1. Thermodynamic and Mechanical Focusing: Non-Ideality and Non-Linearity at Interfaces
Luka Pocivavsek1, Kathleen D. Cao1, Steven Danauskas1, Enrique Cerda3,
Ka Yee C. Lee1, Jaroslaw Majewski4, Mati Meron2, Binhua Lin2
1James Franck Institute and Department of Chemistry, University of Chicago, Chicago, IL USA
2James Franck Institute and CARS, University of Chicago, IL 60637, USA
3Departamento de Física and CIMAT, Universidad de Santiago, Av. Ecuador 3493, Santiago, Chile
4Manuel Lujan Neutron Scattering Center, Los Alamos National Laboratory, Los Alamos, NM USA
Introduction
Stability of Enriched Interfacial Glycerol Layer
The Wrinkle to Fold Transition
Non-ideal interfacial mixing
probed by X-ray liquid reflectivity
Interfaces are ubiquitous in biology: from the earliest points in development where a sphere of cells undergoes geometric transitions to form the first germ layers to the inside of our blood vessels, airways, and lungs [1]. One of the key components of biological interfaces are lipid membranes [2]. Our work has centered on elucidating the mechanical response of model cell membranes, lipid monolayers at liquid interfaces, under compression. We developed a generalized continuum mechanics model that captures both the linear and non-linear response of supported membranes from microns thick to nanometers thin [3]. When membrane thickness is reduced to a point that the bending stiffness is O(10-100kbT), the case with lipids, coupling between the molecules in the membrane and underlying fluid subphase effects the mechanical response of the membrane. This coupling can be probed by studying lipid monolayers on different binary subphase solutions like water and glycerol mixtures. Glycerol is the simplest poly-alcohol. The motivation for studying glycerol is that the extracellular environment of all tissues is made up of simple polyols like sugars or larger polymer versions like glycopolymers [4]. It is beginning to be appreciated that the local mechanical environment of a cell plays a key role in its biology [4]. Here we present a detailed physical study of the interesting and complex mechanical and thermodynamic behavior of the lipid/glycerol/ water interface using x-ray and neutron scattering.
 ~ 2cm A1
10 m thick polyester sheet on water A0
Specular XR is limited to probing samples that remain stable on the order of hours. To probe the thermal stability of the lipid/glycerol/water interface we used the Grazing Incidence X-Ray Off-Specular (GIXOS) technique.
Ideal mixing of binary fluids like water and poly- alcohols, e.g. glycerol, is well established in bulk solution. However this ideal behavior breaks down in the interfacial solvation layer next to the lipid monolayer. Using x-ray reflectivity, we probe the compositional degree and spatial extent of non-ideal mixing between water and glycerol next to a model lung surfactant layer.
Specular X-ray Reflectivity Spectra
ChemMatCARS, ID15, APS, ANL
Liquid Surface
Reflection Geometry - q is wavevector transfer [6]
The figure on the left shows the compression of a ten micron thick polyester membrane (8cm  15cm) on water (though any Newtonian fluid, e.g. glycerol, works). At low compressions the membrane responds through continuous sinusoidal low amplitude wrinkles. The wrinkles cause elastic stresses to be evenly distributed throughout the membrane. Upon further compression (middle panel) the symmetry begins to be broken and wrinkles in the middle grow larger while others decay. The final state of the system is a fold (last panel), where all elastic stresses are focused into a very localized region of the membrane. These transitions are geometrically similar to the folding transitions observed in lipid monolayers (figure on right).
A. XR spectra for DPPC:POPG 7:3 compressed to =30mN/m at 25oC on different subphase solutions: H2O (blue), 20:80 H2O:glycerol (magenta), 40:60 (black), and 64:36 (red). Reflectivity was taken in specular condition ( = ) with Oxford point detector. The data is multiplied by Fresnel (qz/qc)1/4 and plotted against qz/qc, qc = Å-1 (H2O), Å-1 (20:80), Å-1 (40:60), Å-1 (64:36).
B. and C. show that the position of the low and high q fringes shifts to the left as glycerol is added to the subphase. The fringes are points of maximal destructive interference between reflected waves at the interfaces. The first minimum is proportional to the lower bound on the total interface thickness: lmin  2/qz. The shift to lower q indicates that the interface on glycerol becomes thicker.
Scaling Relations
U = Energy
B = Bending Stiffness
K = Substrate Stiffness
L = Sheet Length
A = Amplitude
l = Arc Length of Fold
∆ = Displacement
Ø = Angle of the Tangent
to the Horizontal
 = Wavelength
inextensibility
Wrinkles - Linear Case
Fold - Nonlinear Case
GIXOS was performed in an off-specular geometry ( = 0.3o/5.2mrad) using a linear detector at two angle conditions: ==0.09o/1.5mrad and = 0.09o/1.5mrad while =4o/69mrad. With these angles, the entire reflectivity was collected in a few minutes. To compare the specular reflectivity and GIXOS reflectivity, the diffuse components must be divided out:
Our Model System
A lipid monolayer composed of a 7:3 mixture of DPPC (1,2-dipalmitoyl-sn-glycero-3-phosphocholine) and POPG (1-palmitoyl-2-oleoyl-sn-glycero-3-[phospho-rac-(1-glycerol)] (sodium salt)) on an aqueous subphase is used as a model for lung surfactant.
DPPC
POPG
capillary wave contribution dominated by
Transition Energetically Favored
Stochastic Model Independent XR Fitting
fold
wrinkle
dimensionless displacement (d = /)
dimensionless
energy
d ≈ 0.3  ∆c ≈ /3
For model-independent fitting [5], we perform an electron density profile (EDP) search. An initial EDP is generated based on the average scattering length density (SLD) of the sample, and the estimated sample thickness. The EDP is generated by selecting a number of boxes, typically on the order of ~0.5-2 Å per box, a smoothing parameter (σ), and a δ for each box (D. and E.). The contribution of each box to the EDP is then smeared by the Gaussian error function (erf). Each point in the generated EDP is treated as a layer, and the reflectivity is then calculated by iterating through each of the points of the EDP by the Parratt recursion method.
Master Formula Relating Structure Factor to EDP
StochFit EDPs
The normalized GIXOS structure factor at 25oC both on water and 64:36 H2O:glycerol subphases is in agreement with the specular data (A and B). This confirms that the technique is sensitive to the surface structure factor. At 37oC, GIXOS shows that the interface on 64% glycerol remains stable with the spectrum at 37oC overlaying almost identically that at 25oC (B). On water however the interface shows thinning and a shift of the surface structure factor at 37oC to the left (A).
Monolayers like DPPC:POPG 7:3 form folds into the subphase at large compressions. The folding instability is linked to the general wrinkle-to-fold model [3]. Interestingly the amplitude of the folds is very sensitive to subphase composition. On water folds are O(1-10m) and on binary mixtures of water/glycerol an order of magnitude larger O(100m). D. E.
7:3 DPPC:POPG, 25ºC on water,
∏ ~ 71 mN/m
Scale:
References
A wrinkled surface should be stable against further compression by a third of its initial wavelength. Beyond this, the surface geometry becomes unstable towards the new localized folded state, folding is thermodynamically favored at high compression.
Glycerol enriched layer seen at interface!
[1] Biological Physics of the Developing Embryo, G. Forgacs and S.A. Newman, 2005, Cambridge University Press.
[2] Mechanics of the Cell, D. Boal, 2002, Cambridge University Press.
[3] L. Pocivavsek, R. Dellsy, A. Kern, S. Johnson, B. Lin, K. Y. C. Lee, E. Cerda, Science, 320, 912 (2008).
[4] Cell Mechanics, Y.L. Wang and D.E. Discher ed., 2007, Elsevier.
[5] Danauskas et al., StochFit, Journal of Applied Crystallography, 41, 6, (2008) in press. StochFit is available through open source at stochfit.sourceforge.net
[6] O.G. Shpyrko, Experimental X-ray Studies of Liquid Surfaces, Ph.D. Thesis, Harvard University, 2004.
Fast Surface X-ray Diffraction
Conclusions
Interfacial Glycerol Enrichment probed by Neutron Reflectivity
- surface structural studies using surface sensitive X-ray and neutron techniques have shown that though ideally mixed in the bulk simple fluid solutions like water and glycerol phase separate (de-mix) near the lipid interface, with an enrichment of the glycerol.
DPPC d64
Neutrons interact with nuclei unlike X-rays which scatter from atomic electrons. One power of neutrons is the strong dependence of scattering on the type of isotope. In particular deuterons and protons are radically different scatterers. This study probed DPPC at two different deuteration conditions DPPC d64 with tails deuterated and DPPC d72 with the entire lipid deuterated. The subphase was a solution of D2O (SLD = 6.3x10-6 Å-2) and hydrogenated glycerol (SLD = 0.6x10-6 Å-2). The strong contrast between glycerol and D2O allowed us to determine the extent of glycerol enrichement at the interface. The profiles show Chi maps of the best fit lipid parameters with varying amounts of glycerol enrichment in a 10Å layer underneath the headgroups. The solid black (spectrum) and red (profile) lines are the best fits. The dotted red line in the profile is the fit with bulk solvent in the interfacial layer.
DPPC d72
Grazing Incidence X-ray diffraction was performed in a special pinhole geometry using the two-dimensional Pilatus detector.
D2O
Further Reading
H2O
64% glycerol
W. Lu, et al., Phys. Rev. Lett. 89, (2002).
D. G. Schultz, et al., J. Phys. Chem. B 110, (2006).
K.Y.C. Lee, Annu. Rev. Phys. Chem., 59, 771 (2008).
T. A. Witten, Rev. Mod. Phys., 79, 643 (2007).
E. Cerda, L. Mahadevan, Phys. Rev. Lett.,90, (2003)
L. Bourdieu, J. Daillant, et al., Phys. Rev. Lett. 72, 1502 (1992).
A. Gopal et al., J. Phys. Chem. B 110, (2006).
20:80
The work has allowed us to connect our general model for interfacial compaction with interfacial thermodynamics. Continuum mechanics parameters like bending stiffness for thin systems like lipid monolayers are intricately linked to the underlying interfacial structure which as we show includes the solvation shell. This has deep implications about the role of water and simply hydrogen bonding fluids on lipid encapsulated structures like biological cells.
qxy
qxy
Funding
The red curves are on 25oC and the blue on 37oC. Note that in-plane crystalline structure is lost in the case of a water subphase but remains in the case of glycerol at the higher temperature. This is in agreement with the GIXOS data showing thinning of the interface on water at higher temperature.
64:36
The NSF Inter-American Materials Collaboration: Chicago-Chile, University of Chicago Materials Research Science and Engineering Center program of the NSF, US-Israel Binational Foundation, University of Chicago MSTP, March of Dimes, NSF/DOE for ChemMatCARS.
(SPEAR, Lujan Center, LANSCE, Los Alamos National Lab)