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Translational Diffusion of Fluorescent Proteins by Molecular Fourier Imaging Correlation Spectroscopy

Michael C. Fink , Kenneth V. Adair , Marina G. Guenza ^*  and Andrew H. Marcus  

^* Institute of Theoretical Sciences, Department of Chemistry, University of Oregon, Eugene, Oregon; and Oregon Center for Optics, Eugene, Oregon

Correspondence: Address reprint requests to A. H. Marcus, E-mail: ahmarcus{at}uoregon.edu.

	ABSTRACT

TOP ABSTRACT INTRODUCTION THEORETICAL BACKGROUND EXPERIMENTAL METHODS RESULTS AND DISCUSSION CONCLUSIONS APPENDIX A APPENDIX B ACKNOWLEDGEMENTS REFERENCES

 
The ability to noninvasively observe translational diffusion of proteins and protein complexes is important to many biophysical problems. We report high signal/noise (250) measurements of the translational diffusion in viscous solution of the fluorescent protein, DsRed. This is carried out using a new technique: molecular Fourier imaging correlation spectroscopy (M-FICS). M-FICS is an interferometric method that detects a collective Fourier component of the fluctuating density of a small population of fluorescent molecules, and provides information about the distribution of molecular diffusivities. A theoretical analysis is presented that expresses the detected signal fluctuations in terms of the relevant time-correlation functions for molecular translational diffusion. Furthermore, the role played by optical orientational degrees of freedom is established. We report Fickian self-diffusion of the DsRed tetramer at short timescales. The long-time deviation of our data from Fickian behavior is used to determine the variance of the distribution of the protein self-diffusion coefficient. We compare our results to the expected outcomes for 1), a bi-disperse distribution of protein species, and 2), dynamic disorder of the host solvent.

	INTRODUCTION

TOP ABSTRACT INTRODUCTION THEORETICAL BACKGROUND EXPERIMENTAL METHODS RESULTS AND DISCUSSION CONCLUSIONS APPENDIX A APPENDIX B ACKNOWLEDGEMENTS REFERENCES

 
The translational motion of biomolecular species in complex environments is an important intermediate process in many biochemical reactions (1). Examples include the diffusive search preceding the assembly and disassembly of biomolecular complexes, the binding and release of substrates with receptors, and the switching of signal transduction proteins between active and passive states. In the cell, diffusion occurs through a suspension of macromolecules and lipids, many of which form molecular assemblies and compartments of varying complexity (2–4). The cell regulates its activities by coordinating the expression of proteins and other species that ultimately translate to their target sites. Clearly, measurements that monitor the motions of biological molecules through cell compartments can provide insight into subcellular organization and mechanisms of intracellular processes.

Because of the small volumes and opacity of most cell compartments, methods that characterize the movements of intracellular species require high signal sensitivity. Fluorescence methods are well suited to cell studies because they present strategies that allow the signal to be enhanced while suppressing background and noise. Furthermore, constructs of fluorescent proteins are often used to selectively label biomolecular species, which are expressed in living cells. In principle, single-particle tracking techniques can follow the motions of N labeled macromolecules through their local environments, and thus measure the dispersion of the observed motions. Indeed, live cell molecular imaging and tracking experiments have been reported with increasing frequency in recent years (5–9). Yet, despite their power and utility, all single-fluorophore experiments are subject to the disadvantages associated with detecting weak signals barely above ambient noise levels. The problem of low-signal-detection in the presence of noise is complex (10), but for our purposes we take the following simplified view. The limiting factors are: 1), reduced signal/noise (S/N, generally ), which gives rise to 2), reduced temporal resolution (the time interval between successive measurements); and 3), reduced duration of a set of measurements (due to eventual photodegradation of the fluorescent labels). These effects place practical constraints on the dynamic range accessible to a given experiment, as well as to its sensitivity to deviations from dynamically homogeneous behavior. In the cell, proteins interact with a myriad of intracellular species over a wide range of timescales, which ultimately gives rise to a broad distribution of dynamical behaviors. The purpose of this work is to present a new and alternative method to probe such distributions.

In this article, we demonstrate molecular Fourier imaging correlation spectroscopy (M-FICS) to characterize the translational dynamics of a small population of fluorescent proteins. Specifically, we present measurements in viscous solution of freely diffusing DsRed, a fluorescent protein that exists as a stable tetrameric complex at physiological and low salt concentrations, and over a wide pH range (11–13). We have chosen DsRed to demonstrate our method because of its exceptionally high extinction coefficient ( = 3 x 10⁵ M^–1 cm^–1) and fluorescence quantum yield (q_F = 0.79) (14). In our experiments, the tetrameric complex is suspended in 95% glycerol/water mixtures. These studies are carried out using a sensitivity-enhanced version of our recently developed FICS technique (15–17). M-FICS is a complementary method to single-molecule imaging because it exchanges unneeded spatial information for improved S/N, temporal resolution, and dynamic range (16). Similar to single-molecule spectroscopy, M-FICS has the ability to characterize distributions of molecular parameters. Our experiments monitor the emissive fluctuations resulting from the spatial overlap of an optically resonant intensity interference fringe pattern and a small population of diffusing chromophores (see Fig. 1). The phase of the excitation pattern is continuously swept, resulting in a modulated fluorescence signal. We record the phase and amplitude of the signal using a lock-in amplifier to determine the complex-valued Fourier transform of the fluctuating (real-valued) local chromophore density, defined at the wave vector of the optical grating. From these data, we construct the intermediate scattering function (ISF), which provides a quantitative description of the protein molecular displacements. Furthermore, we characterize the departure of the ISF from Fickian behavior in terms of the variance of the distribution of protein diffusivity.

View larger version (31K): [in this window] [in a new window]   FIGURE 1  (A) Schematic of the M-FICS experimental geometry. A small population of optically resonant molecular dipoles is illuminated by a focused laser interference pattern (represented as gray-scale bars). The transverse beam profile is Gaussian with a waist, w 100 µm. The expanded view shows three fringes across the center of the beam profile; the minimal fringe number used is 30. A molecular configuration is specified (in three dimensions) by the set of vector positions {r1,r2,...,rN} and orientations . The fluorescence signal depends on fluctuations of the local orientational density, defined by Eq. 12. (B) At any instant, a static configuration of molecular dipoles is uniquely described by a sum of phasors in the complex plane, as described by Eq. 15.

In all previous FICS studies, the number of fluorescent chromophores that decorate each particle is large (10³). Such systems are optically isotropic since the absorption and emission probabilities are independent of particle orientation. In this case, the fluctuating signal contains contributions due solely to particle center-of-mass motions. A different situation arises when the particles are themselves optically anisotropic molecules. The system then has a continually fluctuating instantaneous anisotropy, due to the tensorial relationship between the polarization of the excited state population and that of the exciting laser field. If the polarization of the excitation and emission fields is specified, the signal can, in principle, contain steady state and time-dependent contributions from molecular orientational degrees of freedom. Future polarization-selective M-FICS studies could be used to separate the effects of transition dipole reorientation from center-of-mass degrees of freedom. In the present work, optical anisotropy fluctuations either occur on a much shorter timescale than the shortest experimental integration period (5 ms), or are effectively removed using a magic-angle polarization scheme. Thus, optical anisotropy fluctuations contribute only a time-averaged constant factor to the signal. In principle, even with magic-angle detection, there could exist signal contributions due to excited state population dynamics, such as intersystem crossing to long-lived nonradiative dark states. Such photoconversion phenomena have been observed to occur in DsRed on sub-millisecond timescales, and at excitation intensities much higher than those applied in the current studies (a factor 10⁴) (20–22). For the experiments presented below, we show that the signal fluctuations are dominated by contributions from molecular center-of-mass displacements and are unaffected by photoconversion processes. Our results suggest that our technique is well suited for studies of molecular diffusion.

The remainder of this article is organized in the following manner. In Theoretical Background, we relate molecular density fluctuations to the relevant time correlation functions for self-diffusion. In Experimental Methods, we explain the M-FICS method, including instrumental and technical details that were implemented to achieve the necessary sensitivity. In Results and Discussion, we present our results.

	THEORETICAL BACKGROUND

TOP ABSTRACT INTRODUCTION THEORETICAL BACKGROUND EXPERIMENTAL METHODS RESULTS AND DISCUSSION CONCLUSIONS APPENDIX A APPENDIX B ACKNOWLEDGEMENTS REFERENCES

 
Molecular fluctuations and the self-intermediate scattering function
In the following M-FICS experiments, we study the fluctuating emission from the fluorescent protein, DsRed. The protein molecules, which are suspended under dilute conditions in an unlabeled host fluid, are at equilibrium and randomly exchange momentum with the surroundings (23). We treat the system as N Brownian particles in a macroscopic (illumination) volume V, with mean number density . Because our measurements are sensitive only to molecular center-of-mass displacements (see Introduction and following sections), we focus our discussion on the center-of-mass coordinates; however, in Experimental Methods we discuss the potential role played by orientational degrees of freedom.

A local fluctuation from the mean number density is given by

(1)

where is the number density operator for fluorescent molecules at time t, and (x) is the Dirac delta function.

As explained below, the measurement observable in an M-FICS experiment is proportional to a Fourier component of the density fluctuation of the fluorescent molecules, n(r,t), with a specified wave vector, k. This k-space number density operator is given by the spatial Fourier transform,

(2)

Since n(r,t) is real-valued, . We may therefore expand n(r,t) in terms of its Fourier components,

(3)

Measurement of allows us to calculate its autocorrelation function, called the intermediate scattering function (ISF),

(4)

In Eq. 4, the angle brackets indicate an average over all starting times, t, and S(k,) remains nonzero for within the time interval over which successive center-of-mass configurations are correlated. The ISF quantifies the degree of correlation between the Fourier component of the local density fluctuation at time t with that at time t + . It is related by Fourier transformation to the van Hove space-time correlation function, G(r,) (24),

(5)

where

(6)

The van Hove function (Eq. 6) represents the conditional probability that a molecule will be found at position r at time , given that either the same or a different molecule was at position r = 0 at time = 0. The ISF (Eq. 4) contains all of the information necessary to characterize the dynamics of the fluid system and is the natural starting point for theories of the liquid state (24).

We now specialize to the current situation: measurements performed under dilute conditions (10 nM). Because the mean separation between molecules is much larger than any natural length scale of the system, the cross-terms in Eq. 4 representing distinct molecule-molecule interactions contribute minimally to the sum, leaving the self-terms to dominate (24–26). Thus, in the dilute limit, Eq. 4 reduces to

(7)

F_s(k,) is called the self-part of the ISF; it describes the temporal correlation of self-displacements of the molecular center-of-mass. Equation 7 can be further simplified by using the Gaussian model for single particle displacements (25). The Gaussian model is a good approximation when the measurement timescale is large compared to the decay time of the velocity autocorrelation function. In this case, the self-displacement vector r() = r(t) – r(t + ) behaves as a Gaussian random variable, and the self-ISF becomes

(8)

where we have assumed the Fickian relationship between the mean-square displacement r²() and the translational self-diffusion coefficient D_s. For a fluid of mono-disperse spherical particles undergoing Fickian dynamics, the self-diffusion coefficient is given by the Stokes-Einstein relation, D_S = k_BT/6a, where k_B is the Boltzmann constant, T is the temperature, is the viscosity of the medium, and a is the molecular hydrodynamic radius.

Equation 8 predicts that the ISF decays exponentially in time and as a Gaussian with increasing wave number. One often observes deviations from Fickian behavior when considering self-diffusion of polydisperse systems (25) or of identical particles immersed in dynamically heterogeneous media (27). Various particles then exhibit a distribution () of dynamical behaviors, which may reflect differences in particle size, shape, or local environment. In this case, the observed ISF reflects a weighted average over this distribution

(9)

where = k²D_s. In principle, Laplace inversion of the measured ISF could determine (), although this would require exceptionally high S/N. It is less demanding to determine low-order moments of the distribution. An expansion of Eq. 9 at short time yields the second-order truncated cumulant approximation (28–30),

(10)

where the parameter is a measure of the dispersion of the underlying distribution. We return to Eqs. 8 and 10 in Results and Discussion.

	EXPERIMENTAL METHODS

TOP ABSTRACT INTRODUCTION THEORETICAL BACKGROUND EXPERIMENTAL METHODS RESULTS AND DISCUSSION CONCLUSIONS APPENDIX A APPENDIX B ACKNOWLEDGEMENTS REFERENCES

 
Molecular-FICS
M-FICS is an extension of the FICS method introduced by Grassman et al. (16,17) to study diffusion of uniformly labeled fluorescent particles. Here, we have greatly improved its sensitivity to detect the center-of-mass fluctuations of a small number of diffusing molecules. An important aspect of M-FICS is that the molecular motions of interest occur on timescales much longer than the excited state lifetime. Fluctuations of the steady-state fluorescence then reflect slow changes in molecular coordinates. We present measurements on dilute viscous solutions ( 10 nM) of DsRed in 95% glycerol/water. The mean number of molecules is 470,000, in a volume V 78,500 µm³ (as described below). Our measurements span a range of time (10^–3–10¹ s) and distance (1.2–3.5 µm) scales sufficient to characterize the translational diffusion under these conditions.

We use a continuous-wave laser to create an intensity interference fringe pattern that illuminates a fluid sample positioned in a fluorescence microscope (Fig. 1 A). The laser resonantly excites an electronic transition of the molecules, which we treat here simply as N molecular dipoles. Molecules are indicated schematically in Fig. 1 A as white disks, whose center-of-mass and transition dipole orientation coordinates are specified by the vectors {r₁,r₂,...,r_N} and , respectively. The vectors collectively represent the absorption ( ) and emission ( ) transition dipole moments, which are fixed in the molecular frame. Since the fringe spacing d_G is small compared to the focused laser beam waist w (100 µm), we approximate the intensity profile as an infinite plane wave (31,32)

(11)

We have chosen a right-handed coordinate system with the wave vector of the optical grating oriented parallel to the y axis, , and the electric field polarization parallel to the z axis, . We also define a time-varying phase (t) = _Gt + _G, where _G is an angular sweep frequency (_G = _G/2 10 MHz), _G is a constant phase, and I₀ is a constant intensity.

A local fluctuation of the chromophore coordinates is described by the function

(12)

where is a slowly fluctuating molecular intensity, proportional to the i^th molecule's excitation/emission probability, and is its time-averaged value. We assume that the molecular intensities fluctuate on timescales longer than the experimental integration period (5 ms) and the fluorescence lifetime (_F 3.5 ns) (33). Furthermore, the intensities are explicit functions of the orientational coordinates, since they depend on transition dipole coupling to the excitation and emission fields. To streamline our notation, we write the intensities as explicit functions of time, A_i(t). Equation 12 thus describes the fluctuation of the optical orientational density of the system, as each molecule contributes a center-of-mass coordinate weighted by its orientation-dependent intensity. On the experimental timescale, the steady-state molecular intensities fluctuate due to slow motions of the molecular transition dipoles.

At any instant, the emitted fluorescence I_f(t,) is proportional to the excited state population generated by the spatial overlap between the laser intensity I_ex(y,) and the local orientational density , defined according to Eq. 12:

(13)

The form of Eq. 13 shows that the instantaneous fluorescence is proportional to , the Fourier component of the local orientational density evaluated at k = k_G.

We sweep the phase of the grating, (t), across the sample at the velocity _G = _G / k_G (10 m s^–1), orders of magnitude faster than the average speed of molecular diffusion. We thus generate a time-varying excited state population that is proportional to the emitted fluorescence. We use the waveform that sweeps the grating as the reference for lock-in detection. The carrier frequency of this reference (10 MHz) sets the upper limit to the experimentally accessible bandwidth. Signal fluctuations that occur within the experimental bandwidth give rise to a slowly varying envelope function that multiplies the carrier signal. We take advantage of the separation in timescales between the signal fluctuations and the inverse modulation frequency by writing the signal as a two-dimensional function of the slow variable t, and the fast carrier phase, (t) = _Gt + _G. The resulting fluorescence intensity has the form (16,17)

(14)

Equation 14 shows that the signal consists of two parts; a stationary (dc) background and a time-varying (ac) component. The background is proportional to the mean number of optically oriented molecules contained in the illumination volume (i.e., . The ac signal has amplitude, , and phase, , proportional to those of the complex Fourier component of the orientational density fluctuation

(15)

We use a lock-in amplifier to demodulate the signal in terms of the in-phase and in-quadrature components, and , respectively (17). As depicted in Fig. 1 B, at any instant, the Fourier component, , is a vector sum of N single molecule terms. Each molecule contributes an intensity, A_i, and a phase, _i = k_G · r_i = k_Gr_y,i, to the measured value of . Because k_G points in the direction of the y-axis, only the y-projections of the r_i values contribute to the molecular phases. Fig. 1 B illustrates the relationship between the detected signal, , and the center-of-mass positions r_i of the labeled species (see Eq. 15). If the molecular positions are randomly distributed, the signal amplitude scales roughly as the end-to-end vector of a Gaussian random variable, i.e., N^1/2. In this limit, the ratio of the signal modulation amplitude to its mean value scales as the number density fluctuation, i.e., .

The vector fluctuates in the complex plane due to collective orientational and translational motions of the molecules. We determine the time- and wave-vector dependence of to construct the ISF defined by Eq. 4. A general expression for the time correlation function is

(16)

We further define the normalized ISF, F(k,) = S(k,)/S(k), where S(k) = S(k, = 0) is called the static structure function.

At this stage, we make the following assumptions: 1), that the center-of-mass positions of different molecules are statistically independent, and 2), that there is no coupling between rotational and translational diffusion. This is consistent with a dilute solution of molecules undergoing Brownian diffusion (far above the glass transition temperature and neglecting pairwise particle interactions). Equation 16 then simplifies to

(17)

Equation 17 shows that the ISF is the product of two terms. The first, A(t) A(t + ), depends on molecular optical orientational dynamics, and is independent of k_G. The second term is the self-part of the ISF, F_s(k_G,) (defined by Eq. 7), which depends only on the molecular center-of-mass positions. For the purposes of this work, we wish to isolate the translational dynamics from possible anisotropy contributions to the signal. This is accomplished by selecting the polarization of the fluorescence using an analyzer oriented at the magic-angle (MA = 54.7°) relative to the excitation polarization. We discuss the functional form of A(t) A(t + ) and its dependence on optical orientational dynamics in the next section.

Orientational time-correlation function
The molecular intensities, A_i(t), are proportional to the square of the projections of the absorption and emission transition dipole moments onto the excitation and detection electric fields (25,34,35), . The intensities therefore depend on the laboratory fixed polarizer and analyzer orientations, and the probability distribution of the molecular transition dipoles. If we used a long focal length lens for detection, the polarized intensity transmitted through the analyzer would be the square of the projections of those dipoles that emit rays parallel to the optic axis. However, we must account for the more complicated situation associated with the high numerical aperture (NA) objective lens used in the M-FICS geometry (see Fig. 2 A). We place the analyzer behind the objective's back-aperture, so that the polarized intensity is the sum of the squared projections of all rays emitted over the solid angle subtended by the lens. Axelrod treated this problem in detail, showing that the polarized emission from a single dipole contains contributions from field projections onto all three orthogonal laboratory frame axes (36,37). A straightforward calculation (see Appendix A) shows that the molecular intensity transmitted through the analyzer, oriented at the magic-angle (MA = 54.7°) relative to , is

(18)

where the intensity at time t is conditionally dependent on the molecule being excited at time t' < t. In Eq. 18, the proportionality constant _MA accounts for the absorption cross-section, fluorescence quantum yield, and light collection efficiency of the experimental setup. The function P₂(x) = (3x² – 1) is the second Legendre polynomial, which appears in the instantaneous fluorescence anisotropy ; its argument is the projection of the absorption transition dipole moment (at the time t' of excitation) onto the emission transition dipole moment (at the detection time t). The constants K_a, K_b, and K_c depend on the numerical aperture of the objective lens:

(19a)

(19b)

(19c)

where the angle ₀ is defined by NA = n₀ sin ₀, and n₀ is the refractive index of the medium. We note that in the small aperture limit, ₀ 0, K_c 1, and K_a,K_b 0, such that the magic-angle intensities become independent of the anisotropy as expected, i.e., (38). For our experiments, NA = 1.4, n₀ = 1.518, and ₀ = 67.3°, for which K_a = 0.2440, K_b = 0.0157, and K_c = 0.7404. Thus, the effect of high NA is to mix into the magic-angle intensity a finite, albeit small, anisotropy contribution.

View larger version (39K): [in this window] [in a new window]   FIGURE 2  (A) Schematic diagram of the experimental setup for M-FICS (described in text). (Abbreviations have the following meanings: APD, amplified photo-diode; M, mirror; AO1(2), Bragg cell 1 (2); BS, beam splitter; PBS, polarizing beam-splitter; PS, periscope; and /2, half-wave plate.) A spatially and temporally modulated interference fringe pattern illuminates a sample positioned at the object plane of a fluorescence microscope. The pattern is produced using a Mach-Zehnder interferometer (MZI) with acousto-optic Bragg cells placed in the two beam paths. The image of the excitation pattern is magnified and relayed through a Ronche ruling, and the resulting temporally modulated intensity is focused onto an APD. The APD signal is used as negative feedback to a servo, which corrects the relative path lengths of the MZI (via a piezo-mounted mirror), to minimize passive phase instabilities. The weak fluorescence from the sample is collected in an epi-configuration, filtered according to frequency and polarization (at the magic-angle), and detected using a second photon-counting APD module. The signal is processed using a time-of-flight histogram generator (described in Appendix B). A lock-in amplifier demodulates the resulting analog output. (B) A typical time-course for the random phase error (RPE) with feedback loop engaged (red, RMS RPE = 4 nm) and disengaged (black, RMS RPE = 300 nm), measured for dG = 1.0 µm.

(20)

For a system characterized by one fluorescence lifetime _F, we have P_c(t – t') = exp[– (t – t')/_F]/_F (35). Because the experimental integration time (5 ms) is much larger than _F (3.5 ns) (33), all absorption/emission events appear instantaneous in comparison. We may therefore take the limit P_c(t – t') (t – t'), and using Eqs. 18–20 obtain

(21)

In Eq. 21, we have used the numerical aperture values given by the expressions in Eq. 19, and have defined

(22)

as the (slowly) fluctuating steady-state fluorescence anisotropy of the i^th molecule. We regard as a stochastic variable that fluctuates about its mean value given by the integral of the P₂ time-correlation function

(23)

For example, a symmetric top molecule undergoing rotational Brownian diffusion has , and Eq. 23 yields the Perrin equation, , where the rotational diffusion coefficient is given by D_R = k_BT/8a³ (23,38).

Using Eq. 21, we calculate the two-point orientational time-correlation function

(24)

where we have adjusted the proportionality factor to make the leading term in the parentheses equal to one. Equation 24 expresses in terms of the (stationary) mean fluorescence anisotropy, , and its two-point time-correlation function . While describes the average rotation time of the molecular transition dipoles on the timescale of the fluorescence lifetime, describes the regression of spontaneous fluctuations of the function about its mean value, i.e., . We note that the two-point time correlation function is more precisely defined as a four-point time correlation function (with t₁ < t₂ << t₃ < t₄),

(25)

over which the time intervals between adjacent excitation and detection events, t₂ – t₁ and t₄ – t₃, have been integrated in Eq. 22. Multitime correlation functions of this type are often discussed in the context of nonlinear spectroscopic measurements, and can potentially contain information about slow intramolecular structural dynamics or even pairwise molecular interactions. As a simple example, if the optical orientational fluctuations were due solely to isotropic rotational diffusion, one can show by solving the rotational Smoluchowski equation that where is the mean-square anisotropy for an initially isotropic distribution (23). Making the above substitutions into Eq. 24, we obtain

(26)

Note that the time-dependent term in Eq. 26 contributes just 0.35% of the total amplitude. Thus, is dominated by the stationary terms, and the effect of high NA is to rescale this value from .

M-FICS instrumentation
M-FICS studies of molecular diffusion require very dilute samples (10 nM). We must therefore detect weak fluorescence signals against a significant noise background. With this in mind, we modified the original FICS apparatus, described by Grassman et al. (16–18), to detect small numbers of diffusing fluorescent molecules. Our modifications include:

1. Phase modulation at high frequency (_G 10 MHz).
   
2. Reduction of our instruments sensitivity to ambient mechanical vibrations.
   
3. Introduction of single-photon-counting into our phase-synchronous-detection scheme.
   

Interferometer
In Fig. 2 A, we show a schematic diagram of the M-FICS apparatus. The excitation fringe pattern is produced at the sample plane of a fluorescence microscope (model No. TE-2000U; Nikon, Tokyo, Japan) using a modified Mach-Zehnder interferometer. The frequency-doubled continuous-wave output (532 nm) of a low-noise, diode-pumped Nd:YAG laser (Compass model No. 215M, 50 mW, shown in green; Coherent, Santa Clara, CA) is isolated from trace fundamental light (1064 nm) using a pair of harmonic separators (not shown; model No. BSR-51-1037, CVI International, Norfolk, VA), and from trace 780-nm diode pump light using a laser line filter (model No. 520DF40; Omega Optical, Brattleboro, VT). The beam is split by a polarizing beam-cube, with the resulting two beams directed along equivalent interferometer arms (indexed 1 and 2). A periscope rotates the polarization of beam-1 times 90° so that it is parallel to that of beam-2. The relative intensity of the two arms is balanced using a half-wave plate. Each beam path contains an acousto-optic (AO) Bragg cell, labeled AO1 (model No. 15150-2; NEOS Technologies, Melbourne, FL) and AO2 (model No. 46200-2, NEOS Technologies) with 100-µm active areas. A digital, phase-locked, dual-channel driver (model No. N64020-250; NEOS Technologies) supplies distinct frequency waveforms to the two AO cells (_BC1 = 175 MHz and _BC2 = 180 MHz). A custom-built waveform mixer generates a well-defined difference frequency signal from the outputs of the driver. This difference signal serves as the reference (with phase _ref) for phase-synchronous-detection. Each AO imparts a time-varying phase shift to its respective beam. The first-order Bragg peak is spatially filtered and retro-reflected for a second pass, effectively Doppler-shifting the optical frequencies by 2_BC1 and 2_BC2. The electric fields of the two beams are given by E₁₍₂₎ = ₀expi[k₁₍₂₎ · r – 2_lasert – 4 _BC1(2)t + ₁₍₂₎], where k₁₍₂₎ is the wave vector of beam-1 (or -2). Cylindrical lenses (not shown) are used to correct for astigmatism introduced by the Bragg cells. The beam arms are made parallel using a beam-splitter mounted to a translation stage, and reflected by a dichroic mirror (model No. C-36159, 96321 M TRITC HQ, Nikon). The collimated beam diameters are adjusted using a telescope (not shown) to underfill the back aperture of a polarization-preserving oil-immersion objective lens (Plan Apo, 100x, NA 1.4, and w.d. 0.13 mm; Nikon), so that the focused beam waist at the sample is w 100 µm. The beams cross at the focal plane of the objective to create an intensity interference fringe pattern that runs parallel to the y axis. The resulting intensity is the square modulus of the total electric field, E₁ + E₂:

(27)

Equation 27 is the excitation profile described by Eq. 11, where I₀ = 2|₀|², k_G = |k₂ – k₁|, (t) = _Gt + _G, _G = 4(_BC2 – _BC1), and _G = ₂ – ₁. Thus, the phase of the intensity pattern is swept at the difference frequency _G = 2(_BC2 – _BC1) = 10 MHz. Translation of the beam-splitter varies the angle between the beams, thereby adjusting the grating fringe spacing, d_G = k_G/2 (17). For the experiments reported below, the range of fringe spacing used is d_G 1.2–3.5 µm, corresponding to wave numbers, k_G = 1.8–5.2 µm^–1 The laser power, measured just before sample incidence, is typically set to 1 µW.

Phase stabilization
An important source of phase noise is ambient room vibrations that ultimately degrade measurement precision. To reduce this noise, we employ an active-feedback servo in closed-loop configuration to stabilize the phase of the optical grating to <1/100 of the fringe spacing (see Fig. 2, A and B). Details of the approach, including circuit diagrams, are given by Knowles et al. (19); we include here a cursory description.

The magnified image of the excitation grating at the sample is projected through a Ronche-ruling (Fig. 2 A), and subsequently focused onto a small-area avalanche photo-diode (APD; Pacific Silicon Sensor, Westlake Village, CA). As the optical fringe pattern is swept across the ruling, the spatially modulated intensity is converted into a time-varying one. The APD output is measured using a phase-sensitive detector referenced to the difference frequency waveform of the AO-driver (_G = 10 MHz). A type 1 servo (19) generates a feedback signal, which is delivered to a Piezoelectric-mounted mirror (model No. STr-25/150/6; Piezomechanik, Munich, Germany), to minimize the relative phase error RPE _ex – _ref between the excitation and reference waveform phases. In Fig. 2 B, we show typical time courses of the RPE taken when the fringe spacing is set to d_G = 1.0 µm and the feedback loop is left open (black curve) and closed (red curve). When the feedback loop is open, passive fluctuations of the RPE occur on timescales ranging from 10^–2–10¹ s with root mean-square variation 300 nm, much larger than the scale of motions we wish to detect. When the feedback circuit is closed, the RPE continues to fluctuate on the same timescales, but in this case 4 nm. Thus, the precision to which our phase measurements are sensitive is (4 nm/d_G) x 360° 1.4° for d_G = 1.0 µm).

Photon counting phase-synchronous detection
The fluorescence signal (indicated in red, Fig. 2 A) is collected by the objective, transmitted by the dichroic beam-splitter, and filtered for polarization using an analyzer oriented at the magic-angle (model No. 10FC16PB.3; Newport, Irvine, CA). The fluorescence is spectrally filtered using a 532-nm holographic notch filter (model No. HNPF-532.0-1.0; Kaiser Optical, Ann Arbor, MI), a 650-nm short pass filter (model No. SPF-650-1.0; CVI), and a 700-nm short pass filter (model No. SPF-700-1.0; CVI). The filtered emission is focused onto a low dark-count (<25 Hz) single photon-counting APD (model No. SPCM-AQR-16, 175-µm active area; Perkin-Elmer Optoelectronics, Singapore) using an ultra-long working distance objective lens (model No. SLCPlanFl, 40x, NA 0.55, w.d. 2.6 mm; Olympus, Hauppauge, NY).

In previous FICS experiments, a lock-in amplifier was used to demodulate relatively strong fluorescence signals, which occur when the number of detected photons per modulation period is large (16). As an analog device, the lock-in is designed to process continuous photocurrent signals such as the one depicted in Fig. 3 A, thus determining the amplitude and phase of the modulated waveform. For the experiments presented in this work, the modulated signal is too weak—one photon, on average, for every thirty 10-MHz cycles—to directly process using an analog device. We therefore employ an efficient photon-counting scheme that measures the arrival time of individual photons relative to the reference phase, and assigns to it a bin increment (labeled j) that is subsequently stored in digital memory. The process is repeated over an adjustable number of cycles (here 1024), until a statistically relevant histogram (S/N > 250) of intensity versus phase is recorded. A cartoon of a histogram waveform, superimposed with the intensity distribution that it approximates (dashed curve), is depicted in Fig. 3 B. For the purpose of illustration, a total number of 10 bin increments is shown. However, the actual number of phase increments used in our device is 64, and the mean number of counts per bin is 30. A detailed description of the time-of-flight histogram generator is given in Appendix B. The completed histogram is converted to analog and output to a lock-in amplifier (model No. 7265 DSP, Signal Recovery, Wokingham, Berkshire, United Kingdom), which is referenced at the carrier frequency _C = _G/(n x 64) = 10 MHz/(16 x 64) = 9.8 kHz with low-pass filter time constant _LI = 5 ms. The number n is adjustable with possible values 1, 2, 4, ..., 32. A computer, which controls an analog-to-digital data acquisition board (National Instruments, Newbury, Berkshire, United Kingdom), records separately the average fluorescence intensity, , the in-phase and in-quadrature components of the demodulated signals, and , and the RPE. Typically, for each experiment 262,144 successive data points are collected at an acquisition frequency _Acq = 512 Hz. (For our control colloid measurements, _Acq = 8.2 kHz.) From these data, the ratio is determined, from which we calculate the function S(k_G, ) according to Eq. 16. In practice, the time-correlation functions are computed from the density fluctuation, , by use of the convolution theorem and taking the inverse Fourier transform of the associated power spectral density

(28a)

where

(28b)

and

(28c)

The normalized time-correlation function F(k,) = S(k,)/S(k) is constructed from Eq. 28a. Individual data sets are repeated 10 times, crosschecked for consistency, and averaged together.

View larger version (27K): [in this window] [in a new window]   FIGURE 3  (A) Schematic of the total fluorescence intensity, If (sig – ref (described by Eq. 14), relative to the reference phase. The modulated signal oscillates at the grating sweep frequency, G = 10 MHz, with slowly varying amplitude and phase . The mean fluorescence background is time-independent. In the strong signal limit (i.e., many detected photons per modulation period), the raw signal is continuous. (B) Rough schematic of a phase-dependent intensity histogram resulting from the compilation of independent measurements of single photon arrival times (relative to the reference) (see Appendix B). The phase is divided into m bin increments, numbered by the index j. (For clarity, m = 10 in the figure; the actual value used is m = 64.) Nj is the number of photons assigned to bin j. The average number of photons per bin shown in the figure is ; 2. The actual value used is 1250 / 64 ; 20.

Control measurements, signal/noise, and collection efficiency
In addition to our measurements on DsRed, we present control measurements on dilute (1 nM) aqueous suspensions of polystyrene latex spheres (FluoroSphere, model No. F8801, radius a = 50 nm; Invitrogen/Molecular Probes, Eugene, OR). The spheres are uniformly labeled with Rhodamine, which has spectral characteristics similar to those of DsRed. Furthermore, we have adjusted the signal level in our control experiments so that the S/N is comparable to that in our DsRed measurements (see Table 1). For both the latex suspensions and DsRed solutions, blanks with identically prepared solvents were used to determine the background count rate 0.5–1.0 kHz. When the laser was blocked, the signal count rate was the same as the measured dark-count rate 25 Hz.

Typical mean signal count rates for latex and DsRed samples are listed in Table 1. As discussed in M-FICS Instrumentation, the histogram generator constructs an analog ac signal at the carrier frequency _C = 9.8 kHz, which is detected as a root-mean square (RMS) voltage using a lock-in amplifier. The lock-in applies a low-pass filter to the signal with bandwidth _LI (= , the inverse lock-in time constant). For our measurements on DsRed, _LI = 5 ms and _LI = 0.2 kHz; and for the colloid samples, _LI = 640 µs and _LI = 1.56 kHz. We estimate the mean number of photons per data point as . The S/N associated with these measurements is (Appendix B), where the factor is the improvement due to the bandwidth of the lock-in amplifier (39) and we have estimated the bandwidth of the input signal by the carrier frequency.

A useful quantity is the mean number of photons detected per data point per molecule, (Table 1). For the DsRed samples, = 10 nM in the illumination volume V = (w/2)² z, with beam diameter w = 100 µm and sample thickness z; 10 µm. The mean number of molecules in the illumination volume is then = 470,000. The corresponding number of particles in the latex samples (with = 1 nM) is N 47,000. It is clear from the values listed in Table 1 that the S/N ratio can be maintained at an exceptionally high value (>250:1), while the number of photons detected per molecule is small (<0.004).

To determine the instrument collection efficiency, we estimate the emission intensity from the known experimental parameters. Using the extinction coefficient of DsRed ( = 3 x 10⁵ M^–1 cm^–1 for the tetramer) (14), we calculate the absorbance . The drop in excitation intensity due to transmission through the sample is I = I₀[1–exp(–A)] I₀A = (1 µW)(3 x 10^–6), corresponding to I = 8.02 x 10⁶ photons s^–1. The expected emission intensity follows from multiplication by the fluorescence quantum yield, q_F = 0.79 (14), and the factor (1–cos₀) = 0.31 due to the finite collection angle (2₀ = 135°) subtended by the objective lens. This suggests a maximal mean signal count rate of 2.0 MHz. We define our collection efficiency as the ratio , which lies in the range . These values are reasonable, considering the filtering that occurs before the emission reaches the detector.

In Fig. 4 we plot the normalized ISF, obtained by M-FICS, on the control latex suspensions described above. The decays (shown in black) correspond to wave numbers k_G = 2.09, 2.64, 3.27, and 3.90 µm^–1, and are calculated based on Eq. 28, expressions a–c. For this dilute fluid system, the decays are expected to follow the Fickian model for self-diffusion of spherical particles given by Eq. 8. Indeed, we find that the time and wave-number dependences of the correlation functions are in excellent agreement with Eq. 8 (shown as gray dashed curves) with self-diffusion coefficient D_S = 4.3 µm² s^–1. This value for D_S is in excellent agreement with the free diffusion coefficient calculated from the Stokes-Einstein equation, where we have used the viscosity of water = 1 cP and the hydrodynamic radius a = 50 nm. We note that for each of the wave numbers investigated, the decays are well described as single-exponential out to two decades below the initial amplitudes. These measurements serve as a control diagnostic to test the working order of the M-FICS instrument under S/N conditions similar to those used for our fluorescent protein measurements.

View larger version (21K): [in this window] [in a new window]   FIGURE 4  Normalized self-ISF FS(k,) determined by M-FICS on 0.1-µm diameter Rhodamine-labeled poly(styrene) beads in aqueous suspension. The ISF is shown for wave numbers kG = 2/dG = 2.09, 2.64, 3.27, and 3.90 µm–1. (Values for the fringe spacings dG are given in the figure.) Dashed blue curves are plots of the Fickian model (Eq. 8) with DS = 4.3 µm2 s–1. Data and models are shown on log-linear (A) and log-log (B) scales.

	RESULTS AND DISCUSSION

TOP ABSTRACT INTRODUCTION THEORETICAL BACKGROUND EXPERIMENTAL METHODS RESULTS AND DISCUSSION CONCLUSIONS APPENDIX A APPENDIX B ACKNOWLEDGEMENTS REFERENCES

(29)

In Eq. 29, the constant . We therefore expect the detected signal to be independent of optical anisotropy transients, and to only reflect translational motions. Nevertheless, it is possible for the molecules to undergo a photoconversion process on measurement timescales, such as photodegradation, spectral diffusion, or intersystem crossing between bright and long-lived dark states (often called flickering) (22). In this hypothetical situation, the fluorescence-detected excited state population would fluctuate due to the photoconversion process. Because the rates of such processes depend on laser excitation intensity, we would expect their presence to be revealed as intensity-dependent contributions to the decay of the ISF.

Previous studies of DsRed and fluorescent protein mutants, both at the single-molecule level (20) and using conventional fluorescence correlation spectroscopy (21,22,40), have reported photoinduced transitions occurring on sub-millisecond timescales. In those studies, the range of excitation intensities examined is 0.4–100 kW cm^–2. For comparison, the maximum excitation intensity used in our experiments is 1 µW/(50 µm)² = 1.3 x 10^–5 kW cm^–2, which is 30,000–8,000,000 times smaller than those used in the photoconversion studies (20–22,40). The relatively small excitation intensity used in our experiments is possible because of the significant number of molecules inside the illumination volume (N 470,000), which is nevertheless small enough to generate a fluctuating M-FICS signal proportional to number density fluctuations.

In Fig. 5, we show the ISF obtained from M-FICS experiments on dilute DsRed solutions, and as a function of the laser excitation intensity. The ISF is measured at the wave numbers k_G = 2.45, 2.96, and 3.59 µm^–1, and is shown on a vertical scale encompassing one factor of e. For each wave number, we show superimposing decays corresponding to three different laser intensities: I₀ = 1.27 x 10^–5 kW cm^–2 (black), 0.16 x 10^–5 kW cm^–2 (gray), and 0.04 x 10^–5 kW cm^–2 (white). Our results indicate that the functional form of the ISF is independent of excitation intensity, although the S/N ratio degrades significantly with decreasing intensity (e.g., S/N 43:1–55:1 for the lowest intensity value). Indeed, for all of our measurements, which are conducted under similar conditions, the ISF (with time resolution 5 ms) does not exhibit a dependence on excitation intensity. This suggests two possible explanations: 1), The photoinduced dynamics associated with the DsRed system occur on timescales outside of the measurement bandwidth (200–0.01 Hz); or 2), for the low excitation intensities used in our experiments, the steady-state fraction of DsRed molecules populating nonradiative (dark) states is negligibly small. Several workers have associated the photoinduced dynamics observed in fluorescent proteins with slow conformational changes in molecular structure (21,22,41–45). If this mechanism is correct, our observations could support the notion that conformational fluctuations linked to photoconversion processes occur on faster timescales than those sampled in our current measurements.

View larger version (29K): [in this window] [in a new window]   FIGURE 5  Normalized self-ISF FS(k,) determined by M-FICS on dilute solutions of DsRed in 95% (by volume) glycerol/water. The different decays correspond to wave numbers kG = 2/dG = 2.45, 2.96, and 3.59 µm–1. (Values for the fringe spacings dG are given in the figure.) For each wave number, measurements are superimposed corresponding to excitation intensities, I0 = 1.27 (black), 0.16 (gray), and 0.04 (white) x 10–4 W µm–2 (x 10–5 kW cm–2).