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A nonlinear elasticity model of macromolecular conformational change induced by electrostatic forces
Y.C. Zhou, Michael Holst and J. Andrew McCammon
Journal of Mathematical Analysis and Applications, Vol. 340, Issue 1, pp. 135-164 (2008)
In this paper we propose a nonlinear elasticity model of macromolecular conformational change (deformation) induced by electrostatic forces generated by an implicit solvation model. The Poisson-Boltzmann equation for the electrostatic potential is analyzed in a domain varying with the elastic deformation of molecules, and a new continuous model of the electrostatic forces is developed to ensure solvability of the nonlinear elasticity equations. We derive the estimates of electrostatic forces corresponding to four types of perturbations to an electrostatic potential field, and establish the existance of an equilibrium configuration using a fixed-point argument, under the assumption that the change in the ionic strength and charges due to the additional molecules causing the deformation are sufficiently small.The results are valid for elastic models with arbitrarily complex dielectric interfaces and cavities, and can be generalized to large elastic deformation caused by high ionic strength, large charges, and strong external fields by using continuation methods.
Springs and Speeds in Free Energy Reconstruction from Irreversible Single-Molecule Pulling Experiments
David D.L. Minh and J. Andrew McCammon
Journal of Physical Chemistry B, Vol. 112, No. 19, pp. 5892-5897 (2008, A. Szabo festschrift)
The nonequilibrium work relation allows for the calculation of equilibrium free energy differences between states based on the exponential average of accumulated work from irreversible transitions. Here, we compare two distinct approaches of calculating free energy surfaces from unidirectional singlemolecule pulling experiments: the stiff spring approximation and the Hummer-Szabo method. First, we perform steered molecular dynamics simulations to mechanically stretch the model peptide deca-alanine using harmonic potentials with different spring stiffnesses and at various constant pulling velocities. We then calculate free energy surfaces based on the two methods and their variants, including the first and second cumulant expansion of the exponentially weighted work and the Gaussian position approximation for the delta function in Hummer and Szabo's expression. We find that with large harmonic force constants, the second cumulant expansion in conjunction with either the stiff spring approximation or the Hummer-Szabo method perform well. When interpreting dynamic force spectroscopy (pullings at different speeds), the second cumulant expansion of the stiff spring approximation performs the best when pulling velocities are similar, but variants of the Hummer-Szabo perform the best when they are spread over a large spectrum. While these conclusion are not definitive for all systems, the insights should prove useful for scientists interpreting nonequilibrium pulling experiments.
Continuum Simulations of Acetylcholine Consumption by Acetylcholinesterase: A Poisson-Nernst-Planck Approach
Y.C. Zhou, Benzhuo Lu, Gary A. Huber, Michael J. Holst and J. Andrew McCammon
Journal of Physical Chemistry B, Vol. 112, No. 2, pp. 270-275 (2008, Casey Hynes festschrift)
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.
Dynamics of the Acetylcholinesterase Tetramer
Alemayehu A. Gorfe, Chia-en A. Chang, Ivaylo Ivanov and J. Andrew McCammon
Biophysical Journal, Vol. 94, Issue 4, pp. 1144-1154 (2008)
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 multiple conformations with subunits fluctuating relative to one another. Here, a multiscale simulation involving all-atom molecular dynamics and Cbeta-based coarse-grained Brownian dynamics simulations was carried out to investigate the large scale inter-subunit dynamics in AChEt. We sampled the ns-micros time scale motions and found that the tetramer indeed constitutes a dynamic assembly of monomers. The inter-subunit 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 of 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.
Computing accurate potentials of mean force in electrolyte solutions with the generalized gradient-augmented harmonic Fourier beads method
Ilja V. Khavrutskii, Joachim Dzubiella and J. Andrew McCammon
Journal of Chemical Physics, Vol. 128, article 044106, 13 pages (2008)
We establish the accuracy of the novel generalized gradient-augmented Harmonic Fourier Beads (ggaHFB) method in computing free-energy profiles or potentials of mean force (PMFs) through comparison with two independent conventional techniques. In particular, we employ umbrella sampling with 1D weighted histogram analysis method (WHAM) and free molecular dynamics simulation of radial distribution functions to compute the PMF for the Na+-Cl- ion pair separation to 16 Å in 1.0 M NaCl solution in water. The corresponding ggaHFB free-energy profile in 6D Cartesian space is in excellent agreement with the conventional benchmarks. We then explore changes in the PMF in response to lowering the NaCl concentration to physiological 0.3 and 0.1 M, and dilute 0.0 M concentrations. Finally, to expand the scope of the ggaHFB method, we formally develop the free-energy gradient approximation in arbitrary nonlinear coordinates. This formal development underscores the importance of the logarithmic Jacobian correction to reconstruct true PMFs from umbrella sampling simulations with either WHAM or ggaHFB techniques when nonlinear coordinate restraints are used with Cartesian propagators. The ability to employ nonlinear coordinates and high accuracy of the computed free-energy profiles further advocate the use of the ggaHFB method in studies of rare events in complex systems.
Molecular surface-free continuum model for electrodiffusion processes
Benzhuo Lu and J. Andrew McCammon
Chemical Physics Letters, Vol. 451, Issues 4-6, pp. 282-286 (2008)
Incorporation of van der Waals interactions enables the continuum model of electrodiffusion in biomolecular system to avoid the artifacts of introducing a molecular surface and the painful task of the surface mesh generation. Calculation examples show that the electrostatics, diffusion-reaction kinetics, and molecular surface defined as an isosurface of a certain density distribution can be extracted from the solution of the Poisson-Nernst-Planck equations using this model. The molecular surface-free model enables a wider usage of some modern numerical methodologies such as finite element methods for biomolecular modeling, and sheds light on a new paradigm of continuum modeling for biomolecular systems.
An Improved Relaxed Complex Scheme for Receptor Flexibility in Computer-Aided Drug Design
Rommie E. Amaro, Riccardo Baron and J. Andrew McCammon
Journal of Computer-Aided Molecular Design, in press (2008)
The interactions among associating (macro) molecules are dynamic, which adds to the complexity of molecular recognition. While ligand flexibility is well accounted for in computational drug design, the effective inclusion of receptor flexibility remains an important challenge. The relaxed complex scheme (RCS) is a promising computational methodology that combines the advantages of docking algorithms with dynamic structural information provided by molecular dynamics (MD) simulations, therefore explicitly accounting for the flexibility of both the receptor and the docked ligands. Here, we briefly review the RCS and discuss new extensions and improvements of this methodology in the context of ligand binding to two example targets: kinetoplastid RNA editing ligase 1 and the W191G cavity mutant of cytochrome c peroxidase. The RCS improvements include its extension to virtual screening, more rigorous characterization of local and global binding effects, and methods to improve its computational efficiency by reducing the receptor ensemble to a representative set of configurations. The choice of receptor ensemble, its influence on the predictive power of RCS, and the current limitations for an accurate treatment of the solvent contributions are also briefly discussed. Finally, we outline potential methodological improvements that we anticipate will assist future development.
Novel Druggable Hot Spots in Avian Influenza Neuraminidase H5N1 Revealed by Computational Solvent Mapping of a Reduced and Representative Receptor Ensemble
Landon, M.R., R.E. Amaro, R. Baron, C.H. Ngan, D. Ozonoff, J.A. McCammon and S. Vajda
Chemical Biology & Drug Design, Vol. 71, Issue 2, pp. 106-116 (2008)
The influenza virus subtype H5N1 has raised concerns of a possible human pandemic threat because of its high virulence and mutation rate. Although several approved anti-influenza drugs effectively target the neuraminidase, some strains have already acquired resistance to the currently available anti-influenza drugs. In this study, we present the synergistic application of extended explicit solvent molecular dynamics (MD) and computational solvent mapping (CS-Map) to identify putative “hot spots” within flexible binding regions of N1 neuraminidase. Using representative conformations of the N1 binding region extracted from a clustering analysis of four concatenated 40-ns MD simulations, CS-Map was utilized to assess the ability of small, solvent-sized molecules to bind within close proximity to the sialic acid binding region. Mapping analyses of the dominant MD conformations reveal the presence of additional hot spot regions in the 150- and 430-loop regions. Our hot spot analysis provides further support for the feasibility of developing high-affinity inhibitors capable of binding these regions, which appear to be unique to the N1 strain.
Electrostatic Free Energy and its Variations in Implicit Solvent Models
Jianwei Che, Joachim Dzubiella, Bo Li and J. Andrew McCammon
Journal of Physical Chemistry B, Vol. 112, Issue 10, pp. 3058-3069 (2008)
A mean-field approach to the electrostatics for solutes in electrolyte solution is revisited and rigorously justified. In this approach, an electrostatic free energy functional is constructed that depends solely on the local ionic concentrations. The unique set of such concentrations that minimize this free energy are given by the usual Boltzmann distributions through the electrostatic potential which is determined by the Poisson-Boltzmann equation. This approach is then applied to the variational implicit solvent description of the solvation of molecules [Dzubiella, Swanson, McCammon, Phys. Rev. Lett. 2006, 96, 087802; J. Chem. Phys. 2006, 124, 084905]. Care is taken for the singularities of the potential generated by the solute point charges. The variation of the electrostatic free energy with respect to the location change of solute-solvent interfaces, that is, dielectric boundaries, is derived. Such a variation gives rise to the normal component of the effective surface force per unit surface area that is shown to be attractive to the fixed point charges in the solutes. Two examples of applications are given to validate the analytical results. The first one is a one-dimensional model system resembling, for example, a charged solute or cavity in a one-dimensional channel. The second one, which is of its own interest, is the electrostatic free energy of a charged sphercal solute immersed in an ionic solution. An analytical formula is derived for the Debye-Hückel approximation of the free energy, extending the classical Born's formula to one that includes ionic concentrations. Variations of the nonlinear Poisson-Boltzmann free energy are also obtained.
Control of cation permeation through the nicotinic receptor channel
Hai-Long Wang, Xiaolin Chen, Palmer Taylor, J. Andrew McCammon and Steven M. Sine
PLoS Computational Biology, Vol. 4, Issue 2, article e41, 9 pages (2008)
We used molecular dynamics (MD) simulations to explore the transport of single cations through the channel of the muscle nicotinic acetylcholine receptor (nAChR). Four MD simulations of 16 ns were performed at physiological and hyperpolarized membrane potentials, with and without restraints of the structure, but all without bound agonist. With the structure unrestrained and a potential of -100 mV, one cation traversed the channel during a transient period of channel hydration; at -200 mV, two cations traversed the channel while the channel was continuously hydrated. With the structure restrained, however, no cations traverse at either membrane potential, even though the channel was continuously hydrated. The overall results show that cation selective transport through nAChR channel is governed by electrostatics interactions to achieve charge selectivity, but relies on trans-membrane potential, channel hydration and protein dynamics to enable ion passage.
Recent Progress in Numerical Methods for the Poisson-Boltzmann Equation in Biophysical Applications
B.Z. Lu, Y.C. Zhou, M.J. Holst and J.A. McCammon
Communications in Computational Physics, Vol. 3, Issue 5, pp. 973-1009 (2008)
Efficiency and accuracy are two major concerns in numerical solutions of the Poisson-Boltzmann equation for applications in chemistry and biophysics. Recent developments in boundary element methods, interface methods, adaptive methods, finite element methods, and other approaches for the Poisson-Boltzmann equation as well as related mesh generation techniques are reviewed. We also discussed the challenging problems and possible future work, in particular, for the aim of biophysical applications.
A novel switch region regulates H-ras membrane orientation and signal input
Daniel Abankwa, Michael Hanzal-Bayer, Nicolas Ariotti, Sarah J. Plowman, Alemayehu A. Gorfe, Robert G. Parton, J. Andrew McCammon and John F. Hancock
EMBO Journal, Vol. 27, pp. 727-735 (2008)
The plasma membrane nanoscale distribution of H-ras is regulated by guanine nucleotide binding. To explore the structural basis of H-ras membrane organization, we combined molecular dynamic simulations and medium-throughput FRET measurements on live cells. We extracted a set of FRET values, termed a FRET vector, to describe the lateral segregation and orientation of H-ras with respect to a large set of nanodomain markers. We show that mutation of basic residues in helix α4 or the hypervariable region (HVR) selectively alter the FRET vectors of GTP- or GDP-loaded H-ras, demonstrating a critical role for these residues in stabilizing GTP- or GDP-H-ras interactions with the plasma membrane. By a similar analysis, we find that the β2-β3 loop and helix α5 are involved in a novel conformational switch that operates through helix α4 and the HVR to reorient the H-ras G-domain with respect to the plasma membrane. Perturbation of these switch elements enhances MAPK activation by stabilizing GTP-H-ras in a more productive signalling conformation. The results illustrate how the plasma membrane spatially constrains signalling conformations by acting as a semi-neutral interaction partner.
Inhibition of Cathepsin B by Au(I) Complexes: A Kinetic and Computational Study
Shamila S. Gunatilleke, Cesar Augusto F. de Oliveira, J. Andrew McCammon and Amy M. Barrios
Journal of Biological Inorganic Chemistry, Vol. 13, No. 4, pp. 555-561 (2008)
Gold(I) compounds have been used in the treatment of rheumatoid arthritis for over 80 years, but the biological targets and the structure-activity relationships of these drugs are not well understood. Of particular interest is the molecular mechanism behind the antiarthritic activity of the orally available drug triethylphosphine(2,3,4,6-tetra-O-acetyl-β-1-d-thiopyranosato-S) gold(I) (auranofin, Ridaura). The cathepsin family of lysosomal, cysteine-dependent enzymes is an attractive biological target of Au(I) and is inhibited by auranofin and auranofin analogs with reasonable potency. Here we employ a combination of experimental and computational investigations into the effect of changes in the phosphine ligand of auranofin on its in vitro inhibition of cathepsin B. Sequential replacement of the ethyl substituents of triethylphosphine by phenyl groups leads to increasing potency in the resultant Au(I) complexes, due in large part to favorable interactions of the more sterically bulky Au(I)-PR3 fragments with the enzyme active site.
Feature-Preserving Adaptive Mesh Generation for Molecular Shape Modeling and Simulation
Zeyun Yu, Michael J. Holst, Yuhui Cheng and J. Andrew McCammon
Journal of Molecular Graphics and Modelling, Vol. 26, Issue 8, pp. 1370-1380 (2008)
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.
Multi-Scale Modeling of Ventricular Myocytes: Contributions of structural and functional heterogeneities to excitation-contraction coupling in the normal and failing rodent heart
S. Lu, A. Michailova, J. Saucerman, Y. Cheng, Z. Yu, T. Kaiser, W. Li, R.E. Bank, M. Holst, J.A. McCammon, T. Hayashi, M. Hoshijima, P. Arzberger and A.D. McCulloch
IEEE Engineering in Medicine and Biolology Magazine, in review (2008)
Electrostatic Interactions
Nathan A. Baker and J. Andrew McCammon
Catalytically Requisite Conformational Dynamics in the mRNA-Capping Enzyme Probed by Targeted Molecular Dynamics
Robert V. Swift and J. Andrew McCammon
Biochemistry, Vol. 47, No. 13, pp. 4102-4111 (2008)
The addition of a N7-methyl guanosine cap to the 5' end of nascent mRNA is carried out by the mRNA-capping enzyme, a two-domain protein that is a member of the nucleotidyltransferase superfamily. The mRNA-capping enzyme is composed of a catalytic nucleotidyltransferase domain and a noncatalytic oligonucleotide/oligosaccharide binding (OB) domain. Large-scale domain motion triggered by substrate binding mediates catalytically requisite conformational rearrangement of the GTP substrate prior to the chemical step. In this study, we employ targeted molecular dynamics (TMD) on the PBCV-1 capping enzyme to probe the global domain dynamics and internal dynamics of conserved residues during the conformational transformation from the open to the closed state. Analysis of the resulting trajectories along with structural and sequence homology to other members of the superfamily allows us to suggest a conserved mechanism of conformational rearrangements spanning all mRNA-capping enzymes and all ATP-dependent DNA ligases. Our results suggest that the OB domain moves quasi-statically toward the nucleotidyltransferase domain, pivoting about a short linker region. The approach of the OB domain brings a conserved RxDK sequence, an element of conserved motif VI, within proximity of the triphosphate of GTP, destabilizing the unreactive conformation and thereby allowing thermal fluctuations to partition the substrate toward the catalytically competent state.
(Thermo)dynamic role of receptor flexibility, entropy, and motional correlation in protein-ligand binding
Riccardo Baron and J. Andrew McCammon
ChemPhysChem, Vol. 9, Issue 7, pp. 983-988 (2008)
The binding of 2-amino-5-methylthiazole to the W191G cavity mutant of cytochrome c peroxidase is an ideal test case to investigate the entropic contribution to the binding free energy due to changes of receptor flexibility. The dynamic and thermodynamic role of receptor flexibility were studied by 50-ns long explicit-solvent molecular dynamics simulations of three separate receptor ensembles: W191G binding a K+ ion, W191G-2a5mt complex with a closed 190-195 gating loop, and apo with an open loop. We employ a method recently proposed to estimate accurate absolute single-molecule configurational entropies and their differences for systems undergoing conformational transitions. We find that receptor flexibility plays a generally-underestimated role in protein-ligand binding (thermo)dynamics and that changes of receptor motional correlation determine such large entropy contributions.
Computer-aided Drug Discovery: Physics-based Simulations from the Molecular to the Cellular Level
J.A. McCammon
Mapping the nucleotide and isoform dependent structural and dynamical features of Ras proteins
A.A. Gorfe, B.J. Grant and J.A. McCammon
Ras GTPases are conformational switches controlling cell proliferation, differentiation and development. Despite their prominent role in many forms of cancer, the mechanism of conformational transition between inactive GDP- and active GTP-bound states remains unclear. Here we describe a detailed analysis of available experimental structures and molecular dynamics simulations to quantitatively assess the structural and dynamical features of active and inactive states and their interconversion. We demonstrate that GTP-bound and nucleotide-free G12V H-ras sample a wide region of conformational space, and show that the inactive to active transition is a multiphase process defined by the relative rearrangement of the two switches and the orientation of Tyr32. We also modeled and simulated N- and K-ras proteins and found that K-ras is more flexible than N- and H-ras. We identified a number of isoform-specific long-range side chain interactions that define unique pathways of communication between the nucleotide binding site and the C-terminus.