Calculation

Destroy the liposome


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.

・Hypothesis
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.
Fig.1 Assumed state of vesicle in the system Fig.2 Assumed relation of each free energy


・Assumed situation
N0 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.
In this time,we use DOPC (C44H84NO8P).
A : Surface area of one molecule =0.6 [nm2].
V : Volume of the water =10-4 [m3].
m : Molecular weight of DOPC =785 [g/mol].
T : Temperature of the system T=300[K].
Boltzmann’s constant and Planck's constant use the follows.

・Calculation result
Free energy “F” of the vesicle is given as follows*.(*Reference)
In this time,
(pf)bulk is a partition function of the single discrete molecule underwater, and (pf)vesicle is a partition function of the vesicle.
Therefore, free energy “F”,
In this calculation, we assume that N, T, and V are constant,
Because of α= 0 at the moment of N0 – N = 0
This time we approximate,
N/n=X means the number of the vesicles in whole of this system. Therefore we calculate free energy by using this,
Here, a vesicle which radius is 100μm changes to each 10 and 100 vesicles, calculate each free energy,
From these results, we get the follow (fig 3).

Fig.3 The relation of each free energy by calculation

From fig 3, free energy is smaller when the radius of vesicle is small.

・Discussion
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

・Reference
Yukio Suezaki "Physics of lipid film"