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TwoState Model of Protein Unfolding
In the twostate model of protein unfolding, a conformationally flexible protein exists in either a native (N) or denatured (D) ensemble, wherein conformers of each ensemble have similar energies. These two ensemble states are represented by an equilibrium constant:
K = [D]/[N] (S1)
The equilibrium constant is directly related to free energy of stability (G, Eq. S2 and Eq. S3). The equilibrium constant is not easily measured directly in the laboratory. Instead, free energy is approached from a calorimetric direction. Free energy has enthalpic (H(T)) and entropic (S(T)) contributions, both of which are temperature dependent functions:
G = RTln(K) (S2)
G = H(T)  TS(T) (S3)
H(T) = HTm + QUOTE Cp(T) dT (S4)
S(T) = HTm/Tm + QUOTE Cp(T) dln(T) (S5)
Where HTm is a reference enthalpy typically obtained from a calorimeter, Tm is a reference temperature (typically the melting temperature of the protein), and Cp(T) is the difference in sample cell heat capacity between the native and denatured ensemble states. If Cp(T) is assumed to be temperature independent (designated by the notation Cp), Eq. S4 and Eq. S5 can be substituted into Eq. S3 to obtain the GibbsHelmholtz equation:
G = HTm(1 T/Tm) + Cp[T Tm T(ln(T/Tm))] (S6)
Eq. S6 enables calculation of protein stability at any temperature (T).
At any point in the unfolding transition, the fraction of native (fN) and denatured (fD) protein can be determined using mass balance for total protein (PT) and the equilibrium constant:
PT = [D] + [N] (S7)
fN = [N]/PT = [N]/([D] + [N]) = 1/(K + 1) (S8)
fD = [D]/PT = [D]/([D] + [N]) = K/(K + 1) (S9)
As a protein thermally denatures in a differential scanning calorimeter (DSC), the heat capacity (Cp(T)) of the sample cell changes as the fraction of unfolded protein changes:
Cp(T) = fD H(T) d/dT (S10)
Application of the Product Rule expands the above derivative into two terms:
Cp(T) = (fD(d/dT))H(T) + fD(H(T)(d/dT)) (S11)
Differentiating Eq. S4, assuming Cp(T) is a temperatureindependent function, simplifies the expression to Cp. Differentiating fD requires further use of the Quotient Rule and substitution of nested temperaturedependent functions (Eq. S2 and Eq. S6 into Eq. S9):
fD (d/dT) = (e(HTm(1 T/Tm) + Cp[T Tm T(ln(T/Tm))])/
e(HTm(1 T/Tm) + Cp[T Tm T(ln(T/Tm))]) + 1) d/dT (S12)
fD (d/dT) = [((HTm/T CpTm/T + Cp)/RT)((K(K + 1))  (K2/RT))]/
((K+1)2) (S13)
Substituting Eq. S13 back into Eq. S11 generates the twostate model used to fit DSC data:
Cp(T) = H(T)([((HTm/T CpTm/T + Cp)/RT)((K(K + 1))  (K2/RT))]/ ((K+1)2)) + (CpK/(K + 1)) + FBLDSC (S14)
Where FBLDSC refers to the folded baseline heat capacity determined with a linear equation.
To obtain a more stringent and complete fit, denaturation data from FLT was simultaneously fit to the same model. FLT selectively monitors intrinsic fluorescence of tryptophan residues (Trp) in a protein. In the native state, the environment surrounding Trp allows for maximum fluorescence. When the protein denatures, the Trp become solvent exposed and lose much of their initial intensity. Throughout the transition, the total signal measured (S(T)) is a composite of native (SN) and denatured (SD) protein fluorescence:
S(T) = SN + SD (S15)
The contribution from both native and denatured states depends on the fraction of each present at a given temperature. The total integrated fluorescence lifetime signal can be expressed as such:
S(T) = SN/(K + 1) + SD (S16)
SN and SD are determined with linear functions before (the folded baseline, FBLFLT) and after (the unfolded baseline, UBL) the unfolding transition:
SN = FBLFLT = (mN)T + intN (S17)
SD = UBL = (mD)T + intD (S18)
Where m and int are the slope and intercept, respectively, for each baseline. To determine SN throughout the course of the denaturation, the signal contributed by SD must be subtracted off the measured composite signal. The model for FLT signal as a function of temperature becomes:
S(T) = (FBLFLT UBL)/(K + 1) + (UBL) (S19)
By substituting a rearranged Eq. S2 and Eq. S6 into Eq. S19, the fitting of FLT and DSC denaturation data are linked and can be subject to nonlinear least squares regression.
Fluctuation Dissipation Theorem
The fluctuation dissipation theorem describes fluctuations about a mean for a given system. In the case of protein denaturation where heat capacity (Cp) is measured, the enthalpy fluctuates and the theorem is ADDIN EN.CITE Hill19866262626Hill, T. L.An introduction to statistical thermodynamics1986TorontoGeneral Publishing Company, Ltd[1]:
Cp = (H2  2)/RT2 (S20)
Where H is enthalpy in a Gaussian distribution about the mean , R is the gas constant, and T is temperature. Cp is thus proportional to the enthalpy deviation (H2  2) of either folded or unfolded states of a protein. The heat capacities for native and denatured states differ due to the extent of hydrophobic residue exposure in aqueous solution. As such, the native H2  2 is much smaller than the denatured H2  2. Using the folded and unfolded heat capacity baselines obtained from the DSC, the relative widths of the folded and unfolded enthalpy distributions can be approximated.
The distance between the mean enthalpies of each distribution is directly measured by the calorimeter and is the change in enthalpy (H) between folded and unfolded states of the protein. By combining estimates for folded Cp and unfolded Cp with the change in enthalpy, enthalpy distribution plots can be sketched (Fig. S1). The assembly of these plots allows for qualitative monitoring of ligandinduced changes in the proteins native and denatured enthalpy distributions.
Purification of Isolated C2B Domain
The C2B purification was similar to C2A and C2AB with respect to plasmid preparation, E. coli transformation and growth, cell lysis, and nucleic acid degradation. However, C2B was expressed with a 6xHistagged maltosebinding protein (MBP). The amino acids linking C2B to MBP formed a cleavage site for a 6xHistagged tobacco etch virus (TEV) protease. After overnight treatment with benzonuclease at 4 C, C2BMBP fusion protein was equilibrated with slurried NiNTA agarose media (Qiagen) in 20 mM MOPS/100 mM KCl/10 mM imidazole/pH 7.5 buffer for 3 hours at 4 C. The 10 mM imidazole prevented nonspecific protein binding. After equilibration, the slurried media was transferred into a column. Complete elution of undesired protein was monitored by measuring the A280 using a Nanodrop (Thermo Scientific). To remove nucleic acid, a 10 column volume wash was carried out with 20 mM MOPS/1M KCl/pH 7.5 buffer. The column was reequilibrated with the 10 mM imidazole buffer. The nucleic acidfree fusion protein (A260/A280<0.8) was then eluted with 20 mM MOPS/100 mM KCl/250 mM imidazole/pH 7.5 buffer. Eluted MBPC2B was dialyzed with TEV protease in 20 mM MOPS/100 mM KCl/pH of 7.5 buffer (3 mg of TEV protease were added for every 150 mg of fusion protein). After >14 hours, cut MBPC2B and TEV protease were passed over the column. C2B was collected in pass through and its purity was assessed with the Nanodrop (A260/A280<0.8) and gel electrophoresis (Fig. S2). Pure C2B was dialyzed in chelexed Fluka grade 20 mM MOPS/100 mM KCl/pH 7.5 buffer prior to use in denaturation studies. Concentrations were determined using an A280 extinction coefficient of 19060 cm1 M1.
NonLinear Least Squares Regression Analysis
The nonlinear least squares regression analysis is a fitting method that matches the output of Eq. S14 and Eq. S19 to the experimental data by manipulating fit parameters (HTm, Cp, Tm, mN, intN, mD, intD) within the model equations. The close fit is accomplished in Excel through use of the Solver, which minimizes the total sum of the square differences ((experimental signal model sig DP h i
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CJOJQJaJh!CJOJQJaJh4dh!oCJH*OJQJaJh4dh$xCJOJQJaJh4dh0CJOJQJaJh4dh'CJOJQJaJh4dh!oCJOJQJaJ**,,,,,,,,,,... .J.Y.v............//)/3/://///I0J0K00000001111³घhtahX{CJH*OJQJaJhCJOJQJaJhtah^c+CJOJQJaJhtah}[CJOJQJaJhtahHfCJOJQJaJhtahbCJOJQJaJhtahX{CJOJQJaJhtahX{6CJOJQJaJ5111K11111N2R2Z2\2b2d2j2l2v2x2~22224XXXXWXoXpXXdYhYYYYYYYYYᙉt(jhtah!eCJOJQJU]aJhtah'';CJOJQJ]aJ"htahX{CJH*OJQJ]aJU"htahX{CJH*OJQJ]aJhtahKCJOJQJ]aJ"htahX{5CJOJQJ]aJhtahX{CJOJQJ]aJhtahX{CJOJQJaJ(nal)2) to fit the data. Error associated with the resulting fit can then be determined with the macrofunctions SolverAid, which estimates error using 1) the scaling factor by with the fit parameters were changed, 2) the global sum of the square difference (for all 8 data sets) after fitting has occurred, and 3) the models numerical outputs for signal (the theoretical heat capacity or fluorescence signal). For more detail on analysis, see: ADDIN EN.CITE De Levie199954545417De Levie, R.Estimating parameter precision in nonlinear least squares with excel's solverJ. Chem. Educ.J. Chem. Educ.15941598761999Streicher200753535317Streicher, W.W.Makhatadze, G.I.Unfolding thermodynamics of Trpcage, a 20 residue miniprotein, studied by differential scanning calorimetry and circular dichroism spectroscopyBiochemistryBiochemistry28762880462007[2,3].
Supporting References
ADDIN EN.REFLIST 1. Hill TL (1986) An introduction to statistical thermodynamics. Toronto: General Publishing Company, Ltd.
2. De Levie R (1999) Estimating parameter precision in nonlinear least squares with excel's solver. J Chem Educ 76: 15941598.
3. Streicher WW, Makhatadze GI (2007) Unfolding thermodynamics of Trpcage, a 20 residue miniprotein, studied by differential scanning calorimetry and circular dichroism spectroscopy. Biochemistry 46: 28762880.
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