Dimensionality-dependent crossover in motility of polyvalent burnt-bridges ratchets

The burnt-bridges ratchet (BBR) mechanism is a model for biased molecular motion whereby the construct destroys track binding sites as it progresses and therefore acts as a diffusing forager, seeking new substrate sites. Using Monte Carlo simulations that implement the Gillespie algorithm, we investigate the kinetic characteristics of simple polyvalent BBRs as they move on tracks of increasing width. We find that as the track width is increased the BBRs remain nearly ballistic for considerable track widths proportional to the span (leg length) of the polyvalent walker before transitioning to near-conventional diffusion on two-dimensional tracks. We find there exists a trade-off in BBR track association time and superdiffusivity in the BBR design parameter space of span, polyvalency, and track width. Furthermore, we develop an analytical model to describe the ensemble-average motion on the track and find it is in good agreement with our Gillespie simulation results. This work offers insights into design criteria for de novo BBRs and their associated tracks, where experimentalists seek to optimize directionality and track association time.

Figure 1

(a) Each bound leg is assigned a span which defines the radius of a circle, shown in green (gray) circles, around the bound location, shown in red (dark gray). The mutual overlap of bound-leg spans, shown in yellow (light gray), marks the feasible binding locations for unbound legs (not shown) during the following kinetic move. (b) Each BBR leg can interact with any available substrate track site (gray triangle) via a rate of attachment, $k_{\mathrm{on}}$. $k_{\mathrm{eff}}$ includes both substrate cleavage and release.

Figure 2

Left column: The start and detachments points for each trajectory are indicated with a triangle and square, respectively; the time noted is the lifetime of that trajectory. (3,3), (12,3), and (12,8) BBRs are less likely to cross previously visited territories and exhibit more directional walks compared to (3,8) BBRs. Right column: Snapshots of ensemble BBR position distributions, taken from Supplemental Movies S1–S4 [36]. Color coding and gray scale represent the number of independent BBRs at that location in time. All BBR designs, except for (3,8), develop ringlike distributions with decreased occupancy at the origin.

Figure 3

(a) ${\mathrm{Log}}_{10}\text{−}{\log }_{10}\mathrm{MSD}(\vec{r}$)-time plots for (3,8) BBRs display the general trend of decreasing slope as a function of increasing track width. (b) ${\alpha }_r$ for (3,8) BBRs reaches a minimum at track widths of 128–256 before subsequently increasing to a width-invariant plateau of ${\alpha }_r\approx 1.1$. A minimum ${\alpha }_r$ at intermediate widths is observed for all BBR designs examined in this work. (c) $\mathrm{MSD}(x$) for (3,8) BBRs scales from ballistic to superdiffusive as the track width increases from one to two dimensions. (d) ${\alpha }_x$ as a function of width for all BBRs displays a monotically decreasing trend. (e) $\mathrm{MSD}(y$) for (3,8) evolves to ${\alpha }_y=0$ for all BBRs that are constrained by track boundaries. (f) For narrow tracks the BBRs move under confinement and display ${\alpha }_y\approx 0$. As track width increases such that the BBRs do not interact with the boundaries, ${\alpha }_y$ takes on the same values as ${\alpha }_x$ for all BBR designs.

Figure 4

(a) Fraction of BBRs remaining bound in one dimension for all BBR designs examined in this work. The inset shows the early time behavior. (b) Fraction of BBRs remaining bound in two dimensions for all BBR designs examined. For both 1D and 2D tracks (3,8) BBRs remain associated to the track for the longest times, whereas (12,3) BBRs associate to the track for the shortest times. (c) Fraction of (12,8) BBRs remaining bound for all track widths. The detachment curves saturate past track widths of 256. (d) $t_{1/2}$ as a function of track width for all BBR designs. (e) The ratio of $t_{1/2}$ values for various BBR designs. (f) ${\alpha }_x$ values from Fig. 3 plotted against their respective $t_{1/2}$ values.

Figure 5

(a) The evolving $x$-position distribution for (3,8) BBRs on a track of width 8. The numbers 1–4 represent the distribution at specific times. Two modes develop that move symmetrically in opposing directions. (b) Excess $\mathrm{kurtosis},{\gamma }_2$, for $x$-displacement distributions for (3,8) BBRs on all track widths. On narrow tracks ${\gamma }_2\left(x\right)$ quickly approaches $-2.0$. As track width increases the excess kurtosis settles to ${\gamma }_2\left(x\right)=-0.25$. The numbers 1–4 correspond to the histograms from (a). (c) The evolving $y$-position distribution for (3,8) BBRs on a track of width 128. (d) For large track widths ${\gamma }_2\left(y\right)$ limits to $-0.25$, the same as ${\gamma }_2\left(x\right)$. Under confinement ${\gamma }_2\left(y\right)$ settles to $-1.2$, the value for a uniform distribution, as illustrated by the numbers 1–4 from part (c).

Figure 6

Results of the bimodal Gaussian model. (a) In the limit of Brownian dispersion (case a) we recover conventional diffusion at long times. For all other cases we find the long-time limit of $\alpha$ to represent ballistic motion. (b) With increasing $\sigma \left(t\right)$ and constant $\mu$ (case a), ${\gamma }_2$ reaches the Gaussian limit of ${\gamma }_2=0$. In cases b and c, which describe distributions that separate faster than they disperse, ${\gamma }_2$ decreases to $-2.0$. When $\sigma$ and $\mu$ increase at the same rate ${\gamma }_2$ remains constant (case d).

Figure 7

(a) Digestion rate, $k_d$, for each BBR design across all track widths. For all BBR designs substrate digestion rates are found to be width-independent for tracks larger than a width of $\sim 8$. (b) Average number of substrate sites digested before detachment (starvation) for all BBRs across all examined track widths. For (3,8) BBRs, average cleavages are reported for tracks only up to width 32, as most (3,8) BBRs remained bound on larger track widths.

Figure 8

$\mu \left(t\right)$ and $\sigma \left(t\right)$ for BBRs with 3 legs, span 8 on a track of width 128. (a) ${\mu }_{\left|x\right|}\left(t\right)$ and ${\mu }_{\left|y\right|}\left(t\right)$ overlap until the ensemble of motors reaches the $y$ boundary at $\sim 1000$ s, at which point ${\mu }_{\left|y\right|}\left(t\right)$ curves over to the expected value of 32 for a uniform distribution bounded by $\left|y\right|=0,64$. (b) $\frac{{\sigma }_{\left|x\right|}\left(t\right)}{{\mu }_{\left|x\right|}\left(t\right)}$ and $\frac{{\sigma }_{\left|y\right|}\left(t\right)}{{\mu }_{\left|y\right|}\left(t\right)}$, the coefficients of variation, indicate that the dispersion in $y$ positions decreases substantially more than the mean $y$ position on interaction with the $y$ boundary (approximate time indicated by the vertical line). $\frac{{\sigma }_{\left|x\right|}\left(t\right)}{{\mu }_{\left|x\right|}\left(t\right)}$ remains relatively constant following the onset of BBR interaction with the $y$ boundary.