Electrodynamics and Ions in Chemistry* an opportunity for applied mathematics

1. Electrodynamics and Ions in Chemistry*an opportunity forapplied mathematics *and biology: ALL of biology occurs in salt solutions

2. Thanks to Christine and Guowei for inviting me! I apologize for missing talks, but have 3 abstracts, 9 authors to deal with for October 1 Biophysics deadline, and posters, papers and reviews overdue.

3. Electrodynamics and Ions in Chemistry* Applied Electric Field ZERO Applied Electric Field NOT ZERO *and biology: ALL of biology occurs in salt solutions

4. The Electric Field is Strong If you were standing at arm’s length from someone and each of you had  One percent more electrons than protons, the force would lift the Entire Earth! slight paraphrase of third paragraph, p. 1-1 of The Feynman: Lectures on Physics, Mainly Electromagnetism and Matter.1963.NY Addison-Wesley, also at http://www.feynmanlectures.caltech.edu/II_toc.html

5. Concentration Fields are Weak One percent  change in concentrations does almost nothing

6. Chemistry has been about ChemicalsandConcentrations not charges and flow

7. Biology and Technology require Charges and Flow Creating an enormous need and opportunity forMathematics

8. Mathematics Creates our Standard of Living*Mathematics replaces Trial and ErrorMolecular Biology, Drug Design, Batteries/Fuel Cellswith Computation *e.g.,Electricity, Computers, Fluid Dynamics, Optics, Structural Mechanics, …..

9. Mathematics describes only a little ofDaily Life But Mathematics* Creates our Standard of Living *e.g.,Electricity, Computers, Fluid Dynamics, Optics, Structural Mechanics, …..

10. Mathematics is Needed to Describe and Understand Devicesof Biology and Technology Devices Implement Equations

11. How can we use mathematics to describe biological systems? Come to next symposium to hear about the “Calcium/sodium transporter, an atomic machine”Jinn Liang Liu and Bob Eisenberg

12. How can we use mathematics to describe chemical systems? I do not know, but I know how to begin

13. Correlation between currentsis in fact ALWAYS 0.999 999 999 999 999 999 because Continuity of Current is Exact Kirchoff Continuity of Current Lawincluding displacement current is another form of Maxwell’s EquationsHeras, J.A.: Am J Phys 75: 652 (2007); Eur J Phys 30: 845 (2009); Am J Ph79: 409 (2011)

14. Maxwell Equationsin vacuum as written by Heaviside, using Gibbs notation Vector Identity for Wave equation of LIGHT Displacement Current in vacuumEverywhere! c is the velocity of light

15. Generalized Current is Conserved in Vacuum Maxwell Equation in vacuum where J=0 Vacuum Displacement Current exists Everywhere! Conservation law Generalized ‘Maxwell’Current Vector Identity Wave equation of LIGHT Vector Identity c is the velocity of light

16. Generalized Current is Conserved Everywhere Conservation law in Matter Maxwell’s Generalized Current is never stored

17. Rate and Markov Models do not Conserve Current‘Current-in’ does not equal ‘Current-out’ because rate constants are independent [X] means the concentration, really activity of species Z, i.e., concentration is the number densityOrdinary Differential Equations in time cannot describe Electric Felds in SpaceOrdinary Differential Equations in time cannot satisfy continuity of current flow in space

18. This is a serious problem In the real world, Continuity of Current is Exact 1% Discontinuity of Current soon (μsec) creates potentials >106 voltsthat destroy molecules and atoms

19. SCIENCE SPEAK:Parameterization is not Possible under more than one condition because rate constants are customarily independent Rate constants chosen at one boundary charge or one potentialcannot work for different charges or potentials Currents in Rate Models are Independent of Charge and Potential but in the real world Currents depend on Charge and Potential MATHSPEAK:Ordinary Differential Equations in time cannot describe Electric Fields in SpaceOrdinary Differential Equations in time cannot satisfy Continuity of Current in Space

20. Parameterization is not Possible under more than one condition because electric field is global. Connect a battery to a resistor meters away Disconnect the resistor meters away. Atomic scale chemistry changes inside the battery!

21. Electric Field is GlobalMathematics of Continuity in Maxwell equations canCreate New Kind of Charge When we unplug a computer power supply, we often CREATE SPARKS, i.e., a PLASMA, a NEW KIND of current flow Pop!

22. even though‘Charge’ is an Abstraction withVERY different Physics in different systems Continuity of Current is Exact Maxwell Equations are Special

23. ‘Charge’ is an Abstraction with different Physics in different systems butContinuity of Current is ExactNo matter what carries the current! Ag AgCl Ag AgCl D = permittivity  E

24. This is interesting physics BUT Why is Electrodynamics Important to Biologists? K+ Ca++ Na+ 3 Å Cl-

25. Why is Electrodynamics important to Biologists? All of Biology occurs in Salt Solutions of definite composition and concentration and that matters! Salt Water is the Liquid of Life Pure H2O is toxic to cells and molecules! Salt Water is a Complex Fluid Main Ions are Hard Spheres,close enough Sodium Na+ Potassium K+ Calcium Ca2+ Chloride Cl- K+ Ca++ Na+ 3 Å Cl-

26. Working Hypothesis: Crucial Biological Adaptation is Crowded Ions and Side Chains Wise to use the Biological Adaptation to make the reduced model! Reduced Models allow much easier Atomic Scale Engineering

27. Active Sites of Proteins are Very Charged 7 charges ~ 20M net charge = 1.2×1022 cm-3 liquidWater is 55 Msolid NaCl is 37 M + + + + + - - - - Selectivity Filters and Gates of Ion Channels are Active Sites Physical basis of function OmpF Porin Hard Spheres Na+ Ions are Crowded K+ Ca2+ Na+ Induced Fit of Side Chains K+ 4 Å Figure adapted from Tilman Schirmer

28. Crowded Active Sitesin 573 Enzymes Jimenez-Morales,Liang, Eisenberg

29. Everything Interacts with Everything Else by steric exclusioninside crowded active sites Everything interacts with macroscopic Boundary Conditions (and much else) through long range electric field ‘Law’ of mass action needs to be generalized

30. Crowded Channels, Crowded Active Sites Nucleic Acids are Complex Fluidslike liquid crystals of LCD displays All atom simulations of complex fluid are particularly challenging because ‘Everything’ interacts with ‘everything’ else on atomic& macroscopic scales

31. Central Result of Physical Chemistry Ionsin a solutionare aHighly Compressible Plasma although the Solution is Incompressible Free energy of an ionic solution is mostly determined by the Number density of the ions Density varies from 10-11 to 101M in typical biological system of proteins, nucleic acids, and channels Learned from Doug Henderson, J.-P. Hansen, Stuart Rice, among others…Thanks!

32. Electrolytes are Complex Fluids Treating a Complex Fluid as if it were a Simple Fluid will produce Elusive Results with parameters that change with every condition

33. Electrolytes are Complex Fluids After 690 pages and 2604 references, properties of SINGLE Ions are Elusivebecause Every Ion Interacts with Everything Hünenberger & Reif (2011) Single-Ion Solvation. Experimental and Theoretical Approaches to ElusiveThermodynamic Quantities. 2011

34. It is not surprising that Inconsistent Treatments of ionic solutionshave been soUnsuccessful despite more than a century of work by fine scientists and mathematicians Werner Kunz: “It is still a fact that over the last decades, it was easier to fly to the moon than to describe the free energy of even the simplest salt solutions beyond a concentration of 0.1M or so.” Kunz, W. "Specific Ion Effects" World Scientific Singapore, 2009; p 11

35. Shielding is a defining property of Complex FluidsGuoy-Chapman, Debye-Hückel, Poisson Boltzmann It is VERY hard to Simulate in Atomic Detail at Equilibrium and (in my opinion) IMPOSSIBLE to Simulate in Atomic Detail in nonequilibrium Like Batteries or Nerve Fibers because flows involve Far Field (macroscopic) boundaries

36. Main Qualitative Result Shielding Dominates Electric Properties of Channels, Proteins, as it does Ionic Solutions Shielding is ignored in traditional treatments of Ion Channels and of Active Sites of proteins Rate Constants Depend on Shielding and so Rate Constants Depend on Concentration and Charge

37. Main Qualitative ResultShielding in Gramicidin Hollerbach & Eisenberg

38. Reconciling Mass Action and Maxwell-Kirchoff will no doubt be a Long Journey

39. “Journeyof a thousand miles starts with a single step” in the right direction, I beg to add to this Chinese saying

40. But is that journey feasibleor is it aQuixotic Quest for perfection?

41. Replacement of “Law of Mass Action” is Feasible for Ionic Solutions using theAll Spheres(primitive = implicit solvent model of ionic solutions)andTheory of Complex Fluids

42. Replacement of “Law of Mass Action” is Feasible for Ionic Solutions using theAll Spheres(primitive = implicit solvent model of ionic solutions)andTheory of Complex Fluids

43. Variational Approach EnVarA ‘Law’ of Mass Action includingInteractions Conservative Dissipative From Bob Eisenberg p. 1-6, in this issue

44. Energetic Variational Approach allows accurate computation ofFlow and Interactions in Complex Fluids like Liquid Crystals Classical theories and Molecular Dynamicshave difficulties with flow, interactions, and complex fluids Engineering needs Calibrated Theories and Simulations Engineering Devices almost always use flow

45. Energetic Variational ApproachEnVarAChun Liu, Rolf Ryham, and Yunkyong Hyon Mathematicians and Modelers: two different ‘partial’ variations written in one framework, using a ‘pullback’ of the action integral Shorthand for Euler Lagrange process with respect to Shorthand for Euler Lagrange process with respect to CompositeVariational Principle Action Integral, after pullback Rayleigh Dissipation Function Euler Lagrange Equations Field Theory of Ionic Solutions: Liu, Ryham, Hyon, Eisenberg Allows boundary conditions and flowDeals Consistently with Interactions of Components

46. Dissipation Principle Conservative Energy dissipates into Friction Hard Sphere Terms Number Density Thermal Energy time Permanent Charge of protein valence proton charge ci number density; thermal energy; Didiffusion coefficient; n negative; p positive; zivalence; ε dielectric constant Note that with suitable boundary conditions

47. Energetic Variational ApproachEnVarA is defined by the Euler Lagrange Process, as I understand the pure math which gives Equations like PNP BUT I leave it to the pure mathematicians to argue/discuss about the purity of the process and the a priori restrictions on the function spaceswhen two variations are involved

48. PNP (Poisson Nernst Planck)for Spheres Non-equilibrium variational field theory EnVarA Nernst Planck Diffusion Equation for number density cnof negative n ions; positive ions are analogous Diffusion Coefficient Coupling Parameters Thermal Energy Permanent Charge of Protein Ion Radii Number Densities Poisson Equation Dielectric Coefficient valence proton charge Eisenberg, Hyon, and Liu

49. Semiconductor PNP EquationsFor Point Charges Poisson’s Equation Drift-diffusion & Continuity Equation Chemical Potential Permanent Charge of Protein Diffusion Coefficient Thermal Energy Cross sectional Area Flux Number Densities Dielectric Coefficient Not in Semiconductor Valence Proton charge valence proton charge

50. All we have to do isSolve Them!with Boundary ConditionsdefiningCharge Carriersions, holes, quasi-electronsGeometry of Protein Structure