Biomod/2012/TeamSendai/Simulation

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{{:Biomod/2012/Tohoku/Team_Sendai/header}}
{{:Biomod/2012/Tohoku/Team_Sendai/header}}
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<html>
 
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<body>
 
<div id="Container">
<div id="Container">
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<h1>Aim of Simulation</h1>
-
<!--目次 -->
+
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.
-
<div id="mokuji">
+
-
<h2>Contents</h2>
+
-
<ol>
+
-
<li><a href="#Numerical Calculation for Electric Potential"> Numerical Calculation for Electric Potential </a></li>
+
-
<ol>
+
-
<li><a href="#Model">Model</a></li>
+
-
<li><a href="#">Results</a></li>
+
-
</ol>
+
-
<li><a href="#MD Simulation">MD Simulation</a></li>
+
<h1>Numerical Analysis of Electrostatic Potential</h1>
-
<ol>
+
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.
-
<li><a href="#DNA Model">DNA Model</a></li>
+
-
<ol>
+
-
<li><a href="#Results">Results</a></li>
+
-
</ol>
+
-
<li><a href="#Comparison of capture ability">Comparison of "capture" ability</a></li>
+
<h2>Model of DNA</h2>
-
<ol>
+
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.{{-}}
-
<li><a href="#Results">Results</a></li>
+
-
</ol>
+
-
<li><a href="#Reference">References</a></li>
+
-
</ol>
+
-
</ol>
 
-
</div>
 
-
<p>
+
Debye–Hückel equation:
-
<br><br>
+
{{-}}
 +
[[Image: Potential_fomula.png |left|300px]]
 +
{{-}}
 +
reference: Debye length
 +
{{-}}
 +
[[Image: Debyelen.png |left|300px]]
 +
{{-}}
 +
Following GIF animation shows the scheme of a point-charge model and potential calculations.{{-}}
 +
(It takes about 7 seconds to watch one cycle of the GIF animation){{-}}
 +
{{-}}
 +
{{-}}
 +
[[Image: Helix.gif |left|500px|thumb]]
 +
{{-}}
-
</p>
 
-
<a name="Numerical Calculation"></a><h2>Numerical Calculation</h2>
 
-
<p>
 
-
A phosphodiester bond make up the backbone of each helical strand of DNA. <br>
+
<h2>Results</h2>
-
The phosphate groups in the phosphodiester bond are negatively-charged.<br>
+
Electric potential was calculated by using C language.{{-}}
-
Because GATE is made of DNA, we can not ignore the influence of the Coulomb force.<br>
+
Under our conditions, the target DNA has 25 electric beads.{{-}}
-
So we calculate the electric potential inside and outside the GATE.
+
All calculation were performed under these condition: Temperature 298[K], Na+ 50mM.{{-}}
-
</p>
+
<h3>Electric potential from the top of view of the GATE</h3>
 +
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, <I>K<sub>B</sub>T</I> (K<sub>B</sub>; Boltzmann constant).{{-}}
 +
Potential energies were shown by a heat map.{{-}}
-
<a name="Model"></a><h2>Model</h2>
+
[[Image: Potential2.png |center|400px| Electric potential from the top of view of the GATE ]]
-
<p>
+
{{-}}
-
The coodinates are set as follows:<br>
+
{{-}}
-
<img src="http://openwetware.org/images/9/90/Cy.png" width="350px" height="300px">
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Potential energy along axis indicated by the arrows are shown as the following figure.{{-}}
-
<img src="http://openwetware.org/images/6/66/Lin.jpg" width="420px" height="300px"><br>
+
Together, electric potential is too high for the target DNA to enter inside the GATE. {{-}}
-
<br><br><br>
+
-
Point-charge model is used.<br>
+
{{-}}
-
Assumesd the phosphate groups negative charge,and<br>
+
[[Image: Potential3.png |center|600px| Electric potential from the top of view of the GATE along an axis]]
-
negative charge circles the axis of the double helix once every 10.4 base pairs like DNA.<br>
+
{{-}}
 +
{{-}}
-
And we use follow fomula to calculate electric potential.<br><br>
+
<h3>Electric potential at the center of hole in the GATE</h3>
-
Debye–Hückel equation<br>
+
-
<img src="http://openwetware.org/images/e/ec/Potential_fomula.png" width="300px" height="90px"><br>
+
-
<br>
+
-
Debye length<br>
+
-
<img src="http://openwetware.org/images/f/f9/Debyelen.png" width="400px" height="250px"><br><br>
+
 +
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.{{-}}
 +
Blue lines shows electric potential in the case of 1.5 fold radius of the GATE.{{-}}
 +
{{-}}
 +
[[Image: Potential_K.png |center|800px| Electric potential at the center of hole in the GATE ]]
 +
{{-}}
 +
{{-}}
 +
By the figure, we concluded {{-}}
 +
i ) Our designed size is suitable size for the GATE function{{-}}
 +
ii) Enlarging radius of the GATE decreases electric repulsion effect{{-}}
-
Add all potential by negative charge DNA which compose gate have.<br>
 
-
(used C language to output the numbers)<br><br>
 
-
<img src="http://openwetware.org/images/4/45/Helix.gif" width="500px" height="290px"><br>
 
 +
<h3>Hybridization energy of Porters overcomes electric potential of the GATE</h3>
 +
We expected the hybridization energy of the Porters overcomes the inhibition of entrance of the GATE by electric repulsion.{{-}}
 +
The Gibbs energies of hybridization was calculated by neighbor-joining methods.{{-}}{{-}}
 +
The result of Gibbs energy of the fixed Porter were defined as the maximum length of each form of Porter. For examples, Porter 1 has three forms: Stretched form, 1 loop form, and 2 loop form. In this Porter 1 case, we defined Porter 1 has three Gibbs energy according to the hybridization state. Since the maximum lengths of each form were different, the minimum energy of possible length was selected.{{-}} Gibbs energies were calculated at the center of hole in the GATE.{{-}}
 +
The following figure shows the calculated energy of Porters and Toehold DNA used as controls in our project.
-
Condition<br>
+
{{-}}
-
Temperature 298[K]<br>
+
[[Image: Porter1.png |center|500px| Three different form ]]
-
Na+ 50mM<br>
+
{{-}}
-
<img src="http://openwetware.org/images/f/fc/Add2.png" width="500px" height="180px"><br>
+
{{-}}
 +
[[Image: Gibbs kai.png |center|800px| Gibbs energies by the Porters in the GATE ]]
 +
{{-}}
 +
{{-}}
 +
The Gibbs energies by the Porters shown above were summed with electric potential of the GATE.{{-}}
 +
The summation of energies were calculated at the center of hole in the GATE.{{-}}
 +
The result was shown in the next figure.
 +
{{-}}
 +
{{-}}
 +
[[Image: Gibbs2.png |center|800px| Electric potential from the top of view of the GATE ]]
 +
{{-}}
 +
The results of calculation indicate that the outside Porter can catch the target DNA, and inner Porter can pull in the GATE step by step. On the other hand, short DNA like the toehold DNA we called n this study cannot catch the target DNA. {{-}}
-
</p>
 
-
<a name="Results"></a><h2>Results</h2>
 
-
<p>
 
-
<br>
 
-
Electric potential changing z-axis at x-axis and y-axis is 0.<br>
 
-
<img src="http://openwetware.org/images/0/09/1014x0y0potential.png" width="620px" height="450px"><br>
+
<h1>MD Simulation</h1>
-
the length of the gate is 88bp, 30nm.
+
To further strengthen feasibility of our design, we examined the capturing mechanism and
 +
the effectiveness of our structure “CELL GATE" by molecular dynamics (MD) simulation.
-
Target base pair 25 を点電荷と仮定する
 
-
</p>
 
 +
<h2>DNA Model</h2>
 +
[[Image: Format_DNA_rasen.jpg |center|638px]]
 +
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.
-
<a name="MD Simulation"></a><h2>MD Simulation</h2>
+
The potential energy of the system includes 5 distinct contributions.
-
<p>
+
-
We carried out molecular dynamics simulation to examine the the mechanism and
+
-
the effectiveness of our structure “Cell Gate”.
+
-
</p>
+
[[Image: Format_suusiki1.jpg |left|711px]]
 +
{{-}}
-
<a name="DNA Model"></a><h2>DNA Model</h2>
+
The first three terms are intramolecular interactions, bonds, bond angles, and dihedral angles. These terms are important for maintaining structure. In order to express the “tether like structure”, only bond interactions are considered in our DNA model.  
-
<p>
+
-
For simplicity, course-grained DNA model is used in our simulation. <br>
+
-
One DNA nucleotide is represented by one bead in the model and each bead can be<br>
+
-
hybridized with complementary bead.<br>
+
-
  <<モデル載せる>><br><br>
+
-
The potential energy of the system includes 5 distinct contributions.<br>
+
-
  <<ポテンシャル載せる>><br><br><br>
+
-
The first three terms are intramolecular interactions , bonds , bond angles, and<br>
+
-
dihedral angles. In order to express “tether like structure”, only bond interactions<br>
+
-
are active in our DNA model.<br>
+
-
And the latter two are non-bonded interactions. Coulomb interactions are taken into<br>
+
-
account using the Debye-Huckel approximation which enables to internalize<br>
+
-
counterions contribution.<br>
+
-
Constants of these potentials are achieved from references.<br>
+
-
The force on bead i is given by a Langevin equation<br><br><br>
+
-
Langevin equation<br><br>
+
And the latter two terms are non-bonded interactions. Coulomb interactions are taken into account using the Debye-Huckel approximation which enables to internalize counter-ions contribution.
-
   
+
-
<img src="http://openwetware.org/images/1/11/Langevin.png" width="220px" height="80px"><br><br>
+
-
<img src="http://openwetware.org/images/2/23/F%3D.png" width="150px" height="80px"><br>
+
-
The first term donates a conservative force derived from the potential U and the<br>
+
Parameters of these potentials were fit to the reference literature ; Thomas A. Knotts ''et al''.  A coarse grain model of DNA .
-
second is a viscosity dependent friction.<br>
+
 
-
The third term is a white Gaussian noise and effects of solvent molecules are<br>
+
The force on bead i is given by a Langevin equation
-
internalized in this term.<br>
+
 
-
Langevin equation is integrated using a Velocity-Verlet method.<br><br><br>
+
[[Image: Format_suusiki2.jpg |left|357px]]
-
Toehold displacement of dsDNA<br>
+
{{-}}
-
In order to test predictive capability of the model, here we carried out a simulation<br>
+
 
-
of Toehold displacement between two strands.<br>
+
The first term donates a conservative force derived from the potential Vtot and the second is a viscosity dependent friction.
-
Length of strands and simulation situation was as follows.<br><br>
+
 
-
Target strand/Toehold A/Toehold B : 25nt / 9nt (+10nt spacer) / 13nt (+10nt
+
The third term is a white Gaussian noise and effects of collision with solvent molecules which causes brownian motion are internalized in this term.
-
spacer)<br>
+
 
-
Temperature : 300K<br>
+
Langevin equation is integrated using a Velocity-Verlet method.
-
Time-step size / simulation length : 0.01ps / 100ns<br>
+
 
-
Ion concentration : 50mM Na+<br><br>
+
 
-
results<br>
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<h2> Verification of the Coarse-grained model: Toehold displacement of dsDNA</h2>
-
<<後ほど>>
+
[[Image: Format_toea_toeb.jpg |right|400px]]
 +
In order to test the model, here we carried out a simulation of Toehold displacement between two strands.
 +
 
 +
Toehold strands are modeled according to designs of [http://openwetware.org/wiki/Biomod/2012/TeamSendai/Experiment#Comparison_of_Porter_and_Toehold the experiment section].
 +
 
 +
 
 +
Length of strands and simulation condition were as follows:
 +
 
 +
Target strand / Toehold A / Toehold B : 16nt / 9nt / 13nt <br>
 +
Temperature : 300K <br>
 +
Time-step / simulation length : 0.01ps / 100ns<br>
 +
Ion concentration : 50mM Na+<br>
 +
 
 +
<html><div style="clear:both;"></div></html>
 +
 
 +
<h3>Results</h3>
 +
<table border="0" align="left" vertical-align: middle;>
 +
<tr>
 +
<td>
 +
<youtube width="450" align="left">FGpemnMLUMk&border=1&color1=0x6699&color2=0x54abd6</youtube>
 +
<html><div style="clear:both;"></div></html>
 +
  </td>
 +
</tr>
 +
<tr>
 +
<td align="center">
 +
Movie1 : MD simulation of toehold strand displacement
 +
</td>
 +
</tr>
 +
</table>
 +
 
 +
<html><div style="clear:both;"></div></html>
 +
 
 +
 
 +
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. The result supported our coarse-grained DNA model is feasible.
 +
 
 +
 
 +
<h2> Comparison of capture ability</h2>
 +
One of constructional features of our structure ”CELL-GATE”is the use of a novel strand displacement method, called porter system.
 +
 
 +
By comparing our porter 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.
 +
 
 +
<h3> Model and Method</h3>
 +
According to the design of [http://openwetware.org/wiki/Biomod/2012/TeamSendai/Experiment#Comparison_of_Porter_and_Toehold the experiment section], we modeled the porter strand and the toehold strand as shown below. <br><br>
 +
[[Image: Format_selector.jpg |left|680px]]
 +
{{-}}
 +
 
 +
<html><div style="clear:both;"></div></html>
 +
 
 +
Hex-cylinder is represented as the assembly of fixed electrically-charged mass points.<br><br>
 +
[[Image: Format_Hexagon.jpg |left|750px]]
 +
 
 +
<html><div style="clear:both;"></div></html>
-
</p>
+
Simulation was carried out at the following condition:
-
<a name="Comparison of capture ability"></a><h2>Comparison of capture ability</h2>
 
-
<p>
 
-
One of constructional features of our structure ”Cell-Gate” is the use of new strand
 
-
displacement method.<br>
 
-
By comparing our selector strand and a toehold strand, the most popular method for<br>
 
-
strand displacement, we show the effectiveness our structure in terms of capture
 
-
ability.<br><br><br>
 
-
Model and Method<br>
 
-
According to the design of experiment section, we designed models as below of the<br>
 
-
selector strand and the toehold strand.<br>
 
-
<<モデル載せる>><br><br><br>
 
-
Hex-cylinder is represented as the assembly of electrically-charged mass points<br>
 
-
fixed on the field.<br>
 
-
<<モデル載せる>><br><br><br>
 
-
Simulation was carried out at the following condition.<br>
 
Temperature : 300K<br>
Temperature : 300K<br>
Ion concentration : Na+ 50mM<br>
Ion concentration : Na+ 50mM<br>
Box size : 20nm×20nm×20nm (periodic boundary condition)<br>
Box size : 20nm×20nm×20nm (periodic boundary condition)<br>
-
Time-step size / simulation length : 0.01ps / 10ns<br>
+
Time-step / simulation length : 0.01ps / 10ns<br>
-
Results<br>
+
{{-}}
-
<<後ほど>><br>
+
-
</p>
 
 +
<h3>Results and Conclusion</h3>
 +
<table border="0" align="left" vertical-align: middle;>
 +
<tr>
 +
<td>
 +
<youtube width="450" align="left">NXo4cYKkrF0&border=1&color1=0x6699&color2=0x54abd6</youtube>
 +
<html><div style="clear:both;"></div></html>
 +
</td>
 +
<td width="20">
 +
</td>
 +
<td>
 +
<youtube width="450" align="left">qidZS1pI0lc&border=1&color1=0x6699&color2=0x54abd6</youtube>
 +
<html><div style="clear:both;"></div></html>
 +
</td>
 +
</tr>
 +
<tr>
 +
<td align="center">
 +
Movie2 : MD simulation of porter strand
 +
</td>
 +
<td width="20">
 +
</td>
 +
<td align="center">
 +
Movie3 : MD simulation of toehold strand
 +
</td>
 +
</tr>
 +
</table>
 +
<html><div style="clear:both;"></div></html>
-
<a name="DNA model"></a><h2>DNA Model</h2>
 
-
<img src="http://openwetware.org/images/0/04/Format_DNA_rasen.jpg" alt="DNA" align="right" width="638px" height="348px">
 
-
<p>
+
Movie2 and 3 shows the result of each simulation, porter-target and toehold-target.
-
For simplicity, course-grained DNA model is used in our simulation. One DNA nucleotide is represented by one bead in the model and each bead can be hybridized with complementary bead.
+
-
<br clear="right">
+
-
</p>
+
-
<p>
+
-
The potential energy of the system includes 5 distinct contributions.
+
-
</p>
+
-
<img src="http://openwetware.org/images/d/d4/Format_suusiki1.jpg" alt="suusiki1" align="left" width="711px" height="132px">
+
-
<br clear="left">
+
-
<p>
+
-
The first three terms are intramolecular interactions, bonds, bond angles, and dihedral angles. In order to express “tether like structure”, only bond interactions  are active in our DNA model. </br>
+
-
And the latter two are non-bonded interactions. Coulomb interactions are taken into account using the Debye-Huckel approximation which enables to internalize  counterions contribution.</br>
+
-
</br>
+
-
Parameters of these potentials were fit to reference literatures.</br>
+
-
</br>
+
-
The force on bead i is given by a Langevin equation
+
-
</p>
+
-
<img src="http://openwetware.org/images/7/70/Format_suusiki2.jpg" alt="suusiki2" align="left" width="357px" height="75px">
+
-
<br clear="left">
+
-
<p>
+
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.
-
The first term donates a conservative force derived from the potential U and the second is a viscosity dependent friction.</br>
+
-
The third term is a white Gaussian noise and effects of solvent molecules are internalized in this term.</br>
+
-
Langevin equation is integrated using a Velocity-Verlet method.
+
One of the advantages of the porter strand is shrinking ability. The porter strand hybridizes to the target making a loop which makes it possible to extend the strand length without changing the final structure's length.
-
</p>
+
-
<h1> Toehold displacement of dsDNA</h1>
 
-
<img src="http://openwetware.org/images/8/85/Format_toea_toeb.jpg" alt="suusiki2" align="right" width="490px" height="251px">
 
-
<p>
 
-
In order to test the model, here we carried out a simulation of Toehold displacement between two strands.</br>
 
-
Length of strands and simulation situation was as follows.</br>
 
-
Target strand/Toehold A/Toehold B : 25nt / 9nt (+10nt spacer) / 13nt (+10nt spacer)</br>
+
Results obtained from this simulation show that the porter 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.
-
Temperature : 300K</br>
+
-
Time-step size / simulation length : 0.01ps / 100ns</br> 
+
-
Ion concentration : 50mM Na+</br>
+
-
</p>
+
-
<br clear="right">
+
-
<h2>Results</h2>
+
We run additional 5 simulations for each capturing mechanism under the same conditions (NOTE: random seeds were different) and very similar results were obtained. Thus, the simulation results are very feasible under our model.
-
<p>
+
-
<<動画>>
+
-
</br>
+
-
Movie 1 shows the trajectory of each strands 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.
+
-
</p>
+
-
<h2> Comparison of capture ability</h2>
+
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 porter strand helps it to get into the cylinder.
-
<p>
+
-
One of constructional features of our structure ”Cell-Gate” is the use of new strand displacement method.</br>
+
-
By comparing our selector strand and a toehold strand, the most popular method for strand displacement, we looked at the effectiveness of our structure in terms of capture ability.</br>
+
-
</p>
+
-
+
-
<h2> Model and Method</h2>
+
-
<img src="http://openwetware.org/images/5/59/Format_selector.jpg " alt="selector" align="left" width="445px" height="369px">
+
-
<p>
+
-
According to the design of experiment section, we designed models of the selector strand and the toehold strand as below.
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Hex-cylinder is represented as the assembly of electrically-charged mass points fixed on the field.
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<img src="http://openwetware.org/images/c/c4/Format_Hexagon.jpg" alt="hexagon" align="left" width="265px" height="279px">
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Simulation was carried out at the following condition.</br>
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Temperature : 300K</br>
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Ion concentration : Na+ 50mM</br>
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Box size : 20nm×20nm×20nm (periodic boundary condition)</br>
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Time-step size / simulation length : 0.01ps / 10ns</br>
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</p>
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<h2>Results</h2>
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Therefore, we concluded that '''our novel porter strand provides a high capture ability to our system “CELL-GATE”.'''
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<p>
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<<動画+グラフ(一応)後ほど>></br>
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Movie2 and 3 shows the result of each simulation, selector-target and toehold-target.</br>
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We note that this simulation was carried out under periodic boundary condition where the size of the box is 20nm×20nm×20nm, the distance between the target strand and the Hex-cylinder is maintained virtually constant.</br>
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One of the advantages of the selector strand is shrinking ability. The selector strand hybridizes to the target with making loop which makes it possible to extend the strand length without changing hybridized structure's length.Z</br>
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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.</br>
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We run 5 simulations for each under the same conditions and results were almost the same as we first obtained.</br>
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</br>
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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.</br>
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Therefore, we concluded that the selector strand, we originally designed, provides a high capture ability to our system “Cell-Gate”.</br>
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<h2>References</h2>
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1. Thomas A. Knotts ''et al''.  A coarse grain model of DNA , J.Chem.Phys 126,084901(2007)
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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)
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<a name="Reference"></a><h2>Reference</h2>
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3. Xhuysn Guo & D.Thirumalai, Kinetics of Protein Folding: Nucleation Mechanism, Time Scales, and Pathways, Biopolymars, Vol.36, 83-102 (1995)
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<p>
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1. Thomas A. Knotts et al. A coarse grain model of DNA , J.Chem.Phys
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126,084901(2007)<br>
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2. Carsten Svaneborg et al. DNA Self-Assembly and Computation Studied with a
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Coarse-Grained Dynamic Bonded Model, DNA 18,LNCS 7433, pp.123-134,
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2012<br>
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3. Xhuysn Guo & D.Thirumalai, Kinetics of Protein Folding: Nucleation
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Mechanism, Time Scales, and Pathways, Biopolymars, Vol.36, 83-102 (1995)<br>
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4. GROMACS manual ()<br>
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5. Cafemol manual ( http://www.cafemol.org/ )<br>
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6. Thomas A. Knotts et al.  A coarse grain model of DNA , J.Chem.Phys 126,084901(2007)
+
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7. 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)
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8. Xhuysn Guo & D.Thirumalai, Kinetics of Protein Folding: Nucleation Mechanism, Time Scales, and Pathways, Biopolymars, Vol.36, 83-102 (1995)
+
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9. GROMACS manual ( http://www.gromacs.org/ )
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10. Cafemol manual  ( http://www.cafemol.org/ )
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4. GROMACS manual ( http://www.gromacs.org/ )
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5. Cafemol manual  ( http://www.cafemol.org/ )
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Current revision



Team Sendai Top


Contents

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.

Numerical Analysis of Electrostatic Potential

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.


Debye–Hückel equation:


reference: Debye length



Following GIF animation shows the scheme of a point-charge model and potential calculations.
(It takes about 7 seconds to watch one cycle of the GIF animation)




Results

Electric potential was calculated by using C language.
Under our conditions, the target DNA has 25 electric beads.
All calculation were performed under these condition: Temperature 298[K], Na+ 50mM.

Electric potential from the top of view of the GATE

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.

Electric potential from the top of view of the GATE



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.


Electric potential from the top of view of the GATE along an axis



Electric potential at the center of hole in 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.
Blue lines shows electric potential in the case of 1.5 fold radius of the GATE.


Electric potential at the center of hole in the GATE



By the figure, we concluded
i ) Our designed size is suitable size for the GATE function
ii) Enlarging radius of the GATE decreases electric repulsion effect


Hybridization energy of Porters overcomes electric potential of the GATE

We expected the hybridization energy of the Porters overcomes the inhibition of entrance of the GATE by electric repulsion.
The Gibbs energies of hybridization was calculated by neighbor-joining methods.

The result of Gibbs energy of the fixed Porter were defined as the maximum length of each form of Porter. For examples, Porter 1 has three forms: Stretched form, 1 loop form, and 2 loop form. In this Porter 1 case, we defined Porter 1 has three Gibbs energy according to the hybridization state. Since the maximum lengths of each form were different, the minimum energy of possible length was selected.
Gibbs energies were calculated at the center of hole in the GATE.
The following figure shows the calculated energy of Porters and Toehold DNA used as controls in our project.


Three different form



Gibbs energies by the Porters in the GATE



The Gibbs energies by the Porters shown above were summed with electric potential of the GATE.
The summation of energies were calculated at the center of hole in the GATE.
The result was shown in the next figure.

Electric potential from the top of view of the GATE


The results of calculation indicate that the outside Porter can catch the target DNA, and inner Porter can pull in the GATE step by step. On the other hand, short DNA like the toehold DNA we called n this study cannot catch the target DNA.


MD Simulation

To further strengthen feasibility of our design, we examined the capturing mechanism and the effectiveness of our structure “CELL GATE" by molecular dynamics (MD) simulation.


DNA Model

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. These terms are important for maintaining structure. 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 counter-ions 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.


Verification of the Coarse-grained model: Toehold displacement of dsDNA

In order to test the model, here we carried out a simulation of Toehold displacement between two strands.

Toehold strands are modeled according to designs of the experiment section.


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+

Results

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. The result supported our coarse-grained DNA model is feasible.


Comparison of capture ability

One of constructional features of our structure ”CELL-GATE”is the use of a novel strand displacement method, called porter system.

By comparing our porter 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 porter 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, porter-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 porter strand is shrinking ability. The porter 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 porter 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 additional 5 simulations for each capturing mechanism under the same conditions (NOTE: random seeds were different) and very similar results were obtained. Thus, the simulation results are very feasible under our model.


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 porter strand helps it to get into the cylinder.

Therefore, we concluded that our novel porter strand provides a high capture ability to our system “CELL-GATE”.


References

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/ )



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