based on the equation above, what is the value of the correct δhvap to use in this experiment?

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Re-evaluation of the reported experimental values of the rut of vaporization of N-methylacetamide

Abstract

The accuracy of empirical force fields is inherently related to the quality of the target data used for optimization of the model. With the estrus of vaporization (ΔH_vap) of N-methylacetamide (NMA), a range of values have been reported as target data for optimization of the nonbond parameters associated with the peptide bond in proteins. In the nowadays work, the original experimental data and Antoine constants used for the determination of the ΔH_vap of NMA are reanalyzed. Based on this assay, the wide range of ΔH_vap values reported in the literature are shown to exist due to incorrect reporting of the temperatures at which the original values were extracted and limitations in the quality of experimental vapor pressure-temperature information over a wide range of temperatures. Taking these problems into business relationship, a consequent ΔH_vap value is extracted from three studies for which experimental data are available. This assay suggests that the most reliable value for ΔH_vap is 13.0±0.1 at 410 One thousand for use in strength field optimization studies. The nowadays results as well indicate that like analyses, including analysis of Antoine constants lone, may exist of utility when reported ΔH_vap values are not consistent for a given neat liquid.

INTRODUCTION

Structure-office studies of peptides and proteins, including protein folding studies,ⁱ consume a significant volume of intellectual and financial resources. Of the many approaches used to written report such systems empirical force field based calculations represent an effective and always increasingly used methodology from which atomic details of construction-function relationships may exist obtained.^{2 , three} In improver, force fields have the potential to be predictive assuasive, for example, the impact of chemic modification of inhibitors on protein binding or the bear upon of mutations on protein activeness to exist fabricated. Of the protein based predictions, accurate prediction of poly peptide structure based on sequence alone,⁴ the then-chosen poly peptide folding problem, represents a grail of empirical force fields and, accordingly, a meaning number of studies take and continue to be performed towards solving this problem.

Central to the success of empirical forcefulness field based methods is the accuracy of the strength fields themselves. Simply put, the force field must correctly represent the change in energy of the organisation equally a office of conformation and environment in guild to effectively reproduce the experimental regimen.^five Accordingly, significant attempt has been and is still being made towards the optimization of empirical forcefulness fields for proteins, besides as other biomolecules. While much of this attempt is based on the reproduction of high-level breakthrough mechanical data, the most important data with respect to the condensed phase is experimental data, including thermodynamic data on small compounds representative of chemical moieties in macromolecules.^{two , 6} For example, the availability of condensed phase information, including the oestrus of vaporization (ΔH_vap) and complimentary energy of solvation, of N-methylacetamide (NMA) is fundamental for the optimization of the force field parameters associated with the nonbond interactions of the peptide bond with the surround. Careful optimization of the not-bail force field parameters for NMA (e.g. the Lennard-Jones 6–12 and electrostatic parameters) to reproduce such condensed phase data lays the groundwork for a poly peptide force field that accurately models energetic differences associated with changes in the environment, such as moving of the peptide backbone from an aqueous surround to the protein interior during poly peptide folding.

Successful evolution of accurate empirical force fields, therefore, requires the availability of experimental thermodynamic data for small molecules that is both authentic and precise. Experimentally, the most authentic style of determining the oestrus of vaporization is through calorimetry performed at the boiling indicate of the neat liquid. An alternative is the employ of vapor pressure – temperature (P-T) data, equally has been performed for NMA. When bachelor, P-T data may exist used directly to fit empirical force field parameters every bit has been done in a number of cases where accurate data over a wide temperature range is available.^{vii , eight , 9 , ten} However, in the case of neat NMA, the corresponding information reported in the literature^{11 – sixteen} is not sufficiently authentic at low temperatures, making it unsuitable for apply as target data for forcefulness field optimization. Presented in row one of Table one are experimental ΔH_vap values for NMA based on the reported Antoine constants from a number of sources.^{12 , 15 , 17 – 19} As may be seen the values range from 12.eight upwardly to fourteen.half-dozen kcal/mol when temperatures of both 373 and 410 Thou are considered. In improver, values of 14.two¹¹ and xvi.5¹² kcal/mol have been reported. No Antoine constants were presented in the former case, while the latter value tin can readily be excluded if one examines the raw experimental data, every bit performed below. Consequent with the range of reported values are the dissimilar values that have been used every bit target data for the optimization of empirical strength fields for NMA and, by extension, peptides and proteins. For example, the OPLS and AMBER force fields were optimized targeting a value of 13.iii kcal/mol at 373 K; ²⁰ a value also used by Caldwell and Kollman,²¹ Gao and coworkers^{22 , 23} and Kaminski et al. ²⁴ for the development of polarizable force fields. CHARMM22²⁵ and GROMOS²⁶ targeted a value of fourteen.ii at 373 Chiliad as did work by Patel and Brooks in the evolution of a polarizable force field based on a fluctuating charge model.²⁷ In addition, a recent study of a polarizable force field based on a classical Drude oscillator cited four values; −12.7, −13.1, −14.two and −14.eight kcal/mol.²⁸ Thus, it is evident that in social club to develop accurate force field parameters for the peptide backbone it is essential to determine the proper experimental ΔH_vap of NMA.

Table 1

Enthalpies of vaporization, kcal/mol, of NMA from various sources and methods of analysis

Source	Aucejo et al., 199315	Manczinger and Krotüm, 197517	Kortüm and Biedersee, 197013	Averages
1) From reported Antoine constantsa
373K	14.6	13.3	12.8c	n.p.
410K	13.0	12.9	12.viii c	north.p.
2) From Antoine constants refit to full experimental rangea,b
373K	xiv.6, xiv.5, xiii.nine	13.three, 13.3, 13.iii	12.3, 12.3, 12.3	due north.p.
410K	13.0, 13.0, 13.vii	13.0, 13.0, xiii.0	12.5, 12.v, 12.five	n.p.
3) From original experimental data via equation 2 (Temperature range)
	13.8 (353-428K)	13.two (333-443K)	12.6 (353–479)	n.p.
4) From original experimental data via equation 2 from 390 to 430 1000
	13.1	12.ix	thirteen.0	xiii.0±0.i
5) From Antoine constants fit to 390 to 430 Ka,b
373K	14.vii, thirteen.6, thirteen.two	xiv.3, xiii.4, 13.0	xiv.iii, 13.4, 13.0	thirteen.7±0.half dozen
410K	13.ane, xiii.one, 13.1	12.nine, 12.ix, 12.9	12.9, thirteen.0, 13.0	13.0±0.1

In the present report nosotros reanalyze the original experimental data used to determine ΔH_vap of NMA, including available experimental vapor pressure-temperature (P-T) curves for the dandy liquid. From this analysis the source of discrepancies in the original information are identified, allowing for an agreement of the source of the range of previously reported ΔH_vap values. This is followed by determination of a consensus value for ΔH_vap of NMA, a value which we suggest should exist the target for future force field development efforts.

Ciphering OF THE Estrus OF VAPORIZATION FROM VAPOR Force per unit area – TEMPERATURE Data

Typically, adding of the heat of vaporization from experimental information is based on the Clausius-Clapeyron equation:

d P d T = Δ H vap ( V − V liq ) T ≈ Δ H vap V T = P Δ H vap R T 2

(i)

where P is the vapor force per unit area of the liquid, T is the temperature, R is the universal gas constant, V is the molecular volume of the gas phase and V_liq is the molecular volume of the liquid phase. Since we deal with a phase transition from liquid to gas phase, V_liq is many times smaller than the volume of the gas stage, Five, which justifies the use of the guess form of the Clausius-Clapeyron equation shown in the right-hand side of Eq (1).

There are ii approaches for determining the rut of vaporization from experimental liquid - vapor pressure level information. Assuming that the heat of vaporization is constant over the selected temperature range, this equation can be integrated by separating the independent variables:

d P P = Δ H vap R · d T T 2 yielding ln P two P 1 = Δ H vap R ( 1 T 1 − 1 T 2 ) .

(2)

This arroyo allows ΔH_vap to be readily determined from the slope of the vapor pressure-temperature (P-T) curve, though express to a situation where the rut capacity, the temperature dependence of the heat of vaporization, of the neat liquid is zip.

An alternative method for solving Eq (ane) assumes obtaining the derivative of the vapor pressure with respect to temperature. Different formulas have been suggested for this purpose. Amidst these, the Antoine equation is used extensively and has been found to be reliable except where the data is express to very small temperature ranges or for low-boiling substances. The Antoine equation²⁹ is a elementary three-parameter fit to experimental vapor pressure measured over a given temperature range:

where, A, B, C are the fitted parameters. This role allows rearrangement of Eq (1) in the following form:

d P PdT = d ( ln P ) d T = Δ H vap R T 2 = B ( T + C ) 2 or Δ H vap = R B T 2 ( T + C ) two

(4)

This equation explicitly takes into consideration the temperature dependence of the heat of vaporization and should be valid for a wider range of temperatures than Eq (2).

Assay OF ANTOINE CONSTANTS AND EXPERIMENTAL VAPOR Force per unit area-TEMPERATURE (P-T) FOR Northward-METHYLACETAMIDE

Step one of the analysis of the discrepancies in the ΔH_vap values of neat NMA was inspection of available experimental P-T data. Presented in Figure 1 is the P-T information in the class of i/T versus lnP from four studies. Immediately axiomatic is the oldest information set from Gopal and Rizvi.¹² Given the significant departure in this data set as compared to the remaining three sets along with the significant difference in ΔH_vap allows the value of 16.five kcal/mol reported in that report to readily exist discarded as tin can the experimental data. The remaining three data sets from Aucejo et al. and Manczinger and Kortüm, and Kortüm and Biedersee appear to be in reasonable agreement, with all the curves sampling a wide range of temperatures and including a big number of information points. However, inspection of Table 1 shows the ΔH_vap values at 373 K from the reported Antoine constants from those studies to differ by over ane kcal/mol.

Experimental inverse temperature versus the natural log of the vapor pressure (P-T) curves for Northward-methylacetamide from Aucejo et al.,^fifteen Manczinger and Kortüm,¹⁷ Gopal and Rizvi,¹² and Kortüm and Biedersee.^thirteen Experimental data converted to common units of lnP and Torr. Lines connecting the data points were included to facilitate visual inspection and do not represent fits of the information.

This deviation suggested that the method of analysis of the original experimental data may be leading to the discrepancy. The original data was treated via the Antoine equation, eq (3) above. Presented in Table 2 are the Antoine equation constants as originally reported in the cited studies as well as following conversion to common units. The Dykyj constants are a refit of the experimental information from Manczinger and Kortüm; those values and the values from Gopal are included for completeness, though they will not be discussed further. As is evident, meaning differences are present including the impact of constraining C to 0 (i.e. assuming the heat capacity = 0). Notable are the differences in the constants from Aucejo et al.¹⁵, Manczinger and Kortüm,¹⁷ and Kortüm and Biedersee,¹³ despite the similarity of the curves shown in Figure 1. The touch of this difference is observed in the ΔH_vap at different temperatures (Table 1, row one), including the lack of temperature dependence due to C being constrained to naught. The dissimilar values of ΔH_vap equally a function of both the particular written report and temperature along with analysis of the discussion of the reported ΔH_vap in the original publications indicates that the discrepancy of the values reported in the more recent strength field evolution literature is due, in part, to a lack of clarity in the original publications on the temperature associated with the reported ΔH_vap. In addition, it appears that when bachelor, the temperatures were often not correctly noted when citing the original ΔH_vap, further compounding the trouble. However, the differences in the Antoine constants bespeak that the data analysis and/or experimental data contribute to the discrepancies.

Table ii

Reported Antione Constants for NMA¹

Reporteda	A	B	C	Equation
Aucejo, 199315	12.49715	2658.377	−148.3473	lnP(kPa) = A–B(1/(T+C))
Manczinger, 197517	7.7377	2043.37	−60.75	logP(Torr) = A–B(ane/(T+C))
Kortüm, 197013	7.8259	2793.3	0	logP (kPa) = A–B(1/(T+C))
Gopal, 196812	xi.063	3606	0	logP(Torr) = A–B(i/(T+C))
Dykyj, 198418, b	6.60575	1868.206	−75.963	logP (kPa) = A–B(one/(T+C))
Reported data converted to lnP and Torr				lnP(Torr) = A–B(1/(T+C))
Aucejo, 1993	xiv.51214	2658.377	−148.3473
Manczinger, 1975	17.81671	4705.0333	−threescore.75
Kortüm, 1970	20.034786	6431.81094	0
Gopal, 1968	25.473499	8303.121845	0
Dykyj, 1984	17.225287	4301.70329	−75.963

To bank check the previous data assay the available experimental data were refit using a modified version of the FITCHARGE^xxx module in CHARMM.³¹ In all cases, each data set was fit three times using the original Antoine constants from the Aucejo et al., Manczinger and Kortüm, and Kortüm and Biedersee studies equally initial guesses (Tabular array 3). Plumbing fixtures was initially performed over the total range of temperatures used in the respective experiments. Analysis of the Antoine constants for the three information sets (Table three) reveals the impact of the initial guesses on the resulting constants. The plumbing fixtures results are not surprising equally the objective function is non-linear and may accept multiple local minima. In this case the parameter prepare showing the least RMSE represents the all-time fit. From the present fitting the lowest RMSE values were 0.0117, 0.0379 and 0.0926, for the Aucejo et al., Manczinger and Kortüm, and Kortüm and Biedersee data, respectively. Comparison of those values with the RMSE values from the original reported Antoine constants, 0.0118, 0.0383 and 0.0988 for the Aucejo et al., Manczinger and Kortüm, and Kortüm and Biedersee data, respectively, show the refitting to yield simply marginal improvement. These results suggest that the original fitting of the Antoine constants was satisfactory, and the noted differences in ΔH_vap originate from the inherent differences in the experimental data sets.

Table 3

Antoine Constants following refitting to the original experimental data.

Fit over the total range of experimental dataa	Initial guessb	A	B	C	RMSE
Aucejo, 1993	i	14.5106	2658.4542	−148.3022	0.0118
	two	fourteen.5888	2696.4742	−146.5925	0.0117
	iii	20.2896	6183.4048	−22.2746	0.0304
Manczinger, 1975	i	18.4156	5107.5211	−47.0957	0.0379
	ii	eighteen.4156	5107.5211	−47.0957	0.0379
	iii	eighteen.4156	5107.5213	−47.0957	0.0379
Kortüm, 1970	i	xx.7029	7157.6343	27.4688	0.0927
	2	twenty.7080	7162.0838	27.6048	0.0927
	iii	20.8027	7245.8239	30.1571	0.0926
Fit over 390 to 430 K
Aucejo, 1993	i	14.6435	2703.0937	−147.3946	0.0071
	2	17.9300	4698.0665	−64.1666	0.0075
	iii	20.2324	6422.3774	−v.8062	0.0084
Manczinger, 1975	i	xiv.9198	2887.2697	−136.9291	0.0077
	ii	17.8465	4706.7286	−61.4594	0.0066
	three	20.1255	6430.2474	−2.6391	0.0067
Kortüm, 1970	i	15.0520	2956.6394	−133.4645	0.0117
	two	17.8602	4704.1907	−61.7037	0.0110
	iii	20.1509	6428.3894	−three.1271	0.0112

To further verify that the original discrepancies in ΔH_vap values were associated with the experimental P-T data, ΔH_vap values were calculated from the P-T data based on the Clausius-Clapeyron equation,²⁹ eq (2), from the slopes of the 1/T versus lnP plots. Information technology should be reiterated that this approach assumes that the heat capacity is nil (i.due east. C = 0 in the Antoine equation). When this arroyo was applied to the information included in Figure 1, it yielded high quality fits (R² > 0.99 in all cases). Based on the resulting slopes ΔH_vap values of 13.8, 13.ii and 12.6 kcal/mol for the three data sets are obtained (Table 1, row three). The level of agreement is similar to the ΔH_vap values obtained from the Antoine equation. Thus, the present assay indicates that the discrepancy in the reported ΔH_vap values is dominated past contributions from limitations in the experimental data.

Inspection of the experimental data in Figure one shows the agreement to exist good for the Aucejo et al, Manczinger and Kortüm, and Kortüm and Biedersee data sets at the college temperatures. Notwithstanding, the data sets diverge at lower temperatures. The presence of such deviation is not unexpected. Given that the pressures at these lower temperatures become quite small information technology may be causeless that the ability to measure out them accurately becomes limiting. Indeed the extremely low temperatures of the Gopal experiments, to a point where the vapor pressures are a fraction of a Torr (Effigy ane), is suggested to contribute to the significant bug with that dataset. Information technology is the divergence of the experimental data sets at the lower temperatures that leads to differences in the refit Antoine constants discussed above (Tables ii and ​ three) and to the significant differences in ΔH_vap values.

Based on the limitations with the experimental data at lower temperatures, the experimental P-T information were reanalyzed over a higher, though limited, range of temperatures (390 to 430 Yard, Figure 1). This assay included 1) adding of the ΔH_vap values using eq (2) over the selected temperature range and 2) refitting the Antoine constants over the selected temperature range (390 to 430 K) following which ΔH_vap values were obtained from eq (four).

Results from these analyses are included in Table ane for the ΔH_vap values and in Table 3 for the Antoine constants. Based on the calculation of ΔH_vap using eq (two) values close to 13 kcal/mol were obtained for all three studies (Table 1, row 4), with an average and standard departure of thirteen.0±0.1 kcal/mol. Next, refitting of the three experimental information sets over the range 390 to 430 K leads to Antione constants that more accurately reproduce the experimental data as compared to fits of the full temperature ranges used in the experimental studies (Table 3, compare the RMSE values for the top and bottom sections), though the RMSE are similar for each of the subset Antoine constants. The corresponding ΔH_vap values using eq (4) for the three fits of the three data sets (Table 1, row 5) shows the values to range over i.vii kcal/mol at 373 K while all the values are in fantabulous agreement at 410 G. Averaging over these values yields a mean ΔH_vap value of 13.0±0.i kcal/mol at 410 K, which is in platonic understanding with that obtained via eq (two) over the same data range. Thus, information technology is evident that limitations in the experimental data at low temperatures contribute to the discrepancies in the ΔH_vap values of NMA reported in the literature. Moreover, the present information analysis indicates that a ΔH_vap value at thirteen.0±0.ane kcal/mol at 410 M is reliable and should be used equally the target value (and temperature) for the development of theoretical models of NMA.

Equally discussed in the introduction a number of force field development efforts accept been based on calculation of the heat of vaporization at 373 K. Accordingly, the Antoine constants fit to 390-430K data were used to predict ΔH_vap at 373 K. The results in row five of Table one show the derived values to range over 1.7 kcal/mol with an average and standard deviation of 13.7±0.6 kcal/mol. Thus, it is non possible to determine a sufficiently accurate value of the heat of vaporization at 373 K for use in force field development due to the inherent limitations in the available experimental P-T data sets.

With many liquids it may be difficult to obtain the original experimental data to perform the analysis presented above; even so, two or more than sets of Antoine constants may be available in many cases. To test the possible utility of the Antoine constants alone, the reported constants for NMA (Table 2) were used to generate P-T data, with the results presented in Figure two. Inspection of the curves shows them to concord well in the range of 390 to 430 Thousand, with significant difference at lower temperatures, consistent with the original experimental data. Such behavior is not unexpected as the Antoine Constants are but fit to the original data, but the beliefs does indicate that if discrepancies exist in ΔH_vap values for a liquid, inspection of the P-T curves calculated from the Antoine constants may exist of utility to select a temperature range where significant agreement betwixt the different experiments occur, from which more reliable ΔH_vap values may be obtained. Applying this blazon of analysis in the present instance using eq (2) applied to the calculated P-T data in Figure 2 yields ΔH_vap values of 13.0, 12,nine, 12,9 and 12.8, respectively, for the four information sets in Figure 2, yielding an average of 12.nine±0.1 kcal/mol. This is within experimental fault of that calculated from the original experimental data (Table i, rows 4 and 5).

Calculated inverse temperature versus the natural log of the vapor pressure level (P-T) curves for N-methylacetamide from the reported Antoine constants of Aucejo et al., ¹⁵ Manczinger and Kortüm, ¹⁷ Dykyj,¹⁹ and Kortüm and Biedersee.¹³ Lines connecting the data points were included to facilitate visual inspection and do not stand for fits of the data.

In summary, the wide range of ΔH_vap values reported in the literature for liquid NMA are shown to be due to 1) inaccuracies in reporting the temperatures at which the experiments were performed and 2) limitations in the experimental P-T data associated with decreased accurateness in the information obtained at lower temperatures due to the low vapor pressures of NMA. Taking these problems into account allows for the extraction of consistent ΔH_vap values from the data for the three studies for which experimental data is bachelor. This analysis suggests that the most reliable value for ΔH_vap is xiii.0±0.one at 410 K, the value and temperature recommended for use in force field optimization studies. The nowadays results also indicate that similar analysis may be appropriate for other swell liquids for which reported ΔH_vap are used for empirical forcefulness field development.

Acknowledgments

Fiscal support from the NIH (GM051501 and GM072558), Dr. Jirí Sponer for access to reference ¹⁸ and Drs. Edward Harder and Benoit Roux for helpful discussions are acknowledged.

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Source: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2863105/

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