Cooperative effects enhance the transport properties of molecular spider teams

Molecular spiders are synthetic molecular motors based on DNA nanotechnology. While natural molecular motors have evolved towards very high efficiency, it remains a major challenge to develop efficient designs for man-made molecular motors. Inspired by biological motor proteins such as kinesin and myosin, molecular spiders comprise a body and several legs. The legs walk on a lattice that is coated with substrate which can be cleaved catalytically. We propose a molecular spider design in which $n$ spiders form a team. Our theoretical considerations show that coupling several spiders together alters the dynamics of the resulting team significantly. Although spiders operate at a scale where diffusion is dominant, spider teams can be tuned to behave nearly ballistic, which results in fast and predictable motion. Based on the separation of time scales of substrate and product dwell times, we develop a theory which utilizes equivalence classes to coarse-grain the microstate space. In addition, we calculate diffusion coefficients of the spider teams, employing a mapping of an $n$-spider team to an $n$-dimensional random walker on a confined lattice. We validate these results with Monte Carlo simulations and predict optimal parameters of the molecular spider team architecture which makes their motion most directed and maximally predictable.

Figure 1

Cartoon of the spider team model and definition of the leash length $d$. (a) Two spiders are attached to a joint cargo with an inelastic string. Both spiders walk on their respective one-dimensional track. Hats indicate the presence of substrate. (b) The finite length of the linking string induces a maximal distance between the spiders' bodies which gives rise to a maximal span of the spider team, characterized by the “leash length” $d$.

Figure 2

Dynamic properties of spider teams. Positions are given in lattice units throughout this work; time is defined by setting the hopping rate from products to 1. Thin shaded lines show data from finite difference approximations; thick lines show smoothing Bezier curves. (a) Probability distributions (histograms) of spiders to be at position $x$ at time $t={10}^6$; simulation data were binned with a box size 1. Depicted are distributions for a single spider ($n=1$) and spider teams comprised of $n=2,3,4$ spiders and leash length $d=8$, and cleavage rate $r=0.01$. While the single spider distribution follows nearly a Gaussian centered close to the origin, the distributions of spider teams are clearly skewed and shifted towards larger $x$. The asymmetry stems from the P-S preparation of the lattice at $t=0$ (products at the left, substrates at the right) [17]. (b) Mean displacement as a function of time (lines). The shaded areas represent the standard deviation around the mean displacement for a single spider and the four-spider team, respectively, and provide a measure for the randomness of the spiders' motion. Note that the visual impression of the standard deviation is rather that of a relative deviation, since the plot is in double logarithmic scale. (c) MSD as a function of time, $\langle x^2\left(t\right)\rangle$. (d) The variance's effective exponent $\alpha \left(t\right)$ [see Eq. (3)]. For diffusion, $\langle {[x\left(t\right)-\langle x\left(t\right)\rangle ]}^2\rangle \propto t^1$, hence $\alpha =1$; superdiffusion corresponds to $\alpha >1.1$ [16], and ballistic motion to $\alpha =2$. The superdiffusive regime of spider teams lasts longer than that of single spiders; large spider teams reach nearly ballistic motion for significantly long times. For a more detailed discussion, see the main text. (e) Mean velocity of the spiders as a function of time. The mean velocity is defined as the time derivative of the mean displacement, $d\langle x\left(t\right)\rangle /dt$. Spider teams outperform single spiders by an order of magnitude. (f) Sample trajectory of a single spider (top), and a four-spider team with $d=8$ (bottom). Periods in which the spider (team) is in the vicinity of the product-substrate boundary are shaded.

Figure 3

Definition of a boundary period. (a) Path of a single spider through a boundary period. The period always starts in state $\left(\mathrm{i}\right)$. From there, the spider can change to $\left(\mathrm{ii}\right)$, and back. When the right leg cleaves the substrate, the spider arrives at $\left(\mathrm{iv}\right),\left(\mathrm{v}\right),\left(\mathrm{vi}\right)$, or $\left(\mathrm{vii}\right)$. Arriving at $\left(\mathrm{vii}\right)$ corresponds to continuing the same boundary period from a new substrate [with “$\left(\mathrm{vii}\right)$ being the new $\left(\mathrm{i}\right)$”], since $\left(\mathrm{vii}\right)$ and $\left(\mathrm{i}\right)$ are equivalent up to translation. Hence, the number of steps is raised by one upon arriving at $\left(\mathrm{vii}\right)$. If, by contrast, the spider reaches $\left(\mathrm{iii}\right)$, the boundary period ends and a diffusive period begins. The probability to make a successful step, i.e., to reach $\left(\mathrm{vii}\right)$ before $\left(\mathrm{iii}\right)$, is the bias $p_{+}$ calculated by Antal and Krapivsky [12]. The number of steps during a boundary period is then the number of transitions $\left(\mathrm{i}\right)\to \left(\mathrm{vii}\right)$, without reaching $\left(\mathrm{iii}\right)$ in between. This is equivalent to the number of cleavages during a boundary period, not counting the very last cleavage (which is not counted since by definition the spider steps away from the boundary after the last cleavage, and we only count forward steps). (b) Example of a boundary period of a two-spider team. $\left(\alpha \right)$ None of the spiders is in a boundary period, hence none of them experiences a bias. Thus, the spider team is in a diffusive period. When the lower spider reaches a substrate $\left(\beta \right)$ it enters a boundary period. Thus, also the spider team enters a boundary period. In succession, the lower spider's right leg happens to cleave the substrate $\left(\gamma \right)$. The lower spider can then find its way to a new substrate $\left(\delta \right)$ what constitutes a $\frac{1}{2}$ successful step for the spider team and preserves the boundary period. If the upper spider, in this case, steps to a substrate $\left(\epsilon \right)$, this does not yet, however, constitute a step. This is because although the spider team is in a boundary period, the upper spider has not been in a boundary period itself during this team's boundary period. Since a step essentially reflects a cleavage, no step can be integrated in this case. If the lower spider steps away from the new substrate $\left(\zeta \right)$, the spider team enters a diffusive period. In analogy to single spiders, the number of steps during a spider team's boundary period is equivalent to the number of cleavages during that period, divided by the number of spiders, and not counting each spider's last cleavage event.

Figure 4

Justification for the equivalence classes in the limit $r\to 0$. Shown are the analytically calculated probabilities that a spider team ($n=d=2$) successfully completes one step during a boundary period, starting from the specific states $①–⑧$ as given in Eq. (15). Each line corresponds to a state of the equivalence class $\left[2_1\right]$ [cf. Eq. (12)]. In the limit $r\to 0$, the probability to step forward for all eight states collapses to a fixed value $\Pi \approx 0.65$.

Figure 5

Validity of the equivalence class formalism. Shown are the simulation results for the mean number of steps $\langle S\rangle$ for a two-spider team and different values of $d$ and cleavage rates $r=0.0001,0.001,0.01$; broken lines are a guide to the eye. The theoretical result derived within the equivalence class formalism for $r\to 0$ (black) is exact for $d=2,3,4$ (Table ), and we assumed $\Pi =\frac{5}{8}$ for $d\ge 5$ [Eq. (17)].

Figure 6

A spider team can be mapped to a random walk in a confined environment: Transitions of a spider's leg correspond to a change of its center-of-mass coordinate $c_i$ of $\pm \frac{1}{2}$. Shown is the mapping of a two-spider team with a leash length $d=2$. The shape of the environment (solid) follows from the leash constraint which confines the span of the spider team. From $d=2$ follows that the leftmost left and the rightmost right legs of the two spiders may be at most two lattice sites apart. With that restriction, the allowed configurations of the team follow directly, as can be seen with some explicit configurations in the left and the top part of the figure.

Figure 7

Substrate in the staircase random walker picture ($n=d=2$ as before). Like in Fig. 6, explicit configurations are shown for some points. In addition, boxes are drawn which correspond to the substrates on spider 1's (vertical blue box), or spider 2's (horizontal red boxes) lane. This can be understood as follows: When a spider is attached to a substrate with its right leg, it can be either in the spread or the relaxed configuration. Hence a substrate at position $c$ has to be indicated at two points in the center of mass space, namely, at $c-\frac{1}{2}$ and $c-1$; therefore the substrate boxes have width 2. Encircled in the figure are the eight states which have spider 2 at $\begin{array}{ccccc}\hfill •& \hfill \circ & \hfill \widehat{•}& \widehat{\circ }& \widehat{\circ }\end{array}$ or $\begin{array}{ccccc}\hfill \circ & \hfill •& \hfill \widehat{•}& \widehat{\circ }& \widehat{\circ }\end{array}$, respectively, and spider 1 in one of the five states $\begin{array}{ccccc}\hfill •& \hfill •& \hfill \circ & \circ & \widehat{\circ }\end{array},\cdots ,\begin{array}{ccccc}\hfill \circ & \hfill \circ & \hfill •& •& \widehat{\circ }\end{array}$. The resulting states correspond clearly to those of Eq. (15) and Fig. 4. In the figure, there are three horizontal red boxes (substrates on spider 2's lane), and only one vertical blue box (substrate on lane 1). Hence, the difference of the product sea's ends is $\Delta =2$. Since the encircled states $①–⑧$ have, by direct reading, only spider 2 at a substrate (i.e., they are only contained in $\sigma =1$ box), they form the equivalence class $\left[{\Delta }_{\sigma }\right]=\left[2_1\right]$.

Figure 8

Diffusion in the staircase environment. (a) Transition rates between the sites of the staircase environment. Along every arrow drawn, the rate is $\frac{1}{2}$ leading to local detailed balance. (b) The staircase can be split into elementary cells, numbered with integers.

Figure 9

Diffusion constants as a function of the leash length $d$ for $n=2$ and 3 spiders. The dashed line shows the theoretical result for $n=2$ [Eq. (25)]; solid lines are asymptotics for $d\to \infty$. Our theoretical approximation is in good agreement with simulation data (points).

Figure 10

Randomness during a diffusive period. Shown is the mean-squared minimal distance to the boundary of a random walker in the staircase environment ($n=2$), Fig. 6. The walkers start randomly along every point which provides an entrance to the diffusive period (for $d=2$, these are the three points below states $⑥–⑧$ in Fig. 7); they are absorbed when they reach the boundary (which is the second substrate box in Fig. 7; note that the lowermost box has been removed when the walker entered the diffusive period). Obviously, the mean-squared distance is greater the smaller $d$ is. Increasing $d$ thus decreases the randomness.

Figure 11

Standard deviation $\sigma$ of the spider teams' movement and the mean number of steps $\langle S\rangle$ as a function of the leash length $d$. Both $\sigma$ and $\langle S\rangle$ show extrema. To emphasize the correspondence between the minimum of $\sigma$ and the maximum of $\langle S\rangle$ (cf. Fig. 5), the $\langle S\rangle$ axis is drawn in reverse (see right scale). $\sigma$ is measured at the time $t^{*}$ when the mean displacement $\langle x\rangle$ equals 1000. This choice is arbitrary; for smaller values the minima of $\sigma$ persist, but are less pronounced [cf. Fig. 2b].