ICCM
幻灯模式
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bzlu 2019-11-15 17:24:36
#=!= Ion conductance in membrane-channel systems: models, meshing, computing, and applications <br> ####=!= Benzhuo Lu (卢本卓) ####=!= Institute of Computational Math and Scientific/Engineering Computing, ####=!= Chinese Academy of Sciences, China <br> ###=!= Beijing, 6/11/2019 !page ###=!= Outline - Background - Models - Molecular meshing - Finite element simulation - Results -- Selectivity -- Rectification -- Gene sequencing using nanopore technology !page ## Background <div class="row"><div class="col-lg-6">![membrane channel](http://data.xyzgate.com/3b8f1307a38700fd1b3f468a63d30889.jpeg "membrane channel")</div><div class="col-lg-6">**细胞内外离子浓度**![](http://data.xyzgate.com/4cd12d3e5c659ce1868e8d4144069ea2.png) </div></div> - **Two feature of ion channel:** ##### gating ##### <font color=#A52A2A size>selective permeation</font> !page - 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 ###=!= A typical continuum model: Poisson-Nernst-Planck equations ```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} ``` <div class="row"><div class="col-lg-5">![](http://data.xyzgate.com/391214b1ad3d1fcc7ad4308cb386000c.png )</div><div class="col-lg-5">![](http://data.xyzgate.com/6e6bc51318a8d619cd85c0a824f5e298.png )</div></div> !page ## A free energy form (for charged system) and derivation of PNP equations - Free energy form ( X. J. Liu, Y. Qiao and B. Z. Lu. *SIAM J. Appl. Math*. 78: 1131, 2018) ```math \begin{aligned} F[c] &= \int_{\Omega}\{\frac{1}{2}\rho(c)\phi(c) + \beta^{-1}\sum_{i=1}^{K}c_{i}[\log(\Lambda^{3}c_{i})-1] + \sum_{i=1}^{K}\mu_{i}^{ex}c_{i} \}dV \\ & + \int_{\Gamma_{N}}\frac{1}{2}\sigma\phi(c)dS -\int_{\Gamma_{D}}\frac{1}{2}\epsilon(c)\frac{\partial \phi(c)}{\partial n}\phi_{0}dS, \end{aligned} ``` $$\phi= \phi(c)$$ satisfy the Poisson equation ```math -\nabla \cdotp (\varepsilon(c)\nabla \phi(c))=\rho(c) \ \ in\ \Omega, ``` ```math \varepsilon(c)\frac{\partial \phi}{\partial n}=\sigma \ \ \ on\ \Gamma_N, ``` ```math \phi=\phi_0 \ \ on \ \Gamma_D. ``` - Derivation Using a **constitutive relation**, the flux is: $$J_i= - D_i c_i \nabla \frac{\delta F}{\delta c_i}$$, and **mass conservation** gives the NP (drift-diffusion) equations: ```math \frac{\partial c_i}{\partial t} = -\nabla \cdot J_i, ``` !page ## Issues - 模型 - 分析 (解的存在唯一性,多解与分叉,渐进性质,整体性质如 I-V特性,等) - 算法 (FD,**FEM**,谱方法,FV,混合等等) -- 分子网格生成 -- Poisson方程中的奇异电荷 -- 电扩散耦合方程 -- NP方程是对流扩散型方程 -- 保物理结构:保正性、流守恒、保能量(能量耗散)、保总质量、保电中性等 -- 高效性、稳定性 !page ## Issues - models - analysis (solution existence and uiqueness, multiple sulotion and bification, aymptotic property, global property as I-V charaterristics ...) - Algorithms (FD,**FEM**,spectral method, FV,hybrid method and so on) -- molecular mesh generation -- Singualr charges in Poisson equation -- Coupled equations -- Convection-dominated diffusion equation -- Structure-preservation:positivity, flux-conservation, energy dissipation, total mass conservation, carge neutrality ... -- Efficiency, robustness !page ## Notes on a class of size modified PB/PNP models ![hard-sphere](http://data.xyzgate.com/979dd6245f8c8980b5169f425571fb04.png "hard-sphere") I. Borukhov, D. Andelman, H. Orland, *PRL*. 79:435, 1997 TL Horng, TC Lin, C. Liu, B Eisenberg, *J. Phys. Chem. B* 116, 11422, 2012 S Ji, W Liu, *J. Dyn. Differ. Equ.* 24, 955, 2012 Y. Qiao, XJ Liu, MX Chen, BZ Lu, *J. Statistical Physics*, 163:156, 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 !page - 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. 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 ## Free energy with Born solvation energy ```math \begin{aligned} F[c] &= \int_{\Omega}\frac{1}{2}\rho(c)\phi(c) dV + \int_{\Gamma_{N}}\frac{1}{2}\sigma\phi(c) dS -\int_{\Gamma_{D}}\frac{1}{2}\epsilon(r)\frac{\partial \phi}{\partial n}\phi_{0}dS \\ &+ \beta^{-1}\sum_{i=1}^{K}\int_{\Omega}c_{i}[\ln(\Lambda^{3}c_{i})-1]dV - \sum_{i=1}^{K}\int_{\Omega}\mu_{i}c_{i}dV\\ &+ \sum_{i=1}^{K}\int_{\Omega}c_{i}\alpha\frac{q_{i}^{2}}{2a_{i}}(\frac{1}{\epsilon(r)}-\frac{1}{\epsilon_{0}})dV, \end{aligned} ``` $$\alpha$$ is an adjustable parameter to control the double count of electrostatic energy. - In equilibrium state, a generalized Boltzmann distribution can be derived: ```math c_i=c_i^b \exp(-\beta \mu_i) = c_i^b \exp \{-\beta q_{i}\phi - \alpha \beta \frac{q_{i}^{2}}{2a_{i}}(\frac{1}{\epsilon(r)}-\frac{1}{\epsilon_{s}})\}. ``` !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 # Mesh generation ### Molecular surface definition - **van der Walls surface**, **solvent-accessible surface **, **solvent-excluded surface(SES)**, **skin-surface**, **minimal energy surface** and **Gaussian surface** ... ![VDW-surface](http://data.xyzgate.com/11b3d3afa4ee10ffdfc12ccc08e6bf27.jpeg "VDW-surface") =!= ** VDW、SAS and SES surfaces** ### Gaussian surface ```math \{ \vec x \in R^3,\phi \left(\vec x \right) = c \},\ \ \ \ \ \ \ \ \ \ \ (1) ``` where ```math \phi \left(\vec x \right) = \sum \limits_{i=1}^{N}e^{-d(\Vert \vec x - \vec x_i \Vert ^2 -r_i^2)},\ \ (2) ``` $$N$$ number of atoms, $$\vec x_i $$ and $$r_i $$ are coordinates and radius of the $$i$$th atom, $$d$$ a parameter controlling decay rate and (the smaller, the smoother and inlated),$$c$$ the isovalue controling how much volume involved. !page ###=!= Gaussian surface mesh by **TMSmesh** !vism 5a5f45a74f022e4c57b1b363 =!= ** Surface mesh of molecule 1MAG** !page #### **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">![](http://data.xyzgate.com/9189a8e0d43ba76fb3b8dc4febe4acd8.png)</div><div class="col-lg-6">![](http://data.xyzgate.com/bdb3fb7d186e11697bf8fcdfa6f1eb4c.png)</div></div> <br> ==! Chen MX, Lu BZ, *J Chem Theory Comput*. 2011, 7, 203–212 !page - Trace the surface and find all topological details. The surface is further dissected into single valued pieces, and then generate manifold triangular mesh ![](http://data.xyzgate.com/233b59d91e407e183e720f2b3963bdb4.png) !page #### **New version TMSmesh2**: Polynomial approxiamtion and adaptive surface partition - Two steps - **1**:**Adaptive estimation and cell partition**. Finally, in each cubic cell, a trilinear approximation is used; - **2**: **trilinear surface** --> single-value pieces along $$x, y, z$$ directions, --> "triangulation"。 ```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) <div class="row"><div class="col-lg-6">![adaptive-cube](http://data.xyzgate.com/fb44826da938907f475bb1375dce5af1.jpeg "adaptive-cube")</div><div class="col-lg-4">![adp-box](http://data.xyzgate.com/7ccea7fab5183c3937ca847e2e6db6d7.png "adp-box")</div></div> ==! TT Liu, M.X. Chen, and B.Z. Lu, *SIAM J Sci. Computing.*, 40:B507-B527, 2018. !page #### Triangulating the trilinear surface ##### 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 #### Triangulation of a trilinear surface <div class="row"><div class="col-lg-5">![point](http://data.xyzgate.com/9c07b05f3c4097b21a9164ce4892d3a6.png "point")</div><div class="col-lg-5">![polygon](http://data.xyzgate.com/63dc3e98923980610f09297c1940dca8.png "polygon")</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">![subdivide](http://data.xyzgate.com/9104c8c1d6ebb79fdce994b4ef09b113.png "subdivide")</div><div class="col-lg-5">![single](http://data.xyzgate.com/4763449aec1fc75ee34457cf9e526388.png "single")</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 | Algorithm 1 | | ------------ | | **input**: polygon vertices, $$\vec V_1, \cdots, \vec V_n$$, $$n$$ is the number of vertices | | **step 1**: store the vertices into a doubly-linked list in sequence. For each vertex $$\vec V_{i}$$, check if <$$\vec V_{i-1}, \vec V_{i}, \vec V_{i+1}$$> is an ear and mark; | | **step 2**:sort the list, and find ear tip $$\vec V_{i}$$, go to step 3. If $$n<3$$, terminate; | | **step 3**:remove vertex $$\vec V_{i}$$ in the list, $$n = n-1$$, and store traingle <$$\vec V_{i-1}, \vec V_{i}, \vec V_{i+1}$$>. Re-check $$\vec V_{i-1}$$ and $$\vec V_{i+1}$$, and mark in the list. | | **output**: set of triangles | ![trianglation](http://data.xyzgate.com/b243f490f2786689f0aa8c7711377f52.jpeg "trianglation") !page ###=!= Volume mesh by Tetgen - 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 <div class="row"><div class="col-lg-6">![fas2-surface](http://data.xyzgate.com/4824f34319012fe93e5931ca23429490.jpeg "fas2")</div><div class="col-lg-6">![fas2-volume](http://data.xyzgate.com/edc8963cbb8315a0317cfab979203bb3.jpeg "fas2-volume") </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 ###=!= 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 <div class="row"><div class="col-lg-4">![](http://data.xyzgate.com/6a5af8351ad734e438ebbcbfe7dad2a1.png)</div><div class="col-lg-6">![](http://data.xyzgate.com/562252ebdc0ef79d325b4099ea681eed.png)</div></div> !page - KcsA mesh <div class="row"><div class="col-lg-6">![](http://data.xyzgate.com/c9b5fcb5cd27ff2fa23e6a3310ad5fdb.png)</div><div class="col-lg-4">![](http://data.xyzgate.com/2bb6a7c20bdc3f9708bfd161770a68b4.png)</div></div> =!= ** A tetrahedral volume mesh and triangular surface mesh.** (total: 102572 vertices and 643832 tetrahedra) !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 ###=!= A SUPG stabilization method The weak form of NP equations: find $$c_i\in H_c^1(\Omega_s)$$,so that ```math A(c_i, v) := \int_{\Omega_s}D_i(\nabla c_i + z_ic_i\nabla u)\cdot\nabla vd\Omega_s = 0,~\forall v\in H_0^1(\Omega_s) ``` Let $$t$$ be an element, a stabilization term is as following: ```math \begin{aligned} S(c_i, v) := \sum\limits_{t}\alpha_2\tau_{i, t}\int_t(-\nabla\cdot D_i(\nabla c_i + z_ic_i\nabla u))(\mathbf{b_i}\cdot\nabla v)dx \end{aligned} ``` where $$\tau_{i, t}$$ is defined as: ```math \tau_{i, t}:=\dfrac{h_t^3}{||\mathbf{b_i}||_{L^2(t)}h_t+D_i} ``` - the stabilization method solve: ```math A(c_i, v) + S(c_i, v) = 0,~\forall v\in H_0^1(\Omega_s) ``` !page ## Simulation results I: K+ channel conductance ###=!= Influence of parameter $$\alpha$$ on cation concentrations Conditions: Mixed electrolyte: Na+, K+, Cl- bulk conc: $$c_{Na}^{b}$$ = 0.1 M, $$c_{K}^{b}$$ = 0.1 M and $$c_{Cl}^{b}$$ = 0.2 M ; radii:$$a_{Na} = 1.62\AA, a_K = 1.95\AA$$ and $$a_{Cl} = 2.26\AA$$. membrane voltage: $$V_0 = -0.2 V$$ ![BPNP alpha](http://data.xyzgate.com/b279e3df222322d9aedfec943a7a48c4.png "BPNP alpha") !page ###=!= Concentration profile =!= K+ and Na+ concentrations along z-axis (membrane voltage: $$V_0 = -0.2 V$$) ![BPNP concentration](http://data.xyzgate.com/94dc02b8f170dcbf46afba7dedf9e98a.png "BPNP concentration") !page ###=!= I-V curves ![BPNP kcsa IV](http://data.xyzgate.com/0b7eb401a9893cbb83956f85e2e45222.png "BPNP kcsa IV") - Selective binding and permeation of K+ over Na+ - Inner current rectification !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 ## =!= Simulation results II: 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 ## =!= Simulation results: II. DNA-nanopore/channel sequencing ![](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 ==! *Nature Nanotechnology*, 5:807, 2010 !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 ![](http://data.xyzgate.com/df9ac5604fd11f30e49b74b27abbd5d3.png) ![](http://data.xyzgate.com/5135b2ca0e2899606c4c17b035f189ab.png) Open-pore currents Is and current amplitudes ∆Is with 20-pb dsDNA in the conical nanopores. (2r = 4 nm) !page ## Online computing on [xyzgate.com](xyzgate.com) - Programs: **TMSmesh, SMOPT, FEPB, ichannel** ![数字之门二维码](http://data.xyzgate.com/daf69e1e46cc6c03e9bfaf433dccab93.png) !page ###Acknowledgements: Xuejiao Liu, Yu Qiao, Tiantian Liu, Shiyang Bai (Institute of Comput. Math, CAS) Minxin Chen, (Soochow Univ., Suzhou, China) Bin Tu (National Center for Nanoscience and Technology, CAS) Fundings: China NSF, CAS, NCMIS, Science Challenge project #=!= **Thanks!** !page !page #=!= 在线计算化学II:钾离子通道选择性通透的连续模型与模拟 ###=!= 卢本卓,刘雪娇 ####=!= 科学院计算数学研究所,国家数学与交叉科学中心 ###=!= 白石阳,吕永亮 ####=!= 杭州, 2018-5-7 ###=!= Xuejiao Liu (刘雪娇), Benzhuo Lu (卢本卓) ####=!= Institute of Computational Math and Scientific/Engineering Computing, ####=!= Chinese Academy of Sciences, China ####=!= National Center for Mathematics and Interdisciplinary Sciences, !page ###Collaborators: Minxin Chen, (Soochow Univ., Suzhou, China) Bin Tu, Shiyang Bai, Tiantian Liu, Xuejiao Liu, Bo Peng (Inst Computational Math, CAS, China) Linbo Zhang (Inst Computational Math, CAS, China) Jingfang Huang (Univ. of North Carolina, US) Bo Zhang (Indiana University, IN, USA) Nikos P. Pitsianis, and Xiaobai Sun (Duke University, US) Xiaolin Cheng (Oak Ridge National Lab, US) !page ![CiCP](http://data.xyzgate.com/1d5726c0f6ffd55785d2124693df7d31.png) !page ```math \Rightarrow I=\int_s \displaystyle\sum_i^K D_i (\nabla c_i +\beta c_i \nabla(q_i \phi)) ``` !page =!= Influence of parameter $$\alpha$$ on cation concentrations in the KcsA channel ![BPNP alpha](http://data.xyzgate.com/b279e3df222322d9aedfec943a7a48c4.png "BPNP alpha") !page #Diffusion coefficient inside the the channel In BPNP model: - A position-dependent diffusion coefficient is adopted: ```math D_{i}(r) = D_{i}(z) = D_{i}^{bulk}(d_{1} + \frac{1.0-d_{1}}{1.0+e^{-\frac{|z|-z_{0}}{\Delta z}}}), ``` $$D_{Na}^{bulk} = 0.133 \text{Å}^{2}/ps$$, $$D_{K}^{bulk} = 0.196 \text{Å}^{2}/ps$$, and $$D_{Cl}^{bulk} = 0.203 \text{Å}^{2}/ps$$. In PNP model: ```math D(r)=\begin{cases} D_{bulk}, \ \ r\in bulk\ region, \\ D_{chan}+(D_{chan}-D_{bulk})f(r), \ \ r\in buffering\ region \\ D_{chan},\ \ \ r\in channel region, \end{cases} ``` ```math f(r)=f(z)=n(\frac{z-z_{chan}}{z_{bulk}-z_{chan}})^{n+1}-(n+1)(\frac{z-z_{chan}}{z_{bulk}-z_{chan}})^n ``` Hwang, H.; Schatz, G. C.; Ratner, M. A. *J. Phys. Chem.* *A **2007**, 111,* 12506–12512 DCl = 0.203Å2/ps, DK = 0.196Å2/ps, n = 9, For the bottom boundary, zchan = 11 and zbulk = 9. For the top boundary, zchan = 32 and zbulk = 34. B Tu, BZ Lu, et al. *J. Comput. Chem.* 2013, 34: 2065 Membrane meshing: TT Liu, SY Bai, B Tu, MX Chen, and BZ Lu, Membrane-Channel Protein System Mesh, Construction for Finite Element Simulations, *Mol. Based Math. Biol*. 2015; 3:128–139 ![BPNP kcsA mesh](http://data.xyzgate.com/b501e561af97def400eda7bce67a4cf9.png "BPNP kcsA mesh") ![BPNP-energyBarrier](http://data.xyzgate.com/756acfc222a1fd01f1641be9dc6f13db.png "BPNP-energyBarrier") ==! Chen MX, Lu BZ, *J Chem Theory Comput*. 2011, 7, 203–212; *J Mol Graphics Modelling*, 38: 411-418, 2012. #### BITS11 July 3-7, 2018 #=!= Continuum modeling of selective ion permeation in potassium channel ###=!= 刘雪娇,卢本卓 ####=!= 中科院计算数学研究所 ###=!= 台湾 惠荪林场, July 5, 2018 !page 补充: When $$\epsilon$$ is not concentration-dependent, but is genrally $$\epsilon(x)$$, then, the energy form is equivalent to: ```math F[c] = \int_{\Omega}\{\frac{1}{2}\epsilon |\nabla \phi|^2 + \beta^{-1}\sum_{i=1}^{K}c_{i}[\log(\Lambda^{3}c_{i})-1] + \sum_{i=1}^{K}\mu_{i}^{ex}c_{i} \}dV + \int_{\Gamma_{N}}\frac{1}{2}\sigma\phi(c)dS -\int_{\Gamma_{D}}\frac{1}{2}\epsilon(c)\frac{\partial \phi(c)}{\partial n}\phi_{0}dS, ``` ## Issues - 模型 - 分析 (解的存在唯一性,多解与分叉,渐进性质,整体性质如 I-V特性,等) - 算法 (FD,**FEM**,谱方法,FV,混合等等) -- 分子网格生成 -- Poisson方程中的奇异电荷 -- 电扩散耦合方程 -- NP方程是对流扩散型方程 -- 保物理结构:保正性、流守恒、保能量(能量耗散)、保总质量、保电中性等 -- 高效性、稳定性
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