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.     

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.     

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.     

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 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.     

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.     

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.     

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.     

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. .     

Open-state conformation of the KcsA K+ channel: Monte Carlo normal mode following simulations

Potassium channels fluctuate between closed and open states. The detailed mechanism of the conformational changes opening the intracellular pore in the K+ channel from Streptomyces lividans (KcsA) is unknown. Applying Monte Carlo normal mode following, we find that gating involves rotation and unwinding of the TM2 bundle, lateral movement of the TM2 helices away from the channel axis, and disappearance of the TM2 bundle. The open-state conformation of KcsA exhibits a very wide inner vestibule, with a radius approximately 5-7 A and inner helices bent at the A98-G99 hinge. Computed conformational changes demonstrate that spin labeling and X-ray experiments illuminate different stages in gating: transition begins with clockwise rotation of the TM2 helices ending at a final state with the TM2 bend hinged near residues A98-G99. The concordance between the computational and experimental results provides atomic-level insights into the structural rearrangements of the channel's inner pore.     

Monte Carlo investigation of electron specific energy distribution in a single cell model

Radiation and Environmental Biophysics - Knowledge of microdosimetric quantities of certain radionuclides is important in radio immune cancer therapies. Specific energy distribution of...     

Structural principles of B-DNA grooves hydration in fibers as revealed by Monte Carlo simulations and X-ray diffraction

Being interested in possible effects of sequence-dependent hydration of B-DNA with mixed sequence in fibers, we performed a series of Monte Carlo calculations of hydration of polydeoxyribonucleotides in B form, considering all sequences with dinucleotide repeat. The computational results allow the ten base-stacking types to be classified in accordance with their primary hydration in the minor groove. As a rule, the minor groove is occupied by two water molecules per base pair in the depth of the groove, which are located nearly midway between the planes of successive base pairs and symmetrically according to the dyad there. The primary hydration of the major groove depends on the type of the given base pair. The coordinates of 3 water molecules per base pair in the depth of the major groove are determined by the type of this pair together with its position and orientation in the helix, and are practically independent on the adjacent base pairs. A/T-homopolymer tracts do not fit into this hydration pattern; the base pair edges are hydrated autonomously in both grooves. Analysis of the Li-B-DNA x-ray diffraction intensities reveals those two water positions in the minor groove. In the major groove, no electronic density peaks in sufficient distance from the base edges were found, thus confirming the absence of any helical invariance of primary hydration in this region. With the help of the rules proposed in this paper it is possible to position the water molecules of the first hydration shell in the grooves of canonical B-DNA for any given sequence.     

Product yields from irradiated glycylglycine in oxygen-free solutions: Monte Carlo simulations and comparison with experiments

The radiation chemistry of photon-irradiated aqueous solutions of biological molecules may be considered under four distinct time regimes: physical transport (≤10^–15 s); prechemical conversion of H₂O⁺, H₂O^*, and subexcitation electrons into free radicals and molecular products (10^–15 s to 10^–12 s); chemical reactions within individual electron tracks (10^–12 s to 10^–6 s); and chemical reactions within overlapping tracks (>10^–6 s). We have previously reported of the use of the Monte Carlo radiation transport/chemistry codes OREC and RADLYS to model the radiolysis of glycylglycine in oxygen-free solution to a time of 1 μs. These simulations successfully predicted the yields of free ammonia, an end product created solely in the reaction of the hydrated electron with the solute within individual tracks. Other measurable products are only partially created during intratrack reactions, and thus one must additionally consider the late, intertrack chemistry of this system. In this paper, we extend our simulations of glycylglycine radiolysis to model for the first time the events which occur during this late chemistry stage. The model considers the product rates of the reactants in bulk solution by using previously available microsecond intratrack yields given by single-track OREC/RADLYS simulations and an x-ray dose rate of 2.80 Gy min^–1 as used in a companion experimental program. These rates are then applied in a series of coupled, differential rate equations that describe the solution chemistry of glycylglycine radiolysis. Product yields are reported as a function of time over a total irradiation period of 10⁴ s. Excellent overall agreement is seen between the theoretical predictions and measurements of five radiolysis end products: free ammonia, acetylglycine, diaminosuccinic acid, aspartic acid, and succinic acid. The model also gives the explicit contributions of intratrack and intertrack reactions to the various end products. For example, the model predicts that ∼56% and 93% of succinic acid and aspartic acid, respectively, are produced during intertrack reactions at a solute concentration of 0.05 M; these contributions drop to 0.07% and 11%, respectively, at 1.2 M. Received: 22 May 1998 / Accepted in revised form: 27 August 1998