Bidirectional sliding of two parallel microtubules generated by multiple identical motors

Journal of Mathematical Biology - It is often assumed in biophysical studies that when multiple identical molecular motors interact with two parallel microtubules, the microtubules will be...     

Obstacle bypass in protein motion along DNA by two-dimensional rather than one-dimensional sliding

Site-specific DNA-binding proteins locate their target sites by facilitated diffusion. Several proteins have been shown to slide along DNA in vitro. However, whereas sliding is often envisaged as one-dimensional tracking of the DNA major groove, such a mechanism would not allow linear diffusion over long distances in vivo, where short stretches of free DNA are delimited by bound proteins. I propose a two-dimensional sliding mechanism, in which the protein diffuses freely on the cylindrical DNA surface, and I present experiments that can distinguish between one- and higher-dimensional diffusion along the DNA contour length. At 100 mm NaCl, translocation of EcoRI restriction endonuclease between sites on two DNA helices connected by a Holliday junction is as efficient as between sites on the same helix, indicating a three-dimensional mechanism. At 25 mm NaCl, translocation between sites on the same DNA helix is more efficient, indicating a role for sliding at low ionic strength. Obstacles attached to the major groove of one face of the DNA helix did not interfere with sliding, regardless of their orientation relative to the cleavage sites. This result is compatible with two-dimensional but not one-dimensional sliding. As illustrated by Monte-Carlo simulation, two-dimensional sliding may not only allow proteins to move around nucleosomes in vivo but also reduce the redundancy of their search for the target site.     

Fluctuations, responses and energetics of molecular motors

A novel equality relating the rate of energy dissipation to a degree of violation of the fluctuation-response relation (FRR) in non-equilibrium Langevin systems is described. The FRR is a relation between the correlation function of the fluctuations and the response function of macroscopic variables. Although it has been established that the FRR holds in equilibrium, physical significance of violation of the FRR in non-equilibrium systems has been under debate. Recently, the authors have found that an extent of the FRR violation is related in a simple equality to the rate of energy dissipation into the environment in non-equilibrium Langevin systems. In this paper, we fully explain the FRR, the FRR violation, and the new equality with regard to a Langevin model termed a Brownian motor model, which is considered as a simple model of a biological molecular motor. Furthermore, applications of our result to experimental studies of molecular motors are discussed, and, as an illustration, we predict the value of a new time constant regarding the motion of a KIF1A, which is a kind of single-headed kinesin.     

The role of thermal activation in motion and force generation by molecular motors

The currently accepted mechanism for ATP-driven motion of kinesin is called the hand-over-hand model, where some chemical transition during the ATP hydrolysis cycle stretches a spring, and motion and force production result from the subsequent relaxation. It is essential in this mechanism for the moving head of kinesin to dissociate, while the other head remains firmly attached to the microtubule. Here we propose an alternative Brownian motor model where the action of ATP modulates the interaction potential between kinesin and the microtubule rather than a spring internal to the kinesin molecule alone. In this model neither head need dissociate (which predicts that under some circumstances a single-headed kinesin can display processive motion) and the transitions by which the motor moves are best described as thermally activated steps. This model is consistent with a wide range of experimental data on the force-velocity curves, the one ATP to one-step stoichiometry observed at small load, and the stochastic properties of the stepping.     

Dwell time symmetry in random walks and molecular motors

The statistics of steps and dwell times in reversible molecular motors differ from those of cycle completion in enzyme kinetics. The reason is that a step is only one of several transitions in the mechanochemical cycle. As a result, theoretical results for cycle completion in enzyme kinetics do not apply to stepping data. To allow correct parameter estimation, and to guide data analysis and experiment design, a theoretical treatment is needed that takes this observation into account. In this article, we model the distribution of dwell times and number of forward and backward steps using first passage processes, based on the assumption that forward and backward steps correspond to different directions of the same transition. We extend recent results for systems with a single cycle and consider the full dwell time distributions as well as models with multiple pathways, detectable substeps, and detachments. Our main results are a symmetry relation for the dwell time distributions in reversible motors, and a relation between certain relative step frequencies and the free energy per cycle. We demonstrate our results by analyzing recent stepping data for a bacterial flagellar motor, and discuss the implications for the efficiency and reversibility of the force-generating subunits.     

A protein friction model of the actin sliding movement generated by myosin in mixtures of MgATP and MgGTP in vitro.

The sliding movement of an actin filament generated by myosin heads with MgGTP bound is much slower than that by those with MgATP bound. Nonetheless, there is a report that the actin sliding velocity at low (11-21 microM) MgATP concentrations is increased by the addition of MgGTP in a range of 1-3 mM, although the actin sliding velocity at these MgATP concentrations is larger than the maximum sliding velocity attained in the presence of MgGTP alone. The convex rise in the velocity was called "mutual sensitization of MgATP and MgGTP" in the report. Here we propose a theoretical model to account for the mutual sensitization of MgATP and MgGTP. The model is an extension of a protein friction model, accommodating the presence of two different substrates and assuming the presence of motile and non-motile myosins. This new model is in accord with the characteristics of the actin/myosin sliding movement experimentally observed in mixtures of MgATP and MgGTP. Comparison of the model with the experimental results implies that the non-motile and motile myosins are those with the "converse and correct" orientations of their heads with respect to the direction of the actin sliding movement in vitro.     

Molecular motors: thermodynamics and the random walk

The biochemical cycle of a molecular motor provides the essential link between its thermodynamics and kinetics. The thermodynamics of the cycle determine the motor's ability to perform mechanical work, whilst the kinetics of the cycle govern its stochastic behaviour. We concentrate here on tightly coupled, processive molecular motors, such as kinesin and myosin V, which hydrolyse one molecule of ATP per forward step. Thermodynamics require that, when such a motor pulls against a constant load f, the ratio of the forward and backward products of the rate constants for its cycle is exp [-(DeltaG + u(0)f)/kT], where -DeltaG is the free energy available from ATP hydrolysis and u(0) is the motor's step size. A hypothetical one-state motor can therefore act as a chemically driven ratchet executing a biased random walk. Treating this random walk as a diffusion problem, we calculate the forward velocity v and the diffusion coefficient D and we find that its randomness parameter r is determined solely by thermodynamics. However, real molecular motors pass through several states at each attachment site. They satisfy a modified diffusion equation that follows directly from the rate equations for the biochemical cycle and their effective diffusion coefficient is reduced to D-v(2)tau, where tau is the time-constant for the motor to reach the steady state. Hence, the randomness of multistate motors is reduced compared with the one-state case and can be used for determining tau. Our analysis therefore demonstrates the intimate relationship between the biochemical cycle, the force-velocity relation and the random motion of molecular motors.     

Thermal fluctuations biased for directional motion in molecular motors

Recently developed single molecule measurements have demonstrated that the mechanisms for numerous protein functions involve thermal fluctuation, or Brownian motion. Protein interactions bias the random thermal noise in a manner such that the protein can perform its given functions. This phenomenon has been observed in molecular motor unidirectional movement where Brownian motion is used to preferentially bind the motor heads in one direction causing directional motility. This is analogous to that used by proteins in which spontaneous structural fluctuations are used to switch function. Seeing that two very different systems implement similar mechanisms suggests there exists a general scheme applied by diverse proteins that exploits thermal fluctuations in order to achieve their respective functions.     

Cell speed is independent of force in a mathematical model of amoeboidal cell motion with random switching terms

In this paper the motion of a single cell is modeled as a nucleus and multiple integrin based adhesion sites. Numerical simulations and analysis of the model indicate that when the stochastic nature of the adhesion sites is a memoryless and force independent random process, the cell speed is independent of the force these adhesion sites exert on the cell. Furthermore, understanding the dynamics of the attachment and detachment of the adhesion sites is key to predicting cell speed. We introduce a differential equation describing the cell motion and then introduce a conjecture about the expected drift of the cell, the expected average velocity relation conjecture. Using Markov chain theory, we analyze our conjecture in the context of a related (but simpler) model of cell motion, and then numerically compare the results for the simpler model and the full differential equation model. We also heuristically describe the relationship between the simplified and full models as well as provide a discussion of the biological significance of these results.     

Exact computation of pattern probabilities in random sequences generated by Markov chains

Observed patterns in macromolecular sequences are often consideredas words and compared with their probabilities of occurringin random sequences. Calculation of these probabilities, however,often lacks rigour. We have developed an algorithm for exactcomputation of such probabilities for stochastic sequences thatfollow a Markov chain model. The method is applicable to thecase that a random sequence contains one out of two given patternsP and Q, or both simultaneously. Another application yieldsthe probability Junction P(x) that a sequence contains patternP exactly x times. An application to patterns that include wild-cardcharacters yields probabilities for homonucleotide clustersof a given length. We prove the probability of multiple runsof single nucleotides in the SV40 genome to be in accordancewith the dinucleotide composition of the sequence, althoughit is in conflict with mononucleotide composition. Received on January 10, 1990; accepted on April 23, 1990     

Demethylation initiated by ROS1 glycosylase involves random sliding along DNA

Active DNA demethylation processes play a critical role in shaping methylation patterns, yet our understanding of the mechanisms involved is still fragmented and incomplete. REPRESSOR OF SILENCING 1 (ROS1) is a prototype member of a family of plant 5-methylcytosine DNA glycosylases that initiate active DNA demethylation through a base excision repair pathway. As ROS1 binds DNA non-specifically, we have critically tested the hypothesis that facilitated diffusion along DNA may contribute to target location by the enzyme. We have found that dissociation of ROS1 from DNA is severely restricted when access to both ends is obstructed by tetraloops obstacles. Unblocking any end facilitates protein dissociation, suggesting that random surface sliding is the main route to a specific target site. We also found that removal of the basic N-terminal domain of ROS1 significantly impairs the sliding capacity of the protein. Finally, we show that sliding increases the catalytic efficiency of ROS1 on 5-meC:G pairs, but not on T:G mispairs, thus suggesting that the enzyme achieves recognition and excision of its two substrate bases by different means. A model is proposed to explain how ROS1 finds its potential targets on DNA.     

Force and motion generation of myosin motors: muscle contraction

Brownian ratchet theory refers to the phenomenon that non-equilibrium fluctuations in an isothermal medium and anisotropic system can induce mechanical force and motion. This concept of noise-induced transport has motivated an abundance of theoretical and applied research. One of the exciting applications of the ratchet theory lies in the possible explanation of the operating mode of biological molecular motors. Biomolecular motors are proteins able of converting chemical reactions into mechanical motion and force. Operating at energy levels only a few times greater than the energy levels of thermal baths, their operating mode has to be stochastic in nature. Here, we review the theoretical concepts of the Brownian ratchet theory and its possible link to the operation of the myosin II motors involved in muscle contraction.     

First and second moment of counts of words in random texts generated by Markov chains

An exact expression for the variance of random frequency thata given word has in text generated by a Markov chain is presented.The result is applied to periodic Markov chains, which describethe protein-coding DNA sequences better than simple Markov chains.A new solution to the problem of word overlap is proposed. Itwas found that the expected frequency and overlapping propertiesdetermine most of the variance. The expectation and varianceof counts for triplets are compared with experimental countsin Escherichia coli coding sequences.     

Geometrical restrictions on cross-bridge motion and sliding filament dynamics

To ensure asynchronous motion and continuous force development in a sliding filament mechanism when the cross-bridge configurational cycle is geometrically constant and uniform, the repeat distances of the cross-bridge set and the active site set must be different. When the two sets overlap they will then form an interference pattern—the elements are in phase in some regions, out of phase in others. The geometrical restrictions that this puts on the cross-bridge cycle are discussed. One consequence may be that only a discrete set of modes of motion are possible in active filament sliding, the number of modes being determined by the energy supply rate and the internal friction in the system.     

Dynamics of asynchronous random Boolean networks with asynchrony generated by stochastic processes

An asynchronous Boolean network with N nodes whose states at each time point are determined by certain parent nodes is considered. We make use of the models developed by Matache and Heidel [Matache, M.T., Heidel, J., 2005. Asynchronous random Boolean network model based on elementary cellular automata rule 126. Phys. Rev. E 71, 026232] for a constant number of parents, and Matache [Matache, M.T., 2006. Asynchronous random Boolean network model with variable number of parents based on elementary cellular automata rule 126. IJMPB 20 (8), 897-923] for a varying number of parents. In both these papers the authors consider an asynchronous updating of all nodes, with asynchrony generated by various random distributions. We supplement those results by using various stochastic processes as generators for the number of nodes to be updated at each time point. In this paper we use the following stochastic processes: Poisson process, random walk, birth and death process, Brownian motion, and fractional Brownian motion. We study the dynamics of the model through sensitivity of the orbits to initial values, bifurcation diagrams, and fixed-point analysis. The dynamics of the system show that the number of nodes to be updated at each time point is of great importance, especially for the random walk, the birth and death, and the Brownian motion processes. Small or moderate values for the number of updated nodes generate order, while large values may generate chaos depending on the underlying parameters. The Poisson process generates order. With fractional Brownian motion, as the values of the Hurst parameter increase, the system exhibits order for a wider range of combinations of the underlying parameters.     

Three novel pigmentation mutants generated by genome-wide random ENU mutagenesis in the mouse

Three mutant mice with pigmentation phenotypes were recovered from a genomewide random mouse chemical mutagenesis study. White toes (Whto; MGI:1861986), Belly spot and white toes (Bswt; MGI:2152776) and Dark footpads 2 (Dfp2; MGI:1861991) were identified following visual inspection of progeny from a male exposed to the point mutagen ethylnitrosourea (ENU). In order to rapidly localize the causative mutations, genome-wide linkage scans were performed on pooled DNA samples from backcross animals for each mutant line. Whto was mapped to proximal mouse chromosome (Mmu) 7 between Cen (the centromere) and D7Mit112 (8.0 cM from the centromere), Bswt was mapped to centric Mmul between D1Mit214 (32.1 cM) and D1Mit480 (32.8 cM) and Dfp2 was mapped to proximalMmu4 between Cen and D4Mit18 (5.2 cM). Whto, Bswt and Dfp2 may provide novel starting points in furthering the elucidation of genetic and biochemical pathways relevant to pigmentation and associated biological processes.     

Natural killing can be independent of interferon generated in vitro

PBL produce interferon in response to culture with the tumor cell line K562, but this production of interferon does not correlate with natural cytotoxicity. A basal level of natural killing is independent of interferon generated in vitro. We base this conclusion on the following findings: (i) natural killing and interferon level are not temporally correlated; (ii) preincubation of lymphocytes at 37 °C greatly reduces their ability to produce interferon but does not affect their lytic capacity against K562; and, (iii) addition of an anti-interferon antibody has no effect on NK. We conclude that NK against K562 is not dependent upon or correlated with the level of interferon generated during in vitro assay.     

Slip sliding away: load-dependence of velocity generated by skeletal muscle myosin molecules in the laser trap

Skeletal muscle's ability to shorten and lengthen against a load is a fundamental property, presumably reflecting the inherent load-dependence of the myosin molecular motor. Here we report the velocity of a single actin filament translocated by a mini-ensemble of skeletal myosin approximately 8 heads under constant loads up to 15 pN in a laser trap assay. Actin filament velocity decreased with increasing load hyberbolically, with unloaded velocity and stall force differing by a factor of 2 with [ATP] (30 vs. 100 muM). Analysis of actin filament movement revealed that forward motion was punctuated with rapid backward 60-nm slips, with the slip frequency increasing with resistive load. At stall force, myosin-generated forward movement was balanced by backward slips, whereas at loads greater than stall, myosin could no longer sustain forward motion, resulting in negative velocities as in eccentric contractions of whole muscle. Thus, the force-velocity relationship of muscle reflects both the inherent load-dependence of the actomyosin interaction and the balance between forward and reverse motion observed at the molecular level.