The discretizable molecular distance geometry problem is easier on proteins

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1 The discretizable molecular distance geometry problem is easier on proteins Leo Liberti, Carlile Lavor, and Antonio Mucherino Abstract Distance geometry methods are used to turn a set of interatomic distances given by NMR experiments into a consistent molecular conformation. In a set of papers (see the survey []) we proposed a Branch-and-Prune (BP) algorithm for computing the set X of all incongruent embeddings of a given protein backbone. Although BP has a worst-case exponential running time in general, we always noticed a linear-like behaviour in computational experiments. In this paper we provide a theoretical explanation to our observations. We show that the BP is fixed-parameter tractable on protein-like graphs, and empirically show that the parameter is constant on asetof proteins from the ProteinData Bank. Introduction We consider the following decision problem [9]: DISCRETIZABLE MOLECULAR DISTANCE GEOMETRY PROBLEM (DMDGP). Given a simple weighted undirected graph G = (V,E,d) where d : E R +, V is ordered so that V =[n]={,...,n}, and thefollowing assumptions hold:. for allv>3 and u V with v u 3, {u,v} E (DISCRETIZATION) L. Liberti LIX, École Polytechnique, 9 Palaiseau, France liberti@lix.polytechnique.fr C. Lavor Dept. of Applied Maths (IMECC-UNICAMP), State Univ. of Campinas, , Campinas - SP, Brazil clavor@ime.unicamp.br A. Mucherino IRISA, Université de Rennes I, France antonio.mucherino@irisa.fr

2 Liberti, Lavor, Mucherino. strict triangular inequalities d v,v <d v,v +d v,v hold for all v> (NON-COLLI- NEARITY), andgivenanembeddingx :{,,3} R 3,isthereanembeddingx:V R 3 extendingx, such that {u,v} E x u x v =d uv? () An embedding x onv extends an embedding x onu V if x, as a function, is the restrictionofxtou;anembeddingisfeasibleifitsatisfies().wealsoconsiderthe following problem variants: DMDGP K, i.e. the family of decision problems (parametrized by the positive integer K) obtained by replacing each symbol 3 in the DMDGP definition by the symbol K ; the K DMDGP, where K is given as part of the input (rather than being a fixed constant as inthe DMDGP K ). In both variants, strict triangular inequalities are replaced by strict simplex inequalities,seeeq.()in[7].weremarkthatdmdgp=dmdgp 3.Otherrelatedproblems also exist in the literature, such as the DISCRETIZABLE DISTANCE GEOM- ETRY PROBLEM (DDGP) [], where the DISCRETIZATION axiom is relaxed to require that each vertex v > K has at least K adjacent predecessors. The original results in this paper, however, only refer to the DMDGP and its variants. Statements such as p P F(p) holds with probability, for some uncountable set P and valid sentence F, actually mean that there is a Lebesgue-measurable Q P with Lebesgue measure w.r.t. P such that p Q F(p) holds. This notion is less restrictive than genericity based on algebraic independence []. We also point out that a statement might hold with probability with respect to a set which has itself Lebesgue measure 0 in a larger set. For example, we will show that the set of K DMDGP instances having an incongruent solution set X with X = l for some l NhasmeasureintothesetofallYESinstances,whichitselfisasetofmeasure 0intheset of all K DMDGP instances. The DISCRETIZATION axiom guarantees that the locus of the points that embed v in R 3 is the intersection of the three spheres centered at v 3,v,v with radii d v 3,v,d v,v,d v,v. If this intersection is non-empty, then it contains two points with probability. The complementary zero-measure set contains instances that do not satisfy the NON-COLLINEARITY axiom, and which might yield loci for v with zero or uncountably many points. We remark that if the intersection of the three spheresisempty,thentheinstanceisanoone.wesolve K DMDGPinstancesusing a recursive algorithm called Branch-and-Prune (BP) [3]: at level v, the search is branchedaccordingtothe(atmosttwo)possiblepositionsforv.thebpgeneratesa (partial) binary search tree of height n, each full branch of which represents a feasible embedding for the given graph. The BP has exponential worst-case complexity. The K DMDGP and its variants are related to the MOLECULAR DISTANCE GE- OMETRY PROBLEM (MDGP), i.e. find an embedding in R 3 of a given simple weighted undirected graph. We denote the K-dimensional generalization of the MDGP (with K part of the input) by DISTANCE GEOMETRY PROBLEM (DGP),

3 The BP algorithm 3 and the variant with K fixed by DGP K. The MDGP is a good model for determining the structure of molecules given a set of inter-atomic distances [, ], which are usually given by Nuclear Magnetic Resonance(NMR) experiments[], a technique which allows the detection of inter-atomic distances below 5.5Å. The DGP has applications to wireless sensor networks [5], statics, robotics and graph drawing among others. In general, the MDGP and DGP implicitly require a search in a continuous Euclidean space []. K DMDGP instances describe rigid graphs [6], in particular Henneberg type I graphs []. TheDMDGPisamodelforproteinbackbones.Foranyatomv V,thedistances d v,v and d v,v are known because they refer to covalent bonds. Furthermore, theangle between v,v andvisknown because itisadjacent totwocovalent bonds, which implies that d v,v is also known by triangular geometry. In general, thedistanced v 3,v issmallerthan5åandcanthereforebeassumedtobeknownby NMR experiments; in practice, there are ways to find atomic orders which ensure that d v 3,v is known [7]. There is currently no known protein with d v 3,v being exactly equal tod v 3,v +d v,v [3]. Over the years, we noticed that the CPU time behaviour of the BP on protein instances looked more linear than exponential. In this paper we give a theoretical motivation for this observation. More precisely, there are cases where BP is actually Fixed-Parameter Tractable(FPT), and we empirically verify on 5 proteins from the PDB [] that they belong to these cases, and always with the parameter value set to the constant. The strategy is as follows: we first show that DMDGP K is NP-hard (Sect.3),thenweshowthatthenumberofleafnodesintheBPsearchtreeisapower of with probability (Sect..), and finally we use this information to construct a Directed Acyclic Graph (DAG) representing the number of leaf nodes in function of the graph edges (Sect. 5). This DAG allows us to show that the BP is FPT on a class of graphs which provides a good model for proteins (Sect. 5.). TheBP algorithm For all v V we let N(v)={u V {u,v} E} be the set of vertices adjacent to v. An embedding of a subgraph of G is called a partial embedding of G. Let X ne thesetofembeddings(modulotranslationsandrotations)solvingagiven K DMDGP instance. Since vertex v can be placed in at most two possible positions (the intersection of K spheres in R K ), the BP algorithm tests each in turn, and calls itself recursively for every feasible position. BP exploits other edges (than those granted by the DIS- CRETIZATION axiom) in order to prune branches: a position might be feasible with respect to the distances to the K immediate predecessors v,...,v K, but not necessarily with distances to other adjacent predecessors. Forapartialembedding xofgand{u,v} E lets x uv bethespherecenteredatx u withradiusd uv.thebpalgorithmisbp(k+,x, /0)(seeAlg.),wherex istheinitial embedding of the first K vertices mentioned in the K DMDGP definition. By the

4 Liberti, Lavor, Mucherino Algorithm BP(v, x,x) Require: A vertexv V [K],apartialembedding x=(x,...,x v ),asetx. : T = S x uv; u N(v) u<v : for p T do 3: x ( x,p) : ifv=n then 5: X X {x} 6: else 7: BP(v+,x,X) : end if 9: end for K DMDGPaxioms, T.Attermination,X containsallembeddings(modulorotationsandtranslations)extendingx [3,9].Embeddingsx X canberepresentedby sequences χ(x) {,} n representingleft/rightchoiceswhentraversingabranch from root to leaf of the search tree. More precisely: (i) χ(x) i = for all i K; (ii) foralli>k, χ(x) i = ifax i <a 0 and χ(x) i =ifax i a 0,whereax=a 0 isthe equation of the hyperplane through x i K,...,x i. For an embedding x X, χ(x) is the chirality [3] of x (the formal definition of chirality actually states χ(x) 0 = 0 if ax i =a 0, but since thiscase holds withprobability 0, we donot consider ithere). The BP (Alg. ) can be run to termination to find all possible embeddings of G, or stopped after the first leaf node at level n is reached, in order to find just one embedding of G. In the last few years we have conceived and described several BP variants targeting different problems [], including, very recently, problems with interval-type uncertainties on some of the distance values [0]. The BP algorithm is currently the only method which is able to find all incongruent embeddings for a given protein backbone. Compared to continuous search algorithms (e.g. [7]), the performance of the BP algorithm is impressive from the point of view of both efficiency and reliability. 3 Complexity AnyclassofYESinstanceswhereeachvertexvonlyhasdistancestotheK immediate predecessors provides a full BP binary search tree (after level K), and therefore shows that the BP is an exponential-time algorithm in the worst case. One remarkable feature of the computational experiments conducted on our BP implementation [0] on protein instances is that the exponential-time behaviour of the BP algorithm was never noticed empirically. Restrictingd toonlytakeintegervalues,thedgp isnp-completebyreduction from SUBSET-SUM,theDGP K is(strongly)np-hardbyreductionfrom3-sat,and the DGP is (strongly) NP-hard by induction on K []. Only the DGP is known

5 Partial reflection symmetries 5 to be NP-complete, because if d takes integer values then the YES-certificate x(the embedding) can be chosen to have integer values too. The DMDGP is NP-hard by reduction from SUBSET-SUM (Thm. 3 in [9]). We generalize thisresult tothe K DMDGP. Theorem. The DMDGP K isnp-hard forall K. Proof. Let a = (a,...,a N ) be an instance of SUBSET-SUM consisting of positive integers, and define an instance of DMDGP K where V = {0,...,KN}, E includes {i,i+ j}forall j {,...,K}and i {0,...,KN j},and: i {0,...,KN } d i,i+ = a i/k () j {,...,K},i {0,...,KN j} d i,i+j = j di+l,i+l (3) l= d 0,KN = 0. () Let s {,} N be a solution of the SUBSET-SUM instance a. We let x 0 = 0 and for all i = K(l )+ j > 0 we let x i = x i +s l a l e j, where e j is the vector with a one in component j and zero elsewhere. Because l N s l a l = 0, if s solves the SUBSET-SUM instance a then, by inspection, x solves the corresponding DMDGP instance ()-(). Conversely, let x be an embedding that solves ()- (), where we assume without loss of generality that x 0 =0. Then (3) ensures that the line through x i,x i is orthogonal to the line through x i,x i for all i >, and again we assume without loss of generality that, for all j {,...,K}, the lines through x j,x j are parallel to the i-th coordinate axis. Now consider the chirality χ of x: because all distance segments are orthogonal, for each j K the j-th coordinate is given by x KN,j = i mod K=j χ i a i/k. Since d 0,KN = 0, for all j K we have 0 = x KN,j = l N χ K(l )+j a l, which implies that, for all j K, s j =(χ K(l )+j l N) isasolution forthe SUBSET-SUM instance a. Corollary. The K DMDGP isnp-hard. Proof. Every specific instance of the K DMDGP specifies a fixed value for K and hence belongs tothe DMDGP K.Hence the resultfollows by inclusion. Partial reflection symmetries The results in this section are also found in [6], but the presentation below, which isbasedongrouptheory, isnew, and(wehope) clearer.wepartitione intothesets E D ={{u,v} v u K} and E P =E E D. We call E D the discretization edges and E P the pruning edges. Discretization edges guarantee that a DGP instance is in the K DMDGP. Pruning edges are used to reduce the BP search space by pruning its tree.inpractice,pruningedgesmightmakethesett inalg.havecardinality0or instead of. Weassume G isayes instance of the K DMDGP.

6 6 Liberti, Lavor, Mucherino. The discretization group LetG D =(V,E D,d)andX D bethesetofembeddings ofg D ;sinceg D has nopruning edges, the BP search tree for G D is a full binary tree and X D = n K. The discretization edges arrange the embeddings so that, at level l, there are l K possible positions for the vertex v with rank l. We assume that T = (see Alg. ) at each level v of the BP tree, an event which, in absence of pruning edges, happens with probability thus many results in this section are stated with probability. LetthereforeT ={x 0 v,x v}bethetwopossibleembeddingsofvatacertainrecursive callofalg.atlevelvofthebptree;thenbecauset isanintersectionofk spheres, x v is the reflection of x 0 v through the hyperplane defined by x v K,...,x v. Denote thisreflection operator by R v x. Theorem (Cor..6 and Thm..9 in [6]). With probability, for all v>k and u < v K there is a set H uv, with H uv = v u K, of real positive values such that for each x X we have x v x u H uv. Furthermore, x X x v x u = R u+k x (x v ) x u and x X, ifx v {x v,r u+k x (x v )}then x v x u x v x u. WesketchtheproofinFig.forK=;thesolidcirclesatlevels3,,5markequidistant levels from. The dashed circles represent the spheres S x uv (see Alg. ). Intuitively,twobranchesfromleveltolevelor5willhaveequalsegmentlengthsbut different angles between consecutive segments, which will cause the end nodes to be atdifferent distances fromthe node at level. ν ν ν 3 ν 3 5 ν ν 9 ν 5 ν 6 ν 6 ν7 ν 0 ν ν 5 ν ν 3 ν Fig. A pruning edge {,} prunes either ν 6,ν 7 or ν 5,ν.

7 Partial reflection symmetries 7 For any nonzero vector y R K let ρ y be the reflection operator through the hyperplanepassingthroughtheoriginandnormaltoy.ifyisnormaltothehyperplane defined by x v K,...,x v, then ρ y =R v x. Lemma. Letx y R K andz R K suchthatzisnotinthehyperplanesthrough theoriginand normal tox,y. Then ρ x ρ y z=ρ ρxy ρ x z. Proof. Fig. gives a proof sketch for K =. By considering the reflection ρ ρxy of x O y ρ x z ρ x y ρ ρx y ρ x ρ y z=ρ ρxy ρ x z ρ y z ρ x z ρ y Fig. Reflecting through ρ y first and ρ x later is equivalent to reflecting through ρ x first and (the reflectionof ρ y through ρ x )later. the map ρ y through ρ x, we get z ρ y z = ρ x z ρ ρxy ρ x z. By reflection through ρ x weget O z = O ρ x z and O ρ y z = O ρ x ρ y z.byreflectionthrough ρ y we get O z = O ρ y z. By reflection through ρ ρxy we get O ρ x z = O ρ ρxy ρ x z. The triangles (z,o,ρ y z) and (ρ x z,o,ρ ρxy ρ x z) are then equal becausethesidelengthsarepairwiseequal.also,reflectionof (z,o,ρ y z)through ρ x yields (z,o,ρ y z)= (ρ x z,o,ρ x ρ y z),whence ρ ρxy ρ x z=ρ x ρ y z. For v>k and x X wenow define partialreflection operators: g v (x)=(x,...,x v,r v x(x v ),...,R v x(x n )). (5) Theg v smapanembeddingxtoitspartialreflectionwithfirstbranchatv.itiseasy to show that the g v s are injective with probability and idempotent. Further, the g v scommute. Lemma. For x X and u,v V such that u,v>k,g u g v (x)=g v g u (x).

8 Liberti, Lavor, Mucherino Proof. Assume without loss of generality u < v. Then: g u g v (x) = g u (x,...,x v,r v x(x v ),...,R v x(x n )) = (x...,x u,r u g v (x) (x u),...,r u g v (x) Rv x(x v ),...,R u g v (x) Rv x(x n )) = (x...,x u,r u x(x u ),...,R v g u (x) Ru x(x v ),...,R v g u (x) Ru x(x n )) = g v (x,...,x u,r u x(x u ),...,R u x(x n )) = g v g u (x), where R u g v (x) Rv x(x w )=R v g u (x) Ru x(x w )foreach w v bylemma. We define the discretization group to be the group G D = g v v>k generated by theg v s. Corollary. With probability, G D is an Abelian group isomorphic to C n K (the Cartesianproduct consisting of n K copies of thecyclic group oforder ). For all v > K let γ v = (,...,, v,..., ) be the vector consisting of one s in the first v components and in the last components. Then the g v actions are naturally mapped onto the chirality functions. Lemma 3. For all x X, χ(g v (x)) = χ(x) γ v, where is the Hadamard (i.e., component-wise) product. This follows bydefinition of g v and of chiralityof an embedding. Because, by Alg., each x X has a different chirality, for all x,x X there is g G D such that x = g(x), i.e. the action of G D on X is transitive. By Thm., the distances associated to the discretization edges are invariant with respect to the discretization group.. The pruning group Consider a pruning edge {u,v} E P. By Thm., with probability we have d uv H uv,otherwisegcannotbeayesinstance(againstthehypothesis).also,againby Thm.,d uv = x u x v g w (x) u g w (x) v forallw {u+k+,...,v}(e.g.the distance ν ν 9 in Fig. is different from all its reflections ν ν h, with h {0,,}, w.r.t.g,g 5 ).Wetherefore define the pruning group G P = g w w>k {u,v} E P (w {u+k+,...,v}). By definition, G P G D and the distances associated with the pruning edges are invariant withrespect to G P. Theorem 3. The action of G P onx istransitivewithprobability. Proof. This theorem follows from Thm. 5. in [5], but here is another, hopefully simpler, proof. Let x,x X, we aim to show that g G P such that x =g(x) with

9 5 Bounded treewidth 9 probability. Since the action of G D on X is transitive, g G D with x = g(x). Now suppose g G P,thenthereisapruning edge {u,v} E P andanl Ns.t.g= l h= g v h ) for some vertex set {v,...,v l > K} including at least one vertex w {u+k+,...,v}.bythm.,asremarkedabove,thisimpliesthatd uv = x u x v g w (x) u g w (x) v withprobability.ifthesetq={v,...,v l } {u+k+,...,v} has cardinality, then g w is the only component of g not fixing d uv, and hence x = g(x) X, against the hypothesis. Otherwise, the probability of another z Q {w} yielding x u x v = g z g w (x) u g z g w (x) v, notwithstanding the fact that g w (x) u g w (x) v x u x v g z (x) u g z (x) v, is zero; and by induction this alsocovers any cardinality of Q. Therefore g G P and theresult follows. Theorem. Withprobability, l N X = l. Proof. Since G D =C n K, G D = n K. Since G P G D, G P divides the order of G D, which implies that there is an integer l with G P = l. By Thm. 3, the action of G P on X only has one orbit, i.e. G P x = X for any x X. By idempotency, for g,g G P, if gx=g x then g=g. This implies G P x = G P. Thus, for any x X, X = G P x = G P = l. 5 Bounded treewidth The results of the previous section allow us to express the the number of nodes at each level of the BP search tree in function of the level rank and the pruning edges. Fig. 3 shows a DAG D uv that represents the number of valid BP search tree nodes in function of pruning edges between two vertices u,v V such that v > K and u < v K. The first line shows different values for the rank of v w.r.t. u; an arc labelled with an integer i implies the existence of a pruning edge {u+i,v} (arcs with -expressions replace parallel arcs with different labels). An arc is unlabelled if there is no pruning edge {w,v} for any w {u,...,v K }. The vertices of the DAG are arranged vertically by BP search tree level, and are labelled with the number of BP nodes at a given level, which is always a power of two by Thm.. A path in this DAG represents the set of pruning edges between u and v, and its incident vertices show the number of valid nodes at the corresponding levels. For example, following unlabelled arcs corresponds to no pruning edge between u and vand leads toafullbinary BP search treewith v K nodes at level v. 5. Fixed-parameter tractable behaviour For a given G D, each possible pruning edge set E P corresponds to a path spanning all columns in D n. Instances with diagonal (Prop. ) or below-diagonal (Prop. ) E P pathsyieldbptreeswhosewidthisboundedbyo( v 0)wherev 0 issmallw.r.t.n.

10 0 Liberti, Lavor, Mucherino v u+k u+k u+k+ u+k+ u+k+3 u+k Fig. 3 Number of valid BP nodes (vertex label) at level u+k+l (column) in function of the pruning edges (path spanning all columns). Proposition. If v 0 > K s.t. v > v 0 u < v K with {u,v} E P then the BP search treewidth isbounded by v 0 K. This corresponds to a path p 0 =(,,..., v 0 K,..., v 0 K ) that follows unlabelled arcsuptolevelv 0 andthenarcslabelledv 0 K,v 0 K v 0 K,andsoon, leading tonodes that areall labelled with v 0 K (Fig., left). Proposition. If v 0 >K suchthateverysubsequencesofconsecutivevertices>v 0 withnoincidentpruningedgeisprecededbyavertexv s suchthat u s <v s (v s u s s {u s,v s } E P ),then the BPsearch treewidth isbounded by v 0 K. This situation corresponds to a below-diagonal path, Fig. (right). In general, for thoseinstancesforwhichthebpsearchtreewidthhasao( v 0logn)bound,theBP has a worst-case running time O( v 0L logn )=O(Ln), where L is the complexity of computing T. Since L is typically constant in n [], for such cases the BP runs in timeo( v 0n). LetV ={v V l N(v= l )}. Proposition 3. If v 0 >K s.t. for all v V V with v>v 0 there is u<v K with {u,v} E P thenthe BPsearch treewidthat level nisbounded by v 0n. This corresponds to a path along the diagonal v 0 apart from logarithmically many vertices in V (those in V ), at which levels the BP doubles the number of search nodes (Fig. 5). For a pruning edge set E P as in Prop. 3, or yielding a path below it, thebp runs ino( v 0n ).

11 5 Bounded treewidth Fig. A path p 0 yieldingtreewidth (above) and another path below p 0 (below). 5. Empirical verification We consider a set of 5 protein instances from the Protein Data Bank (PDB). Since PDB data include the Euclidean embedding of the protein, we compute all distances, then filter all those that are longer than 5.5Å, which is a reasonable estimate for NMR measures []. We then form the weighted graph and embed it with our implementation of the BP algorithm [0]. It turns out that 0 proteins satisfy Prop., and 5 proteins satisfy Prop., all with v 0 = (see Table ). This is consistent with the computational insight [9] that BP empirically displays a polynomial (specifically, linear) complexity on real proteins.

12 Liberti, Lavor, Mucherino Fig. 5 A path yielding treewidth O(n). PDB ID V v 0 Prop. brv 57 a 75 acw 7 ppt 0 bbl erp aqr 0 kv 3 hj 3 ed7 35 dv0 35 crn 3 jkz 3 ahl 7 ptq 50 brz 59 ccq 0 hoe bqx 3 pht 9 as 67 jk 70 PDBID V v 0 Prop. Prop. a ag hsy 3 - acz 3 - poa 35 - fs itm mbn ngl bc 55 0 la a oy ron 76 - dv 79 - rgs qk ezo 0 - m0 - bpm 3 39 nw 60 - mqq 03-3b Table Computation of minimumv 0 in PDB instances. 6 Conclusion In this paper we provide a theoretical basis to the empirical observation that the BP never seems to attain its exponential worst-case time bound on DMDGP in-

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