Tetrameric mouse acetylcholinesterase: continuum diffusion rate calculations by solving the steady-state Smoluchowski equation using finite element methods

Charged Substrate and Product Together Contribute Like a Nonreactive Species to the Overall Electrostatic Steering in Diffusion-Reaction Processes

The Debye-Hückel limiting law is used to study the binding kinetics of substrate-enzyme system as well as to estimate the reaction rate of a electrostatically steered diffusion-controlled reaction process. It is based on a linearized Poisson-Boltzmann model and known for its accurate predictions in dilute solutions. However, the substrate and product particles are in nonequilibrium states and are possibly charged, and their contributions to the total electrostatic field cannot be explicitly studied in the Poisson-Boltzmann model. Hence the influences of substrate and product on reaction rate coefficient were not known. In this work, we consider all the charged species, including the charged substrate, product, and mobile salt ions in a Poisson-Nernst-Planck model, and then compare the results with previous work. The results indicate that both the charged substrate and product can significantly influence the reaction rate coefficient with different behaviors under different setups of computational conditions. It is interesting to find that when substrate and product are both considered, under an overall neutral boundary condition for all the bulk charged species, the computed reaction rate kinetics recovers a similar Debye-Hückel limiting law again. This phenomenon implies that the charged product counteracts the influence of charged substrate on reaction rate coefficient. Our analysis discloses the fact that the total charge concentration of substrate and product, though in a nonequilibrium state individually, obeys an equilibrium Boltzmann distribution, and therefore contributes as a normal charged ion species to ionic strength. This explains why the Debye-Hückel limiting law still works in a considerable range of conditions even though the effects of charged substrate and product particles are not specifically and explicitly considered in the theory.

Diffusion-controlled reaction rates for two active sites on a sphere

BACKGROUND

The diffusion-limited reaction rate of a uniform spherical reactant is generalized to anisotropic reactivity. Previous work has shown that the protein model of a uniform sphere is unsatisfactory in many cases. Competition of ligands binding to two active sites, on a spherical enzyme or cell is studied analytically.

RESULTS

The reaction rate constant is given for two sites at opposite ends of the species of interest. This is compared with twice the reaction rate for a single site. It is found that the competition between sites lowers the reaction rate over what is expected for two sites individually. Competition between sites does not show up, until the site half angle is greater than 30 degrees.

CONCLUSIONS

Competition between sites is negligible until the site size becomes large. The competitive effect grows as theta becomes large. The maximum effect is given for theta = pi/2.

Influence of neighboring reactive particles on diffusion-limited reactions

Competition between reactive species is commonplace in typical chemical reactions. Specifically the primary reaction between a substrate and its target enzyme may be altered when interactions with secondary species in the system are substantial. We explore this competition phenomenon for diffusion-limited reactions in the presence of neighboring particles through numerical solution of the diffusion equation. As a general model for globular proteins and small molecules, we consider spherical representations of the reactants and neighboring particles; these neighbors vary in local density, size, distribution, and relative distance from the primary target reaction, as well as their surface reactivity. Modulations of these model variables permit inquiry into the influence of excluded volume and competition on the primary reaction due to the presence of neighboring particles. We find that the surface reactivity effect is long-ranged and a strong determinant of reaction kinetics, whereas the excluded volume effect is relatively short-ranged and less influential in comparison. As a consequence, the effect of the excluded volume is only modestly dependent on the neighbor distribution and is approximately additive; this additivity permits a linear approximation to the many-body effect on the reaction kinetics. In contrast, the surface reactivity effect is non-additive, and thus it may require higher-order approximations to describe the reaction kinetics. Our model study has broad implications in the general understanding of competition and local crowding on diffusion-limited chemical reactions.

Multi-core CPU or GPU-accelerated Multiscale Modeling for Biomolecular Complexes

Multi-scale modeling plays an important role in understanding the structure and biological functionalities of large biomolecular complexes. In this paper, we present an efficient computational framework to construct multi-scale models from atomic resolution data in the Protein Data Bank (PDB), which is accelerated by multi-core CPU and programmable Graphics Processing Units (GPU). A multi-level summation of Gaus-sian kernel functions is employed to generate implicit models for biomolecules. The coefficients in the summation are designed as functions of the structure indices, which specify the structures at a certain level and enable a local resolution control on the biomolecular surface. A method called neighboring search is adopted to locate the grid points close to the expected biomolecular surface, and reduce the number of grids to be analyzed. For a specific grid point, a KD-tree or bounding volume hierarchy is applied to search for the atoms contributing to its density computation, and faraway atoms are ignored due to the decay of Gaussian kernel functions. In addition to density map construction, three modes are also employed and compared during mesh generation and quality improvement to generate high quality tetrahedral meshes: CPU sequential, multi-core CPU parallel and GPU parallel. We have applied our algorithm to several large proteins and obtained good results.

Multi-Scale Continuum Modeling of Biological Processes: From Molecular Electro-Diffusion to Sub-Cellular Signaling Transduction

This article provides a brief review of multi-scale modeling at the molecular to cellular scale, with new results for heart muscle cells. A finite element-based simulation package (SMOL) was used to investigate the signaling transduction at molecular and sub-cellular scales (http://mccammon.ucsd.edu/smol/, http://FETK.org) by numerical solution of time-dependent Smoluchowski equations and a reaction-diffusion system. At the molecular scale, SMOL has yielded experimentally-validated estimates of the diffusion-limited association rates for the binding of acetylcholine to mouse acetylcholinesterase using crystallographic structural data. The predicted rate constants exhibit increasingly delayed steady-state times with increasing ionic strength and demonstrate the role of an enzyme's electrostatic potential in influencing ligand binding. At the sub-cellular scale, an extension of SMOL solves a non-linear, reaction-diffusion system describing Ca²⁺ ligand buffering and diffusion in experimentally-derived rodent ventricular myocyte geometries. Results reveal the important role for mobile and stationary Ca²⁺ buffers, including Ca²⁺ indicator dye. We found that the alterations in Ca²⁺-binding and dissociation rates of troponin C (TnC) and total TnC concentration modulate subcellular Ca²⁺ signals. Model predicts that reduced off-rate in whole troponin complex (TnC, TnI, TnT) versus reconstructed thin filaments (Tn, Tm, actin) alters cytosolic Ca²⁺ dynamics under control conditions or in disease-linked TnC mutations. The ultimate goal of these studies is to develop scalable methods and theories for integration of molecular-scale information into simulations of cellular-scale systems.

Differential geometry based solvation model II: Lagrangian formulation

Solvation is an elementary process in nature and is of paramount importance to more sophisticated chemical, biological and biomolecular processes. The understanding of solvation is an essential prerequisite for the quantitative description and analysis of biomolecular systems. This work presents a Lagrangian formulation of our differential geometry based solvation models. The Lagrangian representation of biomolecular surfaces has a few utilities/advantages. First, it provides an essential basis for biomolecular visualization, surface electrostatic potential map and visual perception of biomolecules. Additionally, it is consistent with the conventional setting of implicit solvent theories and thus, many existing theoretical algorithms and computational software packages can be directly employed. Finally, the Lagrangian representation does not need to resort to artificially enlarged van der Waals radii as often required by the Eulerian representation in solvation analysis. The main goal of the present work is to analyze the connection, similarity and difference between the Eulerian and Lagrangian formalisms of the solvation model. Such analysis is important to the understanding of the differential geometry based solvation model. The present model extends the scaled particle theory of nonpolar solvation model with a solvent-solute interaction potential. The nonpolar solvation model is completed with a Poisson-Boltzmann (PB) theory based polar solvation model. The differential geometry theory of surfaces is employed to provide a natural description of solvent-solute interfaces. The optimization of the total free energy functional, which encompasses the polar and nonpolar contributions, leads to coupled potential driven geometric flow and PB equations. Due to the development of singularities and nonsmooth manifolds in the Lagrangian representation, the resulting potential-driven geometric flow equation is embedded into the Eulerian representation for the purpose of computation, thanks to the equivalence of the Laplace-Beltrami operator in the two representations. The coupled partial differential equations (PDEs) are solved with an iterative procedure to reach a steady state, which delivers desired solvent-solute interface and electrostatic potential for problems of interest. These quantities are utilized to evaluate the solvation free energies and protein-protein binding affinities. A number of computational methods and algorithms are described for the interconversion of Lagrangian and Eulerian representations, and for the solution of the coupled PDE system. The proposed approaches have been extensively validated. We also verify that the mean curvature flow indeed gives rise to the minimal molecular surface and the proposed variational procedure indeed offers minimal total free energy. Solvation analysis and applications are considered for a set of 17 small compounds and a set of 23 proteins. The salt effect on protein-protein binding affinity is investigated with two protein complexes by using the present model. Numerical results are compared to the experimental measurements and to those obtained by using other theoretical methods in the literature.

Differential geometry based solvation model II: Lagrangian formulation

Solvation is an elementary process in nature and is of paramount importance to more sophisticated chemical, biological and biomolecular processes. The understanding of solvation is an essential prerequisite for the quantitative description and analysis of biomolecular systems. This work presents a Lagrangian formulation of our differential geometry based solvation models. The Lagrangian representation of biomolecular surfaces has a few utilities/advantages. First, it provides an essential basis for biomolecular visualization, surface electrostatic potential map and visual perception of biomolecules. Additionally, it is consistent with the conventional setting of implicit solvent theories and thus, many existing theoretical algorithms and computational software packages can be directly employed. Finally, the Lagrangian representation does not need to resort to artificially enlarged van der Waals radii as often required by the Eulerian representation in solvation analysis. The main goal of the present work is to analyze the connection, similarity and difference between the Eulerian and Lagrangian formalisms of the solvation model. Such analysis is important to the understanding of the differential geometry based solvation model. The present model extends the scaled particle theory of nonpolar solvation model with a solvent-solute interaction potential. The nonpolar solvation model is completed with a Poisson-Boltzmann (PB) theory based polar solvation model. The differential geometry theory of surfaces is employed to provide a natural description of solvent-solute interfaces. The optimization of the total free energy functional, which encompasses the polar and nonpolar contributions, leads to coupled potential driven geometric flow and PB equations. Due to the development of singularities and nonsmooth manifolds in the Lagrangian representation, the resulting potential-driven geometric flow equation is embedded into the Eulerian representation for the purpose of computation, thanks to the equivalence of the Laplace-Beltrami operator in the two representations. The coupled partial differential equations (PDEs) are solved with an iterative procedure to reach a steady state, which delivers desired solvent-solute interface and electrostatic potential for problems of interest. These quantities are utilized to evaluate the solvation free energies and protein-protein binding affinities. A number of computational methods and algorithms are described for the interconversion of Lagrangian and Eulerian representations, and for the solution of the coupled PDE system. The proposed approaches have been extensively validated. We also verify that the mean curvature flow indeed gives rise to the minimal molecular surface and the proposed variational procedure indeed offers minimal total free energy. Solvation analysis and applications are considered for a set of 17 small compounds and a set of 23 proteins. The salt effect on protein-protein binding affinity is investigated with two protein complexes by using the present model. Numerical results are compared to the experimental measurements and to those obtained by using other theoretical methods in the literature.

Differential geometry based solvation model I: Eulerian formulation

This paper presents a differential geometry based model for the analysis and computation of the equilibrium property of solvation. Differential geometry theory of surfaces is utilized to define and construct smooth interfaces with good stability and differentiability for use in characterizing the solvent-solute boundaries and in generating continuous dielectric functions across the computational domain. A total free energy functional is constructed to couple polar and nonpolar contributions to the salvation process. Geometric measure theory is employed to rigorously convert a Lagrangian formulation of the surface energy into an Eulerian formulation so as to bring all energy terms into an equal footing. By minimizing the total free energy functional, we derive coupled generalized Poisson-Boltzmann equation (GPBE) and generalized geometric flow equation (GGFE) for the electrostatic potential and the construction of realistic solvent-solute boundaries, respectively. By solving the coupled GPBE and GGFE, we obtain the electrostatic potential, the solvent-solute boundary profile, and the smooth dielectric function, and thereby improve the accuracy and stability of implicit solvation calculations. We also design efficient second order numerical schemes for the solution of the GPBE and GGFE. Matrix resulted from the discretization of the GPBE is accelerated with appropriate preconditioners. An alternative direct implicit (ADI) scheme is designed to improve the stability of solving the GGFE. Two iterative approaches are designed to solve the coupled system of nonlinear partial differential equations. Extensive numerical experiments are designed to validate the present theoretical model, test computational methods, and optimize numerical algorithms. Example solvation analysis of both small compounds and proteins are carried out to further demonstrate the accuracy, stability, efficiency and robustness of the present new model and numerical approaches. Comparison is given to both experimental and theoretical results in the literature.

Enzymatic activity versus structural dynamics: the case of acetylcholinesterase tetramer

The function of many proteins, such as enzymes, is modulated by structural fluctuations. This is especially the case in gated diffusion-controlled reactions (where the rates of the initial diffusional encounter and of structural fluctuations determine the overall rate of the reaction) and in oligomeric proteins (where function often requires a coordinated movement of individual subunits). A classic example of a diffusion-controlled biological reaction catalyzed by an oligomeric enzyme is the hydrolysis of synaptic acetylcholine (ACh) by tetrameric acetylcholinesterase (AChEt). Despite decades of efforts, the extent to which enzymatic efficiency of AChEt (or any other enzyme) is modulated by flexibility is not fully determined. This article attempts to determine the correlation between the dynamics of AChEt and the rate of reaction between AChEt and ACh. We employed equilibrium and nonequilibrium electro-diffusion models to compute rate coefficients for an ensemble of structures generated by molecular dynamics simulation. We found that, for the static initial model, the average reaction rate per active site is approximately 22-30% slower in the tetramer than in the monomer. However, this effect of tetramerization is modulated by the intersubunit motions in the tetramer such that a complex interplay of steric and electrostatic effects either guides or blocks the substrate into or from each of the four active sites. As a result, the rate per active site calculated for some of the tetramer structures is only approximately 15% smaller than the rate in the monomer. We conclude that structural dynamics minimizes the adverse effect of tetramerization, allowing the enzyme to maintain similar enzymatic efficiency in different oligomerization states.

PROTO-PLASM: parallel language for adaptive and scalable modelling of biosystems

This paper discusses the design goals and the first developments of PROTO-PLASM, a novel computational environment to produce libraries of executable, combinable and customizable computer models of natural and synthetic biosystems, aiming to provide a supporting framework for predictive understanding of structure and behaviour through multiscale geometric modelling and multiphysics simulations. Admittedly, the PROTO-PLASM platform is still in its infancy. Its computational framework--language, model library, integrated development environment and parallel engine--intends to provide patient-specific computational modelling and simulation of organs and biosystem, exploiting novel functionalities resulting from the symbolic combination of parametrized models of parts at various scales. PROTO-PLASM may define the model equations, but it is currently focused on the symbolic description of model geometry and on the parallel support of simulations. Conversely, CellML and SBML could be viewed as defining the behavioural functions (the model equations) to be used within a PROTO-PLASM program. Here we exemplify the basic functionalities of PROTO-PLASM, by constructing a schematic heart model. We also discuss multiscale issues with reference to the geometric and physical modelling of neuromuscular junctions.

Diffusional channeling in the sulfate-activating complex: combined continuum modeling and coarse-grained brownian dynamics studies

Enzymes required for sulfur metabolism have been suggested to gain efficiency by restricted diffusion (i.e., channeling) of an intermediate APS(2-) between active sites. This article describes modeling of the whole channeling process by numerical solution of the Smoluchowski diffusion equation, as well as by coarse-grained Brownian dynamics. The results suggest that electrostatics plays an essential role in the APS(2-) channeling. Furthermore, with coarse-grained Brownian dynamics, the substrate channeling process has been studied with reactions in multiple active sites. Our simulations provide a bridge for numerical modeling with Brownian dynamics to simulate the complicated reaction and diffusion and raise important questions relating to the electrostatically mediated substrate channeling in vitro, in situ, and in vivo.

Exact solution for anisotropic diffusion-controlled reactions with partially reflecting conditions

We investigate a generalization of the model of Solc and Stockmayer to describe the diffusion-controlled reactions between chemically anisotropic reactants taking into account the partially reflecting conditions on two parts of the reaction surface. The exact solution of the relevant mixed boundary-value problem was found for different ratios of the intrinsic rate constants. The results obtained may be used to test numerical programs that describe diffusion-controlled reactions in real systems of particles with anisotropic reactivity.

Parameters for Carbamate Pesticide QSAR and PBPK/PD Models for Human Risk Assessment

Our interest in providing parameters for the development of quantitative structure physiologically based pharmacokinetic/pharmacodynamic (QSPBPK/PD) models for assessing health risks to carbamates (USEPA 2005) comes from earlier work with organophosphorus (OP) insecticides (Knaak et al. 2004). Parameters specific to each carbamate are needed in the construction of PBPK/PD models along with their metabolic pathways. Parameters may be obtained by (1) development of QSAR models, (2) collecting pharmacokinetic data, and (3) determining pharmacokinetic parameters by fitting to experimental data. The biological parameters are given in Table 1 (Blancato et al. 2000). Table 1 Biological Parameters Required for Carbamate Pesticide Physiologically Based Pharmacokinetic/Pharmacodynamic (PBPK/PD) Models.(a).

Continuum simulations of acetylcholine consumption by acetylcholinesterase: a Poisson-Nernst-Planck approach

The Poisson-Nernst-Planck (PNP) equation provides a continuum description of electrostatic-driven diffusion and is used here to model the diffusion and reaction of acetylcholine (ACh) with acetylcholinesterase (AChE) enzymes. This study focuses on the effects of ion and substrate concentrations on the reaction rate and rate coefficient. To this end, the PNP equations are numerically solved with a hybrid finite element and boundary element method at a wide range of ion and substrate concentrations, and the results are compared with the partially coupled Smoluchowski-Poisson-Boltzmann model. The reaction rate is found to depend strongly on the concentrations of both the substrate and ions; this is explained by the competition between the intersubstrate repulsion and the ionic screening effects. The reaction rate coefficient is independent of the substrate concentration only at very high ion concentrations, whereas at low ion concentrations the behavior of the rate depends strongly on the substrate concentration. Moreover, at physiological ion concentrations, variations in substrate concentration significantly affect the transient behavior of the reaction. Our results offer a reliable estimate of reaction rates at various conditions and imply that the concentrations of charged substrates must be coupled with the electrostatic computation to provide a more realistic description of neurotransmission and other electrodiffusion and reaction processes.

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

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.

Dynamics of the acetylcholinesterase tetramer

Acetylcholinesterase rapidly hydrolyzes the neurotransmitter acetylcholine in cholinergic synapses, including the neuromuscular junction. The tetramer is the most important functional form of the enzyme. Two low-resolution crystal structures have been solved. One is compact with two of its four peripheral anionic sites (PAS) sterically blocked by complementary subunits. The other is a loose tetramer with all four subunits accessible to solvent. These structures lacked the C-terminal amphipathic t-peptide (WAT domain) that interacts with the proline-rich attachment domain (PRAD). A complete tetramer model (AChEt) was built based on the structure of the PRAD/WAT complex and the compact tetramer. Normal mode analysis suggested that AChEt could exist in several conformations with subunits fluctuating relative to one another. Here, a multiscale simulation involving all-atom molecular dynamics and C alpha-based coarse-grained Brownian dynamics simulations was carried out to investigate the large-scale intersubunit dynamics in AChEt. We sampled the ns-mus timescale motions and found that the tetramer indeed constitutes a dynamic assembly of monomers. The intersubunit fluctuation is correlated with the occlusion of the PAS. Such motions of the subunits "gate" ligand-protein association. The gates are open more than 80% of the time on average, which suggests a small reduction in ligand-protein binding. Despite the limitations in the starting model and approximations inherent in coarse graining, these results are consistent with experiments which suggest that binding of a substrate to the PAS is only somewhat hindered by the association of the subunits.

Finite element analysis of the time-dependent Smoluchowski equation for acetylcholinesterase reaction rate calculations

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.

Continuum simulations of acetylcholine diffusion with reaction-determined boundaries in neuromuscular junction models

The reaction-diffusion system of the neuromuscular junction has been modeled in 3D using the finite element package FEtk. The numerical solution of the dynamics of acetylcholine with the detailed reaction processes of acetylcholinesterases and nicotinic acetylcholine receptors has been discussed with the reaction-determined boundary conditions. The simulation results describe the detailed acetylcholine hydrolysis process, and reveal the time-dependent interconversion of the closed and open states of the acetylcholine receptors as well as the percentages of unliganded/monoliganded/diliganded states during the neuro-transmission. The finite element method has demonstrated its flexibility and robustness in modeling large biological systems.

Quality Meshing of Implicit Solvation Models of Biomolecular Structures

This paper describes a comprehensive approach to construct quality meshes for implicit solvation models of biomolecular structures starting from atomic resolution data in the Protein Data Bank (PDB). First, a smooth volumetric electron density map is constructed from atomic data using weighted Gaussian isotropic kernel functions and a two-level clustering technique. This enables the selection of a smooth implicit solvation surface approximation to the Lee-Richards molecular surface. Next, a modified dual contouring method is used to extract triangular meshes for the surface, and tetrahedral meshes for the volume inside or outside the molecule within a bounding sphere/box of influence. Finally, geometric flow techniques are used to improve the surface and volume mesh quality. Several examples are presented, including generated meshes for biomolecules that have been successfully used in finite element simulations involving solvation energetics and binding rate constants.

Spatial modeling of dimerization reaction dynamics in the plasma membrane: Monte Carlo vs. continuum differential equations

Bimolecular reactions in the plasma membrane, such as receptor dimerization, are a key signaling step for many signaling systems. For receptors to dimerize, they must first diffuse until a collision happens, upon which a dimerization reaction may occur. Therefore, study of the dynamics of cell signaling on the membrane may require the use of a spatial modeling framework. Despite the availability of spatial simulation methods, e.g., stochastic spatial Monte Carlo (MC) simulation and partial differential equation (PDE) based approaches, many biological models invoke well-mixed assumptions without completely evaluating the importance of spatial organization. Whether one is to utilize a spatial or non-spatial simulation framework is therefore an important decision. In order to evaluate the importance of spatial effects a priori, i.e., without performing simulations, we have assessed the applicability of a dimensionless number, known as second Damköhler number (Da), defined here as the ratio of time scales of collision and reaction, for 2-dimensional bimolecular reactions. Our study shows that dimerization reactions in the plasma membrane with Da approximately >0.1 (tested in the receptor density range of 10(2)-10(5)/microm(2)) require spatial modeling. We also evaluated the effective reaction rate constants of MC and simple deterministic PDEs. Our simulations show that the effective reaction rate constant decreases with time due to time dependent changes in the spatial distribution of receptors. As a result, the effective reaction rate constant of simple PDEs can differ from that of MC by up to two orders of magnitude. Furthermore, we show that the fluctuations in the number of copies of signaling proteins (noise) may also depend on the diffusion properties of the system. Finally, we used the spatial MC model to explore the effect of plasma membrane heterogeneities, such as receptor localization and reduced diffusivity, on the dimerization rate. Interestingly, our simulations show that localization of epidermal growth factor receptor (EGFR) can cause the diffusion limited dimerization rate to be up to two orders of magnitude higher at higher average receptor densities reported for cancer cells, as compared to a normal cell.

The association of tetrameric acetylcholinesterase with ColQ tail: a block normal mode analysis

Acetylcholinesterase (AChE) rapidly hydrolyzes acetylcholine in the neuromuscular junctions and other cholinergic synapses to terminate the neuronal signal. In physiological conditions, AChE exists as tetramers associated with the proline-rich attachment domain (PRAD) of either collagen-like Q subunit (ColQ) or proline-rich membrane-anchoring protein. Crystallographic studies have revealed that different tetramer forms may be present, and it is not clear whether one or both are relevant under physiological conditions. Recently, the crystal structure of the tryptophan amphiphilic tetramerization (WAT) domain of AChE associated with PRAD ([WAT]₄PRAD), which mimics the interface between ColQ and AChE tetramer, became available. In this study we built a complete tetrameric mouse [AChE_T]₄–ColQ atomic structure model, based on the crystal structure of the [WAT]₄PRAD complex. The structure was optimized using energy minimization. Block normal mode analysis was done to investigate the low-frequency motions of the complex and to correlate the structure model with the two known crystal structures of AChE tetramer. Significant low-frequency motions among the catalytic domains of the four AChE subunits were observed, while the [WAT]₄PRAD part held the complex together. Normal mode involvement analysis revealed that the two lowest frequency modes were primarily involved in the conformational changes leading to the two crystal structures. The first 30 normal modes can account for more than 75% of the conformational changes in both cases. The evidence further supports the idea of a flexible tetramer model for AChE. This model can be used to study the implications of the association of AChE with ColQ.

Acetylcholinesterase (AChE) breaks down acetylcholine in the neuromuscular junction and other cholinergic synapses to terminate neuronal signals. AChE exists as tetramers anchored by structural subunits to the cell membranes in the brain or the basal lamina in the neuromuscular junction. Based on a crystal structure of the tetramerization domain of AChE with a proline-rich attachment domain of the anchoring proteins, a symmetric model of the complex of AChE tetramer with the anchoring protein tail was constructed. Block normal mode analysis revealed the presence of several low-frequency, low-barrier normal modes corresponding to inter-subunit motions. Previous crystal structures of AChE tetramer could be rationalized using these normal modes. These low-frequency modes are due to the presence of a flexible hinge in the structure of AChE. This study paints a picture of a flexible AChE tetramer with different conformational states interconverting easily under physiological conditions, which has important implications on the function of AChE. In particular, AChE is not trapped in the compact tetramer structure, for which access of substrate to two of the active sites is somewhat limited. Rather, the tetramer fluctuates to expose all four of its active sites to ensure rapid removal of acetylcholine.