Aim of Simulation
Molecular robots works in meso-scale (from nm to sub-micro meter). In the meso-scale, characters of each molecule and cooperative movement of molecules could not be ignored. Without calculation of potentials or molecular dynamics simulations, it is difficult to predict the behaviors of the molecules in this scale. Thus, we calculated potentials to confirm the GATE effect, and simulated molecular dynamics of DNAs to strength reliability of our design.
Electric potential of the GATE was calculated inside and outside the GATE. The phosphate groups in the backbone of DNA are negatively-charged. Because the GATE is made by DNA origami, the GATE has the Coulomb force.
Model of DNA
A point-charge model is applied to DNA model. Each phosphate of a nucleotide was converted into one electric beads, which has elementary electoric charge (1.602176565 x 10^−19 [C]). In DNA helix, negative charges appear along the axis of the double helix every 10.4 base pairs. Debye–Hückel equation was used to calculate electric potential. Electric potential of each point around the GATE was deduced from summation of all the Electric Potential for every phosphate presented on DNA.
reference: Debye length
Following figure shows the scheme of a point-charge model and potential calculations.
Electric potential was calculated by using C language. Under our conditions, the target DNA has 25 electric beads.
The following figure shows electric potentials between the target DNA and the GATE from the top of view of the GATE. The electric potentials were standardized by thermal energy, KBT (KB; Boltzmann constant). Potential energies were shown by a heat map.
Potential energy along axis indicated by the arrows are shown as the following figure. Together, electric potential is too high for the target DNA to enter inside the GATE.
The difficulty to go through inside hole of the GATE was also indicated by following figures. The next figures shows electric potential at the center of hole in the GATE. Potentials were calculated along the axis indicated by the red arrow.
We carried out molecular dynamics simulation to examine the capturing mechanism and the effectiveness of our structure “CELL GATE".
For simplicity, a course-grained DNA model was used in our simulation. One DNA nucleotide was represented by one bead in the model and each bead can be hybridized with a complementary bead.
The potential energy of the system includes 5 distinct contributions.
The first three terms are intramolecular interactions, bonds, bond angles, and dihedral angles. In order to express the “tether like structure”, only bond interactions are considered in our DNA model.
And the latter two terms are non-bonded interactions. Coulomb interactions are taken into account using the Debye-Huckel approximation which enables to internalize counterions contribution.
Parameters of these potentials were fit to the reference literature ; Thomas A. Knotts et al. A coarse grain model of DNA .
The force on bead i is given by a Langevin equation
The first term donates a conservative force derived from the potential Vtot and the second is a viscosity dependent friction.
The third term is a white Gaussian noise and effects of collision with solvent molecules which causes brownian motion are internalized in this term.
Langevin equation is integrated using a Velocity-Verlet method.
Toehold displacement of dsDNA
In order to test the model, here we carried out a simulation of Toehold displacement between two strands.
Length of strands and simulation condition were as follows:
Target strand / Toehold A / Toehold B : 16nt / 9nt / 13nt
Temperature : 300K
Time-step / simulation length : 0.01ps / 100ns
Ion concentration : 50mM Na+
Movie1 : MD simulation of toehold strand displacement
Movie 1 shows the trajectory of each strand from the simulation. The target strand moves from Toehold A strand to Toehold B strand which are fixed on the field. This result agrees with the energy gradient.
Comparison of capture ability
One of constructional features of our structure ”Cell-Gate” is the use of a novel strand displacement method.
By comparing our selector strand to toehold strand, the most popular method for strand displacement, we looked at the effectiveness of our structure in terms of it's strand capturing ability.
Model and Method
According to the design of the experiment section, we modeled the selector strand and the toehold strand as shown below.
Hex-cylinder is represented as the assembly of fixed electrically-charged mass points.
Simulation was carried out at the following condition:
Temperature : 300K
Ion concentration : Na+ 50mM
Box size : 20nm×20nm×20nm (periodic boundary condition)
Time-step / simulation length : 0.01ps / 10ns
Results and Conclusion
Movie2 : MD simulation of porter strand
Movie3 : MD simulation of toehold strand
Movie2 and 3 shows the result of each simulation, selector-target and toehold-target.
We note that this simulation was carried out under a periodic boundary condition, where the size of the simulation box is 20nm×20nm×20nm. Then, the distance between the target strand and the Hex-cylinder is maintained virtually constant.
One of the advantages of the selector strand is shrinking ability. The selector strand hybridizes to the target making a loop which makes it possible to extend the strand length without changing the final structure's length.
Results obtained from this simulation show that the selector strand can catch the target strand exists outside of the Hex-cylinder and hybridize completely while the toehold strand never hybridize to the target strand in simulation time.
We run 5 simulations for each capturing mechanism under the same conditions and results were almost the same as we first obtained.
By considering results of electrostatic potential calculation around the hex-cylinder and MD simulation, it is clear that the electrostatic field prevents the entrance of DNA strands into the Hex-cylinder and the selector strand helps it to get into the cylinder.
Therefore, we concluded that our novel selector strand provides a high capture ability to our system “Cell-Gate”.
1. Thomas A. Knotts et al. A coarse grain model of DNA , J.Chem.Phys 126,084901(2007)
2. Carsten Svaneborg et al. DNA Self-Assembly and Computation Studied with a Coarse-Grained Dynamic Bonded Model, DNA 18,LNCS 7433, pp.123-134, (2012)
3. Xhuysn Guo & D.Thirumalai, Kinetics of Protein Folding: Nucleation Mechanism, Time Scales, and Pathways, Biopolymars, Vol.36, 83-102 (1995)
4. GROMACS manual ( http://www.gromacs.org/ )
5. Cafemol manual ( http://www.cafemol.org/ )