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
• Download:  Reprint (PDF)

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
• Download:  Reprint (PDF)

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
• Download:

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.

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
• Download:  Reprint (PDF)

Baker NA, McCammon JA. Non-Boltzmann rate distributions in stochastically gated reactions. J Phys Chem B, 103, 615-617, 1999.

Recently, a new mechanism for reaction selectivity, arising from conformational gating of the reactions, has been reported in the acetylcholinesterase system. Fluctuations in the enzyme are thought to greatly slow the access of molecules larger than the normal substrate to the active-site region. By assuming the gate fluctuations occur as a Brownian process in a harmonic well, it is possible to approximate the reaction rates for various limiting cases of substrate size. However, it is not possible to simplify the rates into a ratio which is equivalent to the Boltzmann distribution of states for the gate fluctuations.

• DOI: 10.1021/jp984151o
• Download:  Reprint (PDF)

Feaster SR, Lee K, Baker N, Hui DY, Quinn DM. Molecular recognition by cholesterol esterase of active site ligands: structure-reactivity effects for inhibition by aryl carbamates and subsequent carbamylenzyme turnover. Biochemistry, 35, 16723-34, 1996.

Interactions of mammalian pancreatic cholesterol esterases from pig and rat with a family of aryl carbamates CnH2n+1NHCOOAr [n = 4-9; Ar = phenyl, p-X-phenyl (X = acetamido, bromo, fluoro, nitro, trifluoromethyl), 2-naphthyl, 2-tetrahydronaphthyl, estronyl] have been investigated, with an aim of delineating the ligand structural features which lead to effective molecular recognition by the active site of the enzyme. These carbamates inhibit the catalytic activity of CEase by rapid carbamylation of the active site, a process that shows saturation kinetics. Subsequent slow decarbamylation usually leads to full restoration of activity, and therefore aryl carbamates are transient inhibitors, or pseudo-substrates, of CEase. Structural variation of carbamate inhibitors allowed molecular recognition in the fatty acid binding and steroid binding loci of the extended active site to be probed, and the electronic nature of the carbamylation transition state to be characterized. Optimal inhibitory activity is observed when the length of the carbamyl function is n = 6 and n = 7 for porcine and rat cholesterol esterases, respectively, equivalent to eight- and nine-carbon fatty acyl chains. In contrast, inhibitory activity increases progressively as the partial molecular volume of the aromatic fragment increases. Hammett plots for p-substituted phenyl-N-hexyl carbamates indicate that the rate-determining step for carbamate inhibition is phenolate anion expulsion. Effects of the bile salt activator taurocholate on the kinetically resolved phases of the pseudo-substrate turnover of aryl carbamates were also studied. Taurocholate increases the affinity of the carbamate for the active site of cholesterol esterase in the reversible, noncovalent complex that precedes carbamylation and increases the rate constants of the serial carbamylation and decarbamylation steps. Structural variation of the N-alkyl chain and of the aryl fused-ring system provides an accounting of bile salt modulation of the fatty acid and steroid binding sites, respectively. In that pseudo-substrate turnover of aryl carbamates proceeds by a three-step mechanism that is analogous to that for rapid turnover of lipid ester substrates, these investigations illuminate details of ligand recognition by the extended active site of cholesterol esterase that are prominent determinants of the substrate specificity and catalytic power of the enzyme.

• PubMed ID: 8988009
• Download:  Reprint (PDF)