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The Electrolytic Basis of Life: Variational Mathematics in Complex Ionic Interactions

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vlora
93 diapositives · 1.1K Views · Jul 19, 2014

This article delves into the intricate world of electrolytes that form the basis of life, focusing on the interactions of ions such as Na+, K+, Ca++, and Cl- in biological systems. It explores the critical role of variational mathematics in modeling these complex electrolyte solutions, addressing the limitations of classical theories like Debye-Hückel. Highlighting evidence from biophysical studies, the piece discusses why simple fluid models fail and emphasizes the significance of understanding ion interactions to advance both chemistry and biology.

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  1. Life is in aComplex Mixture of Electrolytes mostly Na+, K+, and Ca++ Cl - Everything Interacts Through the Eelctric FieldIons Come ‘in pairs’i.e., electrically balanced neutral combinations Cl- 0.6 nm = Channel Diameter Na+ Electric Field is so strong that charges balance to ~10-12 % , otherwise system explodes!!

  2. Chemistry and Biology are about Chemically Specific Properties Chemically Specific Properties are the same thing as theirDEVIATION from properties of SIMPLE FLUIDS Life Occurs in a Complex Fluid~200 mM salt solutions mostly Sodium Na+, Potassium K+, and Calcium Ca++ Chloride Cl -

  3. Variational Mathematics: When everything interacts, we need mathematics.We need aVariational Theory of Electrolytes ‘Everything’ Interacts with Everything Else Hünenberger & Reif (2011) Single-Ion Solvation

  4. Stell, G. and C.G. Joslin, exact quotation: Under physiologically appropriate conditions, it is Almost Never Valid to use Debye-Hückel Theoryit is important to take proper account of Ion Size Biophys J, 1986. 50(5): p. 855-859, emphasis Bob Eisenberg

  5. Mathematics describes only a little ofDaily LifeBut Mathematics* Creates our Standard of Living *e.g.,Electricity, Integrated Circuits, Fluid Dynamics, Optics, Structural Mechanics, …..

  6. Mathematics Creates our Standard of LivingMathematics replaces Trial and Errorwith Computation *e.g.,Electricity, Computers, Fluid Dynamics, Optics, Structural Mechanics, …..

  7. Mathematics increases the Efficiency of Experimentation and Efficiency of design by orders of magnitude We can do more with less But you have to know which mathematics!

  8. What mathematics?What is most helpful?

  9. I believe Variational Approach has a Special Value

  10. Variational Approach is Always self-consistent Allows adding components with minimal parameters

  11. Scientific Discussion Converges Rapidly when Self-consistent with minimal parameters

  12. Variational Approach catalyzes Science as a Social Process

  13. Otherwise, …

  14. Complex Schemes produceUnresolved DiscussionandExperimentation

  15. Complex Schemes produceMore Grants than Designs

  16. Complex Schemes need to be replaced by a Variational Field Theoryin my opinionHere we consider Electrolyte Solutionsin general, not just infinitely dilute NaCl

  17. Poisson Boltzmann does not fit Solutions of Divalent Ions When the counter-ions are doubly charged … theClassical Theory Fails Altogethereven for quite low concentrations and charges Torrie and Valleauexact quotation, emphasis Bob Eisenberg: Biological Solutions are Concentrated Biological Solutions Contain Divalent Ions Torrie and Valleau , Journal of Physical Chemistry, 1982: 86: 3251-3257

  18. Good Data

  19. 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.

  20. Bad Theoryeven without flow “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.

  21. Ions in Water are the Liquid of Life They are not ideal solutions Everything Interacts with Everything For Modelers and Mathematicians Tremendous Opportunity for Applied Mathematics because ‘law’ of mass action assumes nothing interactsChun Liu’s Energetic Variational Principle EnVarA

  22. Everything Interacts Mathematics of Chemistry must deal Naturally with Interactions ‘Law of Mass Action’ assumes nothing interacts So this is a great opportunity for new mathematics and applications!

  23. Everything Interacts Mathematics of Chemistry must deal Naturally with Interactions ‘Law of Mass Action’ assumes Nothing Interacts So this is a great opportunity for new mathematics and applications!

  24. ‘New’ Mathematics of Interactions Variational Approach EnVarA ‘Law’ of Mass Action includingInteractions Conservative Dissipative From Bob Eisenberg p. 1-6, in this issue

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

  26. Multi-Scale Issues Journal of Physical Chemistry C (2010 )114:20719, invited review Three Dimensional (104)3 Biological Scales Occur Together so must be Computed Together This may be impossible in simulations

  27. Force Fields are Calibrated Ignoring Interactions with ions but Chemically Specific Properties come from Interactionsin Ionic Solutions Life occurs in Interacting Solutions

  28. Molecular Dynamics Simulations almost always Assume No Interactions Real Solutions Always Have Interactions Electric Field Every ion interacts with every other ion through the Ionic Atmosphere Ionic atmosphere is crowded around the center Steric Effects

  29. Molecular Dynamics Force Fields are Calibrated assuming no interactions with concentrations Force Fields must be REcalibrated in each Biological Solution Just ask the author(s) of CHARMM Chemically Specific PropertiesofIonic Solutions come fromInteractions Ions in Water are the Liquid of Life. They are not ideal solutions

  30. Calibration is Hard Work Force Fields must be RE-calibrated in each Biological Solutionto verify equilibrium potentials (chemical potentials) Fitting Real Experimentsrequires Accurate Chemical Potentials in mixtures Channels are Identified by Equilibrium Potentials If simulations are uncalibrated, even the type of the channel is unknown like Ringer Solution, with Ca2+

  31. Uncalibrated Simulations will not make devices that actually work

  32. In my opinion ‘New’ Mathematicsis needed to deal with theINTERACTIONS that make ionic solutions non-ideal and create theCHEMICAL SPECIFICITY of life Biological Theory and Molecular Dynamics Simulations almost always assume ideal solutions

  33. No theoryis available for Mixtures of Ions In my opinion ‘New’ Mathematicsis needed to dealwith the INTERACTIONS that make ionic solutions non ideal and that can create theCHEMICAL SPECIFICITY of life

  34. No theory is available for Flow of any kind. In my opinion ‘New’ Mathematicsis needed to dealwith the INTERACTIONS that make ionic solutions non ideal and that can create theCHEMICAL SPECIFICITY of life

  35. No theoryis available for Brownian Motion of Ions Brownian Motion theory is for UNcharged particles.Brownian Motion theory ignores Interactions In my opinion ‘New’ Mathematicsis needed to dealwith the INTERACTIONS that make ionic solutions non ideal and that can create theCHEMICAL SPECIFICITY of life

  36. Where to start? Mathematically ? Physically ?

  37. + ~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 matters In ideal solutions K+ = Na+ Flow time scale is 0.1 msec to 1 min Figure of ompF porin by Raimund Dutzler

  38. Channels are Selective Different Types of Channels use Different Types of Ionsfor Different Information

  39. 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

  40. Thousands of Molecular Biologists Study Channels every day,One protein molecule at a timeThis number is not an exaggeration.We have sold >10,000 AxoPatch amplifiers Ion Channel Monthly Femto-amps (10-15 A) AxoPatch 200B

  41. Where to start? ‘Law of Mass Action’ must be Replacedby a Variational Principle

  42. Comparison with Experiments shows Potassium K+ Sodium Na+ Must include Biological Adaptation! Working Hypothesis Biological Adaptation is Crowded Ions and Side Chains

  43. 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

  44. Ionizable ResiduesDensity = 22 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+: Base Density(probably positive) CD_MS-: Acid Density (probably negative) CD_MSt: Total Ionizable density Jimenez-Morales, Liang, Eisenberg

  45. Example: UDP-N-ACETYLGLUCOSAMINE ENOLPYRUVYL TRANSFERASE (PDB:1UAE) Functional Pocket Molecular Surface Volume: 1462.40 A3 Density : 19.3 Molar (11.3 M+. 8 M-) EC2: TRANSFERASESAverage Ionizable Density: 19.8 Molar Crowded Green: Functional pocket residues Blue: Basic = Probably Positive = R+K+H Red: Acid = Probably Negative = E + Q Brown URIDINE-DIPHOSPHATE-N- ACETYLGLUCOSAMINE Jimenez-Morales, Liang, Eisenberg

  46. Example: ALPHA-GALACTOSIDASE (PDB:1UAS) Functional Pocket Molecular Surface Volume: 286.58 A3 Density : 52.2 Molar (11.6 M+. 40.6 M-) EC3: HYDROLASESAverage Ionizable Density: 26.6 Molar Crowded Green: Functional pocket residues Blue: Basic = Probably Positive = R+K+H Red: Acid = Probably Negative = E + Q Brown ALPHA D-GALACTOSE Jimenez-Morales, Liang, Eisenberg

  47. Best Evidence is from the RyRReceptor Gillespie, Meissner, Le Xu, et al, not Bob Eisenberg  More than 120 combinations of solutions & mutants 7 mutants with significant effects fit successfully

  48. Samsóet al, 2005, Nature StructMolBiol12: 539-44 RyRRyanodine Receptor Slide from Dirk Gillespie, with thanks! • 4 negative charges • Cylinder 10 Å long, 8 Å diameter • 13 M of charge! • 8 oxygenswith charge -1/2 • 18% of available volume • Very Crowded! Aspartate

  49. DFT/PNPvsMonte Carlo Simulations Concentration Profiles Misfit Nonner, Gillespie, Eisenberg

  50. Divalents Gillespie, Meissner, Le Xu, et al KCl CaCl2 NaCl CaCl2 Error < 0.1 kT/e Misfit 2 kT/e CsCl CaCl2 KCl MgCl2 Misfit