Day d on 1 Sec ober 1 Oct
Day d on 1 Sec ober 1 Oct 2017 Short Course From Atoms to Axons Bob Eisenberg Fields Institute Sponsor Huaxiong Huang. Se 1
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! 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
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 4
The Cell Defined by a Membrane Note: intra-cellular compartments are defined by their membranes Bob Eisenberg: beisenbe@rush. edu
SIMULATION of GRAMICIDIN CHANNEL Visualization: Theoretical and Computational Biophysics Group Beckman Institute. http: //www. ks. uiuc. edu/Research/vmd Bob Eisenberg: beisenbe@rush. edu
Bio. MOCA: SIMULATION of GRAMICIDIN CHANNEL Umberto Ravaioli and Trudy van der Straaten Univ of Illinois Urbana-Champaign Bob Eisenberg: beisenbe@rush. edu
Ion Channels are the Valves of Cells Ion Channels are the Main Controllers of Biological Function Selectivity Ions in Water Different Ions carry Different Signals Liquid of Life are the 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 8
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 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! 10
How do a few atoms control (macroscopic) Biological Function? Answer, oversimplified: A few atoms control the electric field Much as they do in transistors 11
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. 12
Conservation of Current and Conservation of Charge are EXACT and UNIVERSAL
Maxwell’s Magnetism Current is Conserved PERFECTLY 14
Continuity of Current enforces long range macroscopic correlations that cannot be described with ordinary differential equations in time
Correlation between currents is in fact ALWAYS 0. 999 999 999 Page 16
Continuity of Current is Exact even though Physics of Charge Flow Varies Profoundly Maxwell Equations are Special ‘Charge’ is an Abstraction with VERY different Physics in different systems
‘Charge’ is an Abstraction with different Physics in different systems Ag Ag. Cl D = permittivity E but Continuity of Current is Exact No matter what carries the current!
Continuity of Current is Exact even though Physics of Charge Flow Varies Profoundly ? e b y l b i s s o p t a h t n a c w o H
Electric Field takes on Whatever Value Conserves current, Specifically, E changes the displacement current So total current is always conserved Details and PROOF at https: //arxiv. org/abs/1609. 09175 Page 20
Electric Field is Different in Different Devices Device 1 Device 2 21
Device 1 Device 2 22
Electric Field is Different in Different Devices so Displacement Current is Different in Different Devices Device 1 Device 2 Total Current = Universal Displacement Current + Device Current 23
Conservation of Current is not enforced in classical Chemical Models Page 24
Rate Models Fail because Current-in does not equal Current-out!! (if rate constants are independent of potential) Rate Constants are INDEPENDENT parameters so 25
Rate Models Fail because Current-in does not equal Current-out!! (if rate constants are independent of potential) 26
‘Current-in’ does not equal ‘Current-out’ in Rate Models if rate constants are independent and Currents are Uncorrelated 27
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 28
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 29 − Note: Biology occurs in different solutions from those used in most measurements
Parameterization is not Possible under more than one condition Rate constants chosen at one boundary charge or one potential cannot 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 30
What does this have to do with biology? LIPID METABOLISM is a big deal 31
What does this have to do with biology? METABOLISM is a big deal 32
Question What 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 Sodium Na+ Potassium K+ Calcium Ca 2+ K+ 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+ Calcium Ca 2+ K+ Ca++ Chloride Cl- Cl - 3Å 34
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+ Calcium Ca 2+ K+ Ca++ 3Å Chloride Cl. Cl - 35
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! 36
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! 37
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 38
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 39 − Note: Biology occurs in different solutions from those used in most measurements
Physical Chemists are Frustrated by Real Solutions
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)
“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, Springer, 1998 Original text “… experimental verification often proves to be an unsolvable task” 42
Werner Kurz “It is still a fact that over the last decades, it was easier to fly to the moon Kunz, W. "Specific Ion Effects" World Scientific Singapore, 2009; p 11. than to describe the free energy of even the simplest salt solutions beyond a concentration of 0. 1 M or so. ”
“ …. it is almost never valid to use Debye-Hückel theory … it is important to take proper account of ion size Stell, G. and C. G. Joslin Biophys J, 1986. 50(5): p. 855 -859.
Good Data 45
Good Data Compilations of Specific Ion Effect 1. >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. 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.
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 47
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 48
Main Qualitative Result Shielding in Gramicidin Hollerbach & Eisenberg 49
Reconciling Mass Action and 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 53
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 accurate computation of Flow and Interactions in Complex Fluids like Liquid Crystals Classical theories and Molecular Dynamics have difficulties with flow, interactions, and complex fluids Engineering needs Calibrated Theories and Simulations Engineering Devices almost always use flow 55
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 56
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 57
Energetic Variational Approach En. Var. A is defined by the Euler Lagrange Process, as I understand the pure math from Craig Evans 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 58
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 59
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 60
All we have to do is Solve them! with Boundary Conditions defining Charge Carriers ions, holes, quasi-electrons Geometry 61
Solution* of PNP Equation *MATHEMATICS This solution was actually DERIVED by computing many conditional probability measures explicitly by repeated 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 Page
Please do not be deceived by the eventual simplicity of Results. This took >2 years! Solution was actually DERIVED !! k e e w t x e N with explicit formulae for probability measures from a Doubly Conditioned Stochastic Process involving Analytical Evaluation of Multidimensional Convolution Integrals 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 Page
All we have to do is Solve them! Don’t Despair Semiconductor Technology has Already Done That! 64
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 should be attempted using techniques of Weishi Liu University of Kansas Tai-Chia Lin National Taiwan University & Chun Liu PSU
The End Any Questions? 66
Function of SINGLE isolated Ry. R Channels in Artificial Planar Lipid Bilayers Axo. Patch-Clamp Amplifier Designed at Rush Planar Bilayer Ca Fused Vesicle Single Channel Current open Experimental Chamber closed Teflon Septa 80 -100 µM Diameter Slide from Mike Fill Thanks! 69
Molecular Biologists Study Channels every day, One protein molecule at a time Ion Channel Monthly This number is not an exaggeration. We have sold >10, 000 Axo. Patch amplifiers Axo. Patch 200 B Femto-amps (10 -15 A) 70
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! 72
Energetic Variational Approach En. Var. A New mechanisms* (e. g. , active transport) can be added *if they define an energy and its variation Energy defined by simulations or theories or experiments is OK Full micro/macro treatment is needed for an Atomic Model, with closure, as in liquid crystals Eisenberg, Hyon, and Liu 73
What does the protein do? Channel and Contents form a Self-Organized Structure with Side Chains at position of Minimum Free Energy Protein Fits the Substrate “Induced Fit Model of Selectivity” 74
What does the protein do? (for biologists) Certain MEASURES of structure are Powerful DETERMINANTS of Function e. g. , Volume, Dielectric Coefficient, etc. Induced Fit Model of Selectivity Atomic Structure is not pre-formed Atomic Structure is an important output of the simulation Nonner and Eisenberg 75
What does the protein do? Protein maintains Mechanical Forces* Volume of Pore Dielectric Coefficient/Boundary Permanent Charge * Driving force for conformation changes ? ? 76 Nonner and Eisenberg
for Biologists: a Word Picture How does Calcium Selectivity Work? qualitatively Nonner and Eisenberg 77
How does it work qualitatively? (for biologists) 2 Ca ++ are LESS CROWDED than 4 Na + Nonner and Eisenberg 78
Selectivity from Crowded Charges 2 Ca++ are less crowded than 4 Na+ Ca Channel Filled with Ca++ (not to scale) Na+ ½ ½ ½ ½ Na+ (not to scale) Na+ ½ ½ ½ ½ Ca++ ½ Na+ ½ ½ ½ ½ Ca++ ½ ½ ½ Channel Protein Glutamate Oxygens = 4 e 8 of –½ charge each Volume 0. 38 nm 3 Dielectric Constant 64 Outside the Filter Bulk Solution Na. Cl and Ca. Cl 2
Ionic Selectivity in Protein Channels Crowded Charge Mechanism 4 Negative Charges of glutamates of protein DEMAND 4 Positive Charges nearby either 4 Na+ or 2 Ca++ Nonner and Eisenberg 80
Ionic Selectivity in Protein Channels Crowded Charge Mechanism Simplest Version: MSA 2 Ca++ are LESS CROWDED than 4 Na+, Ca++ SHIELDS BETTER than Na+, so Protein Prefers Ca++ because Ca++ is less crowded Nonner and Eisenberg 81
Binding Sites* are outputs of our Calculations Induced Fit Model of Selectivity Our model has no preformed structural binding sites but Selectivity is very Specific *Selectivity is in the Depletion Zone, NOT IN THE BINDING SITE of the DEKA Na Channel 82
Selectivity comes from Electrostatic Interaction and Steric Competition for Space Repulsion Location and Strength of Binding Sites Depend on Ionic Concentration and Temperature, etc Rate Constants are Variables 83
We can actually compute the Structures that determine Selectivity 84
Electric Field takes on Whatever Value Conserves current, Independent of Matter and Polarization the Displacement Current Changes to make Conservation work f o l e d o m l a t c i n s e r y r h u p C o t n vity n i t e s a i el m R e l e r c cia e e a h p l T p ithout S s i D w Details and PROOF at https: //arxiv. org/abs/1609. 09175 Page 85
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 86
In a series circuit Current is exactly equal everywhere in every location at every time For ANY mechanism of current flow How is that possible? Page 87
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