Universal structures for embedded integral control in biological adaptation

Universal structures for embedded integral control in biological adaptation | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Article Universal structures for embedded integral control in biological adaptation Robyn Araujo, Lance Liotta This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-1571178/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 20 Apr, 2023 Read the published version in Nature Communications → Version 1 posted You are reading this latest preprint version Abstract At the molecular level, the evolution of life is driven by the generation and diversification of adaptation mechanisms. A universal description of adaptation-capable chemical reaction network (CRN) structures has remained elusive until now, since currently-known criteria for adaptation apply only to a tiny subset of possible CRNs. While adaptation is known to require some form of embedded integral control , current approaches can only identify an internal integral structure in simple special cases. Here we identify the definitive structural requirements that characterize all adaptation-capable collections of interacting molecules, however large or complex. We show that these network structures implement a form of integral control in which multiple independent integrals can collaborate to confer the capacity for adaptation on specific molecules. We present a universal method to test for adaptation capacity, and for detecting the adaptation-conferring integrals, in any CRN. Using this new approach, we demonstrate the existence of embedded integrals in a variety of biologically important CRNs that have eluded previous methods, and for which adaptation has been observed experimentally. This definitive picture of biological adaptation at the level of intermolecular interactions represents a blueprint for adaptation-capable signalling networks across all domains of life, and for the design of synthetic biosystems. Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Introduction The capacity for biological systems to adapt to variable and unpredictable conditions, and to maintain certain key survival-requisite properties within tight tolerances, is fundamental to life itself. This ubiquitous property has been studied under a variety of guises, including robust homeostasis 1 and absolute concentration robustness (ACR) 2,3 , all of which are special cases of the keystone phenomenon known as robust perfect adaptation (RPA) 4 . RPA encompasses two essential features: a baseline reference signal, or setpoint , established by the concentrations of one or more key molecules, and which allows the system to distinguish high/increasing signals from low/decreasing signals; and an actuator signal (the ‘adaptation’), which serves as a memory trace for the altered conditions or stimuli to which the system has been exposed over time 4-6 . RPA has been ubiquitously observed at all scales of biological organization from homeostatic control of plasma mineral concentrations 7 to the regulation of cellular signal transduction networks 8 ; from sensory adaptation 9 to neuronal excitation regulation 10 ; from the orchestration of cellular stress responses 11 to the coordination of chemotaxis in single-celled organisms 12,13 , and is thought to play a critical role in robust patterning during organism development 14,15 . Importantly, RPA corresponds to a special case of a defining problem in classical automatic control – namely, the robust asymptotic tracking of a desired trajectory (the system’s setpoint), while rejecting unwanted disturbances. In the 1970s, the landmark studies of Francis and Wonham 16,17 investigated the necessary controller structures to achieve such robust tracking, and established what is now known as the internal model principle (IMP) 18 . By this principle, a control system is able to reject exogenous stimuli or disturbances by incorporating within itself a model of the dynamic structure of the stimulus or disturbance. In the face of persistent ( constant ) disturbances such as a mutation, an altered external environment, or a new network stimulus, the internal model must produce constant signals and is equivalent to the requirement for integral control 18,19 – a requirement that can readily be met in engineering design problems by incorporating special components as integral-computing controllers. In contrast to engineering control systems, signaling networks that evolve in living systems are dynamically assembled via the physical interactions – involving collisions, binding events, and chemical modifications – among discrete entities, or molecules, which must constitute both the signals and their own controllers . How can these complex self-organising collections of chemical reactions manage to embed the integral-computing structures required for adaptation? Until now, integral-computing molecular interactions have only been identified in exceedingly simple chemical reaction networks (CRNs), such as the antithetic integral control motif 6,20 , and highly simplified versions of bacterial metabolic circuits and phosphorelays 3 , where the requisite integral can be identified via a linear change of coordinates. Crucially, many adapation-capable CRNs have been identified, including some extremely simple CRNs (see Cappelletti et al. 3 ), for which no such linear transformation can reveal an adaptation-conferring integral structure, highlighting the fact that complex nonlinear transformations are generally required to detect presence of integral control, in most adaptation-capable CRNs in nature 21 . Here we identify the universal principles by which all possible instances of RPA-capable CRNs – in all living systems on Earth, as well as in synthetic biology – construct an ‘internal model’ 16-18,22 of any possible disturbance or change in conditions, thereby allowing the CRN to implement integral control. We also develop a novel and definitive algorithmic test for RPA capacity, and provide code in the open-source software Singular ( www.Singular.uni-kl.de ) to implement this test for all examples considered here and in our Supplementary Information (SI), which can be tailored to the study of any CRN. Results Universal structural principles for all adaptation-capable CRNs As a prelude to presenting the universal structural principles by which any CRN can orchestrate a robustly adaptive response (see also SI, Section S1), we first briefly describe two simple examples that have eluded all previous systematic methods to detect RPA, and the presence of integral control, and which exemplify the essential structural principles that are common to all RPA-capable CRNs. First, we consider an RPA-promoting CRN (Fig. 1 a) known as antithetic integral control (Fig. 1 b) 6 , 20 - a controller structure that has been identified in the form of sigma/anti-sigma factors in a range of bacterial strains, including E. coli and Salmonella 20 , and has also been implemented in synthetic networks 6 . In the simplest possible version of this control mechanism (Fig. 1 b, highlighted reactions), two proteins \({X}_{1}\) and \({X}_{2}\) bind with very high affinity (i.e. irreversibly, thus ‘annihilating’ each other). One of these proteins ( \({X}_{1}\) ) is synthesized at a rate proportional to the concentration of a transcription factor ( \(R\) ), while the other protein ( \({X}_{2}\) ) is constitutively produced at a constant rate. From the law of mass action, whereby reaction rates are proportional to the concentrations of their reactant molecules, this simple scheme produces the two reaction rates \({\dot{X}}_{1}= {k}_{1}R-{k}_{2}{X}_{1}{X}_{2},\) (Eq. 1) \({\dot{X}}_{2}={k}_{3}-{k}_{2}{X}_{1}{X}_{2}.\) (Eq. 2) A linear change of coordinates, \(\dot{z}={\dot{X}}_{1}-{\dot{X}}_{2}={k}_{1}R-{k}_{3}\) , suffices to identify an internal model , with integral variable \(z={k}_{1}{\int }_{{t}_{0}}^{t}\left(R\left(\tau \right)-\frac{{k}_{1}}{{k}_{3}}\right)d\tau ,\) for any possible persistent disturbance to the system, thereby establishing the capacity for RPA in the molecule \(R\) with setpoint \({k}_{1}/{k}_{3}\) (see Cappelletti et al. 3 ). But suppose that a modification to the CRN is introduced during evolution whereby the production of \({X}_{2}\) is no longer independent of other signaling activity, but is now under the control of another network protein \({X}_{3}\) , a transcription factor, as depicted in Fig. 1 a,b. The reaction rate for \({X}_{2}\) now becomes \({\dot{X}}_{2}={{k}_{4}{X}_{3}+k}_{3}-{k}_{2}{X}_{1}{X}_{2}.\) (Eq. 3) This perturbed controller structure can no longer impose RPA on \(R\) unless \({X}_{3}\) participates in additional regulatory interactions, whose structure satisfies very strict constraints (see SI Sections S1.5 and S4.2.2). In Fig. 1 a,b we provide an example of a suitable auxiliary controller structure for \({X}_{3}\) , which now includes an additional protein \({O}_{1}\) . In Fig. 1 c and SI Section S4.2.2 we demonstrate that for this expanded CRN, one internal model with integral variable \({z}_{1}={O}_{1}={k}_{7}{\int }_{{t}_{0}}^{t}{O}_{1}\left(\tau \right)\left({X}_{3}\left(\tau \right)-\frac{{k}_{8}}{{k}_{7}}\right)d\tau\) imposes RPA on \({X}_{3}\) , provided that \({O}_{1}\) maintains a non-zero concentration – a concept known as constrained integral control 23 . Having thus imposed a steady state value (setpoint) of \({k}_{8}/{k}_{7}\) on \({X}_{3}\) , a second internal model, with integral variable \({z}_{2}={{X}_{1}-{X}_{2}=k}_{1}{\int }_{{t}_{0}}^{t}\left(R\left(\tau \right)-\left(\frac{{k}_{3}+{k}_{4}{X}_{3}\left(\tau \right)}{{k}_{1}}\right)\right)d\tau\) , can now impose RPA on \(R\) . In this way, two internal models - or polynomial invariants (see Fig. 1 c) - may be defined, each obtained through (at most) a linear coordinate change. From these two separate polynomial invariants, an ‘ RPA polynomial ’ of the form \({\rho }_{1}={k}_{1}{k}_{7}{O}_{1}\left(R- \left(\frac{{k}_{3}{k}_{7}+{k}_{4}{k}_{8}}{{k}_{1}{k}_{7}}\right)\right)\) (Eq. 4) can now be constructed through nonlinear combination, via ‘concatenating monomials’ (see Fig. 1 c, and SI Sections S3 and S4), from which it is clear that the network has the capacity for RPA in \(R\) with setpoint ( \({k}_{3}{k}_{7}+{k}_{4}{k}_{8})/{k}_{1}{k}_{7}\) . We demonstrate (see SI Sections S2 and S4.2.2) that the CRN depicted in Fig. 1 conforms to the topological principles of an Opposer Module 4 , see Fig. 1 d. More specifically, the controller structure of this CRN exhibits the special architecture known as a two-node opposing set (see Araujo et al. 4 ). Each linear combination of model variables that produces an opposer invariant, corresponding to an internal model, thereby constructs an independent opposer integral (see Fig. 1 c). The two separate integrals together confer RPA on the ‘sensor’ molecule R, and ultimately, the entire embedded network (see SI, Section S4.2.2). Second, we consider a CRN for the EnvZ-OmpR osmoregulation system in E-coli (Fig. 2 ). A simplified model of this network with a ‘ deficiency ’ 24 of one was first analyzed by Shinar and Feinberg 2 , and could be shown to exhibit absolute concentration robustness (ACR) – a type of RPA (see SI Section S1.3) – by the Shinar-Feinberg theorem 2 . Deficiency is a key integer invariant associated to a CRN 24 , as we discuss in greater detail in the sections to follow. Crucially, the Shinar-Feinberg theorem applies only to deficiency-one CRNs, which severely limits its ability to detect ACR (and thus RPA) in most molecular networks of biological interest, since the deficiencies of known genome-scale signaling networks (eg. in metabolism) in even the simplest organisms frequently exceed one hundred 25 . In Fig. 2 a,b, we consider the more detailed version of the EnvZ-OmpR CRN, which eludes the Shinar-Feinberg theorem, having a deficiency of two , but which can be shown to exhibit RPA (ACR) in the phosphoform, pOmpR (see Supplementary Materials of Shinar and Feinberg 2 ). The CRN in Fig. 2 conforms to the topological principles of an RPA-conferring Balancer Module (see SI Section S3.2). All Balancer modules are characterized by a collection of parallel pathways emanating from a diverter node (here, EnvZ-ATP ) and culminating in a connector node (here, pOmpR ) 4 . One or more balancer nodes (here, EnvZ-ADP ) may be embedded within the parallel pathways 4 . In Fig. 2 c we identify an integral variable \({z}_{1}={a}_{5}{\int }_{{t}_{0}}^{t}\text{EnvZ-ATP}\left(\tau \right)\left(\frac{\text{EnvZ-ATP}\left(\tau \right)}{\text{EnvZ-ATP}\left(\tau \right)}-\beta {K}_{m1}\right)d\tau\) that first confers RPA on the concentration ratio EnvZ-ATP / EnvZ-ADP – a key invariant known as a balancer invariant (see Araujo et al. 4 ). Having established a setpoint of \(\beta {K}_{m1}\) for this balancer invariant, a second integral variable, \({z}_{2}\) , constructs a connector invariant that imposes RPA on pOmpR (see Fig. 2 c). Crucially, both the balancer invariant and the connector invariant are obtained via linear combinations of the CRN’s mass action equations. These two polynomial invariants together produce an RPA polynomial of the form \({\rho }_{2}=\gamma .\text{EnvZ-ATP}\left(\text{pOmpR}- \frac{{k}_{1}{K}_{m4}}{\gamma }\right),\) (Eq. 5) through nonlinear combination via ‘concatenating monomials’ (see Fig. 2 c), thereby explicitly highlighting the CRN’s capacity for RPA in pOmpR , with setpoint \(\frac{{k}_{1}{K}_{m4}}{\gamma }\) . Our transformative step is to prove that all RPA-capable CRNs, regardless of size, complexity, or ‘deficiency’ 2 , necessarily conform to the general principles encapsulated by these two illustrative examples (see SI Section S4). In fact, all RPA-capable CRNs are characterized by a topological hierarchy of polynomial invariants , each obtained by a linear combination of the CRN’s rate equations, and each corresponding one-to-one with a topological feature (a balancer node, a connector node or an opposer node) of the overarching network structure. The topological principles that are known to hold at the network macroscale for all RPA-capable networks are now understood in complete generality 4 and, together with the integral-computing properties of CRNs at the network microscale presented here, constitute definitive design criteria that unify all possible RPA-capable CRNs. There are two distinct but interrelated components to this central result, which we delineate in turn in the sections to follow: (i) We identify the universal algebraic condition that is satisfied by all RPA-capable CRNs, which encodes the fundamental constraints on the ‘flow’ of biochemical information through the CRN (and hence on the overarching topological structure of the CRN). (ii) We show that this algebraic condition always admits a decomposition into a collection of ‘linear’ problems – a decomposition that is governed by a the CRN’s deficiency . Together, these two mathematical results reveal a universal integral control implementation that holds for all possible RPA-capable CRNs, however large, complex or nonlinear in their dynamics. Adaptation Relies on a CRN design strategy called Kinetic Pairing First, we prove (see Theorem 1 , SI Section S1.4) that for all RPA-capable CRNs, with interacting molecules \({x}_{1}, \dots , {x}_{n},\) and corresponding mass-action rate equations \({f}_{1}, \dots , {f}_{n}\) , there always exist polynomials \({\{h}_{1}, \dots , {h}_{n}\}\subset \mathbb{R}[{x}_{1},\dots ,{x}_{n}]\) such that \({h}_{1}{f}_{1}+\dots +{h}_{n}{f}_{n}=g\left({x}_{i},{x}_{j}\right)\left({x}_{i}-c\right)=\rho ,\) (Eq. 6) where \(\rho =g\left({x}_{i},{x}_{j}\right)\left({x}_{i}-c\right),\) in its lowest order form, is the RPA polynomial of the CRN, \({x}_{i}\) is any RPA-capable variable of the CRN, and \({x}_{j}\) is any variable that does not exhibit RPA (i.e., an actuator variable, or a molecule regulated by an actuator variable). The system setpoint, \(c\) , is a rational function of biochemical parameters. From this new mathematical vantage point, we can now recognize that the special structure of the RPA polynomial specified by Theorem 1, being a function of exactly two variables, imposes fundamental structural limitations on RPA-capable CRNs, and encodes the cardinal principle that we call kinetic pairing (see Fig. 3 and SI Section S2). In particular, the functional form of \(\rho\) suggests two possible topological interpretations of Theorem 1, depending on whether the RPA-capable variable, \({x}_{i}\) , is a regulating variable (for \({x}_{j}\) ) or a regulated variable (by \({x}_{j}\) ). As depicted schematically in Fig. 3 a, if \({x}_{i}\) regulates \({x}_{j}\) (the non-RPA-capable variable), then the upregulating and downregulating contributions of \({x}_{j}\) to its own reaction rate must be precisely matched, or ‘ paired ’, via the pairing function \(g\) . At steady state, this form of \(\rho\) satisfies the condition \(\frac{\partial \rho }{\partial {x}_{j}}=0\) referred to in Araujo et al. 4 as opposer kinetics , and must therefore be embedded in an overarching feedback loop which gives rise to the topological structure of an Opposer module. Our previous exhaustive analysis on Opposer module topologies has established that the feedback segment of such modules may contain multiple opposer ‘nodes’, each with its own opposer kinetics, and each contributing to a collection of embedded interlinked feedback loops known as an opposing set (Fig. 3 a). If, on the other hand, \({x}_{i}\) is regulated by \({x}_{j}\) , then the upregulating and downregulating contributions of \({x}_{j}\) to the reaction rate for \({x}_{i}\) must likewise be precisely paired via \(g\) . As illustrated in Fig. 3 b, the pairing function naturally induces a Balancer topology 4 on the CRN in this case (see SI Section S2), with \({x}_{j}\) performing a diverter function 4 . For this topological structure, the steady-state condition \(\frac{\partial \rho }{\partial {x}_{j}}=0\) corresponds to connector kinetics 4 , provided that additional constraints (referred to as balancer kinetics in Araujo et al. 4 ) can be satisfied for the reactions embedded into any parallel pathways linking \({x}_{j}\) to \({x}_{i}\) . Although Theorem 1 holds for any non-RPA-capable variable, \({x}_{j}\) , we show that there exists a ‘natural’ choice of \({x}_{j}\) with respect to the requisite topological structure of the CRN: a diverter variable (in the case of a Balancer module), or an opposer variable (for an Opposer module). Algorithmically, this choice of non-RPA-capable variable simplifies the elimination polynomials \({h}_{1}, \dots ,{h}_{n}\) in Eq. 6 (see SI Section S1.5 for a fully analysed example). Indeed, for such a judicious choice of \({x}_{j}\) , the pairing function \(g\) is frequently zero-order in \({x}_{i}\) , except in the special case of an autoregulatory role for \({x}_{i}\) (see Fig. 3 ). In the case of the antithetic integral control motif, the pairing function is zero-order in both \({x}_{i}\) and \({x}_{j}\) , giving rise to unconstrained integral control 3 , 23 . RPA-permissive topological features are encoded by CRN deficiency Second, we consider a decomposition of the nonlinear algebraic condition for RPA (Eq. 6) into its component linear contributions, to reveal the general mechanism through which kinetic pairing is transacted in CRNs. Indeed, by identifying the connection between the deficiency of an RPA-capable CRN and the presence of feedback loops and/or feedforward segments (parallel pathways), we prove that the RPA polynomial of a CRN can always be decomposed into a collection of subsidiary polynomial invariants, each corresponding to a component of the CRN’s topological structure, and each residing in the rowspan of the CRN’s reaction rates. In other words, each such subsidiary invariant may be obtained via a linear transformation of the system’s reaction equations, \({f}_{1}, \dots ,{f}_{n}\) . The deficiency of a CRN (Fig. 4) is a non-negative integer that encapsulates the extent to which the individual reactions of the CRN are linearly independent given their distribution into linkage classes 24 . With the exception of the trivial RPA-capable CRN consisting only of an isolated connector node, which has a deficiency of zero (see SI Section S4.2.2), all (non-trivial) RPA-capable CRNs require a deficiency of at least one. The Shinar-Feinberg Theorem 2 (see Theorem 2 in SI) pertains to CRNs with a deficiency of exactly one, and states that any such CRN containing two distinct complexes that differ in a single species S, and admitting a steady-state in the positive orthant, necessarily exhibit ACR (and therefore RPA) in the species S. This key theorem follows from Shinar and Feinberg’s more general result that the steady-state ratio of any two monomials associated to non-terminal complexes (see Fig. 4) is independent of the system’s initial conditions 2 . Here we extend Shinar and Feinberg’s arguments to prove a still stronger result (SI Theorem 3) – namely, that all deficiency-one CRNs contain binomials in the rowspan of their reaction rates, of the form , where and are any two mass-action monomials corresponding to non-terminal complexes, and is a pair of rational functions of the CRN rate constants. In other words, there exists some such that . Crucially, we extend the mathematical framework for our Theorem 3 (on deficiency-one CRNs) to a general method for identifying rowspan polynomials in CRNs of arbitrary deficiency. In fact, there are exactly two ways in which a deficiency exceeding one can be accommodated into an RPA-capable CRN (see SI Sections S4.2.1 and S4.2.2). First, ‘extramodular’ chemical reactions that do not contribute to the RPA-conferring mechanism may increase the deficiency of the CRN without pertubing its RPA capacity: for CRNs with Balancer topology, any reactions upstream of the diverter node and downstream of the connector node constitute extramodular reactions; for CRNs with Opposer topology, any reactions outside the feedback segment of the module (the ‘embedded network’, see Fig. 1 a) may be considered extramodular in this context, notwithstanding their contribution to the ‘controlled’ portion of the overarching module. Mathematically, algebraic independence from the RPA-conferring chemical reactions of the CRN may be established by decomposing the CRN into algebraically-independent subnetworks, wherein the sum of the ranks of the subnetworks is equal to the rank of the full CRN. We provide full technical details on these principles SI Section S4.2, along with a set of worked examples. In any case, algebraically independent subnetworks of the CRN that correspond to extramodular reactions have no bearing on the RPA capacity of the CRN as a whole, and may be excluded from analysis. Second, chemical reactions contributing to either feedforward segments (parallel pathways) or additional feedback loops within the RPA-conferring CRN will increase deficiency by one for each additional such element (see Fig. 5 ; see also SI Section S4.2.2). For this reason, all deficiency-one RPA-capable CRNs must necessarily be either a single Balancer module comprising exactly two (incoherent) parallel pathways, or a single opposer node embedded into an Opposer module. The simple antithetic integral control motif (see highlighted reactions in Fig. 1 ), as well as the simple EnvZ-OmpR CRNs considered by Shinar and Feinberg 2 (cf. Figure 2 ) are example of deficiency-one CRNs that exhibit RPA. We demonstrate that CRNs may be decomposed into subsets corresponding to the deficiency-increasing elements; using this decomposition (along with mathematical induction, as needed), our Theorem 3 may be extended to the identification of RPA-promoting polynomial invariants in the rowspan of the CRN (see SI Section S4.2.2). This method makes clear that although the RPA polynomial associated to an RPA-capable CRN generally requires a nonlinear transformation of the reaction equations (Eq. 6), there always exist linear transformations that can extract ‘special’ polynomial building blocks, each corresponding one-to-one with a topological feature of the overarching network structure, from the CRN’s reaction equations. In particular, all RPA-capable CRNs of Balancer type contain a connector polynomial, corresponding to the connector ‘node’, and one or more balancer polynomials, corresponding to balancer node(s), in their rowspans. CRNs of Opposer type, on the other hand, contain one or more opposer polynomials (corresponding to opposer node(s)) in their rowspans. Integral control and the ‘passing’ of invariants Until now strategies for identifying an internal model, and an associated integral, via a nonlinear coordinate change have only been applicable to exceedingly simple CRNs 18 . By contrast, our approach identifies a well-defined nonlinear map between reaction rates of the model variables \({f}_{1}, \dots , {f}_{n}\) , and an internal model, \(\rho\) (Eq. 6), which exists for all adaptation-capable CRNs. In contrast to all prior control theoretic approaches , this alternative viewpoint decomposes all RPA-capable CRNs into a constellation of linear integral control problems, each with an associated invariant (i.e. internal model). These are distributed across well-defined RPA-permissive network topologies 4 , and construct an RPA polynomial, \(\rho\) , through the process of invariant passing (Fig. 6 ; see also SI Sections S3 and S4). Invariant passing describes the process by which polynomial invariants corresponding to adjacent topological features of the CRN’s overarching network structure (as described in the preceding section) are combined so as to systematically eliminate model variables, and ultimately obtain the RPA polynomial of the CRN. As illustrated schematically in Fig. 6 a for CRNs of Opposer type, opposer invariants are passed from the distal opposer node to the proximal opposer node; for CRNs of Balancer type (Fig. 6 b), invariants are passed from the diverter node to the sequence of balancer nodes within each parallel pathway, culminating at the connector node (Fig. 6 b). Crucially, we demonstrate that the ability or inability of these invariants to ‘pass’ within the rowspan of the system is a question of stoichiometric independence of the individual chemical reactions contributing to successive invariants. We outline the significance of this key concept through the analysis of a simple illustrative example (Fig. 7 ). In Fig. 7 a, we depict a deficiency-one CRN comprising three interacting molecules, \(A\) , \(B\) and \(C\) , which exhibits RPA (and, more specifically, ACR) in the molecule \(A\) . It is easy to show (see SI Section S3.1) that this CRN is topologically a Balancer module, where \(A\) is the connector, \(B\) is the diverter, and \(C\) is a balancer. In Fig. 7 b, we present a modified version of this CRN that preserves both its topology and its deficiency of one. Both CRNs contain an identical connector polynomial in their rowspans. In addition, both CRNs contain a balancer polynomial in their respective rowspans, as expected. In the original CRN (Fig. 7 a), however, the balancer and connector polynomials are stoichiometrically independent, in the sense that the variable to be eliminated in the process of invariant passing (ie. \(C\) ) derives from the reactant complex \(B+C\) in the balancer polynomial, and from the reactant complex \(C\) in the connector polynomial. Therefore, the connector polynomial must be multiplied by the concentration of molecule \(B\) (a ‘concatenating monomial’) in order for the balancer polynomial to ‘pass’ to it, and thereby construct the RPA polynomial. By contrast, the single reactant complex \(C\) contributes to both subsidiary polynomials for the modified CRN (Fig. 7 b), guaranteeing their stoichiometric dependence. It is striking to note that the original form of the CRN (Fig. 7 a) eludes the Shinar-Feinberg theorem, even though the CRN exhibits ACR and has a deficiency of one. It is thereby clear that, even for the special case of deficiency-one CRNs, the Shinar-Feinberg theorem cannot provide a comprehensive description of ACR (and hence RPA). By contrast, the modified form of the CRN (Fig. 7 b), with stoichiometrically-dependent balancer and connector polynomials, does satisfy the Shinar-Feinberg theorem. With the stoichiometric dependence of all subsidiary polynomials now delivering the all-important RPA polynomial to the system’s rowspan, this modified CRN necessarily contains two non-terminal complexes ( \(A+B\) , and \(B\) ) that differ in the single ACR-exhibiting molecule, \(A\) . We illustrate the control diagram corresponding to our decomposition into a topological hierarchy of linear controllers in Fig. 8 for the particular case of a single Opposer module, since all Opposer modules necessarily incorporate an overarching feedback structure, and are thus easily described using standard control diagrams. In principle, there should always exist some nonlinear coordinate change to extract a single output-driven internal model (Fig. 8 a) from system’s rate equations, corresponding to a single integral of the system’s tracking error (Fig. 8 b). But in our representation, each subsidiary polynomial invariant corresponds to an independent internal model, each with its own independent setpoint. For a CRN constituting a three-node opposing set (Fig. 8 c), for instance, there will exist three independent internal models and three corresponding opposer integrals, each conferring RPA on a different variable (Fig. 8 d). These three independent linear control systems collaborate to confer RPA on the ‘sensor’ variable of the CRN, and ultimately, the entire embedded network. A universal algorithm for adaptation detection in complex CRNs It is clear from the illustrative example of the EnvZ-OmpR osmoregulatory motif (Fig. 2 ; see also additional CRN examples in SI) that even for exceedingly simple CRNs, constituting a single RPA module, the integral-computing polynomial invariants may be deeply concealed within the chemical reaction structures, and cannot generally be identified by inspection. For this, we introduce here a universal algorithmic method for establishing the RPA capacity of a CRN, which can identify the subsidiary polynomial invariants automatically . Our method is a direct consequence of the fact that the RPA polynomial of any adaptation-capable CRN is a function of two variables, thereby converting the question or RPA capacity to a well-defined elimination problem. This elimination problem corresponds geometrically to the projection of the system onto just two variables (one RPA-capable, and one RPA-incapable) – a task that can accomplished via computation of the Gröbner basis of \({ =\{h}_{1}{f}_{1}+\dots +{h}_{n}{f}_{n}| {h}_{i}\in \mathbb{R}\left[{x}_{1},\dots ,{x}_{n}\right]\}\) with suitable monomial ordering (see SI Section S5). Remarkably, although the problem of computing a Gröbner basis (e.g. by Buchberger’s algorithm 26 ) for general systems of polynomials is well-known to be NP-Hard 27 , the special ‘ almost linear’ structure of RPA capable CRNs (as described above, see also SI Section S4) allows any RPA-capable CRN to yield easily to this approach in polynomial time. Indeed, failure of Buchberger’s algorithm to terminate rapidly for a given CRN is prima-facie evidence that the CRN does not, in fact, exhibit RPA. We provide full details of this method, along with code in the open-source software Singular ( www.Singular.uni-kl.de ) in our Supplementary Information (see SI Section S5), where we also provide a selection of fully-annotated illustrative examples. This code can readily be applied to any CRN. Discussion Identification of a definitive test for the capacity of a network of chemical reactions to exhibit RPA has been the subject of a long quest, and most attempts have considered only the special case of RPA known as Absolute Concentration Robustness (ACR). These diverse attempts have drawn from a range of different mathematical frameworks, which can be broadly divided into two main categories: the chemical reaction network theory (CRNT) viewpoint 2 , 24 , 25 , and the engineering control theory viewpoint 16 – 19 . The pinnacle of CRNT approaches is the Shinar-Feinberg theorem 2 which identifies a sufficient condition for ACR in CRNs of deficiency one 2 , 24 . It is now well known that most CRNs in nature have a deficiency much greater than one 25 and the Shinar-Feinberg theorem is silent on all such CRNs. In addition, the Shinar-Feinberg theorem cannot reveal the setpoint of any ACR-exhibiting molecules as a function of system rate constants, nor how the existence of ACR corresponds to the presence of integral control. Karp et al. 28 , developed an alternative systematic method to identify ‘complex linear’ polynomial invariants, which require only linear combinations of the mass-action equations of a CRN. Using this method, the two subsidiary polynomial invariants given in Fig. 2 c could be identified in an ad-hoc manner. But without recognizing these two invariants as a balancer invariant and a connector invariant, and the general relationship of such invariants to an ‘RPA polynomial’, this approach cannot make the crucial connection to the essential structure that characterises all possible RPA-capable CRNs, and provides no connection to integral control in any such systems. From the control theory viewpoint, Yi et al. 19 use general linear models to demonstrate the necessity for integral control in all robust asymptotic tracking problems (such as RPA), and extract the internal model for the well-known Barkai-Leibler model of bacterial chemotaxis 13 by ad-hoc (linear) algebraic manipulations. A more recent systematic algebraic method developed by Cappelletti et al. 3 can now identify the capacity for RPA in any mass-action CRN for which an RPA polynomial exists in the rowspan of the system. This method explicitly identifies the presence of integral control in all such CRNs, and also reveals the system’s setpoint as a function of biochemical rate constants, but is silent on any CRN requiring a nonlinear coordinate change to reveal an internal model. Until now, general strategies for identifying an internal model via nonlinear coordinate transformations have remained elusive, and specific nonlinear maps have been identified only for exceedingly simple RPA-capable CRNs 18 . Although a nonlinear diffeomorphism that maps the original model variables to a special ‘block’ form - thereby explicitly revealing an internal model - should always exist in principle 18 , 23 , the identification of such a nonlinear map in even the most complex special cases cannot, of itself, clarify the general principles that unify all possible RPA-capable chemical reaction structures. By contrast, our approach identifies a well-defined nonlinear map – distinct from the transformations considered in previous control theoretic approaches 18 , 19 – between the reaction rates of the individual molcules, \({f}_{1}, \dots , {f}_{n}\) , and a key CRN invariant known as the RPA polynomial. This transformation holds for all RPA-capable CRNs, and constitutes a geometric projection of the full set of molecular concentrations onto a particular subset of the model variables, comprising one RPA-capable molecule and one non-RPA-capable molecule (recognizing that all RPA-capable networks require a minimum of one such variable to constitute the ‘adaptation’ 4 ). The major innovative leap that we make from this mathematical cornerstone is to show that this nonlinear map can always be decomposed into a constellation of linear maps , each existing within a topological hierarchy 4 associated to the CRN’s underlying structure, and each corresponding to an independent linear control problem . The integrals that are formulated by these independent subsidiary control systems thereby collaborate to confer RPA on one or more molecules in the CRN. Our approach unifies both the control theory and CRNT viewpoints, and extends prior results on the macroscale topologies 4 of RPA-capable networks to the microscale level of intermolecular interactions within CRNs. The combination of a definitive algebraic condition with the special ‘almost linear’ structure of the underlying control system provide the essential ingredients for a simple algorithmic test for RPA capacity, even in large, high-deficiency CRNs. The only algorithmic method capable of handling CRNs of arbitrary deficiency prior to this work was the necessary condition for ACR identified by Eloundou-Mbebi et al. 25 Being a necessary condition, the Eloundou-Mbebi method can identify a collection of molecules that certainly couldn’t exhibit ACR (and therefore RPA), and thereby reduces the number of molecules that must be analysed in detail for their ACR (RPA) capacity, eg. via extensive numerical simulation. But the Eloundou-Mbebi method is unable to identify, definitively, which molecules do exhibit ACR (RPA) since it fails to capture the essential structural characteristics common to all RPA-capable networks. Indeed, the Eloundou-Mbebi method characteristically overestimates the space of molecules that could potentially exhibit ACR/RPA quite significantly 25 . For the deficiency-two model of the EnvZ-OmpR phosphorelay (Fig. 2 ), for instance, all nine species satisfy the Eloundou-Mbebi condition, even though only pOmpR can actually exhibit ACR/RPA (as we can easily prove by the new method we present here). And in common with other CRNT-based approaches, the Eloundou-Mbebi condition makes no connection to integral control, and cannot identify the setpoint of any RPA-capable species as a function of biochemical parameters. Only a complete and truly general picture of the integral control problem in CRNs, as we present here, can demarcate the evolutionary trajectories along which complex adaptation-capable biological networks can arise from simpler building blocks, and provide a roadmap for either preserving or disrupting the RPA property in natural, diseased or synthetic networks through design alterations or pharmacological interventions. Declarations Acknowledgements Robyn P. Araujo is supported by an Australian Research Council (ARC) Future Fellowship (project no. FT190100645) from the Australian Government. Author Contributions RPA – conceptualization, methodology, software, formal analysis, funding acquisition, writing – original draft, writing – review & editing. LAL – writing – review & editing. References Tang, Z. F. & McMillen, D. R. Design principles for the analysis and construction of robustly homeostatic biological networks. Journal of Theoretical Biology 408 , 274–289, doi: 10.1016/j.jtbi.2016.06.036 (2016). Shinar, G. & Feinberg, M. Structural sources of robustness in biochemical reaction networks. Science 327 , 1389–1391, doi: 10.1126/science.1183372 (2010). Cappelletti, D., Gupta, A. & Khammash, M. A hidden integral structure endows absolute concentration robust systems with resilience to dynamical concentration disturbances. J R Soc Interface 17 , 20200437, doi: 10.1098/rsif.2020.0437 (2020). Araujo, R. P. & Liotta, L. A. The topological requirements for robust perfect adaptation in networks of any size. Nat Commun 9 , 1757, doi: 10.1038/s41467-018-04151-6 (2018). Araujo, R. P., Vittadello, S. T. & Stumpf, M. P. H. Bayesian and Algebraic Strategies to Design in Synthetic Biology. Proceedings of the IEEE (2021). Aoki, S. K. et al. A universal biomolecular integral feedback controller for robust perfect adaptation. Nature 570 , 533–537, doi: 10.1038/s41586-019-1321-1 (2019). El-Samad, H., Goff, J. P. & Khammash, M. Calcium Homeostasis and Parturient Hypocalcemia: An Integral Feedback Perspective. Journal of Theoretical Biology 214 , 17–29, doi: 10.1006/jtbi.2001.2422 (2002). Ferrell, J. E. Perfect and Near-Perfect Adaptation in Cell Signaling. Cell Systems 2 , 62–67, doi: 10.1016/j.cels.2016.02.006 (2016). Kaupp, U. B. Olfactory signalling in vertebrates and insects: differences and commonalities. Nature Reviews Neuroscience 11 , 188–200, doi: 10.1038/nrn2789 (2010). Badimon, A. et al. Negative feedback control of neuronal activity by microglia. Nature 586 , 417–423, doi: 10.1038/s41586-020-2777-8 (2020). Eisner, V., Picard, M. & Hajnóczky, G. Mitochondrial dynamics in adaptive and maladaptive cellular stress responses. Nature Cell Biology 20 , 755–765, doi: 10.1038/s41556-018-0133-0 (2018). Alon, U., Surette, M. G., Barkai, N. & Leibler, S. Robustness in bacterial chemotaxis. Nature 397 , 168–171, doi: 10.1038/16483 (1999). Barkai, N. & Leibler, S. Robustness in simple biochemical networks. Nature 387 , 913–917, doi: 10.1038/43199 (1997). Ben-Zvi, D. & Barkai, N. Scaling of morphogen gradients by an expansion-repression integral feedback control. Proceedings of the National Academy of Sciences 107 , 6924–6929, doi: 10.1073/pnas.0912734107 (2010). Eldar, A. et al. Robustness of the BMP morphogen gradient in Drosophila embryonic patterning. Nature 419 , 304–308, doi: 10.1038/nature01061 (2002). Francis, B. A. & Wonham, W. M. The Internal Model Principle of Linear Control Theory. IFAC Proceedings Volumes 8 , 331–336, doi: 10.1016/s1474-6670(17)67756-5 (1975). Francis, B. A. & Wonham, W. M. The internal model principle of control theory. Automatica 12 , 457–465, doi: 10.1016/0005-1098(76)90006-6 (1976). Sontag, E. D. Adaptation and regulation with signal detection implies internal model. Systems & Control Letters 50 , 119–126, doi: 10.1016/s0167-6911(03)00136-1 (2003). Yi, T. M., Huang, Y., Simon, M. I. & Doyle, J. Robust perfect adaptation in bacterial chemotaxis through integral feedback control. Proceedings of the National Academy of Sciences 97 , 4649–4653, doi: 10.1073/pnas.97.9.4649 (2000). Briat, C., Gupta, A. & Khammash, M. Antithetic Integral Feedback Ensures Robust Perfect Adaptation in Noisy Biomolecular Networks. Cell Systems 2 , 15–26, doi: 10.1016/j.cels.2016.01.004 (2016). Shoval, O., Alon, U. & Sontag, E. Symmetry Invariance for Adapting Biological Systems. SIAM Journal on Applied Dynamical Systems 10 , 857–886, doi: 10.1137/100818078 (2011). Huang, J. et al. in IEEE Conference on Decision and Control. 5370–5390. Xiao, F. & Doyle, J. C. in IEEE Conference on Decision and Control. 4345–4352. Feinberg, M. Foundations of Chemical Reaction Network Theory . Vol. 202 (Springer, 2019). Eloundou-Mbebi, J. M. O. et al. A network property necessary for concentration robustness. Nature Communications 7 , 13255, doi: 10.1038/ncomms13255 (2016). Cox, D. A., Little, J. & O’Shea, D. Ideals, Varieties and Algorithms . 4th Edition edn, (Springer, 2015). Ananth, P. V. & Dukkipati, A. Complexity of Gröbner basis detection and border basis detection. Theoretical Computer Science 459 , 1–15, doi: 10.1016/j.tcs.2012.08.002 (2012). Karp, R. L., Perez Millan, M., Dasgupta, T., Dickenstein, A. & Gunawardena, J. Complex-linear invariants of biochemical networks. J Theor Biol 311 , 130–138, doi: 10.1016/j.jtbi.2012.07.004 (2012). Additional Declarations There is NO Competing Interest. Supplementary Files SupplementaryMaterialsAraujoLiottaNCOMMSrpa2.pdf Supplementary Information for "Universal structures for embedded integral control in biological adaptation" Cite Share Download PDF Status: Published Journal Publication published 20 Apr, 2023 Read the published version in Nature Communications → Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-1571178","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":100702227,"identity":"6c70d2d4-6746-482c-9d4a-7a54b7b47ad1","order_by":0,"name":"Robyn Araujo","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA40lEQVRIiWNgGAWjYFACHhjjAOMDEJePFC3MBiAuGwlaGNgkwCQhDebtvcekC2ruyRscPHys8muOnQwbA/PDRzfwaJE5cy5NesaxYsMNB46l3Zbdlgx0GJuxcQ4eLRISOWbSPGwJjBsOnDG7LbmNGaiFh02asJZ/CfYgLcWS2+qJ1MLblpAI0sL4cdthIrTwnEu25u1LSJ554FiyNOO24zxszIT8wt578DbPtwTbvhuHD378ua3anp+9+eFjfFqAgEUCovkAAzM4jpjxKwcr+QCm+BsYGH8QVj0KRsEoGAUjEAAA1x1GEnPiYJsAAAAASUVORK5CYII=","orcid":"https://orcid.org/0000-0002-3360-2214","institution":"Queensland University of Technology","correspondingAuthor":true,"submittingAuthor":false,"prefix":"","firstName":"Robyn","middleName":"","lastName":"Araujo","suffix":""},{"id":100702228,"identity":"908cd8ab-2d48-4aa0-9be1-3c6c2257fc70","order_by":1,"name":"Lance Liotta","email":"","orcid":"","institution":"George Mason University","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Lance","middleName":"","lastName":"Liotta","suffix":""}],"badges":[],"createdAt":"2022-04-19 05:40:44","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-1571178/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-1571178/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1038/s41467-023-38011-9","type":"published","date":"2023-04-20T04:00:00+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":20744505,"identity":"ad4102c0-4225-4517-b9bd-4d80098cb33c","added_by":"auto","created_at":"2022-04-25 20:35:59","extension":"jpeg","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":245482,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cem\u003eA CRN containing the antithetic integral control motif as a subnetwork.\u0026nbsp;(a) Closed-loop system for an antithetic integral controller (X1, X2), along with an interconnected auxiliary controller (X3, O1). (b) Chemical reactions for the CRN.\u0026nbsp;(c) The CRN implements integral control via two independent internal models, corresponding to two independent polynomial invariants, each obtained by a linear change of coordinates.\u0026nbsp;These are combined nonlinearly (through the concatenating monomial O1 applied to invariant 2), to obtain the RPA polynomial, which reveals the setpoint of the molecule R.\u0026nbsp;(d) Topologically, the CRN is an Opposer module; the controller architecture is a two-node opposing set (see SI, Section S4.2.2);\u003c/em\u003e\u003c/p\u003e\u003cp\u003e\u003cbr\u003e\u003c/p\u003e","description":"","filename":"floatimage1.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-1571178/v1/14bfb713e52d97a97647263b.jpeg"},{"id":20744737,"identity":"5987fe88-f3fc-4d62-9590-8e211004481f","added_by":"auto","created_at":"2022-04-25 20:40:59","extension":"jpg","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":246214,"visible":true,"origin":"","legend":"\u003cp\u003eSee image above for figure legend.\u003c/p\u003e","description":"","filename":"2.jpg","url":"https://assets-eu.researchsquare.com/files/rs-1571178/v1/47b6a919928d4327a2c1a17a.jpg"},{"id":20744503,"identity":"0a3288f6-dd34-41b2-acd4-add35695761d","added_by":"auto","created_at":"2022-04-25 20:35:59","extension":"jpg","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":217899,"visible":true,"origin":"","legend":"\u003cp\u003eSee image above for figure legend.\u003c/p\u003e","description":"","filename":"3.jpg","url":"https://assets-eu.researchsquare.com/files/rs-1571178/v1/e3a7bd963575c4248f9f497f.jpg"},{"id":20744506,"identity":"2cbb4fd9-c812-4605-97aa-9052252616d8","added_by":"auto","created_at":"2022-04-25 20:35:59","extension":"jpeg","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":103532,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cem\u003eDeficiency is a key integer invariant for a CRN. (a) The linkage classes of a CRN are the connected components of the CRN’s graph.\u0026nbsp;The complexes are the vertices of the graph, while the reactions are the directed edges.\u0026nbsp;Strong-linkage classes are the maximal strongly-connected subgraphs of the CRN.\u0026nbsp;A terminal strong-linkage class (noted in green) is one in which no complex reacts to a complex in a different strong linkage class.\u0026nbsp;Complexes belonging to terminal strong-linkage classes (complexes 2, 3 and 5 in this case) are terminal complexes;\u0026nbsp;all other complexes are non-terminal complexes.\u0026nbsp;See SI Section S1.2 for a complete technical overview. (b) Deficiency is calculated from the number of complexes, linkage classes and the rank of the CRN, as shown.\u0026nbsp;The rank of the CRN is the number of linearly independent reactions, ie. the dimension of the stoichiometric subspace of the CRN (see SI Section S1.2).\u003c/em\u003e\u003c/p\u003e","description":"","filename":"floatimage4.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-1571178/v1/0e9726ee7ab56edb0022cc17.jpeg"},{"id":20744511,"identity":"b63d323a-3387-49d5-aab3-67bc4dc2c8f2","added_by":"auto","created_at":"2022-04-25 20:36:00","extension":"jpeg","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":144102,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cem\u003eDeficiency-increasing topological features of RPA-capable CRNs. (a) An opposing set, containing multiple interconnected feedback loops involving opposer nodes (indicated in yellow), all embedded together into the feedback segment of an Opposer module.\u0026nbsp;As shown, each single opposer node contributes a deficiency of one to the CRN.\u0026nbsp;The interspersed feedback loops that connect the individual opposer nodes further increase deficiency (by one or two, see SI Section S4.2.2), as indicated.\u0026nbsp;(b) A Balancer module, containing multiple feedforward segments between the diverter molecule (D) and the connector molecule (C).\u0026nbsp;A Balancer module with just two (incoherent) feedforward segments, without any embedded feedback loop, has a deficiency of one, as shown.\u0026nbsp;Each additional feedforward segment increases deficiency by one, as shown.\u0026nbsp;Any feedback loop embedded into a feedforward segment also increases deficiency by one (not shown).\u003c/em\u003e\u003c/p\u003e","description":"","filename":"floatimage5.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-1571178/v1/f96365a910bc0f37004241d6.jpeg"},{"id":20745182,"identity":"b1a7be8b-49de-4f9e-84aa-7fd76c856d1d","added_by":"auto","created_at":"2022-04-25 20:50:59","extension":"jpeg","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":123981,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cem\u003eThe principle of Invariant Passing.\u0026nbsp;(a) In CRNs of balancer type, polynomial invariants obtained by linear coordinate changes are ‘passed’ downstream from the diverter molecule along parallel pathways to the connector molecule.\u0026nbsp;(b) In CRNs of opposer type, polynomial invariants are ‘passed’ from the distal opposer reactions towards the ‘proximal’ opposer reactions.\u003c/em\u003e\u003c/p\u003e\u003cp\u003e\u003cbr\u003e\u003c/p\u003e","description":"","filename":"floatimage6.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-1571178/v1/148f56c4501734097093079c.jpeg"},{"id":20745044,"identity":"b42970a7-96ba-425a-ae8e-64abf8d0d853","added_by":"auto","created_at":"2022-04-25 20:45:59","extension":"jpeg","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":224509,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cem\u003eStoichiometric independence of chemical reactions, and its consequences for invariant passing. (a) A simple CRN which is topologically a Balancer module (see SI Section S3.1), involving interactions among three molecules: A (connector), B (diverter), and C (balancer).\u0026nbsp;The connector polynomial (dA/dt) and the balancer polynomial (dB/dt) are stoichiometrically independent, since the CRN complexes that contribute the molecule C (which must be eliminated) are ‘C’ for the connector polynomial, and ‘B+C’ for the balancer polynomial.\u0026nbsp;A concatenating monomial (B) is therefore required to reconcile the two invariants to construct the RPA polynomial. (b) A modified version of the CRN, with identical underlying topology.\u0026nbsp;In this case the connector polynomial (dA/dt) and the balancer polynomial (dB/dt) are stoichiometrically dependent, since the complex ‘C’ contributes to both invariants, and is thereby eliminated within the rowspan of the system.\u0026nbsp;As a consequence, the RPA polynomial resides in the system’s rowspan.\u0026nbsp;\u003c/em\u003e\u003c/p\u003e\u003cp\u003e\u003cbr\u003e\u003c/p\u003e","description":"","filename":"floatimage7.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-1571178/v1/23d90bc5136b082d8522f4cc.jpeg"},{"id":20744738,"identity":"fd518d90-2d63-4d5b-9c37-8e267696b6a4","added_by":"auto","created_at":"2022-04-25 20:40:59","extension":"png","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":47583,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cem\u003eThe distribution of internal models into a topological hierarchy.\u0026nbsp;(a) A network with a single integral controller.\u0026nbsp;(b) A standard integral feedback control diagram, in which the error between setpoint and sensor for the system is integrated in a single integral, corresponding to a single internal model.\u0026nbsp;(c) In CRNs, a universal description of integral control is obtained by decomposing the internal model into multiple subsidiary internal models, each corresponding to a linear coordinate change, and each corresponding to a topological feature of the associated CRN. \u0026nbsp;Here, we depict an Opposer module featuring a three-node opposing set.\u0026nbsp;(d) Since all Opposer modules have a feedback architecture, their feedback control diagram can explicitly incorporate the multiple independent integral-computing elements.\u0026nbsp;The feedback control diagram shown corresponds to the opposer module in (c).\u003c/em\u003e\u003c/p\u003e\u003cp\u003e\u003cbr\u003e\u003c/p\u003e","description":"","filename":"Onlinefloatimage8.png","url":"https://assets-eu.researchsquare.com/files/rs-1571178/v1/153ee36eee0f9bf47e45b8f5.png"},{"id":36525419,"identity":"89f0752e-d4bb-42ed-9977-2c97736d9f55","added_by":"auto","created_at":"2023-05-02 13:02:08","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1240548,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-1571178/v1/b4d817f3-85b5-4807-b352-8f8e2cbe63d7.pdf"},{"id":20745043,"identity":"b4b8debe-c724-43bb-b800-918b18d30e2d","added_by":"auto","created_at":"2022-04-25 20:45:59","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"supplement","size":642191,"visible":true,"origin":"","legend":"\u003cp\u003eSupplementary Information for \"Universal structures for embedded integral control in biological adaptation\"\u003c/p\u003e","description":"","filename":"SupplementaryMaterialsAraujoLiottaNCOMMSrpa2.pdf","url":"https://assets-eu.researchsquare.com/files/rs-1571178/v1/e1405e942009ab0acd925c63.pdf"}],"financialInterests":"There is \u003cb\u003eNO\u003c/b\u003e Competing Interest.","formattedTitle":"Universal structures for embedded integral control in biological adaptation","fulltext":[{"header":"Introduction","content":"\u003cp\u003eThe capacity for biological systems to adapt to variable and unpredictable conditions, and to maintain certain key survival-requisite properties within tight tolerances, is fundamental to life itself. \u0026nbsp; This ubiquitous property has been studied under a variety of guises, including robust homeostasis\u003csup\u003e1\u003c/sup\u003e and absolute concentration robustness (ACR)\u003csup\u003e2,3\u003c/sup\u003e, all of which are special cases of the keystone phenomenon known as robust perfect adaptation (RPA)\u003csup\u003e4\u003c/sup\u003e. \u0026nbsp;RPA encompasses two essential features: \u0026nbsp;a baseline reference signal, or \u003cem\u003esetpoint\u003c/em\u003e, established by the concentrations of one or more key molecules, and which allows the system to distinguish high/increasing signals from low/decreasing signals; and an actuator signal (the \u0026lsquo;adaptation\u0026rsquo;), which serves as a memory trace for the altered conditions or stimuli to which the system has been exposed over time\u003csup\u003e4-6\u003c/sup\u003e. \u0026nbsp; RPA has been ubiquitously observed at all scales of biological organization from homeostatic control of plasma mineral concentrations\u003csup\u003e7\u003c/sup\u003e to the regulation of cellular signal transduction networks\u003csup\u003e8\u003c/sup\u003e; from sensory adaptation\u003csup\u003e9\u003c/sup\u003e to neuronal excitation regulation\u003csup\u003e10\u003c/sup\u003e; from the orchestration of cellular stress responses\u003csup\u003e11\u003c/sup\u003e to the coordination of chemotaxis in single-celled organisms\u003csup\u003e12,13\u003c/sup\u003e, and is thought to play a critical role in robust patterning during organism development\u003csup\u003e14,15\u003c/sup\u003e.\u003c/p\u003e\n\u003cp\u003eImportantly, RPA corresponds to a special case of a defining problem in classical automatic control \u0026ndash; namely, the robust asymptotic tracking of a desired trajectory (the system\u0026rsquo;s setpoint), while rejecting unwanted disturbances. \u0026nbsp; In the 1970s, the landmark studies of Francis and Wonham\u003csup\u003e16,17\u003c/sup\u003e investigated the necessary controller structures to achieve such robust tracking, and established what is now known as the \u003cem\u003einternal model principle\u0026nbsp;\u003c/em\u003e(IMP)\u003csup\u003e18\u003c/sup\u003e. \u0026nbsp; By this principle, a control system is able to reject exogenous stimuli or disturbances by incorporating within itself a model of the dynamic structure of the stimulus or disturbance. \u0026nbsp;In the face of persistent (\u003cem\u003econstant\u003c/em\u003e) disturbances such as a mutation, an altered external environment, or a new network stimulus, the internal model must produce \u003cem\u003econstant signals\u003c/em\u003e and is equivalent to the requirement for integral control\u003csup\u003e18,19\u003c/sup\u003e \u0026ndash; a requirement that can readily be met in engineering design problems by incorporating special components as integral-computing controllers.\u003c/p\u003e\n\u003cp\u003eIn contrast to engineering control systems, signaling networks that evolve in living systems are dynamically assembled via the physical interactions \u0026ndash; involving collisions, binding events, and chemical modifications \u0026ndash; among discrete entities, or molecules, which must constitute \u003cem\u003eboth the signals\u003c/em\u003e \u003cem\u003eand their own controllers\u003c/em\u003e. \u0026nbsp;How can these complex self-organising collections of chemical reactions manage to embed the integral-computing structures required for adaptation? \u0026nbsp;Until now, integral-computing molecular interactions have only been identified in exceedingly simple chemical reaction networks (CRNs), such as the antithetic integral control motif \u003csup\u003e6,20\u003c/sup\u003e, and highly simplified versions of bacterial metabolic circuits and phosphorelays\u003csup\u003e3\u003c/sup\u003e, where the requisite integral can be identified via a linear change of coordinates. \u0026nbsp; \u0026nbsp;Crucially, many adapation-capable CRNs have been identified, including some extremely simple CRNs (see Cappelletti et al.\u003csup\u003e3\u003c/sup\u003e), for which no such linear transformation can reveal an adaptation-conferring integral structure, highlighting the fact that complex \u003cem\u003enonlinear\u0026nbsp;\u003c/em\u003etransformations are generally required to detect presence of integral control, in most adaptation-capable CRNs in nature\u003csup\u003e21\u003c/sup\u003e.\u003c/p\u003e\n\u003cp\u003eHere we identify the universal principles by which all possible instances of RPA-capable CRNs \u0026ndash; in all living systems on Earth, as well as in synthetic biology \u0026ndash; construct an \u0026lsquo;internal model\u0026rsquo;\u003csup\u003e16-18,22\u003c/sup\u003e of any possible disturbance or change in conditions, thereby allowing the CRN to implement integral control. \u0026nbsp;We also develop a novel and definitive algorithmic test for RPA capacity, \u0026nbsp;and provide code in the open-source software \u003cem\u003eSingular\u003c/em\u003e (\u003ca href=\"http://www.Singular.uni-kl.de\"\u003ewww.Singular.uni-kl.de\u003c/a\u003e) to implement this test for all examples considered here and in our Supplementary Information (SI), which can be tailored to the study of any CRN.\u003c/p\u003e"},{"header":"Results","content":"\u003cdiv class=\"Section2\" id=\"Sec2\"\u003e\n \u003ch2\u003eUniversal structural principles for all adaptation-capable CRNs\u003c/h2\u003e\n \u003cp\u003eAs a prelude to presenting the universal structural principles by which any CRN can orchestrate a robustly adaptive response (see also SI, Section S1), we first briefly describe two simple examples that have eluded all previous systematic methods to detect RPA, and the presence of integral control, and which exemplify the essential structural principles that are common to all RPA-capable CRNs.\u003c/p\u003e\n \u003cp\u003eFirst, we consider an RPA-promoting CRN (Fig. \u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003ea) known as \u003cem\u003eantithetic integral control\u003c/em\u003e (Fig. \u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003eb)\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e6\u003c/span\u003e,\u003cspan class=\"CitationRef\"\u003e20\u003c/span\u003e\u003c/sup\u003e - a controller structure that has been identified in the form of sigma/anti-sigma factors in a range of bacterial strains, including E. coli and Salmonella\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e20\u003c/span\u003e\u003c/sup\u003e, and has also been implemented in synthetic networks\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e6\u003c/span\u003e\u003c/sup\u003e. In the simplest possible version of this control mechanism (Fig. \u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003eb, highlighted reactions), two proteins \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({X}_{1}\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({X}_{2}\\)\u003c/span\u003e\u003c/span\u003e bind with very high affinity (i.e. irreversibly, thus \u0026lsquo;annihilating\u0026rsquo; each other). One of these proteins (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({X}_{1}\\)\u003c/span\u003e\u003c/span\u003e) is synthesized at a rate proportional to the concentration of a transcription factor (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(R\\)\u003c/span\u003e\u003c/span\u003e), while the other protein (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({X}_{2}\\)\u003c/span\u003e\u003c/span\u003e) is constitutively produced at a constant rate. From the law of mass action, whereby reaction rates are proportional to the concentrations of their reactant molecules, this simple scheme produces the two reaction rates\u003c/p\u003e\n \u003cdiv class=\"gridtable\"\u003e\n \u003cdiv align=\"left\" class=\"colspec\"\u003e\u003cbr\u003e\u003c/div\u003e\u0026nbsp;\u003ctable border=\"1\" id=\"Taba\"\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\dot{X}}_{1}= {k}_{1}R-{k}_{2}{X}_{1}{X}_{2},\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003e(Eq.\u0026nbsp;1)\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\dot{X}}_{2}={k}_{3}-{k}_{2}{X}_{1}{X}_{2}.\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e(Eq.\u0026nbsp;2)\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n \u003c/div\u003e\n \u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n \u003cp\u003eA linear change of coordinates, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\dot{z}={\\dot{X}}_{1}-{\\dot{X}}_{2}={k}_{1}R-{k}_{3}\\)\u003c/span\u003e\u003c/span\u003e, suffices to identify an \u003cem\u003einternal model\u003c/em\u003e, with integral variable \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(z={k}_{1}{\\int }_{{t}_{0}}^{t}\\left(R\\left(\\tau \\right)-\\frac{{k}_{1}}{{k}_{3}}\\right)d\\tau ,\\)\u003c/span\u003e\u003c/span\u003e for any possible persistent disturbance to the system, thereby establishing the capacity for RPA in the molecule \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(R\\)\u003c/span\u003e\u003c/span\u003e with setpoint \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({k}_{1}/{k}_{3}\\)\u003c/span\u003e\u003c/span\u003e (see Cappelletti et al.\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e3\u003c/span\u003e\u003c/sup\u003e). But suppose that a modification to the CRN is introduced during evolution whereby the production of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({X}_{2}\\)\u003c/span\u003e\u003c/span\u003e is no longer independent of other signaling activity, but is now under the control of another network protein \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({X}_{3}\\)\u003c/span\u003e\u003c/span\u003e, a transcription factor, as depicted in Fig. \u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003ea,b. The reaction rate for \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({X}_{2}\\)\u003c/span\u003e\u003c/span\u003e now becomes\u003c/p\u003e\n \u003cdiv class=\"gridtable\"\u003e\n \u003cdiv align=\"left\" class=\"colspec\"\u003e\u003cbr\u003e\u003c/div\u003e\n \u003cdiv align=\"left\" class=\"colspec\"\u003e\u003cbr\u003e\u003c/div\u003e\u0026nbsp;\u003ctable border=\"1\" id=\"Tabb\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\dot{X}}_{2}={{k}_{4}{X}_{3}+k}_{3}-{k}_{2}{X}_{1}{X}_{2}.\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\"\u003e\u003cp\u003e(Eq.\u0026nbsp;3)\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/table\u003e\u003c/div\u003e\u003cp\u003e\u003cbr\u003e\u003c/p\u003e\u003cp\u003eThis perturbed controller structure can no longer impose RPA on \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(R\\)\u003c/span\u003e\u003c/span\u003e unless \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({X}_{3}\\)\u003c/span\u003e\u003c/span\u003e participates in additional regulatory interactions, whose structure satisfies very strict constraints (see SI Sections S1.5 and S4.2.2). In Fig. \u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003ea,b we provide an example of a suitable auxiliary controller structure for \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({X}_{3}\\)\u003c/span\u003e\u003c/span\u003e, which now includes an additional protein \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({O}_{1}\\)\u003c/span\u003e\u003c/span\u003e. In Fig. \u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003ec and SI Section S4.2.2 we demonstrate that for this expanded CRN, one internal model with integral variable \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({z}_{1}={O}_{1}={k}_{7}{\\int }_{{t}_{0}}^{t}{O}_{1}\\left(\\tau \\right)\\left({X}_{3}\\left(\\tau \\right)-\\frac{{k}_{8}}{{k}_{7}}\\right)d\\tau\\)\u003c/span\u003e\u003c/span\u003e imposes RPA on \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({X}_{3}\\)\u003c/span\u003e\u003c/span\u003e, provided that \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({O}_{1}\\)\u003c/span\u003e\u003c/span\u003e maintains a non-zero concentration \u0026ndash; a concept known as \u003cem\u003econstrained integral control\u003c/em\u003e \u003csup\u003e\u003cspan class=\"CitationRef\"\u003e23\u003c/span\u003e\u003c/sup\u003e. Having thus imposed a steady state value (setpoint) of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({k}_{8}/{k}_{7}\\)\u003c/span\u003e\u003c/span\u003e on \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({X}_{3}\\)\u003c/span\u003e\u003c/span\u003e, a second internal model, with integral variable \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({z}_{2}={{X}_{1}-{X}_{2}=k}_{1}{\\int }_{{t}_{0}}^{t}\\left(R\\left(\\tau \\right)-\\left(\\frac{{k}_{3}+{k}_{4}{X}_{3}\\left(\\tau \\right)}{{k}_{1}}\\right)\\right)d\\tau\\)\u003c/span\u003e\u003c/span\u003e, can now impose RPA on \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(R\\)\u003c/span\u003e\u003c/span\u003e. In this way, \u003cspan class=\"BoldItalic\" name=\"Emphasis\" type=\"BoldItalic\"\u003etwo\u003c/span\u003e internal models - or \u003cem\u003epolynomial invariants\u003c/em\u003e (see Fig. \u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003ec) - may be defined, each obtained through (at most) a \u003cem\u003elinear\u003c/em\u003e coordinate change. From these two separate polynomial invariants, an \u0026lsquo;\u003cem\u003eRPA polynomial\u003c/em\u003e\u0026rsquo; of the form\u003c/p\u003e\n \u003cdiv class=\"gridtable\"\u003e\n \u003cdiv align=\"left\" class=\"colspec\"\u003e\u003cbr\u003e\u003c/div\u003e\n \u003cdiv align=\"left\" class=\"colspec\"\u003e\u003cbr\u003e\u003c/div\u003e\u0026nbsp;\u003ctable border=\"1\" id=\"Tabc\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\rho }_{1}={k}_{1}{k}_{7}{O}_{1}\\left(R- \\left(\\frac{{k}_{3}{k}_{7}+{k}_{4}{k}_{8}}{{k}_{1}{k}_{7}}\\right)\\right)\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e(Eq.\u0026nbsp;4)\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n \u003c/div\u003e\n \u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n \u003cp\u003ecan now be constructed through \u003cem\u003enonlinear\u003c/em\u003e combination, via \u0026lsquo;concatenating monomials\u0026rsquo; (see Fig. \u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003ec, and SI Sections S3 and S4), from which it is clear that the network has the capacity for RPA in \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(R\\)\u003c/span\u003e\u003c/span\u003e with setpoint (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({k}_{3}{k}_{7}+{k}_{4}{k}_{8})/{k}_{1}{k}_{7}\\)\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n \u003cp\u003eWe demonstrate (see SI Sections S2 and S4.2.2) that the CRN depicted in Fig. \u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e conforms to the topological principles of an Opposer Module\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e4\u003c/span\u003e\u003c/sup\u003e, see Fig. \u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003ed. More specifically, the controller structure of this CRN exhibits the special architecture known as a \u003cem\u003etwo-node opposing set\u003c/em\u003e (see Araujo et al.\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e4\u003c/span\u003e\u003c/sup\u003e). Each linear combination of model variables that produces an opposer invariant, corresponding to an internal model, thereby constructs an independent \u003cem\u003eopposer integral\u003c/em\u003e (see Fig. \u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003ec). The two \u003cem\u003eseparate\u003c/em\u003e integrals \u003cem\u003etogether\u003c/em\u003e confer RPA on the \u0026lsquo;sensor\u0026rsquo; molecule R, and ultimately, the entire embedded network (see SI, Section S4.2.2).\u003c/p\u003e\n \u003cp\u003eSecond, we consider a CRN for the \u003cem\u003eEnvZ-OmpR\u003c/em\u003e osmoregulation system in E-coli (Fig. \u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e). A simplified model of this network with a \u0026lsquo;\u003cem\u003edeficiency\u003c/em\u003e\u0026rsquo;\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e24\u003c/span\u003e\u003c/sup\u003e of \u003cem\u003eone\u003c/em\u003e was first analyzed by Shinar and Feinberg\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e2\u003c/span\u003e\u003c/sup\u003e, and could be shown to exhibit absolute concentration robustness (ACR) \u0026ndash; a type of RPA (see SI Section S1.3) \u0026ndash; by the Shinar-Feinberg theorem\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e2\u003c/span\u003e\u003c/sup\u003e. Deficiency is a key integer invariant associated to a CRN\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e24\u003c/span\u003e\u003c/sup\u003e, as we discuss in greater detail in the sections to follow. Crucially, the Shinar-Feinberg theorem applies only to deficiency-one CRNs, which severely limits its ability to detect ACR (and thus RPA) in most molecular networks of biological interest, since the deficiencies of known genome-scale signaling networks (eg. in metabolism) in even the simplest organisms frequently exceed one hundred\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e25\u003c/span\u003e\u003c/sup\u003e. In Fig. \u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003ea,b, we consider the more detailed version of the \u003cem\u003eEnvZ-OmpR\u003c/em\u003e CRN, which eludes the Shinar-Feinberg theorem, having a deficiency of \u003cem\u003etwo\u003c/em\u003e, but which can be shown to exhibit RPA (ACR) in the phosphoform, \u003cem\u003epOmpR\u003c/em\u003e (see Supplementary Materials of Shinar and Feinberg\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e2\u003c/span\u003e\u003c/sup\u003e).\u003c/p\u003e\n \u003cp\u003eThe CRN in Fig. \u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e conforms to the topological principles of an RPA-conferring Balancer Module (see SI Section S3.2). All Balancer modules are characterized by a collection of parallel pathways emanating from a \u003cem\u003ediverter node\u003c/em\u003e (here, \u003cem\u003eEnvZ-ATP\u003c/em\u003e) and culminating in a \u003cem\u003econnector node\u003c/em\u003e (here, \u003cem\u003epOmpR\u003c/em\u003e)\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e4\u003c/span\u003e\u003c/sup\u003e. One or more \u003cem\u003ebalancer nodes\u003c/em\u003e (here, \u003cem\u003eEnvZ-ADP\u003c/em\u003e) may be embedded within the parallel pathways\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e4\u003c/span\u003e\u003c/sup\u003e. In Fig. \u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003ec we identify an integral variable \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({z}_{1}={a}_{5}{\\int }_{{t}_{0}}^{t}\\text{EnvZ-ATP}\\left(\\tau \\right)\\left(\\frac{\\text{EnvZ-ATP}\\left(\\tau \\right)}{\\text{EnvZ-ATP}\\left(\\tau \\right)}-\\beta {K}_{m1}\\right)d\\tau\\)\u003c/span\u003e\u003c/span\u003e that first confers RPA on the concentration ratio \u003cem\u003eEnvZ-ATP\u003c/em\u003e/\u003cem\u003eEnvZ-ADP\u003c/em\u003e \u0026ndash; a key invariant known as a \u003cem\u003ebalancer invariant\u003c/em\u003e (see Araujo et al.\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e4\u003c/span\u003e\u003c/sup\u003e). Having established a setpoint of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\beta {K}_{m1}\\)\u003c/span\u003e\u003c/span\u003e for this balancer invariant, a second integral variable, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({z}_{2}\\)\u003c/span\u003e\u003c/span\u003e, constructs a connector invariant that imposes RPA on \u003cem\u003epOmpR\u003c/em\u003e (see Fig. \u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003ec). Crucially, both the balancer invariant and the connector invariant are obtained via \u003cem\u003elinear\u003c/em\u003e combinations of the CRN\u0026rsquo;s mass action equations. These two polynomial invariants together produce an \u003cem\u003eRPA polynomial\u003c/em\u003e of the form\u003c/p\u003e\n \u003cdiv class=\"gridtable\"\u003e\n \u003cdiv align=\"left\" class=\"colspec\"\u003e\u003cbr\u003e\u003c/div\u003e\n \u003cdiv align=\"left\" class=\"colspec\"\u003e\u003cbr\u003e\u003c/div\u003e\u0026nbsp;\u003ctable border=\"1\" id=\"Tabd\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\rho }_{2}=\\gamma .\\text{EnvZ-ATP}\\left(\\text{pOmpR}- \\frac{{k}_{1}{K}_{m4}}{\\gamma }\\right),\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e(Eq.\u0026nbsp;5)\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n \u003c/div\u003e\n \u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n \u003cp\u003ethrough nonlinear combination via \u0026lsquo;concatenating monomials\u0026rsquo; (see Fig. \u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003ec), thereby explicitly highlighting the CRN\u0026rsquo;s capacity for RPA in \u003cem\u003epOmpR\u003c/em\u003e, with setpoint\u0026nbsp;\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\frac{{k}_{1}{K}_{m4}}{\\gamma }\\)\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n \u003cp\u003eOur transformative step is to prove that \u003cem\u003eall\u0026nbsp;\u003c/em\u003eRPA-capable CRNs, regardless of size, complexity, or \u0026lsquo;deficiency\u0026rsquo;\u003csup\u003e2\u003c/sup\u003e, necessarily conform to the general principles encapsulated by these two illustrative examples (see SI Section S4). \u0026nbsp;In fact, all RPA-capable CRNs are characterized by a \u003cem\u003etopological hierarchy of polynomial invariants\u003c/em\u003e, each obtained by a \u003cem\u003elinear combination\u003c/em\u003e of the CRN\u0026rsquo;s rate equations, and each corresponding \u003cem\u003eone-to-one\u003c/em\u003e with a topological feature (a balancer node, a connector node or an opposer node) of the overarching network structure. \u0026nbsp;The topological principles that are known to hold at the network macroscale for all RPA-capable networks are now understood in complete generality\u003csup\u003e4\u003c/sup\u003e and, together with the integral-computing properties of CRNs at the network microscale presented here, constitute definitive design criteria that unify all possible RPA-capable CRNs.\u003c/p\u003e\n \u003cp\u003eThere are two distinct but interrelated components to this central result, which we delineate in turn in the sections to follow: \u0026nbsp;(i) We identify the universal algebraic condition that is satisfied by all RPA-capable CRNs, which encodes the fundamental constraints on the \u0026lsquo;flow\u0026rsquo; of biochemical information through the CRN (and hence on the overarching topological structure of the CRN). \u0026nbsp;(ii) We show that this algebraic condition always admits a decomposition into a collection of \u0026lsquo;linear\u0026rsquo; problems \u0026ndash; a decomposition that is governed by a the CRN\u0026rsquo;s \u003cem\u003edeficiency\u003c/em\u003e. \u0026nbsp;Together, these two mathematical results reveal a universal integral control implementation that holds for all possible RPA-capable CRNs, however large, complex or \u003cem\u003enonlinear\u0026nbsp;\u003c/em\u003ein their dynamics.\u003c/p\u003e\n \u003ch2\u003e\u003cspan class=\"Underline\" name=\"Emphasis\" type=\"Underline\"\u003eAdaptation Relies on a CRN design strategy called\u003c/span\u003e \u003cspan class=\"ItalicUnderline\" name=\"Emphasis\" type=\"ItalicUnderline\"\u003eKinetic Pairing\u003c/span\u003e\u003c/h2\u003e\n \u003cp\u003eFirst, we prove (see \u003cstrong\u003eTheorem 1\u003c/strong\u003e, SI Section S1.4) that for \u003cem\u003eall\u003c/em\u003e RPA-capable CRNs, with interacting molecules \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({x}_{1}, \\dots , {x}_{n},\\)\u003c/span\u003e\u003c/span\u003e and corresponding mass-action rate equations \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({f}_{1}, \\dots , {f}_{n}\\)\u003c/span\u003e\u003c/span\u003e, there \u003cem\u003ealways\u003c/em\u003e exist polynomials \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\{h}_{1}, \\dots , {h}_{n}\\}\\subset \\mathbb{R}[{x}_{1},\\dots ,{x}_{n}]\\)\u003c/span\u003e\u003c/span\u003e such that\u003c/p\u003e\n \u003cdiv class=\"gridtable\"\u003e\n \u003cdiv align=\"left\" class=\"colspec\"\u003e\u003cbr\u003e\u003c/div\u003e\n \u003cdiv align=\"left\" class=\"colspec\"\u003e\u003cbr\u003e\u003c/div\u003e\u0026nbsp;\u003ctable border=\"1\" id=\"Tabe\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({h}_{1}{f}_{1}+\\dots +{h}_{n}{f}_{n}=g\\left({x}_{i},{x}_{j}\\right)\\left({x}_{i}-c\\right)=\\rho ,\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e(Eq.\u0026nbsp;6)\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n \u003c/div\u003e\n \u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n \u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\rho =g\\left({x}_{i},{x}_{j}\\right)\\left({x}_{i}-c\\right),\\)\u003c/span\u003e\u003c/span\u003ein its lowest order form, is the \u003cem\u003eRPA polynomial\u003c/em\u003e of the CRN, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({x}_{i}\\)\u003c/span\u003e\u003c/span\u003e is any RPA-capable variable of the CRN, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({x}_{j}\\)\u003c/span\u003e\u003c/span\u003e is any variable that \u003cem\u003edoes not\u003c/em\u003e exhibit RPA (i.e., an actuator variable, or a molecule regulated by an actuator variable). The system setpoint, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(c\\)\u003c/span\u003e\u003c/span\u003e, is a rational function of biochemical parameters.\u003c/p\u003e\n \u003cp\u003eFrom this new mathematical vantage point, we can now recognize that the special structure of the RPA polynomial specified by Theorem 1, being a function of exactly two variables, imposes fundamental structural limitations on RPA-capable CRNs, and encodes the cardinal principle that we call \u003cem\u003ekinetic pairing\u003c/em\u003e (see Fig. \u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003e and SI Section S2). In particular, the functional form of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\rho\\)\u003c/span\u003e\u003c/span\u003e suggests two possible topological interpretations of Theorem 1, depending on whether the RPA-capable variable, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({x}_{i}\\)\u003c/span\u003e\u003c/span\u003e, is a \u003cem\u003eregulating\u003c/em\u003e variable (for \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({x}_{j}\\)\u003c/span\u003e\u003c/span\u003e) or a \u003cem\u003eregulated\u003c/em\u003e variable (by \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({x}_{j}\\)\u003c/span\u003e\u003c/span\u003e). As depicted schematically in Fig. \u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003ea, if \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({x}_{i}\\)\u003c/span\u003e\u003c/span\u003e regulates \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({x}_{j}\\)\u003c/span\u003e\u003c/span\u003e (the non-RPA-capable variable), then the upregulating and downregulating contributions of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({x}_{j}\\)\u003c/span\u003e\u003c/span\u003eto its own reaction rate must be precisely matched, or \u0026lsquo;\u003cem\u003epaired\u003c/em\u003e\u0026rsquo;, via the pairing function \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(g\\)\u003c/span\u003e\u003c/span\u003e. At steady state, this form of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\rho\\)\u003c/span\u003e\u003c/span\u003e satisfies the condition \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\frac{\\partial \\rho }{\\partial {x}_{j}}=0\\)\u003c/span\u003e\u003c/span\u003e referred to in Araujo et al.\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e4\u003c/span\u003e\u003c/sup\u003e as \u003cem\u003eopposer kinetics\u003c/em\u003e, and must therefore be embedded in an overarching feedback loop which gives rise to the topological structure of an Opposer module. Our previous exhaustive analysis on Opposer module topologies has established that the feedback segment of such modules may contain multiple opposer \u0026lsquo;nodes\u0026rsquo;, each with its own opposer kinetics, and each contributing to a collection of embedded interlinked feedback loops known as an \u003cem\u003eopposing set\u003c/em\u003e (Fig. \u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003ea).\u003c/p\u003e\n \u003cp\u003eIf, on the other hand, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({x}_{i}\\)\u003c/span\u003e\u003c/span\u003e is regulated by \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({x}_{j}\\)\u003c/span\u003e\u003c/span\u003e, then the upregulating and downregulating contributions of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({x}_{j}\\)\u003c/span\u003e\u003c/span\u003e to the reaction rate for \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({x}_{i}\\)\u003c/span\u003e\u003c/span\u003e must likewise be precisely paired via \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(g\\)\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n \u003cp\u003eAs illustrated in Fig. \u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003eb, the pairing function naturally induces a Balancer topology\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e4\u003c/span\u003e\u003c/sup\u003e on the CRN in this case (see SI Section S2), with \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({x}_{j}\\)\u003c/span\u003e\u003c/span\u003e performing a \u003cem\u003ediverter\u003c/em\u003e function\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e4\u003c/span\u003e\u003c/sup\u003e. For this topological structure, the steady-state condition \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\frac{\\partial \\rho }{\\partial {x}_{j}}=0\\)\u003c/span\u003e\u003c/span\u003e corresponds to connector kinetics\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e4\u003c/span\u003e\u003c/sup\u003e, provided that additional constraints (referred to as balancer kinetics in Araujo et al.\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e4\u003c/span\u003e\u003c/sup\u003e) can be satisfied for the reactions embedded into any parallel pathways linking \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({x}_{j}\\)\u003c/span\u003e\u003c/span\u003e to \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({x}_{i}\\)\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n \u003cp\u003eAlthough Theorem 1 holds for any non-RPA-capable variable, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({x}_{j}\\)\u003c/span\u003e\u003c/span\u003e, we show that there exists a \u0026lsquo;natural\u0026rsquo; choice of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({x}_{j}\\)\u003c/span\u003e\u003c/span\u003e with respect to the requisite topological structure of the CRN: a diverter variable (in the case of a Balancer module), or an opposer variable (for an Opposer module). Algorithmically, this choice of non-RPA-capable variable simplifies the elimination polynomials \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({h}_{1}, \\dots ,{h}_{n}\\)\u003c/span\u003e\u003c/span\u003e in Eq. 6 (see SI Section S1.5 for a fully analysed example). Indeed, for such a judicious choice of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({x}_{j}\\)\u003c/span\u003e\u003c/span\u003e, the pairing function \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(g\\)\u003c/span\u003e\u003c/span\u003e is frequently zero-order in \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({x}_{i}\\)\u003c/span\u003e\u003c/span\u003e, except in the special case of an autoregulatory role for \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({x}_{i}\\)\u003c/span\u003e\u003c/span\u003e (see Fig. \u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003e). In the case of the antithetic integral control motif, the pairing function is zero-order in both \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({x}_{i}\\)\u003c/span\u003e\u003c/span\u003eand \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({x}_{j}\\)\u003c/span\u003e\u003c/span\u003e, giving rise to \u003cem\u003eunconstrained\u003c/em\u003e integral control\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e3\u003c/span\u003e,\u003cspan class=\"CitationRef\"\u003e23\u003c/span\u003e\u003c/sup\u003e.\u003c/p\u003e\n \u003c/div\u003e\n \u003cdiv class=\"Section2\" id=\"Sec3\"\u003e\n \u003ch2\u003eRPA-permissive topological features are encoded by CRN deficiency\u003c/h2\u003e\n \u003cp\u003eSecond, we consider a decomposition of the nonlinear algebraic condition for RPA (Eq. 6) into its component linear contributions, to reveal the general mechanism through which kinetic pairing is transacted in CRNs. Indeed, by identifying the connection between the \u003cem\u003edeficiency\u003c/em\u003e of an RPA-capable CRN and the presence of feedback loops and/or feedforward segments (parallel pathways), we prove that the RPA polynomial of a CRN can always be decomposed into a collection of subsidiary polynomial invariants, each corresponding to a component of the CRN\u0026rsquo;s topological structure, and each residing in the \u003cem\u003erowspan\u003c/em\u003e of the CRN\u0026rsquo;s reaction rates. In other words, each such subsidiary invariant may be obtained via a \u003cem\u003elinear transformation\u003c/em\u003e of the system\u0026rsquo;s reaction equations, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({f}_{1}, \\dots ,{f}_{n}\\)\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n \u003cp style=\"margin-top: 0pt; margin-bottom: 0pt; font-size: 12pt;\"\u003e\u003cspan style='font-family: \"Times New Roman\", Times, serif;'\u003eThe deficiency of a CRN (Fig. 4) is a non-negative integer that encapsulates the extent to which the individual reactions of the CRN are linearly independent given their distribution into linkage classes\u003c/span\u003e\u003cspan style='line-height: 200%; font-family: \"Times New Roman\", Times, serif; font-size: 8pt;'\u003e\u003csup\u003e24\u003c/sup\u003e\u003c/span\u003e\u003cspan style='font-family: \"Times New Roman\", Times, serif;'\u003e. \u0026nbsp;With the exception of the\u0026nbsp;\u003c/span\u003e\u003cspan style=\"font-family: 'Times New Roman', Times, serif;\"\u003e\u003cem\u003etrivial\u003c/em\u003e\u003c/span\u003e\u003cspan style='font-family: \"Times New Roman\", Times, serif;'\u003e\u0026nbsp;RPA-capable CRN consisting only of an isolated connector node, which has a \u0026nbsp;deficiency of zero (see SI Section S4.2.2), all (non-trivial) RPA-capable CRNs require a deficiency of\u0026nbsp;\u003c/span\u003e\u003cspan style=\"font-family: 'Times New Roman', Times, serif;\"\u003e\u003cem\u003eat least\u003c/em\u003e\u003c/span\u003e\u003cspan style='font-family: \"Times New Roman\", Times, serif;'\u003e\u0026nbsp;one. \u0026nbsp;The Shinar-Feinberg Theorem\u003c/span\u003e\u003cspan style='line-height: 200%; font-family: \"Times New Roman\", Times, serif; font-size: 8pt;'\u003e\u003csup\u003e2\u003c/sup\u003e\u003c/span\u003e\u003cspan style='font-family: \"Times New Roman\", Times, serif;'\u003e\u0026nbsp;(see Theorem 2 in SI) pertains to CRNs with a deficiency of exactly one, and states that any such CRN containing two distinct complexes that differ in a single species S, and admitting a steady-state in the positive orthant, necessarily exhibit ACR (and therefore RPA) in the species S. \u0026nbsp; This key theorem follows from Shinar and Feinberg\u0026rsquo;s more general result that the steady-state ratio of\u0026nbsp;\u003c/span\u003e\u003cspan style=\"font-family: 'Times New Roman', Times, serif;\"\u003e\u003cem\u003eany\u0026nbsp;\u003c/em\u003e\u003c/span\u003e\u003cspan style='font-family: \"Times New Roman\", Times, serif;'\u003etwo monomials associated to non-terminal complexes (see Fig. 4) is independent of the system\u0026rsquo;s initial conditions\u003c/span\u003e\u003cspan style='line-height: 200%; font-family: \"Times New Roman\", Times, serif; font-size: 8pt;'\u003e\u003csup\u003e2\u003c/sup\u003e\u003c/span\u003e\u003cspan style='font-family: \"Times New Roman\", Times, serif;'\u003e. \u0026nbsp;Here we extend Shinar and Feinberg\u0026rsquo;s arguments to prove a still stronger result (SI Theorem 3) \u0026ndash; namely, that all deficiency-one CRNs contain binomials in the\u0026nbsp;\u003c/span\u003e\u003cspan style=\"font-family: 'Times New Roman', Times, serif;\"\u003e\u003cem\u003erowspan\u003c/em\u003e\u003c/span\u003e\u003cspan style='font-family: \"Times New Roman\", Times, serif;'\u003e\u0026nbsp;of their reaction rates, of the form\u003c/span\u003e\u003cspan style=\"font-family: 'Times New Roman', Times, serif;\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"130\" height=\"22\" alt=\"\"\u003e\u003c/span\u003e\u003cspan style='font-family: \"Times New Roman\", Times, serif;'\u003e, where\u0026nbsp;\u003c/span\u003e\u003cspan style=\"font-family: 'Times New Roman', Times, serif;\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"39\" height=\"21\" alt=\"\"\u003e\u003c/span\u003e\u003cspan style='font-family: \"Times New Roman\", Times, serif;'\u003e\u0026nbsp;and\u0026nbsp;\u003c/span\u003e\u003cspan style=\"font-family: 'Times New Roman', Times, serif;\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"39\" height=\"22\" alt=\"\"\u003e\u003c/span\u003e\u003cspan style='font-family: \"Times New Roman\", Times, serif;'\u003e\u0026nbsp;are any two mass-action monomials corresponding to non-terminal complexes, and\u0026nbsp;\u003c/span\u003e\u003cspan style=\"font-family: 'Times New Roman', Times, serif;\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"87\" height=\"21\" alt=\"\"\u003e\u003c/span\u003e\u003cspan style='font-family: \"Times New Roman\", Times, serif;'\u003e\u0026nbsp;is a pair of rational functions of the CRN rate constants. \u0026nbsp;In other words, there exists some\u0026nbsp;\u003c/span\u003e\u003cspan style=\"font-family: 'Times New Roman', Times, serif;\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAGYAAAAVCAYAAAC0aZsNAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsMAAA7DAcdvqGQAAAW1SURBVGhD7ZhbSNRPFMfNUHqISC2NvN+oIDQLLw8hIV5BfFDfwnzoJXpIKkhKtIR8KUkRBH0QzBQV76JiEFpiEARJaaSViqZ4Ie/368nv2Rn97fpb3eWfufzZDwzOnP3525nznTnnzFqQGVPEwyyMaWIWxkQxC2OimIU5DNbX16m6upoePnxIs7Oz9PbtW0pLS6PBwUHxhIowGxsbVFFRQc3NzcJClJSURDdu3KCRkRFhORxaW1t5HmjG0t/fTy9fvhSjv8/v37/5/Wqtra2N/SpZWlqiiYkJSkhIYHHGx8cpOzub7t69K57QEQbqXbp0iQoKCmhlZUVYNQpbWFho2Q4LDw8Pio+PFyPDgWMqKyvpwoULNDU1Jax/D/gmNjaWHj16xA6Oioqix48f08WLF+nVq1dkb29PHz9+FE9r5nPixAn69u0bj69fv07FxcXc32JHmNXVVXJzc6OGhgZh2QFHDcJsbm4Ky+Fx5swZ+vTpkxgZz4cPH8jZ2ZnXuxdra2t8MuGTU6dOabX5+XnxlDY5OTn05s0bqqmpoczMTLZhrhAI/3PkyBG2gaGhITp37pwYEZ0/f576+vrESCEMTom/v78YaQM1cezA6OgoH8PDwtramkMBNsnXr1+F1TjgcEXY2AXeDQGePHlCMzMztLy8rNX0oSYMCAkJ4b+Wlpb8FyBVREdHcx8+xcbHKUJ02kIjDAxQEwlJDaiJnXb8+HEKDAzkZ3/9+iU+/Xd0dHTwApqamni3oX/58mXxqeEgAtjY2IjRboaHh8nBwUGMDEdNGJlLgFKY/Px8XocEuQY2kYs0wiC3YJH6OHnyJDeoKfMNvvxfc//+fc6B8rvr6+vJ3d2d+8YAZ2ENnZ2dwqLN8+fPydPTk0+LWtMX0pXCPH36lCYnJ+nq1atcbcXExFBPT494cl80woyNjXEiUqO3t5cXIeMfkhxOzPT0NI8lmAQqkIPE19eXT6x0THJyMgUFBXHfWLCmqqoqMdIGwiBk4lSqtYWFBfGkNkphTp8+zYXGjx8/yMrKajvJG8j+wmRkZJCdnd22M5C0MJYsLi5yqefj40MBAQHCejDY2tpSd3c39zEfOLewsJDHxrKXMFlZWXxijEU3lKEfFxfH/oXQ8JWB7Ahz7Ngxtuhy7do13pmSe/fu8c6V4MuQjN+9e3egwiCnwZnypKKqQsyem5vjMcCp/f79O/dRBXV1damGHZx6vKuurk5YtMEuV24+Q1HLMVeuXKGfP39SXl4ehYaGss0ANMJgcZgo4qcuKBdxsZPgOewo3AVaWlqElfQK8+LFC0pNTRUjTTVy8+bNbYemp6fzGA5E/kIfNvDlyxceQ3gsDN8tHY3vR94DtbW1LFx5eTmfXJyixsZGdgQuy7ogueNd+sILErCXlxfdvn3bmF2uJQxu8gAX2+DgYJ730aNH9eY1HTTCYCKYqG6OwG0WdmUFhouSi4sLJzVlrNUnTGRkJFd1kmfPnvE78W7g5+e37XDcHdCXlRaqFoxxMw4PD6fExES2AyRwLBTOLy0tFVbiO4rcMGFhYfT69WvuK8FdzcnJSVZAqmCToqxGfsB6lU3tHoMQ7+rqSrdu3eIqzNHRcfu+hQsx1p2SksIhDdXlPmiEASUlJbzblMDxnz9/lrU1g50O1XUXpU8Y5C55AgCcjHdCBIDQgzGAOOjLcJSbm7stEmz4XyW4x+AOIBkYGOBfBuSpOnv27K4iBURERNCDBw/EaG8wT2VFhibfrwQ+0n1OXmJ1PzPgF5QdYTABb29vKisrExbjUBPm/fv3vKP32pn6wIlAAtYVYy9wSb5z5w73Ef6Qg5SbCrS3t/MJlhvDRNkRBkBJlIOI58aAEIgyFo7AkZW7WG1nGYOxzkPokRsLpx1hEDYZeoqKijhn7vdzjAmgLYwEJSlu+v8nUHn+l9/Y/jEeFlu7usvcTK1tOv0BbK3KmjzibTAAAAAASUVORK5CYII=\" width=\"102\" height=\"21\" alt=\"\"\u003e\u003c/span\u003e\u003cspan style='font-family: \"Times New Roman\", Times, serif;'\u003e\u0026nbsp;such that\u0026nbsp;\u003c/span\u003e\u003cspan style=\"font-family: 'Times New Roman', Times, serif;\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"264\" height=\"22\" alt=\"\"\u003e\u003c/span\u003e\u003cspan style='font-family: \"Times New Roman\", Times, serif;'\u003e.\u003c/span\u003e\u003c/p\u003e\n \u003cp\u003eCrucially, we extend the mathematical framework for our Theorem 3 (on deficiency-one CRNs) to a general method for identifying rowspan polynomials in CRNs of arbitrary deficiency. In fact, there are exactly two ways in which a deficiency exceeding one can be accommodated into an RPA-capable CRN (see SI Sections S4.2.1 and S4.2.2). First, \u0026lsquo;extramodular\u0026rsquo; chemical reactions that do not contribute to the RPA-conferring mechanism may increase the deficiency of the CRN without pertubing its RPA capacity: for CRNs with Balancer topology, any reactions upstream of the diverter node and downstream of the connector node constitute extramodular reactions; for CRNs with Opposer topology, any reactions outside the feedback segment of the module (the \u0026lsquo;embedded network\u0026rsquo;, see Fig. \u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003ea) may be considered extramodular in this context, notwithstanding their contribution to the \u0026lsquo;controlled\u0026rsquo; portion of the overarching module. Mathematically, algebraic independence from the RPA-conferring chemical reactions of the CRN may be established by decomposing the CRN into algebraically-independent subnetworks, wherein the sum of the \u003cem\u003eranks\u003c/em\u003e of the subnetworks is equal to the rank of the full CRN. We provide full technical details on these principles SI Section S4.2, along with a set of worked examples. In any case, algebraically independent subnetworks of the CRN that correspond to extramodular reactions have no bearing on the RPA capacity of the CRN as a whole, and may be excluded from analysis.\u003c/p\u003e\n \u003cp\u003eSecond, chemical reactions contributing to either feedforward segments (parallel pathways) or additional feedback loops within the RPA-conferring CRN will increase deficiency by one for each additional such element (see Fig. \u003cspan class=\"InternalRef\"\u003e5\u003c/span\u003e; see also SI Section S4.2.2). For this reason, all deficiency-one RPA-capable CRNs must necessarily be either a single Balancer module comprising exactly two (incoherent) parallel pathways, or a single opposer node embedded into an Opposer module. The simple antithetic integral control motif (see highlighted reactions in Fig. \u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e), as well as the simple EnvZ-OmpR CRNs considered by Shinar and Feinberg\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e2\u003c/span\u003e\u003c/sup\u003e (cf. Figure \u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e) are example of deficiency-one CRNs that exhibit RPA. We demonstrate that CRNs may be decomposed into subsets corresponding to the deficiency-increasing elements; using this decomposition (along with mathematical induction, as needed), our Theorem 3 may be extended to the identification of RPA-promoting polynomial invariants in the rowspan of the CRN (see SI Section S4.2.2). This method makes clear that although the RPA polynomial associated to an RPA-capable CRN generally requires a nonlinear transformation of the reaction equations (Eq. 6), there always exist linear transformations that can extract \u0026lsquo;special\u0026rsquo; polynomial building blocks, each corresponding one-to-one with a topological feature of the overarching network structure, from the CRN\u0026rsquo;s reaction equations. In particular, all RPA-capable CRNs of Balancer type contain a connector polynomial, corresponding to the connector \u0026lsquo;node\u0026rsquo;, and one or more balancer polynomials, corresponding to balancer node(s), in their rowspans. CRNs of Opposer type, on the other hand, contain one or more opposer polynomials (corresponding to opposer node(s)) in their rowspans.\u003c/p\u003e\n \u003c/div\u003e\n \u003cdiv class=\"Section2\" id=\"Sec4\"\u003e\n \u003ch2\u003eIntegral control and the \u0026lsquo;passing\u0026rsquo; of invariants\u003c/h2\u003e\n \u003cp\u003eUntil now strategies for identifying an internal model, and an associated integral, via a nonlinear coordinate change have only been applicable to exceedingly simple CRNs\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e18\u003c/span\u003e\u003c/sup\u003e. By contrast, our approach identifies a well-defined nonlinear map between reaction rates of the model variables \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({f}_{1}, \\dots , {f}_{n}\\)\u003c/span\u003e\u003c/span\u003e, and an internal model, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\rho\\)\u003c/span\u003e\u003c/span\u003e (Eq. 6), which exists for all adaptation-capable CRNs. \u003cem\u003eIn contrast to all prior control theoretic approaches\u003c/em\u003e, this alternative viewpoint decomposes all RPA-capable CRNs into a constellation of \u003cspan class=\"BoldItalic\" name=\"Emphasis\" type=\"BoldItalic\"\u003elinear\u003c/span\u003e integral control problems, each with an associated invariant (i.e. internal model). These are distributed across well-defined RPA-permissive network topologies\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e4\u003c/span\u003e\u003c/sup\u003e, and construct an RPA polynomial, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\rho\\)\u003c/span\u003e\u003c/span\u003e, through the process of \u003cem\u003einvariant passing\u003c/em\u003e (Fig. \u003cspan class=\"InternalRef\"\u003e6\u003c/span\u003e; see also SI Sections S3 and S4).\u003c/p\u003e\n \u003cp\u003eInvariant passing describes the process by which polynomial invariants corresponding to adjacent topological features of the CRN\u0026rsquo;s overarching network structure (as described in the preceding section) are combined so as to systematically eliminate model variables, and ultimately obtain the RPA polynomial of the CRN. As illustrated schematically in Fig. \u003cspan class=\"InternalRef\"\u003e6\u003c/span\u003ea for CRNs of Opposer type, opposer invariants are passed from the distal opposer node to the proximal opposer node; for CRNs of Balancer type (Fig. \u003cspan class=\"InternalRef\"\u003e6\u003c/span\u003eb), invariants are passed from the diverter node to the sequence of balancer nodes within each parallel pathway, culminating at the connector node (Fig. \u003cspan class=\"InternalRef\"\u003e6\u003c/span\u003eb).\u003c/p\u003e\n \u003cp\u003eCrucially, we demonstrate that the ability or inability of these invariants to \u0026lsquo;pass\u0026rsquo; \u003cem\u003ewithin the rowspan\u003c/em\u003e of the system is a question of \u003cem\u003estoichiometric independence\u003c/em\u003e of the individual chemical reactions contributing to successive invariants. We outline the significance of this key concept through the analysis of a simple illustrative example (Fig. \u003cspan class=\"InternalRef\"\u003e7\u003c/span\u003e). In Fig. \u003cspan class=\"InternalRef\"\u003e7\u003c/span\u003ea, we depict a deficiency-one CRN comprising three interacting molecules, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(A\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(B\\)\u003c/span\u003e\u003c/span\u003eand \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(C\\)\u003c/span\u003e\u003c/span\u003e, which exhibits RPA (and, more specifically, ACR) in the molecule \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(A\\)\u003c/span\u003e\u003c/span\u003e. It is easy to show (see SI Section S3.1) that this CRN is topologically a Balancer module, where \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(A\\)\u003c/span\u003e\u003c/span\u003e is the connector, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(B\\)\u003c/span\u003e\u003c/span\u003e is the diverter, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(C\\)\u003c/span\u003e\u003c/span\u003e is a balancer. In Fig. \u003cspan class=\"InternalRef\"\u003e7\u003c/span\u003eb, we present a modified version of this CRN that preserves both its topology and its deficiency of one. Both CRNs contain an identical connector polynomial in their rowspans. In addition, both CRNs contain a balancer polynomial in their respective rowspans, as expected. In the original CRN (Fig. \u003cspan class=\"InternalRef\"\u003e7\u003c/span\u003ea), however, the balancer and connector polynomials are stoichiometrically independent, in the sense that the variable to be eliminated in the process of invariant passing (ie. \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(C\\)\u003c/span\u003e\u003c/span\u003e) derives from the reactant complex \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(B+C\\)\u003c/span\u003e\u003c/span\u003e in the balancer polynomial, and from the reactant complex \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(C\\)\u003c/span\u003e\u003c/span\u003ein the connector polynomial. Therefore, the connector polynomial must be multiplied by the concentration of molecule \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(B\\)\u003c/span\u003e\u003c/span\u003e (a \u0026lsquo;concatenating monomial\u0026rsquo;) in order for the balancer polynomial to \u0026lsquo;pass\u0026rsquo; to it, and thereby construct the RPA polynomial. By contrast, the single reactant complex \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(C\\)\u003c/span\u003e\u003c/span\u003econtributes to \u003cem\u003eboth\u003c/em\u003e subsidiary polynomials for the modified CRN (Fig. \u003cspan class=\"InternalRef\"\u003e7\u003c/span\u003eb), guaranteeing their stoichiometric dependence.\u003c/p\u003e\n \u003cp\u003eIt is striking to note that the original form of the CRN (Fig. \u003cspan class=\"InternalRef\"\u003e7\u003c/span\u003ea) eludes the Shinar-Feinberg theorem, even though the CRN exhibits ACR and has a deficiency of one. It is thereby clear that, even for the special case of deficiency-one CRNs, the Shinar-Feinberg theorem cannot provide a comprehensive description of ACR (and hence RPA). By contrast, the modified form of the CRN (Fig. \u003cspan class=\"InternalRef\"\u003e7\u003c/span\u003eb), with stoichiometrically-dependent balancer and connector polynomials, \u003cem\u003edoes\u003c/em\u003e satisfy the Shinar-Feinberg theorem. With the stoichiometric dependence of all subsidiary polynomials now delivering the all-important RPA polynomial to the system\u0026rsquo;s rowspan, this modified CRN necessarily contains two non-terminal complexes (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(A+B\\)\u003c/span\u003e\u003c/span\u003e, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(B\\)\u003c/span\u003e\u003c/span\u003e) that differ in the single ACR-exhibiting molecule, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(A\\)\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n \u003cp\u003eWe illustrate the control diagram corresponding to our decomposition into a topological hierarchy of linear controllers in Fig. \u003cspan class=\"InternalRef\"\u003e8\u003c/span\u003e for the particular case of a single Opposer module, since all Opposer modules necessarily incorporate an overarching feedback structure, and are thus easily described using standard control diagrams. In principle, there should always exist some nonlinear coordinate change to extract a single output-driven internal model (Fig. \u003cspan class=\"InternalRef\"\u003e8\u003c/span\u003ea) from system\u0026rsquo;s rate equations, corresponding to a single integral of the system\u0026rsquo;s tracking error (Fig. \u003cspan class=\"InternalRef\"\u003e8\u003c/span\u003eb). But in our representation, each subsidiary polynomial invariant corresponds to an independent internal model, each with its own independent setpoint. For a CRN constituting a three-node opposing set (Fig. \u003cspan class=\"InternalRef\"\u003e8\u003c/span\u003ec), for instance, there will exist three independent internal models and three corresponding opposer integrals, each conferring RPA on a different variable (Fig. \u003cspan class=\"InternalRef\"\u003e8\u003c/span\u003ed). These three independent linear control systems collaborate to confer RPA on the \u0026lsquo;sensor\u0026rsquo; variable of the CRN, and ultimately, the entire embedded network.\u003c/p\u003e\n \u003c/div\u003e\n \u003cdiv class=\"Section2\" id=\"Sec5\"\u003e\n \u003ch2\u003eA universal algorithm for adaptation detection in complex CRNs\u003c/h2\u003e\n \u003cp\u003eIt is clear from the illustrative example of the EnvZ-OmpR osmoregulatory motif (Fig. \u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e; see also additional CRN examples in SI) that even for exceedingly simple CRNs, constituting a single RPA module, the integral-computing polynomial invariants may be deeply concealed within the chemical reaction structures, and cannot generally be identified by inspection. For this, we introduce here a universal algorithmic method for establishing the RPA capacity of a CRN, \u003cem\u003ewhich can identify the subsidiary polynomial invariants automatically\u003c/em\u003e.\u003c/p\u003e\n \u003cp\u003eOur method is a direct consequence of the fact that the RPA polynomial of any adaptation-capable CRN is a function of \u003cem\u003etwo\u003c/em\u003e variables, thereby converting the question or RPA capacity to a well-defined elimination problem. This elimination problem corresponds geometrically to the \u003cem\u003eprojection\u003c/em\u003e of the system onto just \u003cem\u003etwo\u003c/em\u003e variables (one RPA-capable, and one RPA-incapable) \u0026ndash; a task that can accomplished via computation of the Gr\u0026ouml;bner basis of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\u0026lt;{f}_{1}, \\dots , {f}_{n}\u0026gt;{ =\\{h}_{1}{f}_{1}+\\dots +{h}_{n}{f}_{n}| {h}_{i}\\in \\mathbb{R}\\left[{x}_{1},\\dots ,{x}_{n}\\right]\\}\\)\u003c/span\u003e\u003c/span\u003e with suitable monomial ordering (see SI Section S5). Remarkably, although the problem of computing a Gr\u0026ouml;bner basis (e.g. by Buchberger\u0026rsquo;s algorithm\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e26\u003c/span\u003e\u003c/sup\u003e) for \u003cem\u003egeneral\u003c/em\u003e systems of polynomials is well-known to be NP-Hard\u003csup\u003e\u003cspan class=\"CitationRef\"\u003e27\u003c/span\u003e\u003c/sup\u003e, the special \u0026lsquo;\u003cem\u003ealmost linear\u0026rsquo;\u003c/em\u003e structure of RPA capable CRNs (as described above, see also SI Section S4) allows any RPA-capable CRN to yield easily to this approach in polynomial time. Indeed, failure of Buchberger\u0026rsquo;s algorithm to terminate rapidly for a given CRN is \u003cem\u003eprima-facie\u003c/em\u003e evidence that the CRN does not, in fact, exhibit RPA.\u003c/p\u003e\n \u003cp\u003eWe provide full details of this method, along with code in the open-source software Singular (\u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003e\u003ca href=\"http://www.Singular.uni-kl.de\" target=\"_blank\"\u003ewww.Singular.uni-kl.de\u003c/a\u003e\u003c/span\u003e\u003c/span\u003e) in our Supplementary Information (see SI Section S5), where we also provide a selection of fully-annotated illustrative examples. This code can readily be applied to any CRN.\u003c/p\u003e\n \u003c/div\u003e"},{"header":"Discussion","content":"\u003cp\u003eIdentification of a definitive test for the capacity of a network of chemical reactions to exhibit RPA has been the subject of a long quest, and most attempts have considered only the special case of RPA known as Absolute Concentration Robustness (ACR). These diverse attempts have drawn from a range of different mathematical frameworks, which can be broadly divided into two main categories: the chemical reaction network theory (CRNT) viewpoint\u003csup\u003e\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e,\u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e24\u003c/span\u003e,\u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e25\u003c/span\u003e\u003c/sup\u003e, and the engineering control theory viewpoint\u003csup\u003e\u003cspan additionalcitationids=\"CR17 CR18\" citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e\u003c/sup\u003e.\u003c/p\u003e \u003cp\u003eThe pinnacle of CRNT approaches is the Shinar-Feinberg theorem\u003csup\u003e\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e\u003c/sup\u003e which identifies a sufficient condition for ACR in CRNs of \u003cem\u003edeficiency one\u003c/em\u003e\u003csup\u003e\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e,\u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e24\u003c/span\u003e\u003c/sup\u003e. It is now well known that most CRNs in nature have a deficiency much greater than one\u003csup\u003e\u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e25\u003c/span\u003e\u003c/sup\u003e and the Shinar-Feinberg theorem is silent on all such CRNs. In addition, the Shinar-Feinberg theorem cannot reveal the setpoint of any ACR-exhibiting molecules as a function of system rate constants, nor how the existence of ACR corresponds to the presence of integral control. Karp et al.\u003csup\u003e\u003cspan citationid=\"CR28\" class=\"CitationRef\"\u003e28\u003c/span\u003e\u003c/sup\u003e, developed an alternative systematic method to identify \u0026lsquo;complex linear\u0026rsquo; polynomial invariants, which require only linear combinations of the mass-action equations of a CRN. Using this method, the two subsidiary polynomial invariants given in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003ec could be identified in an ad-hoc manner. But without recognizing these two invariants as a balancer invariant and a connector invariant, and the general relationship of such invariants to an \u0026lsquo;RPA polynomial\u0026rsquo;, this approach cannot make the crucial connection to the essential structure that characterises all possible RPA-capable CRNs, and provides no connection to integral control in any such systems.\u003c/p\u003e \u003cp\u003eFrom the control theory viewpoint, Yi et al.\u003csup\u003e\u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e\u003c/sup\u003e use general linear models to demonstrate the necessity for integral control in all robust asymptotic tracking problems (such as RPA), and extract the internal model for the well-known Barkai-Leibler model of bacterial chemotaxis\u003csup\u003e\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e\u003c/sup\u003e by ad-hoc (linear) algebraic manipulations. A more recent systematic algebraic method developed by Cappelletti et al.\u003csup\u003e\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e\u003c/sup\u003e can now identify the capacity for RPA in any mass-action CRN for which an RPA polynomial exists in the rowspan of the system. This method explicitly identifies the presence of integral control in all such CRNs, and also reveals the system\u0026rsquo;s setpoint as a function of biochemical rate constants, but is silent on any CRN requiring a nonlinear coordinate change to reveal an internal model.\u003c/p\u003e \u003cp\u003eUntil now, general strategies for identifying an internal model via nonlinear coordinate transformations have remained elusive, and specific nonlinear maps have been identified only for exceedingly simple RPA-capable CRNs \u003csup\u003e\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e\u003c/sup\u003e. Although a nonlinear diffeomorphism that maps the original model variables to a special \u0026lsquo;block\u0026rsquo; form - thereby explicitly revealing an internal model - should always exist in principle\u003csup\u003e\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e,\u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e23\u003c/span\u003e\u003c/sup\u003e, the identification of such a nonlinear map in even the most complex special cases cannot, of itself, clarify the general principles that unify all possible RPA-capable chemical reaction structures.\u003c/p\u003e \u003cp\u003eBy contrast, our approach identifies a well-defined nonlinear map \u0026ndash; distinct from the transformations considered in previous control theoretic approaches\u003csup\u003e\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e,\u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e\u003c/sup\u003e \u0026ndash; between the reaction rates of the individual molcules, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({f}_{1}, \\dots , {f}_{n}\\)\u003c/span\u003e\u003c/span\u003e, and a key CRN invariant known as the RPA polynomial. This transformation holds for all RPA-capable CRNs, and constitutes a geometric projection of the full set of molecular concentrations onto a particular subset of the model variables, comprising one RPA-capable molecule and one non-RPA-capable molecule (recognizing that all RPA-capable networks require a minimum of one such variable to constitute the \u0026lsquo;adaptation\u0026rsquo;\u003csup\u003e\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e\u003c/sup\u003e). The major innovative leap that we make from this mathematical cornerstone is to show that this \u003cem\u003enonlinear map\u003c/em\u003e can always be decomposed into a constellation of \u003cem\u003elinear maps\u003c/em\u003e, each existing within a topological hierarchy\u003csup\u003e\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e\u003c/sup\u003e associated to the CRN\u0026rsquo;s underlying structure, and each corresponding to an independent \u003cem\u003elinear control problem\u003c/em\u003e. The integrals that are formulated by these independent subsidiary control systems thereby collaborate to confer RPA on one or more molecules in the CRN. Our approach unifies both the control theory and CRNT viewpoints, and extends prior results on the macroscale topologies\u003csup\u003e\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e\u003c/sup\u003e of RPA-capable networks to the microscale level of intermolecular interactions within CRNs.\u003c/p\u003e \u003cp\u003eThe combination of a definitive algebraic condition with the special \u0026lsquo;almost linear\u0026rsquo; structure of the underlying control system provide the essential ingredients for a simple algorithmic test for RPA capacity, even in large, high-deficiency CRNs. The only algorithmic method capable of handling CRNs of arbitrary deficiency prior to this work was the necessary condition for ACR identified by Eloundou-Mbebi et al.\u003csup\u003e\u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e25\u003c/span\u003e\u003c/sup\u003e Being a \u003cem\u003enecessary\u003c/em\u003e condition, the Eloundou-Mbebi method can identify a collection of molecules that certainly couldn\u0026rsquo;t exhibit ACR (and therefore RPA), and thereby reduces the number of molecules that must be analysed in detail for their ACR (RPA) capacity, eg. via extensive numerical simulation. But the Eloundou-Mbebi method is unable to identify, definitively, which molecules \u003cem\u003edo\u003c/em\u003e exhibit ACR (RPA) since it fails to capture the essential structural characteristics common to all RPA-capable networks. Indeed, the Eloundou-Mbebi method characteristically overestimates the space of molecules that could potentially exhibit ACR/RPA quite significantly\u003csup\u003e\u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e25\u003c/span\u003e\u003c/sup\u003e. For the deficiency-two model of the EnvZ-OmpR phosphorelay (Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e), for instance, all nine species satisfy the Eloundou-Mbebi condition, even though only \u003cem\u003epOmpR\u003c/em\u003e can actually exhibit ACR/RPA (as we can easily prove by the new method we present here). And in common with other CRNT-based approaches, the Eloundou-Mbebi condition makes no connection to integral control, and cannot identify the setpoint of any RPA-capable species as a function of biochemical parameters.\u003c/p\u003e \u003cp\u003eOnly a complete and truly general picture of the integral control problem in CRNs, as we present here, can demarcate the evolutionary trajectories along which complex adaptation-capable biological networks can arise from simpler building blocks, and provide a roadmap for either preserving or disrupting the RPA property in natural, diseased or synthetic networks through design alterations or pharmacological interventions.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eAcknowledgements\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eRobyn P. Araujo is supported by an Australian Research Council (ARC) Future Fellowship (project no. FT190100645) from the Australian Government.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAuthor Contributions\u003c/strong\u003e\u003cbr\u003e\u0026nbsp;RPA \u0026ndash; conceptualization, methodology, software, formal analysis, funding acquisition, writing \u0026ndash; original draft, writing \u0026ndash; review \u0026amp; editing.\u003c/p\u003e\n\u003cp\u003eLAL \u0026ndash; writing \u0026ndash; review \u0026amp; editing.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\u003cli\u003e\u003cspan\u003eTang, Z. F. \u0026amp; McMillen, D. R. Design principles for the analysis and construction of robustly homeostatic biological networks. 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J Theor Biol \u003cb\u003e311\u003c/b\u003e, 130\u0026ndash;138, doi:\u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003e10.1016/j.jtbi.2012.07.004\u003c/span\u003e\u003cspan address=\"10.1016/j.jtbi.2012.07.004\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e (2012).\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"nature-portfolio","isNatureJournal":true,"hasQc":false,"allowDirectSubmit":false,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"","title":"Nature Portfolio","twitterHandle":"","acdcEnabled":false,"dfaEnabled":false,"editorialSystem":"ejp","reportingPortfolio":"","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"","lastPublishedDoi":"10.21203/rs.3.rs-1571178/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-1571178/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eAt the molecular level, the evolution of life is driven by the generation and diversification of adaptation mechanisms. A universal description of adaptation-capable chemical reaction network (CRN) structures has remained elusive until now, since currently-known criteria for adaptation apply only to a tiny subset of possible CRNs. While adaptation is known to require some form of embedded \u003cem\u003eintegral control\u003c/em\u003e, current approaches can only identify an internal integral structure in simple special cases. Here we identify the definitive structural requirements that characterize all adaptation-capable collections of interacting molecules, however large or complex. We show that these network structures implement a form of integral control in which multiple independent integrals can collaborate to confer the capacity for adaptation on specific molecules. We present a universal method to test for adaptation capacity, and for detecting the adaptation-conferring integrals, in any CRN. Using this new approach, we demonstrate the existence of embedded integrals in a variety of biologically important CRNs that have eluded previous methods, and for which adaptation has been observed experimentally. This definitive picture of biological adaptation at the level of intermolecular interactions represents a blueprint for adaptation-capable signalling networks across all domains of life, and for the design of synthetic biosystems.\u003c/p\u003e","manuscriptTitle":"Universal structures for embedded integral control in biological adaptation","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2022-04-25 20:35:57","doi":"10.21203/rs.3.rs-1571178/v1","editorialEvents":[],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"nature-communications","isNatureJournal":true,"hasQc":false,"allowDirectSubmit":false,"externalIdentity":"NCOMMS","sideBox":"Learn more about [Nature Communications](http://www.nature.com/ncomms/)","snPcode":"","submissionUrl":"https://mts-ncomms.nature.com/","title":"Nature Communications","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"ejp","reportingPortfolio":"Nature Communications","inReviewEnabled":true,"inReviewRevisionsEnabled":false}}],"origin":"","ownerIdentity":"03749c4a-afcd-40e9-92d5-1dd3be3026be","owner":[],"postedDate":"April 25th, 2022","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"published-in-journal","subjectAreas":[],"tags":[],"updatedAt":"2023-05-02T13:01:10+00:00","versionOfRecord":{"articleIdentity":"rs-1571178","link":"https://doi.org/10.1038/s41467-023-38011-9","journal":{"identity":"nature-communications","isVorOnly":false,"title":"Nature Communications"},"publishedOn":"2023-04-20 04:00:00","publishedOnDateReadable":"April 20th, 2023"},"versionCreatedAt":"2022-04-25 20:35:57","video":"","vorDoi":"10.1038/s41467-023-38011-9","vorDoiUrl":"https://doi.org/10.1038/s41467-023-38011-9","workflowStages":[]},"version":"v1","identity":"rs-1571178","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-1571178","identity":"rs-1571178","version":["v1"]},"buildId":"cBFmMYwuxLRRLfASyISRj","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}