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The impact of the diffusion parameter on the passage time of the folding process
by Marcelo Tozo Araujo, Jorge Chahine, Elso Drigo Filho and Regina Maria Ricotta
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Submission summary
Authors (as registered SciPost users):  Regina Ricotta 
Submission information  

Preprint Link:  scipost_202212_00043v2 (pdf) 
Date accepted:  20230721 
Date submitted:  20230512 05:18 
Submitted by:  Ricotta, Regina 
Submitted to:  SciPost Physics Proceedings 
Proceedings issue:  34th International Colloquium on Group Theoretical Methods in Physics (GROUP2022) 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
Recently, a mathematical method to solve the Fokker Plank equation (FPE) enabled the analysis of the protein folding kinetics, through the construction of the temporal evolution of the probability density. A symmetric tristable potential function was used to describe the unfolded and folded states of the protein as well as an intermediate state of the protein. In this paper, the main points of the methodology are reviewed, based on the algebraic Supersymmetric Quantum Mechanics (SQM) formalism, and new results on the kinetics of the evolution of the system characterized in terms of the diffusion parameter are presented.
Author comments upon resubmission
List of changes
1. The article presents a summary of a theoretical methodology already published (Reference [4]) and new results (Section 3). It is a mathematical model built to understand the protein folding process by solving the Fokker Plank equation. Through a known mapping between this equation and a Schrödingertype equation, both equations have the same spectrum. Thus, the approximate spectrum is solved through the algebraic method of supersymmetric quantum mechanics (SQM) which allows obtaining the time dependent probability distribution, solution of the the Fokker Planck equation for a given free energy V(x). This allows the subsequent analysis of the (theoretical) kinetics of the protein folding. It is therefore an initial model for the protein folding process. Thus, in future work (in progress) the free energy with specific parameters and units obtained from the computational simulation will be treated and the results compared.
2. The Abstract and Introduction were rewritten in order to make clear the points raised by the reviewers. New references were added, References [2] and [3]. We were careful to ilustrate the model in Section 2 with results different from those published in reference [4] (different free energy V(x)).
3. Questions concerning the choice of parameters in Equation (10) are explained below it, in Section 2.1 with the addition of the following: "The choice of parameters was made in order to have several symmetrical potentials with V(0)=0 and different lateral depths of the wells, as in Reference [10]. Thus, only the variation of the parameter "a" in each tristable potential is enough to deal with the depth of the lateral minima."
4. Question concerning the choice of parameter Q is explained in Section 2.2 with the addition of the following: "To illustrate the model, we choose the potential V (x) with a = 3.90456 and fixed diffusion (arbitrary) constant, Q = 0.5, as in Figure 2. The value of Q is arbitrary but it has to be fixed in order to apply the SQM methodology to obtain the spectrum, as it can be seen from the superpotential W1 in equation (7). In Section 3 we vary the diffusion constant and evaluate its impact on the folding kinetics. It should be mentioned that in other works, [10], the quantities are given in units 1/Q."
5. The new results are presented in Section 3. The first paragraph was rewritten, to avoid confusion between the time t and the passage time (τ). "Using the methodology developed in [4] for the protein folding process, the characteristic passage time τ was evaluated for different values of the diffusion parameter Q for the free energy of Figure 2, V (x) = 3.90456x6 − 8.93851x4 + 5.42373x2. For each fixed value of Q in the interval 0.4 < Q < 20, the passage time τ for the evolution of the population to the right well was calculated, starting from the initial position x0 = xmin = −1.05282, as shown in Figure 6 (dotted line). Figure 6 also shows the best fitting for the results (solid line), given as function of 1/Q."
Published as SciPost Phys. Proc. 14, 015 (2023)