Finite Element Analysis of the Time-Dependent Smoluchowski Equation for Acetylcholinesterase Reaction Rate Calculations

Cheng, Yuhui; Suen, Jason K.; Zhang, Deqiang; Bond, Stephen D.; Zhang, Yongjie; Song, Yuhua; Baker, Nathan A.; Bajaj, Chandrajit; Holst, Michael; McCammon, J. Andrew

doi:10.1529/biophysj.106.102533

Cited by 32 publications

(50 citation statements)

References 53 publications

(70 reference statements)

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“…With the SMOL solver (cf. [17]), the linear system converges very well and the outputs obtained are consistent with other studies [17,41]. Figure 12 depicts the ACh concentration at the steady-state Smoluchowski diffusion at 0.150M ionic strength with the external boundary as Dirichlet using OpenDX.…”

Section: Applications: Molecular Simulationsupporting

confidence: 81%

“…The availability of our software toolchain will give researchers in molecular/cellular modelling and simulation areas easy access to high-fidelity geometric models. ACh distribution around the mAChE enzyme at the steady-state diffusion stage simulated by the SMOL solver [17]. Red corresponds to ACh concentration in the bulk, blue to a concentration of zero.…”

Section: Resultsmentioning

confidence: 99%

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Feature-preserving adaptive mesh generation for molecular shape modeling and simulation

Holst

McCammon

2008

Journal of Molecular Graphics and Modelling

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105

122

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We describe a chain of algorithms for molecular surface and volumetric mesh generation. We take as inputs the centers and radii of all atoms of a molecule and the toolchain outputs both triangular and tetrahedral meshes that can be used for molecular shape modeling and simulation. Experiments on a number of molecules are demonstrated, showing that our methods possess several desirable properties: feature-preservation, local adaptivity, high quality, and smoothness (for surface meshes). We also demonstrate an example of molecular simulation using the finite element method and the meshes generated by our method. The approaches presented and their implementations are also applicable to other types of inputs such as 3D scalar volumes and triangular surface meshes with low quality, and hence can be used for generation/improvment of meshes in a broad range of applications.

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Section: Applications: Molecular Simulationsupporting

confidence: 81%

Section: Resultsmentioning

confidence: 99%

Feature-preserving adaptive mesh generation for molecular shape modeling and simulation

Holst

McCammon

2008

Journal of Molecular Graphics and Modelling

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105

122

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“…The Smoluchowski equation has already been shown to be a powerful tool for modeling diffusion-controlled processes in atomistically detailed environments, such as the binding of small molecules to protein receptors given electrostatic interactions [16][17][18][19] or recessed binding sites. 20 Hence, by representing diffuser/obstacle interactions through an arbitrary PMF in the Smoluchowski equation, we can estimate the influence of such interactions on effective rates of transport in crowded molecular systems.…”

Section: Introductionmentioning

confidence: 99%

Predicting the influence of long-range molecular interactions on macroscopic-scale diffusion by homogenization of the Smoluchowski equation

Kekenes‐Huskey

Gillette

McCammon

2014

The Journal of Chemical Physics

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The macroscopic diffusion constant for a charged diffuser is in part dependent on (1) the volume excluded by solute "obstacles" and (2) long-range interactions between those obstacles and the diffuser. Increasing excluded volume reduces transport of the diffuser, while long-range interactions can either increase or decrease diffusivity, depending on the nature of the potential. We previously demonstrated [P. M. Kekenes-Huskey et al., Biophys. J. 105, 2130] using homogenization theory that the configuration of molecular-scale obstacles can both hinder diffusion and induce diffusional anisotropy for small ions. As the density of molecular obstacles increases, van der Waals (vdW) and electrostatic interactions between obstacle and a diffuser become significant and can strongly influence the latter's diffusivity, which was neglected in our original model. Here, we extend this methodology to include a fixed (time-independent) potential of mean force, through homogenization of the Smoluchowski equation. We consider the diffusion of ions in crowded, hydrophilic environments at physiological ionic strengths and find that electrostatic and vdW interactions can enhance or depress effective diffusion rates for attractive or repulsive forces, respectively. Additionally, we show that the observed diffusion rate may be reduced independent of non-specific electrostatic and vdW interactions by treating obstacles that exhibit specific binding interactions as "buffers" that absorb free diffusers. Finally, we demonstrate that effective diffusion rates are sensitive to distribution of surface charge on a globular protein, Troponin C, suggesting that the use of molecular structures with atomistic-scale resolution can account for electrostatic influences on substrate transport. This approach offers new insight into the influence of molecular-scale, long-range interactions on transport of charged species, particularly for diffusion-influenced signaling events occurring in crowded cellular environments.

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“…21,22 Also, a software was developed to solve the PBE using a multigrid method, 26 and this was used with the steady-state SE to study the consumption of ACh by acetylcholinesterase ͑AChE͒. 24,25 The solution of the nonsteady-state SE for the above system is described in another work 27 and in this work. With the exception of Ref.…”

Section: Introductionmentioning

confidence: 99%

Electrodiffusion: A continuum modeling framework for biomolecular systems with realistic spatiotemporal resolution

Zhou

Huber

et al. 2007

The Journal of Chemical Physics

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122

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A computational framework is presented for the continuum modeling of cellular biomolecular diffusion 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 ͑adaptive͒ finite element and boundary element methods is adopted to solve the Smoluchowski equation ͑SE͒, the Poisson equation ͑PE͒, and the Poisson-Nernst-Planck equation ͑PNPE͒ 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 equations solutions. 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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Finite Element Analysis of the Time-Dependent Smoluchowski Equation for Acetylcholinesterase Reaction Rate Calculations

Cited by 32 publications

References 53 publications

Feature-preserving adaptive mesh generation for molecular shape modeling and simulation

Feature-preserving adaptive mesh generation for molecular shape modeling and simulation

Predicting the influence of long-range molecular interactions on macroscopic-scale diffusion by homogenization of the Smoluchowski equation

Electrodiffusion: A continuum modeling framework for biomolecular systems with realistic spatiotemporal resolution

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