Simple methods to determine the dissociation constant, K_d

Abstract

The determination of the dissociation constant (K_d) is pivotal in biochemistry and pharmacology for understanding binding affinities in chemical reactions, which is crucial for drug development and comprehending biological systems. Here, we introduce a single-molecule fluorescence resonance energy transfer–based method for determining K_d, alongside the conventional electrophoretic mobility shift assay method of K_d, offering insights into thermodynamic interactions between proteins and substrates. The single-molecule fluorescence resonance energy transfer approach is highlighted for its ability to accurately measure binding and dissociation kinetics through fluorescence labeling and the intrinsic nature of protein-DNA interactions, representing a significant advancement in the fields of molecular biology and pharmacology.

DESCRIPTION

The dissociation constant (K_d) is an equilibrium constant used universally in biochemistry and pharmacology to represent the binding affinity of substances (Guo et al., 2020). K_d is crucial for comprehending chemical reactions, as well as in the design and development of drugs (Ma et al., 2017). It provides key thermodynamic properties of biological system, such as binding affinity, concentration dependence, specificity and selectivity, binding kinetics, and allosteric regulation mechanisms. K_d is mainly measured by bulk assays, such as isothermal titration calorimetry, surface plasmon resonance (Krishnamoorthy et al., 2020, Ma et al., 2018), and electrophoretic mobility shift assay (EMSA) (Heffler et al., 2012, Seo et al., 2019), or by single-molecule techniques, such as single-molecule fluorescence resonance energy transfer (smFRET) (Marklund et al., 2020, Myong et al., 2009) and single-molecule protein–induced fluorescence enhance (smPIFE) (Hwang et al., 2011, Hwang and Myong, 2014). Among these, EMSA has the advantage of being relatively simple and widely used for measuring equilibrium K_d in the biochemical and pharmacological fields but is not sensitive enough to measure transient binding interactions with very low K_d (Hellman and Fried, 2007). In contrast, smFRET requires specialized equipment and fluorescence labeling (Roy et al., 2008) but it is very sensitive to detect even tens of micromolar levels of affinity (Yoo et al., 2016). Moreover, the smFRET-based method for K_d measurements has advantages over EMSA in that it requires only a single concentration measurement without the need for substrate titration, and it can improve accuracy by measuring the K_d of only folded proteins when the sample contains a mixture between folded and denatured proteins. Detecting individual molecules at the single-molecule level allows for a deeper understanding of complex molecular interactions and heterogeneity, which is crucial for drug development and understanding biological mechanisms.

Here, we provide measurement methodology of the K_d using smFRET and EMSA. As an example for K_d measurement, we use exonuclease III (ExoIII) and its substrate, overhang DNA (Lee et al., 2022, Yoo et al., 2021). To measure K_d, the binding and dissociation must occur under conditions in which the substrate is not altered by enzyme activity and the enzyme and substrate undergo a steady-state interaction. In this context, we use a catalytically dead mutant (D151N) to proceed in the presence of a reaction cofactor (eg, Mg²⁺). Alternatively, the wild type can be used without the reaction cofactor.

EMSA is one of the simplest, fastest, and cost-effective methods to measure K_d, making it valuable for screening protein-DNA interactions quickly. It is based on the difference in mobility between DNA-protein complexes and free DNA in a gel. Fluorescent labeling of DNA is commonly employed for visualization. Taking the K_d measurement of ExoIII as an example (Lee et al., 2022), the concentration of DNA used is fixed at a level lower than the K_d value but sufficiently high to be detected. The protein concentration is titrated and incubated with the DNA. The DNA-protein complex is then resolved and visualized on a native PAGE (polyacrylamide gel electrophoresis) gel (Fig. 1A). For intensity analysis, software, such as ImageJ (Schneider et al., 2012) or Image Lab (https://www.bio-rad.com/), is used to measure the band intensity of each well. In the equilibrium (Fig. 1B, Equation 1) between an enzyme (E) and a substrate (S), the fraction bound is defined as the ratio of the concentration of the enzyme-substrate (ES) complex to the total concentration of the substrate (DNA) (Fig. 1B, Equation 2). Assuming a 1:1 binding interaction, where 1 enzyme binds to 1 substrate, the K_d is given by Equation 3, where [E] is the concentration of the free enzyme, [S] is the concentration of the free substrate, and [ES] is the concentration of the enzyme-substrate complex. Rewriting the K_d equation, we have Equation 3, (Fig. 1B). Given the mass conservation, [S]_total = [S] + [ES] and [E]_total = [E] + [ES]. For most EMSA experiments, [E] is typically much higher than [S], so it can be assumed that [E] ≈ [E]_total. This assumption simplifies the calculation without significantly affecting accuracy. Substituting [ES] of Equation 4 into Equation 2 (Fig. 1B), we get Equation 5 (Heffler et al., 2012, Pollard, 2010). Thus, the fraction bound equation relates the fraction of DNA bound to the enzyme concentration and the K_d. By plotting the fraction bound vs the total enzyme concentration and fitting the data to Equation 5, the K_d value, which represents the concentration of enzyme at which 50% of the DNA is bound, can be accurately determined (Goodrich and Kugel, 2007, Heffler et al., 2012).

Fig. 1.

Dissociation constant measurement by EMSA. (A) EMSA for ExoIII binding to the overhanging DNA. (B) Fraction bound equation derivation and plot of fraction bound vs total enzyme concentration (ExoIII) [E] _total. K_d is the dissociation constant, indicating the concentration of enzyme at which 50% of the DNA is bound. On the left side, [E], [S], and [ES] are the concentrations of free enzyme, free substrate, and the enzyme-substrate complex, respectively. Fraction bound (Equation 5) is calculated by substituting [ES] of Equation 4 into Equation 2. On the right side, the fraction bound is calculated as the intensity of the enzyme-substrate (ES) complex divided by the total intensity. The data points represent experimental values, and the curve represents the best fit to the fraction bound equation, assuming a 1:1 binding interaction between the enzyme and substrate.

However, the conventional ensemble techniques are limited in the kinetic information they provide, underscoring the necessity for more informative technologies. The smFRET and smPIFE are powerful tools to obtain binding dynamic information between proteins and substrates (Lerner et al., 2018, Roy et al., 2008), based on precise and real-time detection of interactions. Briefly, smFRET is a technique that measures the energy transfer between 2 fluorescent molecules (eg, Cy3 and Cy5) at the single-molecule level, allowing for the detection of molecular interactions and conformational changes in real-time (Roy et al., 2008). Similarly, smPIFE occurs when a protein binds the region near a fluorophore on the DNA, causing an increase in its fluorescence intensity without the need for a second fluorophore, making it useful for studying protein-DNA interactions. A more detailed comparison can be found in the review article (Hwang and Myong, 2014).

For the smFRET-based K_d method, labeling the substrate and protein with donor and acceptor fluorescent dyes is required (Fig. 2A). Proteins can be site-specifically labeled with fluorescence using methods, such as cysteine labeling, sortase, or Sfp synthetase (Kim et al., 2008, Theile et al., 2013, Yin et al., 2006). Due to fluorescence background issues, proteins with low K_d values (eg, less than 10 μM) are typically labeled with Cy3 as the donor, while proteins with high K_d values ( eg, greater than 10 μM) are labeled with Cy5 as the acceptor. DNA can be fluorescently labeled with a dye for FRET pairing through site-specific modification. DNA modified with biotin at the 3′ or 5′ end is immobilized on a biotinylated PEG (Polyethylene Glycol)-coated surface through biotin-neutravidin interaction. Next, proteins labeled with the fluorescence dye are introduced to observe binding and dissociation kinetics. Alternatively, smPIFE can be used by labeling Cy3 or Cy5 near the protein-binding region of DNA (Hwang et al., 2011, Hwang and Myong, 2014). In DNA-protein interactions, the advantage of PIFE is that only the DNA can be fluorescently labeled, and there is no need to label the protein, since the Kd can be extracted by quantifying the increase in fluorescence intensity when the protein of interest binds in the vicinity of the fluorescently labeled DNA.

Fig. 2.

Dissociation constant measurement by smFRET. (A) Schematic of smFRET experiment for K_d measurement. Cy5 (red dot) was attached to 6 base pairs into the duplex. When the protein labeled with Cy3 (green dot) binds to the Cy5-labeled DNA, FRET occurs from Cy3 to Cy5. (B) A representative trajectory of ExoIII binding to the overhanging DNA. The binding and dissociation times were collected from the times between binding (blue) and the unbound (orange) events. (C) Distributions of binding and dissociation times were fitted by a single-exponential decay to determine binding (τ_on) and dissociation (τ_off) characteristic decay times when ExoIII at 1 nM binds to DNA. Pseudo–first order binding (k_on) and dissociation (k_off) rate constants were calculated by Equations 6 and 7. The equilibrium dissociation constant (K_d) is the ratio of k_off and k_on as in Equation 8, where [E] indicates the total enzyme concentration.

A representative FRET-time trajectory shows the real-time binding and dissociation events of the protein to DNA (Fig. 2B). The FRET signal appears when the protein binds to DNA and disappears upon dissociation. The criteria for defining “Bound” and “Unbound” times were based on the fluorescence intensities of the donor (Cy3) and acceptor (Cy5). The bound time (the time it takes for the protein to dissociate) and unbound time (the time from protein dissociation to the next protein binding) are measured by vbFRET (Bronson et al., 2009). This vbFRET software allows unbiased quantification of binding and dissociation kinetics based on a Bayesian approach using the maximum likelihood criterion (available open source via http://vbFRET.sourceforge.net). The bound and unbound events are collected and binned to obtain the on and off values of tau (τ) through single-exponential decay fitting for the first-order reaction model (Fig. 2C). The dissociation rate constant (k_off) is represented as the reciprocal of the bound time (τ_on), and the association rate constant (k_on) is expressed as the reciprocal of the product of the unbound time (τ_off) and the protein concentration (E). K_d can be calculated as k_off/k_on. The additional rate constants, k_on and k_off, provide insights into the difference in the rate and duration of binding and dissociation events (Markiewicz et al., 2012), even if K_d values are the same.

Regarding the accuracy of measurements, K_d values obtained from smFRET were consistent with EMSA, reinforcing the reliability of our findings. For example, the K_d values measured using EMSA (9.36 ± 1.39 nM) and smFRET (9.87 ± 0.7 nM) showed close agreement, highlighting the robustness of our experimental approach. Consistently, K_d measurements by smFRET have been increasingly used in studies of nucleic acid-protein interaction (Marklund et al., 2020, Myong et al., 2009) and protein-protein interaction (Jiang et al., 2019, Liao et al., 2021). In short, utilizing smFRET for K_d measurement not only yields more accurate kinetic constants but also provides mechanistic insights into multistep binding reactions in allosteric regulations, enhancing our understanding of biological processes. By following our step-by-step experimental protocols, the reliable K_d can be easily obtained via binding and dissociation trajectories at the single-molecule resolution.

Author Contributions

D.L. and J.K. conducted the experiments, and D.L., J.K., and G.L. wrote the manuscript.

Declaration of Competing Interests

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.