1. Protein Stability N  U K = = = [U] = [U] [N] fU fN 1-fU 1
Chemical Denaturation :
Chemical denaturation can be followed by circular (CD) or fluorescence. Usually, A two-state model are used to describe the equilibrium between native (N) and unfolding (U) state.
N  U K
K = = =
[U]
[N] fU fN
1-fU
And the spectroscopic data can be fitted by :
[U] = 1
1+exp({m [denaturant]- DG}/ RT)

2. The case of BTK SH3:  : refolding CD : unfolding CD
: unfolding Fluorescence 

3. DG = DGH2O - m [Denaturant]
In the transition region, the Gibbs free energy change (DG) can be calculated from the equilibrium constant K :
DG = -RT lnK
The Gibbs free energy change (DG) for the denaturation process can be represented as a function of denaturant concentration :
DG = DGH2O - m [Denaturant]
Where DG is the Gibbs free energy change at various concentration of denaturant, and DGH2O, the extrapolated value of DG at zero molar denaturant, represents the intrinsic stability and thermodynamic spontaneity of the unfolding of the protein in the absence of denaturant.

4. 3D Temperature-Denaturant Denaturation :
A set of parameters necessary for the complete thermodynamic description of protein folding could be obtained by performing thermal denaturation at a variety of concentrations of GuHCl and using a global fitting method. The method involves fitting the three-dimensional surface for the CD signal versus temperature and GuHCl concentration to a function of 10 parameters (Equation 1).
Signal = {(an+bnT+cn[denaturant]) + (ad+bdT+cd[denaturant])e-DGU/RT}/{1+e-DGU/RT} (1)
where, the dependence of the free energy on denaturant concentration is given by equation (2):
DGU(T,[denaturant]) = DGU(T) + m[denaturant] (2)
And the dependence of free energy on temperature is given by Gibbs-Helmholtz equation:
DGU(T) = DHTm (1T/Tm)  DCp [TmT+T ln(T/Tm)] (3)

5. The case of BTK SH3:

7. NMR Hydrogen-Deuterium Exchange :
NMR can detect the stability of each residue. Normally, we lyophilize the protein and then dissolve the protein in D2O. The slow exchanging amide protons can still be observed by NMR, while the fast exchanging ones can’t.

8. The experimental data can be fitted by a single exponential equation.
Y = A e -kt

9. CH OH OD CD Hydrogen Exchange Mechanism : kokrc kex = kc + ko + krc ko
The commonly accepted model for hydrogen exchange is the two-step model described as follows :
CH OH OD CD ko kc
krc
D2O
Where C and O are the closed and open conformations, and kc and ko are the rate of folding and unfolding. If D2O is in great excess,
kex =
kokrc
kc + ko + krc
Under conditions, kc >> ko , and kc >> krc , then ko
kex = = Kkrc
krc kc
DGHD = -RT ln(K) = -RT ln(kex/krc)

10. The case of BTK SH3:

11. krc = k(acid) + k(base) + k(water)
= kA,ref(ALxAR)[D+] + kB,ref(BLxBR)[OD-] + kW,ref(BLxBR)

12. Backbone Dynamics and Heat Capacities :

13. SB = kB ln[ (3  (1+8S)1/2)] Cp = dSB(T)/dlnT
The conformational entropy (SB) of pico- to nanosecond time-scale bond vector fluctuations can be estimated from the order parameters as :
SB = kB ln[ (3  (1+8S)1/2)]
in which SB is the entropy, kB is Boltzmann’s constant, and S is the generalized order parameter. The equation was derived under the assumption that the motions of individual NH groups are independent of each other. The heat capacity then can be estimated from the relationship:
Cp = dSB(T)/dlnT
where SB(T) is the entropy at temperature.

14. : 283K
: 293K
: 303K
: 313K

15. Folding Kinetics : Stopped-Flow :
The kinetics of folding and unfolding could be studied by stopped-flow CD and fluorescence.
In our case of BTK SH3. For refolding experiments, approximate 100 mM BTK SH3 in 3M GuHCl, 100mM Na2HPO4, pH 6.4, was diluted with same buffer contains no GuHCl to the appropriate final GuHCl concentrations ranging from 0.1~2.5 M. The unfolding was initiated by dilution of protein in 100mM Na2HPO4, pH 6.4, with same buffer contains 7M GuHCl. The final [GuHCl] ranged between 1.2 to 6.2 M.
Folding and unfolding rates were determined from single-exponential fits to the kinetic traces.

16. The case of BTK SH3:

17. The case of BTK SH3:
The folding and unfolding rates and the kinetic m values for folding and unfolding process were obtained by fitting kinetic data to:
lnkobs = ln [kf exp(-mf[denaturant]) + ku exp(mu[denaturant])]

18. Quenched-Flow Pulse Labeling :
BTK SH3 was initially lyophilized and then unfolded in 3 M GuDCl/D2O solution at pH 6.4 to replace all exchangeable hydrogen atoms with deuterium. Refolding was initiated by rapid dilution of the denatured protein with 50 mM sodium phosphate in water, pH 4.5 (ten volumes). After the variable refolding period, the solution containing partially refolded protein was diluted with 200 mM sodium phosphate at pH 8.2 (two volumes). After a 10-ms labeling time, most amides of the unstructured region become fully protonated, while the ones of structured region is well protected. The exchange was quenched by dilution with 400 mM sodium phosphate at pH 4.4 (two volumes) and the refolding can proceed to completion.
Syringe A : Totally denatured protein in D-solvent
Syringe B: Refolding buffer , low pH
Syringe C: Quenching buffer , high pH

19. The TOCSY spectra of BTK SH3 with different refolding times
The TOCSY spectra of BTK SH3 with different refolding times. The changes in proton occupancy of different refolding time were fitted to a single exponential function (Y=A e-t/t + C, where t is the time constant for exchange).

21. (A)
(B) b4 b1 b5 b3 C b2 N
The relationship between the DGHD values (Figure A. kcal/mol) and the time constants ,t, (Figure B, ms) in structure.