Pressure-Modulated Differential Scanning Calorimetry: Theoretical

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Anal. Chem. 2006, 78, 991-996

Pressure-Modulated Differential Scanning Calorimetry: Theoretical Background Jo 1 rg Ro 1 sgen*

Department of Biochemistry and Molecular Biology, 301 University Boulevard, University of Texas Medical Branch, Galveston, Texas 77555-1052 Hans-Ju 1 rgen Hinz*

Institut fu¨r Physikalische Chemie, Westfa¨lische Wilhelms-Universita¨t, Schlossplatz 4/7, 48149 Mu¨nster, Germany

We demonstrate in this work and in the accompanying paper that it is possible to measure simultaneously heat capacity and expansibility of biomolecules in a single DSC experiment. In this study, we provide the theoretical basis for this new method based on rigorous statistical thermodynamics. The theoretical treatment presented here demonstrates that there are two additive contributions to the heat capacity at variable pressure, viz. (1) the isobaric heat capacity and (2) an expansibility term, and that these contributions can be experimentally separated to obtain simultaneously both heat capacity and expansibility in continuous DSC temperature scans preformed under pressure modulation. Equations that describe the mixed heat capacity and expansibility signal are derived, and experimental strategies as well as data extraction procedures are discussed. A quantitative thermodynamic characterization of biomacromolecules requires knowledge of thermodynamic properties, including population sizes. Furthermore, it is of great interest to understand how the populations of different macromolecular states react to changes in experimental conditions. This information is contained in the partition function. However, partition function ratios (population sizes) are only easily accessible in a narrow range of environmental conditions, under which significant populations of more than one state are present. For an extrapolation of the partition function beyond this narrow range, one needs information on slope and curvature of the partition function. This is best accomplished with methods that sample directly derivatives of the partition function. The first derivatives of the partition function with respect to the experimentally controllable intensive properties (pressure, temperature, chemical potential) are the volume, internal energy, and particle number (degree of binding), respectively. The second derivatives are the material properties or thermodynamic response functions compressibility, heat capacity, and buffer capacity as well as the mixed derivatives such as expansibility. The measurement of these extensive and response properties is far from being straightforward for biomacromolecules, as they are usually not available in sufficiently large amounts. Especially, * For correspondence. E-mail: [email protected]. [email protected]. 10.1021/ac0516436 CCC: $33.50 Published on Web 01/21/2006

© 2006 American Chemical Society

direct volumetric measurements1-7 are challenging. For reasons of speed and material consumption, an alternative method of measuring expansibility is needed. Some time ago, a calorimetric method was established that measures expansibility as a function of pressure at constant temperature.8 Recently, a similar method was applied to proteins and other biomolecules using a microcalorimeter that is sensitive enough to detect the minute signals in rather dilute solution.9 In this method, small pressure jumps close to atmospheric pressure are applied at constant temperature to the sample to obtain expansibility data. Then the temperature is changed, the system equilibrated, and the experiment is repeated. Common to both methods8,9 is the requirement to keep the temperature constant, since otherwise, heat capacity and expansibility signals are mixed and a deconvolution can become difficult. In the following, we show how to overcome such difficulties by applying a simultaneous pressure and temperature change to obtain both heat capacity and expansibility in a single experiment. Presently, such simultaneous measurements are only possible for pure or highly concentrated substances using transitometry10 that monitors both volume and heat signals. Our new method of pressure-modulated differential scanning calorimetry makes dilute compounds, which are typically used in biochemistry, accessible to simultaneous heat capacity and expansibility measurement. Performing a temperature scan with variable pressure has some analogy to performing a temperature scan at variable chemical activity of a ligand, because both pressure and chemical potential are intensive properties. Previously, for ligand binding proteins, we investigated the influence of experimental conditions that deviate from the intensive properties typical for the given (1) Holcomb, D. N.; vanHolde, K. E. J. Phys. Chem. 1962, 66, 1999-2006. (2) Bull, H. B.; Breese, K. J. Phys. Chem. 1968, 72, 1817-1719. (3) Bull, H. B.; Breese, K. Biopolymers 1973, 12, 2351-2358. (4) Kratky, O.; Leopold, H.; Stabinger, H. Methods Enzymol. 1973, 27, 98110. (5) Hinz, H.-J.; Vogl, T.; Meyer, R. Biophys. Chem. 1994, 52, 275-285. (6) Ro ¨sgen, J.; Hinz, H.-J. Biophys. Chem. 2000, 83, 61-71. (7) Seemann, H.; Winter, R.; Royer, C. A. J. Mol. Biol. 2001, 307, 1091-1102. (8) Randzio, S. L. J. Phys. E 1983, 16, 691-694. (9) Lin, L. N.; Brandts, J. F.; Brandts, J. M.; Plotnikov, V. Anal. Biochem. 2002, 302, 144-160. (10) Grolier, J. P. E.; Dan, F.; Boyer, S. A. E.; Orlowska, M.; Randzio, S. L. Int. J. Thermophys. 2004, 25, 297-319.

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ensemble, on the experimental signal.11 As an example that often occurs in a standard calorimetric experiment, we considered the high-affinity binding of a ligand to a protein. Such system is a grand canonical ensemble, since it can exchange particlessthe ligandswith the environment, i.e., the bulk solution. Therefore, the state of the system is controlled by the intensive property and Lagrange parameter R ) -µβ ) - ln a, where µ is the chemical potential, β is equal to 1/kT, and a is the absolute activity. Equivalently, one may state that the partition function depends on the activity of the free ligand. However, the temperature scanning experiment is not performed at constant free ligand activity but at a constant total ligand number N in the calorimetric cell. This means that the experiment is not performed along the axes, which define the orthonormal space, where the partition function is defined. As a result, the calculations have to be transformed to curved coordinates, which leads to extra terms entering the heat capacity expression at constant particle number, CN, compared to the microscopically expected heat capacity at constant ligand activity, CR.11 A similar phenomenon occurs in experiments, when instead of the Lagrange parameter γ ) pβ, only the pressure p is kept constant; i.e., the pressure is not increased during a temperature scan. Using a more sophisticated pressure modulation pattern, it is possible to simultaneously measure heat capacity and expansibility, as experimentally demonstrated in the accompanying paper.12 In this work, we provide the theoretical basis for this new technique. PHYSICAL BACKGROUND OF THE METHOD The mixed heat capacity signal with contributions from both temperature and pressure variations is

Cf ) Cp - R/pTp′

(1)

where p′ was used as an abbreviation for the derivative of the pressure with respect to temperature at the given pressure modulation f. A detailed derivation of this equation is given in the Supporting Information available for download from the American Chemical Society web site. Here, we provide the basic concept of our new PMDSC method that is based on eq 1. Separability of Heat Capacity and Expansibility. The first question to be addressed is how to separate the Cp from the R/p contribution to Cf (eq 1). Traditionally, this is done by performing the experiment either at constant pressure or at constant temperature. For the sake of discussing the heat capacity signal for both cases, it is useful to consider the signal in terms of watts rather than Joules per kelvin. So, we consider Cf‚r, where r is the heating rate. At constant temperature (r ) 0), Cpr equals zero. The temperature derivative multiplied by the heating rate equals the time derivative of the pressure, which is nonzero in a pressure scan. This means that Cf‚r ) - R/pTp′r does not vanish. Consequently, the only contribution to the signal at constant temperature is the expansibility. At constant pressure, p′ equals zero and the only contribution to the isobaric signal Cf ) Cp is the isobaric heat capacity Cp. (11) Ro ¨sgen, J.; Hinz, H. J. Biophys. Chem. 2002, 96, 109-116. (12) Accompanying paper. Boehm, K.; Ro ¨sgen, J.; Hinz H.-J. Anal. Chem. 2006, 78, 984-990 (Manuscript No. AC0509760).

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Using such an approach requires two sets of measurements to determine Cp and R/p. To be able to determine Cp and R/p simultaneously, the pressure modulation p′ must be significantly different from the temperature modulation. The most straightforward approach is to vary the temperature linearly and vary the pressure periodically. As we will see below, the time per oscillation period should be fast and similar to the instrument response time in order to get an optimal expansibility signal. This guarantees that the signal amplitude is maximal and also that the oscillating expansibility part of the signal is always well distinguishable from the shallow and nonoscillating Cp part. The Cp part of the signal must be very shallow as a function of time in order to avoid a distortion of the output signal due to the calorimeter response time.13 Choosing an appropriate heating rate is necessary in any kind of DSC and makes sure this condition is fulfilled. Sensitivity of the Method. The sensitivity of the PMDSC with regard to the expansibility can be estimated from a comparison of the magnitude of the R/p term compared to the Cp term in eq 1. The best test case of sensitivity are biomacromolecules, because they usually are available in limited amounts only. Therefore, highsensitivity calorimeters have to be used to obtain good Cp signals despite solute concentrations that can be lower than 0.01% by mass.14 We demonstrate now that the expansibility contribution to Cf can be as large as the Cp contribution. For this purpose, we first estimate p′ for reasonable experimental conditions. Then we calculate the magnitude of the second term in eq 1. A sawtooth pressure modulation pattern is the most straightforward case, because the absolute value of p′ is constant. As discussed below, a reasonable duration for each ramp in a sawtooth pattern is three relaxation times, which is about 3 × 6 s in our case.12 Each ramp has ∼5 bar amplitude. Given a typical heating rate of 0.5 K/min, we therefore have p′ ≈ 40 bar/K. Assuming a typical value6,9 for a protein R/p (10 mL/mol‚K) and 300 K for T, the expansibility contribution to Cf is 12 kJ/mol‚K. This is the order of magnitude of Cp of proteins.6 On the basis of the sensitivity with regard to Cp, we can therefore expect that good expansibility data for dilute solutes are obtained at concentrations as low as or lower than 0.1% by mass. This is a drastic increase in sensitivity compared to the other method that is currently used to simultaneously determine heat capacity and expansibility, namely, transitometry. On the basis of recent publications, it appears that this method produces considerable noise even at concentrations close to 50% by mass.10 Higher sensitivity can be achieved, if the expansibility is not measured directly by monitoring the volume, but by using the calorimetric signal.8,9 This has been done in the past, however, for the price of measuring only one instead of two quantities. It has been demonstrated that, in this way, good expansibility data on biopolymers can be produced at concentrations between 0.5 and 2% by mass.9 CALORIMETRIC SIGNAL BEHAVIOR DURING PRESSURE MODULATION The two most simple and straightforward ways of pressure modulation are pressure ramps and pressure jumps. In this section, the signal behavior is discussed for these two cases as (13) Mayorga, O. L.; Freire, E. Biophys. Chem. 1987, 27, 87-96. (14) Plotnikov, V. V.; Brandts, J. M.; Lin, L. N.; Brandts, J. F. Anal. Biochem. 1997, 250, 237-244.

well as for sinusoidal pressure modulation. The data extraction procedures have some similarity with those used for temperature modulated calorimetry (for reference, see for example, Reading15 and references therein). The calorimetric signal, s(t) ) Cf (eq 1), is distorted by the response h(t) of the calorimeter. Usually this response can be expected to be of first order.13 Then the signal s(t) is obtained by a convolution of the undistorted signal g(t) with a simple exponential function h(t)

s(t) ) g(t)Xh(t) )

∫ g(τ)h(t-τ)dτ t

0

(2)

where the first-order response with time constant k is

h(t) ) exp(-kt)

(3)

Due to the additivity of integration the two contributions to the heat capacity given in eq 1, Cf can be separated into the temperature-related signal and the pressure-related signal originating from Cp and -R/pTp′, respectively. For simplicity, we assume that the heating rate is chosen in a reasonable way, i.e., slow enough so that neither the heat capacity Cp nor the expansibility R/p signal become distorted by intrinsic kinetic processes.13 Hence, all distortions of the signal are ascribed to the pressure modulation term p′ ) ∂p/∂T. Signal after a Pressure Jump. A pressure jump can be considered a fast pressure ramp. During a single ramp, the pressure is continuously altered at constant rate. The resulting undistorted signal is a step function θ(t) with an ordinate value of -R/pTp′ (eq 1) during the pressure ramp and zero otherwise. According to eq 1, the signal s(t) is obtained by a convolution of the undistorted signal, g(t), with an exponential function h(t) that is given by eq 3:

{

exp(kt) - 1 ,t>d -R/pT∆p d sj(t) ) exp(-kt) exp(kd) - 1 r ,tgd d

df0

(5)

Figure 1A shows the signal according to eq 4 for different velocities of pressure variation. As expected, f approaches -R/pTp′ exp(-kt) as the time required for the jump becomes smaller than 1/k. Therefore, there is no need to perform the pressure jump much faster than the response of the calorimeter. Increasing the duration of the pressure change to ∼4/k makes possible the detection of the value -R/pTp′, since the value is reached within the pressure variation time. Faster ramps make (15) Reading, M. J. Therm. Anal. Calorim. 2001, 64, 7-14.

-R/pTp′ less and less well-defined, and in the limit of an infinitely fast jump, -R/pTp′ becomes ill-defined since p′ becomes infinity. Signal during Sequential Pressure Ramps. In contrast to the case of a single pressure ramp in which the ideal signal jumps from zero to -R/pTp′ and back to zero, sequential pressure ramps will result in jumps between plus and minus R/pTp′. This step function again will be called θ. The signal distorted by the instrument response is then

sr(t) )

{

] }

[

-R/pT∆p (-1)[t/d] 2 exp[k(d[t/d]-t)] 1- sinit r d 1 + exp(-kd) (6)

(4)

where we used p′ ) ∆p/(rd). ∆p is the difference between upper and lower pressure limits, d is the time duration of the ramp, and r is the heating rate. In the case of an infinitesimally fast jump we have

lim s ) - R/pTk∆p exp(-kt)/r

Figure 1. Signal response during and after a pressure jump (A) and sequential ramps (B). The duration of the pressure jumps and ramps in units of instrument relaxation times (1/k) is indicated next to each curve. The terms after the initial ratio (-Rp/T∆p/r) in eqs 4 and 6 were used for panel A and B, respectively.

where [t/d] is the Gauss bracket (integral number closest to t/d at the lower side). sinit ) -(exp(-kt)/d)[1 - exp(kd)/1 + exp(kd)] is only relevant at the very beginning of the pressure ramp series and can be ignored for practical purposes. This initial decay can be seen in Figure 1B at times close to zero. As in the case of a pressure jump performing ramps shorter than 1/k does not improve the output signal. Sinusoidal Pressure Modulation. In the case of a sinusoidal pressure modulation p ) p0 - ∆p cos(kmt) the input signal contains the derivative ∂p/∂T ) km∆p sin(kmt)/r, where r is the heating rate. A first-order convolution (eqs 2 and 3) results in the output signal

-R/pT∆p ss(t) ) r

km km 1+ k

()

(

2

)

km k

sin(kmt) - cos(kmt)

(7)

where the initial relaxation term exp(-kt)km/k was left out, since it is negligible for practical purposes. Analytical Chemistry, Vol. 78, No. 4, February 15, 2006

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Figure 2. Relative signal amplitudes as a function of pressure jump/ ramp duration (in units of instrument relaxation times). The amplitudes are given as explained in the text for the cases of jumps (eq 8) and sequential ramps (eq 9) as indicated. Curves are shown before (dotted lines) and after (solid lines) taking into account the instrument response.

DATA ANALYSIS There are three possibilities to obtain information from the heat signal. (1) The signal amplitude of the response to the pressure modulation can be determinedseither directly or through a curve fit. (2) The signal can be integrated using the pressure modulation free signal as integration baseline. This is the standard procedure used for isothermal pressure perturbation calorimetry. 9 (3) In the case of a sinusoidal pressure modulation, heat capacity and expansibility are directly extractable from the raw data without the requirement for applying intermediate steps. All three strategies are discussed now. Signal Amplitude. According to eqs 4 and 6, the amplitude of a pressure jump is

-R/pT∆p 1 - exp(-kd) ‚ r d

(8)

(

∆p ( rd

)

Analytical Chemistry, Vol. 78, No. 4, February 15, 2006

∫ [1 - exp(-kt)]dt + ∫ exp(-kt)[exp(kd) - 1]dt) ) -R T∆pr d

0

∞

(9)

For simplicity, in the further discussion we will disregard the term -R/pT∆p/r. In Figure 2, the amplitudes are plotted versus the duration d per ramp or jump together with the undistorted amplitudes 1/d and 2/d (infinitely fast calorimeter response). It is clearly seen that performing jumps or ramps faster than approximately two times the calorimeter response 1/k ends up in a loss of information on the amplitude. At durations d longer than 2/k, an exponential fit of the signal can be expected to reveal the ideal amplitude with sufficient accuracy. A good compromise yielding a comparably large signal together with a signal closely approaching the ideal amplitude should be around the value of d ) 3/k. In this region, the application of a continuous series of ramps of course doubles the signal compared to a single ramp (or jump equivalently) because of the alternating positive and negative pressure changes. Therefore, for amplitude determination, series of ramps should be preferred over single ramps (jumps). 994

Integration. The behavior of the signal during and after a pressure jump of duration d is given by eq 4. Integration of this equation yields

-R/pT

and the amplitude of pressure ramps is

-R/pT∆p 2 2 ‚ 1r d 1 + exp(-kd)

Figure 3. Extraction of expansibility and heat capacity from a calorimetric temperature scan with sinusoidal pressure modulation. (A) Extracted expansibility (crosses) shown together with original expansibility (bold line). (B) Calorimetric signal (black line) and extracted heat capacity (bold line). The Cf curve was simulated using volumetric and heat capacity parameters, which are within the typical range found for the thermal denaturation of small monomeric proteins:6,9 ∆H° ) 374 kJ/mol, Cp,N ) 24.8 kJ/mol‚K, Cp,D ) 30.9 / kJ/mol‚K, ∂TCp,N ) 142 J/mol‚K2, ∂TCp,D ) 33 J/mol‚K2, Rp,N ) 4.7 / / / 2 mL/mol‚K, Rp,D ) 8.1 mL/mol‚K, ∂T Rp,N ) -0.1 mL/mol‚K , ∂T Rp,D ) -35 µl/mol‚K2, and ∆V0 ) -30 mL/mol, all valid at Tm ) 328.3 K. About 1% noise was added to the data. Heat capacity and expansion curve equations were taken from Ro¨sgen and Hinz.6

d

/ p

(10)

This result is independent of the duration d of the jump. As can be seen from Figure 1, a shorter duration d leads to a sharper peak, which makes integration of the experimental data more reliable due to the shorter interpolated integration baseline. A pressure jump much shorter than the response time of the instrument, however, does not improve the signal more. Since a faster jump will mechanically stress the instrument more than a slower one, applied pressure change should be only slightly faster than the instrument response. In this way, the peak will be sharp (well integrable) and the danger of mechanically damaging the instrument is minimized. Integration in the case of a series of ramps (eq 6) is not useful in any obvious way. Direct Data Extraction. In the case of a sinusoidal pressure modulation, the heat capacity can be separated in a very straightforward manner from the expansibility contribution (Figure 3). Averaging points that are half a modulation period apart will yield Cp (Figure 3B). The difference between this Cp curve and the oscillating Cf signal is given by eq 7. Since the modulation constant

km as well as the instrument response k are known, the expansibility is directly obtained

R/p ) -

r ‚ T∆p

1+

() km k

km

2

Cf-Cp sin(kmt) - cos(kmt)

km k

(11)

Figure 3A shows expansibility data extracted from a Cf curve. Because the sinusoidal oscillation in the denominator of eq 11 passes through zero frequently, the noise in those regions is magnified extremely. Data in regions, in which the noise is magnified by more than a factor of 3 were omitted (∼15% of the data). DISCUSSION OF INTEGRATION, AMPLITUDE DETERMINATION, AND DIRECT DATA EXTRACTION Pressure Jumps and Single Ramps. Fast or slow single pressure ramps (jumps) can be used in two different experimental setups, i.e., in isothermal experiments or in temperature scans. Pressure jumps are most appropriate under isothermal conditions. This is because integration of the resulting peaks requires a stable and linear baseline, which is not given in a thermal scan. Also, extraction of a continuous heat capacity function forms a calorimetric signal that is disturbed by a series of peaks is not straightforward. Integration of heat peaks that are obtained from isothermal pressure jumps (eq 10) is currently the usual data extraction procedure.16,9 In principle, according to eqs 5 and 8, the amplitude of the peak also contains information on the expansibility, and even a curve fit would be possible. Such additional information could be used as an independent check of the validity of the integration. Single slow pressure jumps (ramps) are of limited usefulness. As long as the ramp is faster than the instrument response, the resulting signal is essentially indistinguishable from that obtained after an instantaneous jump. It might, however, be mechanically less stressful to the instrument to perform a fast ramp as opposed to an instantaneous pressure jump. In the case of a ramp that is slow compared to the response time 1/k of the instrument, integration still yields -R/pT∆p/r. However, the larger width of the peak associated with a slow-pressure variation makes an integration more difficult. This is due to the longer baseline required for integration, which has to be interpolated between the starting point and the end of the peak. In contrast to the difficulties with integration, the amplitude of the signal jump -R/pT∆p/r is properly approached in an exponential fashion during the slow-pressure variation. This renders possible a reading of the value of -R/pT∆p/r over practically the whole pressure ramp. Although this is a great advantage over the integration method, at the same time the amplitude -R/pT∆p/r decreases as the pressure variation becomes slower. Sequential Pressure Ramps. The signal amplitude can be doubled by performing a series of consecutive pressure ramps in opposite directions. In this way, the pressure ramp method can yield signal amplitudes similar to those of the fast jump method. The information content is continuous in the case of ramps, but (16) Kujawa, P.; Winnik, F. M. Macromolecules 2001, 34, 4130-4135.

it is discontinuous in the case of jumps, since the information on the expansibility is only contained in the part of the signal triggered during the pressure change. The continuous information content in sequential ramps opens the possibility of performing pressure ramps during temperature scans and thereby overcoming the limitations of separate temperature and pressure scans. Sequential ramps are the simplest and most straightforward way of modulating the pressure. There might be, however, one experimental caveat. As judged by the appearance of spikes in isothermal pressure perturbation calorimetry,9,17 sudden pressure changes induce a signal directly in the heat sensor in addition to the signal resulting from the sample. The same effect seems to occur not only for discontinuous pressure changes but also for discontinuous derivatives of the pressure change p′.12 Such a jump in p′ occurs at the point at which two pressure ramps of opposite slope merge. Though it is straightforward to take this effect into account,12 there might be cases in which it obscures the signal of the sample. In such cases, a more sophisticated pressure modulation is needed, which does not have any discontinuities in either the pressure or its time derivative. This is achieved by using a sinusoidal pressure modulation. Sinusoidal Pressure Modulation. The most elegant pressure modulation pattern is a sinusoidal modulation (Figure 3). It allows for a direct and model-independent separation of the heat capacity from the expansibility signal. This model-independent data extraction works best if the signal change of both heat capacity and expansibility is not too large within one oscillation period. This is because data are averaged that are half a period apart. Under conditions, in which this procedure is difficult because the signal changes within a period can be large, a curve fit might be needed to extract Cp and R/p. Examples for such systems are lipids, which have extremely narrow and large transition peaks in both Cp and R/p.18 Proteins have much less cooperative and smaller transition peaks. In general, it is advisable to do the pressure modulation fast enough or the heating rate slow enough that the transition peak is covered by many oscillation periods. This makes possible direct data extraction according to eq 11 combined with the averaging of points that are half a period apart, which yields Cp. CONCLUSIONS We have shown that it is possible to measure heat capacity and expansibility of biomolecules simultaneously by calorimetric temperature scans under conditions of modulated pressure. We also provided strategies of performing such experiments and pointed out that sequential pressure ramps as well as sinusoidal pressure modulations are the best choice for a modulation pattern. Statistical mechanical considerations on the nature of the heat capacity signal enabled us to derive the necessary equations. As noted earlier,11 there is a difference between microscopic fluctuations of the internal energy and of the enthalpy on one hand and the experimental heat capacity signal (“entropy capacity”) on the other hand. Here we showed that the same principles hold for the enthalpy-volume correlations, i.e., the expansibility. The interpretation of the experimental heat capacity as entropy capacity leads to the proper expressions for the heat capacity under various (17) Randzio, S. L. Thermochim. Acta 2003, 398, 75-80. (18) Albon, N.; Sturtevant, J. M. Proc. Nat. Acad. Sci. U.S.A. 1978, 75, 22582260.

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uncommon, but very useful experimental conditions. These include the heat capacity of high-affinity ligand binding proteins11 in conventional DSC and the pressure modulation calorimetry presented in this paper.

SUPPORTING INFORMATION AVAILABLE

ACKNOWLEDGMENT

Received for review September 14, 2005. Accepted December 6, 2005.

The authors gratefully acknowledge support by the DFG (Grant Hi 204, 24-2) and the Fonds der Chemischen Industrie.

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Additional information as noted in text. This material is available free of charge via the Internet at http://pubs.acs.org.

AC0516436