Optimisation of NMR dynamic models I. Minimisation algorithms and their performance within the model-free and Brownian rotational diffusion spaces

The key to obtaining the model-free description of the dynamics of a macromolecule is the optimisation of the model-free and Brownian rotational diffusion parameters using the collected R1, R2 and steady-state NOE relaxation data. The problem of optimising the chi-squared value is often assumed to be trivial, however, the long chain of dependencies required for its calculation complicates the model-free chi-squared space. Convolutions are induced by the Lorentzian form of the spectral density functions, the linear recombinations of certain spectral density values to obtain the relaxation rates, the calculation of the NOE using the ratio of two of these rates, and finally the quadratic form of the chi-squared equation itself. Two major topological features of the model-free space complicate optimisation. The first is a long, shallow valley which commences at infinite correlation times and gradually approaches the minimum. The most severe convolution occurs for motions on two timescales in which the minimum is often located at the end of a long, deep, curved tunnel or multidimensional valley through the space. A large number of optimisation algorithms will be investigated and their performance compared to determine which techniques are suitable for use in model-free analysis. Local optimisation algorithms will be shown to be sufficient for minimisation not only within the model-free space but also for the minimisation of the Brownian rotational diffusion tensor. In addition the performance of the programs Modelfree and Dasha are investigated. A number of model-free optimisation failures were identified: the inability to slide along the limits, the singular matrix failure of the Levenberg–Marquardt minimisation algorithm, the low precision of both programs, and a bug in Modelfree. Significantly, the singular matrix failure of the Levenberg–Marquardt algorithm occurs when internal correlation times are undefined and is greatly amplified in model-free analysis by both the grid search and constraint algorithms. The program relax (http://www.nmr-relax.com) is also presented as a new software package designed for the analysis of macromolecular dynamics through the use of NMR relaxation data and which alleviates all of the problems inherent within model-free analysis

Quantitative Models of the Mechanisms That Control Genome-Wide Patterns of Transcription Factor Binding during Early Drosophila Development

Transcription factors that drive complex patterns of gene expression during animal development bind to thousands of genomic regions, with quantitative differences in binding across bound regions mediating their activity. While we now have tools to characterize the DNA affinities of these proteins and to precisely measure their genome-wide distribution in vivo, our understanding of the forces that determine where, when, and to what extent they bind remains primitive. Here we use a thermodynamic model of transcription factor binding to evaluate the contribution of different biophysical forces to the binding of five regulators of early embryonic anterior-posterior patterning in Drosophila melanogaster. Predictions based on DNA sequence and in vitro protein-DNA affinities alone achieve a correlation of ∼0.4 with experimental measurements of in vivo binding. Incorporating cooperativity and competition among the five factors, and accounting for spatial patterning by modeling binding in every nucleus independently, had little effect on prediction accuracy. A major source of error was the prediction of binding events that do not occur in vivo, which we hypothesized reflected reduced accessibility of chromatin. To test this, we incorporated experimental measurements of genome-wide DNA accessibility into our model, effectively restricting predicted binding to regions of open chromatin. This dramatically improved our predictions to a correlation of 0.6–0.9 for various factors across known target genes. Finally, we used our model to quantify the roles of DNA sequence, accessibility, and binding competition and cooperativity. Our results show that, in regions of open chromatin, binding can be predicted almost exclusively by the sequence specificity of individual factors, with a minimal role for protein interactions. We suggest that a combination of experimentally determined chromatin accessibility data and simple computational models of transcription factor binding may be used to predict the binding landscape of any animal transcription factor with significant precision

Kernel-function Based Primal-Dual Algorithms for

Recently, [Y.Q. Bai, M. El Ghami and C. Roos, SIAM J. Opt. 15 (2004) 101–128] investigated a new class of kernel functions which differs from the class of self-regular kernel functions. The class is defined by some simple conditions on the growth and the barrier behavior of the kernel function. In this paper we generalize the analysis presented in the above paper for P*(κ) Linear Complementarity Problems (LCPs). The analysis for LCPs deviates significantly from the analysis for linear optimization. Several new tools and techniques are derived in this paper

Computational and sensitivity aspects of eigenvalue-based methods for the large-scale trust-region subproblem

The trust-region subproblem of minimizing a quadratic function subject to a norm constraint arises in the context of trust-region methods in optimization and in the regularization of discrete forms of ill-posed problems, including non-negative regularization by means of interior-point methods. A class of efficient methods and software for solving large-scale trust-region subproblems is based on a parametric-eigenvalue formulation of the subproblem. The solution of a sequence of large symmetric eigenvalue problems is the main computation in these methods. In this work, we study the robustness and performance of eigenvalue-based methods for the large-scale trust-region subproblem. We describe the eigenvalue problems and their features, and discuss the computational challenges they pose as well as some approaches to handle them. We present results from a numerical study of the sensitivity of solutions to the trust-region subproblem to eigenproblem solutions.Electrical Engineering, Mathematics and Computer Scienc

Large-scale eigenvalue problems in trust-region calculations

The Trust-Region Subproblem of minimizing a quadratic function subject to a norm constraint arises in the context of trust-region methods in optimization and in the regularization of discrete forms of illposed problems. In recent years, methods and software for large-scale trust-region subproblems have been developed that require the solution of a large bordered eigenvalue problem at each iteration. In this work, we describe the bordered eigenvalue problems, the computational challenges in solving them, and present some approaches for their efficient solution by means of Krylov subspace methods for linear and nonlinear eigenvalue problems.Electrical Engineering, Mathematics and Computer Scienc

Primal-Dual IPMS for Semidefinite Optimization Based on Finite Barrier Functions

In this paper we extend the results obtained for a class of finite kernel functions by Y.Q. Bai M. El Ghami and C.Roos published in SIAM Journal of Optimization, 13(3):766–782, 2003 [3] for linear optimization to semidefinite optimization. We show that the iteration bound for primal dual methods is O ( √ n log n log n ɛ), for large-update methods and O ( √ n log n ɛ), for small-update methods. The iteration complexity obtained for semidefinite programming is the same as the best bound for primal-dual interior point methods in linear optimization

Computational and Sensitivity Aspects of Eigenvalue-Based Methods for the Large-Scale Trust-Region Subproblem - extended version

The trust-region subproblem of minimizing a quadratic function subject to a norm constraint arises in the context of trust-region methods in optimization and in the regularization of discrete forms of ill-posed problems, including non-negative regularization by means of interior-point methods. A class of efficient methods and software for solving large-scale trust-region subproblems is based on a parametric-eigenvalue formulation of the subproblem. The solution of a sequence of large symmetric eigenvalue problems is the main computation in these methods. In this work, we study the robustness and performance of eigenvalue-based methods for the large-scale trust-region subproblem. We describe the eigenvalue problems and their features, and discuss the computational challenges they pose as well as some approaches to handle them. We present results from a numerical study of the sensitivity of solutions to the trust-region subproblem to eigenproblem solutions.Electrical Engineering, Mathematics and Computer Scienc