We employ this similarity metric to improve the accuracy of the alignments. The experimental results on the KEGG database show that when compared with other state-of-the-art methods, in most cases, our method obtains better performance in the node correctness and edge correctness, and the number of the edges of the largest common connected subgraph for one-to-one reaction mappings, and the number of correct one-to-many reaction mappings. Our method is scalable in finding more reaction mappings with better biological relevance in large metabolic pathways.Metabolic pathway alignment has been widely used to find one-to-one and/or one-to-many reaction mappings to identify the alternative pathways that have similar functions through different sets of reactions, which has important applications in reconstructing phylogeny and understanding metabolic functions. The existing alignment methods exhaustively search reaction sets, which may become infeasible for large pathways. To address this problem, we present an effective alignment method for accurately extracting reaction mappings between two metabolic pathways. We show that connected relation between reactions can be formalized as binary relation of reactions in metabolic pathways, and the multiplications of zero-one matrices for binary relations of reactions can be accomplished in finite steps. By utilizing the multiplications of zero-one matrices for binary relation of reactions, we efficiently obtain reaction sets in a small number of steps without exhaustive search, and accurately uncover biologically relevant reaction mappings. Furthermore, we introduce a measure of topological similarity of nodes (reactions) by comparing the structural similarity of the  In the last few decades, the quantity and quality of metabolic data in biological databases such as KEGG (Kyoto Encyclopedia of Genes and Genomes)  and MetaIn the metabolic pathway alignment problem, metabolic pathways are usually represented as directed graphs, where a node denotes a molecule which can be specified as reaction, enzyme, or compound and an edge represents the interactions between molecules. A one-to-one mapping between nodes in two metabolic pathways maps a node from one pathway to a node in the other. A one-to-many mapping between the nodes in two metabolic pathways maps a node from one pathway to a connected subgraph of many nodes in the other . The sizAccordingly, we can categorize existing literature on metabolic pathway alignment into two types. The first type finds one-to-one mappings between molecules of metabolic pathways to identify similar parts in different pathways , 11–20. et al.  = 1 when reaction ri is connected to reaction rj, and Mp = 0 when ri is not connected to rj, i = 1,2,…,N and j = 1,2,…,M. Mpn can be computed recursively by Mp1 = Mp and n≥2. Mp2.In this work, we represent binary relation of reactions in metabolic pathway by zero-one matrix In the following section, we present our method MPBR.Gp =  and Gp′ = , the goal of MPBR is to find the reaction mappings between Gp and Gp′. Without loss of generality, we assume that |Vp|≤|Vp′|, reaction sets RS⊆Vp and RS′⊆Vp′. MPBR consists of three main stages ; (2) Construct a similarity matrix BM by computing the similarity between the reactions in Gp and Gp′ (as detailed in Subsection ‘Second Stage’); (3) Find mapping  such that the similarity score of mapping  is maximized (as detailed in Subsection ‘Third Stage’). A set RSmap of mappings  is the result for aligning Gp and Gp′. For a pair of metabolic pathways RSn = {RS1, RS2,…, RSN} in Gp, and the set RSm′ = {RS1′, RS2′,…, RSM′} in Gp′ respectively. For metabolic pathway Gp, there is a path from r1 to rn if there is a sequence of reactions r1,r2,…,rn with edges , ,…, and  in Gp. Accordingly, we derive theorem 1.In this subsection, we discuss how to exploit the multiplications of zero-one matrices for binary relation of reactions to create the set ri and rj, there is a path of length n from ri to rj in Gp if and only if Mpn = 1, where n is a positive integer, i = 1,2,…,n and j = 1,2,…,n.Theorem 1: For reactions ri to rj in Gp if and only if Mp = 1, so the theorem is true when n = 1.Proof: We will use mathematical induction. There is a path of length one from n. There is a path of length n+1 from ri to rj if and only if there is a reaction rk in Gp such that there is a path of length one from ri to rk in Gp, so Mp = 1, and a path of length n from rk to rj in Gp, that is, Mpn = 1. Consequently, by the induction hypothesis, there is a path of length n+1 from ri to rj in Gp if and only if there is a reaction rk with Mp = 1 and Mpn = 1. But there is such a reaction if and only if Mpn+1 = 1. Therefore, there is a path of length n+1 from ri to rj in Gp if and only if Mpn+1 = 1, i = 1,2,…,n and j = 1,2,…,n. γAssume that the theorem is true for a positive integer RS of size n for Gp.Following from theorem 1, we introduce how to find reaction set Mpn-1. Then we create a reaction pair set NS and iteratively extend NS with the reaction pair  where Mpn-1 = 1. According to theorem 1, for reactions ri and rj with Mpn-1 = 1, there is a path of length n-1 from ri to rj in Gp if and only if Mpn-1 = 1. That is, there are n reactions in this path. Thus, we can construct reaction set RS of size n containing these n reactions. Finally, we search every path of length n between two reactions in each reaction pair of NS in Gp, and create each reaction set RS of size n containing the reactions in each path to construct the set RSn = {RS1, RS2,…, RSN} in Gp. Similarly, we can find each reaction set RS′ of size m and construct the set RSm′ = {RS1′, RS2′,…, RSM′} in Gp′.First, we compute ri, rj) with Mp2 = 1, i = 1,2,3 and j = 1,2,3,4,5. These reaction pairs are , , ,  and . Next we create a reaction pair set NS = {, , , , } by these reaction pairs. Then, we search the paths of length 2 between two reactions in each reaction pair of NS. These paths are Path 1(r1→r2→r3), Path 2 (r2→r3→r1), Path 3 (r3→r1→r2), Path 4 (r2→r3→r4) and Path 5 (r3→r4→r5) respectively. Finally we create reaction sets RS1 = {r1,r2,r3}, RS2 = {r2,r3,r1}, RS3 = {r3, r1, r2}, RS4 = {r2, r3, r4} and RS5 = {r3, r4, r5} with the reactions in Path 1, Path 2, Path 3, Path 4, and Path 5 respectively.Gp, we drive the following property.Based on theorem 1 and the above searching procedure of reaction sets in RS of size n+1 in Gp can be found through performing Mpn.Property 1: Reaction set Gp.Property 1 illustrates the relationship between the search of reaction sets and the multiplications of zero-one matrices for binary relation of reactions in Mpn, we have the following theorem.For t and s with t>s such that Mpt = Mps, then for any positive integer n, it holds that Mpn∈S = {Mp, Mp2, …, Mpt-1}.Theorem 2: If there exist positive integers Proof:n, when n≤t-1, we have Mpn∈S. We want to prove Mpn∈S when n>t-1. When n>t-1, it holds n>s. For n, s and t, there exist positive integers q and r such that n-s = (t-s)q+r . Therefore, n = s+(t-s)q+r and Mpn = Mps+(t-s)q+r. Since 0<r≤t-s-1, so s+r≤t-1 and Mps+r∈S. Since Mpn = Mps+(t-s)q+r, if we can prove Mps+(t-s)q+r = Mps+r, then it follows Mpn = Mps+r∈S. In the following, we will use mathematical induction to prove Mps+(t-s)q+r = Mps+r.For any q = 1, it yields Mps+(t-s)q+r = Mps+(t-s)+r = Mpt+r = Mps+r. Assume that for any q≤k, Mps+(t-s)q+r = Mps+r holds. When q = k+1, we have Mps+(t-s)(k+1)+r = Mps+(t-s)k+r+(t-s) = Mps+(t-s)k+rMpt-s)q+r = Mps+r also holds for q = k+1. By induction, we have proven that Mps+(t-s)q+r = Mps+r for all q>0. Consequently, since Mpn = Mps+(t-s)q+r, Mps+(t-s)q+r = Mps+r and Mps+r∈S, hence Mpn = Mps+r∈S.ϒWhen Gp and theorem 2 leads to the following corollary.The discussion regarding the search of reaction sets in s and t with s<t such that Mps = Mpt, then the reaction set RS of size n≥t in Gp can be found through performing at most Mpt-1.Corollary 1: If there exist positive integers Proof:S = {Mp, Mp2,…,Mpt-1}. When there exist positive integers s and t with s<t such that Mps = Mpt, according to theorem 2, for any positive integer n, it holds that Mpn∈S = {Mp, Mp2,…,Mpt-1}. In other words, when n-1≥t-1, there is a positive integer v≤t-1 such that Mpv = Mpn-1∈S. According to property 1, we need to perform Mpn-1 to find reaction set RS of size n in Gp. Because Mpv = Mpn-1 when n-1≥t-1, so we only need to perform Mpv to find the reaction set RS of size n in Gp instead of performing Mpn-1, where v≤t-1. ϒLet RS of size n through performing Mpn-1. Furthermore, corollary 1 implies that, in order to find reaction set RS of size n, when n≥t, we only need to try at most Mpt-1. Therefore, we can find reaction set RS of size greater than t in metabolic pathway through performing at most Mpt-1.Property 1 implies that we can find reaction set Mpn-1. Secondly, MPBR creates a reaction pair set NS and iteratively extends NS with the reaction pair  where Mpn-1 = 1. Finally, through finding the paths of length n-1 between two reactions in each reaction pair of NS in Gp, MPBR obtains the set RSn = {RS1, RS2,…, RSN} in Gp and the set RSm′ = {RS1′, RS2′,…, RSM′} in Gp′. Thus, in this way, we avoid the exhaustive search of reaction sets, and produce candidate reaction sets in finite steps.In the first stage, MPBR first computes Gp and Gp′, which combines topological and homological similarities of reactions, and construct a |Vp|×|Vp′| similarity matrix BM, where BM  is the similarity value between nodes (reactions) u and v, u∈Vp, v∈Vp′.The second stage aims to compute the similarity values between any two reactions in u is mapped to node v, then their neighbors in the corresponding graphs should also be similar. From this observation, we measure topological similarity of nodes by comparing the structural similarity of the k-neighborhood subgraphs of nodes. Next, we discuss how to compare the k-neighborhood subgraphs of nodes to compute topological similarity of nodes. Nk(u) is defined as the k-neighborhood node set of u in Gp and Nk(u) is a subset of Vp, u∉Nk(u), where k is an integer and k≥0. The shortest distance between u and x∈Nk(u) is defined as the number of edges of the shortest path between u and x, which is not exceeding k. Similarly, Nk(v) is defined as the k-neighborhood node set of v in Gp′.We first introduce how to compute topological similarity of nodes (reactions) in metabolic pathways. Our computation of topological similarity of nodes is based on the following observation. If node u∈Vp and v∈Vp′, the k-neighborhood subgraph of u in Gp is denoted by Ek(u) is defined as the edge set of Nk(u)∪{u} in Gp. The k-neighborhood subgraph of v in Gp′ is denoted by Nk(v)∪{v} in Gp′. Let d(u) be the degree of u in Gp and d(v) be the degree of v in Gp′. Suppose that the degree sequence of the nodes in Nk(u) is d(x1), d(x2),…,d(xi),…,and Nk(v) is d(y1), d(y2),…,d(yj),…,and k-neighborhood subgraph of u with the k-neighborhood subgraph of v to compute the topological similarity T between u and v:Km is the maximal value of k and is determined by the user, Nk(u) and Nk(v) respectively. The impact of edges on the topological similarity T is evaluated by k = 0, Nk(u) = u and Nk(v) = v.For Bsim between u and v is computed by the following equation:ue is the enzyme catalyzing reaction u, ve is the enzyme catalyzing reaction v, Esim is the similarity score between enzyme ue and enzyme ve. We employ the enzyme similarity score defined in 21], we met al.  reportedet al. . On the 5CR-1tomany is the ratio of the number of correct one-to-many reaction mappings to the total number of mappings produced by the alignment [CR-1tomany can be used to investigate the percentage of the correct one-to-many reaction mappings in the alignment. Higher values for CR-1tomany indicate that the alignments for one-to-many reaction mapping are of high quality [lignment . CR-1tom quality .6C-manytomany is the number of correct many-to-many reaction mappings between the first pathway and the second one. By reference to C-1tomany, let  be a many-to-many mapping from an alignment of GX and GX′ where RS = {r1, r2,…,rx}, RS′ = {r1′, r2′,…,ry′}, and P1,…,Pi,…,Px be the pathways that include a reaction in RS and are associated with metabolism 1.m in the species X. Similar to C-1tomany, a many-to-many reaction mapping  is called correct if it satisfies the following two conditions: (1) The start reactions in RS and RS′ share at least one input compound and the end reactions in RS and RS′ share at least one output compound. (2) Every reaction in RS′ is included in at least one of the pathways P1′,…,Pi′,…,Px′ where each Pi′ is a pathway in metabolism 1.m of species X′ and corresponds to Pi of X.MPBR, CAMPways and SubMAP provide the options of one-to-one alignment and one-to-many alignment. We can perform one-to-one alignment of two pathways to find one-to-one reaction mappings between these two pathways. Similarly, we can perform one-to-many alignment of two pathways to find one-to-many reaction mappings between these two pathways. In the experiments, the one-to-many reaction mappings include 1-to-2 and 1-to-3 reaction mappings.EC, NC and ELCCS to measure the quality of one-to-one alignment, and use C-1tomany and CR-1tomany to measure the quality of one-to-many alignment. In addition, we use C-manytomany to evaluate the capability of MPBR for searching many-to-many reaction mappings.In this paper, we use k = 3 and σ = 0.6, and CAMPways and SubMAP were run using their default parameter settings. MPBR, CAMPways and SubMAP were run on the Sugon 5000A computer system of cluster architecture at Guangxi University, using a single computing node with a quad-core Intel(R) Xeon(R) CPU E5620 @ 2.40GHz and 40GB RAM. The operating system is Linux.In the experiments, MPBR was run using The following five subsections will provide our experimental evaluations on the qualities of the alignment results computed by MPBR, CAMPways and SubMAP respectively. Subsections ‘Same-domain One-to-one Alignments’ and ‘Same-domain One-to-many Alignments’ focus on one-to-one alignment and one-to-many alignment between the species within the same domain respectively. Subsections ‘Across-domains One-to-one Alignments’ and ‘Across-domains One-to-many Alignments’ focus on one-to-one alignment and one-to-many alignment between the species from different domains respectively. Subsection ‘Running time and memory requirements’ discusses the performance of each method in terms of the running time and memory requirements. Subsection ‘Many-to-many Alignments’ discusses the experimental results of many-to-many alignments of the pathways. Subsection ‘Case study’ introduces three sample alignments to show how MPBR, CAMPways and SubMAP can be used to analyze metabolic pathways.NC of the alignment results of MPBR, CAMPways and SubMAP for the fifth level of the FGC hierarchy are shown in Tables NC for the first four levels of the FGC hierarchy are shown in The values of In this subsection, we discuss the quality of the same-domain one-to-one alignments produced by MPBR and other comparative methods. Tables EC, NC and ELCCS for 19, 18 and 18 instances respectively, whereas all three methods obtain equal values of EC, NC and ELCCS for 5, 6 and 7 instances respectively. Additionally, MPBR and CAMPways obtain equal values of EC, NC and ELCCS for 4, 4 and 3 instances respectively. These experimental results emphasize that, for the same-domain one-to-one alignment, in most cases, our MPBR method outperforms other comparative methods not only in topological accuracy but also in biological relevance of the results. Thanks to the use of structural similarity among the neighbors of reactions, MPBR is able to improve the topological and biological accuracy of the alignments.As shown in Tables C-1tomany and CR-1tomany of the one-to-many alignment results for the same domain species are shown in Tables In this subsection, we compare the quality of the same-domain one-to-many alignments produced by MPBR and other comparative methods. The values of C-1tomany and CR-1tomany in 22 and 15 out of all 28 instances respectively. For 4 instances, all three methods obtained the same values of C-1tomany and CR-1tomany, while the value of C-1tomany of MPBR is lower than CAMPways for 2 instances and is lower than SubMAP for 1 instance. The value of CR-1tomany of MPBR is lower than SubMAP for 4 instances, and some values of CR-1tomany of MPBR are lower than CAMPways for 7 instances. This means that, in most cases, MPBR is able to return more correct one-to-many reaction mappings than CAMPways and SubMAP in the same-domain one-to-many alignment. On the other hand, when the size of the pathway becomes large, SubMAP is unable to generate a result for 9 instances under the current computing environment while MPBR and CAMPways are not restricted to the size of the pathway in the same-domain one-to-many alignment.From Tables This subsection discusses the quality of the across-domains one-to-one alignments produced by MPBR and other comparative methods. Tables EC, NC and ELCCS for 47, 42 and 51 instances respectively. Some values of EC and NC of MPBR are a bit lower than those of CAMPways for 4 and 3 instances respectively. This demonstrates that, in most cases, MPBR is also capable of achieving better topological accuracy and biological relevance of the alignment results than other two comparative methods in across-domains one-to-one alignment.As can be seen from Tables EC, NC and ELCCS of the same-domain alignments are obviously higher than those values of across-domains alignments. This is also consistent with the analysis that the biological relevance of the species within the same domain is much stronger [In addition, from Tables stronger . Thus, wC-1tomany and CR-1tomany of the one-to-many alignment results for different domain species are shown in Tables In this subsection, we compare the quality of the across-domains one-to-many alignments produced by MPBR and other comparative methods. The values of C-1tomany and CR-1tomany compared to other comparative methods in 44 and 32 out of 56 instances respectively. For only 10 out of 56 instances MPBR fails to be the best. On the other hand, combining Tables C-1tomany and CR-1tomany of MPBR are still high enough without being affected by the size of pathway. While the comparative methods suffer from the size of large-scale pathway, MPBR overcomes this problem and returns more correct one-to-many reaction mappings.From Tables C-1tomany and CR-1tomany shows that, in most instances, MPBR also performs better than the other two methods in across-domains one-to-many alignment. In conclusion, the results from subsection ‘Same-domain One-to-many Alignments’ and subsection ‘Across-domains One-to-many Alignments’ demonstrate that MPBR is an effective method in retrieving one-to-many reaction mappings in the alignment of metabolic pathways.The above analysis of PSI represents the percentage of the solved instances of each method, in one-to-one alignment, ART1 denotes the average running time for the 64 instances solved by all three methods and ART2 represents the average running time for the 84 instances solved by MPBR and CAMPWays, and in one-to-many alignment, ART3 denotes the average running time for the 41 instances solved by all three methods, and ART4 represents the average running time for the 80 instances solved by MPBR and CAMPWays.In the experiments of one-to-one and one-to-many alignments, we have tested a total of 168 instances. In some cases, SubMAP and CAMPWays consumed an unusually long time until running out of memory, As can be seen from For the one-to-many alignment, we can observe from C-manytomany of many-to-many alignment results of MPBR.In addition to one-to-many alignments, we can also reveal alternative pathways that have similar functions by finding many-to-many reaction mappings between different pathways. This subsection discusses whether MPBR can accurately find such mappings in the alignment of metabolic pathways. The many-to-many reaction mappings include 2-to-2, 2-to-3 and 3-to-3 reaction mappings. Both CAMPways and SubMAP do not implement the functionality of many-to-many alignment. Escherichia coli (eco) and Agrobacterium tumefaciens (atc) are single-cell microorganisms, Homo sapiens (hsa) and Mus musculus (mmu) are complex organisms with cell membranes. Both atc and eco. The result is shown in atc and eco. This implies a common ancestral pathway, which is consistent with the theory that pathways for synthesis of proteinogenic amino acids were established before ancient organisms diverged into archaea, bacteria, and eucarya [In this subsection, we present three cases  terms or sequence information, to compute homological similarity of reactions. Another interesting issue is to exploit binary relation of reactions to identify functional motifs in metabolic pathways.S1 Table(DOC)Click here for additional data file.S2 Table(DOC)Click here for additional data file.S3 Table(DOC)Click here for additional data file.S4 Table(DOC)Click here for additional data file.S5 Table(DOC)Click here for additional data file.


Scientific Reports 10.1038/s41598-017-07589-8, published online 04 August 2017Correction to: The original version of this Article contained an error in the title of the paper, where the word “sources” was incorrectly given as “soruces”. This has now been corrected in the PDF and HTML versions of the Article.


P-values are being scrutinized as representing the most widely applied and abused practices. One line of critique is that P-values are inherently unfit to fulfill their ostensible role as measures of credibility for scientific hypotheses. It has also been suggested that while P-values may have their role as summary measures of effect, researchers underappreciate the degree of randomness in the P-value. High variability of P-values would suggest that having obtained a small P-value in one study, one is, ne vertheless, still likely to obtain a much larger P-value in a similarly powered replication study. Thus, “replicability of P-value” is in itself questionable. To characterize P-value variability, one can use prediction intervals whose endpoints reflect the likely spread of P-values that could have been obtained by a replication study. Unfortunately, the intervals currently in use, the frequentist P-intervals, are based on unrealistic implicit assumptions. Namely, P-intervals are constructed with the assumptions that imply substantial chances of encountering large values of effect size in an observational study, which leads to bias. The long-run frequentist probability provided by P-intervals is similar in interpretation to that of the classical confidence intervals, but the endpoints of any particular interval lack interpretation as probabilistic bounds for the possible spread of future P-values that may have been obtained in replication studies. Along with classical frequentist intervals, there exists a Bayesian viewpoint toward interval construction in which the endpoints of an interval have a meaningful probabilistic interpretation. We propose Bayesian intervals for prediction of P-value variability in prospective replication studies. Contingent upon approximate prior knowledge of the effect size distribution, our proposed Bayesian intervals have endpoints that are directly interpretable as probabilistic bounds for replication P-values, and they are resistant to selection bias. We showcase our approach by its application to P-values reported for five psychiatric disorders by the Psychiatric Genomics Consortium group.Increased availability of data and accessibility of computational tools in recent years have created an unprecedented upsurge of scientific studies driven by statistical analysis. Limitations inherent to statistics impose constraints on the reliability of conclusions drawn from data, so misuse of statistical methods is a growing concern. Hypothesis and significance testing, and the accompanying  P-values are often singled out as major culprits, not only because these concepts are easy to misinterpret, but for purported inherent flaws. Variability of P-values appears to be underappreciated in the sense that when a small P-value is obtained by a given study, researchers commonly suppose that a similarly designed independent replication study is likely to yield a similarly small P-value. We will use the term “replication P-values”, introduced by Killeen1, to mean the P-value obtained from subsequent, replicate experiments with the same sample size, taken from the same population. Great variability of replication P-values casts doubt on validity of conclusions derived by a study at hand and implies lack of confidence in possible outcomes of any follow-up studies. In reality, one should expect a greater uncertainty in what a replication P-value will be. Special prediction intervals for P-values, named “P-intervals”, have been employed to characterize that uncertainty6. P-intervals have been presented as an objective measure of P-value variability, as opposed to subjective judgments reported by researchers in surveys, with the conclusion that the subjective estimates are too narrow and, therefore, researchers tend to underestimate randomness of replication P-values3. While P-intervals have been used mainly as a tool to elucidate flaws of P-values, they have also been defended as important additions to P-values themselves in publications supportive of P-values as universal measures that provide useful summary of statistical tests6. It has been suggested that P-intervals may serve the purpose of improving P-value interpretability, especially in large-scale genomic studies with many tests or in other studies utilizing modern high-throughput technologies6. For example, in their discussion of P-values and their prediction intervals, Lazzeroni and colleagues6 argued that the P-values “will continue to have an important role in research” and that “no other statistic fills this particular niche.” They present P-intervals not as a way to expose alleged weaknesses of P-values but rather as a tool for assessing the real uncertainty inherent in P-values.Poor replicability has been plaguing observational studies. The “replicability crisis” is largely statistical and while there are limits to what statistics can do, a serious concern is misapplication of statistical methods. Significance testing and P-intervals for prediction of uncertainty in replication P-values is the lack of clear interpretation of their endpoints due to their frequentist construction. Classical prediction intervals, also known as “prediction confidence intervals”, have statistical properties that are similar to the regular confidence intervals (CI’s). Both types of intervals are random and are constructed to cover the replication value (1 − α)% of the time, referred to as the coverage property % represents the desired confidence level). As Lazzeroni and colleagues5 noted while discussing results of their simulation experiments, “By definition, the coverage rate is an average across the distribution” [of P-values]. This statement can be expanded as follows: given a large number of original studies with different P-values, if (1 − α)% P-intervals were to be constructed in each of these original studies regardless of statistical significance of the obtained P-value, then the average number of replication P-values covered by respective P-intervals is expected to be (1 − α)%. In this model, there are multiple original studies with a single replication P-value for each prediction interval, and it either falls into the interval or it does not. The average is taken over the proportion of times the replication P-value falls into the prediction interval. Caveating Lazzeroni et al.’s5 discussion, the endpoints of an interval constructed around a P-value obtained in any particular study cannot be interpreted in a probabilistic way with regard to a replication P-value, because it is either captured by the interval or not and the endpoints of the interval do not represent the range of possible values.In our view, the major difficulty with the applications of P-interval interpretation arises when it is constructed for a specific P-value. The coverage property of P-intervals as a long-run average is well-defined for random P-values, and the resulting intervals are themselves random. On the other hand, a P-interval constructed for a given P-value, P-values (0 to 1) to a narrow interval around ε. We can think of a process that generates these P-values as being the same as in the unrestricted case, but then we would discard any P-value outside the ε interval and evaluate coverage only for the intervals around P-values that are similar to P-values restricted to a specific range could be much smaller than the nominal (1 − α)% level expected across all possible values of the P-value5. Thus, the endpoints of any particular P-interval constructed around an obtained P-value are not readily interpreted in terms related to the P-value at hand or any future values in replication studies.Another difficulty with 8. Neyman starts by approaching the interval estimation from a Bayesian perspective, and describes a posterior distribution of the parameter θ, given the data x . Neyman writes that this distribution, Pr(μ|x), “permits the calculation of the most probable values of the μ and also of the probability that μ will fall in any given interval,”7 for example, μL ≤ μ ≤ μU. Neyman notes that the calculation of such a posterior interval requires placing a prior probability distribution on μ, something he seeks to avoid through the development of CIs. In the Bayesian set-up, the endpoints μL and μU are fixed numbers, while μ is random. To derive the CI endpoints as functions of random data, L(x) ≤ μ ≤ U(x), Neyman instead proceeds by working with the probability of the data x given the parameter value μ: Pr(x|μ). In contrast with posterior intervals, the value μ is unknown but constant and the interval endpoints L(x) and U(x) are random.It is illustrative to follow the reasoning of Neyman, who developed the theory of CI'sμ is contained in every interval computed based on the data x, the scientist will be correct (1 − α)% of the time in the long-run. He writes that the use of the intervals in practice would consist of collecting data x, calculating the endpoints, and “ stating that the true value of μ lies between [the interval endpoints]”. He stresses that the word ‘‘stating’’ “is put in italics to emphasize that it is not suggested that we can ‘‘conclude’’ that we can ‘‘conclude’’ that [the true value of μ = μ*] is L(x) ≤ μ* ≤ U(x) nor that we should ‘‘'believe’' 'that μ*is actually between L(x) and U(x)” and continues: “the probability statements refer to the problems of estimation with which the statistician will be concerned in the future”, but “once the sample is drawn and the values of L(x), U(x) determined, the calculus of probability adopted here is helpless to provide answer to the question of what is the true value of μ”.Neyman describes unequivocally the operational usage of CI's as “behavioral”: when a scientist consistently adheres to the rule of deciding to accept that after observing the values of the x’s […] we may decide to behave as if we actually knew that the true value [of the parameter μ] were between [L(x) and U(x)]. This is done as a result of our decision and has nothing to do with ‘‘reasoning’' or ‘‘conclusion'’ […] The above process is also devoid of any ‘'belief’' concerning the [true value of μ]”'.Neyman’s description excludes any probabilistic meaning attached to the endpoints of a particular interval: “P-value in the case of P-intervals, and “0” indicates that the replication P-value is outside that interval. Properly constructed intervals applied repeatedly to independent data sets will result in 1’s occurring with (1 − α) frequency. Although it is desirable to have the shortest possible intervals with this property, there is generally no information provided by the interval endpoints about a possible spread of replication P-values. However, interpretation of the classical interval endpoints in a meaningful way is warranted from a Bayesian viewpoint. A Bayesian derivation of a classical interval may reveal the tacitly assumed data generating mechanism. We will refer to that mechanism conventionally as an implicit prior distribution. It allows one to interpret the endpoints of a replication P-interval as (1 − α) probability of capturing the replication P-value. The endpoints of a P-interval are typically interpreted in a probabilistic fashion without specifying implicit prior assumptions. The following quote from Cumming2 gets to the heart of the matter succinctly: “This article shows that, if an initial experiment results in two-tailed P = 0.05, there is an 80% chance the one-tailed P-value from a replication will fall in the interval  […] Remarkably, the interval—termed a P interval—is this wide however large the sample size.” An equivalent statement appears in a Nature Methods letter by Halsey and colleagues: “regardless of the statistical power of an experiment, if a single replicate returns a P-value of 0.05, there is an 80% chance that a repeat experiment would return a P-value between 0 and 0.44.”4 Both statements make use of a specific value, P-values.An important point in the preceding discussion of confidence and prediction intervals is that their coverage property is defined as a long-run average of zeros and ones, where “1” indicates an event that a random interval covers the quantity of interest, i.e., a replication P-interval coverage due to restrictions on the P-value range, consider the following example. Suppose one performs a test for the mean difference between two populations and predicts variability of P-values in a replication study by constructing the corresponding 80% P-interval. If multiple samples are drawn from these populations and 80% P-intervals are constructed each time regardless of whether the observed P-value was significant or not, the results of these multiple experiments can be summarized graphically by Fig. P-value in the original study. The underlying effect size and sample sizes across studies are kept constant so replication values of the test statistic ranges from about negative two to positive two across all simulated experiments. Pink color highlights P-intervals that failed to capture the future value of statistic. Given these results, one can calculate an empirical binomial probability, i.e., the proportion of times a parameter was captured by the interval, which should be close to the stated nominal level. For instance, in Fig. To further illustrate possible issues with P-intervals are constructed only if the experiment returns a P-value close to 0.05, Pobt ≈ 0.05. That is, all experiments with P-values that did not reach statistical significance are discarded and a particular P-interval is constructed only if the obtained P-value is close to 0.05. Would about 80% of the P-intervals constructed around the respective Pobt still capture the future value of statistic? An intuitive way to think about this scenario is in connection to the publication bias phenomenon, where the actual relationships tend to be weaker in reality than what was claimed in publications, and we may suspect that P-intervals should be similarly biased when constructed around non-random, selected subsets of P-values.Now, consider a bit different scenario, in which P-interval is constructed for a particular Pobt, how can one interpret its bounds? If it is constructed based on an 80% classical prediction interval for a normal test statistic as originally suggested by Cumming2, with no regard to prior distribution assumptions, by definition it guarantees that 80% of P-intervals will capture a replication P-value. That is, the lower and the upper bounds of a P-interval do not provide bounds on the range of possible values of a replication P-value.Further, once a P-values based on the Mixture Bayes approach whose endpoints have a clear probability interpretation for any specific interval constructed based on a given data set. Unlike the classical coverage property, Bayesian intervals based on a posterior P-value distribution have the interval endpoints that are directly interpreted as defining a target range to contain a replication P-value with probability (1 − α).Our main goal in this work is to derive prediction intervals for P-values can be viewed as random variables, reflecting variability due to random sampling. This notion goes back to Fisher, whose method of aggregating information from several independent P-values is based on recognizing the fact that their product is itself a random variable (and twice the negative logarithm of that product has a chi-square distribution)9. The distribution of P-value and thus its variability are easily characterized analytically for the basic test statistics and depend on a measure of effect size, such as the value of odds ratio (OR) of disease given exposure vs. nonexposure to a pollutant. The nature of P-value randomness may be viewed from a number of angles12, but the randomness of P-value reflects randomness of the respective test statistic. When the effect size is zero, P-value of a continuous test statistic is uniformly distributed between zero and one, and as the departure from the null hypothesis increases, the shape of the P-value distribution becomes increasingly skewed toward zero (suppl. section 13) in which case the P-value distribution becomes a weighted average, i.e., a marginal distribution over all possible values of the effect with their respective probabilities as weights.P-intervals is that without specifying any prior knowledge on possible values of the effect size, one can take at face value the magnitude of an obtained P-value (Pobt). In other words, there is information about the magnitude of the effect size contained in the magnitude of Pobt, and that information alone can be used to make predictions about a P-value obtained in a replication study, denoted as Prep. As we illustrate with quotes from Cumming2 and Hasley4, practical applications of P-intervals are often factually Bayesian, defaulting to some interpretation about a possible spread of replication P-values. Our next goal is to explore relationships between P-intervals and Bayesian intervals. These relations are important because not explicitly stating a prior distribution amounts to sweeping a potentially unrealistic prior under the rug. In this work, we give explicit expressions for how the influence of the prior on the P-interval diminishes as more data is collected. Eventually, as the sample size increases, the P-interval endpoints approach those of the Bayesian intervals, but the sample size requirements depend on the variance of the prior distribution. In observational research, and especially in genetic association studies, where majority of tested hypotheses are effectually false, we find that sample sizes need to be very large.The idea behind the P-intervals can be viewed as a special case of the intervals that we develop: they correspond to the assumption that the product of the sample size N and the variance of the prior distribution δ) is a “large” number, in the sense that if we consider a normal random variable whose variance is δ = μ/σ). Flat priors are sometimes described as “noninformative,” reflecting lack of researcher’s knowledge or preference about a possible effect size. Yet, far from being uninformative, the flat prior places equal weighting on tiny as well as on large deviations from the null hypothesis. For example, correlation (being the standardized covariance) cannot be outside -1 to 1 interval, so simply acknowledging this range in a prior is already an improvement over an unrestricted prior. If part of the replicability problem lies with the preponderance of tiny effects in reality, the a priori assumption of a flat distribution for the effect size implicit in P-intervals would tend to result in intervals with the left endpoint that is unrealistically close to zero and thus promote false findings.Further, we show that δ = (μ1 − μ2)/σ for a given value of log(OR) cannot exceed δ values is unrealistic. To re-iterate this point, the effect size, for example, as measured by log(OR) can be quite large, but the standardized effect size  ≈ 1/2. Such large ORs are rarely encountered in observational studies14, suggesting that realistic variance values δ for any OR, no matter how large, cannot exceed ≈ 0.663. This bound places further restrictions on realistic and maximum possible values of the prior variance 15 and cancers16 translates into the values of the order 10−6–10−5 for How large can P-intervals unfit as a prediction interval for a replication value’s (Prep) variability and provide a generalization based on the Mixture Bayes approach, which is not constrained to the conjugate model only and provides researchers with the flexibility to specify any desired prior effect size distribution. Our results reveal immunity of the Mixture Bayes intervals to multiple-testing phenomena and to selection bias. When an interval is constructed for a particular value of Prep, its endpoints can be interpreted as a likely range of the Prep values. We contrast the performance of the traditional P-intervals relative to the Bayesian-based prediction intervals using results from the Psychiatric Genomics Consortium (PGS)5 and conclude with a discussion of the implications of our findings.In the next sections, we show how small values of P-interval can be obtained as a classical prediction interval for the normally distributed test statistics, (Z-statistics). The classical interval prediction problem is to probabilistically predict possible values of a future random observation, Xn+1, based on a sample of n values that have been already obtained, X1,…,Xn. In the case of the Z-statistic, the information about the effect  is summarized by the sample average, Zrep is based on a single obtained test statistic, Zobt, that statistic, as well as its corresponding P-value, Pobt, depend on all n sample observations. Moreover, in cases such as this, P-value as the only summary of data, it is possible, at least for standardized effect sizes, to obtain the full Bayesian posterior distribution and to characterize uncertainty about the effect size values. This conversion of statistics or P-values to posterior distributions requires one to augment information contained in P-values with a prior distribution on the standardized effect size. This approach is quite general, because while effect size may be measured by different types of statistics, such as the difference of two-sample means, or the logarithm of the OR, the summary of the effect present in the data is captured by the same Z-statistic, and it is the type of the test statistic that determines the interval properties, rather than a particular measure of the effect size.The Zrep relates to one-sided P-value as Prep = 1−Φ(Zrep) and has a normal distribution, Φ, where Φ(⋅) is the standard normal cumulative distribution function (CDF). Thus, the classical P-interval is constructed as:zα/2), whereZrep is Φ. Therefore, the P-interval based on the distribution Φ is a Bayesian interval that implicitly assumes that s2|Zobt = 1 and the prediction distribution for Zrep equal to Φ. We refer to the resulting intervals as the Conjugate Bayes intervals. The endpoints of these intervals are given byThe prediction distribution for the statistic P-interval in Eq. . However, the conjugate model is restrictive in that a specific prior distribution has to be assumed, which may not provide an adequate representation of external knowledge about the effect size distribution. It also limits construction of the intervals to P-values derived from statistics for which there are known conjugate priors. Here, we introduce a more flexible approach, the Mixture Bayes, without these restrictions. The Mixture Bayes intervals can be constructed for P-values derived from statistics whose distribution is governed by a parameter γ that captures deviation from the usual point null hypothesis, H0, and has the form N × δ2. This includes normal, chi-squared, Student’s t and F-statistics. We partition the prior distribution of δ1, δ2,…, δB with the corresponding prior probabilities, Pr(δi). As an example, let P-value be derived from an F-test for comparison of two-sample means, with the corresponding sample sizes n1 and n2. Let N = 1/(1/n1 + 1/n2). For i-th prior value of effect, a statistic based on sampling values of 1 = 1, df2 = n1 + n2 − 2:f is the density of the noncentral F-distribution. The posterior distribution is a mixture,Derived as a Bayesian prediction interval through the conjugate model, the endpoints of a l in Eq.  can now Trep, asNext, we obtain the CDF of the prediction distribution for the replication statistic, Then, the Mixture Bayes interval endpoints are derived from the quantiles of this CDF that are given by https://github.com/dmitri-zaykin/bayesian-PValue-Prediction-Intervals. The software allows users to construct a Mixture Bayes prediction interval for a P-value from the standard normal, Student’s t, chi-square or an F-statistic. For a P-value based on the standard normal or a t distribution, users have a choice between the conjugate normal model and the tabulated prior effect size distribution. For the F and the chi-squared test, no conjugate model exists, but prior values can be specified in a tabulated manner.We have developed a user-friendly software tool for implementation of our Mixture Bayes approach, available at p1 and p2, in the case of binary variables) as P-values for commonly used test statistics depends on the product of the sample size, , i.e. δ = μ/σ. For example, when the outcome is a case/control classification and the predictor is also binary, the standardized effect size can be expressed in terms of the correlation (R) times the sample size as follows:v is the proportion of cases in the sample. In terms of the logarithm of the odds ratio, OR,Genetic epidemiology and other observational studies routinely test hypotheses conceptually related to a comparison of two-sample means. Effect size is often measured by the log of OR, which can be related to the difference in means  that we obtained by maximizing the right hand side of Eq. (13) as:For a given value of OR, the standardized effect size Pr(OR ≥ x) = β and assuming a symmetric distribution of the effect size around zero, i.e., m0 = 0, we can relate the value δmax to the prior variance of the conjugate model in the following way:Let The maximum spread for the conjugate prior distribution is, therefore, obtained when its variance is equal toδ value for a given value of OR, however, it is not monotone in OR. The maximum possible value of δmax(OR) can be found to be at OR ≈ 121.35. Curiously, this value of OR implies δmax(OR) value equal to the Laplace Limit constant, 0.662743…It should be noted that Eq.  gives thP-interval is equivalent to a Bayesian prediction interval if 16 reported distribution of effect sizes for breast, prostate and colorectal (BPC) cancers in terms of a table, giving the numbers of different loci (Li) with the corresponding values of ORi. Using the same approach, Chen et al.15 provided the effect size distribution for the BP risk loci. Assuming the total number of independent variants to be M = 300,000, proportions of associated loci are wi = Li/M. We assumed the average OR among non-associated loci to be 1.005 (or its inverse for the negative part of the log(OR) distribution). The variance −6 for both cancer and the BP disorder risk loci. Thus, N needs to be about 50,000 for In Eq. , we showP-interval and Mixture Bayes interval performance for the different values of prior variance P-values would be commonly selected. One may also select results with the largest effect sizes as tentatively most promising for a follow-up. This selection induces a “selection bias”, for example, the largest estimated effect size in the original study would tend to be smaller once re-evaluated in a replication experiment, a phenomenon also known as the winner’s curse17. This would reflect the fact that the actual population effect size would tend to be over-estimated due to selection of the best outcome from the original study. An intuitive way of seeing why a selection bias would be present is to imagine a multiple-testing experiment where none of the tested predictors have any relation to the outcome. When one selects a predictor that showed the maximum estimated effect size, there will obviously be a bias, because the true effect size is zero. But this type of bias would also be present if the underlying effect sizes are non-zero for some or all of the predictors. Selection bias is difficult to correct for in the frequentist setting, but Bayesian analysis can be robust to this bias18. It is expected that the performance of frequentist-based P-intervals may suffer under selection while Bayesian-based intervals may not be affected by it. Thus, we investigated several types of selection and the resulting potential bias, measured by the proportion of times an interval captures Prep relative to the stated nominal level .In the next section, we explore P-value selection (simulation study set-up is detailed in Supplementary Information). The observed P-value was based on a two-sample Z-test and was thresholded according to the following selection rules: (i) no selection, i.e., a prediction interval is constructed for a randomly observed P-value; (ii) selection of P-values around a value, e.g., P-values that were close to the 5% significance level; (iii) selection of P-values that are smaller than a threshold, e.g., P-interval proposed by Cumming2.Table σ0/8 for every component and truncated the normal prior at 10−6 and 1−10−6 quantiles. This provided us with sufficient accuracy and resulted in the number of mixture components, B, equal to 76 for all values of Mixture Bayes intervals were included in these simulations to check how well they approximate a continuous prior distribution assumed by the conjugate intervals. We used mixture components with the length P-value ∈. However, selection and small prior variance both impair performance of P-intervals. For instance, if an interval is constructed for a P-value < 0.001 and P-interval may be as low as 17%. This poor coverage is due to a combination of both the selection bias and the implicit assumption that prior variance of P-interval endpoint being too close to zero. However, even for large values of prior P-interval has poor coverage when constructed for P-values around genome-wide significance levels . On the other hand, for large P-values the coverage of P-intervals becomes greater than the nominal (1 − α)% value. This is a consequence of the fact that P-interval’s width depends on the magnitude of P-values and as P-values become larger, the width of the interval increases as well. For example, given the prior variance P-intervals and the Bayesian intervals coincides at P-value = 0.446 . The P-interval around P ≤ 0.995, while the Bayesian interval is 0.110 ≤ P ≤ 0.958.Table P-interval coverage for P-values less than 0.05, we replicated Fig. P-interval is constructed only for P-values close to 0.05, P-intervals are becoming increasingly likely to miss Prep values altogether. The underlying effect size was kept the same in both figures and blue dots that represent values of zrep have similar range. Restricting P-values to be close to 0.05 induces selection bias, causing overestimation of the underlying effect size  and a vertical shift in P-intervals. Bias in coverage can be potentially removed by extending the interval endpoints by a correct amount, but the appropriate size of the interval appears difficult to determine analytically in a general way.To illustrate implications of decrease in entclass1pt{minimaP-value out of L tests  for attention deficit-hyperactivity disorder (ADHD), autism spectrum disorder (ASD), bipolar disorder (BPD), major depressive disorder (MDD), and schizophrenia. The consortium reported four single-nucleotide polymorphisms (SNPs) associated with these psychiatric disorders but, for illustrative purposes, we constructed prediction intervals only for a single SNP, rs2535629. We used four different methods to calculate prediction intervals: (i) the conjugate Bayesian model with the estimated prior variance, 15 (see Methods section); (ii) Mixture Bayes approximation to this continuous conjugate normal prior, using the same variance, P-intervals for one-sided P-values). We note that prediction intervals given in Table P-value) scale. However, follow-up studies target replication of the directional effects. For example, if a study reports a risk allele for a phenotype of interest and a replicatioin study finds the effect to be protective, one can not conclude that the follow-up study replicated the original report. Thus, one-sided tests would be more appropriate in follow-up studies and prediction intervals for one-sided P-values should be of interest. To transform prediction intervals for a two-sided P-value in Table P-value, one needs to subtract logarithm based ten of two  scale to P-value scale by raising ten to the negative logarithm based ten of the bounds power.To illustrate the interpretation of the prediction intervals and contrast the performance of the Bayesian-based intervals to the classical mentclass2pt{minimmentclass2pt{minimP-values based on the approach suggested by Lazzeroni et al.5 are smaller than the ones from the Bayesian-based methods. For instance, Lazzeroni and colleagues concluded that in a similarly powered replication of the original PGC design, a P-value for an association between rs2535629 and ADHD could be as low as P-value of 0.1005. Our interval results portray a less optimistic picture with the P-value lower bound for ADHD equal to 0.023. Similar observations hold for psychiatric disorders with significant observed P-values. For example, in Lazzeroni et al. the BP is concluded to be likely to yield a P-value between 1.69 × 10−13 and 0.04 and thus could reach genome-wide significance  at a replication study. Mixture Bayes prediction interval based on the reported effect size distribution15 suggests a higher lower bound for BP replication P-value of 6.92 × 10−7, concluding that a second, identical implementation of the original PGC design would be unlikely to yield any P-values <10−8. This difference in the spread of possible replication P-values highlights the implicit prior assumption built into the P-intervals that large effect sizes are as likely to be observed as small ones.Table P-interval: /σ, are readily extractable from P-values that originate from many basic and widely used test statistics, including the normal Z-statistics, Student’s t-test statistics, chi-square and F-statistics.Despite their pitfalls, we believe that P-intervals, a careful construction of the prior distribution may be difficult—a common issue in Bayesian analysis that is not unique to our particular method. Among other problems are assumptions of the model used to compute P-values themselves and including possibilities of confounding unaccounted for by the model. Keeping these limitations in mind, our results show that while classical P-intervals are derived without the explicit assumption that all effect sizes are equally likely, such a “flat” prior is assumed implicitly, whenever the endpoints of any given interval are interpreted as related to the range of replication P-values, which may lead to bias. For instance, bias will be present if a P-interval is constructed for a particular value, such as Pobt = 0.05. It should be recognized that in some experimental fields, statistical comparisons can be carefully targeted to investigate only those effects that are very likely to be real. A large probability of nearly zero effect size in the prior is inappropriate in this case. Still, one can argue that the prior should reflect some degree of skepticism toward a proposed hypothesis. On the other hand, under the flat prior assumption, all possible effect sizes are equally likely and hence a classical P-interval neither contemplates “a degree of doubt and caution and modesty”21 toward the hypothesis that the effect is present and substantial, nor acknowledges implausibility for the standardized effect size to take large values. When the effect size distribution is modeled in such a way that allows a proportionally small chance to encounter a large effect size and assumes that the majority of effect sizes would be close to zero, the Mixture Bayes approach would explicitly incorporate higher chances of what may be deemed “a false positive result” and it would adjust prediction interval bounds accordingly. Similarly, the flat prior assumption will lead to an invalid P-interval if it is constructed for a range of P-values . The (1 − α)% nominal coverage of P-intervals can be Bonferroni-adjusted5 for L tests as (1 − α/L)%. While that procedure can restore the long-run coverage property, i.e., (1 − α/L)% empirical binomial probabilities for P-intervals presented in Table P-value. Further, we showed that a flat prior effect size distribution may be incompatible with the bounded nature of the standardized effect size distribution and once again may lead to biased P-intervals. For example, many observational studies are seeking for associations between health outcomes and environmental exposures and can be viewed conceptually as a comparison of two-sample means, δ = μ1−μ2. Presence of a true association in such studies implies a certain difference in mean values of exposure between subjects with and without disease. In such examples, the prior variance reflects the prior spread of the mean of a test statistic, which usually can be related to the spread of the standardized mean difference. The prior spread in units of standard deviation cannot be very large, especially in the fields of observational sciences, that are currently at the focus of the replicability crisis. For example, assuming that effect sizes with the OR greater than three are relatively rare (1% occurrence rate), the prior variance for δ can not exceed approximately −0.66 < δ < 0.66 for any value of OR.Here, we focus on one of many aspects of statistical assessment of replicability; moreover, there are limitations to our approach. While we believe that due to the limited range of possible values for standardized effect size, it is difficult to do worse in terms of mis-specification of the prior distribution than to assume the prior implicit in lue (Eq. ) and wouP-values is present. Reanalysis of the intervals reported by Lazzeroni and colleagues5 shows that P-intervals can be substantially different from Bayesian prediction intervals, even when sample sizes are very large  are deemed to be statistically significant , where F(Z) is the tail area of the normal curve from—∞ to Z. The probability of the null hypothesis given the P-value takes a very similar form, N is the half of the harmonic mean of the sample sizes n1, n2, and ϑ is the variance of a zero-centered prior distribution for the standardized mean. Clearly, the one-sided P-value approaches this posterior probability as N increases.It is notable that a major part of P-values, or some modifications of them can be useful. Rather than adopting the view that P-values should be abandoned because they are poorly suited for what they are used for in practice, we advocate development of statistical methods for extracting information from them in such a way that when augmented with the external (prior) information about the effect size distribution, P-value can be transformed into a complete posterior distribution for a standardized effect size. How small a particular P-value is (its magnitude) does not inform us what to expect in a replication study25. Nevertheless, P-values, as transformations of statistics (such as the Z-statistic) contain summary information about the standardized effect size. Conditional on that information, one can predict a possible spread of future P-values and the respective statistics in replication studies. As part of addressing the multifaceted replicability crisis, researchers would benefit from availability of tools for prediction of variability inherent in commonly used statistics and P-values. In particular, prediction intervals equip researchers with quantitative assessment of what they may expect if they would have repeated their statistical analysis using an independent confirmatory sample.Overall, we share the viewpoint of Lazzeroni et al. that Supplementary Material


Reaching toward a point target has been intensively studied in human motor control. However, little is known about reaching toward a redundant target, such as grasping a bar, in which the grasping point is irrelevant to the achievement of a task. We examined whether humans could solve the target-redundancy and control problems in a serial fashion or control their body without solving the target-redundancy problem. We equalized the target ranges between two reaching tasks: a point-to-point reaching task without target-redundancy and a point-to-bar reaching task with target-redundancy. In the both tasks, we measured hand viscoelasticity at movement end as parameters that reflect the adopted control strategy. As a result, the hand viscoelasticity in the point-to-bar reaching task was smaller than that in the point-to-point reaching task, even under the same kinematics. These results indicate that the hand viscoelasticity was modulated depending on the target-redundancy. Moreover, it is suggested that a human reaches toward a redundant target by effectively utilizing information of target redundancy rather than explicitly solving the target-redundancy problem. When we move our hand to some specific target, we plan how to reach it before performing a hand movement24We hypothesized two types of strategies adopted by the CNS to deal with the target redundancy in the point-to-bar reaching task. One is a strategy that explicitly solves the target redundancy problem. In this strategy, the CNS explicitly determines a specific target point on the bar to transform the point-to-bar reaching problem to the point-to-point one. After solving the redundancy problem, the CNS can use the same control strategy in both point-to-bar and point-to-point reaching movements. An alternative type is a strategy that does not explicitly solve the target redundancy problem. In this case, the CNS temporarily holds information of a target redundancy and utilizes it for the planning and execution of a point-to-bar reaching movement. Therefore, the CNS adopts different strategies to perform the point-to-bar and point-to-point reaching tasks. This implies that the CNS utilizes the information of target redundancy to generate a motor command. In this study, we investigated which type of strategy the CNS adopts to perform the point-to-bar reaching task. In other words, this study investigated whether the CNS explicitly solves the target redundancy problem during movement planning.et al. reported that the limb stiffness was modulated when the shape of the target changedWe have to control movement conditions to compare the difference in control strategies from the kinematics. In this study, we controlled experimental environment, start position of movement, target position, required movement precision, and movement duration between two reaching tasks and equalize kinematics between the tasks. Previous studies have indicated that humans voluntarily control their viscoelasticities  according to the goal of the given task678t-test showed no significant difference in the X-direction: t(9) = 0.26, P = 0.80 or the Y-direction: t(9) = −1.74, P = 0.12). These results indicate that we could properly control the movement kinematics in both reaching tasks.To properly compare control strategies from motor outputs, we controlled the movement kinematics. First we check the movement kinematics in both reaching tasks. The subjects showed similar amplitude of variability of hand position at movement end in both reaching tasks (n = 10), the stiffness component along the X-axis in the point-to-bar task was significantly smaller than that in the point-to-point task (t-test: t(9) = −3.59, P = 0.0058 < 0.01). However, the stiffness component along the Y-axis was not significantly different between the two tasks (t(9) = 1.50, P = 0.17). Moreover, the hand viscosity showed similar results for hand stiffness (n = 10), the viscosity component along the X-axis in the point-to-bar task was significantly smaller than that in the point-to-point task (t-test: t(9) = −3.10, P = 0.013 < 0.05). However, the viscosity component along the Y-axis was not significantly different between the two tasks (t(9) = 0.84, P = 0.42 > 0.05). These results indicate that subjects modulated hand viscoelastic properties along only the X-axis, which is switched from the redundant direction to the control direction depending on the task.Next we check the movement dynamics. tiffness . Specifitiffness . AccordiIn this study, we examined how the CNS deals with the target redundancy, i.e., the target redundancy problem is explicitly solved or not. In the point-to-point reaching, the CNS deals with the hand trajectory redundancyet al. reported that the shape of the target, i.e., the required movement precision, affects the limb stiffnessX-direction, the variability of hand position at movement end in the point-to-point task would be smaller than that in the point-to-bar task due to the small size of the target. However, there was no significant difference in the standard deviation of hand position at movement end between the two tasks  the response to the external perturbation was not modulated based on target redundancy, and (2) no directional modulation of hand stiffness was observed5119Tagliabue, Ten healthy right-handed males participated in the experiments. Their average age was 25.8 (20–33 years). Their dominant hand was assessed by the Edinburgh handedness inventoryWe used a manipulandum system named “twin visuomotor and haptic interface system (TVINS)” to record force and movement kinematics of both hands at 2000 Hz . SubjectY-direction was fed back. Thus, subjects needed to control only the Y-directional movement. The target bar and subject’s hand were displayed by a 1.5 cm × 19 cm rectangle and a 2 mm × 19 cm rectangle (line), respectively. In the point-to-point reaching task, the hand positions in both directions were fed back. Thus, subjects had to control both directional movements. To equalize the target ranges between the two reaching tasks, we determine a target length in the X-direction in the point-to-point task as the standard deviation of hand position at movement end in the point-to-bar task. This enabled us to control the kinematic properties of movements in the point-to-point task to those in the point-to-bar task. Specifically, the target point width along the Y-axis was 1.5 cm, as with the point-to-bar reaching task, but the target length along the X-axis was ± 2 standard deviation of hand position at movement end in the point-to-bar task (mean ± SD: 3.13 ± 0.75 cm). The position of the subject’s hand was displayed by a 4-mm-diameter circle. The start position was displayed by a 1-cm-diameter circle placed 37 cm ahead of the center of gyration of the shoulder. The center of the target in the point-to-bar task was 16 cm ahead of the start position. The center in the point-to-point task was mean hand position at movement end in the point-to-bar task. We defined the times at movement start and end as the points when the tangential hand velocity was above and below 0.02 m/s, respectively.Subjects performed the point-to-bar reaching task with the one-dimensional target for 80 trials first, then the point-to-point reaching task with the two-dimensional target for 80 trials . In the X-direction. They were not explicitly instructed where they should move the hand on the bar. In the point-to-point task, they were instructed to move the point within the box and to apply control of movements in both directions.Under both task conditions, subjects were instructed to move their hand 16 cm toward the target in 0.55 ± 0.1 s. In the point-to-bar task, they were instructed to move the line within the box and not to give attention to hand movements in the To estimate the hand viscoelasticity, we applied a positional perturbation to the subjects’ hands by using the TVINS at movement end  and measured reactive force. The position of the manipulandum handle automatically shifted toward eight directions in a pseudo-random order. Each perturbation had small amplitude (7 mm) and short duration (0.4 s) with ramp-up, hold and ramp-down profiles . To miniPΔ, velocity VΔ, acceleration AΔ, and force FΔ from perturbation onset. Using mean values of PΔand FΔ across a stable constant phase  phase, i.e., at a time when the first peak of hand force appears after perturbation onset .Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.


The differences between β- and γ-actin are deeper than those between the amino acid sequences of these two proteins. Related research article Vedula P, Kurosaka S, Leu NA, Wolf YI, Shabalina SA, Wang J, Sterling S, Dong D, Kashina A. 2017. Diverse functions of homologous actin isoforms are defined by their nucleotide, rather than their amino acid sequence. eLife6:e31661. doi: 10.7554/eLife.31661Protein isoforms – proteins that are similar to each other and perform similar roles within cells – have played an important role in the generation of biological diversity throughout evolution. In some cases a single gene can encode two or more isoforms by exploiting a process called alternative splicing. In other cases two or more closely related genes are responsible for the isoforms. At its simplest, isoform generation provides a mechanism to specialize the properties of a gene or protein at one of three levels – DNA, messenger RNA (mRNA) or the sequence of amino acids in the protein itself . DependiActins are proteins that are essential for a number of fundamental cellular processes such as cell division and muscle contraction. In mammals, there are six actin isoforms, and two of these – β-actin and γ-actin – have been the focus of much research over the past 40 years. It is not in dispute that β-actin and γ-actin perform different functions in mammalian cells, and engage in distinct biological processes . MoreoveThe fact that β-actin and γ-actin are both conserved across birds and mammals is clear evidence for selective pressure to maintain certain features of these two proteins. However, it is not clear why birds and mammals have evolved to use two near-identical proteins to build cell architecture . The nucNow, in eLife, Anna Kashina and co-workers at the University of Pennsylvania and the National Institutes of Health – including Pavan Vedula and Satoshi Kurosaka as joint first authors – report that the differences between β-actin and γ-actin at the level of nucleotides may be as important as the differences at the level of amino acids . In a elBased on current knowledge, one might expect one of these roles to be localizing molecules of β-actin mRNA (but not γ-actin mRNA) to the periphery of cells. However, Vedula et al. draw attention to another feature unique to the β-actin mRNAs: they have a much higher density of ribosomes than γ-actin mRNAs. This does not relate to the relative amounts of the two proteins in cells, but to a large difference in the capacity of cells to make these proteins at any point in time. The fact that a cell can produce a large amount of β-actin protein in a short burst, whereas it would take a longer time to produce the same amount of γ-actin, could be significant. They also report differences in ribosome density between other actin isoforms and isoforms of histones and tubulins.In the mid-1970s the Nobel Prize-winning biologist Francois Jacob eloquently argued that much of evolution is 'tinkering' . But wou


We obtain some conditions for real-valued functions to be a (edge)  BCK-algebras and BCI-algebras were introduced by Iséki and Iséki and Tanaka  selective d-algebras and pre-d-algebras, and they showed that the squared algebra d-algebra d-algebra if and only if d / BCK-algebras. Using such deformations, d-algebras were constructed from BCK-algebras. Kim et al.  d-algebra  (Neggers and Kim X with a constant 0 and a binary operation "(I)(II)(III)For brevity we also call X a d-algebra. In X we can define a binary relation "≤" by A strong d-algebra (V)A d-algebra which is not a BCK-algebra. polynomial real algebras by using the real functions. In "d-algebras. In "d-algebra. We may apply the analytic method discussed here to several algebraic structures, and it may useful for find suitable conditions to construct several algebraic structures and many examples.We constructed some algebras on the set of real numbers by using elementary functions. The notions of (edge)


Nature Communications 10.1038/s41467-018-03540-1, published online: 19 March 2018.Correction to: The original version of this Article contained an error in the first sentence of the Acknowledgements section, which incorrectly referred to the Estonian Research Council grant identifier as “PUTJD618”. The correct version replaces the grant identifier with “PUTJD619”. This has been corrected in both the PDF and HTML versions of the Article.


In this note, we give an elaboration of a basic problem on convergence theorem of  The rapid development of q-calculus has led to the discovery of various generalizations of Bernstein polynomials involving q-integers. A detailed review of the results on q-Bernstein polynomials along with an extensive bibliography is given in [q-Bernstein polynomials are shown to be closely related to the q-deformed binomial distribution [q-boson theory giving a q-deformation of the quantum harmonic formalism [q-analogue of the boson operator calculus has proved to be a powerful tool in theoretical physics. It provides explicit expressions for the representations of the quantum group q-oscillator algebras well known in the earlier physics literature related to the representation theory of single parameter quantum algebras [During the last decades, the applications of given in . The q-Bribution . It playormalism . The q-at}SUq(2) . Meanwhialgebras .p and q versus single parameter q-Bernstein-type operators [Recently, ursaleen . BesidesMirakyan , BaskakoMirakyan  and KantMirakyan  operatormentclasspt{minimaMirakyan . For furMirakyan , 16–18. perators , 20. Theperators ).k and k, n with For example, consider the posed in . Given \p and q with n. We then choose a sequence In general, we expect The convergence theorems for ers . For exers (ii)Since Set We assert that the inverse proposition of (ii) also holds. Indeed, for each sufficiently large Next, we infer that as . ‘⇐’ of , we knowTherefore, (low from  and (2.1low from . □(If In general, from  we have (If (If Similarly, from (n, there exists In Case (Now we utilize Theorem For n→∞. See , Theoremn→∞. See , formulaIn this note, we mainly obtain the sufficient and necessary conditions for


Fully gapped d-wave superconducting states have, however, been proposed in several cuprate systems thanks to subdominant order parameters producing d + is- or d + id′-wave superconducting states. Here we study both magnetic and potential impurities in these fully gapped d-wave superconductors. Using analytical T-matrix and complementary numerical tight-binding lattice calculations, we show that magnetic and potential impurities behave fundamentally different in d + is- and d + id′-wave superconductors. In a d + is-wave superconductor, there are no bound states for potential impurities, while a magnetic impurity produces one pair of bound states, with a zero-energy level crossing at a finite scattering strength. On the other hand, a d + id′-wave symmetry always gives rise to two pairs of bound states and only produce a reachable zero-energy level crossing if the normal state has a strong particle-hole asymmetry.Impurities in superconductors and their induced bound states are important both for engineering novel states such as Majorana zero-energy modes and for probing bulk properties of the superconducting state. The high-temperature cuprates offer a clear advantage in a much larger superconducting order parameter, but the nodal energy spectrum of a pure  The cuprate superconductors with their much high transition temperatures would here be a tantalizing option, if it were not for their d-wave order parameter symmetry which enforces a nodal energy spectrumd-wave superconductor can only host virtual bound states72Cu3Oδ7− a fully gapped spectrum has recently been discovered and attributed to the existence of subdominant order parameters, with the superconducting symmetry likely being either d + id′)22π/2-phase shift relative to the dominant order. The d + id′-wave state is also a chiral state with its non-trivial topology classified by a Chern number N = 2d + is- or d + id′-wave symmetry.One severely limiting factor in all these studies is the low superconducting transition temperature accompanied by the small energy gap associated with conventional d-wave superconductors. More specifically, we investigate the intra-gap bound states due to potential and magnetic impurities using both an analytic continuum T-matrix formulation and numerical tight-binding lattice calculations. We show that impurities create entirely different bound states in d + is-wave and chiral d + id′-wave superconductors, despite both being fully gapped and with a dominant parent d-wave state. For a d + is-wave superconductor we find that a potential impurity does not induce any bound states, while a magnetic impurity gives rise to a pair of bound states, which behaves very similar to the YSR bound states in conventional s-wave superconductors. This includes the behaviour of the energy spectrum when tuning the scattering strength Umag of the magnetic impurity. Since the d + is-wave state is topologically trivial and with a low-energy s-wave gap, this resemblance with a conventional s-wave superconductor is very plausible. More specifically, we find that the magnetic impurity bound states have a zero-energy level crossing at a finite critical scattering d-wave to the subdominant s-wave order parameter. Moreover, through self-consistent tight-binding calculations we find a first-order quantum phase transition at π-phase shift for the s-wave component of the order parameter.In this work we establish the properties of both potential and magnetic impurities in these two fully gapped d + id′-wave state we find a very different behaviour. Here both potential and magnetic impurities induce two pairs of bound states. For superconductors with a particle-hole symmetric normal state, the bound states are two-fold degenerate and there is no level crossings for any finite coupling. Instead, it is only in the unitary scattering limit (Uc → ∞) that the bound states approach the middle of the gap. Doping the normal state away from particle-hole symmetry, the degeneracy is lifted for a magnetic impurity but not for a potential impurity. Moreover, for finite doping there is now a zero-energy level crossing, but for low doping it occurs only at very large scattering strengths. Self-consistent calculations for a single impurity in a d + id′-wave superconductor finds a first-order phase transition at the level crossing, but no local phase shifts in either the dominant or subdominant order parameters. Considering that recent experiments have demonstrated access to adjustable magnetic scattering strengths Umagd + id′-wave state and the likewise time-reversal symmetry breaking but topologically trivial d + is-wave state.For the chiral d-wave state, which persist even above the superconducting transition temperature and consequently, can not be considered to be a good probe of the symmetry of the superconducting stated-wave superconductors offer a tantalizing alternative due to potentially much higher superconducting transition temperatures. Our results show that magnetic impurities in both d + is- and d + id′-wave superconductors are interesting systems in this regard, as they generate zero-energy impurity states with no band degeneracies.These results have several important consequences. In spite of impurities only generating localized imperfections to the lattice structure, we find that they are very suitable for probing and differentiating the symmetry of the superconducting state. Our results show that the symmetry of the order parameter can even be probed by simply counting the number of induced subgap states. This is in sharp contrast to the virtual bound states in the pure d-wave superconductors with a fully gapped energy spectrum. Introducing a subdominant superconducting order parameter will achieve this, since it very generally align with a complex π/2 phase relative to the dominant d-wave state. Here we consider a two-dimensional (2D) Impurity-induced bound states only exist in ħ = 1)The Hamiltonian in the presence of a single impurity  reads . The exact form of ξ(k) is unimportant as we can linearise the spectrum around the Fermi level, setting k| for the order parameter. The dominant order parameter is Δ1(k), while Δ2(k) represents the subdominant order parameter. We assume that the magnetic moment is large enough to ignore quantum fluctuations and thus we treat the impurity as a classical spin. The local moment of the impurity is directed along the z easy axis, but it is straightforward to show that the results are not affected by this assumption, since the electrons pair in the spin-singlet channel. This is different from magnetic impurities in p-wave superconductors, where the direction of the impurity generally affects the bound state energyτi and σi are the Pauli matrices acting in particle-hole and spin spaces, respectively, while τ0 and σ0 are unit matrices. The bare Green’s function for the superconductor iswhere we use the Nambu space spinor magnitude of the wave vector k, but only on its direction with the energy spectrum Fi(ω) asLet us first consider a fully particle-hole symmetric spectrum for the normal state, which imposes ρ = kF/(2πνF) is the density of states of the 2D free electron gas at the Fermi level. The above result is for a particle-hole symmetric normal state, but this symmetry is often broken in reality. We use the chemical potential μ to measure the degree of particle-hole symmetry breaking in the energy spectrum, thus leaving the case μ = 0 to represent full particle-hole symmetry. Considering a small deviation from particle-hole symmetry, such that F3τ3σ0. Up to first order in F0, F1, and F2 remain unchanged from the particle-hole symmetric case.Here F1(ω) = 0, due to the periodicity of the cosine function and the subdominant order parameter Δs not depending on ϕ. Then, in the limit of μ = 0, the bound states can be found as the solutions toFirst, we consider the Umag. In order to find the bound state energies for a magnetic impurity we rephrase F0(ω) and F2(ω) in terms of the complete elliptic integral of the first kind KSince we are interested in real bound states, we only look for solutions s-wave superconductor: d + is-wave superconductor comes as the solution of s-wave superconductor, where the level crossing signals a quantum phase transition between two different ground states. For a magnetic impurity stronger than d + is-wave superconductor. Interestingly, for a d + is-wave superconductor, the critical coupling depends only on d and Δs separately. By setting s-wave parameter is small compared to the d-wave order.where we have defined If we now break the electron-hole symmetry of the normal state, i.e. assume d + is-wave superconductor. Moreover, the modifications of the bound state spectrum induced by a magnetic impurity is of the order It is clear that also in this case, for a purely potential impurity there are no real roots and consequently, potential impurities never induce any bound states in a d-wave or ϕ) and sin(2ϕ), all off-diagonal terms in the summation of the bare Green’s function, i.e. Fi with μ = 0) is then onlyNext, we turn to the impurity bound state formation inside the gap of a chiral In this case the bound states are found as the solutions tod-wave superconductor lead to exactly the same pairs of intra-gap bound states, these states are explicitly shown in μ = 0 we find that for a potential impurity the negative energy branch (those occupied at zero temperature) consist of one spin-up and one spin-down state, and there is thus a Kramers degeneracy of the states. However, for a magnetic impurity the bound states with negative energy are both spin-down quasiparticles. For an impurity with both potential and magnetic scattering effects on the charge carriers (Upot and Umag both non-zero), four non-degenerate bound states are generally present in a chiral d-wave superconductor, such that the two-fold degeneracy in Very interestingly, pure potential or pure magnetic impurities in a chiral F3τ3σ0, where, up to first order in d-wave state compared to the d + is-wave state. For a potential impurity the bound states energies are still two-fold degenerate, but a non-zero μ shifts the energy of the bound states in the unitary limit, as is illustrated in μ > 0, a level crossing now appears for repulsive impurity scattering (Upot > 0), while for a hole doped system, μ < 0, the level crossing instead occurs for an attractive impurity (Upot < 0). Thus doping can be used as a simple means to control the level crossing for potential impurities. A local chemical potential induced by a tunneling probe could even offer in-situ tunability of the level crossingIf the particle-hole symmetry of the normal states is broken, here by setting μ, as illustrated in Umag. In this case, one pair of bound states move away from the middle of the gap (thick red lines). For these states no zero modes are thus expected even in the unitary scattering limit for any positive scattering Umag > 0. The other pair of bound states (thin blue lines) move toward the middle of the gap and consequently a level crossing appears at some Umag, the bound states also do not depend on the sign of μ. Remarkably, the dimensionless critical coupling for reaching a zero-energy state for both magnetic and potential scatterers is the same μ|.For a magnetic impurity the bound state degeneracy is lifted for finite s-wave superconductivity to the chiral d-wave. More precisely, if the order parameter takes the form s′ is k-independent, the energy degeneracy for magnetic impurity bound states is lifted, while the bound states remain degenerate for a potential impurity. Therefore, the degeneracy of the bound states in the presence of a magnetic impurity in a chiral d-wave superconductor is very fragile and can easily be lifted.For the purpose of the forthcoming self-consistent numerical calculation, we mention already here that even at half-filling, the energy degeneracy can still be lifted by adding a small amount of extended d + is- and chiral d + id′-wave superconductor. The effective Hamiltonian for the 2D superconducting host with an impurity with both potential and magnetic scattering elements located at R is refs 31We now turn to discuss the results obtained from tight-binding lattice calculations. In all calculations we use a generic finite-size square lattice in which we consider a single impurity located at the middle site. Again we consider both a i =  represents a site in the square lattice, with the lattice spacing a set to be 1. The dominant d-wave order exists on nearest neighbour bonds, while the subdominant s-wave order is an on-site parameter and the d′-wave order reigns on next nearest neighbour bonds. For the self-consistent calculations (see below) we do not a priori assume any symmetries or conditions for any of these three order parameters. However, in calculations with constant order parameters throughout the sample, i.e. non-self-consistent calculations, we enforce the d-wave order by setting y-directed bonds is equal in magnitude but with opposite sign compared to the order parameter on x-directed bonds. Likewise, the d′-wave state has opposite signs on bonds in the ±(x + y) direction compared to bonds in the ±(x − y) direction. We also enforce the subdominant order parameter (s or d′) to be purely imaginary in the non-self-consistent calculations.Here We solve 3233d-wave superconductor at finite doping, for both the numeric tight-binding method and an analytic T-matrix calculation. In order to be able to do this comparison, we evaluate the summations appearing in the T-matrix formalism for a discrete mesh over the first Brillouin zone of the square lattice using the same form of the kinetic energy. The main difference between the continuum and discrete calculation is that in the former we neglect the radial dependence of the order parameter in reciprocal space while in the latter, we keep both the radial and angular dependence. Therefore, the discrete summation over the full tight-binding model gives us a more relevant solution to compare with the finite-size numerical calculations. We also find an excellent agreement in the unitary scattering limit (Umag → ∞), which in the tight-binding lattice calculation can be implemented by simply removing the impurity site, thus creating a vacancy.We find that the tight-binding calculations fit exactly to the analytical T-matrix results. For instance, in V in each pairing channel and calculate the order parameter(s) explicitly everywhere in the lattice. For a d-wave state we use the self-consistent condition i, j are nearest neighbour sites. In the self-consistent calculation we start by guessing a value for Δd on each bond, solve d on each bond using the self-consistent condition, and repeat until Δd does not change between two subsequent iterations. For the d + is-wave state we also assume a finite Vs in addition to Vd and calculate separately d + id′-wave state i, j are next nearest neighbour sites.Above we simply assumed constant order parameters and enforced the correct condensate symmetries. Now we allow the condensate to appropriately respond to the impurity through a proper self-consistent calculation. Since the superconductor symmetry is important for the properties of the bound states, this is the most accurate way to ensure a correct solution. In these self-consistent calculations we only assume a finite and constant pair potential d on x- and y-directed bonds are treated fully independent and the same applies to the d′-wave state. Thus, we have not a priori assumed any symmetry for any of the pairing states. In the calculations we use Vd/t = 1.7, Vs/t = 1.7 for d + is-wave state and Vd/t = 1.8, Vd′/t = 1.7 for chiral d-wave state, but the results are not sensitive to these particular values. We also set μ/t = −1 for the most general case and to avoid the van Hove singularity at half-filling. We mainly use a 51 × 51 lattice, with similar results obtained with a 31 × 31 lattice, which guarantees that the results are not sensitive to the lattice size.We take the initial guess for the subdominant order parameter to be purely imaginary but through the self-consistency loop it is free to acquire a real component as well. Likewise, we emphasize that Δd + is- and d + id′-wave states in the presence of either magnetic or potential impurities. The self-consistent tight-binding lattice calculations reveal that the results obtained for fixed order parameters are still largely valid. However, important effects appear around the critical scattering strength in the self-consistent calculations. Starting with the d + is-wave state, the self-consistent results show that the intra-gap localized bound states from a magnetic impurity behave largely in a similar way to their non-self-consistent counterpart as seen in s-wave superconductorUsing the self-consistently calculated order parameters, we extract the LDOS at and close to the impurity site for both s-wave and s-wave order parameter. This extended s-wave order resides on the nearest neighbour bonds and appears only very close to the impurity. It is thus a direct consequence of the impurity weakening the d-wave character in favour of the more disorder-robust s-wave symmetry. At the quantum critical point this extended s-wave state even becomes the dominant order parameter but notably only at the impurity site, farther away the d-wave order parameter is still dominant. In addition, both the s-wave and extended s-wave order parameters develop a π-phase shift on the impurity site across the critical coupling. This is in line with previous calculations for pure s-wave superconductors where the s-wave state undergoes a local π-shift1530d-wave state stays constant.The order parameter at the impurity site shed more light on this transition as can be seen in d-wave superconductor, there is only a level crossing for one pair of bound states as seen in d + is case. Considering the order parameter, the self-consistent calculation reveals that the dominant dxy-wave state, as seen in in s-wave components appear on nearest and next-nearest neighbour bonds, defined here as Δs and Δs′, respectively. Interestingly, this means that for a chiral d-wave superconductor, the impurity does not disturb the dominant d-wave orders nearly as much as in the d + is-wave case. Despite the smallness of the s-wave components generated in the self-consistent calculations, we find that they are still responsible for lifting of the degeneracy of the impurity bands in the half-filled lattice case (μ = 0).For the case of a magnetic impurity in a chiral d-wave superconductors. The main results are summarized in d + id′-wave superconductor, while for a d + is-wave superconductor, there is only one (zero) pair of bound states for a magnetic  impurity. As a result, the number of intra-gap bound states becomes a powerful means for establishing the symmetry of the superconducting state in a fully gapped d-wave superconductor, such as that recently established in cuprate nanoislands or at certain cuprate surfaces22In this work we have investigated impurity-induced bound states in fully gapped d + is-wave and chiral d-wave superconductors is the behaviour of the zero-energy level crossings for the impurity-induced states. For a d + is-wave superconductor, increasing the magnetic scattering strength Umag leads to a level crossing of the bound states, which means there always exists a critical coupling d + id′-wave superconductor there is no level crossing for a particle-hole symmetric normal band structure (here indicated by μ = 0) at any finite scattering strength, for either potential or magnetic impurities. Only at significant doping away from μ = 0 there is a zero-energy level crossing at an experimentally achievable scattering strength. It is also important to notice that the impurity bound states in a chiral d + id′-wave superconductor are often twofold degenerate for both potential and magnetic impurities. For a magnetic impurity the degeneracy is lifted by either a finite μ or by ever-present subdominant extended s-wave components, as we find in our self-consistent calculations, nonetheless, there are often two nearly degenerate states.Another important difference between d + is- and d + id′-wave superconductors, have the potential for hosting Majorana fermions. However, in the case of a single impurity, the two zero energy states at s-wave456d + id′-wave31d + id′-wave superconductor the near degeneracy of the impurity bands however possess additional complications for Majorana fermions in an impurity wire. An even number of positive (or negative) near zero-energy states in the single impurity limit will result in two putative Majorana end modes for a wire, which then hybridize and split off from zero energy, losing their Majorana character. Thus, in terms of the potential for generating Majorana modes, our results directly shows that only magnetic impurity wires in a d + is-wave superconductor or in a heavily doped d + id′-wave state are promising systems.The systems with zero energy states and no impurity band degeneracies, i.e. magnetic impurities in both d + is and d + id′-wave superconductors, these are first-order quantum phase transitions, with clear discontinuities in both energy levels and order parameters. For the d + is-wave state we even find a local π-shift at the phase transition for all subdominant order parameters, consistent with the behaviour in conventional s-wave superconductors30d + id′-wave superconductor we, however, do not find any π-shifts.Finally, we emphasize that our self-consistent calculations confirm our analytical results where we have assumed constant order parameters uninfluenced by the impurities. In addition, the self-consistent calculations shed more light on the nature of the zero-energy level crossings and show that for both How to cite this article: Mashkoori, M. et al. Impurity bound states in fully gapped d-wave superconductors with subdominant order parameters. Sci. Rep.7, 44107; doi: 10.1038/srep44107 (2017).Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.