High-order fractional partial differential equation transform for molecular surface construction

Langhua Hu; Duan Chen; Guo-Wei Wei

Molecular Based Mathematical Biology (2013)

  • Volume: 1, page 1-25
  • ISSN: 2299-3266

Abstract

top
Fractional derivative or fractional calculus plays a significant role in theoretical modeling of scientific and engineering problems. However, only relatively low order fractional derivatives are used at present. In general, it is not obvious what role a high fractional derivative can play and how to make use of arbitrarily high-order fractional derivatives. This work introduces arbitrarily high-order fractional partial differential equations (PDEs) to describe fractional hyperdiffusions. The fractional PDEs are constructed via fractional variational principle. A fast fractional Fourier transform (FFFT) is proposed to numerically integrate the high-order fractional PDEs so as to avoid stringent stability constraints in solving high-order evolution PDEs. The proposed high-order fractional PDEs are applied to the surface generation of proteins. We first validate the proposed method with a variety of test examples in two and three-dimensional settings. The impact of high-order fractional derivatives to surface analysis is examined. We also construct fractional PDE transform based on arbitrarily high-order fractional PDEs. We demonstrate that the use of arbitrarily high-order derivatives gives rise to time-frequency localization, the control of the spectral distribution, and the regulation of the spatial resolution in the fractional PDE transform. Consequently, the fractional PDE transform enables the mode decomposition of images, signals, and surfaces. The effect of the propagation time on the quality of resulting molecular surfaces is also studied. Computational efficiency of the present surface generation method is compared with the MSMS approach in Cartesian representation. We further validate the present method by examining some benchmark indicators of macromolecular surfaces, i.e., surface area, surface enclosed volume, surface electrostatic potential and solvation free energy. Extensive numerical experiments and comparison with an established surface model indicate that the proposed high-order fractional PDEs are robust, stable and efficient for biomolecular surface generation.

How to cite

top

Langhua Hu, Duan Chen, and Guo-Wei Wei. "High-order fractional partial differential equation transform for molecular surface construction." Molecular Based Mathematical Biology 1 (2013): 1-25. <http://eudml.org/doc/267473>.

@article{LanghuaHu2013,
abstract = {Fractional derivative or fractional calculus plays a significant role in theoretical modeling of scientific and engineering problems. However, only relatively low order fractional derivatives are used at present. In general, it is not obvious what role a high fractional derivative can play and how to make use of arbitrarily high-order fractional derivatives. This work introduces arbitrarily high-order fractional partial differential equations (PDEs) to describe fractional hyperdiffusions. The fractional PDEs are constructed via fractional variational principle. A fast fractional Fourier transform (FFFT) is proposed to numerically integrate the high-order fractional PDEs so as to avoid stringent stability constraints in solving high-order evolution PDEs. The proposed high-order fractional PDEs are applied to the surface generation of proteins. We first validate the proposed method with a variety of test examples in two and three-dimensional settings. The impact of high-order fractional derivatives to surface analysis is examined. We also construct fractional PDE transform based on arbitrarily high-order fractional PDEs. We demonstrate that the use of arbitrarily high-order derivatives gives rise to time-frequency localization, the control of the spectral distribution, and the regulation of the spatial resolution in the fractional PDE transform. Consequently, the fractional PDE transform enables the mode decomposition of images, signals, and surfaces. The effect of the propagation time on the quality of resulting molecular surfaces is also studied. Computational efficiency of the present surface generation method is compared with the MSMS approach in Cartesian representation. We further validate the present method by examining some benchmark indicators of macromolecular surfaces, i.e., surface area, surface enclosed volume, surface electrostatic potential and solvation free energy. Extensive numerical experiments and comparison with an established surface model indicate that the proposed high-order fractional PDEs are robust, stable and efficient for biomolecular surface generation.},
author = {Langhua Hu, Duan Chen, Guo-Wei Wei},
journal = {Molecular Based Mathematical Biology},
keywords = {Molecular surface generation; High order fractional derivatives; Fractional calculus; Fractional PDE transform; molecular surface generation; high order fractional derivatives; fractional calculus; fractional PDE transform},
language = {eng},
pages = {1-25},
title = {High-order fractional partial differential equation transform for molecular surface construction},
url = {http://eudml.org/doc/267473},
volume = {1},
year = {2013},
}

TY - JOUR
AU - Langhua Hu
AU - Duan Chen
AU - Guo-Wei Wei
TI - High-order fractional partial differential equation transform for molecular surface construction
JO - Molecular Based Mathematical Biology
PY - 2013
VL - 1
SP - 1
EP - 25
AB - Fractional derivative or fractional calculus plays a significant role in theoretical modeling of scientific and engineering problems. However, only relatively low order fractional derivatives are used at present. In general, it is not obvious what role a high fractional derivative can play and how to make use of arbitrarily high-order fractional derivatives. This work introduces arbitrarily high-order fractional partial differential equations (PDEs) to describe fractional hyperdiffusions. The fractional PDEs are constructed via fractional variational principle. A fast fractional Fourier transform (FFFT) is proposed to numerically integrate the high-order fractional PDEs so as to avoid stringent stability constraints in solving high-order evolution PDEs. The proposed high-order fractional PDEs are applied to the surface generation of proteins. We first validate the proposed method with a variety of test examples in two and three-dimensional settings. The impact of high-order fractional derivatives to surface analysis is examined. We also construct fractional PDE transform based on arbitrarily high-order fractional PDEs. We demonstrate that the use of arbitrarily high-order derivatives gives rise to time-frequency localization, the control of the spectral distribution, and the regulation of the spatial resolution in the fractional PDE transform. Consequently, the fractional PDE transform enables the mode decomposition of images, signals, and surfaces. The effect of the propagation time on the quality of resulting molecular surfaces is also studied. Computational efficiency of the present surface generation method is compared with the MSMS approach in Cartesian representation. We further validate the present method by examining some benchmark indicators of macromolecular surfaces, i.e., surface area, surface enclosed volume, surface electrostatic potential and solvation free energy. Extensive numerical experiments and comparison with an established surface model indicate that the proposed high-order fractional PDEs are robust, stable and efficient for biomolecular surface generation.
LA - eng
KW - Molecular surface generation; High order fractional derivatives; Fractional calculus; Fractional PDE transform; molecular surface generation; high order fractional derivatives; fractional calculus; fractional PDE transform
UR - http://eudml.org/doc/267473
ER -

References

top
  1. O. P. Agrawal. Formulation of Euler–Lagrange equations for fractional variational problems. J. Math. Anal. Appl., 272:368–379, 2002. Zbl1070.49013
  2. B. Baeumer, M. Meerschaert, D. Benson, and S. Wheatcraft. Subordinated advection-dispersion equation for contaminant transport. Water Resour.Res., 37:1543–1550, 2001. 
  3. J. Bai and X. C. Feng. Fractional-order anisotropic diffusion for image denoising. IEEE Trans. Image Proc., 16:2492– 2502, 2007. 
  4. P. W. Bates, Z. Chen, Y. H. Sun, G. W. Wei, and S. Zhao. Geometric and potential driving formation and evolution of biomolecular surfaces. J. Math. Biol., 59:193–231, 2009. Zbl1311.92212
  5. P. W. Bates, G. W. Wei, and S. Zhao. The minimal molecular surface. arXiv:q-bio/0610038v1, [q-bio.BM], 2006. 
  6. P. W. Bates, G. W. Wei, and S. Zhao. Minimal molecular surfaces and their applications. Journal of Computational Chemistry, 29(3):380–91, 2008. [Crossref] 
  7. A. L. Bertozzi and J. B. Greer. Low-curvature image simplifiers: Global regularity of smooth solutions and laplacian limiting schemes. Communications on Pure and Applied Mathematics, 57(6):764–790, 2004. [Crossref] Zbl1058.35083
  8. J. Blinn. A generalization of algebraic surface drawing. ACM Transactions on Graphics, 1(3):235–256, 1982. [Crossref] 
  9. P. Blomgren and T. Chan. Color TV: total variation methods for restoration of vector-valued images. Image Processing, IEEE Transactions on, 7(3):304–309, 1998. [Crossref] 
  10. A. Blumen, G. Zumofen, and J. Klafter. Transport aspects in anomalous diffusion: L’evy walks. Phys. Rev. A, 40:3964–3973, 1989. [Crossref] 
  11. M. Caputo. Linear model of dissipation whose w is almost frequency independent. Geophys. J. R. Astr. Soc., 13:529– 539, 1997. 
  12. V. Carstensen, R. Kimmel, and G. Sapiro. Geodesic active contours. International Journal of Computer Vision, 22:61–79, 1997. Zbl0894.68131
  13. A. Chambolle and P. L. Lions. Image recovery via total variation minimization and related problems. Numerische Mathematik, 76(2):167–188, 1997. [Crossref] Zbl0874.68299
  14. T. Chan, A. Marquina, and P. Mulet. High-order total variation-based image restoration. SIAM Journal on Scientific Computing, 22(2):503–516, 2000. [Crossref] Zbl0968.68175
  15. D. Chen, Z. Chen, C. Chen, W. H. Geng, and G. W. Wei. MIBPB: A software package for electrostatic analysis. J. Comput. Chem., 32:657 – 670, 2011. 
  16. D. Chen and G. W. Wei. Modeling and simulation of electronic structure, material interface and random doping in nano-electronic devices. J. Comput. Phys., 229:4431–4460, 2010. Zbl1191.82113
  17. F. Chen, C. M.and Liu, I. Turner, and V. Anh. A fourier method for the fractional diffusion equation describing sub-diffusion. Journal of Computational Physics, 227:886– 897, 2007. Zbl1165.65053
  18. Z. Chen, N. A. Baker, and G. W. Wei. Differential geometry based solvation models I: Eulerian formulation. J. Comput. Phys., 229:8231–8258, 2010. Zbl1229.92030
  19. Z. Chen, N. A. Baker, and G. W. Wei. Differential geometry based solvation models II: Lagrangian formulation. J. Math. Biol., 63:1139– 1200, 2011. Zbl1284.92025
  20. S. Didas, J. Weickert, and B. Burgeth. Properties of higher order nonlinear diffusion filtering. Journal of mathematical imaging and vision, 35(3):208–226, 2009. Zbl1171.68788
  21. T. J. Dolinsky, J. E. Nielsen, J. A. McCammon, and N. A. Baker. PDB2PQR: An automated pipeline for the setup, execution, and analysis of Poisson-Boltzmann electrostatics calculations. Nucleic Acids Research, 32:W665–W667, 2004. [Crossref] 
  22. R. Gabdoulline and R. Wade. Analytically defined surfaces to analyze molecular interaction properties. Journal of Molecular Graphics, 14(6):341–353., 1996. [Crossref] 
  23. W. Geng and G. W. Wei. Multiscale molecular dynamics using the matched interface and boundary method. J Comput. Phys., 230(2):435–457, 2011. Zbl1246.82028
  24. W. Geng, S. Yu, and G. W. Wei. Treatment of charge singularities in implicit solvent models. Journal of Chemical Physics, 127:114106, 2007. 
  25. J. Giard and B. Macq. Molecular surface mesh generation by filtering electron density map. International Journal of Biomedical Imaging, 2010(923780):9 pages, 2010. 
  26. R. Gorenflo, F. Mainardi, E. Scalas, and M. Raberto. Fractional calculus and continuous-time finance.iii,the diffusion limit.mathematical finance(konstanz, 2000). Trends in Math., Birkhuser, Basel, page 171, 18, 2001. Zbl1138.91444
  27. J. Grant and B. Pickup. A Gaussian description of molecular shape. Journal of Physical Chemistry, 99:3503–3510, 1995. 
  28. J. B. Greer and A. L. Bertozzi. H-1 solutions of a class of fourth order nonlinear equations for image processing. Discrete and Continuous Dynamical Systems, 10(1-2):349–366, 2004. Zbl1159.68619
  29. J. B. Greer and A. L. Bertozzi. Traveling wave solutions of fourth order PDEs for image processing. SIAM Journal on Mathematical Analysis, 36(1):38–68, 2004. [Crossref] Zbl1082.35080
  30. P. Guidotti and K. Longo. Two enhanced fourth order diffusion models for image denoising. Journal of Mathematical Imaging and Vision, 40:188–198, 2011. Zbl1255.68235
  31. P. Guidotti and K. Longo. Well-posedness for a class of fourth order diffusions for image processing. NODEANonlinear Differential Equations and Applications, 18:407–425, 2011. Zbl1227.35149
  32. N. Huang, Z. Shen, S. Long, N. Wu, H. Shih, Q. Zheng, N. Yen, C. Tung, and H. Liu. The empirical mode decomposition and the Hilbert spectrum for nonlinear nonstationary time series analysis. Proceedings of Royal Society of London A, 454:903–995, 1998. Zbl0945.62093
  33. Z. M. Jin and X. P. Yang. Strong solutions for the generalized Perona-Malik equation for image restoration. Nonlinear Analysis-Theory Methods and Applications, 73(4):1077–1084, 2010. Zbl1194.35503
  34. M. Lysaker, A. Lundervold, and X. C. Tai. Noise removal using fourth-order partial differential equation with application to medical magnetic resonance images in space and time. IEEE Transactions on Image Processing, 12(12):1579–1590, 2003. [Crossref] Zbl1286.94020
  35. F. Mainardi and R. Gorenflo. On Mittag-Leffler-type functions in fractional evolution processes. Journal of Computational and Applied Mathematics, 118:283 – 299, 2000. [Crossref] Zbl0970.45005
  36. M. Meerschaert. Fractional calculus, anomalous diffusion, and probability. Fractional Dynamics, R. Metzler and J. Klafter, Eds., World Scientific, Singapore, pages 265–284, 2012. Zbl1297.35276
  37. M. Meerschaert and C. Tadjeran. Finite difference approximations for fractional advection-dispersion flow equations. Journal of Computational and Applied Mathematics, 172(1):65–77, 2004. [Crossref] Zbl1126.76346
  38. D. Mumford and J. Shah. Optimal approximations by piecewise smooth functions and associated variational problems. Communications on Pure and Applied Mathematics, 42(5):577–685, 1989. [Crossref] Zbl0691.49036
  39. A. Nicholls, D. L. Mobley, P. J. Guthrie, J. D. Chodera, and V. S. Pande. Predicting small-molecule solvation free energies: An informal blind test for computational chemistry. Journal of Medicinal Chemistry, 51(4):769–79, 2008. [Crossref] 
  40. S. Osher and R. P. Fedkiw. Level set methods: An overview and some recent results. J. Comput. Phys., 169(2):463– 502, 2001. Zbl0988.65093
  41. S. Osher and L. I. Rudin. Feature-oriented image enhancement using shock filters. SIAM Journal on Numerical Analysis, 27(4):919–940, 1990. [Crossref] Zbl0714.65096
  42. S. Osher and J. Sethian. Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations. Journal of computational physics, 79(1):12–49, 1988. Zbl0659.65132
  43. P. Perona and J. Malik. Scale-space and edge-detection using anisotropic diffusion. IEEE Transactions on Pattern Analysis and Machine Intelligence, 12(7):629–639, 1990. [Crossref] 
  44. M. Raberto, E. Scalas, and F. Mainardi. Waiting-times and returns in high-frequency financial data: an empirical study. Physica A, 314:749–755, 2002. Zbl1001.91033
  45. L. I. Rudin, S. Osher, and E. Fatemi. Nonlinear total variation based noise removal algorithms. Physica D, 60(1- 4):259–268, 1992. [Crossref] Zbl0780.49028
  46. L. Sabatelli, S. Keating, J. Dudley, and P. Richmond. Waiting time distributions in financial markets. Eur.Phys.J.B, 27:273–275, 2002. [Crossref] 
  47. M. F. Sanner, A. J. Olson, and J. C. Spehner. Reduced surface: An efficient way to compute molecular surfaces. Biopolymers, 38:305–320, 1996. [Crossref][PubMed] 
  48. G. Sapiro and D. L. Ringach. Anisotropic diffusion of multivalued images with applications to color filtering. Image Processing, IEEE Transactions on, 5(11):1582–1586, 1996. [Crossref] Zbl0852.68113
  49. J. A. Sethian. Evolution, implementation, and application of level set and fast marching methods for advancing fronts. J. Comput. Phys., 169(2):503–555, 2001. Zbl0988.65095
  50. N. Sochen, R. Kimmel, and R. Malladi. A general framework for low level vision. Image Processing, IEEE Transactions on, 7(3):310–318, 1998. [Crossref] Zbl0973.94502
  51. H. Soltanianzadeh, J. P. Windham, and A. E. Yagle. A multidimensional nonlinear edge-preserving filter for magneticresonace image-restoration. IEEE Transactions on Image Processing, 4(2):147–161, 1995. [Crossref] 
  52. Y. H. Sun, P. R. Wu, G. W. Wei, and G. Wang. Evolution-operator-based single-step method for image processing. Int. J. Biomed. Imaging, 83847:1–27, 2006. [PubMed] 
  53. T. Tasdizen, R. Whitaker, P. Burchard, and S. Osher. Geometric surface processing via normal maps. Acm Transactions on Graphics, 22(4):1012–1033, 2003. [Crossref] 
  54. J. A. Wagoner and N. A. Baker. Assessing implicit models for nonpolar mean solvation forces: the importance of dispersion and volume terms. Proceedings of the National Academy of Sciences of the United States of America, 103(22):8331–6, 2006. [Crossref] 
  55. Y. Wang, G. W. Wei, and S.-Y. Yang. Partial differential equation transform – Variational formulation and Fourier analysis. International Journal for Numerical Methods in Biomedical Engineering, 27:1996–2020, 2011. Zbl1254.94012
  56. Y. Wang, G. W. Wei, and S.-Y. Yang. Selective extraction of entangled textures via adaptive pde transform. International Journal in Biomedical Imaging, 2012:958142, 2012. 
  57. Y. Wang, G. W. Wei, and S.-Y. Yang. Iterative filtering decomposition based on local spectral evolution kernel. Journal of Scientific Computing, pages DOI: 10.1007/s10915–011–9496–0, accepted, 2011. [Crossref] 
  58. Y. Wang, G. W. Wei, and S.-Y. Yang. Mode decomposition evolution equations. Journal of Scientific Computing, accepted,2011. Zbl06046075
  59. G. W. Wei. Generalized Perona-Malik equation for image restoration. IEEE Signal Processing Letters, 6(7):165–167, 1999. [Crossref] 
  60. G. W. Wei. Differential geometry based multiscale models. Bulletin of Mathematical Biology, 72:1562 – 1622, 2010. Zbl1198.92001
  61. G. W. Wei and Y. Q. Jia. Synchronization-based image edge detection. Europhysics Letters, 59(6):814–819, 2002. [Crossref] 
  62. G. W. Wei, Q. Zheng, Z. Chen, and K. Xia. Differential geometry based ion transport models. SIAM Review, 54(4), 2012. 
  63. T. P. Witelski and M. Bowen. ADI schemes for higher-order nonlinear diffusion equations. Applied Numerical Mathematics, 45(2-3):331–351, 2003. [Crossref] Zbl1061.76051
  64. A. Witkin. Scale-space filtering: A new approach to multi-scale description. Proceedings of IEEE International Conference on Acoustic Speech Signal Processing, 9:150–153, 1984. 
  65. M. Xu and S. L. Zhou. Existence and uniqueness of weak solutions for a fourth-order nonlinear parabolic equation. Journal of Mathematical Analysis and Applications, 325(1):636–654, 2007. Zbl1107.35038
  66. Y. You and M. Kaveh. Fourth-order partial differential equations for noise removal. IEEE Transactions on Image Processing, 9(10):1723–1730, 2002. Zbl0962.94011
  67. S. N. Yu, W. H. Geng, and G. W. Wei. Treatment of geometric singularities in implicit solvent models. Journal of Chemical Physics, 126:244108, 2007. 
  68. S. N. Yu and G. W. Wei. Three-dimensional matched interface and boundary (MIB) method for treating geometric singularities. J. Comput. Phys., 227:602–632, 2007. Zbl1128.65103
  69. S. N. Yu, Y. C. Zhou, and G. W. Wei. Matched interface and boundary (MIB) method for elliptic problems with sharp-edged interfaces. J. Comput. Phys., 224(2):729–756, 2007. Zbl1120.65333
  70. G. Zaslavsky. Fractional kinetic equation for hamiltonian chaos.chaotic advection, tracer dynamics and turbulent dispersion. Phys.D, 76:110–122, 1994. Zbl1194.37163
  71. Y. Zhang, C. Bajaj, and G. Xu. Surface smoothing and quality improvement of quadrilateral/hexahedral meshes with geometric flow. Communications in Numerical Methods in Engineering, 25:1–18, 2009. Zbl1158.65314
  72. Y. Zhang, G. Xu, and C. Bajaj. Quality meshing of implicit solvation models of biomolecular structures. Computer Aided Geometric Design, 23(6):510–30, 2006. [Crossref][PubMed] Zbl1098.92034
  73. S. Zhao and G. W. Wei. High-order FDTD methods via derivative matching for Maxwell’s equations with material interfaces. J. Comput. Phys., 200(1):60–103, 2004. Zbl1050.78018
  74. Q. Zheng, D. Chen, and G. W. Wei. Second-order Poisson-Nernst-Planck solver for ion transport. Journal of Comput. Phys., 230:5239–5262, 2011. Zbl1222.82073
  75. Q. Zheng and G. W. Wei. Poisson-Boltzmann-Nernst-Planck model. Journal of Chemical Physics, 134:194101, 2011. 
  76. Q. Zheng, S. Y. Yang, and G. W. Wei. Molecular surface generation using PDE transform. International Journal for Numerical Methods in Biomedical Engineering, 28:291–316, 2012. Zbl1244.92024
  77. Y. C. Zhou and G. W. Wei. On the fictitious-domain and interpolation formulations of the matched interface and boundary (MIB) method. J. Comput. Phys., 219(1):228–246, 2006. Zbl1105.65108
  78. Y. C. Zhou, S. Zhao, M. Feig, and G. W. Wei. High order matched interface and boundary method for elliptic equations with discontinuous coefficients and singular sources. J. Comput. Phys., 213(1):1–30, 2006. Zbl1089.65117

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.