Stall force of a cargo driven by N interacting motor proteins

We study a generic one-dimensional model for an intracellular cargo driven by N motor proteins against an external applied force. The model includes motor-cargo and motormotor interactions. The cargo motion is described by an over-damped Langevin equation, while motor dynamics is specified by hopping rates which follow a local detailed balance condition with respect to change in energy per hopping event. Based on this model, we show that the stall force, the mean external force corresponding to zero mean cargo velocity, is completely independent of the details of the interactions and is, therefore, always equal to the sum of the stall forces of the individual motors. This exact result is arrived on the basis of a simple assumption: the (macroscopic) state of stall of the cargo is analogous to a state of thermodynamic equilibrium, and is characterized by vanishing net probability current between any two microstates, with the latter specified by motor positions relative to the cargo. The corresponding probability distribution of the microstates under stall is also determined. These predictions are in complete agreement with numerical simulations, carried out using specific forms of interaction potentials. Introduction. – Motor proteins, by consuming energy released during adenosine triphosphate (ATP) hydrolysis, pull and transport intracellular organelles on the cytoskeletal filaments [1]. Their activity is crucial for many biological processes such as axonal transport of mitochondria in neurons [2], lipid droplet transport in Drosophila [3], pigment granule transport in Xenopus laevis [4] and so on. Broadly, spatial organisation of different biomolecular structures inside eukaryotic cells is made possible due to the cargo transport by motor proteins. When motor proteins stop functioning, major biological activities slow down and often such disruption in transport is lethal for the cell [5]. The biophysics of motor proteins has been an active interdisciplinary area of research for many decades now, specific areas of interest including mechanochemical coupling, free energy transduction, thermodynamic efficiency, force generation, directed motion and many other collective phenomena. Typically, a motor protein on a cytoskeletal filament moves with a speed of hundreds of nanometres per second and exert forces of a few piconewtons on a cargo. The maximum force with which a motor protein pulls an organelle on the filament is called its stall force [1, 6]. In in vitro optical trap experiments, stall force is determined as the opposing external force at which average velocity of the motor-driven cargo becomes zero [7]. Collective force generation by teams of motor proteins has been a subject of interest in the recent past [8–12]. When a cargo is driven by multiple motors, it is generally assumed that the stall force scales linearly with the number of motors, and hence the measured stall force is also used to determine the number of cargo-bound motors in experiments [9, 11, 13]. In reality, motor-cargo as well as motor-motor interactions [14] may be present, and it is unclear how these interactions within the motor-cargo complex affect the stall force. In Campàs et al. [15], it was shown that direct motor-motor interaction captured effectively via a simplified model influences the force-velocity behaviour as well as the stall force of a multiple-motor assembly [15]. The role of motor-cargo interaction has been the subject of many earlier experimental [14] and theoretical [8,16–19] studies. In our recent work [19] on the dynamics of a cargo elastically coupled to an arbitrary number of motor proteins, numerical simulations showed that, while the average velocity and diffusion coefficient of the cargo is a decreasing function of the stiffness of the motor-cargo linker, the stall force is completely independent of it, and is equal to the sum of stall forces of the individual motors. This