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Unconditionally Energy-Stable SAV-FEM for the Dynamics Model of Protein Folding

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Abstract

In this paper, we consider the numerical approximations of a coupled nonlinear Schrödinger equation, which describes the nonlinear dynamics process of protein folding. The main numerical challenge in solving this system is how to discretize nonlinear terms to ensure unconditional energy stability at the discrete level. In order to address this numerical problem, we construct a linear, decoupled, and second-order time-accurate numerical scheme, which is employed by the scalar auxiliary variable approach for the nonlinear terms combined with the finite element method (FEM) for spatial discretization, and the implicit–explicit approach for the highly nonlinear and coupling terms. The unconditional energy stability and unique solvability of the fully discrete scheme are rigorously proved. Especially, using the temporal-spatial error splitting argument, we obtain optimal \(L^{2}\)-error estimates for r-order FEM overcoming time-step restriction. Finally, numerical results are provided to illustrate the theoretical predictions and demonstrate the efficiency of the methods.

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References

  1. Anfinsen, C.B.: The formation and stabilization of protein structure. Biochem. Eng. J. 128, 737–749 (1972)

    Article  Google Scholar 

  2. Anfinsen, C.B.: Principles that govern the folding of protein chains. Science 181, 223–230 (1973)

    Article  Google Scholar 

  3. Akrivis, G.D., Dougalis, V.A., Karakashian, O.A.: On fully discrete Galerkin methods of second-order temporal accuracy for the nonlinear Schrödinger equation. Numer. Math. 59, 31–53 (1991)

    Article  MathSciNet  Google Scholar 

  4. Brooks, C.L., III., Gruebele, M., Onuchic, J.N., Wolynes, P.G.: Chemical physics of protein folding. Proc. Natl. Acad. Sci. 95(19), 11037–11038 (1998)

    Article  Google Scholar 

  5. Berloff, N.G.: Nonlinear dynamics of secondary protein folding. Phys. Lett. A 337(4–6), 391–396 (2005)

    Article  Google Scholar 

  6. Biswas, A., Moran, A., Milovic, D., Majid, F., Biswas, K.C.: An exact solution for the modified nonlinear Schrödinger’s equation for Davydov solitons in \(\alpha \)-helix proteins. Math. Biosci. 227(1), 68–71 (2010)

    Article  MathSciNet  Google Scholar 

  7. Bao, W., Tang, Q., Xu, Z.: Numerical methods and comparison for computing dark and bright solitons in the nonlinear Schrödinger equation. J. Comput. Phys. 235, 423–445 (2013)

    Article  MathSciNet  Google Scholar 

  8. Caspi, S., Ben-Jacob, E.: Conformation changes and folding of proteins mediated by Davydov’s soliton. Phys. Lett. A 272(1–2), 124–129 (2000)

    Article  Google Scholar 

  9. Cui, H., Liu, O., Xu, G.: Controller design and stability analysis for Schrödinger equation subject to a restricted boundary feedback. In: 2018 37th CCC-IEEE, pp. 1197–1201 (2018)

  10. Cui, H., Han, Z., Xu, G.: Controller design to stabilization of Schrödinger equation with boundary input disturbance. Appl. Anal. 99(5), 796–813 (2020)

    Article  MathSciNet  Google Scholar 

  11. Dill, K.A., Chan, H.S.: From Levinthal to pathways to funnels. Nat. Struct. Mol. Biol. 4(1), 10–19 (1997)

  12. Dobson, C.M.: Protein folding and misfolding. Nature 426(6968), 884–890 (2003)

    Article  Google Scholar 

  13. Deng, B., Shen, J., Zhuang, Q.: Second-order SAV schemes for the nonlinear Schrödinger equation and their error analysis. J. Sci. Comput. 88(3), 1–24 (2021)

    Article  Google Scholar 

  14. Gruebele, M.: Protein dynamics: from molecules, to interactions, to biology. Int. J. Mol. Sci. 10(3), 1360–1368 (2009)

  15. Huang, F., Lerner, E., Sato, S., Amir, D., Haas, E., Fersht, A.R.: Time resolved fluorescence resonance energy transfer study shows a compact denatured state of the B domain of protein A. Biochemistry 48(15), 3468–3476 (2009)

    Article  Google Scholar 

  16. Haynie, D.T.: Biological Thermodynamics. Cambridge University Press, Cambridge (2001)

    Book  Google Scholar 

  17. Haran, G.: How, when and why proteins collapse: the relation to folding. Curr. Opin. Struct. Biol. 22(1), 14–20 (2012)

    Article  Google Scholar 

  18. Henning, P., Peterseim, D.: Crank–Nicolson Galerkin approximations to nonlinear Schrödinger equations with rough potentials. Curr. Opin. Struct. Biol. 27(11), 2147–2184 (2017)

    Google Scholar 

  19. Januar, M., Sulaiman, A., Handoko, L.T.: Nonlinear conformation of secondary protein folding. Int. J. Mod. Phys. C 9, 127–132 (2012)

    Google Scholar 

  20. Karakashian, O., Makridakis, C.: A space-time finite element method for the nonlinear Schrödinger equation: the discontinuous Galerkin method. Math. Comput. 67(222), 479–499 (1998)

    Article  Google Scholar 

  21. Karplus, M., McCammon, J.A.: Molecular dynamics simulations of biomolecules. Nat. Struct. Mol. Biol. 9(9), 646–652 (2002)

    Article  Google Scholar 

  22. Lakshmikanth, G.S., Sridevi, K., Krishnamoorthy, G., Udgaonkar, J.B.: Structure is lost incrementally during the unfolding of barstar. Nat. Struct. Mol. Biol. 8(9), 799–804 (2001)

    Article  Google Scholar 

  23. Li, M., Huang, C., Wang, P.: Galerkin finite element method for nonlinear fractional Schrödinger equations. Numer. Algorithms 74, 499–525 (2017)

    Article  MathSciNet  Google Scholar 

  24. Li, M., Huang, C., Ming, W.: A relaxation-type Galerkin FEM for nonlinear fractional Schrödinger equations. Numer. Algorithms 83, 99–124 (2020)

    Article  MathSciNet  Google Scholar 

  25. Li, M., Zhao, J., Wang, N., Chen, S.: Conforming and nonconforming conservative virtual element methods for nonlinear Schrödinger equation: a unified framework. Comput. Methods Appl. Mech. Eng. 380, 113793 (2021)

    Article  Google Scholar 

  26. Mingaleev, S.F., Gaididei, Y.B., Christiansen, P.L., Kivshar, Y.S.: Nonlinearity-induced conformational instability and dynamics of biopolymers. EPL 59(3), 403 (2002)

    Article  Google Scholar 

  27. Ran, M., Zhang, C.: A linearly implicit conservative scheme for the fractional nonlinear Schrödinger equation with wave operator. Int. J. Comput. Math. 93(7), 1103–1118 (2016)

    Article  MathSciNet  Google Scholar 

  28. Sela, M., White, F.H., Jr., Anfinsen, C.B.: Reductive cleavage of disulfide bridges in ribonuclease. Science 125(3250), 691–692 (1957)

    Article  Google Scholar 

  29. Selkoe, D.J.: Cell biology of protein misfolding: the examples of Alzheimer’s and Parkinson’s diseases. Nat. Cell Biol. 6(11), 1054–1061 (2004)

    Article  Google Scholar 

  30. Shen, J., Xu, J., Yang, J.: The scalar auxiliary variable (SAV) approach for gradient flows. J. Comput. Phys. 353, 407–416 (2018)

    Article  MathSciNet  Google Scholar 

  31. Shi, D., Wang, J.: Unconditional superconvergence analysis of a Crank–Nicolson Galerkin FEM for nonlinear Schrödinger equation. J. Sci. Comput. 72(3), 1093–1118 (2017)

    Article  MathSciNet  Google Scholar 

  32. Shen, J., Xu, J., Yang, J.: A new class of efficient and robust energy stable schemes for gradient flows. SIAM Rev. 61(3), 474–506 (2019)

    Article  MathSciNet  Google Scholar 

  33. Tang, Y., Zou, G. A., Li, J.: Unconditionally energy-stable finite element scheme for the chemotaxis-fluid system. J. Sci. Comput. 95(1), (2023)

  34. Wang, J.: A new error analysis of Crank–Nicolson Galerkin FEMs for a generalized nonlinear Schrödinger equation. J. Sci. Comput. 60(2), 390–407 (2014)

    Article  MathSciNet  Google Scholar 

  35. Wang, P., Huang, C.: A conservative linearized difference scheme for the nonlinear fractional Schrödinger equation. Numer. Algorithms 69(3), 625–641 (2015)

    Article  MathSciNet  Google Scholar 

  36. Wang, Y., Shi, X.: Application of Euler–Lagrange equation in one-dimensional wave equation. Phys. Eng. 27(6), 41–44 (2017). (in Chinese)

    Google Scholar 

  37. Wang, N., Li, M., Huang, C.: Unconditional energy dissipation and error estimates of the SAV Fourier spectral method for nonlinear fractional generalized Wave equation. J. Sci. Comput. 88(1), 19 (2021)

    Article  MathSciNet  Google Scholar 

  38. Wang, X., Zou, G.A., Wang, B.: A stabilized divergence-free virtual element scheme for the nematic liquid crystal flows. Appl. Numer. Math. 192, 104–131 (2023)

    Article  MathSciNet  Google Scholar 

  39. Duan, Y., Kollman, P.A.: Pathways to a protein folding intermediate observed in a 1-microsecond simulation in aqueous solution. Science 282(5389), 740–744 (1998)

    Article  Google Scholar 

  40. Yang, L.Q., Sang, P., Tao, Y., Fu, Y.X., Zhang, K.Q., Xie, Y.H., Liu, S.Q.: Protein dynamics and motions in relation to their functions: several case studies and the underlying mechanisms. J. Biomol. Struct. Dyn. 32(3), 372–393 (2014)

    Article  Google Scholar 

  41. Zou, G.A., Wang, B., Yang, X.: A fully-decoupled discontinuous Galerkin approximation of the Cahn–Hilliard–Brinkman–Ohta–Kawasaki tumor growth model. ESAIM M2AN 56(6), 2141–2180 (2022)

    Article  MathSciNet  Google Scholar 

  42. Zou, G.A., Li, Z., Yang, X.: Fully discrete discontinuous Galerkin numerical scheme with second-order temporal accuracy for the hydrodynamically coupled lipid vesicle model. J. Sci. Comput. 95(1), 5 (2023)

    Article  MathSciNet  Google Scholar 

  43. Zou, G.A., Wang, B., Yang, X.: Efficient interior penalty discontinuous Galerkin projection method with unconditional energy stability and second-order temporal accuracy for the incompressible magneto-hydrodynamic system. J. Comput. Phys. 495, 112562 (2023)

    Article  MathSciNet  Google Scholar 

  44. Zheng, Z., Zou, G.A., Wang, B., Zhao, W.: A fully-decoupled discontinuous Galerkin method for the nematic liquid crystal flows with SAV approach. J. Comput. Appl. Math. 429, 115207 (2023)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors are grateful to the reviewers for the constructive comments and valuable suggestions which have improved the paper.

Funding

Bo Wang is supported by the Major Public Welfare Projects in Henan Province (201300311300), and the Natural Science Foundation of Henan Province (232300420109). Guang-an Zou is supported by the Key Scientific Research Projects of Colleges and Universities in Henan Province, China (23A110006).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Henan University, Kaifeng, 475004, People’s Republic of China

    Dan Zhang, Bo Wang, Guang-an Zou & YuXing Zhang

  2. Henan Key Laboratory of Earth System Observation and Modeling, Henan University, Kaifeng, 475004, People’s Republic of China

    Bo Wang & Guang-an Zou

  3. The Academy for Advanced Interdisciplinary Studies, Henan University, Zhengzhou, 450046, People’s Republic of China

    Bo Wang

  4. School of Mathematics and Statistics, Henan Engineering Research Center for Artificial Intelligence Theory and Algorithms, Henan University, Kaifeng, 475004, People’s Republic of China

    Guang-an Zou

Authors
  1. Dan Zhang
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  2. Bo Wang
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  3. Guang-an Zou
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  4. YuXing Zhang
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Contributions

DZ: Formal analysis, writing-original draft, software-original draft. BW: Conceptualization, supervision, writing-review and editing. GZ: Conceptualization, methodology. YZ: Software-review.

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Correspondence to Bo Wang.

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Zhang, D., Wang, B., Zou, Ga. et al. Unconditionally Energy-Stable SAV-FEM for the Dynamics Model of Protein Folding. J Sci Comput 101, 43 (2024). https://doi.org/10.1007/s10915-024-02687-y

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  • DOI: https://doi.org/10.1007/s10915-024-02687-y

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