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.
Similar content being viewed by others
References
Chen M, von Mikecz A (2005) Exp Cell Res 305:51
Nel AE, Mädler L, Velegol D, Xia T, Hoek EMV, Somasundaran P, Klaessig F, Castranova V, Thompson M (2009) Nat Meter 8:543
Conner SD, Schmid SL (2003) Nature 422:37
Dohertyand GJ, McMahon HT (2009) Annu Rev Biochem 78:857
Zhao F, Zhang Y, Liu Y, Chang X, Chen C, Zhao Y (2011) Small 7:1322
Stayton I, Winiarz J, Shannon K, Ma Y (2009) Anal Bioanal Chem 394:1595
Xing X, He X, Peng J (2005) J Nanosci Nanotechnol 5:1688
Cho EC, Xie J, Wurm PA, Xia Y (2009) Nano Lett 9:1080
Limbach LK, Li Y, Grass RN, Brunner TJ, Hintermann MA, Muller M, Gunther D, Stark WJ (2005) Environ Sci Technol 39:9370
Schübbe S, Cavelius C, Schumann C, Koch M, Kraegeloh A (2010) Adv Eng Mater 12:417
Schübbe S, Schumann C, Cavelius C, Koch M, Mueller T, Kraegeloh A (2012) Chem Mater 24:914
Heller MJ (2002) Annu Rev Biomed Eng 4:129
Slowing II, Vivero-Escoto JL, Wu CW, Lin VSY (2008) Adv Drug Deliv Rev 60:1278
Kneuer C, Sameti M, Bakowsky U, Schiestel T, Schirra H, Schmidt H, Lehr CM (2000) Bioconjugate Chem 11:926
Luo D, Han E, Belcheva N, Saltzman WM (2004) J Contr Release 95:333
Radu DR, Lai CY, Jeftinija K, Rowe EW, Jeftinija S, Lin VSY (2004) J Am Chem Soc 126:1321
Roy I, Ohulchanskyy TY, Bharali DJ, Pudavar HE, Mistretta RA, Kaur N, Prasad P (2005) Proc Natl Acad Sci USA 102:279
Bharali DJ, Klejbor I, Stachowiak EK, Dutta P, Roy I, Kaur N, Bergey EJ, Prasad PN, Stachowiak MK (2005) Proc Natl Acad Sci USA 102:11539
Xia T, Kovochich M, Liong M, Meng H, Kabehie S, George S, Zink JI, Nel AE (2009) ACS Nano 3:3273
Slowing II, Trewyn BG, Giri S, Lin VSY (2007) Adv Funct Mater 17:1225
Chen JF, Ding HM, Wang JX, Shao L (2004) Biomaterials 25:723
Vivero-Escoto JL, Slowing II, Trewyn BG, Lin VSY (2010) Small 6:1952
Schumann C, Schübbe S, Cavelius C, Kraegeloh A (2012) J Biophotonics 5:117
Lieber M, Todaro G, Smith B, Szakal A, Nelson-Rees W (1976) Int J Cancer 17:62
Peckys D, de Jonge N (2011) Nano Lett 11:1733
Malvindi MA, Brunetti V, Vecchio G, Galeone A, Cingolanib R, Pompa PP (2012) Nanoscale 4:486
Sackmann E (1994) FEBS Lett 346:3
Foldvari M, Gesztes A, Mezei M (1990) J Microencapsul 7:479
Puyal C, Milhaud P, Bienvenüe A, Philippot JR (1995) Eur J Biochem 228:697
Mohanraj VJ, Barnes TJ, Prestidge CA (2010) Int J Pharmaceut 392:285
Ash WL, Zlomislic MR, Oloo EO, Tieleman DP (2004) Biochim Biophys Acta 1666:158
Berkowitz ML, Bostick DL, Pandit S (2006) Chem Rev 106:1527
Siu SWI, Vácha R, Jungwirth P, Böckmann RA (2008) J Chem Phys 128:125103
Wong-Ekkabut J, Baoukina S, Triampo W, Tang IM, Tieleman DP, Monticelli L (2008) Nat Nanotechnol 3:363
Marrink SJ, Risselada HJ, Yefimov S, Tieleman DP, de Vries AH (2007) J Phys Chem B 111:7812
Shelley JC, Shelley MY, Reeder RC, Bandyopadhyay S, Moore PB, Klein ML (2001) J Phys Chem B 105:9785
Wallace EJ, Sansom MSP (2007) Nano Lett 7:1923
Shinoda W, DeVane R, Klein ML (2010) J Phys Chem B 114:6836
Girifalco LA, Hodak M, Lee RS (2000) Phys Rev B 62:13104
Hodak M, Girifalco LA (2001) Chem Phys Lett 350:405
Cox BJ, Thamwattana N, Hill JM (2007) Proc Roy Soc A 463:461
Cox BJ, Thamwattana N, Hill JM (2007) Proc Roy Soc A 463:477
Baowan D, Cox BJ, Hill JM (2012) J Mol Model 18:549
Qian D, Liu WK, Ruoff RS (2001) J Phys Chem B 105:10753
Liu P, Zhang YW, Lu C (2005) J Appl Phys 97:094313
Patwardhan SV, Patwardhan G, Perry CC (2007) J Mater Chem 17:2875
Still WC, Tempczyk A, Hawley RC, Hendrickson T (1990) J Am Chem Soc 112:6127
Hirschfelder JO, Curtiss CF, Bird RB (1954) Molecular theory of gases and liquids. Wiley, New York
Cruz-Chu ER, Aksimentiev A, Schulten K (2006) J Phys Chem B 110:21497
Dorota N, Leen T, Dominique L, Johan M, Peter H (2010) Part Fibre Toxicol 7:39
Zhuravlev LT (2000) Colloid Surface Physicochem Eng Aspect 173:1
Makimura D, Metin C, Kabashima T, Matsuoka T, Nguyen QP, Miranda CR (2010) J Mater Sci 45:5084
Petrache HI, Feller SE, Nagle JF (1997) Biophys J 70:2237
Chu Z, Huang Y, Taob Q, Li Q (2011) Nanoscale 3:3291
Lin JH, Baker NA, McCammon JA (2002) Biophys J 83:1374
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
Corresponding author
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
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
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
where
and
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
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
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
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
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
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
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
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
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.
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00894-013-1784-1