Embedding dual function into molecular motors through collective motion

Protein motors, such as kinesins and dyneins, bind to a microtubule and travel along it in a specific direction. Previously, it was thought that the directionality for a given motor was constant in the absence of an external force. However, the directionality of the kinesin-5 Cin8 was recently found to change as the number of motors that bind to the same microtubule is increased. Here, we introduce a simple mechanical model of a microtubule-sliding assay in which multiple motors interact with the filament. We show that, due to the collective phenomenon, the directionality of the motor changes (e.g., from minus- to plus- end directionality), depending on the number of motors. This is induced by a large diffusive component in the directional walk and by the subsequent frustrated motor configuration, in which multiple motors pull the filament in opposite directions, similar to a game of tug-of-war. A possible role of the dual-directional motors for the mitotic spindle formation is also discussed. Our framework provides a general mechanism to embed two conflicting tasks into a single molecular machine, which works context-dependently.


Derivation of Eq. (3)
Here, we derive the model equation (3) in the main manuscript in a different manner. We assume that each motor bound to the filament moves via a one-dimensional random walk along the filament, the driving force of which is generated by binding of the leading head to MT associated with the motor step. The driving force is assumed to arise from thermal activation or ATP hydrolysis, and is denoted by η i = η(x i , t), which is given in the main text. We consider the over-damped Langevin equation of the coordinate of each motor x i and filament y as where γ x and γ y are the friction coefficients of the motor with the surrounding medium and the filament respectively, whereas k is the spring constant and ξ y (t) the thermal random force applied to the filament as ⟨ξ y (t)ξ y (t ′ )⟩ = 2D y γ 2 y δ(t−t ′ ). ν i represents the frictional force between the motor and the filament. It satisfies the no-slip condition, i.e., the relative distance between each bound motor and the filament does not change unless η i causes the filament to slide. From these equations, we obtaiṅ The no-slip condition is simply formulated asẋ i −ẏ = (1/γ x + 1/γ y )η i for each i, becauseẋ i −ẏ should be zero for η i = 0. 1 From the no-slip condition, we obtaiṅ We estimate the orders of the parameters as follows: The friction coefficient of the motor head γ x ∼ 10 −7 [pN·s/nm] (the friction of a sphere with the diameter ∼ 10 nm in water); that of the filament γ y ∼ 10 −3 [pN·s/nm] (the friction of a ∼ 20-µm-long microtubule in water for the sliding assay [1]); N = 1-10 2 ; and k = 0.1-1 [pN/nm]. In addition, η i is estimated as |η i /γ x | ∼ 100 [nm/s] from the single molecular assay of Cin8 [2]. Using these values, each term in Eq. (S3) is estimated as follows: which allows us to neglect the third term. After applying further approximations in the form of γ y + γ x ≃ γ y and γ y + N γ x ≃ γ y , we obtain Eq. (3).

Explicit Form of ρ(x) in Eq. (5)
When Using the imaginary error function erfi(x) = 2 ∫ x 0 e t 2 dt/ √ π, the explicit form of ρ(x) is given as follows: , where D = D x + D y for N = 1 and D = D x for infinitely large N . The density distribution of x is then given by

Explanation of the use of D = D x in Eq.4 for infinitely large N
Through transformation of the variable according tox N (say ∼ 100). Thus, the contribution of ξ y can be ignored, except for the shifting of the parameters, ∆ ± → ∆ ± − |z|.

Force-velocity relation for the single motor
Here, we address the force-velocity relation for the single motor under a situation in Fig. S2(a), where the motor with applied force F walks along an immobile filament. Since the filament is immobilized, the no-slip condition is written Considering the balance with the force F = kx, the force-velocity relation for the single motor is obtained as The stall force is calculated as F = v 0 k/a 0 . Figure S2(b) illustrates the force-velocity relation for the parameters in Fig.2 in the main text. Note that, in Fig. S2(b), the speed for the build-in direction of the motor (i.e., minus-end direction) is considered to be plus. The stall force in Fig. S2 . Even for such parameter, we confirmed that our conclusion is not altered as is shown in Fig. S3. In this case, the directionality switches even for N = 2, and the value of dy/dt does not show a monotonic decrease but overshoots at N = 2 (see S3(b)). This overshooting behavior of dy/dt vanishes by introducing exponential dependencies of detachment rate on force, as is described in the next section.

Exponential dependencies of detachment rate
Throughout the paper, we assumed the hard cutoff (i.e., ∆ + and ∆ − ) for the detachment of the motors. In contrast, the exponential dependencies of detachment rate on force (k of f (F ) = k 0 e F/Fc where F c is critical force) has been well established [3,4]. Here we show that all results are not altered by incorporating the exponential detachment rate. Instead of the detachment rule with the hard cutoff, we improve the spontaneous detachment rater asr(x) =r 0 e x/∆ + for x ≥ 0 andr 0 e −x/∆ − for x < 0. The parameter of a 0 is set as a 0 = 2[s −1 ], where the force-velocity relation illustrated in S2(c) is obtained. The results are shown in Figs. S4, which indicate that the essentially same result as in the main text is obtained with regards to the directionality switch. In this case, the overshooting behavior of dy/dt as in S3(b) at N = 2 vanishes. Similar to Fig.2(b) and Fig.4(b), dy/dt monotonically decreases with N . (c)