Description:

    The Kinesin family of motor proteins are responsible for generation of force and transport of materials in many cellular processes.  This includes the transport of neurotrasmitters, produced in the cell body, to vesicles near the synaptic junctions at the end of nerve axons, and the generation of forces in separating the spindle poles in cell division [1].  The Kinesin motor protein moves toward the plus end of microtubules (long polymers of the cytoskeleton) which serve as tracks directing the motor to specific locations within the cell [1,5]. 

    The Kinesin motor protein has two globular domains each consisting of a beta sheet sandwiched between two layers of alpha helices.  The globular domains are referred to as "heads" and are connected by a partially ordered collection of about 12 residues referred to as the "neck-linker".  The residues after the neck-linker become intertwined and form a coiled-coil structure, referred to as the "stalk", which extends like a tether to attach cargo to the motor.  Each head of Kinesin contains a binding site which hydrolyzes ATP to ADP as an energy source for the force generating processes and to bind or unbind the heads from microtubules [10,5].  A number of experimental advances in obtaining crystal structures, NMR data, and in optical trapping of single Kinesin molecular motors has generated information about how the protein may operate. 

    In my research, I am interested in understanding both abstract mechanisms by which chemical energy can be transduced into mechanical work and in building concrete models which attempt to describe the functioning of actual motor proteins.  On this page is a summary of work done on constructing a model of the Kinesin motor protein consistent with optical trap experimental data.  For a more detailed and complete description of this work, see the paper [2].

    In laser trap experiments, optical effects, such as refraction or induced dielectric effects, are exploited to exert forces on nanometer sized particles [8].  The optical effects drive a particle up the intensity gradient of the light.  For example, when the laser has a cylindrical intensity profile a bead can be driven toward the center axis by refractive forces arising from momentum transfer between the light and the object, see the figure below (top), or when the laser is focused at a point a bead is driven toward the center of focus, see the figure below (bottom).  Dielectric effects can also play a role in generating force.  From the scattered light, localization of the bead can be determined with nanometer and microsecond spatial-temporal precision. 

    To probe Kinesin as it operates, in vitro experiments have been performed in which a bead hundreds of nanometer in diameter is attached to the coiled-coil tether structure and monitored as the motor progresses along a microtubule.  The time series data of individual Kinesin motor proteins are then obtained as the bead is displaced from the trap and a load force is exerted on the operating motor protein, see the figure below (right) which is a simulation of the experimental data. 

     Some interesting features of the time series data include discernible 8nm steps (4nm substeps) of the motor, corresponding to the periodicity of tubulin subunits of the microtubule, and forward and backward steps along the microtubule.  From repeated measurements, the mean velocity and the rate of dispersion (rate of growth of the variance) for the position of an ensemble of motors along the track can be estimated as a function of load force on the bead.  This places constraints on possible models of the motor, such as the number of rate limit chemical reactions per mechanical step, see paper [2,9]. 

In other optical trap experiments, the stage on which the microtubule is affixed is moved at a constant velocity V_s while the velocity of the bead V_b and position x_b are measured.  From the ratio r(x_b) of the velocities, an estimate can be obtained of the elasticity of the tether connecting the probe to the Kinesin motor, see paper [2,9].  Below is a schematic of the basic experiment setup (left), and actual ratio data from [9] with multiple experiments at different laser intensities rescaled in terms of a single intensity, see paper [2].

     A mathematical model of the motor protein incorporating features of both the chemical kinetics and three dimensional mechanics of the motor was formulated in [2,7].  The basic model can be cast in terms of a Markov-Chain with an exponential distribution for chemically limited transitions and a hitting time distribution obtained from a Langevin dynamics model for mechanically limited steps [4,6], see the figures below for a summary of the model and paper [2] for a more mathematical discussion.

     The mechanics of the motor was modeled at a coarse scale, where each of the globular domains where taken in account by spherical excluded-volumes connected by harmonic springs to a structure which incorporates the non-linear elasticity of the tether attaching a cargo bead to the motor, see the figures below and paper [2].

    To make a comparison of the proposed model and statistics measured in the optical trap experiments requires that the ensemble of motor states be analyzed.  When using biologically reasonable parameters a wide range of time scales for the chemical events and mechanics makes direct simulation of the model infeasible.  To overcome this difficulty a numerical method was developed for the model exploiting the separation of time scales between the mechanics and chemical reactions for select events.  The method allowed for efficient simulations to estimate statistics measured in the optical trap experiments.  The results were found to capture many of the observed trends for the motor protein.  A more complete discussion of this work can be found in paper [2].

[underconstruction]

[2] Atzberger, P.J. and Peskin, C.S, A Brownian Dynamics Model of Kinesin in Three Dimensions Incorporating the Force-Extension Profile of the Coiled-Coil Cargo Tether, Bull. Math. Biol., (accepted 2005, to appear) [see homepage for preprint].

[3] Cross, R. A., Crevel, I., Carter, N. J., Alonso, M. C., Hirose, K., Amos, L. A., The conformational cycle of kinesin. Phil. Trans. R. Soc. Lond. B 355, 459–464, (2000).

[4] Gardiner, C. W. , Handbook of stochastic methods, Springer Series in Synergetics, (1985).

[6] Oksendal, B. , Stochastic Differential Equations: An Introduction with Applications. Springer, (2000).

[8] Neuman, K. C. and Block,S. M., Optical trapping, Review of Scientific Instruments, volume 75, number 9, (2004).