Mathematics in Biological Systems: Ion Channels and Electric Fields

1. Second Day October 112017 Short Course From Atoms to AxonsBob Eisenberg Fields Institute Sponsor Huaxiong Huang. Se

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

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

4. + ~30 Å Ion Channelsare theValves of CellsIon Channels are the Main Controllers of Biological Function Ions in Waterare the One Ion trajectory Selectivity Different Ions carry Different Signals Liquid of Life 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

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

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

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

8. + ~30 Å Ion Channelsare theValves of CellsIon Channels are the Main Controllers of Biological Function Ions in Waterare the Selectivity Different Ions carry Different Signals Liquid of Life 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

9. 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

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

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

12. 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.

13. Conservation of Current and Conservation of Chargeare EXACTandUNIVERSAL

14. Maxwell’s Magnetism Current is Conserved PERFECTLY

15. Continuity of Current enforces long range macroscopiccorrelationsthat cannot be described with ordinary differential equations in time

16. Correlation between currentsis in fact ALWAYS 0.999 999 999 999 999 999

17. Continuity of Current is Exact Maxwell Equations are Special even though Physics of Charge Flow Varies Profoundly ‘Charge’ is an Abstraction withVERY different Physics in different systems

18. ‘Charge’ is an Abstraction with different Physics in different systems butContinuity of Current is ExactNo matter what carries the current! Ag AgCl Ag AgCl D = permittivity  E

19. Continuity of Current is Exact How can that possibly be? even though Physics of Charge Flow Varies Profoundly

20. Electric Field takes on Whatever Value Conserves current, Specifically, E changes the displacement currentSo total current is always conserved Details and PROOF at https://arxiv.org/abs/1609.09175

21. Electric Field is Different in Different Devices Device 2 Device 1

22. Device 2 Device 1

23. Electric Field is Different in Different Devices so Displacement Current is Different in Different Devices Device 1 Device 2 differs in every devicebecause E(x,t) differs in every device so the total current is exactly equal at every time in every location and every device Total Current = Universal Displacement Current + Device Current

24. Conservation of Current is not enforced in classical Chemical Models

25. Rate Models FailbecauseCurrent-in does not equal Current-out!! (if rate constants are independent of potential) Rate Constants are INDEPENDENT parameters so

26. Rate Models FailbecauseCurrent-in does not equal Current-out!! (if rate constants are independent of potential)

27. ‘Current-in’ does not equal ‘Current-out’ in Rate Models if rate constants are independent and Currents are Uncorrelated

28. 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

29. 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

30. Parameterization is not Possible under more than one condition Rate constants chosen at one boundary charge or one potentialcannot work for different charges or potentials Currents in Rate Models are Independent of Charge and Potential but in the real world Currents depend on Charge and Potential

31. What does this have to do with biology? LIPID METABOLISM is a big deal

32. What does this have to do with biology? METABOLISM is a big deal

33. QuestionWhat does this have to do with biology? Answer All biology involves electricity All biology occurs in solutions that condut electricity A LOT All biology occurs in Ion Solutions K+ Ca++ Na+ 3 Å Sodium Na+ Potassium K+ Calcium Ca2+ Chloride Cl- Cl-

34. All of Biology occurs in Salt Solutions of definite composition and concentration and that matters! Salt Water is the Liquid of Life Pure H2O is toxic to cells and molecules! K+ Ca++ Na+ 3 Å Sodium Na+ Potassium K+ Calcium Ca2+ Chloride Cl- Cl-

35. All of Biology occurs in Salt Solutions of definite composition and concentration and that matters! Salt Water is the Liquid of Life Pure H2O is toxic to cells and molecules! Salt Water is a Complex Fluid Main Ions are Hard Spheres, close enough Sodium Na+ Potassium K+ Calcium Ca2+ Chloride Cl- K+ Ca++ Na+ 3 Å Cl-

36. 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!

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

38. 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

39. 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

40. Physical Chemists are Frustrated by Real Solutions

41. 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)

42. “Poisson Boltzmann theories are restricted to such low concentrations that the solutions cannot be studied in the laboratory” slight paraphrase of p. 125 of Barthel, Krienke, and Kunz Kunz, Springer, 1998 Original text “… experimental verification often proves to be an unsolvable task”

43. Werner Kurz “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.

44. “ …. it is almost never valid to use Debye-Hückel theory … it is important to take proper account ofion size Stell, G. and C.G. Joslin Biophys J, 1986. 50(5): p. 855-859.

45. Good Data

46. Good Data Compilations of Specific Ion Effect • >139,175 Data Points [Sept 2011] 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.

47. Shielding is a defining property of Complex Fluids It is VERY hard to Simulate at Equilibrium and (in my opinion) IMPOSSIBLE to Simulate in nonequilibrium Like Batteries or Nerve Fibers because flows involve Far Field (macroscopic) boundaries

48. 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

49. Main Qualitative ResultShielding in Gramicidin Hollerbach & Eisenberg

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