A New Trigonometric Spline Approach to Numerical Solution of Generalized Nonlinear Klien-Gordon Equation

The generalized nonlinear Klien-Gordon equation plays an important role in quantum mechanics. In this paper, a new three-time level implicit approach based on cubic trigonometric B-spline is presented for the approximate solution of this equation with Dirichlet boundary conditions. The usual finite difference approach is used to discretize the time derivative while cubic trigonometric B-spline is applied as an interpolating function in the space dimension. Several examples are discussed to exhibit the feasibility and capability of the approach. The absolute errors and error norms are also computed at different times to assess the performance of the proposed approach and the results were found to be in good agreement with known solutions and with existing schemes in literature.     

A Generalized Simplest Equation Method and Its Application to the Boussinesq-Burgers Equation

In this paper, a generalized simplest equation method is proposed to seek exact solutions of nonlinear evolution equations (NLEEs). In the method, we chose a solution expression with a variable coefficient and a variable coefficient ordinary differential auxiliary equation. This method can yield a Bäcklund transformation between NLEEs and a related constraint equation. By dealing with the constraint equation, we can derive infinite number of exact solutions for NLEEs. These solutions include the traveling wave solutions, non-traveling wave solutions, multi-soliton solutions, rational solutions, and other types of solutions. As applications, we obtained wide classes of exact solutions for the Boussinesq-Burgers equation by using the generalized simplest equation method.     

Complete Numerical Solution of the Diffusion Equation of Random Genetic Drift

A numerical method is presented to solve the diffusion equation for the random genetic drift that occurs at a single unlinked locus with two alleles. The method was designed to conserve probability, and the resulting numerical solution represents a probability distribution whose total probability is unity. We describe solutions of the diffusion equation whose total probability is unity as complete. Thus the numerical method introduced in this work produces complete solutions, and such solutions have the property that whenever fixation and loss can occur, they are automatically included within the solution. This feature demonstrates that the diffusion approximation can describe not only internal allele frequencies, but also the boundary frequencies zero and one. The numerical approach presented here constitutes a single inclusive framework from which to perform calculations for random genetic drift. It has a straightforward implementation, allowing it to be applied to a wide variety of problems, including those with time-dependent parameters, such as changing population sizes. As tests and illustrations of the numerical method, it is used to determine: (i) the probability density and time-dependent probability of fixation for a neutral locus in a population of constant size; (ii) the probability of fixation in the presence of selection; and (iii) the probability of fixation in the presence of selection and demographic change, the latter in the form of a changing population size.     

Direct Solution of the Chemical Master Equation Using Quantized Tensor Trains

The Chemical Master Equation (CME) is a cornerstone of stochastic analysis and simulation of models of biochemical reaction networks. Yet direct solutions of the CME have remained elusive. Although several approaches overcome the infinite dimensional nature of the CME through projections or other means, a common feature of proposed approaches is their susceptibility to the curse of dimensionality, i.e. the exponential growth in memory and computational requirements in the number of problem dimensions. We present a novel approach that has the potential to “lift” this curse of dimensionality. The approach is based on the use of the recently proposed Quantized Tensor Train (QTT) formatted numerical linear algebra for the low parametric, numerical representation of tensors. The QTT decomposition admits both, algorithms for basic tensor arithmetics with complexity scaling linearly in the dimension (number of species) and sub-linearly in the mode size (maximum copy number), and a numerical tensor rounding procedure which is stable and quasi-optimal. We show how the CME can be represented in QTT format, then use the exponentially-converging -discontinuous Galerkin discretization in time to reduce the CME evolution problem to a set of QTT-structured linear equations to be solved at each time step using an algorithm based on Density Matrix Renormalization Group (DMRG) methods from quantum chemistry. Our method automatically adapts the “basis” of the solution at every time step guaranteeing that it is large enough to capture the dynamics of interest but no larger than necessary, as this would increase the computational complexity. Our approach is demonstrated by applying it to three different examples from systems biology: independent birth-death process, an example of enzymatic futile cycle, and a stochastic switch model. The numerical results on these examples demonstrate that the proposed QTT method achieves dramatic speedups and several orders of magnitude storage savings over direct approaches.     

The Solution Structure of a Locked Nucleic Acid (LNA) Hybridized to DNA

Abstract

LNA (Locked Nucleic Acids) is a novel oligonucleotide analogue containing a conformationally restricted nucleotide with a 2′-0, 4′-C-methylene bridge that induces unprecedented thermal affinities when mixed with complementary single stranded DNA and RNA. We have used two-dimensional'H NMR spectroscopy obtained at 750 and 500 MHz to determine a high resolution solution structure of an LNA oligonucleotide hybridized to the complementary DNA strand. The determination of the structure was based on a complete relaxation matrix analysis of the NOESY cross peaks followed by restrained molecular dynamics calculations. Forty final structures were generated for the duplex from A-type and B-type dsDNA starting structures. The root-mean-square deviation (RMSD) of the coordinates for the forty structures of the complex was 0.32Å. The structures were analysed by use of calculated helix parameters. This showed that the values for rise and buckle in the LNA duplex is markedly different from canonical B-DNA at the modification site. A value of twist similar to A-DNA is also observed at the modification site. The overall length of the helix which is 27.3Å. The average twist over the sequence are 35.9° ± 0.3°. Consequently, the modification does not cause the helix to unwind. The bis-intercalation of the thiazole orange dye TOTO to the LNA duplex was also investigated by ¹H NMR spectroscopy to sense the structural change from the unmodified oligonucleotide. We observed that the bis-intercalation of TOTO is much less favourable in the 5′-CT^LAG-3′ site than in the unmodified 5′-CT^LAG-3′ site. This was related to the change in the base stacking of the LNA duplex compared to the unmodified duplex.     

A New Generalized Logistic Sigmoid Growth Equation Compared with the Richards Growth Equation

A new sigmoid growth equation is presented for curve-fitting,analysis and simulation of growth curves. Like the logisticgrowth equation, it increases monotonically, with both upperand lower asymptotes. Like the Richards growth equation, itcan have its maximum slope at any value between its minimumand maximum. The new sigmoid equation is unique because it alwaystends towards exponential growth at small sizes or low densities,unlike the Richards equation, which only has this characteristicin part of its range. The new sigmoid equation is thereforeuniquely suitable for circumstances in which growth at smallsizes or low densities is expected to be approximately exponential,and the maximum slope of the growth curve can be at any value.Eleven widely different sigmoid curves were constructed withan exponential form at low values, using an independent algorithm.Sets of 100 variations of sequences of 20 points along eachcurve were created by adding random errors. In general, thenew sigmoid equation fitted the sequences of points as closelyas the original curves that they were generated from. The newsigmoid equation always gave closer fits and more accurate estimatesof the characteristics of the 11 original sigmoid curves thanthe Richards equation. The Richards equation could not estimatethe maximum intrinsic rate of increase (relative growth rate)of several of the curves. Both equations tended to estimatethat points of inflexion were closer to half the maximum sizethan was actually the case; the Richards equation underestimatedasymmetry by more than the new sigmoid equation. When the twoequations were compared by fitting to the example dataset thatwas used in the original presentation of the Richards growthequation, both equations gave good fits. The Richards equationis sometimes suitable for growth processes that may or may notbe close to exponential during initial growth. The new sigmoidis more suitable when initial growth is believed to be generallyclose to exponential, when estimates of maximum relative growthrate are required, or for generic growth simulations.Copyright1999 Annals of Botany Company Asymptote,Cucumis melo,curve-fitting, exponential growth, intrinsic rate of increase, logistic equation, maximum growth rate, model, non-linear least-squares regression, numerical algorithm, point of inflexion, relative growth rate, Richards growth equation, sigmoid growth curve.     

一类时滞反应扩散方程的稳态解和行波解

本文讨论了一类造血生物模型在Dirichlet边值条件下稳态解的全局吸引性,并利用上、下解技术和单调迭代方法讨论了行波解的存在性.     

周期时间制约三维Volterra方程的周期解分枝

利用文[1]的结论。对周期时间制约三维Volterra方程产生的周期解分枝进行了讨论,在一定条件下,给出了参数空间(λ1,λ2,λ3,ω)中确定的分枝曲面。     

含有两个时滞项的退化时滞微分方程周期解

讨论了含有两个时滞项退化时滞微分方程的周期解的问题,特别的,给出了此类方程存在非常数周期解的充要条件,并对二维退化微分方程给出了非常数周期解存在性的代数判据,并在最后给出一个例子验证了判据的有效性.     

Preparative Syntheses of Guanylate-Rich Oligonucleotides Using the Phosphotriester Method in Solution

Abstract

The guanylate-rich fragments: 150 mg d(G₄T₄), 180 mg d(T₄G₄), 350 mg d(T₄G₄T₄), 30 mg d(G₄T₄G₄), 85 mg d(T₄G₄T₄G₄)were synthesized using the triester method. By enzymatic ligation of aliquots the 36mer d(G₄T₄G₄T₄G₄T₄G₄T₄G₄) is obtained.     

Numerical Method Using Cubic B-Spline for a Strongly Coupled Reaction-Diffusion System

In this paper, a numerical method for the solution of a strongly coupled reaction-diffusion system, with suitable initial and Neumann boundary conditions, by using cubic B-spline collocation scheme on a uniform grid is presented. The scheme is based on the usual finite difference scheme to discretize the time derivative while cubic B-spline is used as an interpolation function in the space dimension. The scheme is shown to be unconditionally stable using the von Neumann method. The accuracy of the proposed scheme is demonstrated by applying it on a test problem. The performance of this scheme is shown by computing and error norms for different time levels. The numerical results are found to be in good agreement with known exact solutions.     

Numerical Modeling of Skin Tissue Heating Using the Interval Finite Difference Method

Numerical analysis of heat transfer processes proceeding in a nonhomogeneous biological tissue domain is presented. In particular, the skin tissue domain subjected to an external heat source is considered. The problem is treated as an axially-symmetrical one (it results from the mathematical form of the function describing the external heat source). Thermophysical parameters of sub-domains (volumetric specific heat, thermal conductivity, perfusion coefficient etc.) are given as interval numbers. The problem discussed is solved using the interval finite difference method basing on the rules of directed interval arithmetic, this means that at the stage of FDM algorithm construction the mathematical manipulations are realized using the interval numbers. In the final part of the paper the results of numerical computations are shown, in particular the problem of admissible thermal dose is analyzed.     

Solution Phase Synthesis of Dithymidine Phosphorothioate by a Phosphotriester Method Using New S-Protecting Groups

Abstract

ABSTRACT

A phosphotriester method for the synthesis of dithymidine phosphorothioates with eight S-protecting groups has been investigated. Three of the S-protecting groups possesed catalytic activity, however side reactions occurred under deprotection. The best S-protecting group was 4-chloro-2-nitrobenzyl which could be removed with a minimum of side reactions (0.3 %). The coupling reagent PyFNOP (11) gave protected dithymidine phosphorothioate in 96% yield after 15 min coupling.     

A Heuristic Search Approach Using Approximate Solutions to Petri Net State Equations for Scheduling Flexible Manufacturing Systems

This paper proposes a new heuristic search approach based on an analytic theory of the Petri net state equations for scheduling flexible manufacturing systems (FMSs) with the goal of minimizing makespan. The proposed method models an FMS using a timed Petri net and exploits approximate solutions of the net's state equation to predict the total cost (makespan) from the initial state through the current state to the goal. That is, the heuristic function considers global information provided by the state equation. This makes the method possible to obtain solutions better than those obtained using prior works (Lee and DiCesare, 1994a, 1994b) that consider only the current status or limited global information. In addition, to reduce memory requirement and thus to increase the efficiency of handling larger systems, the proposed scheduling algorithm contains a procedure to reduce the searched state space.     

二阶脉冲微分方程边值问题解的存在性

利用Leray-Schauder不动点理论研究了一类二阶脉冲微分方程三点边值问题,将以往所研究的方程的边界条件和脉冲项做了推广,得到了其解的存在性的新结果,最后通过实例说明了结论的应用.     

A Method for Helical RNA Global Structure Determination in Solution Using Small-Angle X-Ray Scattering and NMR Measurements

We report a “top-down” method that uses mainly duplexes' global orientations and overall molecular dimension and shape restraints, which were extracted from experimental NMR and small-angle X-ray scattering data, respectively, to determine global architectures of RNA molecules consisting of mostly A-form-like duplexes. The method is implemented in the G2G (from global measurement to global structure) toolkit of programs. We demonstrate the efficiency and accuracy of the method by determining the global structure of a 71-nt RNA using experimental data. The backbone root-mean-square deviation of the ensemble of the calculated global structures relative to the X-ray crystal structure is 3.0 ± 0.3 Å using the experimental data and is only 2.5 ± 0.2 Å for the three duplexes that were orientation restrained during the calculation. The global structure simplifies interpretation of multidimensional nuclear Overhauser spectra for high-resolution structure determination. The potential general application of the method for RNA structure determination is discussed.     

Numerical Simulation of Broadband Scattering by Coated and Noncoated Metal Nanostructures Using Discrete Dipole Approximation Method

We suggest numerical method to study the optical response of metal nanostructures. The analysis of optical properties such as scattering and absorption by coated and noncoated nanogeometry has been done using discrete dipole approximation (DDA) method. The core-shell nanogeometry supports surface plasmon resonances, which are highly tunable from 400 to 1100 nm. The tunability of surface plasmon resonance (SPR) highly depends on the structural anisotropy and chosen core-shell material. Further, we have observed that aspect ratio is one of the key parameter to decide the nature and position of the plasmonic peaks and magnitude of optical cross section. We have also shown that coated nanospheroid is a more appropriate geometry as compared to coated nanosphere and noncoated nanospheroid in terms of wide tunability of surface plasmon resonance. The wide tunability in SPR is observed for the effective radii 90 nm core-shell (Au@SiO₂) nanospheroid with aspect ratio 0.1.     

结合支持向量机和贝叶斯方法进行蛋白质二级结构预测

组建一个分两个阶段的分类器来进行蛋白质二级结构预测。第一阶段由支持向量机分类器组成,在第二阶段中使用第一阶段已预测的结果来进行贝叶斯判别。预测性能的改进表明了结合支持向量机和贝叶斯方法预测性能优越于单独使用支持向量机的预测性能。同时也证明残基在形成二级结构时是相互影响的。     

The Application of Electromagnetic Theory to Electrocardiology: II. Numerical Solution of the Integral Equations

In an earlier paper exact integral equations were derived for the surface potentials resulting from sources within an irregularly shaped inhomogeneous body. These exact equations cannot usually be solved. In this paper a discrete analogue is constructed which is not straightforward to solve, but which can be treated by careful mathematical methods. In particular a deflation procedure greatly facilitates the iterative solution of the problem and overcomes the divergence encountered by other authors. Numerical solutions obtained for simple geometries are compared to the exact analytic solutions available in such cases. The necessary convergence of the solutions of the discrete analog towards the solution of the continuous problem is shown to occur only if the coefficients of the discrete analogue are carefully evaluated. Calculations are then presented for realistic thoracic geometries, typical results being presented as surface potential maps. Finally the important effect of the internal regional inhomogeneities, particularly a realistic cardiac blood mass, is demonstrated by obtaining vector loops with and without these effects.