Mechanisms of ligand binding

The binding of a ligand to its target is central to many processes in chemistry and biology. For this reason, theoretical treatments of the equilibrium and kinetic underpinnings of ligand binding have occupied researchers for well over a century. The first attempt to rationalize the reversible encounter of a ligand with a biological macromolecule to produce a complex was proposed by Fischer in 1894¹ as a lock-and-key mechanism of recognition, envisioning a nearly perfect shape complementarity between the ligand and its binding site. Structural rigidity of the ligand and its target dictate the rules for specificity in this mechanism and explain changes in affinity among different ligands binding to the same target or of the same ligand binding to different targets. Fischer's original idea remains relevant to this day in drug design, where pharmacological leads are progressively rigidified to promote shape complementarity with their biological target.² It took several decades for the scientific community to realize that biological macromolecules are intrinsically flexible and may affect the energetics of ligand recognition by changing the shape of the binding site. In 1951, Wyman and Allen proposed a radical new explanation for hemoglobin's Bohr effect, i.e., the linkage between pH and oxygen binding, based on alternative conformations accessible to the protein.³ Later on, Eigen envisioned multiple preexisting states for a host controlling ionic interactions in solution⁴ and Koshland extended the lock-and-key mechanism to account for a conformational rearrangement following the initial encounter to produce a more stable complex through a mechanism that he called “induced fit.”⁵ Development of the allosteric concept^6,7 soon led to the celebrated Monod-Wyman-Changeux model of protein allostery based on a preexisting equilibrium between alternative conformations with different ligand binding affinity.⁸ The elegance of this model stems from the complexity of functional behaviors and cooperativity made possible by a simple linkage established between ligand binding and conformational transitions. An equally elegant model developed by Koshland, Nemethy, and Filmer envisioned linkages with conformational transitions taking place at each step of ligation and through nearest-neighbor interactions for multimeric proteins.⁹ The advent of X-ray structural biology and nuclear magnetic resonance (NMR) has firmly established the conformational plasticity of biological macromolecules^10,11 and its relevance to drug design.¹² Valuable extensions of the original allosteric concept have been presented^13–17 and now include a dynamic view of protein conformations generated by folding and binding along an ensemble of accessible states.^11,18–21 The current challenge for the experimentalist is to decipher the signatures of such emerging complexity from analysis of experimental data. The goal is to produce a coherent description of ligand binding that is consistent with both structural and functional data.

This review focuses on the analysis and interpretation of the kinetics of ligand binding mechanisms that are key to establishing a correlation with structure and any linked conformational transition. Our treatment deals with the most common situation encountered in practice, i.e., the reaction of a ligand with a single site on its target. Emphasis is given to the mathematical underpinnings of this interaction when studied through its kinetic components. We show that even this simple interaction may give rise to substantial functional complexity through the interplay of ligand binding and conformational transitions. A rigorous understanding of this complexity is critical for the correct interpretation of experimental data, the validation of structural information, and to advance basic knowledge and translational applications. Our discussion complements and updates excellent contributions available through monographs^22–25 and reviews^26–29 that should be consulted to expand on the topics presented here.

The phenomenological approach to ligand binding to a biological macromolecule at equilibrium, in a closed system under conditions of constant temperature and pressure, is based on the formulation of a partition function as a polynomial expansion in the ligand activity $x$ of degree $N$⁠, equal to the number of binding sites.^25,30 All relevant thermodynamic quantities are related to this “binding polynomial”^31,32 by simple transformations and offer a rigorous interpretation of the underlying energetics, yet provide little insight into the molecular mechanism of recognition. This is because the polynomial takes the same general form once the value of $N$ is defined, regardless of the number of conformations involved. The kinetic treatment of ligand binding to a biological macromolecule depends not only on the number of distinct ligated species in the system but also on their conformations. Both ligation and conformational states define the dimensionality of the system and contribute to the possible reaction trajectories.²⁴ Unlike equilibrium, kinetics may find differences among various mechanisms even for systems that share the same value of ligand binding sites, $N$⁠. Hence, the mathematical treatment of the kinetic properties of a system can be quite complex, even for the simplest case of ligand binding to a biological macromolecule containing a single site (⁠$N$ = 1). From this complexity comes the ability to infer precious information on the mechanism of ligand binding that is not possible to extract from the equilibrium treatment.

We start our discussion with a reaction scheme describing the transition of a system between two states, i.e.,

For the sake of clarity, we shall focus on the case where the “system” represents a biological macromolecule and the “states” refer to distinct conformations, e.g., folded and unfolded, or open and closed. The reaction in Eq. (1) is completely defined by the first-order rate constants $k12$ and $k21$⁠, measured in s⁻¹, reflecting the forward and reverse transitions between $E1$ and $E2$⁠. Under the assumptions that the system is closed and the macromolecule does not change its aggregation state, the time evolution of the two species in Eq. (1) is given by the differential equations

Equilibrium is reached in the limit $t→∞$ where the ratio

defines the dimensionless equilibrium constant, $K12$⁠, as the ratio of the populations of $E2$ and $E1$ at $t=∞$⁠. The value of $K12$ can be the same for different combinations of $k12$ and $k21$⁠, so long as their ratio remains unchanged. Hence, information on the equilibrium properties of the system depends on a single independent constant, $K12$⁠, from which the two independent rate constants, $k12$ and $k21$⁠, that define how the macromolecule transitions between $E1$ and $E2$ cannot be resolved. Once the value of $K12$ is known, the fraction of macromolecules in-state, $E1$ and $E2$⁠, can be calculated as

The information conveyed by $f1$ and $f2$ is purely phenomenological and states which of the two conformations, $E1$ or $E2$⁠, is more populated at equilibrium as a result of $k12$ being faster than $k21$⁠, or vice versa. The exact values of $k12$ and $k21$ remain undetermined and so is the timescale of the transitions $E1→E2$ and $E1←E2$ defined as the inverse value of $k12$ and $k21$⁠.^22,28 An additional relationship between $k12$ and $k21$ is derived from measurements of the time-dependent evolution of the system, i.e.,

where $Ei∞$ and $Ei0$ are the values of $Eit$ at $t=∞$ and $t=0$⁠, respectively (⁠$i=1, 2$⁠). The quantity

defines the rate of relaxation to equilibrium and is equal to the non-zero eigenvalue, with reversed sign, of the matrix of rate constants in Eq. (2). The symmetry of Eq. (6) is noteworthy. Regardless of whether $k12$ is faster or slower than $k21$⁠, equilibrium is always attained with the same rate, or within the same timescale defined as the inverse value of $α$⁠. Furthermore, the rate of approach to equilibrium, $α$⁠, is always faster than the fastest individual transitions $k12$ and $k21$⁠,^22–24 i.e., the rate at which $E1$ and $E2$ reach equilibrium is always faster than the individual forward and reverse transitions $E1→E2$ and $E1←E2$⁠. However, as seen in the case of the equilibrium constant $K12$⁠, the value of $α$ is a single number from which the individual rate constants $k12$ and $k21$ cannot be resolved unambiguously. Knowledge of both $K12$ and $α$ is necessary to resolve the rates for the individual transitions $E1→E2$ and $E1←E2$ and confirm the validity of the mechanism in Eq. (1) through consistency of Eqs. (3) and (6).

The foregoing example details basic differences between equilibrium and kinetics treatments for the scheme in Eq. (1), but shows that both approaches are necessary to resolve the underlying rate constants. Consider next the case where the two-state reaction scheme in Eq. (1) is perturbed by a dimensionless factor $δ$ that changes the rate of the $E1→E2$ transition. Analogous perturbation of the reverse $E1←E2$ transition obeys the same mathematical treatment because of the intrinsic symmetry of Eq. (1). The relevant reaction scheme becomes

and the rate of relaxation to equilibrium is

Introduction of the perturbation generates a strategy to resolve the individual rate constants $k12$ and $k21$ from measurements of $α(δ)$ as a function of $δ$ (Fig. 1). The resulting plot is a straight line, with slope $k12$ and intercept $k21$⁠. Knowledge of these independent rate constants also defines the value of the equilibrium constant $K12$ through Eq. (3) and makes independent measurement of this parameter only needed to confirm validity of the kinetic experiment. In principle, a method that measures the value of $α$ under the effect of a perturbation $δ$ will be able to resolve the independent rate constants defining the mechanism in Eq. (1).

Consider the case where the perturbation factor $δ$ in Eq. (7) measures the activity or concentration $x$ of a ligand $X$ binding to a biological macromolecule $E$ to generate the complex $EX$⁠. For the sake of simplicity, we assume that the ligand is in large excess over the macromolecule and that its concentration does not change significantly as a result of the binding interaction. This is the so-called pseudo-first-order approximation and describes the conditions most commonly encountered in practice.^22,23 We also assume that the ligand binds to a single site and that binding does not change the aggregation state of the macromolecule. Under these assumptions, Eq. (7) describes binding of a ligand to a biological macromolecule according to the scheme

The rate constant $kon$ has dimensions of M⁻¹s⁻¹ and is no longer first-order as in Eq. (1) or (7) but second-order because the perturbation factor depends on the concentration of the ligand. The rate constant $koff$ retains its first-order dimensionality in s⁻¹ and measures the rate of dissociation of the complex $EX$ into the parent species $E$ and $X$⁠. Unlike $k12$ and $k21$ in Eq. (1) or (7), the values of $kon$ and $koff$ have different dimensionality and cannot be compared directly. Additional distinctions must be considered. Processes that are first-order, as those described in Eq. (1), are minimally affected by translational and rotational diffusion that is extremely important in the second-order rate constant of association for binding interactions.²⁷ This is a reason why biological interactions should be studied in solution with techniques like stopped flow,³³ temperature jump,⁴ or NMR,³³ rather than surface plasmon resonance where the macromolecule or the ligand are immobilized to a matrix.³⁴ Elimination of translational and rotational degrees of freedom may bias the measured rate constant for ligand association and may return incorrect values for the equilibrium binding constant. Additional differences between first-order and second-order rate constants is that the latter tend to be extremely sensitive to electrostatics and solution conditions, especially salt concentration.²⁶ Information on the values of $kon$ and $koff$ is critical to arrive at the equilibrium dissociation constant $Kd=koffkon$⁠.³⁰ The value of $Kd$ defines the strength of interaction and establishes the relative affinity of different ligands binding to the same target or of the same ligand binding to different targets. Changes in $Kd$ also give information on how the interaction is affected by changes in solution conditions, temperature, pressure, or mutations introduced in the system.³⁰ Although this is relevant information about the properties of the system, the value of $Kd$ provides a rather “static” interpretation of the interaction. Information of major biological and biophysical interest is contained in the individual rate constants $kon$ and $koff$⁠. Importantly, the value of $kon$ is limited by diffusion to about 6.5 × 10⁸ M⁻¹s⁻¹ under physiological conditions,^27,35 and each system has optimized this value based on evolutionary needs of increased affinity and speed. Relevant examples are values of $kon$ in the 10⁶–10⁸ M⁻¹s⁻¹ range observed in toxins that neutralize the function of ion channels³⁶ or neural transmission³⁷ to cause paralysis in the prey and inhibitors of coagulation factors that cause the prey to bleed to death³⁸ or rapidly neutralize the action of nucleases in degrading RNA.^27,39 Binding affinity can also be optimized by reducing the value of $koff$⁠, especially when the value of $kon$ has maxed out. Although this may not be convenient in signaling pathways that require fast rates of association and rapid switch of on and off signals,⁴⁰ it often becomes desirable in drug design to increase the so-called “residence time” of a therapeutic molecule on its target.²⁸ Indeed, successful drug design is a balancing act between optimizing $kon$ for rapid association and limiting $koff$ to increase stability of the complex with the biological target. In general, when molecules compete for the same target, binding at the diffusion controlled limit provides a source of selection, as shown convincingly by the binding of various psychoactive substances to the membrane transporters for the monoamines serotonin and dopamine,⁴¹ but so does modulation of $koff$ among competing ligands that have optimized their $kon$⁠.⁴² 

The reaction scheme in Eq. (9) offers the simplest possible interpretation of a binding interaction in terms of the so-called lock-and-key mechanism as envisioned by Fischer back in 1894¹ at a time when biological macromolecules were thought of as rigid bodies, in contrast to the current view that they are inherently dynamic.^10,43–48 In this mechanism, ligand and its biological target are preconfigured for optimal binding through a rigid body association. When the properties of the system are studied in the approach to equilibrium using an apparatus for rapid kinetics, the relaxation obeys a straight line, as a function of the ligand concentration $x$ and the values of $kon$ and $koff$ are the slopes and intercept of the plot, i.e.,

which is analogous to Eq. (8). We stress that the range of processes that can be studied experimentally through relaxation kinetics is limited by the time resolution of the system. Another important limiting factor is that the processes accessible to experimental measurements are always the slowest ones in the reaction, relative to the resolution of the instrument. Any conformational transition [Eq. (1)] or binding interaction [Eq. (9)] that takes place on a timescale faster than the range of detection of the instrument will be undetected. A stopped flow is restricted by the dead time of the instrument to values < 500 s⁻¹ but a continuous flow apparatus can detect much faster rates, in the 2000–20 000 s⁻¹ range.⁴⁹ Even faster ranges in the 10⁵–10⁶ s⁻¹ range can be detected by temperature jump if the reaction is linked to sufficient perturbation of affinity linked to temperature changes.^4,15 An example of the plot in Eq. (10) is given in Fig. 2 for the case of the tripeptide H-D-Phe-Pro-Arg-p-nitroanilide (FPR) binding to the active site of the mutant D194A of the clotting protease thrombin.⁵⁰ The binding mechanism is consistent with a lock-and-key, rigid body type of association with values of $kon$ = 1.3 ± 0.1 × 10⁶ M⁻¹s⁻¹ in the diffusion-limited rate range and a relatively fast $koff$= 8.5 ± 0.5 s⁻¹, corresponding to a value of the equilibrium constant $Kd$ = 6.5 ± 0.6 μM. There are several interactions in biology that obey Eq. (9) and give a linear plot of the relaxation to equilibrium as shown in Fig. 2. They are particularly relevant to drug design where binding is optimized by restricting the conformational landscape of the ligand.^2,28,51 Conformational changes may pose challenges in the design of optimal inhibitors of biological targets.¹² However, the widely accepted importance of protein flexibility in biomolecular recognition suggests increasing target flexibility in the bound state by ligand design as a new strategy for drug discovery.⁵² 

The lock-and-key mechanism assumes that the ligand and its target interact without any linked conformational transitions and offers a valuable reference point for the interpretation of more complex binding mechanisms. Whenever binding is involved, there will be a relaxation that increases linearly with the ligand concentration as predicted by Eq. (10). This conclusion holds true regardless of the complexity of the reaction mechanism. As early as 1951, Wyman and Allen drew attention to conformational changes as possible driving forces for binding and linkage effects in hemoglobin,³ and set in motion a conceptual revolution that would culminate with formulation of the celebrated allosteric concept a few years later.^6–8 Conformational changes linked to binding are also relevant to enzyme kinetics when investigating the microscopic pathway for ligand binding in the kinetic mechanism and the overall rate to form the active Michaelis complex. When the binding step is rate-limiting for catalysis, it will depend on the viscosity of the solution. The lack of such dependence suggests the presence of a conformational change and raises the question as to whether it follows or precedes the binding step. The two possible cases linking the basic lock-and-key mechanism describing binding [Eq. (9)] with an elementary conformational transition [Eq (1)] are discussed below. In the first case, the conformational change follows the binding step and defines the so-called induced fit (IF) mechanism first proposed by Koshland.⁵ In the second case, the conformational change precedes the binding step and defines the so-called pre-equilibrium mechanism first proposed by Eigen⁴ and later on cast in a more structural context⁴⁴ as conformational selection (CS). Under the assumptions used for the lock-and-key mechanism in Eq. (9), the relevant reaction scheme for IF is

The first step in the reaction scheme is identical to the lock-and-key mechanism [Eq. (9)]. The second step describes a conformational rearrangement of the complex into a new final state and is analogous to the two-state reaction scheme in Eq. (1). The relevant reaction scheme for CS is

The first step describes a conformational rearrangement analogous to Eq. (1) that preexists in the biological macromolecule before the binding event, and the second step is identical to the lock-and-key mechanism [Eq (9)]. Both IF and CS merge a basic one-step mechanism involving binding [Eq. (9)] and conformational transitions [Eq. (1)] but in reverse order. A challenge for the experimentalist is to establish which mechanism, IF or CS, is at play and how the rates describing the conformational changes can be measured from analysis of experimental data.

The two schemes in Eqs. (11) and (12) feature a similar topology where two one-step reactions are arranged in reversed order. This presages very different kinetics for IF and CS and suggests that the two mechanisms are intrinsically distinct and mathematically irreducible. A number of considerations support such a conclusion. When the ligand concentration decreases (⁠$x→0$⁠), IF collapses toward a single conformation $E$⁠, whereas CS defines the $E*⇄ E$ preexisting equilibrium between the two possible conformations of the macromolecule. The opposite happens when the ligand concentration increases (⁠$x→∞$⁠): IF defines the $EX ⇄ E′X$ equilibrium between the two possible conformations of the complex whereas CS collapses toward the single conformation $EX$⁠. The number of conformations linked to binding progressively increases for IF but decreases for CS, i.e., IF generates new conformations as a result of the binding interaction but CS reduces the number of conformations from the preexisting ones when binding occurs. There are three species that define Eqs. (11) and (12). Because the system is closed and mass is conserved, only two of these species are independent, and there are only two independent non-zero relaxations associated with each reaction scheme: one reflects binding and eventually increases linearly with $x$ as seen for Eq. (10), the other reflects conformational transitions and features a distinct behavior for IF and CS. In the former case, the rate associated with the conformational transition increases with $x$ as the macromolecule transitions from a single free conformation $E$ to an equilibrium between two bound conformations $EX ⇄ E′X$⁠. In the latter case, the opposite is observed, and the rate associated with the conformational transition decreases with $x$ as the macromolecule goes from an equilibrium between two free conformations $E*⇄ E$ to a single bound conformation $EX$⁠. These are intuitive expectations about the kinetics of IF and CS based on simple inspection of the topology of the two reaction schemes in Eqs. (11) and (12). They imply that IF and CS are mutually exclusive mechanisms of recognition, offering distinct and irreducible interpretations of the linkage between binding and conformational transitions.^28,53

Support for the foregoing conclusion comes from analysis of the functional behavior of the two mechanisms when binding takes place on a significantly faster timescale than conformational transitions, i.e., under the so-called “rapid equilibrium approximation.”^22,23 The condition can be understood by comparing the rates of relaxation to equilibrium for a conformational transition [Eq. (6)] and binding [Eq. (10)]. The rate is constant for the former process but increases linearly with the ligand concentration for the latter. At high enough values of $x$⁠, binding will always be faster than any linked conformational change. When the rate of ligand dissociation $koff$ is faster than the rates associated with conformational transitions, binding will reach equilibrium faster than the associated conformational change, and the two relaxations associated with IF and CS become separated by widely different time scales. Binding, if not too fast to detect by a stopped flow apparatus, will follow a straight line, according to Eq. (10). The conformational change will unfold over a slow timescale, typically within the range of detection of the stopped flow apparatus. Under these conditions, the two schemes in Eqs. (11) and (12) contract into

The term $⟨EEX⟩$ denotes an equilibrium distribution of free and bound species. The “contracted” reaction schemes in Eqs. (13a) and (13b) are equivalent to a two-state reaction [Eq. (1)] involving a conformational exchange between two bound forms for IF [Eq. (13a)] or two free forms for CS [Eq. (13b)]. Of the two species in Eqs. (13a) and (13b), only one is independent, and the kinetics of the systems is governed by a single relaxation rate

where $f$ is the fractional saturation of the macromolecule within the $⟨EEX⟩$ equilibrium [see also Eqs. (4a) and (4b)]. It is important to recognize the similarity of Eqs. (14a) and (14b) with Eq. (8). In the case of IF, the value of $k23$ is perturbed by the fraction of macromolecules in the bound form within the $⟨EEX⟩$ equilibrium. In the case of CS, the value of $k21$ is perturbed by the fraction of macromolecules in the free form within the $⟨EEX⟩$ equilibrium. The expressions in Eqs. (14a) and (14b) reduce the independent parameters in Eqs. (11) and (12) from four to three and enable resolution of all relevant constants from analysis of a single relaxation. The relaxation is saturable, and the three independent parameters are resolved as the limiting values at high and low ligand concentration, plus the mid-point of the transition between these values. Another attractive feature of the simplification generated by the rapid equilibrium approximation is that IF and CS can easily be distinguished from analysis of experimental data because they make very different predictions about the properties of the slow relaxation. The expression associated with IF [Eq. (14a)] predicts a saturable hyperbolic increase in $α(x)$ with $x$⁠, from a value of $k32$ to a plateau of $k23+k32$⁠. The expression associated with CS [Eq. (14b)] predicts a saturable hyperbolic decrease in $α(x)$ with $x$⁠, from a value of $k12+k21$ to a plateau of $k12$⁠. The kinetics of a large number of systems have been interpreted using the rapid equilibrium approximation. In turn, the prevalence of systems featuring a saturable relaxation increasing hyperbolically with $x$ has fostered the notion that IF is the dominant mechanism of molecular recognition in biology, with CS documented only in a handful of cases.²⁸ 

As an example of a biologically relevant system obeying the rapid equilibrium approximation, we discuss the case of glucose binding to glucokinase.⁵⁴ Rapid kinetics of glucose binding to the enzyme obey two relaxations over widely separated time scales [Fig. 3(A)]. The fast relaxation increases linearly with glucose concentration with values of $kon$ = 0.53 ± 0.04 × 10³ M⁻¹s⁻¹ much smaller than the diffusion-limited rate and $koff$ = 7.9 ± 0.8 s⁻¹, corresponding to a value of the intrinsic equilibrium dissociation constant $Kd$= 15 ± 1 mM. The slow relaxation increases hyperbolically with the glucose concentration from a value of $k32$ = 0.24 ± 0.02 s⁻¹ to the sum $k23+k32$= 0.46 ± 0.04 s⁻¹. In this interpretation, glucose binds to glucokinase with a very low affinity and then induces a slow conformational change that optimizes the initial encounter into a final complex. IF causes a strengthening of the initial interaction with a resulting apparent equilibrium dissociation constant

that is lower than the intrinsic $Kd$ measured as the ratio $Kd=koffkon$⁠. The value of $Kd,app$ would be measured from titration of glucokinase with glucose at equilibrium and would overestimate the affinity of the initial encounter by a factor of 2. When a binding interaction is interpreted in terms of IF, the affinity measured at equilibrium is the result of an optimized fit and always overestimates the affinity of the initial encounter. Evidence from structural biology has been used to support the IF mechanism for glucokinase [Fig. 3(B)]: the free form assumes an open conformation⁵⁵ and binding of glucose induces a large conformational change that closes the active site region around the ligand.⁵⁶ 

The case of Na⁺ binding to the clotting protease thrombin offers a relevant example of CS under the rapid equilibrium approximation. In this case, the fast relaxation pertaining to Na⁺ binding is too fast to measure by stopped flow and requires alternative experimental approaches.⁴⁹ The slow relaxation can be measured by stopped flow and decreases hyperbolically with the Na⁺ concentration from the sum $k12+k21$ = 570 ± 40 s⁻¹ to the value of $k12$ = 88 ± 7 s⁻¹ [Fig. 4(A)]. Binding takes place to only one of two alternative conformations of the Na⁺ site that preexists in equilibrium and with a value of the equilibrium constant $Kd$ =1.7 ± 0.1 mM. Structural biology supports the CS mechanism for Na⁺ binding to thrombin [Fig. 4(B)]: the pore of access to the buried Na⁺ site is open or closed in the free form, and Na⁺ binds to the open conformation with minimal changes of the structure.^57,58 As a result of the pre-equilibrium, CS produces an apparent equilibrium dissociation constant

that is always higher than the intrinsic $Kd$⁠. As for the case of glucose binding to glucokinase, the value of $Kd,app$ would be measured from titration of Na⁺ with thrombin at equilibrium, but would underestimate the affinity of the binding interaction to the $E$ form by a factor of 9. When a binding interaction is interpreted in terms of CS, the affinity measured at equilibrium is the result of binding to a fraction of the total population of targets and always underestimates the affinity of binding to this population.

Although the rapid equilibrium approximation has attractive features and simplifies the mathematical expressions required to fit experimental data, it has potential limitations that may bias interpretation of the binding mechanism. Consider the case shown in Fig. 5(A) where the tripeptide FPR binds to the W215A mutant of thrombin according to a single saturable relaxation that increases hyperbolically with the ligand concentration $x$⁠.⁵⁹ The absence of a fast relaxation linked to the binding of FPR may be due to values of $kon$ and $koff$ too fast to measure by stopped flow, or to silencing of the spectral signal linked to FPR binding due to mutation of W215, which is known to be a major fluorophore.⁶⁰ When the data in Fig. 5(A) are interpreted according to IF under the rapid equilibrium approximation [Eq. (13a)], the relaxation measures the conformational transition of the thrombin-FPR intermediate to a final complex. The rate constants for this exchange are derived from the lower asymptote of the plot as $k32$ = 6.3 ± 0.5 s⁻¹ and the upper asymptote as the sum $k23+k32$= 90 ± 8 s⁻¹, with $Kd$ = 280 ± 20 μM. In this case, the value of $k32$ is 13-fold higher than $k23,$ and the resulting value of $Kd,app$ = 20 ± 2 μM overestimates the affinity of the initial encounter by nearly 15-fold. When the same mutant interacts with the tripeptide H-D-Phe-Pro-Lys-p-nitroanilide (FPK), which differs from FPR for the presence of a Lys residue at the position occupied by Arg in FPR, a similar dependence of the relaxation is obtained with values of $k32$ = 35 ± 3 s⁻¹ and $k23+k32$ = 105 ± 9 s⁻¹, with $Kd$ = 930 ± 80 μM and $Kd,app$= 310 ± 30 μM. Interpretation according to IF under the rapid equilibrium approximation points to differences between the two ligands, with FPR binding more specifically because of a lower $Kd,app$ that would be measured experimentally at equilibrium by fluorescence titrations or calorimetry. Importantly, the 15-fold difference in $Kd,app$ caused by replacement of Arg with the less specific Lys at the P1 position⁶¹ of the ligand is due mainly to $k32$⁠, which measures the rate of conversion of $E′X$ to $EX$ and has nothing to do with the intrinsic binding affinity $Kd$⁠. Hence, both ligands drive the $EX→E′X$ transition with comparable rates, but FPR optimizes binding by better maintaining the complex in the $E′X$ conformation. After the initial complex is formed, FPR induces a tighter fit toward a more stable final complex than the cognate ligand FPK.

Using IF for the analysis of FPR and FPK binding to the active site of the thrombin mutant W215A produces a compelling interpretation of the underlying mechanism and points to a strategy to optimize the affinity further. The pharmacology literature is full of similar interpretations that have informed the design of better lead compounds.^2,28,42,51 However, when the binding interaction of FPR and FPK with the thrombin mutant W215A is studied at a higher temperature (25 °C), a different scenario emerges [Fig. 5(B)]. FPR binds according to IF as seen at 10 °C [Fig. 5(A)], with values of $k32$ = 38 ± 3 s⁻¹, $k23+k32$ = 180 ± 10 s⁻¹, $Kd$ = 230 ± 20 μM and $Kd,app$ = 49 ± 5 μM. The higher values of the rate constants $k23$ and $k32$ are expected at higher temperature, and so are the lower affinities $Kd$ and $Kd,app$ due to a binding interaction that is enthalpically driven. On the other hand, binding of FPK is no longer consistent with IF but with CS, and the associated relaxation decreases from a value of $k12+k21$= 330 ± 30 s⁻¹ to $k12$ = 150 ± 10 s⁻¹, with $Kd$ = 23 ± 2 μM and $Kd,app$ = 51 ± 5 μM. How can the same protein behave according to two irreducible mechanisms like IF and CS under identical solution conditions? How can $E*$ and $E$ preexist in equilibrium for FPK but not FPR? The value of the saturable relaxation at $x$ = 0 gives the rate constant $k32$ for IF and the sum $k12+k21$ for CS. The former measures the rate for the $E′X→EX$ transition, i.e., a property of the complex that may change with different ligands, as seen in Fig. 5(A). The latter measures the rate for the $E*⇄ E$ transitions to reach equilibrium, i.e., a property of the free macromolecule that must be independent of the ligand used. The data in Fig. 5(B) are inconsistent with IF because FPK clearly obeys CS, but they are also inconsistent with CS because FPR does not. A scenario similar to that shown in Figs. 5(A) and 5(B) for FPR and FPK binding to the W215A mutant of thrombin has been observed in P-type ATPases when using metal-fluoride complexes to induce E2P-like states with the aim of studying the events that occur during E2P hydrolysis.⁶² Although binding of BeF_x produces an increase in fluorescence analogous to Pi, it causes an increase in the value of $α(x)$ with $x$ according to IF, whereas Pi induces a decrease according to CS. Again, how can the P-type ATPase behave according to IF and CS under identical solution conditions? The paradoxical divergence in the kinetic behavior of a macromolecule toward two distinct ligands under identical solution conditions is resolved once we abandon the rapid equilibrium approximation.

How different is the behavior of the system in the general case, i.e., without the rapid equilibrium approximation? This question was first addressed by simulations using the complex kinetics of MANT-ADP binding to DnaC⁶³ and then in general mathematical terms for IF [Eq. (11)] and CS [Eq. (12)].⁶⁴ Consider the properties of IF under the general assumptions used for the lock and key mechanism [Eq. (9)] and with the ligand in excess over the macromolecule. Because the system is closed, only two of the three macromolecular species in Eq. (11) are independent and the time evolution of the system is

or

Here the A's are elements of the eigenvectors of the matrix in Eq. (17); the c's are arbitrary constants that satisfy the initial conditions $E0$⁠, $EX0,$ and $E′X0$⁠; and $E∞$⁠, $EX∞,$ and $E′X∞$ are the values of $Et$⁠, $EXt,$ and $E′Xt$ at $t=∞$⁠. The $α$'s are the rates of approach to equilibrium defined as the two non-zero eigenvalues, with changed sign, $α1(x)$ and $α2(x)$ of the matrix in Eq. (17), i.e.,

The sums $konx+koff$ and $k23+k32$ measure the rates at which the exchanges $E+X ⇄ EX$ and $EX ⇄ E′X$ reach equilibrium. They make a symmetric contribution to Eqs. 19(a) and 19(b), i.e., they can be swapped without affecting the behavior of the system. The term $koffk23$ breaks the symmetry and depends on the two rate constants that deplete the $EX$ intermediate in Eq. (11). The functional behavior of the IF mechanism in the general case depends not on the relative rates at which binding $E+X ⇄ EX$ and conformational transitions $EX ⇄ E′X$ reach equilibrium. Rather, it depends on how fast the $EX$ intermediate disappears as a result of ligand dissociation (⁠$koff$⁠) and conversion to the more stable complex (⁠$k23$⁠). The limiting values of the relaxations in Eqs. (19a) and (19b) are easily calculated as

The fast relaxation, $α1(x)$⁠, always increases with $x$ and eventually grows linearly as $x→∞$⁠. In this limit [Eq. (20b)], the relaxation becomes indistinguishable from that describing the simple lock-and-key mechanism because it is dominated by the binding interaction. The slow relaxation, $α2(x)$⁠, also increases with $x,$ but saturates at a value $α2∞=k23+k32$ that reflects the rate at which the $EX ⇄ E′X$ exchange reaches equilibrium [Eq. (20d)]. This limiting value is identical to that seen in the case of IF under the rapid equilibrium approximation [Eq. (14a)]. Under saturating conditions of ligand, the system partitions between its main processes $E+X→EX,$ reflecting formation of the complex, and $EX ⇄ E′X$⁠, reflecting the conformational transition once the complex is formed. This asymptotic value depends on properties of the complex and is expected to be different for different ligands. In the absence of ligand, the two transitions make overlapping contributions to the kinetics that depend on all rate constants in the kinetic scheme [Eqs. (20a) and (20c)], except $kon$⁠. Again, these values are expected to be influenced by the particular ligand under consideration. We also note that the difference between the lower limit of the fast relaxation and the upper limit of the slow relaxation defines an important quantity $ε$ such that

When the mechanism of binding obeys IF, the value of $ε$ can be derived directly by inspection of the plot of the two relaxations. Once the value of $ε$ is known, the rate constant $koff$ is also derived by inspection as the sum of the lower limit of the slow relaxation and $ε$ [Eq. (21b)]. The value of $ε$ ranges from 0 to $koff$⁠, and its importance will become apparent in Sec. VII.

Both relaxations in the IF mechanism feature a monotonic increase with the ligand concentration $x$⁠, as shown by the derivative

Therefore, IF cannot account for the kinetic profile shown in Fig. 4(A), even in the general case. It is instructive to show how Eqs. (19a) and (19b) change in the rapid equilibrium approximation, which is obtained by assuming that $konx+koff≫k23+k32$ for all values of $x$⁠. Simple algebra yields the expressions

Separation of the time scales for binding (⁠$konx+koff$⁠) and conformational change (⁠$k23+k32$⁠) produces the expected expressions: the fast relaxation is identical to the lock-and-key model [Eq. (10)], and the slow relaxation is identical to Eq (14a). The lower limit of the fast relaxation separates widely from the asymptotic upper limit of the slow relaxation and the value of $ε$ increases. Hence, a necessary condition for invoking the rapid equilibrium approximation is that the two relaxations be widely separated, as seen for the case of glucose binding to glucokinase, shown in Fig. 3(A). However, the condition is not sufficient, as we will see in Secs. VI and VII. In the general case, Eq. (11) depends on four independent parameters that can be resolved only when both $α1(x)$ and $α2(x)$ are measured experimentally. If only the slow relaxation is accessible to experimental measurements as a hyperbolic increase as a function of $x$⁠, then none of the four independent values of the IF mechanism can be resolved unequivocally from Eqs. (20c) and (20d). Parameters estimated through the simplified Eq. (23b) under the rapid equilibrium approximation may not give a correct interpretation of the underlying kinetic mechanism.

The properties of the mechanism of CS [Eq. (12)] in the general case have been discussed only recently,⁶⁴ notwithstanding their mathematical simplicity. Under the assumption that the system is closed and the ligand is in large excess over the macromolecule, only two of the three macromolecular species in Eq. (12) are independent, and the time evolution of the system is

or

The parameters in Eqs. (25a)–(25c) are analogous to those for IF in Eqs. (18a)–(18c), and the relevant expressions for the relaxations $α$'s are

The similarity of Eqs. (26a) and (26b) to Eqs. (19a) and (19b) is evident. There is a symmetric contribution from the sums of rate constants defining the relaxation to equilibrium of the two separate processes $E* ⇄ E$⁠, reflecting the conformational transition, and $E+X ⇄ EX$⁠, reflecting binding, whereas the term $k21konx$ makes an asymmetric contribution that depends on rate constants that deplete the intermediate species $E$ in the reaction scheme [Eq. (12)]. Notably, this term is a constant for IF (⁠$koffk23$⁠), but a function of the ligand concentration $x$ for CS (⁠$k21konx$⁠), which presages important differences in the kinetic properties of CS compared to IF. The expectation that CS may be mathematically and functionally more versatile than IF is at odds with the documented paucity of systems that behave according to CS based on the rapid equilibrium approximation.^28,33 Yet, the unique features of CS are immediately evident once we consider the limiting values of the two relaxations, i.e.,

The upper limit of the fast relaxation, $α1(∞)$⁠, grows linearly as $x→∞$⁠, as seen for IF [Eq. (20b)]. The analogous limit for the slow relaxation, $α2(∞)$⁠, saturates at a value equal to $k12,$ defining the rate constant for the $E*→E$ transition, which is a property of the free macromolecule. This is somewhat counterintuitive, because the behavior of the system under saturating concentrations of ligand is expected to depend on properties of the bound forms of the macromolecule. Indeed, the value of $α2(∞)$ for IF reflects properties of the complex [Eq. (20d)]. Furthermore, the value of $α2(∞)$ for CS unequivocally defines one of the rate constants in the kinetic mechanism [Eq. (12)], which can be derived from inspection of the relaxation plot, without the need for data analysis. The lower limits of the two relaxations, $α1(0)$ and $α2(0)$⁠, can only be assigned unequivocally once the relative values of $koff$ and $k12+k21$ are known. The larger value is assigned to the fast relaxation, and the smaller one to the slow relaxation. This is a consequence of the term $k21konx$ in Eqs. (26a) and (26b), no longer contributing to the two relaxations when $x→0$⁠. When $koff>k12+k21$⁠, the fast relaxation has a lower limit $α1(0)$ that depends entirely on properties of the complex, and the slow relaxation has both limits, $α2(0)$ and $α2∞$⁠, which depend entirely on properties of the free macromolecule. When $koff<k12+k21$⁠, the lower limit of the fast relaxation no longer reflects properties of the complex but rather of the free macromolecule, whereas the value of $α2(0)$ reflects properties of the complex and no longer of the free macromolecule. In this case, the behavior of the slow relaxation reflects properties of the complex when $x→0$ and the macromolecule is mostly in the free form, but it reflects properties of the free macromolecule when $x→∞$ and the macromolecule is mostly in the bound form. The peculiar contribution of the term $k21konx$ in Eqs. (26a) and (26b) becomes fully apparent when the dependence of each relaxation is studied as a function of the ligand concentration [Eqs. (27a)–(27d)].

The fast relaxation of CS is expected to increase as a function of $x$ from the values of the limits in Eqs. (27a) and (27b). Indeed, the derivative

A quite different scenario is observed for the slow relaxation, whose behavior depends on the sign of the expression $koff−k12$⁠. The derivative

We have already established [Eq. (27c)] that the smaller value between $koff$ and $k12+k21$ is assigned to $α20,$ and this introduces variability in the relative values of $α2(0)$ and $α2∞$⁠. The relaxation $α2(x)$ always decreases with $x$ when $α20=k12+k21$⁠, but it may also decrease when $koff>k12$⁠. In addition, the relaxation increases with $x$ whenever $koff<k12$ and is independent of $x$ in the special case when $koff=k12$⁠. The different behaviors of the slow relaxation of CS in the general case are illustrated in Fig. 6, where the value of $k12$ is changed while keeping $koff$ constant [Fig. 6(A)], or vice versa [Fig. 6(B)]. The examples in Fig. 6(B) offer a possible solution to the paradox presented in Fig. 5. The data at 10 °C [Fig. 5(A)] show the two ligands FPR and FPK obeying a single relaxation that increases hyperbolically with $x$ and converging toward a similar upper asymptote. This kinetic profile would be interpreted with IF under the rapid equilibrium approximation, but is also consistent with CS in the general case. The value of $α2∞=k12$ [Eq. (27d)] for CS reflects a property of the free macromolecule and should be independent of the ligand used, as indeed observed for FPR and FPK [Fig. 5(A)], and for BeF_x and Pi binding to the P-type ATPase.⁶² The difference between the ligands in the lower asymptote is explained by the difference in the value of $koff<k12$⁠, i.e., the two ligands bind to the thrombin mutant W215A with the same mechanism of CS but with different values of $koff$⁠. The conclusion is supported by the data collected at 25 °C [Fig. 5(B)] that again converge toward the same value of $α2∞=k12,$ but differ sharply in the value of $koff$⁠. As this value grows above the value of $k12$ that is independent of the ligand, the condition $koff>k12$ takes hold and inverts the dependence of $α2x$ on $x$ from a hyperbolic increase for FPR to a hyperbolic decrease for FPK. The same conclusion applies to the case of BeF_x and Pi binding to the P-type ATPase: both ligands obey CS but differ in the value of $koff$ that is significantly higher in the case of Pi.⁶² The data in Fig. 5 and the simulations in Fig. 6 demonstrate the importance of interrogating the kinetic behavior of the system with different ligands and under different experimental conditions. Data limited to FPR, even at different temperatures, would have supported IF under the rapid equilibrium approximation as a mechanism of binding to the thrombin mutant W215A. The same conclusion would have been drawn from analysis of FPK binding at 10 °C or analysis of BF_x binding to the P-type ATPase.⁶² 

The relevance of CS in the general case is further documented by the simulations shown in Fig. 6(A) that bear direct relevance to ligand binding to the active site of the clotting factor prothrombin⁶⁵ and its two smaller derivatives, prethrombin-1⁶⁶ and prethrombin-2.⁶⁷ Prothrombin is composed of 579 residues and has a modular assembly [Fig. 7(B)] that comprises the Gla-domain, two kringles, and the protease domain housing the active site.⁶⁸ Binding of the ligand FPR to prothrombin produces a single relaxation, decreasing hyperbolically with the ligand concentration that proves unequivocally the validity of CS as a mechanism of binding [Eq. (28a)] and rules out IF [Fig. 7(A)]. The limiting values in the plot are $α2 0$ = 45 ± 5 s⁻¹, defining the smaller between $koff$ or the sum $k12+k21$⁠, and $α2 ∞$ = 6.9 ± 0.3 s⁻¹ defining the value of $k12$⁠. Cleavage at residue R155 between the two kringles of prothrombin removes the Gla domain and kringle-1 and generates prethrombin-1 [Fig. 7(B)]. FPR binding to prethrombin-1 features a single relaxation independent of ligand concentration [Fig. 7(A)] that again proves unequivocally the validity of CS [Eq. (28c)] and leads to the conclusion that $α2 0=α2 ∞$⁠, or else $koff =k12$= 42 ± 2 s⁻¹. The value compares well to $α2 0$ = 45 ± 5 s⁻¹ measured for prothrombin and suggests that this limit in the slow relaxation of prothrombin likely measures $koff$ and that this rate constant does not change significantly upon removal of the Gla domain and kringle-1. What changes in the transition from prothrombin to prethrombin-1 is the value of $k12$⁠, measuring the rate of opening of the active site in the $E*→E$ transition that increases significantly in prethrombin-1. Hence, removal of the Gla domain and kringle-1 affects the intrinsic $E* ⇌ E$ equilibrium, controlling ligand binding to the active site in the protease domain located > 50 Å away. Removal of kringle-2 from prethrombin-1 further reduces the size of the protein and generates prethrombin-2 [Fig. 7(B)]. In this case, FPR binding produces a single relaxation that increases hyperbolically with the ligand concentration [Fig. 7(A)]. It is possible that the drastic structural perturbation produced by transitioning from prothrombin to prethrombin-2 causes the molecular mechanism of recognition of FPR to change from CS to IF. However, it is reasonable to assume that the mechanism of FPR binding does not change and that the data for prethrombin-2 can also be interpreted with CS to give $α2 0=koff$ = 27 ± 2 s⁻¹ and $α2 ∞=k12$ = 170 ± 10 s⁻¹. The value of $koff$ is similar to that measured for prothrombin and prethrombin-1 and the value of $k12,$ measuring the rate of opening of the active site in the $E*→E$ increases even further compared to prethrombin-1. Hence, FPR binding to prothrombin and its smaller derivatives can be explained in terms of CS, with a value of $koff$ that remains fairly constant and a value of $k12$ that becomes progressively faster as the Gla domain and kringles are removed from the structure. This interpretation is consistent with the conformational plasticity of prothrombin documented by X-ray structural biology^69,70 as the driving force producing long-range perturbation of the active site from the auxiliary Gla domain and kringles.

The versatility of CS and Eq. (26b) is lost under the rapid equilibrium approximation, which is obtained under the assumption $konx+koff≫k12+k21$⁠. Simple algebra yields the expressions

The fast relaxation becomes the familiar expression for the lock-and-key model [Eq. (10)] and the slow relaxation becomes identical to Eq. (14b), which always decreases as a function of $x$⁠. The important conclusion from this analysis is that a simple distinction between IF and CS is only possible under the rapid equilibrium approximation. In the general case, CS produces kinetics that are incompatible with IF because $α2∞<α2(0)$⁠, thereby explaining experimental data that can only be interpreted with CS. In addition, CS is consistent with experimental data where $α2∞>α2(0)$ that might be considered unequivocal proof of IF. Finally, CS also accounts for the case of a relaxation that is independent of the ligand concentration, i.e., $α2x=α2∞=α2(0)$⁠, that finds no explanation in terms of either IF or CS under the rapid equilibrium approximation.⁷¹ The properties of CS in the general case [Eqs. (26a) and (26b)] have significant practical implications for the analysis of experimental data. The fast relaxation $α1(x)$ has a behavior compatible with both IF and CS, and eventually increases linearly with $x$ because it monitors the effect of ligand binding. The slow relaxation, on the other hand, saturates at high $x$ to values that convey information on the conformational properties of the complex for IF or the free species for CS. When the slow relaxation decreases hyperbolically with $x$⁠, the system unequivocally obeys CS, and IF is necessarily ruled out. The same conclusion applies to the case of a slow relaxation being independent of $x$⁠. On the other hand, when the slow relaxation increases hyperbolically with $x$⁠, the system may obey IF or CS, and neither mechanism is necessarily ruled out. CS is always sufficient as an explanation for the mechanism of ligand binding and is also necessary when a saturable relaxation decreases hyperbolically with $x$⁠. In contrast, IF is never necessary as an explanation for the mechanism of ligand binding and is only sufficient when a saturable relaxation increases hyperbolically with $x$⁠.

The greater versatility of CS compared to IF has direct relevance to enzyme kinetics. Formation of a catalytically active complex according to CS may show a change in the rate-limiting step from trapping the $E$ intermediate at low $x$ to the rate $k12$ for the $E*→E$ transition at high $x$⁠. The richer functional repertoire of CS and its potential overlap with IF was first uncovered using simulations of a more complex kinetic scheme involving both IF and CS.⁶³ Surprisingly, however, the full mathematical complexity of the simple schemes for IF [Eq. (11)] and CS [Eq. (12)] reported above has been documented only recently.^64,72

In its general form, CS is not only more versatile than IF but it includes IF as a mathematical special case. The entire repertoire of kinetic properties of IF can be recapitulated by CS, but the reverse is not true. Any experimental data set consistent with IF can be interpreted with identical mathematical accuracy with CS. This result is unexpected given that IF and CS have long been assumed to be mutually exclusive mechanisms and mathematically irreducible.^{4,5,22,23,28,33} The result is also counterintuitive because IF and CS are two-step reactions where binding and conformational changes are linked in sequence, but in reverse order. Why should binding preceded by a conformational change be functionally more versatile than the reverse sequence of events? And why should it include the reverse order as a special case? Functional complexity seems to depend on where conformational transitions take place in the kinetic mechanism, which generates hierarchy within the same topology of the reaction scheme. The relaxations for the IF and CS in Eqs. (19a) and (19b) and Eqs. (26a) and (26b) can be rewritten as

to better identify rate constants that apply to each mechanism. Because IF only produces relaxations that increase monotonically with $x$ [Eq. (22)], we will prove that this mechanism is a special case of CS under the condition $koffCS<k12$ that also produces relaxations that increase monotonically with $x$ [Eqs. (26b) and (28c)]. Simple algebra shows that Eq. (30a) of IF is mathematically identical to Eq. (30b) of CS for any value of $x$ under the following transformations:⁵⁹