Biomod/2013/Sendai/calcuation

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        <h2>Calculation</h2>

<h3>Destroy 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 destroy liposomes if we give them energy by trigger. </br>

</br>

・Hypothesis</br> Assume that a vesicle of large radius existing underwater changes to the plural vesicles of small radius (fig 1). The free energy of each sates F1: large radius, and F2: small radius, probably have following relationship (fig 2).
 In this case, the vesicle is basically small states because being a small radius is more stable, but there is the energy gap “δ” that must exceed to change the size of vesicle.
In this time we do not calculate δ, but calculate the free energy in each radius. From these energy and radius, we demand the relationship of them.</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>

・Assumed situation</br> N<sub>0</sub> surfactant molecules units exist in solution, and N units of them participate in the formation of the vesicle. One vesicle consists of n surfactant molecules units.</br> In this time,we use DOPC (C<sub>44</sub>H<sub>84</sub>NO<sub>8</sub>P). </br> A : Surface area of one molecule =0.6 [nm<sup>2</sup>].</br> V : Volume of the water =10<sup>-4</sup> [m<sup>3</sup>]. </br> m : Molecular weight of DOPC =785 [g/mol]. </br> T : Temperature of the system T=300[K].</br> Boltzmann’s constant and Planck's constant use the follows.</br>


<div align="center"><img src="http://openwetware.org/images/2/29/Cal-00.png" width="293" height="102"></div>


</br>

・Calculation result</br> Free energy “F” of the vesicle is given as follows*.(*Reference)</br>


<div align="center"><img src="http://openwetware.org/images/c/c8/Cal-01.png" width="200" height="70"></div>


In this time,</br>


<div align="center"><img src="http://openwetware.org/images/b/bf/Cal-02.png" width="424" height="139"></div>


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



<div align="center"><img src="http://openwetware.org/images/6/6f/Cal-03.png" width="529" height="98"></div>



In this calculation, we assume that N, T, and V are constant,</br>



<div align="center"><img src="http://openwetware.org/images/5/51/Cal-04.png" width="400" height="118"></div>


<div align="center"><img src="http://openwetware.org/images/d/d8/Cal-ima-04-02.png" width="500" height="90"></div>



Because of α= 0 at the moment of N<sub>0</sub> – N = 0</br>




<div align="center"><img src="http://openwetware.org/images/6/6a/Cal-05.png" width="337" height="93"></div>



This time we approximate,</br>


<div align="center"><img src="http://openwetware.org/images/7/7c/Cal-06.png" width="254" height="42"></div>


N/n=X means the number of the vesicles in whole of this system. Therefore we calculate free energy by using this,</br>



<div align="center"><img src="http://openwetware.org/images/7/77/Cal-07.png" width="400" height="42"></div>


Here, a vesicle which radius is 100μm changes to each 10 and 100 vesicles, calculate each free energy,</br>



<div align="center"><img src="http://openwetware.org/images/6/6c/Cal-08.png" width="342" height="107"></div>



From these results, we get the follow (fig 3).</br>

<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>

From fig 3, free energy is smaller when the radius of vesicle is small.</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> Yukio Suezaki "Physics of lipid film"



<br>

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