Exploration of Protein Unfolding by Modelling Calorimetry Data from Reheating

Studies of protein unfolding mechanisms are critical for understanding protein functions inside cells, de novo protein design as well as defining the role of protein misfolding in neurodegenerative disorders. Calorimetry has proven indispensable in this regard for recording full energetic profiles of protein unfolding and permitting data fitting based on unfolding pathway models. While both kinetic and thermodynamic protein stability are analysed by varying scan rates and reheating, the latter is rarely used in curve-fitting, leading to a significant loss of information from experiments. To extract this information, we propose fitting both first and second scans simultaneously. Four most common single-peak transition models are considered: (i) fully reversible, (ii) fully irreversible, (iii) partially reversible transitions, and (iv) general three-state models. The method is validated using calorimetry data for chicken egg lysozyme, mutated Protein A, three wild-types of haloalkane dehalogenases, and a mutant stabilized by protein engineering. We show that modelling of reheating increases the precision of determination of unfolding mechanisms, free energies, temperatures, and heat capacity differences. Moreover, this modelling indicates whether alternative refolding pathways might occur upon cooling. The Matlab-based data fitting software tool and its user guide are provided as a supplement.


Supplement 1. Derivation of formulas for data fitting for reheating Theoretical Basis
In what follows, Cp(T) is the heat capacity from the first scan as a function of temperature, and C R p(T;T') is for the heat capacity for the reheated run after the first run with the scan rate v reached temperature T', was cooled down at the same rate v and then reheated again. For the sake of simplicity, the formulas in this paper are derived based on the cooling rate equal to that of the first run, but the software tool used for fitting includes the option of choosing a different cooling rate or its approximation (see the User Guide for more information). We will now briefly discuss the derivation of models for reheating in the cases of the four models of unfolding discussed in the main text.

(A) Reversible two-state denaturation:
D N K   In this case, the heat capacity should satisfy the following well-known equation: Here Tm is the melting temperature, that is, the temperature at which half of the protein is denatured: K=1, ∆Cp is the constant change in heat capacity between the folded and denatured states. For totally reversible protein unfolding the reheated run should match the first run, thus the following equation holds for any terminal point of the first run T': It is worth noticing that the modelled reheated run should follow the first run in any equilibrium fully reversible model of unfolding, e.g., a multi-step model derived based on the calculation of vant Hoff's enthalpy, given the cooling scan is performed at the same scan rate as the first run. It follows from the fact that the rate of approaching a new equilibrium is the sum of the rates of folding and unfolding. Thus, if a fully reversible model is valid, and equilibria are assumed to take place during heating, the time needed for the protein to refold is exactly the same as the time of unfolding. Hence, there should be no change to the thermogram during reheating as compared to the first run. D N k   This model is often considered as a simplification of the more general Lurmy-Eyring model (see. Models C,D) whenever the intermediate state I is barely populated due to the faster transition to state D during the scan. If we define the relative concentrations of the states as xn and xd=1-xn respectively, the equation for the heat capacity will be as follows (see the derivation below):

(B) Irreversible two-state denaturation
Here T0 is the initial temperature (low enough to ensure that xn=1, i.e. all the protein is in the native state) and Ei is the exponential integral, all the values of which are usually readily available in the modern programming software (it is usually referred to as expint).
It is worth noting that X(T2,T1,v) represents the decay factor of the native state relative concentration from temperature T1 to T2 given the scan rate v. In other words, it shows the ratio of the protein in the native state at temperature T2 to the one at temperature T1 after changing the temperature at a constant rate of v. If the first run is stopped at the temperature T', the terminal amount of protein in the native state will be xn(T')=X(T',T0,v)xn(T0) or if we assume xn(T0)=1, xn(T')=X (T',T0,v). Hence, after cooling down to the temperature T0 at the rate v, this amount will be reduced to .
The subsequent reheating will result in the fraction of the protein in its native state equal to And the heat capacity for the reheated run can be found as

(C) Partially reversible three-state denaturation with equilibrium
This is a more general model with the irreversible step following a reversible unfolding. It is assumed that the rates of the reaction at the first step allow approximation of the step with an equilibrium constant K. As in (B) we define the relative concentrations of the states as xn, xi and xd=1-xn-xi respectively. Then the equation for the heat capacity will be as follows 1 : Where indices R and I denote the respective values for the reversible and irreversible steps. By definition and the proposed scheme 2 : . Now again, we represent the decay factor for the native state from T1 to T2 given the scan rate v as XI(T2,T1,v) Then, the terminal amount of protein in the native state after the first run up to the temperature T' will be xn(T')=XI (T',T0,v)xn(T0), and after cooling down to the temperature T0 at the rate v and reheating we arrive at Here we again assumed xn(T0)=1. Thus, the heat capacity for the reheated run can be found as Unfortunately, the easiest way to calculate model heat capacities in this case is by direct integration of the equations as no explicit solution is currently available.

(D) General partially reversible three-state denaturation
This is a classical Lumry-Eyring model, in which the first step is not approximated with an equilibrium constant as in (C), rather it is parameterized by two rate constants: k1 for the forward reaction and k-1 for the reverse one. Then the equation for the heat capacity will be as follows: where indices 1 and 2 denote the respective values for the first and second steps. By definition and the proposed scheme 3 : In this case, one decay factor will not be enough as both xi and xd are the solutions to the above system. However, the steps for deriving the heat capacity function for the reheated run are the same. Let xi(T') and xd(T') stand for the fractions of the protein in the intermediate and denatured states after the first heating to the temperature T'. Cooling down can be obtained by integrating the above system from T' to T0 and scan rate -v with the starting values xi(T') and xd (T') to the values xi(T0,T') and xd(T0,T'). Reheating is another integration from T0 to T with the starting values xi(T0,T') and xd(T0,T'). Finally, the heat capacity of the reheated run can be calculated as Formulas for reheating in the case of more complicated models can be derived according to the principles similar to the above four models, which is why they are not elaborated on here. Figure S2. Temperature versus time profile for heating, cooling, and reheating obtained from MicroCal VP-Capillary DSC at various scan rates. The recorded temperature followed linear plot precisely without any artifacts at high temperatures. Although there is a delay at the low temperature, which was purposefully added as recommended by the user guide of the device, it could not have resulted in any additional error in the analyses performed as the starting temperature is far lower than that of the transition.