Bidirectional cooperative motion of molecular motors

Abstract

Recently, in a beautiful set of experiments, it has been shown that a Ncd mutant, NK11, which lacks directionality in its individual motion, was able to exhibit a new kind of directed motion in motility assays (Endow, S. A. & Higuchi, H. (2000) Nature (London) 406, 913-916): the filaments keep a given velocity for a while and then suddenly move in the opposite direction with similar velocity. We show that these observations nicely illustrate the concept of dynamic transitions in motor collections introduced earlier in the case of an infinite number of motors. We investigate the experimentally relevant case of a finite number of motors both when directionality is present (kinesins, myosins, Ncd) and absent (NK11). Using a symmetric two-state model, we demonstrate that bidirectional motion is the signature of a dynamic transition that results from the collective behavior of many motors acting on the same filament. For motors exhibiting directional bias individually, an asymmetric two-state model is appropriate. In that case, dynamic transitions exist for motor collections in the presence of an external force. We give predictions for the dependence of motion on ATP concentration, external forces, and the number of motors involved. In particular, we show that the reversal time grows exponentially with the number of motors per filament.

Figure 1

Schematic representation of N rigidly coupled motors. (a) The motors are interacting with a filament via periodic and asymmetric potentials W₁(x) and W₂(x) with period ℓ. In the examples discussed here, the potentials are chosen piecewise linear with the parameter a/ℓ characterizing the degree of asymmetry of the system, which becomes symmetric for a = ℓ/2. (b) The excitation rate ω₁ is localized near the potential minima within a region of size d, centered at the minimum of the potential while the deexcitation rate ω₂ is constant.

Figure 2

Motion of a finite number of motors for an asymmetric and a symmetric system. (a) Position as a function of time for an asymmetric system a/ℓ = 0.2 (see Fig. 1) with N = 300, ω = 25 ms, ω = 2 ms, U/kT = 20, d/ℓ = 0.2, and χ = ω₂ℓ²λ/U = 0.4. (b) Velocity histogram Qτ(v) of the motion shown in a for averaging time τ = 1 ms. (c) Velocity histogram for the same system with an applied external force f_ext = −2.8 pN. (d) Position as a function of time for a symmetric system with a/ℓ = 0.5 and otherwise the same parameters as in a. (e) Velocity histogram for this symmetric system. (f) Velocity histogram of the same system with an applied external force f_ext = −0.2 pN.

Figure 3

Force-velocity relationships for a symmetric and asymmetric system. (a) Symmetric system with N = 300 motors, the other parameters are the same as in Fig. 2. (b) Force-velocity relation for this system with N = 50. (c) Asymmetric system with a/ℓ = 0.2, N = 200 motors and otherwise same parameters as above. (d) Same plot as in c but with N = 40 motors. The symbols denote the positions of peaks in the velocity histograms, error bars denote the width of these peaks. The dashed lines represent the force-velocity relation in the mean field limit for large N.

Figure 4

(a) Velocity as a function of time for a symmetric system with parameters as in Fig. 2 and N = 200 motors. (b) Same system with N = 300. (c) Velocity correlation functions for this system for different numbers N.

Figure 5

The characteristic time of velocity reversals t_rev as a function of the number N of motors for a symmetric system. The different lines correspond to different values of temperature T and the dimensionless parameter χ = ω₂ℓ²λ/U. The other parameters are chosen as in Fig. 2.

Figure 6

Shape of the velocity histograms of a symmetric system in the (N,τ) plane. Two regions can be distinguished: a region of bimodal velocity histograms and a region where this signature of a dynamic transition is lost. The line that separates the two regions was determined numerically at distinct points (marked as squares with error bars). The black solid line connects these points as a guide for the eye. Bimodal distributions occur for N > N_min ≃ 5. For large N, the interval τ_min ≤ τ ≤ τ_max for which bimodal shapes occur widens and the curves τ_max(N) and τ_min(N) follow simple laws are represented by broken lines: ln(τ_max(N)) ∝ N, and τ_min(N) was fitted best by 1/(N − N₀) with N₀ = 45, which asymptotically behaves like 1/N. Insets a–c display examples of velocity histograms for N = 20 and τ = 2.10⁻¹ s, N = 20 and τ = 10⁻³ s, N = 300 and τ = 10⁻³ s respectively. The experiments of ref. correspond to τ ≃ 0.5 s and N ≃ 300.

Figure 7

Simulation of individual motors attached to a spring. This situation corresponds to a single motor laser trap experiment. (a and c) Schematic representation of an asymmetric and a symmetric system. (b) Histogram of displacements generated by an asymmetric system with a/ℓ = 0.2, moving in a trap of stiffness k = 0.08 pN/nm. Parameter values are ω = 1 s, ω = 2 s, λ₁ = 10⁻⁵ kg/s, λ₂ = 10⁻⁷ kg/s, U/kT = 20, and d/ℓ = 0.2. (d) Histogram of displacement for a symmetric system and otherwise same parameters as in b.