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1
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Fully implicit ADI schemes for solving the nonlinear Poisson-Boltzmann equation

100%
Geng W., Zhao S.
Molecular Based Mathematical Biology
|
2013
|
tom 1
109-123
EN
The Poisson-Boltzmann (PB) model is an effective approach for the electrostatics analysis of solvated biomolecules. The nonlinearity associated with the PB equation is critical when the underlying electrostatic potential is strong, but is extremely difficult to solve numerically. In this paper, we construct two operator splitting alternating direction implicit (ADI) schemes to efficiently and stably solve the nonlinear PB equation in a pseudo-transient continuation approach. The operator splitting framework enables an analytical integration of the nonlinear term that suppresses the nonlinear instability. A standard finite difference scheme weighted by piecewise dielectric constants varying across the molecular surface is employed to discretize the nonhomogeneous diffusion term of the nonlinear PB equation, and yields tridiagonal matrices in the Douglas and Douglas-Rachford type ADI schemes. The proposed time splitting ADI schemes are different from all existing pseudo-transient continuation approaches for solving the classical nonlinear PB equation in the sense that they are fully implicit. In a numerical benchmark example, the steady state solutions of the fully-implicit ADI schemes based on different initial values all converge to the time invariant analytical solution, while those of the explicit Euler and semi-implicit ADI schemes blow up when the magnitude of the initial solution is large. For the solvation analysis in applications to real biomolecules with various sizes, the time stability of the proposed ADI schemes can be maintained even using very large time increments, demonstrating the efficiency and stability of the present methods for biomolecular simulation.
2
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Mathematical modeling of semiconductor quantum dots based on the nonparabolic effective-mass approximation

100%
Liu J.-L.
Nanoscale Systems: Mathematical Modeling, Theory and Applications
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2012
|
tom 1
58-79
EN
Within the effective mass and nonparabolic band theory, a general framework of mathematical models and numerical methods is developed for theoretical studies of semiconductor quantum dots. It includes single-electron models and many-electron models of Hartree-Fock, configuration interaction, and current-spin density functional theory approaches. These models result in nonlinear eigenvalue problems from a suitable discretization. Cubic and quintic Jacobi-Davidson methods of block or nonblock version are then presented for calculating the wanted eigenvalues that are clustered in the interior of the spectrum and may have small gaps and degeneracy. These are challenging issues arising from modeling a great variety of semiconductor nanostructures fabricated by advanced technology in semiconductor industry and science. Generic algorithms for many-electron simulations under this framework are also provided. Numerical results obtained within this framework are summarized to three eminent aspects, namely, accuracy of models, physical novelty, and effectivity of nonlinear eigensolvers. Concerning numerical accuracy, important details related to experimental data are also addressed.
3
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A Poisson-Boltzmann Equation Test Model for Protein in Spherical Solute Region and its Applications

100%
Jiang Y., Ying J., Xie D.
Molecular Based Mathematical Biology
|
2014
|
tom 2
|
nr 1
EN
The Poisson-Boltzmann equation (PBE) is one important implicit solvent continuum model for calculating electrostatics of protein in ionic solvent. Several numerical algorithms and program packages have been developed but verification and comparison between them remains an interesting topic. In this paper, a PBE test model is presented for a protein in a spherical solute region, along with its analytical solution. It is then used to verify a PBE finite element solver and applied to a numerical comparison study between a finite element solver and a finite difference solver. Such a study demonstrates the importance of retaining the interface conditions in the development of PBE solvers.
4
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Progress in developing Poisson-Boltzmann equation solvers

81%
Li C., Li L., Petukh M., Alexov E.
Molecular Based Mathematical Biology
|
2013
|
tom 1
42-62
EN
This review outlines the recent progress made in developing more accurate and efficient solutions to model electrostatics in systems comprised of bio-macromolecules and nanoobjects, the last one referring to objects that do not have biological function themselves but nowadays are frequently used in biophysical and medical approaches in conjunction with bio-macromolecules. The problem of modeling macromolecular electrostatics is reviewed from two different angles: as a mathematical task provided the specific definition of the system to be modeled and as a physical problem aiming to better capture the phenomena occurring in the real experiments. In addition, specific attention is paid to methods to extend the capabilities of the existing solvers to model large systems toward applications of calculations of the electrostatic potential and energies in molecular motors, mitochondria complex, photosynthetic machinery and systems involving large nanoobjects.
5
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Mathematical Models for Sensing Devices Constructed out of Artificial Cell Membranes

81%
Hoiles W., Krishnamurthy V., Cornell B.
Nanoscale Systems: Mathematical Modeling, Theory and Applications
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2012
|
tom 1
143-171
EN
This paper presents a review of ion channel based biosensors with a focus on the mathematical modeling of the stateof- the-art ion channel switch (ICS) biosensor and the novel cation specific (CS) sensor. The characteristics of the analyte present in the electrolyte, the ionic transport of chemical species, and the bioelectronic interface present in the ICS biosensor and CS sensor are modeled using ordinary and partial differential equations. The methodologies presented are important for modeling similar bioelectronic devices. Biosensors have applications in the fields of medicine, engineering, and biology. The recent emergence of biomimetically engineered nanomachine devices capable of measuring femto-molar concentrations of chemical species and the detection of channelopathies (ion channel disorders) makes them an attractive tool due to their high sensitivity and rapid detection rates. Beyond the continuum models used for the ICS and CS sensors, we present methods by which firstprinciple approaches such as molecular dynamics combined with stochastic methodologies can be used to obtain macrolevel parameters such as conductance and chemical reaction rates.
6
Content available remote

Quantum optimal control using the adjoint method

63%
Borzì A.
Nanoscale Systems: Mathematical Modeling, Theory and Applications
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2012
|
tom 1
93-111
EN
Control of quantum systems is central in a variety of present and perspective applications ranging from quantum optics and quantum chemistry to semiconductor nanostructures, including the emerging fields of quantum computation and quantum communication. In this paper, a review of recent developments in the field of optimal control of quantum systems is given with a focus on adjoint methods and their numerical implementation. In addition, the issues of exact controllability and optimal control are discussed for finite- and infinitedimensional quantum systems. Some insight is provided considering ’two-level’ models. This work is completed with an outlook to future developments.
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