to Weishi Liu for inviting me And his

to Weishi Liu 刘�世 for inviting me And his kindness in so many interactions through the years 1

Mathematics describes only a tiny part of life, But Mathematics* Creates our Standard of Living *e. g. , Electricity, Computers, Fluid Dynamics, Optics, Structural Mechanics, …. 2

Remarkable Fact nearly ALL electrical technology is based ONLY on Conservation of Current* in one dimensional systems with branches *Kirchoff’s Current Law 3

“ALL electrical technology” includes all Computers Cell Phones Video …. 4

Computers Cell Phones Video, …. . depend only on Conservation of Current in One Dimension with branches 5

Integrated Circuit Technology as of ~2014 IBM Power 8 Too small to see! 6

Amplifier 741 7

Electricity is Different because …. . 8

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. 9

Electricity is Exact Maxwell Equations* Electrostatics Electrodynamics *as written by Heaviside, not Maxwell, using Gibbs notation Magnetostatics Magnetodynamics

Electricity is Different Bob Eisenberg April 18, 2017 In the language of mathematics In the language of physics (1) (2) J is defined from experimental measurements of B and ‘Current’ is conserved exactly, always, everywhere Page 11

Current is Conserved PERFECTLY Because the Electric Field Changes to accommodate ANY physics 12

Mathematics of Continuity in Maxwell equations can Create New Kind of Physics, New Kind of Charge When we unplug a computer power supply, we often CREATE SPARKS, i. e. , a PLASMA, NEW KIND of current flow a Pop!

Maxwell Equations are Special Continuity of Current is Exact even though Physics of Charge Flow Varies Profoundly Current is NOT the flow of charges

Current is Abstract with Different Physics in Different Systems NOT the Flux of Charges Hungarian Journal of Industry and Chemistry 44(1): 1 -28 ar. Xiv: 1502. 07251 Hungarian Journal of Industry and Chemistry (2016) 44 1 -28 ar. Xiv: 1502. 07251 15

How can we use mathematics to describe biological systems? I believe some biology is Physics ‘as usual’ ‘Guess and Check’ But you have to know which biology! 16

Mobile Charges Define Semiconductors and Ionic Solutions 17

Semiconductor PNP Equations For Point Charges Dielectric Coefficient Poisson’s Equation Permanent Charge of Protein Valence Proton charge Cross sectional Area Drift-diffusion & Continuity Equation Flux Number Densities Diffusion Coefficient Chemical Potential valence proton charge Thermal Energy 18

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 PARAMETERS PNP of POINTS is TRANSFERRABLE Analytical - Numerical Analysis should be attempted using techniques of Weishi Liu University of Kansas Tai-Chia Lin National Taiwan University & Chun Liu PSU

Rate Models Fail (until amended) because Current-in does not equal Current-out!! (if rate constants are independent of potential) 20

Rate Models Fail because Current-in does not equal Current-out!! (if rate constants are independent of potential) 21

Cause of Frustration Biochemical Models are Rarely TRANSFERRABLE Do Not Fit Data even approximately in more than one solution* Title Chosen by Editors: Charlie Brenner, Angela Hopp American Society for Biochemistry and Molecular Biology *i. e. , in more than one concentration or type of salt, like Na +Cl− or K+Cl 22 − Note: Biology occurs in different solutions from those used in most measurements

Question What does this have to do with biology? Answer All biology involves electricity All biology occurs in solutions that conduct electricity A LOT All biology occurs in Ion Solutions Sodium Na+ Potassium K+ Calcium Ca 2+ K+ Ca++ 3Å Chloride Cl. Cl -

Ion Channels are the Valves of Cells Ion Channels are the Main Controllers of Biological Function One Ion trajectory Selectivity Ions in Water are the Different Ions carry Different Signals Liquid of Life Hard Spheres Na+ Chemical Bonds are lines Surface is Electrical Potential Red is negative (acid) Blue is positive (basic) Ca++ + 0. 7 nm = Channel Diameter K+ ~30 Å Figure of omp. F porin by Raimund Dutzler 3Å 2 4

The Cell Defined by a Membrane Note: intra-cellular compartments are defined by their membranes Bob Eisenberg: beisenbe@rush. edu

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 *nearly pico-valves: diameter is 400 – 900 x 10 -12 meter; diameter of atom is ~200 x 10 -12 meter K+ ~30 x 10 -9 meter *Device is a Specific Word, that exploits specific mathematics & science 26

Omp. F Biochemist’s View Structure All Atoms View Chemical Bonds are lines Surface is Electrical Potential Red is positive Bob Eisenberg: Blue is negative beisenbe@rush. edu

A few atoms make a BIG Difference Glycine G replaced by Aspartate D Ompf Omp. F 1 M/1 M G 119 D 1 M/1 M Omp. F 0. 05 M/0. 05 M G 119 D Current Voltage relation determined by John Tang in Bob Eisenberg’s Lab Structure determined by Raimund Dutzler in Tilman Schirmer’s lab 28

How do a few atoms control (macroscopic) Biological Function? Answer, oversimplified: A few atoms control the electric field Much as they do in transistors 29

Motivation and Assumption for Fermi-Poisson Largest Effect of Crowded Charge is Saturation cannot be described at all by classical Poisson Boltzmann approach 劉晉良 Jinn Liang Liu Nonlocal Poisson-Fermi APPROXIMATE Models J Comp Phys (2013) 247: 88 J Phys Chem B (2013) 117: 12051 J Chem Phys (2014) 141: 075102 J Chem Phys, (2014) 141: 22 D 532 Physical Review E (2015) 92: 012711 Chem Phys Letters (2015) 637: 1 J Phys Chem B (2016) 120: 2658 Jinn-Liang is first author on our papers 30

Fermi Poisson Approach 31

Motivation Natural Description of Crowded Charge is a Fermi Distribution because it describes Saturation in a simple way used throughout Physics and Biophysics, where it has a different name! Simulating saturation by interatomic repulsion (Lennard Jones) is a significant mathematical challenge to be side-stepped if possible Eisenberg, Hyon and Liu (2010). JChem. Phys 133: 104104 32

Does not Saturate Boltzmann distribution in Physiology Bezanilla and Villalba-Galea J. Gen. Physiol. (2013) 142: 575– 578 Saturates! 33

Fermi Description uses Entropy of Mixture of Spheres from 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 Expressions in other literature are not consistent with this entropy papers J Comp Phys (2013) 247: 88 J Phys Chem B (2013) 117: 12051 J Chem Phys (2014) 141: 075102 J Chem Phys, (2014) 141: 22 D 532 Physical Review E (2015) 92: 012711 34

Fermi Description uses Energy of Mixture of Spheres Under Development by Jinn Liang Liu 劉晉良 and Bob Eisenberg 35

(Electro)Chemical Potential and Voids are Needed It is impossible to treat all ions and water molecules as hard spheres and at the same time have Zero Volume of interstitial Voids between all particles 36

Consistent Fermi Approach is Novel Consistent Fermi approach has not been previously applied to ionic solutions as 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, JCCT 2010. 6(12): p. 3631 -3639. Zhou, Wang & Li Physical Review E, 2011. 84(2): p. 021901. 37

Challenge Can 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 38

Evidence (start) 39

Poisson Fermi Approach to Bulk Solutions Same Fermi Poisson Equations, different model of nearby atoms in Hydration Shells 40

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 Occupancy Rudolph & Irmer, Dalton Trans. (2013) 42, 3919 Mähler & Persson, Inorg. Chem. (2011) 51, 425 41

Parameters One adjustable Chem Phys Ltrs (2015) 637 1 42

Activity Coefficients Na+ Cl‘normalized’ free energy per mole 43

Activity Coefficients Ca 2+ Cl 2¯ ‘normalized’ free energy per mole 44

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. 45

Three Dimensional Theory Comparison with Experiments Gramicidin A 46

Steric Effect is Large in (crowded) Gramicidin PNPF spheres vs PNP points Points Water Occupancy Spheres Current vs Voltage K+ Occupancy Points Spheres Points Three Dimensional Calculation Starting with Actual Structure 47

Cardiac Calcium Channel Ca. V. n Lipkind-Fozzard Model Binding Curve Liu & Eisenberg J Chem Phys 141(22): 22 D 532 48

Signature of Cardiac Calcium Channel Ca. V 1. n Anomalous* Mole Fraction (non-equilibrium) Na Channel Ca Channel *Anomalous because CALCIUM CHANNEL IS A SODIUM CHANNEL at [Ca. Cl 2] 10 -3. 4 Ca 2+ is conducted for [Ca 2+] > 10 -3. 4, but Na+ is conducted for [Ca 2+] <10 -3. Liu & Eisenberg (2015) Physical Review E 92: 012711 49

More Detail COMPUTING FLOW 50

What is PNPF? PNPF = Poisson-Nernst-Planck-Fermi Implemented fully in 3 D Code to accommodate 3 D Protein Structures Flow Three Dimensional computation is facilitated by using 2 nd order equations Force approximates dielectric of entire bulk solution including correlated motions of ions, following Santangelo 20061 with Liu’s corrected and consistent Fermi treatment of spheres. 2, 3, 4 We introduce 3, 4 two second order equations and boundary conditions That give the polarization charge density 1 Phys. Rev E (2006) 73: 041512 2 Phys. Rev Ltrs (2011) 106: 046102 3 JComp. Phys (2013) 247: 88 4 J Phys. Chem B (2013) 117: 12051 51

Poisson-Fermi Analysis is NON-Equilibrium Flows are Essential in Devices & Biology Structure is Essential in Devices & Biology Implemented fully in 3 D Code to accommodate 3 D Protein Structures Flows cease only at death 1) PNPF uses treatment by Santangelo 20061 used by Kornyshev 20112 of near/far fields crudely separated by fixed correlation length 2) PNPF introduces steric potential 3, 4 so unequal spheres are dealt with consistently 3) PNPF force equation reduces 3, 4 to pair of 2 nd order PDE’s and Appropriate boundary conditions that are consistent and allow Robust and Efficient Numerical Evaluation 4) PNPF combines Force Equation and Nernst-Planck Description of Flow 1 Phys. Rev E (2006) 73: 041512 2 Phys. Rev Ltrs (2011) 106: 046102 3 JComp. Phys (2013) 247: 88 4 J Phys. Chem B (2013) 117: 12051 52

Computational Problems Abound are Limiting if goal is to fit real data Scientists must grasp, ……. not just reach, if we want devices to work and models to be transferrable It is very easy to get results that only seem to converge, and are in fact Not Adequate approximations to the converged solutions Jerome, J. (1995) Analysis of Charge Transport. Mathematical Theory and Approximation of Semiconductor Models. New York, Springer-Verlag. Markowich, P. A. , C. A. Ringhofer and C. Schmeiser (1990). Semiconductor Equations. New York, Springer-Verlag. Bank, R. E. , D. J. Rose and W. Fichtner (1983). Numerical Methods for Semiconductor Device Simulation IEEE Trans. on Electron Devices ED-30(9): 1031 -1041. Bank, R, J Burgler, W Coughran, Jr. , W Fichtner, R Smith (1990) Recent Progress Algorithms for Semiconductor Device Simulation Intl Ser Num Math 93: 125 -140. Kerkhoven, T. (1988) On the effectiveness of Gummel's method SIAM J. Sci. & Stat. Comp. 9: 48 -60. Kerkhoven, T and J Jerome (1990). "L(infinity) stability of finite element approximations to elliptic gradient equations. " Numer. Math. 57: 561 -575. 53

Computational Electronics has solved these problems over the last 40 years in thousands of papers used to design our digital devices Devices and calculations work Models are transferrable Vasileska, D, S Goodnick, G Klimeck (2010) Computational Electronics: Semiclassical and Quantum Device Modeling and Simulation. NY, CRC Press. Selberherr, S. (1984). Analysis and Simulation of Semiconductor Devices. New York, Springer-Verlag. Jacoboni, C. and P. Lugli (1989). The Monte Carlo Method for Semiconductor Device Simulation. New York, Springer Verlag. Hess, K. (1991). Monte Carlo Device Simulation: Full Band Beyond. Boston, MA USA, Kluwer. Hess, K. , J. Leburton, U. Ravaioli (1991). Computational Electronics: Semiconductor Transport and Device Simulation. Boston, Kluwer. Ferry, D. K. (2000). Semiconductor Transport. New York, Taylor and Francis. Hess, K. (2000). Advanced Theory of Semiconductor Devices. New York, IEEE Press. Ferry, D. K. , S. M. Goodnick and J. Bird (2009). Transport in Nanostructures. New York, Cambridge University Press. It is very easy to get results that only seem to converge, but are in fact not adequate approximations to the converged solutions. Jerome, J. W. (1995). Analysis of Charge Transport. Mathematical Theory and Approximation of Semiconductor Models. New York, Springer-Verlag. 54

Keys to Successful Computation 1) Avoid errors by checking against analytical solutions of Guowei and collaborators 2) Avoid singularities (i. e. , acid/base charges) on protein boundaries that wreck convergence 3) Use a simplified Matched Interface Boundary s. MIB method of Guowei and collaborators modified to embed Scharfetter Gummel SG criteria of computational electronics (extended to include steric effects). Scharfetter Gummel is REQUIRED to ENSURE CONTINUITY OF CURRENT Charge Conservation is not enough Scharfetter and Gummel, IEEE Trans. Elec. Dev. 16, 64 (1969) P. Markowich, et al, IEEE Trans. Elec. Dev. 30, 1165 (1983). Zheng, Chen, and G. -W. Wei, J. Comp. Phys. 230, 5239 (2011). Geng, S. Yu, and G. -W. Wei, J. Chem. Phys. 127, 114106 (2007). S. M. Hou and X. -D. Liu, J. Comput. Phys. 202, 411 (2005). J. -L. Liu, J. Comp. Phys. 247, 88 (2013). 4) Modified Successive Over-relaxation SOR for fourth order PNPF 55

Poisson Fermi Status Report Nonequilibrium implemented fully in 3 D Code to accommodate 3 D Protein Structures Only partially compared to experiments In Bulk or Channels, so far. 56

Poisson Fermi Status Report • Gramicidin tested with real three dimensional structure, including flow Physical Review E, 2015. 92: 012711 • Ca. V 1. n EEEE, i. e. , L-type Calcium Channel, tested with homology model J Phys Chem B, 2013 117: 12051 (nonequilibrium data is scarce) • PNPF Poisson-Nernst-Planck-Fermi for systems with volume saturation General PDE, Cahn-Hilliard Type, Four Order, Pair of 2 nd order PDE’s Not yet tested by comparison to bulk data J Chem Phys, 2014. 141: 075102; J Chem Phys, 141: 22 D 532 Numerical Procedures tailored to PNPF have been tested J Comp Phys, 2013 247: 88; Phys Rev E, 2015. 92: 012711 NCX Cardiac Ca 2+/Na+ exchanger branched Y shape KNOWN structure. Physical analysis of a transporter using consistent mathematics and known crystallographic structure This is an all atom calculation with polarizable water molecules as outputs J Phys Chem B 120: 2658 57

NCX Sodium Calcium Transporter Crucial* to Cardiac Function strongly implicated in short term memory and learning Green is Sodium Blue is Calcium *More than 1, 000 experimental references in Blaustein & Lederer Physiological Reviews, 1999 Liu, J. -L. , H. -j. Hsieh and B. Eisenberg (2016) J Phys Chem B 120: 2658 -2669 58

More Detail INSIDE CHANNELS 59

Steric Effect is Significant Gramicidin is Crowded Shielding is Substantial Electric Potential Steric Potential Shielding has been ignored in many papers, where Results are often at one concentration or unspecified concentration, as in most molecular dynamics Shielding Channel is often described as a potential profile This is inconsistent with electrodynamics as in classical rate models 60

Gramicidin Two K+ Binding Sites OUTPUTS of our calculations Binding sites are prominent in NMR measurements & MD calculations BUT they VARY with conditions in any consistent model and so cannot be assumed to be of fixed size or location 61

Inside Gramicidin Water Density Dielectric Function an OUTPUT of model Liu & Eisenberg J Chem Phys 141: 22 D 532 62

Inside the Cardiac Calcium Channel Ca. V 1. n Water Density Liu & Eisenberg (2015) Phys Rev E 92: 012711 Dielectric Function An Output of this Model Liu & Eisenberg J Chem Phys 141(22): 22 D 532 63

Inside the Cardiac Calcium Channel Ca. V 1. n Electric Potential Steric Potential Estimator of Crowding Liu & Eisenberg (2015) Phys Rev E 92: 012711 64

The End Any Questions? 65