The Impact of Rate Formulations on Stochastic Molecular Motor Dynamics

Cells are complex structures which require considerable amounts of organization via transport of large intracellular cargo. While passive diffusion is often sufficiently fast for the transport of smaller cargo, active transport is necessary to organize large structures on the short timescales necessary for biological function. The main mechanism of this transport is by cargo attachment to motors which walk in a directed fashion along intracellular filaments. There are a number of models which seek to describe the motion of motors with attached cargo, from detailed microscopic to coarse phenomenological descriptions. We focus on the intermediate-detailed discrete stochastic hopping models, and explore how cargo transport changes depending on the number of motors, motor interaction, system constraints and rate formulations, which are derived from common thermodynamic assumptions. We find that, despite obeying the same detailed balance constraint, the choice of rate formulation considerably affects the characteristics of the overall motion of the system, with one rate formulation exhibiting novel behavior of loaded motor groups moving faster than a single unloaded motor.


Low-viscosity two-motor analytic approximation
We let P ξ be the unnormalized probability for the two motor heads to be at a distance of ξ from each other, and assume a steady state distribution so that dP ξ dt = 0 ∀ ξ . Starting with some unknown P 0 we can iteratively calculate P n+1 as a function of all P m≤n using the detailed balance constraints, . ( Similarly, the velocity of the cargo depending on the current motor configuration can be calculated from the jump rates as and the long-term velocity under an equilibrium motor configuration is then obtained by summing over all motor configurations weighted by their respective probabilities where p ξ is the unit normalized probability such that ∑ ∞ ξ =0 p ξ = 1.

Two-motor velocity with harmonic potential
When considering motors with a high forward jump bias (∆µ 1) and vanishing drag in the unlimited variants of the AsEx models, the velocity can be solved exactly. In both of the AsEx models, setting the w b terms to zero and using the derivation outlined above, it is trivial to show that the normalized probabilities where the sum in the denominator is an elliptic theta special function and can be written θ 3 (0, e − kΘ 2 ). The velocity contribution of a given configuration v ξ in this limit for the P-AsEx model can be simply written Then, putting the velocity and probabilities together, we arrive at the mean velocity in the low γ and high ∆µ limit

2/7
This function monotonically decreases in kΘ, and should act as a maximum velocity for a two-motor system, showing that the P-AsEx model as formulated cannot result in a two-motor speedup relative to the unloaded motors. The D-AsEx velocity can similarly be calculated using the generalized velocity derivation outlined above. However, since the two unlimited AsEx models can be mapped between one another, we can simply write Noting that for kΘ 4 the ratio of the elliptic theta functions is ≈ 1, we can approximate for small kΘ. This clearly allows for speedups for any Θ < 1/4 with the optimal Θ = 1/8.

Two-motor velocity with linear potential
Since the prior work on modeling the RecBCD protein observed a two-motor speedup used a linear potential between the motor heads, it is useful to explore if this is possible using a linear potential but with a cargo intermediary in the unlimited P-AsEx formulation. The potential can be simply written V (δ x) = ε |δ x|, where ε sets the energy scale of a single jump. The forward rate is then w f (δ x) = w 0 e ∆µ e −εΘ(|δ x+1|−|δ x|) . Noting that the forward rate can take on one of three values since the displacement can only take integer and half-integer values and setting the backward rates w b = 0, the mean velocity can be simply calculated using the relations from Equations SI2 and SI4 v = w 0 e ∆µ e −εΘ (1 + 2 sinh(εΘ)) 1 + sinh(εΘ) ≈ 2w 0 e ∆µ 1 + e εΘ .
As with the harmonic potential in the unlimited P-AsEx model, this monotonically decreases in εΘ.
When k ∼ ∆µ in the low drag limit (v 1 ∼ v 0 ), the potential for anti-cooperative behavior between motors arises, even when the motors do not directly interact. In both the D-AsEx and Glauber models, there is clear strong anti-cooperative behavior observed for high stiffness motors k ∼ 6.7pN nm −1 . Here, we assume that the motor heads are not allowed to occupy the same lattice position, and that they are all attached at the some point to some cargo. This is an overly simplistic assumption, since multiple motors will not be able to attach to the same site on the cargo, but a full exploration of the effects of this positioning is beyond the scope of this work. This limit is still however useful to show the large differences in model behaviors when motor collisions are included. Due to the relatively high coil stiffness compared to the jump bias, it's perhaps not surprising that cooperativity only arises in the low stiffness (Fig. 1A) or high drag limits (Fig. 1B).