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Fermi Description of Finite Size Effects in Biological Solution

This paper introduces the Fermi distribution as a way to describe saturation effects in concentrated biological solutions. It discusses the relevance of finite size effects in biology and the limitations of classical approaches. The paper also explores the treatment of non-uniform particle size, volume changes in channels, and the inclusion of voids in the model. The Fermi functionals and their connection to the energy variational formulation are discussed, along with the implementation of nonequilibrium conditions in a 3D code. Specific models for the L-type calcium channel and gramidicin are tested and numerical procedures tailored to the Poisson-Nernst-Planck-Fermi equations are implemented.

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Fermi Description of Finite Size Effects in Biological Solution

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  1. Fermi Description of Finite Size Effects in Biological Solution 1) Biology is in concentrated >0.3 M mixtures of spherical chargesthat flow. Solutions are extraordinarily concentrated >10M where they are most important, near DNA, enzyme active sites, and channels and electrodes of batteries and electrochemical cells. (Solid NaCl is 37M.) 2) Largest effect of finite size is saturation that cannot be described at all by classical Poisson Boltzmann approach and is described in a (wildly) uncalibrated way by present day Molecular Dynamics. 3) Fermi distribution is the simplest way to describe saturation and avoids computation of forces between spheres. 4) We adopt the simplest treatment so we can deal with 3D structures. 5) But require exact consistency with electrodynamics of flow because all life requires flow. Death is the only Equilibrium of Life 6) Can we describe ion channel selectivityand permeation? Can we describe non-ideal properties of bulk solutions?

  2. Fermi Description usesEntropy of Mixture of Spheresfrom the simplest Combinatoric Argument W is the mixing entropy of spheres with N available non-uniform sites Connection to volumes of spheres and voids, and other details are published in 5 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: 22D532; Physical Review E, 2015. 92:012711.

  3. Voids are Needed Electro-Chemical Potentialand Void Volume It is impossible to treat all ions and water molecules as hard spheres and at the same time have Zero Volumeof interstitial Voidsbetween all particles. 劉晉良 Jinn-Liang Liu made this clever analysisBob Eisenberg helped with the applications

  4. Approach is Novel Previous treatments* do not include nonuniform particle size or have inconsistent treatment of particle size and do not reduce to Boltzmann functionals in the appropriate limit Previous treatments do not include discrete water and so cannot deal with volume changes of channels, or pressure/volume in general Previous treatments do not include voids Previous treatments do not include polarizable water with polarization as an output Previous treatments are inconsistent with electrodynamics and nonequilibrium flows including convection Previous treatments Bazant, Storey, and Kornyshev,. Physical Review Letters, 2011. 106(4): p. 046102. Borukhov, Andelman, and 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 and Zhou, Biophysical Journal, 2011. 100(10): p. 2475-2485. Qiao, Tu, and Lu, J Chem Phys, 2014. 140(17):174102 Silalahi, Boschitsch, Harris, and Fenley, JCCT2010. 6(12): p. 3631-3639. Zhou, Wang, and Li Physical Review E, 2011. 84(2): p. 021901.

  5. Fermi (like) Distribution Steric Potential Quantitative Statement of Charge-Space Competition Simulated and compared to experiments in > 30 papers of Boda, Henderson, et al Gibbs Fermi Functional has been defined and published J Comp Phys, 2013 247:88; J Phys Chem B, 2013 117:12051so the Fermi approach can be embedded in the Energy Variational Formulation EnVarA developed by Chun Liu, more than anyone else

  6. Nonequilibrium Implemented fully in 3D Code to accommodate 3D Protein Structures Use Santangelo (2006)1 and Kornyshev (2011)2 treatment of near and far fields separated by (crude fixed) correlation lengthlc Introducesteric potential 3) Reduce to pair of 2nd order PDE’s and appropriate boundary conditions that allow robust and efficient numerical evaluation 1 Phys Rev E (2006)73, 041512 2Phys Rev Letters, 2011. 106: 046102

  7. Nonequilibrium Implemented fully in 3D Code to accommodate 3D Protein Structures Cahn-Hilliard Type Fourth Order PDE is introduced as a crude correlation length to separate near and far fields approximates the dielectric properties of entire bulk solution including correlated motions of ions, following Kornyshev1 using a corrected and consistent Fermi treatment of spheres We2 introduce two second order equations and boundary conditions That give the polarization charge density 3D computation is facilitated by using 2nd order equations 1 Phys. Rev. Lett. 106 046102 (2011) 2 J Comp Phys, 2013 247:88; J Phys Chem B, 2013 117:12051

  8. Nonequilibrium Implemented fully in 3D Code to accommodate 3D Protein Structures But only partially compared to experiments In Bulk or Channels Specific Model of L type Calcium Channel EEEE now called CaV(tested) J Phys Chem B, 2013 117:12051 Specific Model of Gramicidin(tested) Physical Review E, 2015. 92:012711 PNPF Poisson-Nernst-Planck-Fermi for systems with volume saturation General PDE, Cahn-Hilliard Type, Four Order, Pair of 2nd order PDE’s Not yet tested by comparison to bulk data J Chem Phys, 2014. 141:075102; J Chem Phys,141:22D532; 4) Numerical Procedures tailored to PNPF have been implemented (tested) J Comp Phys, 2013 247:88; Phys Rev E, 2015. 92:012711 Calcium Sodium Exchanger NCX , of Heart, branched Y shaped structure. First attempt at physical consistent analysis of a transporter. Testing by comparison to physiological data, mostly complete ; .

  9. Miracle Alan Hodgkin friendly Alan Hodgkin: “Bob, I would not put it that way” Device Approach to Biology is also a

  10. Take Home Lesson Biology is all about Dimensional Reduction to a Reduced Model of a Device

  11. Biology is made of Devicesand they are Multiscale Hodgkin’s Action Potential is the Ultimate Biological Device from Input from Synapse Output to Spinal Cord Ultimate MultiscaleDevice from Atoms to Axons Ångstroms to Meters

  12. Gain Vout Vin Power Supply 110 v Device Amplifier Converts an Input to an Output by a simple ‘law’ an algebraic equation

  13. DEVICE IS USEFULbecause it is ROBUST and TRANSFERRABLE ggainis Constant!! Device converts an Input to an Output by a simple ‘law’

  14. Gain Vout Vin Input, Output, Power Supply are at Different Locations Spatially non-uniform boundary conditions Power is neededNon-equilibrium, with flow Displaced Maxwellian is enough to provide the flow Power Supply 110 v Device Amplifier Converts an Input to an Output

  15. Gain Vout Vin • Power is neededNon-equilibrium, with flow • Displaced Maxwellianof velocitiesProvides Flow • Input, Output, Power Supply are at Different Locations Spatially non-uniform boundary conditions Power Supply 110 v Device Amplifier Converts an Input to an Output

  16. Device is ROBUST and TRANSFERRABLE because it uses POWER and has complexity! Circuit Diagram of common 741 op-amp: Twenty transistors needed to make linear robust device Power Supply INPUTVin(t) OUTPUTVout(t) Power Supply Dirichlet Boundary Condition independent of time and everything else Device converts Input to Outputby a simple ‘law’ Dotted lines outline: current mirrors  (red);differential amplifiers (blue);class Again stage (magenta); voltage level shifter (green);output stage (cyan).

  17. How do a few atoms control (macroscopic) Device Function ? Mathematics of Molecular Biology is aboutHow does the device work?

  18. Ompf G119D A few atoms make a BIG Difference Glycine Greplaced by Aspartate D OmpF 1M/1M G119D 1M/1M OmpF0.05M/0.05M G119D 0.05M/0.05M • Structure determined by Raimund Dutzlerin Tilman Schirmer’s lab Current Voltage relation determined by John Tang in Bob Eisenberg’s Lab

  19. Multiscale Analysis is Inevitable because aFew Atoms Ångstroms control Macroscopic Function centimeters

  20. Mathematics Must Include Structure andFunction Atomic Space = Ångstroms Atomic Time = 10-15 sec Cellular Space = 10-2 meters Cellular Time = Milliseconds Variables that are the function, like Concentration, Flux, Current

  21. Mathematics Must Include Structure andFunction Variables of Function are Concentration Flux Membrane Potential Current

  22. Mathematics must be accurate There is no engineering without numbers and the numbers must be accurate!

  23. Cannot build a box without accurate numbers!!

  24. Uncalibrated Simulations will make devices that do not work Details matter in devices

  25. CALIBRATED simulations are needed just as calibrated measurements and calculations are needed Calibrations must be of simulations of activity measured experimentally, i.e., free energy per mole. Fortunately, extensive measurements are available

  26. FACTS (1) Atomistic Simulations of Mixtures are extraordinarily difficult because all interactions must be computed correctly (2) All of life occurs in ionic mixtures like Ringer solution (3) No calibrated simulations of Ca2+ are available. because almost all the atoms present are water, not ions. No one knows how to do them. (4) Most channels, proteins, enzymes, and nucleic acids change significantly when [Ca2+] changes from its background concentration 10-8M ion.

  27. CONCLUSIONS Multiscale Analysis is REQUIRED in Biological Systems Simulations cannot easily deal with Biological Reality

  28. Scientists must Graspand not just reach. That is why calibrations are necessary. • Poets hope we will never learn the difference between dreams and realities • “Ah, … a man's reach should exceed his grasp,Or what's a heaven for?” • Robert Browning • "Andrea del Sarto", line 98.

  29. Mathematics describes only a little ofDaily Life But Mathematics* Creates our Standard of Living *e.g.,Electricity, Computers, Fluid Dynamics, Optics, Structural Mechanics, …..

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

  31. Physics Today 58:35 • * Calibration! *not so new, really, just unpleasant

  32. Mathematics is Needed to Describe and Understand Devicesof Biology and Technology

  33. 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!

  34. Device Approach enables Dimensional Reduction to a Device Equation which tells How it Works

  35. DEVICE APPROACH IS FEASIBLE Biology Provides the Data Semiconductor Engineering Provides the ApproachMathematics Provides the Tools

  36. Mathematics of Molecular Biology Must be Multiscalebecause a few atoms Control Macroscopic Function

  37. Mathematics of Molecular Biology Nonequilibriumbecause Devices need Power Supplies to ControlMacroscopic Function

  38. ‘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

  39. Physical Chemists areFrustrated by Real Solutions

  40. 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!

  41. Electrolytes are Complex Fluids Treating a Complex Fluid as if it were a Simple Fluid will produce Elusive Results

  42. Electrolytes are Complex Fluids After 690 pages and 2604 references, properties of SINGLE Ions are Elusive*because Every Ion Interacts with Everything *elusive is in the authors’ choice of title! Hünenberger & Reif (2011) Single-Ion Solvation. Experimental and Theoretical Approaches to Elusive*Thermodynamic Quantities. 2011

  43. 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 0.1M or so.” Kunz, W. "Specific Ion Effects" World Scientific Singapore, 2009; p 11.

  44. Good Data

  45. 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.

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