Thanks to Huaxiong and Chun, Glad to be here!

1. Thanks to Huaxiong and Chun, Glad to be here!

3. How does it work? How do a few atoms control (macroscopic) Biological Function? Inverse Problem! Topic in Lecture Courses at Jiaotong, thanks to Zhenli Xuand at Politechnico Milano, thanks to Riccardo Sacco

4. General Theme Mathematics of Molecular Biology Provides Great Opportunity

5. Alan Hodgkin: “Bob, I would not put it that way” Alan Hodgkin friendly *Topic of Lecture Courses at Jiaotong, thanks to Zhenli Xuand at Politechnico Milano, thanks to Riccardo Sacco Success of Physical Models* of Biological Devices like Ion Channels is a Surprise

6. I believe that much of Molecular Biologyshould be about theAtomic-Molecular Engineeringof Biological Devices* For example, the atomic engineering of Ion Channels *Device is a Specific Word, that exploits specific mathematics and science

7. K+ ~30 Å 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 Ion channels coordinate contraction in skeletal muscle Ion channels control all electrical activity in cellsIon channels produce signals of the nervous system Ion channels are involved in secretion and absorption in all cells:kidney, intestine, liver, adrenal glands, etc. Ion channels are involved in thousands of diseases and many drugs act on channels Ion channels are proteins whose genes (blueprints) can be manipulated by molecular genetics Ion channels have structures shown by x-ray crystallography in favorable cases *nearly pico-valves: diameter is 400 – 900 picometers

8. 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

9. How do a few atoms control (macroscopic) Biological Function? Answer, oversimplified: A few atoms control the electric field Much as they do in transistors

10. 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, Leighton, Sands (1963)‘The Feynman Lectures on Physics, Mainly Electromagnetism and Matter’ also at http://www.feynmanlectures.caltech.edu/II_toc.html.

11. + ~3 x 10-9 meters Channels are Devices ValvesandDiodesDifferent Ions Carry Different Signals through Different Channels ompF porin Analysis of Devices must be NONEQUILIBRIUM with spatially non-uniform BOUNDARY CONDITIONS 0.7 10-9 meter = Channel Diameter Flow time scale is 10-4 sec to 1 min Figure of ompF porin by Raimund Dutzler

12. Channels are Devices Channels are (nano) valves Valves Control Flow Classical Theory & Simulations NOT designed for flow Thermodynamics, Statistical Mechanics do not allow flow Rate and Markov Models do not Conserve Current

13. Maxwell’s Equations A different talk* *Topic in Lecture Courses at Jiaotong, thanks to Zhenli Xuand at Politechnico Milano, thanks to Riccardo Sacco

14. Current is Abstract with Different Physics in Different SystemsNOT the Flux of Charges Hungarian Journal of Industry and Chemistry 44(1): 1-28 arXiv:1502.07251 Hungarian Journal of Industry and Chemistry (2016) 44 1-28 arXiv:1502.07251

15. Valves and Devices haveINPUTS OUTPUTS connected by LAWS involving FLOW fromPOWER SUPPLIES so Analysis of Devices must be NONEQUILIBRIUM with spatially non-uniform BOUNDARY CONDITIONS Thermodynamics, Statistical Mechanics, Molecular Dynamics have No inputs, outputs, flows, or power supplies i.e., Power Supply = spatially nonuniform inhomogeneous Dirichlet conditions

16. Thermodynamics, Statistical Mechanics, Molecular Dynamics are UNSUITED for DEVICES Thermodynamics, Statistical Mechanics, Molecular Dynamicshave No inputs, outputs, flows, or power supplies Power supply = spatially nonuniform inhomogeneous Dirichlet conditions Analysis of Devices must be NONEQUILIBRIUM with spatially non-uniform BOUNDARY CONDITIONS

17. Physical ChemistsPhysiologistsBiophysicists are Frustrated by Ionic SolutionsTheories and Simulations Cannot deal with Sea Water at equilibriumand fail even more badly when flow is involved A different talk!

18. Cause of Frustration Biochemical Models are Rarely TRANSFERRABLEDo Not Fit Data even approximatelyin more than one solution* Title Chosen by Editors Editors: Charlie Brenner, Angela HoppAmerican Society for Biochemistry and Molecular Biology *i.e., in more than one concentration or type of salt, like Na+Cl− or K+Cl −Note: Biology occurs in different solutions from those used in most measurements

19. Electrolytes are Complex Fluids‘Everything’ interacts with everything else Treating a Complex Fluid as if it were a Simple Fluid will produce Elusive Results “Single-Ion Solvation… Elusive* Quantities” 690 pages 2604 references Hünenberger & Reif, 2011

20. General Theme Mathematics of Molecular Biology Provides Great Opportunity

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

22. 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 *Topic of Lecture Courses at Jiaotong, thanks to Zhenli Xuand at Politechnico Milano, thanks to Riccardo Sacco

23. 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

24. Where to start? Biological Adaptation Crowded Charge

25. 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

26. Working Hypothesis Crucial Biological Adaptation is Crowded Ions and Side Chains

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

28. Don’t worry! Crowded Charge is GOOD It enables SIMPLIFICATION by exploiting a biological fact (an adaptation) Charges are Crowded where they are important Enzymes, Nucleic Acids, Ion Channels, Electrodes

29. Where do we begin? Crowded Charge enables Dimensional Reduction* to a Device Equation Inverse Problem! Essence of Engineering is knowing What Variables to Ignore! WC Randels in Warner IEEE Trans CT 48:2457 (2001) *Dimensional reduction = ignoring some variables

30. Where do we begin? • Crowded Charge • has • HUGE electric fields • Poisson Equation, i.e., Conservation of Charge • and • LARGE steric repulsionFermi distribution

31. Motivation and Assumption for Fermi-Poisson • Largest Effect of Crowded Charge • is • Saturation • Saturation cannot be described at all by classical Poisson Boltzmann approach • Simulating saturationby interatomic repulsion (Lennard Jones) is a singular mathematical challengeto be side-stepped if possible, • particularly in three dimensions, Eisenberg, Hyon and Liu (2010) JChemPhys 133: 104104

32. A Nonlocal Poisson-Fermi Model for Electrolyte Solutions Jinn Liang Liu 劉晉良 Jinn-Liang is first author on our papers • J Comp Phys (2013) 247:88 J Phys Chem B (2013) 117:12051J Chem Phys (2014) 141: 075102 J Chem Phys, (2014) 141: 22D532 Physical Review E (2015) 92: 012711Chem Phys Letters (2015)637: 1J Phys Chem B (2016) 120: 2658

33. Motivation Natural Description of Crowded Charge is a Fermi Distribution because it describes Saturationin a simple way used throughout Physics and Biophysics, where it has a different name! Simulating saturationby interatomic repulsion (Lennard Jones) is a significant mathematical challengeto be side-stepped if possibleEisenberg, Hyon and Liu (2010). JChemPhys 133: 104104

34. Does not Saturate Boltzmann distribution in PhysiologyBezanilla and Villalba-Galea J. Gen. Physiol. (2013) 142: 575–578 Saturates!

35. Fermi Description usesEntropy of Mixture of Spheresfrom Combinatoric Analysis W is the mixing entropy of UNEQUAL spheres with N available NON-UNIFORM sites Connection to volumes of spheres and voids, and other details are published in 5 papers J Comp Phys (2013) 247:88 J Phys Chem B (2013) 117:12051J Chem Phys (2014) 141: 075102 J Chem Phys, (2014) 141: 22D532 Physical Review E (2015) 92:012711 Expressions in other literature are not consistent with this entropy

36. Fermi Description usesEnergy of Mixture of Spheres Under Development by DexuanXie Jinn Liang Liu Bob Eisenberg Progress Report at Suzhou Thanks to Xingye Yue and Chun Liu

37. Voids are Needed (Electro)Chemical Potentialand Voids It is impossible to treat all ions and water molecules as hard spheres and at the same time have Zero Volume of interstitial Voidsbetween all particles

38. Consistent Fermi Approach is NovelConsistent Fermi approach has not been previously applied to ionic solutionsas far as we, colleagues, referees, and editors know Previous treatments* have inconsistent treatment of particle size They do not reduce to Boltzmann functionals in the appropriate limit Previous treatments often do not include non-uniform particle size Previous treatments* are inconsistent with electrodynamics and nonequilibrium flows including convection Details Previous treatments do not include discrete water or voids. They cannot deal with volume changes of channels, or pressure/volume in general Previous treatments do not include polarizable water with polarization as an output *Previous treatments Bazant, Storey & Kornyshev,. Physical Review Letters, 2011. 106(4): p. 046102. Borukhov, Andelman & Orland, Physical Review Letters, 1997. 79(3): p. 435. Li, B. SIAM Journal on Mathematical Analysis, 2009. 40(6): p. 2536-2566. Liu, J.-L., Journal of Computational Physics 2013. 247(0): p. 88-99. Lu & Zhou, Biophysical Journal, 2011. 100(10): p. 2475-2485. Qiao, Tu & Lu, J Chem Phys, 2014. 140(17):174102 Silalahi, Boschitsch, Harris & Fenley, JCCT2010. 6(12): p. 3631-3639. Zhou, Wang & Li Physical Review E, 2011. 84(2): p. 021901.

39. ChallengeCan Simplest Fermi Approach • Describe ion channel selectivity and permeation? • Describe non-ideal properties of bulk solutions? There are no shortage of chemical complexities to include, if needed! Classical Treatments of Chemical Complexities

40. Evidence (start) *Topic in Lecture Courses at Jiaotong, thanks to Zhenli Xuand at Politechnico Milano, thanks to Riccardo Sacco

41. Poisson Fermi Approach toBulk Solutions Same Fermi Poisson Equations, different model of nearby atoms in Hydration Shells

42. Bulk Solution How well does the Poisson Fermi Approach for Bulk Solutions? Same equations, different model of nearby atoms Occupancy is 6 + 12 Waters*held Constant in Model of Bulk Solution in this oversimplified Poisson Fermi Model Liu & Eisenberg (2015) Chem Phys Ltr 10.1016/j.cplett.2015.06.079 *in two shells: experimental Data on OccupancyRudolph & Irmer, Dalton Trans. (2013) 42, 3919 Mähler & Persson, Inorg. Chem. (2011) 51, 425

43. Parameters One adjustable Chem Phys Ltrs (2015) 637 1

44. Activity CoefficientsNa+ Cl-‘normalized’ free energy per mole

45. Activity CoefficientsCa2+ Cl2¯‘normalized’ free energy per mole

46. Gramicidin A Unusual SMALL Bacterial Channel often simulated and studied Margaret Thatcher, student of Nobelist Dorothy Hodgkin Bonnie Wallace leading worker Validation of PNP Solvers with Exact Solution following the lead of Zheng, Chen & Wei J. Comp. Phys. (2011) 230: 5239.

47. Three Dimensional TheoryComparison with Experiments Gramicidin A

48. Steric Effect is Large in (crowded) Gramicidin PNPF spheresvs PNP points Points Water Occupancy Spheres Current vsVoltage Spheres Points K+ Occupancy Spheres Points Three Dimensional Calculation Starting with Actual Structure

49. Cardiac Calcium Channel CaV.n Binding Curve Lipkind-Fozzard Model Liu & Eisenberg J Chem Phys 141(22): 22D532

50. Signature of Cardiac Calcium Channel CaV1.n Anomalous* Mole Fraction (non-equilibrium) Na Channel Ca Channel *Anomalous because CALCIUM CHANNEL IS A SODIUM CHANNEL at [CaCl2] 10-3.4 Ca2+ is conducted for [Ca2+] > 10-3.4, but Na+ is conducted for [Ca2+] <10-3. Liu & Eisenberg (2015) Physical Review E 92: 012711