Béla Paláncz
Published October 24, 2016
This article illustrates how Mathematica can be employed to model stochastic processes via stochastic differential equations to compute trajectories and their statistical features. In addition, we discuss parameter estimation of the model via the maximum likelihood method with global optimization. We consider handling modeling error, system noise and measurement error, compare the stochastic and deterministic models and use the likelihood-ratio test to verify the possible improvement provided by stochastic modeling. The Mathematica code is simple and can be easily adapted to similar problems. Read More »
Designing siRNAs for Gene Silencing
Todd D. Allen
Published January 29, 2016
Biological systems possess traits that are a product of coordinated gene expression, heavily influenced by environmental pressures. This is true of both desirable and undesirable traits (i.e. disease). In cases of unwanted traits, it would be tremendously useful to have a means of systematically lowering the expression of genes implicated in producing the undesirable traits. This article describes the implementation of a computational biology algorithm designed to compute small interfering RNA sequences, capable of systematically silencing the expression of specific genes. Read More »
Classical and Metric Multidimensional Scaling
Martin S. Zand, Jiong Wang, Shannon Hilchey
Published September 30, 2015
We describe the use of classical and metric multidimensional scaling methods for graphical representation of the proximity between collections of data consisting of cases characterized by multidimensional attributes. These methods can preserve metric differences between cases, while allowing for dimensional reduction and projection to two or three dimensions ideal for data exploration. We demonstrate these methods with three datasets for: (i) the immunological similarity of influenza proteins measured by a multidimensional assay; (ii) influenza protein sequence similarity; and (iii) reconstruction of airport-relative locations from paired proximity measurements. These examples highlight the use of proximity matrices, eigenvalues, eigenvectors, and linear and nonlinear mappings using numerical minimization methods. Some considerations and caveats for each method are also discussed, and compact Mathematica programs are provided. Read More »
Todd D. Allen
Published February 21, 2015
This article describes the implementation of RIFA, a computational biology algorithm designed to identify the genes most directly responsible for creating differences in phenotype between two biological states, for example, tumorous liver tissue versus cirrhotic liver tissue. Read More »
Hiroshi Sekine, Yuki Otani
Published December 18, 2014
Remarkable advances have been made in radiotherapy for cancer. Tumors inside the body that move with respiration can now be tracked during irradiation or can be irradiated at a specific phase of respiration [1]. Since radiation of these wavelengths is invisible to the human eye, it is not possible to see whether a tumor that is moving during respiration is being irradiated accurately. Therefore, simulations that use the actual respiration wave to show the tumor being irradiated are useful for training and to explain the procedure to patients. Read More »
Todd D. Allen
Published November 26, 2013
This article describes the development of a novel program to process Affymetrix microarray files, which are used in the biological sciences to establish differences in gene expression between two conditions (e.g., diseased tissue versus healthy tissue). Read More »
K. Sheshadri, Peter Fritzson
Published December 21, 2011
A package for solving time-dependent partial differential equations (PDEs), MathPDE, is presented. It implements finite-difference methods. After making a sequence of symbolic transformations on the PDE and its initial and boundary conditions, MathPDE automatically generates a problem-specific set of Mathematica functions to solve the numerical problem, which is essentially a system of algebraic equations. MathPDE then internally calls MathCode, a Mathematica-to-C++ code generator, to generate a C++ program for solving the algebraic problem, and compiles it into an executable that can be run via MathLink. When the algebraic system is nonlinear, the Newton-Raphson method is used and SuperLU, a library for sparse systems, is used for matrix operations. This article discusses the wide range of PDEs that can be handled by MathPDE, the accuracy of the finite-difference schemes used, and importantly, the ability to handle both regular and irregular spatial domains. Since a standalone C++ program is generated to compute the numerical solution, the package offers portability. Read More »
Hugh Murrell, Kazuo Hashimoto, Daichi Takatori
Published July 26, 2011
Fisher first introduced the Fisher linear discriminant back in 1938. After the popularization of the support vector machine (SVM) and the kernel trick it became inevitable that the Fisher linear discriminant would be kernelized. Sebastian Mika accomplished this task as part of his Ph.D. in 2002 and the kernelized Fisher discriminant (KFD) now forms part of the large-scale machine-learning tool Shogun. In this article we introduce the package MathKFD. We apply MathKFD to synthetic datasets to demonstrate nonlinear classification via kernels. We also test performance on datasets from the machine-learning literature. The construction of MathKFD follows closely in style the construction of MathSVM by Nilsson and colleagues. We hope these two packages and others of the same ilk will eventually be integrated to form a kernel-based machine-learning environment for Mathematica. Read More »
Bernhard Voelkl
Published May 24, 2011
In finite populations, evolutionary dynamics can no longer be described by deterministic differential equations, but require a stochastic formulation [1]. We show how Mathematica can be used to both simulate and visualize evolutionary processes in limited populations. The Moran process is introduced as the basic stochastic model of an evolutionary process in finite populations. This model is extended to mixed populations with relative fitness differences. We combine population ecology with game theoretic ideas, simulating evolutionary games in well-mixed and structured populations. Read More »
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