4428
J. Phys. Chem. 199498, 4428-4442
Nonstationary Dynamics of Excimer Formation in Two-Dimensional Fluids R. Merkel and E. Sackmann' Biophysics Laboratory, Faculty of Physics, Technische Universitiit Miinchen. James-Franck-Str. 1 , 0-85748 Garching, Federal Republic of Germany Received: August 5, 1993; In Final Form: January 12, 1994'
A Langmuir-type film balance equipped with a fluorescencemicroscope for the near ultraviolet and a spectrometer for time-resolved fluorescence spectroscopic studies (suited for lifetimes 1 5 ns) was built. The aim was to study membrane dynamics of diffusion-controlled bimolecular reactions in two-dimensional fluids. Complete exclusion of molecular oxygen was achieved. Under this condition, the dynamics of excimer formation may be studied for probe concentrations as small as 0.5 mol %. At the same concentration, domain formation due to lateral phase separation may be visually observed. In the first part, the time evolution of the fluorescence of monomer and excimer in monolayers of L-a-dimyristoylphosphatidylcholine (DMPC) is studied. A pyrene-labeled lecithin serves as an excimer-forming probe. Probe concentration and lateral packing density of the lipids are varied. A method is developed by which the fluorescence lifetime of the monomer and the lateral diffusion coefficients are obtained with high accuracy (lifetime better than 7.5%, diffusion coefficient better than 15%) from the decay of the monomer fluorescence. The method is based on the analysis of the excimer formation in terms of the Smoluchowski theory modified for two-dimensional fluids. It is shown that the time course of the excimer formation is determined by a strongly time dependent rate of association k(t). The strong time dependence is a consequence of the fact that there is no stationary solution of the diffusion equation in two-dimensional fluids. The time evolution of the excimer fluorescence is analyzed in a model of so-called "pseudo-excitation". In the application part the variation of lateral diffusivity with lipid packing density is studied. It is shown that it agrees only in the small packing density regime with the predictions of the free volume model. Over the whole range of packing densities investigated here, the viscosity of the monolayer shows an exponential dependence on density, as was predicted for ordinary fluids by Frenkel. Finally, the method is applied to mixtures of DMPC with cholesterol. A phase diagram for high cholesterol concentrations is established. It is shown that by careful analysis of excimer formation dynamics it is possible to detect local phase separation in cases where the classical methods (thermodynamical and fluorescence microscopy) fail. A remarkable influence of a few percent of pyrene-labeled phospholipids on the phase boundaries is established.
I. Introduction Lipid monolayers and bilayers areof interest not only as models of biological membranes (to study, for instance,the physical basis of self-organization of biomembranes') but also as models of two-dimensional fluids. The most extensively studied process is the lateral diffusion, which exhibits highly interesting features. In free lipid layers the long-range diffusion is dominated by the coupling of the two-dimensional flow field in the bilayer around the diffusing particle to the infinite bathing fluid, which leads to the famous Saffman-Delbriick law of the logarithmicdependence of the diffusion coefficient on the diameter of the diffusing particle.2 In membranes adjacent to solid surfaces, the diffusion can be drastically modified by the frictional coupling of the bilayer to the solid via the thin lubrication layer and indeed can lead to a quadratic dependence of the diffusion coefficient on the molecular diameter.' This behavior is a striking demonstration of the unique features of two-dimensional fluids. Frictional coupling provides a method for the measurement of friction between monolayer and bilayer or between adjacent bilayers.4-6 In view of the many scientific and practical implications of lateral diffusion in membranes, a large number of measuring techniques have been developed. The classical and most simple method of diffusion measurement is the so-called FRAP (fluorescence recovery after photobleaching) technique7 which, however, yields only average values of the diffusion coefficient. A very informative method is the incoherent quasi-elasticneutron scatteringwhich allows a simultaneous evaluation of the restricted diffusion of the hydrocarbon chains of the lipid molecule in its
* Author to whom correspondence should be addressed.
Abstract published in Aduance ACS Abstracts, March 15, 1994.
OO22-3654/94 f 2098-4428%04.50/0
solvent cage and the (long-range) diffusional jumps between adjacent sites.8 Another potentially simple and often applied method is the excimer-probe technique.9 A simple version is based on the measurement of the ratio of the quantum yields of the excimer to the monomer fluorescence which is proportional to thecollision rateof the probes and hence thediffusioncoefficient Dlat. It is, however, well suited to measure relative values of Qat only for small probe concentrations (110 mol W). Rigorous measurementof diffusion coefficients by the excimer technique requires direct evaluation of the time evolution of excimer and monomer fluorescence. Severalgroups have applied kinetic techniques to the measurement of diffusion coefficients in lipid bilayer vesicle su~pensions.~O-~~ However, to our knowledge a systematic study of the dynamics of excimer formation in twodimensional fluids has not yet been attempted. For that reason we have built an instrument for the measurement of excimer formation in lipid monolayers. It consists of a Langmuir film balance equipped with a UV fluorescence microscope, a fluorescence spectrometer, and digital storage oscilloscope for data acquisition. In the first part of this work we studiedthe dynamics of excimer formation in monolayers of DMPC (L-a-dimyristoylphophatidylcholine) using pyrene-labeled lecithin. In the second part (to be published separately),we applied the techniqueto elucidate the dynamics of a macrolipid embedded in a two-dimensional fluid. The excimer formation was analyzed in terms of the Smoluchowski theory of bimolecular reactions in two-dimensional fluids. The excimer formationis determined by a time-dependent rate of formation because there is no stationary solution of the diffusion equation in two dimensions. It is shown that although the monomer decay can be represented by a biexponential decay 0 1994 American Chemical Society
Nonstationary Dynamics of Excimer Formation
The Journal of Physical Chemistry, Vol. 98, No. 16. 1994 4419
curve (within experimental resolution), the classical ForsterBirksI5J6theory (which also predicts such a behavior) is not applicable. The present method was furthermore applied to mixtures of DMPC with cholesterol in order to test its applicability to mixed membranes. In particular, it is shown that local phase separation may be detected in cases where other methods (e.&, fluorescence microscopy) fail.
11. Materials and Methods 11.1. Materials. Phospholipids (L-a-dimyristoylphosphatidylcholine (DMPC) and L-a-dipalmitoylphosphatidylcholine (DPPC)) were purchased from Avanti Polar Lipids (Alabaster, AL). Cholesterol was purchased from Fluka (Buchs, Switzerland). Pyrene-labeled phosphatidylcholine (3-palmitoyl-2-( 1pyrenedecanoy1)-L-a-phosphatidylcholine(P-Py-PC)) was purchased from Molecular Probes (Eugene, OR 97402). All substances were used without further purification. The subphase of the film balance consisted of ultrapure water producedby a milliporeapparatus “Milli-Q” (Millipore Molsbeim, France) with a specific resistance of more than 16 Mil cm. Molecular oxygen was removed by extensive argon bubbling ( 5 h or more) before filling the Langmuir trough. 11.2. Film Balance with Time-Resolving Microfluoresceuce Spectrometer for the Near UV Spectral Range. The instrument designed for the experiments consists of three major parts: ( I ) a classical film balance equipped with a Wilhelmy system for measuring surface tension, (2) a fluorescence microscope suited for ultraviolet light, and (3) a time-resolving fluorescence spectrometer with a time resolution of better than 5 ns. The three parts had to be specially designed for studies of pyrene excimer probes: (i) To prevent quenching of pyrene fluorescence by molecular oxygen, the film balance was covered by a flow hood and perfused with argon gas. (ii) Since pyrene has to be excited at 3OlL350 nm and shows fluorescence from 375 to 500 nm,a microscope with UV optics had to be designed. (iii) The relatively long lifetime of the pyrene probe (13&250 ns) allowed the application of a nitrogen laser as a pulsed light source and a digital storageoscilloscopeas data acquisitionsystem. A schematic view of the apparatus is shown in Figure 1. Its specific features are described in more detail below. Film Balance and Monolayer Preparafion. The film balance consistsofatrough(Teflon)of450X900mm2areaandamovable barrier. The surface tension is measured with a Wilhelmy system, as described previously.”Js For temperature control, thermostated water is circulated through copper tubes attached to the bottom of the trough. The bottom of the trough is formed by a Teflon film containing black pigment (Norton Pampus, D-W4165 Willich, FRG) which results in a drastic reduction of the autofluorescence (at least a factor of 10). Theentire filmbalanceiscoveredby ahcodmadeofglass with a quartz-glass window enabling an area of observation of 40 X 70 mm2. The only access to the atmosphere is a bole of diameter 4 mm. Through this hole the aqueous subphase is filled in and the monolayer is deposited. Oxygen is removed by a continuous stream of high-purity argon at a rate of 45 L/h. The monolayer was deposited by spreading of appropriate amounts of solutions of the lipid (solvent:cbloroform/methanol, volume ratio 3: I). Thisoccurred at vanishing pressure. Typically themonolayer was equilibrated for about 10 min before the first compression. Monolayerswerecompressed andexpanded at rates of less than 2 .&’/molecule min). W-Fluorescence Microscope. The central part of the epifluorescencemicroscope is the objective, serving simultaneously as high-power condenser. For this purpose a reflecting objective (Ealing Reflecting Objective X36, Ealing Electro-optics plc, Watford WDZ 4PW Great Britain) is used to combine long
I
light source
Figure 1. Schematic view of the experimental setup. A conventional film balanceis modified toelrcludemolecularoxygenfromthemonalayer.
This was achieved by a closed cover of quartz-glass. The only accm to theatmosphereisaCmm-diameterhole,thmughwhichaconstantstream of argon (45 L/h) removes all oxygen. The monolayer is illuminated by a nitrogen laservia theK6hler principle (cf. text fordetails). Areflecting objective, Obj., serves as objective and condenser. Fluorescence and irradiation light are separated by a dichroic mirror and an optical filter. For microscopic observation, the image is focused on the photosensitive plate of an ISlTcamera (dashed arrow). For spectroscopy,the movable mirror is placed in the optical path. The image is focused on a light guide. A field stop, not shown, in a plane of intermediate image reduces out-of-plane strawlight. Spectral analysis is done by a grating mane chromator. For timeresolved measurements,a fast photomultiplier, PM, and a digital storage oscilloscope are used. The trigger for each measurement is set by a fast SI-PIN-photodiode, PD, and a constantfraction discriminator (abbreviation: CFD, for details see text). Data analysis is performed on an AT-compatible PC (80486); for details of the data analysis, see text. workingdistance(8mm) witha high transmissionat theexcitation wavelength. Illumination was achieved by the Kobler principle, as explained in the following (cf. Figure 1). The aperture stop defines a spot on a diffuser illuminated by the nitrogen laser. This spot is imaged to infinity by a lens (focal length 140 mm). The excitation light passes the field stop and a second lens, which throws the image of the field stop to infinity and at the same time forms an image of the aperture stop in the back focal plane of the objective. Since the objective is adjusted totubelengthinfinity, thisilluminationsystemresultsina focused image of the field stop in the front focal plane of the objective (=object plane) and a focused image of the aperture stop in the back focal plane of the objective. The light source is a nitrogen laser (Lambda Physik K300) emitting at 337 nm. The pulse width is approximately 3.5 ns (FWHM), pulse repetition rate 50 Hz (synchronizedto thevideo camera),and the pulseenergy isapproximately2 mJ. Thespectral purity of the laser pulse is improved by five dichroic mirrors (high-energylaser mirror, 4 5 O incidence,for nitrogen laser, Melles Griot, Irvine, CA) (not shown in Figure 1). Since the excitation and emitted light are transmitted through the same objective,
4430
The Journal of Physical Chemistry. Vol. 98, No. 16, 1994
Figure 2. Fluorescence micrograph showing a monolayer of DPPC containing 1 mol%ofpyreneprobe. Domainstructureisduetacoe~istence of fluid and crystalline phases. The grainy ~ t r u ~ t uresults ~ e from the intensified camera. The dark half-mwn-like structure on the left side is due to inhomogeneous illumination, Length of the bar is 50 pm.
they must be separated by a second dichroic mirror (Newport HL 1) (denoted as dichroic mirror in Figure I). Residual excitation light is suppressed by an additional filter (Schott KV 370, Mainz, FRG) (denoted as filter in Figure 1). Fluorescence micrographs are taken by an ISIT video camera (Hamamatsu C2400-09). An example is given in Figure 2 for a DPPC monolayer containing 1 mol % of pyrene probe (PPy-PC). The formation of dark domains due to lateral phase separation is clearly visible. In principle, phase separation may be clearly distinguished to pyrene probe concentrations as small as 0.5 mol %. Time-ResolvedFluorescence Spectroscopy. All fluorescence decay measurementsona given monolayer were performed during the first compression following the deposition of the monolayer. Duringthat time thepressureofthemonolayerwas keptconstant. Area changes due to this were negligible. After the completion ofall measurements,thefilm was expandedagain to theconditions of the first measurement. The first fluorescence decay measurement was repeated. In no case were significant differences between the first and last measurement found. What is called measurement in the above is indeed four decay measurements: instrument function (see below), decay of monomer at two wavelengths, and decay of excimer. These four measurements were typically done in IO min. To switch the instrument from observation mode to measuring mode, a mirror is placed in the optical path of the microscope (denoted as movable mirror in Figure I). The fluorescencelight emitted from the monolayer is focused on a quartz glass fiber bundle (Schott, Mainz, FRG) which transmits the light to the entrance slit of the monochromator (PTI, Model 01-009). An aperture is placed in an intermediate image plane in order to suppress stray light which does not originate from the object plane. The size of the aperture defines a measuring spot of 100pm diameter in the object plane. The data acquisition was performed as follows: A fast photomultiplier tube is placed behind the exit slit of the monochromator (Hamamatsu H2431, pulse rise time 0.7 ns) to convert the light intensity passing the monochromator into an electrical current. This current is fed into a digital storage oscilloscope (DSA602,Tektronicr. Bevington,OR) withananalog band width of 1 GHz and a sampling rate of 2 X IO9SSI. In order
Merkel and Sackmann to improve the signal-to-noise ratio, the average over 4096 individual measurements is taken. During this averaging procedurespecialcaremustbe taken toavoidjitteringof the trigger.’9 Therefore, theexact trigger timeof each individual measurement is set by a UV-sensitive Si-PIN-Diode (Hamamatsu S3279, pulse rise time 1.5 n). which is placed behind a dichroic mirror in the illumination system (see Figure 1). The output of this diode is fed to a constant-fraction discriminator (Tennelec TC 453, Tennelec, OakRidge Turnpike, TN). The discriminator sets the trigger point at a specific ratio of the laser light intensity with respect to the actual peak height. Triggering at a specific amplitude would transform the pulse energy fluctuations of the laser into jittering of the trigger. Special care had to be taken to measure the instrument time response L(r) (cf. below). This was done immediately after each measurement. For this purpose the monochromator was set to the excitation wavelength (337 nm). The optical filter KV 370 was replaced by a highly scattering diffuser which attenuates the excitation light reflected hy the water surface to a suitable signal intensity. The instrument response function exhibits a width of 3.5 ns (FWHM). Dora Analysis. Thedata wereanalyzed hyXz-fitting following Levenberg and M a r q ~ a r d t . ~Counting ~ . ~ ~ noise arises during the measurement. It is assumed that the error of each individual data point is normally distributed. The variance of each point is calculated assuming a current amplification of IO‘ by the photomultiplier (thevalue given by the manufacturer at the high voltage used). The fitting procedure typically yields values of xz of roughly 1, indicating that the above assumption is valid. Typically the data can be described over a dynamic range of 2.5 orders of magnitude. In addition to the model for time evolution ofthefluorescencelight (either sumsofexponentialsorthemodel described below), corrections have to be made for background fluorescence and the wavelength-dependent time shift of the pbotom~ltiplier.~~ The background fluorescence is measured at a pure water surface and is added with variable amplitude to the model function during the fitting procedure. This background amplitude is a free-fitting parameter. The wavelength-dependent time shift is implemented in the fitting routine followingGrinvaldzl and is also a free-fitting parameter. The resulting values arc typically of the order of 0.5 ns. In general, it is very difficult to determine the accuracy of the results of the fitting procedure and measurement. The scattering of the data is obtained by repeated measurements of the decay of monomer fluorescenceunder the same conditions (here 2 mol %P-Py-PCinDMPC at a pressureof 20mNImauda temperature of 20’ centigrade). The data can be described by a sum of two exponentials (see Figure 3a). The longer lifetime is 143.2 IIS with a standard deviation of 4.5 ns (see Figure 3h), while the short lifetime is 36.0 ns with a standard deviation of 5.0 ns (averages over 17 measurements). The higher relative error in the short lifetime is due to the low relative intensity in the fast decaying part of the decay (=IO%).
IJS. Theoreticnl Basis of Evaluation of Exciner Formations in Two-Dimensional Solutions The recorded time course of the fluorescenceis a convolution of the instrument time response function L(r) and the intrinsic decay I&) of the fluorescence.l9 I&) is the probability that a fluorophore, being excited at time zero, is still in the excited state at timer. Inotherwords,I& isthefluorescencedecaymeasured with an instrument of infinite time resolution. In the following we derive expressions for the intrinsic decay I&) for monomer and excimer fluorescence. The kinetics of the monomer and excimer fluorescence decay is represented by the following reaction scheme:
The Journal of Physical Chemistry, Vol. 98, No. 16, I994 4431
Nonstationary Dynamics of Excimer Formation
a)
7;
0.1
measured values -fitted vaiuos .-cn v)
t
I
I
I
0
200
400
600
time [nsec] 4
7:
0.6d'
.i -m
0.4-:
B
o.o-
2
1
0.2-: I
1
I
I
. . . .
135 I
25
I
I
I
30
35
40
number of experiment
Figure 3. Measurement of monomer fluorescence decay. Conditions: 2 mol 5% pyrene probc, 20 O C ; lateral pressure of 20 mN/m; wavelength is 390 nm. (a) Representation of a single measurement. Upper diagram: measured decay curve, (data points barely visible), the fitted decay curve and the instrument function, L(t). Middle diagram: normalized residues (e.g., thedeviationof measurement and fit normalized by the standard deviation of each measured data point). Lower diagram: autocorrelationof normalized residues. (b) Resultingvaluesfor the longest lifetime obtained by repeated measurements are shown. Each measurement was performed on a freshly prepared monolayer. The mean value of 143.2 ns is marked by the straight line.
M*
+ hVM;
kM = ~ / T M
(MM)*;
r ( t ) = k(t)c
+ hf + hVE;
kE = ~ / T E
--*
M
+ M*
(MM)*
M
M
-
where M, M*, and (MM)* denote the concentrations of the ground-state monomer, the excited monomer and the excimer. The r ( t ) is the rate by which excimers are formed. T M and 78 are the intrinsic lifetimes of the excited monomer and excimer. kM is the sum of the decay rates of all processes depopulating the excited monomer; ke is defined analogously for the excimer. c denotes the concentration of the excimer probe (molecules per area). Only a small fraction of all labeled molecules is excited;
therefore, c and M are practically identical and are not distinguished in the following. An important assumption is that the thermal dissociationof the excimer to an excited and a groundstate monomer is negligible. Fdrster et al.ls have shown that this process becomes essential in organic solutions only at T > 50 OC, and evidence has been provided that it is also negligible for membranes at T < 50 0C.z2 Please note the following distinction: Intrinsic lifetimes are model parameters. It is assumed that an excited state formed at a given time decays monoexponentially with a rate of 1 over the intrinsic lifetime. This rate includes all photophysical processes with the exception of excimer formation. Measured decay times are the results of the fitting procedure for a monolayer of specific composition under specific conditions. They arise from an interplay of diffusion-limited excimer formation and intrinsic decay (see above). Measured decay times depend on probe concentration in contrast to intrinsic lifetimes, which do not. It is important to emphasize that the only time-dependent process is the formation of the excimer, r(t). An excited state formed at a given time (either by absorption of a photon in the case of monomers or by excimer formation) is assumed to decay monoexponentially with decay times T M for the monomer and 73 for the excimer,respectively. This decay includesall photophysical processes (fluorescence,internal conversion,intersystemcrossing) besides excimer formation. Excimer formation is due to diffusion of the probe and is therefore statistically independent of the fluorescence decay. The reason for the time dependence of the formation of the excimer is that this process, being under diffusional control, is described by nonequilibrium thermodynamics (a theory within which memory effects arise). A lattice model of diffusion gives a good illustration of this point. A diffusion-controlledbimolecular reaction is modeled by a single probe particle performing a random walk on the lattice. One specific lattice site is assumed to be a trap, and if found by the probe, the reaction takes place. When the process starts, all lattice sites are unvisited and the probability of finding the trap in the first step is simply the coordination number of the lattice over the lattice size. With increasing step number more and more lattice sites are visited. The probe continues until it visits the trap. If the random-walk crosses itself, e.g., if lattice sites are visited repeatedly, the mean probability of finding the trap decreases with each subsequent step. It is the self-crossing of the random walk that reduces the reaction rate with increasing time. The general form of this reduction is discussed by deGennes.23 The diffusional control is also the reaction for the proportionality of the rate to the concentration of the dye molecule, c. Hauser et al.24have shown that the solutions of the above reaction scheme can be expressed as Za,M(t) = Za exp
- --
( : M I
c k ( 8 ) de
)
where @ denotes the convolution. ZOis the maximum intensity, and all other symbols are defined in the reaction scheme above. (For a discussion of this solution, see Hauser et al.24) The integral over k ( 8 ) in eq 1 accounts for the fact that the rate of excimer formation is time dependent. The functional form of k ( t ) is sensitively dependent on the dimensionality of the solution and can be calculated by the Smoluchowski theory of diffusion-controlled reactions.25 This theory for diffusioncontrolled reactions in two dimensions has been worked out by Naqvi26 in order to explain the triplet-triplet annihilation in membranes. A further discussion can be found in ref 21. The Smoluchowski theory is based on several assumptions: (1) only the relative movement of two reactants is considered (this gives circular symmetry), (2) at the beginning the diffusing reactants
4432 The Journal of Physical Chemistry, Vol. 98, No. 16, I994
Merkel and Sackmann
:41 0
40
80
120
0.00
103
0.05
0.10
0.15
0.20
mdar fraction of probe
1o
- ~ 1 o-2 1 0-1
-1
oo
1 o1
1 o2
Figure 4. Plots of normalized reaction rate, I (defined in eq 3) as a function of normalized time, Fo (defined in eq 5 ) .
are homogeneously distributed within a fluid (here two-dimensional), and (3) the reactants vanish after the reaction. The flow of particles is described by the diffusion equation. The solution of the diffusion equation with the above boundary conditions has been given by Jaeger,28 who treated the analogous heat conduction problem. H e calculated the heat flow in a circular plate of infinite radius. At the beginning the plate exhibits a uniform temperature distribution. A heat sink of diameter R is kept at constant temperature. It is located in the center of the plate and is switched on a t time zero. Following Naqvi and Jaeger, the time-dependent reaction rate is given by
P
0.00
0.05
0.10 0.15
0.20
mdar fractiq of probe Figure 5. Example of data evaluation for a DMPC monolayer at a molecular area of 100 AZ. (a) Variation of the slower decay rate kl with probe concentration. Extrapolation of the kl versus c plot to vanishing concentration yields the intrinsic lifetime TM of monomer. The extrapolationis indicated by the dashed line. (b) Individualdiffusionconstants, D, as determined by eq 7. The D values were calculated by using the values of kl and T M as shown in part a. The dashed horizontal line indicates the error-weightedaverage of the diffusion constant, defined in eq 10.
measured decay calculated values
2Dt F, = -
R2
In these equations, D is the diffusion constant of one reactant, R is the reaction radius resulting from Smoluchowski theory, JO is the zero-order Bessel function of the first kind, and YOis the zero-order Bessel function of the second kind. The function Z(Fo) is discussed by and a table of values is given by Clarke and Jaeger.29 Since the integral in eq 4 is divergent for Fo = 0, the reaction rate shows a divergence (-t-l/*) for t = 0. For t a,Z(Fo)converges to zero as l/ln(t). The above time dependencies are a consequence of the fact that no stationary solution of the two-dimensional diffusion equation exists (in contrast to the three-dimensionalcase). k(t) is therefore time dependent for all times. In Figure 4 we give plots of the function Z(Fo). An enlightening discussion of the different behavior of diffusion controlled reactions in different dimensions is given by deGennes.23 To the best of our knowledge, only one group has applied these expressions (eqs 1-5) for the evaluation of the fluorescence decay of pyrene probes in bi1a~ers.l~ However, they evaluated only the decay of the monomer and made no systematic study of the concentration dependence. III.l. Data Evaluation Based on the Monomer Fluorescence. All our attempts to evaluate the decay of the fluorescence of the monomer by fitting with the theoretical decay law (eq 1, eqs 3-5)
-
0
100
200
300 400 500
time [nsec] Figure 6. Overlay of measured (points) and calculated (open squares, only every 10th data point is shown) monomer fluorescencedecay. For the theoretical decay curve the error-weighted average of D and the intrinsic lifetime are used. Conditions: 70-& molecular area and 5 mol % pyrene probe.
failed. This is due to the divergence at time zero of the reaction rate and to perturbations of the data by background fluorescence (lifetime roughly 5 ns). Both effects tend to sharpen the peak of the decay curve a t short times and could therefore not be distinguished. A more indirect approach to extract D and the intrinsic lifetime 7~ from the data therefore had to be developed. It is based on the comparison of measurements at different probe concentrationsa t otherwise equal conditions and will be presented in the following section. As can be seen in Figures 3 and 6,the decay curve exhibits fast and slow decaying regimes. They can be formally described by a sum of two exponentials (aledrl aze-f/rZ). The time constant of the slowly decaying exponential is in all cases at least 3 times larger than that of the fast decaying component, 7 2 . Moreover, the intensity (number of detected photons) of the slowly
+
The Journal of Physical Chemistry, Vol. 98, No. 16, 1994 4433
Nonstationary Dynamics of Excimer Formation decayingcomponent (alsl)comprises75-90%of the totalintensity (a171 11272). Therefore, the lifetime of the slowly decaying component can be measured with high accuracy, and the evaluation of the data is based on this parameter. As follows immediately from eqs 1 and 3, the rate a(t) at which the monomer vanishes can be expressed as
+
The measured lifetimes 71 and 7 2 obtained as results of the fitting procedure must be somehow related to eq 6. Our evaluation procedure is based on the observation that the decay of the monomer can be described as monoexponential except for short times. The measured apparent decay rate kl ( = 1 / ~ 1 )of the slowly decaying exponential (obtained by the fitting procedure) is a time average of the theoretical decay rate a(t). Therefore
(7) The average is taken in a time window where the slowly decaying exponential clearly dominates the signal. The lower limit of the integral is assumed to be T I since for shorter times the fast decaying exponential (71) dominates the signal. The upper limit is taken as 3 . 5 ~since ~ for larger times the signal intensity is too low to exert a significant influence on the result of the fitting procedure (e.g., on the value of kl). It should be noted that the exact values of the limits of the averaging procedure are an arbitrary choice. Fortunately, the results are not very sensitive to the choice of values (for instance, using 3/47, and 471 as limits changes the results by less than 10%). For a more elaborate approach, the averaging should not be done in a window but with help of a weighting function that has a maximum in the time window used. This would avoid the use of artificial sharp cutoffs. ~ ( ~ , c , R , D , is T aMfunction ) of time and four parameters. Three of these can be determined separately. The area concentration of the probe, c, is known from the molecular fraction of the probe and the measured area per lipid molecule (A). The reaction radius, R , is determined by the assumption that if two probes meet the excimer formation proceeds immediately. Therefore the area of reaction ( r R 2 )must be equal to 2 times the area per lipid molecule:
?rR2= 2A
Equations6 and 7 suggest immediately that theintrinsiclifetime of the monomer, TM, can be determined by extrapolation of kl to concentration zero lim k , = 1 co
/
~
~
(9)
This procedure is carried out graphically by linear extrapolation. Since three of the four parameters in eq 7 can be determined separately, namely, c, R , and TM, eq 7 contains only one unknown parameter, D, and can therefore be solved numerically. The error made by extrapolation of kl to concentration zero is estimated as 0.3 MHz. The errors of the individual lifetimes 71 are f3 ns. The most extreme combinations of 71 and T M compatible with the above estimates of the errors were inserted in eq 7. The resulting values of D were taken as upper and lower bounds for the error of the diffusion constant. The reliability of the above procedure was checked as follows: Decay curves were calculated from the exact decay laws (eqs 1, 3, 5, and 8) for appropriate but arbitrarily chosen values of D, TM, c and A. They were then convoluted with the measured instrument function L(t). Random numbers with normal distribution of variance Q S(t)-Il2 were added to the resultant curve,S(t),to simulate counting noise of the measurement. These curves were than analyzed by the method described above. The calculated curves could indeed be well fitted by a sum of two exponentials. The results for kl were analyzed as described above. The resulting values of 7~ and D were only slighly smaller (3% and 9%,respectively)than thevalues inserted. Thus, the proposed procedure for the determination of the values of D and 7~ is well justified. More evidence for this is reported in the measurement and discussion sections. From the set of values Di and the errors AD1 obtained for the different probe concentrations under otherwise equal conditions, averagevalues of D weighted by errors were determined according to
-
III.2. Evaluation of the Decay of the Excimer Fluorescenceby the Concept of Pseudoexcitation. The measured time evolution of the excimer fluorescence, zE(t), is a convolution of the instrument function, L(t), with the intrinsic decay function la&). Together with eq 2, this leads to the following expression:
(8)
The area per molecule, A, is simply calculated from the area of the water surface and the amount of lipid spread. This assumption is justified since the two probe molecules have to occupy adjacent sites in order to form excimers. The assumption that the actual area occupied by each molecule and not their hard-core areas determines the reaction radius is based on the freevolume model of diffusion? The basic statement of this model is that the diffusion is determined by the rearrangement of free volume within the monolayer whereas the movement within this area is much faster. This model is supported by neutron scattering experiments in membranes.8 Equations 3-5 are based on the implicit assumption that the reaction probability is 1 if an excited and a ground-state monomer collide. This assumption is supported by the facts that (1) this holds for pyrene in organic solventsI6 and (2) the diffusion constants determined with this assumption agree quite well with results of photobleaching experiments (see Experimental Section, Table 1). A reduced value of the reaction probability would lead to a underestimation of the actual diffusion constants. This is not the case. Also, dissociation of the excimer would decrease the yield of excimer and thus lead to an underestimation of actual diffusion constants.
k(t) is defined in eqs 3-5. Formally eq 11 can be written as
Where P ( t ) is called pseudoexcitation and is defined
The pseudoexcitationis a convolutionof the measured instrument function L(t) with a function that can be calculated by using the values determined for D and TM by the procedures described in section 111.1. By use of the pseudoexcitation instead of the instrument function L(t),the excimer fluorescence decay can be fitted by a single exponential (see Experimental Section). The success of this fitting procedure strongly supports the validity of the applied kinetic model. 111.3. Treatment of Stationary Measurements. The quantity of central interest in stationary measurements of the excimer formation is the ratio of excimer quantum yield (@E) to monomer quantum yield (@M):
4434
Merkel and Sackmann
The Journal of Physical Chemistry, Vol. 98, No. 16, 1994
where it is assumed that the amplitude of Z, is normalized to a single excited molecule. km is the rate of fluorescence decay for the excimer, kFM for the monomer. Combining eq 14 with eqs 1-5 yields
0, -=OM
1601 4
~ F M
,fz6M(r)
dt
I
60
k,, J ( I ~ Mc(k~( )t ) ) @ exp(-t/7E) d t
I
I
I
1
70 80 9 0 100 110 area per molecule [A2]
Figure 7. Intrinsic lifetime, TM, of monomer fluorewenceas a function of the area per molecule. The error bars arise from uncertainties in the
extrapolations. The second identity follows from the properties of the convolution and is obtained by partial integration. The quantity km/kFM can be determined only from absolute measurements of quantum yields, which have not yet been performed. Birks et al.30showed that kFE/km has a value of 7.7 for pyrene, regardless of temperature or solvent. With this value and our results for 7 E / T M and D, eq 15 can be evaluated numerically. In this work, the intensities at a wavelength of monomer fluorescence Z (390 nm) and excimer fluorescence Z’ (475 nm) were determined by integration of the decay curves. (The 475 nm is the maximum of the excimer fluorescence, 390 nm is a convenient choice with respect to amplitude and signalto-noise ratio.) As SackmannZZ has shown, the ratio of intensity Z’/Z is proportional to the ratio of quantum yields, @ E / @ M :
The proportionality constant k depends on the apparatus used and is furthermore a function of the type of solvent and the position of the pyrene moiety on the hydrocarbon chain. It has to be determined for each type of system. The comparison of the calcualted values of @E/@M and the measured values of Z’/Z showed no systematic variation and gave a constant k of 0.21 f 0.06. The fact that the ratio of both quantities is constant is another critical check of the validity of the kinetic scheme applied.
IV. Experimental Results IV.1. Excimer Formation in DMPC Monolayers and Measurementof Diffusion Coefficients. In a first series of experiments, the fluorescence decay of the amphiphilic excimer probe PPy-PC in DMPC monolayers was measured a t the wavelengths 379 and 390 nm ( a regime dominated by monomer emission) and at 475 nm (a regime dominated by monomer emission) and at 475 nm (a regime dominated by excimer emission). It should be noted that in stationary experiments at 475 nm a small contribution of monomer signal is still p r e ~ e n t .In ~ our experiments, no indication of such a contribution was found. Fitting of the excimer decay by addition of a component of the corresponding monomer decay did not improve the quality of the fits. In fact, even small amounts of monomer fluorescence should be detected in our experiments since a t low probe concentrations (where the effect should be strongest) the monomer decays slower than the excimer. After sufficient time, only the monomer emission should be present. No evidence for monomer emission at 475 nm was found. Therefore, we conclude either that there is no monmer emission at 475 nm or it is below the sensitivity of our experiments. Measurements were performed at 20 OC for various molar fractionsof the probe (0.5-23.2 mol 5%) and for various molecular areas, namely, 54, 60, 70, 80, 90, 100 and 110 Az. Within
experimental accuracy, all decay curves (for both monomer and excimer) could be formally fitted by a sum of two exponentials. In the following, we report measurements of the monomer lifetimeand of the diffusion coefficientsobtained by the procedure described in section 111.1. In Figure 5 an example of the data evaluation procedure is presented for a molecular area of 100A2. In Figure 5a the smaller decay rate kl is shown as a function of the molar fraction of the excimer probe P-Py-PC. Extrapolation of the rate to concentration zero leads to an intrinsic decay rate of the isolate monomer of 6.5 f 0.3 MHz (or 7 M = 154 f 7 ns). In Figure 5b the diffusivities as determined by the above procedure are shown. The error-weighted average, defined by eq 10, gives a value of the diffusion constant of 28 rmZ/s (with an error of 4 pm2/s). In order to check the above procedure, the decay functions of the monomer, Z ~ Mwere , calculated. The values of D and 7 M as determined above were used. The resulting curve was convoluted with the measured instrument function L(t). An overlay of the measured decay function and the calculated decay is shown in Figure 6. As can be seen, both curves coincide quite well over the entire time range and not only in the time interval on which the evaluation procedure is based (cf. eq 7). The same procedure was performed at all other molecular areas, with results very similar to those presented here. Effect of Monolayer Packing Density on Monomer Lifetime and Diffusion Coefficient. Figure 7 shows the effect of lipid packing density on the intrinsic lifetime of the monomer, 7M. Clearly 7 M decreases drastically with decreasing packing density. The lateral diffusion coefficients for thevarious molecular areas were determined as described above. The results are presented in Table 1. For comparison, some data obtained by the photobleaching technique31 are also presented. The latter measurements were performed at 2 OC higher temperature. This fact explains half of the difference between the results, the remaining discrepancy is within the error limits of both techniques. For molecular areas of less than 90 A2, the diffusivity decreases with decreasing molecular area. However, it is practically constant at molecular areas larger than 90 Az (cf. Discussion section). Measurement of the Excimer Lifetime 7 E by the Method of Pseudoexcitation. The measured excimer decay functions Ze(t) were evaluated with the approach of pseudoexcitation. Pseudoexcitation functions, P(t), were calculated according to eq 13. Intrinsic lifetimes 7 M and diffusion constants as obtained above were used. Using P(t) instead of the instrument function, L(t), the decay functions, Z E ( t ) , could be fitted by a single exponential. Figure 8 is an example for the goodness of the fits. It should be noted, however, that for a good fit the time axis must be shifted to negativevalues by 4-6 ns. The reason for this shift is not clear. It could be due to systematic errors in D, 7 M , and c. Another possible explanation could be the singularity of the reaction rate of excimer formation k(t) at time zero, which is unphysical.26 For
The Journal of Physical Chemistry, Vol. 98, No. 16, 1994 4435
Nonstationary Dynamics of Excimer Formation
TABLE 1: Variation of the Lateral Diffusior Coefficient of Pyrene Probe with the State of Monolayer (Surface Pressure, r, and Area per Molecule, A ) Formed by Pure DMPC (Our Work,20 "C)' molecular area (A2) 54 60 70 80 90 100 110 ("/m) 35.1 25.5 13.5 6.8 2.7 0.25 0.0 1.6 1.3 8.8 & 1.5 12*2 20f3 28 f 4 28 4 27 5 D Otm2/s) photobleaching (pm2/s) 13.1 17.5 27.0 37.0 For comparison, some diffusion coefficients determined by photobleaching are shown (Meller," 22 "C).The errors are calculated from the errors of intrinsic lifetime, TM, and decay rates, kl.
*
10"
*
1;
probe contents:
80
-8 & tu
t.
-I23.2 %
**e10.0 %
-+-5.0 %
75 70
65 60 55
0
100
200 300 400 500 time [nsec]
60
80
100
area per mdecde [A2] Figure 9. Intrinsic lifetime of excimer fluorescence, TE, measured at
different probe concentrations (see text for details).
T 0
I 100
I
I
I
I
200
300
400
500
200
250
time [nsec]
0
50
100
150
time [nsec]
F M L Evaluationof excimer fluorescenceby pseudoexcitation. Upper diagram: measured decay curve, Z&) (data points barely visible); the fitted decay curve;the instrument function,L(t);and the pseudoexcitation function,P(t). Middle diagram: normalized residues (e.&,the deviation of measurement and fit normalized by the standard deviation of each measured data point). Lower diagram: autocorrelation of normalized residues.
a realistic description it should be considered that the reaction probability, p , differs from 1. This enters into the theory by usage of the so-called radiation boundary condition for the diffusion equation instead of the adsorption boundary condition which is normally used in Smoluchowski theory. The radiation boundary condition does not imply a box-likeconcentrationprofile at time zero and therefore no infinite flux of particles occurs at time zero.Z6 A plot of Z(Fo) for varius values of p is given by Jaeger.28 Figure 9 shows the variation of excimer lifetime with lateral packing density. As in the case of the monomer fluorescence, 78 decreases with decreasing packing density. The curves T B versus area measured for different probe concentrations show a striking parallel shift. If one varies the value of probe concentration used in the calculation of the pseudoexcitation within the range of titration error (=lo%), one can remove this shift. The parallel shift can therefore probably be explained by the error in probe concentration. Homogeneity of Probe Distribution. At all conditions, the fluorescence micrographs were homogeneous. However at a molecular fraction of 23.2%of excimer probe and molecular areas
of 170 AZ,the measured decay rates kl were too high to be consistent with the results at lower concentrations. Fitting of the excimer fluorescencedecay with the method of pseudoexcitation is not possible. This leads to the conclusion that under these conditions the probe molecule is no longer homogeneously distributed. This could be due to the fact that one is already near the tricritical point of DMPC under the conditions a~plied.~z IV.2. Excimer Formation in DMPC-Cbolesterol Mixtures. In a second series of experiments, we studied the lateral diffusion in DMPC-cholesterol mixtures in order to study the effect of cholesterol on the local diffusion and to explore the potential application of the excimer technique to study phase separations. We concentrated on the high-concentration regime which was neglected hitherto in most studieg of cholesterol/lecithin monolayers33 and bilayers (cf. ref 34 and 35). For that purpose, the phase boundaries were first determined by analysis of the pressure-area isotherms, and the microscopic organization of the monolayers was inspected by UV microfluorescence. Excimer formation kinetics was studied for two cholesterol concentrations, 40 and 54 mol Q, and for various concentrations of the probe P-Py-PC (seebelow). As will be shown, the mixing behavior of lipid and cholesterol is qualitatively different at these two concentrations. These concentrations were chosen to explore the possible application of the excimer technique to study local phase separation in monolayers. In the case of the high cholesterol content, the measurement of fluorescencedecay will give evidence for a local inhomogeneity that is undetectable by other methods. Pressure-Area Diagrams and Fluorescence Microscopic Findings. Pressurearea diagrams were recorded for various cholesterol contents. All measurements were performed at 20 OC. Figure 10 shows examples of pure cholesterol (Figure loa), a 60:40 cholesterol-DMPC mixture (Figure lob) and pure DMPC (Figure 1Oc). Figure 10a shows that pure cholesterol forms stable monolayers. The area per molecule in the condensed state is 35.5 A2. The most remarkable result is the extremely small lateral compressibility of cholesterol monolayers K~~~
= 6 X lo4 m / m N
(compared to ~b~1 10-3 m/mN for monolayers of lipids). The isotherm of 60:40DMPC:cholesterol (cf. Figure lob) shows two breaks indicated by arrows. Between the breaks the mixture exhibits lateral phase separation into a cholesterol-poor phase
The Journal of Physical Chemisfry, Vol. 98, No. 16,1994
30
0.1
20
0.01
10
Merkel and Sackmann
x
7 I
0.001
..... ...... ~
0.34
0.35 0.36 0.37 Area per Mdecule [nm']
..
_.
j'
0.38
!
'1 a_:
.
breakpoint ! break point
10
I
0.1 0.01
7
5
0.001
0
0.40
0.50 0.60 0.70 Area per Molecule [ m 2 ]
Figure 10. (a) Pressure-area diagram (falling curves, left ordinate) and eompressibility,x (rising curves, right ordinate),of pure cholesterol. (b) Same data for a 3:2 mixture of DMPC and cholesterol. (c) Same data for pure DMPC. Temperature at all measurements are 20 'C.
Fiwurc 11. tIL3rerccnx mlcrJprdph rhuuing thr. ph3rc > r . p d r ~ t i u nm a m m J l i \ e r o l d I.? V\lPC chdtrterol mtxturcat 1 prcrrurc bctucco the two brcak points 0 5 mol '? p)rene probe IC added l.cnprh of the bar
is 50 pm. and a cholesterol-rich phase, as is clearly indicated by the lateral decay of the monolayer into bright and dark patches (see Figure I I). Above the upper break point a condensed homogeneous phase results. With increasingcholesterol content the upper break point moves to lower pressure and the range of area where the inhomogeneous phase is present is decreasing. At a cholesterol content of 54 mol % both break points merge at pressures below 0.5 mN/m, which may be in the gaseous phase. Addition of the pyrene probe to a mixture of 54 mol 56 cholesterol with DMPC leads to a reappearing of the break point as is shown in Figure 18. At this cholesterol content and pressures below 4.7 mN/m bright areasonagray backgroundareobserved (Figure 12a). At pressuresabove 4.7 mY/m, smalldarkareason gray background areseen (Figure 12b). whileat exactly4.7 mN/m bothstructures coexist. It should be noted that the bright structures are quite abundant. On each frame recorded several of them are present.
Figure 12. Fluorescence micrographs taken in monolayers of a 46:54 DMPC:chalesterol mixture with 0.5 mol 9% pyrene probe added at a pressure (a) below 4.1 mN/m and (b) above 4.1 mN/m. Part b shows only one dark domain; it is surrounded by four short black bars. Length of the black bars below the pictures is 50 pm.
In contrast, the dark structures above 4.1 mN/m are very rare. It is roughly twice per minute that one of them crosses the field of view. It is also noteworthy that exactly the same structures are found for all probe contents used. Moreover, the change between the two regimes takes place always a t a pressure of 4.7 mN/m, whereas the breakpoint changesits location. The pressure of 4.7 mN/misabove the breakpoint of thepressure-areadiagram a t all proheconcentrations used. Thus thechange in microscopic appearance and the break point in the pressurearea diagram seem to be unrelated. The change in microscopic structure is reversible, which means there is no difference in compression and expansion. Repetition ofcompression/expansion cycles also yields the same results. FIuorescenceDecayKinetics in DMPC+ 40 mol% Cholesterol. Pyrene probe concentrations of 0.5, 1.5.3.0, and 5.0 mol %were studied. The decay of monomer and excimer fluorescence has been measured and analyzed as described above. The resulting decay curves were fitted by sums of exponentials. At pressures above the upper break point (located a t 8.5 mN/m, according to Figure lob) where the fluorescence micrographs are homogeneous, the decay of monomer fluorescence can be formally described by a sum of two exponentials. In striking contrast, the biexponential fit was not possible below the break point where the fluorescencemicrographsshow inhomogeneity. At least three exponentials were necessary to fit the data in this regime. This indicates that the decay law is too complicated to be analyzed by our present method of data analysis. Tbedecay ofmonomer fluorescence shows a clearly detectable qualitative change with the mixingldemixing transition at the breakpoint. Above the phase boundary (the upper break point), the fluorescence decay of monomer was evaluated as described aboveinorder tomeasure thediffusioncoefficients. Theintrinsic
The Journal of Physical Chemistry, Vol. 98, No. 16, 1994 4437
Nonstationary Dynamics of Excimer Formation
TABLE 2 Relation between the State of Monolayer (Surface Pressure, r, and Area per Molecule, A ) Composed of a 3 2 DMpC:Cholesterol Mixture and Values of the Diffusion Constant D 13.1
17.5
41.2 & 0.6
40.3 0.5 2.7 0.9
3.1 k 1.0
I#
0
1
40
I
I
I
I
1
I
I.
60 80 100 area per mdecule [A2]
Figure 13. Diffusion constant in monolayers of pure DMPC (filled squares) and 3:2 DMPCcholesterol mixture (only in the homogeneous phase, crosses) as a function of mean area per molecule. For the errors of each data point, refer to Tables 1 and 2.
lifetime of monomer TM was for all pressures TM
= 244 f 12 ns
The results for the diffusion constant at different probe contents were consistent within the experimentalerror. The error-weighted averages (eq 10) are shown in Table 2 and plotted in Figure 13 together with thedata for pure DMPC. Thesuccessfulapplication of our model to the monomer decay and the finding that the excimerdecay could be well fitted by the model of pseudoexcitation show that the mixture is homogeneous on the molecular length scale of excimer formation at pressures above the break point. Theoretical decay curves of monomer fluorescence were calculated as described in section 111.1 and fitted by sums of exponentials. At all reasonable choices of parameters T M (150250 ns) and D (1-50 pm2/s), the resulting decays could be fitted with two exponentials. This provides further evidence that the triple-exponentialdecay of monomer fluorescencebelow the break point (phase boundary) is a result of the inhomogeneity of the mixture. Fluorescence Decay Kinetics in DMPC+ 54 mol % Cholesterol. Measurements of monomer and excimer decay were performed at 0.5, 1.7, and 3.0 mol % pyrene probe and various lateral pressures, namely, 0.1, 1.5, 4.7, 9.0, 21.0, and 35.0 mN/m. All decay functions of the monomer could be well fitted by a sum of two exponentials. It is important to note the following: while the microscopicappearance of monolayer changes qualitatively from many bright domains to a few dark domains at a pressure of 4.7 mN/m, there is no change within this pressure domain in the fluorescence decay of monomer and excimer. Decay curves measured above and below 4.7 mN/m cannot be distinguished. The measured lifetimes T~ of the more slowly decaying exponential do not vary with monolayer pressure between 1 and 35 mN/m. The resulting averaged values are 71 = 235, 199, and 195 ns for probe concentrations of 0.5, 1.7, and 3.0 mol %, respectively. Diffusion-controlledexcimer formation plots of kl (= 1/ T ~versus ) prove concentration show an upward deflection from linearity at high probe concentrations. This can be seen in Figure 5a at probe concentrations above 5 mol %. Provided the excimer formation was diffusion controlled within a homogeneous phase, one would expect that 71 should be smaller than 171 ns at a probe concentration of 3 mol %. Thediscrepancy to the measuredvalue of 195 ns cannot be due to experimental errors. The measured lifetimes TI have an uncertainty of less than 5 ns, and a trivial error in probe concentration can be excluded by the results for
*
30.1
38.9 f 0.5 2.5
1.1
the intensity ratio I'/Z of excimer and monomer fluorescence: I'/I = 0.23,0.62, and 0.99 for probe concentrations of 0.5, 1.7, and 3.0 mol %, respectively. Therefore, we must conclude that the monolayer phase is not homogeneous at this cholesterol content, although the decay can be formally described by a sum of two exponentials. A lateral inhomogeneity of the monolayer is seen by fluorescence microscopy. On a pm length scale, this inhomogeneity changes drasticallyduring compression. Despite this, the behavior of the fluorescence decay kinetics is not at all affected by compression. That means the long-range inhomogeneity alone is not the reason for thebreakdown ofour kinetic model; otherwise, the change from plenty of bright domains to a few dark domains should be reflected in the kinetics of the fluorescencedecay. This supports the above conclusion that the patches of the bulk phase, exhibiting a size of many pms below 4.7 mN/m and filling nearly the whole film above, are actually inhomogeneous on a smaller length scale. Obviously this inhomogeneity occurs on a small length scale that is seen by excimer formation kinetics (typical length scale is some nms). We will address this point again in section V.3.
V. Discussion V.1. Excimer Formation in Membranes Does Not Follow FBrster-Birks Kinetics. The fact that monomer and excimer emission can be formally represented by a sum of two exponentials could suggest that the excimer formation may be described by Farster-Birks kinetics.15J6 Farster and Birks explained the biexponential behavior in terms of the interplay of excimer formation and excimer dissociation while the excimer formation rate is time-independent. Their kinetic model holds for the formation of excimers in organic solvents (for a review, see ref 16). It predicts the following: thedecays of monomer andexcimer are linear combinations of two exponentials with equal decay times. Measurements of the decay times, 71 and 72, for monomer and excimer fluorescencemust yield equal results in the formulas
where the amplitude ratio for the excimer fluorescence, aE, is
aE= 1 Both predictions on the lifetimes and the amplitude ratio are in striking contrast with our observations. By fitting the excimer decay with a sum of two exponentials, we find values for aE from 1.2 to 1.6 for all experiments. The average value for 47 measurements is 1.4 with a standard deviation of 0.16. In not a single measurement of a was a value of 1 or below found. Artificially fixing aE to 1 lead to a dramatic decrease of the quality of the fits. This fact cannot be explained by spurious addition of monomer signal at a wavelength of 475 nm. A contribution of monomer signal leading to a~ of 1.2 at 23.2 mol % probe would absolutely dominate the signal at a probe content of 1.4 or 0.5 mol % because the relative excimer quantum yield decreases with probe content. This was not observed. The set of decay times (71, 72) determined for the monomer fluorescence decay is not identical with the corresponding set for the excimer decay. For small probe contents, the decay of excimer is remarkably faster than the monomer decay. At high probe concentrations the opposite holds. Consider as an example the
4438 The Journal of Physical Chemistry, Vol. 98, No. 16, 1994
0
100
200
300
400
time [nsec]
Figure 14. Measured decays of monomer (390 nm), I&), and excimer (475 nm), IE(t),of 23.2 mol 96 pyrene probe in DMPC at a molecular area of 99 A=.
data at a molecular area of 60 A2. At probe contents of 0.5,l S, 5, and 23.2 mol % we find the following values for the longest decay time 71 of the monomer decay: 228, 181, 123, and 21 ns and decay times of 177,169,132, and 67 ns for the excimer. The general behavior is the same for all areas. A striking example for the discrepancy in decay times is shown in Figure 14 with monomer and excimer decay at 23.2 mol % probe concentration and 99 A2 molecular area. Biexponential fits yield decay times of 71 = 56.8 ns for the excimer and q = 15.7 ns for the monomer. It is thus obvious from our experimental findings that the classical Fbrster-Birks kinetics does not hold for excimer formationin two-dimensional fluids, as has already been suggested by the results of section I11 and has already been remarked for excimer formation in lipid vesicles.llJ3 Excimer Formation in Membranes Can Be Described as Diffusion-Controlled Reaction in a Two-Dimensional Fluid. The kinetic model used in this work is supported by the following findings: 1. The calculated values of the ratio of the quantum yields of excimer to monomer, @E/@M, and the ratio Z’/Zof the measured intensities at 475 and 390 nm are proportional (see section 111.4). 2. The measured decay times of the monomer for a given state of the monolayer but at different probe concentrations (0.5-23.2 mol 3‘ %) can be described by uniquevalues of the intrinsic monomer lifetime, T M , and the diffusion constant, D. This is a critical test because the values of the measured decay times reach from 15 ns (23.2% probe and 100 A2 molecular area) to 241 ns (0.5 % and 54 A2), thus covering more than 1 order in magnitude (see section IV.1). 3. The diffusion constants determined at a given molecular area but for different probe concentrations coincide within the experimental error (see section IV.l and Figure 5b). 4. Excimer decays could be fitted by the method of pseudoexcitation. The values of the intrinsic monomer lifetime, T M , and the diffusion constant, D, are already known. The only free parameter is the intrinsic excimer lifetime, T E . If our model would be wrong, it would be quite improbable that we can reproduce a curve with the form shown in Figure 8 with only one free parameter. Moreover, it would be extremely improbable that the resulting values of T E would reasonable agree for all different probe concentrations. 5 . The measured values of the diffusion constant agree with literature data obtained by a different technique (see section V.2 and Table l), and the values of the intrinsic lifetimes agree reasonably well with literature data for model substances (see below). 6. The amplitude ratio aE is substantially larger than 1 (see preceding paragraph). This can be easily explained by the fact that the theoretical expression for the decay of the excimer (eqs 2 and 11) contains an integral over the reaction rate k(t),which is divergent at time zero. This results in a nonvanishing excimer
Merkel and Sackmann fluorescence intensity at time zero. It should be noted that the hypothesis of ground-state dimers is unnecessary to describe the data. Excimer formation kinetics with non-negligible backreaction can be treated analyti~ally.3~ The results are substantially altered in comparison to our model. Therefore, the success of our model also supportsthis assumption. Our model includesthe assumption of a two-dimensional fluid. In reality, the pyrene probe is part of the monolayer, a thin smectic liquid crystal film. The polar headgroups are confined to the membrane surface, and the hydrocarbon chains are oriented preferentially parallel to the membrane normal. Neutron scattering experimentsshowed8 that rotations of the hydrocarbon chains of the lipids take place within nanoseconds. This means that on the time scale of our experiment the pyrene molecules encounter many times with the right orientation, and the monolayer can be described as twodimensional liquid. The situation would be different for reactions taking place on a picosecond time scale. Comparison with Photostationary Experiments. Photostationary measurements of excimer formation are fairly easy to perform. For this reason, excimer probe studies of membrane fluidity are commonly studied by this techniq~e.~2’However photostationary measurements yield only one parameter: the ratio of quantum yield of excimer and monomer fluorescence (@E/@M). Thisquantity has toberelated to thediffusionconstant. The exact dependence of @E/@M on D is given by eqs 1-5 and eq 15. These equations contain several parameters, among them the intrinsic lifetimes of monomer and excimer. In any attempt to measure the diffusion constant as a function of extrinsic parameters by stationary methods, one must carefully ensure that the intrinsic lifetimes do not vary as functions of the same paremeters. Most approaches to evaluate photostationary experiments use the results of Fbrster-Birks kinetics to calculate an apparent rate of excimer formation, kaw This apparent rate is commonly interpreted as a collision frequency of excimer probes, Y-U, which is related to the hopping frequency Y of lipid molecules via
( n ) denotes the mean number of steps a diffusing excimer probe must go on a two-dimensional lattice to find a reaction partner. This quantity has been calculated with different levels of sophisti~ation.~*~* It is instructive to compare the results of this approach with the results of eq 15. In Figure 15 we compare the values of @B/@M as obtained according to the model by Galla et al.9 and our eq 15 as a function of probeconcentration(Figure 15a) and diffusionconstant (Figure 15b). The model of Galla et al. describes the tendency correctly but does not lead to the exact functional forms. In particular, @E/@M is not linearly dependent on D as expected by the model of Galla et al. In our example, the absolute values of D are overestimated by a factor of 3-5. This compares nicely with the results of Sassaroli et al., who performed a study as described above and calculated ( n ) by Monte-Carlo methods. In their work the diffusion constant appeared to be 3 times higher than thevalues determined from photobleachingexperiments. It should be noted, however, that Galla et al. measured values of D that agree quite closely with photobleaching experiments. This comparison shows that photostationary experiments are able to show changes in diffusivity,provided the intrinsiclifetimes of monomer and excimer do not change. However, a determination of absolute values of the diffusion constant by stationary measurements of excimer formation seems to be problematic, especially at high probe concentrations. Effect of Lipid Packing Density on Intrinsic Monomer Lifetime ofP-Py-PC. In our experiments we found a remarkable decrease of the monomer lifetime with increasing area per molecule. This could be due either to changes in residual quenching by molecular
The Journal of Physical Chemistry, Vol. 98, No. 16, I994 4439
Nonstationary Dynamics of Excimer Formation
0
5
10 15 20 25 probe content [“A]
30
10
20 30 40 D~ 2 / s e c ~
50
Figure 15. Comparison of Galla’sprocedure (broken line) and the exact solution (solid line, eq 15) for the ratio of excimer to monomer quantum yield, @E/@.M. Parameters used for the calculation: area per molecule 82 A2, 7.M = 203.2 ns, TE = 65.5 ns k E / k m = 7.7. (a) Variation of @E/@M with probe content (for D = 24.5 km2/s). (b) Variation of @E/ @.M with diffusion constant (for c = 2%).
oxygen or to a different position of the dye moiety with respect to the air-water interface. Comparative measurements of the monomer lifetime, T M , of 6 4 1-pyrenoyl)hexanic acid in solvents of different polarities showed that T M decreases drastically with the solvent polarity: from 250 ns in hexadecane ( e = 2.06, stationary and at the frequency of the sodium D-line) to 130 ns in H20 (e = 78.54 stationary and e = 1.78 at the frequency of the sodium D-line).39 The values for the monomer lifetime of P-Py-PC in DMPC monolayers range from 260 ns at a molecular area of 54 %L2 to 150 ns at 110 A2, thus falling exactly in the same range. This suggests that a similar effect holds in our case. Recent neutron surface scattering experiments on monolayersa show that by expansion of the monolayer the semipolar headgroups, including the glycerol region, become more hydrated. This supports our conclusion that the density dependence of the lifetime is primarily a polarity effect. Judging from recent quasielastic neutron scattering experiments, it is highly improbable that the pyrene molecule actually penetrates into the aqueous phase since the average displacement of the chain ends in the normal direction is only about 5 A.8 It should also be noted, however, that the excited dye molecule is located close to the water surface. Therefore the polarity changes considerablywithin the near field of the oscillating dipole, and it is difficult to determine the polarity in the region of the dye molecule quantitatively. At the current moment there is no theory describing the polarity effects on fluorescence of pyrene in inhomogeneous media. Quenching by oxygen is a rather unlikely explanation of the density effect on 7 M because the lifetimes are not shorter than those in degassed solutions. In a forthcoming paper41 we will report measurements of the fluorescence decay of a pyrene-labeled polymeric lipid dissolved in monolayers of different lipids (DMPC and L-a-dioleylphosphatidicacid (DOPA)). There we will show that the lifetime does not depend on the lipid structure but only on the area per hydrocarbon chain. The monolayer of DOPA (18 C atoms per chain) is much thicker than that formed by DMPC (14 C atoms per chain). It therefore seems rather unlikely that quenching by oxygen should occur at same rates in both layers. This makes oxygen quenching an unlikely mechanism of the density dependence of the lifetime.
Effect of Lipid Packing Density on the Intrinsic Lifetime of Excimer. The intrinsic lifetimes of excimer also show a decrease with increasing area per molecule (see Figure 9). This could be due to viscosity effects as shown in the following. Birks and c o - w o r k e r ~have ~ ~ shown that the rate of internal conversion of the pyrene excimer in solution is temperature dependent. This dependency could be shown to be caused by the viscosity change of the solvent. Internal conversion is enhanced at low viscosities. Motions distorting the symmetrical sandwich configuration of the excimer are enhanced at low viscosities. The presence of these motions enhances the process of internal conversion. This effect could explain why we find high intrinsic excimer lifetimes in highly viscous compressed films and decreasingvalues with expansion of the monolayer. V.2. Lateral Diffusion Coefficient. The diffusion coefficients obtained by the present excimer technique are about 30% smaller than those obtained by FRAP experiments.31 Part of the discrepancy is due to the fact that the latter experiments were performed at a temperature 2 OC higher. Judging from the temperature dependence of the diffusion constant in supported DMPC bilayers? this 2 OC discrepancy accounts for about half of the above difference. So the diffusion constants as obtained by the two completely different techniques disagree by only 15%. This is remarkably good agreement (within the error limits of both techniques) since the FRAP technique measures diffusion over a much larger length scale (some pm) than the excimer technique. The length scale of the excimer technique can be derived as follows. Typical values of 2 0 are 50 pmz/s (the factor of 2 arises because both reactants diffuse). Typical decay times are 150 ns. Inserting these values in ( x z ) = 2Dt gives a value of 4 nm for (( X ~ ) ) O . ~ The . scale of the excimer technique varies with the experimental conditions, but it is always a molecular scale. The present result contrasts with recent measurementsof lateral diffusion in DMPC bilayers by the excimer technique.3’ Those authors found a 3-fold higher value for the diffusivity than that obtained by FRAP. One reason could be that a time-independent excimer formation rate was assumed for the data evaluation. Discussion of Diffusion Coefficient. The dependence of diffusion coefficient on density or temperature of the phospholipid membrane is often interpreted in terms of the free volume mode1.9343344This theory predicts the following dependence of D on the density: aa*
where gand a are geometry factors, a* is the van der Waals area of the lipid, a is the mean area per molecule, and u is the thermal velocity of the phospholipid molecule? Interpretation of the variation of diffusivity with molecular area in terms of the free volume model is only possible for a narrow range of areas (90-70 A2). One can actually fit the D versus density curve over the entire density range, but one arrives at the paradoxical result that the van der Waals area of the lipid molecule, a*,is nearly zero. In other words: the diffusionconstant depends to a good approximation exponentially on monolayer density
D = Doexp(-@a)
(22)
where Do is a suitable chosen prefactor, a is the area per molecule, and B has the value 4 f 0.5 nm-2. This empirical relation describes the Dversus a curve for DMPC and for the 3:2 DMPCcholesterol mixture over the entire range of areas. In the following will provide evidence that this behavior can be related to the dependence of viscosity on the density of an ordinary fluid. The Saffman-Delbruck theory of diffusion predicts for the diffusion in a monolayefl5
4440
Merkel and Sackmann
The Journal of Physical Chemistry, Vol. 98, No. 16, 1994
where 7 denotes the surface viscosity of the monolayer (thickness times bulk viscosity of monolayer), r is the radius of the diffusant, pw the viscosity of water, and y is Euler’s constant. The above equation holds for monolayers, while for bilayers suspended in water the factor of 2 in the argument of the logarithm is absent. This factor arises because the water is only on one side of the diffusant in the case of a monolayer. The surface viscosities of the monolayer were calculated using eq 23. There are only a few published measurements of monolayer thicknesses. From theX-ray surfacescattering data of M6hwald& for DMPE monolayers we deduce the following empirical rule for the variation of the monolayer thickness, d, with the area per lipid, a:
d = 28 A-O.llXa 1
(24)
Since DMPC has a larger headgroup, the numerical factors will be altered, but the difference should be of minor importance. Using eq 24 we can calculate the variation of the monolayer bulk viscosity with volume per molecule. The result is shown in Figure 16. The same procedure has been applied to the 3:2 DMPC:cholesterol mixture. All data (DMPC alone and mixture) can be well described by the empirical relation
.-sr .L
H
10
>
I
I
1 .o
I 1.2
1
I
I
1.4
I 1.6
v d ume [nm3/mdecule] Figure 16. Variationof volumeviscosity(equalssurfaceviscositydivided by thickness) with volume per molecule of monolayers of pure DMPC (filled triangles) and 3:2 DMPC:cholesterol mixture (filled circles) in the homogeneous phase. Shown also is the exponential law (eq 25) obtained for the average value of I9 = 4.4 nm-3 (solid line) and the limits of error of I9 (k0.9nm-3, dashed lines).
-.
30
E
E
20
L
e3 2
e
10
n
0 20 40 60 00 100 chdesterd content [“A] Figure 17. Phase diagram of monolayers formed by mixtures of DMPC with cholesterol at 20 OC for high cholesterol contents and for pressures 0
with 8 = 4.4 f 0.9 nm-3, A law of this form has been established by Frenkel for ordinary fluids.47 The variation of molecular volume in our measurements is remarkably high (60%). This volume dependence of the monolayer viscosity is the reason for the above-mentioned exponential behavior of the diffusion constant (eq 22). That the same law holds for monolayers of pure DMPC and mixtures of DMPC with cholesterol shows that the primary effect of cholesterol on the monolayer is the reduction of area. This important conclusion is also drawn by Galla et aL9 and Almeida et a1.4* for bilayers. V.3. Phase Diagram of Cholesterol-DMPC Mixture. The evaluation of the break points in the pressurearea diagrams and the analysis of excimer formation lead to the phase lines shown in Figure 17. The lower break point that is located always at practically vanishing pressures is interpreted as a transition from the gaseous phase to the fluid phase. The gaseous phase was not further studied and will be neglected in the following. Between the upper and the lower breakpoints in the pressurearea diagrams exists a demixing region (region 3 in Figure 17). This can be seen by fluorescence microscopy and fluorescence decay measurements. At high cholesterol contents (54%) the influence of the probe molecule complicates the situation, as will be discussed further below. The fluorescence micrographs can be rationalized by the assumption that the probe dissolves preferentially in the phase with higher DMPC content. This assumption seems reasonable because the bulky probe finds not enough space between the tightly packed cholesterol molecules. Our data confirm and extend the phase diagram of mixed DMPC-cholesterol monolayers published by Hirshfeld and S e u P to higher cholesterol concentrations. The latter was obtained by means of fluorescence microscopy and pressurearea diagrams. Our experiments show clearly that while regime 1 is homogeneous both regimes 2 and 3 exhibit phase coexistence. In this case, the phase rule would predict that at about x&,,l= 53% the mixture is stoichiometric unless the nearly pure cholesterol precipitates above this limit are squeezed out in the aqueous phase. The latter alternativity seems improbable because pure cholesterol forms stable monolayers.
below the chain freezing transition. Points are measured values. The vertical dashed lines give only a rough estimate of their actual position. Phase 1 is a homogeneous mixture. Phase 2 is a coexistence of a (supposedly)stoichiometricmixtureand (nearly) purecholesterol. Phase 3 coexistenceof cholesterol-rich and cholesterol-poorfluid phases. The regimewhere 1,2, and 3 merge could not be exploreddue to the influence of the pyrene probe on the phase boundaries. Hirshfeld and Seul were not able to determine whether the upper end point (xcbi = 0.27, ?r = 11.5 mN/m) of region 3 (Figure 17) is a critical or a tricritical point. In other words: they could not determine where the phase boundary of 1/2 merges with the boundary of 1/3. Our finding of a homogeneous mixture a t pressures above the break point at 40 mol % cholesterol clearly shows that the phase boundaries of phases 1/2 and 1/3 merge at cholesterol contents of more than 40%. This indicates that the upper end point of phase 3 is a critical point and not a tricritical point. Addition of the excimer probe has a remarkably strong effect on the position of the phase boundary. This is shown in Figure 18. At a cholesterol content of 40%, addition of the excimer probe leads to a decrease of the pressure a t which the upper break point occurs (Figure 18, circles). This is expected because impurities normally favor the homogeneous phase. At a cholesterol content of 54 mol %, addition of the dye molecule complicates the situation. The experimental findings are (1) a reappearance of two breakpoints (see Figure 18-the upper break point moves to higher pressures by increasing the probe content; the lower stays at pressure zero), (2) a dramatic change of the microscopic appearance of the film a t a constant pressure of 4.7 mN/m, and (3) the fluorescencedecay kinetics being independent of the lateral pressure. These facts can be rationalized as follows. The phase diagram as presented in Figure 17 is a phase diagram for a binary mixture. It contains only two-phase regions. The three phases (cholesterolpoor solution, stoichiometricmixture, and nearly pure cholesterol)
The Journal of Physical Chemistry, Vol. 98, No. 16, 1994 4441
Nonstationary Dynamics of Excimer Formation
dependent measurements that showed up in our experiments as a blowing up of the triple point into a small three-phase region. a
VI. General Discussion and Concluding Remarks
0
1
2
3
4
5
6
7
probe content [%]
Figure 18. Influence of the concentration of the pyrene probe on the phase boundaries. Shown are the pressures of the upper break points (compare Figure lob) for a 3:2 DMPC:cholesterol mixture (filled circles) and a 46:54 mixture (filled triangles).
coexist only at the triple point. This is the point where all phase boundaries meet in Figure 17. Addition of the pyrene probe leads most probably to a shift of the inhomogeneous region 3 to higher cholesterolcontents. Furthermore, the triple point is blown up to a three-phase region. This means that near the triple point the phase diagram is changed qualitatively by addition of the pyrene probe. In all other regions the effects should consist only in small shifts of the phase boundaries (as can be seen a t 40 mol %cholesterol, Figure 18). The experimental findings listed above can now be explained as follows: The shift of region 3 to higher cholesterol content explains the reappearance of the upper break point in the pressure-area diagrams. Between the break points the stoichiometric mixture coexists with the cholesterol-poor solution. At the upper break point the three-phase region is entered. The appearance of nearly pure cholesterol domains results in a decreasing compressibility of the film. At pressures of about 4.7 mN/m, the cholesterol-poor solution vanishes and the system goes back in a two-phase region (coexistence of stoichiometric phase and nearly pure cholesterol). The structure seen in fluorescence microscopy can be interpreted as follows: The black domains are formed by nearly pure cholesterol, the gray background is formed by the stoichiometric mixture, and the bright domains are composed of the cholesterolpoor solution. The fact that all three structurescould be observed at the pressure of 4.7 mN/m shows that this is really a threephase region. Most probably the bright domains of cholesterolpoor solution contain many small islands of stoichiometric mixture that are too small to be seen. This would explain the large size of these domains. At 54 mol % cholesterol, the composition of the film is very near the stoichiometric mixture. According to the lever rule for phase diagrams, the overwhelming amount of the material is contained in the stoichiometric complex. This explains why in fluorescence decay measurements no change with pressure is found. The fact that the fluorescencedecay is governed alone by the stoichiometric mixture together with the breakdown of our kinetic model implies that the stoichiometric mixture itself is no homogeneous fluid on the length and time scale of our experiments (compare also last paragraph of section IV.2). The finding that a t 54 mol % cholesterol no homogeneous phase exists is in agreement with SANS (small angle neutron scattering) studies of Knoll et al.,50who studied the phase behavior of mixed DMPC-cholesterol bilayers. In a recent publication,51the twophase region 3 was examined with several different techniques. The authors did not find evidence for a second inhomogeneous phase, 2, neighboring the two-phase region 3. This can be explained by the fact that the triple point is located at nearly zero pressure. The two-phase regions 2 and 3 meet only in one single point. This does not influence their experiments and could have well been unremarked. Moreover, these authors find insolubility of many fluorescent probes in DMPC-cholesterol mixtures above 30% cholesterol. This again illustrates the limitations of probe-
The present analysis of the kinetics of excimer formation in membranes has clearly shown that this bimolecular reaction cannot be interpreted in terms of FGrster-Birks kinetics which holds for excimer formation in three-dimensional fluids of low viscosity. This has indeed already been remarked by several groups studying excimer formation in vesicles.11J3 To our knowledge, only one group14has attempted to interpret their data in a quantitative way by a model of diffusion-controlled reaction in a two-dimensional solution. Unfortunately these authors working with bilayer vesicles could not systematically vary the structural parameters of the lipid layer. The validity of the present analysis in terms of the timedependent rate of excimer formation in two-dimensional fluids, k ( t ) (given by eqs 3-5) is strongly supported by the fact that the decay of the excimer fluorescence can be described by the method of pseudoexcitation. Further support comes from the agreement of the calculated and measured ratio of monomer to excimer quantum yields. The reliability of the present dynamic excimer probe method for the measurement of lateral diffusion coefficients is further demonstrated by the good agreement of the resulting values of D with those from measurements by photobleaching. The present kinetic excimer probe technique was developed with the aim to study the internal dynamics of two-dimensional macromolecules (equivalent to macrolipids dissolved in phospholipid monolayers1*or bilayers52). This type of application will be reported in a forthcoming ~ a p e r . ~ 1
Acknowledgment. This paper was supported by the Deutsche Forschungsgemeinschaft (SFB 266), the Fonds der chemischen Industrie, and the Bund der Freunde der TUM. We are most grateful to Dr. Wolfgang Frey, Hermann Hagn, and Dr. Andreas Zilker for advice and help during the construction of the apparatus. Helpful discussions with E. Evans, D. Pink, and H. MShwald are gratefully acknowledged. References and Notes (1) Sackmann, E. Can. J. Phys. 1990,68,999-1012. (2) Saffman, P. G.;Delbrikk, M. Proc. Natl. Acad. Sci. USA 1975, 72, 31 11-31 13. (3) Evans, E.; Sackmann, E. J. Fluid. Mech. 1988, 194, 553. (4) Merkel, R.; Sackmann, E.; Evans, E. J . Phys. France 1989,50,1989. (5) Pfeiffer, W.; KBnig, S.; Legrand, J. F.; Richter, D.; Sackmann, E.
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