生化-扩散模拟
幻灯模式
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bzlu 2019-06-26 09:20:52
#=!= 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"></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 ## All-atom MD simulation  !page ## Multi-scale and multi-physics Issues  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  !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"></div><div class="col-lg-4"></div><div class="col-lg-3"></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  !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"></div><div class="col-lg-6"></div></div> !page ##=!= Numerical methods - 有限差分 - 边界元 - 有限元  !page ##=!= Molecular mesh generation <div class="row"><div class="col-lg-6"></div><div class="col-lg-6"></div></div> !page ##=!= TMSmesh - Generation of points <div class="row"><div class="col-lg-6"></div><div class="col-lg-6"></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   !page Dengue Virus  !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！**  !== 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  !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"></div><div class="col-lg-6 text-center"> Walk-and-detect algorithm </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"></div><div class="col-lg-6"></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"></div><div class="col-lg-6"></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"></div><div class="col-lg-6"></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"></div><div class="col-lg-4"></div></div> !page <iframe width=100% height=600 src="https://xyzgate.com/pdf?id=5bd903874f022e1d4454e052"></iframe> !page  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  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"></div><div class="col-lg-6"></div></div> ==! B. Tu, SY Bai, BZ Lu, QF Fang, *Scientific Reports*, 8:9097, 2018 !page ## Results: single strand DNA in a nanotube  !page ## Results: double strand DNA in a nanotube  !page ## Influence of membrane thickness <div class="row"><div class="col-lg-7"></div><div class="col-lg-5"></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"></div><div class="col-lg-6"></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"></div><div class="col-lg-2"></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)</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)</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"> </div><div class="col-lg-8">            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}}}) ```  =!= ** Dielectric coefficient profiles** !page ## 网格  !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.  =!= Ion distributions under a fixed membrane voltage $$\phi_0 = 0.20V$$ !page  =!= I-V curves !page ###=!= Electrostatic potential energy profiles of BPNP model <div class="row"><div class="col-lg-6"></div><div class="col-lg-6"></div></div> <div class="row"><div class="col-lg-6"></div><div class="col-lg-6"></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"></div><div class="col-lg-6"></div></div> =!= I-V curves !page ##=!= 酶-底物反应速率计算 <div class="row"><div class="col-lg-6"></div><div class="col-lg-6"></div></div> !page   !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  !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  !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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bzlu 2019-06-26 09:20:52