to TaiChia Lin for inviting me And his

to Tai-Chia Lin for inviting me And his GREAT kindness through the years here and in the USA and Canada. 1

2

Electricity is Different Electricity has VERY different Physics in different systems. Current ALWAYS follows Maxwell exactly, say ± 10 -12 E changes to produce EXACT conservation of ‘Current’

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! 4

Mathematics describes only a tiny part of life, But Mathematics* Creates our Standard of Living *e. g. , Electricity, Computers, Fluid Dynamics, Optics, Structural Mechanics, …. 5

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 6

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 8 *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

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

How do a few atoms control MACROSCOPIC biology? A mathematical multiscale question More than anything else 10

Inputs Forward Problem Concentrations in Baths How does it work? Potentials in Baths V(t) Structure Output Current i(V, t) Open Channel i(V) PNPF F inite size ions Structure of Charge Permanent Gating Structure of Charge Polarization Inputs Inverse Problem i(t) Output Current i(V, t) 11

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 12

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

Integrated Circuit Technology as of ~2014 IBM Power 8 Too small to see! 14

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 15 − Note: Biology occurs in different solutions from those used in most measurements

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 16

How do a few atoms control (macroscopic) Biological Function? Answer, oversimplified: A few atoms control the electric field Much as they do in transistors 17

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. 18

Maxwell Equations as written by Heaviside, NOT Maxwell, using Gibbs notation Electrostatics Electrodynamics’ Magnetostatics Magnetodynamics

Maxwell’s Equations KEY IDEA ‘Current’ is Conserved PERFECTLY For ANY Polarization 20

Maxwell’s Ampere’s Law so Current is Conserved For ANY Property of Matter 21

Current is Conserved PERFECTLY Because the Electric Field Changes to accommodate ANY physics 22

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

‘Charge’ is an Abstraction with VERY different Physics in different systems Physics of Charge Flow Varies Profoundly Current is NOT the flow of charges Maxwell Equations Control ANY Physics of Current and 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 26

Rate Models Fail (until amended) because Current-in does not equal Current-out!! (if rate constants are independent of potential) 27

Current-in Current-out implies Artifactual Charge SERIOUS CONSEQUENCES Feynman One percent more electrons than protons, would lift the Entire Earth! Failed Models have Metastasis of States 28

Failed Models have Metastasis of States Zagotta, Hoshi, Dittman and Aldrich (1994) J Gen Physiol 103 279 -319. 29

Thermodynamics, Statistical Mechanics, Molecular Dynamics are UNSUITED for DEVICES Thermodynamics, Statistical Mechanics, Molecular Dynamics have 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 30

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 31 − 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+ K+ Calcium Ca 2+ Ca++ 3Å Chloride Cl. Cl -

All of Biology occurs in Salt Solutions of definite composition and concentration and that matters! Salt Water is the Liquid of Life Pure H 2 O is toxic to cells and molecules! Sodium Na+ Potassium K+ K+ Calcium Ca 2+ Ca++ Chloride Cl- Cl - 3Å 33

All of Biology occurs in Salt Solutions of definite composition and concentration and that matters! Salt Water is the Liquid of Life Pure H 2 O is toxic to cells and molecules! Salt Water is a Complex Fluid Main Ions are Hard Spheres, close enough Sodium Na+ Potassium K+ K+ Calcium Ca 2+ Ca++ 3Å Chloride Cl- Cl - 34

Central Result of Physical Chemistry Ions in a solution are a Highly 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 101 M in typical biological system of proteins, nucleic acids, and channels. Learned from Doug Henderson, J. -P. Hansen, Stuart Rice, among others…Thanks! 35

Electrolytes are Complex Fluids ‘Everything’ interacts with everything else Treating a Complex Fluid After 690 pages and 2604 references, as if it were a Simple Fluid will produce Elusive Results because Every Ion Interacts with Everything “Single-Ion Solvation … Elusive* ” Hünenberger & Reif, 2011 * ‘elusive’ is in the title! 36

It is not surprising that Inconsistent Treatments of ionic solutions have been so Unsuccessful 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. 1 M or so. ” Kunz, W. "Specific Ion Effects" World Scientific Singapore, 2009; p 11. 37

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 38 − Note: Biology occurs in different solutions from those used in most measurements

Shielding is a defining property of Complex Fluids Mobile Charges Define Semiconductors and Ionic Solutions Far Field (macroscopic) boundaries 39

When a charge is added to an ionic solution, the other charge rearrange to form an Ionic Atmosphere called Shielding or Screening 40

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 41

Main Qualitative Result Shielding in Gramicidin Hollerbach & Eisenberg 42

! s m r e t g n i d l e i h s No 43

Reconciling Mass Action and Shielding Maxwell/Kirchoff will no doubt be a Long Journey

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

That direction needs to include the electric field, calculated and calibrated, global and local if the journey is ever to end, in my view.

Replacement of “Law of Mass Action” is Feasible for Ionic Solutions using the All Spheres (primitive = implicit solvent model of ionic solutions) and Theory of Complex Fluids 47

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 48

Variational Approach En. Var. A ‘Law’ of Mass Action including Interactions From Bob Eisenberg p. 1 -6, in this issue Conservative Dissipative

Energetic Variational Approach allows ‘exact’ computation of Flow and Interactions in Complex Fluids like Liquid Crystals Engineering needs Calibrated Theories and Simulations Engineering Devices almost always use flow Classical theories and Molecular Dynamics have difficulties with flow, interactions, and complex fluids 50

Perhaps the first Consistent Model of a Protein Machine Francisco Bezanilla Allen Tzyy-Leng Horng Chun Liu Bob Eisenberg 洪子倫 51

extracellular intracellular Figure 1. Geometric configuration of the reduced mechanical model including the attachments of arginines to the S 4 segment.

Results: Case 1, V (m. V) changed from -90 to -8 and back to -90 The stochastic trajectory of each arginine is calculated based on its probability of existence at zones 1 and 3 to determine using Zi, CM at zone 1 or 3 respectively.

Energetic Variational Approach En. Var. A Chun 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 Action Integral, after pullback Rayleigh Dissipation Function Composite Variational Principle Euler Lagrange Equations Field Theory of Ionic Solutions: Liu, Ryham, Hyon, Eisenberg Allows boundary conditions and flow Deals Consistently with Interactions of Components 54

Dissipation Principle Conservative Energy dissipates into Friction Number Density time Thermal Energy Permanent Charge of protein valence proton charge ci number density; Hard Sphere Terms thermal energy; Di diffusion coefficient; n negative; p positive; zi valence; ε dielectric constant Note that with suitable boundary conditions 55

Energetic Variational Approach En. Var. A is defined by the Euler Lagrange Process, as I understand the pure math from Craig Evans (UC Berkely) which gives Equations like PNP BUT I leave it to you (all) to argue/discuss with Craig about the purity of the process when two variations are involved 56

PNP (Poisson Nernst Planck) for Spheres Non-equilibrium variational field theory En. Var. A Nernst Planck Diffusion Equation for number density cn of negative n ions; positive ions are analogous Diffusion Coefficient Thermal Energy Coupling Parameters Ion Radii Poisson Equation Number Densities Dielectric Coefficient valence proton charge Eisenberg, Hyon, and Liu Permanent Charge of Protein 57

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 Not in Semiconductor valence proton charge Thermal Energy 58

All we have to do is Solve them! with Boundary Conditions defining Charge Carriers ions, holes, quasi-electrons Geometry 59

Solution* of PNP Equation *MATHEMATICS Explicit expressions for Prob {. |. } are known! This solution was actually DERIVED by computing many conditional probability measures explicitly by several (~4) multidimensional (~6) analytical integrations Eisenberg, Klosek, & Schuss (1995) J. Chem. Phys. 102, 1767 -1780 Eisenberg, B. (2000) in Biophysics Textbook On Line "Channels, Receptors, and Transporters" Eisenberg, B. (2011). Chemical Physics Letters 511: 1 -6

Please do not be deceived by the eventual simplicity of Results. This took >2 years! Solution was actually DERIVED with explicit formulae from probability measures from ~4 Doubly Conditioned Stochastic Process involving Analytical Evaluation of Multidimensional (~6) Convolution Integrals, each Eisenberg, Klosek, & Schuss (1995) J. Chem. Phys. 102, 1767 -1780 Eisenberg, B. (2000) in Biophysics Textbook On Line "Channels, Receptors, Transporters" Eisenberg, B. (2011). Chemical Physics Letters 511: 1 -6

All we have to do is Solve them! Don’t Despair Semiconductor Technology has Already Done That! 62

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

Can we do as well for Ions? In water Channels Nanodevices Batteries Supercapacitors …. using En. Var. A, steric PNP, PNPF Or whatever!!! Help needed!!!

The End Any Questions? 65

Fermi Poisson Approach 66

A Nonlocal Poisson-Fermi Model for Electrolyte Solutions Jinn Liang 劉晉良 Liu Jinn-Liang is first author on our 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 Chem Phys Letters (2015) 637: 1 J Phys Chem B (2016) 120: 2658 67

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 68

Does not Saturate Boltzmann distribution in Physiology Bezanilla and Villalba-Galea J. Gen. Physiol. (2013) 142: 575– 578 Saturates! 69

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 70

Fermi Description uses Energy of Mixture of Spheres Under Development by Jinn Liang Liu 劉晉良 and Bob Eisenberg 71

(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 72

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. 73

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 74

Evidence (start) 75

Poisson Fermi Approach to Bulk Solutions Same Fermi Poisson Equations, different model of nearby atoms in Hydration Shells 76

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 77

Parameters One adjustable Chem Phys Ltrs (2015) 637 1 78

Activity Coefficients Na+ Cl‘normalized’ free energy per mole 79

Activity Coefficients Ca 2+ Cl 2¯ ‘normalized’ free energy per mole 80

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. 81

Three Dimensional Theory Comparison with Experiments Gramicidin A 82

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 83

Cardiac Calcium Channel Ca. V. n Lipkind-Fozzard Model Binding Curve Liu & Eisenberg J Chem Phys 141(22): 22 D 532 84

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 85

More Detail COMPUTING FLOW 86

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 87

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 88

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. 89

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. 90

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 91

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. 92

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 93

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 94

More Detail INSIDE CHANNELS 95

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 96

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 97

Inside Gramicidin Water Density Dielectric Function an OUTPUT of model Liu & Eisenberg J Chem Phys 141: 22 D 532 98

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 99

Inside the Cardiac Calcium Channel Ca. V 1. n Electric Potential Steric Potential Estimator of Crowding Liu & Eisenberg (2015) Phys Rev E 92: 012711 10 0

Biology is made of Devices and they are Multiscale 101

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 1 0 *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