Browsing by Author "Jamrog, Diane C."

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    A Global Optimization Method for the Molecular Replacement Problem in X-ray Crystallography
    (2002-06) Jamrog, Diane C.; Phillips, George N. Jr.; Tapia, Richard A.; Zhang, Yin
    The primary technique for determining the three-dimensional structure of a protein molecule is X-ray crystallography, from which the molecular replacement (MR) problem often arises as a critical step. The MR problem is a global optimization problem to locate an optimal position of a model protein, whose structure is similar to the unknown protein structure that is to be determined, so that at this position the model protein will produce calculated intensities closest to those observed from an X-ray crystallography experiment. Improving the applicability and robustness of MR methods is an important research topic because commonly used traditional MR methods, though often successful, have their limitations in solving difficult problems. We introduce a new global optimization strategy that combines a coarse-grid search, using a surrogate function, with extensive multi-start local optimization. A new MR code, called SOMoRe, based on this strategy is developed and tested on four realistic problems, including two difficult problems that traditional MR codes failed to solve directly. SOMoRe was able to solve each test problem without any complication, and SOMoRe solved a MR problem using a less complete model than the models required by three other programs. These results indicate that the new method is promising and should enhance the applicability and robustness of the MR methodology.
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    A New Global Optimization Strategy for the Molecular Replacement Problem
    (2002-06) Jamrog, Diane C.
    The primary technique for determining the three-dimensional structure of a protein is X-ray crystallography, in which the molecular replacement (MR) problem arises as a critical step. Knowledge of protein structures is extremely useful for medical research, including discovering the molecular basis of disease and designing pharmaceutical drugs. This thesis proposes a new strategy to solve the MR problem, which is a global optimization problem to find the optimal orientation and position of a structurally similar model protein that will produce calculated intensities closest to those observed from an X-ray crystallography experiment. Improving the applicability and the robustness of MR methods is an important research goal because commonly used traditional MR methods, though often successful, have difficulty solving certain classes of MR problems. Moreover, the use of MR methods is only expected to increase as more structures are deposited into the Protein Data Bank. The new strategy has two major components: a six-dimensional global search and multi-start local optimization. The global search uses a low-frequency surrogate objective function and samples a coarse grid to identify good starting points for multi-start local optimization, which uses a more accurate objective function. As a result, the global search is relatively quick and the local optimization efforts are focused on promising regions of the MR variable space where solutions are likely to exist, in contrast to the traditional search strategy that exhaustively samples a uniformly fine grid of the variable space. In addition, the new strategy is deterministic, in contrast to stochastic search methods that randomly sample the variable space. This dissertation introduces a new MR program called SOMoRe that implements the new global optimization strategy. When tested on seven problems, SOMoRe was able to straightforwardly solve every test problem, including three problems that could not be directly solved by traditional MR programs. Moreover, SOMoRe also solved a MR problem using a less complete model than those required by two traditional programs and a stochastic six-dimensional program. Based on these results, this new strategy promises to extend the applicability and robustness of MR.
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    Comparison of Two Sets of First-order Conditions as Bases of Interior-Point Newton Methods for Optimization with Simple Bounds
    (2001-06) Jamrog, Diane C.; Tapia, Richard A.; Zhang, Yin
    In this paper, we compare the behavior of two Newton interior-point methods derived from two different first-order necessary conditions for the same nonlinear optimization problem with simple bounds. One set of conditions was proposed by Coleman and Li; the other is the standard KKT set of conditions. We discuss a perturbation of the CL conditions for problems with one-sided bounds and the difficulties involved in extending this to problems with general bounds. We study the numerical behavior of the Newton method applied to the systems of equations associated with the unperturbed and perturbed necessary conditions. Preliminary numerical results for convex quadratic objective functions indicate that, for this class of problems, Newton's method based on the perturbed KKT formulation appears to be the most robust.
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    On the Equivalence Between a Commonly Used Correlation Coefficient and a Least Squares Function
    (2003-01) Jamrog, Diane C.; Philips, George N. Jr.; Zhang, Yin
    Many objective functions have been proposed in X-ray crystallography to solve the molecular replacement (MR) problem and other optimization problems. In this paper, we establish the equivalence between optimizing two target functions: a commonly used correlation coefficient and a least squares function. This equivalence may be in neighborhoods about the global optima or the entire MR variable space depending on whether the average values of the observed and calculated data are subtracted from observed and calculated data. In addition, we also present an argument that the correlation coefficient between structure factor magnitudes is likely to perform better than the correlation coefficient between intensities. This was confirmed by the MR program SOMoRe, especially when low-resolution data were used.
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    The Effect of the Separation of Variables on the Molecular Replacement Method
    (2000-06) Jamrog, Diane C.; Phillips, George N. Jr.; Tapia, Richard A.; Zhang, Yin
    Traditional approaches for solving the molecular replacement problem separate a six-dimensional optimization problem into two three-dimensional ones in order to reduce the computational cost. There are, however, serious drawbacks in such a separation of the rotational and translational degrees of freedom. In this paper, we present computational experiments indicating that even under ideal conditions the separation can fail to preserve the correspondence between the global minima of a target function and the correct rotations when low resolution data are used. This phenomenon is a reason why only high resolution data are used in traditional approaches for solving the molecular replacement problem. In this paper, we provide a theoretical explanation for this phenomenon. In order to solve difficult molecular replacement problems, we believe that low resolution terms should be utilized because they generate smooth, shape-defining components in a target function, making it more amenable to global optimization. This study indicates that in order to utilize low resolution data in the molecular replacement method, we need to consider all degrees of freedom simultaneously. The full-dimensional optimization formulation, once a prohibitive procedure due to its high computational cost, should now be feasible given the current state of computational resources and algorithms.