Thermodynamic treatment of ligand-receptor interactions

Macromol. Theory Simul. 5, 165-181 (1996) 165

Thermodynamic treatment of ligand-receptor interactions

Vito Di Noto* a), Lisa Dalla Via b), Antonio Toninello b), Maurizio Vidalia)

a)Dipartimento di Chimica Inorganica, Metallorganica ed Analitica, Universita di Padova, via Loredan 4, 1-351 31, Padova, Italy b)Dipartimento di Chimica Biologica, Universita di Padova, Centro di studio Fisiologia Mitocondriale del CNR di Padova, via Trieste 75, 35121, Padova

(Received: August 4, 1995)

SUMMARY This work reports the development of a physico-chemical model for ligand-receptor

interactions. It considers receptors having groups of different binding sites in equilibrium and far from equilibrium where each of these may have multiple occupancies. The obtained results under appropriate simplifications give the well known Scatchard, Hill and Adair relations and provide a very interesting tool for studying receptor-ligand phenomena and for determining the related parameters and interaction energies.

Introduction

Ligand-receptor interactions play a basic role in many biological functions such as enzymatic reactions, interactions of specific receptors with neurotransmitters and with hormones, transport, antigen-antibody reaction, interactions of small molecules 3-8 ) and ions') with macromolecules, etc. In these processes the receptor usually is any entity (cell or macromolecule) which allows another entity called ligand (that can be an ion, a molecule or a macromolecule) to bind to itself.

Because of the importance of this subject a large amount of biochemical and biophysical research has been directed to exploringRhese interactions in depth I, Binding models, developed in the past and used until today17-2'), can be roughly grouped in the two following categories: (a) independent-sites interaction models, and (b) non-independent-sites interaction models. The first category can be divided into two other classes 17-21): (a) identical independent sites, and (b) multiple classes of independent sites. On the other hand, the second category can be divided into two classes defined as cooperativity models: (a) allosteric 17,22 ,23) , and (b) sequen- tial I , 17,211

All these models have been developed using thermodynamical and statistical approaches of reversible ligand-receptor interactions at equilibrium 1-25,26). However, it may happen that not all receptor sites are in equilibrium at the same time and consequently it is not always possible to study binding processes using thermodynamics at equilibrium, except for those systems where there is only one type of site towards which the ligand affinity during the whole binding process does not change.

As it is well known '. 17*24,25), during binding experiments, receptor sites can undergo conformational changes and these phenomena can increase or decrease ligand-receptor interactions which thus can change ligand affinity towards that type of site. In fact from a mechanical point of view the ligand may interact with a site that chemically is

Macromol. Theory Simul. 5, No. 2, March 1996

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166 V. Di Noto, L. Dalla Via, A. Toninello, M. Vidali

identical during the binding process but after some time this site could attract the ligand with a different force and so, compared with the previous situation, from a thermodynamic point of view, this is a different site.

The fact that identical chemical sites can change their affinity toward ligands or that non-active sites can become active during binding processes suggests that really ligand- receptor systems may not be at equilibrium. Moreover, each chemical or energetical type of binding site needs a different time or concentration to reach the equilibrium with a given ligand.

This work reports on the development of a general binding model treating reccptor sites: static, dynamic, in equilibrium and not in equilibrium with the ligand. Those that are different, independent and which don’t change their affinity towards the ligand may be called static sites. However dynamic sites are those that can change their affinity during binding processes. This last situation is usually regarded as a cooperativity phenomenon 1 7 3 2 4 , 2 5 ) .

Model

The proposed model is based on the following assumptions: (a) the binding site acts as a frame for the coordination of ligands; (b) the receptor contains s different types of sites each of which is singled out by the

i-th index; (c) the sites have the possibility to bond more than one ligand, and the maximum

number of ligand unities that may be bonded to each different i site is nj (the maximum multiple coordination of sites);

(d) the multiple coordination possible at each type of site is indicated by the I index; (e) the activity and the concentration of species in solution are treated as equivalent; ( f ) each site in solution may or may not be at equilibrium with a ligand; (g) the driving force in the binding experiment is assumed to be the change of free

energy when one mole of ligand is coordinated or left in solution by the i type of site. In Fig. 1 a schematic picture of a possible experiment between the ligand distributed

in solution and the receptor, is depicted. If we consider the change in free energy when one mole of ligand is coordinated or

left in solution by each i site with the I multiple coordination at constant temperature and pressure, the total change of free energy of the receptor-ligand system may be written

where p f i , ,uf/ and ,Y:/-, indicate the chemical potentials for the i site with I and (I - 1) multiple coordinations. The superscript S refers to the Gibbs free energy variation in the system when one mole of free ligand in solution changes owing to the coordination of the i site with I multiple coordination. The superscript R refers to the same variation due to the molar changes of i site with I or ( I - 1) multiple coordination in the receptor. d( is an infinitesimal amount of substance which may be called the infinitesimal extent of reaction 27).

Thermodynamic treatment of ligand-receptor interactions 161

Fig. 1 . Model of the receptor-ligand interactions. The receptor includes groups of different types of binding sites each of which may exhibit multiple occupancies

In the assumptions fore-mentioned the chemical potential may be written as R pi, /- I = P$!I + RT In [Ai, 1- 1 I

PF/ = ,up;' + R T ~ ~ [ A ~ , , I (2)

,uf/ = p a o + RTIn[F]

where pf;!, , ,up; and p a o are the standard chemical potentials. [A, I - , ] and [A, ,] are the i site concentration with 1 - 1 and 1 multiple coordination, respectively. R is the gas constant, T the absolute temperature and [F] the solution concentration of free ligand.

The substitution of Eqs. (2) in Eq. (1) a t constant temperature and pressure ( P ) gives


with AE;,, = ~ 2 ; : + p[;Ot - Pg'?. Now if we assume that

where for the i site with 1 multiple coordination G, , and ti, , are the free energy and the extent of reaction, respectively, from Eq. (3) it may be obtained

with

where the first term of the right hand side of Eq. ( 5 ) gives the difference from the equilibrium energy,

Now if we assume in agreement with ref. 28) that the reaction rate for the k site with j coordination is

of the k site.

Jk,j = (%) from Eq. (6) and Eq. (5) and using Eq. (4) it is possible to obtain

where f k , ( t ) is given by

Because the free energy changes for sites not at equilibrium are time dependent, we developed G , ( t ) in a Taylor series29) in the neighbourhood of the equilibrium time t;, obtaining:

The time derivative of Eq. (9) after considering that at equilibrium G)k,,j = 0 gives


If we expand also the reaction rate in a Taylor seriesz9), we have

Thus, upon substitution of Eqs. (10) and (11) into Eq. (8)

From this last equation it is possible to show that for a k type of site with j coordination

lim fk , , ( t ) = 1 r r ; ,

lim f k , j ( t ) = 1 1-+ m

It is to be noted that when the time t is approximately at equilibrium or if in this given time the equilibrium is reached, then fk, ( t ) approximates to unity and from Eq. (7) we obtain Kk,j = K i , j . On the other hand when t -+ 0 from Eq. (7) we have

Kk,j = f k , j (0)Ki,j with 0 5 fk , j (0) 5 1 (14)

Moreover when a type of site k with j coordination needs a very long time, ti , j , to reach the equilibrium, then f k , , (0) + 0 and it is impossible to determine Kk, j . In this case, obviously, the site is not involved in the binding experiment. Thus, these results suggest that in a binding experiment which involves cooperative or non-cooperative phenomena it may be observed only those type of sites with t i j in the measurement time range or near to it. In other words the sites which have a favourable rate of reaction, Jk,j ( t ) , and a short equilibrium time, t l j , are measured first.

Now, because

where Kk,j ( t ) is given by Eq. (7) and if we assume that for a particular binding experiment the maximum sites concentration of i type to be coordinated is

(16) [Bm,,;I = [Aj,01 + [Aj,lI + . . . + [A;, , ]


with

i= I

where [Bi] is the concentration of i sites bonded by the ligand, [B,,,] and [B,ax,i] represent, respectively, the maximum and the maximum i sites concentration that may be bonded by the ligand, by combining Eqs. (15)-(17) we obtain

S

C [Bmax,iI = [BI

' + C i= I

j= 1

From this last equation it should be noted that in the receptor the bonded sites concentration depends on the free ligand concentration, on [B,,, ;] and on K , , (i). On the other hand if we consider

where [bi] is the concentration of ligand bonded to the i site, [b,,,] and [bmax,i] are, respectively, the maximum and the maximum i ligand concentration that may be bonded to the receptor, and repeating the same development as for Eq. (18) we find

i = l = P I

j-1

Now these results allow us to emphasize that when we consider a receptor having only one type of site with multiple coordination, from Eq. (20) the well known Adair equation can easily be obtained.

Furthermore a comparison between Eqs. (19) and (16) shows that


Clearly, in the approximation of ,Ii = 0 it is possible to show that Eq. (20) becomes identical with Eq. (18). This remarkable result suggests that, if we are interested in obtaining the binding constants Ki, ( t ) and also the maximum concentration of sites present in the receptor, the binding experiment must be carried out at l j 3 0. Under these conditions we may fit the experimental data using independently Eq. (20) or (1 8).

From Eq. (21) it is possible to establish that ,Ii I 0 in two cases: a) when the receptor contains different types of sites without multiple coordination; b) when the receptor contains different types of sites with multiple coordination and the experimental data [b] and [F] are measured under experimental situations so that the concentration of multiple coordinated sites is very small in comparison with that of mono-coordinated sites. These results suggest that the best information may be obtained planning a suitable binding experiment and using Eq. (18) in the data fitting procedures.

Hence some algebraic manipulations of Eq. (18) lead to

where

j = I

The quantity ci(F) represents an appropriate measure of the extent of multiple coordination on the i site. Moreover, it is easy to show that, if we consider a receptor having only one type of site without multiple coordination, from Eq. (22) it is possible to obtain the well known Scatchard equation 1,2,17,21*22) .

Let us next assume that the mole fraction of the i site that may be bonded in the receptor is

with S

x,(F) = 1 and 0 5 x i (F) 5 1 i= I

1 x;(F) = S

1 + C [v,(F)]-1*v,(F) j = 1 j * i


Now if we expand V; ( F ) and 5 ( F ) in a Taylor series2g) at the neighbourhood of [So], which is the initial concentration of the ligand, we have

Because [So] %- [B], the Thylor expansion may be truncated at the linear term and because for the i site which is filling V; ( S o ) %- ~(S,)([So] - [B]) and 7 ( S o ) Q

l$ (So)([So] - [B]) we may write

j + i

l/pi has the unit of a concentration. Substituting [F] = [So] - [B] in Eq. (25) we obtain

where the quantity p i can be seen as a parameter that describes the influence of the parallel filling of the other j sites in comparison to that of the i site.

The substitution of Eq. (26) in Eq. (22), after some simple algebraic manipulations, yields

It is therefore apparent that for a receptor having only one type of site with a single coordination the expression (27) gives the well known Hill linear 17,22). The Hill constant‘*’’.”), which is used in the semiempirical treatment of binding data to study the cooperative associations of ligands to the receptor, may be obtained from Eq. (27):

Considering a receptor having only two types of sites with K , and K , constants, we may obtain that if K , < K2 then 1 < n,(F) 5 2 (positive cooperativity) and if K , > K , then 0 5 nH(F) < 1 (negative cooperativity). Moreover if we have K , = K2


then nH (F) = 1. These observations are in agreement with refs. j7) andz2) and show that cooperativity does not depend on the absolute binding constants but on their differences and on the mole fractions. These differences could depend on two phenomena:

a) if in the receptor there are only one or more chemically defined types of sites which after coordination of some ligands change their ligand-receptor affinity hence showing different binding constants during the experiment;

b) if there are different types of binding sites which always keep the same ligand- receptor affinity during the binding process.

Moreover, also the continuous variations of mole fractions during the binding experiment are responsible for the Hill constant changes. These changes then depend on the constant pi , cf. Eq. (25).

A further easy algebraic transformation of Eq. (27) gives

which represents the overall binding constants of the ligand-receptor system. This allows us to obtain the overall standard free energy changes during the ligand-receptor experiment

AG = -RTln(L(F)) (30)

and gives us a functional dependence of the free energy changes on the concentration of the free ligand in solution.

Simulations

This section will show the theoretical results developed in this work via computer simulations for several ranges of parameters at a fixed time t and for a ligand-receptor system where the receptor contains two types of sites. The computer simulations were generated for values of free and bonded ligands, respectively, in the ranges 0 I [F] I 5000 nmol/mg protein (prot) and 0.5 I [B] I 22 nmol/mg prot.

Simulation of [B]/[FJ on [B]

Figs. 2a, b, c and d show the dependence of [B]/[F] on [B] for a receptor having two types of sites. The [B]/[F] profiles shown were obtained by using Eq. (22) and assuming for I/Kl,l the value of 0.12 mg prot/nmol and for 1/K2,1 values which vary in the range 4.8 1

Fig. 2a reports the [B]/[F] curve profiles for a receptor having one or two types of receptor sites with single coordination. These were obtained by using the simplified

I I/Kz,, I 0.12 mg prot/nmol.

Eq. (22)

where

174 V. Di Nolo, L. Dalla Via, A. Toninello, M. Vidali

It should be noted that when the receptor has only one site without multiple coordination ( l /Kl , l = 1 /K2,1 = 0.12 nmollmg prot) the classical Scatchard linear plot is observed ' , I 7 ) . Moreover when the receptor exhibits two types of sites with single coordination, the curve profiles of [B]/[F] on [B] are like that of some hyperbolic function and are always upwardly convex. The convexity of these hyperbolic like profiles increases with the decreasing of l/K2,1 .

In Figs. 2 b, c and d we report the simulations of

(32)

1 + - = [Bl [(- [FI KZ,,(f) 1 +PI . [Fl

A h ( t ) ) F,],,Bmw1 - IB1) 1 +

+ ( Kz,, ( ~ ) K z , z ( ~ ) 1 + P I . [Fl where

1 -

1 A h ( t ) =

K,,l (f)KI,Z(f) %,I (t)KZ,Z(t)

which is obtained from Eq. (22) under the above conditions and assuming that each site may exhibit double coordination.

2.5

2.0

1.5 - I u-

2 m I

1 .o

0.5

0.0 0 5 10 15 20

P I

2.5 k o . 1 2 0 0

0 5 10 1 5 2.0 [Bl

Fig. 2a, b.


5.0

4.5

4.0

3.5

3.0 - - - Y . m - 2.5

2.0

1.5

1 .o

0.5

0.0 0 5 10 15 20 0 5 10 15 20

[Bl [El

Fig. 2. Dependence of [B]/[F] on [B] for a receptor having two types of sites. The profiles were calculated assuming that I/K,,, is constant at the value of 0.12 mg/nmol prot and 1/K2,1 is varied in the range 4.8 5 1/K2,, 5 0.12 mg prot/nmol, the values are reported in the figure; [F] and [B] vary, respectively, in the ranges 0 5 [F] I 5 000 nmol/mg prot and 0.5 I [B] 5 22 nmol/mg prot. Simulations: (a) without multiple coordination, see Eq. (31); (b), (c) and (d) were calculated by using Eq. (32) and assuming values of double coordination constants of the sites, respectively, of 0.1 To (b), 1 .O% (c) and 50% (d) of those of the single coordination constants

The curve profiles of Fig. 2 b are calculated by using Eq. (32) for values of the double coordination constants of the two types of sites which are 0.1% of those of the single coordination constant (I/K,,2 = O.OOl/K,,, and l /K2,2 = 0.001/K2,,). From this figure it is possible to note a functional dependence of [B]/[F] on [B] similar to that in Fig. 2a, but a different convexity is observed a t higher values of [B]. In fact the curves are downwardly convex for [B] 2 2.

Further simulations are obtained from Eq. (32) assuming the following values of the double coordination constants: I%, Fig. 2 c ( I /K l ,2 = O.Ol/K,,, and I/K2,2 = 0.O1/K2,,), and SO%, Fig. 2d ( I /K l ,2 = 0.5/K1,, and I/K2*2 = 0.5/K2,,) of those of the single coordination constant.

Figs. 2 c and d show that when the double coordination constant increases significantly in comparison with the single one, the downward convexity of the curves becomes more evident.

I76 V. Di Noto, L. Dalla Via, A. Toninello, M. Vidali

Simulation of In ([B]/([B,,]-[Bl)] on In (FI

The simulations in Figs. 3a, b, c and d are obtained by using Eq. (27) under the above conditions.

In Fig. 3a the ln([B]/([B,,,] - [B])) dependences on In[F] are shown for a receptor having one or two types of sites both with single coordination. The simulations were obtained by using

It is interesting to note that a receptor with only one site without multiple coordination shows the well known Hill linear plot I s 2 , 17) with a slope of unity and a receptor with two types of sites without multiple coordination yields non-linear curve profiles which are characterized by a slope tending to unity at very high and at very low In [F] values (Fig. 3 a).

The profiles of Figs. 3 b, c and d are obtained considering a receptor with two types of sites, double coordination and

2.5 -

2.0 -

- - - m 1.5- I - E a

k m 1.0-

I v

- C -

0.5 -

0.0

0 . 1 2 0 0 4 4 B

1.0 1.5 2.0 2.5 3.0 3.5 In Fl

2.5

2.0

1.5

1 .o

0.5

0.0

1.0 1.5 2.0 2.5 3.0 3.5 In [Fl

Fig. 3a, b.

Thermodynamic treatment of ligand-receptor interactions 1 I1

1.0 1.5 2.0 2.5 3.0 3.5 In [Fl

5 -

4 -

3 -

2 -

-

1 -

0

1.0 1.5 2.0 2.5 3.0 3.5 In IF1

Fig. 3. Dependence of ln([B]/([B,,,] - [B])) on In [F] for a receptor having two types of sites. Simulations: (a) without multiple coordination, see Eq. (33); (b), (c) and (d) were calculated by using Eq. (34) and assuming values of double coordination constants, respectively, of 0.1 'To (b), 1 .OVo (c) and 50% (d) of those of the single coordination constants. Simulation conditions as in Fig. 2

Thus, Figs. 3b, c and d report the curves of ln([B]/([B,,] - [B])) vs. ln[F] calculated assuming, respectively, that I/Ki,2 = O.OOl/K,, 1, I /Ki ,2 = O.O1/Ki, and l /K i ,2 = OS/K, , , with i = 1, 2.

It is seen that the double coordination of the receptor sites produces non-linear plots with slopes different from unity in the linear regions of the plots. These effects become more and more marked when I/K,.* is increased (see Figs. 3 b, c and d).

Simulation of n,(F) on [B]

The profiles of Figs. 4a, b, c, and d were obtained using Eq. (28) for a receptor having

For a system where the receptor sites exhibit only single coordination it is possible two types of sites.

to obtain from Eq. (28)


Fig. 4a reports the simulation obtained using Eq. (35) and assuming that l /Kl , , has a value of 0.12 mg prot/nmol and l/K2,1 varies in the range 4.8 * < i/K2,1 < 0.12 mg prot/nmol. It is to be observed that: (a) for these receptors n,(F) is always in the range 0 5 n,(F) 5 1; (b) when the differences between the constants, AK, become smaller, nH(F) approximates to 1; (c) the difference of n H ( F ) from 1 depends also on P, .

In the case of a receptor with sites that exhibit multiple coordinations we may obtain the following equation for the Hill constant

1 + A h ( t ) )IF]]

(36)

+ 1 + A h ( t ) ) IF]]

K2,j ( t)K2,2(t)

)K2,2(t)

1 + Pi . F l n,(F) = 1 +

1 + PI * [FI

1 .o

0.8

i2 v I c

0.6

0 5 10 15 20 [Bl

,0.1200 /-0.1080

0 5 10 15 20 [BI

Fig. 4a, b.


0 5 10 15 20 [El

0 0 o m N O r l ” 0 0

2.0

1.8

1.6

1.4

1.2

1 .o

5 10 15 20 [BI

Fig. 4. Dependence of nH(F) on [B] for a receptor having two types of sites. Simulations: (a) without multiple coordination, see Eq. (35); (b), (c) and (d) were calculated by using Eq. (36) and assuming values of double coordination constants, respectively, of 0.1 Yo (b), 1 .O% (c) and 50% (d) of those of the single coordination constants. Simulation conditions as in Fig. 2

In Figs. 4b, c and d the computer simulations of Eq. (36) are shown. These simulations were obtained by using the previous conditions ( l /Kl , l = 0.12 mg prot/nmol and 4.8 . < l/K2,1 < 0.12 mg prot/nmol) and changing the double coordination constants in such a way that the values of these are 0.1 Yo, 1 % and 50% of the corre- sponding single coordination constants. The simulations are shown in Figs. 4b, c and d where it is seen that n H ( F ) decreases below 1 at very low [B] concentrations until it reaches a minimum, and then it increases as [B] increases (see Figs. 4b, c and d). Furthermore when the influence of the multiple coordination on sites becomes higher, nH ( F ) increases and at a particular value of [B] it exceeds nH ( F ) = 1. Moreover when the contribution to the binding process of double coordination of sites becomes significant, n H ( F ) is tending asymptotically upward to 2, as Fig. 4 d shows. These theoretical results are in agreement with the experimental observations reported in the literature I * 17,22).


Conclusions

The treatment reported in this work yields a rigorous thermodynamic basis which explains in a coherent way the nonlinear profiles observed in the experimental Scatchard, Hill and Adair plots i , 2 , 17).

Moreover the model shows that the experimental Scatchard, Hill and Adair relations are specific simplifications of the more general Eq. (18) which may also be applied to experiments at equilibrium or not at equilibrium. On the other hand, by means of 'hb. 1, which summarizes the simulation results reported in this paper, we are able to obtain a first interesting preliminary qualitative information simply by observing the curve profiles and the convexity exhibited in the Scatchard and Hill experimental plots.

Tab. 1. Summary of the curve shapes and features obtained with numerical simulations of Eqs. (22), (27) and (28)

Sites Coordination Scatchard Hill n H

1 single linear-plot linear-plot 1

many single hyperbolic like-plot non-linear plot with 0 5 nH 5 1 with upward a slope tending to convexity unity at very high

and at very low In [F]

with downward a slope different from convexity unity in the linear

regions of the plot

many multiple hyperbolic like-plot non-linear plot with 0 5 nH 5 2

In fact when the Scatchard plot shows a hyperbolic-like profile with upward convexity we may say that the receptor has many types of sites without some multiple coordination, on the contrary when the convexity is downward we may say that the receptor has many types of sites with some multiple coordination. Analogous information is obtained and confirmed from the Hill plot and from the behaviour of his slope, n,(F), in the linear part of the curve profiles.

Moreover the model will provide, by using Eqs. (1 8), (22) and (27) and a curve fitting procedure, the parameters [B,,,], [B,,,, ;I, Kj , j ( t ) and the mole fraction dependence on [F] or [B] at the i site. Besides this, it is a very promising thermodynamic tool because it allows us to better understand and plan the binding experiment. In fact from Eq. (21) it is possible to specify that only when 1 = 0 we may obtain by a fitting procedure the correct [Bma.J and [B,,] and, as Eq. (7) suggests, only when all types of sites are at equilibrium we may obtain the true affinity constants K t j . Otherwise, we can determine only [b,,,], which is the maximum concentration of ligand bonded to the receptor, and Kj,i ( t ) , which is smaller than K i i , from a quantity depending on the time that the site i with j coordination requires to reach equilibrium.

Finally, the use of Eq. (30) yields a complete measure of the standard free energy changes during the ligand-receptor binding experiment.


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Thermodynamic treatment of ligand-receptor interactions