Using a second‐order differential model to fit data without baselines in protein isothermal chemical denaturation

Abstract

In vitro protein stability studies are commonly conducted via thermal or chemical denaturation/renaturation of protein. Conventional data analyses on the protein unfolding/(re)folding require well‐defined pre‐ and post‐transition baselines to evaluate Gibbs free‐energy change associated with the protein unfolding/(re)folding. This evaluation becomes problematic when there is insufficient data for determining the pre‐ or post‐transition baselines. In this study, fitting on such partial data obtained in protein chemical denaturation is established by introducing second‐order differential (SOD) analysis to overcome the limitations that the conventional fitting method has. By reducing numbers of the baseline‐related fitting parameters, the SOD analysis can successfully fit incomplete chemical denaturation data sets with high agreement to the conventional evaluation on the equivalent completed data, where the conventional fitting fails in analyzing them. This SOD fitting for the abbreviated isothermal chemical denaturation further fulfills data analysis methods on the insufficient data sets conducted in the two prevalent protein stability studies.

Abbreviations

BSA

bovine serum albumin

CD

circular dichroism

Far‐UV

Far‐ultraviolet

SOD

second‐order differential

Introduction

Protein thermodynamic stability, a classic and essential characteristic of the prevalent biopolymer, is frequently investigated in fundamental research,1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 drug discovery,12, 13, 14, 15, 16 and protein engineering.17, 18, 19, 20 Evaluation on protein unfolding/folding energy is one of primary methods to understand protein conformational stability,21 protein mutation effects,22 and property changes of small compounds binding to protein.23, 24 Protein unfolding/folding commonly follows a canonical two‐state mechanism,25 where protein cooperatively transits from folded state (F or native state) to unfolded state (U or denatured state) during unfolding in a changing environment (or vise versa in protein (re)folding). Meanwhile, under each defined condition, there is concentration equilibrium between folded and unfolded of protein in its buffer (Equilibrium 1):

$$\mathrm{F}\mathrm{o}\mathrm{l}\mathrm{d}\mathrm{e}\mathrm{d} \left(F\right) \mathrm{p}\mathrm{r}\mathrm{o}\mathrm{t}\mathrm{e}\mathrm{i}\mathrm{n}\underset{⟷}{ K_U} \mathrm{U}\mathrm{n}\mathrm{f}\mathrm{o}\mathrm{l}\mathrm{d}\mathrm{e}\mathrm{d} \left(U\right) \mathrm{p}\mathrm{r}\mathrm{o}\mathrm{t}\mathrm{e}\mathrm{i}\mathrm{n}$$ (1)

Protein thermal and chemical denaturations are two typical experimental methods to unfold protein (or refold protein depending on direction of the changing environment). In isothermal chemical denaturation, for instance, proteins are unfolded by non‐reactive denaturants, such as urea or guanidine salts,23, 25, 26, 27, 28 which are nearly independent to Gibbs free‐energy change during protein unfolding.27, 28 In order to reach the equilibrium state, a constant concentration of protein is generally incubated with different concentrations of denaturant for a sufficient period of time at a constant temperature. Under each individual denaturant concentration, the apparent equilibrium constant $K_U$ is determined in the following Eq. (2).

$$K_U=\frac{\left[{Protein}_{unfolded}\right]}{{[Protein}_{folded}]}= \frac{f_U}{f_F}=\frac{f_U}{1-f_U}$$ (2)

where $f_U$ and $f_F$ are fractions of unfolded and folded protein at each measurement point respectively. Gibbs free‐energy change of protein unfolding at each denaturant concentration ( $\Delta G_U$) has a relationship with $K_U$ as indicated in the following Eq. (3).24

$$\Delta G_U\hspace{0.17em}=\hspace{0.17em}-RT\ln \left(K_U\right)$$ (3)

where R is the gas constant (8.314 J·K⁻¹·mol⁻¹) and T is experimental temperature which was selected at 298 K (25°C) in this paper.

To easily elucidate conventional data analysis in a typical chemical equilibrium denaturation of protein, bovine serum albumin (BSA) is used as a protein model under urea denaturation. Many biophysical measurements can detect the protein unfolding transition. Far‐ultraviolet (Far‐UV) circular dichroism (CD) spectroscopy is one of the most frequent methods used to detect the two‐state transition by directly monitoring conformational changes in protein secondary structure. Herein, the far‐UV CD spectroscopy at wavelength 230 nanometers is selected to monitor BSA secondary structure changes upon increase of urea concentration ( $\left[urea\right]$). BSA denaturation in urea [filled circles of Fig. 1(A)] represents a typical protein isothermal chemical equilibrium denaturation where the protein BSA loses its secondary structure when the denaturant $\left[urea\right]$ increases. Pre‐ and post‐transition baselines of the BSA denaturation can be straightforwardly observed at low ( $\left[urea\right]$ between 0 M and 2 M) and high ( $\left[urea\right]$ from 7 M to 8 M) urea concentrations respectively [Fig. 1(A)]. The two transition baselines were attributed to solvent effects on the far‐UV CD signal of BSA in folded (pre‐transition) or unfolded (post‐transition) states respectively.25 The region between the two baselines ( $\left[urea\right]$ ranging from 2 M to 7 M) is defined as protein BSA transition region from its folded state to unfolded state. In each individual CD measurement (filled circles) of different urea concentrations, the observed CD signal of BSA ( $Y$) follows the relationship described in Eq. (4) below.25

$$Y=Y_F{f}_F\hspace{0.17em}+\hspace{0.17em}{Y}_U{f}_U$$ (4)

where $Y_F$ and $Y_U$ are the observed CD signal (ellipticity) of BSA that folded and unfolded BSA would have under each urea denaturation condition.25 The values of $Y_F$ or $Y_U$ can be predicted and obtained from linear fittings on the two baselines [grey solid line and grey dash line in Fig. 1(A) respectively]. The baseline linear relationships are shown in Eqs. (5) and (6).

$$Y_F=a_F\left[urea\right]+b_F$$ (5)

$$Y_U=a_U\left[urea\right]+b_U$$ (6)

where $a_F$ (892 ± 1 deg·cm²·dmol⁻¹·M ⁻¹) and $a_U$ (207 ± 5 deg·cm²·dmol⁻¹·M ⁻¹) are the slopes of the two baselines, and $b_U$ (−3575 ± 38 deg·cm²·dmol⁻¹) and $b_F$ (−15,300 ± 1 deg·cm²·dmol⁻¹) are the baseline Y‐axis intercepts. The independent parameters $a_F$, $a_U$, $b_U$, and $b_F$ are the four essential, baseline‐related fitting parameters in the convention fitting. Because $f_U+f_F=1$, Eq. (4) can also be transformed into

$$f_U=\frac{Y-Y_F}{Y_U-Y_F}$$ (7)

Figure 1.

Chemical denaturation of BSA and fitting comparison of full data set. (A) Mean‐residue ellipticity change of BSA (filled circles) upon urea denaturation was monitored using CD spectroscopy at 25°C. The denaturation data was fitted by conventional data analysis using Eq. (9). Pre‐transition (fitted by grey solid line) and post‐transition (fitted by grey dash line) baselines are indicated. Calculated Δ $G_U^{bf}$ by the conventional fitting is 15.23 kJ·mol⁻¹. (B) Second‐order differentiation (SOD) analysis on the full SOD data set of BSA chemical denaturation was conducted using Eq. (13) with a high agreement to the conventional equation. The calculated Δ $G_U^{bf}$ value by the SOD method is 15.21 kJ·mol⁻¹.

Meanwhile, Eqs. (2) and (3) can be combined to generate Eq. (8) where $f_U$ is related to the unfolding energy under each urea conditions.

$$f_U= \frac{e^{\frac{-\Delta G_U}{RT}}}{1+e^{\frac{-\Delta G_U}{RT}}}$$ (8)

When combining Eqs. (7) and (8), the observed BSA denaturation can be described using Eq. (9).

$$Y=\frac{e^{\frac{-\Delta G_U}{RT}}}{1+e^{\frac{-\Delta G_U}{RT}}}\hspace{0.17em}\times \hspace{0.17em}{(Y}_U-Y_F)\hspace{0.17em}+\hspace{0.17em}{Y}_F$$ (9)

where $\Delta G_U$ has a linear relationship with denaturant concentration as shown below.23, 24, 25, 27

$$\Delta G_U= \Delta G_U^{bf}-m\left[urea\right]$$ (10)

$\Delta G_U^{bf}$ is the unfolding free‐energy of protein in denaturant‐free buffer and is also the central parameter for evaluating protein stability in the certain buffer. $m$ is a positive slope which was originally defined based on experimental observation,23 and it is predicted to be temperature and pressure dependent.7, 29 Therefore, in addition to the four independent parameters from the baselines ( $a_F$, $a_U$, $b_U$, and $b_F$), there are total 6 fitting parameters needed to be calculated in the conventional analysis, including parameters $\Delta G_U^{bf}$ and $m$ which are 15.23 ± 0.02 kJ·mol⁻¹ and 2.90 ± 0.01 kJ·mol⁻¹·M ⁻¹ respectively obtained here in the BSA denaturation.

In many protein thermodynamic stability studies, however, the pre‐ or/and post‐translation baselines were frequently unobtainable not only because naturally occurring proteins have wide‐range of conformational stabilities30, 31—which may intentionally associate with their physiological functions32, 33, 34—but also due to in vitro experimental conditions of protein unfolding.34 For instance, the usage of osmolytes, such as glycerol35 and trimethylamine N‐oxide,36 created pseudo‐physiological environments and promoted protein solubility/stability but prevented accurate determination of post‐transition baselines.37 Previous study had shown that different baseline selections dramatically changed the final fitting results, especially when the baselines were poorly defined,38 which supports the four baseline‐related fitting parameters are essential in the conventional six‐parameters fitting analysis and the absence of baselines prevents the use of the traditional six‐parameters fitting.37 In the past, analysis on thermal denaturation data without baselines was well accomplished by fitting derivative melting curve with first‐order differential van't Hoff equation that was developed by John and Weeks.39 The differential van't Hoff relationship presented an advantage to fitting approximation that was insensitive to baseline changes, which simplified the data fitting without reducing fitting accuracy. This Weeks' method was widely accepted even in full thermal melting data fittings because of simplicity.11, 40, 41

Nevertheless, in protein chemical denaturation experiments, a similar approximation process in the Weeks' method could not be applied, because physical principles of chemical denaturation are different from those in thermal denaturation. Moreover, solvent effects on spectroscopic properties (for instance, CD or fluorescence spectra of one protein) of folded and unfolded protein in chemical denaturation could be significant. A good example is the large ellipticity changes in the pre‐transition baseline of the model protein BSA under urea denaturation in this paper, where pre‐transition baseline has a slope with value around 900 deg·cm²·dmol⁻¹·M ⁻¹ [Fig. 1(A)]. Therefore, in protein chemical denaturation, baseline contributions cannot be neglected in data fitting especially when denaturation curves have partial baselines. In this study, we successfully applied second‐order differential (SOD) data analysis to fit largely truncated chemical denaturation curves without approximation of missing signal contributions from baseline(s). Together with Weeks' method applied in protein melting curve fitting, this SOD analysis can be used in isothermal chemical denaturation studies, which completes the analysis circle for evaluating data sets without baseline(s) in the two common protein stability studies.

Results and Discussion

The four baseline parameters represent two‐third of total required fitting parameters in the conventional fitting. In this study, the strategy to decrease fitting ambiguity on partial chemical denaturation data is to reduce the number of parameters needed for truncated data analysis. We quickly notice that by introducing Eqs. (5) and (6) into Eq. (9), $Y$—for instant secondary structure of the BSA in this study as a function of $\left[urea\right]$—also can be rewritten as the following:

$$Y= f_U\hspace{0.17em}(A\left[urea\right]+B)+a_F\left[urea\right]+b_F$$ (11)

where $A= a_U-a_F$ and $B= b_U-b_F$. The Eq. (11) clearly shows that $a_F$ and $b_F$ can be eliminated by SOD on $Y$ with respect to $\left[urea\right]$, resulting in the two new fitting parameters A and $B$ remained as indicated in the Eqs. (12) and (13)

$$\frac{dY}{d\left[urea\right]}= {Af}_U\hspace{0.17em}+\hspace{0.17em}\left(A\left[urea\right]+B\right)f_U^{\mathrm{\prime }} + a_F$$ (12)

and

$$\frac{d{Y}^2}{d{\left[urea\right]}^2}\hspace{0.17em}=\hspace{0.17em}2A{f}_U^{\mathrm{\prime }}+\left(A\left[urea\right]+B\right)f_U^{\mathrm{″}}$$ (13)

Because $f_U$ is a also function of $\left[urea\right]$, the first‐order and second‐order differentials of $f_U$ with respect to $\left[urea\right]$ can be transformed and shown in Eqs. (14) and (15) respectively.

$$f_U^{\mathrm{\prime }}= {\frac{m}{RT}f}_U\left(1-f_U\right)$$ (14)

$$f_U^{\mathrm{″}}= {\left(\frac{m}{RT}\right)}^2{f}_U\left(1-f_U\right)\left(1-{2f}_U\right)$$ (15)

By combining Eqs. (8), (10), (14), and (15), the Eq. (13) only has four adjustable fitting parameters retained, including $\Delta G_U^{bf}$, $m$, and the two merged baseline‐related parameters A and B. The baseline‐related parameters are therefore successfully reduced by 50%, which lays down a foundation to fit incomplete chemical denaturation using the SOD fitting in this paper.

As a comparison, CD‐monitored chemical denaturation of BSA was subsequently evaluated by the conventional fitting using the Eq. (9) [black solid line in Fig. 1(A)] and the SOD fitting using Eq. (13) [black solid line in Fig. 1(B)]. The least‐squares computer auto‐fittings were conducted using software Prism 5 and the key fitting parameters are shown in Table 1. The SOD analysis on the full SOD differential data set has calculated $m$ and $\Delta G_U^{bf}$ values of 2.90 ± 0.06 kJ·mol⁻¹·M ⁻¹ and 15.21 ± 0.31 kJ·mol⁻¹ respectively, similarly to $m$ and $\Delta G_U^{bf}$ values (2.90 ± 0.01 kJ·mol⁻¹·M ⁻¹ and 15.23 ± 0.02 kJ·mol⁻¹) that the conventional method has. Fitting values of the merged baseline‐related parameters A (−682 ± 85 deg·cm²·dmol⁻¹·M ⁻¹) and B (11760 ± 531 deg·cm²·dmol⁻¹) from SOD fitting are also consistent with the calculated values of ${(a}_U-a_F)$ (−685 deg·cm²·dmol⁻¹·M ⁻¹) and $(b_U-b_F)$ (11,725 deg·cm²·dmol⁻¹) within error in the conventional fitting method (Table 1). This comparison between the conventional method and SOD fitting on the equivalent full data set strongly suggests the SOD analysis has a high reproducibility and fitting agreement compared to the conventional fitting.

Table 1.

Key Fitting Parameters of Conventional and SOD Analyses on Full and Abbreviated Data Sets

		m (kJ·mol−1 M−1)	ΔG bf (kJ·mol−1)	ΔGbf Difference (%)	A (aU – aF) (deg·cm2·dmol−1·M −1)		B (bU – bF) (deg·cm2·dmol−1)
					aU	aF	bU	bF
Conventional fitting	Complete data seta	2.90 ± 0.01	15.23 ± 0.02	N/A	207 ± 5	892 ± 1	−3575 ± 38	−15300 ± 1
SOD fitting		2.90 ± 0.06	15.21 ± 0.31	−0.2	−682 ± 85		11,760 ± 531
	A‐setb	2.92 ± 0.09	15.39 ± 0.66	1.0	−733 ± 150		11,822 ± 488
	B‐setb	2.90 ± 0.13	15.20 ± 0.47	0.2	−684 ± 541		11,780 ± 4052
	C‐setb	2.98 ± 0.12	15.68 ± 0.62	2.9	−687 ± 97		11,256 ± 934

^aCorresponding to data set in Figure 1.

^bCorresponding to data sets in Figure 2(A–C) respectively.

We consequently applied SOD analysis on three truncated data sets obtained from the full BSA data set under urea denaturation. Truncated data set A [shown Fig. 2(A) represents denaturation data without pre‐transition baseline plus initial unfolding region. Data set B (shown in Fig. 2(B)] has no post‐transition baseline plus part of post‐unfolding region ( $\left[urea\right]$ ranging from 5.5 M to 8 M) [Fig. 2(B)]. Moreover, data set C in Figure 2(C) lacks both the pre‐ and post‐transitional baselines and only has main part of transition region ( $\left[urea\right]$ ranging from 3.5 M to 6.5 M). These abbreviated data sets have been verified in the conventional six‐parameter fitting which cannot successfully fit the abbreviated data due to high ambiguities. It is conceivable that these failures of conventional fitting are due to incomplete information on baselines of the data sets. In contrast, SOD is able to fit the three second‐order differentiated data sets (Fig. 2 lower panels), demonstrating that SOD fitting is more suitable in analyzing largely abbreviated data than the conventional six‐parameter analysis. Similar to the SOD fitting on the derived full data set, the fitting results of SOD on the truncated data sets A, B, and C have fitting results within error to the conventional method. For instance, the calculated $\Delta G_U^{bf}$ from data sets A, B, and C are 15.39 ± 0.66 kJ·mol⁻¹, 15.20 ± 0.47 kJ·mol⁻¹, and 15.68 ± 0.62 kJ·mol⁻¹ respectively, compared to 15.23 ± 0.02 kJ·mol⁻¹ of the conventional fitting. Other key parameters of SOD fitting on the three partial data sets are shown in Table 1, where $\Delta G_U^{bf}$ differences between SOD fittings and the conventional method are compared as well. It is apparent that data set C [Fig. 2(C)] has the highest fitting difference compared to the rest SOD fittings. For instance, $\Delta G_U^{bf}$ difference of data set C is around 3%, which is mainly attributed to both of the baselines missing in the data set in addition to less number of available data points that set C has. Interestingly, the data set A has a relatively higher $\Delta G_U^{bf}$ difference (1%) than that of the data set B (0.2%), which is attributed to that the pre‐transitional baseline of the original full data set has high influential urea‐induced effects than the post‐transitional baseline. The BSA denaturation has shown the fitting slope $a_F$ (892 deg·cm²·dmol⁻¹·M ⁻¹) of pre‐transition baseline is quite larger than $a_U$ (207 deg·cm²·dmol⁻¹·M ⁻¹)—the slope of post‐transition baseline. This observation further supports that the baselines contributions to the data fitting cannot be ignored described in the introduction section.

Figure 2.

SOD fittings on abbreviated BSA chemical denaturation data sets. (A) Mean‐residue ellipticity of BSA in the chemical denaturation without pre‐transitional baseline plus part of initial unfolding (filled circles, upper panel) and SOD fitting (solid line) on the SOD data in the upper panel with calculated Δ $G_U^{bf}$ value of 15.39 kJ·mol⁻¹ (lower panel). (B) SOD fitting (solid line) on the SOD data set without the post‐transitional baseline plus part of post‐unfolding data (filled circles, upper panel) with fitted Δ $G_U^{bf}$ value of 15.20 kJ·mol‐1 (low panel). (C) Data set without both baselines (filled circles, upper panel) was fitted using SOD (solid line) and the Δ $G_U^{bf}$ value is 15.68 kJ·mol⁻¹.

Meanwhile, the comparison of all the fitting results have also revealed that SOD has relatively high fitting errors comparing with the conventional fitting, especially in the comparison between SOD and the convention fittings on the full data set. We attribute this error increase to the fact that the SOD enlarges the data distribution resulting in reduced fitting quality. However, different from the fitting obstacles of unobtainable data in missing baseline region(s) in the conventional fitting, this SOD‐caused disadvantage, similar to other experimental measurements in practices, can be improved by enhancement of data collection, for instance, using computer‐controlled auto‐titration system to reduce titration errors, decreasing titration steps to increase numbers of titration data, or more commonly averaging identical data measurements. Nevertheless, differences of the key SOD‐fitted $\Delta G_U^{bf}$ compared to the conventional fitting are all less than 3% in the measurement here, which strongly suggests the SOD analysis has a high agreement with the conventional fitting method, including the data set C that merely has the transition region. This SOD analysis for the abbreviated data analysis in isothermal chemical denaturation further fulfills analysis method on the insufficient data sets. Along with the Weeks' method39 that previously developed for evaluating truncated data of protein thermal unfolding, the SOD completes analytical circle of baseline‐missing measurements conducted in the two common protein stability evaluations in vitro.

Methods and Materials

CD chemical denaturation of BSA

CD measurement on BSA (Sigma A3675, USA) chemical denaturation was conducted using CD spectrometer J815 (JASCO, Japan) at wavelength of 230 nanometers. 1.6 milliliter of BSA at concentration of 0.05 mg/mL (in the buffer containing 200 mM NaCl and 10 mM Na_x(PO4)_y pH7.4) was auto‐titrated with 8.6 M urea (Sigma U5378, USA) containing 0.05 mg/mL of BSA in the same protein buffer (200 mM NaCl and 10 mM Na_x(PO4)_y pH7.4). The urea titration was conducted using computer‐controlled auto‐titrator ATS‐429 (JASCO, Japan) with 0.25 M titration steps and titration interval was set at 5 minute when there was no CD signal change suggesting each titration point was at equilibrium.

Conventional and SOD data fittings

The BSA CD raw data (θ in millidegree) was converted into mean residue ellipticity ([θ]_MR in deg·cm² dmol⁻¹) using software Excel (Microsoft, USA). All data analyses in the paper were conducted using mathematical software Prism 5 (GraphPad software, USA). The BSA data was first auto‐fitted by the convention fitting using non‐linear regression (curve fitting) option of the software with user‐defined equations [Eq. (9) associated with Eqs. (5), (6), and (10)]. SOD data sets were generated using Prism 5 with default setup of the software where 4 neighbor points of each side of points were averaged. The SOD fitting was conducted using Eq. (13) combining with Eqs. (8), (10), (14), and (15) in user‐defined equation function.