J . Phys. Chem. 1993,97, 1125-1 133
1125
Convolution Analysis of the Pyrene Excimer Formation in Membranes Robert B. Pansu' and Keitaro Yoshibara Institute for Molecular Science, Myodaiji, Okazaki 444, Japan
Tatsuo Arai and Katsumi Tokumaru Department of Chemistry, University of Tsukuba, Tsukuba, Ibaraki 305, Japan Received: May 27, 1992; In Final Form: October 27, 1992
The mechanism of excimer formation in rigid and fluid membranes is analyzed from the fluorescence dynamics of both monomer and excimer emissions. We analyze the kinetics of the excimer emission using a convolution formalism. In the rigid state of distearyldimethylammoniumchloride bilayers, we observed a discrepancy between the decay of the monomer and the excimer populations. This shows that a fraction of the pyrene probes is isolated and is not submitted to excimer formation. The convolution analysis enables us to estimate with good precision the fraction of isolated monomers. We have analyzed the monomer emission using transient diffusion models. In the rigid phase, pyrene molecules are gathered in the boundaries of the crystalline microdomains of the membrane, and the collisional quenching occurs along these defects. In the fluid state, pyrene molecules spread out over all the membrane, and the excimer formation process occurs is a two-dimensional space.
1. Introduction
permits theanalysis of thedecay at shorter times, when the signalto-noiseratioisstillsatisfactory~ Weshow thattheratioofisoiated The formation of pyrene excimer in phospholipid membranes to quenchable pyrenes in DODAC suspensions does not agree was studied for the first time by Vanderkooi and Callis.' with the Poisson distribution. Since then, pyrene has been extremely useful to probe the local In section 4, we suggest that, in the rigid state, the quenchable polarity2 and vis~osity~*~ in bilayers. The phase transition in population is located in a membrane defect: the grain boundaries phospholipidsbetween the rigid crystalline-likestate and the fluid of the microcrystalline membrane. liquid crystalline state has been monitored by the use of pyrene Section 3 deals with diffusive dynamics in the membranes. e x ~ i m e r .Some ~ of us have recently used the fluorescence of The kinetics of the population of quenchable monomers are pyrene excimer as well as the shape of the EPR of paramagnetic compared in the fluid and rigid states of the DODAC membranes. probes to evidence the disappearance of the phase transition in The analysis of the kinetics of the excimer formation process distearyldimethylammoniumchloride (DODAC) bilayers after reveals that, in the fluid state of the membrane, the excimer sonication of the sample.6 formation kinetics agree with that of a diffusive process in a However, the mechanism by which pyrene excimer formation is sensitiveto phase transition has been a controversials ~ b j e c t . ' * ~ - ~ ~two-dimensional space. In the rigid state, probes gather in membrane defects and the reaction kinetics are compatible with Upon phase transition, an abrupt change in the viscosity is a one-dimensional diffusion model. expected, and it is of common sense to assume that this change will affect diffusion processes. Simultaneously, phase segregation 2. MaterialsandMetbods of pyrene occurs at the phase transition in most membranes. However, these simple hypotheses are not able to fully describe Two types of DODAC samples were used in this study. At thechangeobserved in the excimer dynamics, neither qualitatively room temperature, DODAC membranes in multilayered strucnor quantitatively. tures are in the gel ~ t a t e . ~We ~ . 'have ~ shown, using EPR probes, To elucidate the mechanism of excimer formation in DODAC that, at the same temperature, membranes in small DODAC membranes, we have analyzed the kinetics of theexcimer emission. disks are in the liquid state.6 These two types of sample were The dynamics of the excimer population in viscous media is used to exemplify the gel-state and fluid-state behavior of the generally overlooked by most authors because of its complex pyrene probe. analytical expression. But, by the use of the convolution DODAC (distearyldimethylammoniumchloride) was purformalism,I2we have been able to analyze the excimer rise and chased from Tokyo Kasei. 250 pL of a l e 2M solution of pyrene decay. This is developed in this section. in toluene was added to 140.5 mg (2.5 X lo4 mol) of DODAC This analysis reveals that the population of pyrene monomer in order to obtained a M fraction of pyrene in DODAC. is not homogeneous. A fraction of the monomers is submitted Ethanol was added until a complete dissolution of DODAC was to the excimer formation process in their excited state, thereafter achieved. The solvent was removed by vacuum distillation at entitled as quenchable. The other monomers are isolated probes, room temperature, leaving a homogeneous lacquer. Distilled not submitted to quenching by other pyrene molecules. water (25 mL)was added and DODAC allowed to swell overnight. The measurement of the fraction of isolated pyrene monomers The sample was sonicated with a Tomy Seiko UR 200P sonifier is an important step in the determination of micelle size using the until a clear solution was obtained. The solution was filtered Poisson distribution of the probe~.l3-~~ In section 2, we analyze through a 0.8-pm sonifier filter to remove metallic particles from the ratio of isolated to quenchablepyrene in DODAC suspensions. the sample. A neat DODAC suspension was prepared in the The contribution of the isolated pyrene molecules is usually same way. The ratio F of pyrene per DODAC molecules was measured at longer times when the quenchable population has varied from F = 5 X IO-)to 5 X IO4. These Fs were obtained disappeared.I3 The convolution analysis of the excimer kinetics by mixing the suspension, F = in DODAC, with the appropriate amount of neat DODAC. Mixing at the molecular On leave from Physico-Chimie des Rayonements CNRS UA75, Univenit€ Paris, Sud 91405, Orsay Cedex, France. level was obtained by freezing the aqueous sample and sonicating 0022-3654/93/2097-1125$04.00/0
Q 1993 American Chemical Society
Pansu et al.
1126 The Journal of Physical Chemistry, Vol. 97, No. 6, 1993
i
e
i
-
1 1 . ' -100
I
100
.
,
*
'
"
t
I
300 500 Time / m
700
900
Figure 1. Fluorescence decays of pyrene monomer &,,, = 390 nm) and excimer (Lm = 400 nm). Pyrene molar fraction in DODAC multilayers = 2 X lo-'. The monomer fluorescencs decay is characterized by an initial steepdccay followed by a slow exponentialdecay. Theinitialsteep decay is associated with the rise and decay of the excimer. Since at long times, the excimer decay should mimic that of its parent population, we show that the pyrene monomer population is heterogeneous, composed of two components, one of which is submitted to the excimer formation process and the other not.
again. We thus obtained a clear suspension of DODAC membrane fragments,'* hereafter called disks. Samples were fdtered again before use to remove pyrene microcrystals that are slowly formed from the DODAC suspension. The samples were flushed with argon to remove traces of oxygen. The clear DODAC disk suspension was studied first. The previous sampleswere frozen in liquid nitrogen and warmed back to room temperature. After the freezing proccss, an homogeneous, jellylike suspension was formed. With this prep aration protocol, at room temperature, the DODAC bilayer was in the gel state (T, = 40 0C).'7,19920 Fluorescence spectra were recorded on a Hitachi spectrofluorometer. Time-resolved fluorescence measurements were done by the time-correlated single-photon-counting method. In the nanosecond time scale, we used a Horiba fluorometer, and on the picosecond time scale, we used a synchronouslypumped dye laser system.*' For the analysis of the data, we used the software available at the Institute for Molecular Science. When using the nanosecond fluorometer, the excitation wavelength was fixed around 349 nm by filters (NiS04 solution, U320 and U340 Toshiba filters). The emission wavelength was fixed by a monochromator (slit width 14 nm). Monomer decay was observed at 380 nm, and excimer decay was observed at 500 nm, where both spectra do not overlap.22 To further protect detection from the excitation light scattered by the sample, Y45 and 0 5 5 filters were added for the study of excimer emission. For the picosecond excitation, we used the second harmonics (310nm)ofa Rhodamine6Gdyelaser. Theemissionwavelength was fixed by a monochromator (slit width 4 nm). The emission was measured through a polarizer fixed at the magic angle.
3. ExcimerKhtia 3.1. F l u o " c e Decay Dynamk& The monomer and excimer fluorescence decays are presented in Figure 1. Both have been measured at room temperature in DODAC bilayers. The monomer decay is complex and nonexponential at short times, as expected for a diffusion-limited process. At longer times, the decay rate tends to a value of 366 ns. The excimer emission rises from a very low value during the pulse. This shows that the excimer formationis a dynamicprocess and that excimers are not formed from ground-state dimers in the bilayer or in microcrystals. Such a contamination by microcrystals can be observed in the samples that have not been filtered before use. The decay of the excimer emission occurs at a constant rate of 56 ns, which is different from the 366-ns rate measured for
the parent populationof monomers. As thelifetimeof theexcimer is short (65 ns)23 compared to the decay of the parent molecule, the quasi-stationary-state hypothesis applies and the decay of the excimer should mimic the decay of the monomer?* Following the proposition of Blackwell et a1.,7 our hypothesis will be that the monomer excited-state population is heterogeneous, composed of two components. These two populations of pyrene are characterized by the behavior of their excited states. One is constituted of isolated pyrene molecules whose excited state is not submitted to the excimer formation process. The excited state of the other component is submitted to the excimer formation process; these excited states arc the origin of the excimer population. The influence of the ground-state pyrene concentration on the excited monomer decay is presented in Figure 2. Decays for increasing molar fractions of the probe in the DODAC membrane in the rigid state are presented. Upon an increase of the concentration, the short component of the monomer decay becomes more important; more molecules form excimers. But the decay rate of the population with a long lifetime remains unaffected. This confirms furthermore the identification of this population with a long lifetime as being composed of isolated molecules. From the qualitative analysis of the decay shape and from the effect of concentration on the decay rates, we conclude that the pyrene population is not homogeneous. But this preliminary analysis can be confirmed by a more quantitative study. 3.2. Qualitative Analysis. The kinetic scheme of pyrene excimer formation in homogeneous media is well-known. Our qualitative analysis has shown that two populations, a and b, coexist in DODAC membranes and that the quenching process is dynamic. In DODAC, the kinetic steps are hu
Ma-Ma*
P(t)
(R1)
where &M is the decay rate of the excited state on a isolated pyrene, &E = kD + kMD is the decay rate of pyrene excimer, and k ~ ~ (is tthe ) excimer formation rate. Three assumptions are implicit in this scheme: (i) Excimers are only formed through the quenching of monomers (and not from theexcitation of ground-stateoligomm). (ii) Except for the excimer formation process, excited monomers and the excimer decay at a constant rate. (iii) We do neglect the revenibility of the excimer formation:
E*
-
M,
+ Mb*
kMD (R6) Recently, Martinho and Berberan-Santos have shown that, at room temperature, this process can be neglected compared to the excimer deactivation pathway (R5).35 The evolution rate of the excimer population is given by d[El/dt = k ~ ~ [ M b ] [ M b -k,[EI *l
v+- v..
(1)
In this differential equation, both &DM and (&*I are functions of time: &DM(r) and [Mb*](r). At short times, before diffusion reaches a stationary state, the quenching constant k ~ ~ (is ta )
Pyrene Excimer Formation in Membranes
0
02
06
0.4
The Journal of Physical Chemistry, Vol. 97, No. 6, 1993 1127
08
10
limo / p i
Figure 2. Decay of the fluorescence of the pyrene monomer (Lm= 390 nm) represented for decreasing amountsof pyrenc in DODAC multilayers. From the bottom to the top, pyrene/DODAC ratio is 3 X lO-3,lO-3, and 5 X loJ. Decay is normalized to the same maximum intensity. For comparison, the limiting decay for a null concentration of pyrene in the membraneis represented (- - -). As thepyrenemolar fraction isincreased, the initial decay becomes steeper, whereas the rate of the slow component remains unaffected. The relative magnitude of the steep decay increases with pyrene molar fraction.
function of time. This transient behavior has already been observed in membrane systems because of their high local vis~osity.5*~~J~ The expression of k ~ ~ depends ( t ) on the choice of a diffusion model and on the values of local friction. In the case of microheterogeneous media, the time dependence of the reaction rate can originate also from the heterogeneity in the local environments.l* For the same reasons, the analytical expression of [Mb*](t) is not well-known. This is why analyticalexpressions for the excimer population are rather awkward.2' The convolution analysis offers an opportunity to have a numerical description of the excimer rise and decay from the monomer fluorescencewithout any assumption on the time course of k ~ ~ ( t[Mb*](t) ). is known experimentally from the measurement of the monomer fluorescence. The excimer timedependent concentration is the result of the convolution of the monomer concentration by the exponential decay of the excimer. At a given time t, the excimer population is the sum of the excimers formed before, at time t - T : V+(t-r)dr. The number of excimers formed at time t - T being reduced by the factor exp(-kEr) due to their deactivation is [El (1) = KkDM(t-r)
[Mbl
lMb*1( f - r )
exp(-kEr) dr (2)
Let us first study the long-times limit. The stationary diffusion state is reached, and k D M ( t ) is a constant term kDMm. As M* varies slowly compared to exp(-kEr), [Mb*](t-r) can be extracted from the integral, leading to kDMm[Mbl[Mb*l(f)K exp(-kEr) dr =
The same result can be obtained using the quasi-stationary-state hypothesis: the decay of the excimer population should mimic the decay of the monomer one. The general case is more complex, but k ( t ) can be expressed as a function of experimental values: d[Mb*l kDM(r)[Mbl [Mb*l = p ( t ) 7 - (kf+ knf)[Mb*] (4) where P ( t ) is the excited light time profile.
0
I
I
I
IO0
200
300
Tlme / ns
Figure 3. Pyrene ucimer kinetics (+) in DODAC disks simulated (-) according to relation 6 assuming that all the excited monomers (.) contribute to the excimer formation process. The weighted residuals arc displayed over the decay curves. Pyrene molar fraction in DODAC = 7.5 X 10-3. There is a qualitative agreement betwccn the experimental and simulated curves.
It is reasonable to assume that, except for the process of excimer formation, the other decay pathways for the monomer (kf knf) have a constant rate:
+
[E] ( t ) =
{P(t--7) - d[M,I -(
t--7)
-
dt
(kf+ k..)[Mb*l(t-r)) exp(-kE7) d-7 ( 5 ) In relation 5 , we have removed the time-dependent quenching constant from the equations. This keeps us from doing an hypothesis on the analytical shape of k ~ ~ ( t ) . We show in the Appendix that the convolution product can be reorganized as lMb*1(?) + ( k ~ - k ~ ) K [ M b * ] ( ! _ 7 )exp(-kET) dT
+
KP(r--7) exp(-kE-7) dr ( 6 ) In this relation, the unknown quantities reduce to a few parameters. The time-dependent functions are d e d u d from the experiment. 3.3. Numerical Analysis. We developed an algorithm based on the Marquard search process to adjust relation 5. &E and &E - k~ as well as the two normalization factors of P(t) and [Mb*]( f ) are adjusted by the best fit algorithm. HomogeneousPopulationHypothesis. First wechose toneglect the population of isolated monomers detected by the qualitative analysis. The results are displayed in Figures 3 and 4. Two samples are compared: the DODAC disks (bilayer in a fluid state) and the DODAC multilayers (bilayer in a gel state). Excimer decays are represented by the dotted curves; the computed simulation of the excimer decay is patterned by a continuous line. The population of quenchable monomers is represented by the continuous (noisy) line.
Pansu et al.
1128 The Journal of Physical Chemistry, Vol. 97, No. 6,1993
I
I
I
'
.. . .:
0
I
I
I
IO0
200
300
Time
I
ns
Figure 4. Pyrene excimer kinetics in DODAC multilayers simulated according to relation 6 assuming that all theexcited monomerscontribute to the excimer formation process. The weighted residuals and the autocorrelation function of the residuals are displayed over the decay curves. Pyrcne molar fraction in DODAC multilayers = 7.5 X lo-). There is a strong discrepancy between the experimental and simulated curves.
In the disks, a simulation of the data can be achieved, where the decay rate of the excimer is the only adjustable parameter (kE-l = 32 ns). There is a qualitative agreement between the data points and the model-fitting curve. Even if we have not yet analyzed the decay of the monomer population, we can conclude that, in DODAC disks, most of the excited monomers are submitted to the same quenching process, yielding the excimer population. The monomer population is homogeneous. On the contrary, as shown in Figure 4 in DODAC multilayers, whatever the value of the adjustable parameter, there is no agreement between the simulation and the data. Thus, we have included the hypothesis that a fraction of the excited pyrene monomers is not submitted to the quenching process. Heterogeneous Population Hyporhesis. To take this population into account, an exponential contribution with a lifetime of 320 ns is removed from the total monomer emission:
[M*l(t) (1 -AJ[Mb*I(t) - 4 [ M * o I exp(-kMt) (7) [M'o] is the maximum intensity of the monomer emission. A2 is adjusted manually in order to achieve the best fit after convolution. l/kM was fixed to 360 ns. The results are displayed in Figures 5 and 6. In addition to the monomer, excimer, and simulated decays, the weighted residuals are displayed at the top of the figures. In the case of the disk sample, perfect adjustment ( x 2 = 1.01 for t > 0) is obtained by taking into account the hypothesis of an inhomogeneous monomer population. In the same way, a much better fit of the data is now obtained in the case of multilayered DODAC (x2 = 1.9 for t > 0). The measured values aregathered inTable I. In DODAC disks,an adequatesimulation of data can be done assuming that all the pyrene excited monomers are submitted to the excimer formation process, whereas in DODAC multilayers, isolated pyrenes constitute a significant fraction of the sample.
0
200
100
Time
/
300
ns
Figure 5. Pyrene excimer kinetics in DODAC disks simulated according to relation 6 assuming that the pyrene population is heterogeneous. It is assumed that the population of isolated monomers has an exponential decay kinetics with a 320-ns lifetime and constitutes 9% of the total population. The excimer emission is designed by the dotted curve; the computed fit of the excimer decay is patterned by a continuous line that cannot be distinguished from the experimental curve. The weighted residuals are displayed over the decay curves. There is perfect agreement between the experimental and simulated curves.
The two values for the excimer decay rate (kE)are close to one another for the two types of samples. The measured value of k~ is close to that reported in isooctane ( k =~ 1.8 X IO7 s-~).~~.** The fraction of isolated monomers is represented by Az. As shown by the error indicated in Table I as well as by Figures 3-6, this measurement appears to be sensitive and precise. These monomers constitute more than 30% of the sample in the case of the multilayered sample, whereas they can be neglected for the disk sample. Thus, the quantitative treatment supports the conclusion of the qualitative analysis. In the multilayered sample, an important fraction of the monomers is not submitted to the quenching process. 4. Concentration Effect on the Proportions of Isolated and Quenchable Pyrene
In the case of DODAC disks, the presence of isolated monomers could be due to the small size of the disks. When disks are small enough, only one or less than one pyrene is present on average in a disk. If the pyrene molecules are distributed randomly and if the disks have a narrow size distribution, probes are distributed among the disks according to the Poisson law. Surfactant aggregates with two or more pyrenes give rise to an exciplex emission. Aggregates with one pyrene give rise to a monomer emission. The decay law of the total monomer fluorescence is known and is given by'3
where kM is the natural decay of the isolated excited state, kE(f) is the quenching rate constant, and A is the average number of quenchers per aggregate.
Pyrene Excimer Formation in Membranes 1
The Journal of Physical Chemistry, Vol. 97, No. 6, 1993 1129 I
TABLE I
I
~
param
A2 kh(9S-I
kE. s-’
meaning isolated monom fract kM fixed kE
.
.-
5 0.010 - 0
’+ E 8 0.008.-
disks 0.090 0.005 2.77 X 106 2.65 x 107
Isolated Population Multi Quenchable Population Multi Disks
multilayers 0.362 0.005 3.12 X IO6 2.75 X 106
+
c)
a
e
g0.006-
8
u 0.0043 m e
2 0.002-
I
+ 0
zg
I+
Figure 6. Pyrene excimer kinetics in DODAC multilayers simulated under the sameconditionsas in Figure 5. The parent monomer population obtained after subtraction of the isolated population (36%of the pyrene) is represented. The weightedresidualsaredisplayedover thedecay curves. There is a good agreement between the experimental and simulated curves.
(9) On a semilogarithmic plot, the decay becomes a line at long times. The extrapolationof this line to time zero gives the fraction of isolated probes and gives A, the number of pyrenes per aggregate, and N = A/F, the number of surfactants per aggregate. In Figure 7A, the measured aggregation number (N) is represented vs the molar fraction of pyrene in the DODAC membrane. The value of N decreases with the molar fraction of pyrene. The sample does not follow a simple Poisson model, but the decrease of N with pyrene molar fraction can be due to the polydispersity of the disks in the samples.2g In Figure 7B, the population of isolated and quenchable monomers is represented vs the molar fraction of pyrene in the membrane. The amount of quenchable monomers increaseswith the pyrene molar fraction, where saturation of the amount of isolated monomers is observed. This is in contradiction with the Poisson model, even if the sample is polydisperse in size. In a random distribution model, the fraction of isolated monomers should decrease when the average number of pyrenes per aggregate increases. DODAC disks are large enough to contain more that one pyrene per aggregate. The size of the disks cannot be deduced from a Poisson analysis.
t
lO0Ol
Time / ns
After excitation, the pairs disappear and the decay becomes exponential with the lifetime of an isolated monomer:
Disks Multilayers
0
t
I
+
0
0.005 0.01 Pyrene Molar Fraction
I
I
Y
I
0.01 5
Figure 7. Relative proportions of the isolated and reacting populations of pyrene monomers. In A, a Poisson distribution of the probes among small DODAC aggregates is assumed. With this assumption, the aggregation number of the DODAC microstructures can be deduced. The measured aggregation number is displayed as a function of the pyrene molar fraction. This aggregation number appears to be strongly sensitive to the pyrene molar fraction. As it should not be, we conclude that the Poisson distribution does not account for the relative proportions of the pyrene populations. In B, the relative proportions of the isolated and parent populations of pyrene are represented as a function of pyrene molar fraction. The population of isolated pyrene exhibits saturation as the pyrene concentration is increased, whereas the parent population of pyrene increases with pyrene molar fraction.
lasts a few tens of nanoseconds.’ Depending on the distribution of the probe molecules in the DODAC bilayers, various decay shapes can be expected. 5.1. Assumption. If the probes are uniformly distributed in the membrane, the diffusion process occurs in two dimensions according to the relati0n~~1O In
(y)= -[kMt+ ZR2p0/2(14.180(;y2+ -
5. MonomerI(iwtics
where D is the diffusion coefficient,PO is the probe density in the volume of the membrane, R is the reaction radius, and Z is the width of the bilayer. If the probes are localized at the border of the disks or in some other one-dimensional defects, the decay law should be30J’
From the analysis of the excimer population dynamics, we evaluate the contribution of the isolated monomers to the fluorescence. This enables us to analyze the decay rate of the other population: the one submitted to the excimer formation. The quenching of the pyrene monomer emission by the groundstate monomers is a diffusion-limited process. Due to the high viscosity of the DODAC bilayers, the transitory rate of diffusion
where PO is the probe density in the volume of the membrane defect and S is the section of the one-dimensional defect. 5.2. Qualitative Analysis. Whatever the distribution of the pyrene in the bilayer, the transient diffusion state is characterized
Pansu et al.
1130 The Journal of Physical Chemistry, Vof.97, No.6,1993 wwi) 1
4
Pyrenc in DODAC nrmbrmcr
10.
-4
Figure 8. Kinetics of the parent population of pyrene monomers. The populationdecayisrepresentedasafunctionof?1/2in thecaseof DODAC disks and multilayers on a semilogarithmic plot. In the multilayers, the decay points line up, except at very short times when the laser pulse is still active. This is characteristic of a one-dimensional diffusion process. In the disk sample, the decay exhibits a constant curvature in agreement with diffusion occurring in a plane.
param
kip,s-I
k2,s-'
80
160
320
Time in ns
TABLE II
x2
111 0
meaning
disks
multilayers fit 1 fit 2
fitquality
1.3 1.9 3 201(ZR2p~/(2x))2(D/R2)1.4 X lo6 5.0 X IO7 6.2 X IO7 6.0 X lo6 7 X lo6 4.2 X lo6 k M + (ZR2po/ (2r))3.1 7 ( D / R 2 )
by a short time scale decay that is dominatd by the square root dependence on time. The diffusion dynamics in one dimension are characterized by a less important contribution of the exponential dependence. In Figure 8, the decay of monomer population has been represented as a function of the square root of time. The decay in multilayers is dominated by the exp(N2) term, whereas for disk samples the decay tends toward an exponential decay at longer times. A more rapid decay is observed in the case of the multilayered sample. This qualitative analysis confirms the importance of the transient diffusion state and suggests that the quenchingprocess in multilayers followsonedimensional kinetics. 5.3. N d d Analysis. For a more accurate analysis, we fit the results by the following relation:
(12) where k2and kl/2are the adjustable parameters whose meaning is given in Table 11. kM is the decay rate constant of the isolated monomers. It was fixed during the adjustment. The value of 320 ns for l / k ~is deduced from the previous analysis of the excimer decay. A2 is the fraction of isolated monomers. The value is fixed and is obtained from the analysis of the excimer decay. The fitting program includes convolution by the instrument response and was obtained by a simple modification of the usual fitting software. The results of the fits are displayed on Figures 9 and 10 for the disks and multilayer samples. In addition to the monomer, excimer, and simulated decays, the weighted residuals are displayed at the top of the figures. If we assume a two-dimensional diffusion, the diffusion coefficient (D) and the local pyreneconcentration can be deduced from the values of klp and k2. For this calculation, the hydrodynamic radius and the reaction radius are taken to be equal to the van der Waals23radius: R = 3.46 A.27 The values are given in Table 111. For the disk samples, our value of D is intermediate between the value of 8 X l e 7 cm2s-I (above the phase transition) and 8
Figure9. Pyrene monomer kinetics in DODAC disks simulated assuming that the pyrene population is heterogeneous. Pyrene molar fraction in DODAC = 7.5 X lo-'. The monomer emission is designed by the dotted curve; the computed fit of the excimer decay is patterned by a continuous line. These two curves cannot be distinguished. The weighted residuals aredisplayed over thedecay curves. There isvery good agreement between the experimental and simulated curves.
at Time / ns
Figure 10. Pyrenc monomer kinetics in DODAC multilayers simulated assuming that the pyrene population is heterogeneous. The monomer emission is designed by the dotted curve; the computed fit of the excimer decay is patterned by a continuous line. These two curves cannot be distinguished. The weighted residuals of the residuals arc displayed over the decay curvc~.Pyrene molar fraction in DODAC multilayers = 7.5 X IO-). There isgood agreement between theexperimental and simulated curves. The periodical behavior o k r v e d on the residuals is due to reflection along the delay line, of the detection. X le8cm2 s-I (below the phase transition) reported for phospholipid vesicles.5 We have shown6 that, because of their small size, the disks are still in a fluid state down to room temperature, As the measurement temperature is below the phase transition temperature, it can be expected that the fluidity and the diffusion coefficient will be smaller in disks than in liquid bilayers as observed. In the multilayered samples, a low value (7.7 X cm2s-l) is measured for the diffusion coefficient. But more significantly,
Pyrene Excimer Formation in Membranes
The Journal of Physical Chemistry, Vol. 97, No. 6, 1993 1131
TABLE III param x2
D. cm2 s-I ZR2po
meaning fit quality diff coeff pyrene local a n c n
disks 1.3 1.5 x 10-7
multilayers
0.05
1.2
1.9 7.7 x 10-9
a huge increase of the local pyrene concentration is observed in the multilayered sample compared to the disk sample. Both samples have the same composition, as the multilayered sample is obtained by freezing the disks sample. Thechange in the local concentration reflects a change in the distribution of the pyrene molecules in the membranes. We propose that the increase in the local concentration is due to the gathering of the pyrene molecules on the boundaries of the microcrystal of the bilayer. These grain boundaries are onedimensional defects, and the diffusion in these spaces is characterized by the absence of the quadratic term in 1'12 in the decay profde. Figure 8 supportsthisconclusion,whereas thequantitative analysis is less ciear on that point. In order to estimate the precision of our numerical analysis, the results of two reasonable fits of the diffusion are gathered in Table I1 ( x 2 = 1.9 and 3). Onedimensional diffusion is characterizedby a linear term (k2)that equals the radiative decay, 3.14 X lo6 s-l. The dispersion on the values is too high, and we cannot conclude that the parameter k2 is not zero. From the numerical analysis,we cannot exclude that the quenching process is be confined to the one-dimensional space constituted by the grain boundaries. These grain boundaries may have some fractal dimensions.I I Evaluation of D cannot be made, in the case of a onedimensional diffusion process,without hypothesis on the value of the defect section (S)and on the local concentration (PO). But we can check that the measured value of k1/2 agrees with reasonable hypothesis. In gur case, PO can be deduced from the measurement on the sample in the disk form. Let us assume, as an example, that S = &/4, where Z is the width of the membrane. This gives D = 6.5 X lo-* cm2s-I,which is reasonable, in the order of magnitude. Mainly from the increase of the local concentration upon crystallization, it can be proposed that pyrene is localized in the grain boundaries. 6. Localization of the Pyrene Molecules
The population of isolated monomersis revealed by the analysis of excimer kinetics. The location of these isolated monomers has to be discussed. In the disk samples, we have shown that fhe fraction of isolated monomer does not follow the prediction of the Poisson model. These monomers are not isolated in smaUPODAC disks. Furthermore, for the same molar fraction of pyrene in the membrane, the percentage of isolated monomers is higher in the multilayered sample than in the disk one. The reverse is expected for the Poisson model. The analysis of theexcimer kinetics, based on an independent measurement, should be more semitive and provide a more precise result for the determination of isolated monomersthan the extrapolation of the long time scale monomer fluorescence. It does not imply any assumption on the production rate or on the uniformity of the reaction rate in the sample. It should be of great help in the determination of the aggregation numbers from the Poisson distribution. The analysis of the excimer kinetics reveals that the local concentration of quenchable pyrene is higher in the multilayered sample compared to the disk one. EPR studies show that the membrane in the disks is in a liquid state at room temperature, whereas in multilayers is in the solid state (Tc = 40 "C). It is a well-documented phenomenon in phospholipidsthat, below the phase transition temperature, a phase separation occurs.32*?3In a similar way, we can assume that, upon crystallization,thepyrene molecules are expelled from the DODAC microcrystals in the membrane and gather in some defects.
0
0.003
0.006 0.009 0.012 0.015
Molar Fraction Figure 11. Initial decay rateof the monomer decay in DODAC multilayers as a function of the pyrene molar fraction. The increase of the rate with pyrcne molar fraction is indicative of a bimolecular process.
The defects can be intrinsic ones: the defects already exist in the membranes and pyrene molecules are dissolved in them. Or they can be induced by the pyrene molecules: they gather to form microdomains, rich in pyrene, and induce a local fusion of membranes. In other words, the phase diagram of the mixture of pyrene and DODAC presents an eutectic one. At room temperature, thegel DODAC, slightly contaminated with pyrene, would be in equilibrium with liquid DODAC rich in pyrene. The concentration effects observed in Figure 11 offer an answer. It is observed that, upon increasing the pyrene concentration, the rapid initial decay of the monomer population increases. This rate is measured from the half time of the initial decay which is obtained after subtraction of the contribution by the long-lived population. Thus, the local concentration of pyrene increases with the molar fraction. If defects were induced by pyrene, the defects would increase their size at the fmed local concentration in pyrene. We conclude that defects are intrinsic ones, of fixed size, and that pyrenes dissolve and accumulate in these defects. The 'isolated" monomers are some pyrene molecules that are soluble in the microcrystals. Due to the reduced solubility of the pyrene in the microcrystals,the efficiency of the excimer formation is reduced. In addition to the reduction of the local concentration, the increase in viscosity can contribute to the reduction of the excimer formation. A ratio of 10 has been reported for the membraneviscosityin the gel and in the fluid states5 The diffusion area of the excited state is similarly reduced by a factor of 10 in the gel domains compared to the liquid disks. Both effects contribute to the reduction of the quenching efficiency of pyrene molecules in microcrystals and make them behave as isolated probes. In disks, we have also observed the presence of a small amount of isolated probes. The disks are in a fluid state; thus, pyrene molecules cannot be isolated in microcrystalline disks. But as the structure of DODAC suspensions is governed by kinetics laws, the solution is not necessarily homogeneous. The presence of isolated pyrene in disk samples is due to some disks that are large enough to be vesiclesl*or be in the gel state. 7. Concl~iOlLo
We have been able to simulate the kinetics of pyrene excimer in micelles without any assumption on a diffusion model. This enablesus to measure the populatien of isolated pyrene monomers in the DODAC bilayers. This computation protocol is a sensitive and precise method to analyze the composition of pyrene population is microheterogeneous media. It provide us a useful method in the analysis of the Poisson distribution of probes in microdomains. In DODAC, the quenching process is controlled by diffusion. The monomer kinetics shows that, in the DODAC disks at room temperature, the pyrene probe diffuses all over the
Pansu et al.
1132 The Journal of Physical Chemistry, Vol. 97, No. 6, 1993
plane in the membrane, whereas in multilayers, pyrene molecules are repelled from the crystallinedomainsand gather in membrane defects where they are still submitted to a diffusion-limited quenching process.
Let us now introduce the response function of the detection, F(r). The measured signal Se(r)is, in fact, the convolution of the signal by the detection response: SB(t) = F( ) @ B( )
AcLaowkdgmeot. R.B.P. thanks the Japan Society for the
Promotion of Science for fellowship.
Appendix Filiation Hypotbcsia Convolution formalism is a convenient way to expresschemicaldynamics.12 Furthermore,some efficient algorithms exist for the numerical estimation of the convolution product by exponential. This has been applied in particular to pyreneexciplex formation for the test of diffusion-limited proaases in three dimensions.35 Recent developments of this approach focus on the introduction of reversibility in pyrene excimer formation into the formalism.34.35 In this section, we show that, for irreversible reactions, it is possible to check that a transient B is produced from a transient A even if the reaction rate between the species is unknown and not Constant in time. Let us assume the following scheme:
X -c A
P(f)
B
k(r)
A-
A
-
products
Because of the associativity of the convolution product, this can
be reorganized as
k,
where P ( t ) is the pulse shape function. In this kinetic scheme, we assume that only the conversion rate, k(?), is a function of time and that the other decay pathways of A have constant rates, k,. The decay rate of the parent compound is given by
dA = P ( t ) - k(t)A - k,A dr
(All
In relation with (A2), the decay of B is expressed as a function of experimental data, Sp(t)and SA(t),and a few adjustable parameters. In addition, the first term is identical to the formula used by the softwares of fluorescence analysis. Most softwares, developed for the analysis of fluorescence decay, perform the convolution of an assumed decay law by the response function of the measurement setup and compare the result with the measured signal. Typically, they assume that the emission decay law is a sum of exponentialsand then perform the convolution and compare it with the experimental data. The formula adjusted by the program is
where SP(T)is the response function of the measurement setup. The first term of relation A2 and all of relation A3 are identical. The decay profile SA(^) takes the place of the response function SP(T),and a good approximation of the Dirac function a(t-7) is given by ko exp(-ko(t-r)), if Ilk0 is short compared to the other characteristic times: sB(t) = J-ksA(.)
and the concentration of B as a function of time is given by
B(t) = J-:(formation B(t)
rate(r))(surviving fraction(t-7)) dr
J-:k(r) A ( r ) exp(-kE(t-r)) dr
According to relation A1 ,
which after integration by parts gives
which can be reorganized as
[-kO exp(-kO(t-r)) (&E
+
- k,) exp(-k~(t-T))] dT + J-:Sp(r)
exp(-kE(r-r)) dr (A4)
where ko is a fixed parameter and k~ and k~ - k, are adjusted by the best fit software. Extensions. The rqsult can be generalized to any type of decay for B. We have assumed that the decay rate ( k ~of) compound B was constant. But it is acknowledge that the decay of most relaxation phenomena can be approximate as a finite sum of exponentials. Relation A4 can be easily extended in the case where the surviving fraction is a sum of exponentials. On the contrary, we have assumed that, except for the conversion rate, all the other decay pathways of the A compound have a constant rate. This hypothesis is a key one and cannot be reduced. This calculation shows that, except for the second term, the usual software developed for the study of fluorescencedecays can be used to analyze the kinetics of the secondary products even if their formation rates are not constant with time.
Refer"
and Notes
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77 1-2. . -
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