Difference between revisions of "Team:Warwick/Modelling1"

Latest revision as of 06:52, 18 September 2015

Warwick iGEM 2015

DNA Origami Glue

___________________________________________________________________________________________________________________________________________________________

NTNU Brixells Modelling Collaboration

The collaboration focused on two elements:

Calculating the probability of formation, including the probability of E. Coli bonding correctly.
Calculating the effect of arm length on structure changes increases. This has been motivated by the observation that the longer the arms the higher the likelihood that the e.coli cells can attach because they are less closely packed and such wouldn't get in each others way.

There are many methods to look at this kind of problems, but the most realistic may require huge computational resources. So it is necessary to find an approach that is experimentally relevant, computationally tractable, and reflecting the biophysics underlying Brixells.

We, Team NTNU Trondheim have offered ideas, modeling frameworks, software simulation, and data analysis to answer these questions, based on information theory and thermodynamics. For Problem 1, we have suggested an approach in terms of binding affinity and specificity where the information entropy is used as a measure of specificity. For Problem 2, we have suggested a thermodynamics approach using the Poisson-Boltzmann theory where the zinc fingers and E. Coli are approximated by a set of beads (one large bead for the E. Coli, small beads for each zinc finger, and a rod of beads for the DNA arm. Team Warwick has been enthusiastic about this approach since it is a stochastic method that they have not considered, and they have provided that us with data related to the geometry of E. Coli-arm formation.

___________________________________________________________________________________________________________________________________________________________

I.1) Probability of bond formation

The probability of bond formation $P$ is equal for all different types of fingers $i$ can be expressed as the product of the probability of formation for each zinc finger $P_{i}$ :

P = \prod_{i} P_{i}

___________________________________________________________________________________________________________________________________________________________

This is a clear measure of how strong the structure is.

The probability of formation for a zinc finger $i$ is expressed as a function of the specificity.

P_{i} = \frac{S_{i}}{W (S_{i})}

where $W$ is the product log function and $S_{i}$ is the specificity which is the information theoretical "mutual information", and is expressed as follows:

S_{i} = \sum_{j} \frac{K_{i, j}}{\sum_{j} K_{i, j}} \ln (\frac{K_{i, j}}{⟨ K_{i, j} ⟩})

That calculates the specificity of one zinc finger, i.e. how well a zinc finger binds specifically to an arm when it is competing with other zinc fingers. $K_{i, j}$ is the binding affinity of zinc finger $i$ specifically to arm $j$ .

The binding affinity is related to the binding probability $P_{i, j}^{bound}$ according to the following equation:

P_{i, j}^{bound} = \frac{[F_{i}]}{[F_{i}] + K_{i, j}}

where $A_{i}$ is the the arm $i$ and $F_{j}$ is the the finger $j$ , which are governed by the following ligand-binding reaction:

A_{i} + F_{j} \Leftrightarrow A_{i} F_{j}

___________________________________________________________________________________________________________________________________________________________

I.2) Effect of arm length

In order to account for the arm length. It is important to express the binding affinity to the geometry of the system consisting of one arm and one zinc finger. We dropp the indices $i$ and $j$ in the following for simplicity.

K_{i, j} = e^{\frac{-Δ G}{RT}}

We approximate the bacteria, the zinc fingers, and the DNA arm by a set of beads. The bacteria becomes one large sphere, the zinc finger one small sphere, and the DNA arm becomes a rod consisting of a cascade of beads.

Using these assumptions, the free energy can be calculated from geometrical and electrochemical considerations. In other words, knowing the positions, radii, and charge magnitudes on the beads, the binding free energy of the system is directly calculated as follows.

Δ G = G^{bound} - G^{unbound}

where $G^{bound}$ is the bound free energy expressed as:

G^{bound} = \sum_{\begin{matrix} m, n = 1 \\ n \neq m \end{matrix}}^{M + N} (S_{m, n} + V_{m, n} + E_{m, n})

and $G^{unbound}$ is the unbound free energy expressed as:

G^{unbound} = \sum_{\begin{matrix} m, n = 1 \\ n \neq m \end{matrix}}^{N} (S_{m, n} + V_{m, n} + E_{m, n}) + \sum_{\begin{matrix} m, n = N + 1 \\ n \neq m \end{matrix}}^{M + N} (S_{m, n} + V_{m, n} + E_{m, n})

where $m$ and $n$ are bead indices, $M$ and $N$ are the number of beads in each subsystem.

R m, n

is the distance between two beads.

ρ_{m}

is the radius of a bead

m

, and

q_{m}

is the charge magnitude of a bead

m

.

where $E_{m, n}$ is the electrostatic energy potential expressed as:

E_{m, n} = \frac{q_{m} q_{n}}{8 π ϵ R_{m, n}^{2}}

$V_{m, n}$ is the van Der Waals force potential expressed as:

V_{m, n} = \frac{- A}{6} (\frac{2 ρ_{m} ρ_{n}}{R_{m, n}^{2} - {(ρ_{m} + ρ_{n})}^{2}} + \frac{2 ρ_{m} ρ_{n}}{R_{m, n}^{2} - {(ρ_{m} - ρ_{n})}^{2}} + \ln (\frac{R_{m, n}^{2} - {(ρ_{m} + ρ_{n})}^{2}}{R_{m, n}^{2} - {(ρ_{m} - ρ_{n})}^{2}}))

and $S_{m, n}$ is the pair solvent energy expressed as:

S_{m, n} = \frac{1}{8 π} (\frac{1}{ε_{0}} - \frac{1}{ε}) \frac{q_{m} q_{n}}{f_{m, n}}

with

f_{m, n} = \sqrt{R_{m, n}^{2} + ρ_{m} ρ_{n} e^{- g_{m, n}}}

and

g_{m, n} = \frac{R_{m, n}}{4 ρ_{m} ρ_{n}}

___________________________________________________________________________________________________________________________________________________________

Table 1 provides the remaining variable values.

Table 1 - System parameters

Description	Symbol	Value
Zinc finger radius	$q_{zinc}$	$2 \cdot 1.60217662 \cdot 10^{- 19} C$
Base pair equivalent bead radius	$R_{bead}$	$1.5 bp (1 bp = 350 pm)$
DNA arm length	$L_{arm}$	$150 bp - 1500 bp$
DNA arm bead charge magnitude	$N_{zinc}$	$- 2 \cdot 1.60217662 \cdot 10^{- 19} C$
Real gas constant	$R$	$8.3144598$
Temperature	$T$	$300 K$
Absolute permittivity	$ε$	$8.854187817 \cdot 10^{- 12} F / m$
Relative permittivity of water	$ε_{r}$	$88$
Hamaker coefficient of nucleotide	$A$	$6 \cdot 10^{- 10}$
Dielectric constant of water	$ϵ$	$80.4$

___________________________________________________________________________________________________________________________________________________________

I.3) Numerical evaluation

We have simulated the effect of arm length on free energy (low free energy implies high binding probability) using the Born salvation free model, pair van der Waals energy, and electrostatic potential, described above. However, there are some parameters that we are not sure about (Equivalent charge and Hamacker coefficient of Zinc fingers, equivalent charge and Hamacker coefficient of base pair) that depend on charge distribution of the DNA strand, which should be determined experimentally as boundary conditions. Alternatively, one could consider each base pair with independent charge.

The systems have been simulated in MATLAB.

Figure 1 shows a rendering of the bacteria - arm compound (E. Coli in blue, zinc fingers in red, arms in orange).

**Figure 1.** Rendering of bacteria connected to DNA arms through zinc fingers expressed superficially.

Figure 2 shows relative free energy against arm length. As expected, the longer the arms, the lower is the free energy, and thus the higher is the bonding probability.

**Figure 2.**Effect of arm length on free energy for a constant zinc finger surface distribution.

Figure 3 shows relative free energy against arm length, where the distribution of zinc fingers is randomized. We notice that the distribution of the zinc fingers plays a very important role in the binding as shown in the figure where the distribution is varied randomly, but the trend is more and less the same as when the distribution is assumed to be deterministic.

**Figure 2.**Effect of arm length on free energy for randomly distributed zinc finger distribution.

___________________________________________________________________________________________________________________________________________________________

I.4) Conclusion

The bead approximation and thermodynamic information theoretical framework we proposed effectively captures the challenge of modeling binding probability where different DNA strands compete in binding with a multitude of zinc fingers. The model takes into account the geometry and charge magnitude distribution on the bacteria and the DNA strands.

___________________________________________________________________________________________________________________________________________________________

I.5) References

Chahibi, Y.; Akyildiz, I.F.; Balasubramaniam, S.; Koucheryavy, Y., "Molecular Communication Modeling of Antibody-Mediated Drug Delivery Systems," in Biomedical Engineering, IEEE Transactions on , vol.62, no.7, pp.1683-1695, July 2015 doi: 10.1109/TBME.2015.2400631
Stormo, Gary D., and Yue Zhao. "Determining the specificity of protein–DNA interactions." Nature Reviews Genetics 11.11 (2010): 751-760. APA

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II. iGEM Matchmaker and LabSurfing (Technische Universität Darmstadt)

Design by Warwick iGEM 2015

Difference between revisions of "Team:Warwick/Modelling1"

Latest revision as of 06:52, 18 September 2015

Open Menu

DNA Origami Glue

NTNU Brixells Modelling Collaboration

I.1) Probability of bond formation

I.2) Effect of arm length

I.3) Numerical evaluation

I.4) Conclusion

I.5) References

II. iGEM Matchmaker and LabSurfing (Technische Universität Darmstadt)

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-			<h4>DNA Origami Glue Sequences</h4>
+			<h4>DNA Origami Glue</h4>
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+		<!-- Services List--></div>
-		<div class="fifteen columns noleftmargin">
-			<p>
-			<img src="https://static.igem.org/mediawiki/2015/7/7d/Warwick3armedglue.png" class="pics" alt="" height="700px" width="980px">
-			</p>
+		<div class="fifteen columns noleftmargin">
+<p><img src="https://static.igem.org/mediawiki/2015/4/4f/Warwickbubbles9.png" height="120px" width="800px" border="1px"></p>
-			<h5 class="sidebartitle">Probability of Formation</h5>
-			<p>
-<p style="float: right;"><img src="https://static.igem.org/mediawiki/2015/2/2c/WarwickProbabilityofformation.png" align="right" height="300px" width="440px" border="1px"></p>
-<p style="float: left;"><img src="https://static.igem.org/mediawiki/2015/9/9f/WarwickTableofprob.png" height="200px" width="200px" border="1px"></p>
-			As you can see the probability of a structure fully forming decreases exponentially as the complexity increases. However, even though for larger number of arms there is a very high chance of a structure forming it is unlikely for all the arms to form. Therefore, for our experiments it would be better to focus on using structures with fewer number of arms to save time and money.
-<br>
-<br>
-<br>
-<br>
 <br>
-<br>
-<br>
-<br>
-<p style="float: right;">
-			</p>
+<p>___________________________________________________________________________________________________________________________________________________________</p>
+<h4>NTNU Brixells Modelling Collaboration</h4>
+<p>The collaboration focused on two elements:
+<ol>
+<li>Calculating the <b>probability of formation</b>, including the probability of E. Coli bonding correctly.</p>
+<li>Calculating the <b>effect of arm length on structure changes</b> increases. This has been motivated by the observation that the longer the arms the higher the likelihood  that the e.coli cells can attach because they are less closely packed and such wouldn't get in each others way.</li>
+</ol>
-<p style="float: right;">
+<p>There are many methods to look at this kind of problems, but the most realistic may require huge computational resources. So it is necessary to find an approach that is experimentally relevant, computationally tractable, and reflecting the biophysics underlying Brixells.</p>
-			</p>
-		</div>
-		<div class="clear">
+<p>We, <a href="https://2015.igem.org/Team:NTNU_Trondheim">Team NTNU Trondheim</a> have offered ideas, modeling frameworks, software simulation, and data analysis to answer these questions, based on <i>information theory</i> and <i>thermodynamics</i>. For Problem 1, we have suggested an approach in terms of binding affinity and specificity where the information entropy is used as a measure of specificity. For Problem 2, we have suggested a thermodynamics approach using the Poisson-Boltzmann theory where the zinc fingers and E. Coli are approximated by a set of beads (one large bead for the E. Coli, small beads for each zinc finger, and a rod of beads for the DNA arm. <a href="https://2015.igem.org/Team:Warwick">Team Warwick</a> has been enthusiastic about this approach since it is a stochastic method that they have not considered, and they have provided that us with data related to the geometry of E. Coli-arm formation.</p>
-		</div>
-		<br>
-		<div class="panel">
-			<p>
+<p>___________________________________________________________________________________________________________________________________________________________</p>
-				<b></b><a href="#menu-toggle" class="btn btn-default" id="menu-toggle">Toggle Menu</a></p>
+<h4>I.1) Probability of bond formation</h4>
-			</p>
-		</div>
+<p>The probability of bond formation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>P</mi></math> is equal for all different types of fingers <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>i</mi></math> can be expressed as the product of the probability of formation for each zinc finger <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>P</mi><mi>i</mi></msub> </math>:</p>
-	</div><!-- end main content-->
+<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
-    <!-- SIDEBAR-->
+       <mrow>
+         <mi>P</mi>
+         <mo stretchy="false">=</mo>
+         <mrow>
+          <munder>
+           <mo stretchy="false">∏</mo>
+           <mi>i</mi>
+          </munder>
+          <msub>
+           <mi>P</mi>
+           <mi>i</mi>
+          </msub>
+         </mrow>
+        </mrow>
+      </math>
+<p>___________________________________________________________________________________________________________________________________________________________</p>
+<p>This is a clear measure of how strong the structure is.</p>
+<p>The probability of formation for a zinc finger <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>i</mi></math> is expressed as a function of the specificity.</p>
+ <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
+       <mrow>
+         <msub>
+          <mi>P</mi>
+          <mi>i</mi>
+         </msub>
+         <mo stretchy="false">=</mo>
+         <mfrac>
+          <msub>
+           <mi>S</mi>
+           <mi>i</mi>
+          </msub>
+          <mrow>
+           <mi>W</mi>
+           <mrow>
+            <mo fence="true" stretchy="false">(</mo>
+            <mrow>
+             <msub>
+              <mi>S</mi>
+              <mi>i</mi>
+             </msub>
+            </mrow>
+            <mo fence="true" stretchy="false">)</mo>
+           </mrow>
+          </mrow>
+         </mfrac>
+        </mrow>
+      </math>
+<p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>W</mi></math> is the product log function and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>S</mi><mi>i</mi></msub></math> is the specificity which is the information theoretical "mutual information", and is expressed as follows:</p>
+<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
+       <mrow>
+         <mrow>
+          <msub>
+           <mi>S</mi>
+           <mi>i</mi>
+          </msub>
+          <mo stretchy="false">=</mo>
+          <mrow>
+           <munder>
+            <mo stretchy="false">∑</mo>
+            <mi>j</mi>
+           </munder>
+           <mfrac>
+            <msub>
+             <mi>K</mi>
+             <mrow>
+              <mi>i</mi>
+              <mi>,</mi>
+              <mi>j</mi>
+             </mrow>
+            </msub>
+            <mrow>
+             <munder>
+              <mo stretchy="false">∑</mo>
+              <mi>j</mi>
+             </munder>
+             <msub>
+              <mi>K</mi>
+              <mrow>
+               <mi>i</mi>
+               <mi>,</mi>
+               <mi>j</mi>
+              </mrow>
+             </msub>
+            </mrow>
+           </mfrac>
+          </mrow>
+         </mrow>
+         <mi>ln</mi>
+         <mrow>
+          <mo fence="true" stretchy="false">(</mo>
+          <mrow>
+           <mfrac>
+            <msub>
+             <mi>K</mi>
+             <mrow>
+              <mi>i</mi>
+              <mi>,</mi>
+              <mi>j</mi>
+             </mrow>
+            </msub>
+            <mrow>
+             <mo fence="true" stretchy="false">⟨</mo>
+             <mrow>
+              <msub>
+               <mi>K</mi>
+               <mrow>
+                <mi>i</mi>
+                <mi>,</mi>
+                <mi>j</mi>
+               </mrow>
+              </msub>
+             </mrow>
+             <mo fence="true" stretchy="false">⟩</mo>
+            </mrow>
+           </mfrac>
+          </mrow>
+          <mo fence="true" stretchy="false">)</mo>
+         </mrow>
+        </mrow>
+      </math>
+<p>That calculates the specificity of one zinc finger, i.e. how well a zinc finger binds specifically to an arm when it is competing with other zinc fingers. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">            <msub>
+             <mi>K</mi>
+             <mrow>
+              <mi>i</mi>
+              <mi>,</mi>
+              <mi>j</mi>
+             </mrow>
+            </msub></math> is the binding affinity of zinc finger <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>i</mi></math> specifically to arm <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>j</mi></math>.</p>
+<p>The binding affinity is related to the binding probability <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">         <msubsup>
+          <mi>P</mi>
+          <mrow>
+           <mi>i</mi>
+           <mi>,</mi>
+           <mi>j</mi>
+          </mrow>
+          <mi mathvariant="italic">bound</mi>
+         </msubsup></math> according to the following equation:</p>
+<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
+       <mrow>
+         <msubsup>
+          <mi>P</mi>
+          <mrow>
+           <mi>i</mi>
+           <mi>,</mi>
+           <mi>j</mi>
+          </mrow>
+          <mi mathvariant="italic">bound</mi>
+         </msubsup>
+         <mo stretchy="false">=</mo>
+         <mfrac>
+          <mrow>
+           <mo fence="true" stretchy="false">[</mo>
+           <mrow>
+            <msub>
+             <mi>F</mi>
+             <mi>i</mi>
+            </msub>
+           </mrow>
+           <mo fence="true" stretchy="false">]</mo>
+          </mrow>
+          <mrow>
+           <mrow>
+            <mo fence="true" stretchy="false">[</mo>
+            <mrow>
+             <msub>
+              <mi>F</mi>
+              <mi>i</mi>
+             </msub>
+            </mrow>
+            <mo fence="true" stretchy="false">]</mo>
+           </mrow>
+           <mo stretchy="false">+</mo>
+           <msub>
+            <mi>K</mi>
+            <mrow>
+             <mi>i</mi>
+             <mi>,</mi>
+             <mi>j</mi>
+            </mrow>
+           </msub>
+          </mrow>
+         </mfrac>
+        </mrow>
+      </math>
+<p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">         <msub>
+          <mi>A</mi>
+          <mi>i</mi>
+         </msub></math> is the the arm <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>i</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">         <msub>
+          <mi>F</mi>
+          <mi>j</mi>
+         </msub></math>  is the the finger <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>j</mi></math>, which are governed by the following ligand-binding reaction:</p>
+	<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
+	       <mrow>
+	         <mrow>
+	          <msub>
+	           <mi>A</mi>
+	           <mi>i</mi>
+	          </msub>
+	          <mo stretchy="false">+</mo>
+	          <msub>
+	           <mi>F</mi>
+	           <mi>j</mi>
+	          </msub>
+	         </mrow>
+	         <mo stretchy="false">⇔</mo>
+	         <msub>
+	          <mi>A</mi>
+	          <mi>i</mi>
+	         </msub>
+	         <msub>
+	          <mi>F</mi>
+	          <mi>j</mi>
+	         </msub>
+	        </mrow>
+	      </math>
+<p> </p>
+<p>___________________________________________________________________________________________________________________________________________________________</p>
+<h4>I.2) Effect of arm length</h4>
+<p>In order to account for the arm length. It is important to express the binding affinity to the geometry of the system consisting of one arm and one zinc finger. We dropp the indices <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>i</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>j</mi></math> in the following for simplicity. </p>
+	  <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
+	         <mrow>
+	           <msub>
+	            <mi>K</mi>
+	            <mrow>
+	             <mi>i</mi>
+	             <mi>,</mi>
+	             <mi>j</mi>
+	            </mrow>
+	           </msub>
+	           <mo stretchy="false">=</mo>
+	           <msup>
+	            <mi>e</mi>
+	            <mfrac>
+	             <mrow>
+	              <mi mathvariant="normal">-Δ</mi>
+	              <msub>
+	               <mi>G</mi>
+	              </msub>
+	             </mrow>
+	             <mi mathvariant="italic">RT</mi>
+	            </mfrac>
+	           </msup>
+	          </mrow>
+	        </math>
+<p>We approximate the bacteria, the zinc fingers, and the DNA arm by a set of beads. The bacteria becomes one large sphere, the zinc finger one small sphere, and the DNA arm becomes a rod consisting of a cascade of beads.</p>
+<p>Using these assumptions, the free energy can be calculated from geometrical and electrochemical considerations. In other words, knowing the positions, radii, and charge magnitudes on the beads, the binding free energy of the system is directly calculated as follows.</p>
+	   <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
+	         <mrow>
+	           <mi mathvariant="normal">Δ</mi>
+	           <mrow>
+	            <mi>G</mi>
+	            <mo stretchy="false">=</mo>
+	            <mrow>
+	             <msup>
+	              <mi>G</mi>
+	              <mi mathvariant="italic">bound</mi>
+	             </msup>
+	             <mo stretchy="false">−</mo>
+	             <msup>
+	              <mi>G</mi>
+	              <mi mathvariant="italic">unbound</mi>
+	             </msup>
+	            </mrow>
+	           </mrow>
+	          </mrow>
+	        </math>
+<p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">	             <msup>
+	              <mi>G</mi>
+	              <mi mathvariant="italic">bound</mi>
+	             </msup></math> is the bound free energy expressed as:</p>
+<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
+       <mrow>
+         <msup>
+          <mi>G</mi>
+          <mi mathvariant="italic">bound</mi>
+         </msup>
+         <mo stretchy="false">=</mo>
+         <mrow>
+          <munderover>
+           <mo stretchy="false">∑</mo>
+           <mtable>
+            <mtr>
+             <mtd>
+              <mrow>
+               <mi>m</mi>
+               <mi>,</mi>
+               <mrow>
+                <mi>n</mi>
+                <mo stretchy="false">=</mo>
+                <mn>1</mn>
+               </mrow>
+              </mrow>
+             </mtd>
+            </mtr>
+            <mtr>
+             <mtd>
+              <mrow>
+               <mi>n</mi>
+               <mo stretchy="false">≠</mo>
+               <mi>m</mi>
+              </mrow>
+             </mtd>
+            </mtr>
+           </mtable>
+           <mrow>
+            <mi>M</mi>
+            <mo stretchy="false">+</mo>
+            <mi>N</mi>
+           </mrow>
+          </munderover>
+          <mrow>
+           <mo fence="true" stretchy="false">(</mo>
+           <mrow>
+            <mrow>
+             <msub>
+              <mi>S</mi>
+              <mrow>
+               <mi>m</mi>
+               <mi>,</mi>
+               <mi>n</mi>
+              </mrow>
+             </msub>
+             <mo stretchy="false">+</mo>
+             <msub>
+              <mi>V</mi>
+              <mrow>
+               <mi>m</mi>
+               <mi>,</mi>
+               <mi>n</mi>
+              </mrow>
+             </msub>
+             <mo stretchy="false">+</mo>
+             <msub>
+              <mi>E</mi>
+              <mrow>
+               <mi>m</mi>
+               <mi>,</mi>
+               <mi>n</mi>
+              </mrow>
+             </msub>
+            </mrow>
+           </mrow>
+           <mo fence="true" stretchy="false">)</mo>
+          </mrow>
+         </mrow>
+        </mrow>
+      </math>
+<p>and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">	             <msup>
+	              <mi>G</mi>
+	              <mi mathvariant="italic">unbound</mi>
+	             </msup></math> is the unbound free energy expressed as:</p>
+ <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
+       <mrow>
+         <msup>
+          <mi>G</mi>
+          <mi mathvariant="italic">unbound</mi>
+         </msup>
+         <mo stretchy="false">=</mo>
+         <mrow>
+          <mrow>
+           <munderover>
+            <mo stretchy="false">∑</mo>
+            <mtable>
+             <mtr>
+              <mtd>
+               <mrow>
+                <mi>m</mi>
+                <mi>,</mi>
+                <mrow>
+                 <mi>n</mi>
+                 <mo stretchy="false">=</mo>
+                 <mn>1</mn>
+                </mrow>
+               </mrow>
+              </mtd>
+             </mtr>
+             <mtr>
+              <mtd>
+               <mrow>
+                <mi>n</mi>
+                <mo stretchy="false">≠</mo>
+                <mi>m</mi>
+               </mrow>
+              </mtd>
+             </mtr>
+            </mtable>
+            <mi>N</mi>
+           </munderover>
+           <mrow>
+            <mo fence="true" stretchy="false">(</mo>
+            <mrow>
+             <mrow>
+              <msub>
+               <mi>S</mi>
+               <mrow>
+                <mi>m</mi>
+                <mi>,</mi>
+                <mi>n</mi>
+               </mrow>
+              </msub>
+              <mo stretchy="false">+</mo>
+              <msub>
+               <mi>V</mi>
+               <mrow>
+                <mi>m</mi>
+                <mi>,</mi>
+                <mi>n</mi>
+               </mrow>
+              </msub>
+              <mo stretchy="false">+</mo>
+              <msub>
+               <mi>E</mi>
+               <mrow>
+                <mi>m</mi>
+                <mi>,</mi>
+                <mi>n</mi>
+               </mrow>
+              </msub>
+             </mrow>
+            </mrow>
+            <mo fence="true" stretchy="false">)</mo>
+           </mrow>
+          </mrow>
+          <mo stretchy="false">+</mo>
+          <mrow>
+           <munderover>
+            <mo stretchy="false">∑</mo>
+            <mtable>
+             <mtr>
+              <mtd>
+               <mrow>
+                <mi>m</mi>
+                <mi>,</mi>
+                <mrow>
+                 <mi>n</mi>
+                 <mo stretchy="false">=</mo>
+                 <mrow>
+                  <mi>N</mi>
+                  <mo stretchy="false">+</mo>
+                  <mn>1</mn>
+                 </mrow>
+                </mrow>
+               </mrow>
+              </mtd>
+             </mtr>
+             <mtr>
+              <mtd>
+               <mrow>
+                <mi>n</mi>
+                <mo stretchy="false">≠</mo>
+                <mi>m</mi>
+               </mrow>
+              </mtd>
+             </mtr>
+            </mtable>
+            <mrow>
+             <mi>M</mi>
+             <mo stretchy="false">+</mo>
+             <mi>N</mi>
+            </mrow>
+           </munderover>
+           <mrow>
+            <mo fence="true" stretchy="false">(</mo>
+            <mrow>
+             <mrow>
+              <msub>
+               <mi>S</mi>
+               <mrow>
+                <mi>m</mi>
+                <mi>,</mi>
+                <mi>n</mi>
+               </mrow>
+              </msub>
+              <mo stretchy="false">+</mo>
+              <msub>
+               <mi>V</mi>
+               <mrow>
+                <mi>m</mi>
+                <mi>,</mi>
+                <mi>n</mi>
+               </mrow>
+              </msub>
+              <mo stretchy="false">+</mo>
+              <msub>
+               <mi>E</mi>
+               <mrow>
+                <mi>m</mi>
+                <mi>,</mi>
+                <mi>n</mi>
+               </mrow>
+              </msub>
+             </mrow>
+            </mrow>
+            <mo fence="true" stretchy="false">)</mo>
+           </mrow>
+          </mrow>
+         </mrow>
+        </mrow>
+      </math>
+<p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"> m</math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">n</math> are bead indices, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"> M</math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">N</math> are the number of beads in each subsystem.</p>
+             <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"> <mi>R</mi>
+            <mrow>
+             <mi>m</mi>
+             <mi>,</mi>
+             <mi>n</mi>
+            </mrow></math> is the distance between two beads.              <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"> 	                     <msub>
+	                      <mi mathvariant="normal">ρ</mi>
+	                      <mi>m</mi>
+	                     </msub></math> is the radius of a bead <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"> m</math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"> 	                     <msub>
+	                      <mi mathvariant="normal">q</mi>
+	                      <mi>m</mi>
+	                     </msub></math> is the charge magnitude of a bead <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"> m</math>.
+<p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">          <msub>
+          <mi>E</mi>
+          <mrow>
+           <mi>m</mi>
+           <mi>,</mi>
+           <mi>n</mi>
+          </mrow>
+         </msub></math> is the electrostatic energy potential expressed as:</p>
+<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
+       <mrow>
+         <msub>
+          <mi>E</mi>
+          <mrow>
+           <mi>m</mi>
+           <mi>,</mi>
+           <mi>n</mi>
+          </mrow>
+         </msub>
+         <mo stretchy="false">=</mo>
+         <mfrac>
+          <mrow>
+           <msub>
+            <mi>q</mi>
+            <mi>m</mi>
+           </msub>
+           <msub>
+            <mi>q</mi>
+            <mi>n</mi>
+           </msub>
+          </mrow>
+          <mrow>
+           <mn>8</mn>
+           <mi mathvariant="normal">π</mi>
+           <mi mathvariant="normal">ϵ</mi>
+           <msubsup>
+            <mi>R</mi>
+            <mrow>
+             <mi>m</mi>
+             <mi>,</mi>
+             <mi>n</mi>
+            </mrow>
+            <mn>2</mn>
+           </msubsup>
+          </mrow>
+         </mfrac>
+        </mrow>
+      </math>
+<p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">	          <msub>
+	           <mi>V</mi>
+	           <mrow>
+	            <mi>m</mi>
+	            <mi>,</mi>
+	            <mi>n</mi>
+	           </mrow>
+	          </msub></math> is the van Der Waals force potential expressed as:</p>
+	<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
+	       <mrow>
+	         <mrow>
+	          <msub>
+	           <mi>V</mi>
+	           <mrow>
+	            <mi>m</mi>
+	            <mi>,</mi>
+	            <mi>n</mi>
+	           </mrow>
+	          </msub>
+	          <mo stretchy="false">=</mo>
+	          <mfrac>
+	           <mrow>
+	            <mo stretchy="false">−</mo>
+	            <mi>A</mi>
+	           </mrow>
+	           <mn>6</mn>
+	          </mfrac>
+	         </mrow>
+	         <mrow>
+	          <mo fence="true" stretchy="false">(</mo>
+	          <mrow>
+	           <mrow>
+	            <mfrac>
+	             <mrow>
+	              <mn>2</mn>
+	              <msub>
+	               <mi mathvariant="normal">ρ</mi>
+	               <mi>m</mi>
+	              </msub>
+	              <msub>
+	               <mi mathvariant="normal">ρ</mi>
+	               <mi>n</mi>
+	              </msub>
+	             </mrow>
+	             <mrow>
+	              <msubsup>
+	               <mi>R</mi>
+	               <mrow>
+	                <mi>m</mi>
+	                <mi>,</mi>
+	                <mi>n</mi>
+	               </mrow>
+	               <mn>2</mn>
+	              </msubsup>
+	              <mo stretchy="false">−</mo>
+	              <msup>
+	               <mrow>
+	                <mo fence="true" stretchy="false">(</mo>
+	                <mrow>
+	                 <mrow>
+	                  <msub>
+	                   <mi mathvariant="normal">ρ</mi>
+	                   <mi>m</mi>
+	                  </msub>
+	                  <mo stretchy="false">+</mo>
+	                  <msub>
+	                   <mi mathvariant="normal">ρ</mi>
+	                   <mi>n</mi>
+	                  </msub>
+	                 </mrow>
+	                </mrow>
+	                <mo fence="true" stretchy="false">)</mo>
+	               </mrow>
+	               <mn>2</mn>
+	              </msup>
+	             </mrow>
+	            </mfrac>
+	            <mo stretchy="false">+</mo>
+	            <mfrac>
+	             <mrow>
+	              <mn>2</mn>
+	              <msub>
+	               <mi mathvariant="normal">ρ</mi>
+	               <mi>m</mi>
+	              </msub>
+	              <msub>
+	               <mi mathvariant="normal">ρ</mi>
+	               <mi>n</mi>
+	              </msub>
+	             </mrow>
+	             <mrow>
+	              <msubsup>
+	               <mi>R</mi>
+	               <mrow>
+	                <mi>m</mi>
+	                <mi>,</mi>
+	                <mi>n</mi>
+	               </mrow>
+	               <mn>2</mn>
+	              </msubsup>
+	              <mo stretchy="false">−</mo>
+	              <msup>
+	               <mrow>
+	                <mo fence="true" stretchy="false">(</mo>
+	                <mrow>
+	                 <mrow>
+	                  <msub>
+	                   <mi mathvariant="normal">ρ</mi>
+	                   <mi>m</mi>
+	                  </msub>
+	                  <mo stretchy="false">−</mo>
+	                  <msub>
+	                   <mi mathvariant="normal">ρ</mi>
+	                   <mi>n</mi>
+	                  </msub>
+	                 </mrow>
+	                </mrow>
+	                <mo fence="true" stretchy="false">)</mo>
+	               </mrow>
+	               <mn>2</mn>
+	              </msup>
+	             </mrow>
+	            </mfrac>
+	            <mo stretchy="false">+</mo>
+	            <mrow>
+	             <mi>ln</mi>
+	             <mrow>
+	              <mo fence="true" stretchy="false">(</mo>
+	              <mrow>
+	               <mfrac>
+	                <mrow>
+	                 <msubsup>
+	                  <mi>R</mi>
+	                  <mrow>
+	                   <mi>m</mi>
+	                   <mi>,</mi>
+	                   <mi>n</mi>
+	                  </mrow>
+	                  <mn>2</mn>
+	                 </msubsup>
+	                 <mo stretchy="false">−</mo>
+	                 <msup>
+	                  <mrow>
+	                   <mo fence="true" stretchy="false">(</mo>
+	                   <mrow>
+	                    <mrow>
+	                     <msub>
+	                      <mi mathvariant="normal">ρ</mi>
+	                      <mi>m</mi>
+	                     </msub>
+	                     <mo stretchy="false">+</mo>
+	                     <msub>
+	                      <mi mathvariant="normal">ρ</mi>
+	                      <mi>n</mi>
+	                     </msub>
+	                    </mrow>
+	                   </mrow>
+	                   <mo fence="true" stretchy="false">)</mo>
+	                  </mrow>
+	                  <mn>2</mn>
+	                 </msup>
+	                </mrow>
+	                <mrow>
+	                 <msubsup>
+	                  <mi>R</mi>
+	                  <mrow>
+	                   <mi>m</mi>
+	                   <mi>,</mi>
+	                   <mi>n</mi>
+	                  </mrow>
+	                  <mn>2</mn>
+	                 </msubsup>
+	                 <mo stretchy="false">−</mo>
+	                 <msup>
+	                  <mrow>
+	                   <mo fence="true" stretchy="false">(</mo>
+	                   <mrow>
+	                    <mrow>
+	                     <msub>
+	                      <mi mathvariant="normal">ρ</mi>
+	                      <mi>m</mi>
+	                     </msub>
+	                     <mo stretchy="false">−</mo>
+	                     <msub>
+	                      <mi mathvariant="normal">ρ</mi>
+	                      <mi>n</mi>
+	                     </msub>
+	                    </mrow>
+	                   </mrow>
+	                   <mo fence="true" stretchy="false">)</mo>
+	                  </mrow>
+	                  <mn>2</mn>
+	                 </msup>
+	                </mrow>
+	               </mfrac>
+	              </mrow>
+	              <mo fence="true" stretchy="false">)</mo>
+	             </mrow>
+	            </mrow>
+	           </mrow>
+	          </mrow>
+	          <mo fence="true" stretchy="false">)</mo>
+	         </mrow>
+	        </mrow>
+	      </math>
+<p>and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">          <msub>
+           <mi>S</mi>
+           <mrow>
+            <mi>m</mi>
+            <mi>,</mi>
+            <mi>n</mi>
+           </mrow>
+          </msub></math> is the pair solvent energy expressed as:</p>
+ <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
+       <mrow>
+         <mrow>
+          <msub>
+           <mi>S</mi>
+           <mrow>
+            <mi>m</mi>
+            <mi>,</mi>
+            <mi>n</mi>
+           </mrow>
+          </msub>
+          <mo stretchy="false">=</mo>
+          <mfrac>
+           <mn>1</mn>
+           <mrow>
+            <mn>8</mn>
+            <mi mathvariant="normal">π</mi>
+           </mrow>
+          </mfrac>
+         </mrow>
+         <mrow>
+          <mo fence="true" stretchy="false">(</mo>
+          <mrow>
+           <mrow>
+            <mfrac>
+             <mn>1</mn>
+             <msub>
+              <mi mathvariant="normal">ε</mi>
+              <mn>0</mn>
+             </msub>
+            </mfrac>
+            <mo stretchy="false">−</mo>
+            <mfrac>
+             <mn>1</mn>
+             <mi mathvariant="normal">ε</mi>
+            </mfrac>
+           </mrow>
+          </mrow>
+          <mo fence="true" stretchy="false">)</mo>
+         </mrow>
+         <mfrac>
+          <mrow>
+           <msub>
+            <mi>q</mi>
+            <mi>m</mi>
+           </msub>
+           <msub>
+            <mi>q</mi>
+            <mi>n</mi>
+           </msub>
+          </mrow>
+          <msub>
+           <mi>f</mi>
+           <mrow>
+            <mi>m</mi>
+            <mi>,</mi>
+            <mi>n</mi>
+           </mrow>
+          </msub>
+         </mfrac>
+        </mrow>
+      </math>
+	<p>with</p>
+    <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
+          <mrow>
+            <msub>
+             <mi>f</mi>
+             <mrow>
+              <mi>m</mi>
+              <mi>,</mi>
+              <mi>n</mi>
+             </mrow>
+            </msub>
+            <mo stretchy="false">=</mo>
+            <msqrt>
+             <mrow>
+              <mrow>
+               <msubsup>
+                <mi>R</mi>
+                <mrow>
+                 <mi>m</mi>
+                 <mi>,</mi>
+                 <mi>n</mi>
+                </mrow>
+                <mn>2</mn>
+               </msubsup>
+               <mo stretchy="false">+</mo>
+               <msub>
+                <mi mathvariant="normal">ρ</mi>
+                <mi>m</mi>
+               </msub>
+              </mrow>
+              <msub>
+               <mi mathvariant="normal">ρ</mi>
+               <mi>n</mi>
+              </msub>
+              <msup>
+               <mi>e</mi>
+               <mrow>
+                <mo stretchy="false">−</mo>
+                <msub>
+                 <mi>g</mi>
+                 <mrow>
+                  <mi>m</mi>
+                  <mi>,</mi>
+                  <mi>n</mi>
+                 </mrow>
+                </msub>
+               </mrow>
+              </msup>
+             </mrow>
+            </msqrt>
+           </mrow>
+         </math>
+		 <p>and</p>
+  	<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
+  	       <mrow>
+  	         <msub>
+  	          <mi>g</mi>
+  	          <mrow>
+  	           <mi>m</mi>
+  	           <mi>,</mi>
+  	           <mi>n</mi>
+  	          </mrow>
+  	         </msub>
+  	         <mo stretchy="false">=</mo>
+  	         <mfrac>
+  	          <msub>
+  	           <mi>R</mi>
+  	           <mrow>
+  	            <mi>m</mi>
+  	            <mi>,</mi>
+  	            <mi>n</mi>
+  	           </mrow>
+  	          </msub>
+  	          <mrow>
+  	           <mn>4</mn>
+  	           <msub>
+  	            <mi mathvariant="normal">ρ</mi>
+  	            <mi>m</mi>
+  	           </msub>
+  	           <msub>
+  	            <mi mathvariant="normal">ρ</mi>
+  	            <mi>n</mi>
+  	           </msub>
+  	          </mrow>
+  	         </mfrac>
+  	        </mrow>
+  	      </math>
+	  <p>___________________________________________________________________________________________________________________________________________________________</p>
+<p>Table 1 provides the remaining variable values.</p>
+<div class="panel panel-default" style="max-width:600px">
+      <!-- Default panel contents -->
+      <div class="panel-heading">Table 1 - System parameters</div>
+      <!-- Table -->
+      <table class="table">
+        <thead>
+          <tr>
+            <th>Description</th>
+            <th>Symbol</th>
+            <th>Value</th>
+          </tr>
+        </thead>
+        <tbody>
+          <tr>
+            <th scope="row">Zinc finger radius</th>
+            <td><math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
+          <msub>
+            <mi>q</mi>
+            <mi mathvariant="italic">zinc</mi>
+           </msub>
+         </math> </td>
+            <td><math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
+          <mrow>
+            <mrow>
+             <mn>2</mn>
+             <mo stretchy="false">⋅</mo>
+             <mn>1.60217662</mn>
+             <mo stretchy="false">⋅</mo>
+             <msup>
+              <mn>10</mn>
+              <mrow>
+               <mo stretchy="false">−</mo>
+               <mn>19</mn>
+              </mrow>
+             </msup>
+            </mrow>
+            <mi>C</mi>
+           </mrow>
+         </math></td>
+          </tr>
+          <tr>
+            <th scope="row">Base pair equivalent bead radius</th>
+            <td><math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
+          <msub>
+            <mi>R</mi>
+            <mi mathvariant="italic">bead</mi>
+           </msub>
+         </math></td>
+            <td><math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
+          <mrow>
+            <mn>1.5</mn>
+            <mi mathvariant="italic">bp</mi>
+            <mrow>
+             <mo fence="true" stretchy="false">(</mo>
+             <mrow>
+              <mrow>
+               <mn>1</mn>
+               <mrow>
+                <mi mathvariant="italic">bp</mi>
+                <mo stretchy="false">=</mo>
+                <mn>350</mn>
+               </mrow>
+               <mi mathvariant="italic">pm</mi>
+              </mrow>
+             </mrow>
+             <mo fence="true" stretchy="false">)</mo>
+            </mrow>
+           </mrow>
+         </math> </td>
+          </tr>
+          <tr>
+            <th scope="row"> DNA arm length </th>
+            <td><math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
+          <msub>
+            <mi>L</mi>
+            <mi mathvariant="italic">arm</mi>
+           </msub>
+         </math> </td>
+            <td><math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
+          <mrow>
+            <mn>150</mn>
+            <mrow>
+             <mi mathvariant="italic">bp</mi>
+             <mo stretchy="false">−</mo>
+             <mn>1500</mn>
+            </mrow>
+            <mi mathvariant="italic">bp</mi>
+           </mrow>
+         </math> </td>
+          </tr>
+          <tr>
+            <th scope="row"> DNA arm bead charge magnitude </th>
+            <td>  <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
+          <msub>
+            <mi>N</mi>
+            <mi mathvariant="italic">zinc</mi>
+           </msub>
+         </math></td>
+            <td><math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
+          <mrow>
+            <mrow>
+             <mrow>
+              <mo stretchy="false">−</mo>
+              <mn>2</mn>
+             </mrow>
+             <mo stretchy="false">⋅</mo>
+             <mn>1.60217662</mn>
+             <mo stretchy="false">⋅</mo>
+             <msup>
+              <mn>10</mn>
+              <mrow>
+               <mo stretchy="false">−</mo>
+               <mn>19</mn>
+              </mrow>
+             </msup>
+            </mrow>
+            <mi>C</mi>
+           </mrow>
+         </math></td>
+          </tr>
+          <tr>
+            <th scope="row"> Real gas constant </th>
+            <td> <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
+          <mi>R</mi>
+         </math></td>
+            <td><math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
+          <mn>8.3144598</mn>
+         </math> </td>
+          </tr>
+          <tr>
+            <th scope="row"> Temperature </th>
+            <td><math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
+          <mi>T</mi>
+         </math> </td>
+            <td><math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
+          <mrow>
+            <mn>300</mn>
+            <mi>K</mi>
+           </mrow>
+         </math>  </td>
+          </tr>
+          <tr>
+            <th scope="row"> Absolute permittivity</th>
+            <td> <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
+          <mi mathvariant="normal">ε</mi>
+         </math></td>
+            <td> <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
+          <mrow>
+            <mrow>
+             <mn>8.854187817</mn>
+             <mo stretchy="false">⋅</mo>
+             <msup>
+              <mn>10</mn>
+              <mrow>
+               <mo stretchy="false">−</mo>
+               <mn>12</mn>
+              </mrow>
+             </msup>
+            </mrow>
+            <mrow>
+             <mi>F</mi>
+             <mo stretchy="false">/</mo>
+             <mi>m</mi>
+            </mrow>
+           </mrow>
+         </math></td>
+          </tr>
+          <tr>
+            <th scope="row"> Relative permittivity of water </th>
+            <td><math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
+          <msub>
+            <mi mathvariant="normal">ε</mi>
+            <mi>r</mi>
+           </msub>
+         </math> </td>
+            <td> <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
+          <mn>88</mn>
+         </math></td>
+          </tr>
+          <tr>
+            <th scope="row">Hamaker coefficient of nucleotide  </th>
+            <td> <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
+          <mi>A</mi>
+         </math> </td>
+            <td> <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
+          <mrow>
+            <mn>6</mn>
+            <mo stretchy="false">⋅</mo>
+            <msup>
+             <mn>10</mn>
+             <mrow>
+              <mo stretchy="false">−</mo>
+              <mn>10</mn>
+             </mrow>
+            </msup>
+           </mrow>
+         </math>  </td>
+          </tr>
+           <tr>
+             <th scope="row">Dielectric constant of water</th>
+             <td> <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
+          <mi mathvariant="normal">ϵ</mi>
+         </math>  </td>
+             <td> <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
+          <mn>80.4</mn>
+         </math> </td>
+           </tr>
+        </tbody>
+      </table>
+    </div>
+<p>___________________________________________________________________________________________________________________________________________________________</p>
+<h4>I.3) Numerical evaluation</h4>
+<p>We have simulated the effect of arm length on free energy (low free energy implies high binding probability) using the Born salvation free model, pair van der Waals energy, and electrostatic potential, described above. However, there are some parameters that we are not sure about (Equivalent charge and Hamacker coefficient of Zinc fingers, equivalent charge and Hamacker coefficient of base pair) that depend on charge distribution of the DNA strand, which should be determined experimentally as boundary conditions. Alternatively, one could consider each base pair with independent charge.</p>
+<p>The systems have been simulated in MATLAB.</p>
+<p>Figure 1 shows a rendering of the bacteria - arm compound (E. Coli in blue, zinc fingers in red, arms in orange).</p>
+<figure>
+  <img class="img-responsive" style="max-width:500px" src="https://static.igem.org/mediawiki/2015/f/f4/NTNU_Trondheim_collaboration_rendering.png">
+  <figcaption><b>Figure 1.</b> Rendering of bacteria connected to DNA arms through zinc fingers expressed superficially.</figcaption>
+</figure>
+<p>Figure 2 shows relative free energy against arm length. As expected, the longer the arms, the lower is the free energy, and thus the higher is the bonding probability.</p>
+<figure>
+  <img class="img-responsive" style="max-width:500px" src="https://static.igem.org/mediawiki/2015/a/a0/NTNU_Trondheim_collaboration_deterministic.png">
+  <figcaption><b>Figure 2.</b>Effect of arm length on free energy for a constant zinc finger surface distribution.</figcaption>
+</figure>
+<p>Figure 3 shows relative free energy against arm length, where the distribution of zinc fingers is randomized. We notice that the distribution of the zinc fingers plays a very important role in the binding as shown in the figure where the distribution is varied randomly, but the trend is more and less the same as when the distribution is assumed to be deterministic.
+</p>
+<figure>
+  <img class="img-responsive" style="max-width:500px" src="https://static.igem.org/mediawiki/2015/e/e8/NTNU_Trondheim_collaboration_stochastic.png">
+  <figcaption><b>Figure 2.</b>Effect of arm length on free energy for randomly distributed zinc finger distribution.</figcaption>
+</figure>
+<p>___________________________________________________________________________________________________________________________________________________________</p>
+<h4>I.4) Conclusion</h4>
+<p>The bead approximation and thermodynamic information theoretical framework we proposed effectively captures the challenge of modeling binding probability where different DNA strands compete in binding with a multitude of zinc fingers. The model takes into account the geometry and charge magnitude distribution on the bacteria and the DNA strands.</p>
+<p>___________________________________________________________________________________________________________________________________________________________</p>
+<h4>I.5) References</h4>
+<p>
+<ul>
+	<li>Chahibi, Y.; Akyildiz, I.F.; Balasubramaniam, S.; Koucheryavy, Y., "Molecular Communication Modeling of Antibody-Mediated Drug Delivery Systems," in Biomedical Engineering, IEEE Transactions on , vol.62, no.7, pp.1683-1695, July 2015
+doi: 10.1109/TBME.2015.2400631</li>
+	<li>Stormo, Gary D., and Yue Zhao. "Determining the specificity of protein–DNA interactions." Nature Reviews Genetics 11.11 (2010): 751-760.
+APA
+</li>
+</ol>
+</p>
+<p>___________________________________________________________________________________________________________________________________________________________</p>
+<h3>II. iGEM Matchmaker and LabSurfing (<i>Technische Universität Darmstadt</i>)</h3>
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