- Email: [email protected]
Analysis of multitemperature isothermal titration calorimetry data at very low c: Global beats van't Hoff
Accepted Manuscript Analysis of multitemperature isothermal titration calorimetry data at very low c: Global 1 beats van't Hoff Joel Tellinghuisen PII:
S0003-2697(16)30262-7
DOI:
10.1016/j.ab.2016.08.024
Reference:
YABIO 12489
To appear in:
Analytical Biochemistry
Received Date: 28 April 2016 Revised Date:
6 August 2016
Accepted Date: 23 August 2016
Please cite this article as: J. Tellinghuisen, Analysis of multitemperature isothermal titration 1 calorimetry data at very low c: Global beats van't Hoff , Analytical Biochemistry (2016), doi: 10.1016/ j.ab.2016.08.024. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT
Analysis of multitemperature isothermal titration calorimetry data at very low c: Global beats van't Hoff1
Joel Tellinghuisen Department of Chemistry
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Vanderbilt University
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by
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Nashville, Tennessee 37235
Submitted to Analytical Biochemistry 10 Manuscript Pages 1 Figure
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1 Table
Short Title: ITC analysis at very low c
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Joel Tellinghuisen Department of Chemistry Vanderbilt University Nashville, TN 37235
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Author:
Phone: (615) 322-4873 FAX:
Email:
(615) 343-1234
[email protected]
Subject Category: Key Words:
Physical Techniques
ITC, Data analysis, Nonlinear least squares, global analysis, statistical errors
ACCEPTED MANUSCRIPT -2-
ABSTRACT Isothermal titration calorimetry data for very low c (≡ K[M]0) must normally be analyzed with the stoichiometry parameter n fixed — at its known value or at any reasonable value if the
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system is not well characterized. In the latter case, ∆H° (and hence n) can be estimated from the Tdependence of the binding constant K, using the van't Hoff (vH) relation. An alternative is global or simultaneous fitting of data at multiple temperatures. In this Note, global analysis of low-c data at
ITC, Data analysis, Nonlinear least squares, global analysis, statistical errors
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Key Words:
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two temperatures is shown to estimate ∆H° and n with double the precision of the vH method.
ACCEPTED MANUSCRIPT -3It was long thought that successful application of isothermal titration calorimetry (ITC) to 1:1 binding processes, M + X
⇌
MX, required that the product K[M]0 (≡ c) of the binding constant
K and the initial concentration of the cell reagent (titrand M) fall in the approximate range 1-1000
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[1]. It is now recognized that this prescription is too limiting, especially on the small-c side, as it is based on the tacit assumption of titration to a molar excess of about 2 in the ratio [X]0/[M]0 of total titrant to titrand in the cell after the last injection [2]. With decreasing c, this ratio must be
increased to yield significant conversion of M to MX. Then ITC can be used successfully down to c
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~ 0.001, provided the titrant concentration can be made sufficiently large. However, as c decreases below 1, the stoichiometry number n and reaction enthalpy ∆H° become highly correlated, making
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it impossible to obtain reliable estimates of both and sometimes producing divergent fits. But K remains well determined in this regime and is almost completely independent of n. Thus, one can freeze n — at its known value in favorable cases or at any reasonable estimate in cases where the properties of the titrand solution are not well known — and then fit the data to just the two parameters, K and ∆H° [3].
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In biological studies, n is often not well known, and I suggested that ∆H° could be estimated reliably from K estimates at two or more temperatures, through the van't Hoff (vH) relation, after which the correct n could be obtained, since the product n×∆H° is well determined.2 Although the
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utility of ITC at very low c is increasingly being recognized, it appears to me that the power of Tdependent data is not yet appreciated [4-8]. In single-T experiments, the need to fix n at its
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presumably known value can lead to systematic errors, from concentration errors related to preparation and materials limitations [9] and from cell volume errors [10]. In this Note, I obtain an expression for the precision with which ∆H° and n can be estimated from data at two Ts, and I show further that simultaneous or global analysis of the same data yields both parameters with precisions a factor of two better than for the 2-step vH method. Consider first the van't Hoff analysis of K at two Ts. If these are sufficiently close together, we can take ∆H° to be constant. Then the vH relation reads
ACCEPTED MANUSCRIPT -41 1 R ln (K2/K1) = ∆H° T – T , 2 1
(1)
where R is the gas constant and the Ts are absolute (in K). If the T uncertainty is negligible, the uncertainty in ∆H° is determined entirely by that in the Ks. Let the K ratio be designated as z,
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giving
σz/z = [(σ1/K1)2 + (σ2/K2)2]1/2 ≈ 2 (σK/K) ,
(2)
where the final approximation assumes that the relative standard errors (RSE) in the Ks are the same. For low-heat, the estimation precision σq for the heats qi becomes constant [2,11]; then
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below c = 1, (σK/K) becomes ~7.5 σq/|qtot| for a 2-parameter fit, where qtot is the sum of the qis for
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the m injections [12]. Since σ(ln z) = σz/z, the SE in ∆H° is T1T2 σq σ(∆H°) ≈ 10 R , ∆T |qtot|
(3)
where ∆T is the absolute difference in the two Ts. The uncertainty in ∆H° is inversely proportional to ∆T, a reasonable expectation. The dependence on (σq/|qtot|) of the absolute SE in ∆H° is the same as that for the relative SEs in all three parameters from the standard treatment — a direct
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consequence of the RSE in Eq. (2). Since qtot is proportional to h = ∆H° × [M]0, the SE in the van't Hoff ∆H° is inversely proportional to [M]0; and for fixed [M]0, the RSE in ∆H° goes as ∆H°−2. In the 2-step vH procedure, the data at each T are first fitted to the 2-parameter model, with
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n frozen at some reasonable value. This yields the two Ks and two estimates of the apparent ∆H°. Because of the very high correlation between n and ∆H° at low c, this ∆H° varies with n in a way
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that preserves n×∆H°. (E.g., doubling n reduces ∆H° by a factor of two, with negligible effect on K.) The two Ks then give the vH estimate of ∆H°, through Eq. (1); and this can be combined with the ∆H° estimates to obtain the correct n, using nassum×∆H°fit = ncorr×∆H°vH. The uncertainty in ncorr is obtained using error propagation, which gives the squared RSE in ncorr equal to the sum of the same for ∆H°fit and ∆H°vH, with the latter given approximately by Eq. (3) and usually dominating. (See the Online Supplement for an illustration of this method.)
ACCEPTED MANUSCRIPT -5To check this procedure, I have used synthetic data generated using a program like those provided in Refs. [2] and [12], for parameter values giving c = 0.01, presented in Table 1. The
(4)
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initial titrant concentration was set using the empirically derived expression [2,12] 6.4 13 Rm = 0.2 + c ; but Rm > 1.1 c
for the X:M concentration ratio in the cell at the end of the titration. For very small c, the 2nd term dominates, and [X]0 becomes dependent on K alone, ~13/K, so the value of n used here doesn't
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affect this calculation [13]. The resulting value of 133 mM (which includes a factor of ~5 for the cell:syringe volume ratio) was used at both Ts, since this would be the usual experimental choice.
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The results in Table 1 for exact data show that the assumption of n = 1 for the 2-parameter fits produces only slight bias in the estimates for K, and gives estimates of ∆H° off by the expected factor ncorr/nassum = 0.7. The vH procedure hardly determines ∆H° and n for the lower ∆H°, but both are well determined at the higher value, thanks to the inverse dependence of the RSEs on ∆H°2. Eq. (3) predicts SEs higher by < 10%. To improve the precisions one might consider
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increasing [M]0 to increase |qtot| (and c). An alternative is the use of a larger temperature interval ∆T; but this might compromise the assumption of constant ∆H°. Note that if the 2-parameter fits are used to estimate the true ∆H° for known n, a realistic estimate of the uncertainty in the latter should be used, through error propagation, to obtain the uncertainty for ∆H°. For example, the
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uncertainty in the "known" n, from concentration and cell volume uncertainties, will generally
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exceed the ~0.004 RSE in the ∆H°fit values in the last column of the table. In the standard analysis of ITC data at a single T, the three fit parameters are just
independent quantities. But thermodynamically, K and ∆H° are related, through ∆G° = −RT ln K° = ∆H° − T∆S° ,
(5)
where K° represents the dimensionless K, referenced to its chosen standard states. Thus, one can replace K° in the analysis by ∆G° or ∆S°. For data at a single temperature these choices are all statistically equivalent, but for data at multiple temperatures they are not. In particular, traditional
ACCEPTED MANUSCRIPT -6analysis of data sets at two Ts requires 2 Ks and 2 ∆H° values; in global analysis just two parameters replace these 4 when ∆H° (and hence ∆S°) can be taken as constant. Then K becomes K° = exp[∆S°/R − ∆H°/(RT)] .
(6)
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This treatment is not limited to just 2 data sets at two Ts, nor to ∆H° and ∆S° that are constant. For T-dependent ∆H° and ∆S°, we obtain results equivalent to those obtained by integrating the differential form of the vH equation (see Online Supplement) [14].
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Global LS fitting has often been used to analyze multiple data sets that share some common parameters, and it is not new to ITC, having been used at least as early as 1995 by Turner, et al. [15], and by several other groups since [16-22]. The present case offers a quantitative comparison
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of global and traditional methods in perhaps the simplest form in which they can be compared. Accordingly, the results in Table 1 show that the global SEs track those from the vH analysis but are a factor of two smaller, meaning 4-fold better statistical efficiency for the global analysis. Note that to obtain the SEs in ∆G°1 and K1 it is necessary to either include correlation in propagating the error from ∆H° and ∆S°, or to define an equivalent fit with ∆G°1 or K1 replacing ∆H° or ∆S° in the
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fit model [23]. (These points are illustrated with a KaleidaGraph routine in the Supplement.) Figure 1 compares the precisions from global analysis with those for the standard 3parameter fits in the normalized form used in Fig. 3 of Ref. [12]. As was noted before, the RSEs
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for ∆H° are no longer universal functions of σq/|qtot|, giving instead different curves that depend on the magnitudes of ∆H° and ∆T (these results being for ∆T = 5 K only). This dependence also
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manifests as variability in the RSE of K in the c range 0.1 - 10. For c > 10, the improved precision in the global analyses is largely due to the factor of ~ 2 from fitting twice as many points. Thus, while the global analysis is transformational at very low c, yielding results for ∆H° and n where these would be impossible to obtain directly from traditional fitting, the gains are only modest at large c, where the traditional analysis yields good precision in all three parameters. Also, since for c > 10 the RSEs for ∆H° and n can drop to ~0.01 or smaller, the fits can become quite sensitive to small systematic errors in, e.g., [M]0 in different runs [9]. These can be compensated by small
ACCEPTED MANUSCRIPT -7changes in n in the individual fits, but lead to deteriorating fit quality from increased sum of squares (χ2) in global fits with a single n.3 All results obtained above have assumed that T is error-free. This is probably a good
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assumption, as much attention has been devoted to good temperature control in commercial ITC instruments, making random T error very small. On the other hand, the systematic error in the T readout can be significant and should be corrected by T calibration. Such a calibration for a widely used ITC instrument revealed a linear error in the readout T that produced a 5% error in any ∆T
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interval, leading to a 5% error in the vH and global estimates of ∆H° [24].
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Global fitting is, to my knowledge, not an option with the analysis software provided by the ITC instrument manufacturers, but it can be done with some freely available packages, including SEDPHAT [18,19]. Also, the KaleidaGraph routines in the Supplement can be used for both traditional and global ITC analysis. Appendix A. Supplementary data.
Funding
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Supplementary data related to this Note can be found at …
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No financial support from agencies in the public, commercial, or nonprofit sectors.
ACCEPTED MANUSCRIPT -8REFERENCES [1] T. Wiseman, S. Williston, J. F. Brandts, and L.-N. Lin, Rapid Measurement of Binding Constants and Heats of Binding Using a New Titration Calorimeter, Anal. Biochem. 179
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(1989) 131-137. [2] J. Tellinghuisen, Designing isothermal titration calorimetry experiments for the study of 1:1 binding: Problems with the "standard protocol," Anal. Biochem. 424 (2012) 211-220. [3] J. Tellinghuisen, Isothermal titration calorimetry at very low c, Anal. Biochem. 373 (2008)
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395-397.
[4] S. A. Liu, P. J. Focke, K. Matulef, X. L. Bian, P. Moenne-Loccoz, F. I. Valiyaveetil, and S.
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W. Lockless, Ion-binding properties of a K+ channel selectivity filter in different conformations, Proc. Nat. Acad. Sci. 112 (2015) 15096-15100. [5] M. Nagar and S. L. Bearne, An Additional Role for the Bronsted Acid-Base Catalysts of Mandelate Racemase in Transition State Stabilization, Biochemistry 54 (2015) 6743-6752. [6] S. Mohanty, C. Jobichen, V. P. R. Chichili, A. Velazquez-Campoy, B. C. Low, C. W. V.
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Hogue, and J. Sivaraman, Structural Basis for a Unique ATP Synthase Core Complex from Nanoarcheaum equitans, J. Biol. Chem. 290 (2015) 27280-27296. [7] S. G. Krimmer, and G. Klebe, Thermodynamics of protein-ligand interactions as a reference for computational analysis: how to assess accuracy, reliability and relevance of experimental
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data, J. Comput. Aided Molec. Design 29 (2015) 867-883. [8] A. J. Situ, T. Schmidt, P. Mazumder, and T. S. Ulmer, Characterization of Membrane Protein
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Interactions by Isothermal Titration Calorimetry, J. Mol. Biol. 426 (2014) 3670-3680. [9] J. Tellinghuisen and J. D. Chodera, Systematic errors in isothermal titration calorimetry: Concentrations and baselines, Anal. Biochem. 414 (2011) 297-299. [10] J. Tellinghuisen, Volume errors in isothermal titration calorimetry, Anal. Biochem. 333 (2004) 405-406. [11] J. Tellinghuisen, Statistical error in isothermal titration calorimetry: Variance function estimation from generalized least squares, Anal. Biochem. 343 (2005) 106-115.
ACCEPTED MANUSCRIPT -9[12] J. Tellinghuisen, Optimizing isothermal titration calorimetry protocols for the study of 1:1 binding: Keeping it simple, Biochim. Biophys. Acta 1860 (2016) 861-867. [13] J. Tellinghuisen, Optimizing Experimental Parameters in Isothermal Titration Calorimetry, J.
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Phys. Chem. B 109 (2005) 20027-20035. [14] J. Tellinghuisen, Van't Hoff analysis of K°(T): How good ... or bad?, Biophys. Chem. 120 (2006) 114-120.
[15] D. C. Turner, M. Straume, M. R. Kasimova, and B. P. Gaber, Thermodynamics of Interaction
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of the Fusion-Inhibiting Peptide Z-D-Phe-L-Phe-Gly with Dioleoylphosphatidylcholine Vesicles: Direct Calorimetric Determination, Biochemistry 34 (1995) 9517-9525.
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[16] B.M. Baker and K.P. Murphy, Dissecting the energetics of a protein–protein interaction: the binding of ovomucoid third domain to elastase, J. Mol. Biol. 268 (1997) 557–569. [17] K.M. Armstrong and B.M. Baker, A comprehensive calorimetric investigation of an entropically driven T cell receptor-peptide/major histocompatibility complex interaction, Biophys. J. 93 (2007) 597–609.
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[18] J. C. D. Houtman, P. H. Brown, B. Bowden, H. Yamaguchi, E. Appella, L. E. Samelson, and P. Schuck, Studying multisite binary and ternary protein interactions by global analysis of isothermal titration calorimetry data in SEDPHAT: Application to adaptor protein complexes
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in cell signaling, Protein Sci. 16 (2007) 30-42. [19] H. Zhao, G. Piszczek, and P. Schuck, SEDPHAT – A platform for global ITC analysis and
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global multi-method analysis of molecular interactions, Methods 76 (2015) 137-148. [20] L.A. Freiburger, O.M. Baettig, T. Sprules, A.M. Berghuis, K. Auclair, A.K. Mittermaier, Competing allosteric mechanisms modulate substrate binding in a dimeric enzyme, Nat. Struct. Mol. Biol. 18 (2011) 288–294. [21] L. A. Freiburger, K. Auclair, and A. K. Mittermaier, Van't Hoff global analyses of variable temperature isothermal titration calorimetry data, Thermochim. Acta 527 (2012) 148-157. [22] L. Freiburger, K. Auclair, and A. Mittermaier, Global ITC fitting methods in studies of protein allostery, Methods 76 (2015) 149-161.
ACCEPTED MANUSCRIPT -10[23] J. Tellinghuisen, Using Least Squares for Error Propagation, J. Chem. Educ. 92 (2015) 864870. [24] J. Tellinghuisen, Calibration in isothermal titration calorimetry: Heat and cell volume from
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heat of dilution of NaCl(aq), Anal. Biochem. 360 (2007) 47-55.
ACCEPTED MANUSCRIPT -11FOOTNOTES 1.
Abbreviations: ITC — isothermal titration calorimetry; LS — least-squares; vH — van't
2.
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Hoff; SE — standard error; RSE — relative standard error. Figure 6 of Ref. [3] illustrates interparameter correlation and shows that n×∆H° is precisely determined at low c, even though the two parameters are individually not determined. One can define different n values for each data set to deal with this problem in a global
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analysis, as one can also define set-dependent values of K and ∆H° to deal with irregularities in these parameters. Of course, the limit for adding such local variables is just the results
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from the traditional single-set analyses.
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3.
ACCEPTED MANUSCRIPT -12Table 1 Results from van't Hoff (vH) and global analyses of ITC data from single experiments at 298.15 K (T1) and 303.15 K (T2) for two different ∆H° values. Synthetic data for cell volume V0 = 1.4 mL, [M]0 = 20 µM, m = 10 injections of titrant having volume 25 µL and concentration [X]0 = 133.3
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mM, data precision σq = 1 µJ, and K1 = 500 M-1. Data were generated assuming n = 0.7. For the vH analyses, the data sets were analyzed individually with 2-parameter fits (K, ∆H°) and n frozen at 1.00; in the global analyses, the data for each ∆H° were fitted simultaneously to ∆H°, ∆S°, and n.a
501.0±19.0
−∆H°1 (kJ/mol)
6.998±0.119
500.96±4.76
27.993±0.119
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K1 (M-1)
∆H° = −40 kJ/mol
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∆H° = −10 kJ/mol
Parameter
−173
−697
467.8
383.2
K2 (M-1)
468.7±17.7
383.81±3.62
−∆H°2 (kJ/mol)
6.998±0.122
27.993±0.131
−172
−673
qtot,1 (µJ) K2 (in, M-1)b
van't Hoff analysis −∆H° (vH, kJ/mol)
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qtot,2 (µJ)
10.02±8.04
40.04 ±2.01
0.70±0.56
0.699±0.035
0.70±0.30
0.700±0.019
−∆H° (kJ/mol)
10.00±4.30
40.00±1.08
∆S° (J mol-1 K-1)
18.1±14.3
−82.49±3.60
−15.406±0.074
−15.406±0.019
500.0±14.8
500.00±3.85
n (corrected)
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n
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global analysis
∆G°1 (kJ/mol)c K1 (M-1)c
a
Parameter standard errors are given with excess precision for mathematical comparisons.
b
Input values, as determined from K1 and stated ∆H° values.
c
SEs obtained from an equivalent fit with ∆G°1 replacing ∆S° as adjustable parameter; see text.
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-13-
Figure 1. Normalized relative standard errors for K and ∆H°, for analysis of ITC data traditionally
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(25°C, lines with points) and globally (25°C and 30°C, lines only). The traditional 3parameter fit results are independent of ∆H°, while the global results show dependence on
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∆H° for ∆H° below c = 5 and for K when 0.1 < c < 10. Results obtained by varying K and setting [X]0 using Eq. (4), with other quantities as in Table 1.
S0003-2697(16)30262-7
DOI:
10.1016/j.ab.2016.08.024
Reference:
YABIO 12489
To appear in:
Analytical Biochemistry
Received Date: 28 April 2016 Revised Date:
6 August 2016
Accepted Date: 23 August 2016
Please cite this article as: J. Tellinghuisen, Analysis of multitemperature isothermal titration 1 calorimetry data at very low c: Global beats van't Hoff , Analytical Biochemistry (2016), doi: 10.1016/ j.ab.2016.08.024. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT
Analysis of multitemperature isothermal titration calorimetry data at very low c: Global beats van't Hoff1
Joel Tellinghuisen Department of Chemistry
SC
Vanderbilt University
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by
M AN U
Nashville, Tennessee 37235
Submitted to Analytical Biochemistry 10 Manuscript Pages 1 Figure
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1 Table
Short Title: ITC analysis at very low c
EP
Joel Tellinghuisen Department of Chemistry Vanderbilt University Nashville, TN 37235
AC C
Author:
Phone: (615) 322-4873 FAX:
Email:
(615) 343-1234
[email protected]
Subject Category: Key Words:
Physical Techniques
ITC, Data analysis, Nonlinear least squares, global analysis, statistical errors
ACCEPTED MANUSCRIPT -2-
ABSTRACT Isothermal titration calorimetry data for very low c (≡ K[M]0) must normally be analyzed with the stoichiometry parameter n fixed — at its known value or at any reasonable value if the
RI PT
system is not well characterized. In the latter case, ∆H° (and hence n) can be estimated from the Tdependence of the binding constant K, using the van't Hoff (vH) relation. An alternative is global or simultaneous fitting of data at multiple temperatures. In this Note, global analysis of low-c data at
ITC, Data analysis, Nonlinear least squares, global analysis, statistical errors
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Key Words:
M AN U
SC
two temperatures is shown to estimate ∆H° and n with double the precision of the vH method.
ACCEPTED MANUSCRIPT -3It was long thought that successful application of isothermal titration calorimetry (ITC) to 1:1 binding processes, M + X
⇌
MX, required that the product K[M]0 (≡ c) of the binding constant
K and the initial concentration of the cell reagent (titrand M) fall in the approximate range 1-1000
RI PT
[1]. It is now recognized that this prescription is too limiting, especially on the small-c side, as it is based on the tacit assumption of titration to a molar excess of about 2 in the ratio [X]0/[M]0 of total titrant to titrand in the cell after the last injection [2]. With decreasing c, this ratio must be
increased to yield significant conversion of M to MX. Then ITC can be used successfully down to c
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~ 0.001, provided the titrant concentration can be made sufficiently large. However, as c decreases below 1, the stoichiometry number n and reaction enthalpy ∆H° become highly correlated, making
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it impossible to obtain reliable estimates of both and sometimes producing divergent fits. But K remains well determined in this regime and is almost completely independent of n. Thus, one can freeze n — at its known value in favorable cases or at any reasonable estimate in cases where the properties of the titrand solution are not well known — and then fit the data to just the two parameters, K and ∆H° [3].
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In biological studies, n is often not well known, and I suggested that ∆H° could be estimated reliably from K estimates at two or more temperatures, through the van't Hoff (vH) relation, after which the correct n could be obtained, since the product n×∆H° is well determined.2 Although the
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utility of ITC at very low c is increasingly being recognized, it appears to me that the power of Tdependent data is not yet appreciated [4-8]. In single-T experiments, the need to fix n at its
AC C
presumably known value can lead to systematic errors, from concentration errors related to preparation and materials limitations [9] and from cell volume errors [10]. In this Note, I obtain an expression for the precision with which ∆H° and n can be estimated from data at two Ts, and I show further that simultaneous or global analysis of the same data yields both parameters with precisions a factor of two better than for the 2-step vH method. Consider first the van't Hoff analysis of K at two Ts. If these are sufficiently close together, we can take ∆H° to be constant. Then the vH relation reads
ACCEPTED MANUSCRIPT -41 1 R ln (K2/K1) = ∆H° T – T , 2 1
(1)
where R is the gas constant and the Ts are absolute (in K). If the T uncertainty is negligible, the uncertainty in ∆H° is determined entirely by that in the Ks. Let the K ratio be designated as z,
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giving
σz/z = [(σ1/K1)2 + (σ2/K2)2]1/2 ≈ 2 (σK/K) ,
(2)
where the final approximation assumes that the relative standard errors (RSE) in the Ks are the same. For low-heat, the estimation precision σq for the heats qi becomes constant [2,11]; then
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below c = 1, (σK/K) becomes ~7.5 σq/|qtot| for a 2-parameter fit, where qtot is the sum of the qis for
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the m injections [12]. Since σ(ln z) = σz/z, the SE in ∆H° is T1T2 σq σ(∆H°) ≈ 10 R , ∆T |qtot|
(3)
where ∆T is the absolute difference in the two Ts. The uncertainty in ∆H° is inversely proportional to ∆T, a reasonable expectation. The dependence on (σq/|qtot|) of the absolute SE in ∆H° is the same as that for the relative SEs in all three parameters from the standard treatment — a direct
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consequence of the RSE in Eq. (2). Since qtot is proportional to h = ∆H° × [M]0, the SE in the van't Hoff ∆H° is inversely proportional to [M]0; and for fixed [M]0, the RSE in ∆H° goes as ∆H°−2. In the 2-step vH procedure, the data at each T are first fitted to the 2-parameter model, with
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n frozen at some reasonable value. This yields the two Ks and two estimates of the apparent ∆H°. Because of the very high correlation between n and ∆H° at low c, this ∆H° varies with n in a way
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that preserves n×∆H°. (E.g., doubling n reduces ∆H° by a factor of two, with negligible effect on K.) The two Ks then give the vH estimate of ∆H°, through Eq. (1); and this can be combined with the ∆H° estimates to obtain the correct n, using nassum×∆H°fit = ncorr×∆H°vH. The uncertainty in ncorr is obtained using error propagation, which gives the squared RSE in ncorr equal to the sum of the same for ∆H°fit and ∆H°vH, with the latter given approximately by Eq. (3) and usually dominating. (See the Online Supplement for an illustration of this method.)
ACCEPTED MANUSCRIPT -5To check this procedure, I have used synthetic data generated using a program like those provided in Refs. [2] and [12], for parameter values giving c = 0.01, presented in Table 1. The
(4)
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initial titrant concentration was set using the empirically derived expression [2,12] 6.4 13 Rm = 0.2 + c ; but Rm > 1.1 c
for the X:M concentration ratio in the cell at the end of the titration. For very small c, the 2nd term dominates, and [X]0 becomes dependent on K alone, ~13/K, so the value of n used here doesn't
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affect this calculation [13]. The resulting value of 133 mM (which includes a factor of ~5 for the cell:syringe volume ratio) was used at both Ts, since this would be the usual experimental choice.
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The results in Table 1 for exact data show that the assumption of n = 1 for the 2-parameter fits produces only slight bias in the estimates for K, and gives estimates of ∆H° off by the expected factor ncorr/nassum = 0.7. The vH procedure hardly determines ∆H° and n for the lower ∆H°, but both are well determined at the higher value, thanks to the inverse dependence of the RSEs on ∆H°2. Eq. (3) predicts SEs higher by < 10%. To improve the precisions one might consider
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increasing [M]0 to increase |qtot| (and c). An alternative is the use of a larger temperature interval ∆T; but this might compromise the assumption of constant ∆H°. Note that if the 2-parameter fits are used to estimate the true ∆H° for known n, a realistic estimate of the uncertainty in the latter should be used, through error propagation, to obtain the uncertainty for ∆H°. For example, the
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uncertainty in the "known" n, from concentration and cell volume uncertainties, will generally
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exceed the ~0.004 RSE in the ∆H°fit values in the last column of the table. In the standard analysis of ITC data at a single T, the three fit parameters are just
independent quantities. But thermodynamically, K and ∆H° are related, through ∆G° = −RT ln K° = ∆H° − T∆S° ,
(5)
where K° represents the dimensionless K, referenced to its chosen standard states. Thus, one can replace K° in the analysis by ∆G° or ∆S°. For data at a single temperature these choices are all statistically equivalent, but for data at multiple temperatures they are not. In particular, traditional
ACCEPTED MANUSCRIPT -6analysis of data sets at two Ts requires 2 Ks and 2 ∆H° values; in global analysis just two parameters replace these 4 when ∆H° (and hence ∆S°) can be taken as constant. Then K becomes K° = exp[∆S°/R − ∆H°/(RT)] .
(6)
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This treatment is not limited to just 2 data sets at two Ts, nor to ∆H° and ∆S° that are constant. For T-dependent ∆H° and ∆S°, we obtain results equivalent to those obtained by integrating the differential form of the vH equation (see Online Supplement) [14].
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Global LS fitting has often been used to analyze multiple data sets that share some common parameters, and it is not new to ITC, having been used at least as early as 1995 by Turner, et al. [15], and by several other groups since [16-22]. The present case offers a quantitative comparison
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of global and traditional methods in perhaps the simplest form in which they can be compared. Accordingly, the results in Table 1 show that the global SEs track those from the vH analysis but are a factor of two smaller, meaning 4-fold better statistical efficiency for the global analysis. Note that to obtain the SEs in ∆G°1 and K1 it is necessary to either include correlation in propagating the error from ∆H° and ∆S°, or to define an equivalent fit with ∆G°1 or K1 replacing ∆H° or ∆S° in the
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fit model [23]. (These points are illustrated with a KaleidaGraph routine in the Supplement.) Figure 1 compares the precisions from global analysis with those for the standard 3parameter fits in the normalized form used in Fig. 3 of Ref. [12]. As was noted before, the RSEs
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for ∆H° are no longer universal functions of σq/|qtot|, giving instead different curves that depend on the magnitudes of ∆H° and ∆T (these results being for ∆T = 5 K only). This dependence also
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manifests as variability in the RSE of K in the c range 0.1 - 10. For c > 10, the improved precision in the global analyses is largely due to the factor of ~ 2 from fitting twice as many points. Thus, while the global analysis is transformational at very low c, yielding results for ∆H° and n where these would be impossible to obtain directly from traditional fitting, the gains are only modest at large c, where the traditional analysis yields good precision in all three parameters. Also, since for c > 10 the RSEs for ∆H° and n can drop to ~0.01 or smaller, the fits can become quite sensitive to small systematic errors in, e.g., [M]0 in different runs [9]. These can be compensated by small
ACCEPTED MANUSCRIPT -7changes in n in the individual fits, but lead to deteriorating fit quality from increased sum of squares (χ2) in global fits with a single n.3 All results obtained above have assumed that T is error-free. This is probably a good
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assumption, as much attention has been devoted to good temperature control in commercial ITC instruments, making random T error very small. On the other hand, the systematic error in the T readout can be significant and should be corrected by T calibration. Such a calibration for a widely used ITC instrument revealed a linear error in the readout T that produced a 5% error in any ∆T
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interval, leading to a 5% error in the vH and global estimates of ∆H° [24].
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Global fitting is, to my knowledge, not an option with the analysis software provided by the ITC instrument manufacturers, but it can be done with some freely available packages, including SEDPHAT [18,19]. Also, the KaleidaGraph routines in the Supplement can be used for both traditional and global ITC analysis. Appendix A. Supplementary data.
Funding
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Supplementary data related to this Note can be found at …
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No financial support from agencies in the public, commercial, or nonprofit sectors.
ACCEPTED MANUSCRIPT -8REFERENCES [1] T. Wiseman, S. Williston, J. F. Brandts, and L.-N. Lin, Rapid Measurement of Binding Constants and Heats of Binding Using a New Titration Calorimeter, Anal. Biochem. 179
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(1989) 131-137. [2] J. Tellinghuisen, Designing isothermal titration calorimetry experiments for the study of 1:1 binding: Problems with the "standard protocol," Anal. Biochem. 424 (2012) 211-220. [3] J. Tellinghuisen, Isothermal titration calorimetry at very low c, Anal. Biochem. 373 (2008)
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395-397.
[4] S. A. Liu, P. J. Focke, K. Matulef, X. L. Bian, P. Moenne-Loccoz, F. I. Valiyaveetil, and S.
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W. Lockless, Ion-binding properties of a K+ channel selectivity filter in different conformations, Proc. Nat. Acad. Sci. 112 (2015) 15096-15100. [5] M. Nagar and S. L. Bearne, An Additional Role for the Bronsted Acid-Base Catalysts of Mandelate Racemase in Transition State Stabilization, Biochemistry 54 (2015) 6743-6752. [6] S. Mohanty, C. Jobichen, V. P. R. Chichili, A. Velazquez-Campoy, B. C. Low, C. W. V.
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Hogue, and J. Sivaraman, Structural Basis for a Unique ATP Synthase Core Complex from Nanoarcheaum equitans, J. Biol. Chem. 290 (2015) 27280-27296. [7] S. G. Krimmer, and G. Klebe, Thermodynamics of protein-ligand interactions as a reference for computational analysis: how to assess accuracy, reliability and relevance of experimental
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data, J. Comput. Aided Molec. Design 29 (2015) 867-883. [8] A. J. Situ, T. Schmidt, P. Mazumder, and T. S. Ulmer, Characterization of Membrane Protein
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Interactions by Isothermal Titration Calorimetry, J. Mol. Biol. 426 (2014) 3670-3680. [9] J. Tellinghuisen and J. D. Chodera, Systematic errors in isothermal titration calorimetry: Concentrations and baselines, Anal. Biochem. 414 (2011) 297-299. [10] J. Tellinghuisen, Volume errors in isothermal titration calorimetry, Anal. Biochem. 333 (2004) 405-406. [11] J. Tellinghuisen, Statistical error in isothermal titration calorimetry: Variance function estimation from generalized least squares, Anal. Biochem. 343 (2005) 106-115.
ACCEPTED MANUSCRIPT -9[12] J. Tellinghuisen, Optimizing isothermal titration calorimetry protocols for the study of 1:1 binding: Keeping it simple, Biochim. Biophys. Acta 1860 (2016) 861-867. [13] J. Tellinghuisen, Optimizing Experimental Parameters in Isothermal Titration Calorimetry, J.
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Phys. Chem. B 109 (2005) 20027-20035. [14] J. Tellinghuisen, Van't Hoff analysis of K°(T): How good ... or bad?, Biophys. Chem. 120 (2006) 114-120.
[15] D. C. Turner, M. Straume, M. R. Kasimova, and B. P. Gaber, Thermodynamics of Interaction
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of the Fusion-Inhibiting Peptide Z-D-Phe-L-Phe-Gly with Dioleoylphosphatidylcholine Vesicles: Direct Calorimetric Determination, Biochemistry 34 (1995) 9517-9525.
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[16] B.M. Baker and K.P. Murphy, Dissecting the energetics of a protein–protein interaction: the binding of ovomucoid third domain to elastase, J. Mol. Biol. 268 (1997) 557–569. [17] K.M. Armstrong and B.M. Baker, A comprehensive calorimetric investigation of an entropically driven T cell receptor-peptide/major histocompatibility complex interaction, Biophys. J. 93 (2007) 597–609.
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[18] J. C. D. Houtman, P. H. Brown, B. Bowden, H. Yamaguchi, E. Appella, L. E. Samelson, and P. Schuck, Studying multisite binary and ternary protein interactions by global analysis of isothermal titration calorimetry data in SEDPHAT: Application to adaptor protein complexes
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in cell signaling, Protein Sci. 16 (2007) 30-42. [19] H. Zhao, G. Piszczek, and P. Schuck, SEDPHAT – A platform for global ITC analysis and
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global multi-method analysis of molecular interactions, Methods 76 (2015) 137-148. [20] L.A. Freiburger, O.M. Baettig, T. Sprules, A.M. Berghuis, K. Auclair, A.K. Mittermaier, Competing allosteric mechanisms modulate substrate binding in a dimeric enzyme, Nat. Struct. Mol. Biol. 18 (2011) 288–294. [21] L. A. Freiburger, K. Auclair, and A. K. Mittermaier, Van't Hoff global analyses of variable temperature isothermal titration calorimetry data, Thermochim. Acta 527 (2012) 148-157. [22] L. Freiburger, K. Auclair, and A. Mittermaier, Global ITC fitting methods in studies of protein allostery, Methods 76 (2015) 149-161.
ACCEPTED MANUSCRIPT -10[23] J. Tellinghuisen, Using Least Squares for Error Propagation, J. Chem. Educ. 92 (2015) 864870. [24] J. Tellinghuisen, Calibration in isothermal titration calorimetry: Heat and cell volume from
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heat of dilution of NaCl(aq), Anal. Biochem. 360 (2007) 47-55.
ACCEPTED MANUSCRIPT -11FOOTNOTES 1.
Abbreviations: ITC — isothermal titration calorimetry; LS — least-squares; vH — van't
2.
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Hoff; SE — standard error; RSE — relative standard error. Figure 6 of Ref. [3] illustrates interparameter correlation and shows that n×∆H° is precisely determined at low c, even though the two parameters are individually not determined. One can define different n values for each data set to deal with this problem in a global
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analysis, as one can also define set-dependent values of K and ∆H° to deal with irregularities in these parameters. Of course, the limit for adding such local variables is just the results
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from the traditional single-set analyses.
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3.
ACCEPTED MANUSCRIPT -12Table 1 Results from van't Hoff (vH) and global analyses of ITC data from single experiments at 298.15 K (T1) and 303.15 K (T2) for two different ∆H° values. Synthetic data for cell volume V0 = 1.4 mL, [M]0 = 20 µM, m = 10 injections of titrant having volume 25 µL and concentration [X]0 = 133.3
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mM, data precision σq = 1 µJ, and K1 = 500 M-1. Data were generated assuming n = 0.7. For the vH analyses, the data sets were analyzed individually with 2-parameter fits (K, ∆H°) and n frozen at 1.00; in the global analyses, the data for each ∆H° were fitted simultaneously to ∆H°, ∆S°, and n.a
501.0±19.0
−∆H°1 (kJ/mol)
6.998±0.119
500.96±4.76
27.993±0.119
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K1 (M-1)
∆H° = −40 kJ/mol
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∆H° = −10 kJ/mol
Parameter
−173
−697
467.8
383.2
K2 (M-1)
468.7±17.7
383.81±3.62
−∆H°2 (kJ/mol)
6.998±0.122
27.993±0.131
−172
−673
qtot,1 (µJ) K2 (in, M-1)b
van't Hoff analysis −∆H° (vH, kJ/mol)
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qtot,2 (µJ)
10.02±8.04
40.04 ±2.01
0.70±0.56
0.699±0.035
0.70±0.30
0.700±0.019
−∆H° (kJ/mol)
10.00±4.30
40.00±1.08
∆S° (J mol-1 K-1)
18.1±14.3
−82.49±3.60
−15.406±0.074
−15.406±0.019
500.0±14.8
500.00±3.85
n (corrected)
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n
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global analysis
∆G°1 (kJ/mol)c K1 (M-1)c
a
Parameter standard errors are given with excess precision for mathematical comparisons.
b
Input values, as determined from K1 and stated ∆H° values.
c
SEs obtained from an equivalent fit with ∆G°1 replacing ∆S° as adjustable parameter; see text.
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-13-
Figure 1. Normalized relative standard errors for K and ∆H°, for analysis of ITC data traditionally
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(25°C, lines with points) and globally (25°C and 30°C, lines only). The traditional 3parameter fit results are independent of ∆H°, while the global results show dependence on
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∆H° for ∆H° below c = 5 and for K when 0.1 < c < 10. Results obtained by varying K and setting [X]0 using Eq. (4), with other quantities as in Table 1.