Force–velocity relationship for multiple kinesin motors pulling a magnetic bead

Abstract

Although the velocity of single kinesin motors against an opposing force F of 0–10 pN is well known, the behavior of multiple kinesin motors working to overcome a larger load is still poorly understood. We have carried out gliding assays in which 3–7 Drosophila kinesin-1 motors moved a microtubule at 200–700 μm/s against a 0–31 pN load at saturating [ATP]. The load F was generated by applying a spatially uniform magnetic field gradient to a superparamagnetic bead attached to the (+) end of the microtubule. When F was scaled by the average number of motors 〈n〉, the force–velocity relationship for multiple motors was similar to the force–velocity relationship for a single motor, supporting a minimal load-sharing model. The velocity distribution at low load has a single mode consistent with rapid fluctuations of n. However, against a load of 2.5–4.7 pN/motor, additional modes appeared at lower velocity. These observations support the Klumpp–Lipowsky model of multimotor transport [Proc Natl Acad Sci USA 102. 17284–17289 (2005)].

Appendix

Conversion of {x,y,t} tracks to {x″,y″,t} data for which x″ is parallel to the path of the microtubule and y″ is perpendicular to that path.

Consider a microtubule traveling at constant velocity in a straight path, x = x ₀ + v _x t, y = y ₀ + v _y t, with v _x and v _y independent of t.

Step 1. Translocate the (x,y,t) frame to a new coordinate frame (x′,y′,t) in which the microtubule starts at (0,0,0) with equations x′ = x − x₀, y′ = y − y₀. In the x′y′ plane, y′ = (v _y /v _x )x′, a straight line through the origin.

Step 2. Rotate the (x′,y′) coordinate system by angle θ = tan⁻¹(v _y /v _x ) to new coordinates (x″,y″):$\left[ {\begin{array}{*{20}c} {x^{\prime \prime} } \\ {y^{\prime \prime} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\cos (\theta )} & {\sin (\theta )} \\ { - \sin (\theta )} & {\cos (\theta )} \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {x^\prime } \\ {y^\prime } \\ \end{array} } \right].$

In the (x″,y″) coordinate system, $x^{\prime\prime} = \left( {\sqrt {v_{x}^{2} + v_{y}^{2} } } \right)t = vt$ and y″ = 0. Thus, the slope $\frac{{{\text{d}}x^{\prime \prime} }}{{{\text{d}}t}}$ equals the velocity of the particle.

For a microtubule traveling along a curved path, v _x and v _y are functions of t, so θ = tan⁻¹(v _y /v _x ) is also a function of t. We determined θ(t) from the smoothed value of dy/dx. Steps 1 and 2 were then applied to data points 1 and 2. This done, steps 1 and 2 were applied to points 2 and 3, etc. (This is easy in MATLAB.) In that (x″,y″,t) coordinate system, y″(t) = 0 + noise, whereas x″(t) = vt + noise. We tested the algorithm carefully with synthetic data, i.e., data sets {x,y,t} constructed with known values of v _x and v _y, plus Gaussian noise.