The distance k-domination number G is the minimum cardinality among all distance k-dominating sets of G. The first Zagreb index of G is defined as G is n-vertex trees with given distance k-domination number and characterize the extremal trees, which generalize the results of Borovićanin and Furtula (Appl. Math. Comput. 276:208–218, n-vertex tree T, is that a sharp upper bound on the distance k-domination number Let  G. The eccentricity of v is defined as diameter of G is P is called a diameter path of G if the length of P is i from v in G, that is, privatek-neighbor of u with respect to D if pendent vertex is the vertex of degree 1.Throughout this paper, all graphs considered are simple, undirected and connected. Let first Zagreb indexsecond Zagreb indexA chemical molecule can be viewed as a graph. In a molecular graph, the vertices represent the atoms of the molecule and the edges are chemical bonds. A topological index of a molecular graph is a mathematical parameter which is used for studying various properties of this molecule. The distance-based topological indices, such as the Wiener index , 3 and tfound in –25. Manymentclass2pt{minimfound in –40.dominating set of G if, for any vertex domination numberG is the minimum cardinality of dominating sets of G. For distancek-dominating set of G if, for every vertex distancek-domination numberG is the minimum cardinality among all distance k-dominating sets of G )IfTis ann-vertex tree, different from the starthenforIn what follows, we present two graph transformations that increase the Zagreb indices.T be an n-vertex tree (u of v of w to the u (=v)  andThenu and v. ThenIt is obvious that Letuandvbe two distinct vertices inG. are the pendent vertices adjacent touandare the pendent vertices adjacent tov. Defineandas shown in Figure Then eitherorentclass1pt{minimaFor a connected graphGof ordernwithG be a connected graph of order n. If G of order n if Let LetTbe ann-vertex tree with distancek-domination numberThenD be a minimum distance k-dominating set of T,Suppose that k-dominating set of T, which contradicts Clearly, k-dominating set according to the definition of k-dominating set of the tree If k-dominating set of T. Let k-dominating set of T, so the claim is true.We claim that In view ofk-domination number of a connected graph is an attractive problem. In Lemma k-domination number of a tree is characterized. Namely, if T is an n-vertex tree with distance k-domination number Determining the bound on the distance n-vertex trees with distance k-domination number k-domination number mentioned in the above paragraph is sharp. Let k-domination number n-vertex tree with domination number were studied in [The Zagreb indices of mentclass2pt{minimTbe a tree onvertices. Thenif and only if at least one of the following conditions holds: Tis any tree onvertices;for some treeRonvertices, whereis the graph obtained by taking one copy ofRandcopies of the pathof lengthand then joining theith vertex ofRto exactly one end vertex in theith copy ofLetTbe ann-vertex tree with distancek-domination numberIfthenandwith equality if and only ifR on n-vertex tree with vertex set When LetGbe a graph which has a maximum value of the Zagreb indices among alln-vertex connected graphs with distancek-domination number andIfthenu and v are two distinct vertices in u and v, where D be a minimum distance k-dominating set of G. If k-dominating set of T. Hence, we assume that Suppose that n and distance k-domination number n-vertex tree T, then denote by In this section, we give upper bounds on the Zagreb indices of a tree with given order LetTbe ann-vertex tree andThenandThe equality holds if and only ifLet IfTis ann-vertex tree with distancek-domination numberthenwith equality if and only ifwhereAlso,with equality if and only ifwhereT. If k-dominating set of T, a contradiction to Assume that n-vertex tree If Since Let treeThenandwith equality if and only ifn. By Lemma Assume that D be a minimum distance k-dominating set of T and T. Then k-dominating set, a contradiction. Note that T and Let k-dominating set of the tree Assume that If Since k-dominating set, a contradiction. It follows that D. If k-dominating set, a contradiction. Therefore, If Considering s be the unique vertex adjacent to i.e., Let Note that mentclass2pt{minimLetTbe a tree of ordernwith distancek-domination numberThenandwith equality if and only ifT. Define k-dominating set D of T such that Let T by applying Transformation Let By Lemma D is a minimum distance k-dominating set of k-neighbors of x with respect to D in k-dominating set of tree k-dominating set of Suppose that By the definition of the first Zagreb index, we getIf If In what follows, we prove In view of Thus, However, the second condition is impossible. If Note that Hence, if Analogous to the analysis of the first Zagreb index, the second condition above is impossible. Thus,Besides, if k-domination number for k-domination number for T be an n-vertex tree with distance k-domination number Borovićanin and Furtula  proved\mentclass2pt{minim


T, the boundary term B and a scalar field arXiv:1309.2698, Teleparallel theory of gravity and its modifications have been studied extensively in literature. However, gravitational waves has not been studied enough in the framework of teleparallelism. In the present study, we discuss gravitational waves in general theories of teleparallel gravity containing the torsion scalar  Therefore, GR and Teleparallel Equivalent General Relativity (TEGR) result in the same equations of motion.The observed late expansion of the universe can be described by either introducing an exotic form of energy (dark energy) or modifying gravity. In this framework, several modifications have been proposed –4 and, aT or R are taken into account [T and R, i.e. f [f, where B is the boundary term [However, difference between them arise in modified Lagrangians, where scalar fields coupled nonminimally to gravity or arbitrary functions of  account . Such mo account . In a mo f , 9. Thisary term , 11.In both GR and TEGR, gravitational waves (GW) have two independent polarizations, usually denoted as plus and cross modes. However, extra polarizations appear in modified theories. The perturbation theory in the post-Minkowski limit is a way to study the number of GW polarizations.z-direction as the propagation direction of GWs, they are plus b), longitudinal (l), vector-x (x) and vector-y (y) modes. These modes can be described by the independent NP quantities f(T), and in its scalar-tensor representation, are equivalent to that in GR and TEGR [f(R) gravity, where the Lagrangian is an arbitrary function of Ricci scalar, three modes exist  [f, and have shown that the number of GW-modes depends on the form of it [The other way is the Newman-Penrose (NP) formalism , 13. Ado.g. Ref. , 15). As.g. Ref. , GWs in and TEGR . In f(R)s exist  –20. Mode in Ref. , where \f(R), or of TEGR, like f(T). The main task is explaining dynamics by further degrees of freedom of gravitational field (with respect to GR or TEGR) instead of invoking dark components [f(R),because, in such a case, it is possible to reproduce the so-called Sanders potential with better precision. In this case, by a conformal transformation, it is shown that a model like T, but also boundary terms B and a scalar field An important remark is necessary at this point. Modified theories of gravity are taken into account to achieve a comprehensive picture of cosmic dynamics ranging from early inflation, up to large scale structure formation and current acceleration of the universe –4. The amponents . Howevermponents , the flaentclass1pt{minimassion in , could bThe present paper is organized as follows. The field equations of our modified teleparallel theory are derived in Sect. B coupled to the scalar field [B is second order in perturbations; therefore, up to first order we have In teleparallel theories, vierbein fields describe gravity. Considering a set of orthonormal basis in each point of a generic manifold, the metric is given byar field . Let us F and E are two arbitrary functions of scalar field. For i.e.z as the direction of wave traveling. Taking l and b modes can be of the same order. Considering the weak field limit, we can adopt the NP formalism. To obtain the independent NP quantities, one can use the solution  being an analytic function of T and B, one can expand it as followsIf from Eq. , we get F(R) gravity can be easily recovered.In order to respect the local Lorentz symmetry invariance, we have to consider he paper , 62\docuthe mass  reduces Furthermore, the action . These sor . Equatie.g. torsion tensor becomes Under local Lorentz transformation m appear , i.e.95\n fields , up to fone gets 97\documeFinally, let us consider the action containing the following termB and a further scalar field T. The conclusions we reached are the following. There is no extra polarization in TEGR and in f(T) theory with respect to GR as already shown in [B, beside the standard two modes of GR. One can also write the Lagrangian as a function of scalar torsion T and Ricci scalar R, however in order to study GW polarizations, it is better to decouple the Ricci scalar f. In f, extra massless and massive modes arise when the scalar torsion and the boundary term are non-minimally coupled as in the theory of The number of GW polarizations depends on the considered theory of gravity. In present work we have studied GWs in extended teleparallel gravity where a boundary term shown in . Here wehes (see  for a dif(T) gravity, showing the same gravitational modes as GR [In this perspective, the GW170817 event  has set es as GR , should es as GR , 15). In


Here, we provide a first-principle derivation of this HLL based on the gauge-theory approach. We start by considering massless Dirac fermions confined on the one-dimensional boundary of the topological insulator and interacting through a three-dimensional quantum dynamical electromagnetic field. Within these assumptions, through a dimensional-reduction procedure, we derive the effective 1 + 1-dimensional interacting fermionic theory and reveal its underlying gauge theory. In the low-energy regime, the gauge theory that describes the edge states is given by a conformal quantum electrodynamics (CQED), which can be mapped exactly into a HLL with a Luttinger parameter and a renormalized Fermi velocity that depend on the value of the fine-structure constant  Further spatial  symmetries have been proposed in order to extend the periodic table of topological free-fermion systems4, and more recently inversion symmetry has also gathered attention6. However, time-reversal-invariant topological insulators are certainly the most studied so far8. These time-reversal-invariant topological insulators were theoretically proposed to occur in two-dimensional models involving a strong spin-orbit interaction10, and were then experimentally observed in HgTe quantum wells11. The spin-orbit interaction locks the spin and the chirality together and produces counter-propagating edge currents, giving rise to the quantum spin Hall effect. These topologically protected edge modes are right-handed and left-handed Dirac modes that always come in pairs, in agreement with the time-reversal symmetry of the bulk. Their dynamics is consistently described by a 1 + 1-dimensional massless Dirac theory.Topological insulators represent a large family of materials characterized by gapped bulks and metallic edge states. The topological quantum numbers associated to the bulk depend on the discrete symmetries of the microscopic Hamiltonians, such as time-reversal, particle-hole and chiral symmetries13. In this picture, the strength of the interactions is encoded in the Luttinger parameter K, which depends on the value of the coupling constant g of the four-fermion term. Although many studies have pointed out for which values of K the interactions are relevant, it is still unclear how the constant g is related to the microscopic properties of the Dirac edge modes, such as their spin, electric charge, etc. The relevant open question is whether there is any fundamental way to derive the HLL from the universal properties of topological insulators.Moreover, it has been shown that local four-fermion interactions on the edge can transform the free-fermion phase into a new non-Fermi-liquid phase, called helical Luttinger liquid (HLL)U(1) gauge field are free to propagate in all the three spatial dimensions that represent the physical space where the topological insulator is embedded, see Fig. 15 plus the 1 + 1-dimensional massless QED, also known as the Schwinger model17.The main goal of this paper is to provide an answer to this question. Firstly, we consider massless Dirac fermions constrained in one spatial dimension (the boundary), while the quantum excitations  of the U gauge fi19 and related 2D massive Dirac systems, such as silicene and transition metal dicalcogenides20, but to the best of our knowledge, it has not yet been employed in the description of one-dimensional systems, such as the edge currents of topological insulators. Notice that in ref.21, a projection of QED in 3 + 1-dimensions to a 1 + 1-brane was performed. However, a finite-size regulator was introduced to avoid ultraviolet divergences that appear when confining the system to one dimension. Therefore, the effective theory obtained is not scale invariant and cannot be conformal. In our approach, we found an explicit way to deal with the divergences, such that we obtain the CQED without any regulator. Furthermore, by integrating out the CQED gauge field in the corresponding partition function, we find that this gauge theory gives rise to a 1 + 1-dimensional Thirring model22. We then demonstrate that the bosonized version of the interacting-fermion Hamiltonian describes exactly a HLL with a Luttinger parameter K and a renormalized Fermi velocity that depend on the value of the fine-structure constant α.In this work, we focus on the CQED because describes a massless mode along the whole edge and is dominant in the low-energy regime. It also preserves the dimensionality of both, the electric charge and the gauge field of the 3 + 1-dimensional QED from which the CQED will be derived by using a dimensional reduction procedure. This method has been already used in studies of graphene2. They have a gapped bulk and topologically protected Dirac edge modes. These systems realize the quantum spin Hall effect, i.e. the chirality of the Dirac edge modes is locked to the spin, which is preserved due to the time-reversal symmetry. Thus, the dynamics of the edge modes can be described by a 1 + 1-dimensional massless Dirac theory with a two-component Dirac spinor ψ = T, where ψR and ψL are the right-handed spin-up and left-handed spin-down chiral modes, respectively. It was theoretically proposed in refs13 and experimentally confirmed in ref.23 that these topological insulators can support HLLs on the boundary due to the presence of unavoidable electron-electron interactions. These non-Fermi liquid phases fully preserve the time-reversal symmetry and are formally described by the free Dirac theory plus suitable four-fermion interactions. We now show that this model and the corresponding HLL can be derived from a gauge theory by simply assuming that the electrically charged propagating Dirac fermions on the edge interact through a quantum dynamical electromagnetic field Aρ. The essential point of our approach is that the massless Dirac fermions are confined on the one-dimensional boundary, whereas the quantum excitations (i.e. photons) of the electromagnetic field are free to propagate in all the three spatial dimensions, as shown in Fig. d2r = vdxdt and d4r = cdxdydzdt, with v and c the Fermi velocity and the speed of light, respectively. ħ is the Planck constant divided by 2π, e is the electric charge carried by each fermion, ε0 is the vacuum dielectric constant, γμ are 2 × 2 Dirac matrices with μ = 0, 1 and {γν, γμ} = 2gμν, where gμν = diag, Fρβ = ∂ρAβ − ∂βAρ is the field-strength tensor, ρ, β = 0, 1, 2, 3. The effective interaction felt by the massless Dirac fermions due to the gauge field can be obtained by integrating out the Aρ-field in the partition function Seff = SD + Sint is the effective action, with SD the free Dirac action, given by the first term in Eq. δ(x − x′) and δ(t − t′) are two Dirac delta functions and 24, a finite-size regulator for the Dirac delta function in Eq. , and obtain22, whereas the Lagrangian for the 25. The pseudo-differential operator in the kinetic term of the e remains dimensionless, while The replacement of the terms in Eq.  in the e27. However, in the low-energy limit, i.e. It is known that the Schwinger-Thirring model leads to a massless and a massive bosonic mode29, i.e., a (2 + 1)D QED with higher-order derivatives in the Maxwell term  even for the interacting phase by deriving the corresponding HLL in the following section. This CQED shares some properties with PQED. In fact, in both theories the electric charge e is a dimensionless parameter, as in usual (3 + 1)D QED. The fact that the coupling constant remains dimensionless makes perturbative studies more reliable. Moreover, just like in the Luttinger-liquid case, in PQED and in CQED the excitations are collective modes and there are no quasi-particles because the Green’s function has branch cuts instead of poles24.We want to emphasize that the dimensional reduction procedure performed here has been already employed in the study of two-dimensional materials, such as graphene. In this case, the corresponding effective field theory is the so-called Pseudo QED (PQED)., a 2 + D QED wit22, with the coupling constant g = e2/4πε0c. The corresponding Hamiltonian is then calculated by employing a Legendre transformation,ψ = T, with the fermion operators satisfying usual anti-commutation relations. The bosonization of Eq. , passing through non-interacting (K = 1), to attractive (K > 1)31. Nonetheless, how this parameter relates to fundamental properties of the materials was still unclear. In refs33, a formula that connects K with α is derived by employing perturbation theory with either the Kondo or the backscattering interaction. Here, we have presented a gauge-principle derivation of the Luttinger parameter, which is found to depend on the strength of the electron-electron interaction α.The parameter 33, d is the distance from the quantum wells to a closeby metallic gate, and K on α was obtained at the level of perturbation theory on the HLL Hamiltonian, i.e. additional interaction terms had to be taken into account, such as the Kondo or the backscattering interaction. Although our approach is non-perturbative, there are implicit approximations based on the theoretical description of the edge states in terms of QED. The presence of metallic gates in realistic experiments, for instance, could have crucial influence on the field lines of the virtual photons and would modify the effective action in a non-trivial way. Using the values of the parameters reported experimentally for HgTe quantum wells, v ≈ 5.5 × 105 m/s34, εr = 1535, d = 150 nm and 23, the authors in ref.33 find K ≈ 0.8. Within our model, which depends only on α, we obtain K ≈ 0.84.Now, we compare our results with a prior theoretical prediction proposed in refsK, as seen in the case of InAs/GaSb quantum wells23. This implies that our theoretical prediction applies to materials that have weak backscattering and high Fermi velocities, such as HgTe36. Nevertheless, the backscattering term can be obtained within our approach upon considering the massive Thirring model. The corresponding bosonization is discussed in the supplemental material with the Klein factors defined as in ref.37. Other possible 2D topological insulators that would be good candidates to test our theoretical proposal are plumbene monolayers38 and germanene films39. The Fermi velocity in these materials has the same order of magnitude as that in HgTe, indicating that backscattering might not be so relevant.Notice that our approach does not involve the backscattering term, which induces further corrections to the parameter K in order to obtain different regimes of interaction. From Eq. , it would be possible to reach the value of K = 1. For smaller velocities v, the minimum value of the dielectric constant for which K becomes real increases, i.e., for v = 5 × 105 m/s, e.g., v = 106 m/s. On the other hand, if we consider negative values of the dielectric constant by placing the topological insulator on top of a meta-material, then it is possible to switch from repulsive to attractive interactions, i.e., x = 2α/π. We illustrate this situation in Fig. 40.Furthermore, we show how to tune From Eq. , we noti42, as done for topological insulators.In this paper, we derived a gauge theory on the boundary of two-dimensional time-reversal-invariant topological insulators. Our starting point was to assume that the interactions between the charged one-dimensional Dirac fermions at the edge are mediated by a quantum dynamical electromagnetic field, where the virtual photons are free to propagate in all the three spatial dimensions. By implementing a dimensional-reduction procedure, we derived the corresponding CQED, which describes the HLL. We emphasize that our approach is non-perturbative, and has a more vast applicability in condensed-matter physics. Indeed, the one-dimensional effective theory derived here also works in the case of nanowires deposited on a substrate, in which the HLL phase can be easily obtainedK by modifying the dielectric constant of the substrate on which the one-dimensional system might be deposited. Interestingly, we find that upon the use of a meta-material as a substrate, it is possible to change the interactions from repulsive into attractive. These results might have profound implications for transport properties in nanostructures in particular, and nanotechnology in general.In our work, we provide, to the best of our knowledge, a field-theory derivation of the Thirring model, which opens the path to the manipulation of the Luttinger parameter Supplemental Material


Single-cell analysis of the three-dimensional (3D) chromosome structure can reveal cell-to-cell variability in genome activities. Here, we propose to apply recurrence plots, a mathematical method of nonlinear time series analysis, to reconstruct the 3D chromosome structure of a single cell based on information of chromosomal contacts from genome-wide chromosome conformation capture (Hi-C) data. This recurrence plot-based reconstruction (RPR) method enables rapid reconstruction of a unique structure in single cells, even from incomplete Hi-C information. Through the development of measurement techniques, we are getting the glimpse that chromosomal structures in individual cells may be differentially organized to exhibit distinct genome and chromatin activities. Such cell-to-cell variability might be an important aspect of cell physiology during differentiation or cellular responses. Genome-wide chromosome conformation capture (Hi-C) analysis124ad hoc computational methods have been proposed6In a mathematical sense, reconstruction of the 3D chromosome structure from Hi-C data can be considered a problem of geometry. Therefore, many Here, to overcome such difficulties, we propose a new mathematical method using recurrence plots8A recurrence plot8xi is a state corresponding to time i, and ε is a predefined threshold. We plot a point at  when R = 1 and nothing there if R = 0  = 1} is a set of indices plotted in the ith column of the recurrence plot and thus the corresponding Hi-C data (see i and j when they are close to each other). Second, we obtain the shortest distance by the Dijkstra methodε for the recurrence plot is defined intrinsically by the experimental limits of detectable contacts by the sparse Hi-C method as well as the regular Hi-C method. When the coverage depth is too high, the regular Hi-C data may be pre-processed to reduce the number of counts. But, presumably this pre-processing will not be necessary because the proposed RPR method can be used even in the case where as high as 70% of elements for the binary contact map are 1  = R = 1 when the parts corresponding to i and (i + 1) are on the same chromosome. Using this technique, we can ensure that the proposed reconstruction is possible in most cases. If, instead, we modify d as a constant value when regions i and j have a contact or |i−j| ≤ 1, the RPR method becomes more similar to the method of Paulsen et al.A necessary condition for our previous methodThus, as similar to reconstructing the three dimensional structure of strange attractors for deterministic chaos as shown in Single-cell Hi-C cannot report all the chromosomal interactions within a cell. In addition, sequencing data sometimes include errors. We therefore tested by toy models whether the RPR method is tolerant to the contamination of false positive noise and/or to the lack of information, as is often found in actual Hi-C data. Here we used the Lorenz modelH1 cells from Hi-C data for different single cells and from an ensemble (regular) Hi-C data .The Th1 ensemble Hi-C data and single-cell Hi-C data were acquired from the GEO database  the proposed method coverts a contact map to a distance matrix in a deterministic way, and (ii) the classical multidimensional scaling implemented in Matlab gives a unique configuration for a spatial arrangement of points given their distance matrix.et al.et al.et al.13In addition, the RPR method has other advantages in comparison with the conventional methods27172021222324252627How to cite this article: Hirata, Y. et al. Three-dimensional reconstruction of single-cell chromosome structure using recurrence plots. Sci. Rep.6, 34982; doi: 10.1038/srep34982 (2016).


Scientific Reports 10.1038/s41598-018-22914-5, published online 16 March 2018Correction to: The original version of this Article contained an error in the title of the paper, where the word “availability” was incorrectly given as “available”. This has now been corrected in the PDF and HTML versions of the Article, and in the accompanying supplementary material.


Genome median and genome halving are combinatorial optimization problems that aim at reconstruction of ancestral genomes by minimizing the number of evolutionary events between them and genomes of the extant species. While these problems have been widely studied in past decades, their solutions are often either not efficient or not biologically adequate. These shortcomings have been recently addressed by restricting the problems solution space.We show that the restricted variants of genome median and halving problems are, in fact, closely related. We demonstrate that these problems have a neat topological interpretation in terms of embedded graphs and polygon gluings. We illustrate how such interpretation can lead to solutions to these problems in particular cases.This study provides an unexpected link between comparative genomics and topology, and demonstrates advantages of solving genome median and halving problems within the topological framework. The basic case of three given genomes represents the genome median problem (GMP), which asks for reconstruction of a single ancestral genome, called median genome.One of the key computational problems in comparative genomics is the reconstruction of ancestral genomes based on gene species . Since mwhole genome duplication (WGD) events that simultaneously duplicate each chromosome of a genome. WGD events are known to happen in evolution of yeasts [doubled genomes, i.e., genomes immediately resulted from a WGD in the course of evolution. This problem is often posed for input genomes that have all genes present either in a single copy (ordinary genomes) or in two copies . In the simplest form, it is known as the genome halving problem (GHP), which asks for an ordinary genome for a given all-duplicated genome such that the distance between them is minimized. In the case of a given all-duplicated genome and an ordinary genome, the problem, called the guided genome halving problem (GGHP), asks for an ordinary genome at the minimal total distance from both given genomes.Since genome rearrangements preserve the gene content, it must be restricted to genes present in all input genomes with the same multiplicity. To account for genes appearing different number of times in different genomes, one need to consider other types of evolutionary events. One of important sources of duplicated genes in genomes are the f yeasts , fishes f yeasts , plants f yeasts , and evef yeasts , which iWhile the GHP admits a polynomial solution –11, its intermediate genome median problem (IGMP), restricts its solutions to the intermediate genomes, i.e., genomes appearing in a shortest rearrangement scenario between two of the three given genomes [restricted guided genome halving problem, RGGHP) that restricts the constructed doubled genomes to the GHP solution space [A recently introduced variant of the GMP, called the  genomes . Similaron space . It is won space , 15 to tIn this study, we show that the IGMP and RGGHP are, in fact, closely related, and put them into the framework of embedded graphs and polygon gluings . This frn genes as a graph cycle with n directed edges (encoding genes and their strands) alternating with n undirected edges (connecting the extremities of adjacent genes), called P-edges   operation breaks a genome at two positions and glue the resulting fragments in a new order, which model common types of genome rearrangements [P corresponds in P-edges with a different pair of P-edges2 on the same set of four vertices.A ngements , 24. A DP and Q composed of the same set of genes, the breakpoint graphP-edges and Q-edges, forming P-matching and Q-matching. Then P-edges and Q-edges, called PQ-cycles (or QP-cycles). Similarly, the breakpoint graph can be defined for three or more genomes [For genomes entclass1pt{minima genomes .Fig. 2ADCJ scenario between genomes P and Q is a sequence of DCJs transforming P into Q. A shortest such scenario has the following property:A P and Q, each DCJ splits some PQ-cycle in their breakpoint graph into two and thus increases the number of PQ-cycles by one.In a shortest DCJ scenario between genomes DCJ distance  between two genomes:From Lemma P and Q on n genes is given by the formula c is the number of PQ-cycles in the breakpoint graph The DCJ distance between genomes A be an all-duplicated genome and contracted genome graphA-edges form cycles (since each vertex is incident to two A-edges), called A-cycles. For a doubled genome 2R resulted from a WGD3 of an ordinary genome R, the contracted genome graph R-edges, called 2R-edges. It is clear that 2R-edges form a matching in R-edges with R-edges in The definition of breakpoint graph based on edge gluing can be easily extended to genomes with duplicated genes as follows. Let mentclass2pt{minimA and an ordinary genome R composed of the same genes, the contracted breakpoint graphA-edges and R-edges in A-cycles and R-matching, respectively.For an all-duplicated genome mentclass2pt{minimAR-cycles, called an AR-cycle decomposition. We remark that there exists an exponential number of AR-cycle decompositions of AR-cycle decompositions. One is maximalAR-cycle decompositions, which have the maximum possible number of AR-cycles, denoted AR-cycle decompositions is constructed as follows. For each A-cycle in A-edge becomes a directed edge. We decompose AR-cycles such that each R-edge in an AR-cycle connects the head of one A-edge and the tail of another. We call such AR-cycle decomposition an ht-decomposition of The graph mentclass2pt{minimLet us recall the formulation of the GHP and discuss the structure of its solutions.A, find an ordinary genome R minimizing dDCJ.For a given all-duplicated genome R maximizing In other words, the GHP asks for an ordinary genome AR, n is half the number of A-edges in A), and k is the number of even A-cycles in For any all-duplicated genome R such that R-noncrossing as defined below.It was shown in  that theR-edge connecting vertices of distinct A-cycles is called R-interedge. An R-edge connecting vertices of same A-cycles is called R-intraedge. We represent vertices and edges of each A-cycle in R-intraedges as straight chords inside these circles.For the graph A and an ordinary genome R, the contracted breakpoint graph R-noncrossing  and noncrossing R-intraedges (as chords within the corresponding circle); ora single even A-cycles  with single R-interedge and noncrossing R-intraedges.a pair of odd For a given all-doubled genome mentclass2pt{minimR-noncrossing guarantees that the genome R yields a solution to the GHP for an all-doubled genome A, this condition is not necessary, and there exist other genomes R solving the GHP .Given an all-duplicated genome intermediate genome from [We recall the definition of an ome from  Fig. 2)intermediI is intermediate between genomes P and Q iff dDCJ+dDCJ=dDCJ.An intermediate genome between two genomes is any genome appearing in a shortest DCJ scenario between them. In other words, a genome R-noncrossing contracted breakpoint graphs, for ordinary genomes P, Q, I, the breakpoint graph I-noncrossing if every its connected component is formed by a single PQ-cycle and noncrossing I-intraedges (as chords inside each PQ-cycle) .the total number of For ordinary genomes Similarly to the GHP, one can restrict the solution space of the GMP to intermediate genomes and pose the following problem:P, Q, and an outgroup genome R, find an intermediate genome I between genomes P and Q that minimizes dDCJ.Given genomes I plays in the IGMP a similar role to those of the ordinary genome R in the GHP. Indeed, let PQ be an artificial all-duplicated genome formed by the union of genomes P and Q. Then the breakpoint graph PQ-cycles. If I-noncrossing, then I-noncrossing, and k=c is the number of cycles in I such that I-noncrossing. Thus, the IGMP can be viewed as a particular case of the RGGHP, where all cycles are even. We remark that Lemma 5 for the IGMP can be refined as follows: the ht-decomposition with respect to any orientation of PQ-cycles in PQ-cycles are even), and each cycle in this decomposition is either a PI-cycle or a QI-cycle.From Theorem 7, one can observe that the intermediate genome Below we will show that both RGGHP and IGMP can be formulated within the framework of embedded graphs and polygon gluings.We recall the following definition from the topological graph theory:GΣ is a graph whose vertices and edges are points and arcs on a surface4Σ such that the edges do not intersect (except at the vertices);GΣ in Σ represents a collection of regions , and each face is a polygon.5the complement of A (2-cell) embedded connected graph m connected components is defined as the union m connected embedded graphs An embedded graph with GΣ in Σ can be viewed as the result of cutting Σ along the edges of GΣ. Conversely, GΣ can be obtained by gluing the sides of its faces, which are polygons. Let us denote this collection of polygons by GΣ has two sides on Σ, the total number of sides in GΣ, and the edges of GΣ define a (perfect) matching on the sides in Σ is orientable, we can orient sides of each face clockwise. Then the matched sides of GΣ head-to-tail.We remark that the complement of the connected embedded graph orientable gluing as the head-to-tail gluing of sides in each matched pair  matching on their sides Fig. a, we deair Fig. b. It ispartial. It results in an embedded graph GΣ on a surface Σwith boundary. Connected components of the boundary are called holes. In this case, some edges of GΣ represent glued pairs of sides, while the others represent non-glued sides and form holes.A polygon gluing according to a non-perfect matching is called GΣ with v vertices, e edges, and f faces, the Euler formula states that For a connected embedded graph h(Σ) is the number of holes in Σ and g(Σ) is the topological genus (number of handles) of Σ. Unless GΣ is the result of a partial gluing, we have h(Σ)=0.where We start with establishing a correspondence between contracted breakpoint graphs and embedded graphs.A, the A-edges in A-cycles. Let us fix some orientation o of these A-cycles. For each A-cycle with k edges, we assign a k-gon whose sides correspond to the cycle vertices (such that adjacent sides correspond to adjacent vertices). Then the sides of each polygon inherit labels from the corresponding cycle vertices, and the polygon itself inherits the orientation from the cycle. We denote the collection of these labeled oriented polygons by Recall that for an all-duplicated genome R, the R-edges in R-matching on the vertices of A-cycles and thus on the sides of G=Go  Fig. a, b. It,R) Fig. d.Fig. A be an all-duplicated genome, R be an ordinary genome, and o be some orientation of the A-cycles. Then the vertices of Go are in one-to-one correspondence with the AR-cycles in the ht-decomposition of o.Let A-edges in G is an image of some vertices of A-edges belong to the same AR-cycle in the ht-decomposition of A-edge  in W1,V1), {V1,U1}, , {U2,V2},  in AR-cycle in the ht-decomposition of W1,V1), ,  are A-edges and {V1,U1}, {U2,V2} are (undirected) R-edges in Go the side V1 is glued with U1 and the side V2 is glued with U2 (in head-to-tail fashion), and so the vertex corresponding to , which is the head of the side U1 and the tail of the side U2, is glued with the vertices corresponding to  (the tail of V1), and  (the head of V2). Conversely, since every gluing of matched sides implies gluing of vertices that correspond to A-edges from the same AR-cycle, vertices that correspond to A-edges from distinct AR-cycles can not be glued. By transitivity we obtain the statement of the lemma. □Recall that the vertices of 2R) Fig. c, d. Cok polygons with an even number of sides (even-gons) and 2l polygons with an odd number of sides (odd-gons). Then the graph obtained by gluing the sides of n+k vertices, and this upper bound is achieved by the embedded graphs on k+l spheres.Let G, we get that the total number of vertices in G is n is half the number of sides in m is a number of connected components in G. We remark that in order to maximize v we need to maximize m and minimize m is k+l, and it is achieved iff each connected component of G is a result of gluing of either one even-gon or two odd-gons. The minimum value of g(Σi) is achieved iff Σi is a sphere (so that g(Σi)=0).Let mentclass2pt{minimG has a maximal number of vertices  iff it has k+l connected components (each on a sphere). □So, We remark that Lemmas 9 and 10 provide a topological interpretation of the GHP and essentially give a new proof of Theorem 3, which is much simpler than previous ones , 26.A be an all-duplicated genome, R be an ordinary genome, and o be some orientation of the A-cycles. Then a DCJ on the genome R corresponds in the embedded graph Go to cutting two edges and gluing the resulting four sides in a new order .Let R′ be the result of a DCJ on R. Then the R-matching and R′-matching on the sides of Go cuts the two pairs of sides matched in R and glues the resulted four sides according to R′. □Let Lemmas 9, 10, and 11 inspire us to pose the following problem:G, find a shortest sequence of DCJ-surgeries that results in an embedded graph G′ on a maximum number of spheres.Given an embedded graph A and an ordinary genome B is equivalent to the GSP for Go, where o is some orientation of A-cycles.The RGGHP for an all-duplicated genome P, Q, and an outgroup genome T is equivalent to the GSP for Go, where o is any orientation of PQ-cycles.The IGMP for ordinary genomes R be a solution to the RGGHP for an all-duplicated genome A and an ordinary genome B. Let B and R. By Lemma 5, there exists an orientation o of A-cycles such that the ht-decomposition of Go is an embedded graph on a maximum number of spheres. By Lemma 11, the DCJ scenario Go into Go. Thus, the RGGHP for the genomes A and B is equivalent to the GSP for the embedded graph Go.(1) Let PQ-cycles in PR- and QR-cycles for any orientation o of PQ-cycles. Thus, the IGMP for genomes P, Q, T is equivalent to the GSP for Go with any orientation o of PQ-cycles. □(2) Since all A. For each solution R, there exists some orientation o such that Go is an embedded graph on the maximum number of spheres. This inspires us to define a maximal gluing as a polygon gluing that results in an embedded graph on the maximum number of spheres. By Lemma 10, each connected component of this graph has either one even-gon face or two odd-gon faces.Let us enumerate all the solutions to the GHP for a given all-duplicated genome We remark that there exists a method  that fork-gon equals the k-th Catalan number The number of ways to obtain a sphere by gluing the sides of a 2n+1)-gon and a (2m+1)-gon equals The number of ways to obtain a single sphere by gluing the sides of a -gon and a (2m+1)-gon. By Euler formula -gon and the (2m+1)-gon, respectively. So, any maximal gluing can be obtained in the following way: for some l, n−l pairs of the (2n+1)-gon sides are glued and m−l pairs of the (2m+1)-gon sides are glued (transforming each of these polygons into a sphere with one hole), and the remaining 2l+1 sides from one polygon are glued with the remaining 2l+1 sides from the other (resulting in a sphere).Let  of a 2n+-gon and n+1)-gon and a (2m+1)-gon. This is equivalent to enumeration of the pairs  and the ways to glue them into a sphere. Let 2l+1 be the length of the holes in G1 and G2. It is known [k+1)-gon by gluing k−l pairs of its sides. Hence, for each l, there exist G1,G2). If l=0, then there is exactly one way to glue G1 and G2 together. If l>0, then there are 2(2l+1) ways to glue them into a single sphere (the factors 2l+1 and 2 account respectively for rotations and reflections of the holes in G1 and G2 with respect to each other). Combining these results together, we get that the number of maximal gluings of a (2n+1)-gon and a (2m+1)-gon equals Let us enumerate all the maximal gluings of a :P and Q, the number of intermediate genomes equals n1,…,2nk are the lengths of the PQ-cycles in For given ordinary genomes A and ordinary genome B is equivalent to the GSP for G=Go, where o is some orientation of A-cycles. In this section, we show how one can solve the GSP in the case of G being an embedded graph with a single face on a torus .Consider a simple cycle of length 2y 2 Fig. a, b. Afl=1, we apply a DCJ-surgery that cuts the edges of this cycle, resulting in a sphere with two holes of length 2, and then glues each of these holes, resulting in a sphere. So, we have transformed G into an embedded graph on a sphere with l DCJ-surgeries. □If G be an embedded graph on a torus with one face. If G contains two simple odd cycles that have the total length 2l and share exactly one vertex, then G can be transformed into an embedded graph on a sphere with l DCJ-surgeries.Let l−1 DCJ-surgeries on G and obtain two loops (cycles of length 1) that share the vertex. We then apply a DCJ-surgery that cuts these loops, resulting in a sphere with a hole of length 4, and then glues this hole, resulting in a sphere. So, we have transformed G into an embedded graph on a sphere with l DCJ-surgeries. □Similarly to Lemma 17, we can apply G be an embedded graph on a surface with holes. g be the genus of the surface of G and G′ be obtained from G by gluing a pair of sides from different holes. Then the surface of G′ has genus g′=g+1.Let G has one face and can be glued into an embedded graph on a sphere, then G is an embedded graph on a sphere with holes of even length. Furthermore, all simple cycles in G are holes.If Let G have v vertices, e edges, f faces and h holes. Let C1 and C2 be the holes that contain the pair of sides we are gluing. If at least one of the holes C1, C2 has length greater than 1, then G′ has v′=v−2 vertices, e′=e−1 edges, f′=f faces, and h′=h−1 holes. If both C1 and C2 have length 1, then G′ has v′=v−1 vertices, e′=e−1 edges, f′=f faces, and h′=h−2 holes. By the Euler formula (g′=g+1 in both cases.1) Let  Let G haG has one face, it results from a partial gluing of a polygon. Obviously, any partial gluing resulting in a sphere with holes of even length can be extended to a gluing resulting in a sphere. Let us prove that any other gluing can not be extended in such a way. Let g the genus of the surface of G. Consider a gluing of G into an embedded graph on a sphere. If g>0, such gluing does not exist, since the genus cannot be decreased by such gluing. Hence, g=0 and thus G is on a sphere with holes. If there are holes of odd lengths, then some side from one of these holes has to be glued with a side from some other hole, which would increase the genus. So, all holes must be of even length.(2) Since G are holes. Let L be the total length of the holes, and v and e be the number of vertices and edges of G, respectively. Consider the embedded graph G′ resulting from contraction of the edges belonging to holes in G. Then G′ is an embedded graph on a sphere, which has v+h−L vertices, e−L edges, and one face. From the Euler formula  matching G′ can be glued into each of G and G′ is an embedded graph on a sphere with holes of even length. Let 2m be the total length of these holes. Note that every non-glued edge in G′ represents a side of an edge in G that should be cut by some DCJ-surgery from Denote the face of b be a bridge  in G such that its sides s1, s2 are not glued in G′. We will show that gluing of these sides into b in G′ transforms this graph into another embedded graph b is a bridge, s1 and s2 cannot belong to distinct holes in G′. Let C be a hole in G′ that contains both sides s1 and s2. In C is transformed into two holes C1 and C2 (possibly empty) connected by the edge b. It is clear that the lengths of C1 and C2 have the same parity. It remains to show that both lengths are even. Assume that they are odd. Since b is a bridge, no side of C1 is glued with a side of C2 in G. Hence, at least one side from C1 is glued with a side from a hole different from C1 and C2. Similarly, at least one side from C2 is glued with a side from a hole different from C1 and C2. By Lemma 19, gluing of two sides from different holes creates a handle, implying that G should contain at least two handles, a contradiction to G being an embedded graph on a torus . Thus, both holes C1 and C2 in G′. This proves that Let H′ be an embedded graph obtained from G′ by gluing all non-glued sides of bridges in G. Then H′ is on a sphere with holes of even lengths. Note that any edge in G, whose sides are non-glued in H′, is not a bridge and thus belongs to some simple cycle in G.Let H′ into G. A handle in G can be created by gluing either two sides from distinct holes, say C1 and C2, or from one hole, say C, in H′. In the former case, sides from C1 and C2 cannot be glued with sides from any other holes . The sides from Ci  cannot be glued with any other side from Ci, since this would result in a bridge missing in H′. Thus, the sides from C1 and C2 are glued into edges that form a simple cycle in G of length 2l . Since |C1|+|C2|≤2m, we have 4l≤2m. In the latter case, we claim that the edges resulted from gluing of the sides of C form two simple cycles in G, which share a vertex. Indeed, let 2p be the length of C, and H′ have V+2p vertices, E+2p edges, and h holes. After gluing the sides of C (as in G), we obtain a graph on a torus with V+v vertices, E+p edges, and h−1 holes, where v vertices and p edges are obtained from vertices and edges in C and form a (possibly non-simple) cycle G. By the Euler formula . 6 orientation o on its A-cycle.Construct o, construct the embedded graph Go.From Go, find a shortest even cycle C in Go.Using the breadth-first search (BFS) starting at each vertex in C|/2 DCJ-surgeries that cut the edges of C and transform Go into an embedded graph on a sphere.Construct a sequence of |B and return the resulting genome as a solution to the RGGHP.Apply the corresponding DCJs to the genome Theorem 20 inspires us to design the following algorithm for solving the RGGHP for given all-duplicated genome Go. Since in Go the number of edges equals n=|B|=|A|/2 and the number of vertices equals n−1, this step runs in O(n2) time.We remark that our algorithm runs in polynomial time. Indeed, the most time-consuming step is the BFS starting at each vertex of In the present study we establish a somewhat unexpected link between the restricted variants of genome median and halving problems and embedded graphs. We provide a new simple proof for existence of the GHP solutions as well as completely describe the structure of the GHP solution space and determine its cardinality. We also show how the topological framework can be applied for solving the restricted guided genome halving problem (and the intermediate genome median problem) in a particular case. In further development we plan to address the topological problem of an embedded graph surgery (GSP) on an arbitrary orientable surface , which may provide better heuristic solutions for the RGGHP and IGMP.double distance problem asking for a maximal cycle decomposition of the contracted breakpoint graph of a given all-duplicated genome and an ordinary genome. Also, embedded graphs on surfaces with boundaries (holes) can be related to models including genome rearrangements along with gene insertions and deletions [We remark that similar topological interpretations exist for other comparative genomics problems and can provide intuition for their solution. For example, analysis of non-orientable surfaces (such as Klein bottle) seems to be relevant to the eletions , 30.1 Some studies base their analysis on synteny blocks rather than genes. We will use the term “gene” to refer to an actual gene or a synteny block.2 Here we view genome P as being transformed and P-edges as changing.3 A WGD event can simultaneously duplicate each circular chromosome in genome Q either into a single circular chromosome or into two identical circular chromosomes, which have the same contracted genome graph [R may contain duplicated chromosomes of both types.me graph . We assu4 Under a surface we understand a 2-dimensional compact orientable manifold without boundary . We distinguish surfaces up to homeomorphisms.5 Under a polygon (n-gon) we understand a topological disc, whose boundary is formed by a collection of n sides.6 There exist two orientations of the A-cycle in


However, there are several areas that would benefit from a more stringent definition of “unique”, requiring that these sub-sequences of length W differ by more than k mismatches (i.e. a Hamming distance greater than k) from any other sub-sequence, which we term the k-disjoint problem. Examples include finding sequences unique to a pathogen for probe-based infection diagnostics; reducing off-target hits for re-sequencing or genome editing; detecting sequence  insertions; and multiple substitution mutations. Since both sensitivity and specificity are critical, an exhaustive, yet efficient solution is desirable.A common challenge in bioinformatics is to identify short sub-sequences that are unique in a set of genomes or reference sequences, which can efficiently be achieved by k-disjoint) sequences of up to 100 nucleotides in length. On a number of simulated and real data sets ranging from microbe- to mammalian-size genomes, we show that microTaboo is able to efficiently find all sub-sequences of a specified length W that do not occur within a threshold of k mismatches in any other sub-sequence. We exemplify that microTaboo has many practical applications, including point substitution detection, sequence insertion detection, padlock probe target search, and candidate CRISPR target mining.We present microTaboo, a method that allows for efficient and extensive sequence mining of unique  contains supplementary material, which is available to authorized users. There are several areas in bioinformatics and biomedical research that benefit from identifying short sub-sequences among a (large) pool of reference sequences that are as unique as possible, i.e. the next most similar sub-sequence differs by a given number of mismatches or more. For example, in infection diagnostics, pathogen strains can be identified via padlock probes , which aW that differ by at least k mismatches compared to any other sequence in a set, termed k-disjoint, is defined as follows: given two sets A and B containing sequences of length W, find all sequences X in A such that X is more than k mismatches away from any sequence Y in B. This formulation of the problem essentially finds the so-called k-disjoint set of A and B. An alternative formulation is: given two integer values W and k as well as two strings P and T with lengths n and m, respectively, the k-disjoint problem consists of finding all positions i (1 ≤ i ≤ n) which have the property that a W length substring of P beginning at i is at a Hamming distance greater than k away from any W long substring of T. The complement of the k-disjoint is the k-intersection. The k-disjoint problem is closely related to the k-difference primer problem [Mathematically, the problem of identifying a set of sequences or strings of length  problem .k-disjoint is an extension of the exact-match problem, since for k = 0, the k-disjoint problem collapses into identifying truly unique sequences, even if they only differ by one single letter. For this much simpler problem, common solutions include the analysis of k-mers, short sub-sequences of a fixed size k. There, simple lookup, hash-tables, or suffix arrays provide efficient solutions, with a number of implementations available, among them bloom filter based methods and suffix array based programs, such as Jellyfish [k-mer spectra can thus be used to estimate repeat content or heterozygosity in a genome or sequence reads [Conceptually, ellyfish , and Talellyfish . The resce reads , 11. In ce reads ), which k-disjoint problem based on a mismatch matrix and a dictionary tree for fast lookup and search (see Implementation). The output of this method, all sequences that avoid “taboo”, can either be used directly, or serve to pre-filter data for software for specific tasks, such as CRISPR-finder or SMS [ad hoc solution to a niche problem concerning the identification of unique sequences common amongst coagulase negative Staphylococci. Here, we have generalized, refined, and optimized the algorithm and data workflow, as well as packaged the solution into a fully functional software package, microTaboo.Here, we present microTaboo, a general, exhaustive, and efficient solution to the r or SMS . microTar or SMS . BrieflyA and targets B, each containing one or more FASTA files; ii) an integer value, W, indicating the sequence lengths to be found in A; iii) an integer value k representing the mismatch threshold; and iv) a parameter  specifying whether the results should present only the k-disjoint, the k-intersection, or both. microTaboo outputs a two column file for each FASTA file in the query directory, containing sequences satisfying the input parameters, together with their sequence-relative positions.microTaboo requires the following inputs: i) two directories representing queries i)N-coding first maps all sequences of size N onto unique numbers, and then pre-computes a mismatch matrix by calculating the number of mismatches between two N-sequences, allowing for fast look-up; N-coding converts all unique sequences of a fixed length, N, to a unique number , N is limited to 5. If W has multiple delimiters, microTaboo chooses the largest value for N. Next, a mismatch matrix M is constructed, the elements M being the number of mismatches between two N-code sequences. Both N-coding sequences and the mismatch matrix enable us to calculate mismatches between two sequences more efficiently, such that the mismatches between two W-long sequences can be calculated in W/N look-ups in the mismatch matrix.see Fig. . This enii)Fragmentation extracts all sequences of length W (W is a multiple of N) in B, reverse complement strand included, and converts them into a list of vectors of length W/N using N-coding scheme. The output of this process is a list of N-coded sequences.iii)Partitioning divides the above list into sub-lists with respect to the first two elements of each vector, reducing comparisons. The output of this process is a hybrid dictionary-tree-list data structure, called here a Dictree, which is then fed into the filter engine module (see below). The Dictree can be viewed as being an indexed list, where all W-long sequences (in N-code format) from the B FASTA-files are stored and indexed with respect to their first two vector elements. For instance, all N-code vectors of the form <62, 3,… > would be placed into the sub-list with the address: <62, 3 > .iv)Filter engine search is the module that filters sequences in A against the Dictree of B through their respective W/N vectors. That is, the filter engine sequentially extracts all W long substrings in A, N-codes these and searches the Dictree of B for a match. Thus, querying a vector of the same form for k = 0 in the Dictree requires merely a comparison to the list with exact indices as an exact match to the query vector could theoretically only exist in that particular list address. For k > 0, microTaboo looks into multiple lists until either the number of mismatches exceeds k, or all sub-lists that could contain a match has been searched in and not found. Using the mismatch matrix, the addresses of all sub-lists that could possibly contain a sequence within k-mismatches of a given sequence in N-code format, X = {x1, x2, x3, x4, …}, is obtained in the following way:The underlying algorithm that powers mircoTaboo involves several steps see Fig. :Fig. 1Dai, such that: M ≤ kAddress 1 = j, such that: M ≤ k − MAddress 2 = i and j that satisfies these conditions could possibly contain a sequence within k mismatch from the query sequence X.Thus, only the lists in the Dictree with addresses W), N-code size (N), genome length of query organism (LA) and genome length of “taboo” organism (LB). Construction of the Dictree is done in two steps: extracting and converting all words of length W from the “taboo” organism to N-code format and inserting the N-code vectors into the Dictree. The number of operations required to convert a single word into an N-code vector is:There are two computationally intensive procedures involved when running microTaboo, namely, constructing the Dictree, and searching the query organism in the Dictree. To evaluate their respective complexities, we define the following: word length  comes from the fact that microTaboo uses a five letter DNA alphabet and that we have (5N)2 leafs in the Dictree, each pointing to a sub-list with address i, j.The second computationally heavy procedure is searching the Dictree for all words of length W long sequence is first converted into a vector in N-code format, which takes 2 operations. Given that this sub-list contains Lb/5N2 elements, in the worst case, an equal number of vector-to-vector comparisons will be made. Each vector-to-vector comparison will consist of 3, the complete computational complexity for one word isSince any The computational operation burden to filter a query organism against a “taboo” organism is:Combining the complexities for the search and the construction of the Dictree gives:A and LB are the dominating factors in terms of size. Therefore, the term involving the product of these would be the determining factor, making the theoretical complexity:In practice, LN5N2 gives a reduction in the number of computations needed resulting in vastly reduced runtimes, e.g. for any word that is divisible by 5, N becomes 5 which makes the denominator be 511. Taking into account k mismatches, we have empirically determined that microTaboo scales in the following fashion: O(10k).However in practice, the denominator Escherichia coli O157 Sakai [k = 5 mismatches of any W = 30 bp long sequence in 25 other bacteria, including four other E. coli strains , E. coli (~5.5Mbps), Saccharomyces cerevisiae [Candida albicans [k = 0 and W = 30, microTaboo successfully recovered all mutated sites albicans  , 1120 (TbLCJV), and three 60 bp long regions on chromosomes 1, 2, and 5 in S. cerevisiae. microTaboo detected all the sequences in and around the vicinity of the inverted regions. We repeated the experiment by introducing additional point substitutions and achieved identical results , following the pigeon hole principle. While the suffix array runs efficiently with larger values of S (20 and up), albeit slower than microTaboo in all but one case , performance dramatically drops with lower values  sequences and k ranging from 0 to 5. In a microbial genome, we show that the method is capable of quickly finding padlock probes that allow for uniquely identifying a specific strain, and in three simulated experiments, we demonstrate that microTaboo can identify mutations, inversions, and insertions, even in presence of single-point mutations. For pre-filtering regions that could serve as CRISPR targets at different stringencies of uniqueness, we applied microTaboo to a fungal, fly, and mammalian genome. While the genomic territory at three mismatches leaves only less than 1% of the genome accessible to genome editing at high stringency, examining this fraction as a function might be useful to predict the number of expected off-target hits.We present microTaboo, a fast, efficient, and general tool for directly solving the The software is available for the Windows, MAC OS and Linux operating systems and runs both on standard laptop computers for smaller tasks, as well as larger computer clusters for more profound tasks. We also demonstrate that microTaboo can be used for several applications and that it serves especially well as a powerful pre-filtering tool for further analysis of sequence data.Additional file 1: Table S1.Padlock Probe target search. (DOCX 119 kb)Additional file 2:Section 1 - Results, Section 2 - Runtime Comparisons. (DOCX 27 kb)Additional file 3: Table S2.Point mutation detection. (DOCX 39 kb)Additional file 4: Table S3.Inversion detection and virus incorporation. (DOCX 54 kb)Additional file 5:Table S4.Runtime comparisons -microTaboo vs. BLAST and a suffix array method. (DOCX 39 kb)Additional file 6: Table S5.Result coverage comparison — BLAST vs microTaboo. (DOCX 86 kb)Additional file 7: Table S6.Runtime comparisons for exact string matching algorithms. (DOCX 50 kb)


We have examined 35 pair-wise, sequence-specific hydrophobic interactions over the temperature range of 273–333 K, based on data reported by Nemethy and Scheraga in 1962. A closer look at a single example, the pair-wise hydrophobic interaction of leucine-isoleucine, will demonstrate the significant differences when the data are analyzed using the Nemethy-Scheraga model or treated by the Planck-Benzinger methodology which we have developed. The change in inherent chemical bond energy at 0 K, ΔH°(T0) is 7.53 kcal mol compared with 2.4 kcal mol, while ‹ts› is 365 K as compared with 355 K, for the Nemethy-Scheraga and Planck-Benzinger model, respectively. At ‹tm›, the thermal agitation energy is about five times greater than ΔH°(T0) in the Planck-Benzinger model, that is 465 K compared to 497 K in the Nemethy-Scheraga model. The results imply that the negative Gibbs free energy minimum at a well-defined ‹ts›, where TΔS° = 0 at about 355 K, has its origin in the sequence-specific hydrophobic interactions, which are highly dependent on details of molecular structure. The Nemethy-Scheraga model shows no evidence of the thermodynamic molecular switch that we have found to be a universal feature of biological interactions. The Planck-Benzinger method is the best known for evaluating the innate temperature-invariant enthalpy, ΔH°(T0), and provides for better understanding of the heat of reaction for biological molecules.We have shown in our published work the existence of a thermodynamic switch in biological systems wherein a change of sign in ΔCp°(T)


Double-slit imaging is widely used for verifying the resolution of high-resolution and super-resolution microscopies. However, due to the fabrication limits, the slit width is generally non-negligible, which can affect the claimed resolution. In this paper we theoretically calculate the electromagnetic field distribution inside and near the metallic double slit using waveguide mode expansion method, and acquire the far-field image by vectorial Fourier optics. We find that the slit width has minimal influence when the illuminating light is polarized parallel to the slits. In this case, the claimed resolution should be based on the center-to-center distance of the double-slit. With the development of high-resolution and super-resolution microscopies, the resolution of optical microscopy can reach the sub-wavelength scale. In order to verify the resolving capability of a proposed microscope, it is very common to fabricate a sub-wavelength metallic double-slit sample to carry out imaging experiment12345678123567We theoretically investigate how the polarization of the illuminating light and the slit width will affect the imaging result. In traditional diffraction theory, the intensity distribution at the output port of a slit illuminated by a uniform plane wave is still considered as uniform. However, this approximation is not applicable for sub-wavelength structure. In this paper, we use waveguide mode expansion method to rigorously calculate the electric field inside and near the double-slit, and acquire the far-field image by vectorial Fourier optics. After the effect of slit width is revealed, we propose a criteria for resolution verification that can provide more consistent results in double-slit microscopic imaging experiments.The imaging system model is shown in In order to rigorously solve the electric field distribution inside and near the double-slit, we first expand the electric field inside the metallic slits into waveguide modes. The metal is regarded as perfect electric conductor.α =  is a four dimensional subscript, where α1 indicates which slit the mode field is located in, α2 indicates TE or TM mode, α3 indicates the mode order, and α4 indicated the forward or backward propagating direction. Eα is the electric field of mode α inside the waveguide. The scattered field outside the double-slit can be expressed asHere, ETE is the TE-polarized plane wave outside the slits, and ETM is the TM-polarized plane wave outside the slits. FTM and FTE are the Fourier spectrum of TM-polarized and TE-polarized electric field, respectively. And kx is the x-component of the wave vector. Because the slits are assumed to be infinitely long, the diffracted field will not propagate in the direction where ky ≠ 0, thus there is an assumption in ky = 0. It is easy to acquire the boundary conditions at the input port and output port of the double-slit.Here, Ei is the incident field and Er is the reflected field when there is no slit on the metallic board. Subscript t represents the tangential component. Σin and Σout are the whole planes of the double-slit structure’s input port and output port. Sin and Sout are the range of the slits in the planes of the input port and the output port.Here, aα. Thus we acquire the electric field inside the double-slit. Then we can acquire the electric field outside the double-slit by projecting the electric field inside the double-slit to ETE(kx) or ETM(kx) and matching the boundary conditions.According to the waveguide mode expansion method of ref. kx to be , where k0 is the wave number of the incident light. The calculation has been verified with Comsol simulation to ensure the validity of this method. The calculation result is shown in In our numerical calculation, we make the highest waveguide mode order to be 10, and make the range of E|2 in the x-z cross section. When the incident light is y-polarized, only TE modes can exist in the slits, and TE1 is dominant. The diffracted field is TE-polarized, i.e., only y-polarized electric field component exists in the free space behind the double-slit. E|2 in the x-z cross section. When the incident light is x-polarized, only TM modes can exist in the slits, and TM0 is dominant. The diffracted field is TM-polarized, i.e., both x-polarized and z-polarized electric field components exist in the free space behind the double-slit. Note that the profiles in Next, we use vectorial Fourier optics11kx) is a window function related to the NA of the objective lens. Note that in obtaining ky ≠ 0, FTE = FTM = 0, is utilized. If this condition is not satisfied, (e.g. finitely long double-slit), Here, Π(F represents the Fourier transform.The tube lens transform this spectrum into image on the CCD. FTM(kx) at the output port is plotted in FTE(kx) is plotted in When the incident light is y-polarized, the scattered field is totally TE-polarized. The spectrum Ipit/Imax, where Ipit is the intensity at the pit between the two peaks in the image, and Imax is the maximum intensity in the image. Hence, the smaller this ratio is, the more likely the double-slit image can be resolved.Then we vary the spacing between two slits and their width. In order to evaluate the resolution of the image, we introduce a ratio dcc = 0.79λ, dcc = 0.81λ, dcc = 0.87λ and dcc = 0.95λ, and observe the change of Ipit/Imax when the slit width w varies from 0.05dcc to 0.95dcc. The result is shown in Ipit/Imax is insensitive to w/dcc; on the contrary, when the incident light is x-polarized, this ratio is very sensitive to slit width.In some researches, the resolution is claimed based on the center-to-center distance of the double-slit. Here we choose a fixed center-to-center distance from the following values: dss = dcc − w. Here we choose a fixed side-to-side distance from the following values: dss = 0.45λ, dss = 0.55λ, dss = 0.65λ and dss = 0.75λ, and observe the change of Ipit/Imax when the slit width w varies from 0.05dss to 0.95dss. The result is plotted in Ipit/Imax is quite sensitive to w/dss.While in some other researches, the resolution is claimed based on the side-to-side distance of the double-slit, which is λ/NA = 0.68λ. This contradiction is because the Rayleigh’s criteria is for incoherent systems. Generally, for a coherent system, if the two point sources are in phase, it has worse resolution than an incoherent systemIt is convenient to mark 0.735 as the critical condition when the two slits can just be discerneddcc = 0.95λ and the blue curve of dcc = 0.81λ. The energy distribution accounts for this phenomenon. dcc = 0.95λ, where the slit width is w = 0.85dcc and w = 0.95dcc respectively. The energy distribution is concentrated at the edges of two slits. However, if the two slits are too close, the energy is more concentrated at the outer edges rather than the inner ones. dcc = 0.81λ, where the slit width is w = 0.65dcc and w = 0.95dcc respectively. The energy distribution is concentrated at the center of each slit. If the two slits are too close, the peak of energy distribution profile will slightly shift outwards.The general tendency of the curves in dss = 0.75λ for y-polarized illumination. This is because of a small bulge between the two well-separated main peaks. dss = 0.75λ, where the slit width is w = 0.6dss and w = 0.9dss respectively. A bulge can be clearly seen between the two well-separated main peaks for w = 0.9dss.The general tendency of the curves in Ipit/Imax is insensitive to the slit width. That is to say, even though a pair of very thin slits are challenging for fabrication, using relatively wide slits in experiment can still well reproduce the result from thin slits, so long as the polarization of incident light is parallel to the direction of slits and the center-to-center distance is taken as the resolution criteria. These conditions can be adopted for more consistent resolution verification in similar high resolution microscopic experiments.An important observation we get from In practical experiment, if possible, the slit width has better to be greater than half a wavelength to enable a propagating TE mode within the slits. Another thing should be noticed is that there will be no TM light induced if the criteria are adopted, hence the proposed criteria is not suitable to some experiments where TM light is necessary.In conclusion, we rigorously study the scattering and imaging process in a double-slit microscopic imaging experiment and propose a criteria for more consistent resolution verification: The illuminating light polarization should be parallel to the extension direction of the slits, and the verified resolution should be based on the center-to-center distance of the double-slit.How to cite this article: You, S. et al. Resolution criteria in double-slit microscopic imaging experiments. Sci. Rep.6, 33764; doi: 10.1038/srep33764 (2016).


Nature Communications 10.1038/s41467-018-03154-7, published online: 02 Mar 2018Correction to: y-axis originally incorrectly went form ‘0’ to ‘1.10−4’. The correct scale goes from ‘0’ to ‘0.4’. In Fig. 5f, the x-axis labels were incorrectly shifted to the right. The correct version of Fig. 5 is:The original version of this Article contained an error in Fig. 5. In Fig. 5e, the scale on the which replaces the previous:This has been corrected in both the PDF and HTML versions of the Article.