On the relationship between inhibition and receptor occupancy by nondepolarizing neuromuscular blocking drugs
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Hoshino and Furutani Theoretical Biology and Medical Modelling (2021) 18:15
https://doi.org/10.1186/s12976-021-00147-w
RESEARCH Open Access
On the relationship between inhibition
and receptor occupancy by nondepolarizing
neuromuscular blocking drugs
Hikaru Hoshino1* and Eiko Furutani1,2
Abstract
Background: Nondepolarizing neuromuscular blocking drugs (NDNBs) are clinically used to produce muscle
relaxation during general anesthesia. To better understand clinical properties of NDNBs, comparative in vitro
pharmacologic studies have been performed. In these studies, a receptor binding model, which relies on the
assumption that the inhibition, i.e., the effect of an NDNB, is proportional to the receptor occupancy by the drug, has
been effectively used to describe obtained experimental data. However, it has not been studied in literature under
which conditions the above assumption can be justified nor the assumption still holds in vivo. The purpose of this
study is to explore the in vivo relationship between the inhibition and the receptor occupancy by an NDNB and to
draw implications on how in vitro experimental results can be used to discuss the in vivo properties of NDNBs.
Methods: An ordinary differential equation model is employed to simulate physiologic processes of the activation of
receptors by acetylcholine (ACh) as well as inhibition by an NDNB. With this model, the degree of inhibition is
quantified by the fractional amount of receptors that are not activated by ACh due to the presence of an NDNB. The
results are visualized by plotting the fractional amounts of the activated receptors as a function of the receptor
occupancy.
Results: Numerical investigations reflecting in vivo conditions show that the degree of inhibition is not proportional
to the receptor occupancy, i.e., there is a nonlinear relationship between the inhibition and the receptor occupancy.
However, under a setting of high concentration of ACh reflecting a typical situation of in vitro experiments, the
relationship between the inhibition and the receptor occupancy becomes linear, suggesting the validity of the
receptor binding model. Also, it is found that the extent of nonlinearity depends on the selectivity of NDNBs for the
two binding sites of the receptors.
Conclusions: While the receptor binding model may be effective for estimating affinity of an NDNB through in vitro
experiments, these models do not directly describe in vivo properties of NDNBs, because the nonlinearity between
the inhibition and the receptor occupancy causes the modulation of the resultant concentration-effect relationships
of NDNBs.
Keywords: Anesthesia, Neuromuscular blockade, Receptor binding models, Nonlinear relationship
*Correspondence: hoshino@eng.u-hyogo.ac.jp
1
Department of Electrical Materials and Engineering, University of Hyogo,
Hyogo, Japan
Full list of author information is available at the end of the article
© The Author(s). 2021 Open Access This article is licensed under a Creative Commons Attribution 4.0 International License,
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Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made
available in this article, unless otherwise stated in a credit line to the data.Hoshino and Furutani Theoretical Biology and Medical Modelling (2021) 18:15 Page 2 of 14
Background sites of an AChR, respectively. Since the right-hand side
Nondepolarizing neuromuscular blocking drugs of Eq. (2) represents the fraction of free receptors, i.e., the
(NDNBs) inhibit neuromuscular transmission by com- fraction of receptors that are not occupied by the drugs,
peting with acetylcholine (ACh) for binding sites at the model (2) implies that the potency of an NDNB can
the post-junctional nicotinic acetylcholine receptors be completely characterized by the affinity of the drug.
(AChRs). They are widely used during general anesthesia However, in general, there may exist other factors that
to produce muscle relaxation for facilitating tracheal determine the potency of a drug. The underlying assump-
intubation and for providing optimal surgical working tion for the use of the model (2) is that the inhibition,
conditions [1]. To describe in vivo properties of NDNBs i.e. the effect of an NDNB, is proportional to the recep-
and to understand the molecular mechanisms behind tor occupancy by the drug. While the model (2) has been
clinical observations, many in vitro studies have been con- statistically tested in [4–8], the key factors affecting the
ducted (see e.g., [2–10]). In particular, in [7, 10], in vitro validity of this assumption have not been fully discussed in
experiments have been conducted through macroscopic literature, and it has not been examined if the assumption
current recordings from outside-out patches of BOSC23 holds in vivo.
cells or Xenopus oocytes where human adult (rather than The purpose of this study is to explore the in vivo
mouse adult or embryonic) muscle-type AChRs were relationship between the inhibition and the receptor occu-
expressed. In [7], synergistic effects of pairs of NDNBs pancy by an NDNB and to draw implications on how
have been studied. Although it successfully identified in vitro experimental results can be used to discuss the
evidence for synergy between many pairs of NDNBs at in vivo pharmacologic properties of NDNBs. An ordi-
the receptor level, not all the results correlated with syn- nary differential equation model based on [5, 13–15] is
ergism observed in vivo. In [9, 10], it was found that the employed to simulate physiologic processes of the acti-
IC50 , the drug concentration needed to produce a 50% vation of receptors by acetylcholine (ACh) as well as
inhibition of the ACh-induced current, decreases with inhibition by an NDNB. Based on the model, we examine
the increase in the concentration of ACh. Although it was the conditions under which the relationship between inhi-
concluded in [9, 10] that this demonstrated the existence bition and receptor occupancy becomes nonlinear and
of a noncompetitive action of NDNBs, some researchers discuss its clinically relevant implications.
raised concerns [11] over the insufficiency in quantitative
analysis to draw the conclusion. Thus, more investigations Methods
and considerations are needed to describe the clinical Model of competition between acetylcholine and NDNB
observations and to clarify underlying mechanisms of At the neuromuscular junction, electrical impulses from
inhibition based on in vitro experimental results. motor nerve cause the release of transmitter, ACh, to the
While there are several measures identifying in vitro synaptic cleft. Then, part of the released ACh molecules
properties of NDNBs, potency [12] is one of the most bind to AChRs on the muscle membrane and results in
widely used measures for studying NDNBs. It is usually a change in the conductance of the membrane due to
the channel opening. This change causes movement of
quantified by IC50 estimated by regression analysis using
ions into the muscle cell that produces an action poten-
the Hill equation for the relative current Iantag /I0 given by tial spreading over the surfaces of skeletal muscle fibers
Iantag IC50 nH and causing muscle contraction. NDNBs act by compet-
= , (1) ing with ACh for binding to the two sites of an AChR
I0 IC50 nH + [D]nH
and preventing changes in the membrane conductance.
where I0 and Iantag stand for the currents measured in To describe the competition between ACh and NDNB
the absence and presence of the drugs, respectively. The molecules for binding to AChRs, this paper uses the model
parameter nH stands for the Hill coefficient, and [D] developed in [15] with a modification to incorporate the
for the drug concentration. Also, affinity [12], which is dynamics of channel opening as described in [5, 13, 14].
defined as the extent or fraction of drug binding to recep- The complexes formed by binding of ACh, denoted by
tors at any given drug concentration, is used for char- A, and NDNB, by D, to AChR, by R, are represented by
3-letter symbols as shown in Fig. 1. The first and last let-
acterizing the strength of the binding of a ligand to its
ters denote the first and second ligands occupying the
receptors. In [4–8], the relative current versus concentra- sites 1 and 2, respectively, and the middle letter repre-
tion curves were analyzed based on the two-site receptor sents the receptor R. Unoccupied sites are denoted by O,
binding model given by the following equation: and ORO stands for free AChR. The kinetics of ACh and
Iantag KD1 KD2 NDNB are represented using association rate constants
= , (2) kassocAi and kassocDi , for site #i (i = 1, 2), respectively.
I0 KD1 KD2 + KD1 [D] + KD2 [D] + [D]2 Similarly, kdissAi and kdissDi stand for the dissociation rate
where KD1 and KD2 stand for the dissociation equilib- constants for ACh and NDNB. The symbol ARA stands
rium constants for NDNBs binding to the first and second for AChRs bound with two ACh molecules but in theHoshino and Furutani Theoretical Biology and Medical Modelling (2021) 18:15 Page 3 of 14
closed state, and the symbol ARA* for AChRs in the open [ORO] =[R]total −[ ARA∗ ] − [ARA] − [DRD]
state and thus activated due to the conformational change − [ARD] − [DRA]
of AChRs. The rate constants of opening and closing of
AChRs are represented by kopen and kclose , respectively. − [ARO] − [ORA] − [DRO] − [ORD], (4)
Then, the concentrations of these complexes can be cal-
where [R]total stands for the concentration of the post-
culated as a function of time by solving the following
junctional AChRs in the synaptic cleft.
ordinary differential equations:
Here we provide an example of simulation results of
d the model (3). The setting of the parameters is shown
[A] = −kdecay [A] in Table 1. Following [15], the concentration [R]total
dt
+ kdissA1 ([ARA] + [ARD] + [ARO]) of AChRs was calculated using the number of AChRs
(2.1 × 107 ) at the end plates of human deltoid muscle
− kassocA1 [A]([ORA] + [ORD] + [ORO])
reported in [16] and the volume of the synaptic cleft
+ kdissA2 ([ARA] + [DRA] + [ORA])
(4.5 × 1013 L) of rat diaphragm reported in [17]. The ini-
− kassocA2 [A]([ARO] + [DRO] + [ORO]), (3a) tial concentration [A]init of the free ACh was calculated by
d assuming that the number of ACh molecules released is
[ARA∗ ] = −kclose [ARA∗ ] + kopen [ARA], (3b)
dt one tenth of the number of AChRs as discussed in [18, 19].
d This means that the number of ACh molecules released
[ARA] = kassocA1 [ORA] [ A] − kdissA1 [ARA]
dt is 2.1 × 106 corresponding to the release of 300 vesicles
+ kassocA2 [ARO] [ A] − kdissA2 [ARA] [20] with 7000 ACh molecules in each vesicle. Thus, while
+kclose [ARA∗ ] − kopen [ARA], (3c) the concentration of ACh itself is sufficiently higher than
d needed for neuromuscular transmission, due to the con-
[DRD] = kassocD1 [ORD] [ D] − kdissD1 [DRD] siderable excess of AChRs, only a small fractional amount
dt
+ kassocD2 [DRO] [ D] − kdissD2 [DRD], (3d) of AChRs will be activated during neuromuscular trans-
d mission. The rate constant kdecay was determined such
[ARD] = kassocA1 [ORD] [ A] − kdissA1 [ARD] that the half-life of free ACh becomes 58μs as calculated
dt
+ kassocD2 [ARO] [ D] − kdissD2 [ARD], (3e)
in [15]. The dissociation and association rate constants for
ACh were tentatively set to the values obtained in [13, 14]
d
[DRA] = kassocD1 [ORA] [ D] − kdissD1 [DRA] from experiments using mouse adult AChRs. Similarly,
dt
the rate constants for NDNBs were set to tentative values
+ kassocA2 [DRO] [ A] − kdissA2 [DRA], (3f)
based on experiments in [4, 5] using mouse embryonic
d and adult AChRs. Since the values of these rate constants
[ARO] = kassocA1 [ORO] [ A] − kdissA1 [ARO]
dt may be different between human and mouse AChRs, these
+ kdissA2 [ARA] − kassocA2 [ARO] [ A] constants will be varied in a systematic way.
+ kdissD2 [ARD] − kassocD2 [ARO] [ D], (3g) Figure 2 shows the time courses of the concentrations
d of free ACh, [A], of diliganded AChRs at the closed state,
[ORA] = kassocA2 [ORO] [ A] − kdissA2 [ORA]
dt [ARA], and of the activated AChRs, [ ARA∗ ], after the
+ kdissA1 [ARA] − kassocA1 [ORA] [ A] arrival of an electrical impulse from motor nerve and the
+ kdissD1 [DRA] − kassocD1 [ORA] [ D], (3h) release of ACh at t = 0. The initial concentrations for
d
the nine complexes at t = 0 were defined by the steady
[DRO] = kassocD1 [ORO] [ D] − kdissD1 [DRO] state concentrations before the release of ACh. The con-
dt
centration of the NDNB was set to one of [D] = 0,
+ kdissD2 [DRD] − kassocD2 [DRO] [ D]
3.0 × 10−8 M, and 1.0 × 10−7 M. As shown in the upper
+ kdissA2 [DRA] − kassocA2 [DRO] [ A], (3i)
panel of Fig. 2, the concentration of ACh rapidly decreases
d due to the hydrolysis of ACh by acetylcholinesterase and
[ORD] = kassocD2 [ORO] [ D] − kdissD2 [ORD]
dt binding of ACh to AChRs. The time course of [ARA] is
+ kdissD1 [DRD] − kassocD1 [ORD] [ D] shown in the middle panel of Fig. 2. The concentration
+ kdissA1 [ARD] − kassocA1 [ORD] [ A], (3j) [ARA] raises rapidly and decays after reaching the peak at
around t = 0.04 ms. Subsequently, as shown in the lower
panel, the concentration [ ARA∗ ] raises and reaches its
where [D] stands for the concentration of NDNB at the peak at around t = 0.3 ms. The highest [ ARA∗ ], denoted
effect site, and kdecay for the rate constant of the decay of by [ ARA∗ ]max , is attained in the absence of NDNB, i.e.,
free ACh due to hydrolysis of ACh by acetylcholinesterase [D] = 0, and the peak concentration of the activated
and diffusion of ACh away from the synaptic cleft. The AChRs [ ARA∗ ]peak decreases with the increase in the
concentration [ORO] of the unoccupied AChR is given by concentration of [D].Hoshino and Furutani Theoretical Biology and Medical Modelling (2021) 18:15 Page 4 of 14
Fig. 1 Diagram of the interactions between acetylcholine (A), NDNB (D), and the postsynaptic receptor (R). The complexes formed by binding of
ACh and NDNB to AChR are represented by 3-letter symbols, where the first and last letters denote the first and second ligands occupying the sites
1 and 2, respectively. The state ARA* represents the activated AChR
With this model, the effect of an NDNB can be Theoretical analysis of the competition model
quantified by a fraction of activated AChRs given by To conduct the subsequent numerical analysis in a sys-
[ ARA∗ ]peak /[ ARA∗ ]max . This definition corresponds to tematic way, here we theoretically analyze the model
the relative current Iantag /I0 used in in vitro experiments (3). Specifically, we provide a simplified representation
under the assumption that the membrane conductance is of the model under the assumption that the dissociation
proportional to the number of activated AChRs, i.e. the rate constants kdissD1 and kdissD2 of an NDNB are much
number of opened ion channels: smaller than other rate constants such as kdissA1 , kdissA2
and kdecay . For cisatracurium, the dissociation rate con-
stants are reported in [5] as 13s−1 and 34s−1 for mouse
Iantag [ ARA∗ ]peak adult and embryonic AChRs, respectively. Also, for (+)-
= . (5)
I0 [ ARA∗ ]max tubocurarine and pancuronium, these values are reported
Table 1 Setting of parameters for numerical simulation in base case
Symbol Meaning Value
[R]total Concentration of AChRs in the synaptic cleft 7.75 × 10−5 M [15]
[A]init Initial concentration of ACh immediately after the stimulus 7.75 × 10−6 M [15]
kdecay Rate constant of the decay of the concentration of free ACh 1.2 × 104 s−1 [15]
kdissA1 Dissociation rate constant for ACh with site1 of AChR 1.8 × 104 s−1 [13, 14]
kdissA2 Dissociation rate constant for ACh with site2 of AChR 1.8 × 104 s−1 [13, 14]
kassocA1 Association rate constant for ACh with site1 of AChR ∗ 1.1 × 108 M−1 .s−1 [13, 14]
kassocA2 Association rate constant for ACh with site2 of AChR ∗ 1.1 × 108 M−1 s−1 [13, 14]
kdissD1 Dissociation rate constant for NDNB with site1 of AChR ∗ 10s−1 [4, 5]
kdissD2 Dissociation rate constant for NDNB with site2 of AChR ∗ 10s−1 [4, 5]
kassocD1 Association rate constant for NDNB with site1 of AChR ∗ 1.0 × 108 M−1 s−1 [4, 5]
kassocD2 Association rate constant for NDNB with site2 of AChR ∗ 1.0 × 108 M−1 s−1 [4, 5]
kclose Rate constant of channel closing of AChR 1.2 × 103 s−1 [14]
kopen Rate constant of channel opening of AChR 5.0 × 104 s−1 [14]
∗ These dissociation and association rate constants are varied in numerical analysisHoshino and Furutani Theoretical Biology and Medical Modelling (2021) 18:15 Page 5 of 14
Fig. 2 An example of simulation results of the model (3). a The time course of the concentration [A] of free ACh. b The time course of the
concentration [ARA] of the complex ARA representing the activated AChRs
in [4] as 2.1s−1 and 5.9s−1 , respectively. Whereas, the [ ARA∗ ] [DRD]
x∗ARA := , xDRD := . (7)
dissociation rate constants of ACh and the rate constant [R]total [R]total
of hydrolysis are estimated as 18000s−1 and 12000s−1 ,
respectively [13, 15]. Similarly, the dimensionless variables xARA , xARD , xARD ,
xDRA , xARO , xORA , xDRO , and xORD for the remaining
The simplified model is given in a dimensionless rep- seven complexes can be defined by dividing by [R]total . By
resentation. For example, a dimensionless time τ and a using these dimensionless variables, the simplified model
concentration of ACh, xA , can be defined as is given as follows (see Appendix for its derivation):
[A] d
xA = −xA + κA1 λA1 {xARA − xA (OORD (δ) − xARD )}
τ := t · kdecay , xA := , (6) dτ
KA1 + κA2 λA2 {xARA − xA (ODRO (δ) − xDRA )}
+ κA1 λA1 {xARD + xARO − xA (1 − xARA − xARO − Ototal (δ))}
where KA1 stands for the dissociation equilibrium con-
+ κA2 λA2 {xDRA + xORA − xA (1 − xARA − xORA − Ototal (δ))},
stant for ACh with the site #1 given by kdissA1 /kassocA1 . (8a)
Dimensionless variables x∗ARA and xDRD for the concentra- d ∗
x = −κclose x∗ARA + κopen xARA , (8b)
tions of the complex ARA∗ and DRD are given by dτ ARAHoshino and Furutani Theoretical Biology and Medical Modelling (2021) 18:15 Page 6 of 14
d
xARA = κA1 (xORA xA − xARA ) + κA2 (μA xARO xA − xARA )
This fact implies that even in the simulations of the
dτ original model (3), the results will be almost unchanged
+κclose x∗ARA − κopen xARA , (8c)
if the parameter μD kept constant and the parameters
d
xARO = κA1 {xA (1 − xARA − xARO − xORA − Ototal (δ)) − xARO } κD1 := kdissD1 /kdecay and κD2 := kdissD2 /kdecay are small
dτ
+ κA2 (xARA − μA xARO xA ) , (8d) enough to validate the simplification (see Appendix for
d details of the validity of the simplification). This obser-
xORA = κA2 {μA xA (1 − xARA − xARO − xORA − Ototal (δ)) − xORA }
dτ vation facilitates the numerical analysis of the original
+ κA1 (xARA − xORA xA ) , (8e) model (3) by varying the parameters of the model in a
d systematic way as demonstrated in the rest of the paper.
xARD = κA1 {xA (OORD (δ) − xARD ) − xARD } , (8f)
dτ
Second, from the model Eq. 8, it can be seen that the
d
xDRA = κA2 {μA xA (ODRO (δ) − xDRA ) − xDRA } , (8g) entire states during competition can be viewed as divided
dτ
where κA1 , κA2 , κopen and κclose are the dimensionless into 4 parts as shown in Fig. 3. Note that owing to an
parameters representing normalized rate constants given appropriate derivation of the dimensionless representa-
by tion of the model, not only the terms for dissociation of
NDNBs but also association terms are eliminated from the
kdissA1 kdissA2 original model. As a result, the state DRD and the pairs
κA1 := , κA2 := ,
kdecay kdecay of states (ORD, ARD) and (DRO, DRA) can be viewed
kopen kclose as separated from the states describing the activation of
κopen := , κclose := , (9) AChRs, while interaction between these states may occur
kdecay kdecay
through the dynamics of free ACh given by (8a). Specif-
The parameters λA1 , λA2 , and μA are the dimensionless
ically, although the fraction of the state DRD will not
parameters representing affinities of ACh to the binding
change during the competition, the fractions of ORD and
sites of the AChR given by
DRO will change due to association and dissociation with
[R]total [R]total KA1 ACh molecules. Therefore, even if the total occupancy
λA1 := , λA2 := , μA := ,
KA1 KA2 KA2 Ototal is the same, the fraction of the activated AChRs will
(10) decrease with the increase in the occupancies OORD and
with KA2 := kdissA2 /kassocA2 , and finally, μD and δ are the ODRO because free ACh will decrease due to association
dimensionless parameters representing the site-selectivity with ORD and DRO. Note that the ratio of ORD and DRO
and concentration of NDNB, respectively, given by to Ototal is determined by the parameter μD . Finally, the
difference between the receptor binding model (2) and the
KD1 [D] competition-based models (3) and (8) can be clarified. By
μD := , δ := , (11)
KD2 KD1 using the notation introduced above, the model (2) can be
with KD1 := kdissD1 /kassocD1 and KD2 := kdissD2 /kassocD2 . rewritten as follows:
Furthermore, ODRD , OORD and ODRO are functions of δ Iantag 1
standing for the fractions of the complex DRD, ORD, and = = 1 − Ototal (δ) (14)
I0 1 + δ + μD δ + μD δ 2
DRO before the release of ACh given by
μD δ 2 Thus, the model (2) is clearly based on the assumption
ODRD (δ) := , (12a) that the inhibition is proportional to the total occupancy
(1 + δ)(1 + μD δ)
Ototal , while the models (3) and (8) describe the effect of
μD δ
OORD (δ) := , (12b) the partial occupancies OORD and ODRO on the activation
(1 + δ)(1 + μD δ) of AChRs during competition.
δ
ODRO (δ) := . (12c)
(1 + δ)(1 + μD δ) Method of numerical analysis on the relationship between
The total occupancy Ototal of the AChRs by NDNB is inhibition and receptor occupancy
given by The relationship between the inhibition and the recep-
tor occupancy by an NDNB is studied via numeri-
Ototal (δ) := ODRD (δ) + OORD (δ) + ODRO (δ). (13a)
cal simulations of the original model (3). To visualize
From the simplified model (8), several insights can the results, the fraction of activated AChRs given by
be obtained on the relationship between the receptor [ ARA∗ ]peak /[ ARA∗ ]max is calculated under various con-
occupancy by NDNBs and the dynamics of activation centrations of the NDNB (100 values between [D] =
of AChRs. First, it can be seen that the model (8) has 1.0 × 10−14 M and 1.0 × 10−5 M that are spaced equidis-
only one parameter μD characterizing NDNBs. Thus, the tantly on a logarithmic scale, and [D] = 0) and plotted
properties of NDNBs are completely determined by the versus the total occupancy Ototal . For calculating the peak
parameter μD representing the site-selectivity of NDNBs. concentration of ARA, the ordinary differential equationHoshino and Furutani Theoretical Biology and Medical Modelling (2021) 18:15 Page 7 of 14 Fig. 3 Diagram of the interactions between acetylcholine (A), NDNB (D), and the postsynaptic receptor (R) based on the simplified model (8). The state DRD and the pairs of states (ORD, ARD) and (DRO, DRA) can be viewed as separated from the states describing the activation of AChRs, while interaction between these states may occur through the dynamics of free ACh given by (8a) model (3) is numerically solved using the Fortran solver by changing the value of kassocD2 , while kdissD1 , kdissD2 and LSODA provided by the python package SciPy (Version kassocD1 kept constant at the values shown in Table 1. Also, 1.5.2). the values of the dissociation constants kdissD1 and kdissD2 The parameters of the model (3) are varied in a sys- are varied to explore the difference between the original tematic way based on information provided by literature model (3) and the reduced-order model (8). and the findings of the theoretical analysis as explained Furthermore, we examine the effect of varying the ini- in the following. First, we investigate the effect of vary- tial concentration [A]init representing the concentration ing the kinetic constants for ACh and NDNB. For ACh, of ACh immediately after the release of ACh at t = 0. the values of kdissA1 , kdissA2 , kassocA1 , and kassocA2 pre- It has been known that the number of ACh molecules sented in Table 1 were determined by experiments using released in vivo is one tenth of the number of AChRs at mouse adult AChRs. However, it is known that the EC50 , the synaptic cleft [15]. In this paper, following [15], the the concentration of ACh where the half of the AChRs concentrations of AChRs and the initial ACh were set as are activated, for human adult AChRs is smaller than [R]total = 7.75 × 10−5 M and [A]init = 7.75 × 10−6 M, that for mouse adult AChRs: EC50 = 8.48 × 10−6 M or respectively. Under this setting, only a small fraction (at 7.91 × 10−6 M for human adult AChRs [10] and EC50 = least less than one tenth) of the total receptor popula- 1.6 × 10−5 M for mouse adult AChRs [13]. Since this tion will be activated in vivo. However, in many in vitro implies that the affinity of ACh is different between experiments, a quite high concentration of ACh is used human and mouse AChRs, we investigated the effect of [4–8] such that nearly 50% (nearly EC50 ) of the AChRs varying the constants KA1 and KA2 by changing kassocA1 are activated or more than 90% of AChRs are activated. and kassocA2 while kdissA1 and kdissA2 kept constant. Fur- Thus, we investigated the effect of varying [A]init in a wide thermore, since it has been reported in [13] that the two range of concentrations from [A]init = 0.0562 × [R]total to binding sites of mouse adult AChR have similar affini- 5.62 × [R]total . ties, we assume that it is also the case for human adult AChR and the parameters KA1 and KA2 are equal. Thus, Results we investigate the effect of varying the values of KA1 = The upper panel of Fig. 4 shows the relationship KA2 = KA , which corresponds to κA1 = κA2 and μA = 1. between the fraction of activated AChRs given by For the kinetic constants of an NDNB, the effect of vary- [ ARA∗ ]peak /[ ARA∗ ]max and the receptor occupancy ing the parameter μD is investigated, since this parameter Ototal evaluated under various settings of the dissocia- completely characterizes the properties of the NDNB as tion equilibrium constants for ACh. The values of KA1 = far as the dissociation constants kdissD1 and kdissD2 are KA2 = KA is one of 1.0 × 10−4 M, 3.16 × 10−5 M and small enough. The value of parameter μD was assigned 1.0 × 10−5 M, and shown by red, green and blue lines,
Hoshino and Furutani Theoretical Biology and Medical Modelling (2021) 18:15 Page 8 of 14 Fig. 4 Effect of varying kinetic constants for ACh and NDNB on the relationship between the fraction of activated AChRs, [ ARA∗ ]peak /[ ARA∗ ]max , and the receptor occupancy Ototal . a Results of varying the equilibrium dissociation constants for ACh under KA = KA1 = KA2 (κA1 = κA2 and μA = 1). b Results of varying the site-selectivity μD = KD1 /KD2 of an NDNB respectively. The dotted line in the figure shows the lin- AChRs, [ ARA∗ ]peak /[ ARA∗ ]max , and the receptor occu- ear relationship corresponding to the model (2). The pancy Ototal under various settings of the initial concen- results clearly show that there are nonlinear relation- tration of ACh, [ A]init . In the upper panel, the initial ships between [ ARA∗ ]peak /[ ARA∗ ]max and Ototal in all concentration [ A]init is decreased from 5.62 × [ R]total the cases. Furthermore, it can be confirmed that the extent to 0.316 × [ R]total , and it can be seen that the extent of nonlinearity increases with the decrease in the equilib- of nonlinearity increases with the decrease in [ A]init . rium constant KA . In the subsequent analyses, to highlight Also, in the middle panel, the initial concentration is the effects of varying other parameters, we assigned the further decreased to 0.0562 × [ R]total . The extent of value of 1.0 × 10−5 M to KA , where the extent of non- nonlinearlity slightly decreases in this range of parame- linearity was most prominent. Then, the lower panel of ter setting, and thus the nonlinearity is most prominent Fig. 4 shows the results under various settings of μD rep- at [A]init = 0.316 × [ R]total . Interestingly, the relation- resenting the site-selectivity of an NDNB. The red, green ship between [ ARA∗ ]peak /[ ARA∗ ]max and Ototal becomes and blue lines show the results for μD = 1, 10, and 100, almost linear when the value of [ A]init is larger than 3.16× respectively. It can be seen that the relationships between [ R]total . To clarify the meaning of this result, the lower [ ARA∗ ]peak /[ ARA∗ ]max and Ototal are nonlinear in all the panel of Fig. 5 shows the concentration-response relation- cases, and the extent of nonlinearity decreases with the ship between the concentration [ A]init and the activated increase in μD . AChRs [ ARA∗ ]max . Note that the [ ARA∗ ]max is defined Next, the upper and the middle panels of Fig. 5 for each setting of [A]init as the highest [ ARA∗ ] under show the relationship between the fraction of activated various settings of the concentration of an NDNB, and
Hoshino and Furutani Theoretical Biology and Medical Modelling (2021) 18:15 Page 9 of 14 Fig. 5 Effect of varying the initial concentration [A]init of ACh on the numerical results. a The relationship between the fraction of activated AChRs, [ ARA∗ ]peak /[ ARA∗ ]max , and the receptor occupancy Ototal under various settings of [A]init . b The concentration-effect relationship between the activated AChRs in the absence of NDNBs and the initial concentration [A]init it is attained in the absence of NDNB. At the setting of relationship between the fraction of activated AChRs, [A]init = 0.1 × [ R]total , which corresponds to the in vivo [ ARA∗ ]peak /[ ARA∗ ]max , and the receptor occupancy situation [15], only a fraction of AChRs are activated even Ototal under the setting of [A]init = 0.1 × [ R]total corre- in the absence of NDNBs, and result in a highly non- sponding to in vivo concentration. With this setting, the linear relationship between [ ARA∗ ]peak /[ ARA∗ ]max and results are almost unchanged even when the rate constant Ototal . However, at the setting of [A]init ≥ 3.16 × [ R]total , kdissD is as large as kdissA1 , kdissA2 , and kdecay . This implies more than 92% of AChRs are activated in the absence that the model simplification performed in this paper can of NDNB, and in this case, the relationship between be validated for a wide range of settings of kdissD . The [ ARA∗ ]peak /[ ARA∗ ]max and Ototal becomes linear. lower panel of Fig. 6 shows the results under the setting Finally, the effect of varying the dissociation rate con- of [A]init = 10 × [ R]total corresponding to in vitro con- stants kdissD1 and kdissD2 of NDNB was examined under centration. In this case, it can be seen that the results the setting of identical rate constants, i.e. kdissD1 = are highly affected by the setting of the dissociation rate kdissD2 = kdissD . The upper panel of Fig. 6 shows the constant kdissD . Furthermore, the relationships between
Hoshino and Furutani Theoretical Biology and Medical Modelling (2021) 18:15 Page 10 of 14
Fig. 6 Effect of varying the dissociation rate constants kdissD1 and kkdissD2 on the relationship between the fraction of activated AChRs,
[ ARA∗ ]peak /[ ARA∗ ]max , and the receptor occupancy Ototal . a Results under the setting of [A]init = 0.1 × [ R]total corresponding to in vivo
concentration. b Results under the setting of [A]init = 10 × [ R]total corresponding to in vitro concentration
[ ARA∗ ]peak /[ ARA∗ ]max and Ototal are no longer linear problems, an ordinary differential equation model was
when the rate constant kdissD becomes large even if the introduced based on [5, 13–15] to simulate the physio-
concentration [A]init is large. This implies that the model logic processes of activation of receptors by ACh as well
simplification can be validated only for small values of the as inhibition by an NDNB. The theoretical analysis per-
parameter kdissD . formed in this paper clarified that under the assumption
that the dissociation rate constants for an NDNB are small
Discussion and with an appropriate nondimensionalization, the char-
We theoretically and numerically investigated the rela- acteristics of an NDNB can be completely determined
tionship between inhibition and receptor occupancy by by a single parameter μD representing the site-selectivity
NDNBs. While the two-site receptor binding model (2), of the NDNB for two binding sites of AChRs. We then
which assumes a linear relationship between the inhibi- performed numerical analysis of the model by plotting the
tion and the receptor occupancy by an NDNB, has been fractional amounts of the activated AChRs as a function of
statistically tested in [4–8] for several in vitro exper- the receptor occupancy. The numerical results show that
imental settings, it has not been studied in literature under a setting of parameters reflecting in vivo environ-
under which conditions the above assumption holds nor ment, there is a nonlinear relationship between the inhibi-
if the assumption remains valid in vivo. To consider these tion and the receptor occupancy, indicating limitation ofHoshino and Furutani Theoretical Biology and Medical Modelling (2021) 18:15 Page 11 of 14
the applicability of the receptor binding model. Further- sufficiently large and the dissociation rate constants of
more, it has been shown that the extent of nonlinearity an NDNB were sufficiently small. This finding may pro-
depends on the parameters representing kinetic constants vide a reasonable justification of the use of the two-site
for ACh or NDNBs and the concentration of ACh. model (2) in the analysis of kinetic constants for NDNBs
Regarding the nonlinear relationship between the effect through in vitro experiments. At least, the condition of
and the receptor occupancy by an NDNB, it has been large concentration of ACh is satisfied in the experi-
well known that the twitch strength (the degree of mus- ments reported in [4–8], where concentrations that opens
cle contraction) observed in vivo is not proportional about 93% to 95% of the AChRs were used. However, fur-
to the receptor occupancy due to the high margin of ther consideration is needed to identify the conditions
safety [21]. The origin of the safety margin is a copious needed for the justification of the model (2), because
density of AChRs on the post-synaptic membrane and the present study did not take into account the effect
the fact that only a small fraction of AChRs needs to be of desensitization of AChRs, which is the main cause of
activated to trigger the occurrence of an action potential the decay in a measured current observed during in vitro
and the contraction of the associated muscle fiber. Thus, experiments.
the nonlinearity due to the safety margin means that the
response of muscle fiber is not proportional to the fraction Conclusion
of activated AChRs, and the extent of nonlinearity would The relationship between the inhibition and the receptor
not be affected by the properties of an NDNB. However, occupancy by an NDNB was theoretically and numer-
this paper focused on the nonlinearity in the relationship ically investigated. While a receptor binding model,
between the receptor occupancy and the fraction of acti- which assumes a linear relationship, may be effec-
vated AChRs, and the extent of nonlinearity is affected tive for estimating affinity of an NDNB through in
by the properties of an NDNB. In particular, it has been vitro experiments, the model do not directly describe
revealed in this paper that the effect of an NDNB on in vivo pharmacologic properties of NDNBs, because
the extent of nonlinearity can be characterized by a sin- the nonlinearity between the inhibition and the recep-
gle parameter μD representing the site-selectivity of the tor occupancy causes the modulation of the resul-
NDNB. tant concentration-effect relationships of NDNBs. It was
Although the model used and simulations performed in found that the effect of characteristics of an NDNB
this paper are intended to describe in vivo situations, our on the extent of nonlinearity can be identified by a
finding that the extent of nonlinearity is affected by the single parameter representing the site-selectivity of an
concentration of ACh is consistent with results observed NDNB.
through in vitro experiments. In [9, 10], it was found that
the IC50 for several clinically used NDNBs, cis-atracrium, Appendix
rocuronium, and vecronium, decreased with the increase This appendix provides a detailed derivation of the sim-
in the concentration of ACh. With these results, it was plified model (8). Specifically, we derive a dimensionless
concluded in [9, 10, 22] that this phenomenon indicates representation of the original model based on the tech-
a noncompetitive component of inhibition under the idea nique of scaling [23] and perform model-order reduc-
that the enhancement of the inhibition was caused by a tion based on singular perturbation theory [24, 25].
mechanism different from competitive one occurred by The scaling of the model (3) can be done by introduc-
NDNBs in combination with high concentration of ACh. ing dimensionless variables. For example, a dimension-
However, such a shift in the values of IC50 can also be less time τ and a concentration of ACh, xA , can be
explained by a change in the relationship between the defined as
receptor occupancy and the fraction of activated AChRs. [A]
As demonstrated in the upper panel of Fig. 5, the receptor τ := t · kdecay , xA := , (15)
KA1
occupancy Ototal at which [ ARA∗ ]peak /[ ARA∗ ]max (which
where KA1 stands for the dissociation equilibrium con-
corresponds to the relative current) takes the value of
stant for ACh with the site #1 given by kdissA1 /kassocA1 .
0.5 increases with the decrease in the concentration of
Dimensionless variables x∗ARA and xDRD for the con-
ACh, meaning that the IC50 increases with the decrease
centrations of the complex ARA∗ and DRD are
in the concentration of ACh. This shift in the IC50 is
given by
consistent with the observation in [9, 10] and occurs
under a totally competitive mechanism of inhibition by [ ARA∗ ] [DRD]
x∗ARA := , xDRD := . (16)
an NDNB. [R]total [R]total
Interestingly, it was found that the relationship between Similarly, the dimensionless variables xARA , xARD , xARD ,
the fraction of activated AChRs and the receptor occu- xDRA , xARO , xORA , xDRO , and xORD for the remain-
pancy became linear when the concentration of ACh was ing seven complexes can be defined by dividing byHoshino and Furutani Theoretical Biology and Medical Modelling (2021) 18:15 Page 12 of 14
[R]total . By substituting these dimensionless variables with KA2 := kdissA2 /kassocA2 , and finally, μD and δ are the
to the model (3), the following equations can be dimensionless parameters representing the site-selectivity
derived: and concentration of NDNB, respectively, given by
d
xA = −xA + κA1 λA1 {xARA + xARD + xARO KD1 [D]
dτ
μD := , δ := , (20)
− xA (xORA + xORD + xORO )} KD2 KD1
+ κA2 {λA1 (xARA + xDRA + xORA )
− λA2 xA (xARO + xDRO + xORO )}, (17a) with KD1 := kdissD1 /kassocD1 and KD2 := kdissD2 /kassocD2 .
d ∗ Note that due to the scaling performed here, the num-
x = −κclose x∗ARA + κopen xARA , (17b) ber of parameters characterizing the properties of NDNB
dτ ARA
d can be reduced from four to three: from (kdissD1 , kasoocD1 ,
xARA = κA1 (xORA xA − xARA ) + κA2 (μA xARO xA − xARA ) kdissD2 , kassocD2 ) to (κD1 , κD2 , μD ).
dτ
+κclose x∗ARA − κopen xARA , (17c) Furthermore, the order of the model can be reduced
d using the technique of singular perturbation [24, 25] based
xDRD = κD1 (xORD δ − xDRD ) + κD2 (μD xORD δ − xDRD ) , on an inherent multiple-timescale property of the model.
dτ
(17d) Such a multi-scale property arises when the dissociation
d rate constants kdissD1 and kdissD2 of an NDNB are much
xARD = κA1 (xORD xA − xARD ) + κD2 (μD xARO δ − xARD ) , smaller than other rate constants such as kdissA1 , kdissA2
dτ
(17e) and kdecay . By considering the limit κD1 , κD2 → 0, the
d following equations hold from the Eq. 17:
xDRA = κD1 (xORA δ − xDRA ) + κA2 (μA xDRO xA − xDRA ) ,
dτ
(17f)
d d d
d xDRD = (xORD + xARD ) = (xDRO + xDRA ) = 0.
xARO = κA1 (xORO xA − xARO ) + κA2 (xARA − μA xARO xA ) dτ dτ dτ
dτ (21)
+ κD2 (xARD − μD xARO δ) , (17g)
d
xORA = κA2 (μA xORO xA − xORA ) + κA1 (xARA − xORA xA ) Thus, the values of xDRD , xORD + xARD and xDRO + xDRA
dτ
do not change in the reduced-order model, or almost
+ κD1 (xDRA − xORA δ) , (17h)
unchanged in the original model (3), from their initial val-
ues at τ = 0. When the initial conditions are defined by
the steady state concentrations under a given value of δ,
d
xDRO = κD1 (xORO δ − xDRO ) + κA2 (xDRA − μA xDRO xA ) the initial values of xDRD , xORD , xDRO , xARD and xDRA are
dτ
given by
+ κD2 (xDRD − μD xDRO δ) , (17i)
d
xORD = κD2 (μD xORO δ − xORD ) + κA1 (xARD − xORD xA ) μD δ 2
dτ xDRD |τ =0 = ODRD (δ) := , (22a)
+ κD1 (xDRD − xORD δ) , (17j) (1 + δ)(1 + μD δ)
μD δ
xORD |τ =0 = OORD (δ) := , (22b)
where κA1 , κA2 , κD1 and κD2 are the dimensionless param- (1 + δ)(1 + μD δ)
eters representing the rates of the both dissociation and δ
association of the ligands given by xDRO |τ =0 = ODRO (δ) := , (22c)
(1 + δ)(1 + μD δ)
kdissA1 kdissA2 xARD |τ =0 = xDRA |τ =0 = 0 (22d)
κA1 := , κA2 := ,
kdecay kdecay
kdissD1 kdissD2 where ODRD , OORD and ODRO stand for the fractions of
κD1 := , κD2 := , (18) the complex DRD, ORD, and DRO, respectively, at the
kdecay kdecay
steady state. Also, the total occupancy Ototal of the AChRs
and λA1 , λA2 , and μA are the dimensionless parameters by the NDNB is given by
representing affinities of ACh to the binding sites of the
AChR given by
Ototal (δ) := ODRD (δ) + OORD (δ) + ODRO (δ). (23)
[R]total [R]total KA1
λA1 := , λA2 := , μA := ,
KA1 KA2 KA2 By using the equations from (21) to (23), the model (3) can
(19) be reduced to the following form:Hoshino and Furutani Theoretical Biology and Medical Modelling (2021) 18:15 Page 13 of 14
d
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