Mathematics describes only a little of Daily Life

1. Mathematics describes only a little ofDaily Life But Mathematics* Creates our Standard of Living *Electricity, Computers 109 in 50 years, Fluid Dynamics, Optics, Structural Mechanics, …..

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

3. 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!

4. All biology occurs in Ionic Solutions‘Seawater’ is the liquid of lifeSeawater and liquids of life are Concentrated Mixtures of sodium, potassium, calcium ions

5. Physical Chemists areFrustrated by Ionic Solutions

6. Physical Chemists areFrustrated by Real Solutionsbecause IONIC SOLUTIONSare COMPLEX FLUIDSnot ideal gases

7. Central Result of Physical Chemistry Ionsin a solutionare aHighly Compressible Plasma although the Solutionitself isIncompressible Learned from Henderson, J.-P. Hansen, Berry, Rice, and Ross…Thanks!

8. 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.

9. + ~30 Å Ion Channelsare theValves of CellsIon Channels are Devices* that Control 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 Figure of ompF porin by Raimund Dutzler 0.7 nm = Channel Diameter *Devices as defined in engineering , with inputs and outputs, and power supplies.

10. 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

11. 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

12. 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!

13. Where do we start? Physics ‘As Usual’‘Guess and Check’ Stochastic ‘Derivation’ Later will include biological adaptation of Correlations and Crowded Charge

14. We start with Langevin equations of charged particles Opportunity and Need Simplest stochastic trajectories are Brownian Motion of Charged Particles Einstein, Smoluchowski, and Langevin ignored chargeand therefore do not describe Brownian motion of ions in solutions We useTheory of Stochastic Processesto gofrom Trajectories to Probabilities Once we learn to count Trajectories of Brownian Motion of Charge, we can count trajectories of Molecular Dynamics Schuss, Nadler, Singer, Eisenberg

15. Langevin Equations Positivecation, e.g., p= Na+ Negativeanion, e.g., n= Cl¯ Electric Forcefrom all charges including Permanent charge of Protein, Dielectric Boundary charges,Boundary condition charge Schuss, Nadler, Singer, Eisenberg

16. Electric Force fromPoisson Equationnot assumed Electric Forcefrom all charges including Permanent charge of Protein, Dielectric Boundary charges,Boundary condition charge,MOBILE IONS Excess ‘Chemical’ Force Implicit Solvent‘Primitive’ Model Total Force Schuss, Nadler, Singer, Eisenberg

17. Nonequilibrium EquilibriumThermodynamics Theory of Stochastic Processes StatisticalMechanics Schuss, Nadler, Singer & Eisenberg Configurations Trajectories Fokker Planck Equation Boltzmann Distribution Finite OPEN System Device Equation Thermodynamics

18. From Trajectories to Probabilities in Diffusion Processes ‘Life Work’ of Ze’ev Schuss Department of Mathematics, Tel Aviv University Theory and Applications of Stochastic Differential Equations. 1980 John Wiley Theory And Applications Of Stochastic Processes: An Analytical Approach 2009 Springer Singular perturbation methods for stochastic differential equations of mathematical physics. SIAM Review, 1980 22: p. 116-155. Schuss, Nadler, Singer, Eisenberg

19. From Trajectories to Probabilities Sum the trajectories Sum satisfies Fokker-Planck equation Main Result of Theory of Stochastic Processes Jointprobability density of position and velocity with Fokker Planck Operator Coordinates are positions and velocities of N particles in 12N dimensional phase space Schuss, Nadler, Singer, Eisenberg

20. Conditional PNP Electric Force depends on Conditional Density of Charge Closure Needed ‘Guess and Check’ Permittivity, Dielectric Coefficient, Charge on Electron ChannelProtein Nernst-Planck gives UNconditional Density of Charge Mass Friction Schuss, Nadler, Singer, Eisenberg

21. Probability and Conditional Probability areMeasures on DIFFERENT Sets that may be VERY DIFFERENT Considerall trajectories that end on the right vs. all trajectories that end on the left

22. Everything Interacts Here is where we guess! Theory of Stochastic Processes and Thermodynamics Do not deal easily with strong interactions because Interactions are not perturbations Usual Stochastic Processes and Law of Mass Action are not good enough

23. Here is where we do Science, not Mathematics Here we GUESS and CHECK

24. Everything Interacts Here is where WE GUESS to make “apparently simple” theory! Theory of Stochastic Processes and Thermodynamics Do not deal easily with strong interactions because Interactions are not perturbations Usual Stochastic Processes and Law of Mass Action are not good enough: as presented in Chemistry Talk

25. Poisson-Nernst-Planck (PNP) Channel Protein Drift-diffusion & Continuity Equation Chemical Potential Chemical Correlations Poisson’s Equation Diffusion Coefficient Number Densities Flux Thermal Energy Thermal Energy Dielectric Coefficient Permittivity Proton charge Valence Valence Proton charge

26. Semiconductor Equations: One Dimensional PNP Poisson’s Equation Drift-diffusion & Continuity Equation Chemical Potential Permanent Charge of Protein Diffusion Coefficient Thermal Energy Cross sectional Area Flux Number Densities Dielectric Coefficient Valence Proton charge valence proton charge

27. Counting at low resolution gives ‘Semiconductor Equations’ Poisson-Nernst-Planck (PNP) Ions are Points in PNPPNPcontains only the Correlations of Means Gouy-Chapman, (nonlinear) Poisson-Boltzmann, Debye-Hückel, are siblings with similar resolutionbut at equilibrium, without current or flux of any speciesDevices do not exist at equilibrium

28. No Steric Interactions!

29. Comparison with Experiments shows Potassium K+ Sodium Na+ Must Study a Biological Adaptation! Selectivity

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

31. 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 MathematicsChun Liu’s Energetic Variational Principle EnVarA

32. IONIC SOLUTIONSare COMPLEX FLUIDSnot ideal gases

33. Energetic Variational Analysis EnVarAbeing developed by Chun LiuYunkyong Hyon and Bob Eisenberg creates a Field Theory of Ionic Solutions that allows boundary conditions and flow and deals with Interactions of Components Self-consistently

34. 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 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 with Interactions of Components self-consistently

35. Variational Analysis of Ionic Solution EnVarA Generalization of Chemical Free Energy Dielectric Coefficient from Poisson Eq. Number Densities Lagrange Multiplier Eisenberg, Hyon, and Liu

36. EnVarA Dissipation Principle for Ions Hard Sphere Terms Number Density Thermal Energy time Permanent Charge of protein valence proton charge cinumber density; thermal energy; Di diffusion coefficient; n negative; p positive; zivalence Eisenberg, Hyon, and Liu

37. Field Equations with Lennard Jones Spheres Non-equilibriium 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

38. Energetic Variational ApproachEnVarA across biological scales: molecules, cells, tissuesdeveloped by Chun Liuwith (1) Hyon, Eisenberg Ions in Channels (2) Bezanilla, Hyon, Eisenberg Conformation Change of Voltage Sensor (3) Ryham, Eisenberg, Cohen Virus fusion to Cells (4) Mori, Eisenberg Water flow in Tissues Multiple Scales creates a newMultiscale Field Theory of Interacting Components that allows boundary conditions and flow and deals with Ions in solutions self-consistently

39. Energetic Variational Analysis EnVarA Not yet available in Three Dimensions Structures are 3D

40. J Wu et al. Nature 1–6 (2016) doi:10.1038/nature19321 The ion permeation path of Cav1.1

41. The ion permeation path of Cav1.1 J Wu et al. Nature 1–6 (2016) doi:10.1038/nature19321

42. Poisson Fermi Approach to Ion Channels Bob Eisenberg Jinn-Liang Liu Jinn-Liang is first author on our papers 劉晉良

43. Motivation Natural Description of Crowded Charge is a Fermi Distributiondesigned to describe saturation Simulating saturationby interatomic repulsion (Lennard Jones) is a significant mathematical challengeto be side-stepped if possible

44. Motivation Largest Effect of Crowded Chargeis Saturation Saturation cannot be described at all by classical Poisson Boltzmann approach and is described in a (wildly) uncalibrated way by present day Molecular Dynamics when mixtures and divalents are present

45. Motivation Fermi Description is designed to deal with Saturation of Concentration Simulating saturation by interatomic repulsion (Lennard Jones) is a significant mathematical challengeto be side-stepped if possible

46. Fermi DescriptionofSaturation of Volumeby Spherical Ions Fermi (like) Distribution J.-L. Liu J Comp Phys (2013) 247:88

47. Fermi (like)Distribution depends on Steric Factor of System works surprisingly well despite crudeness of molecular model Algebraic Model of Calcium Channel J Comp Phys (2013) 247:88 Algebraic Model of Bulk Solution, e.g. Calcium Chloride

48. Fermi Descriptionof Crowded Charge and Saturation 4) We adopt the simplest treatment so we can deal with 3D structures many chemical complexities are known to us and have been left out purposely for this reason 5) We require exactconsistency with electrodynamics of flow because Key to successful modelling of ions Electric forces are so largethat deviations from consistency do not allow transferrable models and can easily wreck models all together Flow is EssentialDeath is the only Equilibrium of Life

49. Exact consistency with electrodynamics of flow is THE key to successful modelling of ions in my opinion Electric forces are so largethat deviations from consistency do not allow transferrable models and can easily wreck models all together Flow is Essential Death is the only Equilibrium of Life

50. ChallengeCan Simplest Fermi Approach • Describe ion channel selectivity and permeation? • Describe non-ideal properties of bulk solutions? There are no shortage of chemical complexities to include, if needed! Classical Treatments of Chemical Complexities