Effects of target binding kinetics on in vivo drug efficacy: k_off, k_on and rebinding

Abstract

Background and Purpose

Optimal drug therapy often requires continuing high levels of target occupancy. Besides the traditional pharmacokinetic contribution, target binding kinetics is increasingly considered to play an important role as well. While most attention has been focused on the dissociation rate of the complex, recent reports expressed doubt about the unreserved translatability of this pharmacodynamic property into clinical efficacy. ‘Micro’‐pharmacokinetic mechanisms like drug rebinding and partitioning into the cell membrane may constitute a potential fix.

Experimental Approach

Simulations were based on solving differential equations.

Key Results

Based on a selected range of association and dissociation rate constants, k_on and k_off, and rebinding potencies of the drugs as variables, their effects on the temporal in vivo occupancy profile of their targets, after one or multiple repetitive dosings, have here been simulated.

Conclusions and Implications

Most strikingly, the simulations show that, when rebinding is also taken into account, increasing k_on may produce closely the same outcome as decreasing k_off when dosing is performed in accordance with the therapeutically most relevant constant [L_max]/K _D ratio paradigm. Also, under certain conditions, rebinding may produce closely the same outcome as invoking slow diffusion of the drug between the plasma compartment and a target‐containing ‘effect’ compartment. Although the present simulations should only be regarded as a ‘proof of principle’, these findings may help pharmacologists and medicinal chemists to devise ex vivo and in vitro binding kinetic assays that are more relevant and translatable to in vivo settings.

Abbreviations

k_on, k_off

second‐ and first‐order rate constants for association and dissociation

k_a, k_e

first‐order rate constants for the inflow and elimination/clearance of the drug

k_eo

first‐order rate constant for the drug's equilibration between the plasma and effect compartments

[L], [L_e]

concentration of drug in plasma (with [L]_t at given time point) and in tissue ‘effect’ compartment (with [L_e]_t at given time point)

k_on.[L]_t (or [L_e]_t for two‐compartment model)

the rate coefficient that refers to the actual formation of new RL complexes at any time point

[L_max] (= C_max in pharmacokinetics)

maximal concentration of drug near its target

[RL], [RL_max], [RL_max]_eq

concentration of drug–target complex, its maximal value after dosing and theoretical maximal value in case of instant equilibrium binding

Diss t_1/2

dissociation t _1/2 (=0.69/k_off)

Introduction

Drug candidates have traditionally been optimized in terms of efficacy and potency (i.e. pharmacodynamic, PD, properties), while the duration of their in vivo pharmacological activity was linked to how their free concentration changes over time (i.e. a pharmacokinetic, PK, property). Except for irreversible/covalent binding mechanisms, the lifetime/residence time of drug–target complexes (also a PD property; Copeland et al., 2006) was deemed to be too short to significantly contribute to the overall assessment. Accordingly, only a limited number of studies (e.g. Leysen and Gommeren, 1984; Vanderheyden et al., 2000) paid attention to binding kinetics until its importance was highlighted by the seminal reviews by Swinney (2004) and Copeland et al. (2006). Their viewpoint is now corroborated by an increasing amount of observations (e.g. Bradshaw et al., 2015). Also, an important theoretical argument in favour of moving from binding affinity to binding kinetics is that, while drug concentrations are kept constant in in vitro experiments (so that equilibrium binding can be reached after a sufficiently long incubation time), this is not the case in the human body because it is an ‘open system’ in where the concentration of free drug changes over time (Copeland, 2016).

The role of binding kinetics has become increasingly recognized and is now covered in many review articles and even in a recent book (Keserü and Swinney, 2015). Most attention therein has been focused on slow drug dissociation (i.e. a low dissociation rate constant, k_off) because this parameter is widely accepted to represent a key property of many marketed drugs. In this respect, simulations revealed that the clinical action of a drug lasts longer if it dissociates slower from its target than its in vivo PK elimination (Vauquelin and Van Liefde, 2006; Tummino and Copeland, 2008; Lu and Tonge, 2010; Dahl and Akerud, 2013). Yet this focus on k_off may be too restricted. First, some authors have also drawn attention to the utility of fast drug association (i.e. a high association rate constant, k_on) in clinical therapy (Yin et al., 2013; Schoop and Dey, 2015). More disturbingly, Dahl and Akerud (2013) recently even expressed concern about the relevance of including k_off measurements in lead optimization programmes. Indeed, they found that the dissociation of many drugs and drug candidates from their target proceeds faster than their in vivo PK elimination, not slower. Based thereon, they concluded that PK usually prevails over binding kinetics.

In compliance with Occam's razor, this conclusion is based on the simplest PD and PK models according to which drug–target interactions are represented as a reversible single‐step bimolecular process and PK elimination rates rely on drug concentrations in the plasma, not in the vicinity of the target itself (Figure 1A). As a potential alternative explanation for why a drug's therapeutic effect often lags behind its plasma concentration, a more complex ‘two‐compartment’ PK model with slow equilibration of the drug between the plasma compartment and a hypothetical target‐bearing ‘effect compartment’ has already been introduced some time ago (Holford and Sheiner, 1982) (Figure 1A). To illustrate this lag, the drug's effect is often represented as a function of its plasma concentration (Danhof et al., 2008; Gabrielsson et al., 2009) where it gives rise to a counterclockwise hysteresis loop with a depressed effect when the plasma concentration initially increases and an uplifted effect when the concentration subsequently declines. With respect to the PD models, multi‐step ‘induced fit’‐type binding mechanisms are now rather considered to play a key role for achieving high affinity and clinical efficacy of many drugs (Copeland, 2016). Yet despite of this increased complexity, radioligand washout experiments (Figure 1B and Vauquelin, 2012) most often yield mono‐exponential dissociation curves so that the overall dissociation process can still be satisfactorily described by a ‘macroscopic’ k_off (Neubig et al., 2003; Tummino and Copeland, 2008; Vauquelin et al., 2016). Regardless of the binding mechanism, it is of note that such in vitro experiments still essentially focus on obtaining ‘genuine’ k_off values and, to this end, an excess of unlabelled competing ligand has been routinely added to the washout medium to prevent ‘rebinding’ of the radioligand (Figure 1B).

Figure 1.

Schematic representation of compartment models (A), kinetic parameters for rebinding (B) and dosing paradigms that have been used in previous simulations (C). (A) Living organisms are ‘open systems’ wherein, after a single dosing, the concentration of free drug near the target first increases and then declines. In the simplest ‘one‐compartment’ body model, the first‐order rate constants, k_a and k_e, correspond to the inflow/input and elimination/clearance of the drug respectively. Those govern how the concentration of free drug [L] near the target, R, changes over time. Binding proceeds according to a reversible bimolecular mechanism; k_on and k_off are the association and dissociation rate constants thereof. The more elaborate ‘two‐compartment’ body model allows slow equilibration of the drug between the plasma compartment and a hypothetical target‐bearing ‘effect compartment’ within tissues (Holford and Sheiner, 1982; Gabrielsson et al., 2009). The rate by which the drug transfers between both compartments is governed by a single first‐order constant, k_eo, for the present simulations (for the sake of simplicity). When the equilibration of free drug between both compartments is sufficiently slow, its concentration in the effect compartment, [L_e], will fluctuate at a slower pace than its concentration, [L], in the plasma compartment. (B) Top left: inherent to the rate constants, k_on and k_off, that govern a reversible mass action‐type bimolecular binding mechanism, is the assumption that the drug and the target molecules are homogenously distributed in the solute and also that they are able to reach and leave one another by free three‐dimensional diffusion. Top right: in biological systems, targets like ion channels, receptors and enzymes are embedded in cell membranes and/or are present within the cell at high local density (Copeland et al., 2006). Cell walls and other obstacles that hinder free three‐dimensional diffusion of the drug (for example, as shown here, receptors that face a synaptic cleft) will promote drug rebinding; that is, the ability of freshly dissociated drug molecules to experience several binding–unbinding sequences to the original target and/or to those nearby before drifting away (Perry et al., 1980; Goldstein et al., 1989; Goldstein and Dembo, 1995; Coombs and Goldstein, 2004; Gopalakrishnan et al., 2005). Research in different life sciences disciplines gave rise to the same mathematical formulation of this mechanism: namely, k_on is replaced by the ‘effective’ forward rate coefficient k_f = k_on/(1 + k_on.[R]/k), and k_off is replaced by the ‘effective’ reverse rate coefficient k_r = k_off/(1 + k_on.[R]/k). These reformulated equations do not affect the drug's K _D. k_f and k_r are not constants because [R] varies with the extent of target occupancy; the k_on.[R]/k product, which constitutes a metric for rebinding, is maximal when the targets are all free (i.e. when [R] = [R_tot]) and decreases when the occupancy increases (see the Supporting Information Fig. S1). The parameter, ‘k’, depends on the free drug's diffusion rate and on the geometric characteristics of the target's micro‐environment (Coombs and Goldstein, 2004; Vauquelin and Charlton, 2010). Bottom left: the ability of unlabelled competitive ligands/drugs to speed up the dissociation of a pre‐bound radioligand in a concentration‐dependent fashion (Vauquelin and Van Liefde, 2012) represents the most commonly reported experimental manifestation of rebinding. To this end, targets are pre‐incubated with radioligand and subsequently (preferably with an intermediary wash step to remove free radioligand) incubated in a fresh washout medium alone (lane a) as well as in a medium containing an excess of unlabelled competitive ligand (lane b) for different periods of time after which binding is measured (Vauquelin, 2012). A similar approach is theoretically also applicable when binding is measured by spectrophotometrically such as in BRET‐ (Robers et al., 2015) and FRET‐based assays. Bottom right: simulated decline of radioligand binding when the washout is performed in the presence of an excess of unlabelled ligand (red curve, no rebinding – reflecting the genuine k_off of the radioligand) or in a naïve washout medium (black curves account for the increasing values of the rebinding factor, k_on.[R_tot]/k, a metric for the radioligand's susceptibility to experience rebinding; see also in the Supporting Information Fig. S1). Dots refer to simulated data points. (C) Venn diagrams picturing the different in vivo dosing paradigms that were used to compare how the target occupancy by different drugs changes over time. Some of the earlier published simulations relied on a ‘constant [L_max]’ paradigm such as in Figure 2D–F. The peak concentration of all the drugs (denoted as [L_max] in this article) was the same whatever their binding rate constants and their K _D. Other simulations relied on a, from the clinical perspective more representative, ‘constant [L_max]/K _D ratio’ paradigm such as in the ensuing figures of this article. The peak concentration of the drugs was adapted to permit the same maximal occupancy of the target by all the drugs (in the supposition that association and dissociation are too fast for binding kinetics to play any role). Both diagrams overlap when only drugs with the same K _D are compared with one another such as in Gabrielsson et al. (2009).

Because rebinding has already been covered in previous review articles (Vauquelin, 2010; Vauquelin and Charlton, 2010; Swinney et al., 2015; Vauquelin, 2015), it is only its most essential characteristics that are mentioned below and illustrated in Figure 1B. Its initial definition referred to the establishment of a new mass action‐type equilibrium binding between free targets and freshly dissociated drug molecules, implying that those are able to disperse rapidly all over the washout medium. However, subsequent research in different life sciences disciplines led to a distinct conclusion, that is, that it is a highly localized process that takes place at the (sub)microscopic scale because of the presence of obstacles that hinder free three‐dimensional diffusion of drug molecules away from their targets along with a high local density of those targets. Those morphological properties prompt freshly dissociated drug molecules to go through multiple encounters with their initial target and/or targets nearby before drifting further away (Figure 1B). This will result in prolonged ‘apparent’ target occupancy (Figure 1B). The morphological complexity of intact cells and tissues suggests that such ‘hindered diffusion’‐related rebinding is commonplace in vivo. Indeed, the cell represents a confined space (Copeland, 2016), and in intact tissues, receptors can be concentrated in microdomains of the cell membrane (Pike, 2003) and also most often face cavities with little convective stirring, such as synapses and interstitial spaces. Early on, ex vivo experiments have already highlighted the physiological relevance of drug rebinding, and special attention was also drawn to the need to use ‘still‐living’ brain slices in order to observe this phenomenon (Perry et al., 1980; Sadée et al., 1982; Frost and Wagner, 1984; Gifford et al., 1998). Because they mimic the complexity of intact tissues to a reasonable extent, confluent plated cell monolayers can be used as surrogate model systems to study rebinding phenomena (Spivak et al., 2006). Using such recombinant cell monolayers, rebinding of several radiolabelled antagonists to their receptors has already been documented (see Vauquelin and Charlton, 2010). Interestingly, the radioligands that we tested in the washout experiments did not all experience rebinding to the same extent. This disparity is clearly illustrated by the two examples shown in the Supporting Information Fig. S2 and commented upon in the Discussion section. Hence, although rebinding is considered to constitute a nuisance when the purpose is to determine genuine dissociation rates of a drug in, for example, radioligand washout experiments (Figure 1B), it may be regarded as representing a missing link between the measured k_off values in vitro and the duration of the drug's therapeutic effect in vivo (Zhang, 2015).

Although the potential contribution of hindered diffusion‐based rebinding to longer‐lasting target occupancy in vivo was already succinctly addressed elsewhere (Vauquelin, 2010, 2015; Swinney et al., 2015), the present study provides for the first time an extended, systematic evaluation of this factor. The present simulations, in where administration/dosing of the drug is performed daily and in accordance with the therapeutically most relevant paradigm (Figure 1C), reveal that invoking rebinding may prolong its therapeutic effect in closely the same way as by decreasing k_off (i.e. increasing the lifetime of the drug–target complex itself) as well as by invoking slow diffusion of the drug between the plasma compartment and a target‐containing ‘effect’ compartment. Moreover, whereas review articles about binding kinetics essentially focus on k_off, the present findings also suggest that the association rate, k_on, merits closer attention in drug design because of its important contribution to robust rebinding.

Methods

Equations for a ‘closed system’

A closed system refers here to a system in where the total drug and target concentrations remain steady with time. We will here focus on the simplest mechanism of drug–target complex, LR, formation, that is, a reversible bimolecular interaction that obeys the law of mass action. The equilibrium K _D, K _D = k_off/k_on (in where k_on is the second‐order association rate constant and k_off the first‐order dissociation rate constant), constitutes the traditional metrics for the drug's potency (Hulme and Trevethick, 2010). The differential equations 1 and 2 (Table 1) were consecutively solved over very small time intervals to simulate how [R] and [RL] change over time for such closed system according to Vauquelin et al. (2001).

Table 1.

Differential equations to follow the time (t)‐dependent changes in target occupancy in a closed system and in an in vivo‐like and open system with dosing according to the constant [L_max] and constant [L_max]/K _D ratio paradigms

General:		for rebinding: append:
d[R]/dt = −d[RL]/dt with [Rtot] = [R] + [RL] for all			1
Bimolecular binding process with [L] constant:
d[RL]/dt = (kon.[R][L] − koff.[RL])		/(1 + kon.[R]/k)	2
In vivo dosing according to constant [Lmax] paradigm, multiplying kon or koff by factor X:
For X.kon:	d[RL]/dt = ({X.kon}.[L].[R] − koff.[RL])	/(1 + {X.kon}.[R]/k)	3
For X.koff:	d[RL]/dt = (kon.[L].[R] − {X.koff}.[RL])	/(1 + kon.[R]/k)	4
In vivo dosing according to constant [Lmax]/K D paradigm
X.kon:	d[RL]/dt = ({X.kon}.{[L]/X}.[R] − koff.[RL])	/(1 + {X.kon}.[R]/k)	5
X.koff:	d[RL]/dt = (kon.{X.[L]}.[R] − {X.koff}.[RL])	/(1 + kon.[R]/k)	6
X.koff and X.kon:	d[RL]/dt = ({X.kon}.[L].[R] − {X.koff}.[RL])	/(1 + {X.kon}.[R]/k)	7
Equilibration of the drug between the two compartments
d[Le]/dt = keo.[L] − keo.[Le]			8

Part of equations at the right side account for rebinding. For equation 8, it is assumed that the equilibration of the drug between the two compartments does not affect [L].

Equations for a single‐compartment in vivo setting for different dosing paradigms

Living organisms are ‘open systems’ wherein, after a single dosing, the concentration of free drug near the target, [L], first increases and then declines. In the simplest ‘one‐compartment’ body model (Figure 1A), this bell‐shaped pattern can be described by the Bateman function (Garret, 1994), that is,

$$\left[\mathrm{L}\right]=\mathrm{f}.\left({\mathrm{k}}_{\mathrm{a}}/\left({\mathrm{k}}_{\mathrm{a}}-{\mathrm{k}}_{\mathrm{e}}\right)\right).\left({\mathrm{e}}^{‐\mathrm{ke}.\mathrm{t}}-{\mathrm{e}}^{‐\mathrm{ka}.\mathrm{t}}\right).$$

The first‐order rate constants, k_a and k_e, correspond to the inflow and elimination/clearance of the drug respectively. They remain the same throughout this study (i.e. 0.0115 and 0.00575 min⁻¹, for t _1/2 = 60 and 120 min). The maximal concentration of free drug near the target, [L_max] (denoted as C_max in PKs), is attained 120 min after dosing, and because only the amplitude of those plots differs, [L_max] and the concentration of free drug at each time point, [L]_t, will thus vary alike. The parameter, ‘f’, in the Bateman function accounts for the drug's dose, bioavailability and volume of distribution. For the present simulations, f was set to obtain a value of [L_max] that allows the desired amount of maximal target occupancy, [RL_max]_eq, under instant equilibrium conditions (i.e. via [L_max] = K _D/(([R_tot]/[RL_max]_eq) − 1). Of note is that the value of [RL_max]_eq and its ability to be attained at the same moment as [L_max] are only theoretical. In practice, the actual ‘peak binding’, [RL_max], will be less and will be attained later on because the association and dissociation events are not instantaneous.

Simulations that take account of binding kinetics to mimic how [L] and [RL] change over time have already been performed. In some of the comparative studies, [L_max] was kept the same irrespective of their binding kinetic parameters and K _D, while in others, it was the [L_max]/K _D ratio that remained the same for all the drugs (Figure 1C). As illustrated in the Results section, the dosing paradigm has an important effect on how k_on and k_off affect how [RL] evolves over time. Equations 3 and 4 in Table 1 apply when dosing complies with the constant [L_max] paradigm (i.e. mimicking an in vivo situation in where the dose is kept the same irrespective of the ligand's K _D), and equations 5 to 7 in Table 1 apply when dosing complies with the constant [L_max]/K _D ratio paradigm (i.e. mimicking an in vivo situation in where the dose varies in par with the K _D of each drug in question). For the first paradigm, changing k_on or k_off will affect the drug's K _D but not [L_max] (and thus also not [L]_t). Accordingly, changing k_on triggers an equivalent change of the k_on.[L]_t product, that is, the rate coefficient that refers to the actual formation of new RL complexes at any time point. For the second paradigm, changing k_on changes K _D, [L_max] and [L]_t in the opposite way. Consequently, the k_on.[L]_t product remains unchanged. On the other hand, changing k_off now triggers a similar change of the k_on.[L]_t product.

Equations for a two‐compartment in vivo setting

A two‐compartment model with slow equilibration of the drug between the plasma compartment and a hypothetical target‐bearing ‘effect compartment’ has been introduced (Holford and Sheiner, 1982) to explain why a drug's therapeutic effect often lags behind its plasma concentration (Figure 1A). The rate by which the drug transfers between both compartments is here governed by a single constant, k_eo. The Bateman function still accounts for concentration of free drug in the plasma compartment, [L], while its concentration in the effect compartment, [L_e], is calculated by solving equation 8 in Table 1. Replacing [L] by [L_e] in equations 5 to 7 allows the temporal evolution of [RL] to be simulated.

Equations for rebinding

The equations that govern hindered diffusion‐related rebinding have already been extensively documented upon in a previous review article (Vauquelin and Charlton, 2010). In short (more explicit information is provided in the legend of Figure 1B), k_on is replaced by the ‘effective’ forward rate coefficient (k_f = k_on/(1 + k_on.[R]/k) and k_off by the ‘effective’ reverse rate coefficient (k_r = k_off/(1 + k_on.[R]/k). The robustness of rebinding at any level of target occupancy, k_on.[R]/k, is maximal when all the targets are free, that is, when [R] = [R_tot]. This maximal value will be referred to as the ‘rebinding factor’. k_on.[R]/k can then be calculated by multiplying this rebinding factor by [R]/[R_tot]. As shown in the Supporting Information Fig. S1, k_f and k_r are lowest when all the targets are free and do gradually increase till they equal k_on and k_off when all the targets are occupied. Accordingly, rebinding is more prominent at low than high levels of target occupancy. To account for rebinding, the differential equations in Table 1 were appended as indicated.

Relevance of kinetic parameters and the rebinding factor

With the chosen order rate constants for the inflow and elimination/clearance of the drugs (k_a = 0.0115 and k_e = 0.00575 min⁻¹), the evolution of [L] with time is very similar as in Dahl and Akerud (2013). Also, the chosen range of k_on (1.10⁶ to 1.10⁸ M⁻¹·min⁻¹) and k_off values for the binding process applies to the majority of the drugs that were investigated by those authors (Figure 6 in their publication). In accordance with extrapolations from intact cell‐based radioligand washout experiments (please see the Supporting Information Fig. S2 for extremes), the rebinding factor, k_on.[R_tot]/k, was allowed to vary between 3 and 100 in the present simulations.

Figure 6.

Effect of rebinding and k_off after repeated dosings. Simulated [RL] versus time plots by drugs with different k_off (vertical, corresponding to Diss t _1/2 from 30 to 1000 min) according to the constant [L_max]/K _D ratio paradigm: effect of rebinding (horizontal, rebinding factors 10 and 30) on repeated daily administrations of the same dose (differential equations 5 and 6 in Table 1). k_on (=1.10⁷ M⁻¹·min⁻¹) remains constant; free ligand parameters, k_a and k_e, are the same as in Figure 2 and [L_max] = 9 × K _D for all drugs. After each dosing, [L] is adapted to include the free drug that remains 24 h after the previous dosing. Color code for the consecutive dosings is given in panel (A). If the occupancy profile no longer changes after a given day, then the curves after that color‐coded day are not shown. occup, occupancy.

Results

Simulations without rebinding

After a recapitulation about how [RL] changes over time when [L] decreases exponentially, subsequent simulations show how [RL] evolves with time when the [L] versus time plots adopt a bell‐shaped pattern, such as after in vivo dosing. Besides the effect of the drug's k_on and k_off thereon, attention is (for the first time) also paid to the different dosing paradigms that have previously been used for related previous studies.

For a simple reversible bimolecular binding mechanism, [RL] will decline mono‐exponentially with time when free ligand is abruptly removed or prevented from associating (Figure 2A). Simulated data in Figure 2B, C show that this no longer applies when [L] also declines mono‐exponentially. Here, even when the dissociation is very fast (i.e. for high k_off), [RL] declines slower than the drug's elimination because of the hyperbolic relationship between [RL] and [L] (Vauquelin and Charlton, 2010). An initial lag phase can be observed, especially when [RL] is high at the start. In agreement with earlier findings (Vauquelin and Van Liefde, 2006; Tummino and Copeland, 2008; Lu and Tonge, 2010; Vauquelin and Charlton, 2010; Dahl and Akerud, 2013), Figure 2A–C also shows that slow dissociation only substantially prolongs the occupancy when L is eliminated more swiftly.

Figure 2.

Simulated target occupancy versus time plots by drugs with different binding kinetics upon washout of free drug (panel A), exponential decrease of the free drug concentration (panels B, C) and after a single administration in vivo according to the constant [L_max] paradigm (panels E, F) and the constant [L_max]/K _D ratio paradigm (panels G‐I). Simulations are based on solving the differential equations 1 to 7 in Table 1 as earlier described (Vauquelin et al., 2001), and occupancy curves are based on 50 – not presented – data points. Diss t _1/2 values (provided in the panels) are in min and k_on values in M⁻¹·min⁻¹. (A) Simulated target occupancy, [RL], in washout conditions without rebinding in where, after removal of free drug, L, pre‐formed LR complexes are exposed to a naïve medium for different time intervals (abscissa) till 24 h. [RL] is 90% of [R_tot] at the start and decreases mono‐exponentially. The rate thereof is independent of the initial target occupancy. Other parameters: k_on = 1.10⁷ M⁻¹·min⁻¹. (B, C) The same representation and drugs as in panel (A) but [L] decreases mono‐exponentially with t _1/2 of 120 min. Black solid line stands for theoretical continuous equilibrium; the same color code as in panel (A) applies to the drugs. At the start, [RL] amounts 50 or 90% of [R_tot], and [L] and [RL] are at equilibrium. To facilitate comparison, [L] (dotted line) is normalized to reach the same apex as [RL]. (D) Simulated [L] versus time plots after in vivo bolus administration. [L] evolves with time during 24 h according to the Bateman function (Equations for a single‐compartment in vivo setting for different dosing paradigms section) with k_a = 0.0115 min⁻¹ (t _1/2 = 1 h) for inflow and k_e = 0.00575 min⁻¹ (t _1/2 = 2 h) for elimination. ‘f’ (=12.4 nM) was adjusted to yield [L_max] = 9 × K _D (for 90% occupancy at equilibrium) of a drug with k_on = 1.10⁷ M⁻¹·min⁻¹ and k_off = 0.0069 min⁻¹ (Diss t _1/2 = 100 min). (E, F) Simulated [RL] versus time plots after in vivo bolus administration of different drugs according to the constant [L_max] paradigm (equations 3 and 4 in Table 1). The drug that was utilized for the curve in panel (D) serves as ‘median’ here (solid black line). ^&The other drugs have different k_on or k_off values; those values are normalized with respect to this median. In accordance with the present dosing paradigm, the [L] versus time curve shown in panel (D) (and thus also the value of f) applies to all the drugs here. Open circles in panel (F) refer to theoretical [RL_max]_eq values when they are attained. (G, H) Simulated [RL] versus time plots according to the constant [L_max]/K _D ratio paradigm (differential equations 5 and 6 in Table 1). All parameters are the same as for panel (E) (color code as inserted in panel E) except that ‘f’ is now adapted to yield [L_max] = 9 × K _D (for 90% occupancy at equilibrium) for all the drugs. Panel (G): all curves overlap when changing k_on only. Panel (H): effect of changing k_off only. (I) Alternative, [RL] versus [L] representation of the occupancy data shown in panel (H). [L] reaches the same apex for all drugs when expressed in K _D units. occup, occupancy; equil., equilibrium.

Simulations where [L] varies with time according to the Bateman function, with a rapid increase followed by a slower decline such as after in vivo dosing (Figure 2D), are more relevant from the clinical perspective. In such studies, most attention has been focused on how k_off affects the evolution of [RL] over time. Here also, it was noticed that k_off has to drop below k_e to produce longer‐lasting occupancy (Tummino and Copeland, 2008; Vauquelin, 2010, 2015; Dahl and Akerud, 2013). On the other hand, such studies also led Yin et al. (2013) to conclude that k_on also contributes to important pharmaceutical properties in ‘subtle but different ways’. However, only scant attention has hitherto been paid to the fact that such studies were not all based on the same dosing paradigm. In some, [L_max] (and thus the dose) was kept the same irrespective of the drug's K _D (Copeland et al., 2006; Tummino and Copeland, 2008; Yin et al., 2013), while in others, [L_max] was allowed to vary in par with the K _D of each drug in question; that is, the [L_max]/K _D ratio was kept constant (Dahl and Akerud, 2013; Vauquelin, 2010, 2015). The next simulations aim to provide a better insight into the potential effect of these dosing paradigms on how k_on and k_off affect the occupancy pattern.

Simulations for the constant [L_max] paradigm (in where the [L] vs. time plot shown in Figure 2D applies to all the drugs tested) show that increasing k_on or decreasing k_off produces a quite comparable increase in peak occupancy (Figure 2E). This is to be expected because both actions decrease the drug's K _D. In line with the different effect of k_on and k_off on the equations that govern the occupancy profile (please see equations 3 and 4 in Table 1 and the Equations for a two‐compartment in vivo setting section for further explanation), changing k_on or k_off will not affect the occupancy profile in exactly the same way. Varying k_on has the largest effect on the initial ascending phase as shown by the already high levels of occupancy at early time points at high k_on (Figure 2F) and the delayed attainment of peak occupancy at low k_on (Figure 2E). Eventually, when peak occupancy only shows up when [L]_t has already dropped substantially, its magnitude, [RL_max], will also start to drop. On the other hand, the effect of k_off is more obvious on the subsequent decline in target occupancy and this decline will become substantially slower when k_off drops below k_e (Figure 2E). A handicap of this dosing paradigm is that, by changing the theoretical peak occupancy, any change in k_on or k_off will also indirectly affect the rate of this decline (see also Figure 2B, C).

This latter source of interference is avoided when the simulations are based on the constant [L_max]/K_D ratio paradigm. The influence of k_on and k_off on the occupancy profile becomes now also clearly distinct. All occupancy curves now overlap when only k_on is allowed to vary (Figure 2G). As outlined in the Equations for a two‐compartment in vivo setting section and shown in equation 5 in Table 1, this invariance stems from the fact that any change of k_on is now cancelled out by the inverse change of [L]_t, so that the k_on.[L]_t product remains unaffected. In contrast, only varying k_off has an obvious effect on the occupancy profile (Figure 2H). In particular, decreasing k_off will not only prolong the occupancy when it drops below k_e but will eventually also sizeably delay peak binding and lower [RL_max]. This stems from the fact that any change of k_off also produces an equivalent change of the k_on.[L]_t product (equation 6 in Table 1). The different repercussions of decreasing k_off will also act together to exacerbate the counterclockwise hysteresis loop in the corresponding occupancy versus [L] representation (Figure 2I).

Interestingly, the occupancy curves remain exactly the same as those shown in Figure 2G when both k_off and k_on are changed alike to yield an iso‐K _D situation in simulations that are based on the constant [L_max] paradigm (Figure 1C). Comparing equations 6 and 7 in Table 1 reveals that this similarity stems from the equivalent contribution of k_on and [L]_t to the k_on.[L]_t product.

Including rebinding in the simulations

The next simulations compare the effect of ‘diffusion‐limited’ rebinding (please see Figure 1B for basic information) and k_off on the occupancy profile under the in vivo dosing conditions that are the most relevant ones from the clinical perspective. To this end, we will from now on only focus on the constant [L_max]/K _D ratio paradigm. Indeed, real‐life dosing is still highly influenced by the drug's potency (Dahl and Akerud, 2013). An additional advantage of this paradigm is that k_on contributes to the robustness of rebinding but does not affect the occupancy profile by itself (Figure 2G). This greatly simplifies the re‐evaluation about how k_on affects the drug's occupancy profile when rebinding is allowed to take place.

As stipulated in the Equations for rebinding in Figure 1B and shown in the Supporting Information Fig. S1, ‘diffusion‐limited’ rebinding will exert a maximal effect on the association and dissociation kinetics when all the targets are free, and this will gradually decline to zero when the occupancy increases. As a convenient single metric for the rebinding intensity, we will use its highest value, k_on.[R_tot]/k, and refer to it as the ‘rebinding factor’. The rebinding intensity at any level of target occupancy can then be easily calculated by multiplying this factor by [R]/[R_tot].

The simulations that are shown in Figure 3A–F compare the influence of this rebinding factor on the occupancy profile of drugs with fast (high k_off, panel A) to very slow dissociation (low k_off, panel F). In all instances, increasing the rebinding factor (here obligatorily by increasing [R_tot] and/or decreasing k to deal with the fact that k_on is kept constant) will gradually prolong the occupancy and, concomitantly, delay peak binding and lower its magnitude. While barely perceptible at high k_off, these phenomena become gradually more pronounced when k_off decreases. Interestingly, decreasing k_off or increasing the rebinding factor by the same amount results in quite similar occupancy profiles (more situations are shown in the Supporting Information Fig. S3). This similarity is also evident when comparing the 24 h post‐administration AUC values. As shown in Figure 3G, the effect of decreasing the dissociation t _1/2 (Diss t _1/2) (abscissa) on the AUCs is effectively counteracted by increasing the rebinding factor (from red to blue) by the same amount, or in other words, rebinding allows the same effect to be obtained with faster dissociation.

Figure 3.

Simulated [RL] versus time plots by drugs with different binding kinetics after in vivo dosing according to the constant [L_max]/K _D ratio paradigm: effect of rebinding (differential equations 5 and 6 in Table 1). Free ligand parameters, k_a and k_e, are the same as in Figure 2 and [L_max] = 9 × K _D for all drugs. (A–F) Effect of increasing the rebinding factor (color code in panel A) for drugs with k_on = 1.10⁷ M⁻¹·min⁻¹ and, from one panel to the next, decreasing values of k_off (given as Diss t _1/2 in each panel). (G) AUC values for the 24 h post‐administration period (in % of continuous full occupancy) as a function of the Diss t _1/2 of each drug (abscissa). Color code for the rebinding factor is given in panel (A). (H) Effect increasing k_on from 1.10⁶ to 1.10⁸ M⁻¹·min⁻¹ for drugs with constant k_off = 0.0069 min⁻¹ (Diss t _1/2 = 100 min) and constant rebinding factor = 10; all curves do overlap. (I–L) Open circles: effect of increasing k_on (which also produces a proportional increase in the rebinding factor; starting from 1 at k_on = 1.10⁶ M⁻¹·min⁻¹) for drugs with constant Diss t _1/2 = 100 min. Solid lines: effect of increasing the Diss t _1/2 for drugs with constant k_on = 1. 10⁷ M⁻¹·min⁻¹ and constant rebinding factor = 10. Log(k_on) (for open circles) and Diss t _1/2 values (for solid lines) are given in each panel. The curve for the ‘median’ drug is shown in panel (H). occup, occupancy.

Similar to the simulations without rebinding (Figure 2G), varying k_on does not affect the occupancy curve either if the rebinding factor is kept constant (Figure 3H). Here again, this can be explained by the ability of [L]_t to cancel out any change of k_on. Yet it is reasonable to assume that the target's concentration and micro‐environment remain fixed in comparative in vivo studies so that the rebinding factor is proportional to k_on. This should allow k_on to modify the occupancy profile via the intermediary of rebinding. Based on a drug with k_on = 1.10⁷ M⁻¹·min⁻¹, k_off = 6.9.10^{− 3} min⁻¹ (Diss t _1/2 = 100 min) and rebinding factor = 10 as median (Figure 3H), Figure 3I–L shows that changing k_on (open circles) or Diss t _1/2 (solid lines) by the same extent affects the occupancy profile in closely the same way. Taken together, while k_on has no effect on the occupancy profile without rebinding (Figure 2G), increasing a drug's k_on will affect this profile in closely the same way as decreasing k_off when rebinding is also taken into consideration. This situation is most likely to apply for tissues in living organisms.

Comparison of the effect of rebinding with that of the two‐compartment PK model

Slow equilibration of the drug between the plasma compartment and a hypothetical target‐containing ‘effect compartment’ was initially invoked to explain that the effect of a drug often lags behind its plasma concentration (Sheiner et al., 1979; Holford and Sheiner, 1982). The next simulations reveal that rebinding may produce closely the same outcome for drugs that dissociate sufficiently slowly.

Simulations in Figure 4 refer to a relatively slow dissociating drug (Diss t _1/2 = 100 min). Its occupancy profile without rebinding or equilibration is shown as control (black curve) in occupancy versus time plots in Figure 4A–C and the corresponding occupancy versus plasma concentration, [L], plots in Figure 4D–F. Although this drug already generates a moderate counterclockwise hysteresis by itself, introducing rebinding (red curves) and slow equilibration (governed by the rate constant, k_eo, blue curves) will both progressively exacerbate this pattern by decreasing peak occupancy as well as by delaying its attainment and prolonging the occupancy. For the sake of comparison, k_eo values were chosen to yield closely the same peak occupancy as with rebinding in each panel. Although not strictly overlapping, the plots for rebinding and slow equilibration are quite comparable in each panel.

Figure 4.

Effect of two‐compartment equilibration and rebinding and on the occupancy profile of a slow dissociating drug. (A–C) Simulated [RL] versus time plots by the same drug (k_on = 1.10⁷ M⁻¹·min⁻¹ and Diss t _1/2 = 100 min, black line as control) and with, from left to right, increasing the rebinding factor (red line) and decreasing the equilibration k_eo (blue line, k_eo values are in min⁻¹ and chosen to yield the same [RL_max] as with rebinding). Evolution of [L] with time in the single (for rebinding) or plasma compartment (for two‐compartment model) is calculated by using the Bateman function for [L_max] = 9 × K _D and with the same values of k_a and k_e, as for Figure 2. Evolution of [L_e] in the effect compartment with time is calculated by using equation 8 in Table 1. (D–F) Simulated [RL] versus (plasma) concentration, [L], plots of the same drug and conditions as in panels (A–C) (the same color code also applies). (G–I) The same [RL] versus time plots with rebinding (red line) as in panels (A–C). To mimic ex vivo experiments, simulations allow free drug to be removed (i.e. [L] is set to 0) either after 240 min or after 840 min (arrows) and the subsequent washout phase to be carried out in a medium only (closed circles; [L] remains 0) or in a medium containing a large excess of competing ligand to block rebinding (open circles; [L] remains 0, and k_on as well as the rebinding factor are also set to 0). occup, occupancy; reb, rebinding.

Such simulations were also carried out for a very fast dissociating drug (Diss t _1/2 = 0.1 min). To better compare the observations with those of the previous drug, the k_eo's and rebinding factors were kept the same. Curves are here also in black for the control situation without rebinding or equilibration. Figure 5A, D depicts how the concentrations of free drug in the plasma, [L] (black curves), and in the effect compartment, [L_e] (coloured curves), change over time. For the control situation, the purely hyperbolical shape of the occupancy versus [L] plot in Figure 5C, F is typical for equilibrium binding. Here again, introducing slow equilibration gives rise to counterclockwise hysteresis loop and progressively exacerbates this pattern by decreasing peak binding, delaying its attainment and prolonging the occupancy (Figure 5B, C). In contrast, no change in the temporal occupancy profile can be perceived when introducing increasing degrees of rebinding instead (Figure 5E). Also, the more sensitive occupancy versus [L] representation reveals very little change, except for the highest rebinding factor and only at very early time points (Figure 5F). Taken together, while (under the presently examined conditions) rebinding and slow equilibration may affect the occupancy profile of a slowly dissociating drug quite similarly (Figure 4), only the latter mechanism will be able to do so for a fast dissociating drug (Figure 5).

Figure 5.

Effect of two‐compartment equilibration (panels A–C) and rebinding (panels D–F) on the occupancy profile of a fast dissociating drug. Drug binding parameters: k_on = 1.10⁷ M⁻¹·min⁻¹ and k_off = 6.9 min⁻¹ (Diss t _1/2 = 0.1 min). [L_max] = 9 × K _D, k_a and k_e are the same as for Figure 2. (A, D) [L] versus time plot (black line in both panels) in the plasma/single compartment and of the [L_e] versus time plot in the effect compartment (coloured lines in panel A) for the same k_eo values as in Figure 4A–C. (B, C) [RL] versus time plots (panel B) and the corresponding [RL] versus [L] plots (panel C) for instantaneous equilibration between the plasma and effect compartments (control, black line) and equilibration with the k_eo values given in panel (A) (coloured lines). (E, F) [RL] versus time plots (panel E) and the corresponding [RL] versus [L] plots (panel F) without rebinding (control, black line) or with the rebinding factors given in panel (D) (coloured lines are only shown in panel (F), where they are sufficiently separated from the control). occup, occupancy; reb, rebinding.

For the slowly dissociating drug, a potential approach to discern between its rebinding and slow equilibration in ex vivo radioligand binding experiments relies on the rationale (Vauquelin and Charlton, 2010; Vauquelin and Van Liefde, 2012) that it is only in case of rebinding, that adding an excess of competing drug in the washout medium should be able to enhance the ‘apparent’ dissociation of the radioligand. This phenomenon is more clearly illustrated in Figure 4G–I. Setting [L] and k_on = 0 at any time after the dosing (to mimic removal of free radioligand and subsequent washout in the presence of an excess of competitor) yields indeed a faster decline in the occupancy than by setting [L] = 0 alone (to mimic removal of free radioligand only). Interestingly, such observations have already been made for real‐life situations (Perry et al., 1980; Sadée et al., 1982; Frost and Wagner, 1984; Gifford et al., 1998).

Repetitive dosing and increasing the dose

The previous simulations only dealt with a single dosing. The next simulations extend this exploration to repeated daily dosings for up to 8 days. Relevant curves are shown in Figure 6; their corresponding AUCs for days 1 to 8 are shown in Figure 7A–L and their peak‐to‐trough occupancy ratios at day 8 in Figure 7M–O (black curve). For fast dissociation and low rebinding, trough occupancy is nearly insignificant so that the curves after consecutive dosings overlap (Figure 6A, B, D, G). The AUCs remain constant, and the peak‐to‐trough ratios remain equally high. Gradually decreasing k_off and/or increasing the rebinding factor will progressively break this pattern, first by enabling a moderate increase in peak occupancy on day 2 only (Figure 6C, E, J) and then also on day 3 and so on. As shown in Figure 6I–L, peak occupancy still increases on day 8 for the slowest dissociation/highest rebinding combinations. This offers an at least partial compensation for the much‐reduced peak binding of these latter drugs at day 1 (as also shown in Figure 3E, F). Also, their AUCs (Figure 7I–L) will gradually increase during the ensuing days. Finally, the appreciably lower daily fluctuations in their target occupancy is also reflected by the smaller peak‐to‐trough ratios (Figure 7N, O).

Figure 7.

Quantitative aspects of the [RL] versus time plots for the drugs shown in Figure 6 (in black, [L_max] = 9 × K _D) and also of similar plots (not shown) for [L_max]/= 27 × K _D (in red) and 90 × K _D (in blue). k_a and k_e are the same as in Figure 2. (A‐L) 24 h AUCs (in % of continuous full occupancy) are presented as a function of the day of treatment (abscissa). Panels are presented in the same order as in Figure 6. The same color code (shown in panel A) in all panels. (M–O) Peak‐to‐ensuing‐trough ratios at day 8 are presented as a function of the drug's Diss t _1/2. Data correspond to [L_max] = 9 ×, 27 × and 90 × K _D. Data points also include peak‐to‐trough ratios for faster dissociating drugs. The same color code (shown in panel A) in all panels.

The effect of increasing the daily dose (i.e. [L_max]/K _D ratio) on the occupancy profile at day 1 is depicted for an extended range of k_off/rebinding combinations in the Supporting Information Fig. S3. Taken together, increasing the dose will increase the AUCs (Figure 7A–L) and decrease the peak‐to‐trough ratios (Figure 7M–O) for all the drug/rebinding combinations. However, the most notable consequence of this initiative resides in the earlier attainment of the maximal AUCs.

Discussion

The present simulations were aimed to evaluate the effect of k_off and k_on, as well as hindered diffusion‐based rebinding of a drug, on how the occupancy of its target changes over time after in vivo dosing. In this respect, the hitherto prevailing review articles on binding kinetics principally focused on k_off. At odds therewith, open system‐based simulations and earlier findings (Bairy and Wong, 2011) led Yin et al. (2013) to conclude that k_off alone does not provide a complete picture and that a relatively fast association rate is needed to design a good competitive inhibitor (sic). Of note is that their simulations were based on the constant [L_max] paradigm (see the Equations for a single‐compartment in vivo setting for different dosing paradigms section). While the present simulations also show that k_on affects the occupancy profile under such dosing conditions (Figure 2E, F), this is no longer the case when dosing complies with the clinically more relevant constant [L_max]/K _D ratio paradigm where any change of k_on is compensated for by the dosing (Dahl and Akerud, 2013). In contrast, the effect of k_off on the occupancy profile becomes even more pronounced for this latter dosing paradigm (Figure 2G–I). Besides triggering longer‐lasting target occupancy, decreasing k_off does eventually also delay the attainment of peak occupancy and even produce a decline. These comparative simulations thus shed light on an important issue, namely, that the effect of a drug's binding kinetic properties on its occupancy profile needs to be appraised in light of the dosing paradigm that is used.

Further simulations that were based on the constant [L_max]/K _D ratio dosing paradigm revealed that allowing dissociated drugs to bind again to the same or surrounding targets before drifting away (i.e. ‘rebinding’) does affect the occupancy profile in a quite similar (but not strictly identical; see the Equations for rebinding section) way as by decreasing k_off (Figures 3 and 6). Moreover, via the intermediation of such rebinding, increasing k_on becomes now also able to affect the occupancy profile in much the same way as decreasing k_off. This aspect is especially interesting in light of the recent claim by Dahl and Akerud (2013) that many therapeutic drugs have a slower elimination than their dissociation rate, which implies that dissociation kinetics alone should not contribute to the long‐lasting therapeutic effect of these drugs. However, because increasing rebinding may act like a surrogate for decreasing the drug's k_off, the potential limitation of such ‘ineffective’ dissociation rate (and also if the target's local concentration and micro‐environment do not optimally contribute to rebinding) should be offset partially by very large k_on values. In this respect, although the largest k_on that was used for the present simulations is sufficient to embrace most of the drugs that were taken in consideration by Dahl and Akerud (2013), this value is still about 100‐fold less than what is permitted by the diffusion limit. In the same vein, as shown in Figure 2A, B (and also in the Supporting Information Fig. S3 for in vivo dosing), increasing the [L_max]/K _D ratio could also prolong the clinical benefit of such ‘suboptimal’ drugs provided that this does not exacerbate potential detrimental side effects. Taken together, although alternative mechanisms like tissue accumulation and the formation of active metabolites may also contribute, the present simulations thus suggest that prominent rebinding (and if necessary a large k_on) may already be sufficient to reinstate the causal link between slow dissociation and the therapeutic efficacy of ‘borderline’ drugs such as candesartan (Diss t _1/2 of about 2 h vs. elimination t _1/2 from the plasma of about 3.5 h) (Delacrétaz et al., 1995; Fierens et al., 1999) and some of those that were listed by Dahl and Akerud (2013).

Interestingly, not all drugs seem to experience rebinding to the same extent. To deal with this issue, the rebinding factors (i.e. k_on.[R_tot]/k) that were used for the present simulations (from 3 to 100) were chosen to embrace those that were extrapolated from intact cell‐based radioligand washout experiments (for illustrative examples, please see the Supporting Information Fig. S2). Based on ex vivo washout experiments with [³H]‐diprenorphine, Perry et al. (1980) also reported a value of 6. The results presented here suggest it is hazardous to advance numerical values for [R_tot] and k. For instance, the likely heterogenous distribution of the targets at the submicroscopic level, such as the accumulation of receptors in lipid rafts and other microdomains of the membrane (Pike, 2003), constitutes one obstacle for advancing relevant values of [R_tot]. Similarly, k is likely to be a complex function because it depends not only on the drug's local diffusion rate but also on the geometry of the target's local micro‐anatomical environment (Coombs and Goldstein, 2004). The reason for the relatively large variance between the rebinding intensity of distinct drugs is presently unknown, and merits further investigation. In this respect, it is noteworthy that hydrophobic GPCR‐binding radioligands like [³H]‐spiperone and [³H]‐taranabant experience more rebinding than those that display less non‐specific binding to plated cells like [³H]‐candesartan, [³H]‐olmesartan, [³H]‐telmisartan, [³H]‐rimonabant and [³H]‐raclopride (see Vauquelin and Charlton, 2010). Besides examining the drug–target interaction itself (e.g. the influence of ligand structure, target mutation, expression level and micro‐anatomical distribution), it should therefore also be of interest to evaluate the potential effect of membrane lipids as well as of the contiguous ‘unstirred water layer’ (on, for instance, the drug's conformation and mode of approach to the target) on the robustness of rebinding (Sargent and Schwyzer, 1986; Abdiche and Myszka, 2004; Szczuka et al., 2009; Vauquelin and Packeu, 2009; Fotakis et al., 2011; Loftsson, 2012).

Increasing the overall/macroscopic residence time by decreasing k_off and/or increasing rebinding may be impractical for certain therapeutic indications (Copeland, 2010; Núñez et al., 2012) because it will eventually go along with lower peak occupancy, a delay in its attainment and a reduction in the average daily occupancy of the target (quantified by a lower post‐dosing AUC). On the positive side, the occupancy will fluctuate less with time (quantified by a lower peak‐to‐trough occupancy ratio). This may also bring about a higher peak occupancy and AUC after each consecutive dosing until those parameters eventually level off (Figures 6 and 7). This maximal AUC will exceed that of a fast‐dissociating drug, and by administering a higher dose, this parameter increases even more and requires less consecutive administrations to be attained (Figure 7).

The effect of an administered drug often lags behind its plasma concentration. From the early PK viewpoint in where target binding was supposed to reach rapid equilibrium, this delay was attributed to a slow equilibration of the drug between the plasma compartment and a hypothetical target‐containing ‘effect compartment’ within the tissue of interest (Holford and Sheiner, 1982; Derendorf and Meibohm, 1999; Danhof et al., 2008). In view of the now well‐recognized effect of binding kinetics on the in vivo occupancy profile and the increasing awareness about the supplementary effects of rebinding, it was of interest to compare the ability of slow equilibration and rebinding to delay the occupancy profile of different drugs. While only slow equilibration was efficient for the fast dissociating drug, moderate rebinding could already produce a comparable delay for the slow dissociating drug (Figures 4 and 5). These simulations raise the question whether slow equilibration between two compartments can be replaced by rebinding. Slow equilibration may still be important in solid tumours with long distances between the blood capillaries and the target cells and for drugs that slowly cross the blood–brain barrier and cell membranes in general (Smith et al., 2010). On the other hand, the leaky pores in the blood capillaries in many other tissues allow a rapid equilibration between the drug's concentration in the plasma and in the extracellular fluid that is in direct contact with major targets such as GPCRs and ion channels. In this situation, rebinding is likely to play a paramount role, as shown by the ability of unlabelled competitors to accelerate the release of, for example, [³H]‐N‐methyl‐scopolamine from isolated guinea pig atria and of [³H]SCH 23390 from brain slices in which the blood–brain barrier is absent (Lullmann et al., 1988; Gifford et al., 1998).

In conclusion, drug rebinding may be regarded to be a natural consequence of hindered three‐dimensional diffusion because of the morphological properties of our tissues and cells as well as of local target accumulation within cells or on their membranes. Taking account of this phenomenon could therefore be of help to design in vitro binding kinetic assays that are more relevant and translatable to in vivo settings (Cusack et al., 2015; Zhang, 2015). Although the present simulations should only be regarded to provide a ‘proof of principle’, the results obtained suggest that k_on merits more consideration in drug design because of its important contribution to robust rebinding. This parameter is presently often neglected or dismissed in the currently prevailing view about the link between a drug's binding kinetics and its therapeutic efficiency. Incidentally, because increasing k_on and decreasing k_off affect the occupancy profile in much the same way in the presence of rebinding (Figures 3 and 6), it is reasonable to deduce that a high affinity of the drug for its target (i.e. low K _D) has a positive effect on the duration of a drug's in vivo pharmacological activity.

Author contributions

G.V. contributed to all parts of this study.

Conflict of interest

The author declares no conflicts of interest.

Declaration of transparency and scientific rigour

This Declaration acknowledges that this paper adheres to the principles for transparent reporting and scientific rigour of preclinical research recommended by funding agencies, publishers and other organisations engaged with supporting research.

Supporting information

Figure S1 Simulated effect of rebinding on drug dissociation in washout experiments.

Figure S2 Real‐life examples of drug rebinding in washout experiments.

Figure S3 Effect of dosing, koff and the rebinding factor on the target occupancy by different drugs.

Supporting info item

Vauquelin, G. (2016) Effects of target binding kinetics on in vivo drug efficacy: k_off, k_on and rebinding. British Journal of Pharmacology, 173: 2319–2334. doi: 10.1111/bph.13504.