Thank you for the chance to visit many times and work together!

1. Thanks to PSU Math particularly Jinchao Xu, Xiantao Li, and Chun Liu Thank you for the chance to visit many times and work together!

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

3. Mathematics* Creates our Standard of LivingMathematics replaces Trial and Errorwith Computation *e.g., Electricity, Computers, Fluid Dynamics, Optics, Structural Mechanics, ….. Thousands of Molecular Biologists Study Ion Channels Everyday,One protein molecule at a time using amplifiers like the AxoPatch

4. So there is an enormous opportunity for MATHEMATICAL MOLECULARBIOLOGY! But you have to know which molecules That is where I can help, ……. I hope!

5. General Theme Mathematics of Molecular Biology Provides Great Opportunity Biology Provides the Data Engineering Provides the Approach Mathematics Provides the Toolsparticularly variational methods that allow ‘everything’ to interact with ‘everything’ else

6. 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 Euler Lagrange Equations Field Theory of Ionic Solutions that allows boundary conditions and flow and deals Consistently with Interactions of Components

7. 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

8. 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

9. Mathematics of Molecular Biology is (mostly) Reverse Engineering i.e., solving specific Inverse Problems How does it work? How do a few atoms control Biological Function?

10. Ompf G119D A few atoms make a BIG Difference Glycine replaed by Aspartate Structure determined by Raimund Dutzler in Tilman Schirmer’s lab Current Voltage relation by John Tang in Bob Eisenberg’s Lab

11. Mathematics of Molecular Biology How does it work? How do a few atoms control (macroscopic) Biological Function? Inherently multiscale Inherently nonequilibrium

12. How do a few atoms control (macroscopic) Biological Function? Inherently multiscale. Inherently nonequilibrium. Inherently involves macroscopic boundary conditions. Inherently involves nonideal ionic solutions.

13. + ~30 Å Ion Channelsare theValves of CellsIon Channels are the Main Controllers of Biological Function Ions in Water* are the Selectivity Different Ions carry Different Signals Liquid of Life *Pure H2O is toxic to cells & proteins Na+ Hard Spheres Ca++ Chemical Bonds are lines Surface is Electrical Potential Redis negative (acid) Blueis positive (basic) K+ 3 Å 0.7 nm = Channel Diameter Figure of ompF porin by Raimund Dutzler

14. Valves Control Flow Classical Theory: 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)

15. Tutorial What is the biological data?

16. The Membrane The Cell

17. Closed Channel Open Channel ION CHANNELS – Biological Role Ion channels coordinate contraction of cardiac muscle making the heart a pumpIon channels coordinate contraction in skeletal muscleIon channels control all electrical activity and produce nerve signalsIon 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 with genes (blueprints) manipulated by molecular genetics Ion channels have structures shown by x-ray crystallography in favorable cases

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

19. SIMULATION of GRAMICIDIN CHANNEL Visualization: Theoretical and Computational Biophysics Group Beckman Institute. http://www.ks.uiuc.edu/Research/vmd Bob Eisenberg: beisenbe@rush.edu

20. BioMOCA: SIMULATION of GRAMICIDIN CHANNEL Umberto Ravaioli and Trudy van der Straaten Univ of Illinois Urbana-Champaign Bob Eisenberg: beisenbe@rush.edu

21. Single Channel Current open closed Slide from Mike Fill Thanks! Function of SINGLE isolated RyR Channels in Artificial Planar Lipid Bilayers AxoPatchPatch-Clamp Amplifier Designed at Rush Planar Bilayer Ca Fused Vesicle Experimental Chamber Teflon Septa 80-100 µM Diameter

22. Channel Structure Does Not Change once the channel is open Amplitude vs. Duration Current vs. time Open Closed Open Amplitude, pA 5 pA 100 ms Open Duration /ms Lowpass Filter = 1 kHz Sample Rate = 20 kHz Typical Raw Single Channel Records Ca2+ Release Channel of Inositol Trisphosphate Receptor: slide and data from Josefina Ramos-Franco. Thanks!

23. Single Channel Currents have little variance John TangRush Medical Center

24. Goal: Understand Selectivity well enough to Make a Calcium Channel using techniques of molecular genetics, site-directed Mutagenesis

25. Channels are Selective because Diameter Matters Ions are NOT Ideal Potassium K+ = Na+ Sodium / K+ Na+ 3 Å Ideal Ions are Identical if they have the same charge

26. + ~30 Å Channels are SelectiveDifferent Ions Carry Different Signals through Different Channels ompF porin Ca++ Na+ K+ 0.7 nm = Channel Diameter 3 Å Diameter mattersDiameter is the Only Difference between K+ and Na+ In ideal solutions K+ = Na+ Flow time scale is 0.1 msec to 1 min Figure of ompF porin by Raimund Dutzler

27. Experiments have builtTwo Synthetic Calcium Channels MUTANT ─ Compound Calcium selective Unselective Wild Type As density of permanent charge increases, channel becomes calcium selectiveErev ECa in0.1M1.0 M CaCl2 built by Henk Miedema, Wim Meijberg of BioMade Corp.,Groningen, Netherlands Miedema et al, Biophys J 87: 3137–3147 (2004) Mutants of ompF Porin Designed by Theory Glutathione derivatives Atomic Scale || Macro Scale

28. Channels are only Holes Why can’t we understand and build them? Where to start? Why not compute all the atoms?

29. Multiscale Issues Journal of Physical Chemistry C (2010 )114:20719 Three Dimensional (106)3 Biological Scales Occur Together so must be Computed Together This may be impossible in simulations Physicists and Engineers rarely try

30. Why can’t we understand and build channels? Uncalibrated Simulations will not make devices that actually work Calibration is Hard Work particularly for Non-Ideal systems with Correlations, Finite Size effects, and Flows

31. Where do we start? Physics ‘As Usual’‘Guess’, Calculate andCheck Crowded Charges

32. 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

33. Charge Density22 M EC#: Enzyme Commission Number based on chemical reaction catalyzed #AA: Number of residues in the functional pocket MS_A^3: Molecular Surface Area of the Functional Pocket (Units Angstrom^3) CD_MS+: Charge Density (positive) CD_MS-: Charge Density (negative) CD_MSt: Total Charge density Jimenez-Morales, Liang, Eisenberg

34. Working Hypothesis Biological Adaptation is Crowded Ions and Side Chains Everything interacts

35. Working Hypothesis Interactions in Channels come mostly from Finite Size Effects Chemically Specific Propertiescome from Diameter and Charge learned from Doug Henderson, J.-P. Hansen, Stuart Rice, among others…Thanks!

36. Bulk Solutions: Interactions come mostly from Finite Size Effects Chemically Specific Properties of ions (e.g. activity = free energy per mole) are known to come from interactions of their Diameter and Charge and dielectric ‘constant’ of ionic solution Atomic Detail ‘All Spheres’ Model’= Primitive Implicit Solvent Modellearned from Doug Henderson, J.-P. Hansen, Stuart Rice, among others…Thanks!

37. Na+ Three Channel Types RyR, CaV= EEEE, andNav= DEKA analyzed successfully* in a wide range of solutions by the ‘All Spheres’ Primitive Model Implicit solvent model of open channel ½ ½ ½ ½ ½ ½ ½ Na+ Na+ Na+ ionsandproteinside chains are hard spheres in this model * Many methods have been used in more than 30 papers since Nonner and Eisenberg, 1998 ½

38. Solved with Many Methods with similar results Metropolis Monte Carlo MSA(mean spherical approximation SPM (primitive solvent model) DFT (density functional theory of fluids), MC-loc(MC with localized side chains) Non-equilibrium Multiscale DFT-PNP (Poisson Nernst Planck) EnVarA (Energy Variational Approach) DMC Dynamic Monte Carlo NP-LEMC (Nernst Planck Local Equilibrium Monte Carlo) Steric PNP Fermi-Poisson (fourth order PDE); etc.

39. Best Evidence is from the RyRReceptor Dirk GillespieDirk_Gillespie@rush.edu Gerhard Meissner, Le Xu, et al, not Bob Eisenberg  More than 120 combinations of solutions & mutants 7 mutants with significant effects fit successfully

40. The Geometry • Selectivity Filter • is 10 Å long and 8 Å in diameter • confines four D4899negative amino acids • Four E4900positive amino acids are onlumenal side,overlapping D4899 • Cytosolic distributedcharge Protein Cytoplasm Lumen Protein D. Gillespie et al., J. Phys. Chem. 109, 15598 (2005).

41. Ryanodine Receptor Pore Fig 3 The RyR1 conduction pathway from Zalk et al, Nature, 2014, 10.1038/nature13950 “b, Scheme … of all the negatively charged residues in the ionic pathway (red dots) and the [other] negatively charged residues”

42. 1. Gillespie, D., Energetics of divalent selectivity in a calcium channel: the ryanodine receptor case study. Biophys J, 2008. 94(4): p. 1169-1184. 2. Gillespie, D. and D. Boda, Anomalous Mole Fraction Effect in Calcium Channels: A Measure of Preferential Selectivity. Biophys. J., 2008. 95(6): p. 2658-2672. 3. Gillespie, D. and M. Fill, Intracellular Calcium Release Channels Mediate Their Own Countercurrent: Ryanodine Receptor. Biophys. J., 2008. 95(8): p. 3706-3714. 4. Gillespie, D., W. Nonner, and R.S. Eisenberg, Coupling Poisson-Nernst-Planck and Density Functional Theory to Calculate Ion Flux. Journal of Physics (Condensed Matter), 2002. 14: p. 12129-12145. 5. Gillespie, D., W. Nonner, and R.S. Eisenberg, Density functional theory of charged, hard-sphere fluids. Physical Review E, 2003. 68: p. 0313503. 6. Gillespie, D., Valisko, and Boda, Density functional theory of electrical double layer: the RFD functional. Journal of Physics: Condensed Matter, 2005. 17: p. 6609-6626. 7. Gillespie, D., J. Giri, and M. Fill, Reinterpreting the Anomalous Mole Fraction Effect. The ryanodine receptor case study. Biophysical Journal, 2009. 97: p. pp. 2212 - 2221 8. Gillespie, D., L. Xu, Y. Wang, and G. Meissner, (De)constructing the Ryanodine Receptor: modeling ion permeation and selectivity of the calcium release channel. Journal of Physical Chemistry, 2005. 109: p. 15598-15610. 9. Gillespie, D., D. Boda, Y. He, P. Apel, and Z.S. Siwy, Synthetic Nanopores as a Test Case for Ion Channel Theories: The Anomalous Mole Fraction Effect without Single Filing. Biophys. J., 2008. 95(2): p. 609-619. 10. Malasics, A., D. Boda, M. Valisko, D. Henderson, and D. Gillespie, Simulations of calcium channel block by trivalent cations: Gd(3+) competes with permeant ions for the selectivity filter. Biochim Biophys Acta, 2010. 1798(11): p. 2013-2021. 11. Roth, R. and D. Gillespie, Physics of Size Selectivity. Physical Review Letters, 2005. 95: p. 247801. 12. Valisko, M., D. Boda, and D. Gillespie, Selective Adsorption of Ions with Different Diameter and Valence at Highly Charged Interfaces. Journal of Physical Chemistry C, 2007. 111: p. 15575-15585. 13. Wang, Y., L. Xu, D. Pasek, D. Gillespie, and G. Meissner, Probing the Role of Negatively Charged Amino Acid Residues in Ion Permeation of Skeletal Muscle Ryanodine Receptor. Biophysical Journal, 2005. 89: p. 256-265. 14. Xu, L., Y. Wang, D. Gillespie, and G. Meissner, Two Rings of Negative Charges in the Cytosolic Vestibule of T Ryanodine Receptor Modulate Ion Fluxes. Biophysical Journal, 2006. 90: p. 443-453.

43. DFT/PNPvsMonte Carlo Simulations Concentration Profiles Misfit Nonner, Gillespie, Eisenberg Different Methods give Same Results NO adjustable parameters

44. Error < 0.1 kT/e ChannelPREDICTION 62 measurementsThanks to Le Xu! Note Break in Axis AMFEfor Na+/Cs+ mixturesPredicted before measurements AMFE had not been previously observed Mean ± Standard Error of Mean Bulk Solution 2% error Gillespie, Meissner, Le Xu, et al

45. Divalents KCl CaCl2 NaCl CaCl2 Misfit CsCl CaCl2 KCl MgCl2 Misfit

46. KCl Gillespie, Meissner, Le Xu, et al Error < 0.1 kT/e 4 kT/e Misfit

47. Theory fits Mutation with Zero Charge Theory Fits Mutant in K + Ca Theory Fits Mutant in K Error < 0.1 kT/e 1 kT/e Protein charge densitywild type* 13M Solid Na+Cl- is 37M *some wild type curves not shown, ‘off the graph’ 0M in D4899  1 kT/e Gillespie et alJ Phys Chem 109 15598 (2005)

48. Calcium Channel More than 35 papers are available at ftp://ftp.rush.edu/users/molebio/Bob_Eisenberg/reprints http://www.phys.rush.edu/RSEisenberg/physioeis.html

49. Selective Binding CurveL type Ca channel Wolfgang Nonner

50. ‘All Spheres’ Model Side Chains are Spheres Channel is a Cylinder Side Chains are free to move within Cylinder Ions and Side Chains are at free energy minimum i.e., ions and side chains are ‘self organized’, ‘Binding Site” is induced by substrate ions Nonner & Eisenberg