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Linear Approximations

For a differentiable function of two variables, linear approximation estimates values near a known point by replacing the curved surface with its tangent plane. Consider the function\begin{equation*}f(x,y)=x^2+3y^2\end{equation*}near the point (2, 1). The exact value at this point is f(2, 1) = 22 + 3(1)2 = 4 + 3 = 7.The linear approximation of f(x, y)) near (a, b) is\begin{equation*}L(x,y)=f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b)\end{equation*}First, compute the partial derivatives: fx(x, y) = 2x and...
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Efficient molecular surface rendering by linear-time pseudo-Gaussian approximation to Lee-Richards surfaces (PGALRS).

Herbert J Bernstein, Paul A Craig

    Journal of Applied Crystallography
    |March 20, 2010
    PubMed
    Summary

    The new PGALRS algorithm efficiently approximates molecular surfaces for large macromolecules. This computational method speeds up the identification of surface residues, enabling analysis of larger structures.

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    Area of Science:

    • Computational biology
    • Structural bioinformatics
    • Molecular modeling

    Background:

    • Calculating molecular surfaces is computationally intensive, especially for large macromolecules.
    • Existing algorithms struggle with the scale of modern biological datasets.

    Purpose of the Study:

    • To introduce the pseudo-Gaussian approximation to Lee-Richards surfaces (PGALRS) algorithm.
    • To enable efficient computation of molecular surfaces for very large macromolecules.
    • To reduce the computational burden of surface-based analyses.

    Main Methods:

    • Modeling electron density using pseudo-Gaussian atoms to approximate the Lee-Richards surface.
    • Utilizing a NearTree-based nearest neighbor search for efficient identification of surface-accessible atoms and residues.
    • Applying standard Lee-Richards algorithms to a reduced set of identified surface residues for high-quality surface generation.

    Main Results:

    • The PGALRS algorithm approximates the Lee-Richards surface with a time complexity approximately linear in the number of atoms.
    • Identification of closest atoms and residues is achieved in average time O[n(2/3)log(n)].
    • Overall average time complexity for surface computation is reduced to O(n), significantly enhancing efficiency for large molecules.

    Conclusions:

    • The PGALRS algorithm significantly reduces computational cost for rendering meaningful molecular surfaces.
    • This method extends the applicability of molecular surface software to macromolecules exceeding 50,000 atoms.
    • PGALRS serves as a foundation for advanced surface-based motif identification in large biomolecular systems.