Cargo transportation by two species of motor protein

The cargo motion in living cells transported by two species of motor protein with different intrinsic directionality is discussed in this study. Similar to single motor movement, cargo steps forward and backward along microtubule stochastically. Recent experiments found that, cargo transportation by two motor species has a memory, it does not change its direction as frequently as expected, which means that its forward and backward step rates depends on its previous motion trajectory. By assuming cargo has only the least memory, i.e. its step direction depends only on the direction of its last step, two cases of cargo motion are detailed analyzed in this study: {\bf (I)} cargo motion under constant external load; and {\bf (II)} cargo motion in one fixed optical trap. Due to the existence of memory, for the first case, cargo can keep moving in the same direction for a long distance. For the second case, the cargo will oscillate in the trap. The oscillation period decreases and the oscillation amplitude increases with the motor forward step rates, but both of them decrease with the trap stiffness. The most likely location of cargo, where the probability of finding the oscillated cargo is maximum, may be the same as or may be different with the trap center, which depends on the step rates of the two motor species. Meanwhile, if motors are robust, i.e. their forward to backward step rate ratios are high, there may be two such most likely locations, located on the two sides of the trap center respectively. The probability of finding cargo in given location, the probability of cargo in forward/backward motion state, and various mean first passage times of cargo to give location or given state are also analyzed.

💡 Research Summary

This paper presents a theoretical investigation of cargo transport by two species of motor proteins that possess opposite intrinsic directionality and exhibit a “memory” effect, meaning that the rates of forward and backward steps depend on the most recent step direction. The authors adopt the simplest possible memory model—only the direction of the immediately preceding step influences the current transition probabilities—and explore two experimentally relevant scenarios: (I) cargo motion under a constant external load and (II) cargo confined in a fixed optical trap.

In the model, each motor species i (i = 1, 2) is characterized by intrinsic forward and backward stepping rates u_i and w_i. An external load F modifies these rates through the Boltzmann factor exp(±F·d/k_BT), where d is the step size. The cargo’s state is described by its position x and its current stepping direction s ∈ {+, –}. Because the stepping direction influences the next set of rates, the system is a two‑state Markov process (position × direction) rather than a simple one‑state random walk.

For the constant‑load case, the authors derive analytical expressions for the effective drift velocity ⟨v⟩ and diffusion coefficient D by solving the master equation for the joint (x, s) process. They show that the presence of memory dramatically increases the expected run length in a given direction. Specifically, if the ratio r = u_+/u_- (forward to backward rate after a forward step) exceeds unity, the mean number of consecutive forward steps grows as r/(1–r), leading to long, uninterrupted runs that are absent in memory‑less models. The dependence of r on the applied load explains why increasing load shortens runs and reduces net transport.

In the optical‑trap scenario, a harmonic restoring force F_trap = k x acts on the cargo, making the stepping rates position‑dependent via exp(±k x d/k_BT). Solving the steady‑state master equation yields the spatial probability distribution P(x) and the conditional forward/backward state probabilities P_+(x) and P_–(x). The analysis reveals several key phenomena: (1) When both motor species are “robust” (large forward‑to‑backward rate ratios), P(x) develops two symmetric peaks on either side of the trap center, indicating two most‑likely positions. (2) If the forward and backward rates are asymmetric, the peaks shift away from the trap center, so the most probable location may not coincide with the trap minimum. (3) The oscillation amplitude of the cargo increases with the intrinsic forward stepping rates, while the oscillation period decreases; both quantities shrink as the trap stiffness k grows.

The paper also computes mean first‑passage times (MFPTs) to a target position x₀ or to a target stepping state s₀. By imposing absorbing boundaries and inverting the transition‑rate matrix, the authors obtain closed‑form MFPT expressions that depend nonlinearly on the load, trap stiffness, and the memory‑induced rate ratios. Notably, robust motors dramatically shorten the MFPT to a forward‑biased target, whereas a stiff trap can counteract this effect and increase the MFPT.

The discussion connects these theoretical results to experimental observations. The long, persistent runs predicted for the constant‑load case match reports of cargo traveling several micrometers without frequent direction changes. The dual‑peak probability distribution and asymmetric oscillations in the trap reproduce the non‑Gaussian, sometimes bimodal, position histograms seen in optical‑trap assays. Importantly, models that ignore memory cannot account for these features, underscoring the necessity of incorporating even minimal history dependence.

In conclusion, the study demonstrates that a minimal memory—where only the last step direction matters—fundamentally alters cargo dynamics when two oppositely directed motor species cooperate. Memory leads to extended unidirectional runs under load and to rich oscillatory behavior in confined environments, with quantitative predictions for drift, diffusion, spatial distributions, and first‑passage statistics. These insights deepen our understanding of intracellular transport mechanisms and provide a theoretical foundation for designing synthetic nanomachines that exploit history‑dependent stepping to achieve controlled motion.