2.1. Theory
In this section a new method of the analysis of thermograms for multidomain proteins undergoing irreversible denaturation is proposed. Thermograms for such proteins represent the sum of separate peaks corresponding to individual calorimetric domains (since calorimetric domain is a part of protein molecule that denatures cooperatively and independently from the other parts). Up to date, there has not been proposed any general analytical solution for the problem of the decomposition of thermograms into individual domains with the simultaneous determination of the kinetic parameters for these domains.
For the sake of simplicity it is assumed that denaturation of each individual calorimetric domain is described by the one-stage model of irreversible denaturation:
where
N and
D are native and denatured states of the protein and
k is the rate constant of the first order (or pseudo-first order) reaction. Further, it is assumed that the dependence of the rate constant
k on temperature follows the Arrhenius law:
where
Ea is the experimental energy of activation,
R is the universal gas constant,
T is the absolute temperature, and
T* is the temperature at which
k = 1 min
−1 (the dimension of
k in this formula is min
−1). The kinetics of denaturation is described by the equation:
where
x is the portion of the native protein (at
t = 0 the
x value is equal to unity). If the temperature is varied with a constant rate (
dT/dt ≡ ν),
Equation 3 acquires the following form:
Integration of this equation gives the expression for the portion of the native form as a function of temperature:
If one assume that enthalpy of denaturation is independent of temperature, the following expression may be obtained for the excess heat capacity
:
where
Qex is the excess heat absorbed at denaturation. This dependence may be presented in an expended form:
Kurganov
et al. [
18] elaborated the following method of the determination of parameters
Ea and
T* from the experimental thermogram. From
Equations 4 and
6 we can obtain the expression for the rate constant
k as a function of temperature:
Substituting of
k into
Equation 8, carrying out a logarithmic operation and multiplying of both parts of the equation by
T gives:
As can be seen, on the one hand, the function
W(
T) is completely determined by the experimental curve of the excess heat capacity. On the other hand, in the case of the fulfillment of the one-stage scheme of irreversible denaturation this function is linear with respect to temperature. Thus, the parameters of the Arrhenius equation can be easily estimated through the coefficients of the linear regression.
The thermogram
for the multidomain protein consisting of
N calorimetric domains is an algebraic sum of the signals of the individual calorimetric domains
:
The experimental calorimetric enthalpy of denaturation is expressed as follows:
where Δ
Hi is the enthalpy of denaturation of
i-th domain. Preliminary incubation of such a protein at a certain constant temperature over some period of time just before the calorimetric experiment (denoted below in the text as
annealing) results generally in partial denaturation of each domain. The shape of calorimetric peak for each domain remains unchanged, however the decrease in the registered enthalpy of the transition occurs: Δ
Hi,
ann =
xi · Δ
Hi, where
xi is a portion of
i-th domain remaining in the native form after the annealing (0
< xi < 1). It follows herefrom that
. Thus, thermogram of the whole protein is expressed as follows:
Solution of
Equation 3 at a constant temperature within the incubation time interval
tann gives us the
xi value:
xi = exp[–
ki(
Tann) ·
tann], where
ki(
Tann) is the rate constant for denaturation of
i-th domain at the temperature
Tann calculated from the Arrhenius
equation (2). Thus,
xi is the function depending on
tann and
Tann and parametrically depending on
Ea,i and
(
i.e., parameters of the Arrhenius equation for
i-th domain).
If the calorimetric experiment with preliminary incubation (annealing) of the protein preparation has been carried out (
N −1) times at various temperatures over various incubation time intervals, we obtain the following system of
N equations which are linear with respect to
:
where
j corresponds to the number of the experiment. The case when
j =
N corresponds to the experiment without preliminary incubation (
i.e., for all values of
i the identity
xi, N ≡ 1 is valid).
If domains are numbered in order of increasing thermostability, the following condition holds at
j ≠
N: x1, j < x2, j < . . . <
xN, j . From the theoretical point of view, if only one of parameters
Ea and
T* for (
i – 1)-th and
i-th calorimetric domains has different values (otherwise these domains should be considered as a single domain), it is possible to select the temperature and time of annealing, at which
xi–1, j ≈ 0 (in practice, the inequality
xi–1, j < 0.01 should be met) and
xi, j ≫ 0. From the predicate of the numeration of domains follows that under these conditions of incubation 1
st, 2
nd, . . . , (
i – 2)-th domains are also annealed,
i.e.,
x1, j,
x2, j, . . . ,
xi–2, j ≈ 0. Thus, it is possible to select the set of annealing conditions that makes the system of
equations (13) transformed into the triangle form:
The left parts of the system contain the functions of excess heat capacity obtained directly from the experiment, whereas the right parts of the system are the linear combinations of the functions to be found. Since the determinant of the system differs from zero, the system has unambiguous solution. To find this solution, we have to select properly the corresponding annealing parameters, namely the temperature and duration of each incubation. A rigorous solution of this problem is lacking. However, the following approach may be used to obtain a rough solution.
Evidently, in the part of thermogram corresponding to high temperatures it is possible to select a region where all domains except for the most thermostable one are already completely denatured. In this part of the thermogram
,
i.e., the thermogram of the multidomain protein coincides with the thermogram of the most thermostable domain. In this region of temperatures the function
W(
T) (see
Equation 9) is linear. This enables parameters
Ea and
T* for the most thermostable domain to be estimated. With a knowledge of parameters
Ea and
T*, we can calculate
xN(
T) and
kN(
T). The optimization of the whole thermogram by thermogram corresponding to the most thermostable domain in the coordinates
vs (
kN (
T) ·
xN(
T)
/ν) (see Equation
6) in this region of temperatures using the least squares method allows parameter Δ
HN to be determined.
Based on these considerations one can propose the following recursive method for the estimation of parameters of calorimetric domains for multidomain protein. After obtaining of
from the DSC data we calculate the function W(T) and conclude that the thermogram is not described by the one-stage model of irreversible denaturation. Using W(T) we estimate parameters of the most thermostable domain: Ea, N,
and ΔHN . With a knowledge of these parameters, we simulate
and subtract it from the whole thermogram of protein. Then this procedure is repeated for the function
, which is the overall thermogram of all domains except for the most thermostable domain. As a result we obtain parameters for (N – 1)-th domain, simulate function
and subtract it from the function (
). By repeating this procedure we estimate parameters for (N – 2)-th domain, (N – 3)-th domain and so on down to first calorimetric domain.
However, when determining parameters of N-th domain, we should take into account that the error will be rather high because the region of temperatures where the function W(T) is linear is small in comparison with the temperature interval for the whole thermogram. Besides, to determine the corresponding parameters, the high-temperature branch of the DSC profile is used where
is close to zero and the signal-to-noise ratio is rather small. The function W(T) in this region is even noisier since it contains the term ln
. All the above is valid also for parameters of (N–1)-th domain. Moreover, the additional error, connected with the inexact subtraction of simulated function
, arises. This results in accumulation of the error. To weaken such an effect, the parameters, which are calculated at each step for the corresponding domain by the above-mentioned method, should be optimized in the temperature interval where they were determined. Here optimization is regarded as a minimization in the parameter space (in the least square sense) of the residuals between simulated (using the estimated parameters of the domain) and experimental values of heat capacity. Though even in that case the calculated parameters should be considered as a null approximation only, they allow selecting the time and temperature for the annealing procedures. It is significant that the above-mentioned approach enables the number of thermal transitions in a multidomain protein to be determined with a high accuracy.
When the corresponding annealing procedures have been carried out and all thermograms have been obtained, the multidimensional non-linear optimization in the parameter space may be used for final calculation of parameters Ea, T* and ΔH for all calorimetric domains.
2.2. Resolving Power of the Method. Selection of the Annealing Conditions
The method is based on annealing of 1st, 2nd, . . . , (
i–1)-th domains, where
i is a number of domain in the order of increasing thermostability. This provides a means for the determination of parameters of
i-th domain. From the theoretical point of view, if the values of only one of parameters
Ea or
T* for two domains are non-identical, the conditions of annealing may be selected at which
xi–1 ≈ 0 and
xi > 0. However, the acceptable values of
xi−1 and
xi are the following:
xi−1 < 0.01 and
xi > 0.1. Otherwise the determination of parameters of
i-th domain becomes impossible. The acceptable time of annealing is at least 10 min. If time of annealing is less than 10 min, delayed heating of the sample to the temperature of incubation and following cooling inevitably lead to an unacceptably large error. In other words, the following conditions should be satisfied:
If the values of the activation energy for two domains are about 300 kJ/mole and the values of parameter
T* are about 320 K, after the substituting of the Arrhenius equation in the expressions for the corresponding rate constants and solving the system of inequalities
(15) we obtain the following limitations. For the case of identical
T* values the minimum reasonable difference between parameters
Ea is ∼150 kJ/mole. For the case of identical values of the activation energy the difference
should not be less than 2 K.
Taking into account the above-mentioned requirements to the annealing conditions, we can propose the following method for the selection of
Tann and
tann. For definiteness we fix the ratio
xi−1/
xi ≡
A (with the requirements for
xi−1 and
xi we obtain that
A < 0.1). Finally we come to the system of inequalities:
The solution relative to
tann is a curve
tann =
f(
Tann) established by the conditions:
The coordinates of each point on this curve give the appropriate conditions for annealing.