Determination of Equilibrium Constant and Relative Brightness in

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Determination of Equilibrium Constant and Relative Brightness in FRET-FCS by Including the Third-Order Correlations Lingyi Meng, Shanshan He, and Xin Sheng Zhao J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.7b09229 • Publication Date (Web): 20 Nov 2017 Downloaded from http://pubs.acs.org on December 1, 2017

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The Journal of Physical Chemistry

Determination of Equilibrium Constant and Relative Brightness in FRET-FCS by Including the Third-Order Correlations Lingyi Meng1,2‡, Shanshan He1,3‡ and Xin Sheng Zhao1,3* 1

Biodynamic Optical Imaging Center (BIOPIC); 2School of Life Sciences; and 3Beijing

National Laboratory for Molecular Sciences, State Key Laboratory for Structural Chemistry of Unstable and Stable Species, and Department of Chemical Biology, College of Chemistry and Molecular Engineering; Peking University, Beijing 100871, China.

ABSTRACT Fluorescence correlation spectroscopy (FCS) encodes the information of the equilibrium constant (K), the relative fluorescence brightness of fluorophore (Q), and the forward and backward reaction rate constants (k+ and k-) on a physical or chemical relaxation. However, it has been a long-standing problem to completely resolve the FCS data to get the thermodynamic and kinetic information. Recently, we have solved the problem for fluorescence auto-correlation spectroscopy (FACS). Here, we extend the method to fluorescence cross-correlation spectroscopy (FCCS), which appears when FCS is coupled with fluorescence resonance energy transfer (FRET). Among total 12 second-order and third-order pre-exponential factors in a relaxation process probed by the FRET-FCS technique, 3 are independent. We presented and discussed 3 sets of explicit solutions to use these pre-exponential factors to calculate K and Q. Together with the relaxation time, the acquired K will allow people to obtain k+ and k-, so that the goal of deciphering the FRET-FCS data will be fully reached. The theory is verified by extensive computer simulations and tested experimentally on a system of oligonucleotide hybridization.

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I. INTRODUCTION Fluorescence correlation spectroscopy (FCS) is widely used in the study of molecular diffusions and physical and chemical relaxations under equilibrium, where a collective motion of the system may occur simultaneously.1-6 However, the conventional second-order FCS alone is incapable of determining necessary thermodynamic and kinetic properties encoded in the data, including the equilibrium constant (K), the relative fluorescence brightness (Q), and the forward and backward rate constants (k+ and k-) of a physical or chemical relaxation, which has hindered FCS to provide quantitative information on studied systems.7,8 Higher-order correlations7-15 and complementary photon intensity statistics analysis6, such as photon counting histogram (PCH)16 and fluorescence intensity distribution analysis (FIDA)17, have been investigated to deal with this problem and to reveal other properties. With regard to the problem related to the physical and chemical relaxations, the approach of higher-order correlations appear to be more suitable.7,8,15 Very recently, inspired by the work of Melnykov and Hall,15 our laboratory7 and Abdollah-Nia et al.8 have proposed specific algorithms to derive the thermodynamic and kinetic parameters (K, Q, k+ and k-) from the chemical and physical relaxation terms by combining the conventional second-order FCS and higher-order correlations. In our work, the explicit formula to calculate K and Q (so that k+ and k- as well) from the second- and third-order auto-correlations are presented.7 Abdollah-Nia et al.8 have provided multi-detector and sub-binning approach with the higher-order correlations to eliminate the detector artifact and to achieve higher time resolution. With the settlement of the issue on the fluorescence auto-correlation spectroscopy (FACS), how to solve K and Q from the fluorescence cross-correlation spectroscopy (FCCS) data becomes naturally the next task, which typically occurs when an FCS experiment is coupled with fluorescence resonance energy transfer (FRET). FRET-FCS is a convenient probe for biophysical and biochemical reaction. Here, we 2


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only recall a few representative papers among plenty publications that are related to the FRET-FCS technique. As early as 1978, Hass et al. used FRET to study the introchain reactions.18 In 1997, Schwille et al. proposed and experimentally demonstrated the idea of FCCS.19 In 2000, Wallace et al. studied the FRET-labeled hairpin DNA folding by applying FACS to the proximity ratio, which combines FCS with FRET to probe a biochemical reaction.20 Later, Hom and Verkman demonstrated the power of FRET-FCS in a straight sense of FCCS.21 The theoretical treatments on FCCS have been evolving along with the development of the FRET-FCS technique.19,21 The work by Torres and Levitus is among recent examples of the theoretical work on FCCS.22 In their paper a complete set of the second-order correlation functions in a FRET-FCS setting are derived, and the quantitative relations between the experimentally obtainable pre-exponential factors in the relaxation terms and the desired molecular properties (forward and backward rates, reactant and product FRET efficiencies) are presented. Nevertheless, as the same as previous treatments, they were unable to rigorously extract the wanted thermodynamic and kinetic parameters by merely considering the conventional second-order FCS, due to the exact same reason as that appears in FACS: there are more un-known quantities than known relations. In this paper, we extend our methodology on FACS7 to the case of FCCS. We investigated all the second-order and third-order correlation functions occurring in a double dye-labeled system. Under equilibrium, 3 among total 12 pre-exponential factors are independent. We present 3 sets of explicit combinations to calculate the wanted molecular properties: K, QA (the relative brightness of the acceptor) and QD (the relative brightness of the donor). In addition, in comparison with our previous work, new forms of the third-order correlations are proposed, which are more stable and feasible for the purpose of calculating K and Q. We verified our theory by extensive computer simulations, and we also tested it experimentally on a system of oligonucleotide hybridization. Together with the relaxation 3



rate constant, the acquired K will allow people to obtain k+ and k-, so that the goal of deciphering the FRET-FCS data will be fully reached. Our results will prompt FRET-FCS to be more powerful in quantitative investigation of biophysical and/or biochemical processes.

II. THEORETICAL CONSIDERATIONS 2.1 Model We are interested in a physical or chemical relaxation on a double dye-labeled molecule (Fig. 1). It does not matter whether the fluorescently labeled molecules are freely diffusing in solution, or immobilized on a surface but moving collectively with the same speed v with respect to the probing laser (we call the collective motion with the same speed with respect to the probing laser as “scanning”), or diffusing and scanning simultaneously in solution, or fixed at spatial points (such as immobilized on a surface and no scanning). This result comes from the fact that these modes are mutually independent and the statistics are time-independent, so that the kinetic processes are simply used as a mean to obtain identical, independently distributed measurements. Consider a simple reaction from the reactant state 1 to the product state 2 with a forward reaction rate constant k+ and a backward reaction rate constant k-. So, the equilibrium constant of the reaction is K=

k+ k−

2.1.1

In Fig. 1, A and D are different fluorophores labeled on the molecule and are referred as the acceptor and donor, respectively. A1, D1 are the fluorescent states on the reactant and A2, D2 are the fluorescent states on the product. The fluorescence brightness, QλS (S=A, D; λ=1, 2), is the fluorescent photon counting rate of species Sλ under unit concentration and unit laser power in unit time.7 The relative brightness of fluorophore S, QS, is the ratio of Q2S over 4


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Q1S . The forward direction of the reaction is so defined that 0 ≤ QA < 1 . Under this convention,

the range of QD will be 0 ≤ QD < ∞, QD ≠ 1. One may use apparent FRET efficiency (i.e., proximity ratio) to characterize the fluorescent relation between the donor and acceptor: E1 =

Q1A QA , E2 = A 2 D D Q + Q1 Q2 + Q2

2.1.2

A 1

If we define QT =

Q2A + Q2D Q1A + Q1D

2.1.3

it is easy to derive that E1 =

QD − QT QD − QA

2.1.4

E2 =

QD − QT QA QD − QA QT

2.1.5

Or, vice verse QA =

QD =

QT E2 E1

2.1.6

QT (1 − E2 ) 1 − E1

2.1.7

We emphasize that all Q and E defined here are quantities that are affected by the experimental conditions.

2.2 The second-order FCS When the system is linearized, the conventional second-order FRET-FCS is well-known.4,5,22,23 Here we merely review it with our language. The second-order correlation functions by using the difference of the photon counting are defined by Gij(2) (t ) =

δ ni (0)δ n j (t )

2.2.1

ni n j

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where i and j are detector channel A or D. In Eq. (2.2.1) if i=j we take t ≠ 0,

L

stands for

the ensemble average, nl (l=A, D) is the number of detected photons (photon counting) in each time bin ∆t of respective channel, δnl = nl −nl , and nl = nl . For the system described by Fig. 1, when diffusion and scanning are included, one finds Gij(2) (t ) = GD(2) (t )GS(2) (t )GR(2),ij (t )

2.2.2

where GD(2) (t) and GS(2) (t) are terms due to diffusion and scanning respectively, and GR(2),ij (t ) is the term due to the physical or chemical relaxation. If the laser intensity is described by a Gaussian profile with ωxy and ωz being the widths of the laser focus in the xy plane and along the z axis respectively,7 The mathematical forms of GD(2) (t) and GS(2) (t) are4,5 G D( 2 ) (t ) =

1 N

−1

 t   t  1 +  1 + 2  τ ω τD   D  

−

1 2

2.2.3

where τ D = ω xy2 / 4 D , ω = ω z / ω xy , and N is the average number of molecules in the sampling 3

volume as N=VC, where V = π 2 ω xy2 ω z and C is the sample concentration, and   t 2  G (t ) = exp  −     τS    

2.2.4

(2) S

where τ S = ωxy/|v|. GR(2),ij (t) has the form22 GR(2),ij (t) = 1+ αij e−(k+ +k− )t = 1+ αij e−kt

2.2.5

where k = k+ + k- is the relaxation rate constant. There are 4 αij, and under the equilibrium they are22 α AA =

(1 − QA )2 K (1 − QD )2 K (1 − QA )(1 − QD )K , = , α AD =α DA = , α DD 2 2 (1 + QA K )(1 + QD K ) (1 + QA K ) (1 + QD K )

2.2.6

If the assumptions in ref. 22 are fulfilled, terms in Eq. (11) of ref. 22 are identical to those in Eq. (2.2.6). However, our results are more general. Firstly, Eq. (2.2.6) is directly applicable to broad situations, while Eq. (11) in ref. 22 has to make respective corrections when, for 6


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instance, the fluorescence quantum yields of a fluorophore are different between the reactant and product states. This happens when the fluorophores at the reactant and product states face different fluorescence quenching or enhancing environments. Secondly, when molecules are fixed and there are background photons, correction has to be made for Eq. (11) of ref. 22, whereas Eq. (2.2.6) is still directly applicable as long as the background photons are counted as part of the signals (Supporting Information S3). Thirdly, Eq. (11) of ref. 22 needs modification if there is cross talk between A and D channels, whereas Eq. (2.2.6) still holds. Then, the photons due to the cross talk can be considered as part of the signals of respective channels (Supporting Information S3). It is easy to verify that in Eq. (2.2.6) there is a relation 2 αAAαDD = α AD

2.2.7

So, among 4 αij only 2 are independent, but we have 3 unkown, QA, QD and K. Therefore, similar to the case of FACS,7 second-order FRET-FCS alone cannot solve QA, QD and K. One may also define the second-order correlations by using the total photon counting (nl) instead of the difference of the photon counting ( δnl = nl −nl ), that is G '(2) R,ij (t ) =

ni (0)n j (t )

2.2.8

ni n j

The relation between G 'ij(2) (t ) and Gij(2) (t ) is22 (2) G '(2) ij (t ) = 1+ Gij (t )

2.2.9

A very interesting and important phenomenon is worth pointing out. Since we are only interested in physical or chemical relaxation, an ideal experiment is to let the molecules not move at all, and then the interference from diffusion and/or scanning will be eliminated completely.24 Now, if one tracks the fluorescence trajectory from one FRET pair, the second-order correlations by using the total photon counting is (Supporting Information S2)

7



ni (0)n j (t )

G '(2) R,ij (t ) =

ni n j

= 1 + αij e− kt

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2.2.10

while the second-order correlations by using the difference of the photon counting, according to Eq. (2.2.9), is GR(2),ij (t ) =

δ ni (0)δ n j (t ) ni n j

= αij e−kt

2.2.11

In both Eqs. (2.2.10) and (2.2.11), αij are identical to that in Eq. (2.2.6). We emphasize that for an experiment with fixed molecules, Eq. (2.2.6) holds only when one molecule is detected at a time. If more than one molecules are detected, corrections has to be made on Eq. (2.2.6) (Supporting Information S4).25

2.3 The third-order FCS We extend the results for the second-order FRET-FCS to the third-order FRET-FCS by using the methodology presented in ref. 7. The third-order correlation functions by using the difference of the photon counting are defined by Gijk(3) (t1 , t ) =

δ ni (0)δ n j (t1 )δ nk (t )

2.3.1

ni n j nk

where, again, i, j and k are A or D detector channel. In Eq. (2.3.1) t1 is fixed. ∆t should be larger than the time scale of photon emission kinetics, if the photon emission kinetics are not interested. We propose two strategies to choose t1. The first strategy is used when in an experimental setup the fluorescence from a fluorophore is detected by one detector. In this case, if i=j we take t1 ≠ 0 (normally t1=∆t). Similarly, if k=i we take t ≠ 0, and if k=j we take t ≠ t1. When adopting this first strategy we also impose the condition that ∆t

Determination of Equilibrium Constant and Relative Brightness in

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