The Experts below are selected from a list of 360 Experts worldwide ranked by ideXlab platform
Nathan A Baker - One of the best experts on this subject based on the ideXlab platform.
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numerical calculation of protein ligand binding rates through solution of the Smoluchowski Equation using smoothed particle hydrodynamics
BMC Biophysics, 2015Co-Authors: Wenxiao Pan, Michael D Daily, Nathan A BakerAbstract:The calculation of diffusion-controlled ligand binding rates is important for understanding enzyme mechanisms as well as designing enzyme inhibitors. We demonstrate the accuracy and effectiveness of a Lagrangian particle-based method, smoothed particle hydrodynamics (SPH), to study diffusion in biomolecular systems by numerically solving the time-dependent Smoluchowski Equation for continuum diffusion. Unlike previous studies, a reactive Robin boundary condition (BC), rather than the absolute absorbing (Dirichlet) BC, is considered on the reactive boundaries. This new BC treatment allows for the analysis of enzymes with “imperfect” reaction rates. The numerical method is first verified in simple systems and then applied to the calculation of ligand binding to a mouse acetylcholinesterase (mAChE) monomer. Rates for inhibitor binding to mAChE are calculated at various ionic strengths and compared with experiment and other numerical methods. We find that imposition of the Robin BC improves agreement between calculated and experimental reaction rates. Although this initial application focuses on a single monomer system, our new method provides a framework to explore broader applications of SPH in larger-scale biomolecular complexes by taking advantage of its Lagrangian particle-based nature.
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finite element analysis of the time dependent Smoluchowski Equation for acetylcholinesterase reaction rate calculations
Biophysical Journal, 2007Co-Authors: Yuhui Cheng, Yuhua Song, Yongjie Zhang, Chandrajit L Bajaj, Nathan A Baker, Jason K Suen, Deqiang Zhang, Stephen D Bond, Michael Holst, Andrew J MccammonAbstract:This article describes the numerical solution of the time-dependent Smoluchowski Equation to study diffusion in biomolecular systems. Specifically, finite element methods have been developed to calculate ligand binding rate constants for large biomolecules. The resulting software has been validated and applied to the mouse acetylcholinesterase (mAChE) monomer and several tetramers. Rates for inhibitor binding to mAChE were calculated at various ionic strengths with several different time steps. Calculated rates show very good agreement with experimental and theoretical steady-state studies. Furthermore, these finite element methods require significantly fewer computational resources than existing particle-based Brownian dynamics methods and are robust for complicated geometries. The key finding of biological importance is that the rate accelerations of the monomeric and tetrameric mAChE that result from electrostatic steering are preserved under the non-steady-state conditions that are expected to occur in physiological circumstances.
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tetrameric mouse acetylcholinesterase continuum diffusion rate calculations by solving the steady state Smoluchowski Equation using finite element methods
Biophysical Journal, 2005Co-Authors: Deqiang Zhang, Yuhua Song, Yongjie Zhang, Chandrajit L Bajaj, Nathan A Baker, Jason K Suen, Michael Holst, Zoran Radic, Palmer Taylor, Andrew J MccammonAbstract:The tetramer is the most important form for acetylcholinesterase in physiological conditions, i.e., in the neuromuscular junction and the nervous system. It is important to study the diffusion of acetylcholine to the active sites of the tetrameric enzyme to understand the overall signal transduction process in these cellular components. Crystallographic studies revealed two different forms of tetramers, suggesting a flexible tetramer model for acetylcholinesterase. Using a recently developed finite element solver for the steady-state Smoluchowski Equation, we have calculated the reaction rate for three mouse acetylcholinesterase tetramers using these two crystal structures and an intermediate structure as templates. Our results show that the reaction rates differ for different individual active sites in the compact tetramer crystal structure, and the rates are similar for different individual active sites in the other crystal structure and the intermediate structure. In the limit of zero salt, the reaction rates per active site for the tetramers are the same as that for the monomer, whereas at higher ionic strength, the rates per active site for the tetramers are ∼67%–75% of the rate for the monomer. By analyzing the effect of electrostatic forces on ACh diffusion, we find that electrostatic forces play an even more important role for the tetramers than for the monomer. This study also shows that the finite element solver is well suited for solving the diffusion problem within complicated geometries.
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continuum diffusion reaction rate calculations of wild type and mutant mouse acetylcholinesterase adaptive finite element analysis
Biophysical Journal, 2004Co-Authors: Yuhua Song, Yongjie Zhang, Chandrajit L Bajaj, Nathan A BakerAbstract:As described previously, continuum models, such as the Smoluchowski Equation, offer a scalable framework for studying diffusion in biomolecular systems. This work presents new developments in the efficient solution of the continuum diffusion Equation. Specifically, we present methods for adaptively refining finite element solutions of the Smoluchowski Equation based on a posteriori error estimates. We also describe new, molecular-surface-based models, for diffusional reaction boundary criteria and compare results obtained from these models with the traditional spherical criteria. The new methods are validated by comparison of the calculated reaction rates with experimental values for wild-type and mutant forms of mouse acetylcholinesterase. The results show good agreement with experiment and help to define optimal reactive boundary conditions.
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finite element solution of the steady state Smoluchowski Equation for rate constant calculations
Biophysical Journal, 2004Co-Authors: Yuhua Song, Yongjie Zhang, Tongye Shen, Chandrajit L Bajaj, Andrew J Mccammon, Nathan A BakerAbstract:This article describes the development and implementation of algorithms to study diffusion in biomolecular systems using continuum mechanics Equations. Specifically, finite element methods have been developed to solve the steady-state Smoluchowski Equation to calculate ligand binding rate constants for large biomolecules. The resulting software has been validated and applied to mouse acetylcholinesterase. Rates for inhibitor binding to mAChE were calculated at various ionic strengths with several different reaction criteria. The calculated rates were compared with experimental data and show very good agreement when the correct reaction criterion is used. Additionally, these finite element methods require significantly less computational resources than existing particle-based Brownian dynamics methods.
Hermann Grabert - One of the best experts on this subject based on the ideXlab platform.
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Strong friction limit in quantum mechanics: the quantum Smoluchowski Equation.
Physical review letters, 2001Co-Authors: Joachim Ankerhold, Philip Pechukas, Hermann GrabertAbstract:For a quantum system coupled to a heat bath environment the strong friction limit is studied starting from the exact path integral formulation. Generalizing the classical Smoluchowski limit to low temperatures, a time evolution Equation for the position distribution is derived and the strong role of quantum fluctuations in this limit is revealed.
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quantum Smoluchowski Equation ii the overdamped harmonic oscillator
Journal of Physical Chemistry B, 2001Co-Authors: Philip Pechukas, And Joachim Ankerhold, Hermann GrabertAbstract:The quantum Smoluchowski Equation is an Equation for the coordinate-diagonal elements of the density operator, and it is identical to the classical Smoluchowski Equation. Here, the leading finite-friction correction to the quantum Smoluchowski Equation is derived, for the particular case of the overdamped harmonic oscillator. It is in fact a quantum correction, different from the well-known classical correction, and it dominates the classical correction in the strong friction regime.
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Quantum Smoluchowski Equation
Annalen der Physik, 2000Co-Authors: Philip Pechukas, Joachim Ankerhold, Hermann GrabertAbstract:The strong friction limit of the Caldeira-Leggett model for quantum Brownian motion is analyzed. In this Smoluchowski limit the density operator of the Brownian particle is essentially diagonal in coordinate, and the diagonal element satisfies the classical Smoluchowski Equation. This result cannot be obtained from the master Equations that have been proposed for quantum Brownian motion, whether or not these Equations are of Lindblad form.
Andrew J Mccammon - One of the best experts on this subject based on the ideXlab platform.
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electrodiffusion a continuum modeling framework for biomolecular systems with realistic spatiotemporal resolution
Journal of Chemical Physics, 2007Co-Authors: Y C Zhou, Michael Holst, Stephen D Bond, Gary A Huber, Andrew J MccammonAbstract: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 (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 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
Biophysical Journal, 2007Co-Authors: Yuhui Cheng, Yuhua Song, Yongjie Zhang, Chandrajit L Bajaj, Nathan A Baker, Jason K Suen, Deqiang Zhang, Stephen D Bond, Michael Holst, Andrew J MccammonAbstract:This article describes the numerical solution of the time-dependent Smoluchowski Equation to study diffusion in biomolecular systems. Specifically, finite element methods have been developed to calculate ligand binding rate constants for large biomolecules. The resulting software has been validated and applied to the mouse acetylcholinesterase (mAChE) monomer and several tetramers. Rates for inhibitor binding to mAChE were calculated at various ionic strengths with several different time steps. Calculated rates show very good agreement with experimental and theoretical steady-state studies. Furthermore, these finite element methods require significantly fewer computational resources than existing particle-based Brownian dynamics methods and are robust for complicated geometries. The key finding of biological importance is that the rate accelerations of the monomeric and tetrameric mAChE that result from electrostatic steering are preserved under the non-steady-state conditions that are expected to occur in physiological circumstances.
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tetrameric mouse acetylcholinesterase continuum diffusion rate calculations by solving the steady state Smoluchowski Equation using finite element methods
Biophysical Journal, 2005Co-Authors: Deqiang Zhang, Yuhua Song, Yongjie Zhang, Chandrajit L Bajaj, Nathan A Baker, Jason K Suen, Michael Holst, Zoran Radic, Palmer Taylor, Andrew J MccammonAbstract:The tetramer is the most important form for acetylcholinesterase in physiological conditions, i.e., in the neuromuscular junction and the nervous system. It is important to study the diffusion of acetylcholine to the active sites of the tetrameric enzyme to understand the overall signal transduction process in these cellular components. Crystallographic studies revealed two different forms of tetramers, suggesting a flexible tetramer model for acetylcholinesterase. Using a recently developed finite element solver for the steady-state Smoluchowski Equation, we have calculated the reaction rate for three mouse acetylcholinesterase tetramers using these two crystal structures and an intermediate structure as templates. Our results show that the reaction rates differ for different individual active sites in the compact tetramer crystal structure, and the rates are similar for different individual active sites in the other crystal structure and the intermediate structure. In the limit of zero salt, the reaction rates per active site for the tetramers are the same as that for the monomer, whereas at higher ionic strength, the rates per active site for the tetramers are ∼67%–75% of the rate for the monomer. By analyzing the effect of electrostatic forces on ACh diffusion, we find that electrostatic forces play an even more important role for the tetramers than for the monomer. This study also shows that the finite element solver is well suited for solving the diffusion problem within complicated geometries.
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finite element solution of the steady state Smoluchowski Equation for rate constant calculations
Biophysical Journal, 2004Co-Authors: Yuhua Song, Yongjie Zhang, Tongye Shen, Chandrajit L Bajaj, Andrew J Mccammon, Nathan A BakerAbstract:This article describes the development and implementation of algorithms to study diffusion in biomolecular systems using continuum mechanics Equations. Specifically, finite element methods have been developed to solve the steady-state Smoluchowski Equation to calculate ligand binding rate constants for large biomolecules. The resulting software has been validated and applied to mouse acetylcholinesterase. Rates for inhibitor binding to mAChE were calculated at various ionic strengths with several different reaction criteria. The calculated rates were compared with experimental data and show very good agreement when the correct reaction criterion is used. Additionally, these finite element methods require significantly less computational resources than existing particle-based Brownian dynamics methods.
Fraydoun Rezakhanlou - One of the best experts on this subject based on the ideXlab platform.
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pointwise bounds for the solutions of the Smoluchowski Equation with diffusion
Archive for Rational Mechanics and Analysis, 2014Co-Authors: Fraydoun RezakhanlouAbstract:We prove various decay bounds on solutions (fn : n > 0) of the discrete and continuous Smoluchowski Equations with diffusion. More precisely, we establish pointwise upper bounds on nlfn in terms of a suitable average of the moments of the initial data for every positive l. As a consequence, we can formulate sufficient conditions on the initial data to guarantee the finiteness of \({L^p(\mathbb{R}^d \times [0, T])}\) norms of the moments \({X_a(x, t) := \sum_{m\in\mathbb{N}}m^a f_m(x, t)}\), (\({\int_0^{\infty} m^a f_m(x, t)dm}\) in the case of continuous Smoluchowski’s Equation) for every \({p \in [1, \infty]}\). In previous papers [11] and [5] we proved similar results for all weak solutions to the Smoluchowski’s Equation provided that the diffusion coefficient d(n) is non-increasing as a function of the mass. In this paper we apply a new method to treat general diffusion coefficients and our bounds are expressed in terms of an auxiliary function \({\phi(n)}\) that is closely related to the total increase of the diffusion coefficient in the interval (0, n].
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moment bounds for the solutions of the Smoluchowski Equation with coagulation and fragmentation
Proceedings of The Royal Society A: Mathematical Physical and Engineering Sciences, 2010Co-Authors: Fraydoun RezakhanlouAbstract:We prove various bounds on moments where f m is a solution of the discrete (respectively, continuous) Smoluchowski coagulation–fragmentation Equations with diffusion. In a previous paper we proved similar results for all weak solutions to the discrete Smoluchowski Equation provided that there is no fragmentation and certain moments are bounded in suitable L q -spaces initially. We prove the corresponding results in the case of the continuous Smoluchowski Equation. When there is also fragmentation, we need to assume that the solution f is regular in the sense that f can be approximated by solutions to the Smoluchowski Equation for which the coagulation and fragmentation coefficients are 0 when the cluster sizes are large. We also need suitable assumptions on the coagulation rates to avoid gelation. On the fragmentation rate β we assume that sup n sup m ≤ l β( m, n )/n l , and that there exist constants a 0 ≥ 0 and c 0 such that β( n,m ) ≤ c 0 ( n + m )a 0 a0 .
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coagulation diffusion and the continuous Smoluchowski Equation
Stochastic Processes and their Applications, 2009Co-Authors: Mohammad Reza Yaghouti, Fraydoun Rezakhanlou, Alan HammondAbstract:Abstract The Smoluchowski Equations are a system of partial differential Equations modelling the diffusion and binary coagulation of a large collection of tiny particles. The mass parameter may be indexed either by positive integers or by positive reals, these corresponding to the discrete or the continuous form of the Equations. For dimension d ≥ 3 , we derive the continuous Smoluchowski PDE as a kinetic limit of a microscopic model of Brownian particles liable to coalesce, using a method similar to that used to derive the discrete form of the Equations in [A. Hammond, F. Rezakhanlou, The kinetic limit of a system of coagulating Brownian particles, Arch. Ration. Mech. Anal. 185 (2007) 1–67]. The principal innovation is a correlation-type bound on particle locations that permits the derivation in the continuous context while simplifying the arguments of the cited work. We also comment on the scaling satisfied by the continuous Smoluchowski PDE, and its potential implications for blow-up of solutions of the Equations.
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coagulation diffusion and the continuous Smoluchowski Equation
arXiv: Probability, 2008Co-Authors: Mohammad Reza Yaghouti, Fraydoun Rezakhanlou, Alan HammondAbstract:The Smoluchowski Equation is a system of partial differential Equations modelling the diffusion and binary coagulation of a large collection of tiny particles. The mass parameter may be indexed either by positive integers, or by positive reals, these corresponding to the discrete or the continuous form of the Equations. In dimension at least 3, we derive the continuous Smoluchowski PDE as a kinetic limit of a microscopic model of Brownian particles liable to coalesce, using a similar method to that used to derive the discrete form of the Equations in Hammond and Rezakhanlou [4]. The principal innovation is a correlation-type bound on particle locations that permits the derivation in the continuous context while simplifying the arguments of [4]. We also comment on the scaling satisfied by the continuous Smoluchowski PDE, and its potential implications for blow-up of solutions of the Equations.
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moment bounds for the Smoluchowski Equation and their consequences
arXiv: Analysis of PDEs, 2006Co-Authors: Alan Hammond, Fraydoun RezakhanlouAbstract:We prove uniform bounds on moments X_a = \sum_{m}{m^a f_m(x,t)} of the Smoluchowski coagulation Equations with diffusion, valid in any dimension. If the collision propensities \alpha(n,m) of mass n and mass m particles grow more slowly than (n+m)(d(n) + d(m)), and the diffusion rate d(\cdot) is non-increasing and satisfies m^{-b_1} \leq d(m) \leq m^{-b_2} for some b_1 and b_2 satisfying 0 \leq b_2 < b_1 < \infty, then any weak solution satisfies X_a \in L^{\infty}(\mathbb{R}^d \times [0,T]) \cap L^1(\mathbb{R}^d \times [0,T]) for every a \in \mathbb{N} and T \in (0,\infty), (provided that certain moments of the initial data are finite). As a consequence, we infer that these conditions are sufficient to ensure uniqueness of a weak solution and its conservation of mass.
Yuhua Song - One of the best experts on this subject based on the ideXlab platform.
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finite element analysis of the time dependent Smoluchowski Equation for acetylcholinesterase reaction rate calculations
Biophysical Journal, 2007Co-Authors: Yuhui Cheng, Yuhua Song, Yongjie Zhang, Chandrajit L Bajaj, Nathan A Baker, Jason K Suen, Deqiang Zhang, Stephen D Bond, Michael Holst, Andrew J MccammonAbstract:This article describes the numerical solution of the time-dependent Smoluchowski Equation to study diffusion in biomolecular systems. Specifically, finite element methods have been developed to calculate ligand binding rate constants for large biomolecules. The resulting software has been validated and applied to the mouse acetylcholinesterase (mAChE) monomer and several tetramers. Rates for inhibitor binding to mAChE were calculated at various ionic strengths with several different time steps. Calculated rates show very good agreement with experimental and theoretical steady-state studies. Furthermore, these finite element methods require significantly fewer computational resources than existing particle-based Brownian dynamics methods and are robust for complicated geometries. The key finding of biological importance is that the rate accelerations of the monomeric and tetrameric mAChE that result from electrostatic steering are preserved under the non-steady-state conditions that are expected to occur in physiological circumstances.
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tetrameric mouse acetylcholinesterase continuum diffusion rate calculations by solving the steady state Smoluchowski Equation using finite element methods
Biophysical Journal, 2005Co-Authors: Deqiang Zhang, Yuhua Song, Yongjie Zhang, Chandrajit L Bajaj, Nathan A Baker, Jason K Suen, Michael Holst, Zoran Radic, Palmer Taylor, Andrew J MccammonAbstract:The tetramer is the most important form for acetylcholinesterase in physiological conditions, i.e., in the neuromuscular junction and the nervous system. It is important to study the diffusion of acetylcholine to the active sites of the tetrameric enzyme to understand the overall signal transduction process in these cellular components. Crystallographic studies revealed two different forms of tetramers, suggesting a flexible tetramer model for acetylcholinesterase. Using a recently developed finite element solver for the steady-state Smoluchowski Equation, we have calculated the reaction rate for three mouse acetylcholinesterase tetramers using these two crystal structures and an intermediate structure as templates. Our results show that the reaction rates differ for different individual active sites in the compact tetramer crystal structure, and the rates are similar for different individual active sites in the other crystal structure and the intermediate structure. In the limit of zero salt, the reaction rates per active site for the tetramers are the same as that for the monomer, whereas at higher ionic strength, the rates per active site for the tetramers are ∼67%–75% of the rate for the monomer. By analyzing the effect of electrostatic forces on ACh diffusion, we find that electrostatic forces play an even more important role for the tetramers than for the monomer. This study also shows that the finite element solver is well suited for solving the diffusion problem within complicated geometries.
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continuum diffusion reaction rate calculations of wild type and mutant mouse acetylcholinesterase adaptive finite element analysis
Biophysical Journal, 2004Co-Authors: Yuhua Song, Yongjie Zhang, Chandrajit L Bajaj, Nathan A BakerAbstract:As described previously, continuum models, such as the Smoluchowski Equation, offer a scalable framework for studying diffusion in biomolecular systems. This work presents new developments in the efficient solution of the continuum diffusion Equation. Specifically, we present methods for adaptively refining finite element solutions of the Smoluchowski Equation based on a posteriori error estimates. We also describe new, molecular-surface-based models, for diffusional reaction boundary criteria and compare results obtained from these models with the traditional spherical criteria. The new methods are validated by comparison of the calculated reaction rates with experimental values for wild-type and mutant forms of mouse acetylcholinesterase. The results show good agreement with experiment and help to define optimal reactive boundary conditions.
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finite element solution of the steady state Smoluchowski Equation for rate constant calculations
Biophysical Journal, 2004Co-Authors: Yuhua Song, Yongjie Zhang, Tongye Shen, Chandrajit L Bajaj, Andrew J Mccammon, Nathan A BakerAbstract:This article describes the development and implementation of algorithms to study diffusion in biomolecular systems using continuum mechanics Equations. Specifically, finite element methods have been developed to solve the steady-state Smoluchowski Equation to calculate ligand binding rate constants for large biomolecules. The resulting software has been validated and applied to mouse acetylcholinesterase. Rates for inhibitor binding to mAChE were calculated at various ionic strengths with several different reaction criteria. The calculated rates were compared with experimental data and show very good agreement when the correct reaction criterion is used. Additionally, these finite element methods require significantly less computational resources than existing particle-based Brownian dynamics methods.
