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Energetics of liposomes encapsulating silica nanoparticles

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Abstract

Nanoparticles may be taken up into cells via endocytotic processes whereby the foreign particles are encapsulated in vesicles formed by lipid bilayers. After uptake into these endocytic vesicles, intracellular targeting processes and vesicle fusion might cause transfer of the vesicle cargo into other vesicle types, e.g., early or late endosomes, lysosomes, or others. In addition, nanoparticles might be taken up as single particles or larger agglomerates and the agglomeration state of the particles might change during vesicle processing. In this study, liposomes are regarded as simple models for intracellular vesicles. We compared the energetic balance between two liposomes encapsulating each a single silica nanoparticle and a large liposome containing two silica nanoparticles. Analytical expressions were derived that show how the energy of the system depends on the particle size and the distance between the particles. We found that the electrostatic contributions to the total energy of the system are negligibly small. In contrast, the van der Waals term strongly favors arrangements where the liposome snugly fits around the nanoparticle(s). Thus the two separated small liposomes have a more favorable energy than a larger liposome encapsulating two nanoparticles.

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Acknowledgments

This work was supported by a postdoctoral fellowship to DB by the Alexander von Humboldt Foundation. The authors thank Dr. Tihamér Geyer for many helpful discussions and Dr. Michael Hutter for helpful comments on the manuscript.

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Mahidol University, Rama VI Rd, Bangkok, Thailand

    Duangkamon Baowan

  2. Centre of Excellence in Mathematics, CHE, Si Ayutthaya Rd, Bangkok, 10400, Thailand

    Duangkamon Baowan

  3. Nano Cell Interactions Group, INM-Leibniz Institute for New Materials, Campus D2 2, 66123, Saarbruecken, Germany

    Henrike Peuschel & Annette Kraegeloh

  4. Center for Bioinformatics, Campus E2 1, Saarland University, 66123, Saarbruecken, Germany

    Volkhard Helms

Authors
  1. Duangkamon Baowan
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  2. Henrike Peuschel
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  3. Annette Kraegeloh
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  4. Volkhard Helms
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Corresponding author

Correspondence to Duangkamon Baowan.

Appendix

Appendix

The expressions of the interaction energy for both Coulombic and Lennard-Jones potentials are given in this appendix.

A: Electrostatic energy

The electrostatic energy between the inner and the outer head groups and the nanoparticle, utilizing double surface integrals for concentric spheres or for offset of concentric spheres, is given by

$$ \begin{array}{*{20}c} {{Q_1}\left( {a,b} \right)} \hfill & {=\tfrac{{D{\eta_{sh }}(b){e^2}}}{{4\pi }}\left[ {\tfrac{1}{{{\epsilon_0}{\epsilon_r}}}-\tfrac{1}{2}\left( {1-\tfrac{1}{{{\epsilon_r}}}} \right)} \right]\left[ {\tfrac{1}{2}\left( {0.3} \right)(1)L_{1/2}^Q\left( {a,b} \right)} \right.} \hfill \\ {} \hfill & {+\tfrac{1}{2}\left( {-0.3} \right)(1)L_{1/2}^Q\left( {a+0.161,b} \right)+\tfrac{1}{2}\left( {0.3} \right)\left( {-1} \right)L_{1/2}^Q\left( {a,b+0.05} \right)} \hfill \\ {} \hfill & {+\tfrac{1}{2}\left( {-0.3} \right)\left( {-1} \right)L_{1/2}^Q\left( {a+0.161,b+0.05} \right)+\tfrac{1}{2}\left( {0.3} \right)(1)L_{1/2}^Q\left( {a,b+4.336} \right)} \hfill \\ {} \hfill & {+\tfrac{1}{2}\left( {-0.3} \right)(1)L_{1/2}^Q\left( {a+0.161,b+4.336} \right)+\tfrac{1}{2}\left( {0.3} \right)\left( {-1} \right)L_{1/2}^Q\left( {a,b+4.286} \right)} \hfill \\ {} \hfill & {\left. {+\frac{1}{2}\left( {-0.3} \right)\left( {-1} \right)L_{1/2}^Q\left( {a+0.161,b+42.86} \right)} \right],} \hfill \\ \end{array} $$
(A1)

where e denotes an elementary charge, D represents a dangling atom density of 1 nm−2 for both silicon and oxygen and \( L_{1/2}^Q\left( {a,b} \right) \) is given by (4). The rational coefficients come from the proportional content of 1/2 choline and 1/2 phosphate groups in the head group. The charge values are as given in Table 1.

The electrostatic energy between two offset spheres for the SiO2 nanoparticles is given by

$$ \begin{array}{*{20}c} {{Q_2}(a)} \hfill & {=\tfrac{{{D^2}{e^2}}}{{4\pi }}\left[ {\tfrac{1}{{{\epsilon_0}{\epsilon_r}}}-\tfrac{1}{2}\left( {1-\tfrac{1}{{{\epsilon_r}}}} \right)} \right]\left[ {\left( {0.3} \right)\left( {0.3} \right)K_{1/2}^Q\left( {a,a} \right)} \right.} \hfill \\ {} \hfill & { + \left( {-0.3} \right)\left( {-0.3} \right)\mathrm{K}_{1/2}^Q\left( {a+0.161,a+0.161} \right)} \hfill \\ {} \hfill & {\left. {+2\left( {0.3} \right)\left( {-0.3} \right)K_{1/2}^Q\left( {a,a+0.161} \right)} \right],} \hfill \\ \end{array} $$
(A2)

where \( K_{1/2}^Q\left( {a,b} \right) \) is given by (3), and a and a + 0.161 denote the radii of the probability distribution of silicon and oxygen atoms, respectively.

The electrostatic energy between two offset spherical liposomes is given by

$$ \begin{array}{*{20}c} {{Q_3}\left( {a,b} \right)} \hfill & {=Q_3^{*}\left( {b,b+0.05} \right)+Q_3^{*}\left( {b+4.286,b+4.336} \right)} \hfill \\ {} \hfill & {Q_3^{*}\left( {b,b+0.05} \right)+Q_3^{*}\left( {b+4.286,b+4.336} \right)} \hfill \\ {} \hfill & { + 2\mathrm{Q}_3^{**}\left( {b,b+0.05,b+4.286,b+4.336} \right),} \hfill \\ \end{array} $$
(A3)

where

$$ \begin{array}{*{20}c} {Q_3^{*}\left( {a,b} \right)} \hfill & {=\tfrac{{{\eta_{sh }}(a){\eta_{sh }}(b){e^2}}}{{4\pi }}\left[ {\tfrac{1}{{{\epsilon_0}{\epsilon_r}}}-\tfrac{1}{2}\left( {1-\tfrac{1}{{{\epsilon_r}}}} \right)} \right]\left[ {\tfrac{1}{4}(1)(1)K_{1/2}^Q\left( {a,a} \right)} \right.} \hfill \\ {} \hfill & {\left. {+\tfrac{1}{4}\left( {-1} \right)\left( {-1} \right)K_{1/2}^Q\left( {b,b} \right)+2\left( {\tfrac{1}{4}} \right)(1)\left( {-1} \right)K_{1/2}^Q\left( {a,b} \right)} \right],} \hfill \\ \end{array} $$

and

$$ \begin{array}{*{20}c} {Q_3^{**}\left( {a,b} \right)} \hfill & {=\tfrac{{{\eta_{sh }}(a){\eta_{sh }}(d){e^2}}}{{4\pi }}\left[ {\tfrac{1}{{{\epsilon_0}{\epsilon_r}}}-\tfrac{1}{2}\left( {1-\tfrac{1}{{{\epsilon_r}}}} \right)} \right]\left[ {\tfrac{1}{4}(1)(1)K_{1/2}^Q\left( {a,d} \right)} \right.} \hfill \\ {} \hfill & { + \frac{1}{4}\left( {-1} \right)\left( {-1} \right)K_{1/2}^Q\left( {b,c} \right)+\frac{1}{4}(1)\left( {-1} \right)K_{1/2}^Q\left( {a,c} \right)} \hfill \\ {} \hfill & {\left. {+\frac{1}{4}\left( {-1} \right)(1)K_{1/2}^Q\left( {b,d} \right)} \right],} \hfill \\ \end{array} $$

and \( K_{1/2}^Q\left( {a,b} \right) \) is defined by (3).

B: van der Waals energy

The van der Waals energy between a head group and the nanoparticle utilizing a surface integral for SiO2 of radius a and a volume integral for the head group of the inner radius b and of the thickness 0.4 nm is given by

$$ \begin{array}{*{20}c} {{P_1}\left( {a,b} \right)} \hfill & {={\eta_{silica }}{\eta_{vh }}(b)\left[ {\tfrac{1}{6}\left( {-{A_{{Si-{Q_a}}}}{I_3}\left[ {N_n^{LJ}\left( {a,b,0.4} \right)} \right]+{B_{{Si-{Q_a}}}}{I_6}\left[ {N_n^{LJ}\left( {a,b,0.4} \right)} \right]} \right)} \right.} \hfill \\ {} \hfill & {+\tfrac{1}{6}\left( {-{A_{{Si-{Q_o}}}}{I_3}\left[ {N_n^{LJ}\left( {a,b,0.4} \right)} \right]+{B_{{Si-{Q_o}}}}{I_6}\left[ {N_n^{LJ}\left( {a,b,0.4} \right)} \right]} \right)} \hfill \\ {} \hfill & {+\tfrac{2}{6}\left( {-{A_{{O-{Q_a}}}}{I_3}\left[ {N_n^{LJ}\left( {a,b,0.4} \right)} \right]+{B_{{O-{Q_a}}}}{I_6}\left[ {N_n^{LJ}\left( {a,b,0.4} \right)} \right]} \right)} \hfill \\ {} \hfill & {+\left. {\frac{2}{6}\left( {-{A_{{O-{Q_o}}}}{I_3}\left[ {N_n^{LJ}\left( {a,b,0.4} \right)} \right]+{B_{{O-{Q_o}}}}{I_6}\left[ {N_n^{LJ}\left( {a,b,0.4} \right)} \right]} \right)} \right],} \hfill \\ \end{array} $$
(B1)

where A 1−2 and B 1−2 are the Lennard-Jones attractive and repulsive constants, respectively, obtained by the mixing rule. The function \( N_n^{LJ}\left( {a,b,0.4} \right) \) is defined by (11) where n is a positive integer corresponding to the power of the polynomials appearing in integrals I 3 and I 6 defined by (6) and (7). Again, the rational coefficients come from the proportional content of 1/2 choline and 1/2 phosphate groups in the head group, and of 1/3 silicon and 2/3 oxygen atoms in the silica nanoparticle.

The van der Waals energy between the intermediate layer and the nanoparticle utilizing double surface integrals for concentric spheres where the radius of SiO2 (intermediate layer) is assumed to be a (b) is given by

$$ \begin{array}{*{20}c} {{P_2}\left( {a,b} \right)} \hfill & {={\eta_{silica }}{\eta_{si }}(b)\left[ {\tfrac{1}{3}\left( {-{A_{{Si-{N_a}}}}{I_3}\left[ {L_n^{LJ}\left( {a,b} \right)} \right]+{B_{{Si-{N_a}}}}{I_6}\left[ {L_n^{LJ}\left( {a,b} \right)} \right]} \right)} \right.} \hfill \\ {} \hfill & {+\left. {\frac{2}{3}\left( {-{A_{{O-{N_a}}}}{I_3}\left[ {L_n^{LJ}\left( {a,b} \right)} \right]+{B_{{O-{N_a}}}}{I_6}\left[ {L_n^{LJ}\left( {a,b} \right)} \right]} \right)} \right],} \hfill \\ \end{array} $$
(B2)

where \( L_n^{LJ}\left( {a,b} \right) \) is given by (9) and n is a positive integer corresponding to the power of I 3 and I 6 defined by (6) and (7).

The van der Waals energy between the tail group and the nanoparticle utilizing a surface integral for SiO2 of radius a and a volume integral for the tail group of the inner radius b and of the thickness 1.6 nm is given by

$$ \begin{array}{*{20}c} {{P_3}\left( {a,b} \right)} \hfill & {={\eta_{silica }}{\eta_{vt }}(b)\left[ {\tfrac{1}{3}\left( {-{A_{{Si-{C_1}}}}{I_3}\left[ {N_n^{LJ}\left( {a,b,1.6} \right)} \right]+{B_{{Si-{C_1}}}}{I_6}\left[ {N_n^{LJ}\left( {a,b,1.6} \right)} \right]} \right)} \right.} \hfill \\ {} \hfill & {+\left. {\frac{2}{3}\left( {-{A_{{O-{C_1}}}}{I_3}\left[ {N_n^{LJ}\left( {a,b,1.6} \right)} \right]+{B_{{O-{C_1}}}}{I_6}\left[ {N_n^{LJ}\left( {a,b,1.6} \right)} \right]} \right)} \right],} \hfill \\ \end{array} $$
(B3)

where \( N_n^{LJ}\left( {a,b,1.6} \right) \) is defined by (11) with corresponding values of n.

In the case when the inner sphere moves away from the origin to the distance x, the van der Waals energy between a head group and the inner nanoparticle utilizing a surface integral for SiO2 of radius a and a volume integral for the head group of inner radius b and of thickness 0.4 nm is given by

$$ \begin{array}{*{20}c} {{P_4}\left( {a,b} \right)} \hfill & {={\eta_{silica }}{\eta_{vh }}(b)\left[ {\tfrac{1}{6}\left( {-{A_{{Si-{Q_a}}}}{I_3}\left[ {O_n^{LJ}\left( {a,b,0.4} \right)} \right]+{B_{{Si-{Q_a}}}}{I_6}\left[ {O_n^{LJ}\left( {a,b,0.4} \right)} \right]} \right)} \right.} \hfill \\ {} \hfill & {+\tfrac{1}{6}\left( {-{A_{{Si-{Q_o}}}}{I_3}\left[ {O_n^{LJ}\left( {a,b,0.4} \right)} \right]+{B_{{Si-{Q_o}}}}{I_6}\left[ {O_n^{LJ}\left( {a,b,0.4} \right)} \right]} \right)} \hfill \\ {} \hfill & {+\tfrac{2}{6}\left( {-{A_{{O-{Q_a}}}}{I_3}\left[ {O_n^{LJ}\left( {a,b,0.4} \right)} \right]+{B_{{O-{Q_a}}}}{I_6}\left[ {O_n^{LJ}\left( {a,b,0.4} \right)} \right]} \right)} \hfill \\ {} \hfill & {+\left. {\frac{2}{6}\left( {-{A_{{O-{Q_o}}}}{I_3}\left[ {O_n^{LJ}\left( {a,b,0.4} \right)} \right]+{B_{{O-{Q_o}}}}{I_6}\left[ {O_n^{LJ}\left( {a,b,0.4} \right)} \right]} \right)} \right],} \hfill \\ \end{array} $$
(B4)

where \( O_n^{LJ}\left( {a,b,0.4} \right) \) is defined by (12) and n is a positive integer corresponding to the power of the polynomials appearing in integrals I 3 and I 6 defined by (6) and (7).

The van der Waals energy between the intermediate layer and the nanoparticle utilizing double surface integrals for an offset of concentric sphere shown in Fig. 1(d), with the radius of SiO2 (intermediate layer) assumed to be a (b) is given by

$$ \begin{array}{*{20}c} {{P_5}\left( {a,b} \right)} \hfill & {={\eta_{silica }}{\eta_{si }}(b)\left[ {\tfrac{1}{3}\left( {-{A_{{Si-{N_a}}}}{I_3}\left[ {M_n^{LJ}\left( {a,b} \right)} \right]+{B_{{Si-{N_a}}}}{I_6}\left[ {M_n^{LJ}\left( {a,b} \right)} \right]} \right)} \right.} \hfill \\ {} \hfill & {+\left. {\frac{2}{3}\left( {-{A_{{O-{N_a}}}}{I_3}\left[ {M_n^{LJ}\left( {a,b} \right)} \right]+{B_{{O-{N_a}}}}{I_6}\left[ {M_n^{LJ}\left( {a,b} \right)} \right]} \right)} \right],} \hfill \\ \end{array} $$
(B5)

where \( M_n^{LJ}\left( {a,b} \right) \) is given by (10) and n is a positive integer corresponding to the power of I 3 and I 6 defined by (6) and (7).

Once the silica nanoparticle moves away from the origin by distance x, the van der Waals energy between the tail group and the nanoparticle utilizing a surface integral for SiO2 of radius a and a volume integral for the tail group of the inner radius b and of the thickness 1.6 nm is given by

$$ \begin{array}{*{20}c} {{P_6}\left( {a,b} \right)} \hfill & {={\eta_{silica }}{\eta_{vt }}(b)\left[ {\tfrac{1}{3}\left( {-{A_{{Si-{C_1}}}}{I_3}\left[ {O_n^{LJ}\left( {a,b,1.6} \right)} \right]+{B_{{Si-{C_1}}}}{I_6}\left[ {O_n^{LJ}\left( {a,b,1.6} \right)} \right]} \right)} \right.} \hfill \\ {} \hfill & {+\left. {\frac{2}{3}\left( {-{A_{{O-{C_1}}}}{I_3}\left[ {O_n^{LJ}\left( {a,b,1.6} \right)} \right]+{B_{{O-{C_1}}}}{I_6}\left[ {O_n^{LJ}\left( {a,b,1.6} \right)} \right]} \right)} \right],} \hfill \\ \end{array} $$
(B6)

where \( O_n^{LJ}\left( {a,b,1.6} \right) \) is defined by (12) with corresponding values of n.

The van der Waals energy between two offset spheres for the SiO2 nanoparticles is given by

$$ \begin{array}{*{20}c} {{P_7}\left( {a,b} \right)} \hfill & {=\eta_{silica}^2\left[ {\tfrac{1}{9}\left( {-{A_{Si-Si }}{I_3}\left[ {K_n^{LJ}\left( {a,a} \right)} \right]+{B_{Si-Si }}{I_6}\left[ {K_n^{LJ}\left( {a,a} \right)} \right]} \right)} \right.} \hfill \\ {} \hfill & {+\tfrac{4}{9}\left( {-{A_{O-O }}{I_3}\left[ {K_n^{LJ}\left( {b,b} \right)} \right]+{B_{O-O }}{I_6}\left[ {K_n^{LJ}\left( {b,b} \right)} \right]} \right)} \hfill \\ {} \hfill & {+\left. {2\left( {\frac{2}{9}} \right)\left( {-{A_{Si-O }}{I_3}\left[ {K_n^{LJ}\left( {a,b} \right)} \right]+{B_{Si-O }}{I_6}\left[ {K_n^{LJ}\left( {a,b} \right)} \right]} \right)} \right],} \hfill \\ \end{array} $$
(B7)

where \( K_n^{LJ}\left( {a,b} \right) \) is defined by (8) with corresponding values of n appearing in (6) and (7), and a and b represent the radii of the probability distribution of silicon and oxygen atoms, respectively, in silica nanoparticle.

The van der Waals energy between two offset spheres of liposomes encapsulating silica nanoparticles is given by

$$ \begin{array}{*{20}c} {{P_8}\left( {a,b} \right)} \hfill & {={\eta_{sh }}(a){\eta_{sh }}(b)\left[ {\tfrac{1}{4}\left( {-{A_{{{Q_a}-{Q_a}}}}{I_3}\left[ {K_n^{LJ}\left( {a,a} \right)} \right]+{B_{{{Q_a}-{Q_a}}}}{I_6}\left[ {K_n^{LJ}\left( {a,a} \right)} \right]} \right)} \right.} \hfill \\ {} \hfill & {+\tfrac{1}{4}\left( {-{A_{{{Q_o}-{Q_o}}}}{I_3}\left[ {K_n^{LJ}\left( {b,b} \right)} \right]+{B_{{{Q_o}-{Q_o}}}}{I_6}\left[ {K_n^{LJ}\left( {b,b} \right)} \right]} \right)} \hfill \\ {} \hfill & {+\left. {2\left( {\frac{1}{4}} \right)\left( {-{A_{{{Q_a}-{Q_o}}}}{I_3}\left[ {K_n^{LJ}\left( {a,b} \right)} \right]+{B_{{{Q_a}-{Q_o}}}}{I_6}\left[ {K_n^{LJ}\left( {a,b} \right)} \right]} \right)} \right],} \hfill \\ \end{array} $$
(B8)

where \( K_n^{LJ}\left( {a,b} \right) \) is defined by (8) with corresponding values of n appearing in (6) and (7), and a and b denote the radii of choline and phosphate groups, respectively.

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Baowan, D., Peuschel, H., Kraegeloh, A. et al. Energetics of liposomes encapsulating silica nanoparticles. J Mol Model 19, 2459–2472 (2013). https://doi.org/10.1007/s00894-013-1784-1

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