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        <h2>Calculation</h2>

<h3>Destruction the liposome</h3>


<br> Following mathematical formula reveals proves that smaller liposomes are more stable. From this calculation result, we think it is possible to collapse liposomes if we provide with energy by trigger. </br>


<br> ・Hypothesis</br> We assume that a vesicle of large radius existing underwater changes to the plural vesicles of small radius (fig 1).</br> 

&nbsp;The free energy of each states F1: large radius, and F2: small radius, probably have the following relationship (fig 2).</br>
&nbsp;In our case, the vesicle is basically small state because a small radius is more stable.The decisive factor is, however, that there is the energy gap “δ” that must exceed to change the size of vesicle.
In our calculation we have not had the possibility to calculate δ, but to calculate the free energy in each radius. With this values of energy and radius the energy gap "δ" can be deducted.</br>

<table> <tr> <td> <img src="http://openwetware.org/images/7/73/Fig.1.jpg" width="400"> </td> <td> <img src="http://openwetware.org/images/2/2e/Fig.2.jpg" width="400"> </td> </tr> <tr> <td> Fig.1 Assumed state of vesicle in the system </td> <td> Fig.2 Assumed relation of each free energy </td> </tr> </table>


</br></br>

・Assumptions</br> N<sub>0</sub> units of surfactant molecule exist in a solution. We assume that N units of them participate in the formation of the vesicle. One vesicle consists of n units of surfactant molecule.</br> In the Lipo-Hanabi project, we adopt DOPC (C<sub>44</sub>H<sub>84</sub>NO<sub>8</sub>P)as the surfactant molecule. </br> A : Surface area of DOPC molecule =0.6 [nm<sup>2</sup>].</br> V : Volume of water =10<sup>-4</sup> [m<sup>3</sup>]. </br> m : Molecular weight of DOPC =785 [g/mol]. </br> T : Temperature T=300[K].</br> Boltzmann’s constant and Planck's constant are as follows.</br>


<div align="center"><img src="http://openwetware.org/images/2/27/Cal-1.png" width="293" height="102"></div>


</br>

・Results</br> Free energy <i>F</i> of the vesicle is given as follows*.(*Reference)</br>


<div align="center"><img src="http://openwetware.org/images/5/57/Cal-2-1.png" width="200" height="70"></div>


where,</br>


<div align="center"><img src="http://openwetware.org/images/c/c7/Cal-3-1.png" width="424" height="139"></div>


(pf)<sub>bulk</sub> is a partition function of a single discrete molecule underwater, and (pf)<sub>vesicle</sub> is a partition function of the vesicle.</br> Therefore, free energy F is</br>



<div align="center"><img src="http://openwetware.org/images/7/72/Cal-4.png" width="529" height="98"></div>



We assume that N, T, and V are constants:</br>



<div align="center"><img src="http://openwetware.org/images/5/50/Cal-5-1.png" width="400" height="100"></div>


Because α= 0 when N<sub>0</sub>=N</br>




<div align="center"><img src="http://openwetware.org/images/e/e4/Cal-6.png" width="337" height="93"></div>



Here, we approximate,</br>


<div align="center"><img src="http://openwetware.org/images/b/ba/Cal-puls.png" width="254" height="42"></div>


N/n=X is the number of the vesicles in the whole space. Therefore the free energy is derived as follows,</br>



<div align="center"><img src="http://openwetware.org/images/1/1f/Cal-7.png" width="400" height="42"></div>


Here, assume a vesicle with radius 100μm is decomposed into 10 and 100 vesicles. Calculated free energy of each situation is:</br>



<div align="center"><img src="http://openwetware.org/images/a/a9/Cal-8.png" width="342" height="107"></div>



<img src="http://openwetware.org/images/c/c1/Fig.3.jpg" width="400"></br> Fig.3 The relation of each free energy by calculation<br> </br>

</br></br>

・Discussion</br> When all molecules participate in the formation of the vesicle, the free energy becomes a one-tenth time and the number of vesicle increase 10 times. Smaller radius vesicles are more stable.Therefore, it is possible to destroy vesicles using trigger. <br><br>

・Reference</br> Y. Suezaki, "Shishitsu maku no butsuri (Physics of lipid membrane)", Kyusyu Univ. Press, 2007(in Japanese)



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