1788
Langmuir 2003, 19, 1788-1793
Anomalous Subdiffusion in Heterogeneous Lipid Bilayers† Timothy V. Ratto‡ and Marjorie L. Longo*,‡,§ Department of Chemical Engineering and Materials Science, and Biophysics Graduate Group, Division of Biological Sciences, University of California, Davis, California 95616 Received July 1, 2002. In Final Form: September 24, 2002 Fluorescence photobleaching recovery (FPR) is commonly used to measure lipid and protein diffusion in cellular membranes. Typically, a model wherein diffusion is constant with time and the mean-squared displacement is directly proportional to time is used to analyze the results; however, in nonhomogeneous systems such as cellular membranes, anomalous subdiffusion may occur. In anomalous subdiffusion, the diffusion coefficient, D, decreases with time and thus the mean-squared displacement is proportional to some power of time less than 1. Although theory predicts that diffusion can be anomalous through protein interactions or obstruction, the complex composition of cellular membranes has made the actual origin and consequences of anomalous diffusion in phospholipid bilayers unclear. In this study, we use atomic force microscopy to detect and measure the amount of the solid phase in supported bilayers that contain coexisting fluid- and solid-phase lipids. Solid-phase domains in bilayers have been shown to act as obstacles to diffusion. We then use FPR to determine the diffusional behavior of the obstructed bilayers. We find that at a solid-phase area fraction of ∼35% diffusion is anomalous at short times and becomes normal at longer times as predicted by theory and Monte Carlo simulations. Increasing the solid-phase area fraction increases the amount of time diffusion is anomalous before becoming normal. The results of this work imply that accurately measuring diffusion in cell membranes requires an understanding of the heterogeneity of the underlying membrane structure.
Introduction Molecular transport in heterogeneous systems comprises an important area of study due to its significance in understanding a wide variety of phenomena in nature. These phenomena include such diverse processes as island diffusion in metals,1 cosmic ray propagation in the interstellar medium,2 water infiltration in porous building materials,3 and lipid and protein diffusion in cellular membranes.4 The last have earned recent attention due to the discovery that diffusion in these systems often does not obey the Einstein relation,
〈r2〉 ) 4Dt
(1)
where 〈r2〉 is the mean squared displacement, D is the diffusion coefficient, and t is time. Instead, the displacement is proportional to some power of time less than 1,
〈r2〉 ) 4DtR
(2)
where R is known as the anomalous diffusion coefficient. Since R < 1 implies that the diffusion coefficient of the mobile particles is decreasing with time, this type of diffusion is known as anomalous subdiffusion and has, in fact, been seen in a variety of other physical systems as well as having been studied quite extensively theoretic* Corresponding author. E-mail:
[email protected]. Telephone: (530) 754-6348. Fax: (530) 752-1031. † Part of the Langmuir special issue entitled The Biomolecular Interface. ‡ Biophysics Graduate Group. § Department of Chemical Engineering and Materials Science. (1) Schlosser, D. C.; Morgenstern, K.; Verheij, L. K.; Rosenfeld, G.; Bensenbacher, F.; Cosma, G. Surf. Sci. 2000, 465, 19-39. (2) Lagutin, A. A.; Nikulin, Y. A.; Uchaikin, V. V. Nucl. Phys. B, Proc. Suppl. 2001, 97, 267-270. (3) Kuntz, M.; Lavallee, P. J. Phys. D: Appl. Phys. 2001, 34, 25472554. (4) Cherry, R. J.; Smith, P. R.; Morrison, I. E. G.; Fernandez, N. FEBS Lett. 1998, 430, 88-91.
ally.3,5-10 From these studies, it has been shown that a number of phenomena can lead to anomalous diffusion, including broad distributions of particle jump times, correlations between diffusing particles, and/or multiple diffusion rates, rather than one representative diffusion coefficient. It has been proposed that in cellular membranes, protein-protein and protein-lipid binding interactions can lead to multiple jump time distributions, and obstruction, due to the presence of immobile proteins or lipid heterogeneities in the cell membrane, can result in a distribution of diffusivities.4,5,7,11 More than one diffusion coefficient is the result of different paths for diffusion, some relatively free of obstruction and some quite convoluted and hindered. The fluid mosaic model of membranes, first formulated in 1972,12 proposed that the cellular membrane was homogeneous, and therefore component lipids and proteins were free to diffuse laterally. In recent years, it has become obvious that cellular membranes are in fact quite heterogeneous structures and biomolecular diffusion across them is more complex than originally proposed. Singleparticle tracking and fluorescence photobleaching recovery experiments on cell membranes have recently detected anomalous subdiffusion.5,13 Because these membranes contain proteins, which can result in binding interactions, it is not clear what role obstruction alone plays in making diffusion anomalous. Monte Carlo simulations of obstructed diffusion have shown that diffusion due to obstruction should only be anomalous very near the percolation threshold, the point at which the conducting (5) Feder, T. J.; Brustmascher, I.; Slattery, J. P.; Baird, B.; Webb, W. W. Biophys. J. 1996, 70, 2767-2773. (6) Netz, P. A.; Dorfmuller, T. J. Chem. Phys. 1995, 103, 9074-9082. (7) Saxton, M. J. Biophys. J. 1994, 66, 394-401. (8) Bouchaud, J. P.; Georges, A. Phys. Rep. 1990, 195, 127-293. (9) Barta, S.; Dieska, P. Physica A 1995, 215, 251-260. (10) Saxton, M. J. Biophys. J. 2001, 81, 2226-2240. (11) Saxton, M. J. Biophys. J. 1996, 70, 1250-1262. (12) Singer, S. J. N.; Nicholson, G. L. Science 1972, 175, 720-721. (13) Smith, P. R.; Morrison, I. E. G.; Wilson, K. M.; Fernandez, N.; Cherry, R. J. Biophys. J. 1999, 76, 3331-3344.
10.1021/la0261803 CCC: $25.00 © 2003 American Chemical Society Published on Web 11/07/2002
Anomalous Subdiffusion in Lipid Bilayers
Langmuir, Vol. 19, No. 5, 2003 1789
phase of a system becomes disconnected.10 Nonetheless, a recent study of lipid diffusion in a model system containing no proteins, and therefore no possibility of binding, reported anomalous lipid diffusion.14 It was not stated, however, whether the membrane under investigation was near the percolation threshold. Experimentally determined diffusional data from a well-characterized model system displaying only obstruction are required in order to validate the Monte Carlo simulation data and better understand the effects of obstruction on anomalous diffusion. Here, we analyze phase-separated supported lipid bilayers with AFM (atomic force microscopy) and FPR (fluorescence photobleaching recovery) for the purpose of identifying the effect obstruction has on making diffusion anomalous. We prepare these bilayers from mixtures of two phospholipids, 1,2-dilauroylphosphatidylcholine (DLPC) and 1,2-distearoylphosphatidylcholine (DSPC), which ideally phase-separate at room temperature and form a fluid DLPC membrane containing gel-phase DSPC domains. The solid-phase domains serve as obstacles that block diffusing particles. This research was originally motivated by the discovery of a small yet persistent error that was present in our determinations of obstacledependent diffusion coefficients in a previous study.15 In the earlier study, we investigated how obstacles affected diffusional rates; in this paper we examine how obstacles affect the time dependence of diffusion. We found that as we increased the amount of obstruction in our bilayers, the normal diffusion equation fit our data less well at early times. The error was not large enough to affect the determination of the obstacle-dependent long-range diffusion coefficient, which was the intent of the study, but piqued our interest and prompted this study. Here we fit our FPR data with both the normal diffusion model and an anomalous diffusion model. The normal diffusion equation contains only one fitting parameter, D, the diffusion coefficient, which has a constant value for all times. The anomalous diffusion equation contains two fitting parameters, D0, the unhindered diffusion coefficient (i.e., what the diffusion coefficient would be in a fully fluid bilayer without obstruction), and R, the anomalous diffusion exponent. This has the effect of making the diffusion coefficient time dependent as
D(t) ) D0tR-1
(3)
Since for anomalous diffusion R < 1, the rate of diffusion decreases with time; if R ) 1, the rate of diffusion is constant and the equation reduces to normal diffusion. Motivation. Physical proximity is an essential requirement for biomolecular interactions, whether between an enzyme and substrate or a receptor and its ligand, and when anomalous diffusion occurs in cell membranes it can have far-reaching effects for cellular function. For example, since molecular motion is hindered, anomalous subdiffusion can significantly reduce the area a molecule can sample. Figure 1 shows sampling area versus time lines for three different values of R. Additionally, the binding rate between two proteins diffusing in a membrane can be either reduced or enhanced by anomalous subdiffusion since it can slow the initial meeting between the two species but may also slow their separation after an uneventful encounter.16 (14) Schwille, P.; Korlach, J.; Webb, W. W. Cytometry 1999, 36, 176182. (15) Ratto, T. V.; Longo, M. L. Biophys. J. 2002, in press. (16) Saxton, M. J. J. Chem. Phys. 2002, 116, 203-208.
Figure 1. Theoretical comparison of circular areas (in square microns) sampled by a particle moving normally (R ) 1) and subdiffusively (R < 1). Note that after 60 s the particle diffusing normally has sampled an area approximately 4 times that of the particle moving with R ) 0.7.
Of perhaps greater significance to the experimentalist is the fact that anomalous diffusion results in a timedependent diffusion coefficient (i.e., D decreases with time as D ∼ tR-1). If a system that displays anomalous diffusion is analyzed using the normal diffusion model, which assumes that D is constant, the measured diffusion coefficient can depend on the time scale of the measurement. As different techniques for measuring diffusion coefficients operate on quite different time scales, this can result in highly significant deviations between measurements. Materials and Methods Materials. DLPC, DSPC, and 1-palmitoyl-2-[-6[{7-nitro-21,3-benzoxadiazol-4-yl}amino]caproyl]-sn-glycero-3-phosphatidylcholine (NBD-PC) were purchased in chloroform from Avanti Polar Lipids (Alabaster, AL). Small unilamellar vesicles (suv’s) were prepared as outlined in McKiernan et al.17 Briefly, 0.5 mg/mL lipid suspensions were tip-sonicated to clarity using an ultrasonic tip sonifier (Branson sonifier, model 250, Branson Ultrasonics, Danbury, CT). A final sonication was used to heat the vesicles to ∼70 °C before placing the vial containing the suv solution into a 70 °C water bath. All water used in these experiments was purified in a Barnstead Nanopure system (Barnstead Thermolyne, Dubuque, IA) and displayed a resistivity of approximately 18 MΩ. Sample Preparation. We prepared single bilayers on mica using the vesicle fusion technique. The vesicles consist of a mixture of DLPC, DSPC, and NBD-PC, a fluorescent lipid probe that prefers to partition into fluid phases.18 The probe is added at a molar concentration equal to 1 mol % relative to the fluid DLPC concentration. For all samples, 150 µL droplets of the 70 °C vesicle solution were added to freshly cleaved room-temperature mica disks glued to small metal pucks. Giocondi et al. have shown that this thermal quenching process results in the phase separation of the lipid mixture and the formation of small gelphase lipid domains.19 We incubated the lipid mixture on the mica disk for 30 min and then rinsed 10 times with purified water in order to remove unfused vesicles. We waited an additional 120 min in order to ensure that phase separation of the two lipids was complete prior to imaging with the atomic force microscope. AFM Imaging. Samples were imaged with a Digital Instruments NanoScope IIIa (Santa Barbara, CA) in contact mode with either a J or E scan head. Sharpened, coated AFM microlevers, (17) McKiernan, A. E.; Ratto, T. V.; Longo, M. L. Biophys. J. 2000, 79, 2605-2615. (18) Mesquita, R.; Melo, E.; Thompson, T. E.; Vaz, W. L. C. Biophys. J. 2000, 78, 3019-3025. (19) Giocondi, M. C.; Vie, V.; Lesniewska, E.; Milhiet, P. E.; ZinkeAllmang, M.; Le Grimellec, C. Langmuir 2001, 17, 1653-1659.
1790
Langmuir, Vol. 19, No. 5, 2003
model MSCT-AUHW (Park Scientific, Sunnyvale, CA), with nominal spring constants between 0.01 and 0.06 N/m were used for all scans. Hydration of the samples during scanning was preserved using a Digital Instruments AFM Tapping Mode fluid cell model MMTFC. Before imaging, force scans were performed and set points and scan rates were maintained in order to minimize the force between the AFM tip and the sample. Usually, the set points ranged between 0.1 and 0.25 V with scan rates typically between 5 and 8 Hz. After the AFM measurements, the sample was removed from the AFM stage and placed in a Petri dish filled with purified water. Hydration of the bilayer was maintained at all times during the transfer. The Petri dish containing the sample was then moved into the fluorescence recovery after photobleaching (FRAP) apparatus in order to measure the diffusion coefficient of the bilayer. We used the public domain software package Imagetool (University of Texas Health Science Center, San Antonio, TX), which can detect and measure physical parameters of the height images produced from the Digital Instruments AFM software, to analyze the size, shape, and area fraction of the solid-phase domains in our samples. Fluorescence Photobleaching Recovery. Once the AFM imaging is completed, we transfer the sample to a modified Nikon Eclipse 400 Fluorescence microscope (Nikon Inc., Melville, NY) in order to perform the FPR measurements. The initial bleach of the sample is accomplished by sending the full spectrum output of a 100 W xenon lamp through an infrared filter (Edmund Scientific, Barrington, NJ), through an iris that allows us to control the size of the bleach spot, and finally through a 60× water immersion objective that has been focused onto the bilayer. The bleach spot diameter can be varied between 20 and 200 microns, although most measurements are carried out at a bleach spot diameter of 60 microns. Changing the bleach spot size by (20 microns only resulted in changing the length of time until full recovery and did not affect our results. The beam profile, a uniform circular disk, was determined by translating a razor blade through the bleaching beam and measuring the intensity of light transmitted past the edge of the blade.20 To fulfill the mathematical requirement of an infinite reservoir of fluorescent probe molecules, the bleaching time is always less than 10% of the half-time to full recovery. This is a prerequisite of the fitting equation described below. Postbleaching, the lamp output is attenuated 400× using neutral density filters and sent through a 488 nm filter that allows excitation of the NBD-PC probe and is projected onto the sample through the 60× objective. Fluorescence emission from the sample is then collected by the 60× water immersion objective, sent through a filter that removes the excitation frequencies, and gathered by a 20× objective that focuses the filtered light onto a 100 micron pinhole (Edmund Scientific). The spatially filtered light is then collected into a 50× extra long working distance objective (Nikon Inc.) and finally focused onto the 200 micron square active area of a Perkin-Elmer avalanche photodiode (APD) (Perkin-Elmer, Wellesley, MA). The quantum efficiency of the APD is close to 50% at the fluorescence emission frequency of ∼500 nm. The digital pulses output by the APD are counted by a Nanonics Photon counter (Nanonics Ltd. Malcha, Jerusalem, Israel) and binned by a Perkin-Elmer Lockin Amplifier (Perkin-Elmer, Wellesley, MA). We typically bin the signals at 20 ms for the shorter recovery times and 500 ms for the longer recoveries. The APD signal is output into a personal computer and generates an Excel spreadsheet where fluorescent intensity is graphed versus time in order to generate a recovery curve. For normal diffusion, the diffusion coefficient of the sample, D, is measured by fitting the recovery curve with a solution to the differential equation for lateral transport of a molecule by diffusion,20 using the method of Soumpasis.21 For anomalous diffusion, the diffusion coefficient is time dependent and so the fit is modified in the manner shown by Feder et al. whereby the fit is a two-parameter fit that gives both D0, the unhindered diffusion coefficient, and R, the anomalous diffusion exponent.5 This two-parameter fit is an approximation to the actual solution for anomalous diffusion, but it has been shown that the approximation is valid for all t g 1 s.10 The instrument is calibrated and all fits are verified using a phosphate-buffered (20) Axelrod, D.; Koppel, D. E.; Schlessinger, J.; Elson, E.; Webb, W. W. Biophys. J. 1976, 16, 1055-1069. (21) Soumpasis, D. M. Biophys. J. 1983, 41, 95-97.
Ratto and Longo saline (PBS) buffer containing fluorescent fluorescein probe molecules.22 Microliter volumes of the calibration mixture are “sandwiched” between two glass coverslips in order to produce ∼5 micron thick aqueous layers. The diffusion coefficient for these samples is known from other experiments.22 For all lipid mixtures, FPR recoveries were run without a bleaching pulse in order to detect any photobleaching due to the observation beam.
Results and Discussion Atomic Force Microscopy. Supported lipid bilayers containing mixtures of DLPC/DSPC/NBD-PC were imaged using contact-mode AFM. The images shown are subtracted height-deflection data used to show greater contrast between the phases. Figure 2A shows a representative sample displaying phase separation of the two lipid components. The gel-phase DSPC domains are immobile and roughly disk shaped and extend approximately 1.8 nm from the fluid DLPC phase as shown previously.15 Increasing the relative proportion of DSPC to DLPC increases the number of gel-phase domains (Figure 2B) and therefore increases the area fraction of the solid phase. As the domains increase in number, they begin to aggregate at an area fraction of approximately 50% (Figure 2B) and form small corrals. The percolation threshold, the point at which the fluid phase becomes completely discontinuous, occurs at an area fraction of around 70% solid phase (Figure 3). A more in-depth discussion of the percolation behavior of this system is given in our previous work.15 Fluorescent Photobleaching Recovery. Figures 4 and 5 show typical experimental recovery curves for three samples; each curve has been fit with both the normal and anomalous diffusion equations, and the residuals shown at the bottom of each curve illustrate the goodness of fit of each model. Figure 4 shows a recovery curve from a sample that contains no obstacles (i.e., only DLPC and the fluorescent probe are present in the bilayer); note that the normal recovery equation fits the data quite well. However, for the recovery from a sample containing approximately 40% solid-phase lipid (Figure 5A), the anomalous diffusion model with R ) ∼0.95 fits better than the normal model for times of
