Alfredo E. Cardenas

Three-dimensional dynamic Monte Carlo simulations of driven polymer transport through a hole in a wall

Three-dimensional dynamic Monte Carlo simulations of polymer translocation through a cylindrical hole in a planar slab under the influence of an external driving force are performed. The driving force is intended to emulate the effect of a static electric field applied in an electrolytic solution containing charged monomer particles, as is relevant to the translocation of certain biopolymers through protein channel pores embedded in cell membranes. The time evolution of the probability distribution of the translocation coordinate (the number of monomers that have passed through the pore) is extracted from three-dimensional (3-D) simulations over a range of polymer chain lengths. These distributions are compared to the predictions of a 1-D Smoluchowski equation model of the translocation coordinate dynamics. Good agreement is found, with the effective diffusion constant for the 1-D Smoluchowski model being nearly independent of chain length.

Kinetics of cytochrome C folding: Atomically detailed simulations

The vast range of time scales (from nanoseconds to seconds) during protein folding is a challenge for experiments and computations. To make concrete predictions on folding mechanisms, atomically detailed simulations of protein folding, using potentials derived from chemical physics principles, are desired. However, due to their computational complexity, straightforward molecular dynamics simulations of protein folding are impossible today. An alternative algorithm is used that makes it possible to compute approximate atomically detailed long time trajectories (the Stochastic Difference Equation in Length). This algorithm is used to compute 26 atomically detailed folding trajectories of cytochrome c (a millisecond process). The early collapse of the protein chain (with marginal formation of secondary structure), and the earlier formation of the N and C helices (compare to the 60s helix) are consistent with the experiment. The existence of an energy barrier upon entry to the molten globule is examined as well. In addition to (favorable) comparison to experiments, we show that non‐native contacts drive the formation of the molten globule. In contrast to popular folding models, the non‐native contacts do not form off‐pathway kinetic traps in cytochrome c. Proteins 2003;51:245–257.

Unassisted Transport of N-acetyl-L-tryptophanamide through Membrane: Experiment and Simulation of Kinetics

Cellular transport machinery, such as channels and pumps, is working against the background of unassisted material transport through membranes. The permeation of a blocked tryptophan through a 1,2-dioleoyl-sn-glycero-3-phosphocholine (DOPC) membrane is investigated to probe unassisted or physical transport. The transport rate is measured experimentally and modeled computationally. The time scale measured by parallel artificial membrane permeation assay (PAMPA) experiments is ~8 h. Simulations with the milestoning algorithm suggest mean first passage time (MFPT) of ~4 h and the presence of a large barrier at the center of the bilayer. A similar calculation with the solubility-diffusion model yields a MFPT of ~15 min. This permeation rate is 9 orders of magnitude slower than the permeation rate of only a tryptophan side chain (computed by us and others). This difference suggests critical dependence of transport time on permeant size and hydrophilicity. Analysis of the simulation results suggests that the permeant partially preserves hydrogen bonding of the peptide backbone to water and lipid molecules even when it is moving closer to the bilayer center. As a consequence, defects of the membrane structure are developed to assist permeation.

Atomically Detailed Simulations of Helix Formation with the Stochastic Difference Equation

An algorithm is described to compute approximate classical trajectories as a boundary value problem with an integration step in the arc length. High-frequency motions are filtered out when a large integration step is used, maintaining the stability of the algorithm. At the limit of high filtering (large steps), but still offering an accurate description of the continuous path, the trajectory approaches the steepest descent path (SDP). The steepest descent path is widely used as a reaction coordinate in chemical systems. At intermediate step sizes, some inertial motions remain, interpolating between reaction coordinates and exact classical trajectories. Numerical studies of spatial and energetic properties of meta-trajectories are carried out. Two systems are considered here: valine dipeptide and the folding of a small helical protein. Although thermodynamic properties of meta-trajectories are affected by the filtering, the ordering of events remains similar for substantial differences in trajectory resolution.

Membrane permeation of a peptide: It is better to be positive

A joint experimental and computational study investigates the translocation of a tryptophan molecule through a phospholipid membrane. Time dependent spectroscopy of the tryptophan side chain determines the rate of permeation into 150 nm phospholipid vesicles. Atomically detailed simulations are conducted to calculate the free energy profiles and the permeation coefficient. Different charging conditions of the peptide (positive, negative, or zwitterion) are considered. Both experiment and simulation reproduce the qualitative trend and suggest that the fastest permeation is when the tryptophan is positively charged. The permeation mechanism, which is revealed by molecular dynamics simulations, is of a translocation assisted by a local defect. The influence of long-range electrostatic interactions, such as the membrane dipole potential on the permeation process, is not significant.

Modeling kinetics and equilibrium of membranes with fields: Milestoning analysis and implication to permeation

Coarse graining of membrane simulations by translating atomistic dynamics to densities and fields with Milestoning is discussed. The space of the membrane system is divided into cells and the different cells are characterized by order parameters presenting the number densities. The dynamics of the order parameters are probed with Milestoning. The methodology is illustrated here for a phospholipid membrane system (a hydrated bilayer of DOPC (1,2-dioleoyl-sn-glycero-3-phosphocholine) lipid molecules). Significant inhomogeneity in membrane internal number density leads to complex free energy landscape and local maps of transition times. Dynamics and distributions of cavities within the membrane assist the permeation of nonpolar solutes such as xenon atoms. It is illustrated that quantitative and detailed dynamics of water transport through DOPC membrane can be analyzed using Milestoning with fields. The reaction space for water transport includes at least two slow variables: the normal to the membrane plane, and the water density.

Computational study of peptide permeation through membrane: Searching for hidden slow variables.

Atomically detailed molecular dynamics trajectories in conjunction with Milestoning are used to analyse the different contributions of coarse variables to the permeation process of a small peptide (N-acetyl-l-tryptophanamide, NATA) through a 1,2-dioleoyl-sn-glycero-3-phosphocholine membrane. The peptide reverses its overall orientation as it permeates through the biological bilayer. The large change in orientation is investigated explicitly but is shown to impact the free energy landscape and permeation time only moderately. Nevertheless, a significant difference in permeation properties of the two halves of the membrane suggests the presence of other hidden slow variables. We speculate, based on calculation of the potential of mean force, that a conformational transition of NATA makes significant contribution to these differences. Other candidates for hidden slow variables may include water permeation and collective motions of phospholipids.

Measurement of the membrane dipole electric field in DMPC vesicles using vibrational shifts of p-cyanophenylalanine and molecular dynamics simulations.

The magnitude of the membrane dipole field was measured using vibrational Stark effect (VSE) shifts of nitrile oscillators placed on the unnatural amino acid p-cyanophenylalanine (p-CN-Phe) added to a peptide sequence at four unique positions. These peptides, which were based on a repeating alanine-leucine motif, intercalated into small unilamellar DMPC vesicles which formed an α-helix as confirmed by circular dichroic (CD) spectroscopy. Molecular dynamics simulations of the membrane-intercalated helix containing two of the nitrile probes, one near the headgroup region of the lipid (αLAX(25)) and one buried in the interior of the bilayer (αLAX(16)), were used to examine the structure of the nitrile with respect to the membrane normal, the assumed direction of the dipole field, by quantifying both a small tilt of the helix in the bilayer and conformational rotation of the p-CN-Phe side chain at steady state. Vibrational absorption energies of the nitrile oscillator at each position showed a systematic blue shift as the nitrile was stepped toward the membrane interior; for several different concentrations of peptide, the absorption energy of the nitrile located in the middle of the bilayer was ∼3 cm(-1) greater than that of the nitrile closest to the surface of the membrane. Taken together, the measured VSE shifts and nitrile orientations within the membrane resulted in an absolute magnitude of 8-11 MV/cm for the dipole field, at the high end of the range of possible values that have been accumulated from a variety of indirect measurements. Implications for this are discussed.

Analyzing milestoning networks for molecular kinetics: definitions, algorithms, and examples.

Network representations are becoming increasingly popular for analyzing kinetic data from techniques like Milestoning, Markov State Models, and Transition Path Theory. Mapping continuous phase space trajectories into a relatively small number of discrete states helps in visualization of the data and in dissecting complex dynamics to concrete mechanisms. However, not only are molecular networks derived from molecular dynamics simulations growing in number, they are also getting increasingly complex, owing partly to the growth in computer power that allows us to generate longer and better converged trajectories. The increased complexity of the networks makes simple interpretation and qualitative insight of the molecular systems more difficult to achieve. In this paper, we focus on various network representations of kinetic data and algorithms to identify important edges and pathways in these networks. The kinetic data can be local and partial (such as the value of rate coefficients between states) or an exact solution to kinetic equations for the entire system (such as the stationary flux between vertices). In particular, we focus on the Milestoning method that provides fluxes as the main output. We proposed Global Maximum Weight Pathways as a useful tool for analyzing molecular mechanism in Milestoning networks. A closely related definition was made in the context of Transition Path Theory. We consider three algorithms to find Global Maximum Weight Pathways: Recursive Dijkstras, Edge-Elimination, and Edge-List Bisection. The asymptotic efficiency of the algorithms is analyzed and numerical tests on finite networks show that Edge-List Bisection and Recursive Dijkstras algorithms are most efficient for sparse and dense networks, respectively. Pathways are illustrated for two examples: helix unfolding and membrane permeation. Finally, we illustrate that networks based on local kinetic information can lead to incorrect interpretation of molecular mechanisms.

Computer Simulations of the Refolding of Sperm Whale Apomyoglobin from High-Temperature Denaturated State

The refolding mechanism of apomyoglobin (apoMb) subsequent to high-temperature unfolding has been examined using computer simulations with atomic level detail. The folding of this protein has been extensively studied experimentally, providing a large database of folding parameters which can be probed using simulations. In the present study, 4-folding trajectories of apoMb were computed starting from coiled structures. A crystal structure of sperm whale myoglobin taken from the Protein Data Bank was used to construct the final native conformation by removal of the heme group followed by energy optimization. The initial unfolded conformations were obtained from high-temperature molecular dynamics simulations. Room-temperature refolding trajectories at neutral pH were obtained using the stochastic difference equation in length algorithm. The folding trajectories were compared with experimental results and two previous molecular dynamics studies at low pH. In contrast to the previous simulations, an extended intermediate with large helical content was not observed. In the present study, a structural collapse occurs without formation of helices or native contacts. Once the protein structure is more compact (radius of gyration<18 A) secondary and tertiary structures appear. These results suggest that apoMb follows a different folding pathway after high-temperature denaturation.