Thermodynamics of protein folding using a modified Wako-Saitô-Muñoz-Eaton model

Abstract

Herein, we propose a modified version of the Wako-Saitô-Muñoz-Eaton (WSME) model. The proposed model introduces an empirical temperature parameter for the hypothetical structural units (i.e., foldons) in proteins to include site-dependent thermodynamic behavior. The thermodynamics for both our proposed model and the original WSME model were investigated. For a system with beta-hairpin topology, a mathematical treatment (contact-pair treatment) to facilitate the calculation of its partition function was developed. The results show that the proposed model provides better insight into the site-dependent thermodynamic behavior of the system, compared with the original WSME model. From this site-dependent point of view, the relationship between probe-dependent experimental results and model’s thermodynamic predictions can be explained. The model allows for suggesting a general principle to identify foldon behavior. We also find that the backbone hydrogen bonds may play a role of structural constraints in modulating the cooperative system. Thus, our study may contribute to the understanding of the fundamental principles for the thermodynamics of protein folding.

Appendices

Appendix A: The phenomenological parameters

From (6), the free energy equation can be recast as

$$ \Delta G^{\,0}\left( T \right)=\Delta H^{\,0}\left( {T_{1/2}} \right)-T\Delta S^{\,0}\left( {T_{1/2}} \right), $$

(27)

where $\Delta H^{\,0}\left( {T_{1/2}}\right)=T_{1/2} \cdot \Delta S^{\,0}\left( {T_{1/2} } \right)$. Both the enthalpy and entropy differences are constant and are characterized by the transition mid-point T _1/2. These two factors regulate the linear temperature dependence of free energy, thereby facilitating the folding-unfolding behavior. In the statistical mechanical models discussion, we replaced $\Delta S^{\,0}\left( {T_{1/2}}\right)$ and $\Delta H^{\,0}\left( {T_{1/2}}\right)$ with Δs and Δh, respectively, for convenience; thus, Δh can also be represented as T _1/2 · Δs. It should be noted that in the WSME model, entropy is the only concern for individual peptide bond units, as illustrated in (1). However, the M-WSME model uses the free energy equation from (6) to account for the folding of foldons; therefore, for local folding, the entropic cost is balanced by the newly added enthalpic factor for each foldon unit in the system. This is referred to as the “local-folding scheme” in terms of foldon units. In short, the enthalpy part and entropy part both have entropic interpretation and the resulting free energy change is generalized for each effective foldon unit in the system.

Appendix B: The contact-pair treatment—supplement

From (11), we briefly enumerate the configurations for the system as follows.

$$ \begin{array}{rll} Z&=&\left( 1 \right)\cdot \left\{ {\left( {\sum\limits_{x_{\alpha -1} } {\sum\limits_{x_{\alpha +1} } {B_{\alpha -1}^{x_{\alpha -1} } B_{\alpha +1}^{x_{\alpha +1} } } } } \right)\cdots \cdots \left( {\sum\limits_{x_1 } {\sum\limits_{x_N } {B_1^{x_1 } B_N^{x_N } } } } \right)} \right\}\\ &&+\left( {B_\alpha } \right)\cdot \left\{ {\left( {\sum\limits_{x_{\alpha -1} } {\sum\limits_{x_{\alpha +1} } {C_1^{x_{\alpha -1} x_{\alpha +1} } B_{\alpha -1}^{x_{\alpha -1} } B_{\alpha +1}^{x_{\alpha +1} } } } } \right)\cdots } \right.\\ &&\;\qquad\qquad\cdots \left. {\left( {\sum\limits_{x_1 } {\sum\limits_{x_N } {C_{\alpha - 1}^{x_{\alpha -1} x_{\alpha +1} x_{\alpha -2} x_{\alpha +2} \cdots x_1 x_N } B_1^{x_1 } B_N^{x_N } } } } \right)} \right\} . \end{array} $$

(28)

The above equation accounts for the α ^th peptide bond configurations; that is, the first term (see the first curly braces) denotes that x _α  = 0 and the remaining configurations (x _α − 1, x _α + 1, ⋯ , x ₁, x _N ) are to be determined, while the second term (see the second curly braces) denotes that x _α  = 1 and the remaining configurations are to be determined. Note that the first bracket in the second curly braces includes the configuration weights for the first contact pair (1 + B _α − 1 + B _α + 1 + C ₁ B _α − 1 B _α + 1), which denote, respectively, the weights for the configurations (0, 0), (1, 0), (0, 1) and (1, 1). These weights are then divided into two groups: the group with the configuration (1, 1) and the other group with the remaining configurations: (0, 0), (1, 0) and (0, 1). Multiplying them out, it follows that

$$ \begin{array}{rll} Z&=&\left( 1 \right)\cdot \left\{ {\left( {\sum\limits_{x_{\alpha -1} } {\sum\limits_{x_{\alpha +1} } {B_{\alpha -1}^{x_{\alpha -1} } B_{\alpha +1}^{x_{\alpha +1} } } } } \right)\cdots \cdots \left( {\sum\limits_{x_1 } {\sum\limits_{x_N } {B_1^{x_1 } B_N^{x_N } } } } \right)} \right\}\\ &&+\left( {B_\alpha } \right)\left( {1+B_{\alpha -1} +B_{\alpha +1} } \right)\cdot \left\{ {\left( {\sum\limits_{x_{\alpha -2} } {\sum\limits_{x_{\alpha +2} } {B_{\alpha -2}^{x_{\alpha -2} } B_{\alpha +2}^{x_{\alpha +2} } } } } \right)\cdots \cdots \left( {\sum\limits_{x_1 } {\sum\limits_{x_N } {B_1^{x_1 } B_N^{x_N } } } } \right)} \right\}\\ &&+\,B_\alpha C_1 B_{\alpha -1} B_{\alpha +1} \left\{ {\left( {\sum\limits_{x_{\alpha -2} } {\sum\limits_{x_{\alpha +2} } {C_2^{x_{\alpha -2} x_{\alpha +2} } B_{\alpha -2}^{x_{\alpha -2} } B_{\alpha +2}^{x_{\alpha +2} } } } } \right)\cdots } \right.\\ &&\qquad\qquad\qquad\qquad\cdots \left. \left( {\sum\limits_{x_1 } {\sum\limits_{x_N } {C_{\alpha - 1}^{x_{\alpha -2} x_{\alpha +2} \cdots x_1 x_N } B_1^{x_1 } B_N^{x_N } } } } \right)\right\}. \end{array} $$

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where Z ₀ denotes the summation of the first and second terms, as shown in Section 2. Similarly, the same expansion performed on the curly braces of the third term and accounting for the configurations of the second contact pair (α − 2,α + 2), Z ₁ is identified and so forth.

Appendix C: The single foldon unit case

The thermodynamic two-state model is one of the simplest thermodynamic models for protein folding thermodynamics [see Fig. 1]. It correlates with statistical thermodynamics via (7) and (23). In other words, the two-state model in protein thermodynamics corresponds effectively to the single foldon unit case (N = 1) in the M-WSME model.

In the case where N = 1, the interaction terms are excluded and only Δs is specified. Three different values¹¹ (|Δs| = 0.0032, 0.01 and 0.04 kcal/mol K) were used in our investigations, which indicate, respectively, small, middle, and large values assumed for the protein system. Note that Δs < 0 is defined for the WSME model and Δs > 0 for the M-WSME model and the definition does not change the thermodynamic interpretation of $\left\langle {x_i } \right\rangle$, given that their absolute values are consistent. More specifically, Δs = − 0.0032, −0.01 and −0.04 (kcal/mol K), when the WSME model was investigated; while Δs = 0.0032, 0.01 and 0.04 (kcal/mol K), when the M-WSME model was studied. The thermodynamic fraction of the native state compared between the WSME and M-WSME models is given in Fig. 7. Our results show that there is no sigmoidal curve for the WSME model in any case that we studied; instead, a constant value for the fraction was observed and determined using Δs. However, the M-WSME model showed a sigmoidal curve and the Δs can be used to manipulate the sharpness (or smoothness) of the transition; the larger the |Δs| we use, the steeper the transition curve we obtain. The above results suggest that the WSME model cannot be used to describe protein folding in the single unit case due to the imbalance in the entropic cost for a single foldon unit. Therefore, the folding behavior is not shown. However, if the free energy balance is considered for each single foldon unit in the system (the local folding scheme), the folding behavior can be observed for a single foldon. Herein, we emphasize that the sigmoidal behavior from our treatment of a single foldon does not describe the exact form for the folded state in either a local or global sense.

Fig. 8

Comparison of the thermodynamic native state fraction for the WSME (left) and M-WSME (right) models (N = 1). In each panel, the blue, red, and green solid lines denote, respectively, $\left| {\Delta s} \right| = 0.0032$, 0.01 and 0.04 (kcal/mol K). (a) The WSME model. (b) The M-WSME model with T _1/2 = 300 K

In other words, the sigmoid is a thermodynamic behavior, and it describes the folded fraction, which fits the behavior observed from experiments, thus it can be used to trace the equilibrium folding-unfolding process. Thus, this description does not contradict the argument that a single foldon never folds without the aid of interactions with other foldon units. It should be noted that, given |Δs| = 0.04 (kcal/mol K), the M-WSME model showed a complete sigmoidal curve within the temperature range for biological relevance. This value could be used as a reference for other investigations in this paper.

Appendix D: The heat capacity equations

The heat capacity is a thermal-related physical quantity that can be obtained directly from the statistical mechanical relation

$$ \begin{array}{rll} C &=&\left( {\frac{dE}{dT}} \right)\\ &=&k_B \left[ {\frac{1}{Z}\cdot \frac{d}{dT}\left( {T^2\frac{dZ}{dT}} \right)-\left( {\frac{T\cdot {dZ} \mathord{\left/ {\vphantom {{dZ} {dT}}} \right. \kern-\nulldelimiterspace} {dT}}{Z}} \right)^2} \right]\end{array} $$

(30)

where E is the internal energy of the system,¹² Z denotes the partition function of the system and k _B is the Boltzmann constant. Note that $E=k_B T^2\left( {d\,\ln \,Z \mathord{\left/ {\vphantom {Z {dT}}} \right. \kern-\nulldelimiterspace} {dT}} \right)$ is used in (30). According to the set of reduced partition functions derived and shown from (14) through (20), the heat capacity equations for systems with a different N can, in principle, be derived.

For N = 1 (the M-WSME model),

$$ \overline C =\frac{\left( {\Delta h} \right)^2}{RT^2}P_1 \left( {1-P_1 } \right), $$

(31)

where R is the gas constant (in kcal/mol K), Δh = T _1/2 · Δs and $\overline C$ denotes the heat capacity per mole. P ₁ denotes the probability of the folded state (x ₁ = 1); thus, the product P ₁ (1 − P ₁) indicates the transition probability from the folded (x ₁ = 1) to the unfolded state (x ₁ = 0).

For N = 3 (the M-WSME model),

$$ \overline C =\overline C_1 +\overline C_2 +\overline C_3 +\overline C_4, $$

(32)

where

$$ \overline C_1 =\frac{\left( {\epsilon_{13} -\Delta H} \right)^2}{RT^2}\left[ {P_{111} \times \left( {1-P_{111} } \right)}\right], $$

(33)

and ΔH = 3Δh. The product $P_{111} \times \left( {1-P_{111} } \right)$ denotes the transition probability from the state (x ₁, x ₂, x ₃ ) = (1, 1, 1) to (x ₁, x ₂, x ₃ ) = (0, 0, 0). Note that the other components, $\overline C_2$, $\overline C_3$ and $\overline C_4$, which denote certain complicated transitions among all the configurations, are not detailed here.

For N = 3 (the WSME model),

$$ \overline C = \frac{(\epsilon_{13})^2}{RT^2}P_{111}(1-P_{111}). $$

(34)

Herein, we do not intend to show the detailed derivation of $\overline C_2$, $\overline C_3$ and $\overline C_4$ shown in (32). Instead, we have focused on our preliminary derivation, which may be used to distinguish the thermodynamic interpretations of the M-WSME model from those of the WSME model. The detailed derivations are now in progress and will be published elsewhere.