Identifying reliable domain-domain interactions will increase our ability to predict novel protein-protein interactions, to unravel interactions in protein complexes, and thus gain more information about the function and behavior of genes. One of the challenges of identifying reliable domain-domain interactions is domain promiscuity. Promiscuous domains are domains that can occur in many domain architectures and are therefore found in many proteins. This becomes a problem for a method where the score of a domain-pair is the ratio between observed and expected frequencies because the protein-protein interaction network is sparse. As such, many protein-pairs will be non-interacting and domain-pairs with promiscuous domains will be penalized. This domain promiscuity challenge to the problem of inferring reliable domain-domain interactions from protein-protein interactions has been recognized, and a number of work-arounds have been proposed. This paper reports on an application of Formal Concept Analysis to this problem. It is found that the relationship between formal concepts provides a natural way for rare domains to elevate the rank of promiscuous domain-pairs and enrich highly ranked domain-pairs with reliable domain-domain interactions. This piggybacking of promiscuous domain-pairs onto less promiscuous domain-pairs is possible only with concept lattices whose attribute-labels are not reduced and is enhanced by the presence of proteins that comprise both promiscuous and rare domains. Proteins comprise domains which are evolutionary conserved sequence segments with the ability to fold and be functional. An important class of domains mediates protein-protein interactions (PPIs); although not all interactions between proteins can be attributed to interactions between domains, and not all domains in multi-domain proteins play a direct role in protein interaction. Nonetheless, many computational methods which seek to predict PPIs with high accuracy rely on computationally inferred domain-domain interactions (DDIs), e.g. Ideally, the inferred DDIs used to support the predicted PPIs are highly reliable themselves, that is there is a large overlap between the set of inferred DDIs and the set of DDIs confirmed to interact by 3D crystal structures of proteins. This latter set is referred to in this paper as the gold standard domain-domain interactions (GDDIs). This ideal is desirable not only to tease out specific interactions in a protein complex, but also to give predictive power to protein-protein interaction prediction methods (more on this point later in this section).promiscuous domains. Promiscuous domains can occur in many domain architectures However, using GDDIs to predict PPIs generates a large number of false positives (non-interacting protein-pairs predicted as interacting) and thus reduces the accuracy of the prediction method. The large number of false positives stem from the fact that GDDIs are enriched with It is parsimonious to re-use domain-pairs that can interact to facilitate PPIs. Indeed, many DDIs are conserved across organisms by evolution not GDDIs. Promiscuous domains are observed to be heavily involved in PPI mediation promiscuity prevents GDDIs from being highly ranked in computational methods to infer DDIs. Recognizing this domain promiscuity problem, additional measures have been taken to counteract its effects when inferring DDIs from PPIs, e.g. The ‘drift towards rare domain-pairs’ phenomenon in PPI prediction methods has been noted piggy-backing potential of promiscuous domain-pairs. Piggy-backing occurs when a domain-pair improves its score because either one or both of its domain partners happen to occupy the same attribute-label set as one or more rarer domains. Piggy-backing potential is possible only in concept lattices that are not attribute-reduced, and is enhanced by the presence of many mixed architecture proteins. One of the challenges of doing bioinformatics research is the volatility and variety of bioinformatics datasets which exerts a high validation cost for any new method. Therefore, it is useful to understand why a bioinformatics method works and under what conditions. Identifying such conditions also automatically suggests the limits of a method. Towards this end, effort is made to investigate conditions favourable to the proposed concept-based domain-pair ranking method.In this paper, I propose that Formal Concept Analysis (FCA) P be the set of proteins and D the set of domains. Every protein in P comprises one or more domains in D. D(x) denotes the finite set of domains for protein x. If | D(x) | = 1, x is a single-domain protein; if | D(x) | > 1, x is a multi-domain protein. The set of proteins which contains domain a is P(a) = { x ∈ P | a ∈ D(x) }. The frequency of domain a in P is N(a) = | P(a) |. x = D(x) = {a, b, c} where x ∈ P and {a, b, c} ⊂ D denotes protein x is its set of domains D(x) which in turn comprises domains a, b and c.Let P. This relation is symmetric, i.e. if  is an interacting protein-pair, then so is . It is possible for proteins to self-interact. Let 1 denote  ∈ the set of PPIs. 1 implies proteins x and y come from the same organism, i.e.: O(x) = O(y). The set of non-PPIs is also a symmetric relation on 3, and a non-interacting protein-pair 0 also implies O(x) = O(y). O(x) = O(y) implies either 1 or 0. There may be pairs in P × P which are neither interacting nor non-interacting because they do not satisfy the same organism condition.The set of PPIs is a relation on D, and domain self-interaction is possible. A protein-pair  generates domain-pairs, each of which may or may not be reliable, through the cross-product of their domains, i.e. D(x) × D(y). A domain-pair  generates a set of protein-pairs, each of which may or may not be interacting, through the cross-product of their respective protein sets, i.e. P(a) × P(b). For a domain-pair  to be a DDI, it must generate at least one interacting protein-pair.The set of DDIs is a symmetric relation on The Riley dataset x and y implies interactions between the domains of x and the domains of y. The Riley set of proteins, domains and PPIs generate 177,233 putative domain-domain interactions (DDIs). Amongst these possible DDIs are 783 gold standard domain-domain interactions (GDDIs). These GDDIs were obtained from iPfam The proteins are associated with 12,455 Pfam-A and Pfam-B domains. Amongst the set of proteins are 26,032 protein-protein interactions (PPIs). Interactions and non-interactions between protein-pairs are restricted to proteins from the same organism In the following sub-sections, the Riley dataset is examined to support assertions made in this paper. Specifically, the data characteristics of interest are:Highly reliable domain-domain interactions (GDDIs) are enriched with promiscuous domains.GDDIs generate significantly more true positive PPIs and also more false positive PPIs than non-GDDIS. More true positive PPIs agrees with the parsimony or the re-use principle for GDDIs, and more false positive PPIs accords with the promiscuity of gold domains.Protein domain architectures are mostly a mix of rare and promiscuous domains.gold domain is a domain that participates in at least one gold standard domain-domain interaction (GDDI). There are 642 gold domains in the Riley dataset. A domain is more promiscuous if it occurs more frequently in a given set of proteins, i.e. given P and {a, b} ⊂ D, N(a)>N(b) implies a is more promiscuous than b.All domains that participate in at least one GDDI were pooled and their frequencies were compared against the frequency of occurrence of all 12,455 domains. In agreement with previous observation in y for a domain occurrence value of x means y fraction of all domains that occurs x times are gold domains. The tendency is for Gold/All to equal 1 as domain occurrence increases. This supports the notion that GDDIs are significantly enriched with promiscuous domains. This conclusion is not surprising given that the GDDIs are sourced from the iPfam database The bar chart in S. pombe.Because the dataset comprises multi-organisms, there is a possibility that the heavy imbalance between rare and promiscuous domains in the set of all domains may be due to the many organisms with only a handful of proteins  in the da, b) as [N(a)+N(b)]/2. The promiscuity of GDDIs is then compared against the promiscuity of all non-GDDIs. R’s Wilcox.test confirms that GDDIs have significantly larger promiscuity scores than non-GDDIs.The previous test examined the promiscuity of single domains. In the following test, the promiscuity of domain-pairs are examined by quantifying the promiscuity of a domain pair  generates a set of protein-pairs, each of which may or may not be interacting, through the cross-product of their protein sets, i.e. P(a) × P(b). A DDI is either a GDDI or a non-GDDI. The set of DDIs comprise 783 GDDIs and 176,450 non-GDDIs. A PPI prediction for a protein-pair  predicts that protein x interacts with protein y. This prediction is a true positive if 1 can be found in the given set of PPIs and a false positive otherwise.The set of all interacting protein-pairs (PPIs) and the set of all non-interacting protein-pairs (non-PPIs) were generated for each of the 177,233 putative domain-domain interactions (DDIs). A domain-pair (R’s t.test (non-homogeneous variance) and Wilcox.test were used to compare the sizes of the PPI sets generated by GDDIs and by non-GDDIs, and the sizes of the non-PPIs sets produced by GDDIs and by non-GDDIs. The statistical tests confirm that GDDIs generate significantly larger sets of PPIs than non-GDDIs and that GDDIs generate significantly larger sets of non-PPIs than non-GDDIs. Thus, GDDIs generate significantly more true positive and more false positive PPIs than non-GDDIs. More true positive PPIs agrees with the parsimony or the re-use principle for GDDIs, and more false positive PPIs accords with the promiscuity of gold domains.rare domain threshold x, domains which occur ≤ x times in the protein set are classified as rare. While a rare domain may occur in only a handful of proteins, collectively the set of rare domains occur in many proteins. When the rare domain threshold is 2, 54.8% of the proteins in the Riley dataset have at least one rare domain.Promiscuous domains are those which occur frequently in proteins. Rare domains are those which occur infrequently in proteins. The ratio of promiscuous to rare domains varies with the rare domain threshold . At rareonly-rare proteins, i.e. proteins comprising only rare domains, increases, while the proportion of only-promiscuous proteins, i.e. proteins comprising only promiscuous domains, decreases. Both only-rare and only-promiscuous proteins may be single- or multi-domain proteins. Mixed architecture proteins comprise both rare and promiscuous domains and are thus necessarily multi-domain proteins. The proportion of mixed architecture proteins is influenced by the proportions of rare and promiscuous domains. When the rare domain threshold is between 2 and 16, mixed architecture proteins make up the largest proportion (at least a third) of the protein population in the Riley dataset. The peak occurs at rare domain threshold 4 where 37.8% of the proteins have mixed architecture.The concept-based scoring method proposed in this paper does not require or depend on the specification of a rare domain threshold. However, since mixed architecture proteins are the workhorse proteins of the proposed method (Results section), it is reassuring to know that mixed architecture proteins are not rare phenomena in the Riley dataset and that there exists a range of rare domain threshold values where at least a third of the proteins have mixed domain architecture. However for the concept-based scoring method to be feasible at least in principle, the mixed architecture proteins also need to cover gold domains.Up to and including rare domain threshold 11, proteins with mixed architecture provide the largest coverage of all domains  top. HowG by their common attributes and dually a (finite) set of attributes M by their common objects into a (finite) set of partially ordered pairs of sets called  concepts. Implicit is a binary relation I ⊆ G x M which associates objects with attributes.  ∈ I or equivalently g I m denotes object g has attribute m. The triplet  forms the  context within which a FCA is carried out. For small finite examples, a context can be specified completely with a cross-table. The resulting set of concepts, denoted B, forms a concept lattice.Formal Concept Analysis (FCA) G = P, the set of attributes is the set of domains, i.e. M = D, and g I m denotes protein g has domain m, i.e. m ∈ D(g) and dually g ∈ P(m). S. pombe in the Riley dataset. The (fully-labeled) concept lattice depicting this context is given in For the application in this paper, the set of objects is the set of proteins, i.e. This section describes FCA for those unfamiliar with the theory and in enough detail to support the discussion in this paper. The more mathematically inclined are referred to c ∈ B is an ordered pair of sets  such that O ⊆ G, A ⊆ M and the set of all attributes common to all objects in O under relation I is A and the set of all objects with attributes in A under relation I is O. More formally, the last two conditions are A = O′ = {m ∈ M | (µg ∈ O) g I m} and O = A′ = {g ∈ G | (µm ∈ A) g I m} respectively. If this seems a bit chicken-and-egg, these last two conditions can be satisfied by working from the power-set of objects or alternatively from the power-set of attributes; but there exist more efficient FCA algorithms such as Lindig’s C implementation called Colibri-conceptsA concept O is called the extent of concept c, and the set of attributes A is known as the intent of concept c. A concept’s extent is denoted O(c) and its intent as A(c). The prime symbol ′ denotes the mapping from an extent to its intent and vice versa, i.e. let a concept c = , then O′ = A, A′ = O, O = O″, A = O″′, and so on. This pair of maps between the set of extents and the set of intents forms a Galois connection between the two partially ordered sets. The statements O = O″ and A = O″′ are true due to the maximal condition for extents and intents. This implies that if a set of objects (attributes) forms the extent (intent) of a concept, then the set of objects (attributes) uniquely identifies the concept, and conversely a concept unambiguously identifies its extent and its intent.The set of objects B; ≤〉. For two distinct concepts in B ,  ≤  implies O1 ⊂ O2 and dually A1 ⊃ A2. The join (least upper bound) and meet (greatest lower bound) are defined for every pair of non-comparable concepts in 〈B; ≤〉. 〈B; ≤〉 forms a concept lattice. Intuitively, a concept lattice is a two-in-one lattice with a right-side up lattice for the set of extents and an upside down lattice for the set of intents. More formally, a concept lattice is a complete lattice with a top element  and a bottom element . A complete lattice defined on a subset of a power-set is closed under arbitrary joins (in the form of unions) and meets (in the form of intersections) c1 ∨ c2 =  ((O(c1) ∪ O(c2))″, A(c1) ∩ A(c2) ), and the meet (infimum) of two arbitrary concepts c1 ∧ c2 =  ((O(c1) ∩ O(c2), (A(c1) ∪ A(c2))″ ). c1 ∨ c2 is a concept since A(c1) ∩ A(c2) =  (O(c1) ∪ O(c2))′ and  is always a concept. Similarly, c1 ∧ c2 is a concept since O(c1) ∩ O(c2) = (A(c1) ∪ A(c2))′ and  is always a concept. The intersection of any number of extents (intents) always results in an extent (intent). The same is not generally true for unions of extents (intents) c1) ∪ O(c2)) ⊆ (O(c1) ∪ O(c2))″ and (A(c1) ∪ A(c2)) ⊆ (A(c1) ∪ A(c2))″ hold.The set of concepts is ordered by set inclusion 〈c represented as ↓{c} = {x ∈ 〈B; ≤〉 | x ≤ c}. The up-set of c represented as ↑{c} = {x ∈ 〈B; ≤〉 | c ≤ x}. The extent of a concept c is the union of the extent of each concept ∈ ↓{c}. The intent of a concept c is the union of the intent of each concept ∈ ↑{c}. This relationship between concepts makes it possible to reduce the labeling of concepts to objects and attributes specific to a concept (reduced concept lattice.The down-set of  concept . ChanginOA); (ii) with reduced object labels and reduced attribute labels (oa); (iii) with complete object labels and reduced attribute labels (Oa); and (iv) with reduced object labels and complete attribute labels (oA). The oa and oA combinations produce object-reduced concept lattices. The oa and Oa combinations produce attribute-reduced concept lattices. In an object-reduced concept lattice, object labels appear exactly once and the set of object labels (proteins) is finite. Similarly in an attribute-reduced concept lattice, attribute labels appear exactly once and the set of attribute labels (domains) is finite. Each combination is explored in this paper for the proteins and their domains in the Riley dataset. It is possible for two different combinations to produce two different outcomes because the labels of objects and attributes are used instead of the objects and attributes themselves. OL(c) refers to the set of object-labels for O(c), and similarly AL(c) refers to the set of attribute-labels for A(c).There are then four possible ways to label a concept lattice: (i) with complete object labels and complete attribute labels  >1 are Pkinase (5), SAM_2 (2) and Cyclin_N(2)  means a larger extent, and as one goes up in a concept lattice extents increase in size, culminating in the top element whose extent is the entire object set. For lin_N(2)  and theylin_N(2) . In contS. pombe have promiscuous domains (N(d) >1), and these single-domain proteins  reside in concepts one step away from the top element in The position of a protein in an object-reduced concept lattice depends on the promiscuity of its domain(s). Three of the four single-domain proteins for L(c) = ∅ is more likely towards the top of an object-reduced concept lattice. Concepts with multi-domain protein(s) in their extents will tend to have empty attribute-label sets in an attribute-reduced concept lattice unless the multi-domain protein(s) introduces “new” domains. E.g. concept 7 in L(c) = ∅ is more likely towards the bottom of an attribute-reduced concept lattice. The presence of empty object-label sets and empty attribute-label sets influences the number of concept-pairs available to evaluate domain-pairs, and the diversity of CB and PG values produced by the concept-based scoring method.In a reduced concept lattice, there will be concepts with an empty object-label set or an empty attribute-label set. At the very least the object-label sets and attribute-label sets of the top and bottom elements will be, by definition, empty. The more numerous rare domains will “consume” proteins and leave fewer proteins available to fill the object-label sets of concepts with promiscuous domains. Thus an empty object-label set OOne of the earliest methods for detecting over-represented ‘correlated sequence-signatures’ e.g. domain-pairs, in a database of protein-protein interactions used the log-odds of the ratio between observed and expected frequencies to score pairs of sequence-signatures a, b) with the Association method is AM =  log2 [M/(N(a) × N(b))]. M = | {1 | a ∈ D(x) and b ∈ D(y)} | is the number of interacting protein-pairs in the database such that a is a domain of protein x and b is a domain of protein y. N(d) is the number of proteins in the database that has domain d in its domain architecture. The scores are negative in value with a maximum of log2(1) = 0 which is the score for domain-pairs that occur only between interacting protein-pairs, and an undefined minimum of log2(0) which is the score for domain-pairs that occur only between non-interacting protein-pairs. Domain-pairs with larger scores are ranked more highly.Specifically, the score of a domain-pair  = log2 [M/+Z)]. M is defined previously. Z = | {0 | a ∈ D(x) and b ∈ D(y)} | is the number of non-interacting protein-pairs in the database such that a is a domain of protein x and b is a domain of protein y.The Riley dataset comprises proteins from multiple organisms, and both interacting and non-interacting proteins are restricted to those from the same organism. To handle this situation, AM described before, but the scoring is done using pairs of concepts. The concepts used exclude the top and the bottom concepts, and any other concepts with an empty object-label set or an empty attribute-label set. Protein-pairs generated by the object-label sets of a concept-pair, i.e. OL(c1) × OL(c2), are used to compute the log-odds ratio which is then used to evaluate the domain-pairs generated by the attribute-label sets of said concept-pair, i.e. AL(c1) × AL(c2). Essentially, each concept-pair  provides a different context (in the form of protein-pairs) to evaluate a domain-pair. The set of concept pairs {} used to evaluate a domain-pair  is determined by the attribute-label sets, and comprises all distinct concept-pairs where a ∈ AL(ci) and b ∈ AL(cj). Concept-based scoring and ranking is demonstrated in Figures S6, S7, S8 and S9, and Table S2 in The is paper  also useoA and OA), a domain-pair can have more than one score. CB is the largest score found for domain-pair , while PG is the number of unique scores for  which are strictly smaller than CB. Domain-pairs with larger CB scores are ranked more highly. The PG scores help to break ties between domain-pairs with identical CB scores. Given identical CB scores, domain-pairs with larger PG scores are ranked more highly.With concept lattices that are not attribute-reduced (oa concept lattice when many concepts are excluded because of empty object-label or empty attribute label sets), or if its evaluation results in an undefined score, i.e. log2(0). Scoreless domain-pairs are given a negative PG value, and are placed in random order below domain-pairs with CB scores. The Oa scores are identical to those obtained with the Associative method since every domain appears exactly once in an Oa concept lattice and all proteins containing a domain appears in the object-label set of a concept whose attribute-label set has the domain.A domain-pair may be left scoreless if it was not evaluated at all , domains that are more promiscuous have more opportunity to appear in different attribute-label sets. The question then becomes is it better for a promiscuous domain to appear with other promiscuous domains or with rare domains. The results suggest that a promiscuous domain-pair is more likely to improve its score if either one or both of its partner domains appear with rare domains. This is because rare domains tend to produce fewer non-interacting protein-pairs. The Thus, several conditions favourable to the concept-based scoring method arise:The domain-pairs that need to be ranked highly are promiscuous. The GDDIs are promiscuous.not attribute-reduced, i.e. OA and oA. Strictly speaking then, neither oa nor Oa can produce concept-based scores since their PG values are always ≤ 0. The Oa scores are identical to those of the Association method.It must be possible to evaluate a domain-pair with different protein-pairs. This is possible only with concept lattices that are To improve the score of promiscuous domain-pairs through piggy-backing, it must be possible for promiscuous domains to reside with rare domains in one or more attribute-label sets. This is made possible by multi-domain proteins, in particular those which have a mix of promiscuous and rare domains in their architecture, i.e. mixed architecture proteins. Within a sensible range of rare domain thresholds, a substantial proportion of proteins in the Riley dataset are proteins with mixed architecture  and thesThe concept-based scoring method is applied to the Riley dataset using all four types of concept lattices. The algorithm in A, 0.60% in B, 0.21% in C and 0.11% in D. Median promiscuity of GDDIs to DDIs is 7.0∶5.0 in A, 8.5∶5.5 in B, 9.5∶5.5 in C and 15.0∶4.0 in D. With a smaller GDDI to DDI percentage and a much larger GDDI to DDI promiscuity ratio, scenario D is the most difficult of all. The resultant number of PPIs, DDIs, GDDIs and GPPIs for each scenario is summarized in Table S3 in The concept-based scoring method is evaluated under the following four circumstances, all of which are related to input data characteristics. In all scenarios, R’s Wilcox.test confirms that the set of GDDIs is still significantly more promiscuous than the set of non-GDDIs. More promiscuous GDDIs are more likely to survive the changes. GDDIs make up 0.44% of DDIs in Under the default or original circumstance, the complete Riley dataset is used without modification. Since all PPIs in the given input data are used, the probability of including a PPI, Pe is 1.0.PPI data obtained via high-throughput methods are error-prone. To account for the inaccuracies in PPI data, the robustness of computational methods when dealing with PPIs is commonly tested by using Pe <1.0. Ref. The log-odds ratio depends on the set of protein-pairs used and also on the PPI network. To test the robustness of the results against changes in PPIs, the nodes of each organism’s PPI network were shuffled amongst themselves. Node shuffling generates a new set of PPIs, but the number of PPIs per organism and the original PPI network structure  remains unchanged. The new set of DDIs overlaps but is no longer a subset of the original DDIs.Finally, to test the influence of domain architecture on the results, the domains of proteins are shuffled. For this scenario domain repetition within a protein is allowed and a protein becomes a multi-set of domains. The shuffle changes the frequency of domains in a small but still statistically significant way. The new set of DDIs overlaps but is no longer a subset of the original DDIs. Domain shuffling was accomplished with the following steps:Place every instance of a domain in the input data into a sequence, sorted by frequency of occurrence. Domain instances with identical frequency are shuffled amongst themselves.Sort the proteins by their size, i.e. number of domains they contain.Starting from the largest to the smallest protein, assign domains to proteins starting from the least to the most frequently occurring domains. The reason for this is to reduce domain repetition within a protein.The impact of shuffling on domain architecture is shown in a, b) = [N(a)+N(b)]/2 where N(d) is the number of times domain d occurs in a protein set. The GDDI rankings were extracted from the rankings of all DDIs (ranking domain-pairs is described in step 6 of OA and oA ones exhibit a negative correlation between GDDI promiscuity and rank. A negative correlation implies that higher ranking GDDIs tend to be more promiscuous. The relationship is strongly positive (Spearman’s rank correlation rho = 0.88) when the Oa rankings are used. Oa results are identical to the Associative method which is known to penalize promiscuous domain-pairs.Of the four plots in oa to rank promiscuous GDDIs less highly (Spearman’s rank correlation rho = 0.49), but this positive relationship is not as strong as Oa’s because scoreless GDDIs are included in the plot . When the oa concept lattice is used, only 350 of the 783 GDDIs have CB scores. The remaining 55.3% of GDDIs which are scoreless are ranked in random order below the GDDIs with CB scores. But R’s Wilcox.test confirms that the domain-pairs (including non-GDDIs) with CB scores are significantly less promiscuous than the scoreless domain-pairs. Hence, the oa concept lattice tends to ranks more promiscuous domain-pairs less highly.There is also a tendency for oa) concept lattice is the reason many domain-pairs are scoreless. Concepts with an empty object-label set or an empty attribute-label set are ignored when computing scores for domain-pairs (oa concept lattice. The many scoreless domain-pairs means the oa results are unsuitable or too weak for identifying strong interactions in protein complexes. oa performs poorly in the Nye test (Results section). Although oA has empty object-label sets and Oa has empty attribute-label sets, they do not produce any scoreless domain-pairs for the Riley dataset. The number of concepts used by each type of concept lattice for the Riley dataset is given in Table S4 in The increased presence of empty object-label sets and empty attribute-label sets in a reduced , both OA and oA GDDI rankings become positively related with promiscuity, i.e. less promiscuous GDDIs are ranked more highly  and domain frequency (the number of times a domain appears in a protein in a set of input proteins) (oA and OA produce similar results to Oa in Domain shuffling in scenario proteins  and chanroteins) . InsteadD, only 1.69% of the domain-pairs evaluated with oA have PG >0  to evaluate promiscuous domain-pairs, and a decrease in piggy-backing opportunities. ve PG >0  bottom. e highly .GDDI recovery is concerned with finding as many GDDIs as possible while making as few mistakes as possible. The GDDIs are sourced from the iPfam database reliable DDIs, it is preferable to keep the FPR low. Even at FPR = 0.2, there are already 35,290 false positives to eliminate either experimentally or through literature search in scenario A. Further, errors in predicting DDIs can be propagated to other areas that rely on DDI data, such as prediction of protein-protein interactions and identification of protein binding surfaces. Hence, the argument for low FPR or high Specificity. A cost-efficient high-throughput method to eliminate non-interacting domain-pairs with high confidence would render computational efforts to predict reliable DDIs obsolete.For the problem of identifying D, when FPR ≤ 0.2, rankings produced by the non-attribute-reduced concept lattices (OA and oA) have larger TPR or higher Sensitivity than the attribute-reduced concept lattices (Oa and oa). This outcome supports the hypothesis that using concept lattices that are not attribute-reduced is necessary to create a favourable condition for the concept-based scoring method to identify GDDIs. This outcome is robust to changes in the PPI data as evidenced by the results from scenarios B and C. The oa rankings become slightly more competitive when FPR ≥0.25, but the rankings quickly become random as oa is unable to score many domain-pairs due to empty object-label and empty attribute-label sets in the oa concept lattice. For instance, 72.14% of the possible domain-pairs and 55.3% of GDDIs are left scoreless by oa in scenario A. The Oa rankings or Associative method gave the worst GDDI recovery performance.The TPR vs. FPR or ROC graphs are shown in D, up to an FPR as large as 0.5, oA’s GDDI recovery dominates that of OA. A closer look at the top 100 domain-pairs . However, high ranking non-GDDIs do at times interfere with OA’s Nye test results (Results section).Except for the concocted scenario in-pairs  reveals oA domain-pair rankings dominates that of OA is given in oA’s TPR dominates the other three rankings. Of the four rankings , oA contains the most number of GDDIs, DOMINE domain-pairs and 3did domain-pairs in the top 100 domain-pairs  which is not a GDDI and whose interaction is neither recorded in DOMINE nor predicted by 3did. However, a possible 3D structure for the obesity-related protein adipose (adp) involves interaction between TPR and WD40 domains Further evidence that the in-pairs . AmongstoA producing better GDDI recovery results than OA. However, oA is more vulnerable to changes in protein domain architecture than OA. When the domains are shuffled to reduce the number of mixed architecture proteins (scenario D), oA’s TPR which previously has been above 0.2 when FPR ≤ 0.2 drops to less than 0.2. This decline in oA’s GDDI recovery performance supports the notion that mixed architecture proteins play an important role in the success of concept-based scoring.The discussion so far all point to OA’s prioritization of promiscuous domain-pairs, but the effect on GDDI recovery is dampened by OA’s ability to provide many more contexts to evaluate domain-pairs. In scenario D, the oA rankings are produced with only 53.71% of the concepts. In contrast, oA uses 82.38% of the concepts in the other three scenarios . While domain shuffling reduces the range of PG values for OA, the proportion of domain-pairs still able to piggy-back in OA remains above 62%. In contrast, only 1.69% of the domain-pairs evaluated with oA have PG >0  are left scoreless by the highest ranking DDI for a GPPI is a GDDI. It was first performed by Nye et al. This test is concerned with whether OA and oA) produce much higher pass rates than the attribute-reduced concept lattices (Oa and oa). OA’s higher pass rate is attributed to its greater piggy-backing potential  become less able to pass the Nye test. None of the GPPIs that generate more than 50 DDIs pass the Nye test when oa’s ranking is used. For the Oa concept lattice (the Associative method), the cut-off point is even earlier, at 16 DDIs. In contrast, several GPPIs with more than 50 DDIs could still pass the Nye test when ranking is based on concept-based scores computed with the non-attributed-reduced concept lattices (OA and oA).The Nye test results for scenario A is broken down by difficulty in OA ranking than with the oA ranking . This GPPI is supported by only one GDDI, , which has a promiscuity of 100. oA ranks  as the highest domain-pair for GPPI  and so the Nye test is passed. OA ranks  as the second highest, below  which has a promiscuity of 189.5 but is not a GDDI, and so the Nye test is failed. The Nye test is also passed by oa since  is the only domain-pair with a score for GPPI . However, oa’s Nye test performance on a GPPI with this high level of difficulty is more the exception than the norm.With a few exceptions, regardless of difficulty, more GPPIs pass the Nye test with the  ranking . This isy-backed . HoweverOA. These promiscuous Pfam-B domains also suppress GDDI recovery for OA. All domain-pairs in OA’s top 100 that do not involve a Pfam-B domain is either a GDDI, or identified in DOMINE or 3did (Oa). Therefore, there is a need to evaluate the effectiveness of the concept-based scoring method in the absence of Pfam-B domains and on a more current dataset for single organisms. The organisms tested here are yeast (NCBI taxid 559292) and human (NCBI taxid 9606), and the data come from Biogrid (version 3.2.107) A conditions.The previous case with GPPI  illustrates how the presence of promiscuous Pfam-B domains can cloud the Nye test results for  or 3did . HoweverOa is no longer below the y = x line as it was in OA and oA) still dominate that of the attribute-reduced concept lattices (Oa and oa) although now it is no longer as clear as it was in oA’s TPR dominates OA’s TPR , but between attribute-reduced where piggy-backs are impossible and non-attribute-reduced where piggy-backs are possible. The results in OA and oA can still outperform both Oa and oa. The Nye test for yeast and for human is also more successfully passed by both OA and oA than either Oa or oa Click here for additional data file.File S2(XLS)Click here for additional data file.


The mean field model formulated by Liley et al.  describeIn this study, we fix the model parameters to values that have previously been shown to lead to physiologically interesting γ-range activity . We locaAlthough for small system size, spatially homogeneous periodic solutions appear, for system sizes of a few square centimeters we observe centimeter-scale, oscillating patterns, which appear in subcritical Hopf bifurcations and have a strong γ-band component in their power spectrum. An example is shown in Fig. We have implemented a time-stepper in the open-source PETSc environment . It uses


The secondary structure of RNA molecules is intimately related to their function and often more conserved than the sequence. Hence, the important task of searching databases for RNAs requires to match sequence-structure patterns. Unfortunately, current tools for this task have, in the best case, a running time that is only linear in the size of sequence databases. Furthermore, established index data structures for fast sequence matching, like suffix trees or arrays, cannot benefit from the complementarity constraints introduced by the secondary structure of RNAs.We present a novel method and readily applicable software for time efficient matching of RNA sequence-structure patterns in sequence databases. Our approach is based on affix arrays, a recently introduced index data structure, preprocessed from the target database. Affix arrays support bidirectional pattern search, which is required for efficiently handling the structural constraints of the pattern. Structural patterns like stem-loops can be matched inside out, such that the loop region is matched first and then the pairing bases on the boundaries are matched consecutively. This allows to exploit base pairing information for search space reduction and leads to an expected running time that is sublinear in the size of the sequence database. The incorporation of a new chaining approach in the search of RNA sequence-structure patterns enables the description of molecules folding into complex secondary structures with multiple ordered patterns. The chaining approach removes spurious matches from the set of intermediate results, in particular of patterns with little specificity. In benchmark experiments on the Rfam database, our method runs up to two orders of magnitude faster than previous methods.Structator a complete and robust open-source software solution for index-based search of RNA sequence-structure patterns. The Structator software is available at http://www.zbh.uni-hamburg.de/Structator.The presented method's sublinear expected running time makes it well suited for RNA sequence-structure pattern matching in large sequence databases. RNA molecules containing several stem-loop substructures can be described by multiple sequence-structure patterns and their matches are efficiently handled by a novel chaining method. Beyond our algorithmic contributions, we provide with  The discovery of new roles of non-coding RNAs (ncRNAs) has made them of central research interest in molecular biology ,2. Like RNAforrester , 0 ≤ i < n denotes the character of S at position i. Let ε denote the empty sequence, the only sequence of length 0. By n ≥ 0 over ε is denoted by A S = S[0]S[1] ... S[n - 1] and 0 ≤ i ≤ j < n, S[i..j] denotes the substring S[i]S[i + 1] ... S[j] of S. We denote the reverse sequence of S with S-1 = S[n - 1]S[n - 2] ... S[0]. For S = uv, u and u is a prefix of S, and v is a suffix of S. The k-th suffix of S starts at position k, while the k-th prefix of S ends at k. Note that the 0-th suffix of S is S itself and that S[0] is the 0-th prefix of S. The k-th reverse prefix of S is the k-th suffix of S-1. For 0 ≤ k <n, k Sdenotes the k-th suffix of S, and k-th reverse prefix of S.For a sequence RNA alphabet {A, C, G, U}. Its characters code for the nucleotides adenine (A), cytosine (C), guanine (G), and uracil (U). In the following we fix a sequence S over the RNA alphabet S|< 232, such that sequence positions and lengths can be stored in 4 bytes.Let complementary and can form a base pair if and only if non-crossing RNA structure R of length m is a set of base pairs , 0 ≤ i < j < m, stating that the base at position i pairs with the base at position j, such that for all ,  ∈ R: i < i' <j' <j or i' <i < j < j' or i < j < i' <j' or i' <j' <i < j. For the algorithms and methods presented in this paper we only consider this class of structures. For an example of such an RNA secondary structure see Figure stem-loop structure. We call R a stem-loop RNA structure if and only if for all ,  ∈ R : i < i' <j' <j or i' <i < j < j'. Note that due to our definition a stem-loop can contain bulges and interior loops } with an equal number of characters (and ). There is a bijection between the set of (non-crossing) RNA structures R and the set of structure strings H, both of length m, such that for each base pair  ∈ R, H[i] = (and H[j] = ), and H[r] = . for positions r, 0 ≤ r < m, that do not occur in any base pair of R, i.e. r ≠ i ∧ r ≠ j for all  ∈ R. Due to this equivalence we identify both representations.A sequence pattern is a sequence m denote its length |P|. An occurrence of P in a sequence S is a position i, 0 ≤ i < n, such that P[k] = S[i + k] with S[i + k] ∈ φ(P[k]) for all 0 ≤ k < m. An RNA sequence-structure pattern (RSSP) m is a pair of a sequence pattern P and a structure string R, both of length m. A match or occurrence of m in an RNA sequence S is an occurrence i of P in S, such that for all base pairs  ∈ R: S[i + l] and S[i + r] are complementary. Furthermore, define Let Φ = {R, Y, M, K, W, S, B, D, H, V, N} be a set of characters. The IUPAC nucleotide base code introduces the characters in Φ to code nucleotide ambiguity and assigns a specific character class In this paper, structures described by RSSPs are non-branching.affix array is described. This index structure supports efficient unidirectional as well as bidirectional searches and is more space efficient than the affix tree  = 0, and lcpX [i] is the length of the longest common prefix between i ≤ n. Each of the tables lcpF and lcpR requires n bytes and store entries with value up to 255, whereas occasional larger entries are stored in an exception table using 8 bytes per entry [F and lcpR can be accomplished in F and sufR [X . Two important concepts of affix arrays are suffix-intervals and lcp-intervals. An interval [i..j] representing the set of suffixes i ≤ j ≤ n, of width j - i + 1, is a suffix-interval in sufX with depth (prefix length) ℓ ∈ {0,..., n}, or ℓ-suffix-interval, denoted ℓ - [i..j], if and only if the following three conditions hold:For notational convenience, we define nd space . In pracnd space ,33 conster entry . More sper entry ). The coand sufR . In contand sufR  where afX [i] <ℓ;1. lcpX [j + 1] <ℓ; and2. lcpX [k] ≥ ℓ for all k ∈ {i + 1,..., j}.3. lcpℓ - [i..j] in sufX lcp-interval in sufX with lcp-value ℓ ∈ {0,..., n}, or ℓ-interval, if and only if i < j and lcpX [k] = ℓ for at least one k ∈ {i + 1,..., j}.We call a suffix-interval ℓ - [i..j] in sufX , we denote the common prefix of length ℓ of its suffixes Xδ(ℓ - [i..j]) = XS[sufX [i]..sufX [i] + ℓ - 1]. In case of an lcp-interval ℓ - [i..j] in sufX , X δ(ℓ - [i..j]) is the longest common prefix of all suffixes in this interval.For a suffix-interval ℓ - [i..j] in sufX describes simultaneously:In summary, a suffix-interval X by interval borders i and j and depth ℓ. For an example, see the yellow marked region in Figure F.• A location in the index structure sufF in Figure • A  sequence of suffixes SX of length ℓ, namely Xδ(ℓ - [i..j]). That is, for the suffix-interval 4 - [4..6] in sufF in Figure δF(4 - [4..6]) = CUGC.• A substring of XS, namely at positions sufX [i],..., sufX [j]. To give an example, consider Figure F[4] = 10, sufF[5] = 7, and sufF[6] = 4 in SF = AUAGCUGCUGCUGCA.• The occurrences of this substring in S it is sufficient to process lcp-intervals only in sufF. For bidirectional pattern search using affix arrays, described in detail in the next section, we employ information from table sufF as well as sufR. Therefore, we need to associate information of one table to the other. This is done by linking intervals via tables aflkF and aflkR. We observe that there exists a mapping between lcp-intervals in sufF and sufR. This is stated by the following proven lemma [For unidirectional left-to-right search of some pattern in en lemma .Lemma 1 For every lcp-interval q = ℓ - [i..j] in table sufX there is exactly one lcp-interval q-1 = ℓ' - [i'..j'] in table  called reverse lcp-interval of q, such that ℓ' ≥ ℓ and the ℓ - 1-th prefix of  equals (δX(q))-1. The number of suffixes (prefixes) represented by q and q-1 are the same, i.e., j - i = j' - i'.q = (q-1)-1 is not necessarily true. This is stated by the next lemma.We note that the equivalence Lemma 2 If the lcp-interval q-1 with depth ℓ' in  is the reverse of the lcp-interval q with depth ℓ in sufX and ℓ = ℓ', then q = (q-1)-1. Otherwise, if ℓ' > ℓ, then q ≠ (q-1)-1.SF and SR is encoded in tables aflkF and aflkR as follows. Tables aflkF and aflkR store, for each lcp-interval in sufF and sufR respectively, a pointer to the reverse interval in the reverse tables X , defined asThe mapping between intervals in ℓ - [i..j] is an lcp-interval in sufX . Hence, the home position is one of two boundary positions. Strothmann [X ([i..j]) ≠ homeX ([i'..j']) for different lcp-intervals ℓ - [i..j] and ℓ' - [i'..j'].where rothmann  shows thX of string XS$ with total length n + 1 can now be defined as a table in the range 0 to n such that aflkX [homeX (q)] = i', where q is an lcp-interval in sufX and i' is the left border of the reverse interval q-1 = [i'..j'] in X as affix links. Tables aflkF and aflkR occupy 4n bytes each. They can be computed by traversing the lcp-intervals in sufX while simultaneously looking for the corresponding reverse lcp-intervals in et al. , ℓ >0, -m < k < ℓ, describes a substring Xω(v) of X Sof length ℓ - k, defined as the k-th suffix of Xδ(q), i.e. Xω(v) = XS[sufX [i] + k..sufX [i] + ℓ - 1]. At the same time v identifies all occurrences of Xω(v) in XS, namely the positions sufX [i] + k,..., sufX [j] + k.An affix-interval v  = 〈k, q, X〉, we therefore also use the notation X = F and X = R. As an example, consider the affix-interval v = 〈1, 4 - [4..6], F〉 in sufF of the affix array shown in Figure k = 1, q = 4 - [4..6], and X = F. v identifies all occurrences of substring SF at positions sufF[4] + 1 = 11, sufF[5] + 1 = 8, and sufF[6] + 1 = 5. Observe that δF(q) = CUGC due to context k = 1.For Pattern matching using affix arrays means the sequential processing of characters in the pattern guiding the traversal of the data structure. This can be performed in either a traditional left-to-right order resulting in a unidirectional search or in a bidirectional way where character comparison is started at any position of the pattern extending the already matched substring of the pattern to the left or to the right. We will see that bidirectional search using alternating series of left and right extensions is very well suited for fast database search with RNA sequence-structure patterns (RSSPs) containing both paired and unpaired bases. In the following we will explain the two different traversal strategies underlying unidirectional and bidirectional search using affix arrays.S in a unidirectional left-to-right way using information from table sufF only. To search for P , we call the procedure unidir-search of Figure unidir-search. Therefore, in step 0 we start searching for the characters in φ(P[0]) in the suffix-interval q0 = 0 - [0..n] in sufF, which represents all suffixes of S$. In each step k, k ≥ 0, we locate the k + 1-suffix-intervals k qof maximal width, such that P [0..k - 1]d matches δF(kq). For each d ∈ φ(P [k]), this step is performed by a binary search in the suffix-interval qk-1 = ℓ - [i..j] for k q= (ℓ + 1) - [i'..j'], i ≤ i' ≤ j' ≤ j, j' - i' maximal, and S[sufF[i'] + k] = d.Let m steps, if all k qcould be located, δF(mq), m q= m - [r..s], matches the pattern P and the occurrences sufF[r], sufF[r + 1],..., sufF[s] of δF(mq) are reported as occurrences of P in S. Note that in this approach the matched substring of S is extended only to the right and at each step k the occurrences of the already matched prefix are represented by a suffix-interval.After P character by character to the text, where we can freely switch between extending to the left or to the right. Note that as in the case of unidirectional search, ambiguous nucleotides x in the pattern can be handled by enumerating all characters c in the corresponding character class φ(x). We can focus on the situation in the search, whereFor the bidirectional search, we start at some position in r..r' (0 ≤ r ≤ r' <m) of the pattern P is already compared,• a range S matching P[r..r'] are represented by an affix-interval v = 〈k, ℓ - [i..j], X〉 in sufX , and• the occurrences of a substring P[r - 1] or P[r' + 1]). This will result in a new, extended affix-interval xv.• we want to extend X = F and to the right for X = R. However, there are "corresponding" affix-intervals representing the same substring of S but allowing extension to the opposite direction.Like its suffix-interval, an affix-interval directly supports extension of the represented substring in only one direction, namely searching to the left for v, this switch of the search direction requires determining the corresponding affix-interval v' in i = j or v has non-empty context k ≠ 0. There are these two exceptions, since first if i = j, independently of the value of k, Xω(v) is already a unique substring of XS. Second, for a non-empty context k ≠ 0, all occurrences of substring Xω(v) in SX are followed (if k > 0) or preceded (if k < 0) by the same substring If the new search direction differs from the supported search direction of k = 0 and i < j. The affix-interval reverse affix-interval of v = 〈k, ℓ - [i..j], X〉 if and only if j' - i' = j - i, ℓ' ≥ ℓ, and i' and j' of v' are determined via a lookup in table aflkX . We set i' = aflkX [homeX ([i..j])] and j' = i' + (j - i). Observe that ℓ is not necessarily the length of the longest common prefix of all suffixes in [i..j]. For this reason we define ℓlcp = min{lcpX [k] | i < k ≤ j} ≥ ℓ and compute the context of v' as k' = ℓlcp - ℓ. Further, we set ℓ' = ℓlcp. Hence the reverse affix-interval v' is the required corresponding interval of v.Let u of S matching P[r..r'].In our situation, right (left) extension of v by a character c-extension of v, is an operation that computes the affix-interval x vrepresenting all occurrences of a substring uc (cu). It fails, if there is no such substring. We elaborate the cases for right extension. The cases for left extension are symmetric and therefore omitted. For right c-extension of v = 〈k, ℓ - [i..j], X〉, we determine the interval x v= 〈xk, x ℓ- [xi..xj], xX〉 with The X = F and i = j. u is a unique substring S. If S[sufF[i] + ℓ] = c, then x v= 〈k, (ℓ + 1) - [i..j], F〉.• Case X = F and i < j. We determine the minimal x i≥ i and maximal x j≤ j in sufF such that S[sufF[xi] + ℓ] = c and S[sufF[xj] + ℓ] = c by binary search in the suffix-interval ℓ - [i..j]. If x iand x jexist, we set x v= 〈k, (ℓ + 1) - [xi..xj], F〉.• Case The following cases require switching the search direction.X = R, i = j. We evaluate SR[sufR[i] + k - 1]. If SR[sufR[i] + k - 1] = c, set x v= 〈k - 1, ℓ - [i..j], R〉.• Case X = R, i < j, and k = 0. We first determine the reverse affix-interval v' = 〈k', ℓ' - [i'..j'], F〉 of v via a switch of the search direction as described above. Then we compute the minimal x i≥ i' and maximal x j≤ j' via binary search, such that S[sufF[xi] + ℓ'] = c and S[sufF[xj] + ℓ'] = c. If x iand x jexist, we set x v= 〈k', (ℓ' + 1) - [xi..xj], F〉.• Case X = R, i < j, and k >0. We evaluate the (k - 1)-th character of δR(ℓ - [i..j]). That is, if δR(ℓ - [i..j])[k - 1] = c, then we consume the context k by setting x v= 〈k - 1, ℓ - [i..j], R〉.• Case x vcannot be determined.The operation fails if S with an RNA sequence-structure pattern (RSSP) P in S under the complementarity constraints imposed by the structure string R (cf. our definition of RSSP-occurrence). We introduce a search algorithm that checks for complementarity constraints as early as possible in bidirectional search to maximally reduce the search time due to this restriction.Searching a sequence S, each RSSP describes a set of pattern instances, i.e. the set of potential subsequences matching the pattern. Often, there are several patterns that describe the same set of instances. For example, the pattern  ) ) ) describes the same set of instances as  ) )  since the additional base pair  in ( ( ( . . . . ) ) ) does not make the pattern more specific. We will define a pattern to be structure minimal if there is no, in this sense, equivalent pattern containing a true subset of the base pairs. An RSSP structure minimal if and only if for all base pairs  ∈ R it holds thatFor further considerations, we will assume a special 'canonical' form for RSSPs, which we define in the following. Independently of a sequence inconsistent if it does not have any instance. Formally, a pattern is consistent if and only if for each base pair  it holds that P = UAUACACGAN and R = ( ( . . . . . . ) ).R but the bases P[1] = A and P[8] = A are not complementary. An example of a structure minimal and consistent RSSP is  ) ). Note that a pattern can be transformed into an equivalent structure minimal pattern and checked for consistency in Furthermore, a general pattern is called P, R), the non-branching structure R suggests a search strategy, i.e. an order of left and right extensions, which requires switching the search direction at every base pair but makes optimal use of the complementarity constraints due to the base pairs.In this case, one can restrict the search space by comparing the two positions of each base pair immediately after each other. Due to this, the enumeration of characters matching the pattern symbols at each base pair can be restricted to the smaller number of complementary ones. In the search for a sequence-structure pattern this can reduce the number of enumerated combinations of matching characters exponentially. Thus, for structure minimal patterns , where 〈0, 0 - [0..n], F〉 is an affix-interval and r0 is any position in the loop region of the RSSP or any position of a completely unpaired pattern. Then, the procedure traverses the affix-intervals by performing right and left extensions, while at the same time checking base complementarity of paired positions. This verification takes constant time by using a binary table of size To match r0 (or any position if R is empty) for starting our bidirectional search algorithm. However, in order to reduce the combinatorial explosion of the search space due to ambiguous IUPAC characters, it is preferable to match non-ambiguous pattern characters first. To keep the selection simple, we set r0 to the position of the first character c in the possible range such that |φ(c)| is minimal. That is, we start the search with the most specific (least ambiguous) character.In principle, we are free to choose any loop position bidir-search can be extended to support variable-length RSSPs. Such an extended version of bidir-search is provided in Additional file A detailed example of bidirectional RSSP search along with the underlying affix array traversal is provided in Additional file S of length n for an RSSP m < n, where the index structures for S are already computed.We analyze the complexity for searching in a sequence F and sufR, lcpF and lcpR, and aflkF and aflkR. Under our assumption that n <232, each of the four tables sufX and aflkX consumes 4n bytes, and the two tables lcpX are each stored in n bytes . This amounts to a space consumption of 18n bytes for the index structures. The algorithm performs a depth first search, where the depth is limited by m, and therefore requires The bidirectional search algorithm requires tables sufWe assume that F using binary search for locating intervals in z is the number of occurrences of P in S. We remark that this time bound can be lowered at the price of higher memory consumption to Algorithm 1 accomplishes the search for a non-ambiguous pattern ption to  or even tion to , X〉. Consider the following two cases.First, we analyze the time required for performing a single switch of the search direction. Therefore we assume that the current affix-interval is i = j or k ≠ 0. If i = j, S, or, if k ≠ 0, all occurrences of  substring S are followed (if k >0) or preceded (if k <0) by the same substring of length |k| (known as context). Switching the search direction does not require locating the reverse interval of v, because the algorithm can perform the c-extension in the new search direction by consuming context. Therefore, this case requires constant time.(1) Case i < j and k = 0. The algorithm needs to locate the reverse affix-interval v. Interval boundaries i' = aflkX [homeX ([i..j])] and j' = i' + (j - i) of v' are computed in constant time.(2) Case v requires knowing ℓlcp. Then, ℓ' = ℓlcp and k' = ℓ' - ℓ. Without child-tables, we determine ℓlcp by computing the length of the longest common prefix between ℓlcp - ℓ + 1 = k' + 1 character comparisons only, since both suffixes ℓ. With the help of child-tables, ℓlcp is determined in constant time , X〉, with k = 0, i < j, and X = F (X = R), be the current affix-interval. We assume without loss of generality that we perform a left (right) c-extension of v and thus locate the reverse interval t k≥ 0, t ℓ= ℓ + tk, and t j- t i= j - i (see Lemma 1). Observe that t k= 0 implies kt > 0 implies that substring tk, namely To show t vis only located if k = 0, otherwise the context k has to be consumed. Hence there is no reverse interval s ≤ C, s ≠ t, and ks > 0, such that the (s k- 1)-th prefix of t kcan be determined by performing exactly t k+ 1 character comparisons, this implies w in S that matches Note that Therefore, when searching for S, the sequence S must have a certain minimal length. In the case of p = 0, the possible matches are the words in de Bruijn sequence of order m . The order ≪ of the RSSPs in i land i rwe denote the start and end positions of i Ain the multiple alignment, respectively. In practice, An SSD ike Rfam ,4. A matS of length n. A match is represented by a pair p in S. With each collinear, written as chain Consider an RNA SSD i, 1 ≤ i ≤ k - 1.all from global chain score for chain There are two modes to score chains, depending on the nature of the search problem. If the multiple sequence-structure alignment our SSD is derived from and the searched sequences have comparable length, we want the chain to cover as much as possible of the sequence and we define the g and thus the local chain score is defined byThen, the global chaining problem is to find a chain whereO(q log q) time where q is the size of To solve the local chaining problem we use our own implementation of a fast local chaining algorithm described in  with modTo solve the global chaining problem we have developed a new efficient chaining algorithm described next.S1,..., k Sas it occurs when searching a large sequence database. I.e. in case of databases like Rfam k can be in the range of millions. To handle these, we concatenate the single sequences with separator symbols and construct the affix array for the concatenation. For a given SSD i ≤ L, are matched one after the other using fast bidirectional search on the affix array. This results in match sets L is typically in the range of tens while the number of RSSP matches for a particular sequence j Sis in the order of hundreds to thousands if j Sis an mRNA or complete genome sequence. For each match f the following information is recorded:So far our description was based on a single sequence. However, the results described below are based on a large set of sequences i of the RSSP f. This is denoted by f.rssp.• The ordinal number f. This is denoted by f.length.• The length of the RSSP involved in j of the sequence Sj f occurs in. This is denoted by f.seqnum.• The number f in jS. This is denoted by f.pos.• The starting position of f. The weight of f is denoted by f.weight.• The weight i ≤ L, is sorted in ascending order of f.seqnum. Matches with identical sequence numbers are sorted in ascending order of the ordinal number of the RSSP, i.e., by f.rssp. Suppose that b* is the size of b* sequences with at least one RSSP match, the sorting according to the sequence numbers can be done in k* is the number of sequences with at least one RSSP match. As k* ≤ b*, the sorting requires j ≤ k, where jS. As f.rssp = i. In a second sorting step, each j, i bis the size of In an initial sorting step the union lgorithm . Here, kS1, S2,..., k Sone now solves independent chaining problems for sets j ≤ k, of matches sorted according to the ordinal number of the RSSP and the starting position of the matches in jS. Let j be fixed, but arbitrary. For each match f.weight is positive. Hence, an optimal chain ends with a match f such that there is no match f' satisfying f ≪ f'. Similarly, an optimal chain begins with a match f' such that there is no match f satisfying f ≪ f'.For all f'.score of all chains ending with f'. In addition, it computes the predecessor f'.prec of f' in a chain with maximum score ending with f'. To obtain f'.score, one has to maximize over all matches f such that f.rssp < f'.rssp and f.pos + f.length - 1 <f'.pos. This is a two dimensional search problem. As the matches in b is the number of matches in L, and a match for RSSP i. Moreover, matches at position i in the first sequence can be treated as being of equal length because they are matches to the same RSSP i, 1 ≤ i ≤ L, the matches in jS. All these properties allow us to simplify and improve the algorithm of [The chaining problem is solved by a dynamic programming algorithm which tabulates for all matches erved in , and ledrithm of  was deverithm of  in the f• While the algorithm of  requiresj, i bsatisfy • While the algorithm of  requiresL is considered to be a constant, the running time becomes linear in b, where • While the algorithm of  solves ti, 1 ≤ i ≤ L be arbitrary but fixed and assume that all match sets b containing references to the matches. The separation into two loops allows us to get rid of this extra array.To explain our algorithm, let rithm of  where thjS. Let f' be the current element. At this point, all matches f such that f.rssp < f'.rssp have been processed already. In particular, the score f.score and the previous match (if any) in an optimal chain ending with f has been determined. Among the processed matches we only have to consider those matches f satisfying f.pos + f.length - 1 <f'.pos. If there is such a match, one takes the one with maximal score, say f. Then, the optimal chain ending with f' contains the previous match f, and the score is f'.score = f'.weight + f.score. If there is no such match, then the optimal chain ending with f' only consists of f' and f'.score = f'.weight.Now consider the first loop over all elements in f' be the current element to be inserted. As explained in the previous case, one has to make sure that, among the processed matches, one can efficiently determine the match f with the maximum score such that f.pos + f.length - 1 is smaller than some value depending on f'. The processed matches are stored in a linear list which is sorted in ascending order of the position of the matches in jS. Let ≺pos denote this order, that is f ≺pos f" if and only if f.pos + f.length < f".pos + f".length for any matches f and f". If for two processed matches f and f" one has f.pos < f".pos and f.score > f".score, then an optimal chain does not include f". Each chain that uses f" can also use f and increase the chain score. As a consequence, one has to take care that f" is not inserted into the linear list or it is deleted if it was inserted earlier. In this way, f ≺pos f" always implies f.score ≤ f".score for two matches f and f" in the linear list. As the elements to be scored in the first loop and to be inserted in the second loop are ordered in the same way as the elements in the linear list, one can perform the scoring and the insertion loop  by merging two lists of length l1 and l2 in l1 is the number of matches to be scored and inserted and l2 is the length of the linear list involved. Let l1 + l2 ≤ b, one obtains a running time of L such sets, the running time is Now consider the second loop over all elements in BIDsearch), (3) an online variant operating on the plain sequence  for validation of BIDsearch and reference benchmarking, and (4) the efficient global and local chaining algorithms. Algorithm ONLsearch shifts a window of length m = |RSSP| along the sequence of length n to be searched and compares the substring inside the window with the RSSP from left to right until a mismatch occurs. Hence, it runs in BIDsearch and ONLsearch were implemented in the program afsearch. The afconstruct program makes use of routines from the libdivsufsort2 library (see http://code.google.com/p/libdivsufsort/) for computing the sufF and sufR tables in F and lcpR tables we employ our own implementation of the linear time algorithm of [F and aflkR are constructed in F and skpR [We implemented (1) the algorithms necessary for affix array construction, (2) the fast bidirectional search of RSSPs using affix arrays as sketched in Algorithm 2  ) ) ) ) ) ). Allowed base pairs were , , , and . We measured the time needed by BIDsearch and ONLsearch to search for these patterns in the RFAM10 dataset. Results are given in Figure BIDsearch performed very well and was faster than ONLsearch for all parameter combinations. We also investigated the influence of different stem length (data not shown here) and found that the impact on the total running time is negligible. We observe that the advantage of BIDsearch over ONLsearch decreases with increasing loop length l for fixed q. We explain this behavior with the increasing number of affix-intervals that have to be processed for finding all different substrings of the sequences that match the RSSP. However, even for an RSSP with loop length l = 20 containing only structural constraints (q = 0), BIDsearch is still faster than ONLsearch. We further notice that the number of non-ambiguous characters in the loop region has a strong influence on the running time of BIDsearch. That is, by specifying only a few conserved nucleotides in the RSSP's loop region, the running time of BIDsearch is reduced dramatically. For an example of this effect, see the running times of BIDsearch in Figure l = 15 and q ∈ {2, 3, 4}. This renders BIDsearch in particular useful for searching with RSSPs with moderate loop length or existing sequence conservation in the loop region. The speedup factors measured in this experiment were in the range from 1.001 to 78.1 for q = 0 and from 9.28 to 11 × 103 for q = 4. Table BIDsearch over ONLsearch for all investigated combinations of q and l.In a second experiment we investigated the influence of the loop length and the number of non-ambiguous characters in the loop of an RSSP on the running time of BIDsearch for non-artificial real-world RSSPs, we manually compiled a set of 397 RSSPs describing 42 highly structured RNA families taken from the RFAM10 database. These were all families with a consensus secondary structure containing at least 5 stem-loop substructures. We measured the running time needed by BIDsearch, ONLsearch, and the widely used tools RNAMotif [RNABOB [BIDsearch less than 50 seconds to search for the 397 patterns as shown in Table RNABOB and RNAMotif needed more than 2.5 and 3.2 hours respectively to complete the same task. This made for a speedup factor of 196.5 (254.7) for BIDsearch over RNABOB (RNAMotif). Even if we include the time needed for affix array construction, BIDsearch is still faster than RNABOB and RNAMotif.To measure the performance of RNAMotif  and RNAB [RNABOB  to searcBIDsearch when searching for the 397 RSSPs. We observed that BIDsearch is more than 50,000 times faster than RNABOB and RNAMotif for the majority of the patterns and that the total search time required by BIDsearch is dominated by only a small number of patterns. These patterns describe large unconserved loop regions. See Figure S3 in Additional file We also investigated the distribution of speedup factors obtained by BIDsearch and ONLsearch for an increasing size of sequences to be searched. For this, we searched with different RSSPs on random subsets of RFAM10 of different sizes and measured the running time for both algorithms. The results are given in Figure In a further experiment we investigated the scaling behavior of BIDsearch clearly showed a sublinear scaling behavior, whereas ONLsearch scaled only linearly. It took BIDsearch only 566.8 (pattern1), 133.8 (pattern2), and 37.1 (pattern3) milliseconds to search the whole RFAM10 dataset. The obtained speedups of BIDsearch over ONLsearch were in the range from 4.63 (1 MB subset) to 104.79 (full RFAM10) for pattern1, from 12.23 (1 MB subset) to 223.18 (full RFAM10) for pattern2, and from 35.0 (1 MB subset) to 618.37 (full RFAM10) for pattern3. We observe again that the specification of only one or two nucleotides in an RSSP's loop region considerably reduces the running time of the BIDsearch algorithm.In this experiment E. coli [Structator syntax, see Figure Structator delivers 8,619 matches for RSSP1, 1,699 matches for RSSP2, and 142,219 matches for RSSP3. Instead of reporting these matches, Structator computes high-scoring global chains for each sequence containing matches to all three RSSPs. The chains and the sequences they occur in are reported in descending order of the chain score. This procedure resulted in 61 sequences, all belonging to the OxyS family which contains 115 members in total. Hence, by considering only high-scoring chains all the spurious RSSP matches were eliminated. We also described the same three stem-loops in a format compatible with RNAMotif  mode with subsequent global chaining of RSSP matches needed only 3.9 (122.5) seconds to identify all family members, whereas RNAMotif needed 84.7 seconds. The search times for Structator include 0.05 seconds required for the chaining.To demonstrate the effect of global chaining of RSSP matches, we searched with an SSD built for the Rfam family of OxyS RNAs (Acc.: RF00035). OxyS is a small 109-nucleotide long non-coding RNA which is included in response to oxidative stress in E. coli . MembersStructator operating in BIDsearch (ONLsearch) mode and global chaining of RSSP matches it took only 1.3 (138.7) seconds to search RFAM10 with this SSD, where 0.06 seconds were required for the chaining. The computed global chains with a minimum length of 5, computed from the 184,199 single RSSP matches, were ranked according to their global chain score. We observe that the sequences containing the 37 highest scoring chains are exactly all 37 members of the family.We also employed global chaining to detect members of the structurally more complex family of Citrus tristeza virus replication signal (Rfam Acc.: RF00193). Therefore we built an SSD comprising 8 RSSPs, describing 8 of 10 stem-loops the molecule is predicted to fold into. For more information on the molecule's secondary structure and the used descriptor, see Additional file Structator using global chaining for RNA family classification with manually compiled SSDs for 42 Rfam families. For the results of this experiment see Additional file In addition we measured the performance of α(RSSPi) = 10, for 1 ≤ i ≤ 3. Affix array construction for all human chromosomes was accomplished in 12.6 hours by afconstruct. We searched with Structator for the three RSSPs and found 15,090, 1,578, and 14,491 matches for RSSP1, RSSP2, and RSSP3, respectively. For these RSSP matches we computed local high-scoring chains  this task needed 3.1 (633.4) seconds only, including 0.02 seconds for the chaining. RNAMotif also found a single match corresponding to the correct location of the gene, but needed 274.7 seconds. See Figure S5 in Additional file RNAMotif descriptor.As an example of searching a complete genome or whole chromosomes for non-coding RNAs, we searched for the RNA gene Human accelerated region 1F (HAR1F) on both strands of the human genome sequence. HAR1F is one of 49 regions in the human genome that differ significantly from highly conserved regions of the chimpanzee . The conStructator's running time using using BIDsearch with the time needed by a recently published bidirectional pattern search implementation for the same task. The implementation of [BWI, uses a compressed data structure called bidirectional wavelet index. We remark that BWI can only search with a small set of hard-coded patterns, i.e., the user cannot use it to search with his/her own patterns. Moreover, unlike Structator, which provides a full command line interface with many configurable options (see section about the software package), BWI reports neither matching substrings nor matching positions . It onlructures . Neverthurements . BecauseStructator is an open-source software package for fast database search with RNA structural patterns implementing the algorithms and ideas presented in this work. It consists of the command line programs afconstruct and afsearch.afconstruct implements all algorithms necessary for affix array construction, namely a lightweight suffix sorting algorithm for construction of the suffix arrays sufF and sufR, the algorithm for construction of tables lcpF and lcpR [F and aflkR. The program constructs all or if necessary only some of the tables of the affix array for a target database provided in FASTA format and stores them on disk. Therefore the program can also be used to compute only the tables needed for a traditional enhanced suffix array [afconstruct can handle RNA as well as DNA sequences. Moreover, it supports the transformation of input sequences according to user-defined (reduced) alphabets and allows the index construction for transformed sequences. Such personalized alphabets are easily specified in a text file.and lcpR , and theix array . afconstafsearch is the program for performing structural pattern matching. That is, it searches (ribo)nucleic acid sequence databases for entries that can adopt a particular secondary structure. For an overview of the supported RNA sequence-structure patterns (RSSPs), see Figure afsearch, e.g. N standing for nucleotides A, C, G, and U (and T) and R standing for A and G. Besides fixed-length RSSPs with or without ambiguous characters  and maxrightloopextent (mrlex) a variable number of allowed extensions to the left  . This is done by simply specifying a series of RSSPs in the pattern file describing the stem-loop substructures the RNA molecule is composed of in the order of their occurrence in 5' to 3' direction. To incorporate different levels of importance or significance of an RSSP into SSD models and subsequently in the computation of chain scores, RSSP specific weights can be defined in the pattern file. This is particularly useful in the context of RNA family classification where the used SSD may be derived from a multiple sequence-structure alignment or a consensus structure-annotated multiple sequence alignment. Here, it permits the assignment of higher weights to RSSPs describing highly conserved functionally important structural elements occurring in a family of RNAs, and lower weights to RSSPs describing less conserved substructures that occur only in certain members of the family.RSSP matches can be reported directly by afsearch contains all available information of a match or chain of matches, either in a human-readable, or a tab-delimited format. Moreover, afsearch can also report matches in BED format. This allows a direct visualization of the results in e.g. the UCSC genome browser.The output format of Structator software. As part of the software, we give the first publicly available implementation of bidirectional pattern search using the affix array data structure. For the majority of biologically relevant RSSPs, our implementation of BIDsearch shows superior performance over previous programs. In a benchmark experiment on the Rfam database, BIDsearch was faster than RNAMotif and RNABOB by up to two orders of magnitude. Furthermore, in a comparison between BIDsearch and the program of [BIDsearch was faster by up to 2 times. We observed that for RSSPs with long unconserved loop regions, the advantage of BIDsearch over ONLsearch decreases. For such cases, Structator can also employ ONLsearch on the plain sequence data. As a further contribution, we presented for the first time a detailed complexity analysis of bidirectional search using affix arrays. While bidirectional search does not does not improve the worst-case time complexity compared to online search, in practice it runs much faster than online search algorithms and the running time scales sublinearly with the length n of the searched sequences.We have presented a method for fast index-based search of RNA sequence-structure patterns (RSSPs), implemented in the ogram of , which wn bytes of space. This is a significant space reduction compared to the ~ 45n bytes needed for the affix tree. With the program afconstruct we present for the first time a command line tool for the efficient construction and persistent storage of affix arrays that can also be used as a stand-alone program for index construction.Our implementation of the affix array data structure requires only 18Structator, we also introduce a powerful technique to describe RNAs with complex secondary structures. This even allows to effectively describe RNA families containing branching substructures like multi-loops, by decomposition into sequences of non-branching substructures that can be described with RSSPs. Compared to programs like RNAMotif , Structator's pattern description language for RSSP formulation is simple but powerful, in particular in combination with the SSD concept. Beyond the algorithmic contributions, we provide with the Structator software distribution a robust, well-documented, and easy-to-use software package implementing the ideas and algorithms presented in this manuscript.With the new concept of RNA secondary structure descriptors (SSDs) combined with fast global and local chaining algorithms, all integrated into Structator software package including documentation is available in binary format for different operating systems and architectures and as source code under the GNU General Public License Version 3. See http://www.zbh.uni-hamburg.de/Structator for details.The Structator manual. SK developed and implemented the RSSP chaining algorithms and contributed to the manuscript. SW provided supervision and wrote parts of the manuscript. MB initiated the project, provided supervision and guidance, designed/performed the experiments and wrote large parts of the manuscript. RB contributed to the introduction. All authors read and approved the final manuscript.FM implemented the presented algorithms and wrote parts of the manuscript and the Supplemental material. Additional file Click here for file


An error was introduced in the preparation of this article for publication. In the Methods section, there is an error in the fourth equation of the "Statistical Analyses" section. There is a "/" missing between (b-c) and b. The correct equation is N=(b-c)/b x 100.


In a “block-copying paradigm”, subjects were required to copy a configuration of colored blocks from a model area to a distant work area, using additional blocks provided at an equally distant resource area. Experimental conditions varied between the inter-area separation  and the complexity of the block patterns to be copied. Two major behavioral strategies were identified: in the memory-intensive strategy, subjects memorize large parts of the pattern and rebuild them without intermediate visits at the model area. In the acquisition-intensive strategy, subjects memorize one block at a time and return to the model after having placed this block. Results show that the frequency of the memory-intensive strategy is increased for larger inter-area separations  and for simpler block patterns. This strategy-shift can be interpreted as the result of an optimization process or trade-off, minimizing combined, condition-dependent costs of the two strategies. Combined costs correlate with overall response time. We present evidence that for the memory-intensive strategy, costs correlate with model visit duration, while for the acquisition-intensive strategy, costs correlate with inter-area transition  times. Getting around in a constantly changing world relies on contributions from multiple behavioral or cognitive processes competing for common resources such as metabolic energy, information processing capacity, or processing time. The allocation of such resources to the individual processes requires some sort of “decision making” or “executive function” For memorization and processing of visual information, the visual working memory (WM) is an essential resource. WM can be defined as a system for maintaining and processing a certain amount of information temporarily ‘Just-in-time’ acquisition of visual information by repeated looking as opposed to keeping more information in memory Trade-offs between gaze movements and WM use have been studied in a number of tasks which involve looking back and forth between two or more experimental areas In the above studies of the acquisition vs. memory trade-offs, acquisition behavior amounted to gaze shifts carried out by movements of the eyes and/or the head. Clearly, this kind of motor behavior is executed within small spatial scales and within very short periods of time. For example, a saccade is usually performed in less than 100 ms In the present study we approach this question by replacing the gaze-shift component of the block-copying task The goal of the present study was to assess and quantify strategy trade-offs in a walking paradigm including manipulations in acquisition and memorization costs. With overall higher costs for locomotion compared to saccadic motor behavior, we expect a general shift of the task solving strategies towards a greater reliance on memory. Furthermore, we hypothesize that both types of cost manipulations are capable of modulating walking strategies. This modulation can be modeled as a linear optimization of combined costs.48 naïve subjects volunteered to participate in this study . Participants were under- and postgraduate students from the University of Tübingen and their ages ranged from 19 to 36 years (mean 24.6 years). Participants were paid for their participation and gave informed written consent. This research was performed in accordance with the 1964 Declaration of Helsinki and was approved by the ethical committee of the University Hospital of Tübingen.Each block pattern was composed of six quadratic LEGO® duplo® blocks of six different colors, forming a connected pattern. Blocks were placed on a grid where every block covered two by two grid cells. To control memorization demands, we varied the neighborhood rules between adjacent blocks. In “simple patterns”, adjacent blocks shared a complete edge ; in “comThree separate areas arranged in the shape of an equilateral triangle were defined for model (M), resource (R), and workspace (W) operations cf. . SubjectTo record the walking trajectories produced by subjects while operating between the three areas  an infrared light-based tracker system  with 6 degrees of freedom was used. This device tracked a rigid target object  that was fixed on a special helmet participants had to wear. The temporal tracking frequency of the system was 60 Hz. From the trajectories, area visits were detected using a criterion of 0.5 m. For analysis of the areas subsequently visited during a trial, subjects' body-in-space positions were evaluated cf. .In a 2×2 factorial, between subject design, the 48 subjects were randomly assigned to one of four different experimental groups . Across these groups, we varied pattern complexity and locomotion distance in order to quantify the trade-off between memorization and acquisition intensive strategies. Thus, the four experimental groups include the combinations: simple/near, simple/far, complex/near, and complex/far.Subjects were familiarized with the particular task operations in a pre-test where one four-block pattern had to be copied. Afterwards, each subject had to complete ten experimental trials consecutively, each with a different block pattern. Subjects started and finished each trial by standing still for about ten seconds at the central point of the triangular configuration cf. . After eSubjects could visit each of the three areas in any sequence as often as necessary for replicating the current block pattern. They were instructed to do so as quickly and reliably as possible. No feedback was given to subjects during the experiment, neither about their copying performance nor about the walking strategies.In all conditions subjects had to follow three rules throughout the copying task: i) it was forbidden to carry more than one block while walking between the areas ii) once a block was placed at the workspace area no repositioning was allowed, and iii) after placing the last block at the workspace area subjects had to go back immediately to the central point of the area configuration denoting the end of the trial.We analyzed the errors made during copying the block patterns for each of the four experimental groups. Errors were analyzed as pattern errors and block errors. Pattern errors denote the proportion of incorrectly copied patterns. Block errors denote the amount of single blocks copied at a false position or with the false color and were calculated as the proportion of the total number of blocks in the 10 patterns (n = 6 blocks×10 patterns = 60) averaged over subjects. A pattern was considered erroneous if at least one block error occurred.The focus of the present study was to identify and characterize the trade-off between WM load and re-acquisition via locomotion. For that purpose a method was needed that assessed the extent of memory usage quantitatively. Following Ballard et al. A sub-strategy is a section of the walking sequence between two subsequent visits of the workspace area; it usually (except for the ‘control’ sub-strategy W-M-W) corresponds to the placement of one block. As an example, the sequence of visited areas …-W-R-W-M-R-W-R-M-W-… was divided into the sub-strategies W-R-W, W-M-R-W, and W-R-M-W. The W-R-W  sub-strategy is the strategy with the highest memory involvement. Here, all required information concerning (at least) the next block  is retrieved from memory and no additional visit of the model is needed. In contrast, the sequence W-M-R-M-W  denotes the sub-strategy with the lowest memory involvement, where subjects walk to the model area to look for the color of the next block. After picking a suitable block from the resource area, they come back to the model area once again, presumably because they did not remember the position of that block. Thus, color is remembered during the M-R step and position during the M-W step of the sequence. In contrast to the ‘high-memory’ sub-strategy, the associated walking distance is doubled when using the ‘just-in-time’ strategy. The ‘low-memory’ sub-strategy (W-M-R-W) uses an intermediate amount of memory and path length; color has to be remembered during the M-R step, while position has to be remembered during both the M-R and R-W steps. An overview of the different sub-strategies (together with the name that is used throughout the manuscript) and the demands on memory is provided in Starting at the central point, the initial sub-strategy of all subjects for all trials was M-R-W. Since this sequence was a simple consequence of the task design, it was not significant for later analysis of walking strategies and thus excluded. In addition to the analysis of walking trajectories, the time subjects needed for walking and the time spent at each area was recorded. The duration for an area visit was measured from entering until leaving a catchment area defined by a radius of 0.5 meters around each operating area. The time subjects spent at the model area was separated into time for the first and subsequent visits in the course of the analysis of memorization processes.2 = 1.4, p = .71) nor on block errors .Task performance was quantified by the number of pattern errors and block errors. In all conditions participants showed a high level of performance, i.e., on average 9 out of 10 patterns were copied correctly . Furtherp2 = .44) and the distance condition  = 56.81, MSE = 173.54, p<.001, ηp2 = .56). We found no interaction between these two factors  = 1.17, MSE = 173.54, p = .28). Independent of pattern complexity, subjects needed significantly more time to reproduce the patterns in the far as compared to the near distance condition  sub-strategies were ‘high-memory’ (WRW) and ‘low-memory’  and distance  = 4.9, MSE = 187.42, p<.05, ηp2 = .1). Thus, with an increase in distance and a decrease of pattern complexity the trade-off between acquisition and memory is shifted towards the memory-intensive sub-strategy ‘high-memory’. This shift of the trade-off is also reflected in the other main sub-strategy ‘low-memory’. Here, the influence of both factors was inverted  = 15.38, MSE = 160.16, p<.001, ηp2 = .26; distance: F = 7.33, MSE = 160.16, p<.01, ηp2 = .14). A decrease in distance and an increase in pattern complexity induced a shift of the trade-off between acquisition and memory towards the ‘low-memory’ sub-strategy. For both main sub-strategies no significant interaction between the two factors (distance and complexity) was found  = .75, MSE = 187.42, p = .39; ‘low-memory’: F = .44, MSE = 160.16, p = .51). However, based on the estimated parameters of the linear model of the ANOVA on the proportion of ‘high-memory’ (goodness of fit = Radj2 = .35) a difference in distance condition depending on model complexity was apparent, indicating an ordinal interaction.Statistical analysis reveals an influence of both, distance and pattern complexity on the frequency of occurrence of the two main sub-strategies. By calculating a two-factorial ANOVA for the proportion of ‘high-memory’, we identified main effects of pattern complexity  = 22.15, MSE = 187.42, p<.001, ηIn order to assess the degree to which the model is used in the various conditions, we analysed the duration and number of model visits and the number of blocks processed after the initial model visit of each trial.p2 = .31; distance: F = .9, MSE = 39.92, p = .35). Within each experimental condition, subjects spent significantly more time for the first model visit compared to subsequent model visits (p2 = .23). No significant influence of the distance  = .012, MSE = 1.297, p = .91) was found.The total time for all visits subjects spent at the model  dependedl visits . Similarl visits . HoweverTo estimate the amount of memory subjects allocated in the different experimental conditions we analyzed the number of model visits per trial and the number of consecutive ‘high-memory’ cycles after the initial model visit . Initialp2 = .24; distance: F = 2.14, MSE = .55, p = .15). Subjects visited the model for memory refresh more often in the complex conditions (means: 2.77 for the far and 3.19 for near distance) than for simple patterns (means: 2.08 for the far and 2.28 for near distance).The number of model visits per trial was dependent only by pattern complexity but not by the distance  = 13.86, MSE = .55, p<.001, ηp2 = .24) and distance condition  = 5.33, MSE = .89, p<.05, ηp2 = .11). No significant interaction between these two factors was found.We found an overall higher number of consecutive ‘high-memory’ cycles in the simple pattern conditions (means: 3.2 for the far and 2.57 for near distance) than in the complex pattern conditions (means: 2.18 for the far and 1.54 for near distance). Within pattern complexity, also an increase of memorization was found with an increase of the distance. The two-factorial ANOVA with pattern complexity (complex vs. simple) and distance condition (far vs. near) as factors showed significant main effects of pattern complexity  = 14.03, MSE = .89, p<.001, ηIn a trial by trial analysis of blocks processed after the initial model visit, we found moderate correlations between the duration of the initial model visit and the number of consecutive ‘high-memory’ cycles. Correlations reached significance for all conditions .The influence of trial order on the occurrence of used sub-strategies was analyzed by comparing the frequencies of sub-strategies per trial within each experimental group. No influence of the trial order on any of the walking sub-strategies could be found. Working memory (WM) supports many higher cognitive functions by maintaining representations of a limited number of items, and by selecting and attending to those representations which are most relevant for the current task. Memory items include multiple types of information, such as verbal or visuospatial, as well as task rules and are represented in WM through sustained patterns of neural activity The main objective of the present study was to investigate if such a trade-off, already described for the balancing of WM and gaze movements The main result of this paper is that alternative behavioral strategies, which can be used to achieve the same goal, are used to various extents if task parameters are varied. In our block-copying task, the main behavioral strategies are i) initial acquisition of large amounts of information and subsequent operation from memory (“memory-intensive strategy”), and ii) the repeated acquisition or re-acquisition of smaller pieces of data and subsequent processing of these individual packages (“acquisition-intensive strategy”). The memory-intensive strategy is realized by long initial model visits and high proportions of the ‘high-memory’ sub-strategy. The acquisition-intensive strategy, in contrast, uses relatively short initial model visits and a high proportion of the ‘low-memory’ sub-strategy.For more complex patterns, the use of the memory-intensive strategy is reduced, while more re-acquisition steps are performed. A similar but weaker effect was found for walking distance: if walking of larger distances is required, the frequency of the memory-intensive strategy increases.The observed pattern of the dependence of sub-strategy usage  on condiAs compared to the findings of Ballard et al. As measures of memory involvement in a given strategy we analyzed the duration and number of model visits (i.e. the time spent with the encoding of information) and the number of blocks processed , see These findings support the idea that longer and more frequent model visits lead to the build-up of extended memory which in turn is available for the processing of blocks. This idea can be tested directly by calculating within each condition trial-by-trial correlations between the duration of the first model visit and the number of blocks processed consecutively. Here we did indeed find moderate but significant correlations.The first model visit was significantly longer than the second or any later visit, indicating that it plays a special role. Conceivably, the subject could use this first visit to built up a memory of some global features of the pattern which is not necessarily used for immediate block positioning but may be useful for later information intake. Examples of such features are chunks or templates known to reduce working memory load 5 = 210, corresponding to an information content of 10 bits per pattern. A similar calculation for the complex pattern, where 16 positions of the second block are possible, yields 165 = 220 possible patterns corresponding to an information content of 20 bits per pattern. If we assume that the same total memory capacity is used for both cases, we should expect that the number of blocks processed per model visit in the simple conditions is about twice the number processed in the complex conditions. As can be seen from Simple and complex patterns require different amounts of storage capacity. A simple estimate of this capacity can be derived from the following consideration: Suppose the first block of a simple pattern is placed. For the next block, there are four possible positions observing the neighborhood rules described in the The trade-off idea states that behavioral strategies are selected so as to minimize certain task and condition dependent costs. Generally two types of costs are considered: i) processing time and ii) energy consumption or other, non-temporal measures of cognitive effort. The soft constraints hypothesis Gray et al. As revealed by the overall response times , harder a–d in the figure. The model is in good general agreement with the data shown in In this analysis, the individual cost functions are assumed to be stable and known by the trade-off controller. This is in line with the trade-off stability reported in In summary, the analysis presented in