ARTICLE pubs.acs.org/ac
Fluorescence Correlation Spectroscopy in Semiadhesive Wall Proximity Luigi Sanguigno,*,† Ilaria De Santo,† Filippo Causa,‡ and Paolo A. Netti†,‡ † ‡
Center for Advanced Biomaterials for Health Care@CRIB, Istituto Italiano di Tecnologia, P.le Tecchio 80, 80125 Naples, Italy Interdisciplinary Research Centre on Biomaterials, University of Naples Federico II, P.le Tecchio 80, 80125 Naples, Italy
bS Supporting Information ABSTRACT: With examination of diffusion in heterogeneous media through fluorescence correlation spectroscopy, the temporal correlation of the intensity signal shows a long correlation tail and the characteristic diffusion time results are no longer easy to determine. Excluded volume and sticking effects have been proposed to justify such deviations from the standard behavior since all contribute and lead to anomalous diffusion mechanisms . Usually, the anomalous coefficient embodies all the effects of environmental heterogeneity providing too general explanations for the exotic diffusion recorded. Here, we investigated whether the reason of anomalies could be related to a lack of an adequate interpretative model for heterogeneous systems and how the presence of obstacles on the detection volume length scale could affect fluorescence correlation spectroscopy experiments. We report an original modeling of the autocorrelation function where fluorophores experience reflection or adsorption at a wall placed at distances comparable with the detection volume size. We successfully discriminate between steric and adhesion effects through the analysis of long time correlations and evaluate the adhesion strength through the evaluation of probability of being adsorbed and persistence time at the wall on reference data. The proposed model can be readily adopted to gain a better understanding of intracellular and nanoconfined diffusion opening the way for a more rational analysis of the diffusion mechanism in heterogeneous systems and further developing biological and biomedical applications.
F
luorescence correlation spectroscopy (FCS) is a technique that allows the analysis of kinetic and transport processes through the temporal correlation (ACF) of the fluorescence intensity signal collected from a confocal detection volume (DV) defined by adequate optics. The ACF underlines the lifetime of the intensity fluctuations that arise from the motion of the fluorophores in and out of the DV under the action of the Brownian forces. The lifetime of the fluctuations is related to the time needed to travel a distance equal to the characteristic size of the DV. This time is named characteristic diffusion time, td, and corresponds to the lag time at which ACF shows a prominent decay. Originally introduced to observe chemical reactions1 and Brownian motions2 in solution, FCS and related methods have been extended to investigate diffusion in heterogeneous media. Indeed, the FCS technique was recently used to monitor diffusion in crowded biological environments,3 at membrane interfaces,46 as well as cells712 and nanosized systems as nanochannels1315 and thin films.16 Generally, with the examination of diffusion in heterogeneous media, the ACF shows a long correlation tail and the td results are no longer easily evaluated from the ACF.17,7 Several explanations have been proposed to justify such deviations from the standard behavior. Deviations are usually ascribed to the heterogeneity of the environment that can promote sticking events,18 compartmentalization,12 as well as the presence of low mobility populations.19 r 2011 American Chemical Society
Numerical calculations and analytical models have been proposed to clarify the effects of heterogeneities on the ACF. Most of them account for the possibility that fluorophores can be fully or partially confined concluding that the presence of obstacles can lead to erroneous results and misinterpretations if confinement effects are not properly taken into account.7,17,20 Obstacles are usually treated as ideal reflectors, which means that fluorophores can collide with them only elastically. Anyway in some real cases, fluorophores could persist on the obstacle surface for a given finite time experiencing an inelastic hit. Collision timing depends on surface chemistry and chemical affinity. When collision extent becomes appreciable, fluorophores can be considered bound to the surface. Some available models account for spatially distributed binding events18 but become weak when binding occurs on specific regions of space such as obstacle surfaces. Therefore, we modeled the ACF in a nonhomogeneous space where FCS measurements are carried out near to a fixed and impermeable wall, since it represents the simplest model to account for the presence of obstacles on the DV length scale. Our analytical form predicts the ACF changes as a function of the distances between the center of the DV and an impermeable Received: May 16, 2011 Accepted: September 20, 2011 Published: September 20, 2011 8101
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wall. We derive a formulation for rigid and fully adhesive walls and proposed a way to combine them to interpret the effects of temporary adhesion at the wall. Furthermore when sticking effects are not negligible, we highlighted the need to account for slowdowns tangentially to the surface. To validate our approach, the model predictions were compared with reference data relative to FCS experiments carried out in nanochannels and close to a vesicle membrane.
’ STANDARD AUTOCORRELATION FUNCTION FCS is based on the statistical analysis of fluorescence intensity fluctuations due to fluorescent probe trafficking in a small volume of the sample where light is focused, which is named DV. The fluctuations are temporally autocorrelated, and the resulting ACF contains information of probe diffusion time. The ACF can be predicted by averaging all possible trajectories that the probe can cover in a given time interval (lag time). Whenever the environment is homogeneous, Brownian dynamics is adopted to describe fluorophore displacements by time. Brownian motion assumes that fluorophores are considered identical, indistinguishable, and independently moving from the others. Under these assumptions, the ACF provides information that can be interpreted in terms of a single molecule averaged behavior. Briefly, in the simple case of freely diffusing fluorophores, an analytical temporal correlation expression of the intensity fluctuations can be derived by assuming the intensity profile of the DV to be a Gaussian prolate ellipsoid,2 ID ðrÞ ¼ I0 EQ e2ðx
2
þ y2 Þ=rxy 2
e2z =rz 2
2
ð1Þ
where rz and rxy are the radius of the prolate ellipsoid, respectively, along the axial directions, x, y, and z; I0 is the laser intensity at the center of DV, Q is the quantum efficiency of the fluorophore, and E is the collection efficiency of the optical system. Under this further assumption, the ACF of a signal I(t) fluctuating around its mean value ÆIæ is defined as GðtÞ ¼
ÆδIð0ÞδIðtÞæ 1 ¼ ÆNæ ÆIæ2
RR
ID ðrÞID ðr0 Þpðr, r0 , tÞ dr dr0 Z ID 2 ðrÞ dr
ð2Þ
where the averages are taken over time and the stationarity of the system has been further assumed.21,22 In eq 2, the δ operator indicates a quantity deviation from its mean value, ÆNæ is the average number of fluorophore inside the detection volume, and p(r,r0 ,t) represents the probability density for a single fluorophore that started a random walk at time t = 0 at point r0 to be at r at time t. According with the Brownian dynamics description of the fluorophore motion, this probability density can be formulated as ðr r0 Þ2 1 pðr, r0 , tÞ ¼ pffiffiffiffiffiffiffiffiffiffi e 4Dt 4πDt
ð3Þ
where D is the fluorophore diffusion coefficient. Equation 3 represents the probability of being away from the starting position as the result of Brownian steps along three independent directions, which allows one to decouple the double integral in eq 2 along the three axial directions and rewrite the ACF in terms
of its axial contributions, GðtÞ ¼
1 gx ðtÞ gy ðtÞ gz ðtÞ ÆNæ gx ð0Þ gy ð0Þ gz ð0Þ
ð4Þ
with ZZ ðk k0 Þ2 1 2 02 2 2 gk ðtÞ ¼ pffiffiffiffiffiffiffiffiffiffi e2k =rk e2k =rk e 4Dt dk dk0 4πDt
ð5Þ
where k represents an arbitrary axial direction. The double integral in eq 5 can be computed by using different approaches obtaining the following expression pffiffiffi π rk gk ðtÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 þ t=td
ð6Þ
where td is the characteristic diffusion time defined as rk2/(4D).
’ LIMIT SOLUTIONS FOR RIGID AND FULLY ADHESIVE WALLS We represent an obstacle as a rigid wall separating the space into two regions, one fluorophore void and the other characterized by a certain concentration of fluorophore particles. The wall is located at the origin of the reference system adopted, while the center of the DV is placed at a distance d from the wall, see Figure 1a. Furthermore, the wall is assumed perpendicular to an arbitrary axial direction k, as shown in Figure 1a, and considered an ideal reflector. Fluorophore movements across the wall are forbidden, since when colliding, fluorophores are reflected back to the solution. In the case of FCS measurements in finite system size, we already showed that the ACF is obtained by folding the probability density function reported in eq 3, inside the finite space available to diffusion. In addition we proofed that the probability density function folding can be avoided opportunely modifying the intensity profile, eq 1, thus recovering the infinite system size condition, see Figure 1b. Here the same approach is adopted to formulate the ACF expression in case of a single wall in the detection volume proximity, 1 gk ðtÞ ¼ pffiffiffiffiffiffiffiffiffiffi 4πDt Z þ
0
Z
2
e
ðk0 dÞ2 rk 2
0
2
e ∞
∞
Z
∞
2
e
ðk dÞ2 rk 2
e
ðk k0 Þ2 4Dt
dk
0
ð k dÞ2 rk 2
e
ðk k0 Þ2 4Dt
dk dk0
ð7Þ
where k0 and k represent the initial and final positions of the fluorophore path covered in a certain lag time t. In eq 7, the integration domain of k0 is reduced to the interval 0 to ∞, whereas in eq 5 was the entire space, since now fluorophores begin their path only at the wall right side. Possible end points k are extended to the whole space through the reflection of the intensity distribution at the wall, see Figure 2a. The double integral in eq 7 is then computed through the procedure reported 8102
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Figure 1. Geometry of the considered case along an arbitrary direction k. The wall separates space into a region one containing fluorophores with an uniform average concentration (k g 0) and the other containing no fluorophores (k < 0). (a) Section of the prolate ellipsoid representing the intensity profile of DV, the filled part of the Gaussian corresponds to the detection volume portion, whose fluorescence is detected by the optical system. (b) Probability density function for a particle close to a wall, thick red line, resulting from the addition of the unperturbed density function and its reflection at the wall for k > 0.
Figure 2. Graphical representation of the intensity profile, blue line, and probability density function, red filled line, for rigid, sticky, and semiadhesive wall behaviors. (a) Rigid wall case. The intensity profile is reflected at the wall in place of the probability density function. This allows one to recover the open volume shape of the probability density for the ACF computation. (b) Fully adhesive wall case. To all k values less than 0 is assigned the intensity at the wall, and the open volume probability density is used for the ACF. (c) Semiadhesive wall case obtained as a combination of rigid and fully adhesive models. Only a portion of the probability density function is instantaneously reflected to the solution side while its complement is returned back delayed of twall.
in the Supporting Information, 0 c2, i 2 ! c3, i B c þ 2c c 2, i 1, i 5, i 4c e 1, i c4, i erfc pffiffiffiffiffiffi pffiffiffi B 5 2 c1, i π rk B rig B gk ðtÞ ¼ pffiffiffiffiffiffi B c1, i 2 Bi ¼ 1 B @
∑
þ
5
∑ i¼4
c2, i 2 c3, i e4c1, i c
!!1 c2, i C pffiffiffiffiffiffi 4, i erf C 2 c1, i C C pffiffiffiffiffiffi C c1, i C A
wall after colliding. This last state is emulated by modifying the intensity distribution reflection, which is imposed to be constant and equal to the intensity value at the wall, see Figure 2b. As a result, all fluorophores, reaching the wall, emit a unique intensity value corresponding to the one of the DV intensity profile at the wall, see Figure 2b. Under this condition, the ACF contribution along the generic axial direction k is eq 5 and modified as follows Z∞ Z∞ ðk0 dÞ2 ðk dÞ2 ðk k0 Þ2 1 2 r 2 2 k e e rk 2 e 4Dt dk gk ðtÞ ¼ pffiffiffiffiffiffiffiffiffiffi 4πDt 0 0 Z 0 2 ð0 dÞ ðk k0 Þ2 2 þ e rk 2 e 4Dt dk dk0
ð8Þ
where the coefficients cj,i are dimensionless numbers that mainly depend on the ratio t/td. The coefficient expressions are reported in the Supporting Information together with the results of Montecarlo (MC) simulations23 used to verify the procedure reliability. Afterward, we further extended the model in order to account for collisions on a sticky wall. The geometry of the system is kept unchanged, see Figure 1b, though fluorophores now persist at the
∞
ð9Þ which is then computed following the procedure reported before (see the Supporting Information)
gkadh ðtÞ ¼ 8103
pffiffiffi 3 π rk 2 i¼1
∑
! c2, i pffiffiffiffiffiffi 4, i erfc 2 c1, i pffiffiffiffiffiffi c1, i
c2, i 2 c3, i e4c1, i c
ð10Þ
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that return back to the solution side after twall since the collision took place, see Figure 2c. This formulation produces an apparent slower mobility and a sudden correlation drop at twall, see Figure 3, due to the release of adsorbed fluorophores. Generally, fluorophores could experience a range of twall values since their release from the wall is not deterministic. To account for this eventuality, we introduced a further variable σw, which represents the variance of the twall distribution, and by averaging we obtained gksadh ðtÞ ¼ gkadh ðtÞ þ pwall gkr ðtÞ Z ∞ ðt t Þ2 1 w wall þ ð1 pwall Þpffiffiffiffiffiffiffiffiffiffiffiffi2ffi e 2σw 2 gkr ðt tw Þ dtw 2πσw ∞ Figure 3. Semiadhesive wall model predictions for FCS experiment in wall proximity. The laser focus was set at 0.1 rxy. The probability of being reflected back to the solution was 0.5, and the persistence time at the wall was 100 td. Predictions are computed at increasing σwall. As the persistence time distribution becomes broader, the transition between the adhesive, dotted line, and not adhesive behavior, dashed line, becomes smoother.
The expression of the coefficient set and its MC verification are again reported in the Supporting Information. In this context, we neglected the effects of fluorophore bleaching24,25 and of changes in photophysical properties26,27 that could arise because of the long persistence of fluorophores in the adsorbed state. The fully adhesive case is a model extension consisting of an intermediate step necessary to the achievement of a more realistic situation in which fluorophores adsorb on the wall for a finite time before being reflected back to the solution.
’ COMBINING BOTH SOLUTIONS TO SIMULATE A SEMIADHESIVE WALL A more complete scenario of particles diffusion near the walls would account for an intermediate behavior between rigid and fully adhesive walls. Indeed, fluorophores could instantly be reflected after colliding or could stay adsorbed for a given time interval. Therefore, not all fluorophores impacting the wall experience the same dynamics; some are reflected with probability pwall, while others remain absorbed with probability 1 pwall. Those adsorbed return to the solution after a certain time period, twall, related to the wall adhesion strength. To model this complex condition, we acted again on the reflection of the probability density function. The ACF contribution due to the reflected fluorophores is calculated as the difference between the rigid and the fully adhesive wall ACF models derived above, rig
gkr ðtÞ ¼ gk ðtÞ gkadh ðtÞ
ð11Þ
Then, the semiadhesive wall model is obtained as a combination of the fully adhesive wall model eq 10 and the ACF contribution related to reflected fluorophores eq 11, which results in splitting into two parts. gksadh ðtÞ ¼ gkadh ðtÞ þ pwall gkr ðtÞ þ ð1 pwall Þgkr ðt twall Þ
ð12Þ
The term including grk(t) multiplied by pwall represents the contribution due to fluorophores instantaneously reflected back to the solution after collision. The term including grk(t twall) multiplied by 1 pwall accounts for the adsorbed fluorophores
ð13Þ This results in a smoother transition from adhesive to rigid wall behaviors at increasing σw, see Figure 3, making the model more reliable for FCS data analysis. Equations 12 and 13 are the weighted averages between sticky and not sticky wall behaviors. Since both recover the standard ACF at increasing wall distances from the center of DV, this property is transferred to the derived semiadhesive models. Furthermore, it is important to highlight that wall adhesion affects mobility also tangentially to the wall. Indeed, when the fluorophore stops on the wall, it gets immobilized along all directions causing a longer correlation both orthogonally and tangentially. Describing rigorously this further effect is out of the aims of this work. Here, we assume reasonably that the induced lateral mobility reduction is caused by adhesion and is being treated as an increase of the tangential characteristic correlation time. The effect becomes relevant at decreasing wall distances from the center of DV and not negligible when walls lie inside the DV.
’ RESULTS AND DISCUSSION Our theoretical studies, see the Supporting Information, validate the thesis that obstacles strongly affect the ACF shape. Deviations due to the presence of a vertical obstacle along a direction occur even if measurements are carried out away from obstacle surface. Indeed they become effective as soon as the center of DV distance from the wall is 5 times the characteristic DV size. When the center of DV is placed within a distance 1 to 5 times the characteristic DV size from the wall, the fluorophore’s reflection on the obstacle surface leads to longer correlations, since the probability of getting back to the DV results increased. On the other hand, when the center of DV is closer to the wall, which is in the 0:1 range, fluorophore reflection occurs within the DV. The effect is an apparent rise of the characteristic diffusion time, since longer distances have to be covered in order to escape along the only available side. Moreover when the center of DV is located at the wall, the ACF recovers its standard behavior. This is due to the symmetry of the configuration that makes the intensity reflection needed to account for the presence of the wall trivial. Furthermore, adhesion on the wall itself has deep influence on the autocorrelation function, causing again longer correlations. We combined nonadhesive and sticky wall models to interpret the effects of temporary adhesion at the wall. The basic principle of our model is the fluorophore number conservation inside the region where they freely diffuse through the probability density function reflection at the wall. Time delaying and scaling of the probability density function reflection 8104
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Analytical Chemistry have been suggested as a possible approach for the adhesive case, compliant with single molecule features of the FCS technique. Figure 3 shows that near a sticky wall, the ACF strongly deviates from the standard model having a sudden correlation drop at twall. We show that as the persistence time distribution becomes broader, the transition between the adhesive and nonadhesive behavior becomes smoother. The exotic shape is not easily interpreted if wall effects are not fully recognized. In addition, the characteristic parameters related to adhesion, twall, pwall, and σw resulting from our modeling, depend on the magnitude of the fluorophore adsorption at the wall and provide a way to estimate wall adhesiveness. Furthermore, we highlighted the need to account for slowdowns tangentially to the surface, especially when measurements are run within a sticky wall. Indeed, adhesion produces longer correlations along all directions since the fluorophore persists in a fixed position for a finite time. Chaotic sticking events are assumed along the wall tangential directions. Therefore tangentially to the wall, we treated the adhesion as an apparent reduction of the probe mobility according to the FCS stationarity hypothesis and not randomness of sticking events. Consequently, the shape of the correlation function is modified, and the average residence time in the focal volume is no longer linked to the diffusion coefficient by the standard model. In order to validate our model, FCS data from the literature were taken into account. The proposed model was validated through data fitting of some FCS experiments in nanoslits of micrometric width and in proximity of a giant vesicle membrane, both recently reported in the literature. Petrasek13 performed ACF measurements of Alexa 546 water solutions in silicon oxide channels having similar or several times larger width than the horizontal characteristic length of the focal volume. They reported deviations in the ACF shape due to the lateral confinement at several distances from the channel wall. Accounting for the insufficient sensitivity of the optical setup over fluctuations occurring on nanometric lengths, Petrasek reported that 2D ACF standard models were unable to provide physically reasonable motilities approaching the wall. It was shown that the ACF recovers the 2D behavior obtained in larger channels at 1.8 μm from the channel wall, which is about 6 times the focal volume characteristic length, (rxy = 0.32 μm). The other ACF curves were qualitatively interpreted due to a lack of an adequate model for a FCS experiment in semi-infinite systems. Therefore, we use the proposed approximated solutions to fit the ACF of Alexa546 relative to the 0.2, 0.4, 0.6, and 1.8 μm distances from the channel wall. First, we treat the wall as a nonadhesive ideal reflector, eq 8, using the characteristic diffusion time as the unique fitting parameter for each ACF curve, (results not reported). The ideal reflector model well represents the changing in the ACF due to the wall proximity predicting an apparent decrease of the overall fluorophore mobility while approaching the wall, see Figure 5. Though the apparent decrease justifies the ACF measured, its physical explanation is not straightforward since it is recorded at a length scale several times larger than the characteristic fluorophore size. Away from the wall, at 1.8 μm, the model recovers the standard ACF shape and the characteristic diffusion time which becomes equal to the value estimated by the standard model, 0.26 ms. The apparent fluorophore slowdown recorded on the focal volume length scale could be explained assuming that fluorophores reside at the wall before being released back to the detection volume. Thus, we interpreted the same data set accounting
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Figure 4. ACF curves measured for 10 nM solution of Alexa 546 in water, at different distances from the channel wall, fitted to eq 13 and (b) combined with the standard model for the ACF part relative to the wall tangential direction. All experimental data, open symbols, are taken from Petrasek et al.13.
Figure 5. Overall apparent increase of the characteristic diffusion time, open symbols, obtained by eq 8 and progressive increase of the characteristic diffusion time relative to the tangential direction, full symbols, obtained by eq 13.
for wall adhesion through two parameters, the probability of being reflected, pwall, and a characteristic persistence time, twall, at the wall, eq 13. The pwall and twall are simultaneously found through nonlinear regression for all data adopted as a characteristic diffusion time along the orthogonal direction of the one measured away from the wall (at 1.8 μm), whereas the tangential diffusive coefficients were left unconstrained. Fit qualities, Figure 4, were almost the same compared to previous analysis as well as the reliability of estimated diffusion times, 0.61, 0.56, and 0.31 ms at 0.2, 0.4, and 0.6 μm, respectively. The tangential diffusion time now sharply increases at distances comparable with focal volume length, see Figure 5, and the gathered slowdown arises from wall adhesion characterized by a pwall value of 0.30. The persistence time, twall, was 2.3 times bigger then the characteristic diffusion time while the width of the twall distribution, σwall, was 0.8 td. According to those last results, the gathered tangential slowdown is reasonable since once the particles hit the wall, they persist there a time comparable with td, which justifies the characteristic diffusion time doubling along the wall tangential direction. Fradin et al.6 carried out FCS measurements in the equatorial plane of several different giant vesicles having radii in the range 515 μm. During the experiment, the sample was moved step by 8105
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Figure 6. ACF curves measured for a 6.8 nM concentration of Cy3streptavidin outside a vesicle, at different distances from the membrane. Membrane infinitely remote data are fitted by the standard ACF model to estimate the characteristic diffusion time, td. All others ACF are simultaneously fitted, continuous lines, to the semiadhesive wall model, eq 13. All experimental data, open symbols, are taken from Fradin et al.6.
step so that the laser focus would come progressively closer to the membrane. Typical ACF curves of Cy3-streptavidin diffusion at low concentrations approaching the membrane obtained in the cited work are shown in Figure 6. A modified diffusion model was used to interpret the data where a constant baseline was added to account for the ACF plateau recorded very close to the membrane where fluorophore binding or adsorption might become significant. All ACF curves measured far from the membrane were very satisfactorily described adopting a modified free diffusion model. Anyway, including a baseline was not enough to fully resolve the ACF shape in proximity of the vesicle membrane. They reported nominal distances, infinitely remote, 0.44, 0.28, and 0.09 times the characteristic focal volume length. Furthermore, an increase of the characteristic diffusion time at reducing distances from the wall was highlighted. We interpreted the measured ACF at several distances from the vesicle membrane through eq 13. Our results are shown in Figure 6. We fitted the ACF recorded away from the wall obtaining a td value equal to 0.28 ms, and then we performed a nonlinear regression simultaneously on all other data. Center of DV distances from the wall are kept fixed as well as td values. The probability of being reflected back to the focal volume was set equal in all ACF while the tangential slowdown was let free. Considering that the ACF is weakly sampled up to the time scale of seconds, fit quality is satisfactory enough. We are in good agreement with what Fradin et al. measured and analyzed, as our model predicts an apparent increase of DV at those distances as the main wall effect. Our theoretical analysis supplied distinct information about orthogonal and tangential fluorophore motion with respect to the wall. By using this feature, we gathered a trend in the characteristic diffusion time tangentially to the wall. According to our previous results, the progressive fluorophore slowdown approaching the wall was predicted, 10 s, 0.39 ms, and 0.28 ms for 0.09, 0.28, and 0.44 rxy, respectively. It is worth noticing that the ACF tangential contribution attains a characteristic diffusion time almost 4 orders of magnitude bigger than the one measured away from the wall, and this increase was again comparable with the estimated twall, which was 3.52 s. At 0.09 rxy, the DV center is almost located on the wall; therefore, the temporarily blocked fluorophores reside under the highest intensity value inside the focal volume deeply affecting the ACF. As measurements are run
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further from the wall, the tangential ACF part sharply recovers the value associated with freely diffusing fluorophores. The vesicle membrane seems to be strongly adhesive, twall and σwall are, respectively, equal to 3.52 and 0.55 s, although the optimum probability of being reflected back to the DV after colliding was around 0.72. The presence of a strongly adhesive wall is also supported by the trend of the average number of fluorophores inside the focal volume, ÆNæ. Indeed, the real focal volume reduces while approaching the wall yielding an increase of the ACF value at t equal to 0, G(0), which is inversely proportional to ÆNæ. Experimentally, the G(0) value is always lower than what is theoretically predicted accounting only for wall excluded volume effects. Therefore some fluorophores persist a longer time inside the DV as the center of the DV approaches the wall. This behavior enforced the thesis that in proximity of the vesicle membrane wall adhesion occurs. Generally, the parameter likelihood is obtained imposing constraints in the nonlinear regression algorithm used to interpret the data. This means that parameter independence could be weak analyzing particular experimental conditions. With the characteristic diffusion time or the wall distance kept fixed from the center of DV, a higher independence is achieved. This could be recommended more when FCS experiments are carried out at a distance longer than the characteristic DV size where the model predicts deviations along the ACF tail.
’ CONCLUSIONS We have shown that when FCS measurements are carried out close to the walls, the standard expression of the autocorrelation function has to be modified since the average residence time in the DV is no longer trivially linked to the diffusion coefficient. The relevance and occurrence of these deviations depend on the center of DV distance from the wall and wall adhesion strength. We proposed a new ACF form for FCS analysis near impermeable and semiadhesive walls. Three wall characteristic parameters, twall, pwall, and σwall have been introduced to fully characterize fluorophore interactions with the wall. Our model explains the long correlation tail of the ACF, which generally arise in heterogeneous media highlighting the contribution of confinement and adhesion. To illustrate the validity of our approach, we interpreted reference data relative to FCS measurements carried out at several distances from a synthetic wall and a vesicle membrane. An original modeling of the ACF in heterogeneous systems was developed, attaining a rational and reliable analysis of the diffusion mechanism without invoking anomalies in the fluorophore transport. ’ ASSOCIATED CONTENT
bS
Supporting Information. Additional information as noted in text. This material is available free of charge via the Internet at http://pubs.acs.org.
’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected]. Phone: +39 081 7685931. Fax: +39 081 7682404.
’ ACKNOWLEDGMENT This work has been partially funded by the Italian Ministry of Research (MIUR) Project, TriPoDe (Grant DM20160). 8106
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Analytical Chemistry
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dx.doi.org/10.1021/ac201116g |Anal. Chem. 2011, 83, 8101–8107
