We perform a simulation study of the diffusion of small solutes in the confined domains imposed by inverse bicontinuous cubic phases for the primitive, diamond, and gyroid symmetries common to many lipid/water mesophase systems employed in experiments. For large diffusing domains, the long-time diffusion coefficient shows universal features when the size of the confining domain is renormalized by the Gaussian curvature of the triply periodic minimal surface. When bottlenecks are widely present, they become the most relevant factor for transport, regardless of the connectivity of the cubic phase.
REFERENCES
1.
E. M.
Landau
and J. P.
Rosenbusch
, “Lipidic cubic phases: A novel concept for the crystallization of membrane proteins
,” Proc. Natl. Acad. Sci. U. S. A.
93
, 14532
–14535
(1996
).2.
C.
Speziale
, L. S.
Manni
, C.
Manatschal
, E. M.
Landau
, and R.
Mezzenga
, “A macroscopic H+ and Cl− ions pump via reconstitution of EcClC membrane proteins in lipidic cubic mesophases
,” Proc. Natl. Acad. Sci. U. S. A.
113
, 7491
–7496
(2016
).3.
C.
Fong
, T.
Le
, and C. J.
Drummond
, “Lyotropic liquid crystal engineering–ordered nanostructured small molecule amphiphile self-assembly materials by design
,” Chem. Soc. Rev.
41
, 1297
–1322
(2012
).4.
R.
Mezzenga
, P.
Schurtenberger
, A.
Burbidge
, and M.
Michel
, “Understanding foods as soft materials
,” Nat. Mater.
4
, 729
–740
(2005
).5.
X.
Mulet
, B. J.
Boyd
, and C. J.
Drummond
, “Advances in drug delivery and medical imaging using colloidal lyotropic liquid crystalline dispersions
,” J. Colloid Interface Sci.
393
, 1
–20
(2013
).6.
Z. A.
Almsherqi
, S. D.
Kohlwein
, and Y.
Deng
, “Cubic membranes: A legend beyond the flatland* of cell membrane organization
,” J. Cell Biol.
173
, 839
–844
(2006
).7.
E. L.
Snapp
, R. S.
Hegde
, M.
Francolini
, F.
Lombardo
, S.
Colombo
, E.
Pedrazzini
, N.
Borgese
, and J.
Lippincott-Schwartz
, “Formation of stacked er cisternae by low affinity protein interactions
,” J. Cell Biol.
163
, 257
–269
(2003
).8.
Y.
Deng
, M.
Marko
, K. F.
Buttle
, A.
Leith
, M.
Mieczkowski
, and C. A.
Mannella
, “Cubic membrane structure in amoeba (chaos carolinensis) mitochondria determined by electron microscopic tomography
,” J. Struct. Biol.
127
, 231
–239
(1999
).9.
S. T.
Hyde
and G. E.
Schröder-Turk
, “Geometry of interfaces: Topological complexity in biology and materials
,” Interface Focus
2
, 529
(2012
).10.
N.
Garti
, P.
Somasundaran
, and R.
Mezzenga
, Self-Assembled Supramolecular Architectures: Lyotropic Liquid Crystals
(John Wiley & Sons
, 2012
), Vol. 3.11.
J.
Briggs
, H.
Chung
, and M.
Caffrey
, “The temperature-composition phase diagram and mesophase structure characterization of the monoolein/water system
,” J. Phys. II
6
, 723
–751
(1996
).12.
R.
Templer
, J.
Seddon
, N.
Warrender
, A.
Syrykh
, Z.
Huang
, R.
Winter
, and J.
Erbes
, “Inverse bicontinuous cubic phases in 2:1 fatty acid/phosphatidylcholine mixtures. The effects of chain length, hydration, and temperature
,” J. Phys. Chem. B
102
, 7251
–7261
(1998
).13.
D. C.
Turner
, Z.-G.
Wang
, S. M.
Gruner
, D. A.
Mannock
, and R. N.
McElhaney
, “Structural study of the inverted cubic phases of di-dodecyl alkyl-β-d-glucopyranosyl-rac-glycerol
,” J. Phys. II
2
, 2039
–2063
(1992
).14.
J.
Barauskas
, M.
Johnsson
, and F.
Tiberg
, “Self-assembled lipid superstructures: Beyond vesicles and liposomes
,” Nano Lett.
5
, 1615
–1619
(2005
).15.
R.
Negrini
and R.
Mezzenga
, “Diffusion, molecular separation, and drug delivery from lipid mesophases with tunable water channels
,” Langmuir
28
, 16455
–16462
(2012
).16.
R.
Mezzenga
, C.
Meyer
, C.
Servais
, A. I.
Romoscanu
, L.
Sagalowicz
, and R. C.
Hayward
, “Shear rheology of lyotropic liquid crystals: A case study
,” Langmuir
21
, 3322
–3333
(2005
).17.
A. I.
Tyler
, H. M.
Barriga
, E. S.
Parsons
, N. L.
McCarthy
, O.
Ces
, R. V.
Law
, J. M.
Seddon
, and N. J.
Brooks
, “Electrostatic swelling of bicontinuous cubic lipid phases
,” Soft Matter
11
, 3279
–3286
(2015
).18.
R.
Negrini
and R.
Mezzenga
, “ph-responsive lyotropic liquid crystals for controlled drug delivery
,” Langmuir
27
, 5296
–5303
(2011
).19.
I.
Martiel
, N.
Baumann
, J. J.
Vallooran
, J.
Bergfreund
, L.
Sagalowicz
, and R.
Mezzenga
, “Oil and drug control the release rate from lyotropic liquid crystals
,” J. Controlled Release
204
, 78
–84
(2015
).20.
A.
Zabara
and R.
Mezzenga
, “Controlling molecular transport and sustained drug release in lipid-based liquid crystalline mesophases
,” J. Controlled Release
188
, 31
–43
(2014
).21.
R.
Negrini
, W.-K.
Fong
, B. J.
Boyd
, and R.
Mezzenga
, “ph-responsive lyotropic liquid crystals and their potential therapeutic role in cancer treatment
,” Chem. Commun.
51
, 6671
–6674
(2015
).22.
E.
Nazaruk
, P.
Miszta
, S.
Filipek
, E.
Gorecka
, E. M.
Landau
, and R.
Bilewicz
, “Lyotropic cubic phases for drug delivery: Diffusion and sustained release from the mesophase evaluated by electrochemical methods
,” Langmuir
31
, 12753
–12761
(2015
).23.
J.
Clogston
, G.
Craciun
, D.
Hart
, and M.
Caffrey
, “Controlling release from the lipidic cubic phase by selective alkylation
,” J. Controlled Release
102
, 441
–461
(2005
).24.
J.
Clogston
and M.
Caffrey
, “Controlling release from the lipidic cubic phase. Amino acids, peptides, proteins and nucleic acids
,” J. Controlled Release
107
, 97
–111
(2005
).25.
K. W.
Lee
, T.-H.
Nguyen
, T.
Hanley
, and B. J.
Boyd
, “Nanostructure of liquid crystalline matrix determines in vitro sustained release and in vivo oral absorption kinetics for hydrophilic model drugs
,” Int. J. Pharm.
365
, 190
–199
(2009
).26.
S.
Phan
, W.-K.
Fong
, N.
Kirby
, T.
Hanley
, and B. J.
Boyd
, “Evaluating the link between self-assembled mesophase structure and drug release
,” Int. J. Pharm.
421
, 176
–182
(2011
).27.
A.
Zabara
, R.
Negrini
, O.
Onaca-Fischer
, and R.
Mezzenga
, “Perforated bicontinuous cubic phases with ph-responsive topological channel interconnectivity
,” Small
9
, 3602
–3609
(2013
).28.
T. G.
Meikle
, S.
Yao
, A.
Zabara
, C. E.
Conn
, C. J.
Drummond
, and F.
Separovic
, “Predicting the release profile of small molecules from within the ordered nanostructured lipidic bicontinuous cubic phase using translational diffusion coefficients determined by pfg-nmr
,” Nanoscale
9
, 2471
–2478
(2017
).29.
R.
Ghanbari
, S.
Assenza
, A.
Saha
, and R.
Mezzenga
, “Diffusion of polymers through periodic networks of lipid-based nanochannels
,” Langmuir
33
, 3491
–3498
(2017
).30.
M. H.
Jacobs
, Diffusion Processes
(Springer Science & Business Media
, 1967
).31.
B.
Jönsson
, H.
Wennerström
, P.
Nilsson
, and P.
Linse
, “Self-diffusion of small molecules in colloidal systems
,” Colloid Polym. Sci.
264
, 77
–88
(1986
).32.
D. M.
Anderson
and H.
Wennerstroem
, “Self-diffusion in bicontinuous cubic phases, L3 phases, and microemulsions
,” J. Phys. Chem.
94
, 8683
–8694
(1990
).33.
G.
Allaire
, “Homogenization and two-scale convergence
,” SIAM J. Math. Anal.
23
, 1482
–1518
(1992
).34.
R.
Zwanzig
, “Diffusion past an entropy barrier
,” J. Phys. Chem.
96
, 3926
–3930
(1992
).35.
D.
Reguera
and J.
Rubi
, “Kinetic equations for diffusion in the presence of entropic barriers
,” Phys. Rev. E
64
, 061106
(2001
).36.
J.
Kalnin
, E.
Kotomin
, and J.
Maier
, “Calculations of the effective diffusion coefficient for inhomogeneous media
,” J. Phys. Chem. Solids
63
, 449
–456
(2002
).37.
A. M.
Berezhkovskii
, V. Y.
Zitserman
, and S. Y.
Shvartsman
, “Effective diffusivity in periodic porous materials
,” J. Chem. Phys.
119
, 6991
–6993
(2003
).38.
N. S.
Gov
, “Diffusion in curved fluid membranes
,” Phys. Rev. E
73
, 041918
(2006
).39.
Z.
Schuss
, A.
Singer
, and D.
Holcman
, “The narrow escape problem for diffusion in cellular microdomains
,” Proc. Natl. Acad. Sci.
104
, 16098
–16103
(2007
).40.
M.
Wang
and N.
Pan
, “Predictions of effective physical properties of complex multiphase materials
,” Mater. Sci. Eng.: R: Rep.
63
, 1
–30
(2008
).41.
P. S.
Burada
, P.
Hänggi
, F.
Marchesoni
, G.
Schmid
, and P.
Talkner
, “Diffusion in confined geometries
,” ChemPhysChem
10
, 45
–54
(2009
).42.
N.
Ogawa
, “Curvature-dependent diffusion flow on a surface with thickness
,” Phys. Rev. E
81
, 061113
(2010
).43.
S.
Martens
, G.
Schmid
, L.
Schimansky-Geier
, and P.
Hänggi
, “Entropic particle transport: Higher-order corrections to the Fick-Jacobs diffusion equation
,” Phys. Rev. E
83
, 051135
(2011
).44.
C. V.
Valdes
, “Effective diffusion in the region between two surfaces
,” Phys. Rev. E
94
, 022121
(2016
).45.
R.
Hołyst
, D.
Plewczyński
, A.
Aksimentiev
, and K.
Burdzy
, “Diffusion on curved, periodic surfaces
,” Phys. Rev. E
60
, 302
(1999
).46.
D.
Plewczyński
and R.
Hołyst
, “Reorientational angle distribution and diffusion coefficient for nodal and cylindrical surfaces
,” J. Chem. Phys.
113
, 9920
–9929
(2000
).47.
E.
Sanz
and D.
Marenduzzo
, “Dynamic Monte Carlo versus Brownian dynamics: A comparison for self-diffusion and crystallization in colloidal fluids
,” J. Chem. Phys.
132
, 194102
(2010
).48.
H.
Von Schnering
and R.
Nesper
, “Nodal surfaces of fourier series: Fundamental invariants of structured matter
,” Z. Phys. B: Condens. Matter
83
, 407
–412
(1991
).49.
I. L.
Novak
, P.
Kraikivski
, and B. M.
Slepchenko
, “Diffusion in cytoplasm: Effects of excluded volume due to internal membranes and cytoskeletal structures
,” Biophys. J.
97
, 758
–767
(2009
).50.
E. L.
Thomas
, D. M.
Anderson
, C. S.
Henkee
, and D.
Hoffman
, “Periodic area-minimizing surfaces in block copolymers
,” Nature
334
, 598
–601
(1988
).51.
R. M.
Kaufmann
, S.
Khlebnikov
, and B.
Wehefritz-Kaufmann
, “The geometry of the double gyroid wire network: Quantum and classical
,” J. Noncommutative Geom.
6
, 623
–664
(2012
).52.
B.
Halperin
, S.
Feng
, and P. N.
Sen
, “Differences between lattice and continuum percolation transport exponents
,” Phys. Rev. Lett.
54
, 2391
(1985
).53.
G.
Schröder-Turk
, S.
Ramsden
, A.
Christy
, and S.
Hyde
, “Medial surfaces of hyperbolic structures
,” Eur. Phys. J. B
35
, 551
–564
(2003
).54.
A.
Aharony
and D.
Stauffer
, Introduction to Percolation Theory
(Taylor & Francis
, 2003
).55.
R.
Mezzenga
, J.
Ruokolainen
, G. H.
Fredrickson
, E. J.
Kramer
, D.
Moses
, A. J.
Heeger
, and O.
Ikkala
, “Templating organic semiconductors via self-assembly of polymer colloids
,” Science
299
, 1872
–1874
(2003
).56.
L. M.
Antognini
, S.
Assenza
, C.
Speziale
, and R.
Mezzenga
, “Quantifying the transport properties of lipid mesophases by theoretical modelling of diffusion experiments
,” J. Chem. Phys.
145
, 084903
(2016
).57.
D. S.
ViswanathTushar
, T. K.
Ghosh
, D. H. L.
Prasad
, N. V.
Dutt
, and K. Y.
Rani
, Viscosity of Liquids. Theory, Estimation, Experiment, and Data
(Springer
, 2007
).58.
W. M.
Haynes
, D. R.
Lide
, and T. J.
Bruno
, CRC Handbook of Chemistry and Physics
, 95th ed. (CRC Press
, 2014
).59.
J.
Kim
, W.
Lu
, W.
Qiu
, L.
Wang
, M.
Caffrey
, and D.
Zhong
, “Ultrafast hydration dynamics in the lipidic cubic phase: Discrete water structures in nanochannels
,” J. Phys. Chem. B
110
, 21994
–22000
(2006
).60.
L.
Longsworth
, “Diffusion measurements, at 25, of aqueous solutions of amino acids, peptides and sugars
,” J. Am. Chem. Soc.
75
, 5705
–5709
(1953
).61.
C. A.
Lambert
, L. H.
Radzilowski
, and E. L.
Thomas
, “Triply periodic level surfaces as models for cubic tricontinuous block copolymer morphologies
,” Philos. Trans. R. Soc., A
354
, 2009
–2023
(1996
).62.
G. E.
Schröder-Turk
, A.
Fogden
, and S. T.
Hyde
, “Bicontinuous geometries and molecular self-assembly: Comparison of local curvature and global packing variations in genus-three cubic, tetragonal and rhombohedral surfaces
,” Eur. Phys. J. B
54
, 509
–524
(2006
).© 2018 Author(s).
2018
Author(s)
You do not currently have access to this content.