A general model for the motion of multivalent cargo interacting with substrates

Abstract

Multivalent interactions are common in biology at many different length scales, and can result in the directional motion of multivalent cargo along substrates. Here, a general analytical model has been developed that can describe the directional motion of multivalent cargo as a response to position dependence in the binding and unbinding rates exhibited by their interaction sites. Cargo exhibit both an effective velocity, which acts in the direction of increasing cargo-substrate binding rate and decreasing cargo-substrate unbinding rate, and an effective diffusivity. This model can reproduce previously published experimental findings using only the binding and unbinding rate distributions of cargo interaction sites, and without any further parameter fitting. Extension of the cargo binding model to two dimensions reveals an effective velocity with the same properties as that derived for the one-dimensional case.

Keywords: cargo transport; microtubules‌; multivalent cargo; tip tracking.

Figure 1.

Cargo binding model. An individual cargo initially binds with a single leg (interaction point), and then either new legs bind or previously bound legs unbind according to the binding or unbinding rates. Following each binding/unbinding event, the cargo position is updated to be equal to the average bound leg position. Interaction with the substrate ceases upon unbinding of the last leg.

Figure 2.

Dynamics of cargo with uniform binding and unbinding rates. (a) Example bound leg distributions (red) for simulated 10-legged cargo with position-independent binding/unbinding rates, averaged over cargo centre position (10 000 simulated). Fits (green) were calculated as described in electronic supplementary material, §6. (b) Cargo exhibit distinct short- and long-time diffusivities. Distributions from simulations were calculated from the gradients of mean-squared displacement (MSD) distributions, as shown in electronic supplementary material, figure S2 (100 000 simulated for 1 ≤ N ≤ 7; 50 000 simulated for N = 8; 25 000 simulated for N = 9; 10 000 simulated for N = 10), and analytical distributions were obtained using equations (2.1), (2.2) and (2.5). Errors cannot be calculated for the analytical distribution that does not require bound leg distributions, since the motion is purely deterministic. Error bars for other distributions are too small to see.

Figure 3.

Dynamics of cargo with position-dependent binding and unbinding rates. (a) Effective velocities and (b) effective diffusivities, derived by substituting the results of simulations (orange, red; 250 000 simulated) or equations (2.1) and (2.2) (blue, green) into equation (2.4) or equation (2.5), respectively. The corresponding position-dependent binding rate distribution defined in equation (2.8) is also shown (black). Both effective velocity distributions show that cargo are ‘attracted’ towards the central region of increased cargo–substrate binding rate defined in equation (2.8). Distribution smoothing has been outlined in the electronic supplementary material, Methods, and only every fifth data point of each distribution is shown for clarity. Error bars are too small to see.

Figure 4.

Average cargo behaviour. (a) The PDF P(x, t) describing the probability of finding 4-legged cargo at the position x averaged over all simulation time t exhibits a peak in the central region when cargo exhibit the binding rate defined in equation (2.8) and a uniform unbinding rate. Stochastic simulation results (250 000 simulated) agree with those obtained using molecular dynamics simulations (dt = 0.1, t_max = 1 000 000) and by numerically solving equation (2.3) using an analytically derived effective velocity distribution. Only every fifth data point of each distribution is shown for clarity, and error bars for stochastic simulation data are too small to see. (b) Section of an individual 10-legged cargo track (black) showing the positions of each leg while bound (coloured) when subject to ${\overline{k}}_3\mathrm{\Delta }x=0.1L/t_c$. Cargo legs that unbind near x = l_c preferentially rebind within the central region of increased cargo–substrate binding rate defined by equation (2.8) (see displacements near t/t_c ≃ 80).

Figure 5.

Dynamics of cargo simulated using experimental data. (a) The position-dependent effective velocity component $S_{\lambda }^{(1)}(x)$ exhibited by bound cargo permanently associated with N = 10 EBs obtained from stochastic cargo binding simulations (5 × 10⁴ simulated) using input parameters derived from previously published experimental data (see electronic supplementary material, tables S1 and S2). The microtubule edge is set at x_edge = 0 nm, and stable fixed points are exhibited by 10-legged cargo of sizes 2L ≥ 160 nm. Error bars are too small to see. (b) A heatmap showing the maximum effective velocity generated by binding or unbinding events in the direction of the growing microtubule end (100 000 simulated for 2 ≤ N ≤ 8; 50 000 simulated for N = 10; 25 000 simulated for N = 12). Cargo can co-move with growing microtubule ends (asterisks) when ${\overline{k}}_3\hspace{0.17em}\mathrm{\Delta }x-min(S_{\lambda }^{(1)}(x))>57\hspace{0.17em}\mathrm{nm}\hspace{0.17em}{\mathrm{s}}^{-1}$, which qualitatively corresponds to the effective velocity generated by binding and unbinding events being able to counter-balance the microtubule growth velocity.

Figure 6.

Radial dynamics of cargo in two dimensions. Position-dependent components of the effective velocity in the (a) radial direction and (b) tangential direction, derived by substituting the results of two-dimensional simulations (2.5 × 10⁵ simulated) into equation (2.12) and applying a change in coordinate systems (colour bar applies to both plots). The components of the effective velocity in Cartesian coordinates are shown in electronic supplementary material, figure S11. Data were averaged over each local point and its four nearest neighbours before changing coordinate systems. The effective velocity results in cargo being ‘attracted’ towards the central region of increased cargo–substrate binding rate defined in equation (2.13).

Figure 7.

Different types of cargo motion. Schematics showing the three types of motion that the multivalent cargo can display. Increasing grey levels in the substrate sites indicate increasing affinity, i.e. increasing binding and/or decreasing unbinding rate for legs. (a) Diffusive motion for substrates with uniform binding and unbinding rates, (b) directed motility (on average) towards regions of higher affinity, and (c) processive motion of the cargo following the growing substrate edge through preferential unbinding of legs from low-affinity regions and higher probability of rebinding of legs to high-affinity regions on the substrate. Note that (c) assumes that the growing edge has a higher affinity than the rest of the substrate.