Analytical Form of the Fluorescence Correlation Spectroscopy Autocorrelation Function in Chemically Reactive Systems

Fluorescence correlation spectroscopy (FCS) applied to chemically reactive systems provides information about chemical reaction equilibrium constants and diffusion coefficients of reactants. These physical quantities are determined from the FCS-measured autocorrelation function, G(t), as a function of time, t. In most of the studied cases, the analytical form of G(t) is well-known for reactions that are much faster than the diffusion time of reactants across the focal volume probed by FCS or when they are much slower than the diffusion time. Here, we develop an analytical form of G(t) for reactions occurring at an intermediate time scale comparable to the diffusion time. G(t) depends on the reaction rates in such reactions. We focus on reversibly binding a fluorescently labeled small molecule to a macromolecule in a diluted solution in thermodynamic equilibrium. Our approach allows the analysis of FCS data over a wide range of diffusion coefficients, reaction rate constants, and brightness levels of fluorescent labels. Our G(t) is valid even when the fluorescent label changes its brightness upon binding. The easy-to-implement analytical form of the autocorrelation function greatly helps experimentalists study chemical reactions, determining the equilibrium constants of reactions and the reaction rates.


I. REACTION-DIFFUSION EQUATIONS
In the system under study, A and B denote macromolecules and fluorescent dyes, respectively.The complex C is a product of the reaction A + B ⇄ C, and it is usually also fluorescent.The reaction rate constants are k + for the forward reaction and k − for the backward reaction.In applications to the fluorescence correlation spectroscopy (FCS) [1][2][3][4][5][6], the reaction-diffusion (RD) equations for the local concentrations, C i (r, t), are linearised around the equilibrium concentrations, Ci , where i = A, B, C. In open space, the linear equations for concentration fluctuations, δC i (r, t) = C i (r, t) − Ci , are customarily transformed to Fourier components, δ Ci (q, t), where q is the wave vector.δ Ci (q, t) must satisfy the system of ordinary differential equations dδ Ci (q, t) dt = j M ij (q)δ Cj (q, t). ( The matrix below defines the coefficients M ij (q): where k A = k + CB , k B = k + CA , k C = k − and q = |q|.For simplicity, we assume that the diffusion coefficients of the macromolecule and the complex are equal, i.e., D A = D C = D, where D < D B .We express the general solution of Eq. (1) as δ Ci (q, t) = k Gik (q, t)δ Ck (q, 0).Gik (q, t) are Green's functions satisfying the set of equations with the initial conditions Gik (q, 0) = δ ik .The standard method for solving Eqs. ( 1) and ( 3) is diagonalisation of the matrix M (q) [6].However, we do not use this method because of a complicated dependence of the eigenvalues and eigenvectors on q.Instead, we transform the concentration fluctuations as follows The transformation matrix, T ii ′ , is defined by the relations δ Ci = i ′ T ii ′ δ Ci ′ , where i = A, B, C and i ′ = 1, 2, 3.After the transformation, the RD equations assume the following form dδ C1 (q, t) dt = −Dq 2 δ C1 (q, t), (5a) where where i, j = A, B, C and i ′ , j ′ = 1, 2, 3. Gi ′ j ′ (q, t) are Green's functions for Eqs. ( 5) satisfying the initial conditions Gi ′ j ′ (q, 0) = δ i ′ j ′ .The solution of Eq. (5a) is G11 (q, t) = exp(−Dq 2 t).
For convenience, we rewrite Eqs.(5b) and (5c) for Green's functions in the following form where m 2 = D B q 2 + k 23 and m 3 = Dq 2 + k 32 .When q = 0, solving Eqs. ( 7) is trivial.The solutions are for j ′ = 2 and for j ′ = 3, where R = k 23 + k 32 and β = k 23 /R.When q ̸ = 0, the solutions of Eqs.(7) assume the integral form: for j ′ = 2 and for j ′ = 3.I 0 (ζ) and I 1 (ζ) are the modified Bessel functions [7], and γ = √ k 23 k 32 .In general, is a solution of the modified Bessel equation It is worth noting that the modified Bessel functions also appear in the context of the FRAP method [8][9][10][11].When n is an integer hence Applying the last relation and the modified Bessel equation for I 0 (ζ) shows that Eqs.(10) and (11) do define solutions of Eqs.(7) satisfying the initial conditions Gi ′ j ′ (q, 0) = δ i ′ j ′ .At q = 0, these solutions must be compatible with Eqs. ( 8) and (9), respectively, leading to the following identities and

Probability density functions
The dependence of Green's functions on q is Gaussian, and all integrands in Eqs.(10) and (11) are positive.Therefore we can express G22 , G32 , G23 and G33 as follows where + Dρ can take any value between the diffusion coefficients of the macromolecule and the dye.For q = 0, we recover Eqs. ( 8) and ( 9).When and In this case, Green's functions are the same as for unimolecular isomerisation with equal diffusion coefficients of the isomers.
To determine Φ i ′ j ′ (t, ρ), we first change the integration variable from τ to t − τ in Eqs.
In contrast, when Rt → 0, the form of Φ i ′ j ′ (t, ρ) results from the expansion: Then the probability density functions

II. AUTOCORRELATION FUNCTION
The autocorrelation function [6] is defined in the main text.We express it in terms of Green's functions: where Q B and Q C are the quantum yields of the fluorescent components.The effective sampling volume V is defined as and Ni = Ci V .The dimensions H and L define the Gaussian approximation for the light intensity distribution in the focal spot.Their aspect ratio ω = H/L is usually larger than 1.
To calculate the integral in Eq. ( 28), we insert Eq. ( 6)).Using the transformation matrix and its inverse, i.e. where where related to Green's function defined by Eqs. ( 16) and ( 17) as follows Finally, we derive the general expressions for G i ′ j ′ (t): In Eqs.(36), and ⟨G s (t/τ ρ )⟩ 23 = ⟨G s (t/τ ρ )⟩ 32 .The above expressions form the basis of the approximations discussed in the main text.In the next section, we apply them to the case of immobile macromolecules.

A. Pure diffusion and reaction dominant regimes
Approximation (41) is consistent with Ref. [12] in both pure diffusion and reaction dominant regimes, provided that k 23 t ≪ 1.In the first case, β is small, hence G(t) ≈ N −1 G s (t/τ B ).In the second case, e −k 23 t ≈ 1 on the time scale of diffusion, hence

B. Effective diffusion
We discussed this case in the main text in the context of mobile macromolecules.The effective diffusion occurs when (k
Instead of approximation (45), we use the exact expression for G 33 (t) given by Eq. (38d).
It also has an integral form, however, a one-dimensional integral over ρ replaces the threedimensional integral over the wave vector.It is convenient to recast Eq. (38d) using Eq.(36d), hence Then we approximate ⟨G s (t/τ ρ )⟩ 33 by expanding G s (t/τ ρ ) around ⟨ρ⟩ 33 .We showed in the main text that this method can be used in the case of mobile macromolecules if CC / CA CB is the reaction A + B ⇄ C equilibrium constant.Whereas the equilibrium constant of the reaction 2 ⇄ 3 is k 23 /k 32 = C3 / C2 , where C2 = CB and C3 = CA CC /( CA + CC ).We also define the equilibrium concentration C1 = C2 C /( CA + CC ).Then C1 + C3 = CC , and we can express Eq.(31) in the following form i,j=B,C assumed.According to Eqs. (10b) and (11a) G32 /k 23 = G23 /k 32 , hence C2 G32 = C3 G23 .The last relation reduces the number of independent terms in Eq. (32).Combining Eqs.(28) and (32), we express G(t) only in terms of components 1, 2 and 3, i.e.

Figure S1 .
Figure S1.G 33 for the hybrid model as a function of k 32 t for k 23 /k 32 = 10 and a few values of τ B k 23 .G e 33 (solid red line) is calculated numerically from exact expression (38d).G a 33 (circles) is the approximation using the expansion method described in the text.The dashed red line represents the relative deviation, (G a 33 − G e 33 )/G e 33 , and the solid blue line shows exp(−k 32 t).

Figure S2 .
Figure S2.G 33 for the hybrid model as a function of k 32 t for k 23 /k 32 = 100 and a few values of τ B k 23 .The meaning of lines and symbols is the same as in Fig. S1.
The first equation describes free diffusion of component 1.The other two are the RD equations for a binary system whose components participate in a fictitious first-order reaction 2 ⇄ 3 with reaction rate constants k 23 and k 32 .Green's functions transform according to the formula hybrid model (k 23 ≫ k 32 ) and reaction dominant (τ B ≪ 1/k 23 ).The rate constants k 23 and k 32 correspond, respectively, to k * on and k of f in Ref. [12].If we omit the coupling terms in Eqs.(38) then This inequality is equivalent to the condition τ B ≫ 1/(2k 23 ) for immobile molecules.If tk 23 ≫ 1 and tk 32 ≫ 1, only the coupling terms in Eqs.(38) contribute to G(t).Then G 22 (t) ≈ (1 − β)G s (t/τ β ), G 32 (t) ≈ βG s (t/τ β ) and G 33 (t) ≈ βG s (t/τ β ), where 1/τ β = (1 − β)/τ B (see Eqs. (36)).Finally, Thisbehaviour is demonstrated in Figs.S1 and S2, where G 33 (t) for the hybrid model is plotted for k 23 /k 32 = 10 and k 23 /k 32 = 100, respectively.To show the entire period relevant to the FCS experiment, we use 1/k 32 as the time unit.We observe that the expansion method works very well when τ B k 23 ≳ 1.Then the relative deviation of the approximate autocorrelation function from the exact one, ∆ r G 33 = (G a 33 − G e 33 )/G e 33 , is very small.The maximum of |∆ r G 33 | increases when τ B k 23 decreases, but it occurs close to the tail of G 33 .When τ B k 23 ≪ 1, G 33 is virtually indistinguishable from exp(−k 32 t).This case corresponds to the reaction dominant regime (cf.Eq.(42)).