The Structure of Infinitesimal Homeostasis in Input-Output Networks
Yangyang Wang, Zhengyuan Huang, Fernando Antoneli, Martin Golubitsky
TThe Structure of Infinitesimal Homeostasis inInput-Output Networks
Yangyang WangDepartment of MathematicsThe University of IowaIowa City, IA 52242, [email protected] Zhengyuan HuangThe Ohio State UniversityColumbus, OH 43210, [email protected] AntoneliEscola Paulista de MedicinaUniversidade Federal de S˜ao PauloS˜ao Paulo, SP 04039-032, [email protected] Martin GolubitskyDepartment of MathematicsThe Ohio State UniversityColumbus, OH 43210, [email protected] 4, 2020
Abstract
Homeostasis refers to a phenomenon whereby the output x o of a system is approximatelyconstant on variation of an input I . Homeostasis occurs frequently in biochemical networks andin other networks of interacting elements where mathematical models are based on differentialequations associated to the network. These networks can be abstracted as digraphs G with adistinguished input node ι , a different distinguished output node o , and a number of regulatorynodes ρ , . . . , ρ n . In these models the input-output map x o ( I ) is defined by a stable equilibrium X at I . Stability implies that there is a stable equilibrium X ( I ) for each I near I andinfinitesimal homeostasis occurs at I when ( dx o /d I )( I ) = 0. We show that there is an( n + 1) × ( n + 1) homeostasis matrix H ( I ) for which dx o /d I = 0 if and only if det( H ) = 0. Weuse combinatorial matrix theory to factor the polynomial det( H ) and thereby determine a menuof different types of possible homeostasis associated with each digraph G . Specifically, we provethat each factor corresponds to a subnetwork of G . The factors divide into two combinatoriallydefined classes: structural and appendage . Structural factors correspond to feedforward motifsand appendage factors correspond to feedback motifs. Finally, we discover an algorithm fordetermining the homeostatic subnetwork motif corresponding to each factor of det( H ) withoutperforming numerical simulations on model equations. The algorithm allows us to classifylow degree factors. There are two types of degree 1 homeostasis (negative feedback loops andkinetic or Haldane motifs) and there are two types of degree 2 homeostasis (feedforward loopsand a degree two appendage motif). Keywords:
Homeostasis, Coupled Systems, Combinatorial Matrix Theory, Input-Output Net-works, Biochemical Networks, Perfect Adaptation a r X i v : . [ q - b i o . M N ] A ug ontents L . . . . . . . . . . . . . . . . . . . . . . 366.4 Relating S G with L (cid:48) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366.5 Relation between structural homeostasis and L (cid:48) . . . . . . . . . . . . . . . . 37 This paper divides into three parts. Part I, which is just Section 1.1, puts our work in per-spective. Part II, which consists of Sections 1.2-1.12, gives a precise technical description ofour results. Finally, Part III consists of Sections 2-7 and contains the rigorous mathematics,along with the proofs of theorems mentioned in Part II. We note that certain graph theo-retic notions and theorems are needed in the proofs in Part III, but are not needed in thedescription of our results in Part II. 2 system exhibits homeostasis if on change of an input variable I some observable x o ( I )remains approximately constant. Many researchers have emphasized that homeostasis isan important phenomenon in biology. For example, the extensive work of Nijhout, Reed,Best and collaborators [19, 22, 5, 18, 17, 16] consider biochemical networks associated withmetabolic signaling pathways. Further examples include regulation of cell number and size[13], control of sleep [26], and expression level regulation in housekeeping genes [2]. Adaptation is a closely related notion. It is the ability of a system to reset an observable x o ( I ) to its prestimulated output level (its set point ) after responding to an external stimulus I . Adaptation has been widely used in synthetic biology and control engineering (cf. [14,1, 24, 8, 20, 4, 25, 3]). Here, the focus of the research is on the stronger condition of perfectadaptation , where the observable x o ( I ) is required to be constant over a range of externalstimuli I . The literature is huge, and these articles are a small sample.The mathematical formulation of both homeostasis and adaptation is as follows. Startwith a system of ordinary differential equations usually associated to a network of interactingelements. Next define an input-output function that maps the input variable or the externalstimulus I to the output x o ( I ). Then the occurrence of homeostasis or perfect adaptationis a question about the properties of x o ( I ) under (time-dependent) variation of I .For instance, Reed et al. [21] consider biochemical signaling networks whose nodes repre-sent the concentrations of certain biochemical substrates that interact through mass actionkinetics. They identify two homeostatic motifs in three-node networks: the feedforward loop motif (FFL) (Figure 1c) and the kinetic motif (K) (Figure 1b). In related work on three-nodebiochemical networks with Michaelis-Menten kinetics, Ma et al. [14] identify numerically twonetwork topologies that achieve perfect adaptation. To do this, the authors searched 16,038equations in various three-node network topologies over a wide range of parameter space.They found just two motifs that achieved perfect adaptation: the negative feedback loop mo-tif (NFL) (Figure 1a) and the incoherent feedforward loop (IFL) (Figure 1c). The combinedresults of [21] and [14] show that at least three network topologies (K, NFL, IFL ∼ = FFL)emerge as motifs exhibiting homeostasis or perfect adaptation in three-node biochemicalnetworks.Recently, Wang and Golubitsky [11] classified the ‘homeostasis types’ that can occur inthree-node input-output networks based on the notion of infinitesimal homeostasis [10] (seeDefinition 1.2). Using this approach, they were able to reproduce the classification resultsin [14] and [21], within a broader class of systems including, but not limited to, specificmodel systems based on mass action or Michaelis-Menten kinetics. They showed that three-node networks that can exhibit infinitesimal homeostasis are, up to core equivalence (seeDefinition 1.9), the three network topologies mentioned above.This paper generalizes the results of [11] on three-node networks to arbitrarily large input-output networks . We follow [9] and abstract the notion of biochemical network toa ‘math network’ given by a digraph G with a distinguished input node ι and a differentdistinguished output node o . The specific model equations are abstracted into admissible systems of differential equations, namely, one-parameter smooth families of vector fieldscompatible with the network topology of G , such that only the input node depends explicitly3n I . These networks and their associated systems of differential equations are called input-output networks . We show that, under generic conditions (the existence of an asymptoticallystable equilibrium X for a particular parameter value I ), one can always define the input-output function I (cid:55)→ x o ( I ) associated to a given input-output network G .A straightforward application of Cramer’s rule (Lemma 1.5) gives a useful method forcomputing infinitesimal homeostasis points: infinitesimal homeostasis occur at I , namely, dx o d I ( I ) = 0, if and only if det (cid:0) H ( I ) (cid:1) = 0 (Section 1.3). This result motivates the intro-duction of the homeostasis matrix H ( I ) (see equation (1.6)), whose entries are linearizedcoupling strengths and linearized self-coupling strengths associated with the input-outputnetwork. The homeostasis matrix H – which has appeared in the literature under differentnames and notations (cf. [14, 1, 24, 10, 4, 3]) – is the central object in our theory.Our main result states that the homeostasis types that occur in admissible systems ofdifferential equations associated with the network G are classified by the topology of certainsubnetwork motifs of G . Moreover, there is an algorithm (Section 1.8) for determining allthe homeostatic subnetwork motifs and the corresponding homeostasis conditions, whichalso can be used for designing network topologies that display infinitesimal homeostasis.In order to prove our results we introduce new concepts and techniques. The notion of core network (Section 1.4) allows one to go from a general input-output network to a ‘minimalnetwork’ that retains all essential features of homeostasis. We define core equivalence of corenetworks in such a way that the determinant of a homeostasis matrix is determined by its coreequivalence class. Combinatorial matrix theory [7] lets us put H into block upper triangularform and each diagonal block B η is irreducible (no further triangularization is possible)and corresponds to a homeostasis type (Section 1.5). The degree of the homeostasis type isdefined as the size k of the square block B η and we prove that each block B η has either k or k − appendage class and in thesecond structural class (Section 1.6). We characterize combinatorially both homeostasis typesby identifying homeostatic subnetwork motifs and associating a subnetwork motif to eachhomeostasis type (Section 1.7). We also give an algorithm that determines the homeostasisblocks and their respective homeostasis types (Section 1.8).In the biochemical network literature on homeostasis (or adaptation) it is usual to finddesignations attached to the networks, such as negative feedback loop , antithetical integralfeedback , incoherent feedforward loop , etc. [14, 24, 8]. These names refer to the presenceof a certain mechanism that is responsible for the occurrence of homeostasis in a particularnetwork. Ma et al. [14] suggest that studies of these mechanisms can yield design principlesfor constructing network topologies that exhibit homeostasis. This could be called a ‘bottom-up’ approach for constructing homeostasis. It starts by identifying small building blocksthat are associated with homeostasis and then how the blocks can be combined to build-upincreasingly more complex networks that exhibit homeostasis. Here we take a ’top-down’approach. We start with an input-output network G and have an algorithm that shows ushow homeostasis in G can be generated from homeostasis in certain subnetworks.Fundamental to our approach is the discovery that homeostasis in G can be associatedwith only two ‘classes of mechanisms’ that we called structural and appendage , each associ-4ted with certain topological properties (Section 1.7). In addition to classifying homeostasistypes in a given network, these topological constraints also provide insights into the ‘bottom-up’ construction of homeostatic systems. The structural and appendage classes are abstractgeneralizations of the usual ‘feedforward’ and ‘feedback’ mechanisms [14, 8]. More precisely,for each homeostasis type (in each class), there is a corresponding ‘network motif’ and anassociated homeostasis mechanism. For instance, negative feedback loop and antithetical in-tegral feedback are types in the appendage class, and incoherent feedforward loop is a type inthe structural class.The motivation for the term structural homeostasis comes from [21], where the authorsidentify the feedforward loop as one of the homeostastic motifs in three-node biochemicalnetworks. In general, structural homeostasis corresponds to a balancing of two or moreexcitatory/inhibitory sequence of couplings from the input node to the output node; that is,a generalized feedforward loop. There is a degenerate case where the role of the balancingis played by neutral coupling, a transition state between excitation and inhibition. Thishomeostasis type is called Haldane , because Haldane [12] seems to have been the first toobserve this homeostatic mechanism.The intuition behind the term appendage homeostasis is that homeostasis is generatedby a cycle of regulatory nodes; that is, a generalized feedback loop. This loop functionsas controller nodes on a system that does not by itself exhibit homeostasis. There is adegenerate case of appendage homeostasis that we call null degradation where the role of thecontroller is played by a neutral node that balances between degradation and production.A striking outcome of our approach is that we do not need to specify any homeostasisgenerating mechanisms at the outset. However, we find a posteriori that (given the appro-priate generalizations) there are essentially only the two well-known feedback / feedforward types of homeostasis generating mechanisms.Our work is unusual in that it combines ideas from combinatorial matrix theory and graphtheory adapted to input-output networks to determine properties of equilibria of differentialequations. Specifically, the determinant formula (Theorem 3.2) connects the nonzero sum-mands of det( H ) with simple paths from the input node to the output node of the network G . It is reminiscent of the connection between a directed graph and its adjacency matrix [6].These simple paths allow us to identify both structural and appendage homeostasis. Finally,our theoretical results also allow us to derive formulas for determining the chair singularities [15, 21]. We now define the basic objects: input-output networks , network admissible systems of dif-ferential equations , and input-output functions .An input-output network G has a distinguished input node ι , a distinguished output node o (distinct from ι ), and n regulatory nodes ρ = ( ρ , . . . , ρ n ). The network G also has a specifiedset of arrows connecting nodes (cid:96) to nodes j . The associated network systems of differential5quations 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 ) (1.1)where I ∈ R is an external input parameter , X = ( x ι , x ρ , x o ) ∈ R × R n × R is the vectorof state variables associated to the network nodes and F ( X, I ) = ( f ι , f ρ , f o ) is a smoothone-parameter family of G -admissible vector fields on the state space R × R n × R (see [9]for the definition of the space of admissible vector fields attached to a given network G ). Wewrite the network system (1.1) as ˙ X = F ( X, I ) (1.2)Let f j,x (cid:96) denote the partial derivative of the j th node function f j with respect to the (cid:96) th node variable x (cid:96) . We make the following assumptions about the vector field F throughout:(a) F has an asymptotically stable equilibrium at ( X , I ).(b) The partial derivative f j,x (cid:96) can be nonzero only if the network G has an arrow (cid:96) → j .(c) Only the input node coordinate function f ι depends on the external input parameter I and the partial derivative of f ι with respect to I at the equilibrium point ( X , I )satisfies f ι, I ( X , I ) (cid:54) = 0 (1.3)It follows from (a) and the implicit function theorem applied to F ( X, I ) = 0 (1.4)that there exists a unique smooth family of stable equilibria X ( I ) = (cid:0) x ι ( I ) , x ρ ( I ) , x o ( I ) (cid:1) (1.5)such that F ( X ( I ) , I ) ≡ X ( I ) = X . Definition 1.1.
The mapping
I (cid:55)→ x o ( I ) is called the input-output function .Local homeostasis is defined near I when the input-output function x o is approximatelyconstant near I . An important observation is that locally homeostasis occurs when thederivative of x o with respect to I is zero at I . More precisely: Definition 1.2.
Infinitesimal homeostasis occurs at I if x (cid:48) o ( I ) = 0 where (cid:48) indicates dif-ferentiation with respect to I .Terms that involve coupling in network systems are: Definition 1.3.
Let F = ( f ι , f ρ , f o ) be an admissible system for the network G .6a) The partial derivative f j,x (cid:96) ( X , I ) is the linearized coupling associated with the arrow (cid:96) → j at the equilibrium ( X , I ).(b) The partial derivative f j,x j ( X , I ) is the linearized self-coupling of node j at the equi-librium ( X , I ). Remark 1.4.
A notion similar to infinitesimal homeostasis, called perfect homeostasis or perfect adaptation , requires the stronger condition that the derivative of the input-outputfunction be identically zero on an interval. It follows from Taylor’s theorem that infinitesimalhomeostasis implies that the input-output function x o is approximately constant near I , theconverse is not valid in general [21]. This property is called near perfect homeostasis or nearperfect adaptation in the literature (cf. [8, 24]). Hence, infinitesimal homeostasis is anintermediate notion between perfect homeostasis and near perfect homeostasis . As noted previously [10, 21, 11], a straightforward application of Cramer’s rule gives aformula for determining infinitesimal homeostasis points. See Lemma 1.5.We use the following notation. Let J be the ( n + 2) × ( n + 2) Jacobian matrix of (1.2)and let H be the ( n + 1) × ( n + 1) homeostasis matrix given by dropping the first row andthe last column of J : 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 H = (cid:20) f ρ,x ι f ρ,x ρ f o,x ι f o,x ρ (cid:21) (1.6)Here all partial derivatives f (cid:96),x j are evaluated at the equilibrium X . Lemma 1.5.
Let ( X , I ) be an asymptotically stable equilibrium of (1.2) . The input-outputfunction x o ( I ) satisfies x (cid:48) o = ± f ι, I det( J ) det( H ) (1.7) Hence, I is a point of infinitesimal homeostasis if and only if det( H ) = 0 (1.8) at ( X , I ) .Proof. Implicit differentiation of (1.4) with respect to I yields the matrix system J x (cid:48) i x (cid:48) ρ x (cid:48) o = − f ι, I (1.9)7ince X is assumed to be a stable equilibrium, it follows that det( J ) (cid:54) = 0. On applyingCramer’s rule to (1.9) we can solve for x (cid:48) o obtaining x (cid:48) o ( I ) = 1det( J ) det 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 (1.10)which leads to (1.7). By assumption, f ι, I (cid:54) = 0. Hence, the fact that infinitesimal homeostasisfor (1.2) is equivalent to (1.8) follows directly from (1.7). Homeostasis in a given network G can be determined by analyzing a simpler network that isobtained by eliminating certain nodes and arrows from G . We call the network formed bythe remaining nodes and arrows the core subnetwork . Definition 1.6.
A node τ in a network G is downstream from a node ρ in G if there existsa path in G from ρ to τ . Node ρ is upstream from node τ if τ is downstream from ρ .These relationships are important when trying to classify infinitesimal homeostasis. Forexample, if the output node o is not downstream from the input node ι , then the input-output function x o ( I ) is identically constant in I . Although technically this is a form ofinfinitesimal homeostasis, it is an uninteresting form. Definition 1.7.
The input-output network G is a core network if every node in G is bothupstream from the output node o and downstream from the input node ι .Every input-output network G has a maximal core subnetwork; namely, G c is the coresubnetwork whose nodes are the nodes in G that are both upstream from the output anddownstream from the input and whose arrows are the arrows in G whose head and tail nodesare both nodes in G c .The next result concerning core networks follows from Theorem 2.4. Theorem 1.8.
Let G be an input-output network and let G c be the associated core subnetwork.The input-output function associated with G c has a point of infinitesimal homeostasis at I if and only if the input-output function associated with G has a point of infinitesimalhomeostasis at I . It follows from Theorem 1.8 that classifying infinitesimal homeostasis for networks G isequivalent to classifying infinitesimal homeostasis for the core subnetwork G c . Definition 1.9. (a) Two ( n +2)-node core networks are core equivalent if the determinantsof their homeostasis matrices are identical polynomials of degree n + 1.(b) A backward arrow is an arrow whose head is the input node ι or whose tail is the outputnode o . 8 orollary 1.10. If two core networks differ from each other by the presence or absence ofbackward arrows, then the core networks are core equivalent.Proof.
The linearized couplings associated to backward arrows are of form f ι,x k and f k,x o ,which do not appear in the homeostasis matrix (1.8).Therefore, backward arrows can be ignored when computing infinitesimal homeostasiswith the homeostasis matrix H . However, backward arrows cannot be totally ignored, sincethey are involved in the determination of both the equilibria of (1.2) and their stability.Corollary 1.10 can be generalized to a theorem giving necessary and sufficient graphtheoretic conditions for core equivalence. See Theorem 3.3. The previous results imply that the computation of infinitesimal homeostasis reduces tosolving det( H ) = 0, where H is the homeostasis matrix associated with a core network.From now on we assume that the input-output network G is a core network.It is important to observe that the nonzero entries of H are the linearized couplingstrengths f j,x (cid:96) for the network connected nodes (cid:96) → j and the linearized self-couplingstrengths f j,x j . It follows that h = det( H ) is a homogeneous polynomial of degree n + 1in the ( n + 1) entries of H . The assumption that the network is core implies that thispolynomial is nonzero. We use combinatorial matrix theory to show that in general h canfactor and that there is a different type of infinitesimal homeostasis associated with eachfactor.Frobenius-K¨onig theory [7] (see [23] for an historical account) applied to the homeostasismatrix H implies that there are two constant ( n + 1) × ( n + 1) permutation matrices P and Q such that P HQ = B ∗ · · · ∗ B · · · ∗ ... ...0 0 · · · B m (1.11)where the square matrices B , . . . , B m are unique up to permutation. More precisely, eachblock B η cannot be brought into the form (1.11) by permutation of its rows and columns.Hence det( H ) = det( B ) · · · det( B m ) or h = h · · · h m (1.12)is a unique factorization since h η = det( B η ) cannot further factor for each η ; that is, eachdet( B η ) is an irreducible homogeneous polynomial. Specifically: Theorem 1.11.
The polynomial h η = det( B η ) is irreducible (in the sense that it cannot befactored as a polynomial) if and only if the block submatrix B η is irreducible (in the sensethat B η cannot be brought to the form (1.11) by permutation of rows and columns of B η ). roof. The decomposition (1.11) corresponds to the irreducible components in the factoriza-tion (1.12) follows from [7, Theorem 4.2.6 (pp. 114–115) and Theorem 9.2.4 (p. 296)].A consequence of (1.12) and (1.8) is that for each η = 1 , . . . , m there is a defining condition for infinitesimal homeostasis given by the polynomial equation det( B η ) = 0. Recall that theinput-output function is implicitly defined in terms of the external input I and det( B η )is a homogeneous polynomial in the linearized coupling strengths f j,x (cid:96) evaluated at X ( I ).Hence, there are m different defining conditions for infinitesimal homeostasis, h η ( I ) = 0,where each one gives a nonlinear equation that can be solved for some I = I . In practice,for a given model, it is unlikely that these equations can be solved in closed form; however,it is possible that each defining condition can be solved numerically. So, the decompositionof the homeostasis matrix H into blocks B η simplifies the solution of det( H ) = 0. Definition 1.12.
Given the homeostasis matrix H of an input-output network G , we call theunique irreducible diagonal blocks B η in the decomposition (1.11) irreducible components .We say that homeostasis in G is of type B η if det( B η ) = 0 and det( B ξ ) (cid:54) = 0 for all ξ (cid:54) = η . The next results assert that the irreducible components B η of H determine two distinct homeostasis classes (appendage and structural) and that one can associate a subnetwork K η of G with each B η (see Section 4).Let B η be an irreducible component in the decomposition (1.11), where B η is a k × k diagonal block, that is, B η has degree k . Since the entries of B η are entries of H , theseentries have the form f ρ,x τ ; that is, the entries are either 0 (if τ → ρ is not an arrow in G ), self-coupling (if τ = ρ ), or coupling (if τ → ρ is an arrow in G ).Since P and Q in (1.11) are constant permutation matrices, all entries in each row(resp. column) of B η must lie in a single row (resp. column) of H . Hence, B η has the form B η = f ρ ,x τ · · · f ρ ,x τk ... . . . ... f ρ k ,x τ · · · f ρ k ,x τk (1.13)It follows that the number of self-coupling entries of B η are the same no matter whichpermutation matrices P and Q are used in (1.11) to determine B η . In Theorem 4.4 we showthat a k × k submatrix B η has either k or k − Definition 1.13.
The homeostasis class of an irreducible component B η of degree k is appendage if B η has k self-couplings and structural if B η has k − Definition 1.14.
The subnetwork K η of G associated with the homeostasis block B η isdefined as follows. The nodes in K η are the union of nodes p and q where f p,x q is a nonzeroentry in B η and the arrows of K η are the union of arrows q → p where p (cid:54) = q .10heorem 4.7 implies that when B η is appendage, the subnetwork K η has k nodes and B η can be put in a standard Jacobian form without any distinguished nodes ((4.3)). Also,when B η is structural, the subnetwork K η has k + 1 nodes and B η can be put in a standardhomeostasis form with designated input node and output node ((4.4)). Moreover, in thiscase, the subnetwork K η has no backward arrows. That is, K η has no arrows whose head isthe input node or whose tail is the output node. See Remark 4.8 for details. Core input-output networks G have combinatorial properties that we now define and exploit.The key ideas are the concepts of ιo -simple paths and super-simple nodes. Definition 1.15.
Let G be a core input-output network.(a) A directed path connecting nodes ρ and τ is called a simple path if it visits each nodeon the path at most once.(b) An ιo -simple path is a simple path connecting the input node ι to the output node o .(c) A node in G is simple if the node lies on an ιo -simple path and appendage if the nodeis not simple.(d) A super-simple node is a simple node that lies on every ιo -simple path.Nodes ι and o are super-simple since by definition these nodes are on every ιo -simplepath. Lemma 6.1 shows that super-simple nodes are well ordered (by downstream ordering)and hence adjacent super-simple pairs of nodes can be identified. Characterization of appendage homeostasis networks requires the following definitions.
Definition 1.16.
Let G be a core input-output network.(a) The appendage subnetwork A G of G is the subnetwork consisting of all appendage nodesand all arrows in G connecting appendage nodes.(b) The complementary subnetwork of an ιo -simple path S is the subnetwork C S consistingof all nodes not on S and all arrows in G connecting those nodes.(c) Nodes ρ i , ρ j in A G are path equivalent if there exists paths in A G from ρ i to ρ j andfrom ρ j to ρ i . An appendage path component is a path equivalence class in A G .(d) A cycle is a path whose first and last nodes are identical.(e) Let K ⊂ A G be a subnetwork. We say that K satisfies the no cycle condition if forevery ιo -simple path S , nodes in K do not form a cycle with nodes in C S \ K .11n Section 5 we prove that every subnetwork K η of G associated with an irreducibleappendage homeostasis block B η consists of appendage nodes (Theorem 5.2), is an appendagepath component of A G , and satisfies the no cycle condition (Theorem 5.4). The converse isproved in Theorem 7.1. Remark 1.17.
Nodes in the appendage subnetwork A G can be written uniquely as thedisjoint union A G = ( A ˙ ∪ · · · ˙ ∪A s ) ˙ ∪ ( B ˙ ∪ · · · ˙ ∪B t ) (1.14)where each A i is an appendage path component that satisfies the no cycle condition andeach B i is an appendage path component that violates the no cycle condition. Moreover,each A i (resp. B i ) can be viewed as a subnetwork of A G by including the arrows in A G thatconnect nodes in A i (resp. B i ). We call A i a no cycle appendage path component and B i a cycle appendage path component . Corollary 6.10 shows that if B η corresponds to an irreducible structural block, then K η hastwo adjacent super-simple nodes (Theorem 6.9) and these super-simple nodes are the inputnode (cid:96) and the output node j in K η . In addition, it follows from the standard homeostasisform (Theorem 4.7) that the network K η contains no backward arrows. That is, no arrowsof K η go into the input node (cid:96) nor out of the output node j .We use the properties of structural homeostasis to construct all structural homeostasissubnetworks K η up to core equivalence. First, we introduce the following terminology. Definition 1.18.
The structural subnetwork S G of G is the subnetwork whose nodes areeither simple or in a cycle appendage path component B i (see Remark 1.17) and whosearrows are arrows in G that connect nodes in S G .Lemma 5.5 implies that all structural homeostasis subnetworks are contained in S G , whichis an input-output network. That is, G and S G have the same simple, super-simple, input,and output nodes. Lemma 6.2 shows that every non-super-simple simple node lies betweentwo adjacent super-simple nodes. Using this fact, we can define a subnetwork L of S G forevery pair of adjacent super-simple nodes. Definition 1.19.
Let ρ , ρ be adjacent super-simple nodes.(a) A simple node ρ is between ρ and ρ if there exists an ιo -simple path that includes ρ to ρ to ρ in that order.(b) The super-simple subnetwork , denoted L ( ρ , ρ ), is the subnetwork whose nodes aresimple nodes between ρ and ρ and whose arrows are arrows of G connecting nodesin L ( ρ , ρ ).It follows that all L ( ρ , ρ ) are contained in S G . By Lemma 6.3 (d), each appendage nodein S G connects to exactly one L . This lets us expand a super-simple subnetwork L ⊂ S G toa super-simple structural subnetwork L (cid:48) ⊂ S G as follows.12 efinition 1.20. Let ρ and ρ be adjacent super-simple nodes in G . The super-simple struc-tural subnetwork L (cid:48) ( ρ , ρ ) is the input-output subnetwork consisting of nodes in L ( ρ , ρ ) ∪B where B consists of all appendage nodes that form cycles with nodes in L ( ρ , ρ ); that is, allcycle appendage path components that connect to L ( ρ , ρ ). Arrows of L (cid:48) ( ρ , ρ ) are arrowsof G that connect nodes in L (cid:48) ( ρ , ρ ). Note that ρ is the input node and ρ is the outputnode of L (cid:48) ( ρ , ρ ).In Section 6 we prove that every subnetwork K η of G associated with an irreduciblestructural homeostasis block B η is a super-simple structural subnetwork (Theorem 6.11).The converse is proved in Theorem 7.2. Below is an algorithm for enumerating subnetworks corresponding to the m homeostasisblocks. Step 1:
Determining the appendage homeostasis subnetworks from A G . Let A , . . . , A s (1.15)be the no cycle appendage path components of A G (see Remark 1.17). Theorem 7.1 impliesthat these appendage path components are the subnetworks K η that correspond to appendagehomeostasis blocks. In addition, there are s independent defining conditions for appendagehomeostasis given by the determinants of the Jacobian matrices det( J A i ) = 0 for i = 1 , . . . , s . Step 2:
Determining the structural homeostasis subnetworks from S G . Let ι = ρ > ρ >. . . > ρ q +1 = o be the super-simple nodes in S G in downstream order. Theorems 6.11 and7.2 imply that up to core equivalence the q super-simple structural subnetworks L (cid:48) ( ι, ρ ) , L (cid:48) ( ρ , ρ ) , . . . , L (cid:48) ( ρ q − , ρ q ) , L (cid:48) ( ρ q , o ) (1.16)are the subnetworks K η that correspond to structural homeostasis blocks. In addition, thereare q defining conditions for structural homeostasis blocks given by the determinants of thehomeostasis matrices of the input-output networks: det (cid:0) H ( L (cid:48) ( ρ i , ρ i +1 )) (cid:1) = 0 for i = 1 , . . . , q .Therefore, the m = s + q subnetworks listed in (1.15) and (1.16) enumerate the appendageand structural homeostasis subnetworks in G . Here we specialize our discussion to the low degree cases k = 1 and k = 2 where we determineall such homeostasis types (see Figure 1). The first three types appear in three node networksand are given in the classification in [11]. The fourth type has degree k = 2, but can onlyappear in networks with at least four nodes (see Figure 2a). We note that the lowest degreeof a structural homeostasis block with an appendage node (that is, L (cid:48) (cid:41) L ) is k = 3 (seeFigure 2b). 13 ι ο (a) Haldane ( ι → o ); Null-degradation ( τ ) ρι ο (b) 2 Haldane ( ι → ρ ; ρ → o ) ρι ο (c) Feedforward loop Figure 1:
Homeostasis types in three-node networks. (a) three-node core networkexhibiting Haldane and null-degradation homesotasis. (b) three-node core network exhibitingHaldane homesotasis. (c) three-node core network exhibiting degree 2 structural homeostasis.According to [11] this is a list of all three-node core networks up to core equivalence.
Degree 1 no cycle appendage homeostasis (null-degradation)
This corresponds to the vanishing of a degree 1 irreducible factor of the form ( f τ,x τ ). Thesingle node τ is a no cycle appendage path component. Apply Step 1 in the algorithm inSections 1.7-1.8 to Figure 1a. Degree 1 structural homeostasis (Haldane)
This corresponds to the vanishing of a degree 1 irreducible factor of the form ( f j,x (cid:96) ) whoseassociated subnetwork is L (cid:48) ( (cid:96), j ) of the form (cid:96) → j . Apply Step 2 in the algorithm inSections 1.7-1.8 to Figure 1b. Degree 2 structural homeostasis (feedforward loop)
This corresponds to a three-node input-output subnetwork L (cid:48) ( (cid:96), j ) with input node (cid:96) , outputnode j , and regulatory node ρ , where (cid:96) and j are adjacent super-simple and ρ is a simplenode between the two super-simple nodes. It follows that both paths (cid:96) → ρ → j and (cid:96) → j are in L (cid:48) = L . Hence, L (cid:48) is a feedforward loop motif. Homeostasis occurs whendet (cid:0) H ( L (cid:48) ( (cid:96), j )) (cid:1) = f ρ,x (cid:96) f j,x ρ − f j,x (cid:96) f ρ,x ρ = 0Apply Step 2 in the algorithm in Sections 1.7-1.8 to Figure 1c.14 egree 2 no cycle appendage homeostasis This is associated with a two-node appendage path component A = { τ , τ } with arrows τ → τ and τ → τ . Homeostasis occurs whendet (cid:0) J ( A ) (cid:1) = f τ ,x τ f τ ,x τ − f τ ,x τ f τ ,x τ = 0Apply Step 1 in the algorithm in Sections 1.7-1.8 to Figure 2a. ι ο τ τ (a) Haldane ( ι → o ); Degree 2 no cycleappendage ( τ ⇔ τ ) τι ορ (b) Structural with appendage node Figure 2:
Homeostasis types in four-node networks. (a) smallest network exhibitingdegree 2 no cycle appendage homeostasis. (b) smallest network exhibiting appendage nodein structural homeostasis.
We use an artificial example to illustrate the algorithm for enumerating homeostasis blocks.The network shown in Figure 3 has input node ( ι ), output node ( o ), six regulatory nodes( ρ , . . . , ρ ), and four appendage nodes ( τ , τ , τ , τ ). The input-output network G in Figure 3has four ιo -simple paths (see Table 1) and six homeostasis subnetworks that can be foundin two steps using the algorithm in Section 1.8.Table 1: Four ιo -simple paths for network in Figure 3 Simple path ( S ) Complementary subnetwork ( C S ) ι → ρ → ρ → ρ → ρ → ρ → o { τ , τ , τ , τ , ρ } ι → ρ → ρ → ρ → ρ → ρ → o { τ , τ , τ , τ , ρ } ι → ρ → ρ → ρ → ρ → o { τ , ρ , τ , τ , τ , ρ } ι → ρ → ρ → ρ → ρ → o { τ , τ , τ , τ , ρ , ρ } The input-output network G in Figure 3 has six homeostasis subnetworks, which can befound in two steps: 15 I ι ορ ρ ρ ρ τ τ τ τ ρ Figure 3:
The 12-node example.
Input node ι , output node o , six regulatory nodes ρ , . . . , ρ , and four appendage nodes τ , . . . , τ . The network has four ιo -simple paths (seeTable 1) and five super-simple nodes ι, ρ , ρ , ρ , o . The six homeostasis blocks are listed inTable 2. Step 1: G has three appendage path components ( A = { τ } , A = { τ , τ } , B = { τ } ) in A G . Among these, A and A satisfy the no cycle condition, whereas B does not since τ forms a cycle with simple node ρ . Hence, there are two appendage homeostasis subnetworksgiven by A and A . Step 2: G has five super-simple nodes (in downstream order, they are ι, ρ , ρ , ρ , o ). Thefive super-simple nodes lead to four structural homeostasis subnetworks given (up to coreequivalence) by L (cid:48) ( ι, ρ ), L (cid:48) ( ρ , ρ ), L (cid:48) ( ρ , ρ ), L (cid:48) ( ρ , o ).Table 2 lists the six homeostasis subnetworks in G , which give the six irreducible factorsof det( H ) where H is the 11 ×
11 homeostasis matrix of G . The factorization of the degree11 homogeneous polynomial det( H ) is given bydet( H ) = ± f τ ,x τ det( B ) f ρ ,x ι det( B ) f ρ ,x ρ det( B )where B = (cid:20) f τ ,x τ f τ ,x τ f τ ,x τ f τ ,x τ (cid:21) B = (cid:20) f ρ ,x ρ f ρ ,x ρ f ρ ,x ρ f ρ ,x ρ (cid:21) B = f ρ ,x ρ f ρ ,x ρ f ρ ,x ρ f ρ ,x ρ f ρ ,x τ f τ ,x ρ f τ ,x τ f o,x ρ f o,x ρ Nijhout, Best and Reed [15] observed that homeostasis often appears in models in the formof a chair . That is, as I varies, the input-output function x ( I ) has the piecewise linear16able 2: Homeostasis subnetworks in Figure 3.
Class Homeostasis subnetworks Nameappendage A = { τ } null-degradationappendage A = { τ ⇔ τ } no cycle appendagestructural L (cid:48) ( ι, ρ ) = { ι → ρ } Haldanestructural L (cid:48) ( ρ , ρ ) = { ρ , ρ ρ } feedforward loopstructural L (cid:48) ( ρ , ρ ) = { ρ → ρ } Haldanestructural L (cid:48) ( ρ , o ) = { ρ , ρ , ρ , τ , o } degree 4 structuraldescription: increases linearly, is approximately constant, and then increases linearly again.Golubitsky and Stewart [10] observed that it follows from elementary catastrophe theorythat smooth chair singularities have the normal form I , defining conditions x (cid:48) o ( I ) = x (cid:48)(cid:48) o ( I ) = 0and nondegeneracy condition x (cid:48)(cid:48)(cid:48) ( I ) (cid:54) = 0. Moreover, [11] noted that if x (cid:48) ( I ) = g ( I ) h ( I ),where g ( I ) (cid:54) = 0, then the defining conditions for a chair singularity are equivalent to h ( I ) = h (cid:48) ( I ) = 0 and h (cid:48)(cid:48) ( I ) (cid:54) = 0 (1.17)It follows from Lemma 1.5 and (1.12) that a chair singularity for infinitesimal homeostasisis of type B η if h η ( I ) satisfies (1.17) at I = I . In Section 2 we show that infinitesimal homeostasis in the original system (1.1) occursin a network if and only if infinitesimal homeostasis occurs in the core network for theassociated frozen system. See Theorem 2.4. We discuss when backward arrows can beignored when computing the determinant of the homeostasis matrix and the limitations ofthis procedure. See Corollary 1.10. In Section 3 we relate the form of the summands ofthe determinant of the homeostasis matrix H with the form of ιo -simple paths of the input-output network. See Theorem 3.2. In Theorem 3.3 we also discuss ‘core equivalence’. InSection 4 we prove the theorems about the appendage and structural classes of homeostasis.See Definition 4.3, Theorem 4.4, and the normal form Theorem 4.7. In Section 5 we provethe necessary conditions that appendage homeostasis must satisfy. See Theorem 5.4. InSection 6, specifically Section 6.5, we introduce an ordering of super-simple nodes that leadsto a combinatorial definition of structural blocks. See Definition 1.19 and Definition 1.20.The connection of these blocks with the subnetworks K η obtained from the homeostasismatrix is given in Corollary 6.7 and Theorem 6.11. In Section 7 we summarize our algorithmfor finding infinitesimal homeostasis directly from the input-output network G . It also givesa topological classification of the different types of infinitesimal homeostasis that the network G can support. 17 Core networks
Let G be an input-output network with input node ι , output node o , and regulatory nodes ρ j . We use the notions of upstream and downstream nodes to construct a core subnetwork G c of G .The stable equilibrium ( X , I ) of the system of differential equations (1.1) satisfy asystem of nonlinear equations (1.4), that can be explicitly written as f ι ( x ι , x ρ , x o , I ) = 0 f ρ ( x ι , x ρ , x o ) = 0 f ι ( x ι , x ρ , x o ) = 0 (2.1)We start by partitioning the regulatory nodes ρ into three types: • those nodes σ that are both upstream from o and downstream from ι , • those nodes d that are not downstream from ι , • those nodes u that are downstream from ι and not upstream from o .Based on this partition, the system (2.1) has the form f ι ( x ι , x σ , x u , x d , x o , I ) = 0 f σ ( x ι , x σ , x u , x d , x o ) = 0 f u ( x ι , x σ , x u , x d , x o ) = 0 f d ( x ι , x σ , x u , x d , x o ) = 0 f o ( x ι , x σ , x u , x d , x o ) = 0 (2.2)In Lemma 2.1 we make this form more explicit. Lemma 2.1.
The definitions of σ , u , and d nodes imply the admissible system (2.2) has theform ˙ x ι = f ι ( x ι , x σ , x d , x o , I )˙ x σ = f σ ( x ι , x σ , x d , x o )˙ x u = f u ( x ι , x σ , x u , x d , x o )˙ x d = f d ( x d )˙ x o = f o ( x ι , x σ , x d , x o ) (2.3) Specifically, arrows of type σ → d , ι → d , u → d , o → d , u → σ , u → o , u → ι do not exist.Proof. We list the restrictions on (2.2) given first by the definition of d and then by thedefinition of u . σ (cid:54)→ d If a node in σ connects to a node in d , then there would be a path from ι to a nodein d and that node in d would be downstream from ι , a contradiction. Therefore, f d isindependent of x σ . 18 (cid:54)→ d Similarly, the node ι cannot connect to a node in d , because that node would then bedowntream from ι , a contradiction. Therefore, f d is independent of x ι . o (cid:54)→ d If there is an arrow o → d , then there is a path ι → σ → o → d . Hence there is apath ι → d and that is not allowed. Therefore, f d is independent of x o . u (cid:54)→ d Note that nodes in u must be downstream from ι . Hence, there cannot be a connectionfrom u to d or else there would be a connection from ι to d . Therefore, f d is independentof x u . u (cid:54)→ σ if a node in u connects to a node in σ , then there would be a path from u to o and u would be upstream from o , a contradiction. Therefore, f σ is independent of x u . u (cid:54)→ o Suppose a node in u connects to o . Then that node is upstream from o , a contradic-tion. Therefore, f o is independent of x u . u (cid:54)→ ι Finally, if u connects to ι , then u connects to o , a contradiction. Therefore, f ι isindependent of x u .The remaining types of connections can exist in G c . Arrows that can exist in G c are shownin Figure 4. d ι οσ u Figure 4:
Core network.
Arrows indicate paths connecting nodes in partitioned corenetwork.
Lemma 2.2.
Suppose X = ( x ∗ ι , x ∗ σ , x ∗ u , x ∗ d , x ∗ o ) is a stable equilibrium of (2.3) . Then thecore admissible system (obtained by freezing x d at x ∗ d ) ˙ x ι = f ι ( x ι , x σ , x ∗ d , x o , I )˙ x σ = f σ ( x ι , x σ , x ∗ d , x o )˙ x o = f o ( x ι , x σ , x ∗ d , x o ) (2.4) has a stable equilibrium at Y = ( x ∗ ι , x ∗ σ , x ∗ o ) . roof. It is straightforward that Y is an equilibrium of (2.4). Reorder coordinates ( ι, σ, u, d, o )to ( ι, σ, o, d, u ). Then Lemma 2.1 implies that the Jacobian J of (2.3) has the form J = f ι,x ι f ι,x σ f ι,x d f ι,x o f σ,x ι f σ,x σ f σ,x d f σ,x o f u,x ι f u,x σ f u,x u f u,x d f u,x o f d,x d f o,x ι f o,x σ f o,x d f o,x o (2.5)and on swapping the u and o coordinates we see that J is similar to J = f ι,x ι f ι,x σ f ι,x o f ι,x d f σ,x ι f σ,x σ f σ,x o f σ,x d f o,x ι f o,x σ f o,x o f o,x d
00 0 0 f d,x d f u,x ι f u,x σ f u,x o f u,x d f u,x u (2.6)It follows that the eigenvalues of J at X are the eigenvalues of f d,x d , f u,x u , and the eigenvaluesof the Jacobian of (2.4) at Y . Since the eigenvalues of J have negative real part, theequilibrium Y is stable. Lemma 2.3.
Suppose that G is an input-output network with core network G c . Suppose thatthe core admissible system ˙ x ι = f ι ( x ι , x σ , x o , I )˙ x σ = f σ ( x ι , x σ , x o )˙ x o = f o ( x ι , x σ , x o ) (2.7) has a stable equilibrium at Y = ( x ∗ ι , x ∗ σ , x ∗ o ) and a point of infinitesimal homeostasis at I .Then the admissible system for the original network G (obtained by lifting x u and x d ) ˙ x ι = f ι ( x ι , x σ , x o , I )˙ x σ = f σ ( x ι , x σ , x o )˙ x d = − x d ˙ x u = − x u ˙ x o = f o ( x ι , x σ , x o ) (2.8) has a stable equilibrium at X = ( x ∗ ι , x ∗ σ , , , x ∗ o ) and infinitesimal homeostasis at I . Theorem 2.4.
Let x o ( I ) be the input-output function of the admissible system (2.3) and let x co ( I ) be the input-output function of the associated core admissible system (2.4) . Then theinput-output function x co ( I ) associated with the core subnetwork has a point of infinitesimalhomeostasis at I if and only if the input-output function x o ( I ) associated with the originalnetwork has a point of infinitesimal homeostasis at I . More precisely, x (cid:48) o ( I ) = k ( I ) x c (cid:48) o ( I ) (2.9) where k ( I ) (cid:54) = 0 . roof. It follows from Lemma 1.5 that x (cid:48) o ( I ) = 0 if and only ifdet f σ,x ι f σ,x σ f σ,x d f u,x ι f u,x σ f u,x u f u,x d f d,x d f o,x ι f o,x σ f o,x d = 0if and only if det( f u,x u ) det f σ,x ι f σ,x σ f σ,x d f d,x d f o,x ι f o,x σ f o,x d = 0if and only if det( f u,x u ) det( f d,x d ) det (cid:20) f σ,x ι f σ,x σ f o,x ι f o,x σ (cid:21) = 0Both matrices f u,x u and f d,x d are triangular with negative diagonal entries and thus havenonzero determinants. It then follows from Lemma 1.5 that x c (cid:48) o ( I ) = 0 if and only ifdet (cid:20) f σ,x ι f σ,x σ f o,x ι f o,x σ (cid:21) = 0 (2.10)is satisfied.It follows from Theorem 2.4 and Lemma 2.3 that classifying infinitesimal homeostasisfor networks G is identical to classifying infinitesimal homeostasis for the core subnetwork G c . Specifically, an admissible system with infinitesimal homeostasis for the core subnetworkyields, by Lemma 2.3, an admissible system with infinitesimal homeostasis for the originalnetwork which in turn yields the original system for the core subnetwork with infinitesimalhomeostasis by Theorem 2.4. Remark 2.5.
Corollary 1.10 implies that backward arrows can be eliminated when com-puting zeros of det( H ). These arrows cannot be eliminated when computing equilibria ofthe network equations or their stability. See (2.13) in Example 2.6. Example 2.6.
Consider the network in Figure 5. Assume WLOG that an admissible vectorfield for this network ˙ x ι = f ι ( x ι , x ρ , I )˙ x ρ = f ρ ( x ι , x ρ )˙ x o = f o ( x ρ , x o ) (2.11)has an equilibrium at the origin ( X , I ) = (0 , f ι (0 , ,
0) = f ρ (0 ,
0) = f o (0 ,
0) = 0 . Begin by noting that the Jacobian of (2.11) is J = f ι,x ι f ι,x ρ f ρ,x ι f ρ,x ρ f o,x ρ f o,x o (2.12)21 ι ρ Figure 5:
Backward arrow.
Network with a (dashed) backward arrow.The origin is a linearly stable equilibrium if and only if f o,x < f ι,x ι + f ρ,x ρ < f ι,x ι f ρ,x ρ − f ι,x ρ f ρ,x ι > f ι,x ρ = 0. However, whether infinitesimal homeostasis ( x (cid:48) o (0) = 0) occurs is independent ofthe backward coupling sincedet (cid:20) f ρ,x ι f ρ,x ρ f o,x ι f o,x ρ (cid:21) = det (cid:20) f ρ,x ι f ρ,x ρ f o,x ρ (cid:21) = f ρ,x ι f o,x ρ = 0 Let H be the ( n + 1) × ( n + 1) homeostasis matrix (1.6) of the input-output network G withinput node ι , n regulatory nodes ρ j , and output node o , and admissible system (1.1). Lemma 3.1.
Every nonzero summand of det( H ) corresponds to a unique ιo -simple path andhas all coupling strengths within this ιo -simple path as its factors.Proof. Each nonzero summand in det( H ) has n + 1 factors and each factor is the strengthof a coupling arrow or of the linearized internal dynamics of a node. We can write H as H = ι ↓ ρ ↓ ρ n − ↓ ρ n ↓ f ,ι f , · · · f ,n − f ,n f ,ι f , · · · f ,n − f ,n ... ... ... ... ... f n,ι f n, · · · f n,n − f n,n f o,ι f o, · · · f o,n − f o,n ← ρ ← ρ ← ρ n ← o (3.1)The columns of H correspond to n + 1 nodes in the order ι, ρ , . . . , ρ n and the rows of H correspond to n + 1 nodes in the order ρ , . . . , ρ n , o . The entry f j,ι = f ρ j ,x ι in column ι isthe linearized coupling strength of an arrow ι → ρ j . The entry f o,k = f o,x ρk in row o is thelinearized coupling strength of an arrow ρ k → o . The entry f j,k = f ρ j ,x ρk is the linearizedcoupling strength of an arrow ρ k → ρ j . If j = k , the entry f k,k = f ρ k ,x ρk is the linearizedinternal dynamics of node k . Note that each summand in the expansion of det( H ) has onefactor associated with each column of H and one factor associated with each row of H .22ix a summand. By assumption there is a unique factor associated with the first column.If this factor is f o,ι , we are done and the simple path is ι → o . So assume the factor in thefirst column is f k,ι , where 1 ≤ k ≤ n . This factor is associated with the arrow ι → ρ k .Next there is a unique factor in the column called ρ k and that factor corresponds to anarrow ρ k → ρ j for some node ρ j . If node ρ j is o , the summand includes ( f k,ι f o,k ) and theassociated simple path is ι → ρ k → o . Hence we are done. If not, we assume 1 ≤ j ≤ n .Since there is only one summand factor in each row of H , it follows that k (cid:54) = j . Thissummand is then associated with the path ι → ρ k → ρ j and contains the factors ( f k,ι f j,k ).Proceed inductively. By the pigeon hole principle we eventually reach a node that con-nects to o . The simple path that is associated to the given summand is unique because westart with the unique factor in the summand that has an arrow whose tail is ι and the choiceof ρ i is unique at each step. Moreover, every coupling within this simple path is a factor ofthe given summand.The determinant formula (3.2) for det( H ) in Theorem 3.2 is obtained by indexing thesum by the ιo -simple paths of G as described in Lemma 3.1. Theorem 3.2.
Suppose G has k ιo -simple paths S , . . . , S k with corresponding complemen-tary subnetworks C , . . . , C k . Then(a) The determinant formula holds: det( H ) = k (cid:88) i =1 F S i G C i (3.2) where F S i is the product of the coupling strengths within the ιo -simple path S i and G C i is a function of coupling strengths (including self-coupling strengths) from C i .(b) Specifically, G C i = ± det( J C i ) (3.3) where J C i is the Jacobian matrix of the admissible system corresponding to the com-plementary subnetwork C i . Generically, a coupling strength in G cannot be a factor of G C i .Proof. (a) Let S i be the r + 2 node ιo -simple path ι → j → · · · → j r → o and let F S i = f i ,x ι f i ,x i · · · f i r ,x ir − f o,x ir be the product of all coupling strengths in S i . By Lemma 3.1, det( H ) has the form(3.2). We claim that G C i is a function depending only on the coupling strengths(including self-coupling strengths) from the complementary subnetwork C i . Since eachsummand in the expansion of det( H ) has only one factor in each column of H andone factor in each row of H , the couplings in G C i must have different tails and headsfrom the ones that appear in the simple path. Hence, G C i is a function of couplings(including self-couplings) between nodes that are not in the simple path S i , as claimed.23b) Next we show that up to sign G C i is the determinant of the Jacobian matrix of theadmissible system for the subnetwork C i (see (3.3)). To this end, relabel the nodes sothat the ιo -simple path S i is ι → → · · · → r → o and the nodes in the complementary subnetwork C i are labeled r + 1 , . . . , n . Then F S i = ( − χ f ,x ι f ,x · · · f r,x r − f o,x r where χ permutes the nodes of the ιo -simple path S i to 1 , . . . , r . The summands ofdet( H ) associated with S i are F S i G C i , where G C i = (cid:88) σ ( − σ f r +1 ,x σ ( r +1) · · · f n,x σ ( n ) (3.4)and σ is a permutation of the indices r + 1 , . . . , n . Observe that the right hand side of(3.4) is just det( J C i ) up to sign.Lastly, we show that no coupling strength in G can be a factor of det( J C i ). The couplingstrengths correspond to the arrows and the self-coupling strengths correspond to thenodes. The self-coupling strengths are the diagonal entries of J C i , which are genericallynonzero. If we set all coupling strengths to 0 (that is, assume they are neutral), thenthe off-diagonal entries of det( J C i ) are 0 and det( J C i ) (cid:54) = 0. Now suppose that onecoupling strength is a factor of det( J C i ), then det( J C i ) = 0 if that coupling is neutraland we have a contradiction. It follows that no coupling strength can be a factor ofdet( J C i ). Theorem 3.3.
Two core networks are core equivalent if and only if they have the same set of ιo -simple paths and the Jacobian matrices of the complementary subnetworks to any simplepath have the same determinant up to sign.Proof. ⇒ Let G and G be core networks and assume they are core equivalent. Therefore,det( B ) = det( B ) and by Theorem 3.2det( B ) ≡ k (cid:88) i =1 F S i G C i = (cid:96) (cid:88) j =1 F T j G D j ≡ det( B )If a simple path of G were not a simple path of G , the equality would fail; that is, thepolynomials would be unequal. Therefore, we may assume (cid:96) = k and (by renumbering ifneeded) that T i = S i for all i . It follows that k (cid:88) i =1 F S i ( G C i − G D i ) = 024ince the F S i are linearly independent it follows that G C i = G D i for all i ; that is, det( J C i ) = ± det( J D i ) where J C i and J D i are the Jacobian matrices associated with G and G . Hencethe Jacobian matrices of the two complementary subnetworks have the same determinantup to sign. ⇐ The converse follows directly from Theorem 3.2.
Corollary 3.4.
Two core networks are core equivalent if they have the same set of ιo -simplepaths and the same complementary subnetworks to these simple paths.Proof. Follows directly from Theorem 3.3.
In this section we prove that there are two classes of infinitesimal homeostasis: appendage and structural . See Definition 4.3 and Theorem 4.4. The section ends with a description ofa ‘normal form’ for appendage and structural homeostasis blocks. These ‘normal forms’ aregiven in Theorem 4.7.Section 5 discusses graph theoretic attributes of appendage homeostasis and Section 6.5discusses graph theoretic attributes of structural homeostasis. This material leads to theconclusions in Section 7 where it is shown that each structural block is generated by twoadjacent super-simple nodes and each appendage block is generated by a path componentof the subnetwork of appendage nodes.Recall from (1.11) that we can associate with each homeostasis matrix H a set of m irreducible square blocks B , . . . , B m where P HQ = B ∗ · · · ∗ B · · · ∗ ... ...0 0 · · · B m (4.1)and P and Q are ( n + 1) × ( n + 1) permutation matrices. Lemma 4.1.
Let H be an ( n + 1) × ( n + 1) homeostasis matrix and let P and Q be ( n + 1) × ( n + 1) permutation matrices. Then the rows (and columns) of P HQ are the same as therows (and columns) of H up to reordering. Moreover, the set of entries of H are identicalwith the set of entries of P HQ .Proof.
The set of rows of
P H are identical to the set of rows of H . A row of HQ containsthe same entries as the corresponding row of H —but with entries permuted. The secondstatement follows from the first.Recall that the entries of the homeostasis matrix H , defined in (1.6) for an admissiblesystem of a given input-output network G , appear in three types: 0, coupling, and self-coupling. The following lemma is important in our discussion of homeostasis types.25 emma 4.2. The number of self-coupling entries in each diagonal block B η is an invariantof the homeostasis matrix H .Proof. Suppose H is transformed in two different ways to upper triangular form (4.1). Thenone obtains two sets of diagonal blocks B , . . . , B m and ˜ B , . . . ˜ B ˜ m . Since one set of blocksis transformed into the other by a permutation, it follows that the number of blocks in eachset is the same. Moreover, the blocks are related by˜ B M ( ν ) = P ν B ν Q ν where M is a permutation of the index sets and for each ν , P ν and Q ν are permutationmatrices. It follows from Lemma 4.1 that the size and the number of self-coupling entries ofthe square matrices ˜ B M ( ν ) and B ν are identical.We can now define the two homeostasis classes. Definition 4.3.
Let B η be an irreducible k × k square block associated with the ( n +1) × ( n +1) homeostasis matrix H in (4.1). The homeostasis class associated with B η is appendage if B η has k self-coupling entries and structural if B η has k − Theorem 4.4.
Let H be an ( n + 1) × ( n + 1) homeostasis matrix and let B η be a k × k squarediagonal block of the matrix P HQ given in (4.1) , where P and Q are permutation matricesand k ≥ . Then B η has either k − self-couplings or k self-couplings.Proof. Note that either
P HQ = (cid:20) B η D E (cid:21) or P HQ = A B C B η D E (4.2)where A is an nonempty square matrix. In the first case in (4.2) B η has single self-couplingentries in each of either k − k columns.We assume the second case in (4.2). From Lemma 4.1 it follows that P HQ has exactlyone row and exactly one column without a self-coupling entry. Hence, if B η has more than k self-couplings, then B η and hence H have a row with at least two self-couplings, which isnot allowed.We show by contradiction that B η has at least k − B η has (cid:96) ≤ k − (cid:96) self-couplings in B η by assumption,and there are no self-couplings in the 0 block. Let b be the number of self-couplings in B .Then b + (cid:96) is the number of self-couplings in [ B B η t . Now, either every column or everycolumn but one in [ B B η t has a self-coupling. Therefore, k − ≤ b + (cid:96) ≤ k or k − (cid:96) − ≤ b ≤ k − (cid:96) We consider the two cases: 26
Assume b = k − (cid:96) −
1. Then there exists one column in [
B B η t that has noself-couplings. Therefore, every column in A has a self-coupling. Since B has a self-coupling, it follows that one row in [ A B C ] has two self-couplings – a contradiction. • Assume b = k − (cid:96) . Since [ B B η t has self-couplings in every column, it follows that A has a self-coupling in every column save at most one. It then follows that A has aself-coupling in every row save at most one. Since k − (cid:96) ≥
2, at east one row in [
A B C ]has two self-couplings – also a contradiction.Therefore, (cid:96) = k − (cid:96) = k .We build on Theorem 4.4 by putting B η into a standard form of type (4.6). Its proofuses the next two lemmas about shapes and summands. A shape E is a subspace of m × n matrices E = ( e ij ), where e ij = 0 for some fixed subset of indices i, j . A square shape D is nonsingular if det( D ) (cid:54) = 0 for some D ∈ D . A summand of a nonsingular shape D is anonzero product in det( D ) for some D ∈ D . Lemma 4.5.
The nonzero summands of det(
P HQ ) and det( H ) are identical.Proof. Since det( P ) = det( Q ) = ±
1, it follows that det(
P HQ ) = ± det( H ). Hence, thenonzero summands must be identical. Lemma 4.6.
Suppose B and C are nonsingular shapes. Let E be the shape whose size ischosen so that D is the shape consisting of matrices D = (cid:20) B E C (cid:21) where B ∈ B , C ∈ C , E ∈ E . Then each summand of D is the product of a summand of B with a summand of C .Proof. Suppose d is a summand of D . The product d cannot have any entries in the 0 blockof D . Hence, d = bc . Moreover, there is a matrix B ∈ B such that det( B ) = b and amatrix C ∈ C such that det( C ) = c . In fact, we can assume that the nonzero entries of B are precisely the entries in the nonzero product b . Similarly for c . Since det( D ) (cid:54) = 0 anddet( D ) = det( B ) det( C ), it follows that det( B ) = b (cid:54) = 0 and det( C ) = c (cid:54) = 0. Therefore, b and c are summands of B and C , respectively. Conversely, assume that b and c are summandsand conclude that d is also a summand.It follows from Lemma 4.1 that the number of each type of entry in P HQ is the same asthe number in H . Moreover, generically, the coupling and self-coupling entries are nonzero.It follows from (1.6) that the n superdiagonal entries of H are self-coupling entries and theseare the only self-coupling entries in H . In addition, H has one self-coupling entry in eachrow except the last row, and one self-coupling in each column except the first column. ByLemma 4.1 there are exactly n self-coupling entries in P HQ with one in each row but one,and one in each column but one. We use these observations in the proof of Theorem 4.7.27 heorem 4.7.
Let H be an ( n + 1) × ( n + 1) homeostasis matrix. Suppose det( H ) has adegree k ≥ irreducible factor det( B η ) , where B η be a k × k block diagonal submatrix ofthe matrix P HQ given in (4.1) and P and Q are permutation matrices. If B η has k − self-coupling entries, then we can assume that B η has the form f ρ ,x ρ · · · f ρ ,x ρk ... . . . ... f ρ k ,x ρ · · · f ρ k ,x ρk (4.3) and if B η has k self-coupling entries, then we can assume that B η has the form f ρ ,x ρ · · · f ρ ,x ρk − f ρ ,x (cid:96) ... . . . ... ... f ρ k − ,x ρ · · · f ρ k − ,x ρk − f ρ k − ,x (cid:96) f j,x ρ · · · f j,x ρk − f j,x (cid:96) (4.4) Proof.
Theorem 4.4 implies that B η has either k − k self-couplings. Since B η is a k × k submatrix of P HQ (a matrix that has the same set of rows and the same set of columns as H ), B η must consists of k entries of the form B η = f ρ ,x τ · · · f ρ ,x τk ... . . . ... f ρ k ,x τ · · · f ρ k ,x τk (4.5)Since self-couplings must be in different rows and different columns we can use permutationmatrices of the form I p S
00 0 I q where S is a k × k permutation matrix to put B η in the form: ∗ sc · · · ∗∗ ∗ . . . ...... ... ... sc ∗ · · · · · · ∗ or sc ∗ ∗∗ . . . ∗∗ ∗ sc (4.6)where sc denotes a self-coupling entry and ∗ denotes either a 0 entry or a coupling entry.Note that we could just as well have put the self-coupling entries along the diagonal in (4.6)(left).If B η has k − ρ k (cid:54) = τ k and ρ j = τ j for 1 ≤ j ≤ k − B η has k self-couplings, as in (4.6) (right), then we may assume ρ j = τ j for all j . It followsthat the matrices in (4.6) have the form (4.4) or (4.3).28 emark 4.8. We use Theorem 4.7 to associate a subnetwork K η with each homeostasis k × k block B η . This construction implements the one in Definition 1.14 for appendage andstructural homeostasis blocks. The network K η will be an input-output subnetwork with k + 1 nodes when B η is structural and the network K η will be a standard subnetwork with k nodes when B η is appendage.If B η is appendage, then the k nodes in K η will correspond to the k self-couplings in B η and the arrows in K η will be τ i → τ j if h τ j ,x τj is a coupling entry in (4.3).If B η is structural, then the k − K η will correspond to the self-couplings in B η and the input node (cid:96) and the output node j of K η will be given by thecoupling entry in (4.4). The arrows in K η are given by the coupling entries of B η .Note that the constructions of K from H do not require that H is a homeostasis block;the constructions only require that H has the form given in either (4.4) or (4.3). An appendage block B η has k self-couplings and the form of a k × k matrix (4.3), that isrewritten here as: B η = f τ ,x τ · · · f τ ,x τk ... . . . ... f τ k ,x τ · · · f τ k ,x τk (5.1)As discussed in Remark 4.8 this homeostasis block is associated with a subnetwork K η consisting of distinct nodes τ , . . . , τ k and arrows specified by B η that connect these nodes.In this section we show that K η satisfies three additional conditions:(a) Each node τ j ∈ K η is an appendage node (Theorem 5.2).(b) For every ιo -simple path S , nodes in K η do not form a cycle with nodes in C S \ K η (Theorem 5.4(a)).(c) K η is a path component of the subnetwork of appendage nodes of G (Theorem 5.4(b)). Lemma 5.1.
Suppose a nonzero summand β of det( B η ) in (5.1) has f τ j ,x τi as a factor,where τ j (cid:54) = τ i . Then the arrow τ i → τ j is contained in a cycle in K η .Proof. To simplify notation we drop the subscript η below on ˜ H , K , and ˜ K . Let ˜ H be the( k − × ( k −
1) submatrix obtained by eliminating the j th row and the i th column of B η in (5.1). Since τ i (cid:54) = τ j , ˜ H has k − f τ i ,x τi and f τ j ,x τj have been removed when creating ˜ H from B η .It follows from Remark 4.8 that since ˜ H has the form (4.4), we can associate an input-output network ˜ K with ˜ H , where the input node is τ j since it does not receive any inputand the output node is τ i since it does not output to any node in ˜ K . By Lemma 3.1, everynonzero summand in det( ˜ H ) corresponds to a simple path from τ j → τ i . Hence, the nonzerosummand β is given by f τ j ,x τi times a nonzero summand corresponding to a simple path29rom τ j → τ i . Therefore, the arrow τ i → τ j coupled with the path τ j → τ i forms a cycle in K . Theorem 5.2.
Let K η be a subnetwork of G associated with an appendage homeostasis block B η that consists of a subset of nodes τ , · · · , τ k of G . Then B η equals the Jacobian J K η of thenetwork K η and each node τ j is an appendage node.Proof. Admissible systems associated with the network K η have the form˙ x τ = f τ ( x τ , . . . , x τ k )...˙ x τ k = f τ k ( x τ , . . . , x τ k )where the variables that appear on the RHS of each equation correspond to the couplings in(5.1). It follows that the matrix B η in (5.1) equals the Jacobian J K η , as claimed.We show that τ j ∈ K η is an appendage node for each j . More specifically, we show that τ j is in the complementary subnetwork C S of each ιo -simple path S . We now fix τ j and S .We make two claims. First, every nonzero summand α of det( H ) either contains theself-coupling f τ j ,x τj as a factor or a coupling f τ j ,x τi for some i (cid:54) = j as a factor. Second, thisdichotomy is sufficient to prove the theorem.First claim. It follows from Lemma 4.6 that each summand of det( P HQ ) has a summandof det( B η ) as a factor. Therefore, each summand α of det( H ) has a summand β of det( B η )as a factor. The claim follows from two facts. The first is that B η is the Jacobian J K η andhence either the self-coupling is in β or the off diagonal entry is in β ; and the second is thatonce these entries are in β , they are also in α .Second claim. Recall that Theorem 3.2 (the determinant theorem) implies that thesummand α has the form F S g C S where S is an ιo -simple path, C S is the complementarysubnetwork to S , F S is the product of the coupling strengths within S , J C S is the Jacobianmatrix of the admissible system corresponding to C S , and g C S is a summand in det( J C S ).If the summand α has f τ j ,x τj as a factor, it follows that f τ j ,x τj is a factor of g C S since itis a self-coupling and cannot be a factor of F S . Hence, node τ j is a node in C S .If the summand α has f τ j ,x τi as a factor, then f τ j ,x τi is either not a factor of F S or is afactor of F S . In the first case, f τ j ,x τi is a factor of g C S . It follows that τ j is a node in C S . Inthe second case, the arrow τ i → τ j is on the simple path S . Recall that f τ j ,x τi is also a factorof the summand β . It follows from Lemma 5.1 applied to β that τ i → τ j is contained in acycle in K η . This is a contradiction since we show that τ i → τ j cannot be contained in boththe simple path S and a cycle in K η .Since τ i → τ j is contained in a cycle in K η , there exists an arrow τ k → τ i where τ k isa node in K η ( τ k can be τ j ). Since every nonzero summand of det( H ) has a summand ofdet( B η ) as a factor, there exists a summand F S g C S having both f τ j ,x τi and f τ i ,x τk as factors.Note that f τ j ,x τi is a factor of F S and g C S is a summand in det( J C S ). Since τ k → τ i cannotbe contained in S it must be a factor of g C S . However, C S is the complementary subnetworkto S that does not contain any arrow connecting to τ i in the simple path S .30 emma 5.3. Let K be a proper subnetwork of a subnetwork C of G . If nodes in K do notform a cycle with nodes in C \ K , then upon relabelling nodes J C is block lower triangular.Proof. The no cycle condition implies that we can partition nodes in C into three classes:(i) nodes in C \ K that are strictly upstream from K ,(ii) nodes in K ,(iii) nodes in C \ K that are not upstream from K .By definition nodes in sets (i) and (iii) are disjoint from nodes in (ii). Also, nodes in sets (i)and (iii) are disjoint because nodes in K do not form a cycle with nodes in C \ K . Finally,it is straightforward to see that C = (i) ∪ (ii) ∪ (iii). Using this partition of C , we claim thatthe Jacobian matrix of C has the desired block lower triangular form: J C = ∗ · · · ∗ · · · ∗ · · · ∗ · · · ∗ · · · ∗ J K · · · ∗ · · · ∗ ∗ ∗ · · · ∗ ... ... ... ... ... ... ... ∗ · · · ∗ ∗ ∗ · · · ∗ (5.2)Specifically, observe that there are no connections from (i) to (iii) because then a node in (iii)would be strictly upstream from K . By definition there are no connections from (ii) to (iii).Finally, the cycle condition implies that there are no connections from (i) to (ii). Theorem 5.4.
Let K η be a subnetwork of G associated with an appendage homeostasis block B η . Then:(a) For every ιo -simple path S , nodes in K η do not form a cycle with nodes in C S \ K η .(b) K η is a path component of A G .Proof. By Theorem 5.2, K η ⊂ A G is an appendage subnetwork that is contained in eachcomplementary subnetwork C S , B η = J K η and det( J K η ) is a factor of det( H ). To simplifynotation in the rest of the proof, we drop the subscript η and use K to denote the appendagesubnetwork. Proof of (a)
We proceed by contradiction and assume there is a cycle. Let S be an ιo -simple path. Let B ⊂ C S \ K be the nonempty subset of nodes that are on some cycleconnecting nodes in K with nodes in C S \ K . It follows that nodes in K do not form anycycle with nodes in ( C S \ K ) \ B = C S \ ( K ∪ B ). Since
K ∪ B ⊂ C S and nodes in K ∪ B do31ot form a cycle with nodes in C S \ ( K ∪ B ), by Lemma 5.3 we see that the Jacobian matrixof C S has the form J C S = U ∗ J K∪B ∗ ∗ D (5.3)where J K∪B = (cid:20) J K f K ,x B f B ,x K J B (cid:21) Note that f K ,x B (cid:54) = 0 and f B ,x K (cid:54) = 0, since there is a cycle containing nodes in K and B . Weclaim that the polynomial det( J K ) does not factor the polynomial det( J K∪B ). It is sufficientto verify this statement for one admissible vector field.Relabel the nodes so that there is a cycle of nodes 1 → → · · · → p → q nodeas are in K . We can choose the cycle so that the remaining nodes are in B . Anadmissible system for this cycle has the form( f , f , . . . , f p )( x ) = ( f ( x , x p ) , f ( x , x ) , · · · , f p ( x p , x p − ))and all other coordinate functions f r ( x ) = x r . Hence the associated Jacobian matrix is J K∪B = f ,x · · · · · · f ,x p · · · · · · f ,x f ,x · · · · · · · · · · · · · · · ∗ · · · f q,x q − f q,x q · · · · · · · · · · · · · · · f q +1 ,x q f q +1 ,x q +1 · · · · · · · · · · · · · · · f p,x p − f p,x p · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (5.4)where the upper left block is J K . It follows from direct calculation that the determinant ofthe J K∪B is det( J K∪B ) = f ,x f ,x · · · f p,x p + ( − ( p − f ,x f ,x · · · f p,x (5.5)Hence det( J K ) = f ,x f ,x · · · f q,x q is not a factor of det( J K∪B ), given in (5.5). We claim that det( J K ) is also not a factor ofdet( J C S ) because by (5.3), det( J C S ) = det( U ) det( J K∪B ) det( D ). Suppose det( J K ) is a factorof det( J C S ), then it must be a factor of det( J K∪B ), which is a contradiction. Thus det( J K )is not a factor of det( J C S ), which contradicts the fact that K η is an appendage homeostasisblock. Hence, nodes in K cannot form a cycle with nodes in C S \ K .32 roof of (b) We begin by showing that K is path connected; that is, there is a path from τ i to τ j for every pair of nodes τ i , τ j ∈ K . Suppose not, then the path components of K give K a feedforward structure. It follows that we can partition the set of nodes in K into twodisjoint classes: A and B where nodes in B are strictly downstream from nodes in A . Thus,there exist permutation matrices P η and Q η such that P η B η Q η = (cid:20) J A ∗ J B (cid:21) which contradicts the fact that B η is irreducible. Therefore, K is path connected.Next, we show that K is a path component of A G . Suppose that the path component W ⊂ C S of A G that contains K is larger than K . Then there would be a cycle in W ⊂ C S thatstarts and ends in K , and contains nodes not in K . This contradicts (a) and W = K .Recall from Definition 1.18 that S G is a subnetwork of G that can be obtained by removingall appendage path components that satisfy the no cycle condition. Lemma 5.5.
Let G be an input-output network with homeostasis matrix H . Then the struc-tural subnetwork S G is an input-output network with homeostasis matrix H (cid:48) and det( H (cid:48) ) isa factor of det( H ) .Proof. By Theorem 5.2, if B η is an appendage homeostasis block, then the associated subnet-work K η consists of appendage nodes, and B η = J K η . Relabel the blocks so that B , · · · , B p are appendage homeostasis blocks. We can write P HQ = J K ∗ · · · ∗ ∗ ...... 0 J K p ∗ · · · H (cid:48) Hence det( H (cid:48) ) is a factor of det( H ).Recall H is an ( n + 1) × ( n + 1) matrix with n self-couplings. Since the main diagonalentries of J K i are all self-couplings, H (cid:48) is a ( n + 1 − γ ) × ( n + 1 − γ ) matrix where γ is thetotal number of self-couplings in K , · · · , K p . It follows that H (cid:48) has n − γ self-couplings. ByTheorem 4.7 we can assume H (cid:48) has the homeostasis matrix form and is associated with aninput-output subnetwork S G of G .It follows from the upper triangular form of P HQ that S G does not contain any nodein appendage blocks or any coupling whose head or tail is a node in an appendage block.Moreover, a node that is not associated with any appendage block must be contained in S G .Otherwise, the self-coupling of this node will appear in some J K i , which is a contradiction.Hence, S G is an input-output network that consists of all nodes not associated with anyappendage block and all arrows that connect nodes in S G . Remark 5.6.
Suppose K η is an input-output subnetwork of G associated with an irreduciblematrix B η in (4.4). Then, it follows from Lemma 5.5 that B η is a structural block of G ifand only if B η is a structural block of S G . 33 Structural homeostasis blocks
In this section we give a combinatorial description of S G in terms of input-output subnetworksdefined by super-simple nodes. We do this in four stages. § shows that the super-simple nodes in G can be ordered by ι > ρ > · · · > ρ q > o where a > b if b is downstream from a . See Lemma 6.1. § defines the sets L of simple nodes that lie between adjacent super-simple nodes. SeeDefinition 1.19 and Lemma 6.2. § shows how to assign each appendage node in S G to a unique L , thus forming combina-torially the subnetwork L (cid:48) . See Definition 1.20. § shows that the homeostasis matrix of S G can be put in block upper trimngular formwith blocks given by the homeostasis matrices of the L (cid:48) . See Corollary 6.7. Lemma 6.1.
Super-simple nodes in G are ordered by ιo -simple paths.Proof. Let ρ and ρ be distinct super-simple nodes and let S and T be two ιo -simple paths.Suppose ρ is downstream from ρ along S and ρ is downstream from ρ along T . It followsthat there is a simple path from ι to ρ along T that does not contain ρ and a simple pathfrom ρ to o along S that does not contain ρ . Hence, there is an ιo -simple path that doesnot contain ρ contradicting the fact that ρ is super-simple. A super-simple subnetwork L ( ρ , ρ ) is a subnetwork consisting of all simple nodes betweenadjacent super-simple nodes ρ and ρ (see Definition 1.19). The following Lemma showsthat each non-super-simple simple node belongs to a unique L . Lemma 6.2.
Every non-super-simple simple node lies uniquely between two adjacent super-simple nodes.Proof.
Let ρ be a simple node that is not super-simple. By definition ρ is on an ιo -simplepath S and ρ lies between two adjacent super-simple nodes ρ and ρ on S . Suppose ρ isalso on an ιo -simple path T . Then, by Lemma 6.1 ρ and ρ must be ordered in the sameway along T and ρ and ρ must be adjacent super-simple nodes along T . If ρ is downstreamfrom ρ along T , then there would be an ιo -simple path that does not contain ρ , which isa contradiction. A similar comment holds if ρ is upstream from ρ along T . Therefore, ρ isalso between ρ and ρ on T . 34efinition 1.19 implies that if ρ is downstream from ρ then L ( ρ , ρ ) ∩ L ( ρ , ρ ) = (cid:26) ∅ if ρ (cid:54) = ρ { ρ } otherwise (6.1)Lemma 6.3 identifies several properties of the subnetworks L . Lemma 6.3.
Let the pairs of super-simple nodes ρ , ρ and ρ , ρ be adjacent.(a) No arrow connects an upstream node ρ in the subnetwork L ( ρ , ρ ) to a downstreamnode τ in the subnetwork L ( ρ , ρ ) unless ρ = ρ , τ = ρ and ρ and ρ are adjacentsuper-simple nodes.(b) No arrow connects an upstream node ρ in the subnetwork L ( ρ , ρ ) to a downstreamnode τ in the subnetwork L ( ρ , ρ ) unless ρ = ρ or τ = ρ .(c) Suppose that a path of appendage nodes connects L ( ρ , ρ ) to L ( ρ , ρ ) . Then ρ isupstream from ρ .(d) Suppose that the appendage path component B fails the no cycle condition and there isa cycle that connects nodes in B with nodes in C S \ B , where C S is a complementarysubnetwork. Then the nodes in C S \ B that are in the cycle are non-super-simple simplenodes that are contained in a unique super-simple subnetwork.Proof. (a) Suppose an arrow connects a node ρ (cid:54) = ρ in L ( ρ , ρ ) to a node τ in L ( ρ , ρ )where ρ is downstream from ρ . Then there would be an ιo -simple path that connects ρ to ρ to τ to ρ in that order. That ιo -simple path would miss ρ , contradicting thefact that ρ is super-simple. A similar statement holds if τ (cid:54) = ρ or ρ and ρ are notadjacent. This proves (a).(b) Suppose an arrow connects a node ρ (cid:54) = ρ in L ( ρ , ρ ) to a node τ (cid:54) = ρ in L ( ρ , ρ ).Then there would be an ιo -simple path that connects ρ to ρ to τ to ρ in that order.That ιo -simple path would miss ρ , contradicting the fact that ρ is super-simple.(c) Suppose ρ is strictly downstream from ρ . Then there is an ιo -simple path from ι to ρ to some nodes in A G to ρ to o . Therefore, at least one node in A G is not anappendage node. A contradiction.(d) If the cycle contains a super-simple node, then the cycle cannot be in C S . Since thecycle must contain simple nodes that simple node cannot be super-simple.Suppose the cycle contains a simple node τ in L ( ρ , ρ ) and another simple node τ in L ( ρ , ρ ) where ρ is downstream from ρ , then there would be a path connecting τ to τ that does not contain any super-simple node. This would lead to an ιo -simple pathfrom ρ to τ to τ to ρ that misses ρ and ρ . Hence, the simple nodes contained inthe cycle must come from a single super-simple subnetwork.35 emark 6.4. Lemma 6.3 (a,b) implies that two different super-simple subnetworks L ( ρ , ρ )and L ( ρ , ρ ) where ρ is upstream from ρ can only be connected by either having a commonsuper-simple node ( ρ = ρ ) or by having an arrow ρ → ρ where ρ and ρ are adjacentsuper-simple nodes. L By Lemma 6.3 (d) any appendage path component that fails the cycle condition forms cycleswith non-super-simple simple nodes in a unique super-simple subnetwork. We can thereforeexpand a super-simple subnetwork L to a super-simple structural subnetwork L (cid:48) by recruitingall appendage nodes that form cycles with nodes in L (see Definition 1.20).It follows that if ρ is downstream from ρ , then L (cid:48) ( ρ , ρ ) ∩ L (cid:48) ( ρ , ρ ) = (cid:26) ∅ if ρ (cid:54) = ρ { ρ } otherwise (6.2)In particular, each appendage node in G is attached to at most one L . Remark 6.5.
Suppose ρ is downstream from ρ . By Lemma 6.3 (c) and Remark 6.4, noarrow connects a node ρ in L (cid:48) ( ρ , ρ ) \ { ρ } to a node τ in L (cid:48) ( ρ , ρ ) unless ρ = ρ and τ = ρ . S G with L (cid:48) Theorem 6.6.
Let K be an input-output core subnetwork of S G with q +1 super-simple nodes ρ , . . . , ρ q +1 in downstream order in G . Then the homeostasis matrix H K of K can be writtenin an upper block triangular form H K = H L (cid:48) ∗ · · · ∗ H L (cid:48) · · · ∗ ... . . . ... H L (cid:48) q (6.3) where for (cid:96) = 1 , . . . , q , H L (cid:48) (cid:96) is the homeostasis matrix of the super-simple structural subnet-work L (cid:48) (cid:96) = L (cid:48) ( ρ (cid:96) , ρ ( (cid:96) +1) ) .Proof. Since K is an input-output core subnetwork of S G , it follows that K consists of allsimple nodes between adjacent super-simple nodes of K and appendage nodes that formcycles with non-super-simple simple nodes in K . Hence, K consists of nodes and arrows in L (cid:48) ( ρ , ρ ) ∪ · · · ∪ L (cid:48) ( ρ q , ρ q +1 ) plus backward arrows between different super-simple structuralsubnetworks. Hence, for (cid:96) = 1 , . . . , q , nodes in K can be partitioned into disjoint classes:( (cid:96) ) = L (cid:48) (cid:96) \ { ρ (cid:96) +1 } . We claim that the homeostasis matrix H K of K is given by (6.3).It follows from Remark 6.5 that an arrow from a node in one class ( (cid:96) ) to a node in anotherclass (j) where j > (cid:96) can exist only when the two classes are adjacent (that is, j = (cid:96) + 1) and36he head of this arrow is the input node ρ (cid:96) +1 of the downstream class ( (cid:96) + 1). Since entriesbelow H K (cid:96) denote the arrows from nodes in class ( (cid:96) ) to nodes in classes ( (cid:96) + 1) through (q)except the input node ρ (cid:96) +1 in class ( (cid:96) + 1). It follows that all entries below H K (cid:96) are zero andhence H K has the upper block triangular form shown in (6.3). Corollary 6.7.
Suppose that τ , . . . , τ p +1 are the super-simple nodes of G in downstreamorder. Then the homeostasis matrix H (cid:48) of S G can be written in upper block triangular form H (cid:48) = B (cid:48) ∗ · · · ∗ B (cid:48) · · · ∗ ... ... . . . ... B (cid:48) p (6.4) where B (cid:48) (cid:96) is the homeostasis matrix of the super-simple structural subnetwork L (cid:48) ( τ (cid:96) , τ (cid:96) +1 ) for ≤ (cid:96) ≤ p . In addition, p is less than or equal to the number m of structural blocks K η .Proof. It follows from Definition 1.18 that S G has the same super-simple nodes as G and S G is a core subnetwork. By Theorem 6.6, the homeostasis matrix H (cid:48) of S G is given by(6.4). The number of irreducible blocks is the number of K η and that is m . Since m is themaximum number of blocks in H (cid:48) , it follows that m ≥ p by (6.4).If we can show that the number of super-simple nodes in K η is two, then we will showthat K η is core equivalent to one of the L (cid:48) . L (cid:48) This section shows that each structural subnetwork K η is core equivalent to the L (cid:48) having thesame input node. Specifically, we show that the input and output nodes in K η are adjacentsuper-simple and that no other nodes in K η are super-simple. Theorem 6.8.
Let K η be an input-output subnetwork of G associated with an irreduciblestructural homeostasis matrix B η in (4.4) . Then the input and output nodes of K η are super-simple nodes.Proof. We prove this theorem by proving that both the input and output nodes (cid:96) and j of K η are on the ιo -simple path associated with α for all summands α of det( H ). Theorem 3.2(the determinant theorem) implies that α has the form F S g C S where S is an ιo -simple path, C S is the complementary subnetwork to S , F S is the product of the coupling strengths within S , J C S is the Jacobian matrix of the admissible system corresponding to C S , and g C S is asummand in det( J C S ).It follows from Lemma 4.6 that the summands of form (4.2) are the summands of A timesthe summands of B η times the summands of E. Hence, every nonzero summand of det( H )contains a nonzero summand of det( B η ) as a factor. Since (cid:96) and j are the input outputnodes for the homeostasis matrix B η , it follows that every nonzero summand of det( B η ), and37ence det( H ), has both f m,x (cid:96) (where m is one of ρ , . . . , ρ k − , j ) and f j,x n (where n is one of ρ , . . . , ρ k − , (cid:96) ) as factors.From the form of P HQ (and hence H ) we see that f (cid:96),x (cid:96) and f j,x j are not factors ofnonzero summands of det( H ). Suppose the summand α has f m,x (cid:96) as a factor, then f m,x (cid:96) iseither a factor of F S or not a factor of F S . In the first case, it follows that the arrow (cid:96) → m is on the simple path S . Hence, the node (cid:96) is contained in S . In the second case, suppose f m,x (cid:96) is not a factor of F S , then it must be a factor of g C S . That implies that (cid:96) is a nodein C S . It follows that there exists another nonzero summand α (cid:48) of det( H ) which contains f (cid:96),x (cid:96) as a factor, which is is a contradiction. Therefore, we conclude every ιo -simple pathcontains node (cid:96) . By the same type of argument we can also conclude that every ιo -simplepath contains node j . Theorem 6.9.
If a structural block B η of G is irreducible, then K η is an input-output sub-network that has exactly two super-simple nodes.Proof. By Remark 5.6, K η is an input-output subnetwork of S G and K η is a core subnetworkbecause it is irreducible. Suppose in addition to the input and output nodes there are other q > K η , then by Theorem 6.6, the homeostasis matrix B η of K η canbe written in an upper block triangular form with q + 1 > K η is reducible, a contradiction. Corollary 6.10.
The input and output nodes of a structural homeostasis block are adjacentsuper-simple nodes.Proof.
Super-simple nodes can be well-ordered. The proof then follows from Theorem 6.9.
Theorem 6.11. In G , there is a 1:1 correspondence between structural homeostasis blocks K η and super-simple structural subnetworks L (cid:48) and that correspondence is given by havingthe same input node. Moreover, the corresponding K η and L (cid:48) are core equivalent.Proof. By Corollary 6.10, the input and output nodes of each K η are adjacent super-simplenodes and hence each K η leads to a unique L (cid:48) that has the same input node. Therefore, thenumber of K η (equal to m ) is less than or equal to the number p of L (cid:48) . Corollary 6.7 statesthat p ≤ m ; hence, p = m . That is, there is a 1:1 correspondence between K η and L (cid:48) .Let (cid:96) and j be the input and output nodes of the structural block K η . Then the corre-sponding super-simple structural subnetwork is L (cid:48) ( (cid:96), j ). By Definition 1.14, K η consists ofsimple nodes between the two adjacent super-simple nodes (cid:96) and j and appendage nodesthat form cycles with non-super-simple simple nodes in K η . Arrows in K η are non-backwardarrows that connect nodes in K η . It follows from Definition 1.20 that L (cid:48) ( (cid:96), j ) is the union of K η and arrows whose head is (cid:96) or whose tail is j . By Corollary 1.10, K η is core equivalentto L (cid:48) ( (cid:96), j ). 38 Classification and construction
In the Introduction we showed how Cramer’s rule coupled with basic combinatorial matrixtheory can be applied to the homeostasis matrix H to determine the different types ofinfinitesimal homeostasis that an input-output network G can support. Specifically the zerosof det( H ), a homogeneous polynomial in the linearized couplings and self-couplings, canbe factored into det( B ) · · · det( B m ). In this paper we show that there are two types offactors that depend on the number of self-couplings: one we call appendage and the otherwe call structural. Each factor corresponds to a type of homeostasis in subnetworks K η for η = 1 , . . . , m that can be read directly from G . Appendage blocks.
Theorem 5.4 shows that an appendage block B η leads to a subnet-work K η that is a path component of the appendage network A G ⊂ G . Moreover, the nodesin K η do not form a cycle with other nodes in the complementary subnetwork C S for every ιo -simple path S . The factors of det( H ) that stem from appendage nodes are det( J A ), thedeterminant of the Jacobian of the appendage path components A . The converse is alsovalid as shown in Theorem 7.1. Theorem 7.1.
Suppose K η is an appendage path component. If K η satisfies the no cyclecondition, then det( J K η ) is an irreducible factor of det( H ) .Proof. Let C S be the complementary subnetwork of an ιo -simple path S . By Definition1.15(c), K η ⊂ C S . Since nodes in K η do not form a cycle with other nodes in C S , by Lemma5.3, J C S has the following block lower triangular form: J C S = ∗ · · · ∗ · · · ∗ · · · ∗ · · · ∗ · · · ∗ J K η · · · ∗ · · · ∗ ∗ ∗ · · · ∗ ... ... ... ... ... ... ... ∗ · · · ∗ ∗ ∗ · · · ∗ (7.1)Hence det( J K η ) is a factor of det( J C S ), and so a factor of det( H ). Since K η is a pathcomponent and hence is path connected, it follows that J K η is irreducible.It follows that we can construct appendage blocks as follows. First we determine thepath components of the appendage subnetwork of G and second we determine which of thesecomponents K η satisfy the cycle condition in Theorem 5.4. Structural blocks.
Next, we form the subnetwork S G that is obtained from G by deletingthe appendage path components identified above. The last result that is needed is:39 heorem 7.2. Let (cid:96) and j be adjacent super-simple nodes in S G , then det( L (cid:48) ( (cid:96), j )) is anirreducible factor of det( H ) .Proof. It follows from Corollary 6.7 that det( L (cid:48) ( (cid:96), j )) is a factor of det( H (cid:48) ) and hence a factorof det( H ) by Lemma 5.5. Theorem 6.11 states that L (cid:48) ( (cid:96), j ) is core equivalent to a unique K η that is irreducible. Hence, det( L (cid:48) ( (cid:96), j )) is an irreducible factor of det( H ).Next, we compute the super-simple nodes in S G in downstream order, namely, ι = ρ > ρ > · · · > ρ q > ρ q +1 = o It follows that the subnetworks L (cid:48) ( ρ i , ρ i +1 ) are core equivalent to the structural networks K η . Let B i be the homeostasis matrix associated with the input-output networks L (cid:48) ( ρ i , ρ i +1 )and det( B i ) is a factor of det( H ). Acknowledgements.
This research was supported in part by the National Science Foun-dation Grant DMS-1440386 to the Mathematical Biosciences Institute, Columbus, Ohio.
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