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Electrodiffusion: A continuum modeling framework for biomolecular systems with realistic spatiotemporal resolution
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    Affiliations:
    1 Howard Hughes Medical Institute, University of California at San Diego, La Jolla, California 92093-0365, USA and Center for Theoretical Biological Physics, University of California at San Diego, La Jolla, California 92093-0365, USA
    2 Howard Hughes Medical Institute, University of California at San Diego, La Jolla, California 92093-0365, USA, Center for Theoretical Biological Physics, University of California at San Diego, La Jolla, California 92093-0365, USA, and Department of Mathematics, University of California at San Diego, La Jolla, California 92093-0365, USA
    3 Howard Hughes Medical Institute, University of California at San Diego, La Jolla, California 92093-0365, USA
    4 Department of Computer Science, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA
    5 Department of Mathematics, University of California at San Diego, La Jolla, California 92093-0365, USA and Center for Theoretical Biological Physics, University of California at San Diego, La Jolla, California 92093-0365, USA
    6 Howard Hughes Medical Institute, University of California at San Diego, La Jolla, California 92093-0365, USA, Center for Theoretical Biological Physics, University of California at San Diego, La Jolla, California 92093-0365, USA, Department of Chemistry and Biochemistry, University of California at San Diego, La Jolla, California 92093-0365, USA, and Department of Pharmacology, University of California at San Diego, La Jolla, California 92093-0365, USA
    a) Author to whom correspondence should be addressed. Fax: 1-858-534-4974. Electronic mail: blu@mccammon.ucsd.edu
    J. Chem. Phys. 127, 135102 (2007); http://dx.doi.org/10.1063/1.2775933
/content/aip/journal/jcp/127/13/10.1063/1.2775933
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Figures

Image of FIG. 1.

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FIG. 1.

(Color online) An example of conforming and nonconforming 2D meshes. The molecular interior is represented by shading, and the mesh covers the whole domain in each case.

Image of FIG. 2.

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FIG. 2.

(Color) Schematic of problem domain, denoting the boundaries and volumes. denotes the fixed molecular boundary, and is the boundary of the whole volume mesh. If reaction on the molecular surface is considered, according to Song et al.’s treatment (Ref. 24) (similar figure can also be found therein), a small patch around the active site is set to a zero Dirichlet boundary condition (sink boundary) to model the chemical reaction.

Image of FIG. 3.

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FIG. 3.

(Color) An example of mesh generation for a fragment of A-form DNA. (a) Cross section of the whole tetrahedral volume mesh. (b) A close-up view of the fine mesh around the molecule, whose body is colored by green. The edge between green and blue regions lies on the molecular surface. (c) The triangular boundary mesh conforming to the molecular surface.

Image of FIG. 4.

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FIG. 4.

(Color online) The numerical error in the solution of the regular part of the PBE using FEM.

Image of FIG. 5.

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FIG. 5.

(Color) The charge compensation predicted based on the analytical solution of the LPBE for a unit sphere cavity with charge at the center. (a) The compensation charge as a function of radial distance at different ionic strengths represented by . Blue lines correspond to the results from Eq. (21), red marks from Eq. (22). (b) The net charge within as a function of using Eq. (22) (red) and Eq. (21) (blue).

Image of FIG. 6.

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FIG. 6.

(Color) Comparisons of the calculated electrostatic potentials in 1:1 salt around a unit spherical cavity with at the center from different approaches: the analytical LPBE solution (blue square), LPBE (black star), NPBE (red triangle), and PNPE (green circle).

Image of FIG. 7.

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FIG. 7.

(Color) Effects of the addition of a charge flux in the 1:1 salt to (a) the potential, (b) the counterion, and (c) the coion density distributions. In the three figures, the red square marks denote the 1:1 salt case without charge flux, the blue triangle marks denote the case with a negative charge flux ( particle) added in the salt, the black star the case with a positive charge flux added in the salt.

Image of FIG. 8.

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FIG. 8.

(Color) Electrostatic potential and cation density (mM) around a fragment of A-form DNA. (a) Surface electrostatic potential from the BEM LPBE solution in a 1:1 salt. The color scale is from (red) to 10 (blue) . (b) Cross section of the density distribution in 1:1 salt. (c) Density distribution in 2:1 salt (e.g., ). The color scale is doubled for ease of comparison with (b). (d) Density isosurface with a value of in the case (b) from a different orientation. These and all the following density figures are generated using the software OPENDX (Ref. 59).

Image of FIG. 9.

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FIG. 9.

Comparison of calculations of the steady-state reaction rate of AChE monomer with BEM potential, APBS potential, and experimental fitting data. The dotted lines are the results from Ref. 25 with APBS electrostatic potential using unrefined or refined meshes, respectively. The thin solid line is from experimental data (Ref. 3) fit to the Debye-Hückel limiting law.

Image of FIG. 10.

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FIG. 10.

(Color) Visualization of the evolution of substrate concentrations in the diffusion-reaction processes. denotes the one unit positively charged ligand ACh and denotes the neutral ligand . (a)–(g) show the cases of diffusion from an initial pulse in the presence of AChE monomer at ionic strength of . (h) shows the tetramer (1c2b) case with diffusion with an initial condition of ACh pulse and in zero ionic strength. (i) is the same as (h) but with a uniform distribution of ACh as the initial condition. For the sake of visualization, different color scales are used in different subfigures. The largest color scales (red) are 0.000 18 in (a)–(c), 0.0001 in (d), 0.000 05 in (e)–(h), and 1.0 in (i).

Image of FIG. 11.

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FIG. 11.

Substrate consumption processes by AChE monomer at different ionic strength. denotes the positively charged ACh, the neutral . The diffusion starts from a vesicle-sized area containing 10 000 ACh molecules and away from AChE [see Fig. 10(a)]. The bottom figure shows a close-up view of the transient behavior at the beginning.

Image of FIG. 12.

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FIG. 12.

Consumption of bulk ACh by the AChE monomer at different ionic strengths.

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/content/aip/journal/jcp/127/13/10.1063/1.2775933
2007-10-04
2014-04-18

Abstract

A computational framework is presented for the continuum modeling of cellular biomoleculardiffusion influenced by electrostatic driving forces. This framework is developed from a combination of state-of-the-art numerical methods, geometric meshing, and computer visualization tools. In particular, a hybrid of (adaptive) finite element and boundary element methods is adopted to solve the Smoluchowski equation (SE), the Poisson equation (PE), and the Poisson-Nernst-Planck equation (PNPE) in order to describe electrodiffusion processes. The finite element method is used because of its flexibility in modeling irregular geometries and complex boundary conditions. The boundary element method is used due to the convenience of treating the singularities in the source charge distribution and its accurate solution to electrostatic problems on molecular boundaries. Nonsteady-state diffusion can be studied using this framework, with the electric field computed using the densities of charged small molecules and mobile ions in the solvent. A solution for mesh generation for biomolecular systems is supplied, which is an essential component for the finite element and boundary element computations. The uncoupled Smoluchowski equation and Poisson-Boltzmann equation are considered as special cases of the PNPE in the numerical algorithm, and therefore can be solved in this framework as well. Two types of computations are reported in the results: stationary PNPE and time-dependent SE or Nernst-Planck equationssolutions. A biological application of the first type is the ionic density distribution around a fragment of DNA determined by the equilibrium PNPE. The stationary PNPE with nonzero flux is also studied for a simple model system, and leads to an observation that the interference on electrostatic field of the substrate charges strongly affects the reaction rate coefficient. The second is a time-dependent diffusion process: the consumption of the neurotransmitter acetylcholine by acetylcholinesterase, determined by the SE and a single uncoupled solution of the Poisson-Boltzmann equation. The electrostatic effects, counterion compensation, spatiotemporal distribution, and diffusion-controlled reaction kinetics are analyzed and different methods are compared.

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Scitation: Electrodiffusion: A continuum modeling framework for biomolecular systems with realistic spatiotemporal resolution
http://aip.metastore.ingenta.com/content/aip/journal/jcp/127/13/10.1063/1.2775933
10.1063/1.2775933
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