Van Ngo

Representation of Ion–Protein Interactions Using the Drude Polarizable Force-Field

Small metal ions play critical roles in numerous biological processes. Of particular interest is how metalloenzymes are allosterically regulated by the binding of specific ions. Understanding how ion binding affects these biological processes requires atomic models that accurately treat the microscopic interactions with the protein ligands. Theoretical approaches at different levels of sophistication can contribute to a deeper understanding of these systems, although computational models must strike a balance between accuracy and efficiency in order to enable long molecular dynamics simulations. In this study, we present a systematic effort to optimize the parameters of a polarizable force field based on classical Drude oscillators to accurately represent the interactions between ions (K+, Na+, Ca2+, and Cl–) and coordinating amino-acid residues for a set of 30 biologically important proteins. By combining ab initio calculations and experimental thermodynamic data, we derive a polarizable force field that is consistent with a wide range of properties, including the geometries and interaction energies of gas-phase ion/protein-like model compound clusters, and the experimental solvation free-energies of the cations in liquids. The resulting models display significant improvements relative to the fixed-atomic-charge additive CHARMM C36 force field, particularly in their ability to reproduce the many-body electrostatic nonadditivity effects estimated from ab initio calculations. The analysis clarifies the fundamental limitations of the pairwise additivity assumption inherent in classical fixed-charge force fields, and shows its dramatic failures in the case of Ca2+ binding sites. These optimized polarizable models, amenable to computationally efficient large-scale MD simulations, set a firm foundation and offer a powerful avenue to study the roles of the ions in soluble and membrane transport proteins.

Quantum Effects in Cation Interactions with First and Second Coordination Shell Ligands in Metalloproteins

Despite decades of investigations, the principal mechanisms responsible for the high affinity and specificity of proteins for key physiological cations K+, Na+, and Ca2+ remain a hotly debated topic. At the core of the debate is an apparent need (or lack thereof) for an accurate description of the electrostatic response of the charge distribution in a protein to the binding of an ion. These effects range from partial electronic polarization of the directly ligating atoms to long-range effects related to partial charge transfer and electronic delocalization effects. While accurate modeling of cation recognition by metalloproteins warrants the use of quantum-mechanics (QM) calculations, the most popular approximations used in major biomolecular simulation packages rely on the implicit modeling of electronic polarization effects. That is, high-level QM computations for ion binding to proteins are desirable, but they are often unfeasible, because of the large size of the reactive-site models and the need to sample conformational space exhaustively at finite temperature. Several solutions to this challenge have been proposed in the field, ranging from the recently developed Drude polarizable force-field for simulations of metalloproteins to approximate tight-binding density functional theory (DFTB). To delineate the usefulness of different approximations, we examined the accuracy of three recent and commonly used theoretical models and numerical algorithms, namely, CHARMM C36, the latest developed Drude polarizable force fields, and DFTB3 with the latest 3OB parameters. We performed MD simulations for 30 cation-selective proteins with high-resolution X-ray structures to create ensembles of structures for analysis with different levels of theory, e.g., additive and polarizable force fields, DFTB3, and DFT. The results from DFT computations were used to benchmark CHARMM C36, Drude, and DFTB3 performance. The explicit modeling of quantum effects unveils the key electrostatic properties of the protein sites and the importance of specific ion-protein interactions. One of the most interesting findings is that secondary coordination shells of proteins are noticeably perturbed in a cation-dependent manner, showing significant delocalization and long-range effects of charge transfer and polarization upon binding Ca2+.

Estimation of potentials of mean force from nonequilibrium pulling simulations using both minh-adib estimator and weighted histogram analysis method

Nonequilibrium pulling simulations have been a useful approach for investigating a variety of physical and biological problems. The major target in the simulations is to reconstruct reliable potentials of mean force (PMFs) or unperturbed free-energy profiles for quantitatively addressing both equilibrium mechanistic properties and contributions from nonequilibrium processes. While several current nonequilibrium methods were shown to be accurate in computing free-energy profiles in systems with relatively simple dynamics, they have proved to be unsuitable in more complicated systems. To extend the applicability of nonequilibrium sampling, we demonstrate a novel method that combines Minh-Adibs bidirectional estimator with nonlinear WHAM equations to reconstruct and assess PMFs from relatively fast pulling trajectories. We test the method in a one-dimensional model system and in a system of an antibiotic gramicidin-A (gA) channel, which is considered a significant challenge for nonequilibrium sampling. We identify key parameters for efficiently performing pulling simulations to improve and ensure the convergence and accuracy of estimated PMFs. We show that a few pulling trajectories of a relatively fast pulling speed v = 10 Å/ns can return a fair estimate of the PMF of a single potassium ion in gA.

Non-Equilibrium Dynamics Contribute to Ion Selectivity in the KcsA Channel

The ability of biological ion channels to conduct selected ions across cell membranes is critical for the survival of both animal and bacterial cells. Numerous investigations of ion selectivity have been conducted over more than 50 years, yet the mechanisms whereby the channels select certain ions and reject others are not well understood. Here we report a new application of Jarzynski’s Equality to investigate the mechanism of ion selectivity using non-equilibrium molecular dynamics simulations of Na+ and K+ ions moving through the KcsA channel. The simulations show that the selectivity filter of KcsA adapts and responds to the presence of the ions with structural rearrangements that are different for Na+ and K+. These structural rearrangements facilitate entry of K+ ions into the selectivity filter and permeation through the channel, and rejection of Na+ ions. A mechanistic model of ion selectivity by this channel based on the results of the simulations relates the structural rearrangement of the selectivity filter to the differential dehydration of ions and multiple-ion occupancy and describes a mechanism to efficiently select and conduct K+. Estimates of the K+/Na+ selectivity ratio and steady state ion conductance for KcsA from the simulations are in good quantitative agreement with experimental measurements. This model also accurately describes experimental observations of channel block by cytoplasmic Na+ ions, the “punch through” relief of channel block by cytoplasmic positive voltages, and is consistent with the knock-on mechanism of ion permeation.

Parallel-pulling protocol for free-energy evaluation.

Jarzynskis equality (JE) allows us to compute free-energy differences from distributions of work. In molecular dynamics simulations, the traditional way of constructing work distributions is to perform as many pulling simulations as possible. But reliable work distributions are not always produced in a finite number of simulations. The computational cost of using JE is not less than other commonly used methods such as thermodynamic integration and umbrella sampling methods. Here we first show a different proof of JE based on the idea of stepwise pulling procedures that is efficient in computing free energies by using JE. The key point in our proof is that the processes of turning on or off a harmonic potential to perform work are described by double Heaviside functions of time. We then show that the distributions of work performed by the potential can be easily generated from the distributions of a reaction coordinate along a pathway. Based on the proof, we propose sequential and parallel stepwise pulling protocols for generating work distributions that require suitable relaxation time at each pulling step. The criterion for reliable work distributions is that there must be sufficient mutual overlaps between the adjacent distributions of the reaction coordinate along the pathway. We arrive at an alternative formula (besides JE) to compute free-energy differences from the averaged values of the reaction coordinate. The combination of JE and the alternative formula provides a viable way to determine the accuracy of computed free-energy differences. For the stretching of a deca-alanine molecule, our approach requires 21 parallel simulations and relaxation time as small as 0.4 ns for each simulation to estimate free-energy differences with an uncertainty of about 13%.

QM/MM Calculations with deMon2k

The density functional code deMon2k employs a fitted density throughout (Auxiliary Density Functional Theory), which offers a great speed advantage without sacrificing necessary accuracy. Powerful Quantum Mechanical/Molecular Mechanical (QM/MM) approaches are reviewed. Following an overview of the basic features of deMon2k that make it efficient while retaining accuracy, three QM/MM implementations are compared and contrasted. In the first, deMon2k is interfaced with the CHARMM MM code (CHARMM-deMon2k); in the second MM is coded directly within the deMon2k software; and in the third the Chemistry in Ruby (Cuby) wrapper is used to drive the calculations. Cuby is also used in the context of constrained-DFT/MM calculations. Each of these implementations is described briefly; pros and cons are discussed and a few recent applications are described briefly. Applications include solvated ions and biomolecules, polyglutamine peptides important in polyQ neurodegenerative diseases, copper monooxygenases and ultra-rapid electron transfer in cryptochromes.

Is the G-Quadruplex an Effective Nanoconductor for Ions?

We use a stepwise pulling protocol in molecular dynamics simulations to identify how a G-quadruplex selects and conducts Na(+), K(+), and NH4(+) ions. By estimating the minimum free-energy changes of the ions along the central channel via Jarzynskis equality, we find that the G-quadruplex selectively binds the ionic species in the following order: K(+) > Na(+) > NH4(+). This order implies that K(+) optimally fits the channel. However, the features of the free-energy profiles indicate that the channel conducts Na(+) best. These findings are in fair agreement with experiments on G-quadruplexes and reveal a profoundly different behavior from the prototype potassium-ion channel KcsA, which selects and conducts the same ionic species. We further show that the channel can also conduct a single file of water molecules and deform to leak water molecules. We propose a range for the conductance of the G-quadruplex.

Molecular mechanism of flip-flop in triple-layer oleic-acid membrane: correlation between oleic acid and water.

We perform all-atom molecular dynamics simulations to study a pure oleic acid (OA) membrane in water that results in a triple-layer structure. We compute the pressure profiles to examine the hydrophobic and hydrophilic regions, and to estimate the surface tension (≈34.5 mN/m), which is similar to those of lipid membranes. We observe that the membrane of OAs having a large diffusion coefficient (0.4 × 10(-7) cm(2)/s) along the normal to the membrane is an ideal model to study oleic acid flip-flop. In the model, the membrane contains a middle layer serving as an intermediate for water and OAs to easily migrate (flip-flop) from one to other leaflets. Water molecules surrounding OA head-groups help to reduce the barriers at the hydrophobic interface to trigger flip-flop events. Within 500 ns, we observe 175 flip-flop events of OAs and 305 events of water traversing the membrane. The ratio of water passing rate (k(H(2)O) = 0.673 ns(-1)) to OA flip-flop rate (k(OA) = 0.446 ns(-1)) is 3/2. The ratio of the totally correlated water-OA events to the totally uncorrelated water-OA events, n(cor)/n(uncor), is also 3/2. The probability of the totally and partially correlated events is 69%. The results indicate that the trans-membrane movement of water and OAs is cooperative and correlated, and agrees with experimentally measured absorption rates. They support the idea that OA flip-flop is more favorable than transport by means of functional proteins. This study might provide further insight into how primitive cell membranes work, and how the interplay and correlation between water and fatty acids may occur.

K+ block is the mechanism of functional asymmetry in bacterial Nav channels

Crystal structures of several bacterial Nav channels have been recently published and molecular dynamics simulations of ion permeation through these channels are consistent with many electrophysiological properties of eukaryotic channels. Bacterial Nav channels have been characterized as functionally asymmetric, and the mechanism of this asymmetry has not been clearly understood. To address this question, we combined non-equilibrium simulation data with two-dimensional equilibrium unperturbed landscapes generated by umbrella sampling and Weighted Histogram Analysis Methods for multiple ions traversing the selectivity filter of bacterial NavAb channel. This approach provided new insight into the mechanism of selective ion permeation in bacterial Nav channels. The non-equilibrium simulations indicate that two or three extracellular K+ ions can block the entrance to the selectivity filter of NavAb in the presence of applied forces in the inward direction, but not in the outward direction. The block state occurs in an unstable local minimum of the equilibrium unperturbed free-energy landscape of two K+ ions that can be ‘locked’ in place by modest applied forces. In contrast to K+, three Na+ ions move favorably through the selectivity filter together as a unit in a loose “knock-on” mechanism of permeation in both inward and outward directions, and there is no similar local minimum in the two-dimensional free-energy landscape of two Na+ ions for a block state. The useful work predicted by the non-equilibrium simulations that is required to break the K+ block is equivalent to large applied potentials experimentally measured for two bacterial Nav channels to induce inward currents of K+ ions. These results illustrate how inclusion of non-equilibrium factors in the simulations can provide detailed information about mechanisms of ion selectivity that is missing from mechanisms derived from either crystal structures or equilibrium unperturbed free-energy landscapes.

A Novel Electrostatic Regulatory Mechanism in a Calcium Binding Protein, L-plastin

Plastins are a group of highly conserved actin binding proteins. L-plastin is a human isoform of this protein found active in hemopoietic cells. This isoform is also expressed in cancer cells. The N-terminal of this isoform contains a Calcium-binding EF-hand domain that allosterically regulates this protein for binding to its target. It is, however, very difficult to delineate the effects that underlie the allosteric mechanism. To investigate the regulatory mechanisms of the activation of this protein domain, we have used both computational and experimental methods. Working with calcium in MD simulations has previously caused significant problems as classical force fields are not well equipped to deal with calcium. This is because calcium is a divalent ion, which can induce a strong local electrostatic field, and is capable of charge transfer. Although classical force fields are insufficient, a new polarizable force field now known as a Drude force field, provides a more accurate solution to this problem. We describe results from simulations with both Classical and Drude force fields for the calcium-sensitive regions of L-plastin. These ongoing simulations have already provided valuable insight an unexpected and previously unseen electrostatic regulatory mechanism of L-plastin. In addition to computational simulations, to further validate this novel mechanism we obtained promising experimental validation using a number of biophysical methods including isothermal calorimetry (ITC) and differential scanning calorimetry (DSC). Although we are looking into this mechanism in L-plastin, the conserved nature of this protein may indicate that this mechanism is present in a wide range of related proteins, and may help provide us with a deeper understanding of the mechanisms associated with calcium activation and deactivation.