We describe Support Vector Machine (SVM) applications to classification and clustering of channel current data. SVMs are variational-calculus based methods that are constrained to have structural risk minimization (SRM), i.e., they provide noise tolerant solutions for pattern recognition. The SVM approach encapsulates a significant amount of model-fitting information in the choice of its kernel. In work thus far, novel, information-theoretic, kernels have been successfully employed for notably better performance over standard kernels. Currently there are two approaches for implementing multiclass SVMs. One is called external multi-class that arranges several binary classifiers as a decision tree such that they perform a single-class decision making function, with each leaf corresponding to a unique class. The second approach, namely internal-multiclass, involves solving a single optimization problem corresponding to the entire data set (with multiple hyperplanes).Each SVM approach encapsulates a significant amount of model-fitting information in its choice of kernel. In work thus far, novel, information-theoretic, kernels were successfully employed for notably better performance over standard kernels. Two SVM approaches to multiclass discrimination are described: (1) internal multiclass (with a single optimization), and (2) external multiclass (using an optimized decision tree). We describe benefits of the internal-SVM approach, along with further refinements to the internal-multiclass SVM algorithms that offer significant improvement in training time without sacrificing accuracy. In situations where the data isn't clearly separable, making for poor discrimination, signal clustering is used to provide robust and useful information – to this end, novel, SVM-based clustering methods are also described. As with the classification, there are Internal and External SVM Clustering algorithms, both of which are briefly described. SVMs are fast, easily trained, discriminators ,2, for wSVMs use variational methods in their construction and encapsulate a significant amount of discriminatory information in their choice of kernel. In reference  informatThe SVM discriminators are trained by solving their KKT relations using the Sequential Minimal Optimization (SMO) procedure . A chunkBinary Support Vector Machines (SVMs) are based on a decision-hyperplane heuristic that incorporates structural risk management by attempting to impose a training-instance void, or "margin," around the decision hyperplane .ik, where index i labels the M feature vectors (1 ≤ i ≤ M) and index k labels the N feature vector components (1 ≤ i ≤ N). For the binary SVM, labeling of training data is done using label variable yi = ± 1 (with sign according to whether the training instance was from the positive or negative class). For hyperplane separability, elements of the training set must satisfy the following conditions: wβxiβ-b ≥ +1 for i such that yi = +1, and wβxiβ-b ≤ -1 for yi = -1, for some values of the coefficients w1, ..., wN, and b (using the convention of implied sum on repeated Greek indices). This can be written more concisely as: yi(wβxiβ-b) - 1 ≥ 0. Data points that satisfy the equality in the above are known as "support vectors" (or "active constraints").Feature vectors are denoted by xβxiβ - b = 0. The boundary hyperplanes on the two classes of data are separated by a distance 2/w, known as the "margin," where w2 = wβwβ. By increasing the margin between the separated data as much as possible the optimal separating hyperplane is obtained. In the usual SVM formulation, the goal to maximize w-1 is restated as the goal to minimize w2. The Lagrangian variational formulation then selects an optimum defined at a saddle point of L = (wβwβ)/2 - αγyγ(wβxγβ-b) - α0, where α0 = Σγαγ, αγ ≥ 0 (1 ≤ γ ≤ M). The saddle point is obtained by minimizing with respect to {w1, ...,wN,b} and maximizing with respect to {α1, ..., αM}. If yi(wβxiβ-b) - 1 ≥ 0, then maximization on αi is achieved for αi = 0. If yi(wβxiβ-b) - 1 = 0, then there is no constraint on αi. If yi(wβxiβ-b) - 1 < 0, there is a constraint violation, and αi → ∞. If absolute separability is possible the last case will eventually be eliminated for all αi, otherwise it's natural to limit the size of αi by some constant upper bound, i.e., max(αi) = C, for all i. This is equivalent to another set of inequality constraints with αi ≤ C. Introducing sets of Lagrange multipliers, ξγ and μγ(1 ≤ γ ≤ M), to achieve this, the Lagrangian becomes:Once training is complete, discrimination is based solely on position relative to the discriminating hyperplane: wβwβ)/2 - αγ[yγ(wβxγβ-b)+ξγ] + α0 + ξ0C - μγξγ, where ξ0 = Σγξγ, α0 = Σγαγ, and αγ ≥ 0 and ξξ ≥ 0 (1 ≤ γ ≤ M).L =  = α0 -  and αγyγ = 0, where only the variations that maximize in terms of the αγ remain (known as the Wolfe Transformation). In this form the computational task can be greatly simplified. By introducing an expression for the discriminating hyperplane: fi = wβxiβ - b = αγyγxγβxiβ - b, the variational solution for L(α) reduces to the following set of relations : (i) αi = 0 ⇔ yifi ≥ 1, (ii) 0 < αi < C ⇔ yifi = 1, and (iii) αi = C ⇔ yifi ≤ 1. When the KKT relations are satisfied for all of the αγ (with αγyγ = 0 maintained) the solution is achieved.  and the α's associated with the negatives to 1/N(-), where N(+) is the number of positives and N(-) is the number of negatives.) Once the Wolfe transformation is performed it is apparent that the training data  above) enter into the Lagrangian solely via the inner product xiβxjβ. Likewise, the discriminator fi, and KKT relations, are also dependent on the data solely via the xiβxjβ inner product.At the variational minimum on the {wiβxjβ are possible and are usually formulated in terms of the family of symmetric positive definite functions (reproducing kernels) satisfying Mercer's conditions  Substituting the above equations into the Lagrangian and after simplification reduces into the dual formalism:Where all αi,j∑m(δim - αim)(δjm - αjm)(Kij + β) - β∑i,mδimαimMaximize: -1/2∑im, ∑mαim = 1, i = 1...1; m = 1...kConstraint: 0 ≤ αij = xi.xj is the Kernel generalization. In vector notation:Where Ki,j(Δyi - Ai)(Δyj - Aj)(Kij + β) - β∑iΔyiAiMaximize: -1/2∑i, Ai. 1 = 1, i = 1 ...1Constraint: 0 ≤ Ai = Δyi - Ai. Hence after ignoring the constant: -1/2∑i,jτi.τj(Kij + β) + β∑iΔ;yiτi, subject to: τi ≤ Δyi, τi.1 = 0, i = 1 ...l. The dual is solved  using the decomposition method.Let τi,jτim.τjm(Kij + β) - β∑i,mδimτimMinimize: 1/2∑i ≤ Δyi, τi.1 = 0, i = 1 ...lConstraint: τThe Lagrangian of the dual is:i,j,mτim.τjm(Kij + β) - φ∑i,mδimτim - ∑i,muim(δim - τim) - ∑ivi∑mτimL = 1/2∑im ≥ 0Subject to uim:We take the gradient of the Lagrangian with respect to ττm[L] = ∑iτjm(Kij + β) - βδim + uim - vi = 0▼iτjm(Kij + β) - βδim + uim - vi = 0 and fim = ∑iτjm(Kij + β) - βδim, then f(τ) = fim + uim - vi = 0. By KKT conditions we get two more equations:Introducing f(τ) = ∑im(δim - τim) = 0 and uim ≥ 0uim = τim, then uim ≥ 0, hence fim ≤ vi. Case II: if τim < δim, then uim = 0, hence fim = vi. Note: There is atleast one 'm' for all i such that τim < δim is satisfied.Case I: if δTherefore combining Case I & II, we get:m{fim} ≤ vi ≤ minm: τim < δim{fim}maxm{fim} ≤ minm: τim < δim{fim}Or maxm{fim} - minm: τim < δim{fim} ≤ εOr maxim < δim implies that αim > 0. Since ∑mαim = 1, for any i each αim is treated as the probability that the data point belongs to class m. Hence we define KKT violators as:Note: τm{fim} - minm: τim < δim{fim} > ε for all i.maxUsing the method in  to solvei,jτi.τj(Kij + β) + β∑iΔyiτiQ(τ) = -1/2∑i ≤ Δyi, τi.1 = 0, i = 1 ...lSubject to: τExpanding in terms of a single 'τ' vector:p(τp) = -1/2Ap(τp. τp) - Bp.τp + CpQWhere:p = Kpp + βAp = -βΔyp + ∑i≠pτi(Kip + β)Bp = -1/2∑i,j≠pτi.τj(Kij + β) + β∑i≠pτiΔyiCp', we have to minimize:Therefore ignoring the constant term 'Cp(τp) = 1/2Ap(τp. τp) + Bp.τpQp ≤ Δyp and τp.1 = 0Subject to: τThe above equation can also be written as:p(τp) = 1/2Ap(τp + Bp/Ap).(τp + Bp/Ap) - Bp.Bp/2ApQp + Bp/Ap) & D = (Δyp + Bp/Ap) in the above equation. Hence, after ignoring the constant term Bp.Bp/2Ap and the multiplicative factor 'Ap' we have to minimize:Substitute v = (τ2Q(v) = 1/2v.v = 1/2||v||Subject to: v ≤ D and v.1 = D.1 - 1The Lagrangian is given by:2 - ∑mρm(Dm - vm) - σ[∑m(vm - Dm) + 1]L(v) = 1/2||v||m ≤ 0Subject to: ρm = vm + ρm - σ = 0. By KKT conditions we have: ρm(Dm - vm) = 0 & ρm ≥ 0, also vm ≤ Dm. Hence by combining the above in-equalities, we have: vm = Min{Dm, σ}, or ∑mvm = ∑mMin{Dm, σ} = ∑mDm - 1. The above equation uniquely defines the 'σ' that satisfies the above equation AND that 'σ' is the optimal solution of the quadratic optimization problem.  that satisfies the equation: |(σl - σl+1)/σl| ≤ tolerance. The initial value for 'σ' is set to σ1 = 1/K[∑mDm - 1].Update rule for 'τ': Once we have 'σ', τnewm = vm - Bpm/(Kpp + β), or:newm = vm - fpm/(Kpp + β) + τoldmτi} be a data set of 'N' points in Rd. Using a non-linear transformation φ, we transform 'x' to some high-dimensional space called Kernel space and look for the smallest enclosing sphere of radius 'R'. Hence we have: ||φ(xj) - a ||2 ≤ R2 for all j = 1,...,N; where 'a' is the center of the sphere. Soft constraints are incorporated by adding slack variables 'ζj':Let {xj) - a ||2 ≤ R2 + ζj for all j = 1,...,N||φ(xj ≥ 0Subject to: ζWe introduce the Lagrangian as:2 - ∑jβj(R2 + ζj - ||φ(xj) - a ||2) - ∑jζjμj + C∑jζjL = Rj ≥ 0, μj ≥ 0,Subject to: βjζj is a penalty term. Setting to zero the derivative of 'L' w.r.t. R, a and ζ we have: ∑jβj = 1; a = ∑jβjφ(xj); and βj = C - μj.where C is the cost for outliers and hence C∑Substituting the above equations into the Lagrangian, we have the dual formalism as:i,jβiβjKij where 0 ≤ βi ≤ C; Kij = exp(-||xi - xj||2/2σ2)W = 1 - ∑iβi = 1Subject to: ∑jμj = 0 and βj(R2 + ζj - ||φ(xj) - a ||2) = 0.By KKT conditions we have: ζj' if ζj > 0, then βj = C and hence it lies outside of the sphere i.e. R2 < ||φ(xj) - a ||2. This point becomes a bounded support vector or BSV. Similarly if ζj = 0, and 0 < βj < C, then it lies on the surface of the sphere i.e. R2 = ||φ(xj) - a ||2. This point becomes a support vector or SV. If ζj = 0, and βj = 0, then R2 > ||φ(xj) - a ||2 and hence this point is enclosed with-in the sphere.In the kernel space of a data point 'xStaphylococcus aureus, which has a molecule-sized channel opening for partial capture, if not translocation, of biomolecules drawn in by electrophoretic forces (such as DNA)  for all mWhere fCase I:im = 0 for m S.T fm = fmmaxIf αiyi > 0 and hence ζi = 0Implies αyi - fmmax - 1 ≥ 0Hence fCase II:im > 0 for m S.T fm = fmmax and αiyi > αimIf 1 > αi = 0Implies ζyi - fmmax - 1 = 0Hence fCase III:im > 0 for m S.T fm = fmmax and αiyi ≤ αimIf 1 ≥ αi > 0Implies ζyi - fmmax - 1 + ζi = 0Hence fyi - fmmax - 1 < 0Or fThe SVM Decision Tree shown in Fig. simultaneous curves . The problem is that tuning and optimizing a single decision tree is already a large task, even for five species } = fyi, and Let fm = maxm{fm(xi)} for all m ≠ yi, then we define the margin as: (fyi - fm), hence data point xi is dropped if (fyi - fm) = Confidence Parameter. *C was required to achieve 100% accuracy.) The results are shown in Table all species).Suppose we define the criteria for dropping weak data as the margin: For any data point xst 0.0000*C was reThe SVM-Internal approach to clustering was originally defined by . Data poThe minimal enclosing sphere, when mapped back into the data space, can separate into several components; each enclosing a separate cluster of points. The width of the kernel (say Gaussian) controls the scale at which the data is probed while the soft margin constant helps to handle outliers and over-lapping clusters. The structure of a dataset is explored by varying these two parameters, maintaining a minimal number of support vectors to assure smooth cluster boundaries.We have used the algorithm defined in  to identIn each comparison we sub-divide the line segment connecting the two data points into 20 parts; hence we obtain 19 different points on this line segment. The two data points belong to the same cluster only if all the 19 points lie inside the cluster. Given the cost of evaluating utmost 19 points for every comparison, the need to eliminate comparisons that do not have an impact on the cluster connectivity becomes even more important. Finally we have used Depth First Search (DFS) algorithm for the cluster harvest. Results are shown in Tables a priori knowledge of each vector's class. The algorithm works by first running a Binary SVM against a data set, with each vector in the set randomly labeled, until the SVM converges , to a multiclass  SVM, an FSA-based nanopore spike detector, and an HMM-based channel current feature extraction. New, web-accessible, channel current analysis tools, have also been developed for kinetic feature extraction (via channel current sub-level lifetimes), and clustering. The website is designed using HTML and CGI scripts that are executed to process the data sent when a form filled in by the user is received at the web server – results are then e-mailed to the address indicated by the user. The interface to this and all other software described is available via the group Home Page: see Fig. . The SVMAdaptive feature extraction and discrimination, in the context of SVMs, can be accomplished by small batch reprocessing using the learned support vectors together with the new information to be learned. The benefit is that the easily deployed properties of SVMs can be retained while at the same time co-opting some of the on-line adaptive characteristics familiar from on-line learning with neural nets. This is also compatible with the chunking processing that is already implemented. A situation where such adaptation might prove necessary in nanopore signal analysis is if the instrumentation was found to have measurable, but steady, drift (at a new level of sensitivity for example). At the forefront of online adaptation, where the discrimination and feature extraction optimizations are inextricably mixed, further progress may derive benefit from the Information-Geometrical methods of S. Amari -25.In a parallel datarun to that indicated in Fig. If SVM performance on the full HMM parameter set  offers equivalent performance after rejecting weak data, then the possibility for significant improvement with selection on good parameters. An AdaBoost method is being used to select HMM parameters by representing each feature vector component as an independent Naïve Bayes classifier (trained on the data given), that then comprise the pool of experts in the AdaBoost algorithm -34. The • External Multi-class SVM gave best results with Sentropic Kernel while Internal Multi-class SVM gave best results with AbsDiff kernel.• Internal Multi-class approach overcomes the need to search for the best performing tree out of many possibilities. This is a huge advantage especially when the number of classes is large.• Using a margin to define the drop zone for the internal multi-class approach produced far better results i.e. fewer data were dropped to achieve 100% accuracy.• Additional benefit of using the margin is that the drop zone tuning to achieve 100% accuracy becomes trivial.Clustering Methods were also examined. The results show that our SVM-based clustering implementations can separate data into proper clusters without any prior knowledge of the elements' classification. this can be a powerful resource for insight into data linkages (topology).• External and Internal SVM The Nanopore Detector is operated such that a stream of 100 ms samplings are obtained (throughput was approximately one sampling per 300 ms in ).2 transition probabilities, and an a posteriori information from the Viterbi path solution which is, essentially, a de-noised histogram of the bloackade sub-level occupation probabilities . This ii,jβiβjKij where 0 ≤ βi ≤ C; Kij = exp(-||xi - xj||2/2σ2), also ∑iβi = 1. For any data point 'xk', the distance of its image in kernel space from the center of the sphere is given by: R2(xk) = 1 - 2∑iβiKik + ∑i,jβiβjKij. The radius of the sphere is R = {R(xk) | xk is a Support Vectors}, hence data points which are Support Vectors lie on cluster boundaries. Outliers are points that lie outside of the sphere and therefore they do not belong to any cluster i.e. they are Bounded Support Vectors. All other points are enclosed by the sphere and therefore they lie inside their respective cluster. KKT Violators are given as: (i) If 0 < βi < C and R(xi) ≠ R; (ii) If βi = 0 and R(xi) > R; and (iii) If βi = C and R(xi) < R.The dual formalism is: 1 - ∑β {∑i,jβiβjKij - 1}. In the SMO decomposition, in each iteration we select βi & βj and change them such that f(β) reduces. All other β's are kept constant for that iteration. Let us denote β1 & β2 as being modified in the current iteration. Also β1 + β2 = (1 - ∑i = 3βi) = s, a constant. Let ∑i = 3βiKik = Ck, then we obtain the SMO form: f = β21 + β22 + ∑i,j = 3βiβjKij + 2β1β2K12 + 2β1C1 + 2β2C2. Eliminating β1: f(β2) = (s - β2)2 + β22 + ∑i,j = 3βiβjKij + 2(s - β2)β2K12 + 2(s - β2)C1 + 2β2C2. To minimize f(β2), we take the first derivative w.r.t. β2 and equate it to zero, thus f'(β2) = 0 = 2β2(1 - K12) - s(1 - K12) - (C1 - C2), and we get the update rule: β2new = [(C1 - C2)/2(1 - K12)] + s/2. We also have an expression for "C1 - C2" from: R(x12) - R(x22) = 2(β2 - β1)(1 - K12) - 2(C1 - C2), thus C1 - C2 = [R(x22) - R(x12)]/2 + (β2 - β1)(1 - K12), substituting, we have:The Wolfe dual is: f(β) = Min 1new = β1old - [R(x22) - R(x12)]/[4(1 - K12)]βCompute 'C': if percent outliers = n and number data points = N, then: C = 100/(N*n)Initialize β: Initialize m = int(1/C) - 1 number of randomly chosen indices to 'C'iβi = 1Initialize two different randomly chosen indices to values less than 'C' such that ∑2(xi) for all 'i' based on the current value of β.Compute Ri < C; Set II if βi = 0; and Set III if βi = C.Divide data into three sets: Set I if 0 < β2_low = Max{ R2(xi) | 0 ≤ βi < C} and R2_up = Min{ R2(xi) | 0 < βi ≤ C}.Compute RIn every iteration execute the following two paths alternatively until there are no KKT violators:1. Loop through all examples Keep count of number of KKT Violators.2_low - R2_up < 2*tol.2. Loop through examples belonging only to Set I  until RExamine Example Subroutinea. Check for KKT Violation. An example is a KKT violator if:2(xi) > R2_up; choose R2_up for joint optimizationSet II and R2(xi) < R2_low; choose R2_low for joint optimizationSet III and R2(xi) > R2_up + 2*tol OR R2(xi) < R2_low - 2*tol; choose R2_low or R2_up for joint optimization depending on which gives a worse KKT violatorSet I and Rb. Call the Joint Optimization subroutineJoint Optimization Subroutine12) where K12 is the kernel evaluation of the pair chosen in Examine Examplea. Compute η = 4(1 - K2(x2) - R2(x1)]/ηb. Compute D = [R2), β1} = L1c. Compute Min{(C - β1), β2} = L2d. Compute Min{ for all 'i' based on the changes in β1 & β2h. Re-compute R2_low & R2_up based on elements in Set I, R2(x1) & R2(x2)i. Re-compute RThe SVM-clustering software is written in Perl. It runs data on a separate Binary SVM also written in Perl. This SVM uses a C file for kernel calculations. The data run on the SVM is created by running raw data through a tFSA/HMM(written in C), which creates a data set that contains 151 feature vectors for each element. The following is a simple step-by-step description of the basic algorithm used for SVM-clustering on this data:1. Start with a set of data vectors .4. After initial convergence is obtained for the randomly labeled data set, relabel the misclassified data vectors, which have confidence factor values greater than some threshold.5. Rerun the SVM on the newly relabeled data set.6. Continue relabeling and rerunning SVM until no vectors in the data set are misclassified (or there is no improvement).The paper was written by SWH and AY. The external clustering work was contributed by CM. The channel current feature vector extraction used to create the data sets was performed by ML.


In particular, we have chosen log y1 = log y0 + (βL − βH)log d*, holding the continuity of y(d).”


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Jr, AII‐P‐39Fang, Q., AII‐6‐11Fauzi, M., AII‐P‐7Fernando, PHP., AI‐P‐14Fouqueray, P., AI‐6‐6, AII‐6‐13Francisco, C., AI‐6‐15Fujikawa, J., AII‐P‐26Fujimoto, H., AI‐5‐1Fujita, N., AI‐5‐1Fujiwara, T., AI‐P‐39GGao, P., AII‐P‐20Gao, X., AI‐6‐4, AI‐P‐10Ge, X., AI‐P‐44, AI‐P‐5Geng, Y., AI‐6‐2Goh, S‐Y., AI‐P‐16Gonzales, E., AI‐6‐15Guo, J., AI‐P‐27Gushiken, M., AII‐P‐32HHa, IG., AII‐P‐31Ha, KH., AII‐5‐5Hallakou‐Bozec, S., AII‐6‐13Hamada, Y., AII‐P‐30Hamamatsu, K., AI‐5‐1Hamamoto, Y., AI‐6‐3, AI‐6‐5, AI‐P‐29, AII‐P‐23, AII‐P‐24, AII‐P‐26, AII‐P‐35Hamasaki, A., AI‐P‐21, AII‐P‐26Han, C‐J., AII‐5‐3Han, J., AII‐6‐11Han, K., AII‐5‐1Han, SW., AII‐P‐31Han, X., AI‐P‐10, AI‐P‐2Hara, K., AII‐6‐8Hara, Y., AII‐P‐22Haraguchi, T., AI‐6‐5Harashima, S‐i., AII‐P‐13, AII‐P‐27, AII‐P‐9Hatakeyama, S., AII‐P‐1He, R., AI‐P‐36He, X., AI‐P‐20He, Z., AII‐P‐12Herrera L, P., AI‐5‐5Higashiyama, H., AII‐P‐35Himathongkam, T., AII‐P‐28Hirayama, K., AII‐P‐38Hodai, Y., AI‐P‐31Hong, E., AI‐6‐1Hong, J., AI‐P‐9Hong, W‐J., 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JH., AI‐P‐7, AI‐6‐14Kim, JT., AII‐6‐9, AII‐P‐18Kim, J‐W., AII‐P‐2, AII‐P‐3Kim, K., AI‐P‐32, AI‐P‐43Kim, KJ., AI‐6‐12Kim, KS., AII‐P‐18Kim, MJ., AII‐P‐2Kim, S., AII‐P‐29Kim, SH., AII‐P‐31Kim, S‐W., AI‐P‐11Kim, Y., AI‐P‐32, AI‐P‐43Kimura, H., AI‐5‐1Kinoshita, T., AI‐P‐31Kira, D., AII‐P‐38Kirkley G., A., AII‐P‐4Kishida, S., AII‐P‐37Kishimoto, M., AI‐P‐18Kitamura, Y., AII‐P‐40Kitano, N., AII‐P‐24Kitatani, N., AI‐6‐3Ko, R., AII‐P‐27Ko, S‐Y., AII‐5‐3Kobayashi, A., AI‐P‐13Kobayashi, I., AII‐P‐46Kobayashi, M., AII‐6‐8Kobayashi, N., AII‐6‐8Koduka, C., AI‐6‐10Koh, G., AI‐6‐7Koike, M., AI‐P‐33Kondo, Y., AI‐P‐33Koshian, F., AI‐6‐3Koshima, I., AII‐6‐8Kouchi, Y., AII‐P‐38Krittiyawong, S., AII‐P‐28Ku, BJ., AII‐P‐18Kubota, A., AII‐P‐24Kubota, M., AI‐P‐39Kubota, N., AII‐6‐8Kubota, S., AI‐P‐29Kumagai, J., AI‐P‐13Kumar, MS., AI‐6‐11Kumar, PS., AI‐6‐11Kuo, C‐H., AI‐5‐3Kuramoto, H., AII‐P‐38Kurimoto, J., AI‐P‐31Kurita, K., AI‐P‐13Kuroe, A., AII‐P‐24Kurose, T., AI‐6‐3, AI‐6‐5, AI‐P‐29, AII‐P‐23, AII‐P‐24, 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AA., AII‐5‐6Maeda, Y., AI‐P‐17Maejima, Y., AI‐5‐5Makabe, N., AI‐6‐3Maningat, P., AI‐6‐15Maningat, PD., AII‐P‐39Mano, F., AII‐P‐34Marushita, R., AI‐P‐33Masuzaki, H., AI‐6‐10Matsuhisa, M., AII‐P‐37Matsumoto, S., AII‐6‐12, AII‐P‐11McMorran, D., AI‐P‐37Meng, S., AI‐6‐2Meng, X., AII‐P‐45Mim, JM., AII‐P‐18Mimura, M., AI‐P‐17Min, K., AII‐5‐1Mina, A., AI‐6‐2Minami, K., AI‐5‐5Miyamoto, L., AII‐6‐15Miyoshi, H., AII‐P‐21Mizokami, T., AII‐P‐1Monma, N., AI‐P‐33Mopuri, R., AII‐P‐17Mori, M., AII‐6‐12, AII‐P‐11Moriwaki, R., AI‐P‐17Munkhtur, B., AII‐P‐33Murakami, T., AI‐5‐1, AII‐P‐27Murayama, Y., AI‐6‐10NNagai, Y., AI‐P‐18Nagashima, K., AII‐P‐27, AII‐P‐7Nagata, T., AII‐P‐37Nakajima, Y., AII‐P‐46, AI‐P‐23, AII‐6‐12, AII‐P‐11Nakamura, A., AII‐P‐21, AII‐P‐38Nakandakari, M., AI‐6‐10Nakasatien, S., AII‐P‐28Nam, J., AI‐P‐32, AI‐P‐43Nam, MS., AII‐P‐31Nata, K., AII‐P‐1Nemoto, K., AI‐P‐17Nishikawa, T., AI‐P‐17Niu, X., AI‐5‐2Noh, J., AI‐P‐9Noh, Y., AI‐6‐1Nonogaki, K., AII‐6‐14, AI‐P‐35Nukada, H., AI‐P‐37Nukuda, M., AI‐P‐33OOduori S, O., AI‐5‐5Ogasawara, S., AI‐P‐37Ogata, M., AII‐P‐40Ogura, M., AII‐P‐27, AII‐P‐7Oh, MS., AII‐6‐9Oh, SY., AI‐6‐14Ohno, K., AII‐P‐22Oida, K., AII‐P‐24Okada, S., AII‐P‐42, AI‐P‐15, AI‐P‐23, AII‐6‐10, AII‐P‐11Okamoto, S., AI‐P‐29Okamura, E., AII‐P‐26, AI‐P‐21Okamura, K., AI‐P‐29Okamura, T., AII‐6‐12Okazaki, Y., AII‐6‐8Okura, M., AII‐P‐34Omlor, Y., AII‐P‐32Omura, M., AI‐P‐33Osaki, A., AI‐P‐23, AII‐6‐10Ota, K., AI‐P‐33Ozawa, A., AI‐P‐15, AII‐6‐12, AII‐P‐11PPacaira, JPP., AII‐P‐41Pak Kim, Y., AII‐6‐9Pan, J., AII‐P‐45Pan, X., AI‐P‐34, AI‐P‐8, AII‐P‐44Park, J., AI‐P‐32, AI‐P‐43, AII‐P‐25Park, K., AI‐P‐32, AI‐P‐43Park, S., AII‐P‐25Park, TS., AI‐P‐19Park, WH., AII‐6‐9Park, Y., AII‐P‐31Peng, H‐Y., AII‐P‐36Puchi, B., AI‐6‐2RReganit, PF., AI‐6‐15Ren, Q., AI‐6‐2Ren, W., AI‐P‐27Rhee, SY., AII‐P‐31SSagawa, M., AI‐P‐33Saito, K., AII‐P‐40Saito, S., AI‐P‐39Saito, T., AI‐P‐23, AII‐6‐10Saji, H., AI‐5‐1Sakai, S., AII‐P‐38Sakamaki, K., AII‐P‐46Sakamoto, E., AII‐P‐37Sakiyama, Y., AII‐P‐38Sakuramachi, Y., AII‐P‐23, AI‐P‐29Salvana, EM., AI‐6‐15Sanjeewani, AN., AI‐P‐14Sarakom, S., AII‐P‐28Sargis M, R., AII‐P‐4Sarmiento, AJ., AII‐P‐43Sasako, T., AII‐6‐8Sato, H., AI‐P‐33Sato, T., AI‐P‐31, AI‐P‐23, AII‐6‐10Satoh, T., AI‐P‐15, AII‐6‐12, AII‐P‐11, AII‐P‐46Seino, S., AI‐5‐5, AII‐P‐4Seino, Y., AI‐6‐3, AI‐6‐5, AI‐P‐29, AII‐P‐23, AII‐P‐24, AII‐P‐35Shan, AKP., AII‐P‐48Sheen, Y‐J., AI‐P‐47Shen, Y., AI‐6‐8Shen, Z., AII‐P‐16Sheu, WH‐H., AI‐P‐47Shibusawa, N., AI‐P‐15, AII‐6‐12, AII‐P‐11Shibusawa, R., AI‐P‐23, AII‐6‐10Shimabukuro, M., AI‐6‐10Shimoda, Y., AI‐P‐23, AII‐6‐10Shimomura, K., AI‐5‐5Shimomura, Y., AII‐P‐46Shong, M., AII‐P‐18Shu, K‐H., AI‐P‐42Singh, SY., AI‐6‐11Sirakura, T., AI‐6‐10Sison, O., AI‐6‐15Sison‐Pena, CM., AI‐P‐38Son, SM., AI‐P‐24Song, J., AI‐6‐2Song, Y., AII‐P‐19Sonomtseren, S., AI‐P‐22, AI‐P‐40Su, H., AI‐P‐20Sugawara, K., AII‐P‐4Suh, B‐K., AII‐5‐2Sumikawa, M., AI‐P‐39, AII‐P‐37Sumikawa, Y., AI‐P‐39Sumita, K., AI‐6‐5Sun, Y., AI‐5‐2Sunagawa, S., AI‐6‐10Suwanai, 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R., AII‐P‐7, AII‐P‐9WWada, E., AI‐P‐31Wada, Y., AII‐P‐26Wakutsu, M., AI‐P‐33Wang, C., AI‐5‐2, AII‐P‐10Wang, C‐S., AII‐P‐36Wang, J., AII‐P‐14Wang, S., AII‐P‐10, AI‐P‐1Wang, T., AI‐P‐12, AI‐P‐36Wang, W., AII‐P‐49Wang, X., AI‐P‐25Wang, Y., AI‐5‐2, AII‐P‐8, AII‐P‐9, AII‐P‐13, AI‐P‐20Wang, Y‐J., AII‐P‐36Wanothayaroj, E., AII‐P‐28Watanabe, K., AI‐6‐3, AI‐P‐29, AII‐P‐24Watanabe, T., AI‐P‐15, AII‐6‐12Wee, Z., AI‐P‐16Wei, P., AII‐P‐14Wei, PJ., AII‐P‐12Wickramarathne, DBM., AI‐P‐14Wijang, PP., AII‐P‐15Woo, JT., AII‐P‐31Wu, C‐Y., AI‐P‐42Wu, J., AI‐6‐13Wu, M‐J., AI‐P‐42XXi, G., AI‐P‐27Xiao, Y., AI‐6‐8, AII‐P‐44Xin, X., AI‐P‐16Xiong, Q., AI‐6‐8, AI‐P‐34, AI‐P‐8, AII‐P‐44Xu, B., AI‐P‐4Xu, G., AII‐P‐16Xu, W., AII‐6‐15Xu, Y., AI‐6‐8, AI‐P‐34, AI‐P‐8, AII‐P‐44YYabe, D., AI‐6‐5, AII‐P‐23, AII‐P‐27, AII‐P‐7Yagihashi, S., AI‐P‐37Yamada, E., AII‐6‐10, AI‐P‐23Yamada, M., AI‐P‐15, AI‐P‐23, AII‐6‐10, AII‐6‐12, AII‐P‐11, AII‐P‐42, AII‐P‐46Yamaguchi, K., AI‐P‐31Yamamoto, J., AI‐P‐33Yamamoto, K., 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Moreover, we describe two pathways leading to the precipitation of the α-phase mediated by diffusion-based orthorhombic structures, α″lean and α″iso. Via coupling the lattice parameters to composition both phases evolve into α through rejection of Nb. These findings have the potential to promote new microstructural design approaches for Ti–Nb alloys and β-stabilized Ti-alloys in general.Ti-alloys represent the principal structural materials in both aerospace development and metallic biomaterials. Key to optimizing their mechanical and functional behaviour is in-depth know-how of their phases and the complex interplay of diffusive vs. displacive phase transformations to permit the tailoring of intricate microstructures across a wide spectrum of configurations. Here, we report on structural changes and phase transformations of Ti–Nb alloys during heating by in situ synchrotron diffraction. These materials exhibit anisotropic thermal expansion yielding some of the largest linear expansion coefficients (+ 163.9×10 Complex phase transformations in β-stabilised titanium alloys can dramatically change their α and β microstructures, providing tailorability for aerospace or biomaterial applications. Here the authors show that Ti-Nb alloys exhibit giant thermal expansions and identify two new pathways that lead to α phase formation. Certain alloy compositions display low Young’s moduli (E) below 80 GPa after rapid cooling5, providing suitable starting points for the development of novel low-modulus alloys for load-bearing implant applications. Due to their high strength to density ratio β-stabilized Ti-alloys have also become increasingly used for various aerospace applications such as airframes and landing gears3.Ti-alloys are the workhorses in modern aerospace design and engineering of metallic biomaterials. In particular the class of β-stabilized Ti-alloys represents highly promising multifunctional materials with desirable structural and functional properties for various biomedical and engineering applications7. Up to the present day, these features have been stirring ever increasing interest.Due to the thermoelastic martensitic transformation of the body centred cubic (bcc) β-phase to orthorhombic martensite α″ this alloy family also demonstrates shape memory (SM) behaviour and superelasticity (SE)iso precipitates10. The transformation and precipitation pathways occurring during aging decide the morphology, size and arrangement of the precipitating products12.The mechanical and functional properties of these alloys are optimized by controlled adjustment of the microstructural parameters via complex thermomechanical processing paths. For instance, the mechanical behaviour of β and near-β Ti alloys can be significantly improved by uniform dispersions of fine α and/or ωiso was described13. Recently, a unique nanolaminate structure consisting of α″ martensite and planar complexions of a thermal ωath was reported14. Furthermore, for two commercial β-stabilized Ti–Mo–Fe–Al and Ti–Al–Mo–V–Fe–Cr alloys, Ivasishin et al. observed the formation of an intermediate orthorhombic phase exhibiting the same crystal structure as α″ martensite during early stages of α-precipitation15. More recently, employing in situ diffraction methods this phase was also detected upon heat treatment of Ti–Al–Mo–Cr–Sn–Zr, Ti–Al–Mo–V–Cr–(Zr) and Ti–V–Fe–Al alloys used for advanced structural aircraft components18. For some alloys a β-stabilizer lean structure, denoted α″lean, was observed during the decomposition of α″ martensite19. It was reported that decomposition of α″ martensite into α and β phases by aging below the austenite start temperature As involves a spinodal mechanism21 implying continuous changes in chemical composition.The great diversity and complex interplay of diffusion-driven vs. displacive phase transformations in β-stabilized Ti-alloys stimulates their further exploration. For instance, several years ago a novel coupled diffusional-displacive transformation mechanism occurring during formation of isothermal ωMs below RT. Furthermore, β needs to be stable enough against ωath formation and additional α precipitation. Otherwise precipitation of α and/or ωiso as well as martensite formation takes place upon cooling after aging, thus modifying the constitution of phases at the temperature of interest22. These difficulties can be overcome by the use of in situ diffraction methods permitting direct observation of the microstructural evolution caused by heat treatments. In situ diffraction methods, employing, e.g. synchrotron radiation, are therefore the first choice to capture phase reactions and transitions taking place over narrow temperature and time intervals directly at the critical temperature23.Detailed knowledge about transformation and precipitation processes is the foundation for the advancement of tailored thermomechanical treatment routes. However, following these processes through diffraction patterns recorded ex situ at room temperature (RT) is challenging. First, due to the variations in cell parameters caused by thermal expansion and secondly, because of transformations potentially taking place during cooling from the aging temperature to RT. Only when the phases resulting from aging neither decompose nor transform during cooling the microstructure formed at the aging temperature will be observable at RT. For the β-phase this only applies, when it contains sufficient β-stabilizers to suppress the martensite start temperature 7 this system serves as a prototype to study SM and SE in Ni-free Ti-based alloys. Its importance is underlined by the fact that most of the recently developed low-modulus as well as SM and SE β-stabilized Ti-based alloys are modifications of the Ti–Nb system23. Hence, a better understanding of the binary base alloy system will help explain, at least to some extent, the alloys derived from it.Since the discovery of SM in Ti–NbcNb alloys  adapted from previous studies24. At the heating rate employed (10 °C min−1), reversion of α″ martensite followed by substantial ωiso precipitation occurs for c ≥ 28.5  to track these transformations in situ for the same alloy formulations and heating rate as in Fig. c = 16, 21 and 28.5 and of α″ with some austenitic β-phase for c = 36, in agreement with the literature26. Figure 23.The initial microstructure consisted of orthorhombic martensite α″ for iso. In addition, Nb-depleted α″, denoted α″lean, and a thermally formed phase exhibiting the same crystal structure as α″, denoted α″iso, were observed. Using the diffractograms . The aspect ratios influence the shape of the orthorhombic α″ unit cell and determine whether it resembles more closely α′ or β. If bα″/aα″ = α″ are arranged in a hexagonal pattern identical to the atoms on (0001)α′ for hexagonal close-packed (hcp) martensite α′. On the other hand, for bα″/aα″ = cα″/aα″ = 26. Both limiting values are indicated in Fig. α″ through the fractional coordinate y in Wyckoff position 4c of space group Cmcm25.Figure ams Fig.  the lattbα″/aα″ and cα″/aα″ at 50 °C in Fig. c = 16 to c = 36, bα″/aα″ and cα″/aα″, respectively, drop from 1.686 to 1.514 and from 1.567 to 1.460, thus coming closer to the limit of 1.414 corresponding to β. Thus, Nb-lean α″ is structurally closer to hcp α′ whereas Nb-rich α″ is more similar to bcc β.The lattice parameters of α″ martensite are strongly affected by the Nb content. Likewise the shape of the orthorhombic unit cell of α″ varies with Nb content, as seen from c = 16 to 210 °C for c = 36  αL along bα″ is negative, corresponding to a contraction of the bα″ spacing upon heating. The magnitude of αL along aα″ and along bα″ grows with increasing Nb content. αL along cα″ decreases up to c = 28.5, but is largest for c = 36.For α″ martensite in the present alloys aα″ and cα″ spacings expand at a rate of 163.9×10−6 °C−1 and 24.4×10−6 °C−1, respectively, the bα″ spacing contracts by −95.1×10−6 °C−1 between 50 °C and 210 °C. These rates are comparable to or even larger in magnitude than those of materials exhibiting colossal thermal expansion (defined as |αL| ≥ 100×10−6 °C−1) such as Ag3[Co(CN)6]33. The present values for the expansion rates of the bα″ and cα″ spacings fully agree with recently published data for α″ martensite in Ti–35.4Nb30 NiGa alloys31 exhibit similarly strong anisotropic thermal expansion. Still, in most cases their expansion rates are smaller in magnitude than for Ti–36Nb and Ti–28.5Nb 6]29, the fulleride Sm2.75C6028 and the extremely loosely bound solid Xe below 75 K27  are among the largest (both positive and negative) ever reported for solid crystalline metallic systems, be they isotropic or anisotropic. Typical values for 36. In the present case of α″ martensite in Ti–Nb, the expansion and contraction along the unit cell edges partially compensate each other. Yet, the positive αL along aα″ and cα″ dominate over the negative αL along bα″ leading to an expansion rate αV of the bulk between 24.7×10−6 °C−1 and 91.0×10−6 °C−1. The bulk expansion of β-phase in Ti–36Nb turns out slightly smaller (84.3×10−6 °C−1) than that of α″ (91.0×10−6 °C−1).The variation of the unit cell volume due to anisotropic thermal expansion may be positive or negativec =  decomposition of α″ into α and β phases takes place, which is accompanied by an exothermic event  with temperature is inverted relative to the thermal expansion behaviour discussed previously. While upon heating below the temperature interval of decomposition α″ martensite becomes more similar to austenite β, it becomes more α-like during decomposition. This is reflected by the behaviour of bα″/aα″ and cα″/aα″ which first decrease towards α″ martensite is preserved in Ti–16Nb up to 600 °C and in Ti–21Nb up to 510 °C. No precipitation products were detected below these temperatures suggesting that α″ maintains its initial composition present at RT. α″ fully decomposes into α and β by further heating of Ti–16Nb to 630 °C and of Ti–21Nb to 590 °C. During decomposition, the lattice parameters of α″lean with those of α″ martensite at RT suggests that α″lean progressively rejects Nb thereby approaching equilibrium α. In this process aα, cα  and no contributions of α″ are required to fully model the observed intensities. The initial evolution of the {101̅0}α reflection below 560 °C α″ of a second orthorhombic component α″lean,II. Since for both phases the modeled profiles reached comparable agreement with the experimental data it is problematic to unambiguously assign one of these phases.A close look at the diffraction patterns for Ti–21Nb in Fig. Q −1 below 560 °C .5% Fig. . They arc = 16 and 21, α″ reverts martensitically to austenite β0 causing an endothermic event for c = 28.5 and 36  under isothermal conditions37.The refined lattice parameters and unit cell aspect ratios of α″iso with temperature is inverted relative to the structural change of α″ martensite due to thermal expansion. This strongly suggest that compositional changes take place in α″iso. For instance, for a Ti–Mo-based alloy (TIMETAL-LCB) analytical scanning transmission electron microscopy clearly evidenced that the β-stabilizer content of α″iso precipitates was progressively reduced with aging time10.The variation of the unit cell geometry of α″iso unit cell towards a more hcp-like structure  are of particular interest, since they permit designing composites with zero net thermal expansion when combined with positive thermal expansion materials.Aside from reaching giant values in case of Nb-rich alloy formulations, the thermal expansion rates of the present alloys can be adjusted across a wide range by simply modifying their composition. This represents an advantage over many ceramic materials which often show rather limited controllability of their thermal expansion30. For each of the present alloys this process takes place across the entire temperature range in which α″ martensite is present. Irrespective of composition, bα″/aα″ and cα″/aα″ continuously decrease with temperature towards 1.414 α″ take positions which further increase the similarity of α″ to β. This would manifest itself in an increase of the fractional coordinate y of Wyckoff position 4c of space group Cmcm when α″ is heated towards Af25.The polycrystalline nature of the irradiated volume  → α + βα″ → α″lean + α″lean,II (+α″rich + β) → α + βα″ → α″The in situ data clearly evidences that during martensite decomposition α″lean as well as Nb-enriched α″rich form. Nb-enriched α″rich and/or β is expected to take up the Nb expelled from α″lean. α″rich eventually evolves into β. Although α″rich and β were not detected for Ti–21Nb, the presence of at least one of these phases is necessary for mass conservation during Nb-expulsion from α″lean. In addition, in the first pathway α nucleates, whereas in the second pathway a second Nb-lean component α″lean,II nucleates and develops into α.For both pathways Nb-depleted α″c = 16 is very similar to that for c = 21. Again Nb-depleted α″lean forms from the quenched-in α″. However, no evidence for α or α″lean,II was found and the transformation sequence observed for Ti–16Nb during heating was thus:The decomposition process of α″ for When heating continues after α″ decomposition the reflections of α gradually weaken, which corresponds to the endothermic conversion of α to β while approaching the α–β transus Fig. .iso. Comparing the lattice parameters of α″iso  by arc-melting followed by cold-crucible casting22. Analysis of the Nb and O contents of the as-cast alloys by wet chemical analysis and hot gas extraction showed that the deviation from the nominal Nb content and the maximal O content was <0.1 wt.% each. The cast rods were homogenized for 24 h at 1000 °C under Ar and quenched into water (HQ treatment). Prior to quenching the parent β-phase exhibited a grain size of (300 ± 150) µm. Specimens for light microscopy were prepared by mechanical polishing followed by etching in an aqueous solution of 2 vol.% HF and 6.5 vol.% HNO3.Four binary Ti–−1 was achieved with a resistively heated modified Linkam hot stage purged with Ar. Diffraction patterns were recorded by a 2-dimensional (2D) image-plate detector (FReLoN) with 2048 × 2048 pixels centered on the transmitted beam. Example raw 2D-diffraction patterns are shown in Supplementary Fig. λ = 0.20664 Å and a beam cross-section of 50 × 50 µm2 were used. The sample-to-detector distance and zero-point shift were calibrated at RT using standard CeO2 powder. Data collection was done every °C using an exposure time of 0.2 s. Calibration, background subtraction and azimuthal integration of the 2D patterns into 1-dimensional patterns were performed with Fit2D40. Azimuthal integration was carried out over the entire 360° in order to minimize potential effects of texture and graininess. Diffraction angles are reported in terms of the wavevector transfer Q = 4π∙sin(θ)/λ, where θ denotes the semi-angle between the incident and the diffracted beam.In situ SXRD was conducted at the ID11 beamline of the European Synchrotron Radiation Facility (ESRF) in Grenoble, France. Thin rod-shaped samples (diameter ≈ 800 µm) were extracted from the HQ material by electrical discharge machining and subjected to a solution treatment at 1000 °C for 4 h in Ar followed by water quenching. In situ heating of the samples from RT to 760 °C at a rate of 10 °C minlean and α″iso). For each phase detected the lattice parameters were determined depending on temperature by structureless (Le Bail) refinements utilizing the FULLPROF programme suite41. The reflection profiles were simulated with Thompson–Cox–Hastings pseudo-Voigt functions42 and the background was modelled by linear segments, which were included in the refinements.On the patterns obtained phase analysis was carried out. For each phase the space group and occupied Wyckoff position(s) are listed in Table The datasets generated and analyzed during the current study are available from the corresponding author on request.Supplementary InformationPeer Review File


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Scientific Reports6: Article number: 3774010.1038/srep37740; published online 11232016; updated on 05022017The original version of this Article contained errors resulting from the incorrect calculation of formation energy in 2D + 2D + 2D → 21H + 4He + 212 J/m3”.“Now reads:2D + 2D + 2D → 21H + 4He + e- + 15 J/m3”.“In addition, there was an error in the ‘Introduction section’, where“Increase in paraeomagnetic magnitude between 2.7–2.1 billion years”Now reads:“Increase in palaeomagnetic magnitude between 2.7–2.1 billion years”There were also errors in There were also errors in Under the sub-heading “Existence of a D-rich inner core”,21 kg (=100 Mkm3 × 2 Mg/m3 × 4/20 × 1/0.006), the volume of Fe-D crystals up through the present is 4.99 × 1017 m3 (6.67 × 1021/13.36 [Mg/m3]), resulting in an unreacted (Fe-D) crystal volume of 4.37 × 1017 m3 (=4.99 × 1017 − 6.23 × 1016)”.“On the other hand, because the incorporated gross weight of D from the creation of Earth’s core to present time is estimated as 6.67 × 10Now reads:21 kg (=100 Mkm3 × 2 Mg/m3 × 4/20 × 1/0.006), the volume of Fe-D crystals up through the present is 4.99 × 1017 m3 (6.67 × 1021/13.36 [Mg/m3]), resulting in an unreacted (Fe-D) crystal volume of 4.39 × 1017 m3 (=4.99 × 1017 − 6.00 × 1016)”.“On the other hand, because the incorporated gross weight of D from the creation of Earth’s core to present time is estimated as 6.67 × 10Finally, there was an error in the ‘Conclusions’ section where:Now reads:These errors have now been corrected in the HTML and PDF versions of this Article.


The Middle East respiratory syndrome (MERS) coronavirus, a newly identified pathogen, causes severe pneumonia in humans. MERS is caused by a coronavirus known as MERS-CoV, which attacks the respiratory system. The recently defined receptor for MERS-CoV, dipeptidyl peptidase 4 (DPP4), is generally expressed in endothelial and epithelial cells and has been shown to be present on cultured human nonciliated bronchiolar epithelium cells. In this paper, a class of novel four-dimensional dynamic model describing the infection of MERS-CoV is given, and then global stability of the equilibria of the model is discussed. Our results show that the spread of MERS-CoV can also be controlled by decreasing the expression rate of DPP4. The Middle East respiratory syndrome (MERS) coronavirus, a newly identified pathogen, causes severe pneumonia in humans, with a mortality of nearly 44%. Human-to-human spread has been demonstrated, raising the possibility that the infection could become pandemic . A colorT(t), I(t), and v(t) denote the concentration of uninfected cells, infected cells, and free viruses at time t, respectively. The constant λ > 0 is the rate at which new uninfected cells are generated (from a pool of precursor cells). The constants d > 0 and β ≥ 0 are the death rate of uninfected cells and the rate constant characterizing infection of the cells, respectively. The constant d1 > 0 is the death rate of the infected cells due to either viruses or immune responses. The infected cells produce new viruses at the rate d1N during their life, on average having the length 1/d1, where N > 0 is some integer number. The constant c > 0 is the rate at which the viruses are cleared, and the average lifetime of a free virus is 1/c.It is well-known that dynamic models are still playing important roles in describing the interactions among uninfected cells, free viruses, and immune responses . A thT˙=λ−βvtTD(t) represents the concentration of DPP4 on the surface of uninfected cells, which can be recognized by surface spike (S) protein of MERS-CoV )v(t)T(t). It is assumed that DPP4 is produced from the surface of uninfected cells at the constant rate λ1 > 0. DPP4 is destroyed, when free viruses try to infect uninfected cells, at the rate β1(βD(t))v(t)T(t), and is hydrolyzed at the rate γD(t). Here, β1 ≥ 0 and γ > 0 are constants. It is assumed that there is no undestroyed DPP4 on the surface of infected cells. All other parameters in model . InfectT˙=λ−βDtvT(0) ≥ 0, I(0) ≥ 0, v(0) ≥ 0, and D(0) ≥ 0. It is not difficult to show that the solution (T(t), I(t), v(t), D(t)) with the initial condition is existent, unique, bounded, and nonnegative for all t ≥ 0  > 0 and D(t) > 0 for all t > 0). If T(0) > 0, I(0) > 0, v(0) > 0, and D(0) > 0, it is easily proven that the corresponding solution (T(t), I(t), v(t), D(t)) is positive for all t ≥ 0.The initial condition of model  is givena < 1, μ = min{d, (1 − a)d1, c}.Furthermore, it can be easily shown that the set to model , where 0The purpose of the paper is to study local and global stability of the equilibria of model  by usingR0 = Nβλλ1/cdγ. Model  = . If R0 > 1, model , where, for β1 = 0, v = v∗ = dγ(R0 − 1)/βλ1, for β1 > 0, v = v∗ > 0 is the positive root of the equation ββ1c2v2 − Ncβ(λ1 + λβ1)v + Ncdγ(R0 − 1) = 0, and The basic reproductive ratio of the virus for model  is R0 = γ. Model  always h1, model  also hasFirst, we have the following result.Ω1 = {∣ ∈ Ω, T > 0, D > 0}, the infection-free equilibrium E0 =  is globally asymptotically stable when R0 < 1 and globally attractive when R0 = 1.With respect to the set T, I, v, D), Jacobian matrix of model (E0 is f(ρ) = (ρ + d)(ρ + γ)[ρ2 + (c + d1)ρ + cd1(1 − R0)] = 0. Clearly, if R0 < 1, all roots of f(ρ) = 0 have negative real parts. Hence, E0 is local asymptotic stability by Routh-Hurwitz criterion. If R0 = 1, f(ρ) = 0 has the zero root ρ = 0 and three negative roots. Hence, E0 is linearly stable.At any equilibrium (of model  is(5)J=−U is continuous on Ω1 and positive definite with respect to E0 and satisfies condition (ii) of Definition 1.1 in [Ω = Ω∖Ω1. Calculating the derivative of U along the solutions of model  ∈ Ω, U < +∞}. Let M be the largest subset in Q which is invariant with respect to the set model ∣ ∈ Ω, T = T0, D = D0}. From the invariance of M and model ∣ ∈ Ω, T > 0, I > 0, v > 0, D > 0}, the infected equilibrium E∗ is locally asymptotically stable when R0 > 1. In addition, if (2dγμa)2 ≥ β1β2λ1λ3N2, where 0 < a < 1, μ = min{d, (1 − a)d1, c}, the infected equilibrium E∗ is globally asymptotically stable.With respect to the set E∗ is ai > 0  . Furthermore, by using Matlab program, it can been shown that Δ3 = a1a2a3 − a12 − a32a0 has 200 items in which all items are positive. Hence, E∗ is local asymptotic stability by Routh-Hurwitz criterion.The characteristic equation at of model  at E∗ isW is continuous on Ω2 and positive definite with respect to E∗ and satisfies condition (ii) of Definition 1.1 in [Ω = Ω∖Ω2. Calculating the derivative of W along the solutions of model 2/8γ ≥ 0 and γ/2D − T(ββ1v)2/8β1d ≥ 0 are equivalent to the inequality 4dγ ≥ β1β2TDv2. Since T(t) ≤ T0, D ≤ D0, and v(t) ≤ λN/μa for all t ≥ 0, we have that the inequality 4dγ ≥ β1β2TDv2 holds, if the condition (2dγμa)2 ≥ β1β2λ1λ3N2 in dW/dt ≤ 0 on Ω2.On the other hand, notice the inequality in : (12)−xzQ = {dW/dt = 0∣ ∈ Ω, W < +∞}. Let M be the largest subset in Q which is invariant with respect to the set of model ∣ ∈ Ω, T = T∗, D = D∗}. From the invariance of M and model  and R0 = 0.90909 < 1. E0 is asymptotically stable.Let us first give some numerical simulations on the orbits of model . Take thγ = 0.05, and all the other parameters are the same as above. We can also compute the values of the infection-free equilibrium E0, the infected equilibrium E∗, and the basic reproductive ratio, E0 = , E∗ = , and R0 = 2 > 1. E∗ is asymptotically stable. We would like to point out here that, based on the numerical simulations, the condition (2dγμa)2 ≥ β1β2λ1λ3N2 may be further weakened or even removed.Let us take io, E0 = 0,0, 0,20R0 = Nβλλ1/cdγ, let us give some simple discussions on the interactions between the protein DPP4 and the virus infection. Usually, in the absence of any drug treatment, all the parameters in model (R0 can be regarded as relatively fixed constants. If some drug treatment measures are taken, the effectiveness of the treatment can be reflected in the regulation of the parameter γ. For example, by increasing the value of γ, the value of the basic reproductive ratio of R0 can be changed from greater than 1 to less than 1. In the numerical simulations in this section, γ = 0.05 and R0 = 2 > 1. If increasing γ from γ = 0.05 to γ = 0.11, R0 = 0.9090 < 1.Finally, by using the basic reproductive ratio in model  and the