分子几何研究
幻灯模式
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bzlu 2020-11-13 17:44:03
##### <br> <br> #=!= Molecular surface, meshing and related studies #=!= (分子几何相关的一些研究及应用) <br> ###=!= Benzhuo Lu (卢本卓) ####=!= Institute of Computational Math and Scientific/Engineering Computing, ####=!= Chinese Academy of Sciences, China ####=!= (科学院计算数学研究所, 国家数学与交叉科学中心) ####=!= <br> ####=!= 人民大学, Nov. 14, 2020 !page - 介绍 - 分子表面及三角网格生成（TMSmesh） - 通道蛋白及膜体系的表面三角化 - 分子及表面的稀疏表示 - 应用（静电、输运计算等...） !page ##=!= Background - 几何形状： **连续介质模型计算**，分子对接，化学信息中的面积体积空腔等，结构比对，形状比较分析 ... <div class="row"><div class="col-lg-5 text-center">Electrostatics potential map of a drug-targeted protein</div><div class="col-lg-6 text-center">Ion transport in a channel</div></div> <div class="row"><div class="col-lg-5"></div><div class="col-lg-6"></div></div> !page <div class="row"><div class="col-lg-6 text-center">A real view of protein molecule</div><div class="col-lg-6 text-center">Surface mesh</div></div> <div class="row"><div class="col-lg-6"><iframe frameborder="no" border="0" marginwidth="0" marginheight="0" width=100% height=400px src="/scene?id= 59f83a764f022e388f25e817"></iframe></div><div class="col-lg-6"><iframe frameborder="no" border="0" marginwidth="0" marginheight="0" width=100% height=400px src="/scene?id= 59fbc6524f022e672773978b"></iframe></div></div> !page ### Molecular surface definition - **van der Walls surface**, **solvent-accessible surface **, **solvent-excluded surface(SES）**, **skin-surface**, **minimal energy surface** and **Gaussian surface** ...  =!= ** VDW、SAS and SES surfaces** !page ### Gaussian surface The Gaussian surface is defined as a level set of the summation of Gaussian kernel functions, ```math \left\{ \mathbf{x} \in \mathbb{R}^3,\phi \left( { \mathbf{x}} \right) = c \right\}, ``` where ```math \phi ( \mathbf{x}) = \sum \limits _{i = 1}^N e^{-d(\| \mathbf{x} - \mathbf{x}_i \|^2 - r_i^2)}, ``` the parameter $$d$$ is positive and controls the decay rate of the kernel functions, $$\mathbf{x_i}$$ and $$r_i$$ are the location and radius of atom $$i$$, $$c$$ is the isovalue, and it controls the volume enclosed by the Gaussian kernel. While $$|d|$$ is increased, the Gaussian surface is closer to the VDW surface.  !page ##=!= Surface triangulation in TMSmesh #### Old version TMSmesh1 - ray-casting --> intersecting points on the surface --> trace on surface to dissect local topology <div class="row"><div class="col-lg-6"></div><div class="col-lg-6"></div></div> <br> Chen MX, Lu BZ, *J Chem Theory Comput*. 2011, 7, 203–212 !page - trace the surface and find all topological details -- and connect points to form polygons; Using predict-correct algorithm  Connect points  !page - Trace the surface and find all topological details. The surface is further dissected into single valued pieces, and then generate manifold triangular mesh  !page #### **New version TMSmesh2**: Polynomial approxiamtion and adaptive surface partition  **A 2D adaptive cell division** <div class="row"><div class="col-lg-6"></div><div class="col-lg-3"></div></div> <font size="4"> Tiantian Liu, M.X. Chen, and B.Z. Lu, Efficient and qualified mesh generation for Gaussian molecular surface using adaptive partition and piecewise polynomial approximation, ***SIAM J Sci. Computing.***, 40：B507-B527, 2018 </font> !page ```math \phi \left(\vec x \right) = \sum \limits_{i=1}^{N}e^{-(\Vert \vec x - \vec x_i \Vert ^2 - r_i^2 )}=\sum \limits_{i=1}^{N}e^{r_i^2}e^{-\left( x-x_i \right)^2}e^{-\left( y-y_i \right)^2}e^{-\left( z-z_i \right)^2}. ``` In the cubic cell $$[a,b]\times [c,d]\times [e,f]$$, Eq(3) can be approximated as： ```math P(x,y,z) = \sum \limits_{i=1}^{N}e^{r_i^2}P_n(x,x_i,a,b)Q_n(y,y_i,c,d)R_n(z,z_i,e,f), ``` ```math P_n(x,x_i,a,b) = \sum\limits_{j=0}^n \alpha_j(x_i,a,b)L_j(\frac{2x-(a+b)}{b-a}), ``` ```math Q_n(y,y_i,c,d) = \sum\limits_{j=0}^n \beta_j(y_i,c,d)L_j(\frac{2y-(c+d)}{d-c}), ``` ```math R_n(z,z_i,e,f) = \sum\limits_{j=0}^n \gamma_j(z_i,e,f)L_j(\frac{2z-(e+f)}{f-e}), ``` $$L_j(\cdot)$$ is $$j$$-th degree Legendre polynomial, $$n$$ is bigger when cell length is larger ( ~ 2Å in our case) !page #### Trilinear approximation - **Extreme points, fold curves and critical points** The extreme points on a cube face： ```math \left \{ \begin{aligned} g(x,y,z) = c \\ \frac{\partial g(x,y,z)}{\partial \alpha} = 0 \\ \alpha = a \end{aligned} \right. , \quad\alpha \in \{x,y,z\} , a \in \{1,-1\}. ``` The fold curve (connecting the extreme points) in the x, y, and z directions is defined as： ```math \{g \left(x,y,z\right)= c, \frac{\partial g(x,y,z)}{\partial \alpha}=0\},\quad\alpha \in \{x,y,z\}. ``` Criticla point: intersections of two fold curves: ```math \left\{ \begin{aligned} g(x,y,z) = c \\ \frac{\partial g(x,y,z)}{\partial \alpha} = 0 \\ \frac{\partial g(x,y,z)}{\partial \beta} = 0 \end{aligned} \right. , \quad\alpha, \beta \in \{x,y,z\}, \alpha \neq \beta. ``` !page #### Construct the surface in a cell <div class="row"><div class="col-lg-5"></div><div class="col-lg-5"></div></div> <div class="row"><div class="col-lg-6 text-center">(a)Computing the intersection points and extreme points on the faces </div><div class="col-lg-6 text-center">(b)Connecting intersection points and extreme points, form a closed loop </div></div> <div class="row"><div class="col-lg-5"></div><div class="col-lg-5"></div></div><div class="row"><div class="col-lg-6 text-center">(c) Computing the fold curves and critial points </div><div class="col-lg-6 text-center">(d) divide into single-valued pieces through the fould curves </div></div> !page #### Triangulation by Ear Clipping  !page ## Meshing results <div class="row"><div class="col-lg-4"></div><div class="col-lg-6"></div></div> <div class="row"><div class="col-lg-5 text-center">(a)</div><div class="col-lg-6 text-center">(b)</div></div> =!= ** Surface meshes of molecules 2JM0 (589 atoms) and 30S ribosome (88431 atoms)** !page ### Computational efficiency =!= ** CPU times using TMSmesh、TMSmesh 2.0 and MSMS **  Two vertex densities: 1/ Å$$^2$$ and 2/ Å$$^2$$. Run on 64 bit linux machine of Intel$$^\circledR$$ Xeon$$^\circledR$$ CPU E5-4650 v2 2.4GHz and 126 GB memory . !page ##=!= Meshing for membrane-protein system - Difficulty: distinguish the tetrahedra in membrane region and pore region, which may be connected by holes or crevices. <div class="row"> <div class="col-lg-5"></div> <div class="col-lg-6"></div> </div> !page ##=!= 分子形状的稀疏表示 #### 能用更少的“原子”来近似一个分子吗？ <br/>  <font size="4"> Sheng Gui, Zhaodi Chen, Benzhuo Lu, Minxin Chen. Molecular Sparse Representation by 3D Ellipsoid Radial Basis Function Neural Network via L1 Regularization, ***J. Chemical Information and Modeling***, 2021 （in press）https://pubs.acs.org/doi/10.1021/acs.jcim.0c00585 </font> !page ##=!= Background - 几何形状： **连续介质模型**，分子模拟，对接，化学信息中的面积体积空腔等，结构比对，形状比较分析 ... <div class="row"><div class="col-lg-6 text-center">A real view of protein molecule</div><div class="col-lg-6 text-center">Surface mesh</div></div> <div class="row"><div class="col-lg-6"><iframe frameborder="no" border="0" marginwidth="0" marginheight="0" width=100% height=400px src="/scene?id= 59f83a764f022e388f25e817"></iframe></div><div class="col-lg-6"><iframe frameborder="no" border="0" marginwidth="0" marginheight="0" width=100% height=400px src="/scene?id= 59fbc6524f022e672773978b"></iframe></div></div> !page ### Molecular "density" map and Gaussian surface Molecular "density" map can be simulated as a summation of a set of Gaussian kernel functions, ```math \phi ( \mathbf{x}) = \sum \limits _{i = 1}^N e^{-d(\| \mathbf{x} - \mathbf{x}_i \|^2 - r_i^2)}, ``` the Gaussian surface is defined as a level set of the density function, ```math \left\{ \mathbf{x} \in \mathbb{R}^3,\phi \left( { \mathbf{x}} \right) = c \right\}, ``` where the parameter $$d$$ is positive and controls the decay rate of the kernel functions, $$\mathbf{x}_i$$ and $$r_i$$ are the location and radius of atom $$i$$, $$c$$ is the isovalue, and it controls the volume enclosed by the Gaussian kernel. While $$|d|$$ is increased, the Gaussian surface is closer to the VDW surface.  !page ### 曲面的稀疏表示 （用径向基函数 RBF） Li M, Chen F, Wang W, et al. Sparse RBF surface representations [J]. *Computer Aided Geometric Design*, 2016, 48:49-59 <br> - 想法： (1)优化RBF 的系数, (2)RBF的中心放置在中轴线上  - 算法流程 （内部中轴点红色，外部中轴点蓝色）  !page ### 椭球径向基函数神经网络 (Ellipsoid RBF Neural Network) - **Structure of the ellipsoid RBF neural network** - an input layer - a hidden layer with a nonlinear activation function (**Ellipsoid RBF**) - a linear output layer The ellipsoid RBF neural network $$\Psi(\cdot): \mathbb{R}^3 \rightarrow \mathbb{R}$$ is as follows. ```math \Psi(\mathbf{x}) = \sum \limits _{i = 1}^{N} w_i \psi_i(\mathbf{x}) = \sum \limits _{i = 1}^{N} w_i e^{-\|\mathbf{D}_i^{1/2}\Theta_i(\alpha_i, \beta_i, \gamma_i)\left(\mathbf{x} - \mathbf{c}_i\right)\|^2} ``` where $$\psi_i(\cdot)$$ is nonlinear activation function(ellipsoid RBF) of $$i$$th hidden neuron, $$\mathbf{c}_i = \left[c_{i1}, c_{i2}, c_{i3}\right]^{\top} \in \mathbb{R}^3$$ is the $$i$$th ellipsoid RBF center of hidden layer, $$\mathbf{D}_i = diag(d_{i1},d_{i2},d_{i3})$$ represents the lengths of corresponding ellipsoid RBF along three main axes of hidden layer, $$\Theta_i(\alpha_i, \beta_i, \gamma_i)$$ is a rotation matrix of the $$i$$th neuron. $$w_i$$ is the output weight between the $$i$$th hidden neuron and the output node. And $$||\cdot||$$ is the $$L_2$$ norm of vector.  =!= **Structure of the ellipsoid RBF neural network** !page ### Our model ```math \begin{array}{c} \mathop {\min }\limits_{\mathbf{X}} {\rho_s} \cdot E_s(\mathbf{X}) + {\rho_l} \cdot E_{l1}\left(\mathbf{w}, \mathbf{d}\right)\\ s.t.\left\{ \begin{array}{ll} w_i \ge 0 \qquad &i = 1,2,\cdots,N, \\ d_{i} \ge 0 . \end{array} \right. \end{array} ``` $$E_s(\mathbf{X})$$ is the error between $$\Psi$$ and $$\phi$$ at constrained points $$\mathbf{y}_k,k=1,2, \cdots, M$$. $$\Psi$$ is a summation of (ellipsoid) Gaussian RBFs. $$\phi(\mathbf{x})$$ is the implicit function in the definition of Gaussian molecular surface. It is to be approximated by $$\Psi(\mathbf{x})$$. ```math E_s(\mathbf{X}) = \sum \limits _{k=1}^{M}\left[\Psi(\mathbf{y_k};\mathbf{X}) - \phi(\mathbf{y_k})\right]^{2} = \sum \limits _{k=1}^{M}\left[ \sum \limits _{i = 1}^N w_i e^{-d_i||\mathbf{y}_k - \mathbf{c}_i||^2} - \phi(\mathbf{y_k})\right]^{2}, ``` The $$\mathbf{X}$$ is the optimization variable, the formula is ```math \mathbf{X} = \left[\mathbf{w}, \mathbf{d}, \mathbf{c}\right]^\top, i = 1,2,\cdots,N, ``` where $$\mathbf{w}=\left[w_{1}, w_{2}, \cdots, w_{N}\right]$$, $$\mathbf{d}=\left[d_{1}, d_{2}, \cdots, d_{N}\right]$$, $$\mathbf{c}=\left[\mathbf{c}_{1}, \mathbf{c}_{2}, \cdots, \mathbf{c}_{N}\right]$$. The second term of objective function is the sparse $$L_1$$ norm of $$\mathbf{d}$$ and $$\mathbf{c}$$. ```math E_{l1}(\mathbf{w},\mathbf{d}) = \|\mathbf{w}\|_1 + \|\mathbf{d}\|_1 = \sum \limits _{i = 1}^N |w_i| + \sum \limits _{i = 1}^N |d_{i}| . ``` The $$\rho_s > 0$$ and $$\rho_l > 0$$ are parameters which balances the two targets: accuracy of solutions $$E_s$$ and sparsity $$E_{l1}$$. - If using ellipsoid RBF neural network, the model is: ```math E_s(\mathbf{X}) = \sum \limits _{k=1}^{M}\left[\Psi(\mathbf{y_k};\mathbf{X}) - \phi(\mathbf{y_k})\right]^{2} = \sum \limits _{k=1}^{M}\left[ \sum \limits _{i = 1}^N w_i e^{-\|\mathbf{D}_i^{1/2}\Theta_i(\alpha_i,\beta_i,\gamma_i)(\mathbf{y}_k - \mathbf{c}_i)\|^2} - \phi(\mathbf{y_k})\right]^{2}, ``` And the second term of objective function is the sparse $$L_1$$ norm of $$\mathbf{d}_p$$ and $$\mathbf{w}$$. ```math E_{l1}(\mathbf{w},\mathbf{d}_p) = \|\mathbf{w}\|_1 + \sum \limits _{p = 1}^3 \|\mathbf{d}_p\|_1 = \sum \limits _{i = 1}^N |w_i| + \sum \limits _{i = 1}^N |d_{i1}| + \sum \limits _{i = 1}^N |d_{i2}| + \sum \limits _{i = 1}^N |d_{i3}|. ``` !page ### Optimization flow chat <br/>  !page ### Sparse optimization algorithm  !page =!= 粗粒化的优化过程 <video src='https://data.xyzgate.com/794040cabb410b0ccdda4dad521a6308.mp4 'controls='controls'/> !page ### Results  !page <br/>  !page =!= Sparse optimization results | PDBID | NATOM | SparseRBF(Medial Axis) | SparseRBF(Atom Center) | **Our Method** | | :------------: | :------------: | :------------: | :------------: | :------------: | | DIALA | 20 | 249 | 13 | **4** | | ADP | 39 | 690 | 8 | **7** | | 2LWC | 75 | 792 | 51 | **11** | | 3SGS | 94 | 1910 | 56 | **17** | | 1GNA | 163 | 1376 | 108 | **28** | | 1V4Z | 266 | 2629 | 198 | **41** | | 1BTQ | 307 | 2271 | 252 | **54** | | 6BST | 478 | 2702 | 316 | **84** | | 1MAG | 552 | 2959 | 502 | **87** | | 1BWX | 643 | 4555 | 537 | **123** | | FAS2 | 906 | 3667 | 722 | **142** | | 3SJ4 | 1283 | 4335 | 953 | **233** | | 3LOD | 2315 | 4706 | 1810 | **530** | | 1RMP | 3514 | 4822 | 2871 | **701** | | AChE | 8280 | 9365 | 4438 | **1636** | !page =!= Number of RBF vs #. atoms <div class="row"><div class="col-lg-6"></div><div class="col-lg-6"></div> !page =!= Shape preservation and further results analysis  !page =!= Volume/area vs. No. atoms <div class="row"><div class="col-lg-6"></div><div class="col-lg-6"></div> </div> !page ### Solvation Energy - Solving the Poisson-Boltzmann equation ```math -\nabla \cdot \varepsilon \nabla \phi -\lambda \displaystyle\sum_j c_j q_j e^{-\beta q_j \phi} - \displaystyle\sum_k q_k \delta(r-r_k)=0 ``` | PDBID | Natom | Original Energy（kcal/mol） | Sparse Energy （kcal/mol） | Relative Error | | ------------ | :------------: | :------------: | :------------: | :------------: | | ADP | 39|-225.992 |-230.075| 0.0181| | 2FLY |355|-238.927 |-242.670 | 0.0157| | 6BST |478|-916.715 |-920.137|0.0037| | 2O3M |714| -3034.82| -3056.04 |0.0070| | 2IJI |929| -659.502 |-665.894 |0.0097| !page ##=!= Applications ### 1. Poisson-Boltzmann静电计算 (AFMPB：自适应快速多极子边界元PB求解器)  ==! B Zhang, JF Huang,BZ Lu et al, *Computer Phys Commun*, 190: 173, 2015; *Commun in Comput Phys*, 25: 1235, 2019 !page ### 2. 有限元计算离子通道的导电性 - Poisson-Nernst-Planck 方程组: ```math \begin{cases} \frac{\partial c_i(r,t)}{\partial t}=\nabla \cdot {D_i(\nabla c_i + \beta c_i \nabla(q_i \phi))} + R({\bf c},r,t),\ i=1,\cdots,K,\ \\ \nabla \cdot \varepsilon \nabla \phi(r,t)=-\rho^f(r) -\lambda \displaystyle\sum q_i c_i(r,t) \end{cases} ```  =!= (a) Fuel cell (b) Nanofluidic channel (c) Ion channel =!= 燃料电池、纳米流、纳米孔、离子通道、半导体器件等 !page - K+ channel: (KcsA structure, PDB code 1BL8) ------ MacKinnon et al, *Science.* 280:69, 1998 <div class="row"><div class="col-lg-10"></div><div class="col-lg-2"></div></div> =!= Molecular surface and pore radius of the KcsA channel Xuejiao Liu and Benzhuo Lu, Incorporating Born Solvation Energy into the 3D Poisson-Nernst-Planck Model to Study Ion Selectivity in KcsA K+ Channels, *Phys. Rev. E*, 96: 062416, 2017 !page ####=!= I-V curves (from Born-PNP model)  =!= $$c_{Na}^{b}$$ = 0.1 M, $$c_{K}^{b}$$ = 0.1 M, and $$c_{Cl}^{b}$$ = 0.2 M - Selective binding and permeation of K+ over Na+ - Inner current rectification !page ### 3. 纳米孔基因测序 （求解离子输运的 PNP 方程组） =!= A dsDNA in a 4-nm-diameter nanopore. <div class="row"><div class="col-lg-5"></div><div class="col-lg-6"></div></div> ==! B. Tu, SY Bai, BZ Lu, QF Fang, *Scientific Reports*, 8:9097, 2018 !page =!= Influence of membrane thickness <div class="row"><div class="col-lg-6"></div><div class="col-lg-6"></div><div class="col-lg-6 text-center"></div><div class="col-lg-6 text-center"></div></div> !page - Conic pore works better! =!= Influence of conic pore diameter and shape   Open-pore currents Is and current amplitudes ∆Is with 20-pb dsDNA in the conical nanopores. (2r = 4 nm) !page <br> <br> #=!= **Thanks!** <br> <br> ###Collaborators: ####陈旻昕（苏州大学），刘田田，桂升，涂斌，刘雪娇，白石阳，张林波，Jingfang Huang（UNC）等 <br> !page ##=!= 讨论 - （1）什么是好的问题？ - （2） 分子几何的其它问题 - （3） 解方程，比如，不同尺寸的物理系统对数值求解 PNP 的影响 计算区域: 立方体 $$[-\frac{L}{2},\frac{L}{2}]^3$$. ```math \begin{aligned} -\nabla \cdot\epsilon\nabla u =\widetilde{\lambda}(c_p-c_n), \quad\quad in\ \Omega,\\ \nabla \cdot D_p(\nabla c_p + c_p \nabla u)+f_p = 0 ,\quad\quad in\ \Omega,\\ \nabla \cdot D_n(\nabla c_n - c_n \nabla u)+f_n = 0 ,\quad\quad in\ \Omega,\\ u = u(\vec{r}), \quad\quad on\ \Gamma_D,\\ c_p = c_{p}(\vec{r}), \quad\quad on\ \Gamma_D,\\ c_n = c_{n}(\vec{r}), \quad\quad on\ \Gamma_D. \end{aligned} ``` 其中 $$\widetilde{\lambda}=\frac{e_c^2 \beta}{\epsilon_0}$$, $$f_p=-\nabla \cdot D_p(\nabla c_{p}^\* + c_{p}^\* \nabla u^\*)$$, $$f_n=-\nabla \cdot D_n(\nabla c_{n}^\* - c_{n}^\* \nabla u^\*)$$. **解析解:** ```math \begin{aligned} u^*&=&u_a^*+u_b^*\\ u_a^*&=&\lambda\cos{\frac{\pi x}{L}}\cos{\frac{\pi y}{L}}\cos{\frac{\pi z}{L}}\\ u_b^*&=&u^{bulk}\frac{z+L/2}{L}\\ c_{p}^*&=&c^{bulk}+\frac{c^{bulk}}{2}\cos{\frac{\pi x}{L}}\cos{\frac{\pi y}{L}}\cos{\frac{\pi z}{L}}\\ c_{n}^*&=&c^{bulk}-\frac{c^{bulk}}{2}\cos{\frac{\pi x}{L}}\cos{\frac{\pi y}{L}}\cos{\frac{\pi z}{L}} \end{aligned} ``` 其中 $$u^{bulk}=e_c\beta V_m$$, $$\lambda=\frac{L^2}{3\pi^2\epsilon}c^{bulk}\widetilde{\lambda}$$. ```math \begin{aligned} \nabla u^*= \left(\begin{array}{c}{ -\lambda \frac{\pi}{L}\sin(\frac{\pi x}{L})\cos(\frac{\pi y}{L})\cos(\frac{\pi z}{L})} \\ { -\lambda \frac{\pi}{L}\cos(\frac{\pi x}{L})\sin(\frac{\pi y}{L})\cos(\frac{\pi z}{L})} \\ { -\lambda \frac{\pi}{L}\cos(\frac{\pi x}{L})\cos(\frac{\pi y}{L})\sin(\frac{\pi z}{L})+\frac{u^{bulk}}{L}} \end{array}\right) \end{aligned} ``` ```math \begin{aligned} \nabla c_{p}^*= -\nabla c_{n}^*= \left(\begin{array}{c}{ -\frac{c^{bulk}}{2} \frac{\pi}{L}\sin(\frac{\pi x}{L})\cos(\frac{\pi y}{L})\cos(\frac{\pi z}{L})} \\ { -\frac{c^{bulk}}{2} \frac{\pi}{L}\cos(\frac{\pi x}{L})\sin(\frac{\pi y}{L})\cos(\frac{\pi z}{L})} \\ { -\frac{c^{bulk}}{2} \frac{\pi}{L}\cos(\frac{\pi x}{L})\cos(\frac{\pi y}{L})\sin(\frac{\pi z}{L})} \end{array}\right) \end{aligned} ``` $$L=20$$(mesh unit) 如果 $$V_m=1V$$, 有$$u^{bulk}=e_c \beta V_m = 3.894\times10^1 \times 1(V_m \text{的数值})$$. 如果$$c^{bulk}=1M$$, $$c^{bulk}=1 \times \frac{N_A}{dm^3}=1\times \frac{10^3 N_A}{m^3}=1\times 6.022\times 10^{26}\frac{1}{m^3}$$ $$\widetilde{\lambda}=\frac{e_c^2 \beta}{\epsilon_0}=\frac{(1.602\times 10^{-19})^2\times 2.431\times 10^{20}}{8.854\times 10^{-12}}=7.046\times10^{-7}m$$ $$\lambda=\frac{L^2}{3\pi^2\epsilon}c^{bulk}\widetilde{\lambda}=\frac{400(L\text{的数值})^2} {3\pi^2\times 80(\epsilon \text{的数值})}\times 1\times 6.022\times 10^{26}\times 7.046\times10^{-7} =7.164\times 10^{19}\frac{\text{mesh unit}^2}{m^2}$$. $$\lambda \frac{\pi}{L} = 1.125 \times 10^{19}\frac{\text{mesh unit}^2}{m^2}\frac{1}{\text{mesh unit}}$$. !page <br> <br> #=!= **Thanks!** <br> <br> !page  !page !page Refs: Lu, BZ, Cheng, XL, Huang, JF, and McCammon, JA. *Proc. Natl. Acad. Sci. USA*, 103: 19314, 2006. BZ Lu, XL Cheng, JF Huang, JA McCammon, *Computer Physics Communications*, 181 (2010) 1150 Bo Zhang, Jingfang Huang,Benzhuo Lu et al, *Computer Physics Communications*, 190: 173, 2015 - ### Source code: **AFMPB**: http://cpc.cs.qub.ac.uk/summaries/AEGB_v1_0.html **pAFMPB**: http://cpc.cs.qub.ac.uk/summaries/AEGB_v1_1.html (共享内存) **pAFMPB2**: （submitted，可达上万核并行） （分布式内存） - TMSmesh, SMOPT, AFMPB: xyzgate.com !page ##Born energy-modified PNP model (BPNP) XJ Liu and BZ Lu, *Phys. Rev. E, 96: 062416*, 2017 ```math -\nabla\cdot(\epsilon(r)\nabla\phi) = \rho^{f} + \sum_{i=1}^{K}q_{i}c_{i} \quad in \;\Omega, ``` ```math \frac{\partial c_{i}}{\partial t}=\nabla \cdot (D_{i}[\nabla c_{i}+\beta c_{i}\nabla(q_{i}\phi + \alpha\frac{q_{i}^{2}}{2a_{i}}(\frac{1}{\epsilon(r)}-\frac{1}{\epsilon_{0}}))]), \quad in \;\Omega_{s}, i=1, 2, \cdots, K. ``` !page ### Molecular surface mesh generation **Softwares**: SMART (Zauhar, 1995), SIMS (Vorobjev and Hermans 1997), NURBS (Bajaj et al, 1997), LSMS (Can 2006), GAMer (Yu Z et al, ), Skin surface (Edelsbrunner, Shi et al, 1996, 2011), MSMS (Sanner M, 1996) **But, still a challenge for mathematical modeling**: - Quality: -- fidelity (to the original geometry)   X -- manifold mesh   X -- uniformness - Robustness: handle arbitrary molecule?   X - Efficiency   X !page ### Non-manifold mesh  !page ### Convergences of mesh property (area and volume) <div class="row"><div class="col-lg-4"></div><div class="col-lg-4"></div><div class="col-lg-4"></div><div class="col-lg-4 text-center">(a)</div><div class="col-lg-4 text-center">(b)</div><div class="col-lg-4 text-center">(c)</div></div> <div class="row"><div class="col-lg-4"></div><div class="col-lg-4"></div><div class="col-lg-4"></div><div class="col-lg-4 text-center">(d)</div><div class="col-lg-4 text-center">(e)</div><div class="col-lg-4 text-center">(f)</div></div> !page - (2) trace the surface and find all topological details -- and connect points to form polygons; Using predict-correct algorithm  Connect points  The surface is further dissected into single valued pieces to guarantee a Manifold Mesh  - (3) triangulation =!= Chen MX, Lu BZ, *J Chem Theory Comput*. 2011, 7, 203–212; *J Mol Graphics Modelling*, 38: 411-418, 2012. !page ### Computational efficiency of TMSmesh2.0  !page ### Manifoldness =!= ** Number of nonmanifold errors in meshes produced by TMSmesh 2:0 and MSMS**  !page ### Quality improvements - Remove redundant triangles with tiny area - Smoothing !page  Walk-and-detect algorithm  !page ### Volume mesh <div class="row"><div class="col-lg-6"></div><div class="col-lg-6"></div></div><div class="row"><div class="col-lg-6 text-center">(a)</div><div class="col-lg-6 text-center">(b)</div></div> =!= **Surface mesh and volume mesh of FAS2 molecule** !page  !page =!= Shape preservation and further results analysis  !page =!= Sparse optimization | Algorithm 1 | | :------------ | | **Input**: PQR file containing coordinates of centers and radii of atoms| | **Output**: The list of parameters of ellipsoid RBFs| | **Step** 1. initialize optimization variable $$\mathbf{X}$$ | | **Step** 2. select constrained points $$\mathbf{y}_k$$ | | **Step** 3. set the number of maximum iteration $$\tt MaxNiter$$ and number of sparse optimization iteration $$\tt SparseNiter$$ | | **Step** 4. initialize variable of iteration and set tolerance: $$\tt Niter = 0, \tt tol =1e-3$$ | | **Step** 5. optimization of $$E_s$$ and $$E_{l1}$$ alternatively | | $$\ \ \ \ \ \ \ \ $$ **WHILE** $$\tt Niter < \tt MaxNiter$$ **do**| | $$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \tt Niter = Niter + 1$$| | $$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $$ **Step** 5.1 delete useless Gaussian basis function $$\tt abs(c_i) < \tt tol$$ every 20 steps | | $$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $$ **Step** 5.2 calculate $$\tilde{\phi}(\mathbf{y}_{k})$$ for all constrained points by $$\mathbf{X}$$ | | $$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $$ **Step** 5.3 calculate the accuracy term $$E_{s}$$ and sparse term $$E_{l1}$$ | | $$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $$ **Step** 5.4 calculate the adaptive coefficients $$w_s$$ and $$w_l$$ | | $$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $$ $$w_{s}=\max \left\\{\frac{E_{s}}{E_{s}+E_{l 1}}, \varepsilon\right\\}$$, $$w_{l}=\frac{E_{l 1}}{E_{s}+E_{l 1}}$$ | | $$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $$ **Step** 5.5 check the maxium of error between $$\tilde{\phi}$$ and $$\phi$$ and correct the coefficients| | $$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $$ **IF** $$\mathop {\max } _{1 \leq k \leq M} abs(\tilde{\phi}(\mathbf{y}_k) - \phi(\mathbf{y}_k)) > 0.5$$| | $$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $$ $$w_s= 1$$,$$w_l = 0$$| | $$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $$ **ENDIF** | | $$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $$ **Step** 5.6 accucacy optimiztion for $$E_s$$ by set coefficients $$w_s$$ and $$w_{l}$$| | $$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $$ **IF** $$\tt Niter > SparseNiter$$| | $$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $$ $$w_s= 1$$,$$w_l = 0$$| | $$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $$ **ENDIF** | | $$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $$ **Step** 5.7 calculate the gradient of object function| | $$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $$ $$\nabla f = \nabla \cdot (w_s \cdot E_{s} + w_{l} \cdot E_{l1})$$ | | $$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $$ **Step** 5.8 updated $$\mathbf{X}$$ ($$X=X-\tau\nabla f$$), $$\tau$$: step for inexact line search of $$-\nabla f$$| |$$\ \ \ \ \ \ \ \ \ \ \ \ $$**ENDWHILE**| !page - Surface potential of dengue virus System: ~$$10^6$$ atoms Mesh: ~$$10^7$$ nodes， $$2 \times 10^7$$ triangles， Edge length: ~ 1 \AA  ==! B Zhang, JF Huang,BZ Lu et al, *Computer Phys Commun*, 190: 173, 2015; *Commun in Comput Phys*, 25: 1235, 2019 !page =!= K+ and Na+ concentrations along z-axis (fixed membrane voltage: $$V_0 = -0.20 V$$)  中轴可以近似看做一种2D骨架表达方式，中轴包含了描述曲面形状的所有几何拓扑信息。 <font size="3">[2] T. Liao, G. Xu, and Y. J. Zhang, Atom simplifcation and quality T-mesh generation for multi-resolution biomolecular surfaces, *in Isogeometric Analysis and Applications 2014, Lecture Notes in Computational Science and Engineering*, 2015 </font>
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bzlu 2020-11-13 17:44:03