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J Phys Chem B. Author manuscript; available in PMC 2010 May 14.

Published in final edited form as:

PMCID: PMC2692695

NIHMSID: NIHMS111050

The publisher's final edited version of this article is available at J Phys Chem B

See other articles in PMC that cite the published article.

The kinetics of bio-molecular conformational transitions can be studied by two-dimensional (2D) magnetic resonance and optical spectroscopic methods. Here we apply polarization-modulated Fourier imaging correlation spectroscopy (PM-FICS) to demonstrate a new approach to 2D optical spectroscopy. PM-FICS enables measurements of conformational fluctuations of fluorescently labeled macromolecules on a broad range of time scales (10^{-3} - 10^{2} s). We examine the optical switching pathways of DsRed, a tetrameric complex of fluorescent protein subunits. An analysis of PM-FICS coordinate trajectories, in terms of 2D spectra and joint probability distributions, provides detailed information about the transition pathways between distinct dipole-coupled DsRed conformations.

A remarkable feature of proteins and nucleic acids is their unique ability to undergo cooperative rearrangements in structure as part of mechanisms to regulate biological activity. Rather than exist as a single, stable conformation, biological macromolecules often exhibit a broad, heterogeneous distribution of sub-states in thermal equilibrium ^{1}^{, }^{2}. Activation and inter-conversion between sub-states can span many decades over time ^{3}. Such systems exhibit complex spectra of relaxations, with principle time scales determined by transformations between sub-states, and exchange kinetics between different transition pathways ^{4}.

The kinetics of conformational transitions and chemical exchange can be studied by two-dimensional (2D) NMR spectroscopy ^{4}. Transitions between conformations of biological macromolecules are reflected by the magnitudes of diagonal and off-diagonal peaks in 2D NMR spectra. In recent years, chemical exchange spectroscopy has been applied at infrared and visible frequencies ^{5}. 2D optical methods can investigate the inter-conversion between populations of chemical species that are spectroscopically non-equivalent. Such 2D optical experiments measure equilibrium chemical kinetics on the time scales of the excited state lifetimes of vibrational or electronic transitions.

In the current work, we show how polarization modulated Fourier imaging correlation spectroscopy (PM-FICS) can probe the pathways of optical switching conformations of the fluorescent protein complex, DsRed, over a broad range of time scales much longer than the excited state lifetime (10^{-3} - 10^{2} s). The PM-FICS method, and its application to DsRed, is described in the previous article (reference I). Similar to 2D optical and NMR methods, PM-FICS provides a phase-dependent optical signal that determines four-point time-correlation functions and the associated 2D spectra. Moreover, the information obtained from DsRed is sufficient to construct joint probability distributions of time-dependent conformational coordinates.

From a kinetic perspective, the DsRed protein is a dynamically complex heterogeneous system. Unlike monomeric variants of the green fluorescent protein (GFP) ^{6}, DsRed is an obligate tetramer of FP subunits ^{7}. FPs are single chains of ~ 230 amino acid residues, which form an 11-stranded β-barrel with dimensions ~ 3 nm × 4 nm. An α-helix inside the barrel contains the sequence of three residues that form the p-hydroxy-benzylidene-imidazolidinone chromophore. In DsRed, the π -π* electron system of the chromophore is chemically and irreversibly extended to include an acylimine substituent, adjacent to the imidazolidinone ^{8}^{, }^{9}. This so-called maturation process occurs over the course of several days and can be followed by a gradual gain in red photo-luminescence accompanied by a loss of green emission ^{8}^{, }^{10}. Nevertheless, the maturation reaction does not run to completion, even after prolonged aging, so that a given DsRed molecule likely contains at least one `immature' green chromophore ^{8}^{, }^{11}^{-}^{13}. The red chromophore itself undergoes switching transitions, or `flickering', between optical conformations of different emission wavelengths and intensities, and on time scales ranging between milliseconds and tens-of-seconds ^{14}^{-}^{17}.

Although many studies have focused on the reversible, light induced pathways between `bright' and `dark' conformations ^{11}^{, }^{14}^{-}^{17}, thermally driven transitions between ground states are also possible ^{13} ^{16}. Detailed spectroscopic studies reveal that the red chromophore can reversibly interconvert, either through excited-state or ground-state pathways, between two brightly fluorescent red conformations (called `red' and `far-red'), and a relatively dim `green' conformation ^{12}^{, }^{13}. The energetic barriers mediating these transitions are on the order of ~ 1000 cm^{-1} (or 12 kJ mol^{-1} 5 × k_{B}*T* at ambient temperatures). Although the physical nature of the inter-conversion processes remains unclear, possible mechanisms include isomerization of the protein-chromophore hydrogen-bonded network, cis-trans photo-isomerization, and ground state bond rotation ^{16}^{, }^{18}.

For our current purposes, we invoke a simplified model to interpret the conformational dynamics that influence excited state energy transfer between adjacent chromophores in the DsRed complex. Figure 1 depicts the DsRed molecule as four cylinders (arbitrarily labeled 1 - 4) with principle axes oriented approximately as in the crystal structure ^{7}. Crystallographic data suggest that there are three possible relationships between any pair of adjacent chromophores, given by the relative transition dipole orientations (θ^{24} = 41°, θ^{23} = 47°, and θ^{34} = 21°) and the inter-dipole distances (*r*^{24} = 43Å, *r*^{23} = 38Å, and *r*^{34} = 22Å). By symmetry, the relationships between paired transition dipoles 2-4, 2-3, and 3-4 are the same as those between 1-3, 1-4, and 1-2, respectively ^{7}.

Optical conformational transitions of the `mature' red chromophores in DsRed. DsRed is a tetrameric complex of cylindrically shaped fluorescent protein subunits, with relative orientations approximated in the figure. Each subunit has at its center an **...**

As previously mentioned, most DsRed tetramers contain at least one immature green chromophore, which does not mature to red over the course of a PM-FICS measurement (~ 10 minutes). Of the sites that have matured to red (with absorption maximum λ_{max} ~ 563 nm), these undergo reversible inter-conversion to the far-red conformation (λ_{max} ~ 577 nm), or to the weakly fluorescent green conformation (λ_{max} ~ 484 nm), on time scales of tens-of-milliseconds and longer. Figure 1 depicts a DsRed molecule with a single static (immature) green site at position 1 (shaded green), and three dynamically inter-converting red sites at positions 2 - 4. In the experiments presented below, the red optical transitions (λ_{ex} ~ 532 nm) are selectively excited, and the integrated emission from both red and far-red states is detected. Both red and far-red conformations are considered bright states (shaded red in Fig. 1), while green states of the mature chromophore are dark (shaded gray). Thus, the immature green site at position 1 is pinned, while the red sites at positions 2 - 4 undergo reversible switching between bright and dark states. Because the distances and orientations between resonant optical transition dipoles are relatively small, an excited red chromophore can transfer its energy to one of its unexcited red or far-red neighbors by an energy transfer mechanism. When two sites in the DsRed complex are thus optically coupled, the emission polarization rotates by the angle θ^{ae}, which subtends the absorption dipole moment of the initially excited chromophore and the emission dipole moment of the emitting chromophore ^{14}^{, }^{17}. Figure 1 illustrates the three possible pair-wise couplings between bright chromophore sites (for a molecule with one site pinned), and the associated depolarization angles. Also indicated are the six possible angular displacements Δθ^{ae} associated with conformational transitions between the three optically coupled states. Similar pair-wise couplings and transitions are possible for a molecule with all of its sites red (zero sites pinned). However, for a molecule with two or more of its sites pinned, transitions between distinct pair-wise coupled conformations are not possible, and such species are not expected to contribute to the fluctuating emission signals.

The model depicted in Fig. 1 is consistent with available experimental data for DsRed. However, little is known about the details of such thermally activated switching transitions, such as whether they occur at random or in a cooperative manner due to interactions between adjacent FP subunits. For example, a cooperative mechanism could involve a series of optical conformations, dynamically connected along multiple kinetic pathways. The following work demonstrates how 2D PM-FICS can determine such information about the optical transitions of multi-colored FPs by monitoring the coordinate fluctuations of a finite population of molecules.

The equilibrium coordinate fluctuations of a finite population of DsRed molecules in 95% glycerol/water solution were monitored by the PM-FICS method, as described in reference I. The measurement observables are the number density ${Z}_{{k}_{G}}^{ND}\left(t\right)\infty \langle \mathrm{exp}i\left[{k}_{G}x\left(t\right)\right]\rangle $, and the anisotropy density ${Z}_{{k}_{G}}^{AD}\infty \langle \mathrm{exp}i[{k}_{G}x\left(t\right)-2{\theta}^{ae}\left(t\right)]\rangle $, where the angle brackets indicate a sum over the ~ 10^{6} molecules in the illuminated sample volume. In these expressions, *x*(*t*) and θ^{ae} (*t*) are time-dependent position and conformation coordinates, respectively. We interpret these signals using the first order cumulant approximation (discussed in reference I).

Four-point time correlation functions (TCF) for the number density and anisotropy density fluctuations were constructed from products of four sequential data points:

$${C}_{ND}^{\left(4\right)}({t}_{43},{t}_{32},{t}_{21})\equiv \langle {Z}_{{k}_{g}}^{ND\ast}\left(0\right){Z}_{{k}_{g}}^{ND}\left({t}_{21}\right){Z}_{{k}_{g}}^{ND}({t}_{32}+{t}_{21}){Z}_{{k}_{g}}^{ND\ast}({t}_{43}+{t}_{32}+{t}_{21})\rangle $$

(1)

and

$${C}_{AD}^{\left(4\right)}({t}_{43},{t}_{32},{t}_{21})\equiv \langle {Z}_{{k}_{g}}^{AD\ast}\left(0\right){Z}_{{k}_{g}}^{AD}\left({t}_{21}\right){Z}_{{k}_{g}}^{AD}({t}_{32}+{t}_{21}){Z}_{{k}_{g}}^{AD\ast}({t}_{43}+{t}_{32}+{t}_{21})\rangle .$$

(2)

Equations (1) and (2) define the time intervals *t*_{43} (= *t*_{4} -*t*_{3}), *t*_{32} , and *t*_{21} with *t*_{4} ≥ *t*_{3} ≥ *t*_{2} ≥ *t*_{1} ≥ 0. In the first-order cumulant approximation, Eqs. (1) and (2) can be written

$${C}_{ND}^{\left(4\right)}({t}_{43},{t}_{32},{t}_{21})=\langle \mathrm{exp}[-{\scriptstyle \frac{1}{2}}{k}_{\mathrm{G}}^{2}\delta \Delta {\stackrel{\u2012}{x}}_{N}^{2}\left({t}_{21}\right)]\rangle \langle \mathrm{exp}[-{\scriptstyle \frac{1}{2}}{k}_{\mathrm{G}}^{2}\delta \Delta {\stackrel{\u2012}{x}}_{N}^{2}\left({t}_{43}\right)]\rangle \times \langle \mathrm{exp}i{k}_{\mathrm{G}}[\Delta {\stackrel{\u2012}{x}}_{N}\left({t}_{43}\right)-\Delta {\stackrel{\u2012}{x}}_{N}\left({t}_{21}\right)]\rangle $$

(3)

and

$${C}_{AD}^{\left(4\right)}({t}_{43},{t}_{32},{t}_{21})={C}_{ND}^{\left(4\right)}({t}_{43},{t}_{32},{t}_{21}){C}_{A}^{\left(4\right)}({t}_{43},{t}_{32},{t}_{21}),$$

(4)

where

$${C}_{A}^{\left(4\right)}({t}_{43},{t}_{32},{t}_{21})=\langle \mathrm{exp}[-2\delta \Delta {\stackrel{\u2012}{\theta}}_{N}^{ae2}\left({t}_{21}\right)]\rangle \langle \mathrm{exp}[-2\delta \Delta {\stackrel{\u2012}{\theta}}_{N}^{ae2}\left({t}_{43}\right)]\rangle \times \langle \mathrm{exp}i2[\Delta {\stackrel{\u2012}{\theta}}_{N}^{ae}\left({t}_{43}\right)-\Delta {\stackrel{\u2012}{\theta}}_{N}^{ae}\left({t}_{21}\right)]\rangle .$$

(5)

Equation (3) defines the time-ordered displacements of the mean center-of-mass, $\Delta {\stackrel{\u2012}{x}}_{N}\left({t}_{21}\right)[={\stackrel{\u2012}{x}}_{N}\left({t}_{2}\right)-{\stackrel{\u2012}{x}}_{N}\left({t}_{1}\right)]$ and $\Delta {\stackrel{\u2012}{x}}_{N}\left({t}_{43}\right)$, which occur during successive time intervals *t*_{21} and *t*_{43}, respectively. Similarly, Eq. (4) defines the time-ordered displacements of the mean depolarization angle, $\Delta {\stackrel{\u2012}{\theta}}_{N}^{ae}\left({t}_{21}\right)$ and $\Delta {\stackrel{\u2012}{\theta}}_{N}^{ae}\left({t}_{43}\right)$. Equation (4) suggests that the anisotropy TCF can be determined from the ratio ${C}_{A}^{\left(4\right)}({t}_{43},{t}_{32},{t}_{21})={C}_{AD}^{\left(4\right)}({t}_{43},{t}_{32},{t}_{21})\u2215{C}_{ND}^{\left(4\right)}({t}_{43},{t}_{32},{t}_{21})$. These four-point TCFs have similar mathematical forms to those employed in two-dimensional optical and magnetic resonance spectroscopy^{19}. Four-point TCFs contain information about correlations of events that occur during the intervals *t*_{43} and *t*_{21}, and decay on time scales for which the magnitudes of the collective phase displacements deviate by an amount ~ π/4 . Such correlations diminish with increasing waiting period *t*_{32}, so that they appear indistinguishable from the products of functionally independent two-point TCFs. It is therefore useful to focus on the *t*_{32} -dependence of the difference correlation functions, ${C}^{\left(4\right)}({t}_{43},{t}_{32},{t}_{21})-{C}^{\left(2\right)}\left({t}_{43}\right){C}^{\left(2\right)}\left({t}_{21}\right)$.

It is convenient to represent the four-point TCFs in the frequency domain, through their partial Fourier transform, with respect to *t*_{43} and *t*_{21}

$${S}^{\left(4\right)}({v}_{43},{t}_{32},{v}_{21})={\int}_{o}^{\infty}d{t}_{43}{\int}_{o}^{\infty}d{t}_{21}[{C}^{\left(4\right)}({t}_{43},{t}_{32},{t}_{21})-{c}^{\left(2\right)}\left({t}_{43}\right){C}^{\left(2\right)}\left({t}_{21}\right)]{e}^{i{v}_{21}{t}_{21}+i{v}_{43}{t}_{43}}.$$

(6)

The 2D spectral density is given by the absolute value $\mid {S}^{\left(4\right)}({v}_{43},{t}_{32},{t}_{21})\mid $, plotted in the ν_{21} - ν_{43} plane. The 2D spectral density is related to the joint probability that the system undergoes two successive coordinate displacements at the transition rates ν_{21} and ν_{43}, separated in time by the interval *t*_{32} . Such 2D spectra are similar to those obtained by magnetic resonance and optical techniques, and can provide information about the rates of chemical processes.

Information about weights and magnitudes of correlated displacements can be obtained from four-point distribution functions (DFs). We define ${P}^{\left(4\right)}[\Delta {\stackrel{\u2012}{x}}_{N}\left({t}_{43}\right);\Delta {\stackrel{\u2012}{x}}_{N}\left({t}_{21}\right)]$ as the joint probability density of sequentially sampling N molecules whose mean center-of-mass undergo two successive displacements, $\Delta {\stackrel{\u2012}{x}}_{N}\left({t}_{21}\right)$ and $\Delta {\stackrel{\u2012}{x}}_{N}\left({t}_{43}\right)$, during the intervals t_{21} and t_{43}, respectively. We construct the joint distributions by sampling four-point products of the form ${Z}_{{k}_{G}}^{ND\ast}\left(0\right){Z}_{{k}_{G}}^{ND}\left({t}_{21}\right){Z}_{{k}_{G}}^{ND}({t}_{32}+{t}_{21}){Z}_{{k}_{G}}^{ND\ast}({t}_{43}+{t}_{32}+{t}_{21})=\mathrm{exp}[-{\scriptstyle \frac{1}{2}}{k}_{G}^{2}\delta \Delta {\stackrel{\u2012}{x}}_{N}^{2}\left({t}_{21}\right)]\mathrm{exp}[-{\scriptstyle \frac{1}{2}}{k}_{G}^{2}\delta \Delta {\stackrel{\u2012}{x}}_{N}^{2}\left({t}_{43}\right)]\times \mathrm{exp}i{k}_{G}[\Delta {\stackrel{\u2012}{x}}_{N}\left({t}_{43}\right)-\Delta {\stackrel{\u2012}{x}}_{N}\left({t}_{21}\right)]$. Such products are used to calculate 2D histograms of the joint probability to observe mean center-of-mass displacements during consecutive time intervals. As discussed in reference I, if the center-of-mass displacements are uncorrelated, then the joint distribution can be factored into a product of two-point DFs, i.e., ${P}^{\left(2\right)}\left[\Delta {\stackrel{\u2012}{x}}_{N}\left({t}_{43}\right)\right]{P}^{\left(2\right)}\left[\Delta {\stackrel{\u2012}{x}}_{N}\left({t}_{21}\right)\right]$. For such Brownian systems of diffusing molecules, the joint distribution is expected to be a two-dimensional Gaussian centered about the coordinate $\Delta {\stackrel{\u2012}{x}}_{N}\left({t}_{43}\right)=\Delta {\stackrel{\u2012}{x}}_{N}\left({t}_{21}\right)=0$.

2D distributions are similarly defined for displacements of the mean depolarization angle. ${P}^{\left(4\right)}[\Delta {\stackrel{\u2012}{\theta}}_{N}^{ae}\left({t}_{43}\right);\Delta {\stackrel{\u2012}{\theta}}_{N}^{ae}\left({t}_{21}\right)]$ is the joint distribution associated with consecutive displacements of conformation, which is constructed from four-point products of the anisotropy density ${Z}_{{k}_{G}}^{AD\ast}\left(0\right){Z}_{{k}_{G}}^{AD}\left({t}_{21}\right){Z}_{{k}_{G}}^{AD}({t}_{32}+{t}_{21}){z}_{{k}_{G}}^{AD\ast}({t}_{43}+{t}_{32}+{t}_{21})={Z}_{{k}_{G}}^{ND\ast}\left(0\right){Z}_{{k}_{G}}^{ND}\left({t}_{21}\right){Z}_{{k}_{G}}^{ND}({t}_{32}+{t}_{21}){z}_{{k}_{G}}^{ND\ast}({t}_{43}+{t}_{32}+{t}_{21}){Z}^{A\ast}\left(0\right){Z}^{A}\left({t}_{21}\right){Z}^{A}({t}_{32}+{t}_{21}){Z}^{A\ast}({t}_{43}+{t}_{32}+{t}_{21})$. It is possible to divide the above expression by the contributions from the number density to isolate the anisotropy effects alone, i.e., ${Z}^{A\ast}\left(0\right){Z}^{A}\left({t}_{21}\right){Z}^{A}({t}_{32}+{t}_{21}){Z}^{A\ast}({t}_{43}+{t}_{32}+{t}_{21})=\mathrm{exp}[-2\delta \Delta {\stackrel{\u2012}{\theta}}_{N}^{ae2}\left({t}_{21}\right)]\mathrm{exp}[-2\delta \Delta {\stackrel{\u2012}{\theta}}_{N}^{ae2}\left({t}_{43}\right)]\times \mathrm{exp}i2[\Delta {\stackrel{\u2012}{\theta}}_{N}^{ae}\left({t}_{43}\right)-\Delta {\stackrel{\u2012}{\theta}}_{N}^{ae}\left({t}_{21}\right)]$. As we discuss below, such four-point DFs contain detailed information about correlated changes in the conformation of coupled dipoles of the DsRed complex.

In Fig. 2, we present results for the 2D spectral density and the joint DF of the mean center-of-mass displacements. In Fig. 2A is shown the logarithm of $\mid {S}_{ND}^{\left(4\right)}({v}_{21},{t}_{32},{v}_{43})\mid $ in the ν_{21} - ν_{43} plane, for *t*_{32} = 10 ms. Because the TCFs for DsRed translational motion decay exponentially, the Fourier transform-related spectral density [see Eq. (6)] is Lorentzian. For *t*_{32} ≤ 20 ms, a minor feature is observed along the diagonal line ν_{21} = ν_{43} , indicating correlated motion on these relatively short time scales. For *t*_{32} > 20 ms, the feature along the diagonal disappears (data not shown). In Fig. 2B is shown the joint distribution ${P}_{N}^{\left(4\right)}[\Delta {\stackrel{\u2012}{x}}_{N}\left({t}_{21}\right);\Delta {\stackrel{\u2012}{x}}_{N}\left({t}_{43}\right)]$ evaluated at *t*_{21} = *t*_{32} = *t*_{43} = 10 ms . For all values of *t*_{32} investigated, the joint DF appears as a two-dimensional Gaussian consistent with Brownian motion. These results support the view that cooperative center-of-mass displacements do not play a significant role in DsRed dynamics on the time scales of the current measurements.

Contour diagrams of the two-dimensional spectral density, and the joint distribution function of center-of-mass displacements. In panel (A) is shown the logarithm of $\mid {S}_{ND}^{\left(4\right)}({v}_{21},{t}_{32},{v}_{43})\mid $ versus ν_{21} and ν_{43} for a single **...**

Having established an accurate picture of the center-of-mass dynamics for DsRed, it is possible to apply the factorization procedure outlined in Sec. 2.2 to determine the 2D spectrum of anisotropy fluctuations. In Fig. 3 is shown the logarithm of the two-dimensional spectral density $\mid {S}_{A}^{\left(4\right)}({v}_{21},{t}_{32},{v}_{43})\mid $ as a contour diagram in the ν_{21} - ν_{43} plane, with *t*_{32} = 20 ms. Features that appear on the line diagonal to the spectrum (ν_{21} = ν_{43}) indicate sampled populations that maintain their rate of conformational transitions over the duration of the waiting period. Features that lie off the diagonal line, i.e. ν_{21} ≠ ν_{43}, represent populations that undergo transitions between distinct regions of the spectrum during the waiting period. In Fig. 3, the magnitude of the spectrum evaluated at the diagonal line is projected onto the horizontal and vertical axes. The sampled populations are broadly distributed among transition rates ranging from 0 - 25 Hz, and are roughly partitioned into two peaks centered at ~ 10 and 14 Hz. In Fig. 3, these peaks are labeled “slow” and “fast”, and are indicated by vertical and horizontal dashed lines. At the intersections of the dashed lines are diagonal features associated with “slow” and “fast” populations. Off-diagonal features are labeled “slow-to-fast” and “fast-to-slow,” to indicate molecular sub-populations that make transitions between the two spectral regions. Because the 2D spectrum is narrow in the direction of the anti-diagonal (ν_{21} = -ν_{43}), the sampled populations do not readily exchange between fast and slow spectral regions on the time scale of ~ 20 ms.

Logarithm of the two-dimensional spectral density of the mean depolarization angles $\mid {S}_{A}^{\left(4\right)}({v}_{21},{t}_{32},{v}_{43})\mid $, for waiting period *t*_{32} = 20 ms. Features along the diagonal line (labeled “fast” and “slow”) indicate **...**

We next determine joint DFs that contribute to the spectral line shape. In Figs. 4A - 4D are shown contour diagrams of ${P}^{\left(4\right)}[\Delta {\stackrel{\u2012}{\theta}}_{N}^{ae}\left({t}_{43}\right);\Delta {\stackrel{\u2012}{\theta}}_{N}^{ae}\left({t}_{21}\right)]$ corresponding to each of the four labeled points in the 2D spectrum shown in Fig. 3. Features in the joint DF can establish the existence of “pathways” between adjacent conformational transitions. For each of the DFs shown in Fig. 4, the values of the intervals are *t*_{32} = 20 ms, and *t*_{21}, *t*_{43} {70 ms, 100 ms} , which were chosen to correspond to the labeled points in the spectral density. Along the horizontal and vertical axes are shown the projected magnitudes, which span the range ± 30°. These DFs were constructed from histograms of ~ 35,000 four-point products. The procedure was repeated to insure reproducibility of independent data sets, and the results were averaged together to produce the DFs shown in Fig. 4.

(A - D) Joint distributions of the sampled mean displacements of depolarization angles ${P}^{\left(4\right)}[\Delta {\stackrel{\u2012}{\theta}}_{N}^{ae}\left({t}_{43}\right);\Delta {\stackrel{\u2012}{\theta}}_{N}^{ae}\left({t}_{21}\right)]$, where the waiting period *t*_{32} = 20 ms, and *t*_{21}, *t*_{43} {70 ms,100 ms}, as shown. **...**

For the two DFs representing diagonal features of the spectral density (labeled “fast” and “slow”), both exhibit mirror plane symmetry with respect to the diagonal line $[\Delta {\stackrel{\u2012}{\theta}}_{N}^{ae}\left({t}_{21}\right)\simeq \Delta {\stackrel{\u2012}{\theta}}_{N}^{ae}\left({t}_{43}\right)]$. For the two DFs representing off-diagonal features (labeled “slow-to-fast” and “fast-to-slow”), each exhibits the projections of the “fast” and “slow” DFs onto one another. The joint DFs exhibit numerous peaks and shoulders, which undoubtedly reflect the conformational transitions of a complex heterogeneous system. In our current analysis, we focus on a subset of these peaks (indicated by vertical and horizontal dashed lines in Fig. 4). For the DF representing “slow” displacements (Fig. 4C), there are peaks centered at the coordinates $[\Delta {\stackrel{\u2012}{\theta}}_{N}^{ae}\left({t}_{21}\right),\Delta {\stackrel{\u2012}{\theta}}_{N}^{ae}\left({t}_{43}\right)]$ = (-6°, -6°), (+2°, -6°) and (-6°, +2°). For the distribution representing “fast” displacements (Fig. 4B), there are peaks at the coordinates (+2°, +2°), (+16°, +16°), (+2°, +16°), (+2°, -22°), (+16°, +2°) and (-22°, +2°). These peaks indicate correlated events, in which a change in molecular conformation of a given magnitude is temporally correlated to that of another. The diagonal symmetry of the “fast” and “slow” DFs suggests that for the relatively short waiting period of *t*_{32} = 20 ms, there is no temporal bias to indicate which of the two correlated events precedes the other.

The above results suggest that there are two significant optical conformation pathways in DsRed; (i) a “slow” pathway connecting at least two sequential steps, which involve the angular displacements $\Delta {\stackrel{\u2012}{\theta}}_{N}^{ae}$ =+2° and -6°; and (ii) a “fast” pathway connecting at least three sequential steps, which involve the displacements $\Delta {\stackrel{\u2012}{\theta}}_{N}^{ae}$ = +2°, +16°, and -22°. As discussed above, the coordinate pairings given by the joint DFs indicate the adjacencies between sequential steps in a given pathway. For example, the “fast” pathway appears to contain adjacent conformational transitions with $\Delta {\stackrel{\u2012}{\theta}}_{N}^{ae}$ = -22° and +2°, since the points (+2°, -22°) and (-22°, +2°) are present in the joint DF shown in Fig. 4B. On the other hand, the “fast” pathway does not contain adjacent transitions with $\Delta {\stackrel{\u2012}{\theta}}_{N}^{ae}$ = -22° and +16°, because the joint distribution has no significant magnitude at the points (+16°, -22°) and (+22°, +16°). The distributions representing off-diagonal features in the spectral density (Figs. 4A and and4D)4D) contain information about exchange processes between the fast and slow pathways. For the distribution representing “slow-to-fast” exchange (Fig. 4A), features are present at the coordinates (+2°, +2°), (+2°, +16°), (+2°, -22) (-6°, +2°), (-6°, -22°), and (-6°, +16°), For the distribution representing “fast-to-slow” exchange (Fig. 4D), features are present at the coordinates (+2°, +2°), (+2°, -6°), (+16°, +2°), (+16°, -6°), (-22°, +2°) and (-22°, -6°), Features present in the exchange distributions indicate bridging steps between “fast” and “slow” kinetic pathways.

Information about the time scale for exchange between “fast” and “slow” kinetic pathways is obtained from the *t*_{32} -dependence of the 2D spectrum. In Fig. 5A is shown the logarithm of $\mid {S}_{A}^{\left(4\right)}({v}_{21},{t}_{32},{v}_{43})\mid $ for sequentially increasing values of the waiting period: *t*_{32} = 200 ms, 2 s, 5 s and 10 s. As the value of *t*_{32} is increased, the spectral density broadens in the transverse (off-diagonal) direction on the time scale of a few seconds. This transverse broadening indicates that sub-populations of molecules in the “fast” pathway undergo exchange with molecular populations in the “slow” pathway. The average time scale for the exchange is roughly the same as the τ_{A} = 8 s relaxation time for the anisotropy two-point TCF, reported in reference I. Nevertheless, the behavior of the joint DFs indicates that the elementary steps of the exchange processes occur on sub-second time scales. In Fig. 5B, are shown two sets of the joint distributions, corresponding to *t* = 2 s and 5 s.These DFs exhibit features at the same coordinates as those observed for the *t*_{32} = 20 ms DFs (indicated by dashed lines). As the waiting period is increased, there is a gradual loss of diagonal symmetry for “fast” and “slow” DFs (sub-panels B and C, respectively), such that they broaden in the direction of the vertical axis. As discussed further below, this loss of diagonal symmetry corresponds to the introduction of a temporal bias that indicates which of the two correlated transitions precedes the other. Furthermore, the loss of diagonal symmetry suggests a tendency for molecular population to flow from “fast” to the “slow” pathways during the waiting period.

Millisecond conformational dynamics of freely diffusing DsRed was studied using a four-point analysis of PM-FICS trajectories. The 2D spectrum of conformational transitions, $\mid {S}_{A}^{\left(4\right)}({v}_{21},{t}_{32},{v}_{43})\mid $, is roughly partitioned into “fast” and “slow” kinetic pathways. The slow anisotropy relaxation time τ_{A} = 8 s is characteristic of exchange between “fast” and “slow” molecular sub-populations. Detailed information about the pathways connecting adjacent conformational transitions is contained by the joint distributions, ${P}^{\left(4\right)}[\Delta {\stackrel{\u2012}{\theta}}_{N}^{ae}\left({t}_{43}\right);\Delta {\stackrel{\u2012}{\theta}}_{N}^{ae}\left({t}_{21}\right)]$. For waiting periods much shorter than the exchange time, (t_{32} τ_{A}), there is a clear separation between molecular sub-populations participating in each of the two pathways. For the “fast” sub-population, the angular displacements $\Delta {\stackrel{\u2012}{\theta}}^{ae}$ = +2°, +16°, and -22° are observed, with adjacent pairings +2° ↔ +16° and +2° ↔ -22°. For the “slow” sub-population, the paired displacements $\Delta {\stackrel{\u2012}{\theta}}^{ae}$ = +2° and -6° are observed.

Our results can be combined with the model discussed in Sec. 1 for the possible depolarization angles of DsRed optical conformations. We propose the mechanism illustrated in Fig. 6 to partially account for our observations of the conformational transition pathways. The system is assumed to be at equilibrium, with average steady-state concentrations of species maintained by balanced differential rates of inter-conversion. Based on the crystal structure of DsRed ^{7}, three conformations are possible for which the coupled dipoles have relative orientations ${\theta}_{n}^{ae}$ = 47°, 41°, and 21° (see Fig. 1). Spectral shifts at individual chromophore sites result in conformational transitions. The “fast” pathway, indicated by the blue arrow, consists of three temporally correlated steps: -22° → +2° → +16°. The “slow” pathway, indicated by the gold arrow, consists of two temporally correlated steps: +2° → -6°. For each step is indicated the observed (in parentheses) and the expected angular displacements accompanying the conversion between species. In both transition pathways, observations of the angular displacement $\Delta {\stackrel{\u2012}{\theta}}^{ae}$ = +2° are assigned to conformational transitions in which a red site is converted into a far-red site (purple). The similarities between optical properties of the red and far-red states likely correspond to a very small change in the transition dipole moment orientation. It is also hypothesized that intermediates lacking dipolar coupling, such as the one generically depicted at the center of the diagram, connects adjacent species. While there is a directional bias implied by the proposed mechanism, the time ordering of events is interchangeable. A conformational transition upstream in the pathway is correlated to an adjacent downstream transition. Nevertheless, for *t*_{32} τ_{A}, the time ordering of events is not established. That is, an upstream transition is as likely to occur before a downstream transition, as it is likely to occur after one. Features in the exchange distributions indicate correlations between transitions on separate pathways. Therefore, molecules participating in one reactive pathway can participate in the other pathway at a later time. For *t*_{32} τ_{A}, the exchange processes are symmetric; for each exchange process involving transfer of molecular population from the “fast” to the “slow” pathway, there is an equally weighted exchange process in the opposite direction.

As the waiting period is increased to values exceeding the mean relaxation time, the loss of diagonal symmetry of the “fast” and “slow” joint distributions ${P}^{\left(4\right)}[\Delta {\stackrel{\u2012}{\theta}}_{N}^{ae}\left({t}_{43}\right);\Delta {\stackrel{\u2012}{\theta}}_{N}^{ae}\left({t}_{21}\right)]$ (see Fig. 5B) indicates the introduction of temporal bias. For *t*_{32} ≥ τ_{A}, downstream transitions tend to occur with greater probability after the waiting period. Corresponding inverse processes, in which downstream events occur prior to upstream events, receive less weight. Furthermore, the broadening of the exchange distributions occurs in an asymmetric manner. While the “fast-to-slow” distribution appears to elongate in the direction of the $\Delta {\stackrel{\u2012}{\theta}}_{N}^{ae}\left({t}_{43}\right)$ axis, the “slow-to-fast” distribution does so to a lesser extent. This indicates that exchange processes between “fast” and “slow” pathways are more heavily biased in the “fast-to-slow” direction.

In this work, we have demonstrated a new 2D optical approach to study the kinetics of equilibrium conformational transitions of biological macromolecules, over a wide range of time scales (10^{-3} - 10^{2} s). Polarization-modulated Fourier imaging correlation spectroscopy (PM-FICS) was applied to simultaneously monitor molecular center-of-mass and anisotropy fluctuations. When applied to the system of DsRed molecules undergoing free diffusion, the approach allowed us to isolate the effects of optical switching conformational transitions. The phase-selectivity of PM-FICS measurements enables the calculation of 2D distributions and spectral densities. Similar to established 2D spectroscopic methods, the 2D spectral density determined by PM-FICS is useful to decompose the kinetics of a dynamically heterogeneous system, such as DsRed, into its separate components. A unique feature of PM-FICS is its ability to determine joint probability distributions of coordinate displacements, which contain detailed information about the pathways connecting sequential conformational transitions.

The PM-FICS method shares common attributes with 2D optical spectroscopy, single-molecule spectroscopy, and fluorescence fluctuation spectroscopy. The detailed information provided by PMFICS measurements should be useful to address broad ranging problems in the fields of protein and nucleic acid dynamics, as well as other areas in complex systems. In the current work, the well-defined structure of the DsRed molecule made this an appealing candidate to demonstrate the potential of the approach. The ability to perform such measurements on proteins and nucleic acids of general interest, in solution and in cell compartments, could enable future studies of *in vivo* enzymatic function.

We thank Prof. Jeffrey Cina for useful discussions. We acknowledge support for this research from the National Institutes of Health R01 GM67891 and the National Science Foundation CHE-0303715.

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