GENERALIZED TOPOLOGIES:HYPERGRAPHS, CHEMICAL REACTIONS, AND

BIOLOGICAL EVOLUTION

Christoph Flamm1, Barbel M. R. Stadler2, and Peter F. Stadler1−6

1Inst. f. Theoretical Chemistry, University of Vienna, Wahringerstraße 17,A-1090 Vienna, Austria;

2Max Planck Institute for Mathematics in the Sciences, Inselstraße 22,D-04103 Leipzig, Germany;

3Bioinformatics Group, Department of Computer Science, andInterdisciplinary Center of Bioinformatics, Univ. Leipzig, Hartelstraße

16-18, D-04107 Leipzig, Germany;4RNomics Group, Fraunhofer IZI, Perlickstraße 1, D-04103 Leipzig,

Germany;5Center for non-coding RNA in Technology and Health, University ofCopenhagen, Grønnegardsvej 3, DK-1870 Frederiksberg, Denmark;

6The Santa Fe Institute, 1399 Hyde Park Rd., Santa Fe, New Mexico, USA

ABSTRACT: In the analysis of complex networks, the description of evolutionary pro-

cesses, or investigations into dynamics on fitness or energylandscapes notions such as

similarity, neighborhood, connectedness, or continuity of change appear in a natural way.

These concepts are of an inherently topological nature. Nevertheless, the connection to

the mathematical discipline of point set topology is rarelymade in the literature, presum-

ably because in most applications there is no natural objectcorresponding to an open or

closed set. The link to textbook topology thus cannot made ina straightforward manner.

Many of the deep results of point set topology still remain valid, however, when open

sets are abandoned an generalizations of the closure operator are used a the foundation of

the mathematical theory. Here we survey some applications of such generalized point set

topologies to chemistry and biology, providing an overviewof the underlying mathemati-

cal structures.

1

INTRODUCTION

Investigations into general principles underlying chemical reaction networks,molecular phylogenetics, evolutionary changes, or the folding of biopoly-mers naturally encounter notions such assimilarity, proximity, connected-ness, or continuity of change. To a mathematician, all of these conceptsare intrinsically topological in nature. On the other hand,all these systemsare very far away from the “continuum” picture so familiar inphysics andchemistry that familiar constructions such asε-balls seem to be little use.Even worse,thestarting point of textbook topology, theopen set, does nothave a natural counterpart in the paradigmatic examples that we will con-sider throughout this contribution.

EduardCech’s treatise of Topological Spaces [9], however, showed thatthe classical theory of Point Set Topology can be constructed in a differentway, starting from Kuratowski’s axioms of closure functions or an equiv-alent notion of neighborhoods. As we shall see below, these can be seenas formalizations of a vague notion of accessibility that lies at the heart ofmany of the questions that we are interested in. Before proceeding to themathematical framework, we briefly introduce a few of our model systems.We note, finally, that generalized topological spaces have applications invarious applied domains of computer science, including digital image pro-cessing, information representation, the semantics of modal logic, or hand-writing recognition [21, 66, 62, 31, 59, 56, 22, 60].

Chemical Reaction Networks. We start from a fixed setX of moleculartypes or chemical species. A chemical reactionρ is a transformation ruleof the form

∑x∈X

s−ρ,xx→ ∑x∈X

s+ρ,xx (1)

whereE−ρ := x∈ X|a−ρ,x > 0 are the educts andE+

ρ := x∈ X|a+ρ,x > 0 are

the products of reactionρ. Thus we can interpret reactionρ as a hyperedgeEρ = (E−

ρ ,E+ρ ) in a directed hypergraphH. It may be convenient, even, to

interpretE+ andE− as multi-sets to incorporate the stoichiometric coeffi-cientss±ρx as multiplicities [82]. The coefficientss±ρx are connected to thestoichiometric matrixSvia Sρ,x = s+ρ,x−s−ρ,x. Figure 1 shows a graphicaldescription of small chemical reaction network.

2

EB

B

E EAB C

A

EA

∅

∅

∅

ρ5 ρ7

ρ8

ρ4 ρ6

Figure 1: Hypergraph representation of the reaction network of elemen-

tary reaction steps for the overall enzyme catalyzed reaction A+ BE−→ C

with unordered substrate binding. Reaction nodes are drawn as squares andspecies nodes as gray circles. (Note for clarity the in- and outflow reactionnodes (ρ1−ρ3) have been omitted.)

Recently, the Network of Organic Chemistry (NOC) [23, 6] has beenconstructed from all organic reactions reported in the chemical literaturesince 1779. Despite the apparent complexity of this network, which cap-tures all the knowledge on synthetic organic chemistry, NOCshows an un-expected but well-defined topological structure [37]. In particular, NOCpossesses a densely wired core region of about 300 synthetically importantbuilding blocks and industrial compounds which are mutually connected byshort synthetic pathways. The core region is embedded in a large and sparseperiphery constituting approximately 78% of chemistry. Compounds in theperiphery can be reached on average in 3-7 synthetic steps from the coreregion. Core region and periphery are surrounded by small isolated islandsconstituting about 18% of NOC. Much smaller chemical reaction networksthat represent the metabolic pathways of several organism are compiled indedicated databases, see e.g. [52].

3

In the analysis of such chemical networks, so-calledpathways, i.e., se-quences of reactions/hyperedges that connect chemical input with outputmolecules, and catalytic cycles are of particular interest. Chemical orga-nizations, that is, closed and self-maintaining subsets ofH [27, 18], fur-thermore, are closely related to the limit set of the corresponding reactionkinetics [61]. The most useful of these structural featuresare related to al-gebraic invariants that can be expressed in terms ofS, see [4] for a recentreview.

The topological description of chemical networks centers describingwhat can be produced instantaneously from setA⊆X of molecular species,i.e.,

p(A) =⋃

ρ :E−ρ

x∈ X|sx,ρ > 0

(2)

A speciesx is maintainablein A if no reactionρ with E−ρ ⊆ A inevitably

leads to the depletion ofx. The set of maintainable species inA will bedenoted bym(A). A set A is closed if p(A) = A and self-maintaining ifm(A) = A. So-calledflow systemsmake the additional assumption that ev-ery speciesx can flow out of the system. Bothp andm are set-valued setfunctions, i.e., they are of the formp : P(X)→P(X), whereP(X) denotesthe power-set ofX. One can show that in flow systems bothp andm areisotonic (see below), and hence impose a generalized topological structureonX [5].

Evolutionary Computation and Genetic Evolution. A (combinatorial)optimization problem is usually specified in terms of a setX of configura-tions and a cost functionf : X → R, whereR is an ordered set, or, in thecase of multi-objective optimization [17], a partially ordered set [74]. Alarge class of heuristic algorithms, including Simulated Annealing, GeneticAlgorithms, Evolutionary Strategies, or Genetic Programming, attempt tofind optimal solutions by moving through the setX and evaluating the costfunction at different pointsx ∈ X. This search procedure imposes an im-plicit mathematical structure on the setX that determines how points or,more generally, subsets are mutually accessible. In a more biologicallyinspired setting, thissearch spaceis uniquely determined by the geneticoperators, i.e., by mutation, recombination, duplication, deletion, or rear-rangement of gene order.

4

00001111111100001111111110010000101100000000011100000011000000011000000000000000

A B CD A D B Cmutation mutation +

1−point crossover

cost

fu

nct

ion

Figure 2: Barrier tree of a simple landscape with mutation (left) and muta-tion plus 1-point-crossover. Figure adapted from [26].

A natural way of abstracting the action of these operators isto de-termine for each “population”A ⊆ X, the setc(A) of configurations thatcan be reached fromA by applying a single on of these operator. Thisis most easily visualized for mutations: Each parentx may give rise to asetc(x) of possible offsprings (mutants). In this case,(X,c) defines a(possibly directed) graph. The situation becomes more complicated, how-ever, when recombination (crossover) is considered [33]. The analogue ofthe adjacency relation of the graph is the recombination setR(x,y), whichis defined as the set of all (possible) recombinants of two parentsx andy. Recombination sets are usually required to satisfy (1)x,y ∈ R(x,y),and (2)R(x,y) = R(y,x), Often (3)R(x,x) = x is assumed, which is,however, not satisfied by models of unequal crossover [65, 72]. FunctionsR : X×X →P(X) satisfying these three axioms were also considered un-der different names, e.g.transit functions[10] and asP-structures[77, 75].We note, furthermore, that recombination can be seen as a ternary relation(x,v,y)∈ R ⇐⇒ v∈R(x,y) closely connected with betweenness relations[1].

Similar to the chemical networks, we may base a formal treatment oftopological structures on a set-values set-function that encodes reachability

5

at the level of sets by setting

c(A) =⋃

x,y∈A

R(x,y) (3)

The topological structure defined byc also brings with it a concept ofconnectedness [81, 43], and hence allows a construction of level sets or“basins” for the cost functionf : X → R as connected components (w.r.t.c) of the setsXh = x| f (x) ≤ h. The basin have a hierarchical structure:if A andB are connected components ofXh′ andXh′′, h′ < h′′, resp., theneitherA⊆B or A∩B= /0. This gives rise to a tree structure representing thelandscape [53, 32, 19, 24, 25, 39, 79], called the barrier tree. Local minimaof the cost function correspond to the leaves of the tree, interior nodes aresaddle points that define the fitness barriers between local minima. Thesenotion of a local minimum is clearly a topological concepts,as it can bedefined in terms of the behavior of the cost function in the neighborhoodaround a point:x is a local minimum if there is a neighborhoodN of x suchthat f (y) ≥ f (x) for all x∈ N. The concept of a saddle point is much lessclear. A number of similar but not-equivalent constructions is discussed indetail in [25]. A connection to combinatorial vector fields in drawn in [71].

The Genotype-Phenotype Map. Two distinct and largely independentprocesses drive biological evolution and lie at the heart ofCharles Darwin’stheory: the generation of variation and selection of the variants accordingto their fitness. With the advent of molecular genetics it hasbecome clearthat variation is produced (primarily) at the molecular level by mutation, re-combination, and other rearrangements of the genomic sequence informa-tion. Selection, in contrast, acts on the macroscopic living organism. Thegenotype-phenotype map, which relates genetic information to organismalappearance, properties, and behavior, thus plays a centralrole in moderntheories of evolution [76].

RNA secondary structures have played a major role for the understand-ing of genotype-phenotype maps in general since they can be readily ex-plored computationally [64]. In this setting, the primary sequence of theRNA is the genotype, while its secondary structure takes on the role ofphenotype. At the phenotypic level, evolutionary processes are governed bythe accessibility of phenotypic variants. The biophysicalproperties of RNAmolecules allow at least a qualitative understanding of thelikely structural

6

..........(((...))). .(((...)))

((.....)).

.((...)).. ..((...)).

.((.....))

...((...)) ((...))...

((......)) (((....)))

Figure 3: Fontana-Schuster topology of RNA phenotype space.L.h.s.: Fre-quent transitions between RNA secondary structures in response to pointmutations are the opening or closing of singe base pairs and the openingof constrained stems. The latter transition is not reversible since must se-quences are not not pre-disposed to closing a constrained stem in responseto a point mutation. R.h.s.: Space of secondary structures from GC se-quences of length 10. Arrows indicate accessibility definedas frequenttransitions in the sense of the two rules on the r.h.s. Figureadapted from[12, 68].

effects of simple point mutations and thus the formulation of phenotypicrules of accessibility [28, 29], see Figure 3.

An evolutionary trajectory can be regarded as a functionf from the timeaxis into phenotype space, wheref (t) represents e.g. the dominating phe-notype in a population at timet. Computer simulations of the RNA modelreveal a pattern of periods of stasis with intermittent bursts of adaptive evo-lution. With few exception, consecutive phenotypes arise from each otherthrough one of the easily accessible structural changes of Fig. 3. Thesetransition are indeed continuous in the usual topological sense [13, 73].

A phenotype is usually described by a set of “characters”, that is, prop-erties that can be used to differentiate between different types of organisms.The particular state of a given character (e.g. the presenceof 5 fingers) isinterpreted as an evolutionary adaptation caused by natural selection. Thisexplanation requires the assumption that the character state can be producedby mutation without significantly affecting the functionality and/or struc-ture of the rest of the body, i.e., “quasi-independence” [57]. As argued in[80], this is a statement about the structure of accessible sets in phenotypespace, namely the requirement that the phenotype space can be represented

7

Figure 4: Phenotype space as a product of two characters (“body shape”and “faces”). For illustrative purpose the Cartesian graph product ratherthan the strong graph product, which corresponds to standard product oftopological spaces, is shown. The latter is obtained by adding the two di-agonal edges to each quadrangle. Figure adapted from [70].

time

Figure 5: Sketch ofthe DAG G(V,E).Fat dots indicate thevertices in the extantsample X ⊂ V. Apossible clusteringthat conforms to thephylogenetic relation-ships is indicated. Fordetails we refer to[20].

as a product of generalized topological space corresponding to the individ-ual characters, Fig. 4. Since characters are meaningfully defined only in alocal or regional subset of the entire phenotype space, the product structurealso can only be a local approximation. These ideas have leadto the devel-opment of a theory of “Approximative Graph Products” and research intoalgorithms for local approximate factorization of graphs [47, 48].

8

The “DAG of Everything”. Instead of looking at the potential of evolu-tionary processes it is also of interest to consider only theactual historyof life. This idea was recently explored in [20] using a graphG(V,A)whose vertex setV consists of all individuals that ever lived. A directededgex→ y indicates thatx contributed genetic information toy. Pedigreegraphs are special cases of this construction, where the in-degree is lim-ited to 2, namely and arc from the father and an arc from the mother, seee.g. [78, 63]. This model also easily incorporates horizontal gene transferor the formation of hybrid species, and hence forms also a basis to studyarbitrary complex phylogenetic networks [54]. Given a subset of extantobservable individualsX ⊂ V several notions of connectedness of subsetsof X can be defined. The main result of [20] is that these lead to naturalcollections of clusters that can be interpreted e.g. as phylogenies, Fig. 5.The DAGG(V,A) might also be an attractive starting point to study moregeneral forms of phylogenetic networks.

GENERALIZED TOPOLOGY

Abstract closure functions

Instead of open sets, most approaches to generalized topology start froman abstractclosure function c: P(X)→P(X) that encapsulates a notion ofreachability or accessibility from a given subsetA. The natural conjugatei : P(X) → P(X) defined byi(A) = X \c(X \A) identifies the interior ofAas the part ofA that is not accessible from the complement, i.e., the outside,of A. Closure and, equivalently, interior give rise to a concept of neigh-borhood of points:N is a neighborhood ofx, if its interior containsx. Thecollection of neighborhoods can be viewed as a functionN : x→P(P(X))so thatN ∈ N (x) iff x∈ i(N), i.e., iff x /∈ c(X \N). Closure, interior, andneighborhood are equivalent in specifying the generalizedtopology [15].For later reference we note that the notion of neighborhoodsnaturally ex-tends to sets:N ∈N (A) iff A⊆ i(N), i.e.,N is a neighborhood of allx∈ A.

The notion of continuity lies at the heart of topological theory. In themost abstract setting it comes in two flavors. Let(X,cX) and(Y,cY) by twosets, each endowed with its closure function A functionf : X →Y is

9

closure preserving if for all A∈ P(X), f (cX(A)) ⊆ cY( f (A)) holds;continuous if for all B∈ P(Y), cX( f−1(B)) ⊆ f−1(cY(B)) holds.

One says thatf : X →Y iscontinuous in xif B∈N ( f (x)) implies f−1(B)∈N (x). It can be shown thatf : X →Y is continuous if and only if it is con-tinuous in eachx∈ X [35, Thm.3.1.].

Obviously, the identityı : (X,c)→ (X,c) : x 7→ x is both closure-preservingand continuous. Furthermore, the concatenationh = g( f ) of the closure-preserving (continuous) functionsf : X →Y andg :Y → Z is again closure-preserving (continuous).

Note that at this point we have made no assumption at all on theprop-erties ofc. Almost all approaches to extend the framework of topologyassume at least that the closure functions are isotonic [41,42, 15, 7, 34].The importance of isotony is emphasized by several equivalent conditions[42, Lem.10] listed in Tab. 1 below. A (not necessarily non-empty) collec-tion F ⊆ P(X) is astackif F ∈ F andF ⊆ G impliesG∈ F. The closurefunction c is isotonic if and only if the neighborhood systemN (x) is astack for allx ∈ X. In isotone spaces, continuity and closure preservationare equivalent.

Kuratowski’s axioms

Kuratowski’s axioms for the closure function of a topological space [55]may be seen as specializations of the very general closure functions that wehave considered so far. It is interesting to note that each ofthem can beformulated equivalently for closure, interior, and neighborhoods. Differentcombinations of these axioms, summarized in Table 2 define generalizedtopological structures that have been studied to various degrees in the liter-ature.

Connectedness and Separation

Topological connectedness is closely related to separation. The basic idea isto investigate under which conditions closure or neighborhoods of distinctpoints or sets do or do not intersect. We say thatA andB areseparatedifthey have disjoint neighborhoods, i.e.,N (A)∩N (B) = /0. Two sets aresemi-separatedif there are neighborhoodsN′ ∈ N (A) and N′′ ∈ N (B)such thatA∩N′′ = N′∩B= /0. Consider a continuous functionυ : (X,c)→

10

Table 1: Kuratowski’s axioms.The properties below are meant to hold for allA,B∈ P(X) and allx∈ X,respectively.

closure interior neighborhood

K0’ ∃A : x /∈ c(A) ∃A : x∈ i(A) N (x) 6= ∅

K0 c( /0) = /0 i(X) = X X ∈ N (x)K1 A⊆ B =⇒ c(A) ⊆ c(B) A⊆ B =⇒ i(A) ⊆ i(B) N ∈ N (x) andN⊆N′

isotonic c(A∩B) ⊆ c(A)∩c(B) i(A)∪ i(B) ⊆ i(A∪B) =⇒monotone c(A)∪c(B) ⊆ c(A∪B) i(A∩B) ⊆ i(A)∩ i(B) N′ ∈ N (x)KA c(X) = X i( /0) = /0 /0 /∈ N (x)KB A∪B = X =⇒ A∩B = /0 =⇒ N′,N′′ ∈ N (x) =⇒

c(A)∪c(B) = X i(A)∩ i(B) = /0 N′ ∩N′′ 6= /0K2 A⊆ c(A) i(A) ⊆ A N ∈ N (x) =⇒ x∈ NexpansiveK3 c(A∪B) ⊆ c(A)∪c(B) i(A)∩ i(B) ⊆ i(A∪B) N′,N′′ ∈ N (x) =⇒sub-linear N′ ∩N′′ ∈ N (x)K4 c(c(A)) = c(A) i(i(A)) = i(A) N ∈ N (x) ⇐⇒idempotent i(N) ∈ N (x)K5 N (x) = /0 or∃N(x) :additive

⋃

i∈I

c(Ai) = c(⋃

i∈I

Ai)⋂

i∈I

i(Ai) = i(⋂

i∈I

Ai) N ∈ N (x)

⇐⇒ N(x) ⊆ N

[0,1], where[0,1] denotes the unit interval endowed with the usual topologyof the real numbers. It is called anUrysohn functionseparatingA andBif υ(A) ⊆ 0 and υ(B) ⊆ 1. If such a function exists,A and B arecalled Urysohn-separated.B is completely withinA, B ⋐ A, if B andX \Aare Urysohn-separated. Urysohn-separated implies separated implies semi-separated.

A large number of subtly differentseparation axiomshave been con-sidered in the literature, of which here we just list a few to give the flavor.In the following, we consider conditions for all distinct points x, y and alldisjoint non-empty subsetsA, B in (X,c).

(th0) there isz∈ X andN ∈ N (z) such that for allN′ ⊂ N, N′ ∈ N (z)holdx∈ N′ andy /∈ N′ or vice versa.

(th1) N (x) = N (y) impliesx = y.

(T0) there isN ∈ N (x) such thaty /∈ N or there isN′ ∈ N (y) such that

11

Table 2: Axioms for various types of closure space.Defining axioms are marked by•, further properties that are implied aremarked by.

Axiom c(/0)

=/0

A⊆

B=⇒

c(A)⊆

c(B)

isot

onic

A⊆

c(A)

enla

rgin

g

c(A∪

B)⊆

c(A)∪

c(B)

sub-

linea

r

c(c(

A))

=c(

A)

idem

pote

nt

c(⋃

iAi)

=⋃

ic(A

i)ad

ditiv

e

Ref.(K0) (K1) (K2) (K3) (K4) (K5)

Extended Topology • • [42]Brissaud • • [7]Neighborhood space • • • [44]Closure space (•) • • • [67]Smyth space • • • [66]Binary relation • • [58, 8]Pretopology • • • • [9]Topology • • • • •Alexandroff space • • •Alexandroff topology • • • • [2]

x /∈ N′.

(T0’) y /∈ c(x) or x /∈ c(y).

(T1) there isN ∈ N (x) andN′ ∈ N (y) such thatx /∈ N′ andy /∈ N.

(T1’) c(x) ⊆ x.

(T2) there isN ∈ N (x) andN′ ∈ N (y) such thatN∩N′ = /0.

(T212) there isN ∈ N (x) andN′ ∈ N (y) such thatc(N)∩c(N′) = /0.

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(T2U) Any two distinctx andy are Urysohn-separated.

(R) If x /∈ c(A) then there are disjoint neighborhoods ofx andA.

(R’) For every neighborhoodN of x there is also a neighborhoodU ∈N (x) such thatc(U) ⊆ N.

(CR) For every neighborhoodN of x there is also a neighborhoodU ∈N (x) such thatU ⋐ N.

(QN) If c(A)∩c(B) = /0 thenA andB are separated.

(N) If c(A)∩c(B) = /0 thenc(A) andc(B) are separated.

(UN) If c(A)∩c(B) = /0 thenA andB are Urysohn-separated.

(CN) Any two semi-separated sets are separated.

Several symmetry conditions are close associated with separation

(R0) If x is contained in every neighborhood ofy theny is contained inevery neighborhood ofx

(R0’) x∈ c(y) impliesy∈ c(x).

(S) If x∈ N for all N ∈ N (y) thenN (y) = N (y)

(S’) If c(A)∩c(x) 6= /0 thenx∈ c(A).

(RE) If N∩N′ 6= /0 for all N ∈N (x) andN′ ∈N (x), thenN (y) = N (y).

The interesting point about these axioms is that there are elaborate chains ofimplications among them in (ordinary) topological spaces.Some of thesecarry over to pretopologies, neighborhood spaces, or even isotonic spacessatisfying only (K0) and (K1), see Fig. 6 and its caption. Separation axiomshave a close connections to generalized uniform structures[70] and the ex-istence of metrics that allow the definition ofε-balls as basis for neighbor-hoods of points, see [9].

The separation axiom (T0) implies boththinnessconditions (th0) and(th1), which do not seem to have been studied in detail in the context oftopology. In particular (th1) plays an important role in thetheory of graphproducts [49].

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T5

1/2T3T3T4

1/2T2

T2 T1 T0

isotone neighborhood pretopology topology

T5U

CN

T4U

CNU UNQN RE S R0R

CRN

th1 th0

Figure 6: Relationships among separation and symmetry axioms. Differentarrow types indicate implications that hold in all isotone space, in all neigh-borhood space, in all pretopological space, or only in topological spaces.The (T)-separation axioms are defined by a normality condition and (T1),dashed double-arrow, or by a regularity/symmetry condition and (T0), dot-ted double-arrow, respectively. Most of the proofs for the implications arenon-trivial. They can be found in the book [9], in the supplemental materialto [68], or in [69].

There have been several attempts to use connectedness as theprimitivenotion in topological theory [81, 43, 45]. Here, we employ the more usualapproach to defined connectedness as a derived property in closure spaces:A setZ is connectedin (X,c) if it is not a disjoint union of semi-separatedpairs of non-empty setsA,Z \A. In isotonic spaces, connected sets arecharacterized by the Hausdorff-Lennes condition:

[(A)∩ (Z\A)]∪ [c(Z\A)∩A] 6= /0 (4)

In neighborhood spaces, we arrive at a more familiar definition: A set isconnected if and only if it is not the disjoint union of two non-empty closed(open) sets [38, Thm.5.2].

The collection of connected sets satisfies the following four propertiesin isotonic spaces [68, 38]:

(c1) If Z consists of a single point, thenZ is connected.

(c2) If Y andZ are connected andY∩Z 6= /0 thenY∪Z is connected

(c3) If Z is connected andZ ⊆ c(Z), thenc(Z) is also connected.

(c4) Let I be an arbitrary index set andx ∈ X. SupposeZı is connectedandx∈ Zı for ı ∈ I . ThenW :=

⋃ı∈I Zı is connected.

14

Consider a setA and a pointx∈ A and letA[x] be the union of all connectedsubsets ofA that containx. By (c4),A[x] is itself connected. We call it theconnected component ofA containingx. Note that the well-definedness ofA[x] is required e.g. for the construction of the barrier trees mentioned inthe Introduction.

The relationship of connected components and semi-separation becomescounter-intuitive in non-additive spaces: SupposeA has a finite numberk > 1 of connected components and letQ be such a component. ThenQandA\Q are not necessarily semi-separated unless(X,c) is a pretopology[26]. Without the benefit of additivity, furthermore, it becomes natural toinvestigate alternative definitions of connectedness: In [5], for instance, aless stringent definition of connectivity is introduced that is in particularsuitable for chemical reaction networks:A and B are productively sepa-rated if for all Z ⊆ A∪B holds (1)c(Z∩A)∩B = /0 andc(Z∩B)∩A = /0,and (2)c(Z) = c(Z∩A)∪ c(Z∩B). A set isZ productively connectedifit cannot be written as the disjoint union of two non-empty productivelyseparated subsets.

Path-connectedness is a widely used notion of connectedness that ingeneral is stronger than topological connectedness. From the topologicalpoint of view, apathis acontinuousfunctionp : [0,1]→X whose endpointsarep(0) andp(1). A setA is path-connectedif for any two pointsx,y∈ A,there is a pathp with p(0) = x andp(1) = y.

A useful lemma [26] characterizes connectedness in 2-pointsets: Let(X,c) be a neighborhood space andx,y ⊆ X a 2-point subset. Then thefollowing three statements are equivalent: (1)y∈ c(x) or x∈ c(y); (2)x,y is path-connected; (3)x,y is connected. We will return to this pointlater when considering finite generalized topologies.

Product Spaces

Let (X1,c1) and (X2,c2) be two isotonic spaces. Then the product space(X1×X2,c1×c2) is defined by means of the neighborhood systemN (x1,x2),where

N∈N (x1,x2) ⇐⇒ ∃N1∈N1(x1) andN2∈N2(x2) such thatN1×N2⊆N(5)

For sets of the formA1×A2 this translates toc(A1×A2) = c1(A1)×c2(A2)in the product space, see [34, Thm.8.1]. If(X1,c1) and (X2,c2) satisfy

15

(K2), (K3), or (K4), respectively, then so does their product. We remarkthat the construction of products can be extended to infinitefamilies ofspaces, even uncountable ones. The projectionsπi : ∏ j(Xj ,c j) → (Xi,ci) :x = (. . . ,xi, . . .) 7→ xi are continuous.

The inductive product(X1,c1)¤(X2,c2) has the neighborhoodsN ∈N ¤(x1,x2) iff there isN1 ∈N1(x1) andN2 ∈N2(x2) such thatN1×x2∪x2×N2 ⊆ N. This product is discussed briefly in [9].

Finite Generalized Topologies

For practical applications in computational biology and computational chem-istry, finite spaces are of particular interest. Of course, in this case (K3)implies (K5). The by far best studied finite structure are thefinite pre-topologies: these are the simple directed graphs. More precisely, if (X,c)is a finite pretopological space, the associated graph has vertex setX and(x,y) is a directed edge iffy∈ c(x). Conversely, given a graph this recipedefines a finite pretopology onX [73]. Graphs that correspond to topologi-cal spaces are considered e.g. in [13].

Several of the topological concepts outlined above have independentlyhave been developed in graph theory. Continuous function, for instance, co-incide with graph homomorphisms [46], i.e., functionsf : (X,E) → (Y,F)such that(x,y) ∈ E implies ( f (x), f (y)) ∈ F or f (x) = f (y). The usualgraph-theoretical definition of connectedness is also the same as pre-topologicalconnectedness: a two-element subsetx,y is connected if and only if(x,y)∈E or (y,x)∈E. Furthermore, connectedness and path-connectednessis the same [26]. The (R0) symmetry axiom, furthermore, characterizesundirected graphs. On the other hand, it appears to be unknown if strongconnectedness in digraphs has a straightforward interpretation as a topolog-ical property.

The strong product of graphs coincides with the product of generalizedtopological spaces, while the inductive product is the sameas the Cartesiangraph product. For graphs, products have been studied in particular w.r.t.to the conditions under which a graph has a unique prime factorization.We refer to the book [40] for an extensive discussion of this topic. Inter-estingly, thinness conditions play an important role in this context. Localand approximative product structures have been explored recently [47, 48]motivated by the interpretation of characters as local factor spaces of phe-

16

notype space [80].As mentioned in the introduction, topological spaces are typically stud-

ied in terms of their closed sets, i.e., the collectionC = A|c(A) = A. If(K1) to (K4) are satisfied, we have (I1)X ∈ C , (I2) arbitrary intersectionsof closed sets are closed, and (I3) the union of two closed sets is closed.In lattice theory more general so-calledintersection structuresare consid-ered that fulfill only (I2) see e.g., [14], in graph theory thesame structuresappear as convexities [11]. Since neither idempotency nor additivity of theclosure function readily applies to the examples in the introduction we willnot consider them further in this contribution.

Not much is known about finite neighborhood spaces. They correspondto theN-systemsintroduced in [50]. Consider a pair(X,N) consisting ofa nonempty finite setX and a functionN : X → P(P(X)) that associatesto eachx∈ X a collectionN(x) = N1(x),N2(x), . . . ,Nd(x) of d(x) subsetsof X with the following properties:

(N0) N(x) 6= /0.

(N1) Ni(x) ⊆ N j(x) implies i = j.

(N2) x∈ Ni(x), for 1≤ i ≤ d(x).

N-systems are by construction exactly the finite neighborhood spaces, whentheNi(x) are interpreted a minimal neighborhoods, i.e.,N ∈ N (x) if andonly if there isN′ ⊆N with N′ ∈N(x). We may also interpret them as a spe-cial type of directed hypergraphs with hyperedges of the form (x,Ni(x)).Axiom (N1) ensures that it is simple.

The characterization of connectedness in 2-point sets [26]suggests toassociate a graph~Γ(X,c) with vertex setX with a given finite neighborhoodspace(X,c) such that(x,y) is a (directed) edge ify∈ c(x). If c is addi-tive, than~Γ is exactly the graph representation of the pretopological space(X,c). In finite neighborhood spaces every path consists of a finitenumberof connected 2-point sets consisting of consecutive points. Thus a subset ofX is path-connected if and only if the corresponding induced subgraph of~Γ is path connected [26]. Connectedness is strictly weaker property sincea connected set with three points does not necessarily contain connectedpairs. Finite neighborhood spaces admit a unique prime factorization w.r.t.to the usual topological product under certain conditions on~Γ [50].

17

Finite isotone spaces, finally, have not been considered as combinatorialobjects to our knowledge.

DIRECTED HYPERGRAPHS AS TOPOLOGICAL STRUCTURES

It may come as a surprise that hypergraphs have remained virtually unstud-ied from a topological perspective. While the interpretation of graphs asfinite pretopological spaces is quite natural, we will see below that theredoes not seem to be a unique canonical translation for hypergraphs.

A directed hypergraphH consists of a vertex setX and a setE of di-rected hyperedges, each of which is a pairE = (E−,E+) with E−,E+ ⊆ X.A hypergraph is simple if no edge is properly contained in another one, i.e.,if E− ⊆ F− andE+ ⊆ F+ impliesE = F . One way of defining undirectedhypergraphs as a special case of directed ones is to requireE+ = E− for allE ∈ E .

Given a directed hypergraphH, it appears natural to consider the clo-sure function

c(A) :=⋃

E :E−⊆A

E+ (6)

orc(A) := A∪

⋃

E :E−⊆A

E+ (7)

depending on whether we want to insist thatc is enlarging or not. This ap-proach indeed has been taken in [5] to describe chemical reaction networks.These constructions, however, become trivial for undirected hypergraphssince the conditionE+ = E− for all hyperedges ensures thatc(A) ⊆ A.

As a possible remedy one might identify undirected hypergraphs withdirected ones that satisfy the following symmetry condition:

(SH) For every directed hyperedgeE = (E−,E+) every pair(A,B) of non-empty sets withA∪B = E−∪E+ is also a hyperedge.

In particular, in this case(x,E) with E := E− ∪E+ and anyx ∈ E is ahyper-edge. Thus, for a given hyperedgeE , if E− ⊆ A then therex∈ E∩Aand hence(x,E) also contributes toc(A). Thus equ.(6) can be rephrasedas

c(A) =⋃

E :E∩A6= /0

E (8)

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This expression also makes perfect sense for undirected hypergraphs. Itis equally unsatisfactory, however, since this closure function is obviouslyadditive and hence describes a (symmetric) graph with adjacency relationx∼ y iff there a hyperedgeE such thatx,y∈ E.

An alternative approach is to consider hyperedges as minimal neighbor-hoods in the sense of theN-systems introduced in the previous section. Infinite pretopologies, i.e. directed graphs, the intersection of all N ∈ N (x)is again a neighborhood ofx. It coincides with the “in-neighborhood” ofx,given by

N(x) = y∈ X|x∈ c(y) (9)

For directed hypergraphs, one would analogously postulatethat hyper-edgesthat “produce”x are the minimal neighborhoods ofx, i.e.,

N ∈ N (x) ⇐⇒ ∃E : x∈ E+ andE−∪E+ ⊆ N (10)

It is straightforward to compute the corresponding closurefunction

c(A) =

x|∃E : x∈ E+ implies(E−∪E+)∩A 6= /0

(11)

An undesirable feature remains, however. If we specialize this constructionto directed graphs, i.e., so thatE− andE+ consist of two distinct points foreach hyperedge, thenN ∈ N (x) iff there is an arc(y,x) such thatx,y∈ N.The associated closure function isc(A) = A∪z|Nin(z) ⊆ A. This is ingeneral not additive.

A possibly fruitful alternative is to start from the notation of hypergraphhomomorphisms(X,E ) → (Y, mathcalF), i.e., mapsϕ : X → X′ so thatfor every(E−,E+) ∈ E there is(F−,F+) ∈ F so thatϕ(E−) ⊆ F− andϕ(E+) ⊆ F+. One may then ask if there exist isotonic closure functions sothat the hypergraph homomorphisms are exactly the continuous function asin the case of graphs. To our knowledge, this question is open.

We have to conclude that at present there is no satisfactory interpretationof hypergraphs a topological objects.

DYNAMICAL ASPECTS

The notion oftopological dynamical systems, see e.g. [16] does not seemto have explored in systematic way for generalized topological spaces. Anatural starting point would be flows of the formω : T ×X → X whereT

19

is suitable (topological) semigroup such as the non-negative integers and(ı,x) 7→ x, whereı denotes the neutral element ofT. Orbits, for instance,are well-defined in this setting:Ω(x) = tx|t ∈ T ⊆ X, wheretx denotesthe action of the semigroup elementt on the pointx. Similarly, trajectoriesare simply functions of the formξ : T → X such thatt 7→ ξ (t) = tx0 for afixed “initial condition” x0. In [73], for instance, conditions are exploredunder which trajectories of phenotypic evolution are continuous. It turnsout that this matches well with intuition developed earlierin [28, 29] in thecontext of the evolution of RNA secondary structures.

Combinatorial vector fields offer a more specialized alternative formal-ization. LetM be a simplicial complex constructed over the setK of sim-plices, see e.g. [51]. We writeσ < τ if the simplexσ lies in the bound-ary of the simplexτ. A combinatorial vector fieldon M [30] is a mapη : K → K∪∅ such that

1. If η(σ) 6= ∅ thendimη(σ) = dim(σ)+1, andσ < η(σ).

2. If η(σ) = τ 6= ∅, thenη(τ) = ∅.

3. For allσ ∈ K, |σ−1| ≤ 1.

We remark that combinatorial vector fields can in fact be defined on themuch more general CW-complexes [30].

Combinatorial vector fields come with a natural notion ofη-paths asthe analog of trajectories, namely a a finite sequence of simplices γ =(σ0,τ0,σ1,τ1, . . .σn−1,τn−1,σn) such thatη(σi) = τi for 0 ≤ i < n andσi+1 < τi. A rest point is a simplexσ such thatη(σ) = ∅ andη−1(σ) = /0.The rest points and the closedη-paths together form the so-called chain-recurrent setR of η . These play the role of attractors. All other trajectorieslead towardsR.

A Lyapunov functionof the combinatorial vector fieldη is a functionF : M → R such that

1. if σ /∈ R andτ > σ then

(a) F(σ) < F(τ) if τ 6= η(σ)

(b) F(σ) ≥ F(τ) if τ = η(σ)

2. if σ ∈ R andτ > σ then

20

(a) F(σ) = F(τ) if σ ∼ τ(b) F(σ) < F(τ) if σ 6∼ τ

A combinatorial version of Conley’s theorem ensures that there is Lya-punov function for every combinatorial vector field [30]. Combinatorialvector fields thus form discrete analog for gradient vector fields.

In [71] this observation was used to consider a the collection of com-binatorial vector fields on a graphG (with vertex setX) for which a givenfunction F : X → R is a Lyapunov function. In this context, the functionF is interpreted as energy function. The combinatorial vector field, on theother hand, can be interpreted as a partial orientationP on G so thatx,yis directed fromx to y iff η(x) = x,y. Conversely a system of directededgesP with the properties

1. (x,y) ∈ P impliesx,y ∈ E (consistency withG)

2. (x,y) ∈ P and(x,z) ∈ P impliesy = z (uniqueness)

3. (x,y) ∈ P implies(y,x) /∈ P (antisymmetry)

corresponds to a combinatorial vector field onG. A trajectory inη is thengiven by a sequence of vertices(xi), i = 1, . . .k so that(xi ,xi+1) ∈ P.

These combinatorial vector fields provide a convenient description ofthe system of adaptive (downhill) walks on the energy landscape(G,F) andadmits an alternative approach towards characterizing basins, barriers, andtheir hierarchical structure. In particular, they highlight the complicationsin the analysis of landscapes arising from degeneracies in the energy func-tion. GivenF , denote byGF the subgraph ofG with edgesx,y ∈ E(G)if F(x) = F(y). The connected componentsGF(x), the so called shelves ofthe landscape, provide the main complication for practicalcomputations,Fig. 7. While the vector fields for whichF is a Lyapunov function nec-essarily point downwards between shelves, it is complicated to handle thepossible orientations with a single shelf since edges may beoriented in bothdirections depending on the particular choice ofη .

A direct connection to the topology of the landscape(G,F) is obtainedby mean of the following construction:Definition. A point y is reachable fromx in (G,F) if there is a combina-torial vector fieldη for which F is a Lyapunov function and that admits atrajectory fromx to y. LetC(x) be the set of vertices reachable fromx.

21

F

Figure 7: Shelves, i.e., maximal connected sets on which theenergy func-tion F is constant are indicated by dotted boxes. Edges ofG that are ori-ented by the combinatorial vector fieldη are shown as arrows, the remain-ing edges ofG are shown in gray. Edges between shelves are either notoriented or point downwards in any combinatorial vector field for whichF is a Lyapunov function. Both orientations are possible for edges withinshelves. Figure adapted from [71].

As shown in [71],C can be extended to an additive closure function thatsatisfies all of Kuratowski’s axioms and hence defined a finitetopologicalspace on the vertex setX. This topology is intimately related with the bar-rier structure of the landscape by means of the notion of a valley: A valleyin (G,F) is a maximal connected closed set with respect to the reachabilitytopology. Equivalently a valleyW is a maximal connected subset ofX sothat no vertexy /∈W is reachable from anyx∈W.

CONCLUDING REMARKS: COARSE GRAINING

Accessibility is of particular interest in the context of “constructive sys-tems”, i.e., models comprising a set combinatorial objectsand set of ruleswith which they can be combined. This was pioneered e.g. in the work ofFontana and Buss [27] usingλ -calculus as a modelling platform for ab-stract chemistry. The state spaceX in such models is usually not finite.In simulations of (evolutionary) processes on such models,families of ob-

22

jects emerge that share many regularities and differ essentially in size only[27]. In chemistry, polymers are the most obvious example. Naturally, thequestion arises whether there are coarse-grained representations.

In the analysis of the fine structure of the Network of OrganicChem-istry, it has turned out that so-called one-pot reactions, i.e., reactions thatcan be performed concurrently without the need to purify intermediates andhence are of practical interest in practice, play an important role [36]. Sinceone-pot reaction lumps together individual reactions intosequences theycan be seen as a form of coarse-graining of the chemical space. Recently,we have started to explore [3], from an algebraic perspective, whether com-positions of transformation rules can be employed to capture families suchas homologous series (i.e., groups of chemicals with analogous propertiesthat differ only in the number of simple structural units) based on their re-active capabilities.

Coarse-graining is an import issue also in the analysis e.g. of fitnesslandscapes. A rather simple example is the use of barrier trees, which rep-resent a partition of state spaceX. Quotient spacesX/∼, obtained by iden-tifying ∼equivalent points ofX, thus appear to be a natural formal structureto consider in this context. To our knowledge, this avenue has not been ex-plored systematically for any of the model systems discussed above. It willbe interesting to see if the intuitive connection of rule compositions andquotient spaces can be given a precise topological meaning.

Acknowledgments

This work was supported in part by the Volkswagen Stiftung and theDeutscheForschungsgemeinschaftwithin the EUROCORES Programme EUROGIGA(project GReGAS) of the European Science Foundation.

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