While tree-oriented methods for inferring orthology and paralogy relations between genes are based on reconciling a gene tree with a species tree, many tree-free methods are also available . Recently, the link between orthology relations and gene trees has been formally considered from the perspective of reconstructing phylogenies from orthology relations. In this paper, we consider this link from a correction point of view. Indeed, a gene tree induces a set of relations, but the converse is not always true: a set of relations is not necessarily in agreement with any gene tree. A natural question is thus how to minimally correct an infeasible set of relations. Another natural question, given a gene tree and a set of relations, is how to minimally correct a gene tree so that the resulting gene tree fits the set of relations.We consider four variants of relation and gene tree correction problems, and provide hardness results for all of them. More specifically, we show that it is NP-Hard to edit a minimum of set of relations to make them consistent with a given species tree. We also show that the problem of finding a maximum subset of genes that share consistent relations is hard to approximate. We then demonstrate that editing a gene tree to satisfy a given set of relations in a minimum way is NP-Hard, where “minimum” refers either to the number of modified relations depicted by the gene tree or the number of clades that are lost. We also discuss some of the algorithmic perspectives given these hardness results. Homologous gene are grouped into gene families, usually via sequence similarity methods. Moreover, homologous genes can be orthologous, if their parental origin is a speciation, or paralogous, if it is a duplication. Orthologous gene are considered to be more similar in function than paralogs, a conjecture known as the orthology conjecture S-consistgiven in . The S-cR of relations: minimize the number of modified relations, and maximize the number of genes inducing an S-consistent set of relations. The first problem is shown to be NP-complete, while the second problem is shown to be not approximable within factor In this paper, two criteria are considered for correcting a set X, a tree T for X is a tree whose leafset X. We denote by V(T) the set of nodes and by r(T) the root of T. Given an internal node u of T, the subtree rooted at u is denoted clade ofu. A node u is an ancestor of v if u is on the (inclusive) path between v and the root, and we then call v a descendant of u. If v is a strict descendant of u, and if u and v are connected by an edge of T, then v is a child of u. The lowest common ancestor (lca) of u and v, denoted u and v are separated if and only if U of nodes. Let T to T rooted as T displays All trees considered in this paper are assumed to be rooted. They are not necessarily binary, but we assume that all nodes are of degree at least three, except possibly the root that can be of degree two. Given a set speciation, which is the separation of one species into distinct ones. A species tree S for a species set gene family is a set of genes mapping functiongene tree for Spec) or duplication (Dup) accordingly.Species evolve through DS-tree for G is a tree with G as a duplication or a speciation node  of G with children v can be made to be binary while preserving Let DS-tree v, adding a child v and adding S-consistent. It is readily seen that If v, v has a child v is a speciation, v, adding two children v, then adding S-consistent. As for v, by the definition of v is S-consistent. The species for every other node remaining unchanged, we conclude that S-consistency and does not modify If instead DS-trees in Fig. S-consistent. For example, the speciation node z in v, c, d and w, which are pairwise separated in S. Notice that, from Definition G is a DS-tree, then the lca of two leaves of G belonging to the same species must be a duplication node. The converse is not true. For example, in the S-consistent gene tree G of Fig. We can verify that both mentclass2pt{minimmentclass2pt{minimfull if DS-tree G induces a full set A set of orthology/paralogy relations on relation graphR on a gene family uv of the edge set E(R) of R as an orthology relation between u and v, and each missing edge (non-edge) DS-tree G, denoted by R(G), is the graph with vertex set R in Fig. We adopt the graph representation considered in  for fullmentclass2pt{minimDS-tree for a gene family R is satisfiable if there exists a DS-tree G such that A essed in . The fol A relation graph R is satisfiable if and only if Ris-free, meaning that no four vertices of Rinduce a path of length three. A relatR and DS-tree does not necessarily represent a true history for R should be consistent with the species tree, according to the following definition.For example, in Fig. mentclass2pt{minimS, a relation graph R for S-consistent if and only if R is satisfiable by a DS-tree G which is itself S-consistent.Given a species tree R in Fig. S-consistent. Notice that S-consistency implies satisfiability. Results from [S-consistent graphs through Theorem L of size three. For xy|z the unique triplet T on L for which S-consistency of a relation graph in terms of For example the graph lts from  completeentclass1pt{minimaentclass1pt{minimaentclass1pt{minimaLet G be a DS-tree and S be a species tree. Then for any internal node v of G, there exist leaves x, y of  such that both the following hold: (1)  and (2)x and y of s(x) and s(y), then (1) holds. Thus if we suppose that (1) does not hold, then We first show that (1) must hold for some x and y satisfying (1). Suppose that (2) does not hold for x and y, i.e. z is separated from v (i.e. x and z satisfy both (1) and (2). Otherwise, s(x) from s(y). In this last case, y and z are the leaves of interest, ending the proof. Now, take Let be a satisfiable relation graph and let S be a species tree. Then R is S-consistent if and only if S displays all the triplets of G be an S-consistent gene tree satisfying R, and let S must display the s(x)s(y)|s(z) triplet. Let xy|z must be a triplet of G. Moreover, since s(z) must be separated in S, implying that s(x)s(y)|s(z) is a triplet of S.R is satisfiable by some DS-tree DS-tree G satisfying R, in terms of speciation. That is, if v that has a speciation child w, we obtain v and w (delete w and give its children to v). Note that we have R. We obtain the DS-tree G by repeating this operation until we cannot find such a v and w. We claim that if S displays the triplets of G is S-consistent.v be a speciation node of G, and let v. By the construction of G, v is a speciation while R, which implies that S displays the S displays the S. We deduce that all child pairs of v are from separated species, and hence that G is S-consistent. Let entclass1pt{minimaS-consistent as the path of length 2 containing ac|b, while the triplet displayed by S is ab|c.As an example, the graph entclass1pt{minimasubgraphH is a graph with H and some subgraph ofHinduced byH with vertex-set induced subgraph of H if there is a subset I is another graph, we say H is I-free if there is no I. Finally, for some edge set We end this section with additional notations that will be of use later. A R in order to reach satisfiability and S-consistency. Two optimality criteria are considered: (1) the minimum number of edges that need to be removed; (2) the maximum number of genes that can be kept.We raise the issue of leaving out a minimum of information from a relation graph S does not make the problem of removing the minimum number of edges leading to a Based on the same construction used in paper , we showsertion) , here weMinimum edge-removal consistency problem:Input: A relation graph R for a gene family S and an integer k;Output: “Yes” if and only if there exists an S-consistent subgraph R with The Minimum Edge-Removal Consistency Problem is NP-Complete, even if for any distinctS-consistent. It is also clear that verifying if W, does there exists W ? We assume that Given entclass1pt{minima problem : given aW and Z, we construct R and S by first defining the species set X and Y. Then the species set is S is obtained by first connecting S has exactly g and its species s(g). We then define R with R, where R has Given arbitrary R described above can clearly be done in polynomial time. We now show that W and Z admit an exact 3-cover if and only if R admits an S-consistent DS-tree after the deletion of at most Notice that the construction of W, R in which all edges between W-clique are those belonging to a R missing from S-consistent, we use Theorem i, inducing the S. Therefore there exists an S-consistent gene tree  and we consider the approximation complexity of the corresponding optimization version.Maximum Node Consistency problem:Input: A relation graph R for a gene family S and an integer k;Output: “Yes” if and only if there exists an S-consistent induced subgraph R with n is the number of nodes of R). We refer the reader to [We show that Maximum Node Consistency is hard to approximate within a factor eader to  for a deR. Denote m genes: First, we define the set of genes S. First consider S as any binary tree over m leaves m leaves (thus S has R. We make no distinction between s(g).Now, we define the species tree R. Moreover, for each t with Now, define the relation graph R, S). Denote by Let Let  be two subsets of nodes of a solution of Maximum Node Consistency over instance  such that  and . Then there is no edge with one endpoint in and the other in q such that S contains the triplet Assume on the contrary that there is some Now, we are ready to prove the main result of this section.Let a graph H be an instance Maximum Independet Set with m nodes, and let  be the corresponding instance of Maximum Node Consistencywith  nodes. ThenGiven an independent set  of H, we can compute in polynomial time a solution of Maximum Node Consistency of size at least Given a solution of Maximum Node Consistency on instance  of size at least k m, we can compute in polynomial time an independent set of H such thatH and define a solution of Maximum Node Consistency on instance  of size at least S-consistent.Consider an independent set R, S) of size at least km, and consider the subsets k such sets, otherwise The case j, consider the set j that belong to some subset k nodes, since otherwise k, thus concluding the proof. Given an index AP such that OPT is the optimal value of a solution to the problem  be the corresponding instance of Maximum Node Consistency with H, runs R, S) and returns the independent set resulting from Lemma H on which R. Let We get the following as an immediate corollary, which will be of use later:The decision version of Maximum Node Consistency is NP-Hard, even on instances in which for any distinct S, an S-consistent DS-tree G for G according to C in a minimal way. The goal is thus to find a DS-tree C such that the difference between G and G, S are all binary trees.In this section, we are given a gene family Maximum Homology Correction problem:Input: A species tree S, an S-consistent DS-tree G for a gene family k, a set O of orthology and a set P of paralogy relations;Output: “Yes” if there exists an S-consistent DS-tree The Maximum Homology Correctionproblem is NP-Complete, even if S, G and  are required to be binary.S-consistency can be done in polynomial time, as well as counting the common orthologs/paralogs relations (the set of relations is quadratic in size). For our reduction, we use the Minimum Edge-Removal Consistency problem for the case of a gene family with at most one gene per genome, which is NP-Hard by Theorem S, a relation graph R with V(R) in bijection with k, we construct an instance of the Maximum Homology Correction Problem, i.e. a species tree DS-tree G, an orthologous set O and paralogous set P.The problem is clearly in NP, as verifying G by mimicking S - that is by first copying S and its leaf labels, then replacing each leaf G by the gene S is binary, then so is G. All internal nodes of G are labeled as speciations, so all genes of R(G) is a clique. Finally, let G in order to satisfy P.Let S-consistent subgraph R obtained by removing at most k edges if and only if there is an DS-tree O and P with at most G.We show that that there is an R and S. Then there exists a S-consistent DS-tree k edges from R. By Lemma S is binary, then so is R(G).mentclass2pt{minimP and breaks at most P as non-edges, plus at most k other non-edges. Thus k edges from Maximum clade correction problem:Input: A gene tree G, a species tree S, a set O of orthology and a set P of paralogy relations and an integer k;Output: “Yes” if there exists an S-consistent DS-tree O and P such that G and k clades in common.S, G and G and grafting operations to add leaves to G and satisfy a prescribed relation without breaking other relations.Notice that if connectingx and giving it Grafting a new leaf x to a tree T corresponds to adding x to x as a new child of some node u of T; (2) connecting T with x; (3) subdividing an edge uv and adding x as a child of the newly created vertex.Given two trees Let G be an S-consistent gene tree, for some species tree S. Let x be a gene not in G and y be some gene in G with  Then there exists a gene tree obtained by grafting x to G such that the following conditions are satisfied:x and y are orthologs in  andis S-consistent;x to r(G) under a common parent yields the desired result. So we assume u of G such that adding x as a child of u satisfies the three conditions of the Lemma, then we are done. So assume that there is no node u to which we can add x as a child.If uv be an edge of G, and suppose that we graft  x on uv to obtain p the parent of x on u is the parent of p (i.e. p has children x and v). Note that if S-consistency. We will find such a uv that guarantees this S-consistent), and that S-consistent. This will prove the Lemma.Let g be the lowest ancestor of y in G such that g does not exist was handled in the beginning of the proof. Now suppose that g that is also an ancestor of y. Note that x to p being the parent of x and g the parent of p. Then, S by our choice of S-consistency. Also, Let x as a child of g to obtain a new tree S-consistent. By the choice of g, only the latter is possible, implying that all children of g are from separated species in G, but not in g such that s(x). Note that x to p being the parent of x. We have g remain unchanged in S-consistent, again satisfying all required conditions. So instead suppose that The Maximum Clade Correction Problem is NP-Complete, even if S, G and  are required to be binary.S-consistency and comparing the set of clades from G and R, S and k be the Maximum Node Consistency instance, letting R be the relation graph with S the species tree and k an integer. Let S. Construct G as follows: first consider G as any binary tree with n leaves R. Then, replace each leaf G has exactly Verifying mentclass2pt{minimR admits an S-consistent induced subgraph with at least k nodes if and only if G, O and P admit an S-consistent DS-tree O and P such that G and We show that H be a DS-tree satisfying S-consistent. By Lemma S is binary, then so is H. Now, since H corresponds a subtree G. Then build a DS-tree H by replacing each leaf H by Dup . It is not hard to imagine a branch-and-bound algorithm that solves the problem. Call an induced subgraph H of a relation graph Rbad if it is a S. Each As the problems considered in this paper are all computationally hard, only non-polynomial exact algorithms or approximation algorithms avenues can realistically be explored. Let us generalize the Minimum Edge-Removal Consistency problem to the minimum ear time , and cleplied in  to desigR. The idea is simple: as long as R has a bad subgraph H, remove every edge incident to a vertex of H and continue. Even though this is the best known approximation algorithm so far, it has the undesirable effect of isolating many vertices, motivating the exploration of alternative algorithms. One direction would be to consider existing ideas on the problem of satisfiability, i.e. finding the minimum number of editings required to make a graph As for approximations, an algorithm proposed in  can be dposed in .S and partial sets of relations O and P, verifies if there exists an S-consistent gene tree O and P and if so, constructs one among the set of all possible solutions. In ordre to correct a gene tree G, we can envisage an extension of this algorithm allowing to provide G as input, and pick, among the solutions of the algorithm the one which is the closest to G (either in terms of common homology relations or clades).As for gene tree correction, we have developed in  a polynoA gene tree induces a set of orthology and paralogy relations between members of a gene family, but the converse is not always true. In this paper we have shown that attempting to modify a set of relations as least as possible in order to ensure consistency with a species tree leads to the formulation of NP-Complete problems. Moreover, even assuming that the given relations are error-free, it remains computationally difficult to correct a gene tree in order to fit the given set of relations. As various model-free methods are available to infer orthology and paralogy, these correction problems are of practical biological interest. A future direction would be to explore the exact branch-and-bound algorithms and heuristics mentioned in the last section, and design fast approximation algorithms for the relation graph and gene tree editing problems.


It grows optimally at 88 °C by coupling the reduction of Fe(III) oxides to the oxidation of a wide range of compounds, including long-chain fatty acids, and also grows autotrophically with hydrogen and Fe(III). It is the first archaeon reported to use a direct contact mechanism for Fe(III) oxide reduction, relying on a single archaellum for locomotion, numerous curled extracellular appendages for attachment, and outer-surface heme-containing proteins for electron transfer to the insoluble Fe(III) oxides. Here we describe the annotation of the genome of “G. ahangari” strain 234T and identify components critical to its versatility in electron donor utilization and obligate Fe(III) respiratory metabolism at high temperatures. The genome comprises a single, circular chromosome of 1,770,093 base pairs containing 2034 protein-coding genes and 52 RNA genes. In addition, emended descriptions of the genus “Geoglobus” and species “G. ahangari” are described.The online version of this article (doi:10.1186/s40793-015-0035-8) contains supplementary material, which is available to authorized users. It is an obligate Fe(III)-reducing archaeon isolated from the Guaymas Basin hydrothermal system and grows at temperatures ranging from 65–90 °C, with an optimum at about 88 °C , which have pyruvate kinase and alpha/beta domains and are homologous to an A. fulgidus enzyme with pyruvate kinase activity in vitro CC but lack the LRCXXC motif characteristic of most CcmH proteins. However, they all flank a duplicate CcmF-encoding gene found only in “G. ahangari” (GAH_01093), G. acetivorans [F. placidus [.) oxides . The mosochromes . Archaeas absent . Similarain CcmH , we identivorans , and F. placidus .c biogenesis pathway, the iron-reducing Archaeoglobales, Geoglobus and Ferroglobus species, also have more c-type cytochromes than any other archaeon, and many of these c-type cytochromes have multiple heme groups [“G. ahangari” contains 21 genes ; and G. acetivorans has 16 c-type cytochromes (8 with more than 5 heme groups). By contrast, Archaeoglobus species, which do not use Fe(III) electron acceptors . Other species such as A. profundus and A. sulfaticallidus have only 1 monoheme c-type cytochrome and A. fulgidus has 3 c-type cytochromes (none of which have more than 5 heme groups).In addition to having a distinct cytochrome e groups , 18, 26.c-type cytochromes of “G. ahangari” was also investigated. The ExPASy TMPred program [c-type cytochromes have at least 1 transmembrane helix, consistent with their association to the cytoplasmic membrane. One of these c-type cytochrome proteins (GAH_00504) was predicted to be extracellular. We also identified several c-type cytochromes  with predicted sizes once in mature form  matching those reported for outer-surface heme-containing proteins required for the reduction of insoluble Fe(III) oxides, but not soluble Fe(III) citrate, by “G. ahangari” [c-type cytochromes likely function as the terminal electron carriers between the cells and the oxides.The subcellular localization of the putative  program  revealedhangari”  . We identified a number of ub iquinone/menaquinone biosynthesis proteins in the genome of “G. ahangari”  respiration as their primary metabolism. Fe-S proteins and ferredoxins are regarded as some of the most ancient of electron transfer carriers [c-type cytochromes, some of them localized to the outer surface, is consistent with a mechanism evolved for efficient extracellular electron transfer in hot environments.In addition to carriers  and alsocarriers , which i“G. ahangari” strain 234T is only one of three members of the Archaeoglobales capable of dissimilatory Fe(III) respiration. Furthermore, it is an obligate Fe(III) reducer that grows better with insoluble than soluble Fe(III) species. Consistent with this, the genome contains a large number of c-type cytochromes within and on the cell surface, as well as other redox-active proteins such as thermostable ferredoxin and Fe-S proteins. The paucity of c-type cytochromes within non-Fe(III) respiring members of the Archaeoglobales (Archaeoglobus species) is consistent with the physiological separation between these archaea and F. placidus,G. acetivorans, and “G. ahangari”, which can gain energy for growth from the reduction of Fe(III) electron acceptors. Additionally, some genes required for both dissimilatory sulfate and nitrate metabolisms are absent in “G. ahangari” and G. acetivorans. This supports the physiological separation of Geoglobusspp. from F. placidus, which is capable of Fe(III)-, thiosulfate-, and nitrate respiration, and from Archaeoglobus species which are primarily sulfur-respiring organisms. Genomic data also support the reported physiological similarities between “G. ahangari” and other Archaeoglobales such as autotrophic growth with H2 via the reductive acetyl-CoA/Wood-Ljungdahl pathway and the use of similar electron donors, including short- and long-chain fatty acids. Noteworthy is the fact that genomic evidence supports the synthesis of the methanogenic coenzyme-F420 in “G. ahangari”, which is responsible for the characteristic fluorescence detected in all Archaeoglobusspp. except for “G. ahangari” or F. placidus. Hence, the genome sequence of “G. ahangari” provides valuable insights into its physiology and ecology as well as into the evolution of respiration within the Archaeoglobales.Geoglobus” genus and “Geoglobus ahangari” species was accepted for publication with extenuating circumstances at several culture-collection agencies. Thus, upon the original publication “G. ahangari” strain 234T was accepted only at a single agency. In addition, the G + C mol% determined from the complete genome sequence (53.1 mol%) differs from that originally published (58.7 mol%), representing a discrepancy of over 5 mol%. This publication thus warrants an emended description of the genus Geoglobus and the type species, “Geoglobus ahangari”.The initial publication  of the “Geoglobus” is the one provided by Kashefi et al. [The description of the genus “i et al. , with thi et al. . The G +Geoglobus ahangari” is the one provided by Kashefi et al. [T and has been deposited at three culture collection agencies, which include the Deutsche Sammlung von Mikroorganismen und Zellkulturen (DSM-27542), the Japan Collection of Microorganisms (JCM 12378), and the American Type Culture Collection (BAA-425).The description of the species “i et al. , 2, with


Given a graph G = , the problem is to partition the vertex set V into {V1, V2,…, Vh}, where each Vi is maximized, such that for any two vertices x and y in Vi, there are k edge-disjoint paths connecting them. In this paper, we present an algorithm to solve this problem for all k. The algorithm preprocesses the input graph to construct an Auxiliary Graph to store information concerning edge-connectivity among every vertex pair in O(Fn) time, where F is the time complexity to find the maximum flow between two vertices in graph G and n = ∣V∣. For any value of k, the k-edge-connected components can then be determined by traversing the auxiliary graph in O(n) time. The input graph can be a directed or undirected, simple graph or multigraph. Previous works on this problem mainly focus on fixed value of k.The problem of finding  Graph connectivity is a fundamental problem in computer science, which has many background applications in the real world. For example, reliability is one of the major concerns in communications networks: if a network is reliable, the network would still work when some nodes or edges fail. Reliability in communication networks can be represented by the connectivity between each pair of nodes. In social networks, computing the closeness among people is a very important problem, which also relates to the connectivity of the networks. There are many other applications which are related to the connectivity of networks, e.g., finding web pages of high commonality in internet searching; finding protein complexes and gene clusters in computational biology, etc.In theoretical computer science, graph connectivity has been well studied for more than forty years. It has a strong relationship with the problems of maximal network flow and minimal cut.undirected graph G = , where V is the vertex set and E is the edge set, an edge set Ec, which is a subset of E, is an edge-cut of nodes x, y ∈ V, if removing all edges in Ec disconnects x and y in G. We say that x and y are k-edge-connected in G if there is no edge-cut disconnecting x and y with cardinality strictly less than k. In other words, there are at least k edge disjoint paths connecting x and y. From the definition, if two vertices x and y are i-edge-connected, they must be j-edge-connected for any integer j < i. If any vertex pair is at least k-edge-connected in G, we say that graph G is k-edge-connected. For example, in Given an G is k-edge-connected, we are interested in finding subsets of vertices which are k-edge-connected in G. Note that each vertex is k-edge-connected to itself. For example, in social networks, finding a group of people who are strongly connected is more important than computing the connectivity of the entire social network. In an undirected graph G = , a vertex set V′ ⊆ V is a k-edge-connected component if it is a maximal subset of V such that for any two vertices x, y ∈ V′, x and y are at least k-edge-connected in G. For example, in a, b, c, f, g} is a 3-edge-connected component. It is easy to see that k-edge-connectivity is an equivalence relation in V. Thus, the set of the k-edge-connected components forms a partition of V. The 2-edge-connected and 3-edge-connected components for the example in a, b, c, d, e, f, g} and {a, b, c, f, g}, {d}, {e} respectively. The collection of k-edge-connected components is a partition of V immediately implies that every k-edge-connected component is unique and maximal.Instead of just determining whether the entire graph G = , let u, v ∈ V. Let kuv be the maximum number of edge-disjoint directed paths from u to v, and kvu be the maximum number of edge-disjoint directed paths from v to u. Then, u and v are k-edge-connected, where k = min{kuv, kvu}. Notice that the directed paths from u to v and the directed paths from v to u may not be edge-disjoint. Moreover, kuv and kvu may be different. A k-edge-connected component of G is a maximal subset V′ of V such that ∀u, v ∈ V′, u and v are at least k-edge-connected.For a directed graph edge-cut of u and v is an edge set whose removal destroys either all the directed paths from u to v or all the directed paths from v to u. A min-cut of u and v is an edge-cut of u and v with minimum cardinality. If u and v are k-edge-connected, the cardinality of their min-cut is k. In b, c, f, g} are 2-edge-connected.An Problem Statement: Given a graph G = , find the k-edge-connected components of G for every k ≥ 1.k-edge-connected components of undirected graphs has been extensively investigated, especially for some small k.Finding k = 1, this problem is equivalent to finding the connected components of G, which can be done in linear time by a single breadth first search or depth first search.For k = 2, based on depth-first search, Tarjan presented an algorithm for finding all 2-edge-connected components in linear time in [For  time in .k = 3, the first linear time algorithm was given by Galil and Italiano [G into tree edges and back edges. The tree edges form the DFS-tree while each back edge forms a cycle with a path in the DFS tree. Thus, back edges may be considered one by one to increase the connectivity between vertices.For Italiano . Their mItaliano  were givk, Matula [G =  in O(∣V∣∣E∣) time. He also showed that given k in advance, testing whether a graph is k-edge-connected can be done in O(k∣V∣2) time. Nagamochi and Watanabe [k-edge-connected components in a direct or undirected graph G =  given k in advance. Using the reduction in [k-edge-connected undirected graph G =  has a k-edge-connected spanning subgraph G′ =  with ∣E′∣ = O(k∣V∣). They showed that the time complexity can be reduced to O(∣E∣ + k2∣V∣2) for undirected graphs. This result is also based on an interesting observation: if the cardinality of a cut  is strictly less than k, then the edge connectivity of any two vertices x and y such that x ∈ X and y ∈ V − X is also strictly less than k. In this case, there is no need to consider such vertex pair  since x and y must be in different k-edge-connected components. Thus, O(∣V∣) time for finding min-cut of vertex pairs is sufficient. In database community, k-edge-connectivity problem is also a well studied research topic [O(hl∣E∣) time algorithm, where h is the height of the decomposition tree of the graph, and l is a small number which is less than ∣V∣.For general , Matula  gave an Watanabe  gave an ction in , i.e., ach topic , 10. Basu, v ∈ V can be represented by the cardinality of the minimum -cut in G, if the minimum -cut can be computed efficiently for all pairs of vertices, the edge connectivity problem could be solved efficiently. The cut tree (a.k.a. Gomory-Hu tree) [u, v)-cut for all pairs of vertices. A cut tree is a tree T = , where each edge  ∈ ET has a weight which represents the cardinality of the minimum -cut in G, and the cardinality of the minimum -cut for any two vertices s and t ∈ V is the minimum edge weight on the path connecting s and t in the cut tree T. To construct the cut tree, there are mainly two methods from Gomory and Hu [β cut-tree to show the edge connectivity between vertices in directed graphs. From the β cut-tree, the minimal of the maximal number of edge disjoint paths for any pair of vertices can be found easily. Schnorr [β cut-tree by calculating the maximal flow O(∣V∣ log ∣V∣) times, which was improved to 3∣V∣ − 3 times by Gusfield and Naor [Since the edge connectivity of two vertices Hu tree)  is a gooy and Hu  and Gusfy and Hu , respecty and Hu  introduc Schnorr  construcand Naor .k-edge-connected components when k is given in advance, especially for some small k, e.g., k = 2, k = 3, or the input graph is an undirected graph. Our algorithm can give answers for all possible values of k, and for both directed and undirected, simple graph or multiple graph.Previous works focused on finding the k-edge-connected component can be solved by executing an algorithm for max-flow  time algorithm proposed by Goldberg and Tarjan [O(n5) time algorithm to get the min-cut of any two vertices. The other method is to use the global min-cut algorithm proposed by Stoer and Wagner [O(mn + n2 log n) time. If the min-cut capacity is more than k, the graph is a k-edge-connected component; otherwise, any two vertices separated by the cut cannot be in the same k-edge-connected component. In the worst case, the global min-cut algorithm can be executed n − 1 rounds, leading to an O(n2m + n3 log n) time algorithm.If the capacity of each edge is regarded as one, computing  theorem ). Since d Tarjan , we can d Wagner , which fk-edge-connected components for all k in a directed or undirected, simple or multiple graph. We use an s-t max-flow algorithm as the basic procedure which is executed 2n − 2 rounds to construct an auxiliary graph to store information concerning the edge-connectivity between all vertex pairs of the input graph. The time complexity to construct the auxiliary graph is O(Fn), where F is the time required to compute the maximal flow between two vertices in graph G, e.g., for the maximal flow algorithm of Ford and Fulkerson [F = O(fm), where f is the maximal value of all pairs of maximal flows, and for the algorithm by Goldberg and Tarjan [F = O(n3). Furthermore, any improvement made on F automatically implies improvement on the time complexity of our algorithm.In this paper, we give a simple algorithm to find the ulkerson , F = O time by a simple scan over the auxiliary graph.After the auxiliary graph is constructed, for any value of auxiliary graph, which could be used to efficiently find all k-edge-connected components for all k is introduced. In Section 3, the procedure Construction for constructing the auxiliary graph is presented. The correctness proof and the complexity analysis are given in Section 4 and Section 5, respectively. Finally, we give the concluding remark in Section 6.In Section 2, a data structure, called k-edge-connected components (k ≥ 1) is a partition of V. Moreover, each k-edge-connected component is the union of a collection of (k + 1)-edge-connected components. k-edge-connected components of the graph in k ∈ {1, 2, 3} . Consider k = 2. The 2-edge-connected component {b, c, f, g} is the union of the collection of 3-edge-connected components {b}, {c}, {f}, {g}. For the remaining three 2-edge-connected components, each of them is the union of the collection consisting of the component itself. For k = 1. There is only one 1-edge-connected component {a, b, c, d, e, f, g} which is the union of the collection of 2-edge-connected components {a}, {d}, {e}, {b, c, f, g}.It is well-known that the set of k-edge-connected components, it is possible to store the k-edge-connected components, for all k ≥ 1, in a compact form in a data structure which we call auxiliary graph. The auxiliary graph, henceforth denoted by A, is a weighted undirected tree with vertex set V. Let h be the smallest integers such that the h-connected components are singletons. Let Ah be the edgeless spanning forest of A. Then the collection of all h-edge-connected components of G is the collection of vertex sets of the connected components of Ah (each of which consists of a single vertex). Let Ah−1 be a spanning forest of A obtained from adding the edges of weight h − 1 of A to Ah. Then the collection of all (h − 1)-edge-connected components of G is the collection of vertex sets of the connected components of Ah−1. In general, for k < h, let Ak+1 be a spanning forest of A such that the collection of all (k + 1)-edge-connected components of G is the collection of vertex sets of the connected components of Ak+1. Let Ak be a spanning forest of A obtained from adding the edges of weight k of A to Ak+1. Then the collection of all k-edge-connected components of G is the collection of vertex sets of the connected components of Ak.Owing to this hierarchical structure of a}, {b}, {c}, {d}, {e}, {f}, {g}). The collection of all 2-edge-connected components is represented by the spanning forest induced by the edge set consisting of edges of weight two. The collection of all 1-edge-connected components is represented by the spanning forest induced by the edge set consisting of edges of weight one or two which is the auxiliary graph itself. Notice that in this subgraph, the edges of weight one connect subtrees that represent 2-edge-connected components.Auxiliary Graph:A =  is a weighted undirected tree, which represents the edge connectivity among vertices in G. Two vertices u, v ∈ V is k-edge-connected in G if k is the minimum edge weight on the path connecting u and v in A (see Lemma 4). A is constructed by procedure Construction.k-edge-connected components, for any value of k.In the next section, we shall explain how to construct the auxiliary graph and how to generate the collection of all s-t max-flow algorithm, e.g., Ford and Fulkerson algorithm [To determine the weights of the edges in the auxiliary graph, we can use any lgorithm .G is undirected, then each of its edges will be replaced by two directed edges with opposite orientations.Since the max-flow algorithms run on directed graphs only, if the input graph onstruction (see Algorithm 1), on receiving a graph G = , a vertex s (the source) and a set of available vertices N (vertices that can be chosen as the sink), the algorithm randomly picks a vertex t ∈ N − {s}, and runs the max-flow algorithm to determine the max-flow from s to t. If G is a directed graph, it would also run the max-flow algorithm to determine the max-flow from t to s because the two max-flow values can be different. After this step, if G is directed, we will obtain two min-cuts,  and . Since the connectivity of s, t in a directed graph is the minimum of the s − t max-flow and t − s max-flow, the smaller of these two max-flows, say x, is assigned to edge  as the connectivity between s and t. We also set  to the corresponding min-cut  is already the desired min-cut). Then, an edge  with weight x is added to the auxiliary graph A. The procedure then calls itself recursively, first with S as the set of available vertices and s as the source, and then with T as the set of available vertices and t as the source. The recursive calls terminate when S or T is reduced to a single vertex.In procedure CAlgorithm 1: Construction, s, N)IfN = {s} Return.  t from N − {s}. Randomly pick a vertex x, S, T) ≔ s-t max-flow.  ≔ s-t max-flow.  with weight x to A Add edge  Construction CA is constructed, for each query k, the k-edge-connected components can be easily determined as follows: traverse the auxiliary graph A and delete all edges with weights less than k. Then, each connected component in the resulting graph represents a k-edge-connected component in G. Since the number of edges in the auxiliary graph A is n − 1 (see Lemma 5), a search on A can be done in O(n) time, e.g., running the depth-first-search.After the auxiliary graph Lemma 1. The connectivity of vertices is transitive. Let C denote the connectivity between p and q. Let C = x, C = y, C = z, thenProof. Suppose to the contrary that z < min. Let C be a min-cut for a and c. Then ∣C∣ = z < min ⇒ ∣C∣ < x and ∣C∣ < y. Since ∣C∣ < x, C is not an edge-cut of a and b. Therefore, there is a directed path from a to b and a directed path from b to a after C is removed. Similarly, ∣C∣ < y implies that there is a directed path from b to c and a directed path from c to b after C is removed. It follows that there is a directed path from a to c and a directed path from c to a after C is removed. This contradicts the assumption that C is a min-cut for a and c.onstruction, the vertex set of G is partitioned into subsets S and T such that s ∈ S and t ∈ T.After executing Procedure CLemma 2. For any s′ ∈ S and t′ ∈ T, C ≤ C.Proof. Let C(x → y) to be the value of the minimal cut between x and y in graph G. Let x1 = C(s → t) and x2 = C(t → s). W.l.o.g., x1 ≤ x2, thus, the connectivity between s and t is x1 and from Procedure Construction, the partition S and T are computed according to the cut from s to t such that s ∈ S and t ∈ T. For any s′ ∈ S and t′ ∈ T, we have C(s′ → t′) ≤ C(s → t), thus, the connectivity between s′ and t′ satisfies C = min{C(s′ → t′), C(t′ → s′)} ≤ C(s′ → t′) ≤ C(s → t) = C.Lemma 3. A). ThenLet w denote the weight of an edge  in the auxiliary graph A = C, for every edge . As shown in C = x, C = y, C = z.x first. Since y has not yet been determined, we must have s ∈ S; t, u ∈ T . From Lemma 2, we have z = C ≤ C = x. After y is determined, two cases are to be considered.Without loss of generality, assume we determine the value of y ≥ x: Since z ≤ x, then min = x ⇒ z ≤ min. But from Lemma 1, we have z ≥ min. We thus have z = min ⇒ C = min{C, C} = min, w).If y < x and we need to prove that y ≥ z. By Lemma 1, C ≥ min{C, C} ⇒ y ≥ min{x, z} ⇒ y ≥ z as z ≤ x. It follows that z ≤ y < x which implies that z ≤ min{x, y} = min{C, C} = min{w, w}.Otherwise, Lemma 4. The auxiliary graph A is a tree and the connectivity between any two vertices in G is given by the minimum weight on the path connecting them in A.Proof. First, we shall prove that A is a tree by induction on ∣N∣ .N∣ = 1 or 2, A is clearly a tree. Suppose A is a tree for ∣N∣ < h(≥ 3).When ∣N∣ = h. Procedure Construction is invoked first, resulting in the edge  and the min-cut . Then Construction and Construction are invoked. Since ∣S∣,∣T∣ < h, by the induction hypothesis, the auxiliary graph with vertex set S and that with vertex set T are trees. The two trees and the edge  forms a tree which is the auxiliary graph with the vertex set N(= S∪T).Consider ∣G is given by the minimum weight on the path connecting them in A, we apply induction on the length of the path.Next, to prove that the connectivity between any two vertices in x, y be any two distinct vertices. Since A is a tree, there is a distinct path P connecting them in A. Suppose P consists of h edges and h+1 vertices, such that the length of P is h, and P = (x =)v0v1v2…vh(= y). If h = 1, the lemma clearly holds true. If h = 2, the lemma follows from Lemma 3.Let h < k(≥ 3). Consider h = k. Whenever an edge is added to the auxiliary graph A, it always connects a node with an isolated vertex in the current configuration of A, making the isolated vertex a leaf node of the updated A. Thus, when the path P = (x =)v0v1v2…vh(= y) is constructed, either v0 or vh is the newly added vertex. W.l.o.g., let vh be the newly added vertex. From the induction hypothesis, C = min{w∣1 ≤ i < h − 1}. Consider the three vertices x, vh−1 and vh, similar to the proof of Lemma 3, it is easily verified that C = min{C, C}. It follows that C = min{min{w∣1 ≤ i < h − 1}, C} = min{min{w∣1 ≤ i < h − 1}, w} = min{w∣1 ≤ i < h}.Suppose the lemma holds true for Theorem 1. The algorithm returns the correct auxiliary graph.Proof. Immediate from Lemma 4.onstruction is used to construct the auxiliary graph A. Then, in the query phase, A is used to compute the k-edge-connected components for any query k.Our algorithm works in a preprocessing-query manner. In the preprocessing phase, Procedure CLemma 5. There are n − 1 calls of procedure Construction.Proof. Since each call of procedure Construction adds an edge to A, and there are n − 1 edges in the finished A , there are n − 1 calls of procedure Construction.Theorem 2. The preprocessing phase takes O(Fn) time and the query phase takes O(n) time per query, where F is the time to compute the maximal flow for two vertices in graph G.Proof. By Lemma 5, procedure Construction is called n − 1 times. In procedure Construction, the basic algorithm for finding the maximal flow and runs in O(F) time is executed for n − 1 times. Therefore, the preprocessing phase takes O(Fn) time. , where f is the maximal value of all pair of maximal flows).lgorithm  to compuA, the query time is thus O(m + n). Since the vertex set of A is V and ∣V∣ = n, and m = ∣EA∣ = n − 1 , the query time is O(n).Since each query initiates a DFS traversal over k-edge-connected components of a graph, for all k, is presented. The algorithm performs a preprocessing over the input graph to construct an auxiliary graph which is a tree in O(Fn) time, where F is the time complexity to find the maximum flow in the graph. Clearly, any improvement made on F automatically implies improvement to the time complexity of our algorithm. Every subsequent query asking for the k-edge-connected components for any k(≥ 1) can be answered in O(n) time by traversing the auxiliary graph A. The input graph can be a directed or undirected, simple or multiple graph.In this paper, an algorithm for finding all k-edge-connected problem has been studied by the database community [X is k-edge-connected in G if for any two vertices in X, there are at least k-edge disjoint paths within the subgraph G(X) of G induced by X. This variant was motivated by finding cores (or cliques) in large scale networks, which is a fundamental database problem. Note that k-edge-connectivity remains an equivalence relation under this new definition of k-edge-connectivity on induced subgraph, and the vertex partition is always a refinement of the vertex partition of the k-edge-connected components.There are other interesting problems concerning edge-connectivity that are worthwhile to study. For example, a variant of ommunity . In this


Owing to technical challenges, extant SV discovery algorithms either use one signal in isolation, or at best use two sequentially. We present LUMPY, a novel SV discovery framework that naturally integrates multiple SV signals jointly across multiple samples. We show that LUMPY yields improved sensitivity, especially when SV signal is reduced owing to either low coverage data or low intra-sample variant allele frequency. We also report a set of 4,564 validated breakpoints from the NA12878 human genome.  Differences in chromosome structure are a prominent source of human genetic variation. These differences are collectively known as structural variation (SV), a term that encompasses diverse genomic alterations including deletion, duplication, insertion, inversion, translocation or complex rearrangement of relatively large  segments. While SVs are considerably less common than smaller-scale forms of genetic variation such as single nucleotide polymorphisms (SNPs), they have greater functional potential due to their larger size, and they are more likely to alter gene structure or dosage.Our current understanding of the prevalence and impact of SV has been driven by recent advances in genome sequencing. However, the discovery and genotyping of SV from DNA sequence data have lagged far behind SNP discovery and genotyping because they are fundamentally more complex. SVs vary considerably in size, architecture and genomic context, and read alignment accuracy is compromised near SVs by the presence of novel junctions  between the ‘sample’ and reference genomes. Moreover, SVs generate multiple alignment signals, including altered sequence coverage within duplications or deletions (read-depth), breakpoint-spanning paired-end reads that align discordantly relative to each other (read-pair), and breakpoint-containing single reads that align in split fashion to discontiguous loci in the reference genome (split-read) ,2. TheseAn approach that integrates multiple signals allows for more sensitive SV discovery than methods that examine merely one signal, especially when considering heterogeneous samples and/or low coverage data, because each individual read generally produces only one signal type . The impact of improved sensitivity is particularly acute in low coverage datasets or in studies of heterogeneous cancer samples where any given variant may only be present in a subset of cells. However, even with high coverage data, integration of multiple signals can increase specificity by allowing for more stringent criteria for reporting a variant call.Here, we present a novel and general probabilistic SV discovery framework that naturally integrates multiple SV detection signals, including those generated from read alignments or prior evidence, and that can readily adapt to any additional source of evidence that may become available with future technological advances.Our framework is based upon a general probabilistic representation of an SV breakpoint that allows any number of alignment signals to be integrated into a single discovery process . We define a breakpoint as a pair of bases that are adjacent in an experimentally sequenced ‘sample’ genome but not in the reference genome. To account for the varying level of genomic resolution inherent to different types of alignment evidence, we represent a breakpoint with a pair of probability distributions spanning the predicted breakpoint regions  of the mapped reads Figure A. The siOnce the evidence from the different alignment signals is mapped to breakpoint intervals, overlapping intervals are clustered and the probabilities are integrated . Any clustered breakpoint region that contains sufficient evidence (based on user-defined arguments) is returned as a predicted SV. The resolution of the predicted breakpoint regions can be improved by trimming the positions with probabilities in the lower percentile of the distribution .It is well established that variant calling is improved by integrating data from multiple samples ,6,14,15.We implemented LUMPY in an open source C++ software package (available at ) that isWe compared LUMPY’s performance to three other popular and actively maintained SV discovery packages: GASVPro , DELLY  and PindDetection performance was measured using both simulated data and previously published Illumina sequencing data from the widely studied NA12878 CEPH individual. The first simulation measured each tool’s basic detection capabilities in a prototypical scenario by simulating 2,500 homozygous variants from various SV classes at random genomic locations. The second simulation assessed the power of each tool to detect 5,516 known deletion variants present at varying allele frequencies within a mixed sample, as often occurs in heterogeneous tumors. Lastly, the analysis of SVs in the NA12878 genome assessed the performance of each tool on real data containing sequencing errors and other detection confounders that are difficult to simulate. In addition, the analysis of NA12878 (and her parents) measured the effect of considering multiple samples, prior SV knowledge, and third-party CNV predictions on LUMPY’s performance. In each case, we measured performance in terms of sensitivity and false discovery rate (FDR) by comparing the predicted SV breakpoints to either known breakpoints or split-read alignments from long-reads (Pacific Biosciences (PacBio) and Illumina Moleculo) that span the breakpoint. Predictions that either overlapped known variants or had sufficient long-read support were considered true positives, and all other predictions were considered false positives .The simulated results were base on alignments generated by NOVOALIGN, and the NA12878 results were based on BWA alignments. In both cases, YAHA was used to generate split-read alignments. LUMPY was also tested using alternative read alignment pipelines using either BWA-backtrack or BWA-MEM for paired-end alignment, and BWA-MEM for split-read alignment . In this experiment we compared SV detection performance in the NA12878 individual by analyzing the Illumina Platinum Genomes dataset, which represents approximately 50X coverage of the NA12878 genome . We additionally subsampled this dataset to approximately 5X coverage to assess SV detection in low coverage scenarios.et al.  is the relative probability that position b.l.s + i is one end of the breakpoint (similar for b.r); and b.v is the breakpoint variety. Within the context of this method, breakpoint variety determinations are based on the orientation of the evidence. It is important to note that while a breakpoint may be labeled as a deletion when it contains evidence from a paired-end sequence alignment with a +/-orientation, the breakpoint may in fact be the result of some other event or series of complex events.A breakpoint is a pair of genomic coordinates that are adjacent in a sample genome but not in a reference genome. Breakpoints can be detected, and their locations predicted by various evidence classes such as paired-end sequence alignments or split-read mappings. To support the inclusion of different evidence classes into a single analysis, we define a high-level breakpoint type as a collection of the evidence that corroborates the location and variety  of a particular breakpoint. Since many evidence classes provide a range of possible breakpoint locations, we represent the breakpoint’s location with a pair of breakpoint intervals where each interval has a start position, an end position, and a probability vector that represents the relative probability that a given position in the interval is one end of the breakpoint. More formally, a breakpoint is a tuple b and c in the set of all breakpoints B where b and c intersect , then b and c are merged into interval m, b and c are removed from B, and m is placed into B. The evidence set m.E is the union of the evidence sets b.E and c.E.If there exist two breakpoints m.l and m.r would be to let m.l.s = max and m.l.e = min, similar for m.r. However, if a spurious alignment is merged into a set of genuine breakpoints, the resulting breakpoint interval can be ‘pulled’ away from the actual breakpoint. The impact of an outlier can be minimized or eliminated once the full set of corroborating alignments is collected for a given breakpoint, but collecting the full set is complicated by the fact that alignments are considered in order and outliers typically occur first. To account for this, we define a liberal merge process where m.l.s is the mean start position for the left intervals in m.E, and m.l.e is the mean end position for the left intervals in m.E, similar for m.r.A straightforward method to define breakpoint intervals s  is made for each breakpoint b ∈ B that meets a user-defined minimum evidence threshold . The boundaries of the breakpoint intervals s.l and s.r are the trimmed product of the distributions of the left and right intervals in b.E. Let s.l.s = max({e.l.s | e ∈ b.E}), s.l.e = min({e.l.e | e ∈ b.E}), and s.l.p[i] = ∏e∈b.Ee.l.p[i-o] where o is the offset value e.l.s - s.l.s (similar for s.r). The intervals s.l and s.r can then trimmed to include only those positions that are in the top percentile  based on a user-provided value. Given the liberal merge process, it is possible for b.E to contain non-overlapping distributions that would result in a zero-length product. In this case, we identify the maximum point among the sum of the distributions in b.E, any distribution not intersecting this point is removed, and the resulting subset processed normally. Regardless of the trimming value, LUMPY reports both the intervals that contain 95% of the resulting probability distribution and the maximum position of s.l.p and s.r.p. Summation is another option for calculating the combined distribution boundaries and values. In that case s.l and s.r are the trimmed mixture distributions of the left and right intervals in b.E. Let s.l.s = min({e.l.s | e ∈ b.E}), s.l.e = max({e.l.e | e ∈ b.E}), and s.l.p[i] = ∑e∈b.Ee.l.p[i-o]. The value at s.l.p[i] (or s.r.p[i]) represents the level of agreement among the evidence in b.E that position i is one end of the breakpoint. While summation will give less precise breakpoint predictions, it can be a useful option when considering low-quality data.Once all the evidence has been considered, an SV call To combine distinct SV alignment signals such as read-pair and split-read alignments with the general breakpoint type defined above, we define an abstract breakpoint evidence type. This abstract type defines an interface that allows for the inclusion of any data that can provide the following functions: is_bp determines if a particular instance of the data contains evidence of a breakpoint; get_v determines the breakpoint variety ; and get_bpi maps the data to a pair of breakpoint intervals.To demonstrate the applicability of this abstraction, we defined three breakpoint evidence instances: paired-end sequencing alignments, split-read alignments, and a general breakpoint interface. Read-pair and split-read are the most frequently used evidence types for breakpoint detection, and the general interface provides a mechanism to include any other sources of information such as known variant positions or output from other analysis pipelines . As technologies evolve and our understanding of structural variation improves, other instances can be easily added.x, y⟩, which we will refer to as ‘read-pairs’. Each read is aligned to a reference genome R(x) = ⟨c, o, s, e⟩, where R(x).c is the chromosome that x aligned to in the reference genome, R(x).o = +| - indicates the alignment orientation, and R(x).s and R(x).e delineate the start and end positions of the matching sequence within the chromosome. We assume that both x and y align uniquely to the reference and that R(x).s < R(x).e < R(y).s < R(y).e. While in practice it is not possible to know the position of read x in the sample genome (in the absence of whole-genome assembly), it is useful to refer to S(x) = ⟨o, s, e⟩ as the alignment of x with respect to the originating sample’s genome.Paired-end sequencing involves fragmenting genomic DNA into roughly uniformly sized fragments, and sequencing both ends of each fragment to produce paired-end reads ⟨R(x).o = +, R(y).o = - orientation, and at distance R(y).e - R(x).s roughly equivalent to the fragment size from the sample preparation step. Any read-pair that aligns with an unexpected configuration can be evidence of a breakpoint. These unexpected configurations include matching orientation R(x).o = R(y).o, alignments with switched orientation R(x).o = -, R(y).o = +, and an apparent fragment length (R(y).e - R(x).s) that is either shorter or longer than expected. We estimated the expected fragment length to be the sample mean fragment length l, and the fragment length standard deviation to be the sample standard deviation s from the set of properly mapped read-pairs (as defined by the SAM specification) in the sample data set. Considering the variability in the sequencing process, we extend the expected fragment length to include sizes l + vls, where vl is a tuning parameter that reflects spread in the data.Assuming that genome sequencing was performed with the Illumina platform, read-pairs are expected to align to the reference genome with a x and y align to the same chromosome (R(x).c = R(y).c), the breakpoint variety can be inferred from the orientation of R(x) and R(y). If the orientations match, then the breakpoint is labeled as an inversion, and if R(x).o = - and R(y).o = + then the breakpoint is labeled as a tandem duplication. Any breakpoint with the orientation R(x).o = + and R(y).o = - is labeled as a deletion. When x and y align to different chromosomes (R(x).c ≠ R(y).c), the variety is labeled inter-chromosomal. At present, LUMPY does not explicitly support identification of insertions that are spanned by paired-end reads; however, if desired these can be identified in a post-processing step through assessment of ‘deletion’ calls.When x, y⟩ to breakpoint intervals l and r, the ranges of possible breakpoint locations must be determined and probabilities assigned to each position in those ranges. By convention, x maps to l and y to r, and for the sake of brevity we will focus on x and l since the same process applies to y and r. Assuming that a single breakpoint exists between x and y, then the orientation of x determines if l will be upstream or downstream of x. If the R(x).s = +, then the breakpoint interval begins after R(x).e (downstream), otherwise the interval ends before R(x).s (upstream).To map ⟨L and standard deviation s. Since we assume that only one breakpoint exists between x and y, and that it is unlikely that the distance between the ends of a pair in the sample genome (S(y).e - S(x).s) is greater than L, then it is also unlikely that one end of the breakpoint is at a position greater than R(x).s + L, assuming that R(x).o = +. If R(x).o = -, then it is unlikely that a breakpoint is at a position less than R(x).e - L. To account for variability in the fragmentation process, we extend the breakpoint to R(x).e + (L + vfs) when R(x).o = +, and R(x).s - (L + vfs) when R(x).o = -, where vf is a tuning parameter that, like vl, reflects the spread in the data.The length of each breakpoint interval is proportional to the expected fragment length i in the breakpoint interval l is part of the actual breakpoint can be estimated by the probability that x and y span that position in the sample. For x and y to span i, the fragment that produced ⟨x, y⟩ must be longer than the distance from the start of x to i, otherwise y would occur before i and x and y would not span i (contradiction). The resulting probability is P(S(y).e - S(x).s > i - R(x).s) if R(x).o = +, and P(S(y).e - S(x).s > R(x).e - i) if R(x).o = -. While we cannot directly measure the sample fragment length (S(y).e - S(x).s), we can estimate its distribution by constructing a frequency-based cumulative distribution D of fragment lengths from the same sample that was used to find L and s, where D(j) gives the proportion of the sample with fragment length greater than j.The probability that a particular position X that does not contiguously align to the reference genome. Instead, X contains a set of two or more substrings xi…xj (X = x1x2…xn), where each substring aligns to the reference R(xi) = ⟨c, o, s, e⟩, and adjacent substrings align to non-adjacent locations in the reference genome R(xi).e ≠ R(xi + 1).s + 1 or R(xi).c ≠ R(xi + 1).c for 1 ≤ i ≤ n – 1. A single split-read alignment maps to a set of adjacent split-read sequence pairs , and each pair ⟨xi,xi + 1⟩ is considered individually.A split-read alignment is a single DNA fragment By definition, a split-read mapping is evidence of a breakpoint and therefore the function is_bp trivially returns true.xi, xi + 1⟩. When the orientations match R(xi).o = R(xi + 1).o, the event is marked as either a deletion or a tandem duplication. Assuming that R(xi).o = R(xi + 1).o = +, R(xi).s < R(xi + 1).s indicates a gap caused by a deletion and R(xi).s > R(xi + 1).s indicates a tandem duplication. These observations are flipped when orientations R(xi).o = R(xi + 1).o = -. When the orientations do not match R(xi).o ≠ R(xi + 1).o, the event is marked as an inversion and the mapping locations do not need to be considered. When x and y align to different chromosomes, the variety is marked as inter-chromosomal. LUMPY does not currently attempt to identify insertions that are completely contained within a long read, but this will be supported in future versions. We note that this capability requires long-read aligners to report the number and order of alignments within a read .Both orientation and mapping location must be considered to infer the breakpoint variety for ⟨xi, xi + 1⟩ maps to two breakpoint intervals l and r centered at the split. The probability vectors l.p and r.p are highest at the midpoint and decrease exponentially toward their edges. The size of this interval is a configurable parameter vs and is based on the quality of the sample under consideration and the specificity of the alignment algorithm used to map the sequences to the reference genome.The possibility of errors in the sequencing and alignment processes creates some ambiguity in the exact location of the breakpoint associated with a split-read alignment. To account for this, each alignment pair ⟨l and r are centered on either the start or the end of R(xi) and R(xi + 1). When the breakpoint is a deletion l is centered at R(xi).e and r at R(xi + 1).s, and when the breakpoint is a tandem duplication l is centered at R(xi).s and r at R(xi + 1).e. If the breakpoint is an inversion, l and r are both centered either at the start positions or end positions of R(xi) and R(xi + 1), respectively. Assuming that R(xi).s < R(xi + 1).s, if R(xi).o = + then l and r are centered at R(xi).e and R(xi + 1).e, otherwise they are centered at R(xi).s and R(xi + 1).s. If R(xi).s > R(xi + 1).s, then the conditions are swapped.Depending on the breakpoint variety, the intervals The generic evidence subclass provides a mechanism to directly encode breakpoint intervals using the BEDPE format . BEDPE iBoth simulated and real datasets were used to compare the sensitivity and FDR of LUMPY to other SV detection algorithms . Two types of simulations were performed: one in which homozygous variants of diverse varieties were introduced at random positions throughout the reference genome, and another in which a heterogeneous tumor sample was simulated by mixing reads from a modified ‘abnormal’ human reference genome (containing 1000 Genomes deletions) and an unmodified ‘normal’ human reference genome in varying proportions. We also used publicly available Illumina sequencing data of the NA12878, NA12891, and NA12892 individuals. Two scenarios were considered: the original 50X coverage files, and 5X subsamples of the original data sets.In the case of the homozygous simulation, we used SVsim to create new versions of the human reference genome (build 37) containing 2,500 simulated variants of each variety. For deletions, tandem duplications and inversions we randomly placed 2,500 non-overlapping variants ranging from 100 bp to 10,000 bp in size. To simulate translocations, we randomly inserted 2,500 non-overlapping inter-chromosomal regions of 1,000 bp, derived from random donor sites in the reference genome. Although we note that the true variant variety in this case is actually an insertion, the inserted segment exceeds the insert size of the sequencing library as well as the read length, and thus the breakpoints formed by such insertions accurately simulate a translocation. Each simulated genome was sampled to 40X, 20X, 10X, 5X, and 2X coverage.To simulate a heterogeneous tumor sample, we combined simulated reads from both a modified and unmodified version of the human reference genome (build 37). The modified genome was created using SVsim, and included 5,516 non-overlapping deletions identified by the 1000 Genomes Project. Each simulation combined reads from both the modified and unmodified genomes in varying proportions. We refer to the proportion of reads that were derived from the modified genome as the SV allele frequency. The simulated SV allele frequencies were 5%, 10%, 20% and 50%, and the simulated coverages were 10X, 20X, 40X, and 80X. For example, in the simulation with 5% SV allele frequency and 10X coverage, the modified genome was sampled at 0.5X coverage and the unmodified genome was sampled at 9.5X coverage. The two sets of reads are then pooled into a single 10X coverage sample.For all simulations, WGSIM was used to sample paired-end reads with a 150 bp read length, a 500 bp mean outer distance with a 50 bp standard deviation, and default error rate settings. Paired-end reads were mapped to the reference genome with NOVOALIGN version V2.07.08, using the random repeat reporting and allowing only one alignment per read. From the NOVOALIGN output, all soft-clipped (≥20 bp clipped length) and unmapped reads were realigned with the split-read aligner YAHA using a word length of 11 and a minimum match of 15. The NOVOALIGN output was used as input to DELLY, GASVPro, and Pindel, and both NOVOALIGN and YAHA output were used as input to LUMPY. In all algorithms, the minimum evidence threshold was four. For LUMPY, the tuning parameters min_non_overlap was set to 150, discordant_z was set to 4, back_distance was set to 20, weight was set to 1, and min_mapping_threshold was set to 1. For GASVPro, LIBRARY_SEPARATED was set to all, CUTOFF_LMINLMAX was set to SD = 4, WRITE_CONCORDANT was set to true, and WRITE_LOWQ was set to true. For DELLY, map-qual was set to 1, and the inc-map flag was set. DELLY paired-end (pe) and split-read (sr) calls were combined into a single paired-end and split-read (pe + sr) callset by taking the union of the two sets where the split-read call was retained when a call was common to both sets. For Pindel, minimum_support_for_event was set to 4, all chromosomes were considered, and report_interchromosomal_events was set to true. Since the output of DELLY and Pindel are single-base-resolution intervals, we increased the size of those intervals to match the mean interval size of a similar LUMPY call. Any call that had split-read support  was expanded to a 28 bp interval, and any call that had only paired-end support  was expanded to a 282 bp interval. The intervals for GASVPro were not modified since, like LUMPY, it reports an interval whose size is based on the supporting evidence.For the real data, LUMPY, GASVPro, DELLY, and Pindel considered Illumina sequencing of the NA12878 individual. The original sequencing files were at 50X coverage and were used in the 50X experiments. The 5X experiments considered sequencing files that were created by subsamples 10% of the original paired-end alignments. For all the tools, only deletion predictions on chromosomes 1 though X were considered. All sequencing samples were retrieved from the European Nucleotide Archives (submission ERA172924), and were previously aligned using BWA. Soft-clipped (≥20 bp clipped length) and unmapped reads were realigned with the split-read aligner YAHA using a word length of 11 and a minimum match of 15. In addition to the NA12878 data, the LUMPY trio results also considered sequencing data from that individual’s parents, NA12891 and NA12892. The LUMPY prior result considered all 1000 Genomes variant calls using the generic evidence module. The LUMPY read-depth results considered all deletion calls made by CNVnator  for the et al. study [For the identification of the first truth sets, the Mills l. study  validateThe second truth set consisted of the 4,095 deletions called by at least one tool in the 50X dataset or by the 1000 Genomes Project , and thaDeletion calls were considered to be validated if supported by at least two non-duplicate PacBio split reads or at least one Moleculo split read. A supporting long-read is defined by the following criteria: 1) the long-read is split by the aligner such that at least 20 bp aligned to the flanking sequence on either side of the reported breakpoint; 2) the left and right intervals of the split long-read both intersect with the respective left and right intervals of reported breakpoint, allowing 5 bp of slop space on either side of the long-read breakpoint to account for microhomology or inserted sequence at the novel adjacency; 3) the strand orientation of the split long-read is consistent with the strand orientation at the reported breakpoint; 4) in the case of PacBio, where at least two non-duplicate long-reads are required to validate a call, the two long-reads must not only fill criteria 1 to 3, but must also overlap with each other within 5 bp of slop on either side.We performed Monte Carlo shuffling of the callsets to estimate the number of spurious long-read validations due to random chance. We shuffled each callset 100 times using BEDTools shuffle while retaining identical interval sizes, variant spans, and number of variants for each iteration . RegionsFor structural variation detection with LUMPY and other tools, we excluded regions of the reference genome with consistently high sequencing depth over multiple individuals, since high depth is indicative of artifacts in the reference assembly. To define these regions, we first aligned the 17-member CEPH 1463 pedigree to the GRCh37 human reference genome using BWA-MEM 0.7.5a-r405 (bwa mem -t 32 -M -p) . Each mebp: base pair; CNV: copy number variation; FDR: false discovery rate; PacBio: Pacific Biosciences; PCR: polymerase chain reaction; SNP: single nucleotide polymorphism; SV: structural variation.The authors declare that they have no competing interests.RL developed the algorithms, wrote the source code, performed the experiments and prepared the figures. CC performed the long-read SV validation experiments. AQ and IH conceived the framework, devised the experiments and co-advised the project. RL, AQ and IH contributed equally to the writing of the manuscript. All authors read and approved the final manuscript.This file contains the breakpoints used for the homozygous variant simulation. The format is BEDPE. Each line has a ‘TYPE:’ field that indicates DELETION, DUPLICATION, INVERSION, or TRANSLOCATION.Click here for fileThis file contains the breakpoints used for the heterogeneous tumor simulation. The format is BEDPE. These deletions are based on the variants released by the 1000 Genomes Project in [oject in . We seleClick here for fileThis file contains the breakpoints used for the Mills et al. truth set. The format is BEDPE, which is described by [ribed by . The breribed by .Click here for fileThis file contains the breakpoint intervals for the deletion predictions that were made by LUMPY , GASVPro, DELLY, Pindel, or the 1000 Genomes Project [ and that were validated by long read alignments from PacBio and/or Illumina Moleculo sequencing. The format is BEDPE, and the score field indicates the number of overlapping predictions. Note: in some cases one algorithm made two predictions that contributed to a single call. For example, there are two calls with a score of 8. In both cases Pindel contributed two very similar calls. Project  and thatClick here for fileThis file contains the calls made by LUMPY for NA12878 with paired-end and split-read evidence that were also validated with PacBio/Moleculo data. The format is BEDPE and the score field is the total amount of supporting evidence. This file contains extra fields that are described at [ribed at .Click here for file


Spin Hall effect (SHE) has been discussed in the context of Kubo formulation, geometric physics, spin orbit force, and numerous semi-classical treatments. It can be confusing if the different pictures have partial or overlapping claims of contribution to the SHE. In this article, we present a gauge-theoretic, time-momentum elucidation, which provides a general SHE equation of motion, that unifies under one theoretical framework, all contributions of SHE conductivity due to the kinetic, the spin orbit force (Yang-Mills), and the geometric (Murakami-Fujita) effects. Our work puts right an ambiguity surrounding previously partial treatments involving the Kubo, semiclassical, Berry curvatures, or the spin orbit force. Our full treatment shows the Rashba 2DEG SHE conductivity to be  The time-space gauge field can, in turn be linked to a t-geometric velocity which has the same form as13On the other hand, Sinova On the other hand, a separate body of work15161718We are therefore motivated to provide, in this paper, a gauge-theoretic energy framework that unifies SHE velocity of kinetic, Yang-Mills, k-geometric, and t-geometric origins for any SOC system under one equation of motion (EOM). One unified energy system that merges the two spaces of t and k is derived, debunking any previous suspicion of overlapping energy terms. The energy equation with a merged t-k identity is then used to derive the velocity equation-of-motion (EOM) for all SHE systems. Previous efforts13The main accomplishment in our renewed treatment is that we show that a form-invariant t-k Hamiltonian is a complete energy equation, and can be used to derive a complete EOM that describes spin Hall. The complete EOM reveals in clear-cut, and non-overlapping manner, the velocity components of kinetic, Yang-Mills, and geometric origins. The significance of this result is the prediction of a reversal of sign in the SHE conductivity, suggesting the need to revisit and recalculate previously derived SHE conductivity. The Methods section will describe in details the theoretical techniques used in the process of unification. Unification provides the theoretical basis for a form-invariant, t-k manifestation of the gauge potential, which is summarized in We will now use the t-k form-invariant energy to derive a complete spin Hall EOM for the spin carrier. In the energy physics of the SOC systems, we have shown in Methods and summarized in We would like to note that the velocity expression The kinetic SHE velocity can also be written asnΣ and az having unity magnitude. The ± sign arising from the two eigenstates of implies that energy is low when spin is aligned with the effective B field. Thus in the (+) band of B fields in both East and West. In the (−) band of B field projected to the az. Note that -gauged spin orbit field, and with both tn. The expression Ex producing an effective B field. For illustration in the 2D projected region surrounded by the equator, nΣ points along axis −z in the region of +py, and axis +z in the region of −py  band  band  would shThe kinetic velocity has been identified in previous time-spaceThe One popular form of SHE exists in device or hetero-structure that exhibits the Rashba spin orbit coupling (RSOC). Examples are the GaAs/AlGaAs/GaAs semiconductor heterostructure, or the oxide/Metal/Pt metal multilayer, both with structural inversion asymmetry. Shown above in α is the strength of the Rashba SOC effect.where We will first examine a two-dimensional system, where the SHE current density is obtained from the SHE kinetic velocity as followsg represents spin flux for g =g = e. It has been shown that, for g = g = e) and the spin . The Dirac relativistic quantum mechanics is needed to account for this SOC energy. In the absence of any retardation effect due to scattering, the carrier would accelerate due to the E field. As a result, the carrier will acquire an E-dependent energy where The approach that has been used to derive SHE velocity in refs where the gauge potential σγ.B(k) would become σγ.B(t), where HS = H0 + VS, and note the following:The t-space approach has on the other hand, been adopted in refs HI(t) = H0 + VI(t). In the “Interaction” picture, one has itψI(t) = VI(t)ψI(t). Thus a transformation in time space is appropriate hereor k/∂t ≠ 0, a condition that would be fulfilled when E field is present in the device. We assign as follows: H0 = E.r. On the RHS of Eq. itU†) is also known as the time-space gauge with the dimension of energy. The unitary rotation operator U in local space iswhere and one can quickly obtain that Equation H0 = Eea.r is important for the survival of the gauge H0 = Eea.r. Inverse transformation of iU(∂tU†). Dropping the higher order terms, the following is arrivedleads to one obtaining a gauge expression in the k-space. Note that iU∂kU†, subscripts are subject to rk, subscripts are subject to Note that for expression iU†∂tU) = −σ.At.Note that in the above the convention z), generating a gauge field. Still, the post-transformed Hamiltonian only reveals the stationary eigen-states, but one which is more intuitive now with spin aligned to a total field out of plane. Note however, that in both pre- and post-transformed Hamiltonian, expectation values of observables remain the same. But this is not important as we are not interested in the stationary expectation values. What we need from these Hamiltonians are clues to determining the actual propagation physics of electron. In previous works23z-projection of the total field taken along the two lateral sides of the device.We would, however, note that the Hamiltonian alone, pre- or post-transformed could not reflect the full physical reality of the 2D system in which electron propagates in time. As electron propagates in momentum and spin space, so does the spin-orbit magnetic field, and the actual physics related to such effect can only be reflected in the wave-function that is solved taking into account the path of evolution. This is a separate task that needs to take into account approximations therein. Simply stated, a stationary Hamiltonian cannot reveal how electron propagates in time, as it only reveals the stationary eigen-states, which has spin aligned parallel or anti-parallel. Local gauge transformation locks the spin orbit field to a specific axis .


They concluded there is a ‘surge of p-values between 0.041–0.049 in recent decades’ which ‘suggests (but does not prove) questionable research practices have increased over the past 25 years.’ I show the changes in the ratio of fractions of p-values between 0.041–0.049 over the years are better explained by assuming the average power has decreased over time. Furthermore, I propose that their observation that p-values just below 0.05 increase more strongly than p-values above 0.05 can be explained by an increase in publication bias (or the file drawer effect) over the years . I explain why researchers analyzing large numbers of p-values need to relate their assumptions to a model of p-value distributions that takes into account the average power of the performed studies, the ratio of true positives to false positives in the literature, the effects of publication bias, and the Type 1 error rate (and possible mechanisms through which it has inflated). Finally, I discuss why publication bias and underpowered studies might be a bigger problem for science than inflated Type 1 error rates, and explain the challenges when attempting to draw conclusions about inflated Type 1 error rates from a large heterogeneous set of p-values.In recent years, researchers have attempted to provide an indication of the prevalence of inflated Type 1 error rates by analyzing the distribution of  However, as Thus far, however, no solid data exist on the prevalence of such research practices.’ Some researchers have attempted to provide an indication of the prevalence of inflated Type 1 error rates by analyzing the distribution of p-values in the published literature. The idea is that inflated Type 1 error rates lead to ‘a peculiar prevalence of p-values just below 0.05’ . Two observations in the data they collected could seduce researchers to draw conclusions about a rise of p-values just below a significance level of 0.05. The first observation the authors report is how from 1990 to 2013 p-values between 0.041 and 0.049 rose more strongly than the percentage of p-values between 0.051–0.059. The second observation is that the percentage of p-values that falls between 0.041–0.049 has increased more than the increase in the percentage of p-values between 0.001–0.009, 0.011–0.019, 0.021–0.029, and 0.031–0.039 from 1990 to 2013 1p-values with 2 digits , which reveal similar patterns, but here I focus on the three digit data, which focuses on p-values between (for example) 0.041–0.049 because trailing zeroes  are rarely reported.The authors also analyze . The authors  conclude that: “The fact that p-values just below 0.05 exhibited the fastest increase among all p-value ranges we searched for suggests (but does not prove) that questionable research practices have increased over the past 25 years.”Recently, p-values just below p = 0.05 compared to p-values just above p = 0.05 is not caused by a change over time in the percentage of p-values just below 0.05, but by a change over time in the percentage of p-values above 0.05. Perhaps surprisingly, p-values just above 0.05 increase much less than all other p-values. This might be due to a stricter interpretation of p < 0.05 as support of a hypothesis, and less leniency for ‘marginally significant’ p-values just above this threshold. Second, I will explain why the relatively high increase in p-values between 0.041–0.049 over the years can easily be accounted for by a decrease in the average power of studies. At the same time, I will illustrate why this increase in p-values just below 0.05 is unlikely to emerge due to an inflation of the Type 1 error rate due to optional stopping or trying out multiple analyses until p < 0.05. I want to explicitly note that it was possible to provide these alternative interpretations of the data because p-values.I will explain why the data does not suggest an increase in ‘questionable research practices’. First, I will discuss how the relatively stronger increase in p-value distributions before drawing conclusions about the underlying reasons for specific distributions of p-values extracted from the scientific literature. A model of p-value distributions consists of four different factors. First, the p-value distribution depends on the number of studies where the null-hypothesis (H0) is true, and the number of studies where the alternative hypothesis (H1) is true. Second, the p-values for studies where H1 is true depend upon the power of the studies. Statistical power is the probability that a study yields a statistically significant effect, if there is a true effect to be found. Power is determined by the significance level, the sample size, and the effect size. Third, p-values for studies where H0 is true depend upon the Type 1 error rate chosen by the researcher , and any possible mechanisms through which the Type 1 error rate is inflated beyond the nominal Type 1 error rate set by the researcher. When I talk about inflated Type 1 error rates in this article, I explicitly mean flexibility in dependent tests that are performed on the data  that have the goal to lead to a statistically significant result. When these statistical tests are dependent  the Type 1 error rate inflation has a specific pattern where p-values between 0.041–0.049 become somewhat more likely than smaller p-values.As I have discussed before , it is ep-value distribution in the published literature is influenced by publication bias. Publication bias is the tendency to publish statistically significant results . The threshold at which p-values indicate a statistically significant result, as well as the leniency towards ‘marginally significant’ findings, both influence the frequency of observed p-values in the literature. It is important to look beyond simplistic comparisons between p-values just below 0.05 and p-values in other parts of the p-value distribution if the observed p-values are not explicitly related to a model consisting of the four factors that determine p-value distributions.And finally, the p-values between 0.041–0.049 than in p-values between 0.051–0.059 . The data is clear, but the reason for this difference is not, and it is not explored by the authors. Although all p-values are increasing over time, the real question is whether p-values below p = 0.05 are increasing more, or p-values above p = 0.05 are increasing less. A direct comparison is difficult, because a comparison across the p = 0.05 boundary is influenced by publication bias. If publication bias increases, and less non-significant results end up in the published literature due to the file-drawer problem, the percentage of papers reporting p-values smaller than 0.05 will also increase (even when there is no increase in p-hacking). Indeed, both p-values between 0.041–0.049 and 0.051–0.059 observed by p-values between 0.051–0.059.p-values between 0.031–0.039 and 0.041–0.049 over the years on the one hand, and 0.051–0.059 and 0.061–0.069 on the other hand, we can examine whether there is an increase in p-values between 0.041–0.049 (due to an increase in the Type 1 error rate), or an increase in publication bias (or the file-drawer problem), which leads to a lower percentage of p-values between 0.051–0.059. If there is an increase in the Type 1 error rate due to flexibility in the data analysis, the biggest differences over time should be observed just below p = 0.05 . However, there are reasons to assume the biggest difference will be observed in p-values just above p = 0.05. As p-values just above 0.05 to be published, as indicated by a higher prevalence of p-values between 0.051–0.059 than would be expected based on the power of statistical tests and an equal reduction of all p-values above 0.05 due to the file-drawer problem. If publication bias becomes more severe, we might expect a reduction in the tolerance for ‘marginally significant’ p-values just above 0.05, and the largest changes in ratios should be observed abovep = 0.05.By comparing the relative differences between p-values between 0.031–0.039 to p-values between 0.041–0.049 is pretty stable: 1.13, 1.09, and 1.11, respectively. The ratio of p-values between 0.051–0.059 to p-values between 0.061–0.069 shows a surprisingly large reduction over the years: 2.27, 1.94, and 1.79, respectively. It is important to note that flexibly analyzing data with the goal to be able to report a significant finding leads to a change in the p-value distribution both above as below p = 0.05. However, the ratio of p-values between 0.031–0.039 to p-values between 0.041–0.049 should change much more than the ratio of p-values between 0.051–0.059 to p-values between 0.061–0.069, because p-values are drawn from a relatively larger range above p = 0.05, to a relatively small range just below p = 0.05. This surprisingly large change in ratios over time for p-values 0.051–0.059 to 0.061–0.069 indicates that instead of an increase in the Type 1 error rate of p-values below 0.05, the real change over time happens in the p-values between 0.051–0.059.Across the three time periods  the ratio of p-values just above p = 0.05 might be explained by an increasingly strong effect of the file-drawer problem. Where p-values between 0.051–0.059  might have been more readily accepted as support for the alternative hypothesis in 1990–1997, p-values just above 0.05 might no longer deemed strong enough support for the alternative hypothesis in 2005-2013. This idea is speculative, but seems plausible given the increase in publication bias over the years , but the ratios do increase as you move from bins of low p-values between 0.001–0.009 to bins of high p-values between 0.041–0.049.The first part of the title of the article by p-values are reported over the years. As p-value between 0.051 and 0.059. This increased 3.6-fold to 0.067%  in 2013. Positive results increased 10.3-fold in the same period: from 0.030%  in 1990 to 0.314%  in 2013.”In 1990, 0.019% of papers  reported a p-values are finding their way into more and more abstracts, which points to a possible increase in the overreliance on null-hypothesis testing in empirical articles. This is an important contribution to the literature.The first thing to understand is why none of the observed ratios are close to 1. The reason is that there is a massive increase in the percentage of abstracts of papers in which p = 0.05 can be explained. p-value distribution . The three remaining possibilities are a change in the average power of studies over time, a change in the inflated Type 1 error rate over time, and a change in the ratio of studies where H0 or H1 is true. I will discuss each of these three possible explanations in turn.The main question is how these differences in the ratios across the 5 bins below p-values depends on the statistical power of the studies , which is itself a function of the true effect size, the significance level, and the sample size . It is not difficult to model the ratios observed by p-value distribution across the 5 bins as detailed in p-values falling between 0.001 and 0.009 in 1990, but only 19.93% of p-values falling between 0.001 and 0.009 in 2013. This is just the p-value distribution as a function of the power of the tests.We can relatively easily reconstruct the observed data purely based on differences in the average power across the years. Remember that the distribution of p-values reported in abstracts has increased by 10% over the years, and I chose 0.01% in 1990 and 0.1% in 2013 as the percentage of abstracts that report p-values  by the percentage of articles reporting p-values , multiplied by the percentage of p-values expected in each p-value bin based on the assumed power . The number of reconstructed p-values is presented in p-values observed by The total number of studies analyzed by p-values in each of the five p-value bins can now be calculated by dividing the number (N) of p-values in a specific bin for a specific year  by the total (T) number of p-values  to get the fractions for 1990 and 2015 . The reconstructed fractions  is smaller than the reconstructed ratio of fractions in the 0.031–0.039 bin (9.417), but the pattern is reversed in the observed data . However, if we calculate the same ratios of fractions for 2013 with all other preceding years  we find a smaller ratio of fractions in the 0.021–0.029 bin than in the 0.031–0.039 bin for p-values between 0.031–0.039 should be very similar to those between 0.041–0.049. The predicted 1990/2013 ratio of fractions in the 0.031–0.039 bin is 9.417, and the predicted ratio of fractions for the 0.041–0.049 bin is 9.740, while the difference in observed ratios of fractions is much larger . This can hardly be called a ‘surge’ of p-values between 0.041–0.049. This observation is not in line with the idea that the Type 1 error rate has increased, because an increase in Type 1 error rates due to flexibility in the data analysis typically assumes p-values between 0.041–0.049 increase more strongly than p-values between 0.031–0.039  is not likely to reflect the true state of affairs in the real world. Although we lack data about changes in the ratio of true to false effects examined over time (a worthwhile research question in itself), it seems reasonable to at least entertain the possibility of some changes over time for this factor of the model. For now, the most important conclusion is that a change in power over time can mathematically account for the observed changes in ratios of fractions in the different p-value bins. Moreover, the idea that power decreases over time is theoretically plausible, since such a decrease in power over time has been observed in some disciplines, such as psychology  5% of the studies will observe p-values smaller than 0.05. De Winter and Dodou focus on p-values between 0.041–0.049 in the analysis of p-values with three digits , which means 0.8% of the time (0.049 minus 0.041) a p-value within each of the five p-value bins will be observed2p-values between 0.041–0.049 when the null hypothesis is true, p-values between 0.0405–0.0495, which would mean 0.9% of the time a p-value within a single p-value bin will be observed. Using 0.8% or 0.9% Type 1 error rates in each bin does not influence the reconstruction purely based on changes in power over time, and does not have consequences for any of the conclusions based on the reconstructions.Because researchers round numbers to the nearest 3 digits, one might argue that instead of 0.8% of . By increasing the Type 1 error rate above 0.8%, and doing so more strongly for higher p-value bins, we can attempt to reconstruct the consequences of an increase in false positives in the literature on the observed ratios of p-values over time.Let’s assume the average power has not changed over time, and instead try to reconstruct the observed data by ositives , p. 6. Ap-values between 0.001–0.009 , and substantially higher for p-values between 0.011 and 0.049 . Although this would be substantial , it is not impossible.One attempt to specify the parameters to model the ratios  is presented in p-value distribution for true effects (based on the power of the studies) leads to a lower frequency of p-values in the 0.041–0.049 bin than in the 0.031–0.039 bin. A uniform inflation of the Type 1 error rate  would add the same number of false positive p-values to each bin, but because the frequency of true p-values decreases from bins with low p-values to bins with high p-values, the relative increase is stronger in the 0.041–0.049 bin than in the 0.031–0.039 bin because the extra p-values from false positives constitute a relatively larger increase. As However, such a pattern of Type 1 error rates does not predict the ratio of fractions in the last 20 years, which were on average equal for the 0.031–0.039 and 0.041–0.049 bins. If we would change the Type 1 error rates to reconstruct the very similar ratios in the majority of the years in the 0.031–0.039 and 0.041–0.049 bins only based on a change in Type 1 error rates, the Type 1 error rates should be equal across these bins, or even slightly lower in the 0.041–0.049 bin compared to the 0.031–0.039 bin. This would be opposite to predictions based on flexibility during the data analysis . The p-vp-hack studies where H1 is true. If such behavior increases over time, we can expect the percentage of p-values from true effects in the 0.041–0.049 bin to increase in 2013 compared to 1990. If we would incorporate this effect in the model, the relative increase over time in the 0.041–0.049 bin would be even stronger compared to the 0.031–0.039 bin. Again, we do not see huge differences in the ratios between the 0.031–0.039 and 0.041–0.049 bins, which makes the explanation based on flexibility in the data analysis even less likely.Researchers can (and probably do) p-values smaller than 0.05). This means that p-values are either true positives (when H1 is true) or false positives (when H0 is true). The ratios calculated by The third factor that could influence the ratios is the percentage of studies where true effects are examined. A hypothesis is either true or not. The current analysis focuses on statistically significant findings in the published literature  constant over time, the observed ratios of fractions  can be reconstructed purely based on a change in the H0/H1 ratio over time if we assume it was 10 times more likely to examine a true hypothesis compared to a false hypothesis in 1990, and at the same time assume it is 10 times more likely to examine a false hypothesis compared to a true hypothesis in 2013 . On the other hand, while increases in Type 1 error rates can be used to reconstruct the observed ratios, the pattern of inflated Type 1 errors across the 5 bins of p-values just below 0.05 without providing substantial evidence for such a rise . A better approach seems to be to perform targeted analyses of small sets of similar studies, which might be able to yield support for p-hacking (e.g., p-value distribution across a large number of studies.Analyzing huge numbers of ng e.g., . I do nop-hacking, are the biggest problems in the scientific literature. Without publication bias, Type 1 errors would be quite easily identified as they lead to follow-up research that will not observe the hypothesized effect. Even with publication bias, meta-analyses can identify sets of studies that lack evidential value, indicating the studies are a result of selection and reporting biases (p-value distributions in 1990 and 2013 in the data by p-hacking have been very salient in recent years, but I believe it is worthwhile to point out that designing well-powered studies with high informational value (e.g., Although it is important to control Type 1 error rates when performing statistical tests e.g., , I belieg biases . As the ue e.g.,  and prevue e.g.,  are at l10.7717/peerj.1142/supp-1Supplemental Information 1Click here for additional data file.10.7717/peerj.1142/supp-2Supplemental Information 2Click here for additional data file.