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Evolutionary de Rham-Hodge method

  • Corresponding author: Yiying Tong

    Corresponding author: Yiying Tong; 

    Corresponding author: Yiying Tong* Corresponding author: Guo-Wei Wei

    * Corresponding author: Guo-Wei Wei

This work was supported in part by NSF Grants DMS1721024, DMS1761320, IIS1900473, NIH grants GM126189 and GM129004, Bristol-Myers Squibb and Pfizer. GWW thanks Vidit Nanda for useful discussion

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  • The de Rham-Hodge theory is a landmark of the 20$ ^\text{th} $ Century's mathematics and has had a great impact on mathematics, physics, computer science, and engineering. This work introduces an evolutionary de Rham-Hodge method to provide a unified paradigm for the multiscale geometric and topological analysis of evolving manifolds constructed from a filtration, which induces a family of evolutionary de Rham complexes. While the present method can be easily applied to close manifolds, the emphasis is given to more challenging compact manifolds with 2-manifold boundaries, which require appropriate analysis and treatment of boundary conditions on differential forms to maintain proper topological properties. Three sets of unique evolutionary Hodge Laplacians are proposed to generate three sets of topology-preserving singular spectra, for which the multiplicities of zero eigenvalues correspond to exactly the persistent Betti numbers of dimensions 0, 1 and 2. Additionally, three sets of non-zero eigenvalues further reveal both topological persistence and geometric progression during the manifold evolution. Extensive numerical experiments are carried out via the discrete exterior calculus to demonstrate the potential of the proposed paradigm for data representation and shape analysis of both point cloud data and density maps. To demonstrate the utility of the proposed method, the application is considered to the protein B-factor predictions of a few challenging cases for which existing biophysical models break down.

    Mathematics Subject Classification: Primary:53Z10, 53Z50;Secondary:14F40.

    Citation:

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  • Figure 1.  Discrete de Rham cohomology; $ D_k $ is the combinatorial operators such that $ {D}_{k+1}{D}_{k} = 0 $; $ {S}_k $ is the discrete Hodge stars

    Figure 2.  Illustration of normal and tangential harmonic field extensions. Thick lines are the inputs and thin lines are the extended outputs. Left charts in both (a) and (b) show harmonic fields and their extensions while right charts give meticulous detail of interior parts. (a) Normal harmonic forms. A solid ball with a cavity extends inward to a solid ball without cavity. The outside surface is fixed. (b) Tangential harmonic forms. A torus extends to a solid ball

    Figure 3.  Persistence and progression on benzene

    Figure 4.  Snapshots of evolving manifold with the two-body system. a, b, c and d are snapshots from the beginning to the end. b and c show the transition of the Betti-0 number from 2 to 1

    Figure 5.  Eigenvalues and Betti numbers vs isovalue ($ c $) of the two-body system with $ \eta = 1.19 $ and $ \max(\rho)\approx 1.0 $. i shows the smallest eigenvalues of the $ T $ set. The drops at $ c = 0.6 $ correspond to snapshots in Figs. 4 b and c. ii and iii show the smallest eigenvalues of the $ C $ and $ N $ sets respectively

    Figure 6.  Snapshots of evolving manifolds with the four-body system. a is the initial point of four components; b and c show the transition of a ring formed and the persistent Betti-0 number changes from 4 to 1. g and h show the vanishing of the ring and the persistent Betti-1 number changes from 1 to 0.

    Figure 7.  Eigenvalues and Betti numbers vs isovalue ($ c $) of the four-body system with $ \eta = 1.19 $ and $ \max(\rho)\approx 1.2 $. i shows the smallest eigenvalues of the $ T $ set. At near $ c = 0.80 $, the persistent Betti-0 number changes from 4 to 1. ii shows the smallest eigenvalues of the $ C $ set. At around $ c = 1.02 $, the persistent Betti-1 number changes from 1 to 0. iii shows the smallest eigenvalues of the $ N $ set

    Figure 8.  Snapshots of evolving manifold with the eight-body system. a presents the initial state with eight components. b and c show the formation of 6 tunnels when the persistent Betti-0 number changes from 8 to 1 and the persistent Betti-1 number changes from 0 to 5. d and e illustrate that a cavity appears, so the persistent Betti-1 number drops to 0 and the persistent Betti-2 number increases to 1. f shows a solid volume without cavity. The gray planes cut manifolds to create cross-section views to illustrate the process of the formation of cavity as shown in b', c', d' and e'

    Figure 9.  Eigenvalues and Betti numbers vs isovalue ($ c $) of the eight-body system with $ \eta = 1.53 $ and $ \max(\rho)\approx1.1 $. i shows the Fiedler values of the $ T $ set and persistent Betti-0 numbers. ii shows the Fiedler values of the $ C $ set and persistent Betti-1 numbers. iii illustrates the comparison of $ \lambda_{l,1}^C $ and persistent $ \beta_2 $

    Figure 10.  Manifold evolution of benzene with $ \eta = 0.45\times r_{\rm vdw} $. a through h are snapshots from the start to the end. a and b show the transition of the persistent Betti-0 number from 12 to 6. c and d show the formation of a ring; The Betti-0 number changes from 6 to 1 and remains at one to the end, whereas the Betti-1 number changes from zero to one. d, e, f and g illustrate the deformation of the hexagonal tunnel to a round tunnel. From g to h, the ring disappears and the Betti-1 number changes from 1 back to 0

    Figure 11.  Eigenvalues and Betti numbers vs isovalue ($ c $) of the benzene system with $ \eta = 0.45 $ and $ \max(\rho)\approx1.1 $. i shows the smallest eigenvalues of the $ T $ set. The drops at $ c = 0.12 $ correspond to snapshots in Figs. 10 a and b. The drops at $ c = 0.22 $ correspond to snapshots in Figs. 10 c and d. ii shows the smallest eigenvalue of the $ N $ set. The drops at $ c = 0.9 $ correspond to snapshots in Figs. 10 g and h. iii shows the smallest eigenvalues of the $ C $ set

    Figure 12.  Illustration of fullurene ($ \text{C}_{60} $) manifold evolution with $ \eta = 0.5\times r_{\rm vdw} $. a presents sixty components around carbon atom positions. a and b show that the components connect if they share a pentagonal hole and persistent $ \beta_0 $ changes from 60 to 12 and persistent $ \beta_1 $ changes from 0 to 12. c shows the hexagonal holes are formed, resulting in the change of persistent $ \beta_0 $ to 1 and persistent $ \beta_1 $ to 31. (There are 32 rings, but only 31 are independent in terms of homology.) c and d show that the 12 pentagonal rings disappear and the persistent Betti-1 number drops from 31 to 19. d and e show that the 20 hexagonal rings disappear and a cavity forms inside, so that persistent $ \beta_1 $ drops to $ 0 $ and persistent $ \beta_2 $ increases to 1. The vertical plan cuts the manifolds that gives an illustration of cavity in d' and e'

    Figure 13.  Eigenvalues and Betti numbers vs isovalue ($ c $) of the fullurene ($ \text{C}_{60} $) system with $ \eta = 0.5\times r_{\rm vdw} $ and $ \max(\rho)\approx1.3 $. i gives the Fiedler values of the $ T $ set and persistent $ \beta_0 $. ii presents the comparison of $ \lambda^C_{l,1} $ and persistent $ \beta_1 $. iii shows the Fiedler values of the $ N $ set and persistent $ \beta_2 $

    Figure 14.  Illustration of fullurene ($ \text{C}_{60} $) manifold evolution with $ \eta = 0.8\times r_{\rm vdw} $. a shows 12 initial solid pentagonal components. b and c show the formation and contraction process of the 20 rings. d is the snapshot right after the formation of the cavity. e shows the final stage as a solid ball of this example

    Figure 15.  Eigenvalues and Betti numbers vs isovalue ($ c $) of the fullurene ($ \text{C}_{60} $) system with $ \eta = 0.8\times r_{\rm vdw} $; $ \max{\rho}\approx2.5 $. i gives the Fiedler values of the $ T $ set and persistent $ \beta_0 $. ii presents the comparison of $ \lambda^C_{l,1} $ and persistent $ \beta_1 $. iii shows the Fiedler values of the $ N $ set and persistent $ \beta_2 $

    Figure 16.  Experimental and predicted B-factor values plotted per residue (PDB IDs: 1CLL, 2HQK and 1V70). EXP: experimental values; EDH: evolutionary de Rham-Hodge (10 isovalues) method predicted values; GNM: Gaussian network method predicted values

    Figure 17.  The structure of calmodulin (PDB ID: 1CLL) visualized in Visual Molecular Dynamics (VMD) [33] and colored by experimental B-factors (left), EDH (10 isovalues) predict B-factors (middle) and GNM predicted B-factors (right) with red representing the most flexible regions

    Figure 18.  Illustration of surfaces extracted with different isovalues for EMD-1776. The isovalues for a, b, c and d are 0.14, 0.10, 0.07 and 0.04, respectively. In a, $ \beta_0 $ is 12 and $ \beta_1 $ and $ \beta_2 $ are 0; In b, $ \beta_0 = 4 $, $ \beta_1 = 4 $ and $ \beta_2 = 0 $; In c, $ \beta_0 = 1 $, $ \beta_1 = 13 $ and $ \beta_2 = 0 $; In d, $ \beta_0 = 1 $, $ \beta_1 = 9 $ and $ \beta_2 = 0 $

    Figure 19.  Eigenvalues and Betti numbers vs filtration of the EMD-1776 density map. The filtration goes from 2.68 (the largest isovalue (0.28) subtract by 0.14) to 2.78 (the largest isovalue (0.28) subtract by 0.04). i gives the Fiedler values of the $ T $ set and persistent $ \beta_0 $. ii presents the comparison of $ \lambda^C_{l,1} $ and persistent $ \beta_1 $. iii shows the Fiedler values of the $ N $ set and persistent $ \beta_2 $

    Table 1.  Exterior (odd rows) vs. traditional (even rows) calculus in $ \mathbb{R}^3 $. $ f^0 $, $ {\bf{v}}^1 $, $ {\bf{v}}^2 $ and $ f^3 $ stand for $ 0 $-, $ 1 $-, $ 2 $- and $ 3 $-forms with their components stored in either a scalar field $ f $ or vector field $ {\bf{v}} $

    order $ 0 $ order $ 1 $ order $ 2 $ order $ 3 $
    form $ f^0 $ $ {\bf{v}}^1({\bf{a}}) $ $ {\bf{v}}^2({\bf{a}},{\bf{b}}) $ $ f^3({\bf{a}},{\bf{b}},{\bf{c}}) $
    $ f $ $ {\bf{v}}\cdot {\bf{a}} $ $ {\bf{v}}\cdot({\bf{a}}\times{\bf{b}}) $ $ f[({\bf{a}}\times{\bf{b}})\cdot{\bf{c}}] $
    $ d $ $ df^0 $ $ d {\bf{v}}^1 $ $ d {\bf{v}}^2 $ $ d f^3 $
    $ (\nabla f)^1 $ $ (\nabla \times {\bf{v}})^2 $ $ (\nabla \cdot {\bf{v}})^3 $ $ 0 $
    $ \star $ $ \star f^0 $ $ \star {\bf{v}}^1 $ $ \star {\bf{v}}^2 $ $ \star f^3 $
    $ f^3 $ $ {\bf{v}}^2 $ $ {\bf{v}}^1 $ $ f^0 $
    $ \delta $ $ \delta f^0 $ $ \delta {\bf{v}}^1 $ $ \delta {\bf{v}}^2 $ $ \delta f^3 $
    $ 0 $ $ (- \nabla \cdot {\bf{v}})^0 $ $ (\nabla \times {\bf{v}})^1 $ $ (-\nabla f)^2 $
    $ \wedge $ $ f^0\!\wedge\! g^0 $ $ f^0\!\wedge\!{\bf{v}}^1 $ $ f^0\!\wedge\!{\bf{v}}^2 $, $ {\bf{v}}^1\!\wedge\!{\bf{u}}^1 $ $ f^0\!\wedge\!g^3 $, $ {\bf{v}}^1\!\wedge\!{\bf{u}}^2 $
    $ (fg)^0 $ $ (f{\bf{v}})^1 $ $ (f{\bf{v}})^2 $, $ ({\bf{v}}\!\times\!{\bf{u}})^2 $ $ (fg)^3 $, $ ({\bf{v}}\cdot{\bf{u}})^3 $
     | Show Table
    DownLoad: CSV

    Table 2.  Boundary conditions of tangential and normal form

    type $ f^0 $ $ {\bf{v}}^1 $ $ {\bf{v}}^2 $ $ f^3 $
    tangential unrestricted $ {\bf{v}}\cdot {\bf{n}} = 0 $ $ {\bf{v}} \parallel {\bf{n}} $ $ f|_{\partial M} = 0 $
    normal $ f|_{\partial M} = 0 $ $ {\bf{v}} \parallel {\bf{n}} $ $ {\bf{v}} \cdot {\bf{n}}=0 $ unrestricted
     | Show Table
    DownLoad: CSV

    Table 3.  Pearson correlation coefficients in B-factor predictions using GNM, mGNM and EDH for four proteins. Here, mGNM stands for multiscale GNM with two different kernels [70]. $ N_{C^\alpha} $ is the number of residues. In cases of EDH, three different isovalue sets are applied with 10, 20 and 40 points of equal spaces on the interval of [$ 0.1, 1.0 $]

    PDB ID $ N_{C^\alpha} $ GNM[70] mGNM[70] EDH (10) EDH (20) EDH (40)
    1CLL 292 0.261 0.763 0.789 0.797 0.850
    1V70 105 0.162 0.750 0.754 0.772 0.858
    2HQK 216 0.365 0.833 0.854 0.880 0.886
    1WHI 122 0.270 0.484 0.640 0.711 0.794
     | Show Table
    DownLoad: CSV
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