As discussed in SPEX Experimental Design, two sets of experiments were conducted to verify the random walking mechanism. The first set was performed to detect potential leak reactions in the system, while the second set of experiments were used to verify the random walking mechanism. In short, walker goal with fluorophore is planted the end of the track, and walker with quencher is planted at the various positions on the track. When we input walker trigger, walker starts walking on the track. When it gets to the walker goal, it stops there and quenches the fluorophore. Therefore, by analyzing fluorescent level using SPEX, we can perform a bulk analysis of a percentage of walkers that reach the walker goals.
Detection of Potential Leak Reactions
To verify that the walker walks on the origami using the intended mechanisms, we set up control groups where either tracks (TR), except for the one that the walker is bound to, or walker triggers (WT), or both, were missing in the reactions. Walkers are not expected to perform random walking under such conditions. As shown in Figure 1, no decrease in fluorescent signals were observed unless both tracks and walker triggers were added into the system. Hence potential leak reactions are negligible.
Sample 2 is the one that we want to observe, and sample 1, 3, 4 are all controls with/without tracks, or with/without walker trigger. Origami was annealed with the probes for the tracks and the walker goal, and with a hole for the walker start complex. For sample 1 and sample 2, tracks, a walker goal, and a walker start complex were planted for overnight. For sample 3 and sample 4, a walker goal and a walker start complex were planted for overnight without tracks. Walker start complex is a preannealed complex, (walker - walker inhibitor - track 1 - probe for the start site) which was gel purified. For this experiment, start site was chosen to be the furthest point from the walker goal. Following ratio was applied: scaffold: 1x, regular staples: 4x, probes for tracks: 4x, tracks: 6x, probes for walker goal: 1x, walker goal: 1x, walker start complex: 1x. Since SPEX showed some fluctuation due to a problem of the lamp, fluorescent level of four samples were stabilized for 1 hour. After 1 hour, 6x of the walker trigger was added to sample 2 and sample 4.
Before adding walker trigger, fluorescent level of sample 1 and 2 is slightly lower than that of sample 3 and 4 [figure 1]. Fluorescent level of sample 1 and 2 was ~6*10^5 while that of sample 3 and 4 was ~4.5*10^5. This is 25% loss, which is not ignorable. One possibility is that few walkers might be triggered by itself somehow without the trigger. Since only sample 1 and 2 contain tracks, a few self-triggered walkers could walk on the track, reach the walker goal, and quench a few fluorophores at the walker goal. Since sample 3 and 4 do not contain tracks, those self-trigger walkers cannot go anywhere.
In figure 1, no significant decrease of fluorescent level of sample 4 (without track, with walker trigger) shows three things: (1) there is very few free floating walker start complex. If some walker start complex did not land on the start site, free floating walker would directly go to walker goal after walker trigger is added, and the fluorescent level would go down in sample 4. Therefore it shows that insertion of walker start complex into the hole of the origami was quite successful. (2) There is no long jumping from the start site to the walker goal. Walker could not jump from the start site to walker goal in sample 4 (without track, with walker trigger). If we test the start site nearer to the walker goal, we can test how many tracks a walker can jump. (3) We can ignore the probability of stacking of origami that brings walker and walker goal into close proximity.
Only Sample 2 (with track, with walker trigger) shows gradual decrease of the fluorescent level as we designed. It is quite clear from the control sample 4 that walkers somehow use tracks to get to the walker goal, but we cannot figure out how they use tracks from figure 1. There exist two mechanisms that walker can use tracks to get to the walker goal. First one is walker walking or possibly jumping a little bit on the track. In this mechanism, walker does dissociate from the track on origami. Here, jumping refers to walking two or more steps at a time, possibly because distance between two tracks is too short due to incorrectly calculated geometry. Second mechanism is walker moving from tracks on origami to free floating tracks in solution. Then the walker can float or swim to the walker goal. We will call this 3-D diffusion as “Space walk” in contrast with the “Random walking” which we’re trying to verify. In the next set of experiments, we made a special control group "Space walk" to test the rate of 3D diffusion.
Random Walking with Different lengths of tracks using different starting sites
To test space walk hypothesis, we annealed a control origami with unmodified staples at the track position. When we add tracks to this control origami, since unmodified staples does not contain the probe region, no track is formed on origami while added tracks are free floating in solution. This is called a space walk control, abbreviated as SW. In addition to the space walk control, we varied the lengths of tracks on origami, and observed the completion level with timely manner.
Below is the diagram of partial tracks which all have different lengths.
Figure 2 shows the original unnormalized SPEX data. The walkers reached their destinations faster on shorter tracks.But this is not enough to show that the walkers were randomly walking on the surface of origami, because there was 5-fold excess of tracks in solution that the walker can strand displace onto from origami. We refer to this hypothetical process as 'space walking'. To test for this we designed a playground without any probes except for the one on the starting position of the walker (SP10) and that of the walker goal,but with 5-fold excess of tracks in solution, and we compared this to results of the walker on the regular playground at different starting positions.
As can be seen from the graph, space walking was the slowest process. As the rate of SW depends on the concentration of free floating tracks as well as the concentration of origami, the effect of SW on the overall rate of fluorophore quenching can be reduced by diluting free floating tracks and origami (data not shown). It was shown that the random walks all went to around 50% completion instead of 100%. This phenomenon could be attributed to malformed origami, misplanted tracks on the origami, or failure in planting walkers/walker goals.
This graph was normalized so the point at which the walker trigger was released starts at 1, and the point at where the fluorophores are assumed to be fully quenched is 0 (Figure 3). A similar trend was obtained: the rate of quenching was higher when the distance between starting point and walker goal was decreased (Figure 3). This is characteristic of a two dimensional random walk, but could not be used to confirm random walk, as a random walk is defined to be unbiased. Hence Matlab simulation and a mathematical model were employed to verify the occurrence of random walking on the origami.
Comparison: Random Walking with full length of tracks
Same starting positions that we used in the previous section were tested with full tracks, this time. Full track is longer than the longest track we tried previously. This new set of experiment also included space walk control. Results are plotted with the previous data to compare. Space walk control of two different set of experiment was very consistent, and the completion time of full tracks was slower than that of partial tracks. Full track samples showed similar completion time regardless of different starting position, even though starting position 22, which is a middle length track, showed a little digression; it needs a further discussion. Overall, SPEX data consistently shows that completion time is dependent on track length, suggesting a high probability of random walk on origami.
We used a stochastic MATLAB simulation to estimate half-completion times for walkers starting on truncated tracks of varying lengths:
This data, along with the completion time formula generated via mathematical analysis of the track, can serve as controls for the random walk SPEX data.
Normalized SPEX data for three different lengths of tracks was fitted to their respective random walk formula (Figure 7 (1)-(3)) by varying the time taken from one column to the adjacent one. It was found that experimental data can fit well with the theoretical curves within the range of fluorescent noises. Since the model proposed an unbiased random walk with equal forward and backward movement, it is highly likely that DNA walkers can walk on the origami in a truly random manner.
Furthermore, the rate of random walk (time per step) was estimated using the half completion time in each situation. It was found that walkers needed 10 seconds to take one branch migration on the origami with longest track, 16 seconds on origami with medium length track and 27 seconds on origami with shortest track. However, the rate of branch migration should be the same regardless of the length of tracks. We are in the progress of collecting more data for further analysis. Taking the average of these three values ,the rate of branch migration in our case is calculated to be 18 seconds per branch migration, or 0.3 nm/s. This fits with the reported average speed, which was of the order of 0.1 nm/s .
We made a comparison between the models from Matlab simulation and random walk formulae. Results are summarized in the table below.
Both models fit well with each other in estimating the average rate of branch migration on the origami. An interesting fact is that shorter tracks seemed to 'slow down' branch migration, which could be attributed to the differences in branch migration kinetics among different strands. As both models do not involve kinetics of branch migration, we are currently in the process of refining our models using chemical reaction networks (CRN).
 Wickham, S. F. J., Endo, M., Katsuda, Y., Hidaka, K., Bath, J., Sugiyama, H., & Tuberfield, A. J. (2010). Direct observation of stepwise movement of a synthetic molecular transporter. Nature Nanotechnology, 6, 166–169.