Unfolding Fluctuations in Proteins

Article pubs.acs.org/JPCC

Amplitude and Rate of Folding/Unfolding Fluctuations in Proteins Studied with the Help of Fluorescence Traces of the Attached Single Donor−Acceptor Pairs I. S. Osad’ko* Institute for Spectroscopy, Russian Academy of Sciences, Troitsk, Moscow, 142190 Russia ABSTRACT: The two-state model for a protein molecule with attached single donor−acceptor (D−A) pair is considered. The model uses the intensity fluctuations in D- and A-fluorescence for studying fluctuations of the amplitude and rate of FRET resulting from folding/unfolding fluctuations in the protein molecule. Such approach is in contrast to the ordinary approach which analyses the shape S(E) of the distribution of FRET efficiency E in the fluorescence of single D−A pair. Analysis of the shapes of the photon distribution functions and statistical treating of D- and A-fluorescence traces of single D−A pair enables one to find the amplitude and the rate of changes in the distance RDA between the donor and the acceptor molecules attached to a single protein molecule.

1. INTRODUCTION Single donor−acceptor (D−A) pairs attached to macromolecules of various types, such as proteins,1−4 DNA strands,5−12 dendrimers,13 and even cells14 are actively used as a tool for studying quantum dynamics of these macromolecules. Fluctuations of D- and A-fluorescence excited by CW laser are main source of information concerning quantum dynamics of macromolecules. For instance, folding/unfolding processes in proteins result in fluctuations of interdye distance RDA in such attached D−A pairs. These fluctuations of RDA can be studied with the help of the intensity fluctuations in D- and A-fluorescence. Fluorescence intensities ID of a donor molecule and IA of an acceptor molecule measured in an experiment look as follows15

It depends on the distance RDA between the donor and acceptor molecule via the background light and cross talk corrected intensities: IsD,A. Here γ = ηAΦA/ηDΦD defines the ratio of fluorescence quantum yields ΦD,A and efficiencies ηD,A of the detection in D- and A-channels. Equation 2 for the Förster resonance energy transfer (FRET) can be transformed to the following form: −1 ⎡ ⎛ RDA ⎞6 ⎤ ⎥ ⎢ E = 1+⎜ ⎟ ⎢⎣ ⎝ RF ⎠ ⎥⎦ s

This formula is widely used in practice.16−19 Here RF is the Förster radius. It is related to Förster rate F for the energy transfer by the following equation ⎛ R F ⎞6 F = D⎜ ⎟ ⎝ RDA ⎠

ID = IDs + ηDIn , IA = IAs + ηA In + β

ηA ηD

IDs (1)

They include rates of count of photons emitted by the donor and the acceptor molecule, rate In of photon counts of background light (noise). Besides the detector counting Aphotons will detect also small portion of D-photons. These Dphotons will be counted with rate βIsD/ηD (cross talk). Here ηD,A are efficiencies of the photon detection in D- and Achannels. Efficiency of the energy transfer in a D−A pair is described by the following expression: IAs(RDA) s IA(RDA) + γIDs(RDA) © 2015 American Chemical Society

(4)

Here D is the rate of deactivation of the donor molecule isolated from other molecules. It should be noted that eq 3 is correct only if fluorescence of the isolated acceptor molecule has no blinking phenomena. Blinking fluorescence consists of on-intervals with fluorescence and off-intervals without fluorescence. Off intervals emerge when the molecule occupies so-called “dark” states, for instance, the lowest triplet elecronic state of the molecule. Since triplet states exist in any molecule with covalent type of binding consideretion of the influence of triplet states on FRET

IsD,A

E s(RDA) =

(3)

Received: October 9, 2015 Revised: December 8, 2015 Published: December 16, 2015

(2) 697

DOI: 10.1021/acs.jpcc.5b09892 J. Phys. Chem. C 2016, 120, 697−704

Article

The Journal of Physical Chemistry C efficiency is of greate importance.7,15,20,21 For instance, if the acceptor molecule has blinking fluorescence eq 3 should be substituted by the following equation:15 −1 ⎡ ⎛ RDA ⎞6 ⎤ τoff ⎥ ⎢ +⎜ E = 1+ ⎟ ⎢⎣ τon ⎝ RF ⎠ ⎥⎦ s

Figure 1. Energy diagram for 2D−2A pair with fluctuating Förster rate F. State (D*A*) with excited the donor and the acceptor molecule populated weakly at k ≪ D, A is omitted.

(5)

Here τoff and τon are the average durations of off- and onintervals in fluorescence of the acceptor molecule isolated from other molecules. eq 5 shows that triplet state in A-molecule hampers the energy transfer. Full energy transfer is impossible even at RDA→ 0. Hence, we can conclude that fluorescence of D−A molecules used for the investigation of the dynamics of macromolecules should be tested in trial experiments and value of the ratio (τof f/ τon) in fluorescence of the acceptor molecule has to be evaluated. Generally, it would be desirable to use the acceptor molecules in whose fluorescence we find (τoff/τon) ≪ 1. Since D−A pairs with small value of the ratio τof f/τon are the best candidates for studying fluctuations in macromolecules we will describe D−A pairs invoking only the ground and the first excited electronic state in each molecule of D−A pairs; i.e., we will consider in fact so-called 2D−2A pairs. Usually, rate and amplitude of conformational changes in macromolecules are studied with the help of a distribution function S(E) of FRET efficiency E.4,22−25 Gopich and Szabo25 have derived a theoretical formula for S(E) in the form of a sum of Gaussians. This theoretical expression is rather complicated. For instance, two-peaks distribution S(E) is described by three Gaussians with eight adjustable parameters. Besides, their theoretical expression ignores possible difference in fluorescence quantum yields of the donor and the acceptor molecule and such experimental factors as background fluorescence, cross talk, difference in efficiencies of the detection in D- and Achannels. Therefore, it is desirable to find more simple method for studying rate and amplitude of conformational changes in macromolecules with attached D−A pairs that is free of such shortcoming. Such method is proposed in the present paper. The method does not use the measurement of the distribution function S(E) for FRET efficiency with invoking rather complicated theoretical expressions. The method is based on the statistical analysis of the D- and A-fluorescence traces with finding photon distribution functions in D- and Afluorescence.

ρ0̇ = −(k + B)ρ0 + Dρ1 + Aρ2 + bρ3 , ρ1̇ = kρ0 − (F1 + D)ρ1 , ρ2̇ = F1ρ1 − Aρ2 , ρ3̇ = Bρ0 − (k + b)ρ3 + Dρ4 + Aρ5 , ρ4̇ = kρ3 − (F2 + D)ρ4 , ρ5̇ = F2ρ4 − Aρ5

(6)

Here ρj is population of jth state (j = 0, ..., 5). Decay rates D and A of the donor and the acceptor molecules can be taken either from literature or be measured in single molecule experiment. However, microscopic parameters F1,2, which determine amplitude of fluctuations and parameters B and b that determine rate of folding/unfolding, can be found from analysis of fluorescence traces of the D−A pair. As a rule one try to find values of these parameters from the measured distribution S(E) of FRET efficiency. The most successful method for finding these parameters based on analysis of the measured distribution S(E) was proposed by Gopich and Szabo.25 The method was used in ref 4 for treating measured distribution S(E). Figure 2 shows the distribution

Figure 2. Approximation of the distribution function S(E) for FRET efficiency by three Gaussians realized in ref 4.

measured in ref 4 and approximation of this distribution by three Gauss functions in accordance with the following formula proposed in ref 25: ⎛ (E − E )2 ⎞ i ⎟ 2σi 2 ⎠ ⎝

2

S(E) =

∑ ci(2πσi 2)−1/2 exp⎜− i=0

(7)

Since amplitudes staying a front of three Gaussians obey the following relation: c0 = 1 − c1 − c2, eq 7 includes eight independent parameters numbered by index i. FRET efficiencies are described by the following expression: E1,2 = F1,2/(F1,2 + D) at B and b ≪ k. These eight parameters in eq 7 are expressed via desired microscopic parameters: F1,2, B, and b with the help of the following equations:

2. FINDING MICROSCOPIC PARAMETERS FROM ANALYSIS OF MEASURED FRET EFFICIENCY DISTRIBUTION Consider 2D−2A pair emedded in a protein molecule. Folding/ unfolding of the protein molecule will result in fluctuations of the interdye distance RDA. Therfore, Förster rate F(RDA) will fluctuate as well. Quantum states of such 2D−2A pair with fluctuating Förster rate F are shown in Figure 1. Values F1,2 determine the amplitude of fluctuations of Förster rate F. Rates B and b determine rates of folding and unfolding of the protein molecule. Rate equations for this diagram look as follows:

c1 =

B exp( −BT ), R

c0E0 =

c2 =

b exp(−bT ) R

(8a)

B b [1 − exp(−BT )]E1 + [1 − exp(−bT )]E2 R R (8b)

698


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The Journal of Physical Chemistry C σ1,2 2 = E1,2(1 − E1,2)⟨N −1⟩ c0σ02 =

for D- and A-fluorescence obtained with tbin = 10 ms are shown in Figure 4.

(8c)

BE1 + bE2 ⎛ BE1 + bE2 ⎞ −1 ⎜1 − ⎟⟨N ⟩ ⎝ ⎠ R R +

2Bb(E2 − E1)2

[RT + e−RT − 1]

R2 ⎛ BE + bE2 ⎞2 ⎟ − +⎜ 1 ⎝ ⎠ R

2

1 − ⟨N −1⟩ R2T 2

2

∑ ciEi 2 − ∑ ciσi 2 i=0

i=1

(8d)

−1

Here R = B + b, ⟨N ⟩ is the average reciprocal number of photons measured in an experiment, and T is bin time. Eq 7 and eqs 8a−8d do not take into account background fluorescence, cross talk, fluorescence quantum yields D- and A-fluorescence and efficiencies of photodetectors in D- and Achannels. However, all these experimental factors can be allowed for if we consider distributions of photons in D- and Afluorescence instead of a distribution of FRET efficiency.

Figure 4. Traces of D- and A-fluorescence calculated with bin time 10 ms for rates used in Figure 3. Length of each vertical segment shows number of photons counted during bin time.

Intervals with small and large intensity of A-fluorescence we will call off- and on-intervals, respectively. Values of on- and offintervals can be measured in fluorescence traces presented in Figure 4. They enable one to find distribution of on- and offintervals. Such distributions will be found in section 5. The diagram shown in Figure 1 includes the following set of parameters: D, A, k, F1, F2, B, and b. How these parameters can be found with the help of fluorescence traces JD,A presented in Figure 3 4? Fluorescence rates D and A can be taken from the literature or they can be measured in special experiments. The rest five parameters can be found with the help of fluorescence traces like those shown in Figure 4. A). Photon Distribution in Pure Signal (without Experimental Factors). Traces in Figure 4 were calculated by Monte Carlo technique on the basis of eq 6. They do not take into account experimental factors such as fluorescence quantum yields, efficiencies of photon detection, background fluorescence and cross talk. Let us consider first finding values of parameters k, F1, and F2 from traces shown in Figure 4 as the first step, and then we will find influence of the abovementioned experimental factors. Traces in Figure 4 enable one to find photon distribution functions wsA,D(N, tbin) in D- and Afluorescence. These distributions for tbin = 10 ms are shown in Figure 5.

3. FINDING MICROSCOPIC PARAMETERS FROM ANALYSIS OF PHOTON DISTRIBUTION FUNCTIONS The distribution of FRET efficiency shown in Figure 2 has two distinctly pronounced peaks. This means that amplitude F1,2 of Förster rate fluctuations is large enough and rates B and b of folding/unfolding are considerably less than rate k of photon emission. If these two conditions are not hold two peaks cannot be resolved and analysis of the distribution of FRET efficiency becomes rather complicated procedure.26,27 However, analysis of whole shape of the distribution of FRET efficiency can be carried out even in such case to find values of parameters.28 Nevertheless, in this paper, I restrict my consideration only by the cases when the distribution of FRET efficiency has clear two-peak shape. Distributions of photons in D- and A-fluorescence have been already measured in some works. However, they never have been used as a tool for finding dynamical parameters instead of the distribution of FRET efficiency although all experimental factors can be allowed for in the photon distribution functions. At first we consider dynamics described by eq 6 which ignore all experimental factors. Applying the Monte Carlo technique to eq 6, we find time instants for emission of D- and A-photons. They are marked by vertical segments in Figure 3.

Figure 5. Photon distribution function wsA(N, 10 ms) for A-photons (red line) and wsD(N, 10 ms) for D-photons (blue line) found from traces shown in Figure 4. Dashed lines show Poisson distributions.

Figure 3. Time instants of emission of D- and A-photons calculated with D = 2 × 108 s−1, A = 108 s−1, k = 104 s−1, F1 = 107 s−1, F2 = 108 s−1, B = 6 s−1, and b = 3 s−1.

We see two peaks in the distributions of both A-photons and D-photons. Shape of all peaks is described by Poisson function well. Position of maximum in Poisson function is described by the following simple expression: ⟨N⟩ s = Istbin. Here Is is the intensity of light with Poisson distribution of photons. Applying this formula to four Poisson peaks presented in Figure 5 we find

Long segments show time instants for D- and A-photons emitted for small Förster rate F = 107s−1, and short segments− for large Förster rate F = 108s−1. As a rule, in majority of experiments a signal acquisition time (bin time) is used. By using bin time we can transform time instants shown in Figure 3 to fluorescence traces. Such traces 699


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The Journal of Physical Chemistry C ⟨ND1,2⟩s =

D ktbin , D + F1,2

⟨N1,2 A ⟩s =

F1,2 D + F1,2

ktbin

k=

(9)

Positions of maxima in Figure 5 are ⟨N1A⟩s

= 5,

⟨NA2 ⟩s

⟨ND1 ⟩s

= 33,

= 95,

⟨ND2 ⟩s

1,2 ⎞ 1 ⎛ ⟨ND , A⟩ ⎜ + − In⎟⎟ ⎜ ΦD ⎝ ηDtbin ⎠

= 67 (10)

F1 = 107 s−1,

F2 = 108s−1

5. FINDING RATES B AND B OF FOLDING/UNFOLDING We can find rates B and b of folding/unfolding fluctuations by using two approaches. A). Measurement of Correlation Functions in D- and A-Fluorescence. Autocorrelation functions (AF) in D- and Afluorescence are determined with the help of the following equations:

(11)

Hence, in accordance with eq 4 interdye distance changes 101/6 = 1.47 times in process of folding/unfolding. However, rate of folding/unfolding cannot be found with the help of the photon distribution functions. The way values of rates B and b can be found with the help of the fluorescence traces we consider further. B). Distribution of Photons Allowing for Experimental Factors. Consider now how rates k, F1, and F2 can be found in +n a real experiment. The distribution function wsA,D (N, tbin) for photons of signal + background fluorescence (noise) is related to the distribution function wsA,D(N, tbin) for photons of signal as follows:29

gD(2), A (τ ) =

∑ w s(N − m , tbin)P n(m , tbin) m=0

n

(12)

m

Here P (m, tbin) = (Intbin) exp(−Intbin)/m! describes Poisson distribution of photons in background fluorescence. If distribution wsA,D(N, tbin) of photons in signal is of Poissonian type, the distribution of photons in signal + noise will be of Poissonian type as well. Hence, positions of the maxima for measured D- and A-fluorescence are described by the following formulas:

⟨ND⟩ = ⟨ND⟩s +ηDIntbin ⟨NA⟩ = ⟨NA⟩s +ηA Intbin + β

(13)

ηA ηD

⟨ND⟩s

(14)

gD(2)(τ ) = 1 +

It ⟨N ⟩ ⟨N ⟩ F ktbin = A − (1 + β) n bin − β D F+D ηA ΦA ΦA ηDΦD

(17)

(k1 − k 2)2 Bb (k1b + k 2B)2

exp( −Rτ ) (20)

(k1 − k 2)2 Bb [(k − k1)b − (k − k 2)B]2

exp( −Rτ ) (21)

Therefore, eqs 13 and 14 can be rewritten in the following form: (16)

(19)

Here R = B + b and k1,2 = kF1,2/(D + F1,2). Detailed derivation of these theoretical expressions for the model described by eq 6 is given in the Appendix. Figure 6 shows a view of AF calculated with the help of the following values for rates: D = 2 × 108 s−1, A = 2 × 108 s−1, k = 104 s−1, F1 = 107 s−1, F2 = 108 s−1, B = 6 s−1, and b = 3 s−1. If both AF’s are measured, we can find values of both B and b. These rates characterize folding/unfolding processes. Values B and b are adjustable parameters for two AF curves. Two

(15)

It ⟨ND⟩ D ktbin = − n bin F+D ηDΦD ΦD

⟨ID , A(t )ID , A(t + ∞)⟩

gA(2)(τ ) = 1 +

Here ⟨ND,A⟩ are positions of the maxima in the distribution function measured in real experiment. The positions of the maxima ⟨ND,A⟩sin the distribution function ws(N,tbin) are expressed via parameters of eq 6 as follows: ⟨ND⟩s ⟨N ⟩ F D = ktbin , A s = ktbin ηDΦD F+D ηA ΦA F+D

⟨ID , A(t )ID , A(t + τ )⟩

Sign < > means summation of the intensity product over whole measured time trace. Sign ∞ in the fluorescence trace means that shift of time exceeds all relaxation times in the system. The measurement of AF is a standard method for finding relaxation times.4,30 The intensities in eq 19 include counts in detector created by photons emitted by D−A pair and collected by photodetector (signal Is) and by photons from background fluorescence (noise In). Consider first contribution to AF only from signal. View of the theoretical expression for AF depends strongly on the theoretical model used for emitters. We have two types of emitters: D- and A-emitter. Therefore, we have two types of AF:

N

w s + n(N , tbin) =

(18)

Rate k depends on quantum yields, efficiencies of photon detection and cross talk. By using eq 7 and eqs 15−18 we can find parameters k, F1, and F2 from real experiment. However, rates B and b of folding/unfolding we cannot find with the help of the analysis of photon distribution functions.

By using values described by eq 10, eq 9 and taking into account that tbin = 10 ms and D = 2 × 108 s−1, we find the following values for three rate constants: 1,2 4 −1 k = (⟨N1,2 A ⟩s +⟨ND ⟩s )/ t = 10 s ,

1,2 ⎡ ⎞ ⎛ ⟨N1,2 ⟩ ⎞⎤ 1 ⎢⎛ ⟨ND , A⟩ D,A ⎜⎜ − In⎟⎟ − β ⎜⎜ − In⎟⎟⎥ ΦA ⎢⎣⎝ ηA tbin ⎠ ⎝ ηDtbin ⎠⎥⎦

The left-hand side of eqs 16 and 17 include parameters we want to find. The right-hand sides of these equations depend solely on values measured in real experiment. Summation of eqs 16 and 17 yields the following expression for rate:

Figure 6. Autocorrelation functions in D- and A-fluorescence created by the signal. 700


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The Journal of Physical Chemistry C

Equations 20 and 21 for g(2) s (τ) are true if τ is longer as compared with fluorescence lifetime because photon antibunching is absent for such long time. Therefore, eqs 20 and 21 can be rewritten in the following form:

curves for AF calculated with the help of chosen parameters B and b have to coincide with two AF’s measured in an experiment. If a single AF is measured, we can find value only one rate R. AF functions g(2) D,A(τ) approaches unity if lag time that describes shift traces with each other exceeds all relaxation times of the system. Let us consider now a correlation function determined as follows: CD , A(τ ) =

gs(2)(τ ) = 1 + a exp( −Rτ )

Here a is a contrast. Considering background fluorescence as fluorescence of many uncorrelated emitters, we should take g(2) n (τ) = 1. Then eq 25 for AF that takes into account both signal and noise looks as follows:

⟨ID , A(t )ID , A(t + τ )⟩ ⟨ID2 , A(t )⟩

(22)

⎛ Is ⎞2 ( ) 1 gs(2) τ = + ⎜ ⎟ a exp( −Rτ ) +n ⎝ Is + In ⎠

This function equals unity at τ = 0. By using the expression for two-photon correlator pA(τ), found in Appendix we find the following theoretical expression for correlation function CA(τ): CA(τ ) = +

p A (τ ) p A (0)

=

(k1b + k 2B)2 + (k1 − k 2)2 Bb

(k1b + k 2B)2 + (k1 − k 2)2 Bb

exp( −Rτ ) (23)

By using trace for A-fluorescence presented in Figure 4 we find with the help of the following formula: ⟨JA (t )JA (t + τ )⟩/⟨JA (t )JA (t )⟩ = CA(p) 1600

=

i=0

Poff = B exp( −Bt ),

1600

∑ JA (i)JA (i + p)/ ∑ JA (i)JA (i) i=0

Figure 7. Correlation function CA(τ) measured in A-trace shown in Figure 4 (histogram) and CA(τ) calculated with the help of eq 23 for the following set of parameters: D = 2 × 108 s−1, A = 108 s−1, k = 104 s−1, F1 = 107 s−1, F2 = 108 s−1, B = 6 s−1, and b = 3 s−1 (dashed line). The same set of parameters was used for the calculation of fluorescence traces shown in Figure 4.

calculation of the distributions with the help of eq 28 at B = 6 s−1 and b = 3 s−1 are shown in Figure 8 by dashed lines. Figure 8 shows that rate constants B and b can be found with the help of the distributions of off/on intervals measured in fluorescence trace without measurement of correlation functions in fluorescence traces.

6. CONCLUSION Fluctuations of Förster rate F in a single donor−acceptor pair attached to a protein molecule have been studied without invoking distributions S(E) of FRET efficiency E. We have used photon distribution functions. Two-peaks in photon distribution functions were used to find values F1 and F2 from traces in D- and A-fluorescence in the presence of background fluorescence, cross talk and difference in fluorescence quantum yields ΦD,A and efficiency ηD,A of detection in D- and Achannels. Two methods for finding rates B and b of folding/ unfolding fluctuations have been considered as well. The first method is based on the measurement of (2) autocorrelation functions g(2) A (τ) and gD (τ), or correlation function CA(τ) for which eqs 20, 21, and 23 have been derived

In2gn(2)(τ ) + Is2gs(2)(τ ) + 2InIs (In + Is)2

(28)

Figure 8. Distribution of on/off-times measured in A-trace of 40 s duration are shown by histograms. Functions 20 exp(−3s−1t) for ontimes and 34 exp(−6s−1t) for off-times are shown by dashed lines.

the result of the statistical treating of the A-trace presented in Figure 4 with the help of eq 24. The dashed line in Figure 7 shows correlation function CA(τ) calculated with the help of eq 23. Good coincidence of “theoretical” curve with “experimental” histogram proves correctness of our Monte Carlo simulations and our derivation of eq 23. Let us consider how noise will influence AF. Equation for such AF has been derived in ref 29. It appears as follows: =

Pon = b exp(−bt )

The A-trace shown in Figure 4 enables one to measure durations of on/off intervals and to find distribution of these intervals. Results of such statistical treating of A-trace of 40 s duration are shown in Figure 8 by histograms. Results of

(24)

correlation function relating to this very trace. Here each p differs from p + 1 by 10 ms. The histogram in Figure 7 shows

gs(2) (τ ) +n

(27)

Noise decreases the contrast in the measured AF B). Direct Measurement of On/Off Distributions. The method of finding rates B and b with the help of the measured distributions of on/off intervals does not feel existence of noise. This feature of the method is certainly the advantage as compared with the measurement of AF. If we have measured fluorescence trace like those shown in Figure 4, we can find distribution of on/off times. By using eq 6 we can find the following simple theoretical expressions for the distributions of on/off intervals in A-fluorescence:

(k1b + k 2B)2

(k1 − k 2)2 Bb

(26)

(25)

Here Is,n are intensities of signal and noise averaged over long (2) time interval, τ is lag time and g(2) s (τ), gn (τ) are AF for signal and for noise. A single emitter cannot emit two photons simultaneously. Therefore, gs(2)(0) = 0. However, photon antibunching is absent in signal + noise and we have g(2) s=n(0) > 0. 701


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The Journal of Physical Chemistry C in Appendix. It is shown in section 5 these equations describe correlations found from fluorescence traces well. The second method is based on the measurement of the distribution of on/off times directly from the fluorescence traces. This is the simplest method for finding rates B and b, and it works well as Figure 8 shows. Hence, by using fluorescence traces in D- and A-fluorescence we are able to find values of all five rates: k, F1, F2, B, and b, which determine slow dynamics of folding/unfolding processes in proteins in the presence of all factors of real experiment.

ρ0̇ = −(k1 + B)ρ0 + Aρ2 + bρ3 , ρ2̇ = k1ρ1 − Aρ2 , ρ3̇ = Bρ0 − (k 2 + b)ρ3 + ρ4 /TD + Aρ5 , ρ5̇ = F2ρ4 − Aρ5

Here k1 = k

■

APPENDIX Autocorrelation function (AF) of single emitter with several types of fluorescence photons is described by the following formula31 g(2)(τ ) =

⟨I(t )I(t + τ )⟩ = p(τ )/p(∞) ⟨I(t )I(t + ∞)⟩

(A5)

F1 , D + F1

k2 = k

F2 D + F2

(A6)

The set of equations shown in eq A5 describes the slow dynamics of the donor molecule and fast and slow dynamics of the acceptor molecule. Equations for slow dynamics of the acceptor molecule we find by setting: ρ̇2 = ρ̇5 = 0. Then we arrive at the following relations:

(A1)

k1 k ρ0j , ρ5j = 2 ρ3j (A7) A A Here index j shows which type of photon was emitted at t = 0,i.e. in which conformation was the system protein+D-A pair at t = 0. With the help of eq A7 we transform eq A5 to the following couple of equations: ρ2j =

Here p(τ) is a two-photon correlator, i.e., a probability of detection of two photons separated by time interval, τ. If, for instance, A-fluorescence has two types of photons, as red long and short segments in Figure 3 show, we have four types of correlators pAij (τ). They are shown in Figure A1

ρ0̇ = −Bρ0 + bρ3 , ρ3̇ = Bρ0 − bρ3

They describe slow dynamics of the donor and the acceptor fluorescence. Since photons of 2 and 5 type are emitted after quantum jumps 2 → 0 and 5 → 3, respectively, we can transform matrix for two-photon correlators to the following form:

Figure A1. Sequence of A-photons emitted from levels 2 and 5 shown in Figure 1 and four types of correlators pAij (τ).

⎛ pA pA ⎞ ⎛ ρ22 ρ25 ⎞ ⎛ k1ρ00 k1ρ03 ⎞ 22 25 ⎟ ⎟ = p ̂ = ⎜⎜ A ⎟ = ⎜⎜ ⎜ ⎟ A A⎟ ρ ρ ρ ρ k k ⎠ ⎝ 52 52 ⎝ 2 30 2 33 ⎠ ⎝ p52 p55 ⎠

Photons of type 2 or 5 were emitted from levels 2 or 5, i.e., when the system was in the left or in the right conformation shown in Figure 1. In accordance with Figure A1 dynamics of the system is described by the following matrix consisting of four correlators: ⎛ pA ⎜ 22

A

(A2)

pAij (τ)

Indeces i and j in correlators show type of the second and the first photon of the pair, respectively. In accordance with ref31. correlators pAij (τ) are related to the solution of eq 6 as follows: p2Aj (τ ) = Aρ2j (τ ), p5Aj (τ ) = Aρ5j (τ )

p̂A =

k ρ, D + F1 0

ρ4 =

k ρ D + F2 3

−Rτ −Rt ⎞ ⎛ 1 ⎜ k1(b + Be ) k1B(1 − e ) ⎟ R ⎜⎝ k b(1 − e−Rt ) k (B + be−Rτ )⎟⎠ 2 2

(A10)

Here R = B + b. The correlator relating to the probability of finding A-photon of arbitrary type is given by the following expression: ⎛ n2 ⎞ p A (τ ) = (1 1)p ̂ A ⎜ ⎟ ⎝ n5 ⎠ (A11)

(A3)

Here ρ2j(τ) is the solution of eq 6 with initial condition ρ2j(0) = 1, and ρ5j(τ) is the solution of eq 6 with initial condition ρ5j(0) = 1 (j = 2, 5) Rate constant D in eq 6 describes fast electronic dynamics of the order of 108s−1 in the donor molecule. Folding/unfolding is realized much slower. This slow dynamics can be found with the help of eq 6 in which we set ρ̇1 = ρ̇4 = 0. Then we arrive at the following relations: ρ1 =

(A9)

Initial conditions ρ00(0) = ρ30(0) = 1 describe the system in the left conformation and initial conditions ρ03(0) = ρ33(0) = 1 describe the system in the right conformation. By solving eq A8 with these initial conditions we arrive at the following final expression for the matrix pAij (τ) of two-photon correlators

p25A ⎞ ⎟

p̂A = ⎜ A A⎟ ⎝ p52 p55 ⎠

(A8)

Here n2 =

ρ2 (∞) ρ2 (∞) + ρ5 (∞)

n5 =

(A4)

=

ρ5 (∞) ρ2 (∞) + ρ5 (∞)

k1b , k1b + k 2B =

k 2B k1b + k 2B

(A12)

are the probabilities of finding protein + D−A pair in the left and the right conformation, respectively.

By using these relations we transform eq 6 to the following form: 702


Article

The Journal of Physical Chemistry C Inserting eq A10 and eq A12 to eq A11 we arrive at the final expression for two-photon correlator for A-photons.: p A (τ ) =

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⎞ (k − k 2)2 Bb 1⎛ ⎜k1b + k 2B + 1 exp( −Rτ )⎟ R⎝ k1b + k 2B ⎠ (A13)

By using eq A1, we find the following expression for the autocorrelation function for A-fluorescence: gA(2)(τ ) = 1 +

(k1 − k 2)2 Bb (k1b + k 2B)2

exp( −Rτ ) (A14)

The expression for AF measured in D-fluorescence can be found analogously. The result differs from eq A14 by substitution of ki by k−ki, i.e. result looks as follows: gD(2)(τ ) = 1 +

■

(k1 − k 2)2 Bb [(k − k1)b − (k − k 2)B]2

exp( −Rτ ) (A15)

AUTHOR INFORMATION

Corresponding Author

*(I.S.O.) E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■ ■

ACKNOWLEDGMENTS The work was supported by Grant No. 14-12-01415 from the Russian Science Foundation REFERENCES

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Article

The Journal of Physical Chemistry C Molecule Fluorescence. Phys. Rev. A: At., Mol., Opt. Phys. 2011, 83, 063841.

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Unfolding Fluctuations in Proteins

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