Bdnf gene. Viable and fertile homozygote animals were generated, with the GFP signal marking neuronal cell bodies translating the Bdnf mRNA. Importantly, the distribution of immunoreactive BDNF remained unchanged, as exemplified by its accumulation in mossy fiber terminals in the transgenic animals. GFP-labeled neurons could be readily visualized in distinct layers in the cerebral cortex where BDNF has been difficult to detect with currently available reagents. In the hippocampal formation, quantification of the GFP signal revealed that <10% of the neurons do not translate the Bdnf mRNA at detectable levels, with the highest proportion of strongly labeled neurons found in CA3.While BDNF is receiving considerable attention for its role in synaptic plasticity and in nervous system dysfunction, identifying brain circuits involving BDNF-expressing neurons has been challenging. BDNF levels are very low in most brain areas, except for the large mossy fiber terminals in the hippocampus where BDNF accumulates at readily detectable levels. This report describes the generation of a mouse line allowing the detection of single brain cells synthesizing BDNF. A bicistronic construct encoding BDNF tagged with a P2A sequence preceding GFP allows the translation of BDNF and GFP as separate proteins. Following its validation with transfected cells, this construct was used to replace the endogenous  Bdnf mRNAs using GFP as a surrogate marker. The availability of these transgenic animals will also help in understanding the action of drugs such as ketamine that are thought to act by increasing Bdnf translation.BDNF is a highly conserved growth factor known to be essential for the function of the nervous system. Its very low abundance in the brain has retarded the development of drugs targeting BDNF-expressing neurons in disease-relevant brain areas. The present report describes a novel approach allowing the localization of single cells in the adult mouse brain actively translating Bdnf transcription between different brain regions and from one neuron to the next as long documented by in situ hybridization studies in the adult brain of mice, rats, and pigs  is a secreted growth factor required for the development and function of the nervous system . In humaand pigs . Given t neurons , differe neurons  suggest ding GFP . In addiFP :ttaattaagccaccatgaccatcctgtttctgaccatggtcatcagctacttcggctgcatgaaggccgctcccatgaaggaagtgaacgtgcacggccagggcaacctggcttatcctggcgtgcggacacacggcaccctggaatctgtgaacggccctagagctggcagcagaggcctgaccacaacaagcctggccgacaccttcgagcacgtgatcgaggaactgctggacgaggaccagaaagtgcggcccaacgaggaaaaccacaaggacgccgacctgtacaccagcagagtgatgctgagcagccaggtgcccctggaaccccctctgctgttcctgctggaagagtacaagaactacctggacgccgccaacatgagcatgagagtgcggagacacagcgacccagctagaagaggcgagctgagcgtgtgcgacagcatcagcgagtgggtcacagccgccgacaagaaaaccgccgtggacatgtctggcggcaccgtgaccgtgctggaaaaggtgccagtgtccaagggccagctgaagcagtacttctacgagacaaagtgcaaccccatgggctacaccaaagagggctgcagaggcatcgacaagagacactggaacagccagtgcagaaccacccagagctacgtgcgggccctgacaatggacagcaagaaaagaatcggctggcggttcatcagaatcgacaccagctgcgtgtgcaccctgaccatcaagagaggcagaggatccggcatggtgtctaagggggaggaactgttcaccggcgtggtgcccatcctggtggaactggatggcgacgtgaacggacacaagttcagcgtgtccggcgagggcgaaggcgacgccacatacggaaagctgaccctgaagttcatctgcaccaccggcaagctgcccgtgccttggcctaccctcgtgaccacactgacctacggcgtgcagtgcttcagcagataccccgaccatatgaagcagcacgacttcttcaagagcgccatgcccgagggctacgtgcaggaaagaaccatcttctttaaggacgacggcaactacaagaccagggccgaagtgaagttcgagggcgacaccctcgtgaacagaatcgagctgaagggcatcgacttcaaagaggacggcaacatcctgggccacaagctggagtacaactacaacagccacaacgtgtacatcatggccgacaagcagaaaaacggcatcaaagtgaacttcaagatccggcacaacatcgaggacggctccgtgcagctggccgaccactaccagcagaacacccctatcggcgacggccctgtgctgctgcctgacaaccactacctgagcacccagtccgccctgagcaaggaccccaacgagaagagggaccacatggtgctgctggaattcgtgaccgccgctggcatcaccctgggcatggacgagctgtacaaatgaggcgcgcc;Bdnf-Gfp : nls-Gfp (BamHI and AscI restriction sites are underlined): ggatccggcgccaccaatttcagcctgctgaaacaggccggcgacgtggaagagaaccctggccctccaaagaagaagcggaaggtcatggtgtccaagggcgaggaactgttcaccggcgtggtgcccatcctggtggaactggatggcgacgtgaacggccacaagttcagcgtgtccggcgagggcgaaggcgacgccacctatggcaagctgacactgaagttcatctgcaccaccggcaagctgcccgtgccttggcctaccctcgtgacaaccctgacctacggcgtgcagtgcttcagcagataccccgaccacatgaagcagcacgacttcttcaagagcgccatgcccgagggctacgtgcaggaacggaccatcttctttaaggacgacggcaactacaagaccagggccgaagtgaagttcgagggcgataccctcgtgaaccggatcgagctgaagggcatcgacttcaaagaggacggcaacatcctgggccacaagctggagtacaactacaacagccacaacgtgtacatcatggccgacaagcagaaaaacggcatcaaagtgaacttcaagatcaggcacaacatcgaggacggctccgtgcagctggccgaccactaccagcagaacacccccatcggagatggccccgtgctgctgcccgacaaccactacctgagcacacagagcgccctgtccaaggaccccaacgagaagagggaccacatggtgctgctggaatttgtgaccgccgctggcatcacactgggcatggacgagctgtacaagtgaggcgcgcc.P2a-Sv40The PacI/AscI restricted BDNF–GFP fragment was ligated into the identically restricted pAAV plasmid . The BDNThe biosynthesis and secretion of tagged BDNF proteins were analyzed using HEK293 cells transfected with plasmids encoding wild-type (WT) BDNF, BDNF–GFP, and BDNF-P2A-GFP. The enhanced version of GFP was used throughout. Cultures were maintained in Gibco DMEM supplemented with 10% FBS, 1% GlutaMAX and 1% nonessential amino acids . Transfections were performed in a six-well format using 2 μg of the indicated DNAs combined with 4 μl of Invitrogen Lipofectamine 2000 transfection reagent (Thermo Fisher Scientific) diluted within Gibco Opti-MEM medium (Thermo Fisher Scientific). Five hours after transfection, HEK293 cells were cultured in N2B27 medium consisting of equal volumes of Gibco Neurobasal medium and DMEM-F12 (Thermo Fisher Scientific), 1% B27 supplement (Thermo Fisher Scientific), 1% GlutaMAX, and 1% penicillin-streptomycin . BSA (Sigma-Aldrich) was used at a reduced concentration of 75 μg/ml to facilitate the analysis of the conditioned media by SDS-PAGE. BDNF levels were quantified in conditioned media, and brain lysates by ELISA .d-lysine (Sigma-Aldrich), cells were maintained in Neurobasal medium supplemented with 1% GlutaMAX supplement, 1% Penstrep, and 2% SM1 supplement (Stem Cell Technologies). Neurons were cultured for up to 12 d with 50% media changes performed three times weekly. Subsequent transfections were performed on E14.5 neurons at 5DIV using 0.5 μg of indicated DNAs and 1 μl of Lipofectamine 2000 (see above). Depolarization of E17.5 neurons at DIV11 was achieved by supplementing media with 1 mm 4-aminopyridine  for 24 h.Cortices of mice at embryonic day 14.5 (E14.5) for transfection and TrkB phosphorylation assays and E17.5 for immunostaining studies were collected in Hanks’ buffered salt solution (Sigma-Aldrich) and trypsinized in 1 mg/ml trypsin (Worthington) for 20 min at 37°C. The reaction was then stopped using 1 mg/ml trypsin inhibitor (Sigma-Aldrich) before the addition of 1 mg/ml DNase I (Thermo Fisher Scientific) and gentle dissociation with a 5 ml serological pipette. Cells were then pelleted by centrifugation at 1400 rpm for 5 min and resuspended in DMEM supplemented with 2% FBS, 1% GlutaMAX, and 1% Penstrep. Three hours after plating into wells coated with poly-z-stack images .Twenty-four hours after transfection or treatment with 4-AP, neurons were briefly washed with PBS and fixed with 4% paraformaldehyde  for 15 min. After a 5 min permeabilization with PBS containing 0.1% Triton X-100 , cells were incubated for 1 h in blocking solution [3% donkey serum (Sigma-Aldrich) and 1% BSA in PBS-T] at room temperature (RT). Coverslips were then incubated overnight in primary antibodies diluted in blocking solution at the following concentrations: anti-BDNF mAb #9 7 μg/ml; , chickenm sodium orthovanadate (Sigma-Aldrich) to inhibit phosphatase activity and analyzed by SDS-PAGE for TrkB phosphorylation.The conditioned media of transfected HEK293 cells transfected with BDNF cDNAs were standardized to a BDNF concentration of 25 ng/ml after quantification using a BDNF ELISA . Before m Tris-HCl, 150 mm NaCl, 1 mm EDTA, 0.1% SDS, 0.2% sodium deoxycholate, and 1% Triton X-100) supplemented with phosphatase and protease inhibitor cocktail mixes, 10 μm phenanthroline monohydrate, 10 mm aminohexanoic acid, 10 μg/ml aprotonin, and 2 mm sodium orthovanadate . Lysates and conditioned media were centrifuged at 15,000 rpm to remove insoluble components before analysis by SDS-PAGE. Proteins were separated on 4–12% NuPAGE Bis-Tris gels (Invitrogen) and transferred to GE Healthcare Protran NC nitrocellulose membranes (Thermo Fisher Scientific) using a Trans-Blot semi-dry transfer unit (Bio-Rad). Membranes were subsequently blocked for 1 h in blocking solution [5% blotting-grade blocker (Bio-Rad) and 1% BSA in TBS containing 0.1% Tween ] and then probed overnight at 4°C with antibodies to β-actin , BDNF , BDNF propeptide , GFP , or phosphoTrkA (Tyr674/675)/TrkB  in blocking solution (1:2000). Following three 10 min washes in TBS-T, membranes were incubated at RT with HRP-conjugated anti-goat , anti-mouse, anti-rabbit  or anti-chicken  secondary antibodies within blocking solution (1:2000). After a further three 20 min washes in TBS-T, membranes were developed using WesternBright ECL HRP Substrate (Advansta). Densitometric analysis of all blots was performed using quantification functions on Bio-Rad ImageLab software. For blots requiring BDNF quantification, recombinant BDNF standards (Regeneron/Amgen) between 300 and 18.75 pg were run alongside to create calibration curves, as appropriate.Homogenized brain tissues, HEK293 cells, and cultured neurons were incubated for 20 min on ice in RIPA buffer (50 mBdnf knock-out (−Bdnf/−) animals were generated by crossing mice with two floxed Bdnf alleles  Act, 1986.  alleles  with mic alleles . Bdnf-P2sequence , the SV4sequence  and a GFated see C. After Three-month-old mice killed by pentobarbital injections were transcardially perfused with ice-cold PBS and 4% PFA, and their brains were removed and postfixed at RT for 4 h before cryoprotecting in 30% w/v sucrose solution at 4°C overnight. The following day, brains were embedded in OCT  compound and sectioned at 40 μm using a cryostat. Sections were blocked in blocking solution (3% donkey serum and 4% BSA in PBS-T) for 1 h before incubating overnight with mouse anti-BDNF (mAb #9) and chicken anti-GFP (1:1000). Sections were then washed three times for 10 min with PBS-T before incubating with Alexa Fluor 555 anti-mouse IgG and Alexa Fluor 488 anti-chicken IgY secondary antibodies  for 1 h at RT. After a final wash in PBS-T for 10 min, sections were incubated with DAPI diluted in PBS (1:4000) for 20 min and mounted onto precoated polylysine slides (VWR) with Dako fluorescence mounting media. Images of gross brain regions were acquired on a confocal microscope using a 20× objective. For counts of GFP-positive nuclei, images were captured using a 63× oil-immersion and then analyzed using FIJI  and CellBdnf-P2a-Gfp bodyweights, a Kruskal–Wallis test was used with a Conover-Iman post hoc test for multiple comparisons. TrkB activation by BDNF fusion proteins was compared against that of BDNF-myc and analyzed using a one-sample t test. An adjusted p value (≤0.0125) was considered significant after a Bonferroni correction for multiple comparisons. Differences in BDNF and GFP signal intensities in depolarized Bdnf-P2a-Gfp neurons were analyzed using a Student’s t test. All results were expressed as the mean ± SE, and p ≤ 0.05 was considered to be significant unless otherwise stated.Data were analyzed using Microsoft Excel 2013 and RStudio software. For analysis of Bdnf constructs was first tested using transfected HEK293 cells. Constructs encoding unmodified BDNF, BDNF directly fused with GFP or separated from BDNF by a P2A sequence . In addition, the transgene did not measurably interfere with the fertility of the animals. Coronal brain sections of homozygous animals were then examined by confocal microscopy following perfusion, fixation, and staining with antibodies to GFP, BDNF, as well as nuclear staining with DAPI . Selective GFP labeling can be readily observed in distinct cortical layers, including layers 2, 5, and 6 as well as in distinct nuclei including the amygdala as well as all subdivisions of the hippocampal formation  35.5 ± 2.11 ng/g WT cortex; and 45.4 ± 6.38 ng/g for homozygote Bdnf-P2a-Gfp animals. The corresponding values for the hippocampus were 97.4 ± 6.00 ng/g and 109.3 ng/g ± 17.11 for WT and homozygous animals, respectively. To confirm that GFP is cleaved after the BDNF–P2A sequence in vivo, we analyzed the lysates of cortices from wild-type, heterozygote, and Bdnf-P2a-Gfp homozygote animals by Western blot . Importantly, the distribution of the endogenous BDNF protein remains unchanged when comparing the staining of BDNF in the hippocampal formation  is reduced by ∼50%. Caution should then be exerted when using comparatively large fusion constructs such as BDNF–GFP as they would seem unlikely to efficiently activate TrkB. The results presented in Bdnf gene substitution with GFP directly coupled to the C terminal of BDNF . It is conceivable that GFP may have been cleaved from BDNF in the surviving animals as a functional cleavage site at the C terminal of BDNF has been noted following the isolation of BDNF from brain homogenates , possibly due to the selective inclusion of DAPI-positive cells in the granule cell and pyramidal cell layer. We also note that the results summarized in in situ hybridization studies in the rat (http://mouse.brain-map.org/gene/show/11850).The approach described here now opens the possibility to use the GFP signal to isolate and sort cells from the adult brain based on GFP signal intensity, thus allowing their individual profiling by RNAseq. Such results would help to inform the development of drugs selectively targeting these neurons and may deliver new clues as to the endogenous regulators of BDNF expression. Similar objectives could in principle also be reached by randomly isolating single cells from brain regions of interest without prior cell marking. As such data are indeed available for the adult mouse hippocampus , we comp the rat  and the Bdnf mRNA by allowing the selection of cells based on the intensity of the GFP signal. This should prove useful toward the development of new drugs aiming at selectively increasing the levels of BDNF in brain regions of interest, including rapidly acting depressants such as ketamine thought to act by increasing BDNF translation (In conclusion, the mouse line reported in this study should facilitate the detailed characterization of brain neurons actively translating the nslation .


We prove several unique continuation results for biharmonic maps between Riemannian manifolds. Moreover, let ϕ:M→N be a smooth map. One option of finding such maps is to find extrema of their energyharmonic mapsand are characterized by the vanishing of the so‐called tension field, that isτ(ϕ)∈Γ(ϕ*TN). Many results on harmonic maps have been obtained in the past decades, we refer to Finding interesting maps between Riemannian manifolds is one of the most challenging problems in modern Riemannian geometry. Suppose that biharmonic maps, which were first studied in ϕ:M→N, which is defined asbitension field, that is{ei},i=1,…,m=dimM is a local orthonormal frame field tangent to M and Δ represents the Laplacian on ϕ*TN. For recent results on biharmonic submanifolds we refer to Currently, there is a growing interest in a geometric variational problem that generalizes harmonic maps, the so‐called Theorem 1.1(Sampson) Let ϕ1,ϕ2:M→N be two harmonic maps. If they agree on an open subset, then they are identical; and indeed the conclusion holds if ϕ1 and ϕ2 agree to infinitely high order at some point. In particular, a harmonic map which is constant on an open subset is a constant map.In this article, we want to focus on one particular aspect regarding the qualitative behaviour of harmonic and biharmonic maps, namely the unique continuation property. For harmonic maps, the question of unique continuation was settled by Sampson Z−∇Z and for the (rough) Laplacian on ϕ*TN we use Δ:=Trg(∇∇−∇∇).Throughout this article, we will use the following sign conventions. For the Riemannian curvature tensor field, we use ⟨·,·⟩ to indicate the Riemannian metrics on various vector bundles, and the same symbol ∇ for the corresponding Riemannian connections.In general, we will use the same symbol All manifolds are assumed to be connected and we will work only with smooth objects.i,j,k for indices on the domain ranging from 1 to m and Greek indices α,β,γ for indices on the target which take values between 1 and n. When the range of the indices is from 1 to q, for some positive integer q, we will often denote them by a,b,c.Whenever we will make use of indices, we will use Latin indices We will use the Einstein summation convention, that is, repeated indices that are in the diagonal position indicate the sum.2In this section, we will prove the results obtained in this article.Our strategy of proof is to cleverly rewrite the biharmonic map equation such that we are effectively dealing with a second‐order problem to which we can apply the classical result from Aronszajn 2.1Theorem 2.1A be a linear elliptic second‐order differential operator defined on an open subset D of Rm. Let u= be functions in D satisfying the inequalityu=0 in an open subset of D, then u=0 throughout D.Let We recall the following , p.248Tϕ:M→N be a smooth map and let σ be a section in the pull‐back bundle ϕ*TN. We consider a local chart  on M and a local chart  on N such that ϕ(U)⊂V. The section σ can be written asΔσ is given byuiα=∂uα∂xi and (ϕβ)β is the corresponding expression for ϕ in local coordinates. Then, by a straightforward computation, we get , is empty, then, as M is connected, IntA=A=M, that is, ϕ is harmonic everywhere.If the boundary of p0∈∂(IntA). Furthermore, let U be an arbitrary open subset containing p0. Clearly p0 does not belong to IntA and U∩IntA≠∅.Assume that there exists M∖A is open in M, p0 does not belong to M∖A and U∩(M∖A)≠∅.On the other hand, we havep0 includes an open subset where τ(ϕ) vanishes everywhere and an open subset where τ(ϕ)≠0 at any point.Thus, any open subset containing V be an open subset containing ϕ(p0) and U an open subset containing p0 such that ϕ(U)⊂V. Assume that U and V are the domains of local charts. Consider an open subset D in M, containing p0, such that its closure in M is compact and contained in U. As we have seen, the set D contains an open subset where τ(ϕ) vanishes everywhere and an open subset where τ(ϕ)≠0 at any point.Let U, and so on D,Dgij, ϕα and their derivatives, Γβαθ and their derivatives, are bounded on D. Applying Theorem u=0, which implies that τ(ϕ) vanishes everywhere in D. This contradiction implies ∂(IntA)=∅ and we end the proof.□Denote2.2In this subsection, we will prove Theorem ι:Sn→Rn+1 and form the composite map ϕ:=ι∘ϕ:M→Rn+1.Lemma 2.3ϕ:M→Sn⊂Rn+1 with the constant curvature metric, the equation for biharmonic maps acquires the formRicM denotes the Ricci tensor field on the domain manifold M and {ei},i=1,…,m is an orthonormal frame field.For To this end, we consider the inclusion map 1 the following formula holds trueσ∈Γ(ϕ*TN), then we can also think of σ as a section in the pull‐back bundle ϕ*TRn+1 and the connections along ϕ and ϕ are related viaIt is well known that for a spherical target of constant curvature θ(X)=⟨dϕ(X),τ(ϕ)⟩=⟨dϕ(X),Δϕ⟩, see By a direct calculation one finds that∇=∇ϕ. To this end, we fix p∈M arbitrary and let {Xi},i=1,…,m be a geodesic frame field around p. We calculate at the point pΔ|dϕ|2 we again compute at pTrg∇2dϕ=−Δdϕ+dϕ(RicM), where Δ is the Hodge–Laplacian acting on Rn+1‐valued one‐forms. Due to our sign convention, we have −Δdϕ=dΔϕ=∇Δϕ such that we findp□As a next step, we prove thatv=∇ϕ and w=Δϕ. In terms of a local chart  on M and {ea} denoting the canonical basis of Rn+1, a=1,…,n+1, we can write v=ϕiadxi⊗ea, where ϕia=∂ϕa∂xi.Now, we define new variables Then any solution of ϕ1,ϕ2 and their corresponding new variables v1,v2,w1,w2. We set u:=y1−y2. Note that the function u takes values in R(n+1)(m+2), which can be seen fromu as being defined on the image of U and ui(n+1)(m+1)+a=∂u(n+1)(m+1)+a∂xi.At this point we consider two biharmonic maps Δ(w1−w2) can be expressed as a sum of the components of u and their first order derivatives multiplied by terms that do not contain the components of u or their derivatives. This allows us to obtain the estimate for |Δ(w1−w2)|.Lemma 2.4D⊂U be an open subset such that its closure in M is compact and included in U. Then, the following estimate holds on DC depends on D and the derivatives of ϕ1 and ϕ2 up to third order.Let As a next step, we prove that D is assumed to be compact and both ϕ1,ϕ2 are smooth by assumption.□Using Proof of Theorem 1.4a,b=1,…,(n+1)(m+2).Making use of u=0 on D. We finish the proof by denoting A:={p∈M:ϕ1(p)=ϕ2(p)} and using the same arguments as in the proof of Theorem 1.2.□Applying Theorem 2.3In this section, we prove Theorem ϕ:M→N and a local chart  on M and  on N such that ϕ(U)⊂V. In order to avoid any notational confusion the corresponding expression for ϕ in local coordinates will be denoted by ϕ^, that isLemma 2.5ϕ:M→N be a biharmonic map. In terms of local coordinates, it satisfiesLet First, we consider a biharmonic map uθ=τθ(ϕ)=Δϕθ+⟨dϕα,dϕβ⟩Γαβθ in □Using the local form of the tension field v=ϕiαdxi⊗eα, where ϕiα=∂ϕα∂xi and {eα} denotes the canonical basis of Rn.Now, as in the spherical case, we define new variablesThen, any solution of F3 is given byHere, ϕ1,ϕ2:M→N and let  be a local chart on M and  a local chart on N such that ϕ1(U)⊂V and also ϕ2(U)⊂V.At this point, we consider two biharmonic maps u:=y1−y2. Note that the function u takes values in Rn(m+2), which can be seen fromWe set |Δ(w1−w2)|, we need the image of V in Rn to be convex and with compact closure such that we can apply the mean value theorem for functions defined on the image of V. Then, when applying the mean value inequality, the standard norm of the differential of the function will be bounded on the image of V. So, first, we consider the image of V to be an open ball of radius ε in the Euclidean space Rn and then we shrink it to a ball of radius ε2. Accordingly, we consider a smaller domain U. Thus, we will make use of the mean value theorem for functions defined on the open ball of radius ε2 but which are also defined on the closed ball of radius ε2.In order to obtain the estimate for f:Bn(ε2)→R we will apply the following inequalityy* belongs to the standard segment that joins y1 with y2.More precisely, for Δ(w1−w2) can be expressed as a sum of the components of u and their first‐order derivatives multiplied by terms that do not contain the components of u or their derivatives.Then, as a next step, we prove that Lemma 2.6D⊂U be an open subset such that its closure in M is compact and included in U. Then, the following estimate holds on DC depends on D and the derivatives of ϕ^1 and ϕ^2 up to third order.Let We haveUsing The first term can be controlled as followsFor the second term we rewriteThe third contribution can be manipulated as followsRegarding the fourth term we findFor the fifth term we obtainThe sixth contribution can be estimated as□The seventh term can be estimated asProof of Theorem 1.4a,b=1,…,n(m+2).Making use of u=0 on D. We finish the proof by denoting A:={p∈M:ϕ1(p)=ϕ2(p)} and using the same arguments as in the proof of Theorem □Applying Theorem 2.4Sn.In this section, we prove Theorem Sn be the Euclidean unit sphere and denote by N and S the north and south pole, respectively. It is well known thatLet (ya) be local coordinates on Sn−1,a=1,…,n−1. Then,  are local coordinates on Sn∖{N,S}.Let Sn are given by∼ to indicate objects on Sn−1.In this geometric setup the Christoffel symbols on Sn−1 is given byThe equator  be a local chart on M and we denote the domain of the above local coordinates on Sn∖{N,S} by V. In addition, we assume that ϕ(U)⊂V.Now, let ϕ in local coordinates by ϕ^, that is,We denote the corresponding expression for W is an open subset of U and ϕ(W)⊂Sn−1, that is, ϕ(W)⊂Sn−1∩V. Hence, in W we have ϕn=π2. Now, define f:U→R,f:=ϕn−π2. Clearly, f vanishes when restricted to W.Assume that D be an open subset of U such that its closure in M is compact and included in U, and W⊂D⊂U.Let Lemma 2.7f:U→R defined above satisfies the following estimate on DThe function We will prove the following estimate:Δ2f=Δ2ϕn and we will use θ=n in order to obtain the estimates.We have |⟨∇Δϕb,dϕc⟩Γbcn|. We immediately obtainWe expand the first term as□We write the fourth term in the same way as the first one and, the only term we must estimate isProof of Theorem 1.6u and F. Applying Theorem u=0 on D, that is, ϕ maps the whole of D into Sn−1.As in the proofs of Theorems A:={p∈M:ϕ(p)∈Sn−1} and using the same arguments as in the proof of Theorem □We finish the proof by denoting 2.5 be a local chart on N such thatr denotes the dimension of the submanifold P. Moreover, we assume that the image of V in Rn is convex. For example, we can assume that it is the interior of an n‐cube with side lengths ε/2 that lies inside an n‐cube with side lengths ε.In this section, we prove Theorem  be a local chart on M and assume that ϕ(U)⊂V. We denote the corresponding expression for ϕ in local coordinates by ϕ^, that is,P is totally geodesic in N, we have on the intersection V∩PPΓαβθ denote the Christoffel symbols of the submanifold P for 1⩽θ,α,β⩽r. On V∩P we also haveW is an open subset of U and ϕ(W)⊂P. Hence, on W we havea∼=1,…,n−r. Clearly, we have f:U→Rn−r and also f|W=0. Now, let D be an open subset of U such that its closure in M is compact and included in U, and W⊂D⊂U.In addition, let Lemma 2.8D, the function f:U→Rn−r defined above satisfies the following estimateC depends on the geometry of N.Restricted to We will prove the following estimate:b,c=1,…,r. Since we have⟨∇Δϕb,dϕc⟩Γbcr+a∼. To this end we define a new function ϕ^′:U→Rn byϕ′:U→N takes values in P. We note that, according to Γbcr+a∼=Γbcr+a∼(ϕ)=0 on W, but at the points in D∖W the image of the map ϕ may not be in P such that Γbcr+a∼≠0. However, on U, and thus also on D, we haveD□We make use of Proof of Theorem 1.7u and F. Applying Theorem u=0 on D, that is, ϕ maps the whole of D into P.As in the proofs of Theorems □We finish the proof using the same arguments as in the proof of Theorem Remark 2.9ϕ=ψ∘ϕ of a biharmonic map ϕ and a totally geodesic map ψ is biharmonic. Indeed, for arbitrary maps ψ and ϕ and for any σ∈Γ(ϕ*TN) we haveψ is totally geodesic we get ∇Xϕdψ(σ)=dψ(∇Xϕσ), and then, by straightforward computations we findϕ is biharmonic, ϕ is biharmonic too.As to be expected, similar to the case of harmonic maps, the composition ψ defines a regular, totally geodesic submanifold, we can give an alternative proof of the above fact, similar to the proof of Theorem In the particular case when 2.6Theorem 2.10(Sampson) Assume that ϕ:M→N is a harmonic map, with q=ϕ(p). Let S be a C2 hypersurface in N passing through q, at which point we assume that the second fundamental form is definite. If ϕ is not a constant mapping, then no neighbourhood of p is mapped entirely on the concave side of S.Another geometric application of the unique continuation property of harmonic maps was provided by Sampson [Besides the unique continuation property, the proof of this theorem makes use of the maximum principle. This powerful tool only exists for second‐order elliptic partial differential equations, not for fourth order ones, such that we cannot expect to find a generalization of Sampson's maximum principle for proper biharmonic maps.ϕα is biharmonic if and only ifIndeed, in ϕα is harmonic if and only ifc1 and c2 are real constants.The map α(t) admits a local extremum point at t0 such that α(t0)>0, then t0 is a minimum point. Indeed, consider t0∈I, I being an open interval of R, such that α(t)>0 on I. If t0 is an extremum point, harmonicity implies α′′(t0)=mα(t0)>0 and it follows that t0 is a minimum point.We note that if ϕα is biharmonic if and only ifc1, c2, c3 and c4 are real constants.The map Remark 2.11(1)α(t) that admit a local maximum point at t0 such that α(t0)>0. For example, we may considerϕα) lies in the concave side of the m‐sphere Sm(1me) of radius 1me.In the biharmonic case, we have solutions (2)α(t) that admit a local minimum point at t0 such that α(t0)>0. For example, we haveWe also have solutions


AbstractAnnamanumflavimaculatumsp. nov. is described and illustrated from Guizhou and Guangxi, China. Diagnosis for distinguishing the new species to its close congeners is presented and identification key to the genus is also updated. Annamanum Pic, 1925 is a large genus in the subfamily Lamiinae  with 30 described species , A.lunulatum , and A.magnum Holzschuh, 2017, this is the fourth Annamanum species recorded in the Leigongshan area.The genus  species  distribu species . Of thes species . With thSpecimens were collected by two collecting methods: net sweeping and six level Lindgren funnel traps  with 99% ethanol as lure. Collected specimens were pinned or glued on pinned paper cards. Labels were handwritten or printed in Chinese. Materials from Guizhou are preserved in the School of Life Sciences, Guizhou Normal University, Guiyang, Guizhou, China. Materials from Guangxi are preserved in Collection of Wen-Xuan Bi, Shanghai, China.Specimen examination and dissection were conducted under an AmScope SM-4TZ stereomicroscope. Adults were photographed with Canon EOS 6D digital camera equipped with EOS MP-E 65 lenses. Male genitalia were photographed with Olympus DP22 camera mounted on an Olympus SZX7 stereomicroscope.The collection acronyms used in the text are as follows:CBWX Collection Wen-Xuan Bi, Shanghai, China;GZNULSSchool of Life Sciences, Guizhou Normal University, Guiyang, China.Taxon classificationAnimaliaColeopteraCerambycidaeBF6CC0CF-7653-53CB-975F-C1349AD10D3Chttp://zoobank.org/C4EE8EDD-377D-4625-94DB-24C6FA355C94Queniao Tea Farm, Queniao Village, Leishan County, Guizhou Province, China.GZNULS)], HOLOTYPE / Annamanum / flavimaculatum / Shulin Yang [handwritten red label].Holotype male, glued on paper point, with genitalia in a separate microvial. Original label: “中国贵州省雷山县方祥乡雀鸟村茶场，2015年6月18日，六层漏斗诱捕器，杨书林采”  (GZNULS); 1♀, original labels: “中国贵州省雷山县雷公山国家自然保护区，2017年8月27日，杨绍勇采”  (GZNULS); 1♀, Original label: “中国贵州省雷山县方祥乡雀鸟村茶场，2015年6月18日，杨光祖采”  (GZNULS); 1♀, Original label: “中国贵州省雷山县方祥乡雀鸟村茶场，2019年8月2日，杨书林采”  (GZNULS); 1♀, Original label: “中国贵州省雷山县方祥乡雀鸟村茶场，2019年9月12日，杨书林采”  (GZNULS); 1♂, original labels: “广西猫儿山迴龙寺，1900 – 1700 m, 2012.VII.23，毕文烜”  (CBWX); 1♂, original labels: “广西猫儿山三江源，1950–2000 m, 2014.VII.30，宋晓彬”  (CBWX); 1♀, original labels: “广西临桂广福顶，1350 m, 2018.VII.2”  (CBWX).Paratypes: 2♂♂, original labels: “中国贵州省雷山县雷公山国家级自然保护区，2012年7月22日，2016年7月13日，杨书林采”  , female . Measurements for specimens from Guanxi (Figure Head: black, frons generally densely punctured, vertex densely punctured, both frons and vertex covered with dense yellow hairs. Antennae of males exceed apex of elytra by six antennomeres; of females by five antennomeres; antennal tubercles strongly raised; scape and pedicel black, covered with long hairs, not erect but flat towards apex; cicatrix complete, narrow; rest of the antennomeres reddish brown, sparsely covered with white yellowish hairs; base and apex of each antennomere are covered with darker hairs. Eyes deeply emarginated; lower lobe twice as high as gena and one-fourth as wide as frons width between lower lobes of eyes. Labium with small sparse punctures and sparse long dark brown hairs. Mandibles with dense long yellow hairs at outer side and sparse hairs on the front. Thorax: Pronotum black with coarse granules, covered with yellow hairs, whose thickness varies among individuals; disk slightly raised; a small callus at each side of the apical margin, not extending beyond middle. Lateral spines strong, acute, slightly posteriorly and upwards curved. Sternum reddish brown, covered with dense yellow hairs. Scutellum covered with dense white yellowish pubescence, apex rounded. Mesosternal intercoxal process with a slightly projected antero-ventral tubercle (Figure Elytra: gradually tapered in male, less tapered in female, with irregular coarse granules obliquely protruding backwards and gradually smaller posteriorly. Basal fifth bulged between humeri and scutellum. Basal half black, intermingled with yellow and black hairs; black hairs forming a broad transverse black band at the middle, nearly reaching suture. Apical half reddish brown, without granules but with coarse small punctures, mostly covered by yellow hairs forming a large yellow marking with intermingled black hair dots; some of these dots are near suture on the anterior half of the yellow marking and weakly form a line which extends obliquely towards outer margin of elytron from middle of the marking; black dots larger in female. Apex nearly rounded, slightly truncated in inner half. Legs: with dense white yellowish pubescence; femora dark brown, clubbed, not cylindrical, with sparse small punctures; tibiae reddish brown. Abdomen: reddish brown, ventrites with white yellowish pubescence intermingled with sparse punctures. Pygidium shallowly truncated at apex, deeper at the middle in some males. The sexual dimorphism is not very conspicuous. Male genitalia (Figure Female genitalia (Figure e Figure  13.8–17.e Figure  17.5–20.e Figure . Elytra:a Figure , laterala Figure  moderateThe name refers to the yellow patch on the apical half of elytron.China: Guizhou, Guangxi.


A spreadsheet summary of these case histories are included along with a separate spreadsheet, by which maximum likelihood assessment was performed. These data transparently enable researchers to access case history input parameters and processing details, and to compare the case history processing protocols with the ones of different researchers (e.g.: “The influence of SPT procedures in soil liquefaction resistance evaluations.” This data article provides a summary of seismic soil liquefaction triggering and non-triggering case histories, which were compiled, screened for data completeness and quality, and then processed for the development of triggering relationships proposed in “SPT-based probabilistic and deterministic assessment of seismic soil liquefaction triggering hazard”  Specifications TableValue of the data•USA, Japan, Argentina, China, Philippines.This database presents a summary of 210 liquefaction and non-liquefaction case histories compiled from a number of earthquake events in •This database can be used for the development of new probabilistic and deterministic seismic soil liquefaction triggering relationships.•Additionally, a comparison among case histories and protocol details of similar databases •Critical Depth Lower Bound (ft)•Critical Depth (ft) =Critical depth for liquefaction•Critical Depth (m) =Critical depth for liquefaction•critical depth (ft) =Standard deviation of critical depthσ•Water Depth (ft)•water table (ft) =Standard deviation of water tableσ•1(pcf) =Unit weight above ground water tableγ•γ1 (pcf) =Standard deviation of γ1σ•2 (pcf) =Unit weight below ground water tableγ•γ2 (pcf) =Standard deviation of γ2σ•v (psf) =Vertical total stressσ•σv (psf) =Standard deviation of σvσ•v׳ (psf) =Vertical effective stressσ•σv׳ (psf) =Standard deviation of σσv׳σ•σσ׳= Correlation coefficient between σv׳ and σvρ•d=Shear mass modal participation factor, commonly referred to as cyclic shear stress reduction factorr•rd=Standard deviation ofσ•max (g)= Peak horizontal accelerationa•amax=Standard deviation of amaxσ•MwMagnitude for K•Magnitude Scaling Factor (MSF)•KMw=Standard deviation of KMwσ•σ׳v,Mw=Actual CSR, not normalized to a reference state of σ׳v = 1 atm, Mw =7.5CSR•CSR=Standard deviation of CSRσ׳v,Mwσ•σ׳v=100kPa,Mw=7.5= CSR normalized to σ׳v =1 atm, Mw=7.5CSR•CSRN=Standard deviation of CSRσ׳v=100kPa,Mw=7.σ•50 (mm)=Median grain sizeD•D50=Standard deviation of D50σ•FC %=Fines content•FC %=Standard deviation of FCσ•Sampler (SS: standard sampler or SWL: sampler without liners)•S= SPT correction for sampler configuration detailsC•Rod length•R= SPT correction factor for the rod lengthC•Borehole diameter (mm)•B= SPT correction for borehole diameterC•Energy Ratio •E= SPT correction for hammer energy ratio (ER)C•Number of N values•N (mean) =Field measured standard penetration resistance (blows/30 cm)•N =Standard deviation of Nσ•N=Overburden correction factorC•SPT Correction (Energy + Rod length)•1)60= Standard penetration test blowcount value corrected for overburden, energy, equipment and procedural factors.(N•(N1)60 =Standard deviation of (N1)60σ•Reference•s,40ft (fps) = Shear wave velocity for the upper 40 ftV*•σ=Correction for overburden stressK•Mw=Correction for magnitude (duration) effectsK•Mw=7.5,σ׳v=100kPA= CSR normalized to σ׳v =100 kPa, Mw=7.5CSR•1,60,CS=Fines-corrected N1,60 valueNThe output table in each spreadsheet contains the following columns:1,60 and CSR graphs, a summary page including surface manifestation, location of the site and expertise comments, are given in Additionally, for each case history, a detailed presentation of the borehole data, maps, laboratory data, corrected NOn the basis of the compiled resulting database, the maximum likelihood assessment was performed by using the Microsoft Excel Software Tool: Solver. The spreadsheet, which presents MLE assessments, is presented in


After eCPR, patients commonly present with a combined respiratory and metabolic acidosis . It is c2 .2 and pH with hospital survival in eCPR.The aim of the present study was to correlate arterial paCO2 and pH values were (mean ± standard deviation) 38.3 ± 8.9 mmHg/7.28 ± 0.14 (+ 1 h), 38.5 ± 8.5 mmHg/7.30 ± 0.11 (+ 3 h), 38.72 ± 7.42 mmHg/7.31 ± 0.11 (+ 6 h), 38.62 ± 7.26 mmHg/7.34 ± 0.10 (+ 12 h), and 38.22 ± 5.62 mmHg/7.38 ± 0.09 (+ 24 h), respectively. When comparing patients with paCO2 < 35, 35–45, and > 45 mmHg, survival was statistically similar for all observed time points. There was however a highly significant association between hospital survival and pH when comparing groups with pH < 7.3, 7.3–7.4, and > 7.4 . There was however a significant linear correlation of maximum NSE and pH at 1, 3, and 6 h after eCPR .As secondary endpoint and surrogate for neurological outcome, neuron-specific enolase (NSE) was analyzed. Maximum NSE measured within 72 h after eCPR was 150.8 ± 145.1 μg/l (mean ± standard deviation). When correlating maximum NSE with paCO2. Also elevated NSE as a marker for neural injury did correlate with pH but not with paCO2. Being a retrospective, observational, single-center study, inherent limitations and biases are to be presumed and findings are to be considered hypothesis generating. Until further data are available however, it might be reasonable to correct both respiratory and metabolic acidosis in eCPR patients.In this registry study, we found a strong correlation between hospital survival and arterial pH but no such correlation with paCO


This paper defines a Sahlqvist fragment for relevant logic and establishes that each class of frames in the Routley-Meyer semantics which is definable by a Sahlqvist formula is also elementary, that is, it coincides with the class of structures satisfying a given first order property calculable by a Sahlqvist-van Benthem algorithm. Furthermore, we show that some classes of Routley-Meyer frames definable by a relevant formula are not elementary. Sahlqvist Correspondence Theorem is a celebrated result in modal logic ., 16.8, 1mutatis mutandis, the argument for the Sahlqvist Correspondence Theorem  can be ented in . Moreovebi-approximation semantics (which unfortunately is much more complicated than the Routley-Meyer framework). Other settings without boolean negation where Sahlqvist theorems have been obtained are positive modal logic POS must be the case. Hence, (1). So (1) and (2) are actually equivalent.Next we observe that one may assume that any unary predicate  appearing in POS, appears also in the antecedent of (1). For suppose not, that is, there a unary predicate P1,…,Pn∀x1,…,xk(Up≤(P1)∧…∧Up≤(Pn)∧REL∧(DN)AT∧IMP ⊃¬NEG∨POS).(3) ∀Observe that ¬NEG ∨POS is a positive formula.It is easy to see that (1) is equivalent to π1(xi1),…,πk(xik) are all the conjuncts of (DN)AT and IMP in the antecedent of (3) in which the unary predicate Pi occurs. Then if πj(xij) appears in one of the conjuncts in IMP, it must be a formula of the form ∀yz(Rxijyz ∧∃b(Ob ∧ b ≤ y) ⊃ Piz), in which case we define σ(πj(xij)) = λu.(∃yz(Rxijyz ∧∃b(Ob ∧ b ≤ y) ∧ z ≤ u)). On the other hand if πj(xij) appears in one of the conjuncts in (DN)AT we put σ(πj(xij)) = λu.(xij ≤ u) in case πj(xij) = Pixij and σ(πj(xij)) = λu.(xij∗∗≤ u) in case πj(xij) = ¬¬Pixij∗∗. Note that σ is a well-defined function and that for any πj(xij), if πj(xij)[w] then ∀y(σ(πj(xij)(y) ⊃ Piy)[w]. Next define δ(Pi) = λu.(σ(π1(xi1))(u) ∨… ∨ σ(πk(xik))(u)). Now, if (DN)AT[ww1…wk] and IMP[ww1…wk] then ∀u(δ(Pi)(u) ⊃ Piu)[ww1…wk].Suppose Up≤(δ(Pi)) holds for any Pi. This is so because each σ(πj(xij))(u) is upward closed under ≤, and any union of upward closed sets under ≤ is also upward closed under ≤. Moreover, by the reflexivity of ≤, [δ(P1)/P1…δ(Pn)/Pn](DN)AT and [δ(P1)/P1…δ(Pn)/Pn]IMP will hold trivially. So, from δ(P1)/P1…δ(Pn)/Pn]∀x1,…,xk(Up≤(P1)∧…∧Up≤(Pn)∧REL∧(DN)AT∧IMP ⊃¬NEG∨POS),(4) (¬NEG ∨POS)).(5) ∀Given that any unary predicate appearing in POS also appears in the antecedent of (4), (5) is a first order formula in the signature of Routley-Meyer frames, which contains R,∗, and O as the only non-logical symbols.At this point we should note that P1,…,Pn be arbitrary and suppose further that ww1…wk] and IMP[ww1…wk], it must be the case that ∀u(δ(Pi)(u) ⊃ Piu)[ww1…wk]. Hence, by Lemma 3, We have seen that (1) implies (5). All that is left is to show that (5) implies (1). Recall that (1) is equivalent to (3). Thus it suffices to prove that (5) implies (3). Assume (5), let The procedure in the above proof is better understood by working out some examples, which we will do next for the benefit of the reader.M is any model based on Rwyz and V on V (p) = {x : y ≤ x} and V suffices to show that By the Sahlqvist-van Benthem algorithm, the following list of equivalences must hold: V such that By the Sahlqvist-van Benthem algorithm, x(x∗∗≤ x ∧ x ≤ x∗∗) when we consider validity with respect to all the worlds in O of the frame. This is not the same in general as ∀x(x = x∗∗) which is the condition usually required to validate ∼∼ p → p. However, it is certainly the case that, using the construction from Theorem 5 . Suppose M is an arbitrary model based on Rwyz ∃u1u2(Ru1u2y ∧ Pu1 ∧ Qu2) both hold, u1 ≤ z and u2 ≤ z, and, by the Hereditary Lemma, Pz and Qz as desired. Hence, V on V (p) = {x : u1 ≤ x} and V (q) = {x : u2 ≤ x} suffices to guarantee that y,z(Rxyz ⊃ x ≤ z ∧ y ≤ z) by some manipulations.By the Sahlqvist-van Benthem algorithm, x,y,z(Ox ∧ Rxyz ⊃∃u2u3(Ryu2u3 ∧∃b(Ob ∧ b ≤ u2) ∧ u3 ≤ z)) when we consider correspondence with respect to the worlds in O of a given frame. This condition is actually equivalent to the condition ∀x∃b(Ob ∧ Rxbx) corresponding in  then, then ∀y(Piy ⊃ σ(πj(xij))(y))[w]. Next define δ(Pi) = λu.(σ(π1(xi1))(u) ∧… ∧ σ(πk(xik))(u)). Now, if (T)NAT[ww1…wk] and IMP[ww1…wk] then ∀u(Piu ⊃ δ(Pi)(u))[ww1…wk]. The remainder of the proof is as before but using again the contrapositive formulation of Lemma 3 and noting that the intersection of a collection of upward closed sets under ≤ is also upward closed under ≤. □Finally, let p →t)∧∼t →∼ p. Using our Sahlqvist-van Benthem algorithm, we obtain that the following equivalences hold: Up≤(P), Rwyz, ∀u,v(Ryuv ∧ Pu ⊃∃b(Ob ∧ b ≤ v)) and ¬∃b(Ob ∧ b ≤ y∗), it must be that ∃u2(Ryz∗u2 ∧¬∃b(Ob ∧ b ≤ u2)). Thus if Pz∗, ∃b(Ob ∧ b ≤ u2), which is a contradiction, so ¬Pz∗. Consequently M based on V (p) = {x : z∗≤ x}. Now, if Ryu1u2 and u1 ∈ V (p), that is, z∗≤ u1, we that, by p3, Ryz∗u2, and hence, ∃b(Ob ∧ b ≤ u2). This shows that Consider the dual relevant Sahlqvist implication (𝜃) defined by Θ(ϕ) = 𝜃 → ϕ, and applications of ∨ where the disjuncts share no propositional variable in common.A relevant Sahlqvist formula is any formula built up from  relevant Sahlqvist implications, propositional variables, and negated propositional variables using ∧, the operations on formulas Θ (for any propositional variable free relevant formula Every relevant Sahlqvist formula has a local first order correspondent onRoutley-Meyer frames.Immediate from Lemma 11, Lemma 16 and Lemma 9. The only thing the reader should note is that In this paper we have defined a fragment of relevant languages analogue to the Sahlqvist fragment of modal logic. We then went to establish that every class of Routley-Meyer frames definable by a formula in this fragment is actually elementary. This isolates a modest but remarkable collection of relevant formulas. We also showed that there are properties of Routley-Meyer frames definable by relevant formulas which are not first order axiomatizable.


Nature Communications 10.1038/s41467-019-10107-1, published online 21 May 2019.Correction to: The original version of this Article contained errors in Figs 2 and 3. In Fig. 2b, the scale bar label ‘500 µm’ was originally incorrectly given as ‘200 µm’. In Fig. 2d, the scale bar label ‘500 µm’ was originally incorrectly given as ‘100 µm’. In Fig. 3c, the scale bar label ‘200 µm’ was originally incorrectly given as ‘100 µm’. This has been corrected in the PDF and HTML versions of the Article.


In oncology, the aim of dose-finding phase I studies is to find the maximum tolerated dose for further studies. The use of combinations of two or more agents is increasing. Several dose-finding designs have been proposed for this situation. Numerous publications have however pointed out the complexity of evaluating therapies in combination due to difficulties in choosing between different designs for an actual trial, as well as complications related to their implementation and application in practice.In this work, we propose R functions for Wang and Ivanova’s approach. These functions compute the dose for the next patients enrolled and provide a simulation study in order to calibrate the design before it is applied and to assess the performance of the design in different scenarios of dose-toxicity relationships. This choice of the method was supported by a simulation study which the aim was to compare two designs in the context of an actual phase I trial: i) in 2005, Wang and Ivanova developed an empirical three-parameter model-based method in Bayesian inference, ii) in 2008, Yuan and Yin proposed a simple, adaptive two-dimensional dose-finding design. In particular, they converted the two-dimensional dose-finding trial to a series of one-dimensional dose-finding sub-trials by setting the dose of one drug at a fixed level. The performance assessment of Wang’s design was then compared with those of designs presented in the paper by Hirakawa et al. (2015) in their simulation context.It is recommended to assess the performances of the designs in the context of the clinical trial before beginning the trial. The two-dimensional dose-finding design proposed by Wang and Ivanova is a comprehensive approach that yields good performances. The two R functions that we propose can facilitate the use of this design in practice. The study received regulatory approval in January 2013 and officially started in France in May 2013. The protocol was amended in 2014. The main modification concerned the dose-escalation method to apply the method published by Wang and Ivanova !=nd2 | dim(npts)[2]!=nd1)stop(“Verify dimensions matrix npts (dimension nd2xnd1)”)## ntoxif (is.matrix(ntox)==FALSE)stop(“ntox need to be matrix (dimension nd2xnd1)”)if (dim(ntox)[1]!=nd2 | dim(ntox)[2]!=nd1)stop(“Verify dimensions matrix ntox (dimension nd2xnd1)”)## ntox<=nptstemp <- npts-ntoxif (sum(temp<0)!=0)stop(“Verify matrix ntox and npts (Nb of toxicty greated than nb of patients)”)  # Compute the probabilities pijk pijk_model = function {  pijk <- (1—((1-ai)^alpha*(1-bj)^(Beta+Gamma*(1-ai))))  return(pijk) } #likelihood is the joint conditional density of the observed responses (=f(D_k|theta) in the Ivanova’s paper) likelihood = function {  rst <- 1  for(i in 1:nd1){   for(j in 1:nd2){    pijk <- pijk_model    rst  <- rst * pijk^ntox * (1-pijk)^   }  }  return(rst) }  #Estimation_pijk: to estimate the pijk   Estimation_pijk = function  {   #set.seed(55)   alpha_samp <- rexp   Beta_samp <- rexp   Gamma_samp <- rexp   pijk    <- matrix   num     <- numeric(nsamp)   f_lik    <- likelihood   for(j in 1:nd1)   {    for(k in 1:nd2)    {     pijk_samp = pijk_model     pijk <- sum(pijk_samp*f_lik)/sum(f_lik)    }   }   return(pijk)  }  # Recommended_dose: is the decision rule to recommend the next dose: the dose associated with a probability the closest to the target  Recommended_dose = function  {   next_d1 <- NA   next_d2 <- NA   # Next assignement is to one of the combinations from: , , , , , , et    next_assig <- list,nd1), cur_d2),         #               c,nd1), cur_d2),            #               c,                        #               c,nd2)),            #               c,nd2)),            #               c,nd1), min,nd2)), #               c,nd1), min,nd2))) #   next_assig      <- next_assig[!duplicated(next_assig)]   length_next_assig   <- length(next_assig)   difference      <- c   for(ll in 1:length_next_assig)   {    difference  <- c)   }   ind_min_dist <- which.min(difference)   next_d1    <- next_assig[[ind_min_dist]][1]   next_d2    <- next_assig[[ind_min_dist]][2]   return)  }  colnames(npts) <-paste  rownames(npts) <-paste  colnames(ntox) <-paste  rownames(ntox) <-pastepijk          <- Estimation_pijkif(sum(npts)<ntot)   {cur_d1         <- Recommended_dose[[1]]cur_d2         <- Recommended_dose[[2]]MTD_lv         <- NA   }if(sum(npts)==ntot)   {difference       <- abs(pijk-target)MTD_comb      <- which, arr.ind = T)cur_d1         <- MTD_comb[2]cur_d2         <- MTD_comb[1]MTD_lv       <- apply   }print(“- Current Dose agent A (cur_d1):”)print(cur_d1)print(“- Current Dose agent B (cur_d2):”)print(cur_d2)print(“- Number of patients treated at each dose level (npts):”)print(npts)print(“- Number of DLT observed at each dose level (ntox):”)print(ntox)print(“- Updated estimate of the MTD for each dose level of the agent B (MTD_lv):”)print(MTD_lv)return)}ExampleThe example concerns a phase I trial evaluating two agents, Agent A with five dose levels and Agent B with three dose levels. Thirty patients will be included in the trial, by cohorts of 3 patients. The first cohort will be allocated to dose level . Assuming that no toxicity is been observed at this dose, the next dose for the next cohort of patients is given as follows:SBS_designTo apply our R function, the arguments will be set as:prior_1 ← cprior_2 ← cnsamp ← 1000target ← 0.3nd1 ← 5nd2 ← 3ntot ← 30# Data# matrix presenting the number of patients treated for each combinationnpts ← matrix# matrix presenting the number of toxicities for each combinationntox ← matrix# Cohort 1, cur_d1 ← 1 #agent Acur_d2 ← 1 #agent Bnpts ← 3SBS_design“prior_1 must be a vector (length nd1)”. If there is no problem and the parameters are considered correct, calculations can then be continued, with the following message: “+ Parameters OK”.The program first verifies consistencies in the parameters and error messages can be displayed. For example, if the length of prior_1 is superior or inferior to nd1, the program stops and signals that A summary of the number of patients treated at each dose level, the numbers of DLTs observed at each dose level and the next dose to be explored are then given.The function SIM_design is used to generate simulation replicates of a phase I trial using the two-dimensional dose-finding design developed by Wang and Ivanova, under a specified dose-toxicity configuration.Argumentsnd1: The number of dose levels of Agent And2: The number of dose levels of Agent Btox_true: A specified dose-toxicity configurationprior_1: A vector of initial guesses of toxicity probabilities associated with the doses for Agent Aprior_2: A vector of initial guesses of toxicity probabilities associated with the doses for Agent Btarget: The target DLT ratentot: The number of patients enrolledncoh_st: The number of patients enrolled in the cohort used for the starting rulencoh: The number of patients enrolled in the cohort used for the main designnsamp: The number of iteration needed for the MCMCnsim: The number of simulationsseed: Seed of the random number generatorValueresult_pt: Average number of patients treated at the test doses. If nsim = 1, this is a vector of length n indicating the doses assigned to the patients in the simulated trial.result_tox: Average number of toxicities seen at the test doses. If nsim = 1, this is a vector of length n indicating the toxicity outcomes of the patients in the simulated trial.result_MTD: Distribution of the MTD estimates. If nsim = 1, this is a single numerical value of the recommended MTD in simulated trial. If nsim = 1, this is a single numerical value of the recommended MTD in the simulated trial.result_MTD_lv: Distribution of the MTD estimates for each dose level of Agent B. If nsim = 1, this is a single numeric value of the recommended MTD for each dose level of Agent B, in the simulated trial.R codeSIM_design=function{print(“*****************************************************************”)printprint(“*     in Discrete Dose Space of Wang K and Ivanova A      *”)print             *”)print(“*****************************************************************”)print(“* *Institut Curie                         *”)print(“* 25 rue d’Ulm                           *”)print(“* 75005                              *”)print(“* France                            *”)print(“*****************************************************************”)print# Data Checkprint(“+--------------------+”)print(“|  Data Check  |”)print(“+--------------------+”)quote(“ ”)# For nd1 and nd2## nd1 et nd2 are numericif(is.numeric(nd1)==FALSE) stop(“nd1 must be numeric”)if(is.numeric(nd2)==FALSE) stop(“nd2 must be numeric”)## nd1 nd2 entier>0if(ceiling(nd1)-nd1!=0 | nd1<=0) stop(“nd1 must be Integer (>0)”)if(ceiling(nd2)-nd2!=0 | nd2<=0) stop(“nd2 must be Integer (>0)”)## nd1 >= nd2if(nd1 < nd2) stop(“nd1 must be superior to nd2”)# For prior_1 prior_2## For prior_1if(is.vector(prior_1)==FALSE)stop(“prior_1 must be a vector (Numeric Argument)”)if(length(prior_1)!=nd1)stop(“prior_1 must be a vector (lengh nd1)”)if(min(prior_1)<0 | max(prior_1)>1) stop(“Verify probability prior_1 (<0 or >1)”)## For prior_2if(is.vector(prior_2)==FALSE)stop(“prior_2 must be a vector (Numeric Argument)”)if(length(prior_2)!=nd2)stop(“prior_2 must be a vector (lengh nd2)”)if(min(prior_2)<0 | max(prior_2)>1) stop(“Verify probability prior_2 (<0 or >1)”)# For tox_true## tox_true is matrixif(is.matrix(tox_true)==FALSE)stop(“tox_true must be a matrix (Numeric Argument)”)## Dimension de la matriceif(dim(tox_true)[[1]]!=nd2 | dim(tox_true)[[2]]!=nd1)stop(“tox_true must be a matrix (Dimension: nd2 x nd1)”)## Proba of tox between 0 and 1if(min(tox_true)<0 | max(tox_true)>1) stop(“Verify probability tox_true (<0 or >1)”)# Target between 0 and 1if(min(target)<0 | max(target)>1) stop(“Verify probability of tox (<0 or >1)”)# For ntot, ncoh_st, ncoh## ntot, ncoh_st, ncoh, nsamp are numericif(is.numeric(ntot)==FALSE) stop(“ntot must be numeric”)if(is.numeric(ncoh_st)==FALSE) stop(“ncoh_st must be numeric”)if(is.numeric(ncoh)==FALSE) stop(“ncoh must be numeric”)if(is.numeric(nsamp)==FALSE) stop(“nsamp must be numeric”)if(is.numeric(nsim)==FALSE) stop(“nsim must be numeric”)## ntot, ncoh_st, ncoh entier>0if(ceiling(ntot)-ntot!=0 | ntot<=0) stop(“ntot must be Integer (>0)”)if(ceiling(ncoh_st)-ncoh_st!=0 | ncoh_st<=0) stop(“ncoh_st must be Integer (>0)”)if(ceiling(ncoh)-ncoh!=0 | ncoh<=0) stop(“ncoh must be Integer (>0)”)if(ceiling(nsim)-nsim!=0 | nsim<=0) stop(“nsim must be Integer (>0)”)print(“+ Parameters OK”)result_MTD    <- matrixresult_tox     <- matrixresult_pt      <- matrixset.seed(graine)result_MTD    <- matrixresult_MTD_lv    <- matrixresult_tox     <- matrixresult_pt      <- matrixDesign_Wang_Ivanova = function{ d1_max     <- 0 d2_max     <- 0 ### Start-up Rule cur_d1    <- 1 cur_d2    <- 1 start_up_end <- FALSE ###################Other used functions############  # gene_tox: to generate toxicity data gene_tox = function {  Z_tox = runif(ncoh)  y_tox = as.numeric(Z_tox <= true_tox)  return(y_tox) } #Compute the probabilities pijk pijk_model = function {  pijk <- (1—((1-ai)^alpha*(1-bj)^(Beta+Gamma*(1-ai))))  return(pijk) } #likelihood is the joint conditional density of the observed responses (=f(D_k|theta) in the Ivanova’s paper) likelihood = function {  rst <- 1  for(i in 1:nd1){   for(j in 1:nd2){   pijk <- pijk_model        rst  <- rst * pijk^ntox * (1-pijk)^    }  }  return(rst) } #Estimation_pijk: to estimate the pijk Estimation_pijk = function {  alpha_samp   <- rexp  Beta_samp  <- rexp  if (Interaction==0) {Gamma_samp <- 0}  else {Gamma_samp <- rexp}  pijk     <- matrix  num     <- numeric(nsamp)  f_lik    <- likelihood  for(j in 1:nd1)  {   for(k in 1:nd2)   {    pijk_samp = pijk_model    pijk <- sum(pijk_samp*f_lik)/sum(f_lik)   }  }  return(pijk) } ### The start-up Rule start_up = function {y       <- c npts      <- ntox <- matrix colnames(npts)  <-paste rownames(npts) <-paste colnames(ntox)  <-paste rownames(ntox) <-paste n_rem      <- ntot while(start_up_end == FALSE && n_rem >= ncoh_st) {  d1_stup <- cur_d1  d2_stup <- cur_d2  y_tox <- c  n_rem <- n_rem-ncoh_st  #agent 2 is in rows from 1 to nd2 beginning by the end, as in the paper  npts <- npts + ncoh_st  true_tox <- tox_true  y_tox  <- gene_tox  y    <- c)  ntox <- ntox+sum(y_tox)  # if no tox observed => escalate the first agent  if(sum(y_tox) == 0)  {   if(cur_d1 < nd1)   {    d1_stup <- cur_d1+1   }   if(cur_d1 == nd1 && cur_d2 < nd2)   {    d1_stup <- cur_d1-2    d2_stup <- cur_d2+1   }   if(cur_d1 == nd1 && cur_d2 == nd2)   {    start_up_end <- TRUE   }  }  # if at least one tox is observed  else  {   if(cur_d1 > 2 && cur_d2 < nd2)   {    d1_stup = cur_d1-2    d2_stup = cur_d2+1   }   else   {    start_up_end <- TRUE   }  }  cur_d1 <- d1_stup  cur_d2 <- d2_stup } ### Next dose pijk     <- Estimation_pijk dose_level_1 <- which.min) cur_d1 <- dose_level_1 cur_d2 <- 1 nused <- ntot-n_rem return) } # Recommended_dose: is the decision rule to recommend the next dose: the dose associated with a probability the closest to the target Recommended_dose = function {  next_d1 <- NA  next_d2 <- NA  # Next assignement is to one of the combinations from: , , , , , , et   next_assig <- list,nd1), cur_d2),       #          c,nd1), cur_d2),          #          c,                    #          c,nd2)),          #          c,nd2)),          #          c,nd1), min,nd2)), #          c,nd1), min,nd2))) #  next_assig      <- next_assig[!duplicated(next_assig)]  length_next_assig    <- length(next_assig)  difference      <- c  for(ll in 1:length_next_assig)  {   difference  <- c)  }  ind_min_dist <- which.min(difference)  next_d1   <- next_assig[[ind_min_dist]][1]  next_d2   <- next_assig[[ind_min_dist]][2]  return) } ## nd1 et nd2 are numeric if(is.numeric(nd1)==FALSE) stop(“nd1 must be numeric”) if(is.numeric(nd2)==FALSE) stop(“nd2 must be numeric”) ## nd1 nd2 entier>0 if(ceiling(nd1)-nd1!=0 | nd1<=0) stop(“nd1 must be Integer (>0)”) if(ceiling(nd2)-nd2!=0 | nd2<=0) stop(“nd2 must be Integer (>0)”) ## nd1 >= nd2 if(nd1 < nd2) stop(“nd1 must be superior to nd2”) # For tox_true ## tox_true is matrix if(is.matrix(tox_true)==FALSE)stop(“fu must be a matrix (Numeric Argument)”) ## Proba of tox between 0 and 1 if(min(tox_true)<0 | max(tox_true)>1) stop(“Verify probability tox_true (<0 or >1)”) rst_start_up <- start_up npts    <- rst_start_up$npts ntox    <- rst_start_up$ntox cur_d1  <- rst_start_up$cur_d1 cur_d2  <- rst_start_up$cur_d2 ntot    <- rst_start_up$ntot n_rem  <- rst_start_up$n_rem nused   <- rst_start_up$nused y    <- rst_start_up$y pijk    <- rst_start_up$pijk ### Two-Dimensional Trial for(i in 1:(n_rem %/% ncoh)) {  npts <- npts + ncoh  true_tox          <- tox_true  y_tox           <- gene_tox  y             <- c)  ntox <- ntox+sum(y_tox)  pijk            <- Estimation_pijk  next_RD            <- Recommended_dose  cur_d1            <- next_RD[[1]]  cur_d2            <- next_RD[[2]] } ## target between 0 and 1 if(min(target)<0 | max(target)>1) stop(“Verify probability of tox (<0 or >1)”) ### MTD difference <- abs(pijk-target) MTD_comb <- which, arr.ind = T) MTD   <- c)#c ### MTD by level MTD_lv  <- apply return)}ntot_used = 0for(i in 1:nsim){result = Design_Wang_Ivanovaresult_MTD[(nd2-result$MTD[2]+1),result$MTD[1]]  <- (result_MTD[(nd2-result$MTD[2]+1),result$MTD[1]]+1)result_pt                      <- (result$npts+result_pt)result_tox                      <- (result$ntox+result_tox)MTD_lv                       <- as.numeric(result$MTD_lv)MTD_lv                       <- ifelse(is.na(MTD_lv),0,MTD_lv)ntot_used                      <- ntot_used+sum(result$npts)ntot_mean                      <- ntot_used/i for(ii in 1:nd2) {  if(MTD_lv[ii]!=0)  {result_MTD_lv] <-   }  }}Rst_MTD     <- (result_MTD*100)/nsimRst_MTD_lv   <- (result_MTD_lv*100)/nsimRst_pt      <- ((result_pt/nsim)/ntot_mean)*100Rst_tox      <- ((result_tox/nsim)/ntot_mean)*100# Imput Parametersprint(“+--------------------------+”)print(“| Imput Parameters  |”)print(“+--------------------------+”)print(paste(“ - Number of dose level agent A (nd1):”,nd1))print(paste(“ - Number of dose level agent B (nd2):”,nd2))print(paste(“ - Prior distribution agent A (prior_1):”,paste(c(prior_1),collapse=“ ”)))print(paste(“ - Prior distribution agent B (prior_2):”,paste(c(prior_2),collapse=“ ”)))print(paste(“ - Target DLT Rate (target):”,target))print(paste(“ - Nb of patients enrolled (ntot):”,ntot))print(paste(“ - Nb of patients enrolled in the cohort used for the starting rule (ncoh_st):”,ncoh_st))print(paste(“ - Nb of patients enrolled in the cohort used for the main design (ncoh):”,ncoh))print(paste(“ - Number of iteration needed for the MCMC (nsamp):”,nsamp))print(paste(“ - Number of simulation (nsim):”,nsim))print(“+--------------------------+”)print(“|  Output Parameters |”)print(“+--------------------------+”)print(“ - Average number of patients treated at the test doses in terms of percentage (Rst_pt):”)print)print(“ - Average number of toxicities seen at the test doses in terms of percentage(Rst_tox):”)print)print(“ - Distribution of the MTD estimates in terms of percentage(Rst_MTD):”)print)print(“ - Distribution of the MTD estimates for each dose level of the agent B in terms of percentage (Rst_MTD_lv):”)print)print(“+--------------------------------------------------------------------+”)print(“| Send feed back via e-mail:              |”)print(“|  —RAJOUTER TON MAIL          |”)print(“+--------------------------------------------------------------------+”)return)}ExampleFor example, for scenario 1, the parameters should be be fixed in the R function as:tox_true ← matrix, byrow = T, nrow = 3)prior_1 ← c # 6 dose levels of agent Aprior_2 ← c #3 dose levels of agent Btarget ← 0.2ntot ← 54ncoh_st ← 2ncoh ← 3nsamp ← 1nd1 ← 6nd2 ← 3seed ← 2nsim ← 4000Interaction ← 0# Generate 4000 replicates of the two-dimensional dose-finding design of Wang and Ivanova with 54 subjectsresult_sc = SIM_designTo validate our R scripts, we ran the same scenarios (dose-toxicity relationships) proposed by Wang for the simulation study. The results were compared to those published in their paper. The results using R function were similar to those published by Wang.Using the R functions, the results on the percentage of DLTs observed, and a recommended MTD at the end of the trial can be also obtained. The MTD is defined as the dose combination associated with a probability of DLT closest to the target toxicity (results available on request).As stated by Hirakawa et al., thus far, there are no silver bullet designs to resolve the two-agent dose-finding problem. The operating characteristics of model-based dose-finding methods may be varied depending on prior toxicity probabilities, prior distributions, and number of patients included in trials. It is recommended to assess the performance of the designs in the context of the clinical trial before beginning the trial. The two-dimensional dose-finding design proposed by Wang and Ivanova is a comprehensive approach that yields good performances. The two R functions that we propose can facilitate the use of this design in practice. With the first function, the design can be applied to an actual phase I trial. With the second function, a simulation study can be run in order to calibrate the design before its application for a phase I trial, or in order to compare it with other designs. Both R functions are simple, and each part of the algorithm  can be modified depending on the requirements of the trial: i) the start-rule can be updated depending on the clinical context, ii) even if the design proposes to identify a set of MTDs, one MTD can be chosen from the set. For example, the MTD with a probability of DLT closest to the target toxicity can be chosen, or clinicians may prefer to choose to identify the dose for further trials depending on the clinical context.


Nature Communications; 10.1038/s41467-018-05698-0; published online: 09 Aug 2018Correction to: '', respectively.’ The correct version states ‘σ′′’ in place of ‘σ''’.The original version of this Article contained an error in the seventh sentence of the second paragraph of the ‘TA amplitudes for double pump-pulse excitation” section of the Results, which incorrectly read ‘The absorption cross-sections of the transitions from the single exciton state to the biexciton and triexciton states are denoted by σ′ and σ'' are equal to σ or not. The black dotted curves in Fig. 3b, c show the predicted results of the TA amplitudes by assuming identical cross-sections, i.e., σ = σ ′ = σ'' = .’ The correct version states ‘σ′ and σ′′ are equal” instead of ‘σ′ and σ'' are equal’ and ‘σ = σ′ = σ′′’ in place of ‘σ = σ ′ = σ'' = ’.The ninth and tenth sentences of the same paragraph originally incorrectly read ‘The most important issue of this work is to clarify whether the multiple exciton absorption cross-sections σ′ and σThis has been corrected in both the PDF and HTML versions of the Article.


We give a sufficient condition for a nonnegative integer list to be graphic based on its largest and smallest elements, length, and sum. This bound generalizes a result of Zverovich and Zverovich. A degree sequence is said to be graphic if it corresponds to the set of edge adjacency values of the nodes for some simple graph. In 1960, Erdős and Gallai gave a complete characterization of the set of˝ graphic degree sequences.We denote a finite list of nonnegative integers, Let α =  be a nonincreasing degree sequence. Then the degree sequence α is graphic if and only if the sum of α is even and for each integer k where 1 ≤ k ≤ n,It was later shown by I. Zverovich and E. Zverovich that some degree sequences with bounded largest and smallest elements can be verified to be graphic based on their length.Let α =  be a nonincreasing degree sequence of positive integers with even sum. Ifthen α is graphic.There have been several extensions to the original result of Zverovich and Zverovich. These results have been either specializations, such as when the gaps between consecutive integers in the sequence are bounded , or by sLet α =  be an integer sequence such that n − 1 ≥ α1 ≥ ... ≥ αn ≥ 0 and with even sumwhere nα1> s > nαn. Ifthen α is graphic.1complement of a degree sequence α =  is the sequence α is graphic if and only if Before providing a proof of this result, we state some needed definitions and theorems. The α majorizes a second sequence β = , where both sequence have the same length and sum, if and only ifk from 1 to n.A second definition is that a sequence Majorization is a partial order over the set of degree sequences with identical sum. Our use of majorization stems from the following result.If the degree sequence α is graphic and α majorizes β, then β is graphic.In addition, we use the following two results that reduce the number of Erdős-Gallai inequalities needed˝ to verify that a sequence is graphic. These are found in For the degree sequence α = , define the set of indicessuch that i ∈ if αi> αi+1. For For the degree sequence α = , define the set of indiceswhere i ∈ if αi ≥ i − 1. For We are now ready to prove α1, αn, n, and s. We then define the sequence α′ ∈ α1> γ ≥ αn and s = pα1 + γ + (n − p − 1)αn. From the condition that nα1> s > nαn, it follows that the sequence α′ cannot be regular, i.e., α1> αn. To show that the sequence α′ exists, we construct it by first defining the value of γ as γ = τ + αn where τ is the solution to the congruence equationτ < α1− αn. Since α1> αn, then τ is well-defined. From definition of τ, the value of γ is αn ≤ γ < α1. It follows from the congruence equation that s − nαn − τ = p(α1 − αn) for some integer p. To show that the sequence < p < n. First, we rewrite the value for p asτ < α1− αn, αn ≤ ... ≤ α1, and < p < n and establishing that the sequence We begin by defining the set α′ majorizes all the other sequences in β ∈ α′ does not majorize β then there would exist an index k where k = p + 1, since the first p values of α0 are the maximum possible value α1. If β to be less than αn and violating its membership in the set n similar argument for remainder of the indices, establishing that α′ must majorize all the members of α′ is graphic in order to prove the result.We observe that the sequence α′ that α′ for our argument since We also make the assumption concerning k = p and k = p + 1 in order to prove that the sequence α′ is graphic. First consider the case when p < αn. The resulting EG inequality for k = p isα1> n−1, but we assumed that α1 ≤ n−1 ensuring that this inequality always holds. There is an identical argument for the case when k = p + 1 establishing that the EG inequalities always hold when p < αn.Now let us examine the Erdős-Gallai (EG) inequalities forα′. We apply p ≥ αn. By first multiplying both sides of the Inequality s−nαn)(nα1−s)/(α1−αn)2, we algebraically manipulate this new inequality to derive the following equivalent expression,Now consider when the case when α1 ≥ n − αn − 1 along with s ≥ pα1 + (n − p)αn, this new Inequality pα1 ≤ p(p − 1) + γ + (n − p − 1)αn, which is precisely the EG inequality for α′ at k = p.Combining the facts that k = p + 1, we split the instance into two cases: p ≥ γ and p < γ. If p < γ, then we use k such that αk ≥ k − 1, in order to prove that a sequence is graphic. Thus, if p < γ, then satisfying the EG inequality for k = p already established that the sequence is graphic. Else, if p ≥ γ, then γ ≤ 2p − αn−1 and summing this inequality with the Inequality pα1+γ ≤ (p+1)p+(n−(p+1))αn, which is the EG inequality for k = p+1. Therefore, the premise is established.For 2α is graphic by This result generalizes the bound of Zverovich and Zverovich. We observe that for the right hand side of Inequality α1, define αn = 2, n = α1 + 1, and s = 4α1 − 2. These values satisfy the Inequality A equality, and so any sequence with these parameters is graphic. We notice that modifying any one of these values so that the inequality is no longer satisfied while keeping the other three fixed allows for non-graphic sequences. For example, if we increase α1 by 1 or decrease n by 1, we have sequences, such as αn by 1 or increase the sum by 2 (in order to preserve an even sum), then we have sequences such as In general, this bound is sharp in the sense that there exist sequences that satisfy the inequality, but any modification to parameters that causes the inequality to be not satisfied allows for non-graphic sequences to be formed. For example, consider the following set of degree sequences: for a given value of α1 ≤ n − 1 and this restriction cannot be relaxed. This is because there are sequences where α1> n − 1 but fulfill the inequality ). In contrast to Zverovich and Zverovich result, if that inequality is satisfied, it implies that α1 ≤ n−1 (proof of Theorem 1.2, In comparison to the Zverovich and Zverovich result, there is a slight difference in the requirements for applying the bound. This bound requires that α1 = αn, but we can handle this case by using in conjunction the result that states if α1 − αn ≤ 1, then α is graphic (αi> 0. This is because zeros in a degree sequence do not change whether the entire sequence is graphic or not and can be discarded.It should be noted that this bound cannot be used for sets of regular sequences, i.e.,  graphic , Lemma 1Finally, this result has practical use when coupled with the usual Erdős-Gallai conditions for checking˝ if a sequence is graphic. Before a sequence can be tested using the EG conditions, it must first be sorted. During this sorting step, the values for largest and smallest indices, sequence length, and sequence sum can be easily extracted. Thus, this new condition may save a number of steps in graphic testing for certain sequences by verifying that a sequence is graphic without having to compute the full EG inequalities .


Scientific Reports 10.1038/s41598-018-29880-y, published online 31 July 2018Correction to: This Article contains errors in Tables In addition, in Tables “Absolute”should read:“Mean/Median*”As a result, in the legends of Tables “”should read:“”The correct Tables


Amanita muscaria, has been regarded as a universal non‐selective GABA‐site agonist. Deletion of the AGABA receptor (ARGABA) δ subunit in mice (δKO) leads to a drastic reduction in high‐affinity muscimol binding in brain sections and to a lower behavioral sensitivity to muscimol than their wild type counterparts. Here, we use forebrain and cerebellar brain homogenates from WT and δKO mice to show that deletion of the δ subunit leads to a > 50% loss of high‐affinity 5 nM [3H]muscimol‐binding sites despite the relatively low abundance of δ‐containing ARGABAs (δ‐ARGABA) in the brain. By subtracting residual high‐affinity binding in δKO mice and measuring the slow association and dissociation rates we show that native δ‐ARGABAs in WT mice exhibit high‐affinity [3H]muscimol‐binding sites (DK ~1.6 nM on α4βδ receptors in the forebrain and ~1 nM on α6βδ receptors in the cerebellum at 22°C). Co‐expression of the δ subunit with α6 and β2 or β3 in recombinant (HEK 293) expression leads to the appearance of a slowly dissociating [3H]muscimol component. In addition, we compared muscimol currents in recombinant α4β3δ and α4β3 receptors and show that δ subunit co‐expression leads to highly muscimol‐sensitive currents with an estimated EC50 of around 1–2 nM and slow deactivation kinetics. These data indicate that δ subunit incorporation leads to a dramatic increase in ARGABA muscimol sensitivity. We conclude that biochemical and behavioral low‐dose muscimol selectivity for δ‐subunit‐containing receptors is a result of low nanomolar‐binding affinity on δ‐GABAARs.Muscimol, the major psychoactive ingredient in the mushroom  GABA‐site agonist. Here, we show that δ subunit incorporation leads to a dramatic increase in AGABA receptor (ARGABA) muscimol sensitivity. The biochemical and behavioral low‐dose muscimol selectivity for δ subunit‐containing receptors was because of low nanomolar‐binding affinity on α4/6βδ ARGABAs. This paints a consistent picture in which extrasynaptic δ‐ARGABAs are not only exquisitely sensitive to GABA, but also the GABA analogs gaboxadol and muscimol.Muscimol has been regarded as a universal non‐selective  GABAAR subunits are coded by 19 separate genes, α1‐α6, β1‐β3, γ1‐γ3, δ, ε, π, θ, and ρ1‐ρ3 (Olsen and Sieghart AR complexes formed in the brain are of type αβγ2 (γ2‐GABAAR) with a likely subunit stoichiometry of 2(α):2(β):1(γ2)  reside in extra‐ and perisynaptic membranes where their high GABA sensitivity allows them to be activated by ambient [GABA] to mediate tonic inhibition of the nerve cell , cerebral cortex (α4β2/3δ), hippocampal dentate gyrus granule cells (α4β2/3δ), caudate‐putamen and in the nucleus accumbens (α4β3δ)  is the major inhibitory neurotransmitter in vertebrate brain. The inhibitory action of GABA is mediated via ionotropic GABAARs are quite different from classical γ2‐GABAARs. δ‐GABAARs have much higher affinity for GABA, are insensitive to classical benzodiazepines, show high sensitivity to neurosteroids and Zn2+  when compared to GABA. This is likely because of GABA being a partial agonist on these receptors  mice lose low dose THIP effects on tonic currents in neurons in brain slices and behavioral sensitivity to low doses of THIP  and δKO mouse brains and to several αβγ and αβδ‐type recombinant GABAARs by measuring binding and unbinding kinetics. Subtraction of residual high‐affinity (5 nM) [3H]muscimol binding that is seen on abundant GABAAR subtypes in δKO mice from binding in WT membranes allowed us to isolate a native δ‐GABAAR component. This isolated component showed very slow muscimol dissociation rate with an apparent KD  for muscimol of 1.6 nM for α4βδ receptors in the forebrain and around 1 nM for α6βδ receptors in the cerebellum. Recombinant α4β3δ receptors expressed in oocytes revealed a biphasic response to muscimol with the high‐muscimol affinity (slowly deactivating/dissociating) component showing an approximate EC50 of around 1–2 nM.In this study, we investigated high‐affinity (5 nM) muscimol (22 Ci/mmol) was purchased from PerkinElmer Life and Analytical Sciences . Unlabeled muscimol was from Sigma‐Aldrich . GABA was from Sigma‐Aldrich (Cat. No. A2129).muscimol (5 nM) was measured in assay buffer at 22°C in a total volume of 300 μL. Triplicate technical replicates of mouse forebrain (190–215 μg protein), cerebellar (180–210 μg protein) or HEK cell (92–132 μg protein) membranes for each time point were incubated with shaking for various times (15 s–15 min) to measure association of the binding. Non‐specific binding was determined in the presence of 100 μM GABA. The incubation was terminated by filtration of the samples with a Brandel Cell Harvester  onto Whatman GF/B filters . The samples were rinsed twice with 4–5 mL of ice‐cold assay buffer. Filtration and rinsing steps took a total time of ~15 s. Air‐dried filters were immersed in 3 mL of Optiphase HiSafe 3 scintillation fluid  and radioactivity determined in a Wallac model 1410 liquid scintillation counter . The maximal binding disintegrations per minute (DPM) values (at 15 min in association) for recombinant studies with 5 nM [3H]muscimol were between 700 and 2500 DPMs of specific binding (background subtracted). In native membranes, the maximal DPM values were between 2500 and 3000 for WTs and 1300–1500 for δKOs. Mock transfection with pRK5 plasmid did not produce any specific binding over the background.The binding of [3H]muscimol binding, triplicate technical replicates of each sample of mouse brain or HEK cell membranes for each time point were first pre‐incubated at 22°C in a total volume of 300 μL for 15 min with 5 nM [3H]muscimol in the absence and presence of 100 μM GABA. The dissociation was then started by adding 100 μL of 400 μM or 100 μM (non‐specific binding) cold GABA to the incubation mixtures to reach a final 100 μM GABA concentration in all tubes. The tubes were mixed and incubations at 22°C were terminated at various time points (30 s – 30 min) as described above. Dissociation of [3H]muscimol from recombinant receptors in HEK cell membranes was also measured at 0–4°C (on ice) to evaluate how fast [3H]muscimol dissociates from receptors while washing the filter with ice‐cold assay buffer during filtration.To measure dissociation of [3H]muscimol to WT and δKO mouse forebrain and cerebellar membranes was performed essentially as described by Uusi‐Oukari and Korpi (3H]muscimol (0.1–30 nM) at 0 to 4°C for 30 min in the absence and presence of 100 μM GABA determining the non‐specific binding. The incubations were terminated as described above.Saturation analysis of [nd Korpi . Triplic3H]muscimol to δ‐GABAARs in WT animals, ‘native δ‐GABAARs’, were calculated by subtracting the specific δKO binding values (binding to γ2‐GABAARs) from the corresponding WT values at each time point: native δ‐GABAARs = WT‐ δKO. Because of the lack of low‐affinity binding and the relatively small number of time points in our assays, the binding curves fitted better in ‘one binding site’ model. However, varying fast and slow dissociation components are obvious in the graphs  according to manufacturer's instructions.Bmax and KD values were analyzed with Graph Pad Prism 7 software . Statistical significances between the groups were analyzed using unpaired t‐test and one‐way anova followed by Tukey's post hoc test Graph Pad (Graph Pad Prism 7). p‐values of < 0.05 were considered significant. In this study, no sample calculation, assessment of data outliers and data normality were performed, and experiments were done unblinded.Association and dissociation curves for estimation of association and dissociation rate constants, and saturation binding for estimation of ARs to high‐affinity muscimol binding, we measured the time course of 5 nM [3H]muscimol binding to forebrain and cerebellar membranes from both wild‐ type and δKO mice. Deletion of the δ subunit led to > 50% reduction of 5 nM [3H]muscimol binding at 22°C to both forebrain and cerebellar membranes when compared to WT mice  was around four times higher in the cerebellum when compared to forebrain both in WT as well as in the δKO mice . Binding of 5 nM [3H]muscimol to non‐δ‐GABAARs in δKO forebrain was around 100 fmoles and about 300 fmoles (per mg membrane protein) in the cerebellum  of non‐δ‐GABAARs were occupied by muscimol under our binding conditions  when compared to δ‐GABAARs, which was surprising since faster muscimol association would contribute to higher muscimol affinity in δ‐GABAARs. This slower muscimol association to δ‐GABAARs is reflected in higher forebrain and cerebellar association rate constants (kon) of [3H]muscimol binding to δKOs than to WT mouse membranes .To better illustrate high‐affinity muscimol binding kinetics to δ‐GABAved Figs  and 3. B mostly γ‐GABAARs  mostly γ‐GABAARs 3H]muscimol unbinding for up to 30 min. A comparison of muscimol dissociation between WT and δKO animals shows that almost all of the slow muscimol dissociation is because of δ‐GABAARs, with only a minor component present in both the forebrain and cerebellum of δKO animals, which is because of the high‐affinity muscimol binding to non‐δ‐GABAARs  HmuscimolARs in δKO mice from binding to total GABAARs in WT mice, we were able to determine a KD value based on the equation KD = koff/kon. The calculated KD value for δ‐GABAARs in the fore(mid)brain (predominantly α4βδ) is 1.6 nM, and the KD for δ‐GABAARs in the cerebellum (α6βδ) is 1.1 nM. Therefore, under our binding conditions [5 nM [3H]muscimol and 22°C], the majority of δ‐receptors both in forebrain and cerebellum should be occupied at equilibrium.After subtraction of binding to non‐δ‐GABAARs have low affinity and are therefore not occupied at 5 nM [3H]muscimol. Dissociation rate constants koff of [3H]muscimol binding were higher in δKOs than in WTs in both forebrain (p < 0.001) and cerebellar membranes (p < 0.05) (Table t‐test) indicating faster [3H]muscimol dissociation in δKOs lacking δ‐GABAARs. The Koff values of the calculated native δ‐GABAARs in both forebrain and cerebellum were smaller than those of δKO indicating slower [3H]muscimol dissociation from δ‐GABAARs than from γ2‐GABAARs  conformations which could also depend on subunit composition. We therefore decided to measure association and dissociation on selected recombinant receptor subtypes. As observed for native δ‐GABAARs [3H]muscimol association at 22°C was much slower in α6β2δ receptors when compared to high‐affinity binding to α1β2γ2 and α6β2γ2 recombinant receptors .Measurements in native brain tissues have the advantage that we can measure native receptors. The disadvantages is that the fraction of δ receptors is variable . In addition, because of the low‐muscimol affinity of most γ2‐GABA3H]muscimol from α6β2, α6β2γ2 and especially α1β2γ2 receptor subtypes was very fast   Table d. From a3H]muscimol dissociation kinetics at 22°C with unbinding at lower temperature (0°C) on selected γ2 and δ‐GABAAR subtypes. At 0°C dissociation from α6β2δ and α6β2γ2 were significantly slower than from α1β2γ2 GABAARs with 70% of [3H]muscimol still remaining bound to α6β2δ subtype at 30 min after start of the dissociation (Table p < 0.001). [3H]Muscimol dissociated also significantly slower from α6β2δ when compared to α6β2γ2 GABAARs , a difference that was also noted at 22°C , we decided to compare Hmuscimolsee Fig. c. It canors Fig. d.3H]muscimol binding sites as well as binding kinetics are in the same range as found in the literature , electrophysiological measurements are typically performed at 22°C and in rodent behavioral experiments receptors are studied at body temperature (37°C). Such temperature differences could have a major influence on binding affinities of GABA and GABA analogs. Also, the high‐affinity muscimol binding sites have been interpreted to represent desensitized or otherwise non‐functional high‐affinity conformations  of 1.1 nM in the cerebellum and 1.6 nM in fore/mid‐brain  apparent [3H]muscimol affinities (KD) for these non‐δ‐GABAARs were also around 1 nM  and partly also from higher affinity α6βγ2 receptors, while the extremely slow dissociation from δ‐GABAARs allows the majority of muscimol to be retained as seen in autoradiographs , seems to be a plausible explanation. Another  possibility is that such high‐affinity binding to non‐δ‐GABAARs is due to freezing, since at 22°C room temperature high‐affinity binding was lower when never‐frozen whole brain membranes were used  effects in δKO mice.We show here for the first time that co‐expression of the δ subunit leads to highly muscimol‐sensitive α4β3δ currents. Remarkably, the ECARs is challenging since they generally show biphasic GABA and THIP concentration response curves likely because of incomplete δ subunit incorporation into functional receptors  value −3.60 for GABA, whereas adding hydrophobic ring structures in muscimol (logP = −1.71) and THIP (logP = −0.81) . It seems therefore likely that GABA has the lowest BBB permeability, followed by muscimol and THIP. Given that THIP affinity for δ‐GABAARs is lower when compared to muscimol  permeability usually correlates with lipid‐solubility and is therefore rather poor for highly water‐soluble molecules like GABA, muscimol and THIP. Consistent with a low BBB permeability it has been shown that only around 0.02% 1/5000) of peripherally injected [/5000 of A agonist binding sites in GABAARs are located at the two extracellular β+α‐ interfaces , without the δ subunit actually directly contributing to the GABA‐binding site. This implies that δ increases the GABA‐binding‐site affinity and slows muscimol dissociation in the βαδβα pentamer allosterically. The reciprocal of dissociation rate constant, the drug‐target residence time τ (= 1/koff), has been shown to often predict in vivo efficacy better than binding affinity . For example, muscimol EC50 for α1βγ2 GABAARs is ~1 μM, whereas GABA EC50 is ~100 μM  induces low nanomolar muscimol currents.It appears that in general the GABA analog muscimol is similar to GABA in many aspects, only that it shows about 100–1000 times higher affinity (with THIP having intermediate affinity) across the board for different GABAA analog THIP, which has been shown to be highly selective for δ‐GABAARs  and NIH grant AA021213 to MW. The authors have no conflict of interest to declare.