Theoretical formalism for bead movement powered by single two-headed motors in a motility assay

Kinesins and dyneins are protein motors that can use the free energy of ATP hydrolysis to carry a cargo and move uni-directionally along a microtubule filament. The purpose of this paper is to derive the formalism connecting the ATP-driven translocation reactions of these motors on microtubule filaments and the movement of the bead carried by the motor in a motility assay in which the bead is clamped at an arbitrary constant force. The formalism is thus useful in elucidating the load-dependent kinetic mechanism of the free-energy transduction of the motor using the mechanical data obtained from the motility assay. The formalism is also useful in assessing the effect on the measured motility data of various physical and hydrodynamic parameters of the assay, such as the size of the bead, the viscosity of the medium, the stiffness of the elastic element connecting the motor and the bead, etc. In a previous paper [Biophys. J. 67 (2000) 313] (hereafter referred to as paper I), we have derived the formalism for the case that the motor in the assay has only one head. In this paper we extend the derivation to the case that the motor is two-headed. The formalism is derived based on a simple two-state hand-over-hand model for the movement of the motor on microtubule, but can be easily extended to more complicated kinetic models. Effects of various hydrodynamic parameters on the velocity of the bead are studied with numerical calculations of the model. The difference between the formalism presented in this paper and the widely used "chemical" formalism, in which the movement of the kinesin and the bead is described by pure chemical reactions, is discussed.     

Theoretical formalism for kinesin motility I. Bead movement powered by single one-headed kinesins

The directional movement on a microtubule of a plastic bead connected elastically to a single one-headed kinesin motor is studied theoretically. The kinesin motor can bind and unbind to periodic binding sites on the microtubule and undergo conformational changes while catalyzing the hydrolysis of ATP. An analytic formalism relating the dynamics of the bead and the ATP hydrolysis cycle of the motor is derived so that the calculation of the average velocity of the bead can be easily carried out. The formalism was applied to a simple three-state biochemical model to investigate how the velocity of the bead movement is affected by the external load, the diffusion coefficient of the bead, and the stiffness of the elastic element connecting the bead and the motor. The bead velocity was found to be critically dependent on the diffusion coefficient of the bead and the stiffness of the elastic element. A linear force-velocity relation was found for the model no matter whether the bead velocity was modulated by the diffusion coefficient of the bead or by the externally applied load. The formalism should be useful in modeling the mechanisms of chemimechanical coupling in kinesin motors based on in vitro motility data.     

Monte Carlo simulations of a protein molecule with and without hydration energy calculated by the hydration-shell model

Monte Carlo simulations of a small protein, carmbin, were carried out with and without hydration energy. The methodology presented here is characterized, as compared with the other similar simulations of proteins in solution, by two points: (1) protein conformations are treated in fixed geometry so that dihedral angles are independent variables rather than cartesian coordinates of atoms; and (2) instead of treating water molecules explicitly in the calculation, hydration energy is incorporated in the conformational energy function in the form of g _i A _i, whereA _i is the accessible surface area of an atomic groupi in a given conformation, andg _i is the free energy of hydration per unit surface area of the atomic group (i.e., hydration-shell model). Reality of this model was tested by carrying out Monte Carlo simulations for the two kinds of starting conformations, native and unfolded ones, and in the two kinds of systems,in vacuo and solution. In the simulations starting from the native conformation, the differences between the mean propertiesin vacuo and solution simulations are not very large, but their fluctuations around the mean conformation during the simulation are relatively smaller in solution thanin vacuo. On the other hand, in the simulations starting from the unfolded conformation, the molecule fluctuates much more largely in solution thanin vacuo, and the effects of taking into account the hydration energy are pronounced very much. The results suggest that the method presented in this paper is useful for the simulations of proteins in solution.     

ATPase kinetic characterization and single molecule behavior of mutant human kinesin motors defective in microtubule-based motility

Conventional kinesin is a microtubule-based motor protein that is an important model system for understanding mechanochemical transduction. To identify regions of the kinesin protein that participate in microtubule binding and force production, Woehlke et al. [(1997) Cell 90, 207-216] generated 35 alanine mutations in solvent-exposed residues. Here, we have performed presteady-state kinetic and single molecule motility analyses on three of these mutants [Y138A, loop 11 triple (L248A/D249A/E250A), and E311A] that exhibited a similar approximately 3-fold reduction in both microtubule gliding velocity and microtubule-stimulated ATPase activity. All mutants showed normal second-order ATP binding kinetics, indicating correct folding of the active site. The Y138A and loop 11 triple mutants were defective both in nucleotide hydrolysis and in microtubule-stimulated ADP release rates, the latter suggesting a defect in allosteric communication between the microtubule and the active site. A single molecule fluorescence assay further revealed that the loop 11 mutant is defective in initiating processive motion, suggesting that this loop is important for the initial contact between kinesin and the microtubule. Y138A, on the other hand, can bind to the microtubule normally but cannot move processively. For E311A, neither the rate of nucleotide hydrolysis nor ADP release could account for its slower ATPase and gliding velocity, which suggests that either phosphate release or a conformational transition is rate-limiting in this mutant. The single molecule assay showed that E311A has a reduced velocity of movement, but is not defective in processivity. Thus, while these mutants behave similarly in solution ATPase and multiple motor gliding assays, kinetic and single molecule analyses reveal defects in distinct processes in kinesin's mechanochemical cycle.     

Adsorption into the MFI zeolite of aromatic molecule of biological relevance. Investigations by Monte Carlo simulations

Adsorption of paracresol and water into the silicalite-1 (MFI) zeolite has been investigated using canonical and grand-canonical Monte Carlo simulations. The most stable sites of adsorption of paracresol are found to be located at the channel intersections. Grand-canonical simulations have shown that at low loading, water molecules adsorb preferably at the vicinity of paracresol molecules, whereas they are also located in the sinusoidal channels as the loading increases. In order to explain the experimental adsorption isotherm observed for the coadsorption of water and paracresol in the MFI zeolite we propose a new concept of apparent adsorption enthalpy that varies with the concentration of the solution. The mathematical expression for the apparent enthalpy is introduced in an adsorption isotherm model. We shall refer to this theoretical isotherm as a non-langmuirian isotherm. The non-linear expression for the apparent adsorption enthalpy accounts for a variable accessibility of the sites of adsorption with respect to the concentration of the solution. Figure Co-adsorption of paracresol and water in silicalite-1 zeolite and comparison between experimental and modelled adsorption isotherms.     

Numerical and Monte Carlo simulations of phenolic polymerizations catalyzed by peroxidase

Numerical and Monte Carlo simulations of horseradish peroxidase-catalyzed phenolic polymerizations have been performed. Kinetic constants for the simulations were fit to data from the oxidation and polymerization of bisphenol A. Simulations of peroxidase-catalyzed phenolic polymerization were run as a function of enzyme concentration and radical transfer and radical coupling rate constants. Predictions were performed with respect to conversion vs. time and number average molecular weight and polydispersity vs. conversion. It is shown that the enzymatic polymerization of phenols can be optimized with respect to high molecular weights by employing low enzyme concentrations and phenols with low radical coupling rate constants coupled with relatively high radical transfer rate constants. Such phenols may be identified by using linear free energy relationships that relate radical reactivity to electron donating/withdrawing potential of the phenolic substituent. (c) 1993 John Wiley & Sons, Inc.     

Monte Carlo simulations of structure and entanglements in polymer melts

Atomistic models of short chain branched (SCB) polyethylene melts have been equilibrated at 450 K using a connectivity altering Monte Carlo method. Quantities related to the chain dimensions and entanglements have been determined. The simulated tube diameters, 〈a_pp〉, of SCB melts are found to scale with the backbone weight fraction, ?, as 〈a_pp〉～?^? 0.46, close to the scaling predicted by the binary contact model, 〈a_pp〉～?^? 0.5. Similar relationships are observed experimentally for polymer solutions, and reproduced by the present methods.     

Monte Carlo study of single molecule diffusion can elucidate the mechanism of B cell synapse formation

B cell receptors have been shown to cluster at the intercellular junction between a B cell and an antigen-presenting cell in the form of a segregated pattern of B cell receptor/antigen complexes known as an immunological synapse. We use random walk-based theoretical arguments and Monte Carlo simulations to study the effect of diffusion of surface-bound molecules on B cell synapse formation. Our results show that B cell synapse formation is optimal for a limited range of receptor-ligand complex diffusion coefficient values, typically one-to-two orders of magnitude lower than the diffusion coefficient of free receptors. Such lower mobility of receptor-ligand complexes can significantly affect the diffusion of a tagged receptor or ligand in an affinity dependent manner, as the binding/unbinding of such receptor or ligand molecules crucially depends on affinity. Our work shows how single molecule tracking experiments can be used to estimate the order of magnitude of the diffusion coefficient of receptor-ligand complexes, which is difficult to measure directly in experiments due to the finite lifetime of receptor-ligand bonds. We also show how such antigen movement data at the single molecule level can provide insight into the B cell synapse formation mechanism. Thus, our results can guide further single molecule tracking experiments to elucidate the synapse formation mechanism in B cells, and potentially in other immune cells.     

Confinement effects on the thermodynamics of protein folding: Monte Carlo simulations

The effects of chaperonin-like cage-induced confinement on protein stability have been studied for molecules of varying sizes and topologies. Minimalist models based on Gō-like interactions are employed for the proteins, and density-of-states-based Monte Carlo simulations are performed to accurately characterize the thermodynamic transitions. This method permits efficient sampling of conformational space and yields precise estimates of free energy and entropic changes associated with protein folding. We find that confinement-driven stabilization is not only dependent on protein size and cage radius, but also on the specific topology. The choice of the confining potential is also shown to have an effect on the observed stabilization and the scaling behavior of the stabilization with respect to the cage size.     

Up-and-down movement of a sliding actin filament in the in vitro motility assay

We observed a three-dimensional up-and-down movement of an actin filament sliding on heavy mero-myosin (HMM) molecules in an in vitro motility assay. The up-and-down movement occurred along the direction perpendicular to the planar glass plane on which the filament demonstrated a sliding movement. The height length of the up-and-down movement was measured by monitoring the extent of diminishing fluorescent emission from the marker attached to the filament in the evanescent field of attenuation. The height lengths whose distribution exhibits a local maximum were found around the two values, 150 nm and 90 nm, separately. This undulating three-dimensional movement of an actin filament suggests that the interactions between myosin (HMM) molecules and the actin filament may temporally be modulated during its sliding movement.     

Multi-particle sampling in Monte Carlo simulations on fluids: efficiency and extended implementations

Various technical aspects affecting the efficiency of a recently proposed novel Monte Carlo (MC) simulation scheme based on biased simultaneous displacements/rotations of all particles of the system are investigated using two polarisable models of water, the Chialvo–Cummings and Brodholt–Sampoli–Vallauri models, as a test case. Necessary expressions for polarisable site–site interaction models are derived along with a novel smoothing of the potential at the cut-off distance. In addition to the common thermodynamic and structural properties, the mean-squared displacements, rotation relaxation, speed of equilibration (translational order parameter, TOP) and autocorrelation coefficients have been computed as well, in order to assess the efficiency of the method. Gain in speed by parallelisation has also been examined. Performance of the method is coumpared with both the standard one-particle move method and the available approximate methods. It is shown that the multi-particle move (MPM) method performs about by a factor of 10 faster for the systems considered when compared with the common MC scheme, and several times faster when compared with the approximate methods. Parallelised codes of the MPM method may then perform about 70 times faster than the conventional MC. These conclusions hold true for the system size simulated (N = 256) because the efficiency of the multi-particle method depends on the size of the system: its efficiency even increases with increasing number of particles.     

Current and selectivity in a model sodium channel under physiological conditions: Dynamic Monte Carlo simulations

A reduced model of a sodium channel is analyzed using Dynamic Monte Carlo simulations. These include the first simulations of ionic current under approximately physiological ionic conditions through a model sodium channel and an analysis of how mutations of the sodium channel's DEKA selectivity filter motif transform the channel from being Na(+) selective to being Ca(2+) selective. Even though the model of the pore, amino acids, and permeant ions is simplified, the model reproduces the fundamental properties of a sodium channel (e.g., 10 to 1 Na(+) over K(+) selectivity, Ca(2+) exclusion, and Ca(2+) selectivity after several point mutations). In this model pore, ions move through the pore one at a time by simple diffusion and Na(+) versus K(+) selectivity is due to both the larger K(+) not fitting well into the selectivity filter that contains amino acid terminal groups and K(+) moving more slowly (compared to Na(+)) when it is in the selectivity filter.     

Activation of Bax by joint action of tBid and mitochondrial outer membrane: Monte Carlo simulations

The mitochondrial pathway of apoptosis proceeds when molecules, such as cytochrome c, sequestered between the outer and inner mitochondrial membranes are released to the cytosol by mitochondrial outer membrane (MOM) permeabilization. Bax, a member of the Bcl-2 protein family, plays a pivotal role in mitochondrion-mediated apoptosis. In response to apoptotic stimuli, Bax integrates into the MOM, where it mediates the release of cytochrome c from the intermembrane space into the cytosol, leading to caspase activation and cell death. The pro-death action of Bax is regulated by interactions with both other prosurvival proteins, such as tBid, and the MOM, but the exact mechanisms remain largely unclear. Here, the mechanisms of integration of Bax into a model membrane mimicking the MOM were studied by Monte Carlo simulations preceded by a computer prediction of the docking of tBid with Bax. A novel model of Bax activation by tBid was predicted by the simulations. In this model, tBid binds to Bax at an interaction site formed by Bax helices α1, α2, α3 and α5 leading, due to interaction of the positively charged N-terminal fragment of tBid with anionic lipid headgroups, to Bax reorientation such that a hydrogen-bonded pair of residues, Asp98 and Ser184, is brought into close proximity with negatively charged lipid headgroups. The interaction with these headgroups destabilizes the hydrogen bond which results in the release of helix α9 from the Bax-binding groove, its insertion into the membrane, followed by insertion into the membrane of the α5–α6 helical hairpin. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

Response of SOI microdosimeter in fast neutron beams: experiment and Monte Carlo simulations

In this study, Monte Carlo codes, Geant4 and MCNP6, were used to characterize the fast neutron therapeutic beam produced at iThemba LABS in South Africa. Experimental and simulation results were compared using the latest generation of Silicon on Insulator (SOI) microdosimeters from the Centre for Medical Radiation Physics (CMRP). Geant4 and MCNP6 were able to successfully model the neutron gantry and simulate the expected neutron energy spectrum produced from the reaction by protons bombarding a ⁹Be target. The neutron beam was simulated in a water phantom and its characteristics recorded by the silicon microdosimeters; bare and covered by a ¹⁰B enriched boron carbide converter, at different positions. The microdosimetric quantities calculated using Geant4 and MCNP6 are in agreement with experimental measurements. The thermal neutron sensitivity and production of ¹⁰B capture products in the p+ boron-implanted dopant regions of the Bridge microdosimeter is investigated. The obtained results are useful for the future development of dedicated SOI microdosimeters for Boron Neutron Capture Therapy (BNCT). This paper provides a benchmark comparison of Geant4 and MCNP6 capabilities in the context of further applications of these codes for neutron microdosimetry.     

Coarse-grained Monte Carlo simulations of mucus: structure, dynamics, and thermodynamics

A simple coarse-grained model of mucus structure and dynamics is proposed and evaluated. The model is based on simple cubic, face-centered lattice representation. Mucins are simulated as lattice chains in which each bead of the model chains represents a mucin domain, equivalent to its Kuhn segment. The remaining lattice sites are considered to be occupied by the solvent. Model mucins consist of three types of domains: polar (glycosylated central segments), hydrophobic, and cysteine-rich, located at the terminal part of the mucin chains. The sequence of these domains mimics the sequence of real mucins. Static and dynamic properties of the system were studied by means of Monte Carlo dynamics. It was shown that the model system undergoes sol-gel transition and that the interactions between hydrophobic domains are responsible for the transition and characteristic properties of the dynamic network in the gel phase. Cysteine-rich domains are essential for frictional properties of the system. Structural and dynamic properties of the model mucus observed in simulations are in qualitative agreement with known experimental facts and provide mechanistic explanation of complex properties of real mucus.     

Testing for heterogeneity among phenotypic correlations: a comparison of methods using Monte Carlo simulations

Heterogeneous phenotypic correlations may be suggestive of underlying changes in genetic covariance among life-history, morphology, and behavioural traits, and their detection is therefore relevant to many biological studies. Two new statistical tests are proposed and their performances compared with existing methods. Of all tests considered, the existing approximate test of homogeneity of product-moment correlations provides the greatest power to detect heterogeneous correlations, when based on Hotelling's z*-transformation. The use of this transformation and test is recommended under conditions of bivariate normality. A new distribution-free randomisation test of homogeneity of Spearman's rank correlations is described and recommended for use when the bivariate samples are taken from populations with non-normal or unknown distributions. An alternative randomisation test of homogeneity of product-moment correlations is shown to be a useful compromise between the approximate tests and the randomisation tests on Spearman's rank correlations: it is not as sensitive to departures from normality as the approximate tests, but has greater power than the rank correlation test. An example is provided that shows how choice of test will have a considerable influence on the conclusions of a particular study. This revised version was published online in July 2006 with corrections to the Cover Date.     

Monte Carlo simulations of the chiral recognition of fenoprofen enantiomers by cyclomaltoheptaose (beta-cyclodextrin)

Differential complexation of fenoprofen enantiomers by cyclomaltoheptaose (beta-cyclodextrin) was investigated by Monte Carlo docking simulations. The chiral discrimination of (R)- and (S)-fenoprofen by beta-cyclodextrin was discussed in terms of the difference in the interaction energies and the patterns of molecular interactions. The interaction energies between each enantiomer of fenoprofen and beta-cyclodextrin were consistent with the reported experimental results that showed that the S isomer interacted preferentially with beta-cyclodextrin and was retained longer in a separation process than the R isomer. The thermodynamic preference of inclusion complex formation of (S)-fenoprofen could be explained by the orientation of the phenyl group attached to the chiral carbon, which provided closer contact and thus more favorable intermolecular interactions between the host and guest molecule. The results presented here would be very useful for the prediction of chiral recognition ability of beta-cyclodextrin.     

Prediction of protein conformation in water and on surfaces by Monte Carlo simulations using united-atom method

The united-atom method has been used to model an avian pancreatic polypeptide (APP) in water and the adsorption process of an albumin subdomain (AS) onto graphite surface to observe the capability of this lumped modelling approach to generate structures observed in protein data bank (PDB) and from atomistic modelling. The subdomain structure of a protein is simplified by the united-atom approximation where the side chains and peptide groups are represented by lumped spheres. The total potential energy of the adsorption process involves the interaction between these lumped spheres by means of virtual bond chain interaction and the interaction of the spheres with the graphite surface by means of Lennard-Jones potential. The protein/polypeptide structure has been perturbed by Monte Carlo with energy minimisation to obtain the global minimum. Results on the APP in water showed a near-to-experimental PDB conformation revealing the two α-helix structures of this small protein molecule with the root mean square deviation among carbon backbone atoms of 5.9 Å. Protein adsorption on biosurfaces has been made by modelling AS, which has 60 amino acids. The surface is graphite, which is characterised by its hydrophobicity. Graphite was chosen because of its widely used applications in certain implants that interact with blood. Our simulation results showed final conformation close to that obtained by atomistic modelling. It also proved that the whole pattern of intramolecular hydrogen bonds was distorted. The model also demonstrated the random conformation of the original α-helix secondary structures of AS consistent with experimental and atomistic results. While atomistic simulation works well for simulating individual small proteins, the united-atom model is more efficient when simulating macromolecular and multiple protein adsorption where time and limiting computer capacity are key factors.     

Molecular dynamics simulations of RNA: an in silico single molecule approach

RNA molecules are now known to be involved in the processing of genetic information at all levels, taking on a wide variety of central roles in the cell. Understanding how RNA molecules carry out their biological functions will require an understanding of structure and dynamics at the atomistic level, which can be significantly improved by combining computational simulation with experiment. This review provides a critical survey of the state of molecular dynamics (MD) simulations of RNA, including a discussion of important current limitations of the technique and examples of its successful application. Several types of simulations are discussed in detail, including those of structured RNA molecules and their interactions with the surrounding solvent and ions, catalytic RNAs, and RNA-small molecule and RNA-protein complexes. Increased cooperation between theorists and experimentalists will allow expanded judicious use of MD simulations to complement conceptually related single molecule experiments. Such cooperation will open the door to a fundamental understanding of the structure-function relationships in diverse and complex RNA molecules. .