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J. Phys. Chem. B 2007, 111, 4220-4226
Extracting the Time Scales of Conformational Dynamics from Single-Molecule Single-Photon Fluorescence Statistics Jianyuan Shang and Eitan Geva* Department of Chemistry, UniVersity of Michigan, Ann Arbor, Michigan 48109-1055 ReceiVed: NoVember 17, 2006; In Final Form: January 27, 2007
The quenching rate of a fluorophore attached to a macromolecule can be rather sensitive to its conformational state. The decay of the corresponding fluorescence lifetime autocorrelation function can therefore provide unique information on the time scales of conformational dynamics. The conventional way of measuring the fluorescence lifetime autocorrelation function involves evaluating it from the distribution of delay times between photoexcitation and photon emission. However, the time resolution of this procedure is limited by the time window required for collecting enough photons in order to establish this distribution with sufficient signalto-noise ratio. Yang and Xie have recently proposed an approach for improving the time resolution, which is based on the argument that the autocorrelation function of the delay time between photoexcitation and photon emission is proportional to the autocorrelation function of the square of the fluorescence lifetime [Yang, H.; Xie, X. S. J. Chem. Phys. 2002, 117, 10965]. In this paper, we show that the delay-time autocorrelation function is equal to the autocorrelation function of the square of the fluorescence lifetime divided by the autocorrelation function of the fluorescence lifetime. We examine the conditions under which the delay-time autocorrelation function is approximately proportional to the autocorrelation function of the square of the fluorescence lifetime. We also investigate the correlation between the decay of the delay-time autocorrelation function and the time scales of conformational dynamics. The results are demonstrated via applications to a two-state model and an off-lattice model of a polypeptide.
I. Introduction Many experimental, theoretical, and computational studies over the last several decades were aimed at elucidating the nature of conformational dynamics in biopolymers under different conditions.1,2 More recently, powerful single-molecule (SM) techniques have emerged which make it possible to explore conformational dynamics with an unprecedented level of detail.3-70 These SM techniques often take advantage of the fact that the rate of fluorescence quenching is highly sensitive to the conformational state of the macromolecule. For example, in SM fluorescence resonance energy transfer (SM-FRET) experiments, one labels a macromolecule with one donor and one acceptor dye molecule at specific sites and measures the variation over time of the donor fluorescence lifetime due to quenching by nonradiative energy transfer to the acceptor (cf. Figure 1).39-70 The energy transfer rate constant is often described by Fo¨rster theory,71-75
kET(R) ) kD
() R0 R
6
(1)
where k-1 D is the fluorescence lifetime of the free donor, R is the distance between donor and acceptor, and R0 is a parameter that depends on the choice of donor-acceptor pair, their relative orientation, and other experimental conditions.75 Thus, the rate of quenching via FRET is sensitive to the donor-acceptor distance, which will fluctuate in time due to conformational dynamics. Another example corresponds to quenching by * Address correspondence to this author.
Figure 1. A schematic view of the donor-acceptor photophysics. D/A and D*/A* correspond to the ground and excited donor/acceptor, respectively. It is assumed that only the donor is photoexcited by a pulsed laser with a repetition rate of kex. kET is the donor-to-acceptor energy transfer rate constant, and kD/kA are the free donor/acceptor fluorescence rate constants.
electron transfer.3,8,15,76 The quenching rate constant in this case is often assumed to be exponential,
ke(R) ) k0e e-βR
(2)
where R is the distance between the quencher and the fluorophore. The sensitivity of the fluorescence lifetime to the conformational state implies that the decay of the fluorescence lifetime autocorrelation function contains information on the time scale of conformational dynamics. The conventional way of measuring the fluorescence lifetime autocorrelation function involves
10.1021/jp067657c CCC: $37.00 © 2007 American Chemical Society Published on Web 03/29/2007
Extracting the Time Scales of Conformational Dynamics evaluating it from the distribution of delay times between photoexcitation and photon emission. However, the time resolution of this procedure is limited by the length of the time window required for collecting a large number (∼100) of photons in order to establish this distribution with sufficient signal-to-noise ratio. For example, SM-FRET experiments are often performed by using a dual-channel detection scheme. In this case one detects fluorescence photons from the donor and acceptor in a selective manner. One may excite the donor by a train of short pulses, and obtain the fluorescence lifetime from the distribution of time delays between photoexcitation and photon emission within a given time window of length TW. Alternatively, one may employ CW excitation and measure the fraction of photons detected in the acceptor channel over the same time window, which can be directly related to the fluorescence lifetime (see below). Importantly, the fluorescence lifetime measured in either way represents an average over the conformations visited within this time window. If TW is very short in comparison to the time scale of conformational dynamics, then R remains fixed during the measurement and the measured lifetime is directly related to the instantaneous conformation. On the other hand, if TW is longer than the time scale of conformational dynamics, the lifetime will coincide with the corresponding ensemble average over all conformations. In practice, one is usually in between those two limits. Thus, the accessible time resolution is dictated by TW, with conformational dynamics on time scales shorter than TW being inaccessible. The conformational dynamics of biopolymers is often characterized by a very wide and continuous dynamical range that spans many decades of time scales, from femtoseconds to hours. It is therefore highly desirable to improve the resolution of single molecule techniques so that their dynamical range is as wide as possible. Yang and Xie have recently proposed a new strategy for improving the time resolution by obtaining the lifetime autocorrelation function from the autocorrelation function of the delay time between photoexcitation and single fluorescence photon emission.77 Their analysis is based on the argument that the autocorrelation function of the delay time between photoexcitation and photon emission is proportional to the autocorrelation function of the square of the fluorescence lifetime. In this paper, we present a general relation between the autocorrelation function of the single-photon time-delay and that of the fluorescence lifetime. More specifically, we show that the delay-time autocorrelation function is equal to the autocorrelation function of the square of the fluorescence lifetime divided by the autocorrelation function of the fluorescence lifetime. We also explore the conditions under which the decay of the timedelay autocorrelation function coincides with that of the autocorrelation function of the square of the lifetime. Finally, we demonstrate the validity of the above-mentioned general relationship via kinetic Monte Carlo simulations of photon statistics in the case of a two-state model and an off-lattice model polypeptide that undergoes Langevin dynamics. The organization of the remainder of this paper is as follows. The general relationship between the single-photon time-delay autocorrelation function and fluorescence lifetime autocorrelation function is derived in Section II. Applications to model systems are presented in Section III. A summary of the main results is provided in Section IV. II. A General Relationship between the Single-Photon Time-Delay and Fluorescence Lifetime Autocorrelation Functions For the sake of concreteness, we will present the derivation in the context of SM-FRET experiments. It should be empha-
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Figure 2. A schematic view of the experimental setup. A single macromolecule is subjected to a train of short pulses with a repetition rate kex. τ(tj) is defined as the time interval between photoexcitation of the donor at time tj and the emission of a fluorescence photon by either the donor (green) or the acceptor (red).
sized, however, that the final result is general and will also hold for other quenching mechanisms such as electron transfer. In the case of SM-FRET, one considers a macromolecule, labeled by a pair of donor-acceptor dye molecules at two specific sites. We assume that the donor is photoexcited by a train of short pulses with a repetition rate of kex (∼108 s-1). The photoexcited donor either fluoresces back to the ground state, with a rate constant kD (∼109 s-1), or is quenched by nonradiatively transferring its energy to the acceptor, with a rate constant kET (cf. Figure 1). In the case where energy transfer (ET) from donor to acceptor takes place, emission of a fluorescence photon by the excited acceptor, with rate constant kA (∼109 s-1), follows. The fluorescence photon from the donor is assumed to be blue-shifted relative to that from the acceptor, so that they can be detected in a selective manner. The rate constants kD and kA are also assumed to be insensitive to the conformational state of the macromolecule. However, kET is assumed to be explicitly dependent on the conformational state of the macromolecule at the time when ET takes place (cf. eq 1). The probability for quenching via ET is given by:
E)
kET kET ≡ kET + kD k
(3)
where k-1 ) (kET + kD)-1 is the donor fluorescence lifetime. The probability for the complimentary donor fluorescence event is given by:
F)1-E)
kD kD ) kET + kD k
(4)
Photoexcitation of the donor by a pulse at time t leads to the emission of a fluorescence photon by either donor or acceptor in a time interval τ(t) after the pulse (cf. Figure 2). The timedelay τ(t) can then be viewed as a dynamical variable. In the case where the fluorescence lifetime is fixed, τ(t) can be sampled from the probability distribution P(τ) ) ke-kτ, and its values at different times are uncorrelated. However, as mentioned above, conformational dynamics can give rise to an explicitly timedependent fluorescence lifetime that will lead to additional, timecorrelated, fluctuations of τ(t). Thus, the trajectory traced by τ(t) can contain information on the time scale of conformational dynamics, which one would like to be able to extract. Following Yang and Xie,77 we consider the autocorrelation function of τ(t) for the subset of excitation-fluorescence events where the fluorescence photon is emitted by the donor (the complimentary autocorrelation function of τ(t) for the subset of excitationfluorescence events where the fluorescence photon is emitted by the acceptor is presented in Appendix A):
4222 J. Phys. Chem. B, Vol. 111, No. 16, 2007
CD(t2 - t1) ≡ 〈τ1τ2〉D )
Shang and Geva
∫0∞ dτ1 ∫0∞ dτ2 PD(τ1,τ2)τ1τ2
(5)
Here, τj ) τ(tj) (j ) 1, 2), where t1 and t2 are any two times along the τ(t) trajectory, and PD(τ1,τ2) is the corresponding joint probability density for τ(t1) ) τ1 and τ(t2) ) τ2. The probability density PD(τ1,τ2) can now be put in the following form
The derivation that led to eq 8 can be easily extended to a more general form of the correlation function. For example, it is straightforward to show that
〈τz11-1 τz22-1〉D )
Γ(z1) Γ(z2) 〈Fz11 Fz22〉 〈F1F2〉
kzD1+z2-2
1 -z2 〈k-z 1 k2 〉
PD(τ1,τ2) )
∫01 dF1 ∫01 dF2 P(F1,F2) PD(τ1|F1)PD(τ2|F2) ∫0∞ dτ1 ∫0∞ dτ2 ∫01 dF1 ∫01 dF2 P(F1,F2) PD(τ1|F1) PD(τ2|F2) (6) where Fj ) F(tj) ) kD/[kD + kET(tj)] ) kD/kj, P(F1,F2) is the joint probability density for F ) F1 at t ) t1 and F ) F2 at t ) t2, and P(τj|Fj) is the conditional probability for a donor fluorescence event at a time-delay τj after photoexcitation, given that the donor fluorescence probability is given by Fj (cf. eq 4). It should be noted that in writing eq 6, we assumed that the conformational state, and therefore F, does not change significantly on the time scale of the fluorescence lifetime. The key quantity in eq 6 is the conditional probability P(τj|Fj). This conditional probability is given by the product of two probabilities: (1) the probability for fluorescence by the donor at time tj, which is given by Fj, and (2) the probability that the time-delay between donor photoexcitation and emission of the fluorescence photon is given by τj, which is given by kj exp(-kjτj) ) (kD/Fj) exp(-kDτj/Fj). Thus,
P(τj|Fj) ) Fj ×
)
( )
( )
kD kD kD exp - τj ) kD exp - τj Fj Fj Fj
(7)
Substituting this expression into eq 6, and explicitly integrating over τ1 and τ2 leads to the following result:
〈τ1τ2〉D )
∫0∞ dτ1 ∫0∞ dτ2 ∫01 dF1 ∫01 dF2 P(F1,F2)e-k τ /F e-k τ /F τ1τ2 ∫0∞ dτ1 ∫0∞ dτ2 ∫01 dF1 ∫01 dF2 P(F1,F2)e-k τ /F e-k τ /F D 1
1
D 1
D 2
1
2
D 2
2
Γ(z1) Γ(z2) -1 -1 (9) 〈k1 k2 〉 where z1,z2 > 0 and Γ(z) is the Gamma function. Comparison of eq 9 with the corresponding results from the work of Yang and Xie (see eq 10 of ref 78) reveals that the latter can be obtained from the former by assuming that 〈k1-1 k2-1〉 ≈ 〈k-1〉2 in the denominator of eq 9. This approximation suggests that the correlation functions 〈τ1τ2〉D and 〈k1-2 k2-2〉 decay on the same time scale. The conditions under which this assumption holds will be considered on the following section. III. Demonstrative Applications A. Conformational Dynamics with Two-State Kinetics. We start by considering a macromolecule that interconverts between two subsets of conformations, a and b, with rate constants κab (from a to b) and κba (from b to a). The donor fluorescence probabilities in states a and b are denoted by Fa and Fb, respectively. It is straightforward to show that for this model,
〈X1X2〉 ) 〈X〉2 + PaPb(Xa + Xb)2e-κ(t2-t1)
Here, X is any property of the system, κ ) κab + κba is the overall rate constant, Pa ) κba/κ and Pb ) κab/κ are the equilibrium probabilities to be in states a and b, respectively, and 〈X〉 ) PaXa + PbXb is the equilibrium average of X. Substituting X ) F and X ) F2 into eq 10 yields explicit expressions for 〈F1F2〉 and 〈F12 F22〉, respectively. Substituting these expressions into eq 8 then yields the following expression for 〈τ1τ2〉D:
〈τ1τ2〉D ≡ CD(t2 - t1) )
2 2 2 2 1 ∫ dF1 ∫ dF2 P(F1,F2)F1 F2 1 〈F1 F2 〉 ) 2 ) 2 ) kD ∫ dF1 ∫ dF2 P(F1,F2)F1F2 kD 〈F1F2〉
〈k1-2 k2-2〉 〈k1-1 k2-1〉
(8)
The last equality in eq 8 is based on the fact that kj ) kD/Fj (cf. eq 4). Equation 8 represents a new general relationship between the donor time-delay autocorrelation function, 〈τ1τ2〉D, and the fluorescence lifetime correlation functions 〈k1-2 k2-2〉 and 〈k1-1 k2-1〉. Importantly, 〈τ1τ2〉D can be measured with a time resolution dictated by the time interval between two subsequent single photon emissions by the donor (gk-1 ex ). This implies that 〈k1-2 k2-2〉/〈k1-1 k2-1〉 can potentially contain information on conformational dynamics with similar time resolution. This represents a significant improvement in time resolution in comparison to the commonly employed practice of evaluating the fluorescence lifetime based on the delay times of ∼100 photons collected over a time window that will be ∼100 times longer than the time interval between two subsequent single photon emissions by the donor.
(10)
2 2 1 〈F 〉 kD2 〈F〉2
( )
1 + P aP b
1 + PaPb
F2a - F2b 〈F2〉 Fa - Fb
(
〈F〉
2
)
2
e-κ(t2-t1) (11) e-κ(t2-t1)
This correlation function can now be normalized as follows:
C h D(t) )
CD(t) - CD(∞) CD(0) - CD(∞)
[
( (
1 + P aP b
)
1 + PaPb
)] )
Fa - F b
〈F〉 F a - Fb 〈F〉
2
2
e-κt (12) -κt
e
Clearly, C h D(t) f e-κt when PaPb((Fa - Fb)/〈F〉)2 , 1. Under these conditions 〈τ1τ2〉D ∝ 〈k1-2 k2-2〉, and the decay rate of 〈τ1τ2〉D does indeed coincide with the time scale of conformational dynamics, which is given by κ-1 in this case. However, the condition PaPb((Fa - Fb)/〈F〉)2 , 1 does not hold for large values of Fa/Fb and/or small values of Pa (cf. Figure 3), which can lead to situations where 〈τ1τ2〉D is not proportional to 〈k1-2 k-2 h D(t) does not reflect the time scale of 2 〉 and the decay of C conformational dynamics. The validity of eq 8 can be tested by calculating 〈τ1τ2〉D directly via kinetic Monte Carlo simulations of the stochastic
Extracting the Time Scales of Conformational Dynamics
J. Phys. Chem. B, Vol. 111, No. 16, 2007 4223
Figure 4. A small segment of a stochastic trajectory of the two-state system. Fluorescence emission events by the donor and acceptor are indicated by green and red dots, respectively, along the trajectory. The trajectory was generated with the following parameter values: kD ) kA ) 1.0, kex ) 0.1, κ ) 0.02, Pa ) 0.5, Fa ) 0.8, and Fb ) 0.4.
Figure 3. Contour plots of C h D(t) as a function of t and Fa/Fb, for the indicated values of Pa, as obtained for the two-state model. Also shown for reference is a plot of e-kt (upper panel). As can be seen C h D(t) ≈ e-kt, except at large values of Fa/Fb and/or small values of Pa.
system kinetics and photon statistics. One starts by generating the stochastic trajectory of the system, which was done by using the algorithm of Makarov and Matiu.79 Within this algorithm, one treats the residence times in states a and b as the stochastic variables. The residence time in state a (b) is then determined by -ln(η)/κab (-ln(η)/κba), where η is a random number between zero and one. Once the stochastic trajectory of the system is established, one can use the very same kinetic Monte Carlo algorithm to generate the stochastic photon streams, which can then be used for calculating 〈τ1τ2〉D. The residence time in state (D*,A) following photoexcitation is given by -ln(η)/(kD + kET), after which the system makes a transition to either the ground state, with probability F ) kD/(kD + kET), or the state (D,A*), with probability E ) kET/(kD + kET). It should be noted that the value of kET at the time of the transition depends on whether the system is in state a or b. In the case where the system ends up in state (D,A*), its residence time in this state is given by -ln(η)/kA, after which it makes a transition back to the ground state. In the simulations reported below, we have assumed that kD ) kA ) 1.0. We have also assumed that the laser repetition rate is given by kex ) 0.1 (
