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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 a single pathway: The potential for the driving force.(a) the Φ-component of the potential, (b) the Ψ-component, and (c) the isolines Φ(r) = const (blue and red lines are, respectively, for the negative and positive values, and the blue-red dashed line is for the zero value), and Ψ(r) = const (black lines). In panel c the Φ(r) = const and Ψ(r) = const isolines are shown with interval of  (see the text). Characteristic values of Ψ(r) in the reaction channel: in region 1 Ψ(r) ≈ −2 × 10−5, and in region 2 Ψ(r) ≈ 5 × 10−5.
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pone.0121640.g002: Two-well landscape with a single pathway: The potential for the driving force.(a) the Φ-component of the potential, (b) the Ψ-component, and (c) the isolines Φ(r) = const (blue and red lines are, respectively, for the negative and positive values, and the blue-red dashed line is for the zero value), and Ψ(r) = const (black lines). In panel c the Φ(r) = const and Ψ(r) = const isolines are shown with interval of (see the text). Characteristic values of Ψ(r) in the reaction channel: in region 1 Ψ(r) ≈ −2 × 10−5, and in region 2 Ψ(r) ≈ 5 × 10−5.

Mentions: The calculated functions Φ(r) and Ψ(r) are shown in Fig. 2a and 2b, respectively. In agreement with the Helmholtz decomposition [20], Φ(r) accounts for the source and sink of the folding flow, and Ψ(r) for the vorticity effects. The surface for the Φ-component is simple; it is characterized by two peaks of different sign—one (positive) peak corresponds to the source of the folding flow, and the other (negative) peak represents the sink of the flow. Poisson’s Equation (9), in which q(r) > 0 in the source region, q(r) < 0 in the sink region, and q(r) = 0 in the rest of the r space, can be viewed as an equation that describes the stationary diffusion of some substance of density Φ(r) between the source and sink. It is evident that the substance ejected from the source will spread over the surface in all directions. Therefore, to canalize the flow between the source to sink, an addition force is required. This force is produced by the Ψ-component of the potential. Two properties of the Ψ(r) surface are noteworthy. The first one is that in the region between the source and sink, where the Φ(r) surface is relatively flat, two parallel ridges of different sign are formed that connect the source and sink. More clearly, it is seen from Fig. 2c, where characteristic values of the Ψ-component in the regions corresponding to the ridges are indicated: the Ψ-component decreases from the periphery of the flow field, where it is zero (Fig. 2b), to approximately −2 × 10−5 in region 1 (the first ridge) and then increases to 5 × 10−5 in region 2 (the second ridge), with a subsequent decrease to zero at the opposite side of the flow field. The second property of the Ψ(r) surface is the presence of (two) peaks of different sign in the region for the unfolded states, which are due to the local vortices formed in this region (Fig. 1f). Such vortices were observed for the α-helical hairpin [12], SH3 domain [15] and three-stranded β-sheet protein [14], as a result of partial folding and unfolding of the protein in the attempts to overcome the barrier to the native state. In should be noted that in contrast to the presence of parallel ridges of different sign in Ψ(r), which is typical of the folding process [17], the appearance of the peaks in Ψ(r) depends on the value of the free energy barrier between the unfolded and the native states. If the barrier is large enough, the system dwells in the basin for the unfolded states performing a circulating motion, as in the present case; then the peaks are formed. However, if the barrier is low, so that the system easily overcomes it, the peaks are absent [17].


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 a single pathway: The potential for the driving force.(a) the Φ-component of the potential, (b) the Ψ-component, and (c) the isolines Φ(r) = const (blue and red lines are, respectively, for the negative and positive values, and the blue-red dashed line is for the zero value), and Ψ(r) = const (black lines). In panel c the Φ(r) = const and Ψ(r) = const isolines are shown with interval of  (see the text). Characteristic values of Ψ(r) in the reaction channel: in region 1 Ψ(r) ≈ −2 × 10−5, and in region 2 Ψ(r) ≈ 5 × 10−5.
© Copyright Policy
Related In: Results  -  Collection

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

pone.0121640.g002: Two-well landscape with a single pathway: The potential for the driving force.(a) the Φ-component of the potential, (b) the Ψ-component, and (c) the isolines Φ(r) = const (blue and red lines are, respectively, for the negative and positive values, and the blue-red dashed line is for the zero value), and Ψ(r) = const (black lines). In panel c the Φ(r) = const and Ψ(r) = const isolines are shown with interval of (see the text). Characteristic values of Ψ(r) in the reaction channel: in region 1 Ψ(r) ≈ −2 × 10−5, and in region 2 Ψ(r) ≈ 5 × 10−5.
Mentions: The calculated functions Φ(r) and Ψ(r) are shown in Fig. 2a and 2b, respectively. In agreement with the Helmholtz decomposition [20], Φ(r) accounts for the source and sink of the folding flow, and Ψ(r) for the vorticity effects. The surface for the Φ-component is simple; it is characterized by two peaks of different sign—one (positive) peak corresponds to the source of the folding flow, and the other (negative) peak represents the sink of the flow. Poisson’s Equation (9), in which q(r) > 0 in the source region, q(r) < 0 in the sink region, and q(r) = 0 in the rest of the r space, can be viewed as an equation that describes the stationary diffusion of some substance of density Φ(r) between the source and sink. It is evident that the substance ejected from the source will spread over the surface in all directions. Therefore, to canalize the flow between the source to sink, an addition force is required. This force is produced by the Ψ-component of the potential. Two properties of the Ψ(r) surface are noteworthy. The first one is that in the region between the source and sink, where the Φ(r) surface is relatively flat, two parallel ridges of different sign are formed that connect the source and sink. More clearly, it is seen from Fig. 2c, where characteristic values of the Ψ-component in the regions corresponding to the ridges are indicated: the Ψ-component decreases from the periphery of the flow field, where it is zero (Fig. 2b), to approximately −2 × 10−5 in region 1 (the first ridge) and then increases to 5 × 10−5 in region 2 (the second ridge), with a subsequent decrease to zero at the opposite side of the flow field. The second property of the Ψ(r) surface is the presence of (two) peaks of different sign in the region for the unfolded states, which are due to the local vortices formed in this region (Fig. 1f). Such vortices were observed for the α-helical hairpin [12], SH3 domain [15] and three-stranded β-sheet protein [14], as a result of partial folding and unfolding of the protein in the attempts to overcome the barrier to the native state. In should be noted that in contrast to the presence of parallel ridges of different sign in Ψ(r), which is typical of the folding process [17], the appearance of the peaks in Ψ(r) depends on the value of the free energy barrier between the unfolded and the native states. If the barrier is large enough, the system dwells in the basin for the unfolded states performing a circulating motion, as in the present case; then the peaks are formed. However, if the barrier is low, so that the system easily overcomes it, the peaks are absent [17].

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.