University:
Boston University
Course:
CAS BI 108 | Biology 2
Academic year:

2022
Views:

234

Pages:

24
Author:

StellarSymphony

Boundary-Integral Methods in Molecular Science and Engineering Lecture 1: Biology is Awesome. Outline for Lectures 1. Biology is awesome. If you can solve Poisson, you can join in the fun! 2. There’s more than one way to skin a cat. Sometimes PDEs can be advantageously reframed as integral equations. 3. Numerical solution of integral equations presents different challenges than PDEs. 4. A diversity of unusual computational challenges will continue to drive biological simulation. Example 1: Bacterial Chemotaxis • “Nutrient receptors” cover cell membrane and guide cell towards higher-concentration areas E. coli No food gradient Mostly “tumbling” Food gradient Mostly “gradient following” Example 2: Nuclear Pore Complex 40 nm Fermi 40 nm Example 3: Electrostatics In Wound Healing Galvani’s “Figure 1”limb! Electric-field induced extra Example 4: Compaction of DNA Sugar-phosphate backbone is highly negatively charged! How do you get it to wrap tightly and compress it?? Fact: Water Makes Life Possible Fox L. Freberg Vander Kass ‘05 Biology’s Multiscale Challenges • Fast computers and MD are not a panacea…  The inverse problem: what kinds of molecules have property X?  Combinatorial number of problems to simulate • Hence: Even with exascale MD, fast approximate physical models have enormous value for science and engineering Turnbull (2007) Agarwal and Alam (2006); Ramasubramaniam (2007) Molecular Machines, To Scale Water Glucose Ribosome ATP Cholesterol Hemoglobin Potassium Phospholipid Myoglobin Gramicidin Porin 20,000 water atoms are needed to provide even a small water box around myoglobin! Mechanosensitive Channel Aquaporin GroES GroEL lac operon DNA F1-F0 ATPase Nucleasome A Crucial Consequence of Solvation • Molecular binding involves sacrificing solute--solvent interactions for solute--solute interactions: solv δ=0 δ=1 This is only a VERY SIMPLE MODEL for molecular binding! Stepping Towards a Simpler Model • To analyze protonation of ionizable groups: HA H+ + A- Assume spherical protein Kirkwood, 1934; Tanford, 1957 The Basic Continuum Electrostatic Model • Mixed-dielectric Poisson problem with point charge sources • Assumes molecule is at infinite dilution--more on that later • Hundreds to thousands of times faster than MD: seconds to a few hours This model can be derived rigorously from sophisticated statistical-mechanical theories (see, e.g., Beglov+Roux, 1996) Biological Fluids are Electrolytes • Mobile ions in solvent redistribute in response to electric fields • Assuming that the charge density in the solvent is Boltzmann-like, we obtain the Poisson-Boltzmann equation L. Freberg Linearize This treatment of ionic solutions is very useful, but incredibly oversimplified--more details in Lecture 4! Applications of the Continuum Model • How dielectric boundaries “focus” electric fields • How ion channel proteins selectively pass one species but not others Acetylcholinesterase Baker (2004) Honig+Nicholls (1995) • Accounting for solvent effects on electronic structure DFT study of photoactive yellow protein Boda, Gillespie, Eisenberg et al (2007) Chiba et al (2008) Continuum Models Capture Important Physics • Linear response means quadratic energy: Assume ligand rigidity, and no charge transfer: ligand • Under our assumptions, this energy function is convex receptor • The optimal charge distribution… • The idea: It always costs energy to remove the water from the receptor volume Energy  … balances the “desolvation penalty” against ligand-receptor interactions  … is a guide for design Charge Solving the PDE Directly is Possible, But… The idea: Just throw down a finite-difference grid or a finite-element mesh and go to town! PDE Complications 1. Boundary conditions are at infinity 2. Point charges must be spread onto the grid 3. The dielectric interface is approximated A Boundary Integral Method + + + + + + - - - + + + - - - - 1. Boundary conditions handled exactly 2. Point charges are treated exactly 3. Meshing emphasis can be placed directly on the interface Differential vs Integral Equations Differential operator Unknown Source/forcing Integral operator Unknown Source/forcing Kernel: for our problems, this will be the Green’s function of the corresponding PDE Must specify boundary conditions BCs automatically included (generally) Operator is local Operators are often nonlocal Kinds of Integral Equations First kind Second kind Fredholm: domain of integration is fixed (above: D) Volterra: domain integration varies with the independent variable Closely related to ODEs Integral equations can be linear or nonlinear. Applications of Integral Equations • Wiener-Hopf integral equations • Radiosity in graphics • Fluid structure in chemistry Boundary Integral Equations: Fluids • Stokes: steady/unsteady incompressible, steady linearized compressible • Wide applications in potential flow (Images courtesy David Willis) ADXL76 accelerometer (Analog Devices; picture courtesy Xin Wang and Jacob White) BIE Application: Electromagnetics VLSI interconnect analysis: extracting parasitic capacitances and inductances in complex geometries Modified Green’s functions allow treatment of planar multi-layer substrates K. Nabors Stochastic integral equation: gives average parasitics Hu, Daniel et al (2007) Z. Zhu (2004) BIE Applications In Bioscience • • • • Protein folding Molecular transport Ion channel selectivity Electron tomography Totrov and Abagyan (2001) Lysozyme Boda, Gillespie, Eisenberg et al (2007) Duraiswamy et al (1998) Aragon (2004) Next: Turning PDEs into BIEs • Advantages and disadvantages of each • The boundary element method (BEM) for solving BIEs numerically