Article pubs.acs.org/JPCB

Protein Oligomerization Equilibria and Kinetics Investigated by Fluorescence Correlation Spectroscopy: A Mathematical Treatment David M. Kanno† and Marcia Levitus*,† †

Department of Chemistry and Biochemistry and the Biodesign Institute, Arizona State University, PO Box 875601, Tempe, Arizona 85287, United States S Supporting Information *

ABSTRACT: Fluorescence correlation spectroscopy (FCS) is a technique that is increasingly being used to investigate protein oligomerization equilibria and dynamics. Each individual FCS decay is characterized by its amplitude and a characteristic diffusion time, both of which are sensitive to the degree of dissociation of the protein. Here, we provide a mathematical treatment that relates these observables with the parameters of interest: the equilibrium constants of the different protein dissociation steps and their corresponding dissociation and association kinetic rate constants. We focused on the two most common types of protein homooligomers (dimers and tetramers) and on the experimental variables relevant for the design of the experiment (protein concentration, fractional concentration of labeled protein). The analysis of the theoretical expectations for proteins with different dissociation constants is a key aspect of experiment design and data analysis and cannot be performed without a physically accurate treatment of the system. In particular, we show that the analysis of FCS data using some commonly used empirical models may result in a serious misinterpretation of the experimental results.

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intensity that arise when fluorescent molecules diffuse into and out of a femtoliter-sized optically defined volume (Figure 1A,B). The temporal behavior of these fluctuations is commonly quantified by the autocorrelation function defined in eq 1, which decays with a characteristic time that can be analyzed in terms of the diffusion coefficients of the particles present in the sample (Figure 1C,D)13−15

INTRODUCTION Self-association of proteins to form homodimers or higherorder homooligomers is very common. For example, about two-thirds of the human enzymes with known stoichiometry are oligomeric, and that among these, homodimers and homotetramers are prevalent.1,2 There are a number of experimental methods that can be used to determine the stoichiometry and the thermodynamic stability of oligomeric proteins. The latter is characterized by the equilibrium dissociation constant (Kd), the experimental determination of which requires measuring some property sensitive to the degree of dissociation of the protein. Regardless of the experimental approach, it is evident that in order to observe significant dissociation, measurements must be made at protein concentrations on the order of Kd. This creates a particular challenge for very stable oligomers, which are characterized by Kd values in the nanomolar range or lower and therefore require measurements that are usually below the limits of detection of most biophysical methods.3 Single-molecule techniques are particularly attractive for investigating the equilibrium and kinetics of dissociation of stable oligomeric proteins because their exquisite sensitivity allows the detection of fluorescently labeled proteins at the low concentrations required to detect significant dissociation. In particular, a technique known as fluorescence correlation spectroscopy (FCS) is increasingly being used to investigate protein oligomerization4−12 because of its ability to determine translational diffusion coefficients, which are good measures of the hydrodynamic size of the protein. FCS relies on the measurement and analysis of the fluctuations in fluorescence © 2014 American Chemical Society

G (τ ) =

⟨I(t )I(t + τ )⟩ −1 ⟨I(t )⟩2

(1)

The quantitative determination of the equilibrium and kinetic rate constants of protein oligomerization reactions requires that FCS data is analyzed using appropriate models to determine the concentrations of all oligomeric species that exist in solution at any given time. So far, most kinetic and equilibrium studies that used FCS to investigate the stability of oligomeric proteins have not been rigorous in terms of the analysis of the data. Empirical functions such as sigmoidal curves (for equilibrium data) or exponential functions (for kinetic data) were used in the analysis, but as we will discuss in this work, these functions are not always appropriate to describe the true relationships between the various variables involved in the FCS experiments. Therefore, although empirical models may on occasion yield reasonable estimates of the parameters of interest, it can be argued that they are not in general appropriate to obtain quantitatively accurate Received: July 31, 2014 Revised: September 23, 2014 Published: September 30, 2014 12404

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Figure 1. Schematic representation of the system investigated in this work. (A) Diagram of the instrument used in FCS. An objective is used to focus the laser into the sample and to collect a fraction of the emitted photons. Fluorescence is detected after passing through a pinhole. (B) Molecules (monomers, dimers, or tetramers) diffuse into and out the 3D Gaussian observation volume (represented by dotted arrows). Particles closer to the focal point are excited and detected with more efficiency and therefore appear brighter (represented as a darker filling color). (C) The measured fluorescence intensity fluctuates around a mean value as molecules move due to Brownian motion. (D) The temporal behavior of the fluctuations is analyzed in terms of the autocorrelation function, characterized by its amplitude (G0) and a characteristic diffusion time (τD) that is inversely proportional to the diffusion coefficient of the fluorescent particles (see eq 7). (E) Tetramer dissociation via a dimeric intermediate. The dissociation and association kinetic rate constants are represented by kd and ka, respectively, and the dissociation equilibrium constants by Kd.

kinetic rate constants). In this manuscript, we introduce the mathematical formalism that accurately describes FCS data of oligomeric proteins in general, and we analyze in detail the two most common mechanisms of oligomerization in proteins: the formation of dimers and the formation of tetramers via dimeric intermediates (Figure 1E).1,2,16

data. A more fundamental reason to favor physically accurate models, however, is that empirical equations are by nature not appropriate to test underlying physical models. We will illustrate this point with an example that shows that fitting experimental equilibrium data with an empirical sigmoidal equation gives seemingly reasonable results but in reality hides the fact that the assumed mechanism of oligomerization is incompatible with the experimental data. In previous work, we derived algorithms to analyze the results of FCS experiments aimed to determine the equilibrium dissociation constants (Kds) of the proteins PCNA (a trimer) and Rubisco activase (a hexamer that aggregates into higherorder species).4,5 Although we deem these algorithms to be accurate, in this manuscript we present a new mathematical treatment that simplifies the analysis of the data significantly and can be easily adapted to a variety of oligomerization mechanisms. Our original approach relied on a rather convoluted sequence of steps that, as we will show here, can be greatly simplified by just assuming that the observation volume is elongated in the axial direction (i.e., z0 > ∼5r0, where z0 and r0 are the axial and radial semiaxes of the Gaussian observation volume). This requirement is in fact already fulfilled in most FCS experiments because the spatial resolution of confocal microscopes is poorer along the axial direction than in the radial direction.14 This simple assumption allowed us to write analytical equations that link the FCS observables (the diffusion coefficient of the mixture and the amplitude of the decay) with the parameters of interest (the dissociation equilibrium and

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RESULTS AND DISCUSSION

The Total Autocorrelation Function of a Partially Dissociated Oligomeric Protein. The autocorrelation function of a solution containing a single fluorescent species is described by13,14 G(τ ) = ⟨N ⟩−1(1 + 4Dτ /r02)−1(1 + 4Dτ /z 02)−1/2

(2)

where τ is the correlation lag time, D is the diffusion coefficient of the fluorescent particle, r0 and z0 are the radial and axial semiaxis of the Gaussian observation volume, respectively, and ⟨N⟩ is the average number of particles present in an effective volume Veff = π3/2r02z0. In this work, we will only consider onephoton excitation measurements, and we will assume that other sources of fluctuations (e.g., triplet blinking)17 are controlled and minimized. The term “particle” will be used to refer to the diffusing object, which can be a monomer, a dimer, or a higherorder oligomer depending on conditions. The total correlation function of a solution containing N species is given by13,14 12405

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N

G (τ ) =

2 ∑i = 1 NB i i g i (τ ) N

2 ( ∑i = 1 NB i i)

efficiency (i.e., the probability that a monomer from the labeled protein stock carries a label) and is included here as an explicit parameter because labeling proteins is often challenging and understanding how this variable affects the analysis of FCS is of particular interest. We note that the product CL f represents the actual concentration of labeled protein in the solution, whereas CL is the total concentration of protein in the “labeled” stock, which in general contains a fraction of unlabeled protein. We also note that the term CL f/C0 represents the fraction of subunits in the mixture that carries a fluorescent label. In general, the parameter Veff can be determined independently in a calibration experiment (although, as we will discuss below, its precise knowledge is not essential), f can be determined using bioanalytical techniques and CL and C0 are known. On the other hand, the concentrations Cn are typically unknown because they depend on the equilibrium and kinetic rate constants that are being determined experimentally. In principle, one can imagine fitting an experimental FCS decay with eq 6 using the variables Cn as fitting parameters and using these concentrations to calculate the desired thermodynamic or kinetic parameters. However, this approach is not feasible in practice because the diffusion coefficients of the individual terms in eq 6 are expected to lie within a narrow range. Stokes− Einstein’s equation predicts that the diffusion coefficient of a protein scales with the number of protomers (n) as n1/3, and therefore the diffusion coefficient of a tetrameric protein is predicted to be only 41/3 ≈ 1.6 times smaller than the diffusion coefficient of its constituent monomers. As a consequence, the experimental autocorrelation function of the multicomponent solution (described by eq 6) is expected to fit well even with a model that assumes a single component in solution (eq 2). Attempting to fit an experimental decay with a model with more free parameters than needed, such as eq 6, can potentially result in overfitting and hence in unreliable values of the parameters of interest (in this case the concentrations Cn). We therefore favor fitting each experimental decay with a single diffusion coefficient (eq 2) and analyzing these values afterward to obtain the concentrations Cn that give rise to the measured value of D. Analysis of Multicomponent Decays in Terms of a Single Apparent Diffusion Time. As we did in previous work,4 we call the diffusion coefficient recovered from the one-component fit “the apparent diffusion coefficient” (DApp) of the polydisperse solution. The value of DApp depends on the diffusion coefficients of the individual species (Dn, 1 ≤ n ≤ m) as well as the relative contributions of each species to G(τ) (given by An, eq 6). We will focus on two types of experimental approaches, which are designed to determine either dissociation equilibrium constants (Kd) or dissociation and association rate constants (kd and ka). In the first case, protein solutions are prepared in a wide range of concentrations, usually keeping the concentration of labeled protein constant, and solutions are allowed to equilibrate before the FCS measurement. The FCS decay acquired at a given protein concentration is then analyzed according to eq 2 to obtain the apparent diffusion coefficient of the mixture, which depends on the degree of dissociation of the protein. The outcome of these measurements is a plot of DApp as a function of total protein concentration (see Figures 2 and 3 for theoretical predictions and refs 4, 5, and 8 for examples of experimental data), which can be analyzed to obtain the desired dissociation constants. In the second case (kinetic measurements), the protein is diluted from a concentrated

(3)

where Ni is the average number of molecules of species i in the observation volume, Bi is their molecular brightness, and gi(τ) is the autocorrelation function of a homogeneous solution of species i normalized to an amplitude of unity (⟨N⟩ = 1 in eq 2). For a mixture of oligomeric species, assuming that each protein subunit carries one fluorescent label: G (τ ) =

1 Veff NAC02

m

∑ n2Cngn(τ) n=1

(4)

Here, C0 is the total concentration of protein expressed in terms of monomers (C0 = ∑mn=1nCn), Cn is the molar concentration of oligomers containing n subunits, NA is Avogadro’s number, m represents the size of the largest oligomer present in solution (e.g., m = 4 for a tetrameric protein), and we assume that the brightness of each species (Bi) is directly proportional to the number of labels present in the particle (equal to n if all subunits are labeled). Because the optimal signal-to-noise ratio in FCS is achieved when the concentration of diffusing fluorescent particles is in the 100 pM to 100 nM range, experiments at higher protein concentrations are typically performed using solutions containing a mixture of labeled and unlabeled protein.5,8 The rationale behind this strategy is that the degree of dissociation of the protein will be determined by the total protein concentration, while the amplitude and noise of the FCS decay will depend on the concentration of fluorescent particles only. This allows the investigation of protein oligomerization at concentrations much higher than the optimal low nanomolar range. To use eq 3, however, it is important to recognize that the equilibration of labeled and unlabeled solutions of an oligomeric protein results in a random redistribution of subunits and therefore in a random number of fluorescent labels per particle. The number of labeled subunits in an oligomeric particle is a binomial random variable that depends on the relative concentrations of labeled (CL) and unlabeled protein (CU) in the mixture. Although gi can be safely assumed to be independent of the degree of labeling (the contribution of a fluorescent tag to the diffusion coefficient of a protein is negligible), oligomers containing different numbers of labeled subunits have different brightnesses (Bi) and therefore need to be treated as separate species in eq 3. Let Nn,b represent the number of oligomers containing n subunits and b labels (0 ≤ b ≤ n) that are present on average in the observation volume. Equation 3 can then be written as m

G (τ ) =

n

∑n = 1 ∑b = 1 Nn , bb2gn(τ ) m

n

(∑n = 1 ∑b = 1 Nn , bb)2

(5)

where we assume that the brightness of each species (Bi) is directly proportional to the number of fluorescent labels (b) present in the particle. Taking into account that Nn,b is a binomial random variable, we obtain (see Supporting Information for the derivation): m

G (τ ) =

∑ A ngn(τ), n=1

An =

nCn[1 + (n − 1)(C Lf /C0)] fVeff NAC LC0 (6)

where C0 = CL + CU is the total concentration of protein (labeled and unlabeled). The variable f represents the labeling 12406

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quantitative model that relates the experimentally measured DApp values with the concentrations of all species in solution (Cn, eq 6). Developing such a mathematical framework is the focus of this work. In previous work,4 we analyzed equilibrium and kinetic FCS data using an iterative algorithm that relies on fitting the curve described by eq 6 with eq 2. For example, to analyze equilibrium data, each iteration starts by assuming values for the equilibrium constants that describe the various dissociation steps of the protein (e.g., tetramer−dimer, dimer−monomer). The concentrations of all species (Cn) are then calculated at each measured protein concentration (C0) using these Kds, and the theoretical autocorrelation functions are computed for each value of C0 using eq 6. Each of these theoretical G(τ) curves is fitted with eq 2 to obtain a DApp, and these values (one for each value of C0) are then compared with the experimental values to evaluate whether the assumed Kds provide a satisfactory description of the experimental data. This algorithm is rather inefficient, and its implementation is not straightforward. In this work, we show that this approach is in most cases unnecessarily convoluted, and we present an improved algorithm that can be easily implemented to obtain equilibrium and kinetic constants from experimental FCS data. We will prove that the assumption that the term involving z0 in eq 2 does not contribute significantly to the measured G(τ) curves allows the derivation of analytical relationships that link the experimental DApp values with the parameters of interest (Kd or kd, for equilibrium or kinetic experiments, respectively). As we will show in the next sections, these analytical expressions can be easily used to analyze protein dissociation FCS data in a quantitative and accurate manner. We start by noting that in a typical FCS instrument z0 > 5 r0,14 and therefore one-component autocorrelation functions (eq 2) can be approximated as

Figure 2. Expected τApp values for dimeric proteins of various Kd values (see legend) as a function of protein concentration (labeled and unlabeled). The concentration of labeled protein was assumed to remain constant through the titration (CL = 0.1 nM). The arrows on the right axis point to the characteristic diffusion times of the pure monomers and dimers used to simulate the data (τD1 = 0.4 ms, τD2 = 0.504 ms). The curves in blue and in red were calculated assuming f = 1 (all subunits are labeled) and f = 0.5 (half of the subunits are labeled), respectively.

stock (where dissociation is negligible) to a final concentration well below the Kd. FCS decays are then measured as a function of time to obtain DApp values as the protein dissociates into its constituent units (e.g., monomers). In this case, the goal is to analyze the rate of decrease of the DApp values to obtain the dissociation and association rate constants. Example of theoretical predictions are shown in Figures 5 and 6, and examples of experimental data can be found in refs 5, 8, and 9. In both types of experiments, estimates of Kd and kd may be obtained using empirical functions to analyze the data. For example, sigmoidal curves have been used to analyze equilibrium DApp values and exponential functions were used to analyze the results of kinetic experiments.8,9 The precise determination of the thermodynamic and kinetic parameters, however, requires a

G(τ ) ≈ G0(1 + 4Dτ /r02)−1 = G0(1 + τ /τ D)−1

(7)

Here, τ is a characteristic diffusion time defined as τ = r02/4D. The validity of this approximation is illustrated in Figure S1 (see Supporting Information), which shows that the difference between curves generated with eqs 2 and 7 is smaller than the typical noise of an experimental FCS decay. D

D

Figure 3. Bottom: Expected τApp values for tetrameric proteins of various Kd values: (A) Kd1 = 10 nM and Kd2 = 100 nM, (B) Kd1 = 1 nM and Kd2 = 1 μM, and (C) Kd1 = 10 nM and Kd2 = 10 μM. The concentration of labeled protein was assumed to remain constant through the titration (CL = 0.1 nM). The dotted lines represent the characteristic diffusion times of the pure monomers, dimers, and tetramers used to simulate the data (τDM = 0.114 ms, τDD = 0.145 ms, τDT = 0.183 ms). Top: Fractional concentrations of monomers (red), dimers (black), and tetramers (blue) for the Kd values indicated above. 12407

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dissociation equilibrium and kinetic rate constants from experimental FCS data, as illustrated in the following sections. Determination of Equilibrium Dissociation Constants. Case I: Monomer−Dimer Equilibria. According to classical thermodynamics, the equilibrium between dimers and monomers is described by a single dissociation constant Kd that relates the degree of dissociation (α1 = C1/C0) with the total protein concentration (C0, expressed in terms of monomers):

Therefore, eq 6 can be expressed in terms of the characteristic diffusion times of the different oligomers (τnD) as m

G (τ ) =

∑ A n(1 + τ /τnD)−1

(8)

n=1

Because, as we established above, the FCS decay of a mixture of oligomers cannot be distinguished experimentally from the decay of a single species, eq 8 can be simply expressed in terms of an apparent characteristic diffusion time τApp as G(τ ) ≈ Go(1 + τ /τApp)−1

(9)

m

∑ A n(1 + τApp/τnD)−1 = n=1

1 2

m

∑ An n=1

α1 = (10)

c=1−2

τApp =

α1 1 + (1 − α1)(C Lf /C0)

(16)

1 (c(τ2D − τ1D) + 2

c 2(τ2D − τ1D)2 + 4τ1Dτ2D ) (17)

which, together with eqs 13 and 16, establishes an analytical relationship between the measured τApp values and the variables τD1,2, Kd, C0, CL, and f. We note that C0, CL, and f are known, and at least one of the values of τD1,2 can be in principle measured experimentally at concentrations well above or below Kd. This leaves Kd and possibly one τD value as the only unknowns, and therefore their values can be determined from a nonlinear fit of the experimental τApp vs C0 data. Figure 2 shows the expected τApp vs C0 data for a dimer−monomer system obtained using τD1 = 0.4 ms, τD2 = 0.504 ms (i.e., 21/3τD1 ), CL = 0.1 nM, and the Kd and f values shown in the figure legend. The value of τD1 is similar to the value we measured for the monomer of PCNA in previous work,5 but we note that the actual value used in this analysis is irrelevant as long as τD2 /τD1 is close to the theoretical value of 21/3. The code used to generate the curves in Figure 2 is provided as Supporting Information as Matlab code, together with a function that can be used to fit τApp vs C0 experimental data. We also provide a Microsoft Excel spreadsheet that can be used for the same purposes to highlight how simple implementing these equations truly is (see Supporting Information). Although we favor more sophisticated platforms such as Matlab, we hope that demonstrating how τApp data can be analyzed accurately with a popular and simple spreadsheet software will encourage scientists to avoid empirical models and use instead rigorous mathematical models such as the one developed in this work. The simulations shown in Figure 2 indicate that results are less sensitive to the efficiency of labeling ( f) when the concentration of labeled protein is significantly lower than the Kd of the protein. This is a fortunate result because the determination of labeling efficiencies is not straightforward, and values of f can only be typically estimated within a ca. 20% uncertainty. At first sight it may seem odd that fluorescence-based measurements

(11)

β3 = τ1Dτ2D + τ1Dτ3D + τ2Dτ3D β4 = a1τ1D(τ2D + τ3D) + a 2τ2D(τ1D + τ3D) + a3τ3D(τ1D + τ2D) β5 = τ1Dτ2Dτ3D αi[1 + (ni − 1)(C Lf /C0)] 3

1 + ∑i = 1 αi(ni − 1)(C Lf /C0)

(15)

The positive root of eq 15 is

β2 = a1τ1D + a 2τ2D + a3τ3D

=

(14)

where

β1 = τ1D + τ2D + τ3D

Ai 3 ∑i = 1 Ai

8C0Kd + Kd2 )

2 τApp − c(τ2D − τ1D)τApp − τ1Dτ2D = 0

where

ai =

1 ( − Kd + 4C T

For a system consisting of monomers and dimers, eq 10 can be manipulated to obtain

which can be manipulated algebraically to obtain an analytical relationship between τApp (measured experimentally) and the parameters of interest: An and τnD. The parameters An are functions of the concentrations of all species Cn (eq 6), which in turn can be expressed in terms of the total protein concentration C0 and the equilibrium dissociation constants (equilibrium experiments) or as a function of time and the kinetic association and dissociation rate constants (kinetic experiments). Equation 10 is, therefore, a key result that ultimately provides a link between the observables (τApp) and the desired thermodynamic and kinetic equilibrium constants. In this work, we will focus on the case m ≤ 3 (no more than three species in equilibrium), although the same ideas can be expanded to analyze more complex systems. Algebraic manipulation of eq 10 for the case m = 3 gives (see Supporting Information): 3 2 τApp + (β1 − 2β2)τApp + (β3 − 2β4 )τApp − β5 = 0

(13)

The degree of dissociation in equilibrium conditions is therefore

where Go = Σmn=1An is the amplitude of the decay. Evaluating eqs 8 and 9 at τ = τApp gives G(τApp) =

2α12C T 1 − α1

Kd =

(12)

Here, αi is the fractional concentration of species i, defined as αi = niCi/C0, and we note that species 1, 2, and 3 do not necessarily need to be monomers, dimers, and trimers. For example, species 3 can describe a tetramer, in which case a3 needs to be computed using n3 = 4. We also note that none of these parameters depend on Veff, and therefore precise calibration of the observation volume is not needed to analyze the experimental results. Although these expressions may seem convoluted, they are actually simple relationships that link the measured τApp values with the concentrations of the three species in solution (C1, C2, C3). These relationships, therefore, provide the mathematical framework that allows the determination of 12408

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for the parameters of interest (in this case Kd) when τD1 or τD2 cannot be determined from the experimental data with precision. Maybe more importantly, fitting with the appropriate equations is critical to test whether the assumed model (in this case a simple dimer−monomer equilibrium) is adequate to describe the experimental observations. For example, eq 14 predicts a change in α1 from 0.1 to 0.9 over 2.86 decades of protein concentration (a concept G. Weber named the “logarithmic span” of the protein).18 A plot of τApp vs log C0 that shows a change steeper than predicted may be accurately fitted with an empirical sigmoidal curve but will not be consistent with the assumed mechanism. This important point will be illustrated with an example in the next section. So far, we have concentrated on how to extract information from the apparent diffusion times, but it is worth noting that the amplitude of the FCS decay may contain useful information as well. For the monomer−dimer system discussed in this section, the amplitude of the autocorrelation function can be expressed as (eq 6):

are insensitive to the degree of labeling of the sample, but this is actually a consequence (and one may argue an added benefit) of working with a fixed low concentration of labeled protein. When CL ≪ C0, most of the protein in solution does not carry labels, and the probability that a dimer contains two fluorescently labeled subunits is small. Under these conditions, the fluorescence signal originates from monomers or singly labeled dimers that exist at concentrations dictated by the dissociation constant. Lowering the labeling efficiency results in a higher fraction of monomers and dimers that contain no labels (and are therefore undetectable), but because the signal still originates from the same proportion of singly labeled monomers and dimers, the apparent diffusion time of the mixture remains unchanged. In contrast, τApp depends strongly on labeling efficiency when a large fraction of the protein in solution contains fluorescent labels. Figure S2 (see Supporting Information) shows an extreme case involving a situation where experiments are performed with a purely labeled sample (i.e., C0 = CL for all protein concentrations, no unlabeled protein added). Because fully labeled dimers contribute to the signal four times more than the monomers, an equimolar mixture of fully labeled dimers and monomers would cause the τApp to be biased toward τD2 . These brightness-related biases may result in an underestimation of the Kd value unless they are taken into account explicitly, as we did in all the equations derived in this work. Inspection of Figure 2 suggests that the value of Kd can be estimated with reasonable accuracy as the concentration of protein at which τApp = τD1 + (τD2 − τD1 )/2 (i.e., the concentration at which the apparent diffusion time is halfway between the dimer and the monomer). Mathematically, in the limit of low labeling efficiency ( f → 0), eq 16 becomes c = 1−2α1, and therefore c = 0 when C0 = Kd (α1 = 1/2). This gives an apparent diffusion time τApp = (τD1 τD2 )1/2 at the point C0 = Kd. A Taylor expansion of (τD1 τD2 )1/2 about τD2 → τD1 shows that τApp can indeed be approximated as the midpoint between τD2 and τD1 when (τD2 /τD1 ) ≈ 1. Because this ratio is expected to be about 21/3 ≈ 1.26 for a dimer−monomer system, it is not surprising that a midpoint estimate provides a very reasonable value of Kd in this simple case. In fact, the same arguments can be used to show that (τApp − τD1 )/(τD2 − τD1 ) = α2 when f → 0 and (τD2 /τD1 ) ≈ 1. In other words, under these conditions, the normalized τApp curve equals the fractional concentration of dimer. This may seem to suggest that the treatment presented in this section is just an overcomplicated way of obtaining a parameter that can be trivially evaluated from the experimental data without even performing a fit. Although this may be in part true in this case given the simplicity of the problem, we will prove that simple estimates like these may be deceiving in more complex cases such as the monomer−dimer−tetramer problem discussed below. In addition, it should be stressed that estimating Kd without using the proper mathematical treatment requires that both τD2 and τD1 are measured experimentally. Figure 2 shows that this would require performing experiments over ca. four decades of protein concentration spanning both sides of the Kd value. Because acquiring FCS decays at concentrations lower than 100 pM is difficult, and many proteins are not soluble or form aggregates at high micromolar concentrations, it follows that measuring both τD1 and τD2 experimentally may not be straightforward in many cases (see for example Figure 2 for the cases Kd = 1 nM and Kd = 100 nM). Fitting the experimental τApp values with the appropriate mathematical models, therefore, is critical to obtain accurate values

G0 =

(C Lf /C0)(1 − α1) + 1 Veff NAC Lf

(18)

Although dimer dissociation leads to an increase in the total number of diffusing particles in solution, the experiments described in this section are performed at constant CL, and the total number of fluorescently labeled particles in solution is nearly independent of C0. Therefore, the amplitudes of the FCS decays are expected to remain fairly constant throughout the whole concentration range regardless of the degree of dissociation of the protein (see Supporting Information Figure S3 for an example). Although we have shown that G0 is not particularly useful in terms of determining Kd, we will demonstrate later that it is an important variable to consider in kinetic experiments aimed to determine dissociation rate constants. Case II: Monomer−Dimer−Tetramer Equilibria. In this section, we will discuss the quantitative properties of the τApp vs C0 curves for a tetramer−dimer−monomer system. Rajagopalan et al. recently used this experimental approach to determine the equilibrium dissociation constants and the dissociation rate constants of the tetrameric protein p53,8 and we will use this example to illustrate how the mathematical models of eqs 11 and 12 can be used to analyze experimental data. In their study with p53, the authors report τApp vs log C0 experimental data with two inflection points (see the red curve in Figure 4 for an approximate representation of these data), which they assign to the tetramer−dimer and dimer−monomer dissociation constants. Although this approach may intuitively seem reasonable, we will show that the reported Kd values are not consistent with the experimental τApp vs log C0 data measured by the authors. All mechanistic studies reported to date have shown that tetrameric proteins assemble via a dimeric intermediate (i.e., trimeric intermediates have never been observed).16 Tetramer dissociation can therefore be described by considering three species with n = 1, 2, 4 in eqs 11 and 12. To avoid confusion with the terminology used in previous sections, where C3 (concentration of species 3) would refer to the concentration of tetramer (n = 4), we will use [M], [D], and [T] for the monomer, dimer and tetramer concentrations in place of C1, C2, and C3, respectively. The concentrations of monomer, dimer, and tetramer at a given total protein concentration are determined by two dissociation constants (Kd1 and Kd2) and a mass balance: 12409

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of τDM only when Kd1 is at least 2 orders of magnitude greater than the lowest measured C0. For example, in the two cases with Kd1 = 10 nM (Figure 3A,C), the apparent diffusion time at C0 = 0.1 nM is within 1% of τDM. On the other hand, this difference is greater than 6% when Kd1 = 1 nM (Figure 3B). An uncertainty of 6% is actually quite significant considering that the difference between τDD and τDM is just 23%. Likewise, to determine τDT experimentally, measurements need to be carried out at values of C0 at least 2 orders of magnitude greater than Kd2 (see Figure 3C for an example where this is not the case). Another important feature of the curves expected in these types of experiments is the existence of a rather “flat” region (intermediate plateau) in the τApp vs log C0 graph at values around τDD whenever the Kd values are separated by at least 2 orders of magnitude. Well-separated Kd1 and Kd2 values result in the accumulation of a significant concentration of dimer at intermediate C0 concentrations, which translates into a region of concentrations where τApp ≈ τDD (see for example the region around 100 nM in Figure 3B). In contrast, when Kd1 and Kd2 are separated by less than 2 orders of magnitude, all three species coexist at all concentrations, and τApp is expected to change between τDT and τDM without a noticeable change in slope around τDD (Figure 3A). The discussion above can be considered rather trivial, but it emphasizes how these simple chemical equilibrium considerations provide the tools necessary to understand the shapes of the concentration-dependent τApp graphs (e.g., whether τApp is expected to plateau at low and/or high concentrations, and whether a change in curvature is expected at intermediate concentrations). For example, these arguments allow us to predict that proteins that show “flat” regions at intermediate concentrations (e.g., Figure 3B,C) have Kds that are well-separated, and as a consequence it should not be possible to measure both τDT and τDM in the six decades of concentration commonly accessible by FCS. Similarly, if τDT and τDM can be determined in this concentration range, it should be practically impossible to observe two different phases corresponding to the tetramer−dimer and dimer−monomer equilibria (e.g., Figure 3A). These simple arguments suggest that, in contrast to our previous discussion on the dissociation of dimeric proteins, obtaining reasonable estimates of the Kds of tetramers is not straightforward and experimental data should fitted with the appropriate mathematical models. As already mentioned, fitting with the appropriate model is not only important to obtain accurate values of the parameters of interest, but maybe more importantly, to test whether the assumed model is adequate to describe the experimental data. This is, in our opinion, one of the most important arguments against using empirical functions to estimate the thermodynamic and kinetic constants that describe the oligomerization equilibria of proteins. To illustrate this point, we will discuss the experimental data reported by Rajagopalan et al. in ref 8. In this work, the authors determined experimental values of τApp for solutions containing 0.1 nM labeled protein and total protein concentrations in the 0.1 nM to 2 μM range. An approximate representation of these data, obtained from the figures reported by the authors, is shown in Figure 4 (red line). Interestingly, the experimental τApp vs log C0 graph shows two well-separated regions, which the authors assign to the tetramer−dimer and dimer−monomer equilibria. This data fits well with a double sigmoidal equation with inflection points located at approximately 1 and 150 nM, which the authors assign to the Kd1 and Kd2 values of the tetrameric protein. Sigmoidal curves are typically used to describe ligand binding data,19,20 and in

Figure 4. Bottom: Black trace is the expected τApp values for a tetrameric protein with Kd1 = 1 nM, Kd2 = 150 nM, τDM = 0.114 ms, τDD = 0.145 ms, and τDT = 0.183 ms. Red trace is an approximated representation of the experimental data reported in ref 8. The strong disagreement indicates that the experimental data is not well-described by the assumed oligomerization mechanism (see text). Top: Fractional concentrations of monomers (red), dimers (black), and tetramers (blue) for the Kd values indicated above.

Kd1 =

[M]2 [D]2 , Kd2 = , [M] + 2[D] + 4[T] = C0 [D] [T] (19)

These equations are enough to get analytical expressions for [M], [D], and [T] in terms of C0, Kd1, and Kd2, and substitution of these expressions in eqs 11 and 12 results in equations relating τApp to C0, CL, f, Kd1, and Kd2. Although in principle analytical solutions can be written, expressions are too complex for practical purposes, and we prefer to solve this system of equations numerically (see Matlab code in Supporting Information). To analyze the results expected for this mechanism, we will consider the case where experiments are performed using 0.1 nM labeled protein (as in ref 8) and total protein concentrations in the 0.1 nM to 100 μM range (C0). This range spans the lowest and highest protein concentrations typically accessible in these measurements. We further assume f = 1 and the diffusion times reported by Rajagopalan et al. for the p53 tetramerization domain: τDT = 0.114 ms, τDD = 0.145 ms, and τDM = 0.183 ms. The absolute values of these diffusion times depend on instrumental factors that cannot be reproduced between different laboratories, but we note that the relevant variables in this analysis are the ratios of the τD values for the different species and not the values themselves. The values reported by Rajagopalan et al. are consistent with the expected ratio of 21/3 for oligomeric species whose mass differ by a factor of 2, so we deem them adequate to illustrate the results expected for tetrameric proteins in general. Figure 3 shows the expected τApp vs C0 graphs for the cases Kd1 = 10 nM and Kd2 = 100 nM (Figure 3A), Kd1 = 1 nM and Kd2 = 1 μM (Figure 3B), and Kd1 = 10 nM and Kd2 = 10 μM (Figure 3C). The fractional concentrations on top of each graph were calculated from the solutions of eq 19 as αM = [M]/C0, αD = 2[D]/C0, and αT = 4 [T]/C0. Inspection of the τApp vs C0 graphs shows that, as expected, the τApp value measured at the lowest protein concentration is a good approximation 12410

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principle it seems reasonable to use this type of fitting model to obtain the Kds from τApp vs log C0 plots. However, if this protein dissociates following the simple tetramer−dimer− monomer mechanism assumed by the authors, we should be able to reproduce the data plotted in red in Figure 4 using the mathematical models developed here with Kd1 = 1 nM and Kd2 = 150 nM. This theoretical trace is shown as a black curve in Figure 4. Changing the Kd values and the individual τDs (within reasonable boundaries consistent with the sizes of the oligomers) did not result in a better fit. In other words, we were unable to reproduce the shape of the experimental τApp vs log C0 curve with the mathematical equations that describe tetramer dissociation. A key aspect of the experimental data is that the authors observed a change in τApp consistent with the theoretical ratio of 41/3 (complete dissociation of a tetramer) over only four decades of protein concentration (C0 = 0.1 nM to 2 μM). However, the fractional concentrations of monomer, dimer, and tetramer calculated using Kd1 = 1 nM and Kd2 = 150 nM (Figure 4, top graph) indicate that the concentration of dimer should still be significant at both ends of the investigated concentration range, and therefore the change in apparent diffusion times should be smaller (Figure 4, black curve). Another important difference is that the theoretical τApp vs log C0 plot for a protein with Kd1 = 1 nM and Kd2 = 150 nM lacks the clear changes in curvature that were observed experimentally in this concentration range. Basically, the values of Kd1 and Kd2 would need to be relatively similar to explain the observed change in τApp of 41/3 over four decades of concentration, but at the same time, the values of Kd1 and Kd2 would need to be significantly different to explain the inflection points and changes in curvature in the τApp vs log C0 plot. From this discussion, it follows that no Kd values exist that can satisfactorily describe the shape of the curve, and therefore, if we rule out any possible experimental artifacts, the system studied by Rajagopalan et al. is not well-described by the assumed oligomerization model. In simple terms, the protein studied in this work seems to dissociate over a range of concentrations much smaller than that predicted by simple chemical equilibrium considerations. We have no expertise in the protein that is the focus of the work of Rajagopalan et al., but we note that G. Weber and collaborators characterized several proteins that dissociate when diluted over concentration ranges significantly smaller than the values predicted by simple equilibrium considerations.18,21−23 Weber explained these observations in terms of what he called a “conformational drift”, where the increase in dissociation constant with dilution arises from a slow change in conformation of the monomers when they become separated from each other. We stress that we are not in a position to elaborate on whether this concept describes Rajagopalan’s data, or even if it does, whether it has any implications or relevance in terms of the biochemical function of the protein. Yet, this example highlights that the analysis of FCS data in terms of a rigorous formalism is not only important to obtain accurate values of the Kds, but can potentially unveil qualitatively unexpected behaviors. In contrast, the determination of Kd values based on empirical equations that were not formally developed to describe FCS data can hide the fact that the results are not consistent with the assumed model, which in turn can hinder the discovery of interesting phenomena. Determination of Dissociation and Association Kinetic Rate Constants. To measure the rate constants of the dissociation reaction, typically the labeled protein is diluted

at t = 0 from a concentrated stock (where dissociation is negligible) to a final concentration where dissociation is believed to be significant. FCS decays are then measured over time (minutes to hours), and dissociation constants are calculated from the rate of decrease of τApp as the oligomeric protein dissociates into its smaller constituents. This approach has been used by Rajagopalan et al. and by Garai et al. to investigate the kinetics of dissociation of the tetrameric proteins p53 and ApoE4, respectively, and by us to investigate the trimeric protein PCNA.5,8,9 In the first two cases, the authors used exponential functions to fit τApp vs t curves to obtain halflives of dissociation. As we will show in this section, the mathematical framework we derived shows that the τApp vs t curves are not accurately described by exponential functions, and fitting experimental data with exponential models may yield quantitatively erroneous results. Case I: Kinetics of Dimer Dissociation. For the experiment described in the previous paragraph, the time-dependent concentration of monomer (C1) is described by the differential equation k dC1 = kd(C0 − C1) − 2 d C12 dt Kd

(20)

where kd/ Kd represents the association rate constant (ka) and (C0 − C1)/2 = C2. Assuming C1(0) = 0, the analytical solution of eq 20 is ⎛ 1⎜ C1(t ) = − Kd + 4 ⎜⎝

Kd2

⎡ ⎛ 8C0 + Kd 1 + 8KdC0 tanh⎢ ⎜⎜ kdt ⎢⎣ 2 ⎝ Kd

⎛ Kd + 2 arctanh⎜⎜ ⎝ 8C0 + Kd

⎞⎞⎤⎞ ⎟⎟⎟⎟⎥⎟ ⎠⎠⎥⎦⎟⎠

(21)

The values of τApp can be then calculated analytically by plugging eq 21 into eqs 16 and 17 (note that α1 = C1/C0, and in this experiment CL = C0). The concentration of monomer at equilibrium, Ceq 1 , is given by (see eq 21): C1(t → ∞) =

1 ( − Kd + 4

8C0Kd + Kd2 )

(22)

If the system remains close to equilibrium at all times (i.e., the initial protein concentration is C0 ≈ Ceq 1 ), C1(t) can be approximated by an exponential function C1(t) ≈ −t/τkin Ceq ) with a relaxation time τkin given by τkin = 1 (1 − e −1 kd (8C0/Kd + 1)−1/2.24 FCS kinetic experiments cannot be carried out under conditions close to equilibrium because they rely on measurements of τApp, and these values are expected to change only by a factor of 21/3 ≈ 1.26 when dissociation is complete. Because measuring small changes in τApp is difficult, to maximize the range of measured τApp values, the protein should ideally be diluted from a concentrated stock solution (C ≫ Kd) to a final concentration where the degree of dissociation is expected to be high (C0 < Kd). Although the assumptions used to derive the expression of τkin do not hold in these far-from-equilibrium experiments, this value still gives an approximate expected time constant for the changes in apparent diffusion time over time. It is worth noting that the acquisition of an individual FCS decay requires at least a couple of minutes, so investigating oligomerization kinetics by FCS is not feasible for systems with relaxation times shorter than approximately 30 min. Because typical protein−protein 12411

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association rate constants (ka = kd/Kd) are in the 105−106 M−1 s−1 range,25 the equilibration of a 1 nM dimeric protein solution is expected to be too fast for FCS measurements if Kd is greater than ca. 5 nM. On the other extreme, measurements over several hours become increasingly challenging due the difficulty in keeping both the measurement conditions (e.g., laser intensity, optical alignment) and the protein solution sufficiently stable. This imposes an upper limit of measurable τkin values of approximately 300 min (5 h), which translates into a lower limit of Kd ≈ 0.5 nM. Association rate constants may vary over several orders of magnitude, so these Kd estimates should be taken with care. Figure 5A shows the expected time-dependent changes in τApp for a hypothetical system with C0 = 0.1 nM, ka = 105 M−1 s−1,

labeling efficiency because the degree of dissociation of a protein with this Kd at equilibrium is (eq 21) C1eq/C0 = 0.85, and therefore, a significant concentration of dimers remain in the equilibrated solution. The doubly labeled dimeric particles obtained at high values of f would bias the measured τApps to higher values when compared to the singly labeled dimers that result from poor labeling. Because the labeling efficiency is considered explicitly in all our equations, our mathematical models can accurately describe these differences. Experimental τApp(t) data can be fitted with the equations developed above using kd as the fitting parameter (we assume that Kd can be evaluated independently as described above). A reasonable estimate can also be obtained using an empirical monoexponential model (τApp = C + A exp(−t/τkin)). We note that the proper relaxation time for an exponential fit should −1/2 be τkin = k−1 , which depends on both the d (1 + 8C0/Kd) dissociation and association rate constants as well as the protein concentration. The empirical exponential fit gives in fact reasonable estimates of kd (30% error for the case Kd = 1 nM with f = 1, see Supporting Information Figure S4), and interestingly, this error decreases with decreasing labeling efficiency. This is due to the existence of doubly labeled dimers when f is large, which bias the τApp values to higher values and therefore distort the curve. Although this may suggest that poorly labeled samples are preferred in these experiments, it is important to keep in mind that the concentration of labeled sample determines the number of photons collected during the experiment, and therefore the signal-to-noise of the autocorrelation functions. As described above, experiments should be carried out at a total concentration of protein well below the Kd to maximize the degree of dissociation (and therefore the extent of change in τApp). Because acquiring FCS decays at concentrations lower than c.a. 100 pM (labeled protein) usually results in poor signal-to-noise decays, working at low f values is not a feasible approach in most cases. In contrast to the equilibrium measurements described in previous sections, where the concentration of labeled particles remains fairly constant during the experiment, the kinetic experiments described in this section involve not only changes in τApp but also measurable changes in FCS amplitudes. Because the amplitude of the autocorrelation decay (G0) of a single diffusing species is inversely proportional to the concentration of fluorescently labeled particles (eq 2), complete dissociation of a dimer is expected to decrease G0 to half its initial value. The expected changes in amplitude are therefore greater than the expected changes in τApp (τDD ≈ 21/3τDM), suggesting that the analysis of the time-dependent changes in G0 is a more sensitive approach to study the kinetics of protein dissociation. The expected changes in G0 are described by eq 18 and shown in Figure 5B for the systems described above. We note that G0 is directly proportional to α1, which is the only time-dependent variable in eq 18. Because α1(t) is approximately mono−t/τkin exponential (α1(t) ≈ αeq )), it is not surprising that 1 (1 − e G0 can be fitted with a single exponential equation with a time constant that is very close to the actual relaxation time of the system. Although the analysis of G0(t) appears to be a promising approach to obtain kd, we note that we typically observe that the values of τApp can be determined experimentally with better reproducibility, especially in experiments that involve changing samples periodically. Kinetic experiments over hours cannot be performed continuously on the same sample because of significant sample evaporation, photobleaching, protein instability on the coverslip surface, etc. We systematically obtain

Figure 5. Expected kinetic behavior of τApp (top) and G0 (bottom) for a 0.1 nM labeled dimeric protein with ka = 105 M−1 s−1, f = 1, τD1 = 0.4 ms, τD2 = 0.504 ms, and the Kd values indicated in the graphs.

f = 1, and Kd values in the 0.1−10 nM range. In all cases, we assumed that the protein is diluted at t = 0 from a stock concentrated enough so that C1 (t = 0) = 0. As anticipated from the discussion above, the lowest and highest Kd values in this example would result in changes in τApp and G0 that are too fast and too slow for FCS measurements, respectively. In contrast to the equilibrium measurements discussed before, these curves are expected to depend on labeling efficiency because the experiment involves purely labeled sample, and doubly labeled dimers contribute four times more to the FCS decay than singly labeled dimers. A comparison of the results expected for two different labeling efficiencies for the case Kd = 1 nM is shown in Figure S4 (see Supporting Information). We note that the τApp measured at long times (equilibrium conditions) depends on 12412

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more reliable and reproducible results by incubating the protein in a large (compared with the μL volumes used in FCS) Eppendorf tube and taking a small aliquot to measure an FCS decay at regular intervals. This procedure, in our experience, increases the variability in G0 more than it affects the reproducibility of τApp. It is of course up to the experimenter to determine the most reliable way of acquiring data and whether amplitudes can be determined with the accuracy needed in this analysis. Regardless of these experimental considerations, we conclude that both G0(t) and τApp(t) can be in principle used individually or globally to determine the dissociation rate constant of the protein provided that the equilibrium dissociation constant is known. An exponential fit will give, in general, an approximated value of the relaxation time (τkin) from which kd can be obtained. A more accurate and quantitative way of analyzing data, however, involves using the equations obtained in this work, which were derived specifically to describe the time-dependent behavior of the FCS parameters G0 and τApp. The Matlab code used to generate the G0(t) and τApp(t) data presented in this section is provided as Supporting Information. As we did in the section devoted to equilibrium measurements, we also provide an Excel file that can be used to simulate and/ or analyze experimental data. Once again, our goal is to demonstrate that using these accurate expressions is straightforward, and therefore using exponential functions as empirical models is not justified. Case II: Kinetics of Tetramer Dissociation via a Dimeric Intermediate. The system of differential equations describing the dissociation of a tetrameric protein via a dimeric intermediate is k d[M] = kd2(C0 − [M] − 4[T]) − 2 d2 [M]2 dt Kd2 2 kd1 ⎛ (C0 − [M] − 4[T]) ⎞ d[T] = −kd1[T] + ⎜ ⎟ ⎠ dt 2 Kd1 ⎝

(23)

where kd1/Kd1 and kd2/Kd2 represent the association rate constants for the formation of the tetramer and the dimeric intermediate, respectively. This system of equations can be solved numerically with initial conditions [T] = C0/4 and [M] = 0, which assumes that the protein is diluted at t = 0 from a concentrated stock solution so that the degree of dissociation can be considered negligible. The time-dependent concentration of dimer can be then obtained from the mass balance as [D] = (C0 − [M] − 4[T])/2. Knowledge of the concentrations of all species at each measured time can then be used to calculate the time-dependent apparent diffusion times (eqs 11 and 12) and amplitudes (G0 = ∑3i=1Ai, eq 6). A Matlab code that allows the calculation of τApp(t) and G0(t) for given values of equilibrium and rate constants is provided in Supporting Information. Figure 6 shows the expected results for a kinetic experiment involving a 0.1 nM solution of a tetrameric protein. As discussed in the previous section, the values of τApp are somewhat dependent on labeling efficiency because they are biased toward the brightest particles. It is important to notice that these experiments are rather limited in terms of the values of kd that can be measured experimentally. As discussed above, each individual FCS acquisition typically requires at least a couple of minutes and experiments over many hours are challenging. This limits the measurable kd values to a range of about 1 order of magnitude (approximately 5 × 10−4 to 5 × 10−5 s−1). Because protein association

Figure 6. (A,B) Expected kinetic behavior of τApp and G0 for a 0.1 nM labeled tetrameric protein with ka1 = ka2 = 105 M−1 s−1, τDM = 0.114 ms, τDD = 0.145 ms, τDT = 0.183 ms, and the Kd values indicated in the graphs. All traces were calculated using f = 1 except for the curve represented as a dashed line ( f = 0.1). The amplitude of the trace calculated with f = 0.1 (inset in B) is significantly larger because poor labeling results in a lower concentration of labeled particles. (C) Expected changes in τApp for the case Kd1 = 10 nM, Kd2 = 1 nM, and the protein concentrations (CL) indicated in the graph. All other parameters were kept constant.

constants usually fall in a rather small range (typically 105− 106 M−1 s−1),25 it is unlikely that both dissociation rate constants will fit in the measurable kd range unless Kd1 and Kd2 are relatively similar. For instance, Figure 6A shows an example where the dissociation of the dimer occurs over measurable time scales but the dissociation of the tetramer is too fast (large kd1) and an example where the dissociation of the tetramer can be measured accurately but the dissociation of the dimer is too slow (small kd2). We note that these curves were generated assuming a protein concentration CL = C0 = 0.1 nM, but because both the relaxation time (τkin) and the degree of 12413

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dissociation depend on this variable, C0 can be actually used as a parameter to modulate the τApp decays to some extent (Figure 6C). Because measurements in the 0.1−10 nM range are reasonable, it is possible to envision performing measurements at more than one protein concentration and fitting all data globally to improve the accuracy of the kinetic parameters of interest. The decays of Figure 6A fit well with an empirical double exponential function (c1 exp(−k1t) + c2 exp(−k2t) + y0), but the recovered rate constants are in general only good estimates of the orders of magnitude of the dissociation rate constants kd1 and kd2. For example, a biexponential fit of the curve generated assuming C0 = 0.1 nM in Figure 6C gives k1 = 0.045 min−1 and k2 = 9.8 × 10−3 min−1, whereas the curve generated using C0 = 0.5 nM gives k1 = 0.043 min−1 and k2 = 5.5 × 10−3 min−1. The actual values of kd1 and kd2 used to simulate the data are 0.06 min−1 and 6 × 10−3 min−1, respectively. Not surprisingly, the rate constants obtained using biexponential fits are reasonable approximations of the actual values because they capture the relevant time scales of the time-dependent changes in τApp. In this regard, one could argue that fitting with an empirical exponential function represents only a small improvement from determining rate constants by visual inspection of the experimental data. It is worth noting that, unlike the actual dissociation rate constants kd1 and kd2, the empirical dissociation rate constants obtained in the biexponential fits (k1 and k2) depend on protein concentration. This is a clear demonstration that the exponential equation does not describe the problem accurately, and as a consequence the empirical parameters k1 and k2 do not behave physically as true dissociation rate constants. As discussed above, the complexity of the time-dependent τApp and G0 curves suggests that, to obtain quantitatively meaningful results, the best strategy is to perform experiments at more than one concentration and analyze all data globally using the physically accurate models presented in this work.

in solution, which is dictated by the efficiency of labeling and the relative concentrations of labeled and unlabeled proteins. Our analysis shows that a low degree of labeling can be beneficial to equalize the contributions of species of different size, but the need to maximize the number of acquired photons during FCS acquisition may render this strategy unfeasible. Finally, we point out that although we focused on the two most common protein oligomerization models, the equations presented in this work can be easily modified to describe other mechanisms or to incorporate other variables of interest.

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ASSOCIATED CONTENT

S Supporting Information *

Derivations of eqs 6 and 11. Validity of the approximation z0 ≫ r0. Concentration-dependent changes in FCS amplitudes. Concentration-dependent changes in FCS amplitudes. Effect of varying the labeling efficiency on the kinetic results expected for of a dimeric protein. Matlab code and Microsoft Excel files containing the algorithms described in the text. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*Phone: +1-480-727-8586. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the National Science Foundation (grant MCB-1157765 to M.L.). We thank Jennifer K. Binder for her help proofreading the manuscript.

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REFERENCES

(1) Ali, M. H.; Imperiali, B. Protein Oligomerization: How and Why. Bioorg. Med. Chem. 2005, 13, 5013−5020. (2) Marianayagam, N. J.; Sunde, M.; Matthews, J. M. The Power of Two: Protein Dimerization in Biology. Trends Biochem. Sci. 2004, 29, 618−625. (3) Ozbabacan, S. E. A.; Engin, H. B.; Gursoy, A.; Keskin, O. Transient Protein−Protein Interactions. Protein Eng., Des. Sel. 2011, 24, 635−648. (4) Chakraborty, M.; Kuriata, A. M.; Henderson, J. N.; Salvucci, M. E.; Wachter, R. M.; Levitus, M. Protein Oligomerization Monitored by Fluorescence Fluctuation Spectroscopy: Self-Assembly of Rubisco Activase. Biophys. J. 2012, 103, 949−958. (5) Binder, J. K.; Douma, L. G.; Ranjit, S.; Kanno, D. M.; Chakraborty, M.; Bloom, L. B.; Levitus, M. Intrinsic Stability and Oligomerization Dynamics of DNA Processivity Clamps. Nucleic Acids Res. 2014, 42, 6476−86. (6) Keller, D.; Orpinell, M.; Olivier, N.; Wachsmuth, M.; Mahen, R.; Wyss, R.; Hachet, V.; Ellenberg, J.; Manley, S.; Gonczy, P. Mechanisms of Hssas-6 Assembly Promoting Centriole Formation in Human Cells. J. Cell Biol. 2014, 204, 697−712. (7) Herrick-Davis, K.; Grinde, E.; Cowan, A.; Mazurkiewicz, J. E. Fluorescence Correlation Spectroscopy Analysis of Serotonin, Adrenergic, Muscarinic, and Dopamine Receptor Dimerization: The Oligomer Number Puzzle. Mol. Pharmacol. 2013, 84, 630−642. (8) Rajagopalan, S.; Huang, F.; Fersht, A. R. Single-Molecule Characterization of Oligomerization Kinetics and Equilibria of the Tumor Suppressor P53. Nucleic Acids Res. 2011, 39, 2294−2303. (9) Garai, K.; Frieden, C. The Association−Dissociation Behavior of the Apoe Proteins: Kinetic and Equilibrium Studies. Biochemistry 2010, 49, 9533−9541. (10) Takahashi, Y.; Okamoto, Y.; Popiel, H. A.; Fujikake, N.; Toda, T.; Kinjo, M.; Nagai, Y. Detection of Polyglutamine Protein Oligomers

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CONCLUDING REMARKS We presented a rigorous mathematical treatment that relates the observables of FCS experiments (diffusion coefficients and amplitudes) to the thermodynamic and kinetic parameters that describe protein oligomerization (dissociation equilibrium and kinetic rate constants). We demonstrated that, although empirical equations may result in good estimates of these parameters in very simple cases (e.g., dimer dissociation), physically accurate models are needed to correctly interpret experimental results in general. This was illustrated with an example that shows that using empirical models in the analysis of FCS data provides seemingly reasonable results but in reality obscures the fact that the assumed dissociation model is incompatible with the experimental data. Physically accurate models, such as the ones derived in this work, involve parameters that are directly related to the physical variables that characterize the system and can therefore be used not only to analyze data but also to simulate possible outcomes under different conditions. This is particularly critical when designing and interpreting complex experiments such as the ones described in this work. A key aspect of FCS measurements is that the contribution of each labeled species is proportional to the square of its brightness, and therefore the larger oligomers dominate the measured FCS decay. This, if ignored, results in a bias of the measured diffusion times. The average brightness of an oligomeric particle depends on the fraction of labeled particles 12414

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in Cells by Fluorescence Correlation Spectroscopy. J. Biol. Chem. 2007, 282, 24039−24048. (11) Samiee, K. T.; Foquet, M.; Guo, L.; Cox, E. C.; Craighead, H. G. Lambda-Repressor Oligomerization Kinetics at High Concentrations Using Fluorescence Correlation Spectroscopy in Zero-Mode Waveguides. Biophys. J. 2005, 88, 2145−2153. (12) Chen, Y.; Wei, L. N.; Muller, J. D. Probing Protein Oligomerization in Living Cells with Fluorescence Fluctuation Spectroscopy (Vol 100, Pg 15492, 2003). Proc. Natl. Acad. Sci. U. S. A. 2004, 101, 1792−1792. (13) Thompson, N. L., Fluorescence Correlation Spectroscopy. In Topics in Fluorescence Spectroscopy; Lakowicz, J. R., Ed.; Plenum Press: New York, 1991; Vol. 1, pp 337−355. (14) Muller, J. D.; Chen, Y.; Gratton, E. Fluorescence Correlation Spectroscopy. Methods Enzymol. 2003, 361, 69−92. (15) Krichevsky, O.; Bonnet, G. Fluorescence Correlation Spectroscopy: The Technique and Its Applications. Rep. Prog. Phys. 2002, 65, 251−297. (16) Powers, E. T.; Powers, D. L. A Perspective on Mechanisms of Protein Tetramer Formation. Biophys. J. 2003, 85, 3587−3599. (17) Widengren, J.; Mets, U.; Rigler, R. Fluorescence Correlation Spectroscopy of Triplet-States in SolutionA Theoretical and Experimental Study. J. Phys. Chem. 1995, 99, 13368−13379. (18) Weber, G. Phenomenological Description of the Association of Protein Subunits Subjected to Conformational DriftEffects of Dilution and of Hydrostatic Pressure. Biochemistry 1986, 25, 3626− 3631. (19) Munson, P. J.; Rodbard, D. LigandA Versatile Computerized Approach for Characterization of Ligand-Binding Systems. Anal. Biochem. 1980, 107, 220−239. (20) Weiland, G. A.; Molinoff, P. B. Quantitative Analysis of Drug− Receptor Interactions 0.1. Determination of Kinetic and Equilibrium Properties. Life Sci. 1981, 29, 313−330. (21) King, L.; Weber, G. Conformational Drift of Lactate Dehydrogenase. Biophys. J. 1986, 49, 72−73. (22) King, L.; Weber, G. Conformational Drift of Dissociated Lactate Dehydrogenases. Biochemistry 1986, 25, 3632−3637. (23) Ruan, K.; Weber, G. Hysteresis and Conformational Drift of Pressure-Dissociated Glyceraldehydephosphate Dehydrogenase. Biochemistry 1989, 28, 2144−2153. (24) Craig, M. E.; Crothers, D. M.; Doty, P. Relaxation Kinetics of Dimer Formation by Self Complementary Oligonucleotides. J. Mol. Biol. 1971, 62, 383−401. (25) Schreiber, G. Kinetic Studies of Protein−Protein Interactions. Curr. Opin. Struct. Biol. 2002, 12, 41−47.

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