We did 3D simulation of the molecular rolling robot over the DNA origami field using molecular dynamics. In this simulation, the robot is represented as mass points and moves by following the Langevin equation. Hybridizing between robot legs and substrates on the field, and cleaving the substrates by legs, are described by a potential shift.
The program code was written entirely in C and OpenGL.
Model and Methods
In this simulation, we described the system as a coarse-grained model(1). In a coarse-grained model, representative points of the structures are extracted. In our case, these representative points were connected by springs and strings, and their potentials associated with them.
For example, see Fig.1, which represents extraction of corresponding points from the molecular spider, and a scheme of potential to maintain structure. Streptavidine structures were maintained by a spring-type potential which keeps length among points and weakly affect angles among points. Deoxyrybozyme legs were connected with a string-type potential, which is taken into consideration when points go beyond the determined length. On the spider simulation, blue points, green points, and yellow lines, represent mass points of the structure, top of spider legs, lines represents bonds, respectively.
Fig.1 Representation of the molecular spider in our simulation
Each mass point is moving under influence of energy V. The energy V is sum of potential energy for maintaining structure, and that for binding substrates on field. The potential of substrates is zero when legs are out of the effective area (as cut-off), and changes by distance between legs and substrates when legs enter effective area. The force from differentiation of the substrate potential is proportional to the distance.
The motion of each mass point is described by the Langevin Equation(2). In this equation, acceleration is determined by sum of force F from the differentiation of energy V, viscosity resistance -βv , and white Gaussian random force η(t). White Gaussian random force was obtained by the box-muller methods(3). Distribution of the obtained random force is shown in Fig.2.
Fig.2 Distribution of white Gaussian random force
Fig.3 Field in simulation
In order to deduce parameters, we simulated movements of DNA spider robot in the DNA spider article (Lund et al. 2010)(4). Red points and blue points represent uncleaved substrates and cleaved substrates, respectively. See the above movie, in which spider goes toward goal with cleaving substrates.
Fig.4 (Left) Percentages of spiders that reached goal (without including the ones that left the field) (Right) Percentages of spiders which did not reach goal (which does not includes robots that was apart from field of DNA origami by brownian motion) These are resulted from 100 DNA spider simulation.
Fig.4 describes how many spiders reached the goal and how many did not at a certain time. We tuned simulation parameters to be consistence with the result in figure 2G of Lund et al. (Nature, 2010)(4). In our simulation, the time for reaching the goal substrate was about 20 times longer than the time that DNAzyme cleaved substrate. The result matched very well to the time in the article (about 21 times longer).
Simulation of the triangular prism robot
We simulated whether our triangular prism robot can reach the goal of the field. The video in the left shows that our robot has high probability to reach the goal. We supposed that the triangular prism robot could reach the goal by only rotary motion. However, our simulation result indicated that the triangular prism moves forward by combination of, both, rotary motion and walking motion. This combined motion did not impede the robot efficiency for moving forward. Anyway, this design is satisfying our initial racing condition of reaching the goal.
Comparison between the spider and triangular prism robot
Fig.5 Speed of spider and triangular prism.
In order to examine which of the robots (our triangular prism or DNA spider) is the fastest in reaching the goal, a simulation was carried out. Locations of the substrates for the molecular spider were determined by following the design of the substrates in the original DNA spider paper, those positions are seen in the video. Locations of the substrate for our triangular prism robot were considered to be the same as our field design. Since the body of the triangular prism robot is bigger than the spider body, we could reduce the number of substrates for the prism on the field. It can be seen from this video that the triangular prism robot reaches the goal faster than the DNA spider.
Data of triangular prism simulation
Fig.6 Goal time and percentage of 1 leg type and 3 leg types robot
To check the effect of the number of leg-types over the time to reach the goal, speeds of triangular prism with 3 types of legs and with 1 type of legs were calculated by our simulation. The right part of Fig.6 shows the results. The goal time of both cases are very similar, but the percentage of the triangle prism was different. 1 leg type robot has a higher goal probability. Thus, we concluded that the 1 leg type robot could be better.
Cleavage rate and spider speed
Fig.7 Correlation between cleavage time and percentage of spiders reached the goal substrate ,and goal time
To verify effects of cleavage rates of DNAzyme, simulation of the molecular spider was done by changing cleavage time. Fig.7 is the result from our simulation. Increase of cleaving rate facilitated speed in reaching the goal, but percentage of spiders which reached goal reduced. Therefore, there is a trade-off between increasing the cleaving rate and the percentage of robots that reaches the goal.
Field design and spider speed
Fig.8 Design of spider fields. All of these fields are based on the field we used in spider simulation (base field). (a)1 Substrate line is removed from the base field every 2 line. (b) 2 Substrate lines are removed every 3 line. (c) Upper substrates are removed. (d) The base field.
Fig.9 Correlations between the number of substrates and moving data of spiders
Fig.8 shows design of fields and moving data of spider over each field. Designs of these fields were based on the field used in the molecular spider article (figure 1d in Lund et al. Nature 2010))(4). Substrates in the field (a), (b), (c) are removed in a different pattern.
The results by the simulation are shown in the right figure. Less substrates gained speed to reach goal, and decreased probability of goal. But the goal probability depends on not only the number of substrates but also on the location of the substrates.
These suggested that it is better to reduce substrates on the field as much as possible to gain robot's speed.
Body size and leg size
Fig.10 These robot's size is the same.
In this section, both robots, triangular prism and a modified molecular spider, were chased in silico on the same field. On this simulation, spider robot has longer legs, while design of triangular prism was the same as the one we used above. Right figure shows moving data of each robot. As a result, the molecular spider with long legs was slower than our robots, although it had more goal probability. It might be due to difference in the body size, leg length and the number of legs. Longer leg increases the probability to reach the goal, making more molecular spiders reach goal. And if the structure has more legs, it can cleave more substrates in the same time. Therefore, when the design of the fields is the same, robots with many legs are fast. This notion explained why the triangular prism was faster when comparing with the molecular spider.