Algebraic Coarse-Graining of Biochemical Reaction Networks
11 Algebraic Coarse-Graining of BiochemicalReaction Networks
Dimitri Loutchko Abstract
Biological systems exhibit processes on a wide range of time andlength scales. This work demonstrates that models, wherein the interactionbetween system constituents is captured by algebraic operations, inherentlyallow for successive coarse-graining operations through quotients of the alge-bra. Thereby, the class of model is retained and all possible coarse-grainingoperations are encoded in the lattice of congruences of the model. We ana-lyze a class of algebraic models generated by the subsequent and simultaneouscatalytic functions of chemicals within a reaction network. Our ansatz yieldscoarse-graining operations that cover the network with local functional patchesand delete the information about the environment, and complementary oper-ations that resolve only the large-scale functional structure of the network.Finally, we present a geometric interpretation of the algebraic models throughan analogy with classical models on vector fields. We then use the geomet-ric framework to show how a coarse-graining of the algebraic model naturallyleads to a coarse-graining of the state-space. The framework developed hereis aimed at the study of the functional structure of cellular reaction networksspanning a wide range of scales.
Introduction rocesses in biology take place on many different time and length scales. Thisis evident already at the level of single cells. In the temporal domain, chemicalreactions catalyzed by enzymes have characteristic scales that range from mi-croseconds to seconds [1], over hours for genetic regulation [2] up to the orderof days for the completion of the cell cycle [3]. The complex organization of The University of Tokyo, Graduate School of Frontier Sciences, Department of Com-plexity Science and Engineering5-1-5 Kashiwanoha, Kashiwa-shi, Chiba-ken 277-8561E-mail: [email protected] a r X i v : . [ q - b i o . M N ] A ug a living cell, however, gives rise to a situation, where even a single observablecan exhibit fluctuations with a continuous 1 /f -spectrum. Such spectra havebeen measured for glycolytic oscillations in yeast [4], in cardiac cells [5] andin electroencephalograms of human brains [6].The spatial structures found in cells range from the subnanometer scale forsmall metabolites over proteins and complexes on the scale of tens of nanome-ters and larger to cell organelles measuring micrometers. In the last decades,the viewpoint emerged that the cytoplasm is a highly structured functionalunit. Experiments have shown that transient formation of protein complexesoccurs frequently and often entire metabolic pathways [7] and cell signalingcascades [8, ? ] are carried out within such complexes under limited exchangeof matter with the environment [9]. Globally, the cytoplasm mediates stronglynon-local effects of cyclic conformational molecular motions on metabolic dif-fusivity [10, ? ] and exhibits glass-like properties impacting all intracellular pro-cesses involving large components [11], providing hints at the large scale spatialand temporal structure of the cytoplasm.From these experimental results the viewpoint emerges that the scales oc-curring in cellular processes do not possess a discrete spectrum, but that itis dense in both the spatial and temporal domain. In particular, this impliesthat a change of scale of a model via coarse-graining based on scale separa-tion might not be, even in principle, possible for models of complex biologicalsystems. With a point of view moving towards systems biology, aiming at aholistic description of biological systems, this can pose a serious obstacle.In this work, we present an approach that circumvents the difficulty of di-rectly coarse-graining the state space in order to achieve a scale-transformationby coarse-graining the space of functions acting on the state space. The ad-vantage of this approach is that the space of functions is endowed with anatural algebraic structure, which descends to the quotients of the functionalalgebra and thus to the corase-grained models. This means that the possi-ble coarse-graining procedures are encoded in the lattice of congruences ofthe functional algebra and that consecutive coarse-graining procedures usingincreasingly coarse congruences lead to a multiscale description of the system.We demonstrate this idea on a class of algebraic models for biochemicalreaction networks. These models are based on the chemical reaction system(CRS) formalism developed by Hordijk and Steel [12]. The main applicationfor CRS has been the extensive and successful study of self-sustaining reactionnetworks [13,14,15,16]. In [17], it was shown that the CRS formalism hasa natural algebraic structure corresponding to subsequent and simultaneouscatalytic events and the respective algebraic models were constructed.In section 1, the semigroup models and their basic properties are reviewed.Section 2 expands the idea of algebraic coarse-graining sketched above andpresents a class of congruences that correspond to coarse-graining of the small-scale structure of the network and a complementary class that corresponds tocoarse-graining of the environment. Finally, in section 3, which is the core ofthis article, we add another layer to the formalism by establishing a “geo-metric” viewpoint of the semigroup models. This approach is motivated by a correspondence to classical models that employ vector fields. It is shownhow the algebraic models are attached to the state space, which is the powerset of all chemicals of the network under consideration, in a compatible man-ner. Thereby, the dynamics is given by a section compatible with the partialorders on state space and on the functional algebra. We show how the coarse-graining procedure by a congruence on the functional algebra descends to thestate space and thereby leaves all structures and compatibilities intact. Asa demonstration, we discuss the geometry of the congruences introduced insection 2. All references to the Supplementary Information are denoted by acapital S. The mathematical background needed for this work is covered insection S1. The formalism of CRSThis introduction to the chemical reaction system (CRS) formalism follows[12]. A classical chemical reaction network (CRN) is a finite set of chemicals X together with a set of reactions R = { r i } i ∈ I indexed by a finite set I eachequipped with a reaction rate constant. A reaction r ∈ R is usually written as a A + a A + ... + a n A n −→ b B + b B + ... + b m B m , (1.1)where a i , b j ∈ N and A i , B j ∈ X , A i = B j for i = 1 , ..., n and j = 1 , ..., m . Wewill only utilize the sets of substrates and products, which we call the domaindom( r ) = { A , ...A n } and range ran( r ) = { B , ..., B m } of a reaction r given by1.1, because the CRS formalism does not employ detailed kinetic information,but instead emphasizes the catalytic function of the chemicals in X . Definition 1.1. A chemical reaction system (CRS) is a triple ( X, R, C ), where X is a finite discrete set of chemicals, R is a finite set of reactions and C ⊂ X × R is a set of reactions catalyzed by chemicals of X . For any pair ( x, r ) ∈ C ,the reaction r is said to be catalyzed by x .Following [18] a CRS can be graphically represented by a graph with twokinds of vertices and two kinds of directed edges. As an example, consider thegraph in Fig. 1. The solid disks correspond to the chemicals X and the opencircles correspond to the reactions from R . The chemicals participating in areaction are shown by solid arrows. If a reaction is catalyzed by some chemical,this is indicated by a dashed arrow.The catalytic function of chemicals can be equipped with a natural alge-braic structure, namely, the subsequent function and the simultaneous func-tion, as well as combinations thereof.The algebraic structure of a CRSThroughout this section, let ( X, R, C ) be a CRS. The state of the CRS isdefined by the presence or absence of the chemicals, i.e. by giving the subset a bc d e
Fig. 1
Example of a graphical representation of a CRS. The CRS consists of five chemicals X = { a, b, c, d, e } and three reactions a + b → c , c + b → d and c + d → e . The first tworeactions are catalyzed by d and a , respectively, whereas the last reaction is not catalyzed. Y ⊂ X of chemicals that are present. Thus the state space of the CRS is thepower set X = { , } X .A reasonable way to define the function of some given chemical x ∈ X isvia the reactions it catalyzes, i.e. by the way it acts on the state space X . Thisdefinition is motivated by the work of Rhodes [19]. Definition 1.2.
Let (
X, R, C ) be a CRS with state space X = { , } X . The function φ r : X → X of a reaction r ∈ R is defined as φ r ( Y ) = ( ran( r ) if dom( r ) ⊂ Y ∅ elsefor all Y ⊂ X . The sum φ + ψ of two functions φ, ψ : X → X is given by( φ + ψ )( Y ) = φ ( Y ) ∪ ψ ( Y ) (1.2)for all Y ⊂ X . The function φ x : X → X of x ∈ X is defined as the sum overall reactions catalyzed by x φ x = X ( x,r ) ∈ C φ r . Two functions φ x and φ y with x, y ∈ X can be composed via( φ x ◦ φ y )( Y ) := φ x ( φ y ( Y )) for any Y ⊂ X. The composition ◦ is the usual composition of maps and therefore associative.The addition is extended to arbitrary functions via the formula (1.2). It isassociative, commutative and idempotent (cf. S1.9). This leads to the definitionof the functional algebra of a CRS. Definition 1.3.
Let (
X, R, C ) be a CRS. Its functional algebra ( S ( X ) , ◦ , +) isthe smallest subalgebra of the full algebra of functions T ( X ) (definition S1.19)that contains { φ x } x ∈ X and the zero function given by 0( Y ) = ∅ for all Y ⊂ X and is closed under the operations ◦ and + . We denote this algebra by S ( X ) = h φ x i x ∈ X . Analogously, for any subset of chemicals Y ⊂ X , the subalgebra S ( Y ) of S ( X )of functions supported on Y is defined as S ( Y ) = h φ x i x ∈ Y and S ( ∅ ) = { } isthe trivial algebra. Definition 1.4.
When we consider only the multiplicative structure on S ( X )or only the additive structure, the resulting objects ( S ( X ) , ◦ ) and ( S ( X ) , +),are semigroups (cf. S1.20-S1.22).As a subalgebra of T ( X ), S ( X ) is finite . The two operations ◦ and + haveobvious interpretations in terms of the function of enzymes on a CRS: Thesum of two functions φ x + φ y , x, y ∈ X describes the joint or simultaneous function of two enzymes x and y on the network - it captures the reactionscatalyzed by both x and y at the same time. The composition of two functions φ x ◦ φ y , x, y ∈ X describes the subsequent function on the network: first y andthen x act by their respective catalytic function. By definition S ( X ) capturesall possibilities of joint and subsequent functions of elements of the network onthe network itself. The following properties follow directly from the definitions. Lemma 1.5.
Let S ( X ) be the algebra of functions of the CRS ( X, R, C ).(I) There is a natural partial order on S ( X ) given by φ ≤ ψ ⇔ φ ( Y ) ⊂ ψ ( Y ) for all Y ⊂ X. (1.3)(II) Any φ, ψ ∈ S ( X ) satisfy φ ≤ φ + ψ. (1.4)To facilitate the discussion in the following sections, we give an explicitrepresentation of the elements of S ( X ). Lemma 1.6 ([17], lemma 4.1) . Any element φ ∈ S ( X ) can be written as anested sum φ = X y ∈ Y φ y ◦ ( X y ∈ Y y φ y ◦ ( ... ◦ ( X y n ∈ Y y y ...yn − φ y n ) ... )) (1.5)for some n ∈ N , where Y, Y y ..., Y y y ...y n − are multisets (possibly empty)of elements in X and each Y y y ...y j with j < n depends on the elements y ∈ Y, y ∈ Y y , ..., y j ∈ Y y y ...y j − . Remark 1.7.
The previous lemma implies that each element φ ∈ S ( X ) canbe represented as a tree with edges labeled by functions φ y and the verticesrepresenting sums over the underlying edges. The sums are then multipliedwith the function on the edge above the respective vertex. Fig. 2A gives anexample of such a representation.The representation of a function by a tree implies a correspondence toreaction pathways in the CRS, where the leafs of the tree correspond to startingreactions and vertices correspond to joining reaction pathways. As an example,Fig. 2B shows the pathways corresponding to the tree from Fig. 2A. However,the mapping of functions to reaction pathways is not always injective. Forexample, the reaction pathway shown in Fig. 2B corresponds to the function φ represented in Fig. 2A, but it is also the reaction pathway of the function φ + φ g . a b cd e fg A B c bfdg ea
Fig. 2
The tree A shows the function φ = φ a ◦ (( φ d ◦ φ g ) + φ e ) + ( φ b ◦ φ f ) + φ c as anexample of an explicit representation of a general element of S as discussed in the text. B visualizes the reaction pathway within a CRS corresponding to the function represented in A . As the root of the tree A has three branches, the pathway has three components thatare not interconnected. Note that the pathway B does not represent a unique function. Forexample, it is also the pathway corresponding to the function φ + φ g . General approachThe following considerations apply to any algebra in the sense of universalalgebra, but for clarity we restrict ourselves to the semigroup ( S ( X ) , +). Theidea of functional coarse-graining is to determine all partitions ρ of the set S ( X ) such that the algebraic operation descends from S ( X ) to operationsbetween the sets of the partition. This means that the functions of S ( X ) shouldbe grouped into classes that behave “similarly” with respect to the algebraicoperation. The advantage of this procedure is that the reduced, i.e. coarse-grained, space of functions tautologically has the same algebraic operation asthe original model and therefore retains the same biological interpretation.Thus the class of models is not changed and then further coarse-graining canbe iteratively performed in the same manner to obtain a description of thesystem on many scales. The lattice of congruences of the algebra characterizesall possibilities for such coarse-graining procedures and moreover is endowedwith a partial order that precisely determines the possibility of consecutivecoarse-graining procedures.Let us now formulate the above discussion in mathematical terms (cf. sec-tion S2 for details and definitions). Being a partition of S ( X ) means that ρ isan equivalence relation . We write φρψ if and only if φ and ψ are in the sameequivalence class. The set of all equivalence classes is denoted by S ( X ) /ρ andthe equivalence class of φ ∈ S ( X ) is denoted by φρ . For the descent of thealgebraic operation from S ( X ) to S ( X ) /ρ to be well-defined, the relation ρ must be a congruence, i.e. satisfy φρψ and φ ρψ ⇒ ( φ + φ ) ρ ( ψ + ψ ) (2.1)for all φ, φ , ψ, ψ ∈ S ( X ). Then the operation + on S ( X ) /ρ is independent ofthe choice of equivalence class representatives. Due to property 2.1, all properties of the operation + on S ( X ) are in-herited by + on S ( X ) /ρ and thus S ( X ) /ρ becomes a semigroup. It is calledthe quotient of ( S ( X ) , +) by ρ . Analogously, if ρ satisfies φρψ and φ ρψ ⇒ ( φ ◦ φ ) ρ ( ψ ◦ ψ ) for all φ, φ , ψ, ψ ∈ S ( X ), then it is a congruence on ( S ( X ) , ◦ ).If ρ is a congruence on ( S ( X ) , +) and ( S ( X ) , ◦ ), then it is a congruence on( S ( X ) , ◦ , +). We note that the lattice of congruences of ( S ( X ) , ◦ , +) is a sub-lattice of both the lattices of congruences of ( S ( X ) , ◦ ) and ( S ( X ) , +) and thusit can be studied by via the lattices on ( S ( X ) , ◦ ) and ( S ( X ) , +) individually.Congruences on Semigroup modelsThis section focuses on congruences on ( S ( X ) , +). Coarse-graining proceduresfor the functions of ( S ( X ) , +) corresponding to small pathways via remark 1.7as well as large pathways and combinations thereof are presented. Congruenceson ( S ( X ) , ◦ ) are treated in section S3. Finally, the natural inverse to coarse-graining via semigroup extensions is discussed. Congruences on ( S ( X ) , +)The length len ( φ ) of a function φ ∈ S ( X ) captures the size of the pathwaycorresponding to φ via remark 1.7 and is defined as follows. Definition 2.1.
For any φ ∈ S ( X ), let len ( φ ) be the largest integer n suchthat there is a non-zero function ψ ∈ S ( X ) n that satisfies ψ ≤ φ . Hereby, S ( X ) n is the ideal of S ( X ) consisting of all elements of the form a ◦ a ◦ ... ◦ a n for a , ..., a n ∈ S ( X ). The length of the zero function is 0.By this definition len satisfies len ( φ + ψ ) = max { len ( φ ) , len ( ψ ) } (2.2)for any φ, ψ ∈ S ( X ). Here, the inequality len ( φ + ψ ) ≥ max { len ( φ ) , len ( ψ ) } follows from lemma 1.5(II). The opposite inequality follows from the fact thatthe sum of two functions cannot have a longer pathway of subsequent catalyzedreactions than the ones already contained within one of summands.This leads to a definition of some special equivalence relations ρ n on S ( X )for any n ∈ N by stipulating that the functions φ, ψ ∈ S ( X ) are in the sameequivalence class if and only if their lengths do not exceed n , i.e. φρ n ψ ⇔ len ( φ ) ≤ n and len ( ψ ) ≤ n (2.3)in addition to φρ n φ for all φ ∈ S ( X ). Equation 2.2 immediately implies thatthe relations ρ n satisfy the property 2.1 and are thus congruences. Moreover,the ρ n are totally ordered by inclusion as subsets of S ( X ) × S ( X ) ρ ( ρ ( ... ( ρ N and ρ N = S ( X ) × S ( X ) for N = max φ ∈S ( X ) { len ( φ ) } . This gives rise to pro-jections π n,k : S ( X ) /ρ n (cid:16) S ( X ) /ρ n + k for 0 ≤ n ≤ N and 0 < k ≤ N − n . The π n,k are naturally homomorphismswith respect to addition. Moreover, there are inclusions ι n,k : S ( X ) /ρ n + k , −→ S ( X ) /ρ n for 0 ≤ n ≤ N and 0 < k ≤ N − n . A priori , the ι n,k are just maps of sets.In general, the obstruction for being homomorphisms is that there can exist φ, ψ ∈ S ( X ) /ρ n + k such that φ + ψ = 0 in S ( X ) /ρ n + k , but ι n,k ( φ )+ ι n,k ( ψ ) = 0in S ( X ) /ρ n . However, the property 2.2 ensures that this does not occur andthus ι n,k are homomorphisms with respect to addition.A biological interpretation of the quotients S ( X ) /ρ n and the maps π n,k and ι n,k now follows from remark 1.7. The elements of S ( X ) /ρ n corresponding topathways of length less or equal to n are set to 0 and are therefore not resolvedanymore. The non-zero elements of S ( X ) /ρ n capture the global structure ofthe network and contain only pathways that have sufficient length. Therefore,taking the quotient of S ( X ) with respect to ρ n corresponds to extraction of thelarger functional structure and neglecting the functional structure up to a givensize n . For n = 0, the whole functionality is resolved. With increasing n , moreand more functions disappear until all functions are set to 0 for n = N and thequotient S ( X ) /ρ N becomes trivial. Thereby, the projections π n,k correspondto the negligence of functions with length between n and n + k and the ι n,k correspond to the inclusion of functions of length at least n + k into S ( X ) /ρ n .The inverse of coarse-grainingFor 0 ≤ k ≤ N , define the congruences ρ k on S ( X ) by φρ k ψ ⇔ len ( φ ) ≥ k and len ( ψ ) ≥ k. and φρ k φ for all φ ∈ S ( X ). ρ k groups all functions of length at least k into oneequivalence class and fully resolves all smaller functions. This corresponds toa coarse-graining of the environment around patches of functions shorter than k . Again, relation 2.2 ensures that ρ k is a congruence with respect to addition.For k, n such that N ≥ k > n ≥
0, we can define the congruences ρ kn = ρ k ∪ ρ n corresponding to a resolution of functions φ ∈ S ( X ) with n ≤ len ( φ ) < k and to the coarse-graining of all shorter functions and all longer functionsinto single equivalence classes in the quotient S ( X ) /ρ kn . Note that this is aninstance of lattice algebra (S1.11) and thus such construction can be carriedfurther using arbitrary combinations of the lattice operations. The inclusion ι kn : S ( X ) /ρ kn , −→ S ( X ) /ρ n is a semigroup homomorphism and we have a shortexact sequence of semigroups (cf. S1.2)0 → S ( X ) /ρ kn ι kn −→ S ( X ) /ρ n π n,k − n −−−−→ S ( X ) /ρ k → , i.e. the semigroup S ( X ) /ρ n is an extension of S ( X ) /ρ k by S ( X ) /ρ kn . Thisis verbatim the biological interpretation: the functions of length at least k extended by the functions of length between k and n give functions of lengthat least n . While S ( X ) /ρ k encodes the large scale functional structure of thenetwork and S ( X ) /ρ kn a strictly lower scale, S ( X ) /ρ n resolves the functionson both scales. This suggests that within the considered algebraic frameworkthe inverse procedure to coarse-graining is given by extensions of the algebra.Under the given biological interpretation, the study of semigroup extensionsbecomes the study of the possibilities to couple two systems on different scalesin an algebraically consistent way. For the semigroup models, a solid theoreticalbasis in terms of the generalization of the homological characterizations ofgroup extensions to semigroups is already known [20]. The congruences ρ n , ρ k , ρ kn and their combinations through the lattice opera-tions join and meet allow to construct coarse-graining procedures more easilyand flexibly compared to the techniques employed by the classical methods.Moreover, the kinds of coarse-grained structures obtained often go beyondwhat is possible with classical approaches. For example, the congruences ρ k correspond to a coarse-graining of the environment. Classically, one would fixsome subsystem and integrate over the other degrees of freedom, i.e. over theenvironment, to obtain a coarse-grained description. In contrast, the elementsof S ( X ) /ρ k still resolve the full network structure, but a given class of functionsis “integrated out”. Whereas the classical procedure is based on a reduction ofthe system’s state space , the algebraic procedure is a reduction of the system’s functional space . In this section, a geometric picture is developed, wherein thealgebraic models are the functional algebras of the state space X = { , } X .From this geometric point of view, we show that there is a natural way forcoarse-graining of the state-space resulting from coarse-graining of ( S ( X ) , +)by any congruence. Dynamics on a semigroup model
In [21], a discrete dynamics on X is introduced to characterize self-sustainingchemical reaction systems. Thereby, for each set Y ⊂ X , its function Φ Y : X → X is defined as Φ Y = X φ ∈S ( Y ) φ. (3.1)Equivalently, Φ Y is the unique maximal element of S ( Y ). If Y ⊂ Z ⊂ X , thenlemma 1.5(II) implies Φ Y ≤ Φ Z . Definition 3.1.
The discrete dynamics on a CRS (
X, R, C ) with initial con-dition Y ⊂ X is generated recursively by the propagator D : X → X via Y Φ Y ( Y ). Analogously, the dynamics can be parametrized by N as Y n +1 = Φ Y n ( Y n ) for all n ∈ N .Geometry of semigroup modelsClassically, the dynamics of a chemical reaction network is modeled on thestate space R N ≥ , which keeps track of the exact concentrations of the chemi-cals from X , where N = | X | is assumed to be finite. The time evolution of theconcentrations through chemical reactions is governed by a set of differentialequations dx/dt = f class ( x ), which are usually derived from mass action kinet-ics. In the parlance of differential geometry, the whole system is described bythe real manifold R N ≥ with a smooth section f class into its tangent bundle T R N ≥ R N ≥ . π f class The physical system is modeled by an appropriate choice of the initialcondition x ∈ R N ≥ and by integration of the differential equation dx/dt = f class ( x ). This yields a trajectory x ( t ) parametrized by the semigroup ( R ≥ , +).Analogously, one can view the algebra of functions S ( X ) as a structure S above the state space X S X , π f where S is defined in analogy to the tangent bundle as S = ‘ Y ∈ X S ( Y ), suchthat the fiber over Y ⊂ X is S ( Y ). The partial order on X given by inclusionof sets induces a compatible partial order on the fibers via S ( Y ) < S ( Y ) for Y ⊂ Y . Denote by ι Y,Y : S ( Y ) , −→ S ( Y ) the inclusion of subalgebras. Thepartial order and this compatibility is the analogue of matching Euclidean thetopologies of R N ≥ and its tangent bundle. The dynamics on X is determinedby a choice of elements f Y ∈ S ( Y ) for each Y . The smoothness of the section f in the classical case is reflected by the compatibility of the partial order on X with the partial order on the semigroup elements, i.e. ι Y,Y ( f Y ) ≤ f Y (3.2)for Y ⊂ Y . The trajectory for any initial condition Y ⊂ X is parametrizedby N via Y n +1 = f Y n ( Y n ). One natural example of a choice of dynamics is f Y = Φ Y by definition 3.1. This geometric viewpoint is illustrated in the leftpart of Fig. 3. All details of this analogy to classical dynamics governed by avector field are shown in Table S1. Y Y Y XY Y S ( X ) S ( Y ) S ( Y ) S ( Y ) S ( Y ) S ( Y ) X S = lattice of congruences 1 ρ ρ ρ ρ ρ ρ ρ Δ coarse-grainingprocedure bychoice of congruenceand dynamics X/ ρ S/ρ = Y / ρ =Y / ρ =Y / ρ X / ρ Y / ρ =Y / ρ S ( X ) / ρ S ( Y ) / ρ S ( Y ) / ρ Fig. 3
A sample representation of the geometric object
S → X . The partial order on X isvisualized in the bottom part of the graph. Over each Y ∈ X , the algebra of functions S ( Y )is represented by a network. Note, however, that according remark 1.7, the algebra containsseveral network functions. The blue part of the network represents the section f : X → S .In the middle, a representation of the lattice of congruences on S ( X ) is shown. Picking acongruence leads to a coarse-graining of the geometric space, whereby certain functionalalgebras and certain elements of X belong to the same partitions and thus are considered asidentical in the coarse-grained description. Coarse-graining in the geometric contextEach point of the state space X has an algebra of functions S ( Y ), which isa subalgebra of S ( X ), attached to it. A congruence ρ on S ( X ) descends toa congruence on S ( Y ), which is also denoted by ρ . This is compatible withthe inclusion maps ι Y,Y , which descend to ι ρY,Y : S ( Y ) /ρ , → S ( Y ) /ρ for Y ⊂ Y . Thereby, a projection S = ‘ Y ∈ X S ( Y ) −→ ‘ Y ∈ X S ( Y ) /ρ to a spaceover X is induced. This shows the coarse-grained functional structure over X and suggests to group elements of the state space according to their function.Note that the functions in the S ( Y ) /ρ are not well-defined on X . However,it is possible to construct a natural equivalence relation ρ X on X such thatthe functions in S ( Y ) /ρ are well-defined on the quotient X /ρ X . In this regard,define an equivalence relation ρ pre X on X via Y ρ pre X Y ⇔ ι ρX,Y ( S ( Y ) /ρ ) = ι ρX,Y ( S ( Y ) /ρ ) . (3.3)Note that actual equality and not just isomorphism is required. The followingpartition of X is the closest one to X /ρ pre X among those with the property thatthe dynamics f : X → S descends in a well-defined manner. Definition 3.2.
Let ρ X be the finest partition of X such that ρ pre X ≤ ρ X andfor each Y ρ X ∈ X /ρ X , the dynamics f ρY ρ X ( Y ρ X ) := g Y ( Y ) ρ X , (3.4)is independent of the choice of coset representative of Y ρ X and of the choiceof g Y , which is any coset representative of f Y ρ . This yields the desired coarse-grained geometric object S /ρ X /ρ Xπ f ρ (3.5)with fibers ( S /ρ )( Y ρ X ) = ‘ Y ρ X Y S ( Y ) /ρ . The coarse-graining approachpresented above is illustrated in Fig. 3.In section S4, we work out the geometric coarse-graining by the congruences ρ n defined in equation 2.3. Essentially, depending on the size of n and thearchitecture of the network, there are three qualitatively different cases. 1) ρ = ρ pre X : The functions and states set on 0 and ∅ play no role in the larger-scale structure of the network. 2) ρ > ρ pre X : Some of the functions of lengthat most n induce larger functionality. 3) ρ = X × X : The geometric model istrivial and functions of length at most n suffice to produce the whole system.This example demonstrates how congruences as simple as the ρ n highlightfunctional aspects of biochemical reaction networks. For small n , it is to beexpected that the geometric coarse-graining procedure follows case 1) andleads with increasing n via case 2) to case 3). It is certainly interesting tostudy the quantitative changes of this behavior in biological reaction networksand compare the results to corresponding results on random networks. The main goal of this paper is to present a systematic algebraic approachto coarse-grain biological systems via their functionality. The technical back-ground behind this approach, besides possible philosophical or aesthetical con-siderations, is that the function of system components on each other inherentlyshould have an algebraic structure. Then the requirement for a functionallyconsistent corase-graining is equivalent to taking quotients of the algebra offunctions. We have worked out this approach for the functional structure ofchemical reaction systems and have shown that it naturally implies a coarse-graining of the state space using a geometrically minded interpretation of thealgebraic models.The presented formalism is mathematically strict and constructive. In ad-dition, all structures are finite and thus all presented methods can be directlyimplemented as a program and applied to experimental biological data. Incomparison to classical models, the dimension of the state space in the alge-braic models is much smaller.Therefore, applications to systems with enough functionality includingmetabolites, enzymes, DNA, RNA and signaling molecules are conceivable.Such large and functionally rich systems are precisely the target systems forthe presented models. It would be very interesting to determine and ana-lyze the lattice of congruences of a large cellular network. One could verify the commonly used functional partitions and interactions between cellular or-ganelles and other components based on the lattice of congruences. It will beexciting to actually see cell organelles automatically emerge after the algebraiccourse-graining of the state space. With the presented methods, one could alsoattempt to recover the causality implied by the central dogma of biology andstudy possible obstructions to it. More importantly, a wealth of new functionalrelationships - even between large-scale structures - could be found and aid inthe discovery of new pharmaceutical applications. In addition, the comparisonof the statistical properties of the lattice of congruences of a biological sys-tem to those of random networks could provide new insight on the functionalorganization in biology, including the organization on large scales.Finally, we note that the CRS formalism has already been applied to macro-scopic problems such as ecology [22] and economy [23] and thus our coarse-graining approach can also be applied to study modularity in these systems. Acknowledgements
I am deeply indebted to Gerhard Ertl for valuable discussions andhis enormous moral as well as financial support at the FHI in Berlin. I am very thankfulto Hiroshi Kori for stimulating discussions and his generous support at the University ofTokyo. I thank J¨urgen Jost and Peter F. Stadler for discussing this work.
References
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Dimitri Loutchko S1 Basic concepts and definitions
The definitions given here follow [1] and [2].S1.1 Algebraic objectsIn this section, an algebra is defined in the sense of universal algebra and re-lated elementary concepts are presented. Then the notions are specialized byapplication to the main objects encountered in the text, i.e. the algebra of func-tions ( S ( X ) , ◦ , +) and its subalgebras ( S ( Y ) , ◦ , +), the semigroups ( S ( X ) , +)and ( S ( X ) , ◦ ) and their subsemigroups as well as the lattice X = { , } X . Definition S1.1. An algebraic type τ = ( O , α ) is a pair, where O is the set ofoperations and α is a map α : O → N . For each f ∈ O , we say that α ( f ) ∈ N is the arity of the operation f . Definition S1.2. An algebra A = ( A ; F ) of algebraic type τ = ( O , α ) is anon-empty set A and a function F on O such that F ( f ) : A n → A with n = α ( f ) for all f ∈ O . If O = { f , ..., f k } is finite, we write A = ( A ; f , ..., f k )and say that A is of type ( n , ..., n k ), where n i = α ( f i ) is the arity of therespective operation for i = 1 , ..., k . Remark S1.3.
Note that the notion of algebra used here is the notion em-ployed in the area universal algebra and is different from the more commonly The University of Tokyo, Graduate School of Frontier Sciences, Department of Com-plexity Science and Engineering5-1-5 Kashiwanoha, Kashiwa-shi, Chiba-ken 277-8561E-mail: [email protected] a r X i v : . [ q - b i o . M N ] A ug used notion of algebra over a ring in the area of commutative algebra. Ex-amples for the latter are matrix algebras or polynomials over a commutativering. Definition S1.4. A direct sum of finitely many algebras { A i = ( A i ; F i ) } ni =1 of the same type τ is an algebra A = L ni =1 A i = ( L ni =1 A i ; L ni =1 F i ) of type τ , where L ni =1 A i is the direct sum of sets and the operations L ni =1 F i ( f ) aredefined componentwise. Definition S1.5. A semigroup is an algebra ( S ; ◦ ) of type (2) such that theoperation ◦ is associative, i.e. a ◦ ( b ◦ c ) = ( a ◦ b ) ◦ c for all a, b, c ∈ S . Example S1.6.
For a finite set A , the full transformation semigroup ( T ( A ) , ◦ )is the set of all maps { f : A → A } . The semigroup operation is the compositionof maps, i.e. ( f ◦ g )( a ) = f ( g ( a )) for all a ∈ A . Definition S1.7. A semigroup with zero is a semigroup ( S ; ◦ ) with an element0 ∈ S such that a ◦ ◦ a = 0 for all a ∈ S . A semigroup with zero is oftenregarded as an algebra ( S ; ◦ ,
0) of type (2 , Definition S1.8. A commutative semigroup is a semigroup ( S ; ◦ ) such that a ◦ b = b ◦ a for all a, b ∈ S . Definition S1.9. A commutative semigroup of idempotents is a commutativesemigroup ( S ; ◦ ) such that a ◦ a = a for all a ∈ S . Example S1.10. A direct sum of two semigroups ( S ; ◦ ) and ( S ; ◦ ) is thesemigroup ( S ⊕ S ; ◦ ), where ( a , b ) ◦ ( a , b ) = ( a ◦ a , b ◦ b ) for all( a , b ) , ( a , b ) ∈ S ⊕ S . In what follows, we omit the information of theoperation ◦ and write S ⊕ S for the direct sum.Now we give a central definition that will be used further in section S2. Definition S1.11. A lattice is an algebra ( L , ∨ , ∧ ) of type (2 ,
2) such thatthe operations satisfy x ∨ ( y ∨ z ) = ( x ∨ y ) ∨ z ; x ∧ ( y ∧ z ) = ( x ∧ y ) ∧ zx ∨ y = y ∨ x ; x ∧ y = y ∧ xx ∨ x = x ; x ∧ x = xx ∧ ( x ∨ y ) = x ; x ∨ ( x ∧ y ) = x for all x, y ∈ L . The operation ∨ is called join and ∧ is referred to as meet .The following proposition gives an equivalent characterization of a latticeas a partially ordered set. Proposition S1.12 (cf. [1],Prop.1.1.11.) . Let ( L , ∨ , ∧ ) be a lattice. Then x ≤ y iff x = x ∧ y defines a partial order on L such thatinf { x, y } = x ∨ y sup { x, y } = x ∧ y. Conversely, if ( L , ≤ ) is a partially ordered set such that x ∨ y := inf { x, y } x ∧ y := sup { x, y } exist for all x, y ∈ L , then ( L , ∨ , ∧ ) is a lattice. Example S1.13.
For any set X , the power set X = { , } X is a lattice. Thepartial order is given by inclusion of sets and the join and meet are given bythe union and intersection of sets, respectively, i.e. Y ∨ Y = Y ∪ Y ,Y ∧ Y = Y ∩ Y for all Y, Y ⊂ X . Definition S1.14. An algebra homomorphism from an algebra ( A ; F ) to analgebra ( B ; G ) of the same type τ = ( O , α ) is a map φ : A → B such that forall f ∈ O and all a , ..., a k f ∈ A φ ( F ( f )( a , ...a k f )) = G ( f )( φ ( a ) , ...φ ( a k f )) , where k f is the arity of f . Definition S1.15. An algebra isomorphism from an algebra ( A ; F ) to ( B ; G )of the same type τ is a homomorphism φ : A → B that is one-to-one. If for any two algebras ( A ; F ) and ( B ; G ), there exists anisomorphism, we say that the algebras are isomorphic and write A ’ B . Example S1.16. A semigroup homomorphism is an algebra homomorphismfrom ( S ; ◦ ) to ( T ; ◦ ), i.e. it is a map f : S → T such that f ( a ◦ b ) = f ( a ) ◦ f ( b )for all a, b ∈ S Example S1.17. A homomorphism of semigroups with zero from ( S ; ◦ ,
0) to( T ; ◦ ,
0) is a semigroup homomorphism f : S → T such that f (0) = 0. Definition S1.18. A subalgebra of A = ( A ; F A ) is an algebra B = ( B ; F B )of the same type τ = ( O , α ) such that B ⊂ A and F B ( f ) is the restriction of F A ( f ) from A α ( f ) to B α ( f ) for all f ∈ O .More naturally, a subalgebra of A = ( A ; F A ) is an algebra B = ( B ; F B ) of thesame type such that there exists an injective algebra homomorphism ι : B → A . We arrive at the main definition of this section.
Definition S1.19.
The full algebra of functions ( T ( X ) , ◦ , + ,
0) on a finite set X is an algebra of type (2 , , T ( X ) , ◦ ) is the full transformationsemigroup on the power set X = { , } X (cf. example S1.6). The operation ofaddition + is defined as ( f + g )( Y ) = f ( Y ) ∪ g ( Y )for all Y ⊂ X and all f, g ∈ T ( X ). Note that + is commutative and idempotent(cf. definition S1.9). The zero element 0 is the constant map 0( Y ) = for all Y ⊂ X . It is the neutral element with respect to addition, i.e. f +0 = 0+ f = 0for all f ∈ T ( X ) and a left-zero with respect to multiplication, i.e. 0 ◦ f = 0 forall f ∈ T ( X ). Whenever we consider the operation + on ( T ( X ) , +), we do notexplicitly mention the information on the zero element, but implicitly assumeits existence. Remark S1.20.
The algebra ( T ( X ) , ◦ ) obtained from ( T ( X ) , ◦ , + ,
0) by dis-carding the operation + and the information on 0 is the full transformationsemigroup on X from example S1.6. Remark S1.21.
The algebra ( T ( X ) , +) obtained from ( T ( X ) , ◦ , + ,
0) by dis-carding the operation ◦ and not explicitly showing the information on 0 is acommutative semigroup of idempotents with zero. Example S1.22.
The algebra of functions ( S ( X ) , ◦ , +) defined in the maintext is a subalgebra of ( T ( X ) , ◦ , + , multiplicative semigroup model ( S ( X ) , ◦ ) is a subsemigroup of ( T ( X ) , ◦ ) and the additive semigroup model ( S ( X ) , +) = ( S ( X ) , + ,
0) is a subsemigroup of ( T ( X ) , +). Remark S1.23.
For any Y ⊂ X , the algebra ( S ( Y ) , ◦ , +) is defined as asubalgebra of ( T ( X ) , ◦ , + , T ( Y ) , ◦ , + ,
0) with Y = { , } Y . Note that we follow the notation from the main text and omit0 from the notation ( S ( Y ) , ◦ , +). By definition, S ( Y ) is generated by thefunctions supported on Y , i.e. by the set { φ x } x ∈ Y . For any Y ⊂ Y ⊂ X ,the inclusion of sets ι SetY,Y : Y , −→ Y induces an inclusion of functions ι F unctionsY,Y : { φ x } x ∈ Y , −→ { φ x } x ∈ Y , which induces an algebra homomorphism of type (2 , , ι Y,Y : ( S ( Y ) , ◦ , +) = h φ x i x ∈ Y , −→ h φ x i x ∈ Y = ( S ( Y ) , ◦ , +)of subalgebras of ( S ( X ) , ◦ , +). We note that the homomorphisms ι Y,Y arecompatible with the partial order on X , i.e. ι Y,Y = ι Y,Y ◦ ι Y ,Y for any Y ⊂ Y ⊂ Y . Moreover, the homomorphisms ι Y,Y descend to homo-morphisms of semigroups ι ◦ Y,Y : ( S ( Y ) , ◦ ) , −→ ( S ( Y ) , ◦ ) and to homomorphisms of commutative semigroups of idempotents with zero ι + Y,Y : ( S ( Y ) , + , , −→ ( S ( Y ) , + , Y ⊂ Y ⊂ X . As set-maps, ι Y,Y , ι ◦ Y,Y and ι + Y,Y are identical and thereforewe denote all of them by ι Y,Y : S ( Y ) , −→ S ( Y )in agreement with definition S1.14, when the algebraic type is clear from thecontext.S1.2 Semigroup extensionsThe following definitions are used in the last part of section 2 of the main text. Definition S1.24. A short exact sequence of semigroups with zero ( S , ◦ , S , ◦ ,
0) and ( S , ◦ ,
0) is an injective homomorphism ι : S → S and a surjec-tive homomorhism π : S → S such that π ( ι ( a )) = 0 for all a ∈ S . A shortexact sequence is represented as0 → S ι −→ S π −→ S → . Definition S1.25. An extension of a semigroup with zero ( S , ◦ ,
0) by ( S , ◦ , S , ◦ ,
0) that fits into a short exact sequence 0 → S ι −→ S π −→S → Definition S1.26.
A short exact sequence of semigroups with zero0 → S ι −→ S π −→ S → splits , if there is a homomorphism s : S → S of semigroups with zero, suchthat π ◦ s = id | S . Such an s is called a section of π . Remark S1.27.
If the sequence0 → S ι −→ S π −→ S → S is isomorphic to the direct sum S ⊕ S . In this case,we say that the extension S is trivial . S2 Congruences
In this section, congruences on algebras and the resulting quotient algebrasare defined and it is shown explicitly how the operations descend to the quo-tient. For semigroups, Rees quotient semigroups are introduced as a specificexample. Finally, the lattice of congruences is discussed.We begin with preliminary definitions. The two following definitions arewell-known, however, we introduce an equivalence relation on a set A as asubset of A × A in addition to the usual viewpoint as a partition of A . Definition S2.1.
Let A be a set. A relation ρ on A is a subset of A × A ρ ⊂ A × A . If ( a, b ) ∈ ρ , we say that a and b are related via ρ and write aρb. Definition S2.2.
Let A be a set. An equivalence relation ρ on A is a relationthat is reflexive, symmetric and transitive, i.e. aρa for all a ∈ A ,aρb ⇒ bρa for all a, b ∈ A ,aρb ∧ bρc ⇒ aρc for all a, b, c ∈ A . Equivalently, ρ can be identified with a partition of the set A , i.e. A = a i ∈ I A i . Thereby, each a ∈ A is contained in exactly one coset A i , which contains allelements b ∈ A such that aρb and only those. The A i are called equivalenceclasses or cosets . We denote the set of equivalence classes of A as A /ρ . Forany a ∈ A , the unique set A i containing a is called the equivalence class of a . We denote the equivalence class of a by aρ . Moreover, for any equivalenceclass A i , any element a ∈ A i is called coset representative . Definition S2.3.
Let A = ( A ; F ) be an algebra of algebraic type τ = ( O , α ).A congruence ρ on A is an equivalence relation on A that is compatible with thealgebraic operations of A , i.e. for all f ∈ O and all a , ..., a α ( f ) , b , ..., b α ( f ) ∈ A the implication a ρb ∧ a ρb ∧ ... ∧ a α ( f ) ρb α ( f ) ⇒ F ( f )( a , ..., a α ( f ) ) ρF ( f )( b , ..., b α ( f ) ) (S2.1)holds.We have the following main lemma . Lemma S2.4.
Let A = ( A ; F ) be an algebra of algebraic type τ = ( O , α ) andlet ρ be a congruence on A . Then the operations F naturally descend to theset A /ρ as ( F/ρ )( f )( a ρ, ..., a α ( f ) ρ ) := F ( f )( a , ..., a α ( f ) ) ρ, which is independent of choice of coset representatives a , ..., a α ( f ) by relationS2.1 and therefore well-defined. Thus A /ρ = ( A /ρ ; F/ρ ) is an algebra of type τ . A /ρ is called the quotient algebra of A by ρ or just the quotient of A by ρ .In the rest of this section, we discuss a specific example of congruences onsemigroups. This requires a preliminary definition. Definition S2.5.
Let ( S , ◦ ) be a semigroup. An ideal I is a proper subset of S such that SI ∪ IS ⊂ I , where the notation AB = { a ◦ b | a ∈ A , b ∈ B} (S2.2)for A , B ⊂ S is used.
Definition S2.6.
Let ( S ( X ) , ◦ ) be a semigroup and I ⊂ S an ideal. Define acongruence ρ I via ρ I = { ( a, b ) | a, b ∈ I} ∪ { ( c, c ) | c ∈ S} . (S2.3)The Rees factor semigroup is the quotient semigroup S /ρ I . It is denoted by S / I . Remark S2.7.
Note that in the Rees factor semigroup S / I , all elements of I are identified, i.e. they are in the same equivalence class, and all elementsof S ( X ) \ I remain in their own separate equivalence classes. Remark S2.8.
It is important to mention that the language congruencesdoes not lead to any new features for groups (and therefore rings, modulesand algebras in commutative algebra), but is crucial in universal algebra, e.g.already for semigroups. Indeed, for any group G , a congruence ρ is uniquelydetermined by a normal subgroup N < G via aρb ⇔ ab − ∈ N and eachnormal subgroup uniquely corresponds to a congruence as the kernel of theprojection G → G /ρ . Thus, the study of congruences is reduced to the study ofnormal subgroups. However, for semigroups, the congruences are not alwaysdetermined by subsemigroups in the same manner as for groups. For example,congruences on finite semigroups can yield congruence classes of different sizes.This is the case for all Rees quotients of a finite semigroup S by a properideal I ⊂ S . Hereby, all elements of
S \ I form separate classes, whereas allelements of I belong to the same class. In contrast, in quotients of groups G / N all congruence classes are in bijection with the respective normal subgroup N and thus necessarily have the same size. For the remainder of this section, we fix an algebra A = ( A ; F ). Let Con( A )be the set of all congruences on A . Each congruence ρ is a subset of A × A and thus Con( A ) is partially ordered by inclusion of sets, i.e. ρ ≤ ρ ⇔ ρ ⊂ ρ for any ρ, ρ ∈ Con( A ). As the intersection of congruences is still a congruence,(Con( A ) , ≤ ) admits arbitrary infima. Because a supremum of a subset is justthe infimum of the set of upper bounds, (Con( A ) , ≤ ) admits arbitrary supremaand is therefore a lattice by proposition S1.12.Let ρ, ρ ∈ Con( A ) be two arbitrary congruences. If ρ < ρ , we say that ρ is finer than ρ and, vice versa, that ρ is coarser than ρ . The lattice ofcongruences has a maximal element = A × A and a minimal element ∆ = { ( a, a ) | a ∈ A} ⊂ A × A .We have the following lemma. Lemma S2.9.
Let ρ be a congruence of A . There is a one-to-one correspon-dence between the congruences ρ of A coarse than ρ and the congruences of A /ρ : Con( A /ρ ) ←−−→ { ρ ∈ Con( A ) such that ρ ≤ ρ } . Moreover, let ρ ∈ Con( A ) be such that ρ ≤ ρ and let ρ be the correspondingcongruence in Con( A /ρ ). Then there is a natural algebra isomorphism A /ρ ’ ( A /ρ ) /ρ . Remark S2.10.
This lemma is central in the coarse-graining procedure viacongruences as it ensures that the final result of subsequent coarse-grainingprocedures by increasingly coarser congruences ρ ≤ ρ ≤ ... ≤ ρ n is independent of the sequence { ρ i } ni =1 , but only depends on the final con-gruence ρ n . Moreover, with the notations as in the lemma, it ensures thatthe lattice Con( A /ρ ) includes all possible coarse-graining procedures that aregiven by the lattice Con( A ) after fixing the congruence ρ . That means thatthe sequence ρ ≤ ρ ≤ ... ≤ ρ n can be either selected in Con( A ) at once orconstructed step by step by iteratively choosing the congruence in Con( A /ρ i )corresponding to ρ i +1 after coarse-graining by ρ i . S3 Congruences on ( S ( X ) , ◦ ) As a supplement to the main text, where we put the focus on congruences on( S ( X ) , +), we discuss a class of congruences on ( S ( X ) , ◦ ) that are similar inspirit to the congruences ρ k from the main text. Here, we present these congru-ences to illustrate the flexibility of our approach to coarse-graining. We write S ( X ) for ( S ( X ) , ◦ ) throughout this section as the operation is understood tobe ◦ .Consider the chain of ideals S ( X ) ) S ( X ) ) ... ) S ( X ) N = S ( X ) N +1 , where the notation S2.2 is used. The sequence stabilizes for some N ∈ N dueto the finiteness of S ( X ). The powers S ( X ) n are proper ideals of S ( X ) for2 ≤ n ≤ N and give rise to congruences θ S ( X ) n via the expression S2.3. Fornotational convenience, we will write θ n := θ S ( X ) n . The quotient semigroups S ( X ) /θ n can be interpreted via the complexity of function defined as follows. Definition S3.1.
Let φ be some function in the semigroup model S ( X ). φ has complexity n if there exists some n ∈ N with 1 ≤ n ≤ N such that φ ∈ S ( X ) n \ S ( X ) n +1 . Constant functions (including 0) have complexity ∞ . The complexity of φ isdenoted by comp ( φ ).The complexity comp ( φ ) of a function φ determines whether the functioncan be decomposed into a product of at most comp ( φ ) functions. For example,a non-constant function φ x of a chemical x ∈ X has complexity 1, becauseit cannot be further decomposed. By remark 7 from the main text, functionscorrespond to reaction pathways within the CRS. Intuitively, comp ( φ ) givesthe length of the shortest pathway described by φ . By definition, any twofunctions φ, ψ ∈ S ( X ) satisfy the inequality comp ( φ ) + comp ( ψ ) ≤ comp ( φ ◦ ψ ) . (S3.1) Example S3.2.
The CRS shown in figure S1 demonstrates that the inequalityS3.1 can be strict. The functions φ = φ x ◦ φ x + φ y and ψ = φ x + φ y ◦ φ y have complexity 1. Their composition can be written as ( φ x + φ y ) ◦ ( φ x + φ y ) ◦ ( φ x + φ y ) and thus has complexity 3. x x a a x a y y b b y b Fig. S1
The functions φ = ( φ x ◦ φ x ) + φ y and ψ = φ x + ( φ y ◦ φ y ) have complexity1, but their composition has complexity 3.0 The quotient semigroups S ( X ) /θ n are the semigroups of functions of com-plexity at most n , i.e. the functions with complexity lower than n are all inseparate congruence classes and the functions with complexity greater or equalto n are in the congruence class of 0.The composition of two functions φ, ψ ∈ S ( X ) /θ n with comp ( φ ) , comp ( ψ ) Illustration of coarse-graining of the environment via the congruence θ . The figureshows three functions φ green , φ blue , φ red colored in green, blue and red via the representationof elements in S ( X ) as pathways in the CRS. The circles indicate the local patches ofcomplexity at most 2. Each of the functions has a local structure of complexity 2 lyingin the respective circles. The functions φ green , φ blue , φ red are nonzero in S ( X ) /θ . Thecomposition φ green ◦ φ blue gives the function in the blue patch. It has complexity ≤ φ blue ◦ φ red has complexity 4 and equals zero in S ( X ) /θ . which results from the coarse-graining procedure in section 3 of the main textby the congruences ρ n (defined in section 2 of the main text). Let the dynamics f : X → S be given by f Y = Φ Y and ρ n be the congruence given by φρ n ψ ⇔ len ( φ ) ≤ n and len ( ψ ) ≤ n. Write ρ := ρ n and denote the respective equivalence relation resulting from ρ n via equation [11] in the main text by ρ pre X and the equivalence relationon X resulting from definition 10 in the main text by ρ X . Moreover, define len ( S ( Y )) := len ( Φ Y ) and len ( Y ) := len ( Φ Y ) for any Y ⊂ X .Recall that ρ n identifies all functions of length less or equal to n with zero.Therefore, all Y ⊂ X with len ( Y ) ≤ n are equivalent to the empty set by therelation ρ pre X . This means that a certain lower part of the lattice X is reducedto a single element ∅ ρ pre X ∈ X /ρ pre X with only the zero function in its algebra.These are all the subsets of X that support only functions of low length. Allother elements of X are in separate congruence classes. Moreover, all algebras S ( Y ) with len ( Y ) > n only have functions longer than n , which are each theirown equivalence class, and the zero function. To construct the relation ρ X and the dynamics f ρ , it is necessary to discuss3 different cases: If for all Y, Y ∈ ∅ ρ pre X and any φ ∈ S ( Y ), we have φ ( Y ) ∈ ∅ ρ pre X , thenby definition 10 from the main text, f ρ is well-defined as f ρ ( ∅ ρ pre X ) = ∅ ρ pre X .For any set Z ∈ X \ ∅ ρ pre X X , there are no changes from the original dynamicsas both Z and Φ Z are in separate equivalence classes and f ρ ( Z ) = Φ Z ( Z ) iswell-defined. In this case, ρ X = ρ X and the coarse-graining of functions of lowlength leads to the contraction of all Y ⊂ X supporting only such functionsand retains all other sets and functions in a manner consistent with the origi-nal dynamics. If there are Y, Y ∈ ∅ ρ pre X such that Φ Y ( Y ) ∈ X \ ∅ ρ pre X , then by defini-tion 10 from the main text Φ Y ( Y ) must be in ∅ ρ X . Iteratively adding all thesets Φ Y ( Y ) for Y, Y ∈ ∅ ρ X to ∅ ρ X until ∅ ρ X is stable under this operationimplies that f ρ ( ∅ ρ X ) = ∅ ρ X is well-defined. As in case 1), all Z ∈ X \ ∅ ρ X andthe corresponding Φ Z are in separate equivalence classes and f ρ ( Z ) = Φ Z ( Z )is well-defined. This means that ∅ ρ X contains all the sets that support onlyshort functions and all sets that can be produced by short functions and thefunctions supported on the produced sets. If, similar to case 2), ∅ ρ X is the whole state space X , then successivecombinations of functionality of length n are enough to generate the wholenetwork. The coarse-grained geometrical model is trivial in this case.This example demonstrates how congruences as simple as the ρ n highlightfunctional aspects of biochemical reaction networks. For small n , it is to beexpected that the geometrical coarse-graining procedure follows case 1) andleads with increasing n via case 2) to case 3). For real biological systems it isalready interesting to study quantitative changes of this behavior in biologicalreaction networks and compare them to random networks. We note that thecongruences ρ n are rather coarse that in many cases leads eventually to thecomplete contraction of the phase space.. However, it is as well possible toconstruct congruences that are finer than ρ n and pay attention to functionalmodularity of the network. In such cases, the procedure yields functional par-titions highlighting the interplay of the respective modules. This will be thetopic of forthcoming work. S5 Analogy between classical and algebraic models Table S1 Analogy between classical models describing the dynamics of a chemical reactionnetwork by a set of ordinary differential equations and the algebraic models. Note that f Y denotes the section f at Y , which is a function X → X and f Y ( Y ) is its value at Y . We alsouse the notations N = | X | and X = { , } X .Classical AlgebraicState space R N ≥ X “Geometry“ of statespace Euclidean topology Partial order by inclusionSpace of functions T R N ≥ = ‘ x ∈ R N ≥ T x R N ≥ S = ‘ Y ∈ X S ( Y )Attachment of func-tions to state space T R N ≥ π −→ R N ≥ S π −→ X Topology of attach-ment Natural topology Compatibility of partial or-dersDynamics Smooth section R N ≥ f class −−−−→ T R N ≥ Section X f −→ S compatiblewith partial orderTrajectory Integration of dx/dt = f class ( x ) with initial condi-tion x ∈ R N ≥ Iteration of Y f Y ( Y )with initial condition Y ⊂ X Parametrization of atrajectory ( R ≥ , +) ( N , +)4 References 1. J. Almeida, Finite semigroups and universal algebra , vol. 3 (World Scientific, 1995)2. J.M. Howie,