to Mimi Dai for inviting me!

1. Thanks! to Mimi Dai for inviting me!

2. Mathematics describes only a little ofDaily Life But Mathematics* Creates our Standard of Living *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. + ~3 x 10-9 meters Channels are Selective Molecular DevicesDifferent Ions Carry Different Signals through Different Channels ompF porin Ca++ Na+ K+ 300 x 10-12 meter 0.7 10-9 meter = Channel Diameter Diameter mattersIonic solutions are NOT ideal Classical Biochemistry assumes ideal solutions. K+& Na+ are identical only in Ideal Solutions. Flow time scale is 10-4 sec to 1 min Figure of ompF porin by Raimund Dutzler

5. ‘Typical’ Cell

6. K+ ~30 x 10-9meter 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 Coordinate contraction in skeletal muscle Control all electrical activity in cells Produce signals of the nervous system Are involved in secretion and absorption in all cells:kidney, intestine, liver, adrenal glands, etc. Are involved in thousands of diseases and many drugs act on channels Are proteins whose genes (blueprints) can be manipulated by molecular genetics Have structures shown by x-ray crystallography in favorable cases Can be described by mathematics in some cases • *Device is a Specific Word, that exploits specific mathematics & science *nearly pico-valves: diameter is 400 – 900 x 10-12 meter; diameter of atom is ~200 x 10-12 meter

7. Channels are Devices Channels are (nano) valves Valves Control Flow Classical Theory & Simulations NOT designed for flow Thermodynamics, Statistical Mechanics do not allow flow Rate and Markov Models do not Conserve Current

8. Thermodynamics, Statistical Mechanics, Molecular Dynamics are derived in ‘Thermodynamic Limit’ 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

9. Multi-Scale Issues are Always Presentin Atomic Scale Engineering • Atomic & Macro Scales are both used by channels just because Channels are Nanovalves • By definition: all valves use small structures to control large flows

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

11. How does it work? How do a few atoms control (macroscopic) Biological Function Mathematics of Molecular Biology is aboutSolving Specific Inverse Problems • Problem for Channels has actually been solvedBurger, Eisenberg, Engl (2007) SIAM J Applied Math 67: 960-989

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

13. Biology is made of Devicesand they are MULTISCALE A different talk! Hodgkin’s Action Potential is the Ultimate Multiscale model from Atoms to Axons Ångstroms to Meters

14. Simulations produce too many numbers 106 trajectories each 10-6 sec long, with 109 samples in each trajectory, in background of 1022 atoms Estimators are Needed Estimators are a kind of Reduced Model

15. Multi-Scale Issues A different talk! Journal of Physical Chemistry C (2010 )114:20719 Three Dimensional (104)3 Atomic and Macro Scales are BOTH used by channels because they are nanovalves so atomic and macro scales must be Computed and CALIBRATED Together This may be impossible in all-atom simulations

16. Where to start? Mathematically ? Physically ?

17. Biology is Easier than Physics Reduced Models Exist* for important biological functions or the Animal would not survive to reproduce *Evolution provides the existence theorems and uniqueness conditions so hard to find in theory of inverse problems. (Some biological systems  the human shoulder  are not robust, probably because they are incompletely evolved,i.e they are in a local minimum ‘in fitness landscape’ .I do not know how to analyze these. I can only describe them in the classical biological tradition.)

18. Engineers:this is reverse engineering For example, Find Charge Distribution in Channel from Current Voltage Relations Problem (with noise and systematic error) has actually been solvedbyTikhonov RegularizationBurger, Eisenberg, Engl (2007) SIAM J Applied Math 67: 960-989 using procedures developed by Engl to study Blast Furnaces and their Explosions Inverse Problems Given the OutputDetermine the Reduced Model

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

20. Guess Working Hypothesis Crucial Biological Adaptation is Crowded Ions and Side Chains Thanks toJie Liang UIC Bioengineering! Wise to use the Biological Adaptation to make the reduced model! Reduced Models allow much easier Atomic Scale Engineering

21. 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 TilmanSchirmerBiozentrum Basel

22. Crowded Active Sitesin 573 Enzymes Jimenez-Morales,Liang, Eisenberg

23. Everything Interacts with Everything Else by steric exclusioninside crowded active sites Everything interacts with macroscopic Boundary Conditions (and much else) through long range electric field ‘Law’ of mass action needs to be generalized

24. Physical ChemistsPhysiologistsBiophysicists are Frustrated by Ionic SolutionsTheories and Simulations Cannot deal with Sea Water at equilibriumand fail even more badly when flow is involved A different talk!

25. Cause of Frustration Biochemical Models are Rarely TRANSFERRABLEDo Not Fit Data even approximatelyin more than one solution* A different talk! 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

26. Electrolytes are Complex Fluids‘Everything’ interacts with everything else Treating a Complex Fluid as if it were a Simple Fluid will produce Elusive Results “Single-Ion Solvation… ElusiveQuantities” 690 pages 2604 references Hünenberger & Reif, 2011

27. Don’t worry! Crowded Charge is GOOD It enables SIMPLIFICATION by exploiting a biological fact (an adaptation) Charges are Crowded where they are important Enzymes, Nucleic Acids, Ion Channels, Electrodes

28. Where do we begin? Crowded Charge enables Dimensional Reduction* to a Device Equation Inverse Problem! Essence of Engineering is knowing What Variables to Ignore! WC Randels in Warner IEEE Trans CT 48:2457 (2001) *Dimensional reduction = ignoring some variables

29. Where do we begin? • Crowded Charge • has • HUGE electric fields • Poisson Equation, i.e., Conservation of Charge • and • LARGE steric repulsionFermi distribution

30. Now some Math ‘Derivation’ of Field Equations

31. Here I start fromStochastic PDE and Field Theory Other methodsgive nearly identical results MSA (Mean Spherical Approximation) SPM (Primitive Solvent Model) Non-equil MMC (Boda, Gillespie) several forms DFT (Density Functional Theory of fluids, not electrons) DFT-PNP (Poisson Nernst Planck) EnVarA (Energy Variational Approach) Steric PNP (simplified EnVarA) Poisson Fermi Chemistry Models MATHField Theory

32. Solved with PNP including Correlations Other methodsgive nearly identical results MMC Metropolis Monte Carlo (equilibrium only) DFT (Density Functional Theory of fluids, not electrons) DFT-PNP (Poisson Nernst Planck) MSA (Mean Spherical Approximation) SPM (Primitive Solvent Model) EnVarA (Energy Variational Approach) Non-equil MMC (Boda, Gillespie) several forms Steric PNP (simplified EnVarA) Poisson Fermi

33. Always start with TrajectoriesZe’ev Schuss Department of Mathematics, Tel Aviv University Always start with Trajectories because • Trajectories are the equivalent of SAMPLES in probability theory • Trajectories satisfy PHYSICAL boundary conditions • Trajectories satisfy classical PHYSICAL ordinary differential equations (we hope)

34. 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 Equations1980 Theory and Applications Of Stochastic Processes: An Analytical Approach 2009 Singular perturbation methods for stochastic differential equations of mathematical physicsSIAM Review, 1980 22: 116-155 Schuss, Nadler, Singer, Eisenberg

35. Trajectories in Condensed Phases are Noisy 1 Volt ~40 kBT/e 10-12 sec Note: Brownian noise looks the same on all scales! Function has unbounded variation, crossing any line an infinite number of times in any interval no matter how small.

36. 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

37. I only count them…., James Clerk Maxwell “I carefully abstain from asking molecules where they start… avoiding all personal enquiries which would only get me into trouble.” slightly reworded fromRoyal Society of London, 1879, Archives no. 188 In Maxwell on Heat and Statistical Mechanics, Garber, Brush and Everitt, 1995

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

39. Electric Force in Ion Channelsnot assumed GLOBAL Electric Forcefrom all charges including Permanent charge of Protein, Dielectric Boundary charges,Boundary condition charge,MOBILE IONS Excess ‘Chemical’ Force ‘All Spheres” Implicit Solvent‘Primitive’ Model Total Force Schuss, Nadler, Singer, Eisenberg

40. 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

41. More MathMany papers • We actually performed the sum and showed it was the same as a MARGINAL PROBABILITY estimator of SINGLET CONCENTRATION defined in chemistry • We actually did a nonequilibrium BBGKY expansion with electrostatic & steric correlations • We  along with everyone else  assume a closure • Nadler, B., T. Naeh and Z. Schuss (2001). SIAM J Appl Math 62: 443-447. • Nadler, B., T. Naeh and Z. Schuss (2003). "SIAM J Appl Math 63: 850-873. • Nadler, B., Z. Schuss and A. Singer (2005). "" Physical Review Letters 94(21): 218101. • Nadler, B., Z. Schuss, A. Singer and B. Eisenberg (2003). Nanotechnology 3: 439. • Nadler, B., Z. Schuss, A. Singer and R. Eisenberg (2004). Journal of Physics: Condensed Matter 16: S2153-S2165. • Schuss, Z., B. Nadler and R. S. Eisenberg (2001). Physical Review E 64: 036116 1-14. • Schuss, Z., B. Nadler and R. S. Eisenberg (2001). "Phys Rev E Stat Nonlin Soft Matter Phys 64(3 Pt 2): 036116. • Schuss, Z, B Nadler, A Singer, R Eisenberg (2002) Unsolved Problems Noise & Fluctuations, UPoN 2002, Washington, DC AIP • Singer, A, Z Schuss, B Nadler, R. Eisenberg (2004) Physical Review E Statistical Nonlinear Soft Matter Physics 70 061106. • Singer, A, Z Schuss, B Nadler, R Eisenberg (2004). Fluctuations & Noise in Biological Systems II V. 5467. D. Abbot, S. M.

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

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

44. 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

45. Conditioning and Correlations are VERY strong and GLOBAL when Electric Fields are Involved, as in Ionic Solutions and Channels so cannot do the probability theorywithout variational methods We had to guess the conditioned sets

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

47. Semiconductor PNP EquationsFor Point Charges 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 Not in Semiconductor Valence Proton charge valence proton charge

48. PNP (Poisson Nernst Planck)for Spheres Non-equilibrium 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

49. All we have to do isSolve them!with Boundary ConditionsdefiningCharge Carriersions, holes, quasi-electronsGeometry

50. Here is where we do Science, not Mathematics Here we have Two Guesses