Sufficient conditions for the additivity of stall forces generated by multiple filaments or motors

Molecular motors and cytoskeletal filaments work collectively most of the time under opposing forces. This opposing force may be due to cargo carried by motors or resistance coming from the cell membrane pressing against the cytoskeletal filaments. Some recent studies have shown that the collective maximum force (stall force) generated by multiple cytoskeletal filaments or molecular motors may not always be just a simple sum of the stall forces of the individual filaments or motors. To understand this excess or deficit in the collective force, we study a broad class of models of both cytoskeletal filaments and molecular motors. We argue that the stall force generated by a group of filaments or motors is additive, that is, the stall force of $N$ number of filaments (motors) is $N$ times the stall force of one filament (motor), when the system is reversible at stall. Conversely, we show that this additive property typically does not hold true when the system is irreversible at stall. We thus present a novel and unified understanding of the existing models exhibiting such non-addivity, and generalise our arguments by developing new models that demonstrate this phenomena. We also propose a quantity similar to thermodynamic efficiency to easily predict this deviation from stall-force additivity for filament and motor collectives.

Figure 1

Biased random walk model for filaments and motors: (a) Multiple $\left(N>1\right)$ filaments against a wall (see Ref. [29] for a detailed study of this model). Polymerization and depolymerization rates are $u$ and $w$, when the filaments are away from wall (hence force free). Note that when more than one filament touches the wall simultaneously, the depolymerization rate becomes independent of the force $\left(w\right)$, as a single depolymerization event cannot cause any wall movement. (b) Multiple motors moving on a track. Note that only the leading motor bears the force $f$, while the trailing motors are experiencing force through hard core interaction. (c) Two filaments at the collective stall force $f_s^{\left(2\right)}$ undergoing simple processes of polymerization and depolymerization. (d) Two motors diffusing on a tilted free-energy landscape. The force is acting only on the leading motor, and the energy landscape of the leading motor changes with the application of force. Due to hard-core interaction, a larger force $[>f_s^{\left(1\right)}]$ is required to stall the system of two motors. It is like moving uphill, with the assistance of a trailing motor, which rectifies the average backward motion of the leader [86].

Figure 2

(a) Random hydrolysis model with three filaments, where individual monomer switches from T to D unidirectionally and randomly. Different processes are shown by arrows. $k_0=3.2\mu {\mathrm{M}}^{-1}{\mathrm{s}}^{-1},c=100\mu \mathrm{M},w_T=24{\mathrm{s}}^{-1}$, and $r=0.2{\mathrm{s}}^{-1}$. (b) ${\mathrm{\Delta }}^{\left(2\right)}$ [red (dark gray) curve] and $\alpha$ [blue light (gray) curve] versus $w_D$. Note that both $\alpha$ and ${\mathrm{\Delta }}^{\left(2\right)}$ are zero only at $w_T=w_D$ and they are highly correlated in sign. (c) The fluxes per filament (defined in the text), at stall, for T and D monomers as a function of the filament number $N$ for $w_D=290{\mathrm{s}}^{-1}(\alpha >0)$. (d) The collective stall force per filament $[f_s^{\left(N\right)}/N]$ against $N$, the number of filaments. For $\alpha >0,w_D=290{\mathrm{s}}^{-1}$ and for $\alpha <0,w_D=5{\mathrm{s}}^{-1}$.

Figure 3

(a) Schematic diagram of a generalized random hydrolysis model with two-way switching (both $\mathrm{T}\to \mathrm{D}$, and $\mathrm{D}\to \mathrm{T})$. Different processes (shown in arrows) are discussed in the text. (b) Schematic depiction of a connected loop in the configuration space of a single filament, within the model. (c) Deviation ${\mathrm{\Delta }}^{\left(2\right)}$ versus $u_T$ [red (dark gray) curve], and $\alpha$ versus $u_T$ [blue (light gray) curve], for the generalized random hydrolysis model. The parameters are $w_T=2{\mathrm{s}}^{-1},k_{TD}=0.3{\mathrm{s}}^{-1},k_{DT}=0.4{\mathrm{s}}^{-1},u_D=3{\mathrm{s}}^{-1}$, and $w_D=1{\mathrm{s}}^{-1}$.

Figure 4

(a) Schematic diagram of single-filament two-state model with switching between states 1 [blue (light gray)] and 2 [red (dark gray)]. Various processes are shown by arrows and corresponding rates, as discussed in the text. (b) Schematic depiction of a connected loop in the configuration space of single-filament two-state model. The configurations are denoted by ordered pairs, whose first element is the instantaneous length and second element is the chemical state (1 or 2). (c) ${\mathrm{\Delta }}^{\left(2\right)}$ [red (dark gray) curve] and $\alpha$ [blue (light gray) curve] as a function of $u_2$ for the two-state model. Parameters are $k_{12}=0.5{\mathrm{s}}^{-1},k_{21}=0.5{\mathrm{s}}^{-1},w_1=0.1{\mathrm{s}}^{-1},u_1=1{\mathrm{s}}^{-1},w_2=0.8{\mathrm{s}}^{-1}$. Data obtained using analytical expression of $f_s^{\left(1\right)}$ and $f_s^{\left(2\right)}$ taken from Ref. [32].

Figure 5

(a) Schematic diagram of the motor model proposed by Campàs et al. [33], where multiple interacting motors push against a cargo with a constant force $f$ acting against their motion. Various processes are shown by arrows and discussed in the text. (b) Schematic depiction of a closed loop of dynamically connected configurations for the model shown in (a). (c) Deviation ${\mathrm{\Delta }}^{\left(2\right)}$ versus $\overline{u}$ within the model of Campàs et al. (data obtained from the exact formula given in Ref. [33]). Parameters are specified in the text. We took the force distribution factor $\delta =1$. (d) Motors walking on a free-energy landscape [33]. The effect of nearest-neighbor interactions is shown schematically.

Figure 6

Multiple step size motor model: (a) Schematic diagram of the motors moving with two distinct step sizes. The kinetic processes (shown in arrows) are explained in the text. (b) A closed loop of connected configurations for the model with one motor. (c) Deviation ${\mathrm{\Delta }}^{\left(2\right)}$ versus $u_2$ [red (dark gray) curve] and $\alpha$ versus $u_2$ [blue (light gray) curve]. The parameters are $u_1=80{\mathrm{s}}^{-1},w_1=8{\mathrm{s}}^{-1}$, and $w_2=1{\mathrm{s}}^{-1}$.

Figure 7

Schematic of the (a) continuous [73] and (b) discrete two-state Brownian ratchet model [104, 105]. (a) A spontaneous motion is expected when the ratio of transition rates $\frac{{\omega }_1\left(x\right)}{{\omega }_2\left(x\right)}$ is far from the reversible value given by the detailed balance condition. (b) Discrete version of the two-state ratchet model with effective transition rates ${\vec{\omega }}_A\left(x\right),{\omega }_A\left(x\right),{\omega }_B\left(x\right)$, and ${\overleftarrow{\omega }}_B\left(x\right)$. (c) Energy landscape corresponding to discrete two-state model.