Partition Function Formalism for Analyzing Calorimetric Experiments Stanley J. Gill,' Kenneth P. Murphy, and Charles H. Robed University of Colorado, Boulder. CO 80309 In most physical chemistry courses partition functions are introduced to evaluate the average molecular properties of molecules distributed over electronic, rotational, vibrational, or translational energy levels. However, the partition function formalism is applicable to determining average properties of systems distributed over a variety of energetic states such as multiple ligation states ( 1 4 , multiple phase equilibria (2, 5-7), multiple aggregation states (8,91,and conformational states of proteins (10-13). In this article we wish to demonstrate how partition functions can be used in the formulation and analysis of calorimetric problems. The essential concept is recognizing that the partition function describes the populations of various molecular states of the system. Furthermore, the dependence of the partition function upon the independent variables of temperature, pressure, and ligand activity gives the average thermodynamic properties of enthalpy, volume, and amount of ligand bound (14). I t is in this context, with temperature as the independent variable, that one obtains the enthalpy change appropriate for calorimetric studies. The partition function formalism has been particularly useful in describing the thermal properties of various systems ranging from hydrophobic solvation (15) to thermal denaturation of proteins (11,12). With the increasing availability and use of titration microcalorimeters and scanning calorimeters, knowledge of this approach should be of iuterest to a larger audience. The often forbidding nature of complex equilibria is overcome by this method in which one need only write down the partition function for the system and then operate on this function to obtain the average properties of the system. Speclflcatlon of the Partition Function The partition function for a system is formulawd by specifying the populations of all accessible equilibrium states of the system. For achemical system the population of aparticular state is conveniently obtained from the Gibbs free euer-
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Journal of Chemical Education
gy of that state relative to a given reference state. The partition function is then the sum of the various populations. Imposing particular constraints on the system leads to various types of partition functions which then determine particular thermodynamic functions. Details are given in texts on statistical thermodynamics, for example Hill (16). Schellman (I) provides a nice discussion of partition functions applied to ligation reactions.
Ligation For ligand binding systems the reference species is the unligated macromolecule, and the other species are described by the number of ligands bound. The stepwise equilibrium reaction processes can be represented by MXi-I X = MX; where M is a macromolecule with t binding sites for ligand X with i running from 0 to t. I t is particularly convenient to express the equilibrium reactions as overall equilibria: M iX = MXi with equilibrium constants Pi. Here, as throughout the paper, the activities of the various species are assumed to be given by their concentrations, i.e., the chemical potentials of the species are determined by ideal dilute solution laws. The population of a species MX; relative to the unligated species M is then [MXJ = Pixi[M].The partition function Q, as the sum of all species, is given as:
+
+
Qiimtim
where
I
-
= [MI + IMXI
+ IMXJ + . . . + [MXtI
1.
To whom requests for reprints should be addressed
(1)
Aggregation In the case of systems with self-association the reference species is the monomer, and the appropriate stepwise equiM = Mj where j is greater than or equal to libria are Mi-, one. The overall equilibrium is j M = Mj with an equilibrium constant A,. The population Mj is [Mj] = A,[M]j and the partition function is
+
with A,
= 1.
Comblnation of Llgatlon and Aggregation One can also consider combinations of different equilibrium processes, such as the situation in which both binding of a ligand and aggregation of the macromolecule occur. The schematic representation of the reactions of this system is then two-dimensional:
and is denoted as q. This heat can be positive or negative (endothermic or exothermic) depending on whether the reaction absorbs or gives off heat to maintain the system at constant temperature. The molar enthalpy change is obtained by applying the van't Hoff derivative relationship to each equilibrium constant in the partition function and summing over all species:
is the average where R is the gas constant and H, molar enthalpy of the system relative to the unligated, non, aggregated reference state. The reference state enthalpy @ is required since the enthalpiesare alldefined relative to this state. In the ligand binding case (eq 6 ) no such constant was necessary because of our definition of the reference species as havinn zero lizands bound. The constant pressure heat capacityof rhe ~acromolecularsystem relativ-e to the standardstate isobtained from the temperaturederivativeof the enthalpy as given in eq 7. Applications to Caiorlmatrlc Processes
Here each ligation equilibrium constant pij refers to the overall reaction of i ligands with a particular unligated, aggregated species composed of j monomers: Mj iX = M,Xj. Similarly, aggregation equilibrium constants Aj refer to the M,. The overall aggregation of unligated monomers: j M population of a given state, [MjXj], is therefore given by Aj[M]j&xi, and the partition function Q by
+
Q=
-
2 AjjlM]j1&;xi
(5)
;
t
aggregation
t
ligation
I t can be seen that if no aggregation equilibria are present eq 5 reduces to eq 2, and similarly, if no ligation equilibriaexist, i t reduces to eq 3. This description can be extended to include processes such as the addition of a second ligand, transition to different allosteric states, or changes of phase. These possibilities, along with the previous examples, are listed in the table. Combination of pairs of basic processes leads t o a two-dimensional array of molecular forms in equilibrium with each other as seen by eq 4. Combination of three processes gives three-dimensional arrays, and so on. The partition function in general reflects the dimensionality of the system. computation of Average, or Thermodynamic, State Properties
The partition function can be used to obtain commonly observed properties of the system (14). For instance, the total amount of bound ligand in the system is just the product of the ligand concentration, x , and the derivative of Q with respect to the ligand concentration (the logarithmic derivative of Q). The amount bound per mole of macromolecule in the system, X,, is obtained by normalizing to the total macromolecule concentration, mt:
Example 1: Enthalpy change of ligatlon of a macromolecule In our first example we consider the ligation of a macromolecule, in this chse the reaction between deoxy-myoglobin (Mb) and carbonmonoxide (CO). We have performed an experiment in a titration microcalorimeter (17)in which the reaction cell was filled with a solution of horse deoxy-myoglobin (Sigma) and the injection syringe with carbonmonoxy-myoglobin. The heat was measured for successive 5-pL injections until the myoglobin in the cell was saturated with CO. The heat for an injection depends on the product of the enthalpy of the reaction and the change in the amount of ligand bound. Although this simple process can be described by equations that can practically be written by inspection, we wish t o illustrate its formulation within the framework of the partition function. The reaction is described by the equation: Mb + CO = MhCO
so the partition function Q is the sum of the accessible states of the macromolecule: (9) Q = [MI + [MXI Here the bracketed terms indicate concentrations, M represents Mb, a n d X represents CO. The species MX is related to the free M by the equilibrium constant 8:
so that Q can be written in terms of the free ligand concentration, the free macromolecule concentration and 8.
Q = [MI(l + B[Xl)
(11)
The average molar enthalpy of the system is obtained from eq 7 as:
, is the molar enthalpy change for the ligation in which & reaction. In the tvnical exneriment we know the stoichiometric. or total, conEentratiin of ligand and macromolecule but hot their free concentrations. Therefore i t is convenient to express the free concentrations in eqs 11-12 in terms of total concentrations. The total concentration of ligand is: x, = [Xl(l+ PIMI)
In calorimetric experiments, heat is the observed quantity
(8)
(13)
and similarly for the macromolecule: Volume 67 Number 11 November 1990
929
myoglobin and carbon monoxide are generally known well enough to be used in the fitting process without further adjustment. The most convenient method for performing these calculations is to use one of the standard nonlinear least square programs commonly available and described in detail by Bevington (19).
,:I -188
+,
3 \
-150
Example 2: Enthalpy Change for Dissolotlon of a Slightly Soluble Solid
-200
Next consider an equilibrium mixture in which a substance is present in equilibrium between a crystalline phase and a solution phase and in which aggregation occurs in solution. From the table this requires combining phase change and aggregation processes. The equilibrium processes can be summarized by the following array:
rT
Injection S t e p
M' Figure 1. Titration of deoxymyogiobin wim carbonmonow-myogiobin. The reaction conditions were 100 p M Mb in 10 mM Tris buffer at pH 8 in bolh soimions with a w b o n monoxide concernration of 790 p M in the injection syringe and an injection volume of 5 pL was used. The size of the squares indicates the standard enor of a point. The solid line is the best fit Uworeticai EUWB to eq 13. The best fit values of the parametersare 0 = 1.22 X 10' M-' and tf= -60 W moi-'.
Utilizing eqs 13-14 we obtain the expression for the free ligand concentration: (-1
[XI=
+ Bz,) * 41 + 28x, + 8'~: - 4P2rnt za
= ceiIfi
+ ~,~,..,(l - D')
Referring all species concentrations in eq 19 to the solid crystalline phase, and expressing in terms of equilibrium constants yields:
(15)
,
which can be substituted into eq 12. In practice two types of titration procedures are used: one where the solution volume increases with each titration step and the other where the reaction volume is fixed. The first instance is typified by the standard buret titration familiar to all student labs. In the second case, which applies to the data in Figure 1,each volume of injected titrant displaces an equivalent volume of solution from the reaction cell, and the concentration of any species in the cell is related to the titration step i by (18): c;
where the superscripts s and aq indicate solid and aqueous phases. The reactions include the dissolution of the molecule itself and any further aggregation processes that occur. The solution phase is described by the partition function:
(16)
in which ci is the concentration after the ith injection step, cell is the initial concentration of the material in the cell, ctieant is the initial concentration of material in the injection syringe, and D is a dilution factor D = (U,.II - u~,,~)/u,.II. The observed heat a t constant pressure is glven by the difference in the enthalpy of the system after and before the injection. The enthalpy of the system is the product of the number of moles in the system and the average molar enthalpy (from eq 121, so we have:
- m;.c i .V d l - [(H, - Hom)i-, .ci-I .(u,,~I - vinj) + (H, - pm)titrant. ctiusnt.uinjl
where K,is the equilibrium constant between the solid phase and the aoueous monomer and hi's are the overall equilibrium constants for the formation of an aggregate of i monomeric units. In the simplest case, where the only significant aggregate is the dimer, eq 20 takes the form Calorimetry reveals the quantitative aspects of these processes. Heat measured in a titration calorimeter from the addition of a small volume u, of pure solvent to an equilibrium mixture of solid and saturated solution reflects both processes from eq 21. Specifically, applying eq 7 to eq 2 1 we ., Then multiplying by the obtain an expression for H,,, - @ amount of solid dissolved (u mt) gives the heat for the addition of a volume v of solvent to the solidlsaturated solution mixture:
.
9i = (Hm
(17)
The results of the experiment and the fit of the data to eq 17 are given in Figure 1. A few details of the fitting procedures are given as follows. The titration data, expressed as the measured heat qi for the ith addition of titrant, is fit utilizing the general eq 17 in conjunction with the combination of relevant expressions eqs 12, 15, and 16. The fitting process which involves the solution to this set of nonlinear equations requires initial guesses of the key parameters for , and 8; the other parameters of the the system, namely & uinj, along with the initial concentrations of the system, 1~~11, 930
Journal of Chemical Education
Bask Molecular ProceUMII* Re~ctlon X Ligand Binding Y Ligand Binding Aggregation of M Phase equilibria between phases a. 8 , y,etc. R, T, etc. forms
Specific Equliibrlab
M=MX=MX2=MX3= M=MY=MY2=MYJ= M=Mp=M3= M" = Mo = Mr = . . . MR=MT=,,,
...
...
...
'Combinationo of Wse linear a m p result In multldlmenslonal m y s such as that g ran n aq 4 of me ten Note Wst Wnaas eq~illbrlsan0 811mter C I01 COnlO.matIOMII equ D W are Iwmam y tns %me n mat tnsy I sb (dsmcal partat on lmctons ' ~aIlmDlcaN . . ot noston. ms amnoon ot laam m caM-b.m.no reactoms and ot monomer in agsregation reactions is assumed but not explicitly shown.
Since the enthalpy change terms always appear as a product with one or more equilibrium constants, it is not possible to resolve the parameters in eq 23 from data a t a single temperature. However, the parameters can be resolved if data over a ranee of temneratures is available. Conseauentlv the temperatu;e depenience of the parameters is requirei. In the simoleat case the enthalov chanees are indeoendent of temperature and the equilibr& constants depend on temperature according to the van't Hoff relationship: a In Kla'l~ = -AHDIR. When AHo is dependent on temperature, as in the diketopiperazine example, a heat capacity change term or terms is required (20). Equations 22 and 23 were used to fit data obtained for the dissolution of diketopiperazine (20) as shown in Figures 2a and 2b. A more complex formulation, in which the aggregate size is unlimited (i.e., infinite aggregation), haa also been given for some diketopiperazine derivatives (21). Summary
We have illustrated the utility of the partition function in the analysis of calorimetric data for several different chemical processes: ligand binding to a macromolecule, and dissolution and aggregation of a slightly soluble compound. The partition function formalism allows for the analysis of such disparate systems in a straightforward manner &ng a common approach. From the simple cases used to illustrate these ideas one can readilv extend the loeic to svstems of virtuallv any degree of complexity. Acknowledgment
Injection Step Figure 2. (a) Experimental heals of dlssolutlon of diketoplperazine as a function of temperature (201. The solld llne Indicates me best flt curve to e q 2 1 In me text. The best lit values at 298 K are & = 0.145 M. = 28.0 k J mol-'. A 6 = 57 J moi-' K-'. A@ = -29 W mol-', and At = 0.19 M-1'2. (b) Experlmemal heats of dliutlon f a a saturated solution of diketopiperazlne tibated Into lnltiallv ,mre water 120).Th8 solid curve is the best fit llne using the paramttsrs given n a. The data in both s ana b (along wnh dilutim data at two add lional tempsratues nm shown) were fol simultaneously Tne injecuon step is related to me concentration in me cell by eq 16 in the text
Ae
.
a a
Here and are the molar heat of solution and dimerization, respectively. Dilution experiments reveal the ag regation effeds and permit determination of A 2 and Afff Using the solution for dimer partition function ([MI &[MI2) yields Hm- Prn the solution. The heat for the titration step i from eq 17 is:
+
This work was made possible by NIH grant HL22325 and NSF grant CHE-8611408. Literature Clted I. 2. 3. 4. 5. 6. 7.
8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.
Following the procedure in the first example the monomer concentration, [MI, at any point is related to the total concentration by:
19.
20. 21.
Schelhan. J. A. Biopolymers 1975,14.999-1018. Wyrnan,J. J.Am. Chem. Soc. 1967,89,2202-2218. Hill, T.L. Cmperolivify Theov i n Biochemistry; Spengsr: New York, 1985. Heas, V. L.;Szabo, A. J. Chem. Edur 1979.56.289-283. Wyman, J.: Gill, S. J. Roc. Not1 Acad. Sci. 1980, 77,52394212, Fig. 1,p 5240. J M d B i o l . 19S0.140.299Gil1.S. J.;S~oksnc,R.:Benedirt,R.C.:Pall,L.;Wyrnan,J. 312. Haire, R. N.; T i d W. A.; Niazi, G.; henberg, A,: Gill. S. J.: Riehey, B. Bioehem. Biophya. Re& Comm. 1981,101.177-182. Herzfeld, J.; Sfanlay. H. E. J. Mol. Bid. 1974.82.231-265. Wyman,J. Qunrf.R~v.Biophy~. 1384.17, 453498. Freire. E.: Biltonen. R. L. Biopolymers 1918.17.463-479. Robert,C. H.;Gill,S. Wyman, J. Bioehemblry 1988,27,68294835. Robert, C.H.; Colasirno, A.: Gill, J. Biopolymer~1989,28,1705-1729. Freire, E. Commanfa Mal. Cell. Biophvs. 1989,6. 123-140. Gill, S. J.; Richey, B.; Bishop, G.; Wyman, J. Biophyaieol Chom. 1985.21.1-14. G.: Wad*. 1.J.Phrs. Chom. 1985,89,3758-3761. Gill. J.:Dee,S. Hill, T.L. Introduction to Sfofisflrol Thermodynomies; Addison-Wesley: Reading, MA, 19M). McKinnon, I. R.; Pall, L.; Pamdy-Morreale, A,: Oili, S. J. A n d Bioehem. 1984, 139. 13P139. Paraly-Momale, A.; Roben, C. H.: Bishop, G. A,: Gi1l.S. J. J. Biol. Cham. 1987,262, 1099P10999. Bevington, P. Dafo Redudion and Enor Anolysia /or the Physical Sciences; McGraw-Hill: Near York. 1969. Murphy, Thermochim. Act. 1989.139.279-290. Murphy, K. P.;Gill, S. J. J , Chem. Thwmodynamira 1989,21,903-913,
J.;
S.
S.
P.:Olofsnon.
K.P.;Gill,S.J.
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