High-order fractional partial differential equation transform for molecular surface construction
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.
Department of Mathematics Michigan State University, MI 48824,
Mathematical Biosciences Institute The Ohio State University,
Columbus, OH, 43210, USA
Department of Mathematics Michigan State University, MI 48824,
Department of Electrical and Computer Engineering Michigan
State University, MI 48824, USA
Department of Biochemistry and Molecular Biology Michigan
State University, MI 48824, USA
- O. P. Agrawal. Formulation of Euler–Lagrange equations for fractional variational problems. J. Math. Anal. Appl.,272:368–379, 2002.
- B. Baeumer, M. Meerschaert, D. Benson, and S. Wheatcraft. Subordinated advection-dispersion equation forcontaminant transport. Water Resour.Res., 37:1543–1550, 2001.
- J. Bai and X. C. Feng. Fractional-order anisotropic diffusion for image denoising. IEEE Trans. Image Proc., 16:2492–2502, 2007.
- P. W. Bates, Z. Chen, Y. H. Sun, G. W. Wei, and S. Zhao. Geometric and potential driving formation and evolutionof biomolecular surfaces. J. Math. Biol., 59:193–231, 2009.
- P. W. Bates, G. W. Wei, and S. Zhao. The minimal molecular surface. arXiv:q-bio/0610038v1, [q-bio.BM], 2006.
- P. W. Bates, G. W. Wei, and S. Zhao. Minimal molecular surfaces and their applications. Journal of ComputationalChemistry, 29(3):380–91, 2008.[Crossref]
- A. L. Bertozzi and J. B. Greer. Low-curvature image simplifiers: Global regularity of smooth solutions and laplacianlimiting schemes. Communications on Pure and Applied Mathematics, 57(6):764–790, 2004.[Crossref]
- J. Blinn. A generalization of algebraic surface drawing. ACM Transactions on Graphics, 1(3):235–256, 1982.[Crossref]
- 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]
- A. Blumen, G. Zumofen, and J. Klafter. Transport aspects in anomalous diffusion: L’evy walks. Phys. Rev. A,40:3964–3973, 1989.[Crossref]
- M. Caputo. Linear model of dissipation whose w is almost frequency independent. Geophys. J. R. Astr. Soc., 13:529–539, 1997.
- V. Carstensen, R. Kimmel, and G. Sapiro. Geodesic active contours. International Journal of Computer Vision,22:61–79, 1997.
- A. Chambolle and P. L. Lions. Image recovery via total variation minimization and related problems. NumerischeMathematik, 76(2):167–188, 1997.[Crossref]
- T. Chan, A. Marquina, and P. Mulet. High-order total variation-based image restoration. SIAM Journal on ScientificComputing, 22(2):503–516, 2000.[Crossref]
- 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.
- D. Chen and G. W. Wei. Modeling and simulation of electronic structure, material interface and random doping innano-electronic devices. J. Comput. Phys., 229:4431–4460, 2010.
- F. Chen, C. M.and Liu, I. Turner, and V. Anh. A fourier method for the fractional diffusion equation describingsub-diffusion. Journal of Computational Physics, 227:886– 897, 2007.
- Z. Chen, N. A. Baker, and G. W. Wei. Differential geometry based solvation models I: Eulerian formulation. J.Comput. Phys., 229:8231–8258, 2010.
- Z. Chen, N. A. Baker, and G. W. Wei. Differential geometry based solvation models II: Lagrangian formulation. J.Math. Biol., 63:1139– 1200, 2011.
- S. Didas, J. Weickert, and B. Burgeth. Properties of higher order nonlinear diffusion filtering. Journal of mathematicalimaging and vision, 35(3):208–226, 2009.
- 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]
- R. Gabdoulline and R. Wade. Analytically defined surfaces to analyze molecular interaction properties. Journal ofMolecular Graphics, 14(6):341–353., 1996.[Crossref]
- W. Geng and G. W. Wei. Multiscale molecular dynamics using the matched interface and boundary method. JComput. Phys., 230(2):435–457, 2011.
- W. Geng, S. Yu, and G. W. Wei. Treatment of charge singularities in implicit solvent models. Journal of ChemicalPhysics, 127:114106, 2007.
- J. Giard and B. Macq. Molecular surface mesh generation by filtering electron density map. International Journalof Biomedical Imaging, 2010(923780):9 pages, 2010.
- R. Gorenflo, F. Mainardi, E. Scalas, and M. Raberto. Fractional calculus and continuous-time finance.iii,the diffusionlimit.mathematical finance(konstanz, 2000). Trends in Math., Birkhuser, Basel, page 171, 18, 2001.
- J. Grant and B. Pickup. A Gaussian description of molecular shape. Journal of Physical Chemistry, 99:3503–3510,1995.
- 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.
- J. B. Greer and A. L. Bertozzi. Traveling wave solutions of fourth order PDEs for image processing. SIAM Journalon Mathematical Analysis, 36(1):38–68, 2004.[Crossref]
- P. Guidotti and K. Longo. Two enhanced fourth order diffusion models for image denoising. Journal of MathematicalImaging and Vision, 40:188–198, 2011.
- P. Guidotti and K. Longo. Well-posedness for a class of fourth order diffusions for image processing. NODEANonlinearDifferential Equations and Applications, 18:407–425, 2011.
- N. Huang, Z. Shen, S. Long, N. Wu, H. Shih, Q. Zheng, N. Yen, C. Tung, and H. Liu. The empirical modedecomposition and the Hilbert spectrum for nonlinear nonstationary time series analysis. Proceedings of RoyalSociety of London A, 454:903–995, 1998.
- Z. M. Jin and X. P. Yang. Strong solutions for the generalized Perona-Malik equation for image restoration. NonlinearAnalysis-Theory Methods and Applications, 73(4):1077–1084, 2010.
- M. Lysaker, A. Lundervold, and X. C. Tai. Noise removal using fourth-order partial differential equation withapplication to medical magnetic resonance images in space and time. IEEE Transactions on Image Processing, 12(12):1579–1590, 2003.[Crossref]
- F. Mainardi and R. Gorenflo. On Mittag-Leffler-type functions in fractional evolution processes. Journal of Computationaland Applied Mathematics, 118:283 – 299, 2000.[Crossref]
- M. Meerschaert. Fractional calculus, anomalous diffusion, and probability. Fractional Dynamics, R. Metzler and J.Klafter, Eds., World Scientific, Singapore, pages 265–284, 2012.
- 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]
- 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]
- A. Nicholls, D. L. Mobley, P. J. Guthrie, J. D. Chodera, and V. S. Pande. Predicting small-molecule solvation freeenergies: An informal blind test for computational chemistry. Journal of Medicinal Chemistry, 51(4):769–79, 2008.[Crossref]
- S. Osher and R. P. Fedkiw. Level set methods: An overview and some recent results. J. Comput. Phys., 169(2):463–502, 2001.
- S. Osher and L. I. Rudin. Feature-oriented image enhancement using shock filters. SIAM Journal on NumericalAnalysis, 27(4):919–940, 1990.[Crossref]
- S. Osher and J. Sethian. Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobiformulations. Journal of computational physics, 79(1):12–49, 1988.
- P. Perona and J. Malik. Scale-space and edge-detection using anisotropic diffusion. IEEE Transactions on PatternAnalysis and Machine Intelligence, 12(7):629–639, 1990.[Crossref]
- M. Raberto, E. Scalas, and F. Mainardi. Waiting-times and returns in high-frequency financial data: an empiricalstudy. Physica A, 314:749–755, 2002.
- L. I. Rudin, S. Osher, and E. Fatemi. Nonlinear total variation based noise removal algorithms. Physica D, 60(1-4):259–268, 1992.[Crossref]
- L. Sabatelli, S. Keating, J. Dudley, and P. Richmond. Waiting time distributions in financial markets. Eur.Phys.J.B,27:273–275, 2002.[Crossref]
- 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]
- G. Sapiro and D. L. Ringach. Anisotropic diffusion of multivalued images with applications to color filtering. ImageProcessing, IEEE Transactions on, 5(11):1582–1586, 1996.[Crossref]
- 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.
- N. Sochen, R. Kimmel, and R. Malladi. A general framework for low level vision. Image Processing, IEEE Transactionson, 7(3):310–318, 1998.[Crossref]
- H. Soltanianzadeh, J. P. Windham, and A. E. Yagle. A multidimensional nonlinear edge-preserving filter for magneticresonaceimage-restoration. IEEE Transactions on Image Processing, 4(2):147–161, 1995.[Crossref]
- 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]
- T. Tasdizen, R. Whitaker, P. Burchard, and S. Osher. Geometric surface processing via normal maps. Acm Transactionson Graphics, 22(4):1012–1033, 2003.[Crossref]
- J. A. Wagoner and N. A. Baker. Assessing implicit models for nonpolar mean solvation forces: the importance ofdispersion and volume terms. Proceedings of the National Academy of Sciences of the United States of America,103(22):8331–6, 2006.[Crossref]
- Y. Wang, G. W. Wei, and S.-Y. Yang. Partial differential equation transform – Variational formulation and Fourieranalysis. International Journal for Numerical Methods in Biomedical Engineering, 27:1996–2020, 2011.
- Y. Wang, G. W. Wei, and S.-Y. Yang. Selective extraction of entangled textures via adaptive pde transform. InternationalJournal in Biomedical Imaging, 2012:958142, 2012.
- 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]
- Y. Wang, G. W. Wei, and S.-Y. Yang. Mode decomposition evolution equations. Journal of Scientific Computing,accepted,2011.
- G. W. Wei. Generalized Perona-Malik equation for image restoration. IEEE Signal Processing Letters, 6(7):165–167,1999.[Crossref]
- G. W. Wei. Differential geometry based multiscale models. Bulletin of Mathematical Biology, 72:1562 – 1622, 2010.
- G. W. Wei and Y. Q. Jia. Synchronization-based image edge detection. Europhysics Letters, 59(6):814–819, 2002.[Crossref]
- G. W. Wei, Q. Zheng, Z. Chen, and K. Xia. Differential geometry based ion transport models. SIAM Review, 54(4),2012.
- T. P. Witelski and M. Bowen. ADI schemes for higher-order nonlinear diffusion equations. Applied NumericalMathematics, 45(2-3):331–351, 2003.[Crossref]
- A. Witkin. Scale-space filtering: A new approach to multi-scale description. Proceedings of IEEE InternationalConference on Acoustic Speech Signal Processing, 9:150–153, 1984.
- 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.
- Y. You and M. Kaveh. Fourth-order partial differential equations for noise removal. IEEE Transactions on ImageProcessing, 9(10):1723–1730, 2002.
- S. N. Yu, W. H. Geng, and G. W. Wei. Treatment of geometric singularities in implicit solvent models. Journal ofChemical Physics, 126:244108, 2007.
- S. N. Yu and G. W. Wei. Three-dimensional matched interface and boundary (MIB) method for treating geometricsingularities. J. Comput. Phys., 227:602–632, 2007.
- S. N. Yu, Y. C. Zhou, and G. W. Wei. Matched interface and boundary (MIB) method for elliptic problems withsharp-edged interfaces. J. Comput. Phys., 224(2):729–756, 2007.
- G. Zaslavsky. Fractional kinetic equation for hamiltonian chaos.chaotic advection, tracer dynamics and turbulentdispersion. Phys.D, 76:110–122, 1994.
- Y. Zhang, C. Bajaj, and G. Xu. Surface smoothing and quality improvement of quadrilateral/hexahedral meshes withgeometric flow. Communications in Numerical Methods in Engineering, 25:1–18, 2009.
- Y. Zhang, G. Xu, and C. Bajaj. Quality meshing of implicit solvation models of biomolecular structures. ComputerAided Geometric Design, 23(6):510–30, 2006.[Crossref][PubMed]
- S. Zhao and G. W. Wei. High-order FDTD methods via derivative matching for Maxwell’s equations with materialinterfaces. J. Comput. Phys., 200(1):60–103, 2004.
- 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.
- Q. Zheng and G. W. Wei. Poisson-Boltzmann-Nernst-Planck model. Journal of Chemical Physics, 134:194101,2011.
- Q. Zheng, S. Y. Yang, and G. W. Wei. Molecular surface generation using PDE transform. International Journal forNumerical Methods in Biomedical Engineering, 28:291–316, 2012.
- Y. C. Zhou and G. W. Wei. On the fictitious-domain and interpolation formulations of the matched interface andboundary (MIB) method. J. Comput. Phys., 219(1):228–246, 2006.
- Y. C. Zhou, S. Zhao, M. Feig, and G. W. Wei. High order matched interface and boundary method for ellipticequations with discontinuous coefficients and singular sources. J. Comput. Phys., 213(1):1–30, 2006.