Brownian dynamics simulations of protein folding: Access to milliseconds time scale and beyond

Protein folding occurs on a time scale ranging from milliseconds to minutes for a majority of proteins. Computer simulation of protein folding, from a random configuration to the native structure, is nontrivial owing to the large disparity between the simulation and folding time scales. As an effort to overcome this limitation, simple models with idealized protein subdomains, e.g., the diffusion–collision model of Karplus and Weaver, have gained some popularity. We present here new results for the folding of a four-helix bundle within the framework of the diffusion–collision model. Even with such simplifying assumptions, a direct application of standard Brownian dynamics methods would consume 10,000 processor-years on current supercomputers. We circumvent this difficulty by invoking a special Brownian dynamics simulation. The method features the calculation of the mean passage time of an event from the flux overpopulation method and the sampling of events that lead to productive collisions even if their probability is extremely small (because of large free-energy barriers that separate them from the higher probability events). Using these developments, we demonstrate that a coarse-grained model of the four-helix bundle can be simulated in several days on current supercomputers. Furthermore, such simulations yield folding times that are in the range of time scales observed in experiments.

DNA transport by a micromachined Brownian ratchet device

We have micromachined a silicon-chip device that transports DNA with a Brownian ratchet that rectifies the Brownian motion of microscopic particles. Transport properties for a DNA 50-mer agree with theoretical predictions, and the DNA diffusion constant agrees with previous experiments. This type of micromachine could provide a generic pump or separation component for DNA or other charged species as part of a microscale lab-on-a-chip. A device with reduced feature size could produce a size-based separation of DNA molecules, with applications including the detection of single-nucleotide polymorphisms.

Brownian flocculation of polymer colloids in the presence of a secondary minimum

Most analyses of Brownian flocculation apply to conditions where London–van der Waals attractive forces cause particles to be strongly bound in a deep interparticle potential well. In this paper, results are reported that show the interaction between primary- and secondary-minimum flocculation when the interparticle potential curve reflects both attractive and electrostatic repulsive forces. The process is highly time-dependent because of transfer of particles from secondary- to primary-minimum flocculation. Essential features of the analysis are corroborated by experiments with 0.80-μm polystyrene spheres suspended in aqueous solutions of NaCl over a range of ionic strengths. In all cases, experiments were restricted to the initial stage of coagulation, where singlets and doublets predominate.

Minimum size limit for useful locomotion by free-swimming microbes

Formulas are derived for the effect of size on a free-swimming microbe’s ability to follow chemical, light, or temperature stimuli or to disperse in random directions. The four main assumptions are as follows: (i) the organisms can be modeled as spheres, (ii) the power available to the organism for swimming is proportional to its volume, (iii) the noise in measuring a signal limits determination of the direction of a stimulus, and (iv) the time available to determine stimulus direction or to swim a straight path is limited by rotational diffusion caused by Brownian motion. In all cases, it is found that there is a sharp size limit below which locomotion has no apparent benefit. This size limit is estimated to most probably be about 0.6 μm diameter and is relatively insensitive to assumed values of the other parameters. A review of existing descriptions of free-floating bacteria reveals that the smallest of 97 motile genera has a mean length of 0.8 μm, whereas 18 of 94 nonmotile genera are smaller. Similar calculations have led to the conclusion that a minimum size also exists for use of pheromones in mate location, although this size limit is about three orders of magnitude larger. In both cases, the application of well-established physical laws and biological generalities has demonstrated that a common feature of animal behavior is of no use to small free-swimming organisms.

Torsional rigidity of single actin filaments and actin–actin bond breaking force under torsion measured directly by in vitro micromanipulation

Knowledge of the elastic properties of actin filaments is crucial for considering its role in muscle contraction, cellular motile events, and formation of cell shape. The stiffness of actin filaments in the directions of stretching and bending has been determined. In this study, we have directly determined the torsional rigidity and breaking force of single actin filaments by measuring the rotational Brownian motion and tensile strength using optical tweezers and microneedles, respectively. Rotational angular fluctuations of filaments supplied the torsional rigidity as (8.0 ± 1.2) × 10−26 Nm2. This value is similar to that deduced from the longitudinal rigidity, assuming the actin filament to be a homogeneous rod. The breaking force of the actin–actin bond was measured while twisting a filament through various angles using microneedles. The breaking force decreased greatly under twist, e.g., from 600–320 pN when filaments were turned through 90°, independent of the rotational direction. Our results indicate that an actin filament exhibits comparable flexibility in the rotational and longitudinal directions, but breaks more easily under torsional load.

Sorting by diffusion: An asymmetric obstacle course for continuous molecular separation

A separation technique employing a microfabricated sieve has been demonstrated by observing the motion of DNA molecules of different size. The sieve consists of a two-dimensional lattice of obstacles whose asymmetric disposition rectifies the Brownian motion of molecules driven through the device, causing them to follow paths that depend on their diffusion coefficient. A nominal 6% resolution by length of DNA molecules in the size range 15–30 kbp may be achieved in a 4-inch (10-cm) silicon wafer. The advantage of this method is that samples can be loaded and sorted continuously, in contrast to the batch mode commonly used in gel electrophoresis.

Protrusive growth from giant liposomes driven by actin polymerization

Development of protrusions in the cell is indispensable in the process of cell motility. Membrane protrusion has long been suggested to occur as a result of actin polymerization immediately beneath the cell membrane at the leading edge, but elucidation of the mechanism is insufficient because of the complexity of the cell. To study the mechanism, we prepared giant liposomes containing monomeric actin (100 or 200 μM) and introduced KCl into individual liposomes by an electroporation technique. On the electroporation, the giant liposomes deformed. Most importantly, protrusive structure grew from the liposomes containing 200 μM actin at rates (ranging from 0.3 to 0.7 μm/s) similar to those obtained in the cell. The deformation occurred in a time range (30 ∼ 100 s) similar to that of actin polymerization monitored in a cuvette (ca. 50 s). Concomitant with deformation, Brownian motion of micron-sized particles entrapped in the liposomes almost ceased. From these observations, we conclude that actin polymerization in the liposomes caused the protrusive formation.

Determination of gene organization in individual haplotypes by analyzing single DNA fragments from single spermatozoa

To determine human Ig heavy chain variable region (VH) gene segment organization on individual homologous chromosomes, an efficient approach has been developed. Single spermatozoa were used as subjects for the study. Upon sperm lysis, VH regions in each sperm were randomly sheared into fragments by the random Brownian force. The fragments were separated from each other by aliquoting the lysate into a certain number of tubes. The gene segments in the VH1 and VH4 families in each tube were identified by denaturing gradient gel electrophoresis after PCR amplification. The polymorphic VH sequences were used to determine the parental origins of the analyzed sperm. VH segment organization in the parental haplotypes was determined by aligning the overlapping fragments from the spermatozoa with the corresponding haplotypes. Based on this comparison between the resulting haplotype maps and the composite map reported previously, the VH region on chromosome 14 could be subdivided into four portions. The numbers and compositions of the VH gene segments differ considerably among the maps in two portions, but are highly conserved in the other two. The data also indicate that the VH region on chromosome 15 may contain a large duplicated block with copy number varying among haplotypes. The approach used in the present study may be used to construct high-resolution haplotype maps without molecular cloning.

Electrostatic steering and ionic tethering in enzyme–ligand binding: Insights from simulations

To bind at an enzyme’s active site, a ligand must diffuse or be transported to the enzyme’s surface, and, if the binding site is buried, the ligand must diffuse through the protein to reach it. Although the driving force for ligand binding is often ascribed to the hydrophobic effect, electrostatic interactions also influence the binding process of both charged and nonpolar ligands. First, electrostatic steering of charged substrates into enzyme active sites is discussed. This is of particular relevance for diffusion-influenced enzymes. By comparing the results of Brownian dynamics simulations and electrostatic potential similarity analysis for triose-phosphate isomerases, superoxide dismutases, and β-lactamases from different species, we identify the conserved features responsible for the electrostatic substrate-steering fields. The conserved potentials are localized at the active sites and are the primary determinants of the bimolecular association rates. Then we focus on a more subtle effect, which we will refer to as “ionic tethering.” We explore, by means of molecular and Brownian dynamics simulations and electrostatic continuum calculations, how salt links can act as tethers between structural elements of an enzyme that undergo conformational change upon substrate binding, and thereby regulate or modulate substrate binding. This is illustrated for the lipase and cytochrome P450 enzymes. Ionic tethering can provide a control mechanism for substrate binding that is sensitive to the electrostatic properties of the enzyme’s surroundings even when the substrate is nonpolar.

Direct observation of extension and retraction of type IV pili

Type IV pili are thin filaments that extend from the poles of a diverse group of bacteria, enabling them to move at speeds of a few tenths of a micrometer per second. They are required for twitching motility, e.g., in Pseudomonas aeruginosa and Neisseria gonorrhoeae, and for social gliding motility in Myxococcus xanthus. Here we report direct observation of extension and retraction of type IV pili in P. aeruginosa. Cells without flagellar filaments were labeled with an amino-specific Cy3 fluorescent dye and were visualized on a quartz slide by total internal reflection microscopy. When pili were attached to a cell and their distal ends were free, they extended or retracted at rates of about 0.5 μm s−1 (29°C). They also flexed by Brownian motion, exhibiting a persistence length of about 5 μm. Frequently, the distal tip of a filament adsorbed to the substratum and the filament was pulled taut. From the absence of lateral deflections of such filaments, we estimate tensions of at least 10 pN. Occasionally, cell bodies came free and were pulled forward by pilus retraction. Thus, type IV pili are linear actuators that extend, attach at their distal tips, exert substantial force, and retract.

Conformational transitions monitored for single molecules in solution.

Phenomena that can be observed for a large number of molecules may not be understood if it is not possible to observe the events on the single-molecule level. We measured the fluorescence lifetimes of individual tetramethylrhodamine molecules, linked to an 18-mer deoxyribonucleotide sequence specific for M13 DNA, by time-resolved, single-photon counting in a confocal fluorescence microscope during Brownian motion in solution. When many molecules were observed, a biexponential fluorescence decay was observed with equal amplitudes. However, on the single-molecule level, the fraction of one of the amplitudes spanned from 0 to unity for a collection of single-molecule detections. Further analysis by fluorescence correlation spectroscopy made on many molecules revealed a process that obeys a stretched exponential relaxation law. These facts, combined with previous evidence of the quenching effect of guanosine on rhodamines, indicate that the tetramethylrhodamine molecule senses conformational transitions as it associates and dissociates to a guanosine-rich area. Thus, our results reveal conformational transitions in a single molecule in solution under conditions that are relevant for biological processes.

The kinesin walk: a dynamic model with elastically coupled heads.

Recently individual two-headed kinesin molecules have been studied in in vitro motility assays revealing a number of their peculiar transport properties. In this paper we propose a simple and robust model for the kinesin stepping process with elastically coupled Brownian heads that show all of these properties. The analytic and numerical treatment of our model results in a very good fit to the experimental data and practically has no free parameters. Changing the values of the parameters in the restricted range allowed by the related experimental estimates has almost no effect on the shape of the curves and results mainly in a variation of the zero load velocity that can be directly fitted to the measured data. In addition, the model is consistent with the measured pathway of the kinesin ATPase.