Determination of Equilibrium Constant and Relative Brightness in

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Determination of Equilibrium Constant and Relative Brightness in Fluorescence Correlation Spectroscopy by Considering Third-Order Correlations Zhenqin Wu, Huimin Bi, Sichen Pan, Lingyi Meng, and Xin Sheng Zhao J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.6b07953 • Publication Date (Web): 24 Oct 2016 Downloaded from http://pubs.acs.org on November 6, 2016

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The Journal of Physical Chemistry

Determination of Equilibrium Constant and Relative Brightness in Fluorescence Correlation Spectroscopy by Considering Third-Order Correlations Zhenqin Wu1,2‡, Huimin Bi1,2‡, Sichen Pan1,2, Lingyi Meng2,3, and Xin Sheng Zhao1,2* 1

Beijing National Laboratory for Molecular Sciences, State Key Laboratory for Structural

Chemistry of Unstable and Stable Species, Department of Chemical Biology, College of Chemistry and Molecular Engineering, 2Biodynamic Optical Imaging Center (BIOPIC), and 3

School of Life Sciences, Peking University, Beijing 100871, China.

ABSTRACT Fluorescence correlation spectroscopy (FCS) is a powerful tool to investigate molecular diffusion and relaxations, which may be utilized to study many problems, such as molecular size and aggregation, chemical reaction, molecular transportation and motion, and various kinds of physical and chemical relaxations. This paper focuses on a problem related to using the relaxation term to study a reaction. If two species with different fluorescence photon emission eﬃciencies are connected by a reaction, the kinetic and equilibrium properties will be manifested in the relaxation term of the FCS curve. However, the conventional FCS alone cannot

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simultaneously determine the equilibrium constant (K) and the relative fluorescence brightness (Q), both of which are indispensable in the extraction of thermodynamic and kinetic information from the experimental data. To circumvent the problem, an assumption of Q = 0 is often made for the weak fluorescent species, which may lead to numerous errors when the actual situation is not the case. We propose to combine third-order FCS with the conventional second-order FCS to determine K and Q without invoking other resources. The strategy and formalism are verified by computer simulations and demonstrated in a classical example of hairpin DNA folding process.

1. INTRODUCTION Since its invention,1-4 fluorescence correlation spectroscopy (FCS) has found countless applications, such as the studies of molecular size and aggregation, chemical reaction, molecular transportation and motion, and various kinds of physical and chemical relaxations. This paper focuses on a problem related to using the relaxation term to study a reaction. In this circumstance, one problem within the FCS technique has severely undermined its application. To illustrate it, let us take the simplest example. Consider a reaction A

k k

B

(1.1)

where A and B are two states or species with different photon emission efficiencies under laser excitation, and k+ and k- are the forward and backward reaction rate constants. Suppose the photon emission efficiency from A is larger than that from B. We call A as a bright state and B as a dark state and call the ratio of photon emission eﬃciency of B with respect to that of A as the relative brightness of B and denote it as Q. The equilibrium constant of Reaction 1.1 is K

k k

(1.2)


2

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Parallel to the nomenclature of the spatial correlation functions in statistical mechanics, we call the conventional FCS the second-order FCS, defined by5 G (2) (t ) 

where

 n(0) n(t ) n2

(1.3)

stands for the ensemble average, n is the number of detected photons (photon

counting) in each time bin,  n  n  n , and n  n . In the case of Reaction 1.1 in solution, if the laser intensity at the focus is described by  x2  y 2 2 z 2  I (r, t )  I 0  t  exp  2  2   xy2 z  

(1.4)

where xy and z are the widths of the laser focus in the xy plane and along the z axis, respectively, a well-known formula for the conventional FCS function is derived as5 (1.5)

G(2) (t )  GD(2) (t )  GR(2) (t )

where

GD(2) (t )

is the contribution from the diffusion. Assuming that A and B have the same

diffusion constant D, 1

(2) D

G

1 t   t  (t )  1   1  2  V  D    D 

where



1 2

(1.6)

 D  xy2 /  4D  , V   3 / 2xy2 z ,

and   z / xy . GR(2) (t ) is due to Reaction 1.1 with the following

form GR(2) (t ) 

 t 1 (2) 1   exp   C  R

   

(1.7)

where C is the average concentration of A and B together (so that N  CV is the total number of molecules in the detection volume V) and  (2) 

K (1  Q) 2 (1  KQ) 2

(1.8)


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R 

1 k  k

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(1.9)

By carrying out FCS experiments, one may obtain  (2) and  R . If Q is known, K can be found, and then k+ and k- can be calculated. Unfortunately, without additional information the conventional FCS does not allow people to simultaneously determine K and Q. Although there are various ways to estimate Q, the assumption of Q = 0 is often made to circumvent this problem. Clearly this will lead to serious error when the actual Q is not zero.6-11 In the past, people have considered various kinds of higher-order time correlations to seek for new information.12-14 For example, Palmer III and Thompson looked into the molecular aggregation by using higher-order FCS,15 Qian and Elson pointed out that the third-order and dual-color time correlations can be used to probe non-equilibrium steady states,16 and Ridgeway et al. developed general formulism of fluorescence triple correlation spectroscopy and applied it to the study of the ribosomal assembly intermediates.17,18 Especially to our interest, Melnykov and Hall proposed to use higher-order correlations to tackle several problems, including the relative brightness in a chemical reaction.19 However, no explicit equations were presented in their paper to facilitate the application of this idea to solve K and Q easily. In order to utilize higher-order time correlations to solve K and Q with a docile manner, we implement the idea of Melnykov and Hall by extending the formulism of the conventional FCS presented in Ref. 5 to a special form of the third-order FCS. We found that when the photon counting is high enough, our third-order correlation function is no longer dependent on the absolute brightness of the molecules, as in the case of the conventional FCS. Then, all the concerned molecular parameters, K, Q, k+ and k-, can be determined easily by fitting the conventional FCS and the third-order FCS simultaneously. In the following sections, we first develop an explicit formulism of the specific third-order correlation function. Then, we verify it


4

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by computer simulations. Finally, we apply the formulism to a classical example of hairpin DNA folding process to demonstrate our strategy. The settlement of the problem with an easy solution paves a road for the convenient FCS application of various biophysical and biochemical problems.

2. THEORETICAL FORMULISM

Parallel to the ideas in Ref. 5 for the second-order FCS and following the convention on the symbols used there as closely as possible, we consider that in a solution m freely diffusing species are fluctuating around the equilibrium connected through reversible chemical reactions which are governed by the chemical master equations. We consider both diffusion in traditional FCS5 and linear motion in other applications.4,15,20 For simplicity we assume that all species have the same diffusion constant D and additional linear motion perpendicular to the laser beam at a constant speed v. Thus, the time evolution of the concentration of the jth species in the laser focus is described by4,5  C j (r, t ) t

 D2 C j  r, t   v  C j  r, t    kkj  Ck  r, t    k jl  C j  r, t  k

(2.1)

l

where kjl is the linearized rate constant from species j to l, and  C j (r, t )  C j (r, t )  C j

where C j

 Cj

(2.2)

.

Hereby, we calculate the third-order fluorescence correlation function G (3) (t ) 

1  n  0  n  0  n t  n3

(2.3)


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where the symbols are the extension of Eq 1.3. n(t )   n j (t ) if the quantum interference does not j

come in to play a role, where nj(t) is the detected photon number from species j. Within a certain time resolution (or bin time) t, the detected photon number of the jth species is

n j t   Δt  drI r, t  Q j t  C j r, t 

(2.4)

where I(r,t) is the laser intensity described by Eq 1.4, and Qj is the photon counting rate of unit concentration of the jth species at the unit laser intensity and in unit time. By introducing an assumption that the fluctuations of concentration and photon counting are independent, we may write

 n j t   Δt  drI r, t  Qj (t ) C j r, t   C j   Δt  drI r, t  Q j (t ) C j r, t 

(2.5)

Substituting Eq 2.5 into Eq 2.3, we have G (3) (t ) 

1 Δt 3  drdr 'dr " I (r, 0) I (r ', 0) I (r ", t )  Q j (0)Qk (0)Ql (t ) n3 j , k ,l

 C j  r, 0   Ck  r ', 0   Cl  r ", t 

(2.6)

Taking the same assumption as that in Ref. 5 that the fluctuation of the molecular concentration follows the Poisson distribution and that molecules at different positions are not correlated, the following initial condition is derived

 Ci r,0  C j r ',0  Ck r ",0  Ciijik  r  r ' r  r "

(2.7)

The Fourier transform of Eq 2.1 reads c j (q, t ) t

  M kj ck  q, t 

(2.8)

k

where c j  q, t    2 

3/ 2

 dre

iqr

 C j  r, t 

(2.9)

and


6

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  M kj  kkj   Dq 2  iv  q   k jl   kj l  

(2.10)

Following the same procedure in Ref. 5, the solution of Eq 2.8 is derived as m

m

s 1

p 1

cl (q, t )   X l( s ) exp( ( s ) t ) ( X 1 )(ps ) c p (q,0)

(2.11)

so that  C j  r,0   Ck  r ',0   Cl  r ", t    2 

3

m

 iqr " iqr  X l(s) exp( (s)t)( X 1 )(js) r  r ' jk C j  dqe

(2.12)

s 1

Substituting Eq 2.12 into Eq 2.6, we have G(t ) 

m 1 Δt 3  dqI 2 (q, 0) I (q, t ) Q j 2 (0)Ql (t )C j  X l( s ) exp( ( s ) t )( X 1 )(js ) 3 n j ,l s 1

(2.13)

where I (q, t ) 

1

dr "e  2   3/ 2

I 2 (q, 0)=

1

 2 

3/ 2

 iq  r "

I (r ", t )

(2.14)

iq  r 2  dre I (r, 0)

By taking I (0, t )=

1

 2 

3/ 2

 drI (r, t ) 

2 I 0 (t )xy z

(2.15)

8

we find n  Δt  drI  r, t   Q j (t )C j



j

2V 4

n

j ,0

Cj

(2.16)

j

where n j ,0  n j ,0 (t ) , n j ,0 (t )  ΔtI 0 (t )Q j (t )

(2.17)

n j ,0 is time independent because it is the ensemble average at the equilibrium.

Plugging Eqs 2.14 – 2.16 into Eq 2.13, a final result is derived to be G(3) (t )  GD(3) (t )  GS(3) (t )  GR(3) (t )

(2.18)


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where the factor due to the diffusion is 1

 4 t   4 t  G (t )  1   1  2 2  3 3V  3  D   3   D  8

(3) D

1/ 2

(2.19)

the factor due to the linear motion is 1  4  t 2  4 t   GS(3) (t ) = exp     1     3  S   3 D    

(2.20)

where S = xy/v. Eqs 2.19 and 2.20 have been obtained previously from more general higherorder correlation functions,15 and our derivation reconfirms their results. The factor due to the reaction GR(3) (t ) 

1     n j ,0 C j   j 

3

m



(2.21)

n 2j ,0 nl ,0 C j  X l( s ) exp( '( s ) t )( X 1 )(js ) s 1

j ,l

is our new result, where  '( s )   ( s )  Dq2  iv  q does no longer contain q. When the simple case of Reaction 1.1 is considered, following a similar procedure as that in Ref. 5 Eq 2.21 becomes G (t )  (3) R



2 2 2 nA,0 / nA,0 1  K  1  K nB,0

C nA,0 1  KQ  2

2

2

 1    

(3)

 t exp    R

   

(2.22)

with  (3) 

2 / 1  Q  1  nB,0 2 1  KQ  1  K nB,0

2 nA,0



2 / nA,0



K

(2.23)

Eqs 2.22 and 2.23 show that GR(3) (t ) is dependent on the absolute photon counting. In general, the fluctuation of the photon counting comes from the fluctuations of the molecular absorption and emission efficiencies, of the apparatus detection efficiency, and of the laser intensity. As indicated by Eq 2.17, the fluctuation is also related to t and all the contributions could be entangled together. If the statistics of n j ,0 all together follow the Poisson distribution, we have


8

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(2.24)

n2j ,0  n j2,0  n j ,0

then GR(3) (t ) 

  (3)

(1  K ) 1  KQ 2  (1  KQ) / nA,0    t  (3) 1   exp     2 2 C (1  KQ)  R   (1  Q)2 1  Q  1/ nA,0  K

(2.25)

(2.26)

(1  KQ) 1  KQ 2  (1  KQ) / nA,0 

It is worth noticing that the Poisson distribution becomes a Gaussian distribution when n j ,0 is high. According to Eq 2.24, 2 nA,0

2 nA,0  nA,0



2 nA,0

2 nA,0

2  nA,0 1  nA,0   2 1  1   nA,0  nA,0 

and 2 nB,0 2 nA,0



2 nB,0  nB,0 2 nA,0  nA,0



2 nB,0 1  1/ nB,0 

2 nA,0 1  1/ nA,0 



B,0 A,0  

n

,n

2 nB,0 2 nA,0

 Q2 ,

Then, it is reasonable to take the limit that when nj,0, Eqs 2.22 and 2.23 become G (t )  (3) R

1  K  1  KQ2   C (1  KQ) 2

2

1  Q  1  Q  K  1  KQ  1  KQ2 

 t  (3) 1   exp      R  

(2.27)

2



(3)

(2.28)

no matter what distribution of nj,0 is. Under such a limit, GR(3) (t ) is dependent only on the relative brightness but not the absolute photon counting. We have found that in the range of 0  Q  1 and 0  K  , there is unique solution for Q and K from (2) of Eq 1.8 and (3) of Eq 2.28. That is Q

p  p2  4 2

(2.29)


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where p



(3)

  (2)    (2)    (2)  (3)  2

2

2

(2.30)

 (2)  (3)   (2) 

and  2 (2) Q  (1  Q) 2  (1  Q) (1  Q) 2  4 (2) Q  , 2 (2) Q 2  K   2 (2) Q  (1  Q) 2  (1  Q) (1  Q) 2  4 (2) Q ,  2 (2) Q 2 

 (3)  2 (2)

(2.31a)

 (3)  2 (2)

(2.31b)

Then, by fitting experimental data, K, Q, k+, and k- can be derived. If the Poisson statistics for the photon counting is assumed, one can derive that  (3)  nA,0   

Q(1  K ) (3)

nA,0 1  KQ 2  1  KQ  / nA,0  (1  Q)

  (3)

(2.32)

Enlightened by Eq 2.32, we propose to get (3) in Eq 2.28 through appropriate extrapolation, even though the photon counting may not strictly follow the Poisson statistics. Also, we derive from Eq 2.16 that in the case of Reaction 1.1 nA,0 

4n 1  K 

(2.33)

2 N 1  KQ 

Eq 2.33 allows us to estimate nA,0 , which is needed in the extrapolation. Before verifying the formulism, let us elaborate it a bit further for a given 0 < Q < 1. According to Eqs 1.8 and 2.28, both (2) and (3) have maxima with respect to K with (2)  max 

(1  Q) 2 4Q

at K  1/ Q

1  Q  1  Q 

(2.34)

2



(3) max



Q 1  Q1/ 2 

2

at K  1/ Q

3

respectively. Defining 

 (3) 1  Q  KQ  KQ 2   (2) 1  KQ 2

(2.35)


10

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we have  Q  Q3  0 K 1  KQ 2 2

(2.36)

Eq. 2.36 tells us that if Q is fixed  increases monotonously with K from (K=0) = 1+Q to

(K) = 1+1/Q. Since the separation point of (3) = 2(2) in Eq 2.31 is exactly at K = 1/Q, (2) where  (2)   max , Eq 2.34 tells us that Eq 2.31a works for the region of KQ < 1 and Eq 2.31b

works for the region of KQ > 1.

3. METHODS 3.1 Simulation method To verify the theory and formula, we conducted computer simulations on a model system composing of diffusion and Reaction 1.1. The simulations were carried out in a 3×3×15 m3 cuboid box, about 150 times bigger than the laser focus volume which sits at the center. Periodic boundary conditions were implemented to maintain the concentration. We put 600 diffusing and reacting molecules into the volume, representing a concentration of 7.4 nM. The movement started with 600 molecules randomly located inside this box. Then, in each 10 s step the motion governed by the simultaneous Brownian diffusion and the first order forward and backward chemical reactions were calculated. Each molecule was propagated for 2000 s. The photon counting of the signals or background was chosen to match the condition of typical FCS experiments. The laser field was described by Eq 1.4 with xy = 0.3 m and z = 1.5 m. The fluorescence photon counting that subjected to Poisson distribution when necessary was generated. Prior to the computer simulations, we assigned K, Q, nA , k+ and k- to the system, based on which one may predict what values of  (2) , 

(3)

n  A,0

should be. All the assigned


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quantities are called the input values. With these input values, the computer simulations were carried out to let the system evolve in time to generate molecular trajectories. From the trajectories, the photon counting of all molecules were added together to get the fluorescence trace. From the fluorescence trace the second- and third-order FCS functions were obtained. The FCS functions were fitted to get  (2) , 

(3)

 n  , D and R as well as other quantities, from which A,0

K, Q, nA , k+ and k- can be calculated. All the quantities obtained from the analysis of the computer simulated fluorescence traces are called simulated values. For each set of input molecular parameters, three fluorescence traces were simulated independently, so that the average and the error of the simulated quantities could be evaluated. Because the input parameters are comparable with typical experimental conditions, the simulation errors offer some clue to what if an experiment were carried out. From the comparison between simulated and input values, both the correctness of the equations and the accuracy of the simulations were tested and verified. 3.2 Experimental setup The FCS measurements were conducted on a home-built inverted fluorescence confocal microscope.20 The laser beam from a continuous-wave laser (532 nm) was collimated before entering the microscope. Laser power could be regulated by an attenuator. The laser beam was focused into the sample solution by an oil immersion objective (Nikon, 100×, NA = 1.4). The emitted fluorescence was spatially filtered by a 50 m pinhole, and then divided into two channels using a polarized beam splitter (Daheng Optics, GCC-402102), and finally detected by two avalanche photodiodes (APDs) (Perkin-Elmer, SPCM-AQR-14). The fluorescence traces with a time resolution of 10 s bin time were recorded by a photon counter (Becker & Hickl GmbH, PSM 400) during the FCS measurements.


12

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3.3 Hairpin DNA design Hairpin DNA samples are the same as those used in Ref. 20. The sequence of experimental group (hp) is 5’-linker -GGGG-(T)21-CCCC-TMR-3’, and the sequence of control group (polyT) is 5’-linker-GGGG-(T)21-TTTT-TMR-3’. The linker sequence from 5’ to 3’ reads: biotinTCTCTCTCTCTCTCTCTCTTTTTT. Both the hp and polyT DNAs have the same length and similar sequence, therefore almost identical diffusion constants. In the hairpin hp, when the hairpin is unfolded, TMR is fluorescent, so that the unfolded state corresponds to the bright state A. Once the hairpin is folded, TMR is quenched by the four nearby guanines, and the folded state is the dark state B. Increasing NaCl concentration will shift the equilibrium towards the direction of more folded state. In the control polyT, there is no hairpin folding-unfolding reaction, and its FCS data enable us to remove contributions other than the folding-unfolding reaction from the FCS data of hp.20 3.4 Data acquisition The experimental procedure and detail are reported previously.20 The DNA samples, including the experimental group (hp) and the control group (polyT), were prepared from the stock solution (10 M, in 1× TE buffer). The buffer for data acquisition contained 20 mM TrisHCl (pH 8.0), 1 mM EDTA, 1 mM Trolox, 0.01 % (v/v) Tween-20, and variable concentrations of NaCl. The final concentration of DNA samples was 10 nM. The coverslips (Fisher Scientific, Cover Glass 22 × 22 mm) were cleaned by piranha solution at 95C for 3 hours. Hybridization chambers (Life technologies, Secure-Seal) were used to seal the sample solution. One fluorescence trace file lasted for 10 s. To get reasonable S/N ratio, 60 to 100 files were accumulated for each fluorescence trace depending on the photon counting rates of TMR-labeled DNA under different NaCl concentrations. All experiments were conducted at 25C.


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4. RESULTS AND DISCUSSION 4.1 Verification of the equations We first verified Eq 2.19 through simulations by only allowing diffusion to occur with τD = 650 s and by keeping the photon counting as constant without Poisson distribution. Table 1 (upper two rows) presents the comparison between the input and simulated diffusion times. It is seen that the simulated quantities agree well with the input values. Table 1. Comparison between simulated and input characteristic diffusion and relaxation times Trace 1

Trace 2

Trace 3

Input

 D (G (2) ) /s

649

649

649

650

3 D (3) (G ) /s 4

488

487

489

488

 R (G (2) ) /s

100

100

99

100

 R (G (3) ) /s

103

100

100

100

Then, we considered the Reaction 1.1 with τR = 100 s upon diffusion, still assuming that the photon counting is constant without Poisson distribution. In this case, both α(2) and α(3) should only depend on the relative brightness instead of absolute photon counting. The simulated quantities for the diffusion part maintained good consistency with the input values as in the case of diffusion alone. The accuracy and relative error in simulated R was at the same level as the diffusion time (Table 1, lower two rows). Table 2 presents the results and comparison for α(2) and α(3) under different K, Q conditions. The relative deviations of simulated α(2) and α(3) from the input values were larger than the case of characteristic times but all were less than 5%, typically seen for α(2) in an FCS experiment.


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Table 2. Simulated α(2) and α(3) and their relative deviations from the input values α(2)

α(2) deviation (%)

α(3)

α(3) deviation (%)

K = 3/7, Q = 0.02

0.410±0.010

1.3

0.415±0.006

-0.4

K = 3/7, Q = 0.1

0.316±0.010

-0.9

0.377±0.004

3.5

K = 3/7, Q = 0.5

0.074±0.010

2.0

0.119±0.012

-0.1

K = 1, Q = 0.02

0.924±0.017

0.1

0.965±0.016

0.5

K = 1, Q = 0.1

0.677±0.014

1.2

0.822±0.005

2.5

K = 1, Q = 0.5

0.112±0.012

1.1

0.203±0.007

1.8

K = 7/3, Q = 0.02

2.049±0.044

0.1

2.197±0.042

0.7

K = 7/3, Q = 0.1

1.235±0.022

-0.7

1.661±0.010

0.8

K = 7/3, Q = 0.5

0.125±0.011

0.8

0.260±0.009

1.9

4.2 Getting (3) by extrapolation If the fluctuation of nj,0 follow the Poisson distribution, α(3) varies with

nA,0 ,

while α(2) still

depends only on relative brightness. Experimentally, this can be checked in two ways. The first way is to keep all conditions fixed except that the laser power changes, and the second one is to keep all conditions fixed except that the time resolution (t in Eq 2.17) varies. We simulated the FCS functions in both ways. The results from both methods showed that α(2) is independent of nA,0 and

α(3) varies with

nA,0

according to Eq 2.26 quantitatively. Fig. 1 presents an example

when the laser power was changed. In principle, the asymptotic α(3) at

nA,0   can

be obtained

by varying either laser power or time resolution. However, we found experimentally that neither of them is effective because the experimental errors among different fluorescence traces are larger than the trend to be searched, as it is shown in later discussion of the experimental data.


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Also, even if it is feasible, it is time consuming, because multiple fluorescence traces of either different laser powers or different time resolutions have to be taken.

Figure 1. Relationship between α(3) and

1 / nA,0 .

Red circle, without Poisson distribution; Black

square, with Poisson distribution. Solid line is the fit to the simulated points. Dashed line is calculated using Eq 2.32. In this simulation, the input values were D = 650 s, R = 100 s, K = 1, Q = 0.1.

We thought of an alternative way of getting the asymptotic α(3). That is, multiple fluorescence traces of different time resolutions are obtained by binning from one original fluorescence trace. Suppose initially the fluorescence trace was recorded with a time resolution of t with corresponding average photon counting of nA,0 . One can successively bin the data i = 2, 3,…, times and calculate their FCS to get α(3)s at the time resolution of 2t, 3t,…, t with respective photon counting of

2nA,0

, 3nA,0 ,…,  nA,0 . Based on Eq 2.32, α(3)(i) varies with i

according to


16

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 (3)  i   

   (3) 2 i 1  KQ  1  KQ  /  inA,0 





(4.1)

where constant  is a fitting parameter. When either i or nA,0 is large, the plot of  (3) (i) against 1/i should be linear and the intercept on the vertical axis is the extrapolated α(3). In such a way, the asymptotic α(3) is extrapolated from a single fluorescence trace. In order to be consistent, R and

D can be globally fitted for all binnings. In the fitting, K and Q are adjusted iteratively. That is, put trial K and Q in Eq 4.1 to fit the data to get α(3). Use this new α(3) (together with α(2)) to calculate new K and Q. Repeat the procedure until the K and Q come out equal the K and Q put in. An example on simulated fluorescence traces of identical input values is shown in Fig. 2. As expected, the second-order FCS curve during the binning does not change within the error of simulation (Fig. 2a), while the third-order FCS curve changes with the binning (Fig. 2b). Fig. 2c shows the extrapolation of  (3) (i) against 1/i. In this example, the extrapolated values of α(3) from three parallel traces are 0.807, 0.818, and 0.802, respectively. The average is 0.809, which is +0.9% off from the input value. In the simulations, we found that at some point of binning, α(2) starts to vary with the binning as well, indicating that then both α(2) and α(3) are wrong due to various sorts of distortions from the binning. It is suggested that the longest binning time should be less than 1/2 of the shortest decay time constant of the FCS curve. Because of the extrapolation, we do not use and do not need an assumption that at the experimental condition the photon count is high. For instance, in Fig. 1 nA,0 is  1 at the smallest nA,0 ,

and the extrapolation works well. In any event, one should check the experimental photon

counts to make sure that the extrapolation is feasible. Under typical experimental conditions, nA,0 is often larger than a few photons before binning, so that at the last binning point nA,0 may reach tens of photons. Then, the extrapolation can be safely carried out. In practice, whether the


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extrapolation is feasible or not can be checked by looking into the trend of the data points to be extrapolated.

Figure 2. Extrapolation of α(3). (a) The binning of G (2) (t ) ; (b) The binning of G (3) (t ) ; (c) The plot of binned α(3) vs. 1/i for three independently simulated fluorescence traces of identical input values. In this simulation, the input values were D = 800 s, R = 300 s, K = 1, Q = 0.1,

nA,0 =

2.1.

4.3 The influence of background noise One factor that may undermine the effectiveness of this method is the background noise. Although the noise of all sorts inevitably influences the correlation function, we found that the noise does not cause α(2) and α(3) to deviate from their noiseless values. Taking G(3)(t) as the example. Denoting the photon counting from the noise to be nb (t ) , the formula for G(3)(t) is modified to

  n(0)  n (0)     n(t )  n (t )   2

G (t )  (3)

b

b

n  n 

3

(4.2)

b


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According to the definition, the noise is neither correlated among different times, nor to the signal. Hence, the numerator in Eq 4.2 is exactly the same as that in Eq 2.3, while the denominator is enlarged. Therefore, although the overall amplitude of third-order correlation is reduced to G (3)  G (3) no noise 

n3

 n  nb 

3

(4.3)

the key coefficient α(3) is unchanged. Similarly, α(2) does not change, while G(2)(t) varies according to G (2)  G (2) no noise 

n2

 n  nb 

2

(4.4)

To verify this conclusion, we incorporated different levels of background noises subject to a Poisson distribution into our simulation. By setting the photon counting of the noise from 1:50 to 1:1 of the signal, we simulated the second-order and third-order correlations of noise-included traces. As shown in Fig. 3, α(2) and α(3) indeed do not change as the noise to signal ratio changes, while the overall amplitudes varies according to Eqs 4.3 and 4.4.


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Figure 3. Influence of noise on α(2) and α(3) at fixed signal photon counting. (a) G (2) (t ) at different noise to signal ratios; (b)

G (3) (t ) at

different noise to signal ratios; (c) G (2) (t ) after scaled according

to Eq 4.4; (d) G (3) (t ) after scaled according to Eq 4.3. In (c) and (d) all lines overlap together within the simulation errors. In this simulation, the input values were D = 667 s, R = 100 s, K = 1, Q = 0.1,

nA,0 =

4.2.

4.4 Experimental data analysis and results The experimental data analysis followed the same procedure as that in Ref. 20. For each trace, the second-order and third-order correlations of experimental group (hp) and control group


20

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(polyT) were calculated. We binned each fluorescence trace from two to ten times, obtaining ten FCS curves of different photon countings for each G (2) (t ) and G (3) (t ) (Fig. 4a and Fig. 4b). Because of the similar sequences, the diffusion constants of hp and polyT are equal, so as the diffusion part in the correlation (GD(t)). The correlation curve of the hp group was divided by the correlation curve of the respective polyT group to remove the same GD(t), so as to obtain the correlation of the reaction (GR(t)).20 GR(t) was successfully fitted with single exponential decay function to get α(2),α(3) and R. We delineated α(3) against the binning time and fitted it by Eq 4.1 iteratively to get the intercept α(3). Figure 4c depicts the plot of α(3) against 1/i of three fluorescence traces acquired at different laser powers. The experimental results on R, α(2) and extrapolated α(3) under different NaCl concentrations and laser powers are collected in Table 3. α(2) should not change with respect to the laser power. So, the variation of α(2) at different laser powers is a good measurement of the intrinsic error due to the experimental method. We see from Table 3 that the relative experimental error on α(2) is about 5%. As for the relative experimental error on α(3), it is often larger than that of α(2). The original (without binning) α(3) should grow as the laser power increases, but the extrapolated α(3) should be the same for all the laser powers. In the experiments we found that the variation of the original α(3) on the laser power is not clear at all, because the experimental error on α(3) conceals the possible trend. Therefore, it is not practical to extrapolate α(3) by varying the laser power. Due to the same reason and proven by our experimental trials, to extrapolate α(3) by taking data at different time resolutions is neither feasible. It turns out that the binning method is a much simple, easy, and robust method. With α(2) and α(3) in hands, we can easily calculate K and Q using Eqs 2.29 – 2.31. The calculated equilibrium constant and relative brightness of the dark state in the hairpin folding


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reaction under different NaCl concentrations are listed in Table 4, where the bright A state corresponds to the unfolded open state and the dark B state corresponds to the folded close state. Table 3. Experimental results of R, α(2), and extrapolated α(3) CNaCl = 50 mM Laser Power

20 W

50 W

100 W

200 W

Mean

SD

RSD

R / s

486

513

473

510

495

19

3.8%

α(2)

0.307

0.311

0.288

0.297

0.301

0.010

3.5%

α(3)

0.356

0.341

0.292

0.331

0.330

0.027

8.3%

Laser Power

20 W

50 W

100 W

200 W

Mean

SD

RSD

R / s

410

385

416

393

401

15

3.6%

α(2)

0.522

0.512

0.514

0.489

0.509

0.014

2.8%

α(3)

0.650

0.583

0.612

0.587

0.594

0.016

2.6%

CNaCl = 75 mM

CNaCl = 100 mM Laser Power

20 W

50 W

100 W

Mean

SD

RSD

R / s

330

332

302

321

17

5.2%

α(2)

0.693

0.729

0.650

0.691

0.039

5.7%

α(3)

0.968

1.027

0.893

0.963

0.067

7.0%


22

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Figure 4. Extrapolation of α(3) using experimental data under 100 mM of NaCl. (a) The binning of G (2) (t ) ; (b) The binning of G (3) (t ) ; (c) The plot of binned α(3) against 1/i at three different laser powers.

Table 4. Equilibrium constant and relative brightness of hairpin folding reaction under different NaCl concentrations. NaCl Conc.

K

Q

50 mM

0.37±0.06

0.07±0.04

75 mM

0.76±0.08

0.11±0.02

100 mM

1.53±0.13

0.16±0.02

One obvious and interesting result from Table 4 is that the relative brightness varies with the NaCl concentration. The higher the NaCl concentration is, the higher the relative brightness of the dark state will be. The sensible explanation is that as the NaCl concentration increases, the random coil DNA of the unfolded bright state becomes more collapsed so that the fluorescence dye is quenched more by G nucleotides, while the configuration of the folded hairpin stem (the dark state) is less affected by the NaCl concentration. Therefore, the relative brightness of the dark state goes up as the NaCl concentration increases. In Ref. 20, we assumed that the folded hairpin has identical Q from 0 M to 500 mM of NaCl concentrations. Based on such an assumption, Q  0.224  0.006 is obtained. Because Q is a more sensitive function of α(2) and α(3) in the range of KQ > 1, the data from high NaCl concentrations weighted more in the global fitting. It is acceptable that Q reported in Ref. 20 is larger than any Q in Table 4 and is actually expected from the trend of variation.


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Our primary goal is to get K. In order to get K, we have to know Q. In current hairpin folding system, if one takes the assumption that Q = 0, the equilibrium constants would be α(2) in the traditional treatment, that would be 0.30 ± 0.01, 0.51 ± 0.02, and 0.69 ± 0.04 at the NaCl concentrations of 50, 75, and 100 mM, respectively. If one takes the assumption that Q = 0.224 ± 0.006 throughout all NaCl concentrations, the equilibrium constants by using Eq 1.8 would be 0.45 ± 0.05, 1.01 ± 0.19, and 1.82 ± 0.20 at the NaCl concentrations of 50, 75, and 100 mM, respectively. The comparison of above values with Table 4 tells us that not only the assumption of Q = 0 is wrong, the assumption of one species having a fixed Q under different environments also induces unacceptable errors. The more the K and Q deviate from 0, the severer the error of K is if Q is not precise.9 The necessity of having a method to acquire instantaneous Q is well demonstrated by this example. In this work, we focused on the simplest A B reaction. The extension of the formulism to other types of reactions is straight forward. A special example is the reaction of C+D CD. Suppose C is the species that has different fluorescence efficiencies before and after the reaction and D is a nonfluorescent species, then if the experiment is done at a concentration condition of [C]

Determination of Equilibrium Constant and Relative Brightness in

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