生化-扩散模拟
幻灯模式
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bzlu 2022-02-24 00:36:41
#=!= 有限元模拟3D离子通道/纳米孔等的静电作用和导电性质 <br> ####=!= 卢本卓 ####=!= 科学院计算数学与科学工程计算研究所 ####=!= 国家数学与交叉科学中心 <br> <br> ####=!= 北航,2019/6/26 !page ## 内容 - 介绍 - 静电及扩散模型 - 有限元模拟: -- 生物分子表面/立体网格生成 -- 有限元计算 -- 应用举例 -- 钾离子通道的选择性通透模拟 -- 纳米管的电流特性及基因测序 -- 酶-底物扩散反应速率计算 - 可视化及在线计算 !page ##=!= Background <div class="row"><div class="col-lg-6 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-6">![](http://data.xyzgate.com/1193ce98688a0603c4ccdccd9cded64e.png)</div><div class="col-lg-6">![membrane channel](http://data.xyzgate.com/3b8f1307a38700fd1b3f468a63d30889.jpeg "membrane channel")</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 ## All-atom MD simulation ![](http://data.xyzgate.com/c387b5b9037f19b17dd769b384210dcf.png) !page ## Multi-scale and multi-physics Issues ![](http://data.xyzgate.com/8a7161214c1bd5fcc7b18208b9862af7.png) Model: not idea gas For ionic solution, PB-like theories fail for > 0.1 M. Thank for the slide to Bob Eisenberg !page ## Methods for Modeling of Ion Permeation ![](http://data.xyzgate.com/193a8d82d609bc750acc4d5601bb5435.png) !page ##=!= Continuum models **Poisson-Nernst-Planck** (非平衡): ```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 + \mu^{ex}_i))} ,\ i=Na^+, Cl^-, K^+, ... \ \\ \nabla \cdot \varepsilon \nabla \phi(r,t)=-\rho^f(r)-\lambda \displaystyle\sum q_i c_i(r,t) \end{cases} ``` **Poisson-Boltzmann 方程** (平衡态): ```math -\nabla \cdot \varepsilon \nabla \phi -\lambda \displaystyle\sum_j c_j^b q_j e^{-\beta q_j \phi} - \rho^f=0 ``` <div class="row"><div class="col-lg-5">![](http://data.xyzgate.com/391214b1ad3d1fcc7ad4308cb386000c.png )</div><div class="col-lg-4">![](http://data.xyzgate.com/6e6bc51318a8d619cd85c0a824f5e298.png )</div><div class="col-lg-3">![](http://data.xyzgate.com/8155a2f609aacace37975c8291c71d3c.png)</div></div> !page ###=!= PNP模型用于其它领域: 燃料电池、纳米孔、半导体等 #### (a) Fuel cell (b) Nanofluidic channel (c) Ion channel ![](http://data.xyzgate.com/9f4a475ce442f40965ad548a1d3ffd1c.png) !page <div class="row"><div class="col-lg-6">纳米孔</div><div class="col-lg-6">MOSFET-SiO2</div></div> <div class="row"><div class="col-lg-6">![](http://data.xyzgate.com/d989e4054df959f8e1be90169e8d4065.png)</div><div class="col-lg-6">![](http://data.xyzgate.com/51ad3f8f1ae5152b1b253c6a615e5023.png)</div></div> !page ##=!= Numerical methods - 有限差分 - 边界元 - 有限元 ![FD-BEM-FEM](http://data.xyzgate.com/6d45a0a470a0e7544d1be43cf0d20a35.png "FD-BEM-FEM") !page ##=!= Molecular mesh generation <div class="row"><div class="col-lg-6">![](http://data.xyzgate.com/1a43bce46d141a1b69b519a69952931e.png)</div><div class="col-lg-6">![](http://data.xyzgate.com/999ef4e0e78705314baff19767dc0f8e.png)</div></div> !page ##=!= TMSmesh - Generation of points <div class="row"><div class="col-lg-6">![](http://data.xyzgate.com/9189a8e0d43ba76fb3b8dc4febe4acd8.png)</div><div class="col-lg-6">![](http://data.xyzgate.com/bdb3fb7d186e11697bf8fcdfa6f1eb4c.png)</div></div> ==! M. X. Chen and B. Z. Lu. *J. Chem. Theory Comput.*, 7:203-212, 2011. ==! TT Liu, MX Chen, BZ Lu, *SIAM J Sci. Comput*, 40:B507-B527, 2018 !page - Find local geometry and triangulation ![](http://data.xyzgate.com/233b59d91e407e183e720f2b3963bdb4.png) ![](http://data.xyzgate.com/f446f48af89489938e4f0c8f9ec84327.png) !page Dengue Virus ![](http://data.xyzgate.com/32752b672c50d640d1be6be23d2f25fc.png) !page ###=!= AFMPB (Adaptive fast multipole PB solver) #####=!= Surface potential of dengue virus - System: ~$$10^6$$ atoms $$\ \ \ \ \ \ \ \ \ \ \ \ \ \ $$ Mesh: ~$$10^7$$ nodes, $$2 \times 10^7$$ triangles, Edge length: ~ 1 \AA - **用最新的并行快速边界元求解程序 pAFMPB,3位精度,12288核 可在几秒内完成计算!** ![](http://data.xyzgate.com/d86f3f6fd5cda43d1ea5f8eeee4c6036.png) !== Bo Zhang, Jingfang Huang,Benzhuo Lu et al, *Computer Physics Communications, 190: 173, 2015* !== B. Zhang, J. DeBuhr, D. Niedzielski, S. Mayolo, B. Lu, T. Sterling, DASHMM Accelerated Adaptive Fast Multipole Poisson-Boltzmann Solver on Distributed Memory Architecture, *Computer Phys Commun.* 25: 1235, 2019 !page ###=!= Volume mesh generation - Using Tetgen to generate volume mesh based on TMSmesh surface mesh - Sometime it needs to further reduce/smooth the surface mesh before volume mesh generation ![](http://data.xyzgate.com/6b5e125e70b95ff6d89bc06db9f91e81.png) !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-6">![](http://data.xyzgate.com/b7e126d09612416c42f1a25eb95776f9.png)</div><div class="col-lg-6 text-center"> Walk-and-detect algorithm ![](http://data.xyzgate.com/569a76aa36eb5d7dc8162696e0995f78.png)</div></div> !page <iframe width=100% height=600 src="https://xyzgate.com/ngldatashow?id=5b6b8df84f022e24b055d977#"></iframe> !page ##Connexin (Cx26) <div class="row"><div class="col-lg-6">![](http://data.xyzgate.com/6a5af8351ad734e438ebbcbfe7dad2a1.png)</div><div class="col-lg-6">![](http://data.xyzgate.com/562252ebdc0ef79d325b4099ea681eed.png)</div></div> !page ##=!= Finite element solution - Parallel adaptive Finite element method (based on PHG) - Interface problem (jump dielectric coefficient) by using conforming mesh (body-fitted mesh generation) - Singular charges treatment using potential decomposition technique - Solving the coupled systems -- Gummel iteration (decoupled approach, relaxation)/Newton method (coupled approach) -- Continuation - Initial guess - Stablization method (SUPG) Zhang LB, Numer. Math. Theor. Meth. Appl. 2, 65 (2009). Cheng IL et al. 2003 JH Chaudhry, J Comer, A Aksimentiev, LN Olson, Commun. Comput. Phys. 15: 93, 2014 Lu BZ, et al. 2007; 2008; 2009; 2010; 2011; 2013,2015 !page ##=!= 应用研究 - **有限元模拟离子通道, 软件ichannel** (可达 上千至上万核 并行!) ==! Bin Tu, Minxin Chen, Yan Xie, Linbo Zhang, Bob Eisenberg, and Benzhuo Lu, *J. Comput. Chem.* 34: 2065, 2013. <iframe frameborder="no" border="0" marginwidth="0" marginheight="0" width=100% height=500px src="https://data.xyzgate.com/65b1d5677c801cea93818d386bbd00f0.mp4"></iframe> !page ### Simulating the current-voltage characteristic of ion channels <div class="row"><div class="col-lg-6" style="text-align:center">VDAC1 dimer</div><div class="col-lg-6" style="text-align:center">α-HL dimer </div></div> <div class="row"><div class="col-lg-6">![](http://data.xyzgate.com/7b5082de43631531e25bffbe1e84c8aa.png)</div><div class="col-lg-6">![](http://data.xyzgate.com/0c79cd214f489cf6a26ce2b14590a602.png)</div></div> !page <div class="row"><div class="col-lg-6" style="text-align:center">VDAC1 dimer</div><div class="col-lg-6" style="text-align:center">α-HL dimer </div></div> <div class="row"><div class="col-lg-6">![](http://data.xyzgate.com/69ba5e901a8ce9d672b8f952572a9a3a.png)</div><div class="col-lg-6">![](http://data.xyzgate.com/c4d4c3bfbfaa77dad20415442b95419c.png)</div></div> =!= BD (black), PNP (blue), SM PNP (red) Lee KI; Jo S; Rui H; Egwolf B; Roux B; Pastor RW; Im W. *J. Comput. Chem. 2012,* 33:331. B Tu, BZ Lu, et al. *J. Comput. Chem.* 2013, 34: 2065 !page ## =!= DNA-nanopore/channel sequencing <div class="row"><div class="col-lg-6">![](http://data.xyzgate.com/5912fc7417876d7847f175428702c76f.png)</div><div class="col-lg-4">![](http://data.xyzgate.com/d01a9ea5977ed9b188230f235115dfa8.png)</div></div> !page <iframe width=100% height=600 src="https://xyzgate.com/pdf?id=5bd903874f022e1d4454e052"></iframe> !page ![](http://data.xyzgate.com/b5ed45342aa8f53971570dd1a2b2db02.png) Concatenated sets of 200 translocations of 3-kb linear dsDNA through 4-nm-diameter pores fabricated in membranes with different h values; heff is the nanopore effective thickness used in the geometric model discussed in the text. On decreasing h from 60 to 6 nm, the open-pore current increased and the DNA signal amplitude increased !page ![](http://data.xyzgate.com/2378246e278aa79a92a2ecedc96df1b5.png) Discrimination among small nucleic acids using thin nanopores. a, Continuous current versus time traces from a 3-nm-diameter pore in a 7-nm-thick membrane measured at 0 C, 500 mV (TEM image of pore is shown). !page ##Cylindrical pore =!= A dsDNA in a 4-nm-diameter nanopore. The SiN membrane is shown in gray <div class="row"><div class="col-lg-5">![](http://data.xyzgate.com/2386ff702b74793988ac1e401bd75d46.png)</div><div class="col-lg-6">![](http://data.xyzgate.com/a5961765fa329d820cb61cd0263609de.png)</div></div> ==! B. Tu, SY Bai, BZ Lu, QF Fang, *Scientific Reports*, 8:9097, 2018 !page ## Results: single strand DNA in a nanotube ![](http://data.xyzgate.com/c1871340883e39ad8ec9001e08b1e014.png) !page ## Results: double strand DNA in a nanotube ![](http://data.xyzgate.com/de2782db594bec24656256cc6d3f9f68.png) !page ## Influence of membrane thickness <div class="row"><div class="col-lg-7">![](http://data.xyzgate.com/e40ced820f46034d6b0d9f09aeccd477.png)</div><div class="col-lg-5">![](http://data.xyzgate.com/f19726fd9222dfa753b436487344a805.png)</div></div> Left: Dependence of average experimental Io (without DNA, black circles) and the most probable DNA current amplitude Ip (with DNA, red triangles) on h. The black dashed line is a fit using equation (1) to the average Io data from the combined data of 20 pores, which yields an effective pore thickness. Right: computational results. !page ##=!= Conic pore works better! ###--Influence of conic pore diameter and shape <div class="row"><div class="col-lg-6">![](http://data.xyzgate.com/df9ac5604fd11f30e49b74b27abbd5d3.png)</div><div class="col-lg-6">![](http://data.xyzgate.com/5135b2ca0e2899606c4c17b035f189ab.png)</div></div> Open-pore currents Is and current amplitudes ∆Is with 20-pb dsDNA in the conical nanopores. (2r = 4 nm) !page ##=!= K+ channel selectivity - Potasium channel: KcsA structure (PDB code 1BL8) ------ MacKinnon et al, *Science.* 280:69, 1998 <div class="row"><div class="col-lg-10">![BPNP kcsa](http://data.xyzgate.com/b095cb620d357471fbc482cb6c347781.png "BPNP kcsa")</div><div class="col-lg-2">![](http://data.xyzgate.com/cac4dce793b34fc485354413e749ce6c.png)</div></div> =!= Molecular surface and pore radius of the KcsA channel - <font color=#A52A2A size> **K+ ~1000 times selective over Na+ ! ---> why ?**</font> !page ## Notes on a class of size modified PB/PNP model(s) I. Borukhov, D. Andelman, H. Orland, PRL. 79:435, 1997 Y. Qiao, XJ Liu, MX Chen, BZ Lu, *J. Statistical Physics*, 163:156-174, 2016. ```math \mu_{i}^{ex} = \beta^{-1} (1-\displaystyle\sum_k a_k^3 c^k(r)) [\log\Lambda^{3}(1-\displaystyle\sum_k a_k^3 c^k(r)) -1] ``` - It is a low order (local) approximation of the Fundermental Measure Theory in classical DFT - In equilibrium --> size-modified PB - In non-equilibrium, --> ** (asymmetric/non-uniform) size-modified PNP** (Lu B. et al. *Biophysical J*. 2011) ```math \nabla \cdot \varepsilon(r) \nabla \phi(r)=-\rho^f(r)-\displaystyle\sum_i q^i c^i(r), \ \ r\in \Omega, ``` ```math -\nabla \cdot \{ {D^i(r) \nabla c^i(r)+ \frac{D^i(r)v^i(r)c^i(r)}{1-\displaystyle\sum_k a_k^3 c^k(r)} \displaystyle\sum_k a_k^3 \nabla c^k(r)+ \beta D^i(r)c^i(r)q^i \nabla \phi(r)} \}=0,\ \ r\in \Omega, \ i=1, \cdots, K, ``` !page - **Conclusion/Observation: The size effects in these models are hard to predict apparant selective transport in a channel** <div class="row"><div text="cndnc" class="col-lg-6">Current in gA channel at 2.0M, 0.5M, 0.1M bulk concentrations(Yu Q., Lu B., *J Chem Phys.* 2014)![](http://data.xyzgate.com/9387ba16625178a943e25ab46daf8ab8.png)</div><div class="col-lg-6">cDFT predicted concentration selectivity of Na+ over K+ with surface potential 0.1V in a nanopore (Yu Q. and Lu B et al. *Chem Phys Lett*, 709:116,2018)![](http://data.xyzgate.com/9ae81268785b5b2ac1edee64c2505fb9.png)</div></div> !page ## Born energy-modified PNP model (BPNP) - Born solvation energy: $$u_i^{ex}= G_{Born}=\frac{q_i^2}{a_i}(\frac{1}{\varepsilon(r)}-\frac{1}{\varepsilon_o})$$ <div class="row"><div class="col-lg-3"> ![](http://data.xyzgate.com/ea41e5b2b3c14a91a44ef3141220aa5d.png)</div><div class="col-lg-8"> ![](http://data.xyzgate.com/8bf93fa3ecb96a2d2127972c16ac7fc4.png)           XJ Liu and BZ Lu, *Phys. Rev. E*, 96: 062416, 2017 </div> !page - BPNP model ```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. ``` - Position-dependent dielectric coefficient ```math \epsilon(r) = \epsilon(z) = \epsilon_{s}(a_{1} + \frac{1.0-a_{1}}{1.0+e^{-\frac{|z|-z_{0}}{\Delta z}}}) ``` ![BPNP epsilon_r](http://data.xyzgate.com/b3aef5b1747e72d09e1226b6c3fec81f.png "BPNP epsilon_r") =!= ** Dielectric coefficient profiles** !page ## 网格 ![](http://data.xyzgate.com/024a4c13552ac663d7a55feced752edd.png) !page ## K+选择性及内向整流 Conditions: Mixed electrolyte: Na+, K+, Cl- c_bulk= 0.1M for Na+ and K+, cbulk= 0.2M for Cl-, radii:a_Na = 1.0A, $$a_K$$ = 1.50A and a_Cl = 2.0A. ![](http://data.xyzgate.com/8e79153e7986e0e204125c21e32b9054.png) =!= Ion distributions under a fixed membrane voltage $$\phi_0 = 0.20V$$ !page ![](http://data.xyzgate.com/f98fd08520fa12d984106a7b4f518675.png) =!= I-V curves !page ###=!= Electrostatic potential energy profiles of BPNP model <div class="row"><div class="col-lg-6">![](http://data.xyzgate.com/f81805072b49e250e36dcd4c457fcaba.png)</div><div class="col-lg-6">![](http://data.xyzgate.com/1f6a9e409576978cf6bd9a3b0f9181c9.png)</div></div> <div class="row"><div class="col-lg-6">![](http://data.xyzgate.com/2a3aaf8772fe1a3316d09fabd6aa5093.png)</div><div class="col-lg-6">![](http://data.xyzgate.com/c8d71c9e747247a8f87354ba01055632.png)</div></div> =!= Membrane voltages: 0.20 V (left) and −0.20 V (right) in the intracellular region !page ##=!= 纳米孔中的周期扫描膜电压(含时模拟):hysteresis 现象 <div class="row"><div class="col-lg-6">![](http://data.xyzgate.com/d948f4c8472b7ba6265b48d0946824bc.png)</div><div class="col-lg-6">![](http://data.xyzgate.com/1705a4aa05bf5b714e0103eafd404f25.png)</div></div> =!= I-V curves !page ##=!= 酶-底物反应速率计算 <div class="row"><div class="col-lg-6">![](http://data.xyzgate.com/7aa3f5f017f286a673d4d5e5a867980c.png)</div><div class="col-lg-6">![](http://data.xyzgate.com/670ca249f6107d73d546f862ae85bbcf.png)</div></div> !page ![](http://data.xyzgate.com/b4b1ff03790ed603fd69cd6485a9117c.png) ![](http://data.xyzgate.com/4f205328f609299ada3c1e22b4a85ecc.png) !page ##=!= 在线计算及可视化平台 =!= xyzgate.com - Programs: **TMSmesh, SMOPT, FEPB, ichannel** 等 - 例子: 静电计算、网格、及可视化 等 ![数字之门二维码](http://data.xyzgate.com/daf69e1e46cc6c03e9bfaf433dccab93.png) !page ###Acknowledgments 计算数学所: 白石阳,谢妍,乔瑜,刘田田,刘雪娇,许竞劼,桂升,马召灿,张波 张林波研究员 陈旻昕 (苏州大学) 涂斌 (国家纳米中心) 经费:CAS, NSFC, NCMIS #Thanks !page !page ## Method in our TMSmesh - Gaussian surface ```math \{\bar{x} \in R^3 | \phi(\bar{x}) =c\} ``` where ```math \phi(\bar{x})=\displaystyle\sum_{i=1}^n e^{-b({\| x-x_i\|}^2 -r_i^2)} ``` !page ## Model extension - **Size effects** (Andelman et al, PRL. 79:435, 1997; Lu B. *Biophysical J*. 2011 ) ```math u_{ex}^{water}=c_0(1-\displaystyle\sum_i c_i a_i^3) ln(c_0(1-\displaystyle\sum_i c_i a_i^3)a_0^3-1) ``` $$\to$$**SMPNP** ```math \nabla \cdot \varepsilon(r) \nabla \phi(r)=-\rho^f(r)-\displaystyle\sum_i q^i c^i(r), \ \ r\in \Omega, ``` ```math -\nabla \cdot \{ {D^i(r) \nabla c^i(r)+ \frac{D^i(r)v^i(r)c^i(r)}{1-\displaystyle\sum_k a_k^3 c^k(r)} \displaystyle\sum_k a_k^3 \nabla c^k(r)+ \beta D^i(r)c^i(r)q^i \nabla \phi(r)} \}=0,\ \ r\in \Omega, \ i=1, \cdots, K, ``` - **Variable dielectric coefficient model ** !page - Born solvation energy ```math u_{ex}:\ G_{born}=\frac{q_i^2}{a_i}(\frac{1}{\varepsilon(r)}-\frac{1}{\varepsilon_o}) ``` $$\to$$ **BPNP** ```math \nabla \cdot \varepsilon(r) \nabla \phi(r)=-\rho^f(r)-\displaystyle\sum_i q^i c^i(r), \ \ r\in \Omega, ``` ```math \nabla\cdot[D^i(r)(\nabla c^i(r)+ \beta c^i(r) \nabla(q_i \phi(r)+ \alpha \frac{q_i^2}{2a_i}( \frac{1}{\varepsilon(r)-\frac{1}{\varepsilon_o}}))=0, \ \ r\in \Omega_s, \ \ i=1, \cdots, K, ``` ==! XJ Liu and BZ Lu, *Phys. Rev. E*, 96: 062416, 2017 ## 介电系数 ![](http://data.xyzgate.com/f7c6d1364160668ed02b9d725019f081.png) !page #=!= 离子、分子系统中的电扩散反应过程模拟 ####=!= 卢本卓 ####=!= 科学院计算数学与科学工程计算研究所, ####=!= 国家数学与交叉科学中心,北京 ####=!= 沈阳,03/22/2018 #=!= 有限元模拟电扩散过程及其在化学生物等领域的应用 ####=!= 科学院理化所,2018 !page
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