Nathan A. Baker

Cheng Y, Suen JK, Zhang D, Bond SD, Zhang Y, Song Y, Baker NA, Bajaj CL, Holst MJ, McCammon JA. Finite element analysis of the time-dependent Smoluchowski equation for acetylcholinesterase reaction rate calculations. Biophys J, 92, 3397-406, 2007.

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 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 the- oretical 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.

• PubMed ID: 17307827
• PubMed Central ID:  PMC1853150
• DOI:  10.1529/biophysj.106.102533
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Zhang D, Suen J, Zhang Y, Song Y, Radic Z, Taylor P, Holst MJ, Bajaj C, Baker NA, McCammon JA. Tetrameric mouse acetylcholinesterase: continuum diffusion rate calculations by solving the steady-state Smoluchowski equation using finite element methods. Biophys J, 88, 1659-1666, 2005.

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.

• PubMed ID: 15626705
• PubMed Central ID:  PMC1305222
• DOI:  10.1529/biophysj.104.053850
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Song Y, Zhang Y, Bajaj C, Baker NA. Continuum diffusion reaction rate calculations of wild type and mutant mouse acetylcholinesterase: adaptive finite element analysis. Biophys J, 87, 1558-66, 2004.

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.

• PubMed ID: 15345536
• DOI:  10.1529/biophysj.104.041517
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Song Y, Zhang Y, Shen T, Bajaj CL, McCammon JA, Baker NA. Finite element solution of the steady-state Smoluchowksi equation for rate constant calculations. Biophys J, 86, 2017-29, 2004.

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.

• PubMed ID: 15041644
• DOI:  10.1016/S0006-3495(04)74263-0
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Tai K, Bond SD, MacMillan HR, Baker NA, Holst MJ, McCammon JA. Finite element simulations of acetylcholine diffusion in neuromuscular junctions. Biophys J, 84, 2234-41, 2003.

A robust infrastructure for solving time-dependent diffusion using the finite element package FEtk has been developed to simulate synaptic transmission in a neuromuscular junction with realistic postsynaptic folds. Simplified rectilinear synapse models serve as benchmarks in initial numerical studies of how variations in geometry and kinetics relate to endplate currents associated with fast-twitch, slow-twitch, and dystrophic muscles. The flexibility and scalability of FEtk affords increasingly realistic and complex models that can be formed in concert with expanding experimental understanding from electron microscopy. Ultimately, such models may provide useful insight on the functional implications of controlled changes in processes, suggesting therapies for neuromuscular diseases.

• PubMed ID: 12668432
• DOI:  10.1016/S0006-3495(03)75029-2
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Baker N, Tai K, Henchman R, Sept D, Elcock A, Holst M, McCammon JA. Mathematics and molecular neurobiology. Computational Methods for Macromolecules: Challenges and Applications. Gan HH, Schlick T, eds., 2002.

Baker N, Holst M, Wang F. Adaptive multilevel finite element solution of the Poisson-Boltzmann equation II: refinement schemes based on solvent accessible surfaces. J Comput Chem, 21, 1343-52, 2000.

We apply the adaptive multilevel finite element techniques (Holst, Baker, and Wang 21) to the nonlinear Poisson–Boltzmann equation (PBE) in the context of biomolecules. Fast and accurate numerical solution of the PBE in this setting is usually difficult to accomplish due to presence of discontinuous coefficients, delta functions, three spatial dimensions, unbounded domains, and rapid (exponential) nonlinearity. However, these adaptive techniques have shown substantial improvement in solution time over conventional uniform-mesh finite difference methods. One important aspect of the adaptive multilevel finite element method is the robust a posteriori error estimators necessary to drive the adaptive refinement routines. This article discusses the choice of solvent accessibility for a posteriori error estimation of PBE solutions and the implementation of such routines in the “Adaptive Poisson–Boltzmann Solver” (APBS) software package based on the “Manifold Code” (MC) libraries. Results are shown for the application of this method to several biomolecular systems.

• DOI: <a href="http://dx.doi.org/10.1002/1096-987X(20001130)21:153.0.CO;2-K”>10.1002/1096-987X(20001130)21:15<1343::AID-JCC2>3.0.CO;2-K
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Quinn DM, Feaster SR, Nair HK, Baker NA, Radić Z, Taylor P. Delineation and decomposition of energies involved in quaternary ammonium binding in the active site of acetylcholinesterase. J Am Chem Soc, 122, 2975-80, 2000.

The quaternary ammonium binding locus in the active site of mammalian acetylcholinesterase is subtended by the side chains of Trp86, Tyr133, Glu202, and Tyr337. Linear free-energy relationships define the interactions involved in molecular recognition by mouse acetylcholinesterase of the quaternary ammonium moiety of ligands. For substrates CH₃C(O)XCH₂CH₂Y [X = O, Y = CHMe₂, or CH₂CH₃; X = S, Y = H, NH⁺Me₂, or N⁺Me₃ ] and trifluoroacetophenone transition state analogue inhibitors m-YC₆H₄C(O)CF₃ [Y = H, Me, Et,iPr, tBu, CF₃, NH₂, NO₂, NMe₂, or N⁺Me₃], log(k_cat/K_m) and pK_i depend linearly on the molar refractivity, but not the hydrophobicity, of the substituents Y. These correlations indicate that, in the acylation stage of catalysis, interactions in the quaternary ammonium binding locus stabilize the tetrahedral intermediate (as modeled by transition state analogue affinity) by (5 × 10⁵)-fold (ΔΔG^TI = −32.5 kJ mol^-1) and the transition state by (2 × 10⁴)-fold (ΔΔG^‡ = −24.5 kJ mol^-1). To evaluate the contribution of cation-π interactions, Trp86 was converted into Tyr, Phe, and Ala by site-specific mutagenesis. For this set of enzymes, a linear free-energy relationship is observed between the pK_i values for inhibitions by the respective neutral and cationic transition state analogue inhibitors, m-tert-butyltrifluoroacetophenone and m-(N,N,N-trimethylammonio)trifluoroacetophenone, which indicates that the free energy released on interaction of the quaternary ammonium moiety with Trp86 arises about equally from cation-π and charge-independent interactions.

• DOI: 10.1021/ja9933588
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Baker NA, Helms V, McCammon JA. Dynamical properties of fasciculin-2. Proteins, 36, 447-53, 1999.

Fasciculin-2 (FAS2) is a potent protein inhibitor of the hydrolytic enzyme acetylcholinesterase. A 2-ns isobaric-isothermal ensemble molecular dynamics simulation of this toxin was performed to examine the dynamic structural properties which may play a role in this inhibition. Conformational fluctuations of the FAS2 protein were examined by a variety of techniques to identify flexible residues and determine their characteristic motion. The tips of the toxin “finger” loops and the turn connecting loops I and II were found to fluctuate, while the rest of the protein remained fairly rigid throughout the simulation. Finally, the structural fluctuations were compared to NMR data of fluctuations on a similar timescale in a related three-finger toxin. The molecular dynamics results were in good qualitative agreement with the experimental measurement

• PubMed ID: 10450086
• DOI:  10.1002/(SICI)1097-0134(19990901)36:4<447::AID-PROT8>3.0.CO;2-E
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Malany S, Baker N, Verweyst M, Medhekar R, Quinn DM, Velan B, Kronman C, Shafferman A. Theoretical and experimental investigations of electrostatic effects on acetylcholinesterase catalysis and inhibition. Chem-Biol Interact, 120, 99-110, 1999.

The role of electrostatics in the function of acetylcholinesterase (AChE) has been investigated by both theoretical and experimental approaches. Second-order rate constants (kE = k(cat)/Km) for acetylthiocholine (ATCh) turnover have been measured as a function of ionic strength of the reaction medium for wild-type and mutant AChEs. Also, binding and dissociation rate constants have been measured as a function of ionic strength for the respective charged and neutral transition state analog inhibitors m-(N,N,N-trimethylammonio)trifluoroacetophenone (TMTFA) and m-(t-butyl)trifluoroacetophenone (TBTFA). Linear free-energy correlations between catalytic rate constants and inhibition constants indicate that kE for ATCh turnover is rate limited by terminal binding events. Comparison of binding rate constants for TMTFA and TBTFA attests to the sizable electrostatic discrimination of AChE. Free energy profiles for cationic ligand release from the active sites of wild-type and mutant AChEs have been calculated via a model that utilizes the structure of T. californica AChE, a spherical ligand, and energy terms that account for electrostatic and van der Waals interactions and chemical potential. These calculations indicate that EA and EI complexes are not bound with respect to electrostatic interactions, which obviates the need for a ‘back door’ for cationic ligand release. Moreover, the computed energy barriers for ligand release give linear free-energy correlations with log(kE) for substrate turnover, which supports the general correctness of the computational model.

• PubMed ID: 10421443
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