PEDS Advance Access originally published online on August 8, 2005
Protein Engineering Design and Selection 2005 18(9):445-456; doi:10.1093/protein/gzi046
© The Author 2005. Published by Oxford University Press. All rights reserved. For Permissions, please e-mail: journals.permissions@oupjournals.org
Quantitative measurement of protein stability from unfolding equilibria monitored with the fluorescence maximum wavelength
Elodie Monsellier and
Hugues Bedouelle1
Unit of Molecular Prevention and Therapy of Human Diseases (CNRS FRE 2849), Institut Pasteur, 28 rue Docteur Roux, 75724 Paris Cedex 15, France
1 To whom correspondence should be addressed. E-mail: hbedouel{at}pasteur.fr
 |
Abstract
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The fluorescence of tryptophan is used as a signal to monitor
the unfolding of proteins, in particular the intensity of fluorescence
and the wavelength of its maximum
max. The law of the signal
is linear with respect to the concentrations of the reactants
for the intensity but not for
max. Consequently, the stability
of a protein and its variation upon mutation cannot be deduced
directly from measurements made with
max. Here, we established
a rigorous law of the signal for
max. We then compared the stability
G(H
2O) and coefficient of cooperativity
m for a two-state equilibrium
of unfolding, monitored with
max, when the rigorous and empirical
linear laws of the signal are applied. The corrective terms
involve the curvature of the emission spectra at their
max and
can be determined experimentally. The rigorous and empirical
values of the cooperativity coefficient
m are equal within the
experimental error for this parameter. In contrast, the rigorous
and empirical values of the stability
G(H
2O) generally differ.
However, they are equal within the experimental error if the
curvatures of the spectra for the native and unfolded states
are identical. We validated this analysis experimentally using
domain 3 of the envelope glycoprotein of the dengue virus and
the single-chain variable fragment (scFv) of antibody mAbD1.3,
directed against lysozyme.
Keywords: cooperativity/denaturant/dengue virus/envelope glycoprotein/free energy/scFv antibody fragment/unfolding
|
Introduction
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The possibility of measuring the stability of proteins with
precision finds many applications in fundamental and applied
research. It has allowed one to understand and quantify the
forces that contribute to the conformational stability of proteins
in their aqueous environment and the effects of sequence changes
on this stability (Alber, 1989
; Pace
et al., 1996
). The data
on the stability of proteins and their mutants are important
for developing reliable energy functions for proteins (Guerois
et al., 2002
; Bava
et al., 2004
). These force fields are used
in algorithms to predict the structure or docking of proteins
and to design new proteins and stabilizing changes. The precise
measurement of stability is also important to understand and
describe the unfolding and folding of proteins at an atomic
resolution, by a combination of experimental and theoretical
approaches, i.e. the analysis of the
values and molecular dynamics
(Fersht and Daggett, 2002
).
By definition, the thermodynamic stability G of a protein is equal to the variation of free energy between its native and unfolded states. It can be deduced from the constant of equilibrium between these two conformational states and thus from the measurement of concentrations. The stability depends on the physico-chemical conditions and must therefore be given in standard conditions, e.g. G(H2O) in aqueous buffer at 20°C. The concentration of the unfolded state is usually very low in physiological conditions; therefore, the values of the stability are measured in variable physico-chemical conditions and extrapolated to the standard conditions. A physical quantity that is sensitive to the conformational state of the protein, is used for the measurement of concentrations.
The fluorescences of tryptophan and tyrosine residues are sensitive to their electronic environment. Therefore, the intrinsic fluorescence of proteins is widely used to measure the concentrations of their different molecular states in a reaction of unfolding. Only very low concentrations of protein are needed, which minimizes protein aggregation. The most useful fluorescence signals are the intensity Y of the emitted light and the wavelength max at which this intensity is maximal. These two parameters are usually measured after excitation at a fixed wavelength (Eftink, 1994).
The use of the fluorescence intensity Y as a signal to measure the stability of proteins may present difficulties. The Y signal is a function of the protein concentration and is therefore sensitive to volumetric errors. The Y signals of the native state N and of the unfolded state U of a protein generally vary with the concentration of the denaturant and the description of this variation requires at least two parameters (Santoro and Bolen, 1988). The precise determination of these parameters requires a large number of experimental points and thus large amounts of protein material. The Y signals of states N and U are not always sufficiently different for precise measurements (Tan et al., 1998; Dumoulin et al., 2002; Ewert et al., 2003). For example, the denaturation of different domains in a protein can lead to variations of Y that compensate each other. The use of the max signal avoids many of the above difficulties. This signal does not depend on the concentration of protein and increases monotonically during the unfolding. The max signals of states N and U are often independent of the denaturant concentration. Therefore, the description of an unfolding profile requires less parameters and protein material when it is monitored with max as compared with Y.
The quantitative analysis of the unfolding profiles is easier when the recorded signal is a linear function of both concentrations and specific signals of the component molecular species. The Y signal satisfies these conditions of linearity because it depends only on the light absorbed by the molecules and on their quantum yields of fluorescence (emitted photons/absorbed photons). In contrast, there is no simple law for the composition of the max signals. Numerous authors ignore this physical difficulty, apply a linear law of additivity to max and attempt, by this empirical approach, to derive the stability G(H2O) of proteins or the concentration x1/2 of denaturant that gives half unfolding. Some authors justify such an empirical approach by the observation that either the intensities or quantum yields of fluorescence for states N and U of the protein under study are identical (Tan et al., 1998; Ewert et al., 2003). A theoretical study has shown that the error on G(H2O), calculated empirically from measurements of max,can reach 50% (Eftink, 1994). Several experimental studies have compared the values for the thermodynamic parameters of unfolding at equilibrium, calculated rigorously from Y data and empirically from max data. These values are close in some studies and differ widely in others (Jager and Pluckthun, 1999; Jung et al., 1999; Martineau and Betton, 1999; Dumoulin et al., 2002).
Hence the wavelength max is a robust signal for monitoring the unfolding of proteins, but whether it allows one to derive reliable values of their stabilities remains to be demonstrated. In this study, we rigorously derived a law of composition for max from that for Y. From this law, we could determine the correction that must be applied to the empirical value G'(H2O) of the stability, calculated by applying a linear law of the signal to max. The corrective term depends on the curvatures of the emission spectra for states N and U at their respective max. It can be easily determined and is not negligible in general.
We validated our theoretical analysis with two proteins. Domain 3 of the envelope glycoprotein E from serotype 1 of the dengue virus (E3.1, residues 296–400) has been implicated in the interactions between the virus and its cellular receptors (Mukhopadhyay et al., 2005). The single-chain variable fragments scFv of antibodies are widely used in fundamental and applied research. Many studies aim at increasing the stability of scFvs, which is often limiting for applications. Such studies on scFv fragments require methods to compare precisely and reliably their stabilities and the recourse to the max signal is often necessary and has been extensively used (Worn and Pluckthun, 2001). The scFv fragment of antibody mAbD1.3, directed against hen egg-white lysozyme, is a model system for fundamental studies and the development of new methodologies. Many structural and thermodynamic data are available on this system (Sundberg and Mariuzza, 2002).
|
Theory
|
|---|
Equilibrium of unfolding
Let P be a monomeric protein, N its native folded state and U its unfolded state. Let us assume that this protein unfolds according to the equilibrium
 | (1) |
In
physiological conditions, the protein is almost entirely in
its native form N and the concentration of state U cannot be
detected. To be studied, the equilibrium of unfolding is generally
shifted with a denaturing agent, such as urea or guanidinium
chloride (GdmCl). A new equilibrium forms for each concentration
x of denaturant. An unfolding profile is obtained by measuring
a signal of the protein, sensitive to its conformational state,
as a function of
x. The equations derived in this and the following
paragraphs allow one to determine the concentrations of N and
U for each value of
x.
The laws of mass action and conservation give the two following equations, where K is an equilibrium constant and C (M) the total concentration of the protein:
 | (2) |
 | (3) |
Generally, it is more convenient
to reason on molar fractions:
 | (4) |
Equations 2 and
3 can be rewritten as
 | (5) |
 | (6) |
The variation of free energy
G between states
N and U is given by
 | (7) |
where
R is
the gas constant and
T is the temperature (K). By definition,
G is the stability of protein N. Generally, one assumes that
the variation of free energy between two conformational states
is a linear function of
x (Pace, 1986
; Myers
et al., 1995
):
 | (8) |
Note that parameters fn, fu, K and G are functions of x. Let x1/2 be the concentration of denaturant that results in half-advancement of the unfolding reaction, i.e. fn(x1/2) = 0.5. Under these conditions, Equations 5
–7 show that the stability G of the protein is zero and Equation 8 shows that the value of x1/2 is given by
 | (9) |
Law of the signal: fluorescence intensity
Let us assume that the intensity of fluorescence, for a set excitation radiation, is used to monitor the unfolding equilibrium of Equation 1. If Yt(, x) is the global signal of the unfolding mixture, the law of additivity of the signals applies:
 | (10) |
where
is the wavelength at which the fluorescence
emission is measured and
Yn,
Yu and
Yd are the molar signals
of state N, state U and the denaturant, respectively. The signal
of the denaturant alone is generally measured in a separate
experiment and only the protein signal
Y(
,
x) is considered:
 | (11) |
Equation 11 can be rewritten
with molar fractions as follows:
 | (12) |
Experimentally, one observes that
Y(
,
x) is a linear function
of
x at low and high concentrations of denaturant (Santoro and
Bolen, 1988
) and therefore one can write for every
x
 | (13) |
 | (14) |
where
hn =
mn/
yn and
hu =
mu/
yu are intrinsic parameters of the protein.
Generally, one monitors the unfolding reaction at the wavelength
such that
 | (15) |
where
xmax is the
highest concentration of denaturant attainable.
Law of the signal: max
For a given concentration x of denaturant and variable values of the wavelength , Y(, x) represents the emission spectrum of protein P. The wavelength at which the intensity Y(, x) of the emitted light is maximum is denoted max(x). Then, if Y(, x) is approximated by a continuous differentiable function of :
 | (16) |
The differentiation of
Equation 12 gives
 | (17) |
Let
n and
u be the
max values for states N and U of protein P, respectively. If
both
Yn and
Yu are increasing functions of
for
<
n and
decreasing functions for
u <
, then
Equations 16 and
17 imply
 | (18) |
Hence
max remains within the
interval of wavelengths [
n,
u] for any mixture of states N and
U.
The Y(, x) function can be written as a Taylor expansion about = max (Weisstein, 2002; http://mathworld.wolfram.com/TaylorSeries.html). For many proteins, the fourth-order remainder of the Taylor expansion is negligible and their fluorescence spectra can be approximated over a wide interval of wavelengths by the following cubic function (see Results):
 | (19) |
Equation 19 can be simplified if one takes
Equation 16 into
account:
 | (20) |
with
a(
x) =
Y(
max,
x),
b(
x) = (
2Y/
2)(
max,
x) and
c(
x) = (
3Y/
3)(
max,
x). In particular,
the fitting of
Equation 20 to the spectra of states N and U
enables one to determine precise values of
n and
u, respectively,
based on an extended portion of each spectrum (see Results and
Figure 2).
Once the values of n and u are known with precision, the molar spectra of states N and U can generally be approximated on the interval [n, u] by the following quadratic functions, obtained by neglecting the third-order remainder of a Taylor expansion (see Results and Figure 3):
 | (21) |
 | (22) |
With these approximations,
which should be checked in each particular case and for
belonging
to [
n,
u],
Equations 12,
21 and
22 give
 | (23) |
According to
Equations 16,
18 and
23,
max(
x) is a solution
of the equation
 | (24) |
This solution
is given by
 | (25) |
Comparison of the approximate and empirical equations
The stability of a protein P, unfolding according to Equation 1, is often deduced from the following set of empirical equations, drawn by homology with Equations 5–8 and 12 (see Introduction):
 | (26) |
 | (27) |
 | (28) |
 | (29) |
where the corresponding empirical (or apparent) parameters
are labeled with a prime. Comparison of
Equations 29 and
25 shows that the empirical parameters
f'
n and
f'
u are related
to the molar fractions
fn and
fu of states N and U by
 | (30) |
From
Equations 26,
30 and
5, one deduces
 | (31) |
and from
Equations 28,
31 and
7, for every
value of
x:
 | (32) |
In particular,
for
x = 0:
 | (33) |
Equation 33 shows
that the stability of protein P is not equal to the empirical
parameter
G'(H
2O) and there is a corrective term in general.
If G'(x) and G(x) in Equation 32 are replaced by their expressions in Equations 28 and 8 and G(H2O) in Equation 8 by its expression in Equation 33, one obtains
 | (34) |
Equation 34 allows one to calculate the cooperativity of unfolding
m from the empirical parameter
m'. Let
x'
1/2 be the concentration
of denaturant that gives
f'
n(
x'
1/2) = 0.5. From
Equations 26–
28,
it follows that
 | (35) |
Equations 9,
33 and
35 allow one to compare
x'
1/2 with
x1/2:
 | (36) |
Hence the empirical concentration
x'
1/2 is
not equal to the concentration
x1/2 of denaturant that results
in half-advancement of the unfolding reaction.
Geometric interpretation
The curvature 
of any curve Y() in the plane is given by (Weisstein, 2002; http://mathworld.wolfram.com/Curvature.html)
 | (37) |
Equations 16 and
37 imply, if
Y(
,
x) is a fluorescence
intensity,
 | (38) |
Equations 20 and
38 imply that parameters
bn(
x) and
bu(
x) are the curvatures
of the fluorescence spectra for states N and U of protein P
at their respective
max and at a concentration
x of denaturant.
Therefore, the correcting factor in
Equation 33 depends on the
ratio of these curvatures in the absence of denaturant. Note
that the ratio
bu/
bn does not depend on the concentration of
the protein or the spectrofluorimeter or its setup. Below, we
give methods for determining the values of
bn and
bu experimentally.
Curvature versus concentration in urea
For simplicity, we can rewrite Equation 33 as follows:
 | (39) |
 | (40) |
The corrective
term
E involves the curvature
bu(0) of the spectrum for state
U in the absence of denaturant, which cannot be measured directly.
Equation 40 can be rewritten, for every
x, as follows:
 | (41) |
Let us assume that the curvature
bu(
x) of the
spectrum for state U varies linearly with
x over the whole interval
of the concentration in denaturant and that the curvature
bn(
x)
of the spectrum for state N varies linearly with
x in the pre-transition
region (see Results for justifications). Then
 | (42) |
 | (43) |
where
kn and
ku are intrinsic parameters of the protein. The combination
of
Equations 41 and
42 gives an expression for
E where
xmax is the maximum concentration of denaturant attainable (e.g.
8 M urea) and whose every term can be measured experimentally:
 | (44) |
The combination of
Equations 34,
42 and
43 gives, for
x belonging to the pre-transition region,
 | (45) |
As ln(1 +
kx)
kx in the neighborhood
of
x = 0,
Equation 45 gives
 | (46) |
The combination of
Equations 36,
40 and
46 gives
 | (47) |
Note that
Equation 46 does not depend on the
exact form of
Equation 43.
|
Materials and methods
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Bacterial strains, plasmids and media
The Escherichia coli strains HB2151 (Carter et al., 1985) and RZ1032 (Kunkel et al., 1987) and plasmid pMR1 (Renard et al., 2002) have been described. mAbD1.3 is a murine monoclonal antibody, directed against hen egg-white lysozyme. pMR1 codes for a single-chain scFv fragment of mAbD1.3, in the format NH2–VH–(Gly4Ser)3–VL–H6–COOH, where VH and VL are the variable domains of the heavy chain and light chain, respectively, and H6 represents a hexahistidine tag. In pMR1, the expression of the scFvD1.3–H6 gene is under control of the tet promoter and ompA signal sequence from E.coli. The sequence of the recombinant scFvD1.3–H6 gene differed slightly from the published sequences at the 5'- and 3'-ends of the constitutive VH and VL genes, as a result of the cloning steps (Figure 1). Plasmid pLB11 is a derivative of the pET20b+ vector (Novagen) and codes for a hybrid E3.1–H6 between domain 3 (residues 296–400) of the envelope glycoprotein E from the dengue virus (serotype 1) and a hexahistidine tag (Despres et al., 1993; H.Bedouelle et al., in preparation). The E3.1–H6 domain comprises a unique disulfide bridge between residues Cys302 and Cys333. Buffer A was 50 mM Tris–HCl, pH 7.9, 150 mM NaCl. Ultrapure urea and guanidine hydrochloride (GdmCl) were purchased from MP Biochemicals. Solutions of urea and GdmCl were freshly prepared daily. The concentrations of urea or GdmCl were measured with a refractometer with a precision of 0.01 M.
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Fig. 1.. Modifications of the scFv genes in plasmid pMR1. The DNA sequences of the first eight residues and last eight residues of the VH and VL genes in plasmid pMR1, coding for scFvD1.3–H6, differed slightly from the published sequences as a result of the cloning steps (England et al., 1999).
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Proteins and general conditions
The E3.1–H6 and scFvD1.3–H6 recombinant proteins were produced from plasmids pLB11 and pMR1, respectively, in the periplasmic space of strain HB2151. They were purified by nickel ion chromatography as described (Renard et al., 2002; H.Bedouelle et al., in preparation). The protein fractions were analyzed by SDS–PAGE in denaturing conditions. The concentration of acrylamide–bisacrylamide (29:1) was 15% for scFvD1.3–H6 and 17% for domain E3.1–H6. The fractions that were homogeneous at >95% were pooled, dialyzed against buffer A, snap frozen in liquid nitrogen and stored at –70°C. The concentration of protein in the purified preparations was measured by absorbance spectrometry. The extinction coefficients were calculated as described (Pace et al., 1995): 
280nm(E3.1–H6) = 9530 mM–1 cm–1 and 280nm(scFvD1.3–H6) = 51130 mM–1 cm–1.
Unfolding with urea was performed as described (Pace, 1986). Each reaction mixture (1 ml) contained purified protein (10 µg/ml; 0.80 µM for E3.1–H6 and 0.37 µM for scFvD1.3–H6) and varying concentrations of urea (0–9 M) in buffer A. Control reactions were prepared by replacing the protein with buffer. The mixtures were incubated for 14 h at 20°C to enable the reactions of unfolding to reach equilibrium. To test the reversibility of the unfolding reaction, a protein sample (10 µg) was denatured in 7 M urea and buffer A for 4 h. The denatured protein was diluted with buffer A to reach a final concentration of urea between 7 and 1 M. The diluted mixture was then incubated for 14 h at 20°C to enable the reaction to reach equilibrium as above. The concentration of urea was measured in each reaction mixture after the completion of each experiment, as described above.
Fluorescence measurements
Fluorescence experiments were performed at 20°C with a Perkin-Elmer LS-5B spectrofluorimeter. The proteins were excited at 278 nm and the amino acid tryptophan at 290 nm; the slit width was 2.5 nm for excitation and 5 nm for emission. The fluorescence spectra were recorded in the interval 320–370 nm for scFvD1.3–H6 and E3.1–H6 and 310–374 nm for tryptophan. The signal was acquired for 2 s at each wavelength and the increment of wavelength was 0.5 nm. The fluorescence signal for the protein or tryptophan was obtained by subtraction of the signal for the solvent alone. In a first step, each spectrum Y(,x), where x was fixed and variable, was approximated over the whole interval of wavelength [n – 20 nm, u + 20 nm] by the fitting of Equation 20 to the experimental data, with a, b, c and max as floating parameters. In particular, n = max(0) and u = max(xmax) were determined in this way for x equal to 0 M and xmax M of denaturant, respectively. In a second step, the Y(, x) spectrum was approximated on the narrower interval [n – 2 nm, u + 2 nm] by the fitting of Equation 21 or 22 with a and b as floating parameters and max(x) set to the value that had been determined in the first step. This procedure allowed us to optimize the bn(x) and bu(x) parameters in this narrower interval of wavelength to which max(x) necessarily belonged (Equation 18).
Analysis of the unfolding profiles
The solution of Equations 5 and 6 is given by
 | (48) |
The combination of
Equations 6 and
12–
14 gives
 | (49) |
The combination of
Equations 7,
8 and
48 gives
 | (50) |
Equation 49,
where
fn is developed as in
Equation 50 and which relates
the intensity of fluorescence to the concentration
x of urea,
was fitted to the unfolding data with
yn,
mn,
yu,
mu,
m and
G(H
2O) as floating parameters (see below).
Similarly, the combination of Equations 26–29 gives
 | (51) |
 | (52) |
where
u is
larger than
n. One generally observes experimentally that
n and
u do not vary with
x.
Equation 51, where
f'
u is developed
as in
Equation 52 and which relates
max to
x, was fitted to
the unfolding data with
n,
u,
m' and
G'(H
2O) as floating parameters
(see below).
Calculations
The curve fits were performed with the Kaleidagraph program (Synergy Software), which uses a Levenberg–Marquardt algorithm. We used the general curve fit routine and the corresponding Pearson's coefficient of correlation, RP. The three-dimensional structures of the variable fragment FvD1.3 (PDB 1vfa; Bhat et al., 1994) and of domain E3.2 from serotype 2 of the dengue virus (PDB 1oan; Modis et al., 2003) were analyzed with the WHAT IF program as described (Vriend, 1990; Renard et al., 2002).
|
Results
|
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Fluorescence of tryptophan in solution
The concentration of the unfolded state U of a protein is generally negligible and undetectable in the absence of a denaturing agent. Therefore, the properties of state U are extrapolated from measurements performed at high concentration of denaturant. The residues of tryptophan are exposed to the solvent in the U state of proteins. Therefore, we assumed that their properties of fluorescence in state U could be mimicked by those of the amino acid tryptophan in solution. We therefore determined the fluorescence properties of tryptophan and their variations with the concentration x of the denaturant, either urea or guanidine hydrochloride (GdmCl).
Solutions of the amino acid tryptophan were prepared in x M urea, with x varying between 0 and 8 M. Tryptophan was excited at 290 nm and its fluorescence emission spectrum was recorded at 20°C for each value of x. The maximal fluorescence emission intensity, maxY(x, ) = Y[x, max(x)] and the wavelength max(x) of this maximum were determined by fitting the cubic function of Equation 20 to the spectrum on the interval of wavelengths 310–374 nm. The Pearson's coefficient for the fitting was RP > 0.9985 for every x (Figure 2a). We found that max(x) did not vary significantly with x and its value was equal to 354.17 ± 0.02 nm (mean ± SE) in these experiments with urea. In contrast, Y[x, max(x)] increased with the concentration of urea (see Figure 4) according to the linear law
 | (53) |
with
hW,urea = 0.050 ± 0.001 M
–1 (mean ± SE in the curve fit). Such a linear variation
of
Y[
x,
max(
x)] has already been reported for tryptophan and
N-acetyl-
L-tryptophanamide, with similar values of
hW,urea (Schmid,
1989
; Eftink, 1994
). This variation justifies the assumption
of linear baselines for the transitions of unfolding, induced
with urea and monitored by fluorescence (
Equations 13 and
14).
The value of the relative slope
hW,urea for the amino acid tryptophan
was consistent with those for the baselines of the two proteins
that we studied here (see the values of
hn and
hu in
Table I).
The quadratic function of
Equation 22 was then fitted to each
spectrum over the narrower interval 328–356 nm, with the
value of
max(
x) fixed at 354.17 nm. We chose this interval because
it contains the values of
max for most proteins, whatever their
folding state. The approximation of the tryptophan spectrum
by
Equation 22 on this interval was excellent for every
x (
RP > 0.9985;
Figure 3a). We found that the curvature
bW(
x) of
the spectrum at
max(
x) varied significantly with the concentration
x of urea (
Figure 4), according to the linear law
 | (54) |
with
kW,urea = 0.0485 ± 0.0011 M
–1.
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Fig. 2.. Determination of the max value by the fitting of Equation 20 to the emission spectra. Circles, 0 M urea; diamonds, 8 M urea; (a) amino acid tryptophan; (b) E3.1; (c) scFvD1.3. Excitation was at 290 nm for tryptophan and 278 nm for the proteins.
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|
Similarly, solutions of the amino acid tryptophan were prepared
in
x M GdmCl, with
x varying between 0 and 6 M. We found that
the maximal intensity
Y[
x,
max(
x)], wavelength
max(
x) and curvature
bW(
x) did not vary significantly with the concentration
x of
GdmCl (
hW,GdmCl = 0.0014 ± 0.0009 M
–1,
max = 355.28
± 0.03 nm and
kW,GdmCl = –0.001 ± 0.002
M
–1, respectively). The negligible variation of
Y[
x,
max(
x)]
with GdmCl has already been reported for both tryptophan and
N-acetyl-
L-tryptophanamide (Schmid, 1989
; Eftink, 1994
).
N-Acetyl-
L-tryptophanamide
is sometimes used to avoid the charged NH
2 and COOH groups that
are present in the amino acid tryptophan but not in the corresponding
protein residues.
Unfolding profiles of two model proteins
Domain E3.1 from serotype 1 of the dengue virus and the antibody fragment scFvD1.3 were produced in the periplasmic space of E.coli and purified by affinity chromatography on a nickel ion column, through a C-terminal hexahistidine tag (see Materials and methods). The purified preparations of proteins were homogeneous at >95%, as checked by SDS–PAGE. The proteins were incubated in increasing concentrations x of urea, used as a denaturant, and their fluorescence properties were characterized. Each experiment was repeated 4–6 times from independent preparations of protein. The reversibility of the unfolding reactions was verified.
We followed the unfolding with both Y(x), the intensity of fluorescence emission by the reaction mixture at a fixed wavelength and max(x), the wavelength of the maximal intensity (Figure 5). The Y(x) signal was measured at the emission wavelength for which the difference between states N (0 M urea) and U (8 M urea) was maximal (Equation 15). The values of max(x) were determined by fitting Equation 20 to the emission spectra over the interval [n – 20 nm, u + 20 nm], where n and u were the values of max(x) for x = 0 and 8 M urea, respectively (RP > 0.99; Figure 2). The number of terms in Equation 20 and the interval of wavelengths that are used in the fitting should be adjusted for each particular protein. Here, we found that the use of wider or narrower intervals increased the error on max(x) in the fitting. The fitting of a second-power polynomial over the same interval of wavelength decreased the RP coefficient whereas that of a fourth-power polynomial left RP unchanged but increased the errors on the fitting parameters. The characteristic parameters of states N and U are summarized in Table I. We found that max(x) remained constant outside the transition region for the two proteins under study, i.e. its value remained equal to n in the pre-transition region and to u in the post-transition region (Figure 5).
The ratio
Yn(0)/
Yu(8) of the
Y(
x) signal for states N and U
of domain E3.1 was important (2.5-fold) whereas the difference
u –
n of the
max signals for the two states was moderate
(10 nm). Both signals allowed us to monitor the unfolding of
E3.1 with sensitivity. The wavelength
n of state N had a high
value, 340 nm, and the molar intensity
Yn of this state increased
strongly with the concentration of urea (
Figure 5). The high
value of
n was consistent with the known structural data. Indeed,
domain E3.1 comprises only one Trp residue, which is conserved
between the four serotypes of the dengue virus and partially
exposed (20.7%) to the solvent in the crystal structure of glycoprotein
E from serotype 2 (Modis
et al., 2003
).
Both values of Yn(0)/Yu(8) and u – n for the scFvD1.3 fragment were moderate, 1.5-fold and 12 nm respectively. The wavelength n of state N had also a high value, 339 nm. The scFvD1.3 fragment comprises six Trp residues. H-Trp52 in the variable domain VH of the heavy chain and L-Trp92 in the variable domain VL of the light chain are located in hypervariable loops and partially exposed to the solvent in the crystal structure of the free FvD1.3 fragment (38.8 and 39.8% exposure, respectively) (Bhat et al., 1994). The four other Trp residues are conserved in all the molecules of immunoglobulins and buried in the structure (0.0, 0.2, 2.2 and 10.8% exposure). The high value of n was thus consistent with the partial exposures of H-Trp52 and L-Trp92.
The rigorous Equation 49 and the empirical Equation 51 were fitted to the experimental values of Y(x) and max(x), respectively, to obtain parameters G(H2O), m and x1/2 from Y(x) and G'(H2O), m' and x'1/2 from max(x) (Figure 5, Table II; see Materials and methods). The coefficients of cooperativity m and m', determined from Y and max, respectively, were identical if the SE values were taken into account. The stabilities G(H2O) and G'(H2O) were significantly different for domain E3.1 but not for the scFvD1.3 fragment if the SE values were taken into account. The values of x1/2 and x'1/2 were significantly different for E3.1 (Table II).
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Table II.. Comparison of the thermodynamic parameters obtained with the Y and max signals for the two proteins under study
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Determination of the spectral curvatures
The high standard errors on the values of G(H2O) and x1/2, deduced from the Y signal, and the differences between these values and those of G'(H2O) and x'1/2, deduced empirically from the max signal, stressed the importance of having a rigorous method for the calculation of the thermodynamic stability from max. We showed in the Theory section that such a method exists when the emission spectra of the protein under study can be approximated by a quadratic function on the interval of wavelength [n, u]. It requires the determination of parameters bn(0) and bu(0), which are the curvatures of the emission spectra for state N at wavelength n and state U at u, respectively, in a medium without any denaturant.
We fitted the quadratic function of Equation 21 (or equivalently Equation 22) to the spectra of domain E3.1 and fragment scFvD1.3 in x M urea on the interval [n – 2 nm; u + 2 nm] and found that the fittings were excellent for every value of x (RP > 0.95; Figure 3b and c; see Materials and methods). From these fittings, we determined the values of bn(x) for x in the region of pre-transition and the values of bu(x) for x in the region of post-transition. The values of the ratio bn(0)/bu(8) were very different for the two proteins under study (Table I) and the difference in curvature between the spectra of states N and U for domain E3.1 are clearly visible in Figure 2. We observed that bn(x) varied linearly with the concentration of urea in the pre-transition region but with a proportionality factor kn whose value was low in each individual experiment and not significantly different from zero on average (Equation 43; Table I). We observed that bu(x) also varied with x. The relation of dependence was imprecise because of the small number of experimental data points in the post-transition region, but consistent with that of the amino acid tryptophan. To obtain greater precision, we assumed that the curvature bu(x) followed the same variation as that of tryptophan, i.e. that bu(x) followed Equation 42 with a factor of proportionality ku = kW,urea (Equation 54).
Quantitative parameters of stability obtained from max
From Equation 46 and the factors of proportionality given above, kn = 0 and ku = kW,urea, we evaluated the corrective term for m'. This term was equal (in kcal/mol·M) to –0.026 for E3.1 and –0.020 for scFvD1.3. It was substantially below the SE value on m' in every case (Table II). From Equation 44, the values of bn(0)/bu(8) and ku = kW,urea, we calculated the corrected value G''(H2O) of G'(H2O). Finally, we calculated the corrected value x''1/2 of x'1/2 as G''(H2O)/m' since we found that m m'. Table II gives the values of G(H2O), m and x1/2, calculated from the Y signal, those of G'(H2O), m' and x'1/2, calculated empirically from the max signal and those of G''(H2O) and x''1/2, obtained after correcting the values of G'(H2O) and x'1/2. The corrections brought the empirical values obtained from max closer to those obtained from Y in every instance.
If the standard errors were taken into account, the rigorous value m and the empirical value m' were equal in our two examples. Similarly, the rigorous value x1/2 and its corrected value x''1/2 were equal in our two examples. The values G(H2O) and G''(H2O) were equal for scFvD1.3; they were very close for E3.1, with intervals of error within 0.2 kcal/mol. The remaining differences might be due to the theoretical and experimental approximations that we performed. Alternatively, the experiments that used the intensity Y might be theoretically more rigorous but experimentally less precise.
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Discussion
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The theory and results presented here allowed us to obtain rigorous
values of the stability from unfolding profiles, monitored with
the wavelength
max, for a domain of a viral protein and the
scFv fragment of an antibody. We discuss the validity of the
theory and its application, in general and for the two studied
proteins.
Precise determination of max
The use of the wavelength max as a signal to monitor the unfolding of proteins requires a precise method to determine its value. This determination is not trivial because the fluorescence emission spectrum Y() of proteins is complex in nature. Many methods have been proposed. The fitting of a polynomial, written as a Taylor expansion about max (Equation 20), has the following advantages. Such a function is continuous and differentiable and its fitting avoids the smoothing of the experimental data. It enables one to obtain directly the value of max as a fitting parameter and the SE value on max in the fitting, whatever the number of terms in the polynomial. We found an excellent fitting of a cubic function to our experimental data over a wide interval of wavelengths, 320–370 nm, with residuals lower than 1% of Y(max) on average. The SE value on max in the fitting was typically 3–6% of the max value. However, the number of terms in the Taylor expansion and the interval of wavelengths that is used for the fitting should be optimized for each particular protein.
Composition of the max signals
We showed that the global max signal for a mixture of unfolding is a linear function of the specific max signals, n and u, for the constitutive states N and U, if their individual spectra can be represented by quadratic functions (Equation 25). The max signal of the mixture is not a linear function of the molar fractions fn and fu of states N and U as for the Y signal. However, we showed that it is possible to define apparent molar fractions f'n and f'u such that the max signal of the reaction mixture is a linear function of both f'n and f'u and the wavelengths n and u (Equation 30). We also showed that max for the unfolding mixture is between n and u if the spectra of N and U show regular behaviors (Equation 18). Therefore, the above theoretical treatment still applies if the spectra of N and U can be approximated by quadratic functions only over the interval [n,u] and not over the whole scale of wavelengths (Figure 3). First, we determined precise values of n and u with cubic functions (see previous paragraph). Then, we fitted the quadratic function that is constituted by the first three terms of the Taylor expansion of Y to the spectrum of N (or U) over the [n – 2 nm, u + 2 nm] interval and found that the fittings were excellent in our two experimental examples (RP 
0.95). Three parameters characterize the portion of spectrum that is approached by a parabola: the value of max, the intensity of fluorescence at max and the curvature of the spectrum at max (Equations 21, 22 and 38).
Is it always possible to approximate the spectra of states N and U by quadratic functions over the interval of wavelengths [n, u]? The protein spectra that we report here and those that are available in the literature indicate that such an approximation is possible in many cases: for proteins that contain only one Trp residue as domain E3.1 or several Trp residues as fragment scFvD1.3; and for proteins that have various folds, including all ß-proteins (E3.1 and scFvD1.3, this work; the E.coli CspA protein, Vu et al., 2001) and 
/ß-proteins (barnase, Sancho and Fersht, 1992; the E.coli CheY protein, Filimonov et al., 1993; Protein L, Scalley et al., 1997). We found that n and u did not vary as a function of the denaturant concentration for the two proteins under study and mentioned that this behavior is quite general. However, such variations of n and u have been reported in a few cases (Ewert et al., 2002). The above theoretical treatment remains valid in such cases if quadratic functions can be fitted to the spectra over the interval [min(max), max(max)] described by the max signal of the reaction mixture during the unfolding.
Implications for the empirical use of max
Our theoretical analysis showed that the use of an empirical law of additivity for the max signal leads to inexact values of the stability parameters G'(H2O), m' and x'1/2 for a monomeric protein that unfolds according to a two-state equilibrium. We gave the corrective terms that allow one to obtain the rigorous values G(H2O), m and x1/2 from the empirical values (Equations 33, 34 and 36). For G(H2O) and x1/2, the corrective terms involved the curvatures bn(0) and bu(0) of the spectra for states N and U at their respective max and a concentration of denaturant x = 0 (Equations 33 and 36). For m, the corrective term involved the laws of variation for the curvatures bn(x) and bu(x) as a function of x (Equation 34). We found that the curvature bn(x) varied linearly with x in the pre-transition region for the two proteins studied, with very small coefficients kn of linear variation (Equation 43; Table I) and that bu(x) varied linearly with x in the post-transition region, with coefficients ku (Equation 42). We also found that the curvature bW(x), at max(x), of the spectrum for the amino acid tryptophan varied linearly with x over the whole range of denaturant concentration, with a well-defined coefficient kW of linear variation (Figure 4; Equation 54), and we proposed to use this linear law of variation for the U state of any protein.
The following relations then result from Equation 46 and the value ku = kW. If the denaturant is urea, kW,urea = 0.0485 ± 0.0011 M–1 at 20°C and
 | (55) |
If the denaturant is GdmCl,
kW,GdmCl = –0.001 ±
0.002 M
–1 at 20°C and
 | (56) |
As 0.15
RT 
0.1 kcal/mol.M for
T 338 K (65°C),
Equations 55 and
56 show that the difference between the empirical value
m' and rigorous value
m of the cooperativity parameter is lower
than the experimental error on
m' if the variation of the curvature
bn(
x) in the pre-transition region remains within wide limits.
We found that such was the case for the two proteins st
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Abstract
Introduction
