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Protein folding as a complex reaction: a two-component potential for the driving force of folding and its variation with folding scenario.

Chekmarev SF - PLoS ONE (2015)

Bottom Line: Chem.The Ψ-component is more complex and reveals characteristic features of the process of folding.The present approach is potentially applicable to other complex reactions, for which the transition from the reactant to the product can be described in a space of two (collective) variables.

View Article: PubMed Central - PubMed

Affiliation: Institute of Thermophysics, 630090 Novosibirsk, Russia and Department of Physics, Novosibirsk State University, 630090 Novosibirsk, Russia.

ABSTRACT
The Helmholtz decomposition of the vector field of probability fluxes in a two-dimensional space of collective variables makes it possible to introduce a potential for the driving force of protein folding [Chekmarev, J. Chem. Phys. 139 (2013) 145103]. The potential has two components: one component (Φ) is responsible for the source and sink of the folding flow, which represent, respectively, the unfolded and native state of the protein, and the other (Ψ) accounts for the flow vorticity inherently generated at the periphery of the flow field and provides the canalization of the flow between the source and sink. Both components obey Poisson's equations with the corresponding source/sink terms. In the present paper, we consider how the shape of the potential changes depending on the scenario of protein folding. To mimic protein folding dynamics projected onto a two-dimensional space of collective variables, the two-dimensional Müller and Brown potential is employed. Three characteristic scenarios are considered: a single pathway from the unfolded to the native state without intermediates, two parallel pathways without intermediates, and a single pathway with an off-pathway intermediate. To determine the probability fluxes, the hydrodynamic description of the folding reaction is used, in which the first-passage folding is viewed as a steady flow of the representative points of the protein from the unfolded to the native state. We show that despite the possible complexity of the folding process, the Φ-component is simple and universal in shape. The Ψ-component is more complex and reveals characteristic features of the process of folding. The present approach is potentially applicable to other complex reactions, for which the transition from the reactant to the product can be described in a space of two (collective) variables.

No MeSH data available.


Two-well landscape with two pathway: The potential for the driving force.General notations are as in Fig. 2. In panel c the Φ(r) = const and Ψ(r) = const isolines are shown with interval of . Characteristic values of Ψ(r) in the reaction channel: Ψ(r) ≈ −1 × 10−4 (region 1), Ψ(r) ≈ 1.3 × 10−4 (region 2), Ψ(r) ≈ 3.8 × 10−5 (region 3), and Ψ(r) ≈ 8.2 × 10−5 (region 4).
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pone.0121640.g004: Two-well landscape with two pathway: The potential for the driving force.General notations are as in Fig. 2. In panel c the Φ(r) = const and Ψ(r) = const isolines are shown with interval of . Characteristic values of Ψ(r) in the reaction channel: Ψ(r) ≈ −1 × 10−4 (region 1), Ψ(r) ≈ 1.3 × 10−4 (region 2), Ψ(r) ≈ 3.8 × 10−5 (region 3), and Ψ(r) ≈ 8.2 × 10−5 (region 4).

Mentions: One complication of the folding scenario described in the previous section is the presence of multiple folding pathways [39, 40]. We will consider the simplest case when there are two independent pathways, as, e.g., in the B domain of protein A due to the symmetrical backbone structure of the protein (a three-helix bundle) [41, 42], or in the fyn SH3 domain, where the fast folding trajectories are organized in two essentially independent routes due to the reverse order of formation of the contacts between the β1 and β5 strands and the RT-loop and the β4 strand [15]. The values of the parameters determining the potential energy function in the present case are given in Table 2, and the corresponding PES is shown in Fig. 3a. The temperature at which the simulations were performed was T = 1.5. The native state was associated with the point xn = 0.65, yn = 0.2, and the MD trajectories were initiated in the vicinity of the point xu = 0.35, yu = 0.8. The MFPT is . Figs. 3b-3f and 4a-4c show the simulation results, akin to the corresponding panels of Figs. 1 and 2.


Protein folding as a complex reaction: a two-component potential for the driving force of folding and its variation with folding scenario.

Chekmarev SF - PLoS ONE (2015)

Two-well landscape with two pathway: The potential for the driving force.General notations are as in Fig. 2. In panel c the Φ(r) = const and Ψ(r) = const isolines are shown with interval of . Characteristic values of Ψ(r) in the reaction channel: Ψ(r) ≈ −1 × 10−4 (region 1), Ψ(r) ≈ 1.3 × 10−4 (region 2), Ψ(r) ≈ 3.8 × 10−5 (region 3), and Ψ(r) ≈ 8.2 × 10−5 (region 4).
© Copyright Policy
Related In: Results  -  Collection

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Show All Figures
getmorefigures.php?uid=PMC4388825&req=5

pone.0121640.g004: Two-well landscape with two pathway: The potential for the driving force.General notations are as in Fig. 2. In panel c the Φ(r) = const and Ψ(r) = const isolines are shown with interval of . Characteristic values of Ψ(r) in the reaction channel: Ψ(r) ≈ −1 × 10−4 (region 1), Ψ(r) ≈ 1.3 × 10−4 (region 2), Ψ(r) ≈ 3.8 × 10−5 (region 3), and Ψ(r) ≈ 8.2 × 10−5 (region 4).
Mentions: One complication of the folding scenario described in the previous section is the presence of multiple folding pathways [39, 40]. We will consider the simplest case when there are two independent pathways, as, e.g., in the B domain of protein A due to the symmetrical backbone structure of the protein (a three-helix bundle) [41, 42], or in the fyn SH3 domain, where the fast folding trajectories are organized in two essentially independent routes due to the reverse order of formation of the contacts between the β1 and β5 strands and the RT-loop and the β4 strand [15]. The values of the parameters determining the potential energy function in the present case are given in Table 2, and the corresponding PES is shown in Fig. 3a. The temperature at which the simulations were performed was T = 1.5. The native state was associated with the point xn = 0.65, yn = 0.2, and the MD trajectories were initiated in the vicinity of the point xu = 0.35, yu = 0.8. The MFPT is . Figs. 3b-3f and 4a-4c show the simulation results, akin to the corresponding panels of Figs. 1 and 2.

Bottom Line: Chem.The Ψ-component is more complex and reveals characteristic features of the process of folding.The present approach is potentially applicable to other complex reactions, for which the transition from the reactant to the product can be described in a space of two (collective) variables.

View Article: PubMed Central - PubMed

Affiliation: Institute of Thermophysics, 630090 Novosibirsk, Russia and Department of Physics, Novosibirsk State University, 630090 Novosibirsk, Russia.

ABSTRACT
The Helmholtz decomposition of the vector field of probability fluxes in a two-dimensional space of collective variables makes it possible to introduce a potential for the driving force of protein folding [Chekmarev, J. Chem. Phys. 139 (2013) 145103]. The potential has two components: one component (Φ) is responsible for the source and sink of the folding flow, which represent, respectively, the unfolded and native state of the protein, and the other (Ψ) accounts for the flow vorticity inherently generated at the periphery of the flow field and provides the canalization of the flow between the source and sink. Both components obey Poisson's equations with the corresponding source/sink terms. In the present paper, we consider how the shape of the potential changes depending on the scenario of protein folding. To mimic protein folding dynamics projected onto a two-dimensional space of collective variables, the two-dimensional Müller and Brown potential is employed. Three characteristic scenarios are considered: a single pathway from the unfolded to the native state without intermediates, two parallel pathways without intermediates, and a single pathway with an off-pathway intermediate. To determine the probability fluxes, the hydrodynamic description of the folding reaction is used, in which the first-passage folding is viewed as a steady flow of the representative points of the protein from the unfolded to the native state. We show that despite the possible complexity of the folding process, the Φ-component is simple and universal in shape. The Ψ-component is more complex and reveals characteristic features of the process of folding. The present approach is potentially applicable to other complex reactions, for which the transition from the reactant to the product can be described in a space of two (collective) variables.

No MeSH data available.