thermodynamic descriptors, profiles and driving forces in membrane receptor-ligand interactions
TRANSCRIPT
Introduction
From the bioenergetical point of view, nature and dynamics of any biochemical interaction is governed by driving forces resulting from interrelationships between contributions of reaction enthalpy (ΔH) and entropy (ΔS) to the changes of standard free (Gibbs) energy (ΔG). Moreover, changes of these three thermodynamic state descriptors may generate additional information regarding structure, mechanism of action and certain functions of the biological system in question. They are, therefore, commonly investigated in ligand-receptor and similar biological interaction studies; numerous data are reported in the literature (1). Under suitable circum-stances, two of them can be assessed experimentally: ΔH by isothermic titration calorimetry (ITC) (2–7), ΔG
by conversion of the equilibrium dissociation constant K
d obtained in a binding experiment:
(1)∆ =G RT K dln
(R is the gas constant, 8.314 J·K−1mol−1); ΔS at a constant absolute temperature T follows from the Gibbs–Helmholtz equation
∆ = ∆ − ∆G H T S . (2)
Moreover, ITC measurements of ΔH at various tem-peratures or the other calorimetric alternative, the dif-ferential scanning calorimetry, enable reliable estimates of the reaction heat capacity change, ΔC
p. Their use in
the reaction thermodynamics of ligand-macromolecule
Journal of Receptors and Signal Transduction, 2010; 30(6): 454–468
Address for Correspondence: Vladimir Pliska, Collegium Helveticum, ETH Zurich STW-GLA, Schmelzbergstrasse 25, CH-8092 Zürich, Switzerland. E-mail: [email protected]
R e s e a R c h a R T I c L e
Thermodynamic descriptors, profiles and driving forces in membrane receptor-ligand interactions
Vladimir Pliska
Collegium Helveticum, Swiss Federal Institute of Technology (ETH) and The University of Zurich, Zurich, Switzerland
abstractExtension of the (isothermal) Gibbs–Helmholtz equation for the heat capacity terms (ΔC
p) allows formulating
a temperature function of the free (Gibbs) energy change (ΔG). An approximation of the virtually unknown ΔC
p temperature function enables then to determine and numerically solve temperature functions of ther-
modynamic parameters ΔH and ΔS (enthalpy and entropy change, respectively). Analytical solutions and respective numeric procedures for several such approximation formulas are suggested in the presented paper. Agreement between results obtained by this analysis with direct microcalorimetric measurements of ΔH (and ΔC
p derived from them) was approved on selected cases of biochemical interactions presented
in the literature. Analysis of several ligand-membrane receptor systems indicates that temperature profiles of ΔH and ΔS are parallel, largely not monotonic, and frequently attain both positive and negative values within the current temperature range of biochemical reactions. Their course is determined by the reaction change of heat capacity: temperature extremes (maximum or minimum) of both ΔH and ΔS occur at ΔC
p = 0,
for most of these systems at roughly 285–305 K. Thus, the driving forces of these interactions may change from enthalpy-, entropy-, or enthalpy-entropy-driven in a narrow temperature interval. In contrast, thermo-dynamic parameters of ligand-macromolecule interactions in solutions (not bound to a membrane) mostly display a monotonic course. In the case of membrane receptors, thermodynamic discrimination between pharmacologically defined groups—agonists, partial agonists, antagonists—is in general not specified and can be achieved, in the best, solely within single receptor groups.
Keywords: Membrane-coupled receptors; thermodynamics/ligand-receptor interaction; thermodynamic descriptors; heat capacity change; agonism/antagonism; enthalpy/entropy change
(Received 02 August 2010; accepted 10 August 2010)
ISSN 1079-9893 print/ISSN 1532-4281 online © 2010 Informa Healthcare USA, Inc.DOI: 10.3109/10799893.2010.515594 http://www.informahealthcare.com/rst
Journal of Receptors and Signal Transduction
2010
30
6
454
468
02 August 2010
00 00 0000
10 August 2010
1079-9893
1532-4281
© 2010 Informa Healthcare USA, Inc.
10.3109/10799893.2010.515594
RST
515594
LRST
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Thermodynamics of ligand-receptor interactions 455
interactions in homogenous media, mostly in aqueous solutions, became a common routine.
Microcalorimetric measurements in membrane-bound components are, however, still impractical. Although the heat change detection limit of the ITC method lies for new microcalorimeters at 0.4–4 μJ (8) that would allow measurements at a receptor concentra-tion in the 10−11 m range for interactions with ΔH around ±40 kJ·mol−1, calorimetry in membrane suspensions will scarcely yield reliable results. Thus, the thermodynamic analysis of ligand-membrane receptor interactions still relies on the use of the van’t Hoff isochore defining the temperature change of K
d as
d ln
d.
2
K
T
H
RTd = −
∆
(3a)
Its integrated form
ln1 ,K
H
R T
S
Rd =∆
−∆
(3b)
is formally identical with the combination of eqs. 1 and 2. Most popularly, ΔH (in this case frequently denoted as “van’t Hoff enthalpy”) and ΔS at a temperature T are estimated from this equation under the assumption that their change within a certain temperature range is negligible, and thus, that ΔG is in that range a linear function of T (the so-called van’t Hoff plot analysis). However, the Gibbs–Helmholtz equation defines the proportion of energy terms ΔH, TΔS at a constant temperature T; thus, T is not a variable in eqs. 2 and 3, and the two thermodynamic terms involved are not temperature independent.1 Conversion of eq. 2 into a temperature function applicable within an interval T
1, T
2 requires insertion of corresponding temperature
dependencies of ΔHT and ΔS
T . The former follows from
the Kichhoff’s law
∆ = ∆ + ∆H H C T TT T p
T
T
2 1
1
2
d ,( )∫
(4)
the latter is expressed as
∆ = ∆ +∆
S SC T
TTT T
p
T
T
2 1
1
2
d ,( )
∫
(5)
where ΔCp is the change of heat capacity (at a constant
pressure) from the initial (non-bound) to the final (bound) state of the system, ΔC
p(T) its temperature
dependence, subscripts T1, T
2 refer to the initial and final
temperatures. The temperature-dependent Gibbs free enthalpy, ΔG(T), near an arbitrarily selected reference temperature τ is then expressed by
∆ = ∆ − ∆ + ∆ −∆
τ τ
τ τ
G T H T S C T T TC T
TTp
Tp
T
( ) ( ) ( )∫ ∫d d .
(6)
Obviously, temperature changes of heat capacity may in numerous instances considerably influence the course of ΔG(T) even in a narrow temperature range around τ. In consequence, the use of eq. 3b with temperature as a variable is essentially incorrect; it may yield unreliable ΔH, ΔS estimates in cases of interactions in that only weak chemical bonds are reshaped.
An analytical form of the temperature function ΔC
p(T) is virtually unknown. It can be constructed form
the calorimetric ΔH data by using a stepwise selec-tion of the optimal regression model (corresponding routines are usually parts of standard computerized statistical packages), or by using a polynomial ΔC
p(T)
approximation, as it is common in similar physico-chemical instances. This latter procedure is suitable for thermodynamic studies of receptor-ligand interactions in which independently obtained ΔH data are not avail-able. Solutions based on polynomial approximations of the extended Gibbs–Helmholtz equation (eq. 6) are subjects of this communication. They are demon-strated on several sets of data presented in the recent literature.
Materials and methods
Data sets
Computation procedures were used for analysis of data taken from literature sources cited below (see references). Most of them were revised in the recently published com-pilation (1). The reference temperature (τ) for thermody-namic data was 298 K.
Interpolation routines
In order to use nonlinear models in the data analysis, incomplete ΔG(T) values (e.g. in cases of large temperature intervals between two neighboring values) were extended for computational purposes by interpolated auxiliary values. Nonlinear and/or regression routines were then employed to analyze these sets. Several routines available in the commercial software packages were employed for interpolation. The function “InterpolatingPolynomial” in the MATHEMATICA 5 software2 symbolically computes
1ΔH and ΔS may appear constant usually only in “highly” exothermic or endothermic chemical reactions but scarcely in ligand-macromolecule interactions in which mainly weak bonds are reshaped.
2http://mathworld.wolfram.com/LagrangeInterpolatingPolynomial.html
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456 V. Pliska
the interpolating Lagrange polynomial that can then be used for the input of missing values. Other routines (spline, cubic spline, piecewise cubic Hermite interpo-lation) are parts of the MATLAB 7 software (function “interp1”).3 This latter routine was preferred in the com-putations presented here.
Nonlinear data analysis
An analysis of temperature functions was customarily carried out by nonlinear and multiple regression rou-tines of the software package SYSTAT 124 for several polynomial degrees. Only those models were accepted whose all regression coefficients were significant at least at a significance level α ≤ 0.05. The F-test for selection of the optimal polynomial degree was based on differ-ences between regression variances of two subsequent polynomials (degree i and i+1, respectively) expressed in terms of the corresponding multiple correlation coefficients r
i, r
i+1 (9)
Fr r
rn pi i
ii= +
++
12 2
12 11
( ),
(7)
where n is the number of data pairs, p the number of (all) polynomial coefficients. Numbers of degrees of freedom ν for the F-value are (1; n-p
i+1); the error prob-
ability α was evaluated by the MATLAB function “fcdf”. The critical significance level was arbitrarily set to α = 0.01 (the generally recommended level in these instances is between 0.01 and 0.05). The first polynomial that appears insignificantly different from the next higher one can be considered as optimal. When n-p is small (usually when the size of the data set is small as it is frequently the case here), the use of a multiple correlation coefficient R
adj
adjusted for the given degrees of freedom (n-p) instead of r in eq. 7 is indicated (10),
R rn
n padj2 21 1
1= − −
−−( ) .
(10)
Selection of the regression model
Stepwise regression analysis (11,12) was used to mini-mize the number of temperature terms in regression models described by multiple T-terms. The SYSTAT procedures used to these aims (13) allow an automatic variable reduction in both forward and backward direc-tion (i.e. by subsequent adding a variable beginning with no variables in the model, or stepwise removing single variables beginning with a complete model).
Results
Mathematical models
General solution of thermodynamic functionsThe universal form of the ΔG(T) function, irrespective of a particular form of ΔC
p(T), results from a resolution
of the second integral term TC T
TTp
T ( )d
∫ in eq. 6 (by
applying rules of per partes integration)
∆ ∆ ∆G T G T1
TC T T Tp( ) = − ( )
∫∫τττ
2 d d ,TT
(11)
where ΔGτ is the Gibbs energy at the reference tempera-ture τ. The corresponding temperature functions of ΔH and ΔS are generalized eqs. 4 and 5,
∆ ∆ ∆H T H C T Tp( ) = ( )∫ττ
+ dT
(12)
∆ ∆∆
S T ST
T( ) = +( )
∫ττ
CTp
T
d .
(13)
Eqs. 12 and 13 for computation of ΔCp(τ) (i.e. at a single
temperature τ) were recently used by Yoo and Lewis (14). The temperature T
e of their prospective extremes—maxima
or minima—within a given temperature interval are identi-cal with the temperature at ΔC
p(T) = 0, since obviously
d
d
d
d.
∆ ∆∆
H
TT
S
TCp≡ =
(14)
The entropic contribution to the Gibbs free energy, TΔS, displays an extreme at the temperature for which ΔC
p = −ΔS and is, usually only slightly, shifted against
the other two extremes. The nature of these extremes is determined by the sign of the second derivative at T
e,
d
d
d
d
2
2
∆H
TT=
∆Cp
(15)
d
d
1 d
d
d
d.2
2∆ ∆ ∆S
T T T
S
T= −
Cp
(16)
Thus, when the slope of the heat capacity curve ΔCp(T) = 0
at temperature Te is negative, ΔH(T) reaches a maximum,
otherwise a minimum. As for ΔS(T), a maximum occurs when d
d
d
d
C
TS
Tp < , a minimum when d
d
d
d
C
TS
Tp > .
Approximation of heat capacity-temperature functionsIn terms of the rule for summation of internal energies (15), the overall heat capacity of a mixture of components in diluted water solutions equals to the sum of molar heat capacities times molar fractions of the respective
3http://www.mathworks.com/access/helpdesk/help/techdoc/ref/interp1.html4http://www.systat.com/products.aspx
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Thermodynamics of ligand-receptor interactions 457
components. However, temperature functions neither of individual C
p’s of reactants and products, nor of their
change in an interacting system (ΔCp) can be derived from
thermodynamic relationships; a physically well-founded form of ΔC
p(T) remains virtually unknown. Pragmatically,
a temperature change of Cp for numerous substances is
commonly approximated by a polynomial in T,
C T = a a T a T a Tp ( ) 0 1 22
33 ...+ + + +
(17)
or in a broader temperature range (16,17) to
C T = a bT cTp ( ) + + −2 .
(18)
Since the summation rule (see above) is obviously valid also for linear combinations of eqs. 17 or 18, analogous temperature functions are applicable also for changes of the heat capacity ΔC
p (from an initial to equilibrium
state) in an interacting biological system.In order to select the “best general” approximation, the
significance of single terms in eqs. 17 and 18 (independ-ent variables T, T2, T−2 and constant terms) upon ΔC
p was
investigated by stepwise regression analysis of several ΔCp
data sets obtained either directly by calorimetry (18–20) or derived from calorimetric measurements of ΔH (21–23). Models based on eq. 17 comprising one (constant term a
0) to three (linear or 2nd order polynomial) a-terms were
detected as optimal ΔCp approximations. The T−2-term
was eliminated in all investigated sets, just confirming the assumption that eq. 18 is impractical as an approximation of ligand-macromolecule interactions, and perhaps also others, in a narrow temperature range.
However, rather contradictory physical and formal mathematical requirements restrict the free choice of the final approximation formula:
1. Since molar heat capacities at zero temperature equal zero, the limit value of any heat capacity change is zero: lim
T pC→
=0
0 . (The limit value of enthalpy change at zero temperature is generally different from zero (24) and reflects the overall sum of bond enthalpies of interacting components at equilibrium.) Thus, the con-stant term in the physically relevant ΔC
p temperature
function should be zero (a0 = 0 in eq. 17).
2. Above this boundary condition, ΔCp may reach an
almost constant value over a certain temperature range, as it is frequently reported for manifold bio-chemical interacting systems. In general, and con-trary to the previous condition, a
0 ≠ 0.
Obviously, the first model is represented by a polynomial
∆C T a T a T a Tp nn( ) = + + +1 2
2 ....... . (19)
Although eq. 19 fulfils the “zero” condition, limT pC
→=
00 ,
it can approximate solely the temperature range around
the accessible experimental (and interpolated) data. The second model may therefore offer a better numeric solu-tion. For computational reasons, temperature values can be related to a selected reference temperature τ as (T-τ); the constant term is then equal to the heat capacity at the reference temperature, ΔC
p,τ:
∆ = ∆ + − τ + − τ + + − ττC T C b T b T ... b Tp p, z
z( ) ( ) ( ) ( )1 2
2.
(20)
Temperature change of the Gibbs free energy: ξ-transformation5
Inserting eq. 19 into eq. 11 results in
∆ ∆ ∆G T RT K H Td( ) +=∑≡ = −τ τln ,S aii
z
j11
(21)
where
j t tii i
i ii i=
++ − −( )+1
11 1
( )( )T T i+1 .
(22)
Usually, computations comprises maximally the first three transformed temperature ξ
i terms,
j t t
j t t
j t t
12 2
22 3 3
33 4 4
1
21
2
1
3
1
2
1
3
1
4
1
3
= − +
= − +
= − +
T
T
( )T
T
T T
.
(23)
Constant coefficients ai obtained by regression analysis
using ξ-transformed T values determine then temperature functions of enthalpy and entropy changes,
∆ ∆H T Ha
iTi
i
ni i( ) ( )= +
+−
=
+ +∑τ τ11
1 1
(24a)
∆ = − ττ=
S T S( ) ( )∆ + ∑ a
iTi
i
ni i
1
(24b)
Temperature change of the Gibbs free energy: ψ-transformationSimilarly, when using eq. 20 for ΔC
p approximation, the
temperature function of ΔG is
∆ ∆ ∆ ∆G T H T S C bp ii
z
i( ) ,= − +=∑τ τ τ +c c0
1 (25)
where for any i (from zero to z)
5SYSTAT commands (as text files) for computation of thermodynamic descriptors and some additional criteria from T, ΔG data sets can be required by the author.
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458 V. Pliska
c i
i i
i= −
+
− ττ
+τ
+
τ
T Ti
T
i j
j
ii1
1 1 1
1
1
( ) −
+( )
−( ) − +=
−∑
ln
11
jT
j−( )
τ
.
(26)
The first three terms, commonly used in regression analysis, are
c
c
c
0
1
2
1
1 2
= − τ − +τ
τ τ τ ττ
TT
T
TT
T
T
ln
ln
= − −( ) − −
−
2
== − −( ) − −( )+ −
+
TT T
T
T
3.
1
22 1 3
2 2 2τ τ τ τ τ ττ
ln
(27)
Using computed bi, temperature functions ΔH(T) and
ΔS(T) become
∆ = ∆ + ∆ −τ ++
− ττ τ=
+H T H C T
b
iTp,
i
zi( ) ( ) ( )∑ 1
1
1
1,
(28a)
∆ = ∆ + ∆τ
+ −τ τ+
=
S T S CT
bp,
i
i
z
i( ) ( )∑ln 1 .1
11c
(28b)
Although ψ-transformation, compared to ξ-transformation, of temperature T is more complex, it is preferred to the previous one for several reasons: ΔC
p,τ values result straightforward from the regression analysis, smaller numeric values (T-τ) better ensure its accuracy.
Relation between ξ- and ψ-transformation proceduresObviously, coefficients a
i and b
i in eqs. 19 and 20 are
interdependent and mutually linked by rather sim-ple formulas. These contain binomial coefficients,
Cj
i j iij =
−!
!( )!, since powers of (T-τ)z follow the binomial
expansion:
a C bij i
j i
z
ij
ij i= τ−
=
−( )−∑ 1
(29a)
bi = τ −
=
C aij
jj i
j i
z
∑ ;
(29b)
in eq. 29b, b0 ≡ ΔC
p,τ.Correspondence of the two transformation modes is
apparent in the limit case τ = 0, i.e. when the integrations in eqs. 6, 11–13 cover the entire temperature range from zero to T. Under this condition ΔC
p,0 = 0, and eq. 20 con-
verts to eq. 19 with a new set of polynomial coefficients
a0,i
. Since also limits of the entropic contribution at T = 0, TΔS
0, in eq. 21 equals zero, one obtains
∆ = ∆ −+=
+G T Ha
iT,i
i
zi( ) ∑0
0
1
1
1.
(30)
Thus, when the lower integration limit is shifted to zero, the resulting ΔG(T) equation is analogical to the empiric polynomial relationship currently used for esti-mation of thermodynamic functions, as suggested by Edelhoch and Osborne (25) (denoted in this article as π-transformation),
∆ = + + + +G T c c T c T c T( ) 0 1 22
33 .......;
(31)
When the “complete” ΔCp approximation by eq. 17 is
employed, i.e. a0 ≠ 0, lower summation limit in eq. 30 is
i = 0 and eqs. 30 and 31 become identical.
Correspondence between extended Gibbs–Helmholtz analysis and calorimetric measurements
Calorimetric data on ligand—membrane receptor inter-actions suitable for comparison with results of the pre-sented analysis were not found in the current literature. Thence, in order to estimate the extent of agreement between direct calorimetric ΔH and ΔC
p measurements
on the one hand and the ΔH(T), ΔCp(T) relationships
resulting from the corresponding ΔG(T) analysis on the other hand, reported data on several biochemical interactions in homogenous solutions were used.
Case 1: ribonuclease S’Figure 1 shows the case of ribonuclease S’: interactions of the S-protein with the S-peptide (upper panels) and a modified S-peptide analog, Met(O
2)-13-S-peptide (lower
panels) (18). The temperature dependence of ΔH shows at a first inspection a curvilinear form and ΔC
p is linearly
decreasing within the investigated temperature range. Clearly, optimal transformation functions ψ (one b-term), ξ (two a-terms), and π (eq. 31, three terms) yield ΔH(T) curves identical with the polynomial function fitted to the ΔH data (for details see caption to Figure 1); lower degrees of the transformation functions (exemplified by the 2nd order polynomial in Figure 1) result in an insufficient fit. ΔC
p(T) functions, too, display within the physically relevant
temperature range almost an identity of experimental and transformation profiles, for the modified S-peptide (lower right panel), and at least the same trend for the other lig-and, the S-peptide (upper right panel). The course of the ΔC
p(T) functions was drawn here in an extended tem-
perature range (from 200 to 325 K), in order to show that both curves obtained by T-transformations and by fitting experimental data display similar trends even outside of the measurement range. In this instance, the correspondence of the two modes can be classified as very tight.
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Thermodynamics of ligand-receptor interactions 459
Case 2: Taq DNA polymeraseThermodynamic analysis of the interaction between the DNA polymerase from Thermus aquaticus (Taq) to the DNA primed-template is presented in Figure 2 (19,26). The interaction is accompanied with conformational rear-rangements of both the DNA and the protein, as inferred from the circular dichroism spectra. ΔG values result from the ITC at various temperatures. The left-hand panel shows calorimetrically determined ΔH values (closed circles) fitted by a second-order polynomial (bold line: exp) and computed temperature function of the van’t Hoff enthalpies. In this instance, all methods used (eqs. 21, 25, 31) yield practically overlapping curves for corresponding z-degrees of approximation polynomials (corresponding triplets for ψ, ξ, and π are z, z+1, z+2, respectively). Due to a midsized scatter of the data, the approximations do not yield a full overlap with the calorimetric curve: significant
correlations of lower z-degrees slightly decline in the mid-dle temperature range, those of higher n follow better the trend of the experimental curve but deviate in the lower range. As for ΔC
p(T) on the left-hand panel, the suitabil-
ity of a fit cannot be unambiguously assessed since the comparative curve was estimated as a derivative dΔH/dT of the calorimetric ΔH data in the left-hand panel. In any case, the negative slope of the calorimetric ΔC
p(T) curve
is reflected in the approximations of higher z-degree.
Case 3: trp repressor/operatorA thermodynamic analysis of the trp repressor/opera-tor system from Escherichia coli is presented in Figure 3. Ladbury et al. (21) identified by isothermal titration calorimentry of the (dimeric) repressor protein two independent binding sites on the 20 base-pair operator. Calorimetric data show a low but distinct scatter around
S-peptide
0
−50
−100
−150
−200
−250
−300
−350
∆H [k
J m
ol−1
]
∆Cp
[kJ
K−1
mol
−1]
T [k] T [k]260 270 280 290 300 310 320
π2
π2
π3π3
ψ1 ψ1
exp
expξ2
ξ2
15
10
5
0
−5
−10
−15180 200 220 240 260 280 300 320 340
Met(O2)-13-S-peptide
−50
−100
−150
−200
−250
−300
∆H [k
J m
ol−1
]
∆Cp
[kJ
K−1
mol
−1]
T [k]
π2
π3ψ1
ξ2
T [k]
π2
π3
ψ1
exp
ξ2
15
10
5
0
−5
−10
−15180 200 220 240 260 280 300 320 340
0
260 270 280 290 300 310 320
Figure 1. Ribonuclease S’: interaction S-protein with S-peptide (upper panels) and Met(O2)-13-S-peptide (lower panel). Calorimetric data (18)
(circles) were fitted by second-order polynomials (bold lines), curves resulting from optimal ξ, ψ- and polynomial (π) transformations by thin lines. Subscripts at ξ, ψ, π symbols denote the degree of the corresponding transformation function. Dash-dotted lines display strongly deviating fits. Shaded fields in right panels depict experimental temperature range.
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460 V. Pliska
∆H [k
J m
ol−1
]
∆Cp
[kJ
K−1
mol
−1]
T [k]
π2
π2
π3
π3
ψ1
ψ0
ψ1ψ0 exp
exp
150
100
50
0
−50
−100
−150
−200
−250 −8
−7
−6
−5
−4
−3
−2
−1
0
1
260 280 300 320 340 360T [k]
260 280 300 320 340 360
Taq DNA polymerase
Figure 2. Taq-polymerase binding to the DNA primed-template. Data from (19,26). Symbols see Figure 1.
trp repressor/operatorsite 1
50
25
0
−25
−50
−75
−100
−125
−150
−50
−100
−1
−2
−3
−4
−5
∆H [k
J m
ol−1
]
∆Cp
[kJ
K−1
mol
−1]
ψ1
ψ0
ψ0
ψ1
ψ2
ψ1
ψ2
exp
exp
exp
exp1
exp3
T [k]
260
250
200
150
100
50
0
270 280 290 300 310 320
T [k]
260 270 280 290 300 310 320
T [k]
260 270 280 290 300 310 320
T [k]
260 270 280 290 300 310 320
∆H [k
J m
ol−1
]
∆Cp
[kJ
K−1
mol
−1]
π2
π2
−2.0
−2.5
−3.0
−3.5
−4.0
−4.5
−5.0
p2
p3
p3
p2
1
0
site 2
Figure 3. trp Repressor/operator system (21). ΔH data were fitted by a 3rd order power function A + BT3 (stepwise regression); both this and a linear function are highly significant. Differences between the two fits are reflected in the course of ΔC
p(T) function (symbol exp
3 for the power
function, exp1 for the linear one). Other symbols cf. Figure 1.
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Thermodynamics of ligand-receptor interactions 461
Table 1. Thermodynamic parameters (298 K) of membrane-bound receptor systems.
Receptora Ligandb Responsec Ref.ΔH kJ mol−1
TΔS kJ K−1mol−1
ΔCp kJ
K−1mol−1 Te
A1
BA ag (27) −13.15 27.81 −8.94 286.2 −0.72
CHA −35.39 15.93 −13.03 283.3 −0.84
CPA −29.11 31.243 −20.12 286.1 −1.62
LPIA −10.64 41.68 −14.18 287.8 −1.34
MECA 5.18 45.96 −8.68 287.5 −0.80
NECA −10.52 35.47 −5.16 283.1 −0.33
PAA −19.97 13.66 −8.11 280.3 −0.43
8-PT ant (27) −41.98 −1.69 −3.57 284.5 −0.25
CPT −55.97 −10.97 −0.11 301.9 0.03
DPCPX −30.97 −19.02 −3.88 289.2 −0.43
Theophylline −39.68 −11.03 −0.90 326.8 0.03
AChE Reversed acetylcholine (28) −17.57 −3.43 −5.27 292.5 −0.95
2-Furyl(furthrethonium)-TMA 2.75 22.92 −1.99 295.9 −0.76
2-Thienyl-TMA −27.25 −7.55 0.48 296.8 0.37
5-Methyl-2-furyl (methyl- furthetho-nium)-TMA
−9.17 9.54 0.34 289.4 −1.02
l-β-Methylacetylcholine 9.30 25.33 0.28 290.2 −0.12
Cycloheptyl-TMA −18.15 −0.29 1.85 291.5 0.28
β−AR Adrenaline ag (29) −44.39 −11.65 −0.36 297.4 −0.54
Isoprenaline −42.83 −2.92 −0.93 294.5 −0.26
Noradrenaline −112.77 −76.92 −6.51 284.8 −2.12
β-AR IHYP ant (29) −20.08 40.17 −1.25 269.4 −0.04
Practolol 42.00 71.045 2.17 299.8 −1.34
Propranolol 10.12 61.29 −0.52 297.4 −0.91
β-AR (solubilized) Isoprenaline ag (30) −27.60 14.13 6.20 291.3 0.93
Propranolol ant −32.17 19.74 6.84 294.4 1.90
D2
(±)ADTN ag (32) 4.06 51.44 2.46 294.1 0.62
DOPAmine −13.94 26.32 −5.270 338.4 0.15
(+)Butaclamol ant (32) 73.13 118.84 2.63 291.3 0.38
[3H]Spiroperidol 41.61 98.21 1.32 289.5 0.15
6,7-ADTN ag (31) −52.28 −12.41 −6.98 290.9 −0.95
Apomorphine 0.81 40.10 0.81 291.3 0.12
(+)Butaclamol ant (31) 41.46 89.36 3.39 292.2 0.58
(±)Sulpiride −85.04 −40.95 −4.04 307.6 0.43
[3H]Spiperone −14.91 41.95 −1.54 301.0 0.51
Cis-fluopenthixol −4.89 41.17 −1.64 305.6 0.22
Haloperidol −13.17 37.76 1.42 288.5 0.14
Piquindone −79.82 −33.16 2.70 292.3 0.46
GABAA
4-PIOL ag (33) −3.35 25.54 0.80 290.3 0.10
GABA 11.45 46.97 0.07 298.8 −0.09
Imidazole-4-AA 4.09 32.33 0.59 288.4 0.06
Isoguvacine 5.83 38.80 0.49 296.9 0.42
Muscimol 26.75 65.41 0.62 299.2 −0.54
THIP 0.75 29.99 −0.42 282.8 −0.20
Bicuculine ant (33) −18.10 10.93 0.70 292.7 0.17
Pitrazepin −5.51 31.92 0.60 298.9 0.69
R 5135 −17.62 29.47 −0.04 302.7 −0.045
SR 95103 −20.10 14.07 −0.97 284.3 −0.47
SR 95531 −20.10 22.06 −0.09 294.1 −0.09
Isulin-lymphocytes Insulin ag (34) −19.76 34.22 −2.14 307.0 0.24
nAChR (nicotinic) [3H]Cytisine ag (35) −46.90 2.92 −5.90 286.1 −0.48
Carbachol −39.16 −4.12 −2.25 282.0 −0.13
Nicotine −39.31 4.67 −3.54 284.8 −0.26
Table 1. continued on next page
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462 V. Pliska
Table 1. Continued.
Receptora Ligandb Responsec Ref.ΔH kJ mol−1
TΔS kJ K−1mol−1
ΔCp kJ
K−1mol−1 Te
Dihydro-β-erythroidine ant (35) 41.20 77.42 −0.88 295.1 −0.30
Hexamethonium 6.16 26.82 0.89 290.9 0.12
Succinylcholine 17.20 44.22 0.36 312.6 −0.03
Tubocurarine 17.43 46.61 −0.01 297.2 −0.01
OP Etorphine/−Na ag (36) 7.37 62.72 −1.72 16.6 −0.01
Etorphine/+Na 2.59 57.38 −0.72 288.3 −0.07
Diprenorphine/−Nad ant (36) −5.14 48.12 −1.20 291.6 −0.19
Diprenorphine/+Na3 −14.11 40.83 −3.30 268.3 −0.11
OXTR high Kaff
Oxytocin ag (16) 45.02 96.71 1.39 291.8 0.22
Oxytocin-GTPe 147.76 196.35 7.38 300.2 −3.83
OXTR high Kaff
Atosiban ant (16) −116.26 −77.59 −8.80 287.5 −0.81
OXTR low Kaff
Oxytocin ag (16) −17.89 27.18 1.60 281.2 0.10
Oxytocin-GTP4 −174.14 −139.28 −10.03 284.2 −4.17
OXTR low Kaff
Atosiban ant (16) −120.81 −92.79 2.00 294.9 0.64aReceptor groups: abbreviations see text.bSubstance names and chemical nature see cited references.cag: agonist; ant: antagonist.dNa+ in the medium absent: −Na; present: +Na.eGTP in the medium present.
an optimized cubic power function obtained by the step-wise regression (see caption to Figure 3). In case of the binding site 1, however, the estimated course of the van’t Hoff enthalpy by the commonly used Edelhoch–Osborne method (25) strongly deviates for both experimental ΔH and derived ΔC
p data. On the other hand, ξ- and
ψ-transformations generate acceptable approximations. As for binding site two, stepwise regression indicates again the best fit of ΔH with the 3rd order power func-tion, but there is no significant difference with respect to a simple linear fit. Differences, however, are visible in the ΔC
p curves, and it cannot be a priori decided, which
temperature function is physically more appropriate. Generally, the values of ΔC
p resulting from any transfor-
mation procedure are somewhat less negative then those derived from the derivatives dΔH/dT.
Thermodynamic profiles of ligand—membrane-bound receptor interactions
Descriptors of temperature functionsThe functions ΔH(T) and ΔS(T), constructed from data in the literature (cf. Table 1), usually display a curvilinear non-monotonic course, with an extreme at the tempera-ture T
e (eq. 14). The nature of the extreme is, as mentioned
above (eqs. 15 and 16), determined by the sign of the ΔCp
slope at this temperature,
sT p Tee
C T= ∆ /d d .( )
(32)
Obviously, as follows from eq. 11 and from corresponding transformed eqs. 24 and 28, the courses of the two tem-perature functions, and those of the entropic contribution,
are parallel. By solving eqs. 21 and 25 numerically, the two descriptors can—up to a certain polynomial degree—be computed from the ξ and ψ coefficients. Thus, for a 2nd order polynomial ΔC
p approximation in the former case
we obtain
Ta
aa
e
Te
= −
= −
1
2
1 ;s
(33a)
for the 3rd order polynomial two pairs of solutions results:
Ta
a a a a
a a T a T
e
T e ee
= − ± −
= + +
1
24
2 3 .3
2 22
1 3
1 2 32
( )s
(33b)
Similar relationships for the ψ-transformations are
TC
bb
ep,
Te
= τ −∆
=
τ
1
1 ,s
(34a)
for the ΔCp approximation with a single b-term, and
Tb
b b b b b b b
b b
e p,
Te
= − ± τ ± − τ − ∆ − τ ± τ
= ±
τ1
22 ( 2 ) 4 ( C )
22
1 2 1 22
2 1 22
1 2
( )s (( ) .Te − τ
(34b)
for two b-terms.Profiles of selected receptor groups and ligands are
presented in Figure 4. A characteristic feature of mem-brane-bound receptor systems is a non- monotonic course of the ΔH(T), ΔS(T) curves. In contrast to the majority of biochemical interactions in liquid media
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Thermodynamics of ligand-receptor interactions 463
(cf. Figures 1–3), they display an extreme at a temper-ature T
e, the type of which (maximum/minimum) is
determined by the sign of Te. Obviously, the profiles
are neither characteristic within individual groups nor for biological effects elicited by the correspond-ing ligands (agonists or antagonists). Driving forces
(enthalpy, entropy, and enthalpy-entropy), depicted in Figure 4 by differently shaded areas, frequently change within the experimental temperature range (0–37–42°C). In these instances, it is problematic to unambiguously assign a specific driving force to the respective interaction.
∆H, T
∆S [k
J m
ol−1
]∆H
, T∆S
[kJ
mol
−1]
∆H, T
∆S [k
J m
ol−1
]
∆H, T
∆S [k
J m
ol−1
]∆H
, T∆S
[kJ
mol
−1]
∆H, T
∆S [k
J m
ol−1
]
T [K]
∆Cp [kJ m
ol −1K−1]
∆Cp∆Cp
∆Cp
∆Cp
∆Cp
∆Cp
100
25
0
0
−100
−25
−50
−75
−100
−200
−300260 270 280 290 300 310 320
T [K]260 270 280 290 300 310 320
T [K] T [K]260 270 280 290 300 260 280 320300310
T [K]260 270 280 290 300 310 320
T [K]260 270 280 290 300 310 320
−30
−20
−10
0
10
20
∆Cp [kJ m
ol −1K−1]
∆Cp [kJ m
ol −1K−1]
∆Cp [kJ m
ol −1K−1]
∆Cp [kJ m
ol −1K−1]
−20
−10
−1
−2
−3
−4
0
5
0
10
10 4
2
0
20
∆Cp [kJ m
ol −1K−1]
−5.0
−2.5
0.0
2.5
5.0
∆H
∆H
∆H
∆H
∆H
∆H
T∆ST∆S
T∆S
T∆S
T∆S
T∆S
HS
-driv
en
HS
-driv
enS-driven
S-drivenS-driven S-driven
S-driven
H-driven
H-driven H-driven
H-driven
HS-driven
HS-driven
HS-driven
CHAagonists antagonists
ADTN250
200
150
100
50
0
−50
−5 −2
−4−10
−25
−50
−75
−100
−125
75
50
25
0
−25
Haloperidol
DPCPX
4
3
2
1
0
Nicotine50
80
60
40
20
0
25
0
Hexamethonium
A1
D2
nAChR
Figure 4. Thermodynamic profiles of some agonists and antagonists. For abbreviations see text and references in Table 1. Driving forces (H-, S-, HS- for enthalpy, entropy and enthalpy-entropy driven) in the temperature range depicted by different shading tones.
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464 V. Pliska
Thermodynamic parametersTemperature data of individual groups of membrane receptors, and membrane-bound enzymes (acetylcho-linesterase) were collected from the literature:
• A1: adenosine receptors, rat brain (27);
• AChE: acetylcholinesterase, bovine erythrocytes (28)
• β-AR: β adrenergic receptors, turkey erythrocyte membrane (29);
• β-AR: β adrenergic solubilized, L6 myoblasts (30);• D
2: dopamine receptors, rat striatum (31,32);
• GABAA: γ-aminobutyric acid type A receptors, synap-
tosomal membranes of rat whole brain (33);• Insulin receptors, cultured human lymphocytes (line
IM-9) (34);• nAChR: acetylcholine receptor (nicotinic), human
thalamus (35);• OP: opioid receptors, rat brain (μ, δ, κ agonists) (36);• OXTR: oxytocin receptors, sheep myometrial cell
membranes; high (∼10−9 mol/l) and low (∼10−7 mol/l) affinity binding site (16).
In addition to interactions of non-receptor nature already mentioned (18,19,21,26), dimerization of α-chymotrypsin (37), inhibition of fumarase (38,39), self-association of glucagon (40) and ApoA-II (human high density lipopro-tein complex) (41) were evaluated.
Measurements of ΔG or Kd at 4–9 temperatures were
reported in numeric or graphic form (graphs were digitized). If necessary, the data set was completed by interpolation between two neighboring temperatures (maximal four values) which enables employing trans-formed formulas of higher order and reducing asymptotic errors in the computational procedure, without changing trends obtained by using solely experimental values for computation. Results are summarized in Tables 1 and 2.
Data analysis
Relationships between thermodynamic parameters: factor analysisPrincipal component analysis (two factors, no rotation) was used to determine mutual relationships between parameters ΔH, TΔS, ΔC
p, T
e, and Te
, which seem to sufficiently characterize thermodynamic interaction profiles. Figure 5 shows the factor loading plot. The first two parameters dominate within the first factor and compose together with ΔC
p about 85 p.c. of its
loadings. The second factor is dominated by ca 60 p.c. by the parameters T
e and Te
but 25 p.c. of the loadings are covered by ΔC
p which is hence similarly reflected
in both factors (17 p.c. in the first one). A high degree of correlation appears only between the parameters ΔH and TΔS, as predicted by eq. 11. The parameters ΔH and ΔC
p are slightly correlated as well; any other
correlations are insignificant.
Group differencesSince the usual F-test for single parameters indicate significant (α = 0.057 for TΔS) to highly significant (α = 0.000 for ΔC
p and T
e) variances between “recep-
tor” and “non-receptor” groups, the modified t-test procedure (separate variances) was employed. Except for ΔH (−15 and −49 kJ·mol−1 for the former and the lat-ter group, respectively), differences in parameter mean values were less significant. However, the parameter T
e
requires attention: its values lie obviously within the range of the experimental temperature range from 0°C to about 42°C for interactions within the membrane-receptor group, and on both sides outside of this tem-perature range for non-membrane-bound interactions (cf. Tables 1 and 2). This is demonstrated in Figure 6 showing on the left-hand panel the correlation T
e vs.
ΔH, and on the right-hand panel cumulated mean
Table 2. Thermodynamic parameters (298 K) of some non-receptor biochemical interactions.
Target/processa Ligand/substrate1 Effectb Ref.ΔH
kJ·mol−1
TΔS kJ·K−1mol−1
ΔCp
kJ·K−1mol−1 Text
CGA+IP3 Chromogranin A/−Cac (14) −130.40 −33.15 5.65 0
Chromogranin A/+Ca2 −55.96 55.51 −6.11 0
Dimerization Chymotrypsin (37) −6.97 18.21 −3.05 269.3 −0.11
DNA–Taq Taq-polymerase (19) 21.70 67.90 −2.04 274.4 −0.09
DNA 13 bp duplex Integrase (23) −28.82 10.52 −4.11 285.4 −0.87
Fumarase Transaconitate inh (39) 39.92 58.00 2.50 330.0 −0.08
Succinate 47.81 55.81 5.69 279.7 0.31
Malonate 24.44 32.97 2.90 676.2 −0.01
Ribonuclease S’ S-peptide (18) −169.10 −124.97 −6.20 257.0 −0.13
Met(O2)-S-peptide −143.06 −110.46 −4.44 267.6 −0.13
Self-association Glucagon (40) −104.81 −70.32 −2.88 233.6 −0.04
Self-association ApoA-II (41) −120.07 123.17 −50.24 356.3 0.86
trp-operator (Kass,1
) trp-repressor (21) −93.14 −42.65 −6.79 297.0 1.56
trp-operator (Kass,2) trp-repressor (21) 10.43 42.08 −2.58 0aInteraction systems and substance names see cited references.binh: enzyme inhibitor.cCa2+ in the medium absent: −Ca; present: +Ca.
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Thermodynamics of ligand-receptor interactions 465
values of single instances within these groups. Thus, it can be inferred that processes associated with a membrane-bound component display a change of the ΔC
p sign within the temperature span of common
biological processes.
Agonist/antagonist differencesStatistical analysis (analysis of variance (ANOVA)) was carried out on a group of membrane receptors, which included A
1, β-AR (membrane bound only), D
2, GABA
A,
and nAChR. Adjusted sums of squares and Kolmogorov–Smirnov normality test were used as a computational option. First, significance of response differences (ago-nist vs. antagonist) in individual thermodynamic param-eters between agonists (ag) and antagonists (ant) was assessed in the pooled sample of all receptor groups. The α-values, as a measure of significance (bold face numbers for significant α-values), are summarized as follows:
Except for Te, the thermodynamic parameters of ago-
nists diverge from those of antagonists; moreover, the low α-value indicates a significant effect of the receptor type upon ΔC
p. Insignificant T
e differences just reflect,
and confirm, the circumstances depicted in Figure 6. It should be pointed out that all α-values for the interac-tion group × response are highly significant, meaning that response differences are particularly strong within single receptor groups.
The significance of the response mode was then tested within individual groups by the one-way ANOVA followed by the Bonferroni and Tuckey’s pairwise mean comparison test.6 ANOVA was preferred to a simple F- and t-test procedure in order to estimate the “pure”—last square—variance explained by the response. The results are summarized in Figure 7. A detailed view indicates that agonistic and antagonistic responses significantly influence the mean values of ΔH and TΔS within all receptor groups except for D
2 recep-
tors where, too, a clear-cut difference is evident but due to a substantial variance not tested as significant. This is indeed in accordance with the ANOVA results of the pooled group discussed above. However, the signs and the trends of these parameters in individual groups and response types are not uniform. As far as the ΔC
p parameter is concerned, it is clear that the sig-
nificance of the response effect goes on account of a
single receptor group, A1, where the rather numerous
agonists really display systematically high negative ΔCp
values. Similarly as for Te and Te
, however, the ΔCp
values seem to be rather uniformly distributed within all investigated receptor groups.
Discussion
This contribution aims at thermodynamic features of interactions between ligands and membrane-bound receptors and their specificity among the family of biologically relevant processes in general. The first question which can be posed in this context is about the suitability of their descriptors, the thermodynamic parameters, and the methods of their evaluation. Since microcalorimetry still does not offer safe possibilities for measurements in heterogeneous suspensions, and since doubts may be raised as to whether solubilized receptors sufficiently mimic the particular cellular system, most of the experimental methods used here consist in assessment of temperature dependencies of the corresponding equilibrium constants. Their analysis consists in application of the Gibbs–Helmholtz equa-tion correlating the free energy change of the interac-tion, ΔG, with its enthalpic and entropic components. However, the equation relates to isothermic conditions
1.0
1.0
0.5
0.5
0.1863
0.0
0.0
0.85610.41530.17960.0632
0.05030.07230.0907
−0.5
−0.5−1.0
−1.0
−0.0250−0.0797
Factor(1)
Fact
or(2
)
σTe
σTe
∆Cp
∆Cp
∆Cp
∆H
∆H
T∆S
T∆ST∆S
Te
Te
Te
Correlation coefficients
Figure 5. Factor analysis of thermodynamic descriptors. Factor load-ing graph (factors 1 and 2) and coefficients of correlation between individual parameters.
ΔH TΔS ΔCp
Te
Receptor group 0.2795 0.3051 0.0001 0.4174
Response (ag/ant) 0.0354 0.0503 0.0002 0.2205
Interaction group × response
0.0023 0.0012 0.0146 0.1061
6http://www.upa.pdx.edu/IOA/newsom/da1/ho_posthoc.doc
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466 V. Pliska
and thus, the frequent use of the so-called van’t Hoff plot is generally not indicated, in particular not in cases of a curvilinear temperature course of ΔG. Nonlinearity of ΔG goes on account of the temperature-dependent heat capacity contribution to the enthalpy term ΔH: hypothetically, its value at zero absolute temperature, ΔH(0), is not necessarily zero (24,42,43) since H(0) of a particular reaction component represents the (nonzero) potential energy of its thermodynamically optimal con-figuration, and since the equilibrium constant of an interacting system at that temperature may differ from unit. Any temperature value of ΔH is then determined by Kirchhoff’s law (eq. 4) with the lower temperature limit T
1 = 0. Thus, the temperature contribution to the
heat capacity Cp determines the temperature course of
other thermodynamic parameters.Several procedures were suggested for estimation
of the ΔCp temperature function: they are based on an
analysis of an empiric ΔG(T) function (16,25,35), or on inserting a ΔC
p(T) approximation into the Kirchhoff
equation (16). Another T-transformation method (ψ-transformation) was suggested in this communica-tion. It appears that the best ΔC
p(T) approximation is a
polynomial function in T without a constant term, and that in the great majority of cases a polynomial of sec-ond order is fully sufficient (higher orders may indeed yield physically irrelevant results). Under these circum-stances, ΔC
p(T) is roughly linear or linear. At any rate,
thermodynamic analysis of a ligand-receptor interac-tion, and of any other biological interaction as well, is by far not a standard routine. “Well-balanced” compu-tational procedure is critical for trustworthy estimates of thermodynamic descriptors. Their values obtained by computations based the so-called van’t Hoff plot analysis (eq. 3b) may—sometimes—display considerable deviations from a supposed true value (16,44).
The occurrence of an extreme on the parallel ΔH(T) and TΔS(T) profiles is attributed to the change ΔC
p(T)
sign within the accessible temperature range: maxi-mum when the slope of ΔC
p(T) in the zero point, Te
, is negative, minimum when positive. The temperature T
e of an extreme in membrane-bound interactions
occurs mostly between 5°C and 35°C. Comparison with a group of selected biochemical interactions in a homogenous environment (aqueous solutions) shows that their T
e is mostly beyond this range. This may bring
about a notion that the bilayer membrane contributes, as a carrier, appreciably to the thermodynamics of the interaction, most likely by influencing the heat capac-ity changes of the system. Indeed, phospholipid and similar bilayer membranes frequently change their fluidity in similar temperature ranges (45–47). But also temperature-dependent conformational changes of either receptor protein or ligand (or both), changes in receptor coupling to other membrane substructures etc., may stand for an alternative explanation of this phenomenon.
Finally, a comment should be added to the different thermodynamics of agonists and antagonists recep-tor interactions. Distinct conformational changes, as already mentioned, may account for the clear-cut dif-ferences in ΔH(T) and TΔS(T) between agonists and antagonists in single receptor groups (29,48,49). This is also reflected in the highly significant ANOVA interac-tion “group × response.” When pooled over all groups, thermodynamic driving forces of individual ligand-receptor interactions do not display any recognizable systematic trend for agonists and antagonists within the relevant temperature range. Thus, different thermody-namic profiles of agonistically or antagonistically acting ligands can scarcely provide a general explanation of their antipodal biological behavior.
400
350
300
250
200
320
273
T e
∆H
−200 −100 1000 200
membra
ne
recep
tors
non-r
ecep
tor
intera
ction
s
Figure 6. Parameter Te within the group of membrane-coupled receptors (circles) and non-receptor interactions (crosses). Left-hand panel: plot T
e [K] vs.
ΔH [kJ·mol−1] (no significant correlation, r = 0.180); right-hand panel: scatter plot. Broken lines delimit experimental temperature range.
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Thermodynamics of ligand-receptor interactions 467
Acknowledgements
The critical reading and comments to this article by Dr. Paul Pliska, Dr. Vojtĕch Spiwok and Professor Gerd Folkers are gratefully acknowledged.
Declaration of interest
The author declares no conflict of interest.
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40
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0
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A1 D2 GABAA nAChRβ-AR
A1 D2 GABAA nAChRβ-AR
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**
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* * *
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