Dimensions, energetics, and denaturant effects of the protein unstructured state

Abstract

Determining the energetics of the unfolded state of a protein is essential for understanding the folding mechanics of ordered proteins and the structure–function relation of intrinsically disordered proteins. Here, we adopt a coil‐globule transition theory to develop a general scheme to extract interaction and free energy information from single‐molecule fluorescence resonance energy transfer spectroscopy. By combining protein stability data, we have determined the free energy difference between the native state and the maximally collapsed denatured state in a number of systems, providing insight on the specific/nonspecific interactions in protein folding. Both the transfer and binding models of the denaturant effects are demonstrated to account for the revealed linear dependence of inter‐residue interactions on the denaturant concentration, and are thus compatible under the coil‐globule transition theory to further determine the dimension and free energy of the conformational ensemble of the unfolded state. The scaling behaviors and the effective θ‐state are also discussed.

Abbreviation

C state

maximally collapsed denatured state

GdmCl

guanidinium chloride

m‐value

the slope of the empirical linear relation between the free energy of unfolding and the denaturant concentration

N state

native or folded state

R_g

radius of gyration

SASA

solvent accessible surface area

SAXS

small‐angle X‐ray scattering

smFRET

single‐molecule fluorescence resonance energy transfer

U state

unfolded or denatured state

Introduction

Most proteins can spontaneously fold into a specific conformation, called the native (folded) state (N), as a prerequisite to perform their biological functions.1, 2, 3 The knowledge of how the native state is stabilized over the unfolded (denatured) state (U) is important in both understanding the molecular mechanism of biological systems and in developing therapies against protein‐misfolding diseases such as Alzheimer's, Parkinson's, Huntington's, and amyotrophic lateral sclerosis.4, 5, 6 Traditionally, chemical denaturants such as guanidinium chloride (GdmCl) and urea were used widely as probes to study protein stability. The free energy of unfolding ( $\Delta G_{\mathrm{N}\to \mathrm{U}}$) has been well documented to depend linearly on denaturant concentration (D):7

$$\Delta G_{\mathrm{N}\to \mathrm{U}}\left(D\right)=\Delta G_{\mathrm{N}\to \mathrm{U}}^{\left(0\right)}-mD,$$ (1)

where the intercept $\Delta G_{\mathrm{N}\to \mathrm{U}}^{\left(0\right)}$ represents the free energy in the absence of denaturant, and the slope m reflects the extent of the influence of the denaturant on the specific protein. However, the exact microscopic mechanism is still in debate.8, 9, 10 The most fundamental disagreement lies in whether denaturant molecules interact directly with proteins or not. In the binding model, for example, it was assumed that denaturant molecules bind directly to the protein at specific sites with certain association constants,11 which results in a logarithmic dependence between the free energy and the denaturant activity (a):

$$\Delta G_{\mathrm{N}\to \mathrm{U}}\left(a\right)=\Delta G_{\mathrm{N}\to \mathrm{U}}^{\left(0\right)}-RT{\displaystyle \sum _i\Delta n_i\ln \left(1+K_{i}a\right)},$$ (2)

where K _i is the effective association constant of the type‐i binding site, Δn _i is the effective number of the binding sites, R is the gas constant, and T is the temperature. In the indirect interacting approach, for example, the transfer model, it is assumed that the denaturant molecules change the water structure (aqueous environment) and the effective properties of the solvent linearly depend on D, which immediately gives Eq. (1) based on the proposition of group additivity.12, 13, 14 In practice, the debate is difficult to adjudicate because the denaturation ability of usual denaturants is weak and their working concentration is so high that denaturant molecules will certainly find themselves in close proximity to the protein chain.8

The recent development of single‐molecule fluorescence resonance energy transfer (smFRET) makes it possible to determine the property of the denatured state under equilibrium conditions of coexistence with the native state, and thus provides a unique opportunity to reexamine the molecular mechanism of the denaturant effect in protein folding.8, 15, 16 In 2009, Nettels et al. showed that the radius of gyration (R _g) of the denatured state as a function of the denaturant activity can be well fitted with a binding‐model equation and a sole effective binding constant K.17 If such an empirical fitting reflects the microscopic process, the obtained K can be combined with the unfolding free energy from conventional experiments to determine the binding‐site number Δn in Eq. (2), which is difficult, if not impossible, to determine previously. On the other hand, in the same year, Ziv and Haran proposed that the denatured state can be described by a coil‐globule transition theory of a polymer and suggested that the expansion of the denatured state should be incorporated into the transfer model.18 Remarkably, their analysis on smFRET data of the denatured state showed that the free energy of the coil‐to‐globule collapse depends linearly on the denaturant concentration and the resulting slope agrees very well with the m‐value in Eq. (1) of protein folding. This is quite astonishing when taking into consideration that m is expected to be related to both native and denatured states while only the denatured state was measured in smFRET.

Although there have been many studies on the interplay between smFRET and the denaturant effect,14, 15, 16, 17, 18, 19, 20, 21, 22, 23 some basic questions remain to be answered. For example, are the binding model and the transfer model compatible with each other in interpreting experimental smFRET data? What is the crucial information that can be extracted from smFRET of the denatured state? In addition, there is partial contradiction in the studies from different groups. For example, the results of Ziv and Haran suggest that the θ‐state (where intrachain and chain‐solvent interactions balance such that the polymer appears as an ideal chain) occurs at a denaturant concentration of D = 2∼6M,18 while Hofmann et al. claim that this value occurs at $D\approx 0$ M.20 To clarify these questions, more elaborate studies are necessary.

In this article, we adopt the coil‐globule transition theory to develop a general scheme to extract interaction and free energy information from smFRET data. Experimental data of a series of proteins are analyzed in detail. The free energy difference between specific and nonspecific interactions reveals the driving force of protein folding. Both the transfer model and the binding model are found to explain the extracted intrachain interactions, and are therefore compatible under the coil‐globule transition theory in describing protein behaviors in the denatured state.

Results

m‐Value can be extracted from smFRET data with a coil‐globule transition theory

The behaviors of the denatured state are described by a modified coil‐globule transition theory. Details were provided in Models and Methods section, and a flow chart of data analysis is summarized in Figure 1. With a set of smFRET data on different denaturant concentration D, both the dimensions and energetics can be readily extracted as explained in the follows.

Figure 1.

The flow chart of data analysis under a coil‐globule transition theory. Measured smFRET data ( $\overline{E}\sim D$), total and labeled lengths of protein (N and n) and generic model parameters (ρ ₀ and bσ ³) are included in dashed frames as known quantities. Equations used are shown in rectangles. Input or obtained relations between various quantities are shown in ellipses. Core results are highlighted with shadows. See Models and Methods section for details.

In our model, the protein is regarded as a polymer chain with excluded volume (bσ ³) and effective short‐range interactions (ε, positive for attraction) among residues (monomers). When the effects of intrachain attraction and solvent–chain interaction cancel each other completely, the system reaches the θ‐state as polymers and the resulting ideal chain is specified by a parameter, ρ ₀, in its scaling law $R_{\mathrm{g},\theta }={\rho }_0{N}^{1/2}$, where R _g,θ is the root mean squared radius of gyration and N is the residue number of the protein. ρ ₀ and bσ ³ are fixed parameters for various proteins while ε depends on both the protein sequence and environment conditions such as denaturant concentration and temperature. If ρ ₀ and bσ ³ are known, ε can be extracted from smFRET data and be further used to calculate the free energy of the system (see Models and Methods section).

Using six smFRET datasets of proteins (F1−F5 and G1, see Table 1) in the literature, we optimized the parameters ρ ₀ and bσ ³ in the coil‐globule transition theory by making the resulting free energy to be consistent as possible with the m‐value of proteins from usual thermodynamic measurements (see Fig. S1, Supporting Information). As a result, we obtain ρ ₀ = 0.34 nm and bσ ³ = 0.030 nm³. Ziv and Haran derived bσ ³ from R _g of the native (collapsed) state recorded in the Protein Data Bank (PDB) by $R_{\mathrm{g},\text{native}}=Nb{\sigma }^3$ and thus bσ ³ varied with protein.18 In our approach, we assume ρ ₀ and bσ ³ to be constant and calculated the R _g of the collapse state and θ‐state from the scaling law, which is convenient for the analysis of proteins whose PDB structure is not available, for example, intrinsically disordered proteins (IDPs). For proteins discussed in the work by Ziv and Haran, our parameters are close to their derived values (see Figs. S2−S3, Supporting Information). The results of our analysis are summarized in Figure 2. A linear dependence between ε and D is found, confirming the observation of Ziv and Haran.18 Similar linearity is also found in other datasets we studied. The slope and intercept of ε ∼ D for all 37 datasets are listed in Table 1. Although we assumed ρ ₀ and bσ ³ to be constant among different proteins, the resulting free energy based on smFRET data (scattering symbols in the right columns in Fig. 2) is linear over a very broad range of D, and the slope agrees well with the m‐value from conventional thermodynamic measurements (solid lines in the right columns in Fig. 2) for all six proteins, supporting the validity of our approach. Examination on the parameter sensitivity indicates that the calculated slope decreases with increasing ρ ₀ and bσ ³, but the dependence on bσ ³ is relatively weaker (see Fig. S5, Supporting Information). Hofmann et al. also adopted the scaling law in estimating R _g of the collapse state and θ‐state, but with their corresponding parameters, the slope of the resulting free energy based on smFRET data deviates from the m‐value (see Fig. S4, Supporting Information). In addition, a protein can be labeled with a FRET donor–acceptor pair at different positions. We find that different labels give consistent ε values when the sequence length between the labeled positions is used for N to fit smFRET data (see Figs. S7–S8).

Table 1.

List of the Datasets and the Determined Properties

Number	Protein	N	n	Q +	Q −	m ref	Ref.	kε	ε 0	m cal	R g (D ≈ 6 M)
A1	IN	60	57	6	14		22	−0.0727	1.40	1.76	2.88
A2	ProTαN	129	55	5	23		22	−0.0452	1.19	2.46	3.21
A3	ProTαC	129	55	5	36		22	−0.0366	1.21	2.03	3.25
A4	CspTm	67	54	8	14		22	−0.101	1.48	3.04	2.87
B1	hCyp	167	164	20	22		20	−0.0431	1.36	3.26	4.31
B2	hCyp	167	153	18	21		20	−0.0434	1.36	3.43	4.18
B3	hCyp	167	123	12	18		20	−0.0469	1.38	4.46	3.83
B4	hCyp	167	112	12	16		20	−0.0478	1.37	4.22	3.82
B5	hCyp	167	97	10	13		20	−0.0425	1.39	3.53	3.44
C1	IN	60	57	6	14		20	−0.0655	1.39	1.58	2.76
C2	ProTαN	129	55	5	23		20	−0.0364	1.20	1.89	3.14
C3	ProTαC	129	55	5	36		20	−0.0327	1.22	1.82	3.22
D1	CspTm	71	67	11	15		20	−0.0622	1.51	1.63	2.82
D2	CspTm	70	59	8	14		20	−0.0755	1.52	1.85	2.65
D3	CspTm	70	46	5	12		20	−0.0690	1.54	1.72	2.48
D4	CspTm	72	34	5	10		20	−0.0571	1.59	1.61	2.14
D5	CspTm	34	34	5	10		20	−0.0794	1.59	1.00	2.08
E1	R17‐93	116	94	11	23		20	−0.0515	1.35	2.39	3.51
E2	R15‐93	114	94	13	20		20	−0.0787	1.44	3.29	3.30
E3	R17‐60	116	61	10	14		20	−0.0510	1.42	2.33	2.93
E4	R15‐60	114	61	10	13		20	−0.0534	1.47	2.10	2.86
F1	Barstar	90	78	6	18	3.13	24	−0.116	1.82	2.72	2.66
F2	Protein L	65	65	6	14	3.22	25	−0.117	1.57	3.34	3.09
F3	CspTm	66	66	11	15	2.93	25	−0.109	1.54	3.06	3.08
F4	Protein L	64	64	6	14	2.50	23	−0.0777	1.73	1.52	2.54
F5	RNaseH	155	133	15	23	5.61	21	−0.103	1.46	6.17	4.09
G1	CspTm	67	66	11	15	2.43	19	−0.124	1.42	3.68	3.26
G2	CspTm	67	58	8	14		19	−0.123	1.65	3.33	2.92
G3	CspTm	67	47	5	12		19	−0.113	1.54	3.17	2.67
G4	CspTm	67	46	5	12		19	−0.106	1.59	2.89	2.59
G5	CspTm	67	34	5	10		19	−0.084	1.52	2.28	2.39
a1	IN	60	57	6	14		22	−0.062	1.27	1.64	2.90
a2	ProTαN	129	55	5	23		22	−0.060	0.96	3.71	3.60
a3	ProTαC	129	55	5	36		22	−0.160	0.22	10.1	4.51
a4	CspTm	67	54	8	14		22	−0.083	1.79	1.43	2.37
b1	Barstar	90	78	7	18	1.90	24	−0.060	1.60	1.51	2.63
b2	Im9	86	59	5	16	1.69	26	−0.072	1.50	2.64	2.88

Number: the index number for datasets. Datasets using the denaturant GdmCl are labeled in capital letters, while those using urea are labeled in lowercase letters. Protein: the abbreviated name of the protein as in the original references. N: the total length of the protein in number of residues. n: the sequence length of the protein between the labeled dyes. Q ₊: the positive charge of the sequence between the labeled dyes. Q ₋: the negative charge of the sequence between the labeled dyes. The charges of the dyes are also included in Q ₊ and Q ₋. m _ref: the m‐values determined by usual thermodynamic measurements as reported in original references (refer to Ziv and Haran18). k _ε and ε ₀: the slope and intercept of the ε ∼ D linear fit, that is, ε = ε ₀ + k _ε D. m _cal: the m‐values determined by the coil‐globule transition theory. ε and ε ₀ are measured in units of k _B T, while m _ref, m _cal, and m _cal are measured in units of RT/M. R _g (D ≈ 6M): R _g (in units of nm) for the sequence between dyes obtained from smFRET experiments under a denaturant concentration of D ≈ 6M.

Figure 2.

Coil‐globule transition properties of six protein datasets: (from top to bottom) F1 (Barstar), F2 (Protein L), F3 (CspTm), F4 (Protein L), F5 (RNaseH), and G1 (CspTm). References for the datasets are given in Table I. The properties are extracted from smFRET data of the denatured state by the coil‐globule transition method with the optimized parameters of ρ ₀ = 0.34 nm and bσ ³ = 0.030 nm³. The properties are given as a function of the GdmCl concentration D (in units of M). The left panels give the root mean squared R _g, that is, ${〈R_{\mathrm{g}}^2〉}^{1/2}$, in units of nm, where the value of R _g,θ is indicated by solid lines. Middle panels give the mean interaction energy ε in k _B T units, where linear fits to scattering data points are shown as solid lines. The right panels give the protein stability $\Delta G_{\mathrm{N}\to \mathrm{U}}/RT$, where $\Delta G_{\mathrm{N}\to \mathrm{U}}$ is calculated by Eqs. (15–19) with the intercept being adjusted to match the conventional thermodynamic measurements, and the linear behaviors determined by the conventional thermodynamic measurements are shown in solid lines for comparison.

Expansion of the denatured state has small contribution to the m‐value

Now we examine the origin of the m‐value. The linear dependence between ε and D may be regarded as a direct consequence of the transfer model. In most applications of the transfer model, the solvent accessible surface area (SASA) of the denatured state is assumed to be independent of D. In the language of the coil‐globule transition theory we studied, the average value of volume occupation fraction, $\overline{\varphi }$, would remain approximately constant with changing D (see Fig. S9, Supporting Information). Under such an approximation, the differentiation of the unfolding free energy [refer to Eqs. ((15), (16), (17), (18), (19)) in Models and Methods section] with respect to D immediately gives

$$\frac{\partial \Delta G_{\mathrm{N}\to \mathrm{U}}\left(\epsilon \right)}{\partial D}=\frac{1}{2}N\left(1-\overline{\varphi }\right)\frac{\partial \epsilon }{\partial D},$$ (3)

which is obviously independent of D and equal to −m. However, as Ziv and Haran have indicated,18 the denatured state exhibits a continuous expansion (which reflects in the ${〈R_{\mathrm{g}}^2〉}^{1/2}$ curve in the left panels of Fig. 2) with increasing D. Therefore, one can define the D‐dependent m from Eqs. ((15), (16), (17), (18), (19)) as:

$$\begin{array}{c}m\left(D\right)=-\frac{\partial \Delta G_{\mathrm{N}\to \mathrm{U}}\left(\epsilon \right)}{\partial D}\\ =-\frac{1}{2}N\left(1-\overline{\varphi }\right)\frac{\partial \epsilon }{\partial D}+\frac{1}{2}N\epsilon \frac{\partial \overline{\varphi }}{\partial D}-\frac{\partial }{\partial D}\left[N{k}_{\mathrm{B}}T\frac{1-\overline{\varphi }}{\overline{\varphi }}\ln \left(1-\overline{\varphi }\right)\right]\\ \equiv m_{\epsilon }+m_{\overline{\varphi },H}+m_{\overline{\varphi },S},\end{array}$$ (4)

where $m_{\epsilon }$ denotes the contribution from the variation of ε with D when the variation of $\overline{\varphi }$ is ignored, while $m_{\overline{\varphi },H}$ and $m_{\overline{\varphi },S}$ denote the enthalpic and entropic contributions from the variation of $\overline{\varphi }$ with D. Taking the protein CspTm as an example and adopting the linear ε ∼ D relation previously obtained in Figure 2, different contributions to m (D) are analyzed in Figure 3. The magnitude of $m_{\overline{\varphi },H}$ and $m_{\overline{\varphi },S}$ changes significantly with D; that is, it will cause a nonlinear dependence of $\Delta G_{\mathrm{N}\to \mathrm{U}}$ on D if $m_{\overline{\varphi },H}$ or $m_{\overline{\varphi },S}$ acts alone [Fig. 3(a)]. However, they adopt negative and positive signs, respectively, and thus compensate each other to leave a small contribution to m. In contrast, $m_{\epsilon }$ changes negligibly with D [owing to the small $\overline{\varphi }$ within the considered D range, refer to Fig. 3(b) and the expression for $m_{\epsilon }$ in Eq. (4)], and forms the main contribution to m and the linear dependence of $\Delta G_{\mathrm{N}\to \mathrm{U}}$ on D. [It is also noted that m decreases at small D in Fig. 3(a), which actually results from both the variation of $m_{\epsilon }$ due to the variation of $\overline{\varphi }$ and the incomplete compensation between $m_{\overline{\varphi },H}$ and $m_{\overline{\varphi },S}$.] An examination of the correlation between $m_{\epsilon }$ and m (Fig. 4) further confirms that m is dominated exclusively by the contribution of $m_{\epsilon }$. Therefore, although the denatured state exhibits an expansion ( $\overline{\varphi }$ decreases with D), its enthalpic and entropic contributions to the folding free energy compensate each other and the overall effect can be neglected. That is the reason why the transfer model can be safely applied under a fixed denatured state in usual cases.

Figure 3.

Origin of the m‐value. The protein CspTm is examined and the linear ε ∼ D relation obtained in the dataset F3 is adopted in the calculation. (a) Different contributions to m (D) (in units of RT), which are calculated by Eq. (4). (b) The average value of the volume occupation fraction, $\overline{\varphi }$, as a function of D. This property can be widely observed in different proteins (Fig. S8, Supporting Information).

Figure 4.

Agreement between the calculated m and $m_{\epsilon }$. Six systems in Fig. 1 are analyzed with the resulting data shown in squares. $m_{\epsilon }$ is calculated with representative $\overline{\varphi }$ for each system. The solid line is the ideal case of $m_{\epsilon }=m$.

Difference between specific and nonspecific interactions can be extracted by combining smFRET and conventional equilibrium experiments

The coil‐globule transition is regarded as an important process during the initial stage of protein folding.27, 28, 29 In relating the coil‐globule transition to the folding stability, the maximally collapsed denatured state (C) with $\overline{\varphi }=1$ is introduced as a thermodynamic reference state and its free energy difference with U can be readily calculated by Eqs. ((15), (16), (17)). Although C has similar compactness with N, it is not a unique conformation, but is composed of an ensemble of difference conformations all with $\overline{\varphi }=1$, and is dominated by nonspecific interactions. On the other hand, N is a unique conformation and is stabilized by specific interactions as hinted by the success of the Gö‐like model.30, 31 In conventional equilibrium methods, the free‐energy difference between C and N, $\Delta G_{\mathrm{C}\to \mathrm{N}}$, is impossible to measure because C is usually inaccessible in experiments. Since the $\Delta G_{\mathrm{C}\to \mathrm{U}}$ can now be extracted from smFRET data with the help of the coil‐globule transition theory, $\Delta G_{\mathrm{C}\to \mathrm{N}}$ can be determined via $\Delta G_{\mathrm{C}\to \mathrm{N}}=\Delta G_{\mathrm{C}\to \mathrm{U}}-\Delta G_{\mathrm{N}\to \mathrm{U}}$, where $\Delta G_{\mathrm{N}\to \mathrm{U}}$ is the protein stability determined in conventional equilibrium methods. This may provide valuable information about interplay between specific and nonspecific interactions in protein folding, which is crucial in incorporating non‐native interactions into native‐centric chain models.30, 32, 33

The determined free energy difference between C and N is given in Figure 5. The G _C state is found to have similar variation with G _N as a function of D [Fig. 5(a)]. As a result, the free energy difference $\Delta G_{\mathrm{C}\to \mathrm{N}}$, is nearly independent of D, as expected, because C and N have similar SASA. For limited systems with available data in our analysis, $\Delta G_{\mathrm{C}\to \mathrm{N}}$ is approximately proportional to the chain length [Fig. 5(b)], with an expression of $\Delta G_{\mathrm{C}\to \mathrm{N}}/RT=-0.36N+1.58$. In all cases, G _C is much higher than G _N and G _U, so the maximally collapsed state cannot exist in equilibrium systems. The properties of G _C can only be determined via approximate extrapolation as used here.

Figure 5.

Free energy difference between C and N. (A) The free energy of C, N, and U states for protein CspTm as a function of D. G _U and G _C are calculated by Eqs. (15, 16) with smFRET dataset F3. G _N is calculated as $G_{\mathrm{N}}=G_{\mathrm{U}}-\Delta G_{\mathrm{N}\to \mathrm{U}}$ where $\Delta G_{\mathrm{N}\to \mathrm{U}}$ is expressed as Eq. (1) whose parameters are determined from equilibrium experiments. (B) $\Delta G_{\mathrm{C}\to \mathrm{N}}$ of the six proteins correlate with the chain length N. The proteins are the same as Fig. 1. $\Delta G_{\mathrm{C}\to \mathrm{N}}$ is nearly independent of D, so each protein is presented by a data point. A linear fit is shown as a solid line.

θ‐State and scaling behaviors

The θ‐state is important for both the coil‐globule transition and protein folding.20, 34, 35, 36 Hofmann et al. suggested that the θ‐state is achieved for the denatured proteins at around the aqueous cellular milieu, i.e., $D\approx 0$.20 By contrast, it was suggested from the results of Ziv and Haran that the θ‐state occurs within $D=2\sim 6$ M.18 Under the framework of the coil‐globule transition theory, the θ‐state occurs at $\epsilon \approx 1$ since the excess free energy with respect to that of the ideal chain can be expanded up to the linear term of ϕ as (see Fig. S10, Supporting Information):

$$g\left(\varphi ;\epsilon \right)\approx -\frac{1}{2}\left(\epsilon -1\right)\varphi ,$$ (5)

which becomes zero at $\epsilon =1$, resulting in a scaling law of the θ‐state of $R_{\mathrm{g}}={\rho }_0{N}^{1/2}$, as mentioned above. (When higher order terms are taken into account, the θ‐state occurs at a ε value slightly larger than 1.) To simplify the analysis, we use an approximated scaling law of

$$R_{\mathrm{g}}={\rho }_0{N}^{\nu },$$ (6)

to extract the scaling exponent ν from R _g with the optimized parameter of ρ ₀ = 0.34 nm as explained above. Here ν varies with solvent conditions. Three critical points of ν are highlighted: 3/5 for the expanded coil state,37 1/2 for the θ‐state and 1/3 for the most compact globule state. The determined results for all 31 datasets with GdmCl as denaturant are summarized in Figure 6. ν increases with D in a similar manner for different systems (except a few systems with large net charge at small D). Most datasets end at ν = 0.5 − 0.55, and none can reach the good solvent point of ν = 0.6 at the highest denaturant concentration. Therefore, GdmCl is not sufficiently strong as a denaturant for most proteins even at the highest concentration. The θ‐state, defined as ν = 0.5, occurs at a range of D = 2∼6M for most systems. This is fully consistent with Ziv and Haran,18 while this range is markedly different from the result obtained by Hofmann et al.20 The reason for such a discrepancy lies in the different parameter ρ ₀ adopted. Hofmann et al. used an effective ρ ₀ value of 0.22 nm, smaller than our optimized value of 0.34 nm as explained above. A direct fit to the data of Ziv and Haran (see Fig. S2, Supporting Information) gives ρ ₀ = 0.33 nm. As we have discussed above, with the parameters of Hofmann et al., one cannot reproduce the experimental m‐values (Fig. S4, Supporting Information). Therefore, ρ ₀ = 0.34 or 0.33 nm may be more reasonable in interpreting smFRET data.

Figure 6.

Scaling exponents ν as a function of the GdmCl concentration D for 31 protein datasets (A1−G5). ν is calculated from R _g via $R_{\mathrm{g}}={\rho }_0{N}^{\nu }$ with ρ ₀ = 0.34 nm, i.e., from each R _g datapoint, we obtained a ν as $v=\ln \left(R_{\mathrm{g}}/{\rho }_0\right)/\ln N$.

To gain more insight about the prefactor ρ ₀ of the scaling law, we have calculated R _g of the systems A1−G5 at a high GdmCl concentration of D ≈ 6M, and plotted the R _g values as a function of N in Figure 7. Data from small‐angle X‐ray scattering (SAXS) studies38 are also plotted for comparison. It is clearly shown that R _g values obtained from smFRET are larger than that from SAXS under similar N values [Fig. 7(a)]. As indicated by Kohn et al.,38 the SAXS data can be well described by a scaling law in good solvent conditions, that is, $\nu \approx 0.6$ [see Fig. 7(b)]. For smFRET data, on the other hand, we find that a direct fit gives $\nu \approx 0.4$. If we fix ν to be the value of the θ‐state (i.e., 0.5), we find that the fitting result [solid line in Fig. 7(c)] can also describe the data quite well. The prefactor ρ ₀ obtained in this way, ρ ₀ = 0.37 nm, is close to the value we used (0.34). A smaller value of ρ = 0.22 nm [dotted line in Fig. 7(c)] is inconsistent with the data points.

Figure 7.

Scaling properties of R _g as a function of N. (a) R _g of different proteins measured from SAXS (adopted from Kohn et al.38) and smFRET (datasets A1−G5, D ≈ 6 M, see Table I) experiments. (b) Data from SAXS shown in logarithmic scale and the solid line represents the scaling law fit of $R_{\mathrm{g}}=0.193N^{0.598}$ nm.38 (c) Data from smFRET and the solid line represent the fitting scaling law of $R_{\mathrm{g}}=0.37N^{0.5}$ nm by fixing $\nu =0.5$, while the dotted line represents the scaling law of $R_{\mathrm{g}}=0.22N^{0.5}$ nm.

It is noted that there is long‐standing discrepancy between smFRET and SAXS data as demonstrated in Figure 7(a). The key may lie in the fact that smFRET does not measure R _g, not even R _ee, but just measures $\overline{E}$ which is thus translated into R _ee and R _g with the help of certain theoretical models. Some theoretical approaches adopted in data analysis, for example, the relation of $〈R_{\mathrm{g}}^2〉=〈R_{\text{ee}}^2〉/6$, may cause systematic discrepancy. Molecular simulations may be helpful in clarifying the problem, which were not pursued in this study.

Screened charge effect can be incorporated into the coil‐globule transition theory

Most proteins are electriferous. GdmCl is an electrolyte that would eliminate charge effects in the denatured state when D is not too small. However, for urea which is a nonelectrolyte, or for GdmCl with too small a D value to eliminate the charge effects, the electrostatic interactions would cause the denatured protein to expand in size, and such effects should be taken into account in the theoretical description. Here, we adopt an approach previously used by Müller‐Späth et al.,22 where the effect of screened electrostatic interactions is described in terms of the effective excluded volume (see Models and Methods section). It is noted that the approach assumes a random distribution of charges, while the distribution of charges in a protein is fixed by the sequence and might be very inhomogeneous along the chain. Therefore, it should be regarded as a first approximation.

Taking this effect into consideration, we determine the parameter σ in the previous bσ ³ = 0.030 nm³ to be σ = 0.8 nm based on all 31 GdmCl datasets (A1−G5, see Fig. S6, Supporting Information). The results for four example systems are shown in Figure 8. ProTαC and ProTαN possess significant net charges, and their R _g increases rapidly at low D [Fig. 8(a,b)]. With the corrected charge effect in the coil‐globule transition theory, a linear ε ∼ D relation is sufficient to produce results (blue lines) consistent with the experiments. If the charge effect is not incorporated into the theory, a linear ε ∼ D relation produces results (red lines) that seriously deviate from the experiments at low D. For Protein L with minimal charge, whether to include the charge effect or not does not result in any discernible difference [Fig. 8(c)]. For CspTm with significant opposite charges, the attraction between opposite charges causes R _g to decrease at low D [Fig. 8(d)], which is partially captured by the theory that incorporates the charge effect (blue lines).

Figure 8.

Charge effects on R _g for four proteins (datasets): (a) ProTαC (C3), (b) ProTαN (A2), (c) Protein L (F2), and (d) CspTm (A4). Results calculated from the experiment smFRET data are shown as circles. The determined ε data points are fitted with a linear ε ∼ D relation, which are further used to recalculate R _g with (blue line) and without (red line) charge effects. With the charge terms, the model fits better with GdmCl dimension data, mainly at very low concentrations.

Linearity versus nonlinearity: lessons from the binding model

The most straightforward explanation for the linear dependence between ε and D is provided by the transfer model where the denaturant changes the solvent environment in a linear‐dependent way with D. However, the binding model can also account for the ε ∼ D linearity even if it defines a logarithmic dependence on the denaturant activity a. The key for the transition from linear to nonlinear lies in the fact that a of GdmCl depends on D in a highly nonlinear manner:11

$${\log }_{10}a=-0.5191+1.4839{\log }_{10}D-0.2562{\left({\log }_{10}D\right)}^2+0.5884{\left({\log }_{10}D\right)}^3.$$ (7)

Under such a nonlinear a ∼ D, the experimental transfer free energies (the solvation free energy difference with denaturant added) $\delta g_i$ of the side chain and backbone group of amino acids39 can be satisfactorily fitted by a linear dependence on D as $\delta g_i=m_{i}D+b_i$,14 or by a binding formula on a as $\delta g_i=RT{n}_i\ln (1+K_{ia})$.20 In a similar spirit, we find that the ε data obtained in Figure 2 can be nicely refitted with a binding model as $\epsilon \left(a\right)={\epsilon }_0+\Delta \epsilon \ln \left(1+Ka\right)$ (Fig. 9). It should be noted that although ε can be fitted with the binding model, the obtained parameter K is quite arbitrary. For example, for Protein L (dataset F2), different K values (0.3, 0.59, 1.0) describe the ε data to nearly the same accuracy (see solid lines with different colors in the panel for F2 in Fig. 9). Such parameter uncertainty is also consistent with the fact that the number of independent parameters in the binding‐model formula [Eq. (2)] is more than those in the linear formula [Eq. (1)] by one. Therefore, the transfer model and the binding model are both effective in interpreting the coil‐globule transition revealed by smFRET data. It is impossible to discriminate these two model based on smFRET results.

Figure 9.

Fitting of ε data with the binding model. The scattered symbols are ε data extracted from smFRET data with the coil‐globule transition theory. Blue solid lines are the fitting to the scatterings with a formula of $\epsilon ={\epsilon }_0+\Delta \epsilon \ln \left(1+Ka\right)$. For Protein L (F2), except for the fitting results with optimized K = 0.59 (blue line), we also draw the results with deviating values of K = 0.3 (red line) and 1.0 (green line) for comparison.

Sometimes, R _g data are fitted with a “binding‐model‐like” formula as:17, 20, 22

$$R_{\mathrm{g}}\left(a\right)=R_{\mathrm{g},0}+\Delta R_{\mathrm{g}}\frac{Ka}{1+Ka}$$ (8)

where quotation marks are used to emphasize the fact that a binding model does not necessarily result in an expression of R _g as seen in Eq. (8); for example, a binding model to describe ε as described above will not result in Eq. (8) for a coil‐globule transition. The phenomenological applications may be misleading. For example, the fitting K values using Eq. (8) on smFRET data are smaller than 1,17, 22 and inconsistent with any K values fitted in the transfer free energies of amino acids (K ≈ 3 − 7). Actually, Eq. (8) can be used to fit various data with diverse mechanisms. For example, an improper binding model can be tried as:

$$R_{\mathrm{g}}\left(D\right)=R_{\mathrm{g},0}+\Delta R_{\mathrm{g}}\frac{KD}{1+KD},$$ (9)

which is not expected to reflect the nature of smFRET data with denaturant GdmCl because a should be used instead of D. The phenomenological fitting with Eq. (9) to R _g data works remarkably well (Fig. S11, Supporting Information). In another example, we theoretically calculate R _g as a function of ε using the coil‐globule transition theory and found that the resulting data for ε < 1.5 (which covers most data in the smFRET experiments) can also be well fitted with an improper binding model formula (see Fig. S12, Supporting Information). Therefore, the application of Eq. (8) in analyzing smFRET data should be applied cautiously and the meaning of the resulting parameter K is highly suspicious.

It is noted that we use the transfer model and the binding model to account for the linearity of ε on D, but not ΔG. The characteristics of ΔG and R _g are explained with the coil‐globule transition theory based on the ε ∼ D linearity. Therefore, the transfer model and the binding model become compatible under the coil‐globule transition theory in describing the denaturant effects of proteins.

Discussion

In contrast to protein folding that is usually highly cooperative, the coil‐globule transition is noncooperative. This offers significant advantages to smFRET studies, because the properties of the system vary gradually with conditions (such as D) and abundant valuable information (such as ε and G) can be extracted under a wide range of conditions. Therefore, the smFRET technique combined with the coil‐globule transition theory has numerous applications. For example, ε and m can be obtained at different temperatures and at high temperatures when the native state is unstable. The enthalpy and entropy components can then be reliably determined to provide clues about protein folding and the denaturant effect. In analyzing the data under different temperatures, it is noted that parameters such as the Förster radius as R ₀ in Eq. (20) may depend on the temperature.40 Another interesting topic is the urea effect. Urea is not an electrolyte and does not separate into ions in solutions. Unlike the case in GdmCl, a depends on D in a roughly linear manner in urea. Thus, urea is a useful system to test the validity of the binding model. At present, smFRET data with urea are limited.22, 24, 26 In addition, urea is not capable of screening electrostatic interactions, which hinders the direct application of our theoretical scheme when the charge effect cannot be ignored. Therefore, the analysis on the urea effect is not pursued in this article. For future smFRET experiments with urea, adding ionic salts such as KCl to screen the charge effect would be helpful.

The interpretation of smFRET data requires the use of theoretical models. The coil‐globule transition theory adopted here is based on a mean field approximation. The validity of the model is supported by the consistent m‐value obtained by various approaches; however, support from other approaches, for example, molecular dynamics (MD) simulations,41, 42, 43 would be helpful. MD simulations are also helpful in clarifying some other questions regarding smFRET results, for example, why the R _g measured from smFRET is usually larger than the value obtained from SAXS (Fig. 7).

The smFRET technique is also useful in studying the thermodynamics and energetics of IDPs. IDPs are abundant in all species and possess some advantages in playing essential functions.44, 45, 46 Intriguingly, IDPs do not have ordered native structures, so standard techniques used to characterize conventional proteins may be not applicable for IDPs. SmFRET, with its power in probing the denatured state, becomes an ideal technique in studying IDPs. With the absence of the native state, the application of smFRET in IDPs is even simpler than that in ordered proteins because it is not necessary to remove the signal disturbance from the native state and thus even conventional FRET on the ensemble level is also applicable.

Conclusions

In this work, we have applied a coil‐globule transition theory to smFRET data of the denatured state of a series of proteins to investigate the denaturant effects in proteins. A modified theory with universal parameters for a GdmCl solution was generated to extract interaction and free energy information from smFRET data with no requirement for PDB structural data. By combining the protein stability data, we have determined the free energy difference between the native state and the maximally collapsed denatured state, providing clues on the specific/nonspecific interactions in protein folding. It was demonstrated that most proteins reach the θ‐state within $D=2\sim 6$ M, and barely reach the good‐solvent point with a scaling exponent of ν = 0.6. With respect to the mechanism of the denaturant effects, both the transfer model and the binding model are able to account for the observed linear ε ∼ D relation, and thus are compatible under the coil‐globule transition theory.

Models and Methods

Coil‐globule transition theory

The conformational ensemble of the denatured state of a protein is described by the coil‐globule transition theory by Ziv and Haran,18 which was modified from the mean‐field Sanchez theory of polymers.47 In the spirit of Sanchez's pristine derivation, we also introduce a few modifications to produce a more applicable scheme, as will be explained below.

The denatured protein is regarded as a polymer chain with excluded volume and short‐range attractive interactions among residues (monomers). The probability distribution function of R _g is given as:

$$P\left(R_{\mathrm{g}}\right)=P_0\left(R_{\mathrm{g}}\right)\exp \left(-\frac{Ng\left(\varphi ;\epsilon \right)}{k_{\mathrm{B}}T}\right),$$ (10)

where N is the number of residues of the protein. $P_0\left(R_{\mathrm{g}}\right)$ is the distribution when the excluded volume and attractive interactions are absent, that is, an ideal chain. It is approximated to be the Flory–Fisk empirical distribution:37

$$P_0\left(R_{\mathrm{g}}\right)\propto R_{\mathrm{g}}^6\exp \left(-\frac{7R_{\mathrm{g}}^2}{2R_{\mathrm{g},\theta }^2}\right),$$ (11)

where R _g,θ is the root mean‐squared R _g of the θ‐state, which depends on the chain length in a scaling law of

$$R_{\mathrm{g},\theta }={\rho }_0{n}^{1/2},$$ (12)

where ρ ₀ is a universal parameter for various proteins. $g\left(\varphi ;\epsilon \right)$ in Eq. (10) is the excess free energy per monomer of the conformations of R _g with respect to those in the ideal chain, and is given as:

$$g\left(\varphi ;\epsilon \right)=-\frac{1}{2}\varphi \epsilon +k_{\mathrm{B}}T\left[\frac{1-\varphi }{\varphi }\ln \left(1-\varphi \right)+1\right],$$ (13)

where ε is the effective attraction (ε > 0) or repulsion (ε < 0) interactions among residues. ε varies with the denaturant concentration for a protein. ϕ in Eqs. (10) and (13) is the effective volume fraction occupied by the chain, which is defined in terms of R _g as:

$$\varphi =\frac{Nb{\sigma }^3}{R_{\mathrm{g}}^3},$$ (14)

where σ is the average excluded volume of a residue, and b is a unit‐less proportional coefficient between ϕ and $N{\sigma }^3/R_{\mathrm{g}}^3$. Nbσ ³ is related to the volume of the native state. In our study, σ and b are assumed to be universal constants for various proteins. It is noted that according to Sanchez's pristine derivation,47 ϕ is proportional to $N{\sigma }^3/R_{\mathrm{g}}^3$, but the proportional coefficient b is not necessary equal to 1. Although one can define an effective volume as ${\stackrel{\sim }{\sigma }}^3=b{\sigma }^3$ to remove b and simplify the symbol $\stackrel{\sim }{\sigma }$ to σ (which is actually adopted in most studies without any explicit statement), we keep the pristine form because σ will also appear in the charge effect as will be discussed below. It is also noted that the relation between R _g of the native state and the θ‐state required by Landau's theory of phase transition18, 20 does not apply here because N is not necessarily large and the system is not necessarily at the phase transition state. Therefore, ρ ₀ and $b{\sigma }^3$ are independent parameters in our model.

Following Ziv and Haran,18 the excess free energy of the denatured state with respect to that of an ideal chain is calculated as:

$$G_{\mathrm{U}}\left(\epsilon \right)=Ng\left(\overline{\varphi };\epsilon \right)=-\frac{1}{2}N\epsilon \overline{\varphi }+N{k}_{\mathrm{B}}T\left[\frac{1-\overline{\varphi }}{\overline{\varphi }}\ln \left(1-\overline{\varphi }\right)+1\right],$$ (15)

where $\overline{\varphi }=〈\varphi 〉$ is the average value of ϕ over the distribution at ε. It is noted that the last term (which contributes a constant) is needed to ensure that $G=0$ when $\epsilon =1$ and $\sigma \to 0$, i.e., the excess free energy of the ideal chain is zero [see also Eq. (5) for linear expansion version]. The maximally collapsed denatured state (C) is introduced as a thermodynamic reference state. By definition, this state has $\overline{\varphi }=1$ and

$$G_{\mathrm{C}}\left(\epsilon \right)=Ng\left(\varphi =1;\epsilon \right)=-\frac{1}{2}N\epsilon +N{k}_{\mathrm{B}}T$$ (16)

Therefore, the free energy of collapse is given as:

$$\Delta G_{\mathrm{C}\to \mathrm{U}}\left(\epsilon \right)=\frac{1}{2}N\epsilon \left(1-\overline{\varphi }\right)+N{k}_{\mathrm{B}}T\frac{1-\overline{\varphi }}{\overline{\varphi }}\ln \left(1-\overline{\varphi }\right).$$ (17)

The free energy of unfolding is related to that of the collapse state as:

$$\Delta G_{\mathrm{N}\to \mathrm{U}}\left(\epsilon \right)=\Delta G_{\mathrm{N}\to \mathrm{C}}\left(\epsilon \right)+\Delta G_{\mathrm{C}\to \mathrm{U}}\left(\epsilon \right).$$ (18)

Since the solvent accessible surface area (SASA) of C and N is similar, their interaction with solvent is also similar and thus their free energy difference $\Delta G_{\mathrm{N}\to \mathrm{C}}\left(\epsilon \right)$ is approximately independent of the denaturant concentration D and ε, which is a function of D for a protein. Therefore, Eq. (18) is rewritten as:

$$\Delta G_{\mathrm{N}\to \mathrm{U}}\left(\epsilon \right)=\Delta G_{\mathrm{C}\to \mathrm{U}}\left(\epsilon \right)+\Delta G_{{}_{\mathrm{N}\to \mathrm{C}}}^0,$$ (19)

where the superscript is used to indicate that $\Delta G_{{}_{\mathrm{N}\to \mathrm{C}}}^0$ is a constant independent of ε (D). Eqs. ((15), (16), (17), (18), (19)) can be used to investigate the dependence of protein‐folding free energy on D (except a constant $\Delta G_{{}_{\mathrm{N}\to \mathrm{C}}}^0$, which is inessential in determining the m‐value) if ε can be extracted from smFRET experimental data.

In smFRET experiments, the measured average FRET efficiency ( $\overline{E}$) of the denatured state is ascribed to the following average over the end‐to‐end (donor‐to‐acceptor) distance distribution $P_{\text{ee}}\left(R_{\text{ee}}\right)$, as:

$$\overline{E}={\displaystyle \int \frac{R_0^6}{R_{\text{ee}}^6+R_0^6}{P}_{\text{ee}}}\left(R_{\text{ee}}\right)d{R}_{\text{ee}},$$ (20)

where R ₀ is the Förster radius of the FRET pair (5.4 nm when AlexaFluor 488 is used as a donor and AlexaFluor 594 as an acceptor chromophore20). To relate $\overline{E}$ to R _g, the relation between $P_{\text{ee}}\left(R_{\text{ee}}\right)$ and $P\left(R_{\mathrm{g}}\right)$ should be considered. Different schemes have been proposed for this relationship, e.g., a conditional‐probability approach.18, 22 The simplest approach is to assume $P_{\text{ee}}\left(R_{\text{ee}}\right)$ obeys a Gaussian distribution and requires $〈R_{\mathrm{g}}^2〉=〈R_{\text{ee}}^2〉/6$.22 It has been shown that the results from different schemes are similar.22 Therefore, ε can be readily calculated from $\overline{E}$ at each D using Eqs. ((10)−20) (i.e., via comparing $〈R_{\mathrm{g}}^2〉\sim \overline{E}\sim D$ and $〈R_{\mathrm{g}}^2〉\sim \epsilon$; see also the flow chart in Fig. 1), which can be further used to calculate $\Delta G_{\mathrm{C}\to \mathrm{U}}$ using Eq. (17). If the protein stability $\Delta G_{\mathrm{N}\to \mathrm{U}}$ is known from independent conventional methods, $\Delta G_{\mathrm{N}\to \mathrm{C}}$ can be determined as $\Delta G_{\mathrm{N}\to \mathrm{C}}=\Delta G_{\mathrm{N}\to \mathrm{U}}-\Delta G_{\mathrm{C}\to \mathrm{U}}$.

Charge Effects on Coil‐Globule Transition

To account for the long‐range electrostatic interactions between the charges in the chain and the screening of charges by the ionic denaturant GdmCl or ionic salts such as KCl, following Müller‐Späth et al.,22 we have adopted an approach that was originally developed by Higgs and Joanny to extract Flory‐like dimensions for polyampholytes. In this approach, the effect of electrostatic interactions is described in terms of the effective excluded volume as:

$${\left({\sigma }^{*}\right)}^3={\sigma }^3+\frac{4\pi l_{\mathrm{B}}{\left(f-g\right)}^2}{{\kappa }^2}-\frac{\pi l_{\mathrm{B}}^2{\left(f+g\right)}^2}{\kappa },$$ (21)

where σ and ${\sigma }^{*}$ are the excluded volume without and with electrostatic interactions, respectively. f and g are the probabilities for the occurrence of a positive charge and a negative charge in a monomer, respectively, which are determined from the amino acid sequence. The second term on the right‐hand‐side of Eq. (21) represents repulsive interactions due to the net charge of a protein, which results in an increase of the excluded volume, whereas the third term leads to a reduction in the excluded volume through attractive interactions between opposite charges. l _B is the Bjerrum length given in:

$$l_{\mathrm{B}}=\frac{e^2}{4\pi {\epsilon }_0{\epsilon }_{\mathrm{r}}{k}_{\mathrm{B}}T},$$ (22)

where e is the elementary charge, ε ₀ is the permittivity of vacuum, and ε _r is the dielectric constant. κ is the reciprocal of the Debye length adopted from the Debye–Hückel theory:

$$\kappa =\sqrt{8\pi l_{\mathrm{B}}I},$$ (23)

where I is the ionic strength of the solution. By replacing $\sigma$ in Eq. (14) with ${\sigma }^{*}$ defined in Eq. (21), the screening electrostatic interactions can be incorporated into the coil‐globule transition theory above. Unless the concentration of GdmCl and ionic salts is small, ${\sigma }^{*}\approx \sigma$ and the electrostatic interactions can be ignored.

Experimental Data from the Literature

Experimental smFRET data were collected from the literature. Thirty‐seven datasets were measured on 10 different proteins (see Table 1 and Table SI, Supporting Information), including eight foldable proteins (cold shock protein, CspTm; cyclophilin A, hCyp; spectrin domains R15 and R17; Ribonuclease inhibitor protein, Barstar; immunity protein colicin E9; IgG binding domain B1 of protein L, Protein L; Ribonuclease HI, RnaseH) and two more highly charged IDPs (prothymosin α, ProTα; the N‐terminal domain of HIV Integrase, IN). Some references have provided suitable R_g data directly.20, 22, 23 In some other references,19, 21, 24 ^– 26 raw $\overline{E}$ data are available, so we have transformed them into R_g data through Eq. (20) (the Förster radius of the FRET pair R ₀ is 5.4 nm here when AlexaFluor 488 is used as a donor and AlexaFluor 594 as an acceptor chromophore20, see also the flow chart in Fig. 1) and a Gaussian distribution with $〈R_{\mathrm{g}}^2〉=〈R_{\text{ee}}^2〉/6$. Dye linkers were not taken into account. Most datasets were obtained using GdmCl as denaturant and their numbers are labeled in capital letters in Table 1. Only six out of 37 datasets were obtained using denaturant urea whose number in Table 1 are labeled in lowercase letters instead. In this study, we have focused on the properties under GdmCl because the electrostatic effect in the presence of urea is more complicated.

Supporting information

Supporting Information