Homeostasis in Networks with Multiple Input Nodes and Robustness in Bacterial Chemotaxis
HHomeostasis in Networks with Multiple Input
Nodes and Robustness in Bacterial Chemotaxis
João Luiz de Oliveira MadeiraUniversidade de São PauloInst. Matemática e EstatísticaSão Paulo, 05508-090, [email protected] Fernando AntoneliUniversidade Federal de São PauloEscola Paulista de MedicinaSão Paulo, 04039-032, [email protected]
Abstract
A biological system is considered homeostatic when there is a regulatedvariable that is maintained within a narrow range of values. A closely relatednotion of adaptation can be defined as a process where a system initially re-sponds to a stimulus, but then returns to basal or near-basal levels of activityafter some period of time. Adaptation of the chemotaxis sensory pathway ofthe bacterium
Escherichia coli is essential for detecting chemicals over a widerange of background concentrations, ultimately allowing cells to swim towardssources of attractant and away from repellents. Since bacterial chemotaxis is aparadigm for adaptation or homeostasis, we use it as a ‘model system’ to basethe development of a general theory for the analysis of homeostasis in coupleddynamical systems associated with ‘input-output networks’ with multiple in-put nodes. This theory allows one to define and classify ‘homeostasis types’ ofa given network in a ‘model independent’ fashion, in the sense that it dependsonly on the network topology and not on the specific model equations. Each‘homeostasis type’ represents a possible mechanism for generating homeostasisand is associated with a suitable ‘subnetwork motif’. Within this framework itis also possible to formalize a notion of ‘generic property’ of a coupled dynami-cal system, generalizing the notion of ‘robustness’. Finally, we apply our theoryto a model of
Escherichia coli chemotaxis and show that, from this abstractpoint of view, adaptation or homeostasis in this model is caused by a mecha-nism that we called input counterweight . This new homeostasis mechanism isgenerated by a balancing between the several input nodes of the network andrequires the existence of at least two input nodes. a r X i v : . [ q - b i o . M N ] N ov ontents Escherichia coli
Chemotaxis 184 Structure of Infinitesimal Homeostasis 23
A Irreducibility of Homogeneous Polynomials 51Keywords:
Homeostasis, Coupled Systems, Combinatorial Matrix Theory, Input-Output Networks, Biochemical Networks, Perfect Adaptation
A system exhibits homeostasis if on change of an input variable I some observ-able x o ( I ) remains approximately constant. Many researchers have emphasized thathomeostasis is an important phenomenon in biology. For example, the extensive workof Nijhout, Reed, Best and collaborators [7,23–26] consider biochemical networks as-sociated with metabolic signaling pathways. Further examples include regulation ofcell number and size [19], control of sleep [32], and expression level regulation inhousekeeping genes [3]. Adaptation is a closely related notion. It is the ability of a system to reset anobservable x o ( I ) to its pre-stimulated output level (its set point ) after respondingto an external stimulus I . Adaptation has been widely used in synthetic biologyand control engineering (cf. [2, 4, 5, 20, 29]). Here, the focus of the research is on the2tronger condition of perfect adaptation , where the observable x o ( I ) is required tobe constant over a range of external stimuli I .One of the most studied biological models that exhibits perfect adaptation (orperfect homeostasis) is the Escherichia coli chemotaxis [1, 6]. Generally speaking, bacterial chemotaxis refers to the ability of bacteria to sense changes in their ex-tracellular environment and to bias their motility towards favorable stimuli (attrac-tants) and away from unfavorable stimuli (repellents). Understanding the responseof E. coli cells to external attractants has been the subject of experimental work andmathematical models for nearly 40 years.Many models [1, 6] have been formulated and developed to provide a compre-hensive description of the cellular processes and include details of receptor methyla-tion, ligand-receptor binding and its subsequent effect on the biochemical signalingcascade, along with a description of motor driving CheY/CheY-P levels (see [30]).However, including such detail has often led to very complex mathematical modelsconsisting of tens of governing differential equations, making mathematical analysisof the underlying cellular response difficult, if not in many cases, impossible.Extensive mathematical modeling has described different aspects of the chemo-taxis pathway and has mainly focused on explaining the initial response to additionof attractant, as well as robustness of perfect adaptation. For instance, Barkai andLeibler [6] showed that perfect adaptation is robust (insensitive to parameter vari-ations in the pathway), if the kinetics of receptor methylation depends only on theactivity of receptors and not explicitly on the receptor methylation level or externalchemical concentration. Their idea was later extended by others, providing condi-tions for perfect adaptation [21, 33], as well as robustness to noise by the networkarchitecture [17, 18].A model proposed by Clausznitzer et al. [9] has sought to provide a comprehensivedescription of the E. coli response, by coupling a simplified statistical mechanical de-scription of receptor methylation and ligand binding, with the signalling cascade dy-namics. The model consists of five nonlinear ordinary differential equations (ODEs)and is parameterized using data from the literature. The authors were able to showthat the model is in good agreement with experimental findings. However, it is afifth-order nonlinear ODE model, still difficult to treat analytically. More recently,Edgington and Tindall [11] undertook a comprehensive mathematical analysis of anumber of simplified forms of the model of [9] and proposed a fourth-order reductionof this model that has been used previously in the theoretical literature [10].We consider here a fourth-order model of [10, 11]. The model for the regulationof concentration of CheA/CheA-P ( a p ), CheY/CheY-P ( y p ), CheB/CheB-P ( b p ) andthe kinetics of receptor methylation ( m ) as the extracellular ligand concentration ( L )3s varied, is given by the following system of ODEs (in non-dimensional form) dmdt = γ R (1 − φ ( m, L )) − γ B φ ( m, L ) b p da p dt = φ ( m, L ) k (1 − a p ) − k (1 − y p ) a p − k (1 − b p ) a p dy p dt = α k (1 − y p ) a p − k y p db p dt = α k (1 − b p ) a p − k b p (1.1)where γ B , γ R , k , . . . , k are non-dimensional parameters and φ ( m, L ) = 11 + e F ( m,L ) and F ( m, L ) = N (cid:34) − m (cid:32) LK off a LK on a (cid:33)(cid:35) (1.2)The key observation of [11] is that system (1.1) has a unique asymptoticallystable equilibrium X ∗ = ( m ∗ , a ∗ p , y ∗ p , b ∗ p ) , with a ∗ p , y ∗ p and b ∗ p positive, for the non-dimensional parameters obtained from the parameter values originally used in [9].Furthermore, [11] were able to show that some pairs of parameters might yield oscil-latory behavior, but in regions of parameter space outside that observed experimen-tally. This was done by carrying out the stability analysis for pair-wise parametervariations, whereby for each case the occurrence of at least two non-zero imaginaryparts was recorded as indicating possible oscillatory dynamics. The steady-state X ∗ can be easily found by numerical integration for parameter values that are experi-mentally valid, although some combinations of parameters produce a large stiffnesscoefficient [9].More importantly, the stability of X ∗ persists as L is varied in the range (0 , + ∞ ) .By standard arguments (see subsection 2.1) this implies that there is a well-definedsmooth mapping L (cid:55)→ X ∗ ( L ) = (cid:0) m ∗ ( L ) , a ∗ p ( L ) , y ∗ p ( L ) , b ∗ p ( L ) (cid:1) . Since the values of a ∗ p , y ∗ p and b ∗ p are independent of L (see [11]), it follows that the individual componentfunctions a ∗ p ( L ) , y ∗ p ( L ) and b ∗ p ( L ) are actually constant functions with respect to L .In Figure 1 we show the time series of the three variables a p , y p and b p andhow they are perturbed when the parameter L is varied by sudden jumps. Aftera transient, which depends on the contraction rates at the equilibrium point, eachvariable returns to its corresponding steady-state value. Moreover, when the param-eters γ B , γ R , k , . . . , k are changed the equilibrium X ∗ also changes, but the newfunctions a ∗ p ( L ) , y ∗ p ( L ) and b ∗ p ( L ) remain constant with respect to L , possibly withdifferent values. This is the robust perfect adaptation that was shown to occur inother models [1, 6, 17, 18, 21, 33]. 4igure 1: Time series of the model (1.1), showing perfect adaptation of the threevariables a p (green), b p (cyan), y p (blue) at the non-dimensional equilibrium given by a ∗ p = 5 . × − , b ∗ p = 0 . , y ∗ p = 0 . . Input parameter L is given by a step function(red curve). Other parameters were set to non-dimensional values of [11, Table 2].Time series were computed using the software XPPAut [12].Even though this result is completely expected, it raises an interesting question: how come that several distinct mathematical models, with varying degrees of detail anddistinct parameters are capable of exhibiting the same behavior, as far as homeostasis(or adaptation) is concerned?
From the biological point of view it is a trivial question, since all these modelsare supposed to describe the same phenomenon, and hence, should yield the samepredictions. Nevertheless, it is not a trivial mathematical issue. One could argue thatthese models are related among themselves by certain mathematical procedures, forinstance, ‘reduction of variables’, ‘averaging’, etc., but it is not easy to show a priori ,that any of these procedures preserve the property in question. Of course, after theprocedure is applied, one can always check a posteriori if the property in questionwas lost or not.Our goal in this paper is to understand how homeostasis (or adaptation) occursin (1.1) – and, more generally, in a large class of systems containing it – by apply-ing a newly developed theory for studying homeostasis in coupled systems. Morespecifically, Wang et al. [31], motivated by Reed et al. [27], introduced the notionof ‘abstract input-output network’ and developed a theory to analyze homeostasisin networks with a ‘single input node’ (see Section 2) based on the notion of ‘in-finitesimal homeostasis’ proposed by Golubitsky and Stewart [14,15]. Unfortunately,results of [31] cannot be directly applied to (1.1), as this system has two input nodes.5ne of the motivations of this paper is to extend the theory of [31] to input-output networks with multiple input nodes. The main outcome of this effort is anextension of the ‘classification of homeostasis types’ of [31]. In the original schemeof [31], ‘homeostasis types’ correspond to ‘mechanisms’ that are responsible for theoccurrence of homeostasis in a network and are associated to certain ‘subnetworkmotifs’. Moreover, these ‘homeostasis types’ are divided in two ‘homeostasis classes’,called appendage and structural . In our extended version of the theory, the samestructure of the classification is maintained, with the addition of a new ‘homeostasisclass’, that does not exist in the single node case, called input counterweight . Thisnew homeostasis mechanism is generated by a balancing between the several inputnodes of the network and requires the existence of at least two input nodes.The other motivation is to show how to recast a specific model, such as (1.1),on the abstract framework developed here and to study the ‘robustness’ of certainproperties. Undoubtedly, an important concern when analyzing a theoretical modelgiven by a dynamical system is whether the properties of interest are preservedunder perturbations (of a certain type). For instance, in systems biology and controlengineering phenomena are usually modeled by a parametric family of ODEs and arelevant concept ‘robustness’ means that a property is robust if it is preserved bychanges in the parameters of the model. This is exactly the notion of robustnessthat is employed in the literature on perfect adaptation of chemotaxis [1, 6, 9, 11,21, 33]. In singularity theory one is concerned with parametric families of maps ordynamical systems, as well. However, the role of ‘robustness’ is played by genericity – structure preservation by all small perturbations (with respect to an appropriatefunction space). In this case the small perturbations are as general as possible, andnot restricted to variation of parameters within a fixed family. Of course, in orderconsider robustness in this wider sense one is required to go beyond the analysis ofa fixed parametric family and work with whole classes of systems.Returning to the analysis of model (1.1) using the theory developed here, weobtained the following results (see Section 3):(1) Using the classification of homeostasis types we provide a new perspective atperfect adaptation exhibited by (1.1) and show that the mechanism responsiblefor it is the novelty of our theory, the input counterweight homeostasis.(2) There is an infinite dimensional ‘space of admissible perturbations’ of (1.1) that exhibit perfect adaptation. By ‘admissible perturbation’ we mean a vector fieldthat is compatible with the same network structure of (1.1). This goes far beyondthe concept of ‘robustness’ in the strict sense of preservation by changes in theparameters. 63) There is an infinite dimensional ‘space of admissible perturbations’ of (1.1) thatdestroy perfect adaptation. Again, by ‘admissible perturbation’ we mean a vec-tor field that is compatible with the same network structure of (1.1). This isillustrated in Figure 4.From the singularity theory point of view, item (3) above is expected. It is well-known that an exactly flat (constant) function has ‘infinite codimension’. However,in biology, homeostasis is an emergent property of the system, not a preset targetvalue, thus it is expected that the input-output function is only approximately flat.Many of the more recent models of homeostasis do not assume a preset target value;instead, this emerges from the dynamics of the model. This means that we expecttypical singularities to have finite codimension, and the approach based on singularitytheory is then potentially useful. In fact, it is shown in [14, 15] that ‘infinitesimalhomeostasis’ is a finite codimension property.Notwithstanding that perfect adaptation is not a generic property, the weakernotions related to it, namely ‘infinitesimal homeostasis’ and ‘near perfect adaptation’,appear to be persistent in the model (1.1), even for the perturbations that destroyperfect adaptation, as mentioned in item (3) above (see Figure 4). Based on theseobservations we finish this introduction with a general conjecture about homeostasisin general multiple input nodes input-output networks
Conjecture.
Fix a multiple input nodes input-output network G and an appropri-ate functional space of vector fields X G compatible with the network G .(1) Near perfect adaptation occurs generically in X G , in the sense that it holds onan open subset X G .(2) Infinitesimal homeostasis occurs generically in X G , in the sense that it holds onan open subset of finite-parameter families in X G . Structure of the paper.
The remaining of this paper is divided in three parts.In Section 2 we present the general theory of homeostasis for networks with multipleinput nodes and state its main results. In Section 3 we apply the general theory tothe model equations (1.1) after appropriately recasting it on our context. In Section4 we provide detailed proofs of the theorems that support the main results of thegeneral theory. 7
Homeostasis in Coupled Systems
In this section we define the basic objects of the theory: multiple input nodes input-output networks, network admissible systems of differential equations, input-outputfunctions and infinitesimal homeostasis points. Then we introduce the generalizedhomeostasis matrix and show how to use it to find infinitesimal homeostasis pointsand to classify homeostasis types. Finally, we relate the classification of homeostasistypes with the topology of the network by associating ‘subnetwork motifs’ to theirreducible factors of the determinant of the generalized homeostasis matrix andpresent an algorithm to find all these factors in terms of the ‘subnetwork motifs’constructed before.
Golubitsky and Stewart proposed a mathematical method for the study of home-ostasis based on dynamical systems theory [14, 15] (see the review [16]). In thisframework, one consider a system of differential equations ˙ X = F ( X, I ) (2.3)where X = ( x , · · · , x k ) ∈ R k and parameter I ∈ R represents the external input tothe system.Suppose that ( X ∗ , I ∗ ) is a linearly stable equilibrium of (2.3). By the implicitfunction theorem, there is a function ˜ X ( I ) defined in a neighborhood of I ∗ such that ˜ X ( I ∗ ) = X ∗ and F ( ˜ X ( I ) , I ) ≡ . The simplest case is when there is a variable, let’ssay x k , whose output is of interest when I varies. Define the associated input-outputfunction as z ( I ) = ˜ x k ( I ) .The input-output function allows one to formulate two of the most used defini-tions that capture the notion of homeostasis [2, 20, 29]. Definition 2.1.
Let z ( I ) be the input-output function associated to a system ofdifferential equations (2.3). We say that the corresponding system (2.3) displays(a) Perfect Homeostasis (Adaptation) on the interval ] I , I [ if dzd I ( I ) = 0 for all I ∈ ] I , I [ (2.4)That is, z is constant on ] I , I [ . 8b) Near-perfect Homeostasis (Adaptation) relative to the point I on the interval ] I , I [ if for a fixed δ | z ( I ) − z ( I ) | (cid:54) δ for all I ∈ ] I , I [ (2.5)That is, z stays within z ( I ) ± δ on ] I , I [ .It is clear that perfect homeostasis implies near-perfect homeostasis, but theconverse does not hold. Inspired by Reed et al. [7,22], Golubitsky and Stewart [14,15]introduced another definition of homeostasis that is essentially intermediate betweenperfect and near-perfect homeostasis. Moreover, this new definition allows the toolsfrom singularity theory to bear on the study of homeostasis. Definition 2.2.
Let z ( I ) be the input-output function associated to a system ofdifferential equations (2.3) . We say that the corresponding system (2.3) displays Infinitesimal Homeostasis at the point I on the interval ] I , I [ if dzd I ( I ) = 0 (2.6)It is obvious that perfect homeostasis implies infinitesimal homeostasis. Moreover,it follows from Taylor’s theorem that infinitesimal homeostasis implies near-perfecthomeostasis in a neighborhood of I . It is easy to see that the converse to bothimplications is not generally valid (see [27]).When combined with coupled systems theory [13] the formalism of [14–16] be-comes very effective in the analysis of model equations. A multiple input-node input-output network is a network G with n distinguished input nodes ι = { ι , ι , . . . , ι n } , all of them associated to the same input parameter I , one distinguished output node o , and N regulatory nodes ρ = { ρ , . . . , ρ N } . Theassociated network systems of differential equations have the form ˙ x ι = f ι ( x ι , x ρ , x o , I )˙ x ρ = f ρ ( x ι , x ρ , x o )˙ x o = f o ( x ι , x ρ , x o ) (2.7)where I ∈ R is an external input parameter and X = ( x ι , x ρ , x o ) ∈ R n × R N × R isthe vector of state variables associated to the network nodes.9e write a vector field associated with the system (2.7) as F ( X, I ) = ( f ι ( X, I ) , f ρ ( X ) , f o ( X )) and call it an admissible vector filed for the network G .Let f j,x (cid:96) denote the partial derivative of the j th node function f j with respect tothe (cid:96) th node variable x (cid:96) . We make the following assumptions about the vector field F throughout:(a) The vector field F is smooth and has an asymptotically stable equilibrium at ( X ∗ , I ∗ ) . Therefore, by the implicit function theorem, there is a function ˜ X ( I ) defined in a neighborhood of I ∗ such that ˜ X ( I ∗ ) = X ∗ and F ( ˜ X ( I ) , I ) ≡ .(b) The partial derivative f j,x (cid:96) can be non-zero only if the network G has an arrow (cid:96) → j , otherwise f j,x (cid:96) ≡ .(c) Only the input node coordinate functions f ι m depend on the external inputparameter I and the partial derivative of f ι m , I generically satisfies f ι m , I (cid:54) = 0 . (2.8) Definition 2.3.
Let G be an input-output network with n input nodes and F be afamily admissible vector field with equilibrium point ˜ X ( I ) = (cid:0) x ι ( I ) , x ρ ( I ) , x o ( I ) (cid:1) .The mapping I (cid:55)→ x o ( I ) is called the input-output function of the network G , relativeto the family of equilibria ˜ X ( I ) . As noted previously [14, 16, 27, 31], a straightforward application of Cramer’s rulegives a formula for determining infinitesimal homeostasis points. This has a straight-forward generalization to multiple input networks.Let J be the ( n + N + 1) × ( n + N + 1) Jacobian matrix of an admissible vectorfield F = ( f ι , f σ , f o ) , that is, J = f ι,x ι f ι,x ρ f ι,x o f ρ,x ι f ρ,x ρ f ρ,x o f o,x ι f o,x ρ f o,x o (2.9)The ( n + N + 1) × ( n + N + 1) matrix (cid:104) H (cid:105) obtained from J by replacing the lastcolumn by ( − f ι, I , , t , is called generalized homeostasis matrix : (cid:104) H (cid:105) = f ι,x ι f ι,x ρ − f ι, I f ρ,x ι f ρ,x ρ f o,x ι f o,x ρ (2.10)10ere all partial derivatives f (cid:96),x j are evaluated at (cid:0) ˜ X ( I ) , I (cid:1) . Lemma 2.1.
The input-output function x o ( I ) satisfies x (cid:48) o ( I ) = det (cid:0) (cid:104) H (cid:105) (cid:1) det( J ) (2.11) Here det( J ) and det (cid:0) (cid:104) H (cid:105) (cid:1) are evaluated at (cid:0) ˜ X ( I ) , I (cid:1) . Hence, I is a point of in-finitesimal homeostasis if and only if det (cid:0) (cid:104) H (cid:105) (cid:1) = 0 (2.12) at the equilibrium (cid:0) ˜ X ( I ) , I (cid:1) .Proof. Implicit differentiation of the equation f ( ˜ X ( I ) , I ) = 0 with respect to I yields the linear system J x (cid:48) i x (cid:48) ρ x (cid:48) o = − f ι, I (2.13)Since ˜ X ( I ) is assumed to be a linearly stable equilibrium, it follows that det( J ) (cid:54) = 0 .On applying Cramer’s rule to (2.13) we can solve for x (cid:48) o ( I ) obtaining (2.11).By expanding det( (cid:104) H (cid:105) ) with respect to the last column and each ι k (input) rowone obtains det (cid:0) (cid:104) H (cid:105) (cid:1) = n (cid:88) m =1 ± f ι m , I det( H ι m ) (2.14)Note that when there is a single input node, i.e. n = 1 , Lemma 2.1 gives thecorresponding result obtained in [31]. In this case, there is only one matrix H ι m = H ,called the homeostasis matrix , that played a fundamental role in the theory developedin [31]. Hence, it is expected that the matrices H ι m should play a similar role in thegeneralization of [31] to the multiple input node case. Definition 2.4.
Let G be an input-output network with n input nodes and f bean admissible vector field, with a family of equilibrium points ˜ X ( I ) . The partialhomeostasis matrix H ι m of f is obtained from the Jacobian matrix J of F by droppingthe last column and the ι m row (see formula (4.51)).11 .4 Classes and Types of Homeostasis The classification of homeostasis types proceed as in [31]. The first step is to applyFrobenius-König theory [8, 28] to the generalized homeostasis matrix (cid:104) H (cid:105) . Moreprecisely, Frobenius-König theory implies that there exists (constant) permutationmatrices P and Q such that P (cid:104) H (cid:105) Q = B ∗ · · · ∗ ∗ B · · · ∗ ∗ ... ... . . . ... ... · · · B s ∗ · · · C (2.15)where each diagonal block B , . . . , B s and C is fully indecomposable (in the senseof [8]), that is, det( B ) , . . . , det( B s ) and det( C ) are irreducible polynomials. As P and Q are constant permutation matrices, we have that det (cid:0) (cid:104) H (cid:105) (cid:1) = ± det( B ) · · · det( B s ) · det( C ) (2.16)In order to simplify nomenclature, we will call B , . . . , B s and C irreducible home-ostasis blocks , although in the literature the term irreducible matrix may have adifferent meaning (see [28]).A direct comparison of factorization (2.16) with expansion (2.14) suggests thatthe irreducible factors det( B j ) are the common factors of det( H ι m ) and det( C ) isa weighted alternating sum of f ι , I , . . . , f ι n , I . Indeed, as we show in Section 4, thematrix C in (2.15) contains all the functions f ι (cid:96) , I as entries, that is, it is a homoge-neous polynomial of degree on f ι , I , . . . , f ι n , I , whereas the matrices B , . . . , B s donot contain any of them.The next step is to classify the irreducible homeostasis blocks of (cid:104) H (cid:105) accordingto their number number of self-couplings. Indeed, we show that each block B j oforder k j has exactly k j or k j − self-couplings (see Section 4). But, unlike [31], inthe multiple input nodes case we find three classes of irreducible homeostasis blocksthat may occur in core networks with multiple input nodes. Definition 2.5.
Let B j be an irreducible homeostasis block of order k j which doesnot contain any partial derivatives f ι m , I with respect to I . We say that the home-ostasis class of B j is appendage if B j has k j self-couplings and structural if B j has k j − self-couplings. Definition 2.6.
We say G exhibits appendage homeostasis if there is an appendageirreducible homeostasis block B j such that det( B j ) = 0 . In an analogous way, we12ay G exhibits structural homeostasis if there is an structural irreducible homeostasisblock B j such that det( B j ) = 0 .As shown in Wang et al. [31], appendage and structural homeostasis occur in corenetworks with one input node. Nevertheless, networks with multiple input nodes alsoexhibit a new class of homeostasis that is not found in networks with only one inputnode. Definition 2.7.
Let C be an irreducible homeostasis block whose determinant det( C ) is an homogeneous polynomial of degree on the variables f ι , I , . . . , f ι n , I .We say that the homeostasis class of C is input counterweight . Moreover, we saythat G exhibits input counterweight homeostasis when det( C ) = 0 .The final step in our theory is to associate a ‘subnetwork motif’ of G to eachhomeostasis block of (cid:104) H (cid:105) in such a way that each class of homeostasis corresponds to adistinguished class of subnetworks. Because of the appearance of a third homeostasisclass, the extension of the results of [31] to the multiple input nodes require severalnew ideas. We start with the basic combinatorial definitions needed to understand the construc-tion of ‘subnetwork motifs’ associated with the homeostasis blocks in networks withmultiple input nodes.
Definition 2.8.
Let G be a network with input nodes ι , . . . , ι n and output node o .We call G a core network if every node in G is upstream from o and downstream fromat least one input node. Analogously, we define the core subnetwork G m between ι m and o as the subnetwork composed by nodes downstream ι m and upstream o .Recall that every node is downstream and upstream from itself, which means thatthe input nodes and the output node must be present in a core network.We classify the nodes in a core network G according to their role in the topologyof the network. Definition 2.9.
Let G be a core network with input nodes ι , ι , . . . , ι n and outputnode o .(a) A directed path connecting nodes ρ and τ is called a simple path if it visits eachnode on the path at most once. 13b) An ι m o -simple path is a simple path connecting the input node ι m to the outputnode o .(c) A node is ι m -simple if it lies on an ι m o -simple path.(d) A node is ι m -appendage if it is downstream from ι m and it is not an ι m -simplenode.(e) A node is absolutely simple if it is an ι m -simple node, for every m = 1 , . . . , n .(f) A node is absolutely appendage if it is an ι m -appendage node, for every m =1 , . . . , n .Wang et al. [31] introduced the concept of path equivalent classes in appendagesubnetworks of networks with only one input node. As we need this definition inother contexts, we generalize it to every subnetwork of G . Definition 2.10.
Let K be a nonempty subnetwork of G . We say that nodes ρ i , ρ j of K are path equivalent in K if there are paths in K from ρ i to ρ j and from ρ j to ρ i .A K -path component is a path equivalence class in K . Definition 2.11.
Let G be a core subnetwork with multiple input nodes ι , . . . , ι n and output node o and let G m be the core subnetwork between ι m and o .(a) The G m -complementary subnetwork of an ι m o -simple path S is the subnetwork C m S consisting of all nodes of G m not on S and all arrows in G m connecting thosenodes.(b) The G -complementary subnetwork of an ι m o -simple path S is the subnetwork CS consisting of all nodes of G not on S and all arrows in G connecting those nodes.We start with the ‘subnetwork motifs’ associated with appendage homeostasis. Definition 2.12.
Let G be a core subnetwork with multiple input nodes ι , . . . , ι n and output node o and let G m be the core subnetwork between ι m and o .(a) For every m = 1 , . . . , n , we define the ι m -appendage subnetwork A G m as the sub-network of G composed by all ι m -appendage nodes and all arrows in G connecting ι m -appendage nodes.(b) The appendage subnetwork A G is the subnetwork of G composed by all absolutelyappendage nodes and all arrows in G connecting absolutely appendage nodes, i.e., A G = A G ∩ · · · ∩ A G n
14y Definition 2.10, each path component of a network is a path equivalence classof this network. Therefore, we can partition A G in different A G -path components.We still need another concept to associate a component of this partition with theappendage homeostasis blocks. Definition 2.13.
Let A i be an A G -path component. We say that A i satisfies the generalized no cycle condition if the following holds: for every m = 1 , . . . , n , forevery ι m o -simple path S m , nodes in A i are not CS m -path equivalent to any node in CS m \ A i .The condition in Definition 2.13 is the correct generalization of the ‘no cycle con-dition’ of Wang et al. [31]. Finally, it is shown in Section 4, that each appendagehomeostasis block corresponds exactly to an A G -path component A i satisfying thegeneralized no cycle condition is an irreducible appendage homeostasis block (seeTheorems 4.11 and 4.13). Moreover, this is equivalent to the assertion that eachappendage homeostasis block is an appendage homeostasis block of each core sub-network G m (see Theorems 4.10 and 4.12).The topological characterization of appendage homeostasis in networks with mul-tiple input nodes is similar to the topological characterization of appendage home-ostasis in single input node networks. This is not the case for the other homeostasisclasses. Indeed, in single input node networks there are only two classes of home-ostasis, appendage and structural, while in networks with multiple input nodes thereis also the input counterweight homeostasis. Moreover, single input node networksalways support structural homeostasis, which is not always the case with networkswith multiple input nodes (see Section 4).Now we consider the ‘subnetwork motifs’ associated with structural homeostasis. Definition 2.14.
Let G be a core subnetwork with multiple input nodes ι , . . . , ι n and output node o .(a) An ι m - super-simple node is an ι m -simple node that lies on every ι m o -simple path.(b) An absolutely super-simple node is an absolutely simple node that lies on every ι m o -simple path, for every m = 1 , . . . , n . In particular, an absolutely supersimple-node is an ι m -super-simple node, for every m = 1 , . . . , n .It is straightforward that every core network G with multiple input nodes ι , . . . , ι n and output node o has at least one absolutely super-simple node: the output node o . However, in contrast to core networks with only one input node where both theinput the output nodes are super-simple, core networks with multiple input nodes15ay exhibit the output node as their only super-simple node (see Figure 2). In fact,as we will see later, this is the reason why abstract core networks with multiple inputnodes do not necessarily support structural homeostasis. 𝜊𝜄 𝜎𝜄 % Figure 2: A core network with input nodes ι and ι and output node o . The onlyabsolutely super-simple node is o . Node σ is an absolutely simple node, but it isnot between two absolutely super-simple nodes, as it would be expected for corenetworks with only one input node (see Lemma 4.17).Similarly to what happens in networks with single input node there is a naturalway of ordering the absolutely super-simple. Indeed, the absolutely super-simplenodes can be uniquely ordered by ρ > ρ > · · · > ρ p > o , where a > b when b isdownstream a by all ι m o -simple paths (see Lemma 4.15). Through this ordering, wesay that two absolutely super-simple nodes ρ k , ρ k +1 are adjacent when ρ k +1 is thefirst absolutely super-simple node which appears after ρ k . Definition 2.15.
Let ρ k > ρ k +1 be adjacent absolutely super-simple nodes.(a) An absolutely simple node ρ is between ρ k and ρ k +1 if there exists an ι m o -simplepath that includes ρ k to ρ to ρ k +1 in that order, for some m ∈ { , . . . , n } (b) The absolutely super-simple subnetwork , denoted L ( ρ k , ρ k +1 ) , is the subnetworkwhose nodes are absolutely simple nodes between ρ k and ρ k +1 and whose arrowsare arrows of G connecting nodes in L ( ρ k , ρ k +1 ) .In addition to the A G -path components that satisfy the generalized no cyclecondition there may be A G -path components B i that do not satisfy this property.More precisely, for every m = 1 , . . . , n , there is an ι m o -simple path S m such thatnodes in B i are CS m -path equivalent to an absolutely simple node in CS m \ B i whichbelongs to an absolutely super-simple subnetwork L ( ρ k , ρ k +1 ) , where ρ k , ρ k +1 are16djacent absolutely super-simple nodes. There is an unique correspondence betweeneach A G -path component B i and the absolutely super-simple subnetwork to which B i is CS m -path equivalent, for some ι m o -simple path S m (see Lemma 4.20). Theunion of the A G -path components B i and the corresponding absolutely super-simplesubnetworks generate the primary subnetworks associated to structural homeostasis. Definition 2.16.
Let ρ k and ρ k +1 be adjacent absolutely super-simple nodes in G .The absolutely super-simple structural subnetwork L (cid:48) ( ρ k , ρ k +1 ) is the input-outputsubnetwork consisting of nodes in L ( ρ k , ρ k +1 ) ∪ B , where B consists of all absolutelyappendage nodes that are CS m -path equivalent to nodes in L ( ρ k , ρ k +1 ) for some ι m o -simple path S m , for some m ∈ { , . . . , n } , i.e., B consists of all A G -path components B i that are CS m -path equivalent to nodes in L ( ρ k , ρ k +1 ) for some S m , for some m ∈{ , . . . , n } . Arrows of L (cid:48) ( ρ k , ρ k +1 ) are arrows of G that connect nodes in L (cid:48) ( ρ k , ρ k +1 ) .Note that ρ k is the input node and that ρ k +1 is the output node of L (cid:48) ( ρ k , ρ k +1 ) .Each absolutely super-simple structural subnetwork L (cid:48) ( ρ k , ρ k +1 ) is a single nodeinput-output network with ρ k as the input node and ρ k +1 as the output node. There-fore, the homeostasis matrix H ( L (cid:48) ( ρ k , ρ k +1 )) is well defined. Indeed, we show thatthe homeostasis matrix of each absolutely super-simple structural subnetwork cor-responds to an irreducible structural homeostasis block and, conversely, each irre-ducible structural homeostasis block is given by the homeostasis matrix of an abso-lutely super-simple structural subnetwork (see Theorems 4.22 and 4.23).Finally, we define the ‘subnetwork motif’ associated with input counterweighthomeostasis. Definition 2.17.
Let the absolutely super-simple nodes of G be ρ > · · · > ρ s > o .The input counterweight subnetwork W G of G is the subnetwork composed by: (1) theinput nodes ι , . . . , ι n , (2) the absolutely super-simple node ρ , (3) nodes τ for whichthere exists an m ∈ { , . . . , n } such that there is an ι m o -simple path that passesat ι m , τ and ρ in that order, (4) the nodes that are not absolutely appendage norabsolutely simple, and (5) nodes in C , where C consists of all absolutely appendagenodes that are CS m -path equivalent to nodes that are not absolutely appendage andthat are not between two absolutely super-simple nodes, for some ι m o -simple path S m ( m ∈ { , . . . , n } ). Arrows of W G are the arrows of G that connect nodes of W G . The classification of homeostasis types obtained in this paper allows us to we writedown an algorithm for enumerating subnetworks corresponding to the r = p + q + 1 homeostasis blocks. 17 tep 1: Determine the A G -path components A , . . . , A p that satisfy the generalizedno cycle condition. By Theorems 4.11 and 4.13, these are the appendage homeosta-sis subnetworks of G , and their corresponding Jacobian matrix J A i is an irreducibleappendage homeostasis block that appears in the normal form of (cid:104) H (cid:105) . Moreover,there are p independent defining conditions for appendage homeostasis based on thedeterminants det( J A i ) = 0 , for i = 1 , . . . , p . Step 2:
Determine the absolutely super-simple nodes of G . If the only absolutelysuper-simple node of G is o , then G does not support structural homeostasis. On theother hand, if there is more than one absolutely super-simple node, consider theirnatural order ρ > · · · > ρ q > ρ q +1 = o and determine the corresponding abso-lutely super-simple structural subnetwork L (cid:48) ( ρ k , ρ k +1 ) . By Theorems 4.22 and 4.23,the corresponding homeostasis matrix H ( L (cid:48) ( ρ k , ρ k +1 )) is an irreducible structuralhomeostasis block that appears in the normal form of (cid:104) H (cid:105) . Moreover, there are q in-dependent defining conditions for structural homeostasis based on the determinants det (cid:0) H ( L (cid:48) ( ρ k , ρ k +1 )) (cid:1) = 0 , for k = 1 , . . . , q . Step 3:
Determine the input counterweight subnetwork W G of G . Then, the gener-alized homeostasis matrix of (cid:104) H (cid:105) ( W G ) is, up to permutation of rows or columns, theinput counterweight homeostasis block C that appears in the normal form of (cid:104) H (cid:105) .Furthermore, there is one defining condition for input counterweight homeostasisbased on the determinant det (cid:0) (cid:104) H (cid:105) ( W G ) (cid:1) = 0 . Escherichia coli
Chemotaxis
In order to apply the theory developed in this paper to the model for the
Escherichiacoli chemotaxis of [11], we first must rewrite the system (1.1) in the standard form(2.7). We make the following correspondence between variables: m ↔ x ι , a p ↔ x ι , b p ↔ x σ , y p ↔ x o , L ↔ I . This gives the following system of ODEs ˙ x ι = γ R (1 − φ ( x ι , I )) − γ B x σ φ ( x ι , I )˙ x ι = φ ( x ι , I ) k (1 − x ι ) − k (1 − x o ) x ι − k (1 − x σ ) x ι ˙ x σ = α k (1 − x σ ) x ι − k x σ ˙ x o = α k (1 − x o ) x ι − k x o (3.17)with the function φ given by (1.2) and the input parameter I . Note that the ODEsystem (3.17) is an admissible system for the abstract input-output network shown18n Figure 3(a). In fact, the general form of an admissible system for this network is ˙ x ι = f ι ( x ι , x σ , I )˙ x ι = f ι ( x ι , x ι , x σ , x o , I )˙ x σ = f σ ( x ι , x σ )˙ x o = f o ( x ι , x o ) (3.18) 𝜊𝜄 𝜎𝜄 % 𝜊𝜄 𝜎𝜄 % 𝜊𝜄 𝜎𝜄 % 𝜊𝜄 𝜎𝜄 % (a) (b)(c) (d) Figure 3: Abstract network and the ‘subnetwork motifs’ associated to the types ofhomeostasis that this network supports. (a) Abstract network G corresponding tothe admissible ODE system in (3.17). It is a core network with input nodes ι and ι and output node o . (b) The only absolutely appendage node in G is σ (in red), andfor every ι m o -simple path S m , σ is not CS m -path equivalent to any other node, whichmeans that σ is an A G -path component. (c) The absolutely super-simple nodes of G are ι > o . The corresponding absolutely super-simple structural subnetwork is L (cid:48) ( ι , o ) (in green). (d) The greatest absolutely super-simple node of G (accordingto the natural ordering) is ι . The input counterweight subnetwork W G is composedby nodes ι and ι (in blue).From the theory developed in this paper, it is clear that the network G is a corenetwork with input nodes ι and ι and output node o . This core network is the union19f networks G (the core subnetwork between ι and o ) and G (the core subnetworkbetween ι and o ).The Jacobian matrix of network G is: J = f ι ,x ι f ι ,x σ f ι ,x ι f ι ,x ι f ι ,x σ f ι ,x o f σ,x ι f σ,x σ f o,x ι f o,x o (3.19)The generalized homeostasis matrix of network G is: (cid:104) H (cid:105) = f ι ,x ι f ι ,x σ − f ι , I f ι ,x ι f ι ,x ι f ι ,x σ − f ι , I f σ,x ι f σ,x σ f o,x ι (3.20)Even though one can easily find the factorization of matrix (cid:104) H (cid:105) above, by directcalculation, it is very instructive to apply the algorithm described in Subsection2.6 to factorize det (cid:0) (cid:104) H (cid:105) (cid:1) to reveal the structure of the underlying network motifsassociated with homeostasis types supported by network G We start by observing that σ is the only absolutely appendage node of G . More-over, for every ι m o -simple path S m , σ is not CS m -path equivalent to any other node.Therefore, G supports appendage homeostasis at A = { σ } , and the correspondingirreducible factor is det( J A ) = f σ,x σ (see Figure 3(b)).The absolutely super-simple nodes of G are ι > o , that define the absolutelysuper-simple structural subnetwork L (cid:48) ( ι , o ) = { ι ←→ o } . As this subnetwork is com-posed by only nodes, its homeostasis matrix is a degree homeostasis block andthe corresponding irreducible factor is det (cid:0) H ( L (cid:48) ( ι , o )) (cid:1) = f o,x ι (see Figure 3(c)).We have already determined the appendage and structural blocks of (cid:104) H (cid:105) . Nowwe describe the input counterweight block. In fact, as the absolutely super-simplenodes of G are ι > o then W G = { ι → ι } (see Figure 3(d)). Furthermore, byLemma 4.25, W G is a core network with input nodes ι and ι , and output node ι and thus (cid:104) H (cid:105) ( W G ) = (cid:18) f ι ,x ι − f ι , I f ι ,x ι − f ι , I (cid:19) (3.21)Hence, the complete factorization of det (cid:0) (cid:104) H (cid:105) (cid:1) is det (cid:0) (cid:104) H (cid:105) (cid:1) = f σ,x σ f o,x ι ( − f ι , I f ι ,x ι + f ι , I f ι ,x ι ) (3.22)Summarizing, network G (Figure 3) generically supports three types of homeostasis:(1) appendage (null-degradation) homeostasis associated with { σ } , (2) structural20Haldane) homeostasis associated with { ι ←→ o } and (3) input counterweight home-ostasis associated with { ι → ι } Now, specializing to the model equations (3.17), we observe that, although theabstract network supports appendage and structural homeostasis, the model equa-tions (3.17) cannot exhibit these types of homeostasis. In fact, at equilibrium wehave that f σ,x σ = − α k x ι − k < (3.23) f o,x ι = α k (1 − x o ) (cid:54) = 0 (3.24)The first inequality follows from the fact that all parameters are positive and x ι ispositive at equilibrium. The last inequality follows from that fact that if α k (1 − x o ) = 0 ⇒ x o = 1 , at equilibrium, then one would have ˙ x o = − k (cid:54) = 0 .This leaves the only remaining possibility: input counterweight homeostasis. Toverify that the model equations (3.17) indeed exhibit input counterweight homeosta-sis we compute, assuming that both φ x ι and φ I (see Remark 3.1) are non-zero: f ι , I = − φ I ( x ι , I ) ( γ R + γ B x σ ) f ι ,x ι = φ x ι ( x ι , I ) k (1 − x ι ) f ι , I = φ I ( x ι , I ) k (1 − x ι ) f ι ,x ι = − φ x ι ( x ι , I ) ( γ R + γ B x σ ) (3.25)Thus, for any C function φ , we have − f ι , I f ι ,x ι + f ι , I f ι ,x ι ≡ (3.26)In particular, x (cid:48) o ( I ) = 0 for all I and hence, the model equations (3.17) exhibits,not only infinitesimal homeostasis (of the input counterweight type), but perfecthomeostasis.A consequence of the above calculation is that the property of perfect homeostasisin the model (3.17) for E. coli chemotaxis is robust , in the sense that it does notdepend on the values of the parameters of the model. Even more, it holds for amuch larger set of perturbations than just parameter change. Consider the space ofall sufficiently regular vector fields X G given by the right-handed side of (3.18), withan appropriate topology (for example, the C vector fields with the C topology).Now consider the subspace Y ⊂ X G of vector fields whose first two components arethe same as in (3.17), with an arbitrary function φ (of class C ), and arbitrary lasttwo components. The it is clear that Y is a closed infinite dimensional subspace of X G that contain the model equations (3.17) and satisfy (3.26). In other words, anyperturbation of (3.17) contained in the infinite dimensional space Y displays perfecthomeostasis. 21 emark 3.1. To show that in the model equation (3.17), both φ x ι and φ I arenon-zero, recall the correspondence between the variables m ↔ x ι and L ↔ I anddifferentiate the equations in (1.2), obtaining φ x ι ( x ι , I ) = N e F ( x ι , I ) e F ( x ι , I ) ) φ I ( x ι , I ) = N e F ( x ι , I ) (cid:16) K on a − K off a (cid:17) (1 + e F ( x ι , I ) ) (cid:16) I K on a (cid:17) (cid:16) I K off a (cid:17) (3.27)As K on a (cid:54) = K off a , we conclude by (3.27) that in the studied model φ x ι (cid:54) = 0 and φ I (cid:54) = 0 . Remark 3.2.
As can be seen in Figure 1 it looks like the time series of all threevariables a p , b p and y p exhibit homeostatic behavior. We can use our results to showthat this is indeed the case. By interchanging the roles of the nodes ι and σ withthe output node o , we can compute new generalized homeostasis matrices H ( ι ↔ o ) and H ( σ ↔ o ) and show that the corresponding input-output functions have the sameirreducible factor associated with input counterweight homeostasis (see (3.26)) which,in turn, causes all of them to vanish simultaneously. The matrices H ( ι ↔ o ) and H ( σ ↔ o ) can be obtained from the Jacobian matrix (3.19) by replacing the second and thirdcolumns, respectively, by ( − f ι, I , , t . The corresponding determinants are det( H ( ι ↔ o ) ) = f σ,x σ f o,x o ( − f ι , I f ι ,x ι + f ι , I f ι ,x ι )det( H ( σ ↔ o ) ) = f o,x o f o,x ι ( − f ι , I f ι ,x ι + f ι , I f ι ,x ι ) We have shown that the persistence of perfect homeostasis in the model (3.17)for
E. coli chemotaxis goes far beyond parameter change. It holds on an infinitedimensional ’space of perturbations’. In what follows we show that there is aninfinite dimensional ’space of perturbations’ that destroy perfect homeostasis.Consider the following -parameter family of perturbations of (3.17), for (cid:15) (cid:62) , ˙ x ι = γ R (1 − φ ( x ι , I )) − γ B x σ φ ( x ι , I ) − (cid:15) ψ ( x ι )˙ x ι = φ ( x ι , I ) k (1 − x ι ) − k (1 − x o ) x ι − k (1 − x σ ) x ι ˙ x σ = α k (1 − x σ ) x ι − k x σ ˙ x o = α k (1 − x o ) x ι − k x o (3.28)where ψ is a C function. 22t is clear that (3.28) is admissible for the network G for all (cid:15) (cid:62) and coincideswith (3.17) when (cid:15) = 0 . The expressions for f ι , I , f ι , I and f ι ,x ι are independent of (cid:15) and thus are the same as in the original system (3.25). The expression for f ι ,x ι is f ι ,x ι = − φ x ι ( x ι , I ) ( γ R + γ B x σ ) − (cid:15) ψ x ι ( x ι ) (3.29)Since the equations of f σ,x σ and of f o,x ι are independent of (cid:15) , the same argument asbefore shows that (3.28) do not exhibit appendage or structural homeostasis, for all (cid:15) (cid:62) . As for the input counterweight homeostasis factor, we get − f ι , I f ι ,x ι + f ι , I f ι ,x ι = − (cid:15) k φ I ( x ι , I ) (1 − x ι ) ψ x ι ( x ι ) (3.30)Recall that φ I ( x ι , I ) (cid:54) = 0 . Generically, (1 − x ι ) (cid:54) = 0 , as well. Otherwise, x ι = 1 atequilibrium, and so x o and x σ must satisfy the over-determined linear system − k (1 − x o ) − k (1 − x σ )0 = α k (1 − x σ ) − k x σ α k (1 − x o ) − k x o (3.31)Therefore, if the function ψ z ( z ) does not vanish on an interval (say, ψ ( z ) = z ),the model (3.28) do not generically, exhibit perfect homeostasis for all (cid:15) > . It isclear that this construction can be carried out for an arbitrary number of independentparameters and functions ( (cid:15) n , ψ n ) n , showing that the set of perturbations that destroyperfect homeostasis is not contained in any finitely parametric family of vector fields.This is a manifestation of the well-known phenomenon in singularity theory, that anexactly flat function has ‘infinite codimension’.However, numerical simulations suggests that, at least when (cid:15) > is small, themodel (3.28) displays near perfect homeostasis, see Figure 4. Finally, it seems tobe possible to show that if ψ z ( z ) vanishes at some point z (say, ψ ( z ) = z ) theninfinitesimal homeostasis occurs for an open set of parameters in the model (3.28). In this section we provide the proofs of all results behind the classification of home-ostasis types and the algorithm in subsection 2.6. We follow Wang et al. [31] andprovide the appropriate generalizations of each corresponding result. Nevertheless,it is important to remark that there are new difficulties that arise in the multipleinput node context that do not have a single input node counterpart. Completelynew arguments were required to overcome these difficulties.23igure 4: (Upper) Time series of the model (3.28), with ψ ( z ) = z and (cid:15) = 0 . ,showing near perfect homeostasis of the original variables a p ↔ x ι (green), b p ↔ x σ (cyan), y p ↔ x o (blue) at the corresponding non-dimensional equilibrium. Inputparameter L ↔ I is given by a step function (red curve). Other parameters wereset to non-dimensional values of [11, Table 2]. (Lower) Input-output functions of theoriginal variables a p ↔ x ι (green), b p ↔ x σ (cyan), y p ↔ x o (blue), as functions ofinput parameter L ↔ I , for the model (3.28). (Left) For (cid:15) = 0 , which reduces tothe original model (1.1), we have perfect homeostasis (constant input-output func-tions) and (Right) for (cid:15) = 0 . we have near perfect homeostasis. Time series werecomputed using the software XPPAut and input-output functions were computedby numerical continuation of an equilibrium point using
Auto from
XPPAut [12].24 .1 Core Networks
We present an analogous argument as the one by Wang et al. [31] to construct corenetworks in networks with multiple input nodes.The linearly stable equilibrium ( X , I ) of (2.7) satisfies the system of equationsthat can be explicitly written as f ι ( x ι , x ι , . . . , x ι n , x ρ , x o , I ) = 0 f ι ( x ι , x ι , . . . , x ι n , x ρ , x o , I ) = 0 ... f ι n ( x ι , x ι , . . . , x ι n , x ρ , x o , I ) = 0 f ρ ( x ι , x ι , . . . , x ι n , x ρ , x o ) = 0 f o ( x ι , x ι , . . . , x ι n , x ρ , x o ) = 0 (4.32)Following Wang et al. [31] we partition the regulatory nodes ρ in three typesdepending if they are upstream from the output node or/and downstream from atleast one input node. More precisely:(1) those nodes σ that are both upstream from o and downstream from at least oneinput node ι m ,(2) those nodes d that are not downstream from any input node ι m ,(3) those nodes u which are downstream from at least one input node ι m , but notupstream from o .We can now rewrite equations in (4.32) as f ι ( x ι , x ι , . . . , x ι n , x σ , x u , x d , x o , I ) = 0 f ι ( x ι , x ι , . . . , x ι n , x σ , x u , x d , x o , I ) = 0 ... f ι n ( x ι , x ι , . . . , x ι n , x σ , x u , x d , x o , I ) = 0 f σ ( x ι , x ι , . . . , x ι n , x σ , x u , x d , x o ) = 0 f u ( x ι , x ι , . . . , x ι n , x σ , x u , x d , x o ) = 0 f d ( x ι , x ι , . . . , x ι n , x σ , x u , x d , x o ) = 0 f o ( x ι , x ι , . . . , x ι n , x σ , x u , x d , x o ) = 0 (4.33)25s already proved by Wang et al. [31], (4.33) may be simplified to f ι ( x ι , x ι , . . . , x ι n , x σ , x d , x o , I ) = 0 f ι ( x ι , x ι , . . . , x ι n , x σ , x d , x o , I ) = 0 ... f ι n ( x ι , x ι , . . . , x ι n , x σ , x d , x o , I ) = 0 f σ ( x ι , x ι , . . . , x ι n , x σ , x d , x o ) = 0 f u ( x ι , x ι , . . . , x ι n , x σ , x u , x d , x o ) = 0 f d ( x d ) = 0 f o ( x ι , x ι , . . . , x ι n , x σ , x d , x o ) = 0 (4.34)Figure 5 shows the connections which can be found in G . 𝜊𝜄 𝜎𝑢 𝑑𝜄 ’ ⋮ Figure 5: The possible connections in G . The corresponding core network G c iscomposed by the input nodes ι , . . . , ι n , the nodes σ that are upstream from theoutput node and downstream of at least one input node, and the output node o .We call the core subnetwork G c from G the subnetwork composed by the inputnodes ι , . . . , ι n , the nodes σ that are upstream from the output node and downstreamof at least one input node, and the output node o . Note that if we fix x d at somevalue, it is trivial to obtain an admissible system to G c from (4.34).26 emma 4.1. Suppose X = ( x ∗ ι , x ∗ ι , . . . , x ∗ ι n , x ∗ σ , x ∗ u , x ∗ d , x ∗ o ) is a linear stable equilib-rium of (4.34) . Then the core admissible system (obtained by freezing x d at x ∗ d ) ˙ x ι = f ι ( x ι , x ι , . . . , x ι n , x σ , x ∗ d , x o , I )˙ x ι = f ι ( x ι , x ι , . . . , x ι n , x σ , x ∗ d , x o , I ) ... ˙ x ι n = f ι n ( x ι , x ι , . . . , x ι n , x σ , x ∗ d , x o , I )˙ x σ = f σ ( x ι , x ι , . . . , x ι n , x σ , x ∗ d , x o )˙ x o = f o ( x ι , x ι , . . . , x ι n , x σ , x ∗ d , x o ) (4.35) has a linear stable equilibrium in Y = ( x ∗ ι , x ∗ ι , . . . , x ∗ ι n , x ∗ σ , x ∗ o ) .Proof. It is trivial that Y is an equilibrium of (4.35). We start by verifying that Y is linearly stable. Indeed, the Jacobian matrix J of (4.34) evaluated at X is J = f ι ,x ι · · · f ι ,x ιn f ι ,x σ f ι ,x d f ι ,x o ... . . . ... ... ... ... ... f ι n ,x ι · · · f ι n ,x ιn f ι n ,x σ f ι n ,x d f ι n ,x o f σ,x ι · · · f σ,x ιn f σ,x σ f σ,x d f σ,x o · · · f d,x d f u,x ι · · · f u,x ιn f u,x σ f u,x d f u,x u f u,x o f o,x ι · · · f o,x ιn f o,x σ f o,x d f o,x o (4.36)and therefore the eigenvalues of J are the same eigenvalues of f d,x d , f u,x u and of thematrix J c , where J c = f ι ,x ι · · · f ι ,x ιn f ι ,x σ f ι ,x o ... . . . ... ... ... f ι n ,x ι · · · f ι n ,x ιn f ι n ,x σ f ι n ,x o f σ,x ι · · · f σ,x ιn f σ,x σ f σ,x o f o,x ι · · · f o,x ιn f o,x σ f o,x o (4.37)Notice now that J c is the Jacobian matrix of (4.35) calculated at Y , and therefore,if X is a linearly stable equilibrium, then so it is Y . Theorem 4.2.
Let x o ( I ) be the input-output function of the admissible system (2.7) and let x co ( I ) be the input-output function of the associated core admissible system (4.35) . Then x co has a point of infinitesimal homeostasis at I if and only if x o hasa point of infinitesimal homeostasis at I . roof. By Lemma 2.1 the input-output function of the admissible system (2.7) isgiven by dx o d I = det (cid:0) (cid:104) H (cid:105) (cid:1) det( J ) (4.38)where (cid:104) H (cid:105) = f ι ,x ι · · · f ι ,x ιn f ι ,x σ f ι ,x d − f ι , I ... . . . ... ... ... ... ... f ι n ,x ι · · · f ι n ,x ιn f ι n ,x σ f ι n ,x d − f ι n , I f σ,x ι · · · f σ,x ιn f σ,x σ f σ,x d · · · f d,x d f u,x ι · · · f u,x ιn f u,x σ f u,x d f u,x u f o,x ι · · · f o,x ιn f o,x σ f o,x d (4.39)Likewise, by Lemma 2.1 the input-output function of the core admissible system(4.35) is given by dx co d I = det (cid:0) (cid:104) H c (cid:105) (cid:1) det( J c ) (4.40)where (cid:104) H c (cid:105) = f ι ,x ι · · · f ι ,x ιn f ι ,x σ − f ι , I ... . . . ... ... ... f ι n ,x ι · · · f ι n ,x ιn f ι n ,x σ − f ι n , I f σ,x ι · · · f σ,x ιn f σ,x σ f o,x ι · · · f o,x ιn f o,x σ (4.41)From Lemma 4.1, we have det( J ) = det( f d,x d ) · det( f u,x u ) · det( J c ) (4.42)From (4.39) and (4.41) we get det (cid:0) (cid:104) H (cid:105) (cid:1) = det( f d,x d ) · det( f u,x u ) · det (cid:0) (cid:104) H c (cid:105) (cid:1) (4.43)Hence dx o d I = det (cid:0) (cid:104) H (cid:105) (cid:1) det( J ) = det( f d,x d ) · det( f u,x u ) · det (cid:0) (cid:104) H c (cid:105) (cid:1) det( f d,x d ) · det( f u,x u ) · det( J c ) = det (cid:0) (cid:104) H c (cid:105) (cid:1) det( J c ) = dx co d I (4.44)and so the theorem is proved. 28heorem 4.2 allows us to analyze the core subnetwork in the search for infinitesi-mal homeostasis points. Therefore, given a network G with output node o and inputnodes ι , ι , . . . , ι n , all of them associated to the same input parameter I , we candetach from G nodes d that are not downstream from any input node and nodes u which are not upstream from o to obtain the core subnetwork G c . We can alsoanalyze G c from a different point of view.Given a network G described above, denote by V the set of nodes of G and by E the set of arrows of G . Consider the core subnetwork G c and V c and E c the setsof nodes and of arrows of G c , respectively. For every m = 1 , . . . , n , consider thesubnetwork G m generated by ι m , o and all nodes downstream from ι m and upstreamfrom o . Proposition 4.3.
Given an input-output network G with multiple input nodes ι , ι , . . . , ι n , all of them associated to the same input parameter I and output node o , then its core subnetwork G c is the union of all core networks G m from ι m to o G c = G ∪ · · · ∪ G n (4.45) i.e., considering networks as directed graphs, we define V m and E m as the sets ofnodes and arrows of G m , respectively, for every m = 1 , . . . , n , and V c and E c as thesets of nodes and arrows of G c , respectively. Then: V c = V ∪ · · · ∪ V n and E c = E ∪ · · · ∪ E n (4.46) Proof.
The proof is straightforward as every node in G c is upstream o and downstreaman input node ι m .For example, consider the abstract network G in Figure 6(a). In this example,the subnetwork G composed by nodes downstream ι and upstream o is ι → σ → σ → o (4.47)On the other hand, the subnetwork G is ι → σ → o (4.48)By Proposition 4.3, we conclude that the corresponding core network is the oneshown in Figure 6(d). 29 𝜄 𝜎 𝑢𝑑 𝜄 ’ 𝜎 ’ 𝜊𝜄 𝜎 𝑢𝑑 𝜄 ’ 𝜎 ’ 𝜊𝜄 𝜎 𝑢𝑑 𝜄 ’ 𝜎 ’ 𝜊𝜄 𝜎 𝜄 ’ 𝜎 ’ (a) (b)(c) (d) Figure 6: (a) An abstract network G with output node o and input nodes ι and ι . (b) Nodes and arrows that belong to the core subnetwork G between ι and o are highlighted in blue. (c) Nodes and arrows that belong to the core subnetwork G between ι and o are highlighted in red. (d) The core network G c is obtained bythe union between G and G . Nodes and arrows of G c that belong to both G and G are highlighted in purple. Nodes and arrows which belong to G but not to G are highlighted in blue and nodes and arrows which belong to G but not to G arehighlighted in red. Consider a core network G with input nodes ι , ι , · · · , ι n , output node o and regula-tory nodes σ which are upstream o and downstream at least one of the input nodes.The generalized homeostasis matrix of G is given by (cid:104) H (cid:105) = f ι ,x ι · · · f ι ,x ιn f ι ,x σ − f ι , I ... . . . ... ... ... f ι n ,x ι · · · f ι n ,x ιn f ι n ,x σ − f ι n , I f σ,x ι · · · f σ,x ιn f σ,x σ f o,x ι · · · f o,x ιn f o,x σ (4.49)30xpanding the determinant det (cid:0) (cid:104) H (cid:105) (cid:1) with respect to the last column gives det (cid:0) (cid:104) H (cid:105) (cid:1) = n (cid:88) m =1 ± f ι m , I det( H ι m ) (4.50)where, according to Definition 2.4, H ι m = f ι ,x ι · · · f ι ,x ιn f ι ,x σ ... . . . ... ... f ι m − ,x ι · · · f ι m − ,x ιn f ι m − ,x σ f ι m +1 ,x ι · · · f ι m +1 ,x ιn f ι m +1 ,x σ ... . . . ... ... f ι n ,x ι · · · f ι n ,x ιn f ι n ,x σ f σ,x ι · · · f σ,x ιn f σ,x σ f o,x ι · · · f o,x ιn f o,x σ (4.51) Remark 4.1.
Note that when the network G has only one input node ι = ι , (4.51)is exactly the homeostasis matrix defined in Wang et al. [31].Let G m be the subnetwork of G consisting of ι m , o and the nodes downstream ι m and upstream o , i.e., the core subnetwork between ι m and o . Denote by H cι m thehomeostasis matrix of G m , considered as an input-output network, i.e., det( H cι m ) = (cid:12)(cid:12)(cid:12)(cid:12) f ρ,x ιm f ρ,x σ f o,x ιm f o,x ρ (cid:12)(cid:12)(cid:12)(cid:12) (4.52)By the definition of G , there may be nodes in G that are not downstream ι m (butdownstream other input nodes).The vestigial subnetwork (with respect to ι m ) D m of G consists of nodes d m thatare not downstream ι m and the arrows that connect these nodes, that is, D m = G \ G m . As nodes in D m must not be downstream from ι m and, consequently notdownstream from any node ρ , we have, by (4.51), after a permutation of rows andcolumns that det( H ι m ) = ± (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f ρ,x ιm f ρ,x ρ f ρ,x D m f o,x ιm f o,x ρ f o,x D m f D m ,x D m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (4.53)The Jacobian J D m of the subnetwork D m is defined as J D m ≡ (cid:2) f D m ,x D m (cid:3) . Therefore,by (4.52) and (4.53) we have det( H ι m ) = ± f ι m , I det( J D m ) det( H cι m ) (4.54)31n order to simplify notation, for m = 1 , . . . , n such that G and G m have the samenodes, i.e., G = G m ⇒ D m = G \ G m = ∅ , define J D m ≡ [1] . With this convention, itfollows from Eqs. (4.50) and (4.54) that det (cid:0) (cid:104) H (cid:105) (cid:1) = n (cid:88) m =1 ± f ι m , I det( J D m ) det( H cι m ) (4.55) Before proceeding with the topological characterization of homeostasis types we needto show that, for every m = 1 , . . . , n , det( J D m ) does not share common factors with det (cid:0) (cid:104) H (cid:105) (cid:1) . We use Frobenius-König theory to factorize det( J D m ) and then we showthat none of these factors of det( J D m ) is a factor of det (cid:0) (cid:104) H (cid:105) (cid:1) .By Frobenius-König theory, for every m = 1 , . . . , n , there are permutation matri-ces P m and Q m such that P m J D m Q m = D m, ∗ ∗ · · · ∗ D m, ∗ · · · ∗ D m, · · · ∗ ... ... ... . . . ... · · · D m,n m (4.56)where det D m,j is an irreducible polynomial, for every ≤ j ≤ n m . We study thenumber of self-couplings in each matrix D m,j . For this purpose, consider that eachmatrix D m,j is a square matrix of order k j and suppose that J D m is a square matrixof order N , i.e., that the subnetwork D m has N nodes. It is straightforward that N = n m (cid:88) j =1 k j (4.57) Lemma 4.4.
Each matrix D m,j has exactly k j self-couplings.Proof. Suppose that D m is composed by nodes ρ , ρ , . . . , ρ N . Therefore, as everynode is self-coupled, then one of the summands of det J D m is f ρ ,x ρ · f ρ ,x ρ · · · · · f ρ N ,x ρN (4.58)As proved in Wang et al. [31, Lem 4.6], by Frobenius-König theory, (4.58) is theproduct of a summand of each of the determinants det D m,j , which means that in allof these matrices two different self-couplings cannot share the same line or column.By the pigeon hole principle, each matrix D m,j has exactly k j self-couplings.32ote that as the self-couplings must have different rows and columns from eachother, we may choose permutation matrices P m and Q m such that the product P m J D m Q m have all the self-couplings in the main diagonal, i.e, we may assume thatfor every j = 1 , . . . , n m , D m,j has the form: D m,j = f τ ,x τ f τ ,x τ · · · f τ ,x τkj f τ ,x τ f τ ,x τ · · · f τ ,x τkj ... ... . . . ... f τ kj ,x τ f τ kj ,x τ · · · f τ kj ,x τkj (4.59)where τ , τ , . . . , τ k j are nodes of D m , i.e., D m,j is the Jacobian of the subnetwork D m,j of D m composed by nodes τ , τ , . . . , τ k j . Lemma 4.5.
Let K be a proper subnetwork of D m . If nodes in K are not D m -pathequivalent to any node in D m \ K , then upon relabelling nodes, J D m is block lowertriangular.Proof. The proof is exactly the same as in Wang et al. [31, Lem 5.3].The following theorem fully characterizes the subnetworks D m,j . Theorem 4.6.
Let D m,j be the subnetwork of D m associated to the matrix D m,j .Then:(a) Nodes in D m,j are not D m -path equivalent to any node in D m \ D m,j (b) D m,j is a path component of D m Proof. ( a ) Suppose that there are nodes in D m,j which are D m -path equivalent tonodes D m \ D m,j . Let B ⊂ D m \ D m,j be the non-empty set of nodes that are D m -path equivalent to nodes in D m,j . Notice that D m -path equivalence is an equivalencerelation, and therefore, nodes in D m,j and B are not D m -path equivalent to nodes in ( D m \ D m,j ) \ B = D m \ ( D m,j ∪ B ) . We now have two possibilities. If D m \ ( D m,j ∪B ) = ∅ , then D m = D m,j ∪ B . However, det( J D m,j ) = det( D m,j ) is not a factor of det( J D m,j ∪B ) , as proved by Wang et al. [31, Thm 5.4], which is a contradiction. Onthe other hand, if D m \ ( D m,j ∪ B ) (cid:54) = ∅ , then, by Lemma 4.5, J D m is a block lowertriangular matrix of the form: J D m = U ∗ J D m,j ∪B ∗ ∗ V (4.60)33here J D m,j ∪B = (cid:32) f D m,j ,x D m,j f D m,j ,x B f B ,x D m,j f B ,x B (cid:33) (4.61)Again, det( J D m,j ) = det( D m,j ) is not a factor of det( J D m,j ∪B ) , which is a contradic-tion. Therefore, we conclude that nodes in D m,j are not D m -path equivalent to nodesin D m \ D m,j . ( b ) It is enough to show that D m,j is path connected, as, by item ( a ) of this lemma, ifa D m -path component contains nodes in D m \ D m,j , then this path component doesnot contain any node in D m,j . Suppose that D m,j is not path connected. In thatcase, we may split in two subnetworks A and B such that J D m,j = (cid:18) J A ∗ J B (cid:19) (4.62)This is however a contradiction as by hypothesis det D m,j is irreducible. Therefore D m,j is a D m -path component. Proposition 4.7.
For every j = 1 , . . . , n m , D m,j contains an ι k -simple node, forsome k = 1 , , . . . , n , k (cid:54) = m .Proof. Consider a subnetwork D m,j . Nodes in D m,j are not downstream from ι m , andtherefore, as G is a core network, for every node ρ in D m,j , there is k ∈ { , . . . , n } , k (cid:54) = m , such that ρ is downstream from ι k and ι k is not downstream from ι m . If ρ is an ι k -simple node, the corollary is proved. On the other hand, suppose that ρ is an ι k -appendage node. By definition of ι k -appendage node, there is a path ι k → σ → · · · → σ p → ρ → σ p +1 → · · · → o such that at least one node τ inthis path appears before and after ρ . Moreover, nodes in the path between τ and ρ (and vice-versa) must not be downstream from ι m , and therefore, they are pathsof D m . If ι k satisfies this condition, then ρ is D m -path equivalent to ι k , which is an ι k -simple node. If ι k does not satisfy, consider the smallest r such that σ r satisfies thecondition. Then σ r is an ι k -simple node and ρ is D m -path equivalent to σ r . In bothcases, there is an ι k -simple node which belongs to the same D m -path equivalenceclass of ρ . By theorem . , as D m,j is a path component of D m , this ι k -simple nodebelongs to D m,j . Proposition 4.8.
For every m = 1 , . . . , n such that D m is not empty, factors of det( J D m ) are not factors of det (cid:0) (cid:104) H (cid:105) (cid:1) .Proof. We prove this statement by contradiction. Suppose that there is m such that D m is not empty and there is an irreducible factor det( D m,j ) of det( J D m ) which is also34 factor of det (cid:0) (cid:104) H (cid:105) (cid:1) . By equation (4.55), det (cid:0) (cid:104) H (cid:105) (cid:1) can be seen as an homogeneouspolynomial with variables f ι p , I and respective coefficients ± det( J D p ) det( H cι p ) , forevery p = 1 , . . . , n . By Lemma A.1, det( D m,j ) must be a factor of det( J D p ) det( H cι p ) ,for every p = 1 , . . . , n . Consider now a node ρ in D m,j . As G is a core network,there is k ∈ { , . . . , n } , k (cid:54) = m , such that ρ is downstream from ι k and ι k is notdownstream from ι m . As det( D m,j ) is irreducible, we conclude that det( D m,j ) mustbe a factor of det( J D k ) or of det( H cι k ) . As ρ is in the core network between ι k and o , det( D m,j ) must be a factor of det( H cι k ) . We have already proved in Lemma 4.4 thatthe number of self-couplings in D m,j is equal to the order of D m,j . As shown in Wanget al. [31, Thm 5.2], this means that det( D m,j ) is an appendage block and thereforeall nodes in D m,j should be ι k -appendage. However, by Proposition 4.7, D m,j has an ι k -simple node, which is a contradiction. Our aim now is to associate the factorization of det (cid:0) (cid:104) H (cid:105) (cid:1) with the network topology.In order to do this, recall that, by equation (4.55), we have det (cid:0) (cid:104) H (cid:105) (cid:1) = n (cid:88) m =1 ± f ι m , I det( J D m ) det( H cι m ) and hence we can consider the expression of det (cid:0) (cid:104) H (cid:105) (cid:1) as an homogeneous polynomialof degree on variables f ι m , I and respective coefficients ± det( J D m ) det( H cι m ) for all m = 1 , . . . , n .By Lemma A.1, to factorize det (cid:0) (cid:104) H (cid:105) (cid:1) , we must search for common factors ofthe coefficients ± det( J D m ) det( H cι m ) . On the other hand, Proposition 4.8 impliesthat, for all m = 1 , . . . , n , det( J D m ) does not share common factors with any term det( J D p ) det( H cι p ) , for all p ∈ { , . . . , n } , p (cid:54) = m . Therefore, in order to look forcommon factors between the coefficients, we must search for common factors of det( H cι m ) .Bearing all the facts above in mind and applying Frobenius-König theory to det( H cι m ) , there are square matrices B , B , . . . , B s such that det( B j ) is irreduciblefor j = 1 , . . . , η and for every m = 1 , . . . , n , there exists a square matrix C ι m suchthat det( H cι m ) = ± (det( B ) · det( B ) · · · · · det( B s )) · det( C ι m ) (4.63)where det( C ι ) ,. . . , det( C ι n ) do not share common factors. In case det( H cι ) , . . . , det( H cι n ) do not share common factors, we can consider that H cι m = C ι m .35rom Eqs. (4.55) and (4.63) it follows that det (cid:0) (cid:104) H (cid:105) (cid:1) = (det( B ) · det( B ) · · · det( B s )) (cid:32) n (cid:88) m =1 ± f ι m , I det( J D m ) det( C ι m ) (cid:33) (4.64)By Corollary A.2, the expression n (cid:88) m =1 ± f ι m , I det( J D m ) det( C ι m ) ≡ det( C ) (4.65)is irreducible. Substituting (4.65) into (4.64) gives det (cid:0) (cid:104) H (cid:105) (cid:1) = det( B ) · det( B ) · · · det( B s ) · det( C ) (4.66)where each of the matrices B , . . . , B s and C is an irreducible homeostasis block. Asexplained in Subsection 2.5, C is the input counterweight homeostasis block . Observethat the difference between the matrices B , B , . . . , B s and C is that the terms f ι m , I do not appear in any of the matrices B j . Corollary 4.9.
Every core network G with multiple input nodes supports input coun-terweight homeostasis.Proof. The proof is straightforward as det (cid:0) (cid:104) H (cid:105) (cid:1) is always a multiple of an irreduciblehomogeneous polynomial of degree on variables f ι , I , . . . , f ι n , I Remark 4.2.
Although input counterweight homeostasis does not occur in networkswith only one input node, it is interesting to note that in these networks there isa corresponding matrix C . In fact, from equation (2.10) and considering networkswith only one input node ι , we have C = [ − f ι, I ] ⇒ det( C ) = − f ι, I (cid:54) = 0 (4.67)As, by hypothesis, f ι, I (cid:54) = 0 , the input counterweight homeostasis is never present innetworks with only one input node.In Wang et al. [31] the classification of the irreducible homeostasis blocks wasbased on the number of self-couplings. The same arguments can be used in our con-text. Initially, as H cι m is the homeostasis matrix of G m , we conclude that B , . . . , B s are irreducible homeostasis blocks of each of the core networks G m with one inputnode. Thus, we can conclude that, for every j = 1 , . . . , η , B j has exactly k j self-couplings or k j − self-couplings, where k j is the order of B j . In order to maintainthe same terminology employed in [31], we call B j an appendage homeostasis block when B j has exactly k j self-couplings, and a structural homeostasis block otherwise(see Definition 2.5). 36 .4.1 Appendage Homeostasis Recall that each appendage homeostasis block B j of order k j has exactly k j self-couplings. In an analogous way of equation (4.59), B j must be the Jacobian matrixof a subnetwork K j . Theorem 4.10.
Let K j be a subnetwork of G associated with an appendage home-ostasis block B j . Then the following statements are valid:(a) Each node in K j is an ι m -appendage node, for every m = 1 , . . . , n .(b) For every ι m o -simple path S , nodes in K j are not C m S -path equivalent to anynode in C m S \ K j , for all m = 1 , . . . , n .(c) K j is a path component of A G m , for all m = 1 , . . . , n .Proof. The diagonal block B j must be an appendage homeostasis block of each coresubnetwork G m . By [31, Thm 5.2 and Thm 5.4], the theorem follows.Now we can characterize K j with respect to the whole core network G . Theorem 4.11.
Let K j be a subnetwork of G associated with an appendage home-ostasis block B j . Then the following statements are valid:(a) Each node in K j is an absolutely appendage node.(b) For every ι m o -simple path S , nodes in K j are not CS -path equivalent to any nodein CS \ K j , for all m = 1 , . . . , n .(c) K j is a path component of A G .Proof. ( a ) Each node in K j is ι m -appendage, for all m = 1 , . . . , n , which means thateach node in K j is absolutely appendage (see definition 2.9). ( b ) Suppose that for some m = 1 , . . . , n , there is an ι m o -simple path such that thereis a node ρ in K j that is CS -path equivalent to a node τ in CS \ K j , which meansthat there are paths in CS such that ρ → τ and τ → ρ . By item ( a ) above, ρ is ι p -appendage for all p = 1 , . . . , n , and, in particular, ρ is ι m -appendage ⇒ every nodedownstream from ρ is downstream from ι m and belongs to G m ⇒ there are paths ρ → τ and τ → ρ in C m S , and therefore ρ is C m S -path equivalent to τ ∈ C m S \ K j ,which is a contradiction by Theorem 4.10. ( c ) By statement ( a ) above, K j ⊆ A G m , for all m = 1 , . . . , n , which means thatall nodes in K j belong to the same A G -path equivalence class. Consider the pathcomponent T of A G such that K j ⊆ T . We have T ⊆ A G ⇒ T ⊆ A G m , for all m = 1 , . . . , n . As K j is a path component of A G m , for all m = 1 , . . . , n , we concludethat K j = T . 37ang et al. [31] proved that the conditions of Theorem 4.10 are not only necessary,but also sufficient to determine subnetworks associated to appendage homeostasis innetworks with only one input node. Theorem 4.13 generalizes this result to networkswith multiple input nodes. Proposition 4.12.
Suppose K j is a subnetwork of G such that(a) K j is an A G m -path component, for all m = 1 , . . . , n .(b) For every ι m o -simple path S , nodes in K j are not C m S -path equivalent to anynode in C m S \ K j , for all m = 1 , . . . , n .then det( J K j ) is an irreducible factor of det (cid:104) H (cid:105) .Proof. By [31, Thm 7.1], statements ( a ) and ( b ) mean that det( J K j ) is an irreduciblefactor of det H cι m , for every m = 1 , . . . , n . Therefore, by (4.55), det( J K j ) is anirreducible factor of det (cid:0) (cid:104) H (cid:105) (cid:1) . Theorem 4.13.
Suppose K j is a subnetwork of G such that(a) K j is an A G -path component.(b) For every ι m o -simple path S , nodes in K j are not CS -path equivalent to any nodein CS \ K j , for all m = 1 , . . . , n .then det( J K j ) is an irreducible factor of det (cid:0) (cid:104) H (cid:105) (cid:1) .Proof. We begin proving that assertion ( b ) above implies assertion ( b ) of Proposition4.12. In fact, for every m = 1 , . . . , n and every ι m o -simple path S , C m S ⊆ CS , andtherefore 4.13 ( b ) ⇒ ( b ) . On the other hand, if K j is an A G -path component, then,as A G = A G ∩ · · · ∩ A G n , the nodes in K j belong to the same A G m -path equivalenceclass, for all m = 1 , . . . , n . Consider the A G m -path component T m such that K j ⊆ T m .If there is a m such that K j (cid:54) = T m , then for this m , for every ι m o -simple path S , as T m ⊆ C m S , nodes in K j are C m S -path equivalent to nodes in C m S \ K j , which is acontradiction. Therefore, K j is a path component of A G m , for all m = 1 , . . . , n . Asboth assumptions of Proposition 4.12 are satisfied, we conclude that det( J K j ) is anirreducible factor of det (cid:0) (cid:104) H (cid:105) (cid:1) . 38 .4.2 Structural Homeostasis In order to characterize structural homeostasis in networks with multiple input nodeswe start with the absolutely super-simple nodes. In networks with one input node,the super-simple nodes may be ordered by simple paths. We can then apply thisresult to the subnetworks G m . Lemma 4.14.
For every m = 1 , . . . , n , the ι m -super simple nodes in G m can beuniquely ordered by ι m > ρ m, > ρ m, > · · · > ρ m,p m > o , where a > b when b isdownstream from a by all ι m o -simple paths.Proof. For every m = 1 , . . . , n , G m is a core subnetwork with only one input node ι m . Hence, we can apply the result about ordering obtained by Wang et al. [31, Lem6.1] to conclude that the ι m -super-simple nodes may be uniquely ordered by all ι m o -simple paths.We now extend Lemma 4.14 to the absolutely super-simple nodes of G . Lemma 4.15.
The absolutely super-simple nodes in G can be uniquely ordered by ρ > ρ > · · · > ρ p > o , where a > b when b is downstream from a by all ι m o -simplepaths.Proof. By Lemma 4.14, for each m = 1 , . . . , n , we can order the absolutely super-simple according to the ι m o -simple paths. Suppose there are m , m ∈ { , . . . , n } , m (cid:54) = m , such that there are absolutely super-simple nodes ρ i , ρ j , ρ i (cid:54) = ρ j , suchthat ρ i > ρ j according to ι m o -simple paths and ρ j > ρ i according to ι m o -simplepaths. This means that there are an ι m o -simple path S : ι m → · · · → ρ i →· · · → ρ j → · · · → o and an ι m o -simple path S : ι m → · · · → ρ j → · · · → ρ i →· · · → o . By definition of S , there is an ι ρ i -simple path which does not pass by ρ j , and by definition of S there is an ρ i o -simple path which does not pass by ρ j .Therefore, we conclude that there must be an ι o -simple path which does not passby ρ j , contradicting the fact that ρ j is an absolutely super-simple node. Corollary 4.16. If G is a core network with input nodes ι , . . . , ι n and output node o , then at most one input node of G is an absolutely super-simple node.Proof. Suppose that G has two input nodes ι i and ι j which are absolutely super-simple nodes. Notice that if we order the super-simple nodes according to the ι i -simple paths, we have ι i > ι j , and if we order them according to the ι j -simple paths,we have ι j > ι i . The different orderings contradict Lemma 4.15.39et ρ k > ρ k +1 be adjacent ι m -super-simple nodes for some m ∈ { , . . . , n } . An ι m -simple node ρ is between ρ k and ρ k +1 if there exists an ι m o -simple path thatincludes ρ k to ρ to ρ k +1 in that order. Lemma 4.17.
Every ι m -simple node, which is not ι m -super-simple, lies uniquelybetween two adjacent ι m -super-simple nodes.Proof. Since G m can be seen as the core subnetwork between ι m and o , i.e., G m is acore subnetwork with one input node, the result follows from [31, Lem 6.2].Observe that each network A G m can be partitioned in ( p m + q m ) A G m -path com-ponents A m, , . . . , A m,p m , B m, , . . . , B m,q m , where nodes in components A m,i are not C m S m -path equivalent to any node in C m S m \ A m,i , for every ι m o -simple path S m .Whereas this is not the case for nodes in the A G m -path components B m,i , i.e., forevery B m,i , there is an ι m o -simple path S m such that nodes in B m,i are C m S m -pathequivalent to nodes in C m S m \ B m,i . Clearly, for every m = 1 , . . . , n one has that A G m = ( A m, ˙ ∪ · · · ˙ ∪A m,p m ) ˙ ∪ ( B m, ˙ ∪ · · · ˙ ∪B m,q m ) (4.68) Lemma 4.18.
Consider the A G m -path component B m,i and suppose there is an ι m o -simple path S m such that nodes in B m,i are C m S m -path equivalent to nodes in C m S m \B m,i . Then there is at least an ι m -simple node ρ in C m S m \ B m,i such that nodes in B m,i are C m S m -path equivalent to ρ . Moreover, the ι m -simple nodes in C m S m \ B m,i that are C m S m -path equivalent to B m,i , including ρ , are not ι m -super-simple and arecontained in a unique ι m -super-simple subnetwork.Proof. This is proved in [31, Lem 6.3].Let ρ k > ρ k +1 be adjacent ι m -super-simple nodes, for some m ∈ { , . . . , n } . The ι m -super-simple subnetwork , denoted L m ( ρ k , ρ k +1 ) , is the subnetwork whose nodes are ι m -simple nodes between ρ k and ρ k +1 and whose arrows are arrows of G m connectingnodes in L m ( ρ k , ρ k +1 ) .Let ρ k and ρ k +1 be adjacent ι m -super-simple nodes in G m . The ι m -super-simplestructural subnetwork L (cid:48) m ( ρ k , ρ k +1 ) is the input-output subnetwork consisting ofnodes in L m ( ρ k , ρ k +1 ) ∪B m , where B m consists of all appendage nodes that are C m S m -path equivalent to nodes in L m ( ρ k , ρ k +1 ) for some ι m o -simple path S m . Arrows of L (cid:48) m ( ρ k , ρ k +1 ) are arrows of G m that connect nodes in L (cid:48) m ( ρ k , ρ k +1 ) .As G m is a core subnetwork with only one input node, the homeostasis matrixof each ι m -super-simple structural subnetwork L (cid:48) m ( ρ k , ρ k +1 ) ( H ( L (cid:48) m ( ρ k , ρ k +1 )) ) is anirreducible structural homeostasis block of the homeostasis matrix of G m ( H cι m ) (see[31, Thm 6.11]). We have already proved in Proposition 4.8 that the irreducible40tructural homeostasis blocks of (cid:104) H (cid:105) are also structural homeostasis blocks of eachof the matrices H cι m . Therefore, we now study under which conditions there is ansuper-simple structural subnetwork shared by all the core subnetworks G m . Lemma 4.19.
Let ρ k > ρ k +1 be two adjacent absolutely super-simple nodes. Thenthe following properties are valid:(a) ρ k and ρ k +1 are adjacent ι m -super-simple nodes, for every m = 1 , . . . , n .(b) Every ι m -simple node in L m ( ρ k , ρ k +1 ) is an absolutely simple node, for every m = 1 , . . . , n .(c) L m ( ρ k , ρ k +1 ) = L ( ρ k , ρ k +1 ) , for every m = 1 , . . . , n .Proof. ( a ) Suppose there is m ∈ { , . . . , n } such that there is an ι m -super-simplenode ρ which satisfies ρ k > ρ > ρ k +1 . Then, every ι m o -simple path passes by ρ k , ρ and ρ k +1 , in that order. Suppose now there is j ∈ { , . . . , n } , j (cid:54) = m , such that thereis an ι j o -simple path which does not pass by ρ . That means that there is a path in G between ρ k and ρ k +1 which does not pass by ρ . As there are an ι m ρ k -simple path,a ρ k ρ k +1 -simple path and a ρ k +1 o -simple path such that neither of them passes by ρ ,we can obtain an ι m o -simple path which does not pass by ρ , which is a contradiction. ( b ) By item ( a ) above ρ k , ρ k +1 must be adjacent ι m -super-simple nodes, and so L m ( ρ k , ρ k +1 ) is well defined. For some m ∈ { , . . . , n } , consider an ι m -simple node ρ which is between ρ k and ρ k +1 . By definition, there is an ι m o -simple path S m whichpasses by ι m → · · · → ρ k → · · · → ρ → · · · → ρ k +1 → · · · → o in that order. Let’s take j ∈ { , . . . , n } . Consider any ι j o -simple path S j . S j and S m have some nodes in common (at least ρ k , ρ k +1 and o ), and these nodes must appearin the same order. Therefore, we can build a path S ∗ j taking the S j stretch from ι j to ρ k and the S m stretch from ρ k to o passing by ρ (and ρ k +1 ). By the argumentabove, S ∗ j is an ι j o -simple path, and therefore ρ is an ι j -simple node. As this processmay be done to any j ∈ { , . . . , n } , then ρ is an absolutely simple node. ( c ) For every m = 1 , . . . , n , all absolutely simple nodes between ρ k and ρ k +1 are ι m -simple nodes, and therefore L ( ρ k , ρ k +1 ) ⊆ L m ( ρ k , ρ k +1 ) . On the other hand, bystatement b ) of this lemma, every ι m -simple node between ρ k and ρ k +1 is an abso-lutely simple node, which means that L m ( ρ k , ρ k +1 ) ⊆ L ( ρ k , ρ k +1 ) ⇒ L m ( ρ k , ρ k +1 ) = L ( ρ k , ρ k +1 ) .To verify that the absolutely super-simple structural subnetworks are well de-fined, we must partition the appendage subnetwork A G in a similar way that we41ave done to A G m . Therefore, we partition A G in ( p + q + r ) A G -path components A , . . . , A p , B , . . . , B q , C , . . . , C r , where(1) A G -path components A i satisfy the following condition: for all m = 1 , . . . , n , forevery ι m o -simple path S m , nodes in A i are not CS m -path equivalent to any nodein CS m \ A i .(2) A G -path components B i satisfy the following condition: for all m = 1 , . . . , n ,there is an ι m o -simple path S m such that nodes in B i are CS m -path equivalentto an absolutely simple node in CS m \ B i which belongs to an absolutely super-simple subnetwork L ( ρ k , ρ k +1 ) , where ρ k , ρ k +1 are adjacent absolutely super-simple nodes.(3) A G -path components C i do not satisfy neither of the conditions (a) and (b) above.Again, we have A G = ( A ˙ ∪ · · · ˙ ∪A p ) ˙ ∪ ( B ˙ ∪ · · · ˙ ∪B q ) ˙ ∪ ( C ˙ ∪ · · · ˙ ∪C r ) (4.69)Figure 7 exemplifies the three types of A G -path components in a core network G with multiple input nodes.We now generalize Lemma 4.18 to the A G -path components B i . Lemma 4.20.
Consider the A G -path component B i and suppose there is an ι m o -simple path S m such that nodes in B i are CS m -path equivalent to an absolutelysimple node ρ in CS m \ B i which belongs to an absolutely super-simple subnetwork L ( ρ k , ρ k +1 ) . Then the following statements are valid:(a) Nodes in CS m \ B i that are CS m -path equivalent to B i which are not absolutelyappendage, including ρ , are absolutely simple nodes contained in L ( ρ k , ρ k +1 ) .Furthermore, these nodes are not absolutely super-simple.(b) For every j = 1 , . . . , n , there is an ι j o -simple path S j such that nodes in B i are CS j -path equivalent to ρ in CS j \ B i .(c) Suppose there is another ι m o -simple path ˜ S m such that nodes in B i are C ˜ S m -pathequivalent to nodes in C ˜ S m \ B i . Then, there is an absolutely simple node τ between ρ k and ρ k +1 which is C ˜ S m -path equivalent to B i .Proof. ( a ) By hypothesis, ρ is an absolutely simple node which belongs to L ( ρ k , ρ k +1 ) .Moreover, ρ cannot be an absolutely super-simple node, as S m passes by all absolutelysuper-simple nodes and ρ belongs to CS m . Consider so the ι m o -simple path S ∗ m ι m → · · · → ρ k → · · · → ρ → · · · → ρ k +1 → · · · → o 𝜎 𝜄 % 𝜎 % 𝛾 𝑜𝜎 ( 𝛼𝜄 Figure 7: A core network with input nodes ι and ι and output node o . Nodes σ and σ are absolutely simple while σ and o are absolutely super-simple. The absolutelysuper-simple subnetwork L ( σ , o ) is composed by nodes σ , σ and o . The appendagesubnetwork A G is composed by α, β and γ , and each of these nodes corresponds toa distinct A G -path component. Following the nomenclature proposed above, node α (in blue) corresponds to an A G -path component A , β (in red) corresponds to an A G -path component B and γ (in green) corresponds to an A G -path component C .According to definition 2.16, the subnetwork composed by nodes σ , σ , β and o isan absolutely super-simple structural subnetwork L (cid:48) ( σ , o ) .Take now a node τ of CS m \ B i such that nodes in B i are CS m -path equivalent to τ and suppose that τ is not absolutely appendage. As τ is downstream nodes in B i ,then we conclude that τ is downstream ι j , for every j = 1 , . . . , n . That means thatthere is j ∈ { , . . . , n } such that τ is ι j -simple. Consider the ι j o -simple path S j thatpasses by τ . As ρ k and ρ k +1 are absolutely super-simple nodes, S j must pass by ρ k and ρ k +1 . We shall see that τ is between ρ k and ρ k +1 . In fact, suppose that S j passesby these nodes in the following order ι j → · · · → τ → · · · → ρ k → · · · → ρ k +1 → · · · → o Then, as τ and ρ belong to the same CS m -path component and, as ρ k and ρ k +1 donot belong to CS m (because ρ k and ρ k +1 are absolutely super-simple nodes), then43here is a path from τ → ρ which does not pass by ρ k or ρ k +1 . Taking this pathtogether with S ∗ m and S j , we can obtain an ι j o -simple path that does not pass by ρ k ,which is a contradiction. On the other hand, if S j is of the form ι j → · · · → ρ k → · · · → ρ k +1 → · · · → τ → · · · → o then, in analogous manner, we can obtain an ι j o -simple path that does not pass by ρ k +1 , which is also a contradiction. Therefore, τ must be between ρ k and ρ k +1 , andconsequently, by lemma 4.19, τ is an absolutely simple node of L ( ρ k , ρ k +1 ) . Moreover,as τ lies in CS m , then τ must not be absolutely super-simple. ( b ) For every j = 1 , . . . , n , we can obtain an ι j o -simple path S j such that the pathof S j from ρ k to o coincides with the path of S m from ρ k to o . Consider now CS j .Clearly B i ⊂ CS j . On the other hand, the absolutely appendage nodes that are CS m -path equivalent to nodes in B i also belong to CS j . Consider now the node τ which is absolutely simple and CS m -path equivalent to B i . We need to verify that τ also belongs to CS j . In fact, by statement a ) , τ is an absolutely simple node between ρ k and ρ k +1 , and by the initial hypothesis, S m and S j share the same path between ρ k and ρ k +1 , meaning that τ does not lie in S j , and therefore τ belongs to CS j .By item ( a ) above, nodes that belong to the same CS m -path component as B i areabsolutely appendage or absolutely simple, and therefore nodes that are CS m -pathequivalent to B i are also CS j -path equivalent to B i , including, in particular, ρ . ( c ) Suppose there is an ι m o -simple path ˜ S m such that nodes in B i are C ˜ S m -pathequivalent to nodes in C ˜ S m \ B i . As B i is an A G -path component, there is at leastone node τ ∈ C ˜ S m \ B i such that τ is not absolutely appendage and τ is C ˜ S m -pathequivalent to nodes in B i . In fact, if that was not the case, we would find an A G -pathcomponent that contains B i and which is different from B i , which is a contradiction.Therefore, there is j = 1 , . . . , n such that τ is ι j -simple. By item ( b ) above, there isan ι j o -simple path S j such that ρ (cid:54)∈ S j and nodes in B i are CS j -path equivalent to ρ . Moreover, as τ is ι j -simple, there is an ι j o -simple path S ∗ j that passes by τ . As ρ k and ρ k +1 are super-simple nodes, they are present in ˜ S m , S j and S ∗ j . Suppose that S ∗ j follows the order ι j → · · · → τ → · · · → ρ k → · · · → ρ k +1 → · · · → o In this case, we have the following:(1) By definition of S ∗ j , there an ι j τ -simple path that does not pass by ρ k ,(2) As τ is C ˜ S m -path equivalent to nodes in B i and ˜ S m contains ρ k and ρ k +1 , forevery node from B i , there is a path from τ to this node which does not pass by ρ k or ρ k +1 , 443) By a similar argument, for every node in B i there is a path from this node to ρ which does not pass by ρ k or ρ k +1 ,(4) As ρ is between ρ k and ρ k +1 , there is a ρo -simple path which does not pass by ρ k ,Now it follows from (1)-(4) above that we can obtain an ι j o -simple path that doesnot pass by ρ k , which is a contradiction. On the other hand, if S ∗ j follows the order ι j → · · · → ρ k → · · · → ρ k +1 → · · · → τ → · · · → o, then, in an analogous manner, we can obtain an ι j o -simple path that does not passby ρ k +1 , which is also a contradiction. Therefore, τ must belong to L j ( ρ k , ρ k +1 ) ,which means that, by Lemma 4.19, τ is an absolutely simple node that is between ρ k and ρ k +1 .Lemma 4.20 implies that the correspondence between the A G -path components B i and the absolutely super-simple subnetworks is unique, and therefore the absolutelysuper-simple structural subnetworks are well defined (see Definition 2.16). Lemma 4.21.
Let ρ k > ρ k +1 be two adjacent absolutely super-simple nodes. Then L (cid:48) m ( ρ k , ρ k +1 ) = L (cid:48) ( ρ k , ρ k +1 ) , for every m = 1 , . . . , n .Proof. By Lemma 4.19, we already know that L m ( ρ k , ρ k +1 ) = L ( ρ k , ρ k +1 ) , for every m = 1 , . . . , n . By definition, we also know that L (cid:48) ( ρ k , ρ k +1 ) = L ( ρ k , ρ k +1 ) ∪ B , where B consists of all absolutely appendage nodes that are CS m -path equivalent to nodesin L ( ρ k , ρ k +1 ) for some ι m o -simple path S m , for some m ∈ { , . . . , n } . Consideran A G -path component B i ⊂ L (cid:48) ( ρ k , ρ k +1 ) . By item ( b ) of Lemma 4.20, for every m = 1 , . . . , n , there is an ι m o -simple path S m such that nodes in B i are CS m -pathequivalent to nodes in L ( ρ k , ρ k +1 ) . Moreover, as nodes in B i are downstream from ι m , then all nodes that are CS m -path equivalent to nodes in B i are downstream from ι m , and therefore nodes in B i are C m S m -path equivalent to nodes in L m ( ρ k , ρ k +1 ) ,for every m = 1 , . . . , n . As this property is valid for every A G -path component B i ⊂L (cid:48) ( ρ k , ρ k +1 ) , we conclude that L (cid:48) ( ρ k , ρ k +1 ) ⊆ L (cid:48) m ( ρ k , ρ k +1 ) , for every m = 1 , . . . , n .On the other hand, consider an A G m -path component B m,i ⊂ L (cid:48) m ( ρ k , ρ k +1 ) . As thereis a path between nodes in L m ( ρ k , ρ k +1 ) = L ( ρ k , ρ k +1 ) and nodes in B m,i , then nodesin B m,i are downstream from ι j , j = 1 , . . . , n . Suppose a node τ of B m,i is not anabsolutely appendage node, i.e., that there is j ∈ { , . . . , n } such that τ is ι j -simple.Consider the ι j o -simple path S j that passes by τ . If τ is between ρ k and ρ k +1 , then,45y Lemma 4.19, τ is an absolutely simple node, which is contradiction considering τ is ι m -appendage. Suppose then that S j follows the order ι j → · · · → τ → · · · → ρ k → · · · → ρ k +1 → · · · → o In that case, as there is a path between τ ∈ B m,i and nodes in L m ( ρ k , ρ k +1 ) = L ( ρ k , ρ k +1 ) which does not pass by ρ k and there is a path between ι j and τ whichdoes not pass by ρ k , then, we can obtain an ι j o -simple path that does not pass by ρ k , contradicting the fact that ρ k is absolutely super-simple. By a similar argument,if S j follows the order ι j → · · · → ρ k → · · · → ρ k +1 → · · · → τ → · · · → o then we can obtain an ι j o -simple path that does not pass by ρ k +1 , which is also a con-tradiction. Therefore, we conclude that every node in B m,i is absolutely appendage,and consequently B m,i is an A G -path component. We know that there is an ι m o -simple path S m such that B m,i is C m S m -path equivalent to nodes in L m ( ρ k , ρ k +1 ) = L ( ρ k , ρ k +1 ) , and, consequently, as C m S m ⊂ CS m , nodes in B m,i are CS m -path equiva-lent to nodes in L m ( ρ k , ρ k +1 ) = L ( ρ k , ρ k +1 ) , i.e., B m,i ⊂ L (cid:48) ( ρ k , ρ k +1 ) . As this is validfor every A G m -path component of L (cid:48) m ( ρ k , ρ k +1 ) , we conclude that, for every m =1 , . . . , n , one has L (cid:48) m ( ρ k , ρ k +1 ) ⊆ L (cid:48) ( ρ k , ρ k +1 ) ⇒ L (cid:48) m ( ρ k , ρ k +1 ) = L (cid:48) ( ρ k , ρ k +1 ) .Wang et al. [31] proved that in networks with only one input node, every irre-ducible structural homeostasis block corresponds to the homeostasis determinant ofa super-simple structural subnetwork L (cid:48) ( τ, σ ) , where τ is the input node and σ isthe output node of this subnetwork. On the other hand, the homeostasis determi-nant of each super-simple structural subnetwork L (cid:48) ( τ, σ ) uniquely corresponds to anirreducible structural homeostasis block. Theorem 4.22.
Consider the core network G with multiple input nodes. If thereis an irreducible structural homeostasis block B s such that det( B s ) is an irreduciblefactor of det (cid:0) (cid:104) H (cid:105) (cid:1) , then G has adjacent absolutely super-simple nodes ρ k and ρ k +1 such that det( B s ) = det (cid:0) H ( L (cid:48) ( ρ k , ρ k +1 )) (cid:1) Proof.
By the results above, we know that det( B s ) is an irreducible factor of det (cid:104) H (cid:105) iff det( B s ) is an irreducible factor of each homeostasis determinant det( H cι m ) . Thatmeans, by the results of [31, Thm 6.11], that for every m = 1 , . . . , n , there are ι m -super-simple nodes ρ k m , ρ k m +1 such that det( B s ) = det (cid:0) H ( L (cid:48) m ( ρ k m , ρ k m +1 )) (cid:1) . Inparticular, these implies that all the subnetworks L (cid:48) ( ρ k m , ρ k m +1 ) must share the sameinput and output nodes, i.e., there are absolutely super-simple nodes ρ k , ρ k +1 suchthat det( B s ) = det (cid:0) H ( L (cid:48) m ( ρ k , ρ k +1 )) (cid:1) , for every m = 1 , . . . , n . By Lemma 4.21, det( B s ) = det (cid:0) H ( L (cid:48) ( ρ k , ρ k +1 )) (cid:1) . 46 orollary 4.23. Consider a core network G with multiple input nodes. If G haveabsolutely super-simple nodes other than the output node, then the homeostasis matrixof each absolutely super-simple structural subnetwork corresponds to an irreduciblestructural homeostasis block.Proof. Consider the adjacent absolutely super-simple nodes ρ k , ρ k +1 in G . By Lemma4.21, for every m = 1 , . . . , n , we have L (cid:48) m ( ρ k , ρ k +1 ) = L (cid:48) ( ρ k , ρ k +1 ) . As proved in [31,Thm 7.2], this means that the homeostasis matrix of L (cid:48) ( ρ k , ρ k +1 ) is an irreduciblestructural homeostasis block of each subnetwork G m , and therefore the homeostasismatrix of L (cid:48) ( ρ k , ρ k +1 ) is an irreducible structural homeostasis block of G . Recall that for each core network G , the determinant of the input counterweighthomeostasis block C is unique up to signal and its explicit formula is given by (4.65).We have already proved that det( C ) is an irreducible factor of det (cid:0) (cid:104) H (cid:105) (cid:1) . Itremains to show that C is associated with the input counterweight subnetwork W G (see Definition 2.17). For this purpose, we first verify that in certain sense G can bedivided in subnetworks associated to appendage homeostasis, absolutely super-simplestructural subnetworks (structural homeostasis) and W G . Lemma 4.24.
Consider a core network G with multiple input nodes, with absolutelysuper-simple nodes ρ > · · · > ρ s > ρ s +1 = o , and its associated input counterweightsubnetwork W G . Then, the following statements are valid(a) W G does not share nodes with any of the subnetworks A i , where A i is an A G -path component that satisfy the following condition: for all m = 1 , · · · , n , forevery ι m o -simple path S m , nodes in A i are not CS m -path equivalent to any nodein CS m \ A i .(b) For k (cid:54) = 1 , W G does not share nodes with L (cid:48) ( ρ k , ρ k +1 ) . Moreover, the onlycommon node between W G and L (cid:48) ( ρ , ρ ) is ρ .(c) If a node σ in G is such that it does not belong to any subnetwork A i definedin item ( a ) , neither to any absolutely super-simple structural subnetwork, then σ belongs to W G .Proof. ( a ) The proof is straightforward, as none of the nodes in W G are absolutelyappendage nodes that satisfy the condition of item ( a ) above.47 b ) Consider the subnetwork S composed by the union of all absolutely structuralsuper-simple subnetworks of G : S = L (cid:48) ( ρ , ρ ) ∪ L (cid:48) ( ρ , ρ ) ∪ · · · ∪ L (cid:48) ( ρ s , o ) Therefore, with the exception of ρ , by Lemmas 4.19 and 4.20, every node σ in S satisfies one of the two conditions: σ is absolutely simple and for every m = 1 , . . . , n there is a simple path S m that passes by ι m , ρ and σ in this order; or σ is anabsolutely appendage node which belongs to an A G -path component B i that CS m -path equivalent to an absolutely simple node ρ in CS m \ B i which belongs to anabsolutely super-simple subnetwork L ( ρ k , ρ k +1 ) . In both cases, by definition, σ doesnot belong to W G . Whereas ρ belongs to both W G and S . ( c ) If σ is absolutely simple, then for every ι m o -simple path S m that passes by σ , σ must be upstream from ρ (if this does not happen, then, by Definition 2.15, σ belongsto one absolutely structural super-simple subnetwork, which is a contradiction). Thismeans that there is an ι m o -simple path S m which follows the order ι m → · · · → σ →· · · → ρ , and therefore σ belongs to W G . On the other hand, if σ is absolutelyappendage, then, by the partition of A G explained above, σ must belong to an A G -path component C i for which there is CS m -path equivalent to nodes that are notabsolutely appendage and that are not between two absolutely super-simple nodes,for some ι m o -simple path S m , for some m ∈ { , . . . , n } , i.e., in this case σ alsobelongs to W G . Finnaly, if σ is not absolutely simple nor absolutely appendage, then σ belongs to W G .Lemma 4.24 implies that the network G is basically the union between W G ,the subnetworks associated to appendage homeostasis and the absolutely structuralsuper-simple subnetworks of G . Moreover, these different subnetworks do not sharecommon paths. It is interesting to note that W G contains all the vestigial subnet-works, as these subnetworks are composed by nodes which are not either absolutelysimple nor absolutely appendage. Lemma 4.25.
Consider a core network G with multiple input nodes and its associatedinput counterweight subnetwork W G . Then, the following statements are valid(a) W G is a core network with input nodes ι , . . . , ι n and output node ρ .(b) W G does not support either appendage nor structural homeostasis.(c) det (cid:0) (cid:104) H (cid:105) ( W G ) (cid:1) is a factor of det (cid:0) (cid:104) H (cid:105) (cid:1) . roof. ( a ) As W G is a subnetwork of G , every node in W G is downstream from atleast one of the input nodes. We must now verify that every node in W G is upstreamfrom ρ . In fact, this is true for the input nodes and for ρ . This is also true forevery node τ for which there is m ∈ { , . . . , n } such that there is an ι m o -simple paththat passes at ι m , τ and ρ in that order. On the other hand, take a node σ whichcannot be classified as absolutely appendage nor absolutely simple. There are twopossibilities for σ : (i) σ is not downstream every input node, or (ii) σ is downstreamfrom every input node, but there are m, j ∈ { , . . . , n } such that σ is ι m -simple and ι j -appendage. In the first case, σ must be downstream at least one input node ι m .If σ is ι m -simple, then if by the ι m o -simple path that passes by σ , σ is downstream ρ , then σ would be downstream every input node, which is a contradiction, andtherefore σ must be upstream ρ by this ι m o -simple path. If σ is ι m -appendage, then σ must be G m -path connected to some ι m -simple node which, by a similar argument,must be upstream ρ , and, again, σ is upstream ρ . Considering now the second case( σ is downstream every input node, but there are m, j ∈ { , . . . , n } such that σ is ι m -simple and ι j -appendage), then σ must be upstream ρ by the ι m o -simple paththat passes by σ , as if this does not happen, σ would be absolutely simple. Finally,take an absolutely appendage node σ in C described in Definition 2.17. Then, thereis a path between σ and a node τ in W G such that there is an ι m o -simple path S m passing by τ and τ is not between two absolutely super-simple nodes. If by S m , τ is downstream ρ , then that means that τ must be between two absolutely super-simple nodes, which is a contradiction. Therefore, S m must pass by ι m , τ and ρ in that order, and, as there is a path from σ to τ , then σ is also upstream ρ . Asall nodes in W G , we can see this subnetwork as a core network between with inputnodes ι , . . . , ι n and output node ρ . ( b ) The only absolutely appendage nodes in W G are the absolutely appendage nodesin C . By Lemma 4.20, for every ι m o -simple path S m for which an A G -path component C i ⊆ C is CS m -path equivalent to nodes in CS m \ C i , then not absolutely appendagenodes CS m -path equivalent to C i are not between two absolutely super-simple nodes,meaning that these nodes are also present in W G . Therefore, all A W G -path compo-nents do not follow the necessary conditions to present appendage homeostasis, i.e.,this kind of homeostasis is not supported by W G . On the other hand, as the onlyabsolutely super-simple node in W G is ρ , then W G does not support structural home-ostasis neither. ( c ) We verify this looking at the factorization of det (cid:0) (cid:104) H (cid:105) (cid:1) . Consider permutationmatrices P and Q such that P (cid:104) H (cid:105) Q is the Frobenius-König normal form of the ma-trix (cid:104) H (cid:105) . Recall that each row of P (cid:104) H (cid:105) Q represents the partial derivatives of afunction f j with respect to all other nodes of G , and each column of P (cid:104) H (cid:105) Q repre-49ents the partial derivatives of all the functions that describe the dynamics of nodewith respect to same node j (with the exception of the column composed by zerosand by the terms f ι m , I ). Consider the A G -path components A i which satisfy the fol-lowing condition: for all m = 1 , . . . , n , for every ι m o -simple path S m , nodes in A i arenot CS m -path equivalent to any node in CS m \ A i , and the absolutely super-simplestructural subnetworks L (cid:48) ( ρ , ρ ) , L (cid:48) ( ρ , ρ ) , . . . , L (cid:48) ( ρ p , o ) . We have already verifiedthat the Jacobian J A i of each A G -path components A i and the homeostasis matrix H ( L (cid:48) ( ρ k , ρ k +1 )) of each absolutely super-simple structural subnetwork appear as in-dependent irreducible blocks of P (cid:104) H (cid:105) Q . By equation (4.64) and by our results onthe characterization of appendage and structural blocks, we get det (cid:0) (cid:104) H (cid:105) (cid:1) = det( J A ) · · · det( J A r ) · det (cid:0) H ( L (cid:48) ( ρ , ρ )) (cid:1) · · · det (cid:0) H ( L (cid:48) ( ρ p , o )) (cid:1) · det( C ) (4.70)We need to determine which rows and columns of (cid:104) H (cid:105) appear in C . Recall that thematrices that appear in the right-handed side of equation (4.70) are the blocks thatappear in the normal form of (cid:104) H (cid:105) . Therefore, C consists of rows and columns thatare not present in such matrices. This means, in particular, that C consists of:(1) rows that contain the partial derivatives of the functions f ι m , for every m =1 , . . . , n ,(2) rows that contain the partial derivatives of f ρ ,(3) rows that contain the partial derivatives of the functions f σ , where σ representsnodes in G which do not belong to any subnetwork A i neither to any absolutelysuper-simple structural subnetwork,(4) columns composed by zeros and by the terms f ι m , I ,(5) columns containing the partial derivatives with respect to nodes ι m , for every m = 1 , . . . , n and nodes σ (described in item (3)) above,(6) columns containing the partial derivatives with respect to ρ is not a columns of C , as this column is present in H ( L (cid:48) ( ρ , ρ )) .As shown in item ( a ) above, W G is a core network with input nodes ι , · · · , ι n andoutput node ρ , then, by Lemma 4.24, we conclude that C is equivalent, up topermutation of rows and/or columns, to the matrix (cid:104) H (cid:105) ( W G ) , which means that det (cid:0) (cid:104) H (cid:105) ( W G ) (cid:1) is a factor of det (cid:0) (cid:104) H (cid:105) (cid:1) . 50ow we can finally characterize input counterweight homeostasis in terms ofnetwork topology. Theorem 4.26.
Consider a core network G with multiple input nodes and its asso-ciated input counterweight subnetwork W G . Then, (cid:104) H (cid:105) ( W G ) is, up to permutation ofrows or columns, the irreducible input counterweight homeostasis block C of (cid:104) H (cid:105) .Proof. By Lemma 4.25, det (cid:0) (cid:104) H (cid:105) ( W G ) (cid:1) is a factor of det (cid:0) (cid:104) H (cid:105) (cid:1) . Moreover, as W G does not support neither appendage or structural homeostasis, det (cid:0) (cid:104) H (cid:105) ( W G ) (cid:1) isan irreducible homogeneous polynomial of degree on variables f ι , I , . . . , f ι n , I . ByFrobenius-König theory, we conclude that (cid:104) H (cid:105) ( W G ) must be, up to permutation ofrows or columns, the irreducible input counterweight homeostasis block C of (cid:104) H (cid:105) . AppendixA Irreducibility of Homogeneous Polynomials
In this appendix we prove general results about irreducibility of certain homogeneouspolynomials that are used in Section 4.
Lemma A.1.
Let P ( y , y , . . . , y n ) be an homogeneous polynomial of degree 1, i.e., P ( y , y , . . . , y n ) ≡ a y + · · · + a n y n (A.71) then, a polynomial Q ( y , y , . . . , y n ) divides P iff either Q ( y , y , . . . , y n ) ≡ b and b divides each term a i , for all i = 1 , . . . , n , or there is a coefficient c such that Q ( y , y , . . . , y n ) ≡ b y + · · · + b n y n (A.72) and a i = c · b i , for all i = 1 , . . . , n .Proof. ( ⇐ ) It is straightforward to see that in both cases Q divides P .( ⇒ ) Suppose that a polynomial Q ( y , y , . . . , y n ) divides P . As P is an homogeneouspolynomial of degree 1, then Q must have at most degree , i.e., we can explicitlywrite Q as Q ( y , y , . . . , y n ) = b + b y + · · · + b n y n (A.73)Moreover, there is a polynomial R ( y , y , . . . , y n ) such that P ≡ QR . By the sameargument, we can explicitly write R as R ( y , y , . . . , y n ) = c + c y + · · · + c n y n (A.74)51y (A.73) and (A.74), we have P ( y , y , . . . , y n ) ≡ Q ( y , y , . . . , y n ) R ( y , y , . . . , y n ) a y + · · · + a n y n ≡ ( b + b y + · · · + b n y n ) · ( c + c y + · · · + c n y n ) a y + · · · + a n y n ≡ b c + n (cid:88) i =1 ( b i c + b c i ) y i + (cid:88) ≤ i Let P ( y , y , . . . , y n ) be an homogeneous polynomial of degree 1, i.e., P ( y , y , . . . , y n ) ≡ a y + · · · + a n y n . (A.80) such that a , · · · , a n do not share common factors. Then P is irreducible. roof. Suppose there is a polynomial Q ( y , y , . . . , y n ) which is a divisor of P . ByLemma A.1, there is a coefficient c that divides every term a i . As a , . . . , a n donot share common factors, c = ± , and hence, either Q ( y , y , . . . , y n ) ≡ ± or Q ( y , y , . . . , y n ) ≡ ± P ( y , y , . . . , y n ) . That is, P is irreducible. Acknowledgments. 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