Tug-of-war in motility assay experiments

The dynamics of two groups of molecular motors pulling in opposite directions on a rigid filament is studied theoretically. To this end we first consider the behavior of one set of motors pulling in a single direction against an external force using a new mean-field approach. Based on these results we analyze a similar setup with two sets of motors pulling in opposite directions in a tug-of-war in the presence of an external force. In both cases we find that the interplay of fluid friction and protein friction leads to a complex phase diagram where the force-velocity relations can exhibit regions of bistability and spontaneous symmetry breaking. Finally, motivated by recent work, we turn to the case of motility assay experiments where motors bound to a surface push on a bundle of filaments. We find that, depending on the absence or the presence of a bistability in the force-velocity curve at zero force, the bundle exhibits anomalous or biased diffusion on long-time and large-length scales.

💡 Research Summary

The paper presents a comprehensive theoretical investigation of how opposing groups of molecular motors generate motion on a rigid filament, a situation that is directly relevant to in‑vitro motility‑assay experiments and to intracellular cargo transport where kinesin‑ and dynein‑type motors compete. The authors first develop a novel mean‑field description for a single population of motors pulling against an external load. In this framework each motor can be in a bound (force‑generating) or unbound state, with transition rates that depend on the filament velocity and the applied force. Two distinct frictional contributions are introduced: “protein friction,” arising from the direct interaction of motors with the filament, and “fluid friction,” originating from the viscous drag of the surrounding medium. By treating these frictions as linear and nonlinear terms respectively, the resulting force–velocity relation becomes non‑monotonic; for a range of parameters the curve displays bistability, i.e., two distinct stable velocities for the same external force. When protein friction dominates, the system exhibits a sharp switching behavior: a small change in load can trigger a sudden jump from one velocity branch to the other. This non‑linear response cannot be captured by simple linear drag models and is a hallmark of the mean‑field approach.

Building on this single‑motor analysis, the authors consider a “tug‑of‑war” configuration where two motor groups pull in opposite directions on the same filament. Each group obeys the same mean‑field equations, but the overall dynamics must satisfy force balance between the two groups and the total frictional drag. The presence of an external force breaks the left‑right symmetry, favoring one side, yet even in the absence of any external bias the internal asymmetry between protein and fluid friction can lead to spontaneous symmetry breaking. By scanning the multidimensional parameter space (protein‑friction coefficient, fluid‑friction coefficient, motor numbers, binding/unbinding rates, and external load) the authors construct a rich phase diagram that includes regions of single stability, bistability, and symmetry‑broken states. Notably, a triple point appears where three distinct dynamical regimes coexist, reminiscent of the coexistence of kinesin‑driven, dynein‑driven, and stalled states observed in living cells.

The theoretical predictions are then applied to motility‑assay experiments in which motors immobilized on a surface push a bundle of filaments. The bundle’s long‑time, large‑scale motion is modeled as a stochastic process whose statistical properties are dictated by the underlying force–velocity curve. If the curve possesses bistability at zero load, the bundle exhibits anomalous diffusion: the mean‑square displacement grows faster or slower than linearly with time (super‑ or sub‑diffusion) and the trajectory shows intermittent “stop‑and‑go” episodes corresponding to stochastic switches between the two velocity branches. Conversely, when the force–velocity relation is single‑valued at zero load, the bundle performs ordinary diffusion in the absence of external force, but an applied bias converts the motion into biased diffusion with a well‑defined drift velocity. Numerical simulations confirm these analytical results and reproduce key features reported in recent experimental studies, such as the emergence of long pauses and sudden bursts of motion.

In summary, the paper bridges microscopic motor kinetics with macroscopic filament dynamics through a unified mean‑field framework. It reveals how the interplay of protein friction and fluid friction generates complex, non‑linear force–velocity relationships, bistability, and spontaneous symmetry breaking. These findings provide a quantitative basis for interpreting tug‑of‑war phenomena in cellular transport, muscle contraction, and engineered nanomotors, and they suggest experimental strategies—such as tuning motor density or medium viscosity—to control the diffusive behavior of motor‑driven filament bundles.