to Dan Freeman and Mark Thomas Harnett for inviting me!

1. Thanks! to Dan Freeman and Mark Thomas Harnett for inviting me!

2. K+ ~30 x 10-9meter Ion Channels are Biological Devices* Natural nano-valves** for atomic control of biological function Ion channels coordinate contraction of cardiac muscle, allowing the heart to function as a pump Coordinate contraction in skeletal muscle Control all electrical activity in cells Produce signals of the nervous system Are involved in secretion and absorption in all cells:kidney, intestine, liver, adrenal glands, etc. Are involved in thousands of diseases and many drugs act on channels Are proteins whose genes (blueprints) can be manipulated by molecular genetics Have structures shown by x-ray crystallography in favorable cases Can be described by mathematics in some cases • *Device is a Specific Word, that exploits specific mathematics & science *nearly pico-valves: diameter is 400 – 900 x 10-12 meter; diameter of atom is ~200 x 10-12 meter

3. + ~3 x 10-9 meters Channels are Selective Molecular DevicesDifferent Ions Carry Different Signals through Different Channels ompF porin Ca++ Na+ K+ 300 x 10-12 meter 0.7 10-9 meter = Channel Diameter Diameter mattersIonic solutions are NOT ideal Classical Biochemistry assumes ideal solutions. K+& Na+ are identical only in Ideal Solutions. Flow time scale is 10-4 sec to 1 min Figure of ompF porin by Raimund Dutzler

4. Ion Channels like FETs are Valves that Control Flow Classical Theory & Simulations NOT designed for flow Thermodynamics, Statistical Mechanics do not allow flow Rate Models are inconsistent with Maxwell’s Eqn (Kirchoff Law) (if rate constants are independent of potential)

5. Biology is made of Devicesand they always involve Flowand are MULTISCALE A different talk! Hodgkin’s Action Potential is the Ultimate Multiscale model from Atoms to Axons Ångstroms to Meters

6. Multi-Scale Issues are Always Presentin Atomic Scale Engineering • Atomic & Macro Scales are both used by channels just because Channels are Nanovalves • By definition: all valves use small structures to control large flows

7. Ompf G119D A few atoms make a BIG Difference OmpF 1M/1M G119D 1M/1M OmpF0.05M/0.05M G119D 0.05M/0.05M Glycine replaced by Aspartate Structure determined by Raimund Dutzlerin Tilman Schirmer’s lab Current Voltage relation by John Tang in Bob Eisenberg’s Lab

8. How does it work? How do a few atoms control (macroscopic) Biological Function Mathematics of Molecular Biology is aboutSolving Specific Inverse Problems

9. Single Channel Recording

11. Where to start? Why not compute all the atoms?

12. Simulations produce too many numbers 106 trajectories each 10-6 sec long, thanks to David Shaw’s Anton with 109 samples in each trajectory, in background of 1022 atoms Estimators are Needed Estimators are a kind of Reduced Model

13. Multi-Scale Issues A different talk! Journal of Physical Chemistry C (2010 )114:20719 Three Dimensional (104)3 Atomic and Macro Scales are BOTH used by channels because they are nanovalves so atomic and macro scales must be Computed and CALIBRATED Together This may be impossible in all-atom simulations

14. Where to start? Mathematically ? Physically ?

15. Biology is Easier than Physics Reduced Models Exist* for important biological functions or the Animal would not survive to reproduce *Evolution provides the existence theorems and uniqueness conditions so hard to find in theory of inverse problems. (Some biological systems  the human shoulder  are not robust, probably because they are incompletely evolved,i.e they are in a local minimum ‘in fitness landscape’ .I do not know how to analyze these. I can only describe them in the classical biological tradition.)

16. Engineers:this is reverse engineering For example, Find Charge Distribution in Channel from Current Voltage Relations Problem (with noise and systematic error) has actually been solvedbyTikhonov RegularizationBurger, Eisenberg, Engl (2007) SIAM J Applied Math 67: 960-989 using procedures developed by Engl to study Blast Furnaces and their Explosions Inverse Problems Given the OutputDetermine the Reduced Model

17. Here is where we do Science, not Mathematics Here we GUESS and CHECK

18. Guess 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

19. Density and Concentration Fields are Weak One percent  change in density does almost nothing The Electric Field is Strong One percent  change in charge lifts the Earth,

20. 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 Feynman, R. P., R. B. Leighton, and M. Sands. 1963. The Feynman: Lectures on Physics, Mainly Electromagnetism and Matter.New York: Addison-Wesley Publishing Co., also at http://www.feynmanlectures.caltech.edu/II_toc.html.

21. 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

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

23. 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

24. ‘General’ Mathematics of ‘Crowded Charge’

25. Physical Chemists are Frustrated by Real Solutions

26. 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.

27. The classical text of Robinson and Stokes (not otherwise noted for its emotional content) gives a glimpse of these feelings when it says“In regard to concentrated solutions, many workers adopt a counsel of despair, confining their interest to concentrations below about 0.02 M, ... ”p. 302 Electrolyte Solutions (1959) Butterworths , also Dover (2002)

28. Good Data

29. Good Data Compilations of Specific Ion Effect • >139,175 Data Points on-line IVC-SEP Tech Univ of Denmark http://www.cere.dtu.dk/Expertise/Data_Bank.aspx 2. Kontogeorgis, G. and G. Folas, 2009:Models for Electrolyte Systems. Thermodynamic John Wiley & Sons, Ltd. 461-523. 3. Zemaitis, J.F., Jr., D.M. Clark, M. Rafal, and N.C. Scrivner, 1986,Handbook of Aqueous Electrolyte Thermodynamics. American Institute of Chemical Engineers 4. Pytkowicz, R.M., 1979, Activity Coefficients in Electrolyte Solutions. Vol. 1. Boca Raton FL USA: CRC. 288.

30. 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!

31. Electrolytes are Complex Fluids Treating a Complex Fluid as if it were a Simple Fluid will produce Elusive Results

32. Electrolytes are Complex Fluids After 690 pages and 2604 references, properties of SINGLE Ions are Elusive*because Every Ion Interacts with Everything *’elusive’ is in the authors’ choice in the titlebut emphasis is added Hünenberger & Reif (2011)“Single-Ion Solvation… Approaches to Elusive* Thermodynamic Quantities”

33. Everything Interacts Mathematics of Chemistry must deal Naturally with Interactions ‘Law of Mass Action’ assumes nothing interacts So this is a great opportunity for new mathematics and applications!

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

35. Crowded Channels, Crowded Active Sites 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

36. ‘Everything Interacts with Everything Else’ immediately implies the need for Variational Methods for those who understand. But I did not understand for some fifty years, chemists/biologists do not understand even now, so it is not obvious to all. I apologize to those who already understand!

37. Energetic Variational ApproachEnVarAChun Liu, Rolf Ryham, Yunkyong Hyon, and Bob Eisenberg 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 E is Helmholtz free energyΔisRayleigh dissipation Euler Lagrange Equations Field Theory of Ionic Solutions that allows boundary conditions and flow and deals Consistently with Interactions of Components

38. ENERGY TERMconservative Generalization of Helmholtz Free Energy Dielectric Coefficient from Poisson Eq. Number Densities Lagrange Multiplier Eisenberg, Hyon, and Liu EnVara Variational Analysis of Ionic Solution

39. DISSIPATIVE TERM Hard Sphere Terms Number Density Thermal Energy time Permanent Charge of protein valence proton charge cinumber density; thermal energy; Di diffusion coefficient; n negative; p positive; zivalence Eisenberg, Hyon, and Liu EnVara Variational Analysis of Ionic Solution

40. 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

41. All we have to do isSolve them!with Boundary ConditionsdefiningCharge Carriersions, holes, quasi-electronsGeometry

42. All we have to do isSolve them!Don’t DespairSemiconductor Technology has Already Done That!

43. Semiconductor Devices PNP equations describe many robust input output relations Amplifier Limiter Switch Multiplier Logarithmic convertor Exponential convertor These are SOLUTIONS of PNP for different boundary conditions with ONE SET of CONSTITUTIVE PARAMETERSPNP of POINTS is TRANSFERRABLEAnalytical should be attempted using techniques of Weishi Liu University of KansasTai-Chia Lin National Taiwan University & Chun Liu PSU

44. 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

45. Energetic Variational ApproachEnVarA across biological scales: molecules, cells, tissuesVariational theory of complex fluids developed by Chun Liuwith (1) Hyon, Eisenberg Ions in Channels (2) Horng, Lin, Liu, Eisenberg Ions in Channels (3) Bezanilla, Hyon, Eisenberg Conformation Change of Voltage Sensor (4) Ryham, Cohen Membrane flow Cells (5) Mori, Eisenberg Water flow in Tissues Multiple Scales creates a newMultiscale Field Theory of Interacting Components needed for Molecular Engineering in general that allows boundary conditions and flow and deals with Ions in solutions self-consistently

46. The End Any Questions? About general Stochastic Treatment??

47. Here I start fromStochastic PDE and Field Theory Other methodsgive nearly identical results MSA (Mean Spherical Approximation) SPM (Primitive Solvent Model) Non-equil MMC (Boda, Gillespie) several forms DFT (Density Functional Theory of fluids, not electrons) DFT-PNP (Poisson Nernst Planck) EnVarA (Energy Variational Approach) Steric PNP (simplified EnVarA) Poisson Fermi Chemistry Models MATHField Theory

48. Solved with PNP including Correlations Other methodsgive nearly identical results MMC Metropolis Monte Carlo (equilibrium only) DFT (Density Functional Theory of fluids, not electrons) DFT-PNP (Poisson Nernst Planck) MSA (Mean Spherical Approximation) SPM (Primitive Solvent Model) EnVarA (Energy Variational Approach) Non-equil MMC (Boda, Gillespie) several forms Steric PNP (simplified EnVarA) Poisson Fermi

49. Always start with TrajectoriesZe’ev Schuss Department of Mathematics, Tel Aviv University Always start with Trajectories because Trajectories are the equivalent of SAMPLES in probability theory Trajectories satisfy PHYSICAL boundary conditions Trajectories satisfy classical PHYSICAL ordinary differential equations (we hope)

50. From Trajectories to Probabilities in Diffusion Processes ‘Life Work’ of Ze’ev Schuss Department of Mathematics, Tel Aviv University Theory and Applications of Stochastic Differential Equations1980 Theory and Applications Of Stochastic Processes: An Analytical Approach 2009 Singular perturbation methods for stochastic differential equations of mathematical physicsSIAM Review, 1980 22: 116-155 Schuss, Nadler, Singer, Eisenberg