Origin of the Nonexponential Dynamics of Excited-State Proton

J. Phys. Chem. B 2008, 112, 7203–7210

7203

Origin of the Nonexponential Dynamics of Excited-State Proton Transfer in wt-Green Fluorescent Protein R. Gepshtein, P. Leiderman, and Dan Huppert* Raymond and BeVerly Sackler Faculty of Exact Sciences, School of Chemistry, Tel AViV UniVersity, Tel AViV 69978, Israel ReceiVed: December 2, 2007; ReVised Manuscript ReceiVed: March 18, 2008

We used an inhomogeneous excited-state proton-transfer kinetics model to explain the origin of the nonexponential time-resolved emission of the A-band of wt-green fluorescence protein. The calculated fit is rather good for both H2O and D2O samples in a wide temperature range of 80-229 K. We attribute the inhomogeneous kinetics to the distance dependence of the excited-state proton-transfer rate between the proton donor (the hydroxyl group of the chromophore) and the oxygen of a nearby water molecule. Introduction The green fluorescent protein (GFP) of the jellyfish Aequorea Victoria is used as a biological fluorescence marker and is one of the few known examples of excited-state proton transfer (ESPT) in nature. The chromophore is protectively housed along a coaxial helix threaded through the center of an 11-stranded β-barrel consisting of 238 amino acids.1,2 The chromophore is covalently anchored and effectively secluded from the aqueous solvent surrounding the protein.2,3 Additional, noncovalent, coupling of the chromophore to the protein backbone is facilitated via an extended hydrogen-bonded network.4 At room temperature, wt-GFP exhibits two main absorption peaks with maxima at 398 nm (band A) and 478 nm (band B). The ratio of the absorption bands depends on the pH. As the pH increases, the B band becomes stronger. At room temperature, there are two emission peaks in the steady-state fluorescence spectrum: a very weak band at 460 nm and a very strong band at 510 nm. The excited-state dynamics of wt-GFP have been studied by several groups,5–15 following photoexcitation of each of its two strong absorption bands in the visible part of the spectra by using fluorescence upconversion spectroscopy (about 100 fs time resolution) and other ultrafast techniques like pump-probe spectroscopy and transient infrared spectroscopy. From previous studies, the high energy absorption and emission bands are assigned to the protonated form of the chromophore which we designate by ROH. The low-energy bands are assigned with the deprotonated form which we designate by RO-. Excitation of the A-band leads to a photoprotolytic cycle of proton transfer and subsequently to geminate recombination with the RO-. The excited-state interconversion rate shows a large kinetic isotope effect. In a previous study,16 we found that the fluorescence decay of the wt-GFP ROH band is nonexponential up to 10 ns. Moreover, the long-time wt-GFP fluorescence decays with a power law of t-3/2. This type of long-time decay pattern was also observed in the fluorescence of many ESPT reactions to the solvent. ESPT reactions in solution provide useful models for proton transfer in general.17–22 The development of short pulse lasers has enabled the direct measurement of ESPT * Corresponding author. E-mail: [email protected]. Phone: 972-36407012.

reactions and made it possible to investigate their mechanisms in more detail.23 The photoprotolytic cycle is measured by timeresolved fluorescence techniques. We used the reversible geminate-recombination model to fit the fluorescence data of wt-GFP samples.16 In our first work, we explained the t-3/2 power-law decay of the A-band emission as arising from the ability of the proton to hop from one proton-acceptor site to other sites. Under certain circumstances, its motion can be approximately described by a random walk in three dimensions, and hence, a diffusion constant can be assigned to such motion. The proton can also recombine to its original site, the hydroxyl group, of the wt-GFP chromophore and thus repopulate the protonated form, ROH*. Such a process causes a nonexponential decay and an asymptotic power-law decay with a slope of about -3/2. In a recent study, we measured both the steady-state and the time-resolved emission of wt-GFP in water as a function of temperature. The model of proton geminate recombination was used to analyze the experimental data. We found that the rates of both the proton transfer and the back reaction of excited wtGFP decrease as temperature decreases. The model of intermolecular-vibration-assisted proton tunneling of Trakhtenberg and co-workers24,25 was used to fit the non-Arrhenius behavior of the experimental proton-transfer rate constant as a function of temperature. We found a remarkably good fit of the experimental temperature dependence of the rate constant and the model. In this study, we will explore another direction that may cause the nonexponential fluorescence decay of the ROH band of the chromophore in wt-GFP. The nonexponential short- and longtime fluorescence decay arises from a distance dependence of the ESPT process between the oxygens of the proton donor and the proton acceptor (water molecule, W22 Figure 2 of ref 4). A previous attempt to explain the complex dynamics of wt-GFP were presented by Winkler et al.8 They studied the dynamical process following 400 nm, 30 fs photoexcitation of wt-GFP by pump-probe spectroscopy and found that the ESPT dynamics are nonexponential. They explained the nonexponential optical response of pump-probe signals of GFP by using a dynamic model that included reactive and dissipative processes. In the reactive proton-transfer process, they used an energy-dependent rate coefficient for ESPT. This assumption was used to explain the short-time nonexponential ESPT dynamics. The dissipative

10.1021/jp711372s CCC: $40.75  2008 American Chemical Society Published on Web 05/21/2008

7204 J. Phys. Chem. B, Vol. 112, No. 24, 2008 processes included both intra- and intermolecular excess vibrational-energy transfer. An additional nonradiative decay pathway leading to internal conversion via motion along a torsional coordinate of the bridge between the two rings was suggested to explain the long-time signal. The overall nonexponential nature of the pump-probe signal originated from overlapping time scales of reactive and nonreactive elementary processes following optical excitation. In this study, we suggest that nonexponential fluorescence arises from inhomogeneous kinetics of the proton-transfer process. Nonadiabatic theories of electron and proton transfer show that the rate constant depends on the square of the coupling matrix element, C. For small and intermediate coupling, the matrix element decreases exponentially with distance,26,27

C ) C0 exp[-R(Q - Q0)]

20 Å-1 < R < 30 Å-1 (1)

where C is the coupling matrix element, C0 is the coupling matrix element at the donor-acceptor equilibrium position, Q0 is the equilibrium distance on a one-dimensional potential surface, and Q is the actual donor–acceptor distance. The value of the exponential factor R in proton-transfer reactions is about 30 times larger than the value for the distance dependence of the coupling matrix for electron-transfer reactions, and thus, a small fluctuation of the order of 0.03 Å in the internuclear distances changes the proton-transfer rate. The GFP proton wire operating upon photoexcitation of the internally caged chromophore was investigated by Vendrell et al.28 by means of classical molecular dynamics and multiconfigurational electronic-structure calculations. They found that the structure of the proton wire is ready to operate as soon as the chromophore is photoexcited, leading to a concerted protontransfer process in which a total of three proton translocations occur in the vicinity of the chromophore. Their simulations show that the oxygen-oxygen distances of the oxygen atoms in the proton wire are centered at around 2.7-2.8 Å. From our point of view, the important molecular dynamics simulation issue that could explain the nonexponential dynamics is that the width of the distribution of the oxygen-oxygen distances in the proton wire is fairly large. The distributions are slightly asymmetric, and their full width at half-maximum (fwhm) is ∼0.2 Å. In this study, we used the concept of inhomogeneous kinetics as the main reason for the nonexponential dynamics of the timeresolved emission5,6,16 as well as the pump-probe signal of GFP in both H2O8,16 and D2O.16 The intermolecular proton-transfer reaction strongly depends on the distances between the proton donor and acceptor. In particular, the distance between the two heavy atoms (the oxygens) of the donor and the acceptor determines the proton-transfer rate. This distance in the GFP proton wire has a relatively large spread of about 0.2 Å. In terms of a proton tunneling mechanism in GFP, such a large change in distance between a donor and an acceptor will strongly affect the ESPT rate constant. Experimental Section Time-resolved fluorescence was acquired by using the timecorrelated single-photon counting (TCSPC) technique, the method of choice when sensitivity, large dynamic range, and low-intensity illumination are important criteria in fluorescence decay measurements. For excitation, we used a cavity dumped Ti:Sapphire femtosecond laser, Mira, Coherent, which provides short, 80 fs, pulses of variable repetition rates, operating at the SHG frequency, at the spectral range of 380-400 nm with a relatively

Gepshtein et al. low repetition rate of 500 kHz. A low rate may be important to excite fully relaxed wt-GFP. The TCSPC detection system is based on a Hamamatsu 3809U photomultiplier and Edinburgh Instruments TCC 900 computer module for TCSPC. The overall instrumental response was about 35 ps (fwhm). The excitation pulse energy was reduced to about 10 pJ by neutral density filters. The large dynamic range of the TCSPC system (more than four orders of magnitude) enabled us to accurately determine the nonexponential photoluminescence decay profiles of the wt-GFP fluorescence. We checked the sample absorptions prior to and after timeresolved measurements. We could not find noticeable changes in the absorption spectra due to sample irradiation. The timeresolved emission decay curves of the wt-GFP samples were the same after repeated experiments. We thus concluded that under our irradiation condition, no sample deterioration could be detected. Steady-state fluorescence spectra at a low temperature were taken by using a CVI SM-240 spectrometer with a resolution of about 2 nm. We used wt-GFP gift samples from the laboratory of Prof. S. J. Remington from the Institute of Molecular Biology, University of Oregon at Eugene. wt-GFP samples of 10 mg/mL including 0.3 M NaCl were stored under refrigeration. Samples were prepared by dilution of the stock solution with deionized water. The TCSPC measurement samples were placed in a 3 mm optical path length quartz cell. The absorbance at 397 nm was typically 0.1 OD. The temperature of the irradiated sample was controlled by placing the sample in both a liquid helium cryostat (Janis) for low temperatures and also in a liquid nitrogen cryostat for high temperatures with a thermal stability of approximately (1 K. Inhomogeneous Kinetics Model. In recent papers,29,30 we used a model that accounts for inhomogeneous kinetics arising from a frozen structural medium surrounding an ensemble of excited molecules. With some modifications, the model is also applicable to kinetics in very viscous solvents or semi-frozen matrices, such as the GFP barrel, when only a small structural relaxation takes place during the excited-state lifetime. We wish to use the model to fit the GFP time-resolved emission data in both H2O and D2O samples. The mathematical derivation of the inhomogeneous kinetics model for the proton transfer in GFP is similar to that of ref 29, which deals with the radiationless transition of the GFP chromophore in solution. In the model, which is relevant to the ESPT process in wt-GFP, we propose that the proton is transferred to a water molecule next to the hydroxyl group of the chromophore’s tyrosine. For simplicity, we assume that the O-O distance distribution is Gaussian with a certain width defined by a variance σ and an average distance x0. We shall use a continuous coordinate x (x g 0) to define the distribution. The distribution is given by

p(x) )

1

√2πσ2

[

exp -

(x - x0)2 2σ2

]

(2)

where x0 is the mean (the peak position) of the Gaussian. We assume that the rate constant of proton transfer depends exponentially on the coordinate x. For convenience, we used an arbitrary distance scale (x-axis) that is incompatible with the actual distances between the oxygens involved in the protontransfer process. The Gaussian position is set at x0 ) 5 Å at the bottom of a harmonic potential

Excited-State Proton Transfer in wt-Green Fluorescent Protein

U ) c(x - x0)2

(3)

where c ) 1.4 kJ/(mol · Å2) was also used as an arbitrary parameter. The distance-dependence rate constant is given by

k(x) ) A exp{- [a(x - xo) + b(x - x0)2]}

( )∫ -t τf

∞

0

p(x) exp[-k(x)t] dx

(5)

The first exponential accounts for the homogeneous radiative decay process, whereas the integral of the second exponential represents the inhomogeneous proton-transfer rate that depends on the oxygen-oxygen distance distribution. The decay of P(t) is nonexponential and depends on the excited-state lifetime, τf, the mean value k(x0), the parameters a and b of eq 4, and the population Gaussian width 2σ2. Klafter et al.31 discussed several possibilities of certain spatial distribution functions of acceptors, P(R), and relaxation rates as a function of a donor as a consequence of interaction with the acceptor W(R) that will generate nonexponential decay of the donor population. They found in certain cases that when W(R) ) B exp(-γR), for one dimension, P(t) takes the form of a power law. Combining the Proton Tunneling Model with the Inhomogeneous Kinetics. We wish to provide the physical rationale for the use of the inhomogeneous kinetics model described above. In a recent article, we used a proton tunneling model25 to describe the temperature dependence of the proton-transfer rate constant in wt-GFP. The proton-transfer process can proceed via, over, and/or under a potential barrier mechanism. The proton-transfer process is usually described by a one-dimensional potential surface connecting two potential wells, that of the reactant and that of the product. For a one-dimensional potential surface with an intermolecular distance R between the two heavy atoms (in our case, the oxygens), the tunneling probability per unit time is given by

W(R) ) ν exp[-J(R)]

(6a)

where J(R) is given by

J(R) )

2 p

∫ab dr {2m[U(R, r) - E(R)]}1⁄2

reactions,33–35 the linear term in the expansion contributes much more than the quadratic term. At the equilibrium position of the heavy atoms R0, the temperature-independent rate constant is simply given by

k(R0) ) ν exp[-J(R0)]

(4)

When x < x0, the linear term of (x - x0) promotes a larger rate constant than at x0 and vice versa. The quadratic term decreases the rate constant for distances that deviate on both sides from x0, that is, x * x0. The quadratic term is needed to get a better fit at long times for H2O samples. We get a good fit for D2O samples with the linear term only. The justification for the quadratic term is given from the tunneling theory described in the next section. In the static limit, where the GFP structure next to the hydroxyl group is time-independent with respect to the time of the proton-transfer process, the probability P(t) that the excited state has not transferred the proton by time t after excitation is given by

P(t) ) exp

J. Phys. Chem. B, Vol. 112, No. 24, 2008 7205

(6b)

Here, E(R) is the energy level, U(R, r) is the potential energy of the tunneling particle, r is the intramolecular coordinate, and ν is the frequency of the pre-exponential factor (of the order of 1013 sec-1). The function J(R) is now expanded in powers of the reactant displacement from equilibrium, with terms up to quadratic values being retained.

1 J(R) ) J(R0) + J ′(R0)(R - R0) + J ′′ (R0)(R - R0)2 (7) 2 Calculations of the function J(R) for realistic potential barriers32 show that considering the rate constants of proton tunneling

(8)

co-workers24

Trakhtenberg and derived a simple formula for the temperature dependence of the proton-transfer rate constant when an intermolecular vibration modifies both the potential barrier and the intermolecular distance between the two oxygens,

[

k(T) ) ν exp -J(R0) + 1 (∂J ⁄ ∂R0-0)2[δ0-0] coth(pΩ0 ⁄ 4kBT) 8

]

(9)

where Ω0 is the effective intermolecular frequency, δ0-0 is the total amplitude of the zero level oscillations, kB is the Boltzman constant, and T is the temperature. Equation 7 provides the basis for the derivation of an expression for a rate constant in a system were the distances between the donor and acceptor are not only at equilibrium R0 but exhibit a distribution. The change in the tunneling rate constant from its equilibrium-position value depends exponentially on the difference (R - R0) or, in general, on a power series of (R - R0). The first correction term to the tunneling rate is given by

k1 ) ν exp[-J′(R0)(R - R0)]

(10)

The second correction to the rate constant is given by the quadratic term

1 k2 ) ν exp - J ′′ (R0)(R - R0)2 2

[

]

(11)

The inhomogeneous rate constant k(x) given in eq 4 has the ingredients of the distance-dependent tunneling rate constant given by the first two terms of the right-hand side of eqs 7 and 8 when replacing R0 and R with x0 and x, respectively. A, a, and b are replaced by the parameters ν, J′(R0), and (1/2)J′′(R0), respectively. We used eq 5 to calculate the time-dependent fluorescence of the A-band of wt-GFP at several temperatures in the range 80 < T < 290 K. Results Figure 1, upper panel, shows on a semilog scale a plot of the time-resolved emission of wt-GFP excited by ∼150 fs with a 400 nm pulse and measured at the ROH band at 455 nm at several temperatures. The decay profiles are nonexponential at all temperatures. When these decay curves are multiplied by exp(t/τf) for the finite excited-state lifetime correction and plotted on a log-log plot (Figure 1, lower panel), the long-time fluorescence obeys a power-law decay. The power-law decay t-R changes from a steep decay at high temperatures T ≈ 300 K, R ≈ 3/2, to R ≈ 1/2 at T ≈ 100 K. Figure 2, upper and lower panels, shows similar plots for a wt-GFP sample in D2O. At short times after the pulse excitation, the decay rates provide a good estimate for the ESPT rate. Similar experimental results are also shown in our previous paper.42 Discussion The origin of the nonexponential, time-resolved, fluorescence decay of the A-band of wt-GFP is the topic of this article. In previous studies,16 we attributed the nonexponential behavior of the time-resolved emission of the A-band to the geminate

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Gepshtein et al.

Figure 1. (a) Time-resolved emission of the A-band of wt-GFP at several temperatures in the range of 100-283 K (dots). The solid lines are computerized decay curves generated by using an inhomogeneous kinetics model (see text). The temperatures are (top to bottom) 100, 148, 185, 210, and 283 K. (b) Log-log plot of the (lifetime-corrected) fluorescence decay of the A-band of wt-GFP in H2O. The temperatures are the same as those in the upper panel.

recombination of the transferred proton with the chromophore’s tyrosine hydroxyl group. This process repopulates the ROH excited-state form of the chromophore, and as a consequence, the fluorescence decay deviates from the simple exponential decay kd

ROH* {\} RO-* + H+

(12)

kr

Using simple chemical rate constants for the forward and backward rate constants kd and kr leads to a biexponential decay of the ROH fluorescence. Diffusional motion of the proton within the proton wire38,36 or via several trajectories in the barrel where the proton recombination process is assisted by a proton diffusion mechanism leads to a nonexponential fluorescence decay of the ROH* band. In our recent work,37 we studied the temperature dependence of the long-time fluorescence tail of the A-band (the protonated chromophore) of wt-GFP. We found that the (lifetime-corrected) long-time fluorescence decays at temperatures above 245 K as t-3/2. Below 200 K, the long-time fluorescence undergoes a transition and decays as t-1/2. In the intermediate temperature range of 245-200 K, we noticed a gradual change in the power-law coefficient from t-3/2 to t-1/2. These findings support the model suggested by Agmon38 that the proton might escape from the barrel structure via a conformational change, enabling the rotation of threonine 203. In a subsequent and more recent work, Agmon38 provided a quantitative mathematical model connecting the decay of the fluorescence long-time power law and the proton pathway. The

Figure 2. (a) Time-resolved emission of the A-band of GFP/D2O at several temperatures in the range of 94-325 K. The temperatures are (bottom to top) 325, 287, 253, 187, 135, 115, and 94 K. The solid line is a fit calculated by an inhomogeneous kinetics model. (b) Log-log plot of the (lifetime-corrected) time-resolved emission of the A-band of wt-GFP in D2O. The temperatures are the same as those in the upper panel.

switching from t-3/2 at a high temperature to t-1/2 at a low temperature is explained along the lines of the escape route of the proton from the barrel. Below, we will present the experimental data indicating that the recombination process is not effective in wt-GFP at short times or even within the lifetime of the excited state, τf ≈ 3 ns. Transient IR studies of GFP indeed show that the glutamate 222 is the primary proton acceptor, as suggested by Brejc et al.,4 and reacts with the proton to form the long-lived acid at approximately the rate of the deprotonation of chromophore, kE

E222- + H+ 98 E222 (kE ≈ kd). This fact is in favor of a concerted mechanism of proton transfer via the proton wire connecting the donor with the acceptor. These findings reduce the importance of a diffusional motion of the transferred proton within the various potential acceptors in the barrel. Transient IR studies13–15 also show that the protonated form of E222 does not release the proton at times shorter than about 1 ns. The rate of deprotonation of E222 can also be estimated from the solution chemistry of weak acids. In aqueous solution, an acetate anion reacts with a proton with a large intrinsic rate, and therefore, the overall rate is diffusion-controlled with a reaction rate constant of two counter-charged ions, kD ≈ 5 × 1010 s-1.39 The back reaction is very slow and can be deduced from the forward reaction rate constant, and the equilibrium constant is pka ) 4.7, so that the value of the acid dissociation rate constant of acetic acid will be k ≈ 106 s-1. In aqueous



solution, glutamic acid is also a weak acid, and hence, it is expected that the rate of protonation is fast, whereas that of deprotonation is slow. If this assumption is also correct for the protein environment, the nonexponential fluorescence decay of the ROH band cannot be explained by the proton geminaterecombination model. We therefore propose a different approach to explain the origin of the nonexponential decay of the ROH form of the GFP chromophore. Before presenting our model, we will discuss a previous model, proposed by Winkler et al.,8 based on the energy-dependent proton-transfer rate constant k(E), that explains the nonexponential nature of the complex dynamics of excited wt-GFP. Winkler et al.8 explained the complex decay pattern of the pump-probe signal of wt-GFP in terms of an ESPT model that includes reactive and nonreactive elementary processes following optical excitation. At early times, before equilibration in the A* potential well is complete, ESPT is the dominant decay channel for A*. As intermolecular vibration redistribution and vibrational energy transfer (VET) progressively withdraw excess vibrational energy from the reactive modes, ESPT gradually decelerates, whereas the chromophore bridge torsional barrier crossing with subsequent internal conversion to A becomes increasingly important. At long times, 1/t , k0, the chromophores are located below the threshold for ESPT, leaving internal conversion as the dominant pathway for radiationless deactivation of A*. Consequently, at early times, the rise of the I* emission is correlated with the initial fast decay of the A* emission. However, at later times, A* decays predominantly via internal conversion. This would explain the nonexponential nature of the observed dynamics, the pronounced deuteration dependence of the fast components (seen in the emission from both A* and I*), the lack of a kinetic isotope effect for the slow components (seen only in emission from A*) and, most importantly, the distinctly different behavior of A* and I* on time scales above 100 ps. An important assumption in their model is an energy-dependent ESPT rate constant k(E). VET and redistribution change the energy and thus the rate. As a result, the energy relaxation processes lead to the nonexponential ESPT dynamics. Our approach to the nonexponential decay relies on the concept of inhomogeneous kinetics, which applies also to enzymatic reactions. From solution chemistry, it is known that in this case, the proton-transfer rate strongly depends on the distance between the two heavy atoms, that is, the oxygens of the donor and of the acceptor.26,27 The distance dependence is exponential in the case of nonadiabatic theories or simple tunneling expressions. The primary proton acceptor in wt-GFP is glutamate 222.4 The proton transfer proceeds via a proton wire connecting the hydroxyl group of the chromophore tyrosine and E222. Between the donor and the acceptor, the wire includes a water molecule and serine 205. The donor-wire-acceptor segment consists of four oxygens and three hydrogens. For simplicity, we will discuss a system that includes only two oxygens and a single hydrogen covalently bonded to one of the oxygens O-H · · · O. The linear system can approximately be described by a one-dimensional double-well potential surface. The proton crosses between the wells by either over-the-barrier or under-the-barrier (tunneling) processes. In our model, the reaction coordinate is the distance between the two oxygens. Fit of the Experimental Results with the Inhomogeneous Kinetics Model. We fit the experimental results by using eq 5. Figures 1 and 2 show the fitting curves (solid lines) of the experimental time-resolved emission data (dots) of the ROH band (A-band) of H2O and D2O at several temperatures in the

range of 80-280 K. The fits are rather good at all temperatures. The fitting parameters are the pre-exponential factor of the rate constant k(x), A(T), the distribution P(x) width, σ2 (eq 2), and the exponential distance dependence of the linear term a and the quadratic term b of the rate constant k(x), see eq 4. We also used two additional parameters in the fitting procedure. For the excited-state lifetime of the protonated ROH form, τfROH ≈ 3 ns. We deduce this value from the long-time component of ROH fluorescence of the S205V variant. In a recent study,40 we measured the photophysical properties of the S205V mutant and also the S203V/S205V dual mutant. In S205V, the proton-transfer rate at room temperature is 30 times slower than in wt-GFP, kPT-1 ≈ 300 ps. The temperature dependence of the proton transfer in S205V is large, Ea ) 18 kJ/mol, whereas it is small for wt-GFP, Ea ≈ 4 kJ/mol. Therefore, at about 180 K, the rate of proton transfer of S205V is much smaller than the radiative rate. At T e 180 K, the decay of the A-band deviates only slightly from a pure exponential. The long-time component is τ ) 3.0 ns. The S203V/S205V mutant is incapable of transferring a proton. At pH 7.2, the room temperature quantum yield of the ROH fluorescence is quite large, about 0.3.40 The time-resolved emission of the dual mutant shows a nonexponential decay at room temperature with an average lifetime of about 1500 ps. At 180 K and below, the lifetime is 3 ns, the same as the one that we used in the fit of the experimental data in this study. In conclusion, the data of the time-resolved emission of the ROH of S203V/S205V indicate that the nonradiative processes at room temperature slightly shorten the effective excited-state lifetime. The average lifetime of the nonexponential decay of the ROH of S203V/S205V is 1500 ps, half of that of the pure radiative lifetime, τ ) 3000 ps. At high temperatures, the ROH of wt-GFP has an average decay of only 10 ps, and hence, the long tail decreases to 0.001 of its initial amplitude at about t ) 3.0 ns. The calculation of the signal is almost insensitive to the value used for the lifetime when τ is longer than 1500 ps. For high temperatures, we used τfROH ) 2.9 ns, whereas for low temperatures, T < 170 K, we used τfROH ) 3.3 ns. For the numerical calculation of the computed fit, we used the userfriendly graphic program SSDP (ver. 2.63) of Krissinel and Agmon.41 An additional parameter in the fit, which only slightly affects the fitting quality at long times, is the diffusion constant of the population distribution, D. The initial distribution of P(x) on the excited-state potential surface immediately after a short pulse excitation is determined by the ground-state conformation of the protonated form and its time dependence, as well as by the excess energy dissipation processes in the excited state. We assume that some rearrangement processes that affect the distribution take place on a long time scale. The molecular dynamics simulations by Vendrell et al.28 show that the distances between the oxygens of the proton wire fluctuate on a time scale of picoseconds-nanoseconds. These fluctuations can justify the use of a diffusion constant to the distribution. At room temperature, the relevant time for protein rearrangements affecting the distances between the oxygens is a few picoseconds. At low temperatures, the processes that affect P(x) are probably in the range of tens of picoseconds. The value of the diffusion constant in the high-temperature region is about 3 × 10-8 cm2/s. In the low-temperature region of T < 175 K, D ≈ 1 × 10-8 cm2/s. The use of a diffusion constant allows the population to slightly change within the experimental time window of 0 < t < 10 ns. Thus, the protein configuration is not completely frozen with respect to the proton-transfer rate


Figure 3. Fit by an inhomogeneous kinetic model (solid line) of Chattoraj’s5 experimental up-conversion data (circles), see text.

and the excited-state radiative rate, and therefore, the donoracceptor distance in the one-dimensional model is also not fixed within the experimental observation time of ∼10 ns. The rate of change of the configurations is not ultrafast and thus also permits the diffusion along the x-axis during the excited-state time window. We impose in our model calculation a weak harmonic (parabolic) potential around the equilibrium position x0, U(x) ) c(x - x0)2. We set a value for c ) 1.4 kJ/(mol · Å2). The actual potential might be deduced from the molecular mechanics simulations.28 Figure 3 shows the fit of the time-resolved emission of the A-band measured by the fluorescence up-conversion technique of Chattoraj et al.5 for the inhomogeneous model. The mean rate constant (A in eq 4) is 1.2 × 1011 s-1. As seen in the figure, the fit is rather good at all times, up to about 200 ps. The experimental data were calculated by a three-exponential function by using the data presented in Table 1 of Chattoraj et al.5 In the homogeneous kinetics model, the two short-time components of 3 and 12 ps arise from the distance dependence of a single proton-transfer process. A Gaussian distribution of the oxygen-oxygen distance combined with an exponential distance dependence of the rate constant can accurately follow the time dependence of the fluorescence up-conversion signal. Temperature Dependence of the ESPT Rate Constant. A(T) is the temperature-dependent proton-transfer rate constant at x0, the mean donor-acceptor Gaussian distance distribution. Figure 4 shows an Arrhenius plot of ln[A(T)] versus 1/T (open symbols). The slope of the plot is not constant. At low temperatures, the slope is much smaller than that at high temperatures. In previous studies, we used the geminaterecombination model for the fit of the time-resolved emission. From the fit, we got the proton transfer and recombination rates, kPT and kr. For comparison, a plot of ln[kPT] versus 1/T is also shown in Figure 4 (full symbols). We explained the nonArrhenius behavior of kPT in terms of a vibrational assisted tunneling mechanism.42 The model includes an intermolecular vibration that modulates the distance between the two heavy atoms (the oxygens) and periodically modifies the potential

Gepshtein et al.

Figure 4. Arrhenius plot of A(T) versus 1/T. For comparison, the proton-transfer rate constant is plotted as a function of 1/T. The solid lines are calculations based on the tunneling model (see text).

surface. The proton tunneling rate increases when the distance between the two oxygens reach the minimum distance of the oscillation cycle and vice versa. The amplitude of the oscillation depends on the temperature. The larger the temperature, the larger the amplitude, and hence, the larger the proton-transfer rate. The second term on the right-hand side of eq 9 gives the temperature dependence of such a model. At a sufficiently low temperature (depending on the intermolecular vibrational frequency Ω0), the rate constant is independent of the temperature. In wt-GFP, we found that the frequency, for both H2O and D2O, is about 200 cm-1, and the low-temperature turning point is at T ≈ 70 K. Below 70 K, the rate constant is temperatureindependent. The solid line in Figure 4 is computed by using eq 9. Similar data (full symbols) and computation were already given in our previous paper.42 The parameters for the fit are Ω0 (the intermolecular frequency), δ0-0 (the total amplitude of the zero energy oscillation), T (the temperature), J′(R0), and J(R0). The solid lines in the figure are the calculated fits of the ln[kPT] values (circles) obtained from the time-resolved emission of the A-band. As seen from the figure, the fits are rather good for both H2O and D2O data. There are differences in the values of the proton-transfer rate constant obtained from the geminaterecombination model and A(T), the approximate value for the proton rate constant of the inhomogeneous distribution model, A(T) ) k(T, x0), where x0 is the mean of population distribution of the inhomogeneous kinetics model. The lower the temperature, the larger the difference, especially for the D2O sample. The width of the population distribution P(x), σ2, also depends on the temperature. For a system that is in thermal equilibrium in a harmonic potential, the mean square of the distribution σ2 scales with the force constant, k, of the potential energy, and the width is given by σ2 ) kBT/k. As expected in the fitting of the experimental results, we obtained that the width of the population for H2O samples decreased with the temperature. The width at 240 K was about a factor of 2.5 larger than that at about 100 K, (see Tables 1 and 2). Figure 5, upper panel, shows the population distance distribution between the donor and the acceptor at several selected times



TABLE 1: Temperature Dependence of the Kinetic Parameters of the Model for wt-GFP in H2Oa T [K]

1/2σ2b

A [ns-1]c

b [Å-2]c

283 260 230 210 185 173 160 148 136 124 112 100 88 78

2.8 3.5 3.8 4.0 4.2 5.0 5.0 5.0 6.0 6.0 6.0 7.5 8.5 9.0

60.0 40.0 28.0 21.0 13.0 10.4 9.5 7.4 5.2 6.0 4.3 3.2 3.0 2.5

1.80 1.30 1.20 1.00 0.70 0.60 0.50 0.50 0.45 0.50 0.65 0.40 0.30 0.30

a For all temperatures, the exponential parameter a was fixed, a ) 4 Å-1 (see eq 4). D ) 1.0 × 10-8 cm2 · s-1, τ ) 3.3 ns (see text), and U ) 1.4 kJ/(mol · Å2). b See eq 2. c See eq 4.

TABLE 2: Temperature Dependence of the Kinetic Parameters of the Model for wt-GFP in D2Oa T [K]

1/2σ2b

A [ns-1]c

253 240 217 197 187 173 165 154 125 94

2.0 1.6 3.0 5.0 5.0 5.5 5.0 4.0 4.0 3.0

13.50 8.00 5.20 3.00 2.25 1.75 1.52 1.40 0.75 0.625

a For all temperatures, the exponential parameter a was fixed, a ) 3 Å-1 (see eq 4). D ) 1.6 × 10-8 cm2 · s-1, τ ) 3.1 ns (see text), and U ) 1.4 kJ/(mol · Å2). b See eq 2. c See eq 4.

in the time range of 0-500 ps after a pulse excitation to S1. The excited-state population decays by two channels, the reactive channel and the radiative channel; the relative height of the distribution decays approximately in the same way as the rate of the time-resolved fluorescence because the width at short times changes only slightly as time progresses from its initial position at time zero from x0 ) 5 Å to about 5.8 Å at about 500 ps. At 283 K, the reaction rate at the peak of the distribution is about (10 ps)-1, and the radiative lifetimes is τf ) 3 ns. Figure 5, middle panel, shows similar results as those in the upper panel of Figure 5, but the distributions are taken from the calculation for the low-temperature decay of about 88 K. The decay of the peak height is much slower for the lower temperature. Figure 5, lower panel, shows the normalized distributions. As expected from the inhomogeneous rate constant, the left-hand side of the distribution, x < x0, which shows short distances, disappears much faster than the long distances, x > x0. At long times, the distribution position moves to longer distances and is asymmetric. Another fitting parameter in the calculation of the timeresolved emission is the exponential distance factor, a, of the rate constant given by eq 4. This parameter was held constant at all temperatures. Because we used an arbitrary scale for the distance x between the donor and the acceptor, the value we used, a ) 4 Å-1, is meaningless. For a more realistic distance scale, we expect that a in eq 4 will have values similar to those of R in eq 1; that is, a will be of the same order as R when R ≈ 20 Å-1.

Figure 5. (Upper panel) Population distance distribution between the donor and the acceptor at several selected times in the time range of 0-500 ps. T ) 283K. (Middle panel) Similar results as those in the upper panel, but the distributions are taken from the calculation for the low-temperature decay of about 88 K. (Lower panel) Normalized distributions at selected times for T ) 283 K.

The last fitting parameter is the quadratic distance factor b of the rate constant given in eq 4. This parameter is used only for the water samples. It provides a small correction at long times of the calculated signal of the time-resolved emission of the ROH. Figure 6 shows the time-resolved emission of the ROH band of wt-GFP H2O sample at T ) 185 K along with two calculated fitting curves. The solid line is a fit that includes in the calculation both the linear and the quadratic terms. The broken line shows the contribution of the linear term only. As seen from the figure, the calculation based on the linear dependence of the tunneling on the distance is rather good at short and


Figure 6. Time-resolved emission of the A-band in H2O at 185 K along with the fit calculated by using eq 4. The dashed line is a calculation that does not include the quadratic term.

intermediate times, whereas at long times, the calculated signal decay rate is larger than the experimental signal. Summary The nonexponential origin of the time-resolved emission of the A-band of wt-GFP is the topic of the current study. We used the concept that the proton-transfer rate is very sensitive to the distances between the proton donor and the proton acceptor. For a nonadiabatic proton-transfer theory, the distance dependence between the donor and the acceptor is exponential for small and intermediate coupling matrices. The tunneling theory also supports the exponential dependence on the distance between the proton donor and the proton acceptor. Vendrell et al.28 found in a recent molecular dynamics study on wt-GFP that the oxygen-oxygen distances in the proton wire connecting the chromophore and glutamate 222 (the acceptor) form a distribution that is centered at 2.7-2.8 Å, with the width of the distribution being fairly large. The distributions are slightly asymmetric, and their full width at half-maximum is ∼0.2 Å. The inhomogeneous kinetics model based on the exponential distance dependence of the rate constant is capable of reproducing the fine details of the nonexponential fluorescence decay of the A-band of wt-GFP in a wide range of temperatures. The value of the excited-state proton transfer at the mean of the distance distribution is about the same as the one extracted from the analysis of the time-resolved emission of the A-band by the proton geminate-recombination model.37 From the best fit of the time-resolved emission, we find that the distance distribution width scales approximately with the absolute temperature, as expected from a population of reactants in thermal equilibrium situated in a parabolic potential well. Acknowledgment. We thank S. J. Remington for his generous gift of wt-GFP. We thank J. Klafter, N. Agmon, and S. Stavrov for their helpful and fruitful suggestions and discussions. This work was supported by grants from the Binational US-IsraelScienceFoundationandtheJames-FranckGerman-Israel Program in Laser-Matter Interaction. References and Notes (1) Cubitt, A. B.; Heim, R.; Adams, S. R.; Boyd, A. E.; Gross, L. A.; Tsien, R. Y. Trends Biochem. Sci. 1995, 20, 448. (2) Ormo¨, M.; Cubitt, A. B.; Kallio, K.; Gross, L. A.; Tsien, R. Y.; Remington, S. J. Science 1996, 273, 1392. (3) Yang, F.; Moss, L. G.; Phillips, G. N. J. Nat. Biotechnol. 1996, 14, 1246.

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