principles from control theory to probe stability and feedback in an organism's response to perturbations. We use examples from aquatic (two behaviors performed by electric fish), terrestrial (following of walls by cockroaches), and aerial environments (flight control by moths) to highlight how one can use control theory to understand the way feedback mechanisms interact with the physical dynamics of animals to determine their stability and response to sensory inputs and perturbations. Each case study is cast as a control problem with sensory input, neural processing, and motor dynamics, the output of which feeds back to the sensory inputs. Collectively, the interaction of these systems in a closed loop determines the behavior of the entire system.

Predicting performance and plasticity in the development of respiratory structures and metabolic systems

The scaling laws governing metabolism suggest that we can predict metabolic rates across taxonomic scales that span large differences in mass. Yet, scaling relationships can vary with development, body region, and environment. Within species, there is variation in metabolic rate that is independent of mass and which may be explained by genetic variation, the environment or their interaction (i.e., metabolic plasticity). Additionally, some structures, such as the insect tracheal respiratory system, change throughout development and in response to the environment to match the changing functional requirements of the organism. We discuss how study of the development of respiratory function meets multiple challenges set forth by the NSF Grand Challenges Workshop. Development of the structure and function of respiratory and metabolic systems (1) is inherently stable and yet can respond dynamically to change, (2) is plastic and exhibits sensitivity to environments, and (3) can be examined across multiple scales in time and space. Predicting respiratory performance and plasticity requires quantitative models that integrate information across scales of function from the expression of metabolic genes and mitochondrial biogenesis to the building of respiratory structures. We present insect models where data are available on the development of the tracheal respiratory system and of metabolic physiology and suggest what is needed to develop predictive models. Incorporating quantitative genetic data will enable mapping of genetic and genetic-by-environment variation onto phenotypes, which is necessary to understand the evolution of respiratory and metabolic systems and their ability to enable respiratory homeostasis as organisms walk the tightrope between stability and change.