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What is different about life? it is inherited oberwolfach march 7 1 2018

The document discusses the hierarchy of devices in biological structures and their ability to control and influence vast numbers of atoms, suggesting that life is characterized by complex multiscale interactions. It addresses traditional statistical mechanics' limitations in explaining how a few thousand atoms can influence systems containing 10^25 atoms, emphasizing the structural complexity and electric interactions in biological systems. The insights stem from various studies and experiments, revealing the significant role of ion channels and the implications of crowding on biological functionality.

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1 Life is Different: it is inherited Oberwolfach Workshop, February 2018 Bob Eisenberg Department of Applied Mathematics Illinois Institute of Technology Department of Physiology and Biophysics Rush University Chicago USA u  
Page 2 Oberwolfach Workshop 1809 The Mathematics of Mechanobiology and Cell Signaling 25 February – 3 March 2018 Organizers: Davide Ambrosi, Milano Italy Chun Liu, State College PA USA Matthias Röger, Dortmund Germany Angela Stevens, Münster Germany at the Mathematisches Forschungsinstitut Oberwolfach.
3 Thanks to Chun Liu 柳 春 For a very special Friendship and Collaboration!
Page 4 Life is special because it is inherited from a tiny number of atoms And the central question of biology is How is this possible?
Page 5
Page 6
Page 7 How can a few thousand atoms conceivably control 1025 atoms?
8 Experimental Evidence: A few atoms make a BIG Difference OmpF 1M/1M G119D 1M/1M G119D 0.05M/0.05M OmpF 0.05M/0.05M Structure determined by Raimund Dutzler in Tilman Schirmer’s lab Current Voltage relation determined by John Tang in Bob Eisenberg’s Lab Ompf G119D Glycine G replaced by Aspartate D
Page 9 How can a few thousand atoms conceivably control 1025 atoms? Traditional Statistical Mechanics says this is impossible! 𝑨 𝒙, 𝒕 ≡ 𝒂 𝒙, 𝒕 where 𝒓 𝟐 = 𝒙 𝟐 + 𝒚 𝟐 + 𝒛 𝟐 and 𝑾 𝒙 = 𝑵𝒆 − 𝒓 𝟐 𝑹 𝟐 𝑹 specifies the radius of the small spherical volume over which the spatial average takes place. = 𝑾 𝒙′ 𝒂 𝒙 − 𝒙′, 𝒕 𝒅 𝟑 𝒙′
Page 10 How can a few thousand atoms conceivably control 1025 atoms? The thousand atoms of one gene occupy say 10-27 m3 The volume of a person might be 1m3 Volume of Canada, USA 𝒐𝒓China 1m high is 1013 m3 Fraction of space of a gene is about 10-27 Fraction of Space of One Person in Canada is 10-13 1 m3 has no effect in Canada 1 gene should have no effect
Page 11 How can a few thousand atoms conceivably control 1025 atoms? Biological Answer: Structure: a Hierarchy of Devices Physical Answer: Electrodynamics: Strong and Universal inside atoms to stars Another talk* another day! *Eisenberg, Oriols, and Ferry. 2017. Dynamics of Current, Charge, and Mass. Molecular Based Mathematical Biology 5:78-115 arXiv https://arxiv.org/abs/1708.07400.
12 Everyone knows Biology is made of Structures Working hypothesis: The Structures make Devices that span the scales
13 ATOM 10-10 m Molecule 10-8 m Organelle 10-6m Cell 10-5m Cell 10-5m Tissue 10-3 m Organ 10-1 m ORGANISM 1 m Answer: Biology is made of Devices and they span the scales
Page 14 How can a few thousand atoms conceivably control 1025 atoms? ANSWER: by forming a HIERARCHY of DEVICES
Page 15 Different Kind of Averaging in Device Definition of a Device Output is Perfectly Correlated with Input Averaging in a Device Creates a Perfectly Correlated* Replica of the Input 𝑨 𝒙, 𝒕 ≡ 𝒂 𝒙, 𝒕 ≠ 𝒅 𝟑 𝒙′ 𝑾(𝒙′ )𝒂(𝒙 − 𝒙′, 𝒕) Not equal Precise stochastic definition *Coherence function of Device = 1.0 Coherence function in general = 𝑷 𝒙𝒚 𝒇 𝑷 𝒙𝒙 𝒇 𝑷 𝒚𝒚 𝒇 ; 𝑷 𝒙𝒚; 𝑷 𝒛𝒛 = cross, self-power spectral density = estimator of
Page 16 Structural Complexity so characteristic of life, so daunting to mathematicians is the Hierarchy of Devices
Page 17 Biology is made of Devices and they are Multiscale Structural Complexity of Life is the Hierarchy of Devices
18 One Cell contains many Devices Structural Complexity of Life is the Hierarchy of Devices
19 Real Biological System A Nerve Cell is a Hierarchy of Devices Cell Body, Dendrites, Axon, Terminals Example: Bob, I would not put it that way….. Andrew Huxley with his mentor Alan Hodgkin
20 Nerve Signal is the propagating ‘Action Potential’, a waveform in x and t Purely Local Theory is Impossible because the phenomena involves centimeter scales, as well as Angstroms. All atom Molecular Dynamics is impossible because ~1023 atoms are involved Atomic Properties of the Voltage Sensor are coupled to the macroscopic electric field a centimeter away. The electric field controls the sensor; the sensor controls the electric field. That is how propagation works!
Page 21 Channels are Source of Signal
Page 22
Page 23 Start with Voltage Sensor
24 Vargas, E., Yarov-Yarovoy, V., Khalili-Araghi, F., Catterall, W. A., Klein, M. L., Tarek, M., Lindahl, E., Schulten, K., Perozo, E., Bezanilla, F. & Roux, B. An emerging consensus on voltage-dependent gating from computational modeling and molecular dynamics simulations. The Journal of General Physiology 140, 587-594 (2012). Emerging Consensus …. Voltage Sensor Structure (NOT conduction channel)
25 Vargas, E., Yarov-Yarovoy, V., Khalili-Araghi, F., Catterall, W. A., Klein, M. L., Tarek, M., Lindahl, E., Schulten, K., Perozo, E., Bezanilla, F. & Roux, B. An emerging consensus on voltage-dependent gating from computational modeling and molecular dynamics simulations. The Journal of General Physiology 140, 587-594 (2012). Emerging Consensus …. Voltage Sensor Structure (NOT conduction channel)
26
27 Physiologists§ Mistakenly call a Saturating distribution ‘Boltzmann’ e.g., Bezanilla, Villalba-Galea J. Gen. Physiol (2013) 142: 575 𝑸 𝑽 = 𝑸 𝒎𝒂𝒙 𝟏 + 𝒆𝒙𝒑 −𝑸 𝒎𝒂𝒙 𝑽 − 𝑽 𝟏 𝟐 𝒌 𝑩 𝑻 Physicists: Saturation Fermi distribution Boltzmann* distribution does NOT saturate. Boltzmann is exponential, like 𝒆𝒙𝒑 −𝑽 𝒌 𝑩 𝑻 . *Boltzmann (1904) Lectures on Gas Theory, Berkeley § p.503 of Hodgkin and Huxley. 1952. ‘Quantitative description ...’ J. Physiol. 117:500-544. Bezanilla. How membrane proteins sense voltage Nature Rev Mol Cell Biol (2008) 9, 323 Fermi Distribution not Boltzmann Arginines Internal Dissolved Ions 𝑲+, 𝒐𝒓𝒈𝒂𝒏𝒊𝒄𝒔− External Dissolved Ions 𝑵𝒂+ 𝑪𝒍− Dissolved Ions 𝑲+, 𝒐𝒓𝒈𝒂𝒏𝒊𝒄𝒔− External Dissolved Ions 𝑵𝒂+ 𝑪𝒍− Voltage Clamp = Test Potential Voltage Clamp =0 Holding Potential Current 𝐈 𝐦 to maintain test potential Holding Potenti al Test Potentials Holding Potenti al Test Potentials 𝐈 𝐦
Perhaps the first Consistent Model of a Protein Machine made by a conformation change 28 Francisco Bezanilla Chun Liu 柳 春 Allen Tzyy-Leng Horng 洪子倫 Bob Eisenberg
Figure 1. Geometric configuration of the reduced mechanical model. Voltage sensor works by charge injection through a fluid dielectric of side chains, attachments of arginines to the S4 segment. intracellular extracellular Voltage Sensor works by Charge Injection through a fluid dielectric of side chains, not yet fully known. More work needed!
30
31 𝐸 = 𝑘 𝐵 𝑇 𝑐𝑖 𝑙𝑜𝑔𝑐𝑖 − 𝜀0 𝜀𝑟 2𝑎𝑙𝑙 𝑖 ∇𝜙 2 + 𝑧𝑖 𝑒 𝑎𝑙𝑙 𝑖 𝑐𝑖 𝜙 + ( 𝑉𝑖 + 𝑉𝑏 ) 𝑎𝑟𝑔𝑖𝑛𝑖𝑛𝑒𝑠 𝑐𝑖 𝑉 + 𝑔𝑖𝑗 2 𝑐𝑖 𝑐𝑗 𝑎𝑟𝑔𝑖𝑛𝑖𝑛𝑒𝑠 𝑖,𝑗 𝑑𝑉, Variational Formulation EnVarA because ‘Everything’ interacts with ‘Everything Else’ through the electric field and often through steric interactions of Pauli exclusion principle Poisson Equation and Transport equation are DERIVED, not assumed from variations like 𝛿𝐸 𝛿𝜙 = 0, so cross terms are always consistent, i.e., satisfy all equations with one set of parameters.
30 Defining Laws Charge Creates Electric Field − 1 𝐴 𝑑 𝑑𝑧 Γ𝐴 𝑑𝜙 𝑑𝑧 = 𝑖=1 𝑁 𝑧𝑖 𝑐𝑖 , 𝑖 = Na, Cl, 1, 2, 3, 4 Transport of Mass 𝜕𝑐 𝑖 𝜕𝑡 + 1 𝐴 𝜕 𝜕𝑧 𝐴𝐽𝑖 = 0, 𝑖 =Na, Cl, 1,2,3,4
33 𝑽𝒊 𝒛, 𝒕 = 𝑲(𝒛 − 𝒛𝒊 + 𝒁 𝑺𝟒(𝒕) ) 𝟐 , where K is the spring constant, zi is the fixed anchoring position of the spring for each arginine ci on S4, 𝑍 𝑆4(𝑡) is the center-of-mass z position of S4 by treating S4 as a rigid body. 𝑍 𝑆4(𝑡) follows the motion of equation based on spring-mass system: Conformation Change of Arginines is Described by an Elastic System
Current Carried by Arginines note cross terms 𝟏= − 𝟏 𝑪 𝟏 𝒛 + 𝒛 𝒂𝒓𝒈 𝑪 𝟏 𝒛 + 𝑪 𝟏 𝑽 𝟏 𝒛 + 𝑽 𝒛 + 𝒈𝑪 𝟏 𝑪 𝟐 𝒛 + 𝑪 𝟑 𝒛 + 𝑪 𝟒 𝒛 𝟐 = − 𝟐 𝑪 𝟐 𝒛 + 𝒛 𝒂𝒓𝒈 𝑪 𝟐 𝒛 + 𝑪 𝟐 𝑽 𝟐 𝒛 + 𝑽 𝒛 + 𝒈𝑪 𝟐 𝑪 𝟏 𝒛 + 𝑪 𝟑 𝒛 + 𝑪 𝟒 𝒛 𝟑 = − 𝟑 𝑪 𝟑 𝒛 + 𝒛 𝒂𝒓𝒈 𝑪 𝟑 𝒛 + 𝑪 𝟑 𝑽 𝟑 𝒛 + 𝑽 𝒛 + 𝒈𝑪 𝟑 𝑪 𝟏 𝒛 + 𝑪 𝟐 𝒛 + 𝑪 𝟒 𝒛 𝟒= − 𝟒 𝑪 𝟒 𝒛 + 𝒛 𝒂𝒓𝒈 𝑪 𝟒 𝒛 + 𝑪 𝟒 𝑽 𝟒 𝒛 + 𝑽 𝒛 + 𝒈𝑪 𝟒 𝑪 𝟏 𝒛 + 𝑪 𝟐 𝒛 + 𝑪 𝟑 𝒛
Page 35 Figure 9. (a) Time courses of subtracted gating current [A1] with voltage rising from -90mV to V mV at t=10, holds on till t=150, and drops back to -90mV, where V=-62, -50, … -8 mV. (b) τ2 versus V compared with experiment [7]. Fitting Data
Page 36 Figure 3. (a) QV curve and comparison with [7]. Steady-state distributions for Na, Cl and arginines at (b) V=-90mV, (c) V=-48mV, (d) V=-8mV. Fitting Data
37 Ions Electric Field Current is Conserved OUTPUT Current including 𝜺 𝟎 𝑬 𝒕 INPUT Voltage Clamp t Conservation of Current is an Important Constraint. Rate Models of Chemical Kinetics; Molecular Dynamics do not conserve current A different talk
Page 38 Nerve Signaling is a Hierarchy of Devices
Page 39 Channels are Source of Signal
40 Classical cable theory of transmission lines, telegrapher’s equations, Kelvin, Hodgkin, Noble, including 3D-cable theory, ~10 papers, e.g., Eisenberg and Johnson. 1970. Three dimensional electrical field problems in physiology. Prog. Biophys. Mol. Biol. 20:1-65 Barcilon, Cole, Eisenberg. 1971. Singular Perturbation … SIAM J. Appl. Math. 21:339-354. Another talk, another time! Channels + lipid capacitance Poisson Equation
How do ions move through channels? >75 papers since 1986 41
42 Working Hypothesis bio-speak: Crucial Biological Adaptation is Crowded Ions and Side Chains Biology occurs in concentrated >0.3 M mixtures of spherical charges NOT IDEAL AT ALL Poisson Boltzmann does NOT fit data!! 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
43 Solutions are Extraordinarily Concentrated >10M Solid NaCl is 37M where they are most important, DNA, enzyme active sites, channels and electrodes of batteries and electrochemical cells
Active Sites of Proteins are Very Charged 7 charges ~ 20M net charge Selectivity Filters and Gates of Ion Channels are Active Sites = 1.2×1022 cm-3 - + + + + + - - - 4 Å K+ Na+ Ca2+ Hard Spheres 44 Figure adapted from Tilman Schirmer liquid Water is 55 M solid NaCl is 37 M OmpF Porin Physical basis of function K+ Na+ Induced Fit of Side Chains Ions are Crowded
Crowded Active Sites in 573 Enzymes 45 Enzyme Type Catalytic Active Site Density (Molar) Protein Acid (positive) Basic (negative) | Total | Elsewhere Total (n = 573) 10.6 8.3 18.9 2.8 EC1 Oxidoreductases (n = 98) 7.5 4.6 12.1 2.8 EC2 Transferases (n = 126) 9.5 7.2 16.6 3.1 EC3 Hydrolases (n = 214) 12.1 10.7 22.8 2.7 EC4 Lyases (n = 72) 11.2 7.3 18.5 2.8 EC5 Isomerases (n = 43) 12.6 9.5 22.1 2.9 EC6 Ligases (n = 20) 9.7 8.3 18.0 3.0 Jimenez-Morales, Liang, Eisenberg
Charge-Space Competition 46 More than 35 papers are available at ftp://ftp.rush.edu/users/molebio/Bob_Eisenberg/reprints Monte Carlo Methods Dezső Boda Wolfgang NonnerDoug Henderson Bob Eisenberg
47 Mutants of ompF Porin Atomic Scale Macro Scale 30 60 -30 30 60 0 pA mV LECE (-7e) LECE-MTSES- (-8e) LECE-GLUT- (-8e)ECa ECl WT (-1e) Calcium selective Experiments have ‘engineered’ channels (5 papers) including Two Synthetic Calcium Channels As density of permanent charge increases, channel becomes calcium selective Erev  ECa in 0.1M 1.0 M CaCl2 ; pH 8.0 Unselective Natural ‘wild’ Type built by Henk Miedema, Wim Meijberg of BioMade Corp. Groningen, Netherlands Miedema et al, Biophys J 87: 3137–3147 (2004); 90:1202-1211 (2006); 91:4392-4400 (2006) MUTANT ─ Compound Glutathione derivatives Designed by Theory || Evidence RyR (start)
48 Ca Channel log (Concentration/M) 0.5 -6 -4 -2 Na+ 0 1 Ca2+ Charge -3e Occupancy(number) E E E A Monte Carlo simulations of Boda, et al Same Parameters pH 8 Mutation Same Parameters Mutation EEEE has full biological selectivity in similar simulations Na Channel Concentration/M pH =8 Na+ Ca2+ 0.004 0 0.002 0.05 0.10 Charge -1e D E K A
49 Mutants of ompF Porin Atomic Scale Macro Scale 30 60 -30 30 60 0 pA mV LECE (-7e) LECE-MTSES- (-8e) LECE-GLUT- (-8e)ECa ECl WT (-1e) Calcium selective Experiments have ‘engineered’ channels (5 papers) including Two Synthetic Calcium Channels As density of permanent charge increases, channel becomes calcium selective Erev  ECa in 0.1M 1.0 M CaCl2 ; pH 8.0 Unselective Natural ‘wild’ Type built by Henk Miedema, Wim Meijberg of BioMade Corp. Groningen, Netherlands Miedema et al, Biophys J 87: 3137–3147 (2004); 90:1202-1211 (2006); 91:4392-4400 (2006) MUTANT ─ Compound Glutathione derivatives Designed by Theory || Evidence RyR (start)
Poisson Fermi Approach to Ion Channels 50 . 劉晉良 Jinn-Liang Liu discovered role of SATURATION Bob Eisenberg helped with applications
Motivation Natural Description of Crowded Charge is a Fermi Distribution designed to describe saturation Simulating saturation by interatomic repulsion (Lennard Jones) is a significant mathematical challenge to be side-stepped if possible 51
Motivation Largest Effect of Crowded Charge is Saturation important in channels, DNA, enzymes, and electrochemical devices in general Saturation cannot be described at all by classical Poisson Boltzmann approach and is described in a (wildly) uncalibrated way by present day Molecular Dynamics 52
53 Liu, Eisenberg. 2018. Journal of chemical physics 148:054501 also arXiv:1801.03470 . Calibration in Bulk Solution: New Result
54 Individual activity coefficients of 2:1 electrolytes. Comparison of PF results with experimental data [26] on i = Pos2+ (cation) and Neg− (anion) activity coefficients γi in various [PosNeg2] from 0 to 1.5 M. Calibration in Bulk Solution: New Result
Na Channel 55 Signature of Cardiac Calcium Channel CaV1.n Anomalous* Mole Fraction (non-equilibrium) Liu & Eisenberg (2015) Physical Review E 92: 012711 *Anomalous because CALCIUM CHANNEL IS A SODIUM CHANNEL at [CaCl2]  10-3.4 Ca2+ is conducted for [Ca2+] > 10-3.4, but Na+ is conducted for [Ca2+] <10-3. Ca Channel
56 Three Dimensional Theory Comparison with Experiments Gramicidin A
‘Law’ of Mass Action including Interactions From Bob Eisenberg p. 1-6, in this issue Variational Approach EnVarA 1 2- 0E      6 7 8 6 7 8 r r x u Conservative Dissipative Chun Liu 柳 春
58 Energetic Variational Approach allows accurate computation of Flow and Interactions in Complex Fluids like Liquid Crystals Engineering needs Calibrated Theories and Simulations Engineering Devices almost always use flow Classical theories and Molecular Dynamics have difficulties with flow, interactions, and complex fluids
59 Energetic Variational Approach EnVarA 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 } } 1 2 0 E       '' Dissipative'Force'Conservative Force r r x u Action Integral, after pullback Rayleigh Dissipation FunctionAction Integral, after pullback Rayleigh Dissipation Function Field Theory of Ionic Solutions: Liu, Ryham, Hyon, Eisenberg Allows boundary conditions and flow Deals Consistently with Interactions of Components 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 Composite Variational Principle Shorthand for Euler Lagrange process with respect to r x Shorthand for Euler Lagrange process with respect to r u
2 , = , = , i i i B i i j j B i i n p j n p D c c k T z e c d y dx k T c                    = Dissipative r r% 1 4 4 4 4 4 4 4 4 4 2 4 4 4 4 4 4 4 4 4 3 , = = , , 0 , , = 1 log 2 2 i B i i i i i j j i n p i n p i j n p c k T c c z ec c d y dx d dt                        Conservative r r% 6 4 4 4 4 4 4 4 4 4 4 44 7 4 4 4 4 4 4 4 4 4 4 4 48 Hard Sphere Terms Permanent Charge of proteintime ci number density; thermal energy; Di diffusion coefficient; n negative; p positive; zi valence; ε dielectric constantBk T Number Density Thermal Energy valence proton charge Dissipation Principle Conservative Energy dissipates into Friction = , 0 2 1 22 i i i n p z ec              Note that with suitable boundary conditions 60
61 PNP (Poisson Nernst Planck) for Spheres Eisenberg, Hyon, and Liu 12 , 14 12 , 14 12 ( ) ( ) = ( ) | | 6 ( ) ( ) ( ) , | | n n n nn n n n n n B n p n p p a a x yc c D c z e c y dy t k T x y a a x y c y dy x y                                r r r r r r r r r r r r Nernst Planck Diffusion Equation for number density cn of negative n ions; positive ions are analogous Non-equilibrium variational field theory EnVarA Coupling Parameters Ion Radii =1 or( ) = N i i i z ec i n p               0ρ Poisson Equation Permanent Charge of Protein Number Densities Diffusion Coefficient Dielectric Coefficient valence proton charge Thermal Energy
62 Semiconductor PNP Equations For Point Charges       i i i i d J D x A x x dx    Poisson’s Equation Drift-diffusion & Continuity Equation          0 i i i d d x A x e x e z x A x dx dx            Ρ 0idJ dx         ex * x x x ln xi i i iz e kT              1 2 3 Finite Size Special Chemistry Chemical Potential Thermal Energy Valence Proton charge Permanent Charge of Protein Doping of Semiconductor Cross sectional Area Flux Diffusion Coefficient Number Densities Dielectric Coefficient valence proton charge Not in Semiconductor ( )i x
All we have to do is Solve them! with Boundary Conditions defining Charge Carriers ions, holes, quasi-electrons Geometry 63
64 Mathematical Solutions can be simple and easy to recognize! f b k k L R
Page 65     out inJ J k f k b kJ l k C l k C    6 44 7 4 48 6 44 7 4 48 Unidirectional Efflux Unidirectional Infflux eft ightL R f b k k L R *but potential profile must be computed by solving electrostatics problem. Complex coupled nonlinearities are in the electric field!!! Solution is a Chemical Reaction* exactly
66 Solution* of PNP Equation *MATHEMATICS This solution was actually DERIVED from several conditional probability measures by ANALYTICAL integrations No approximations, no numerics 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 Simple formulas are available for the probabilities   {                       Unidirectional Efflux Unidirectional I ConditionalDiffusion ChannelSource ProbabilityVelocity LengthConcentration 6 4 4 44 7 4 4 4 48 1 4 2 4 31 2 3 1 4 4 2 4 4 3 k k k k k D D CR LJ R L RC L Prob Prob Rate Constant nfflux 6 4 4 44 7 4 4 4 48
Page 67 Please do not be deceived by the eventual simplicity of Results. This took >2 years! Solution was actually DERIVED 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 on-line textbook of USA Biophysical Society, DeFelice editor. Eisenberg, B. (2011) Chemical Physics Letters 511: 1-6
All we have to do is Solve them! Don’t Despair Semiconductor Technology has Already Done That! 68
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 formulas are possible Weishi Liu University of Kansas
Device converts Input to Output by a simple ‘law’ 70 Device is ROBUST and TRANSFERRABLE because it uses POWER and has complexity! Dotted lines outline: current mirrors (red); differential amplifiers (blue); class A gain stage (magenta); voltage level shifter (green); output stage (cyan). Circuit Diagram of common 741 op-amp: Twenty transistors needed to make linear robust device INPUT Vin(t) OUTPUT Vout (t) Power Supply Dirichlet Boundary Condition independent of time and everything else Power Supply
Integrated Circuit A Hierarchy of Devices and Structures Technology as of ~2014 IBM Power8 71 Too small to see!
72 Any Questions ?

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What is different about life? it is inherited oberwolfach march 7 1 2018

  • 1.
    1 Life is Different:it is inherited Oberwolfach Workshop, February 2018 Bob Eisenberg Department of Applied Mathematics Illinois Institute of Technology Department of Physiology and Biophysics Rush University Chicago USA u  
  • 2.
    Page 2 Oberwolfach Workshop1809 The Mathematics of Mechanobiology and Cell Signaling 25 February – 3 March 2018 Organizers: Davide Ambrosi, Milano Italy Chun Liu, State College PA USA Matthias Röger, Dortmund Germany Angela Stevens, Münster Germany at the Mathematisches Forschungsinstitut Oberwolfach.
  • 3.
    3 Thanks to ChunLiu 柳 春 For a very special Friendship and Collaboration!
  • 4.
    Page 4 Life isspecial because it is inherited from a tiny number of atoms And the central question of biology is How is this possible?
  • 5.
  • 6.
  • 7.
    Page 7 How cana few thousand atoms conceivably control 1025 atoms?
  • 8.
    8 Experimental Evidence: A fewatoms make a BIG Difference OmpF 1M/1M G119D 1M/1M G119D 0.05M/0.05M OmpF 0.05M/0.05M Structure determined by Raimund Dutzler in Tilman Schirmer’s lab Current Voltage relation determined by John Tang in Bob Eisenberg’s Lab Ompf G119D Glycine G replaced by Aspartate D
  • 9.
    Page 9 How cana few thousand atoms conceivably control 1025 atoms? Traditional Statistical Mechanics says this is impossible! 𝑨 𝒙, 𝒕 ≡ 𝒂 𝒙, 𝒕 where 𝒓 𝟐 = 𝒙 𝟐 + 𝒚 𝟐 + 𝒛 𝟐 and 𝑾 𝒙 = 𝑵𝒆 − 𝒓 𝟐 𝑹 𝟐 𝑹 specifies the radius of the small spherical volume over which the spatial average takes place. = 𝑾 𝒙′ 𝒂 𝒙 − 𝒙′, 𝒕 𝒅 𝟑 𝒙′
  • 10.
    Page 10 How cana few thousand atoms conceivably control 1025 atoms? The thousand atoms of one gene occupy say 10-27 m3 The volume of a person might be 1m3 Volume of Canada, USA 𝒐𝒓China 1m high is 1013 m3 Fraction of space of a gene is about 10-27 Fraction of Space of One Person in Canada is 10-13 1 m3 has no effect in Canada 1 gene should have no effect
  • 11.
    Page 11 How cana few thousand atoms conceivably control 1025 atoms? Biological Answer: Structure: a Hierarchy of Devices Physical Answer: Electrodynamics: Strong and Universal inside atoms to stars Another talk* another day! *Eisenberg, Oriols, and Ferry. 2017. Dynamics of Current, Charge, and Mass. Molecular Based Mathematical Biology 5:78-115 arXiv https://arxiv.org/abs/1708.07400.
  • 12.
    12 Everyone knows Biology ismade of Structures Working hypothesis: The Structures make Devices that span the scales
  • 13.
    13 ATOM 10-10 m Molecule 10-8 m Organelle 10-6m Cell 10-5m Cell 10-5m Tissue 10-3m Organ 10-1 m ORGANISM 1 m Answer: Biology is made of Devices and they span the scales
  • 14.
    Page 14 How cana few thousand atoms conceivably control 1025 atoms? ANSWER: by forming a HIERARCHY of DEVICES
  • 15.
    Page 15 Different Kindof Averaging in Device Definition of a Device Output is Perfectly Correlated with Input Averaging in a Device Creates a Perfectly Correlated* Replica of the Input 𝑨 𝒙, 𝒕 ≡ 𝒂 𝒙, 𝒕 ≠ 𝒅 𝟑 𝒙′ 𝑾(𝒙′ )𝒂(𝒙 − 𝒙′, 𝒕) Not equal Precise stochastic definition *Coherence function of Device = 1.0 Coherence function in general = 𝑷 𝒙𝒚 𝒇 𝑷 𝒙𝒙 𝒇 𝑷 𝒚𝒚 𝒇 ; 𝑷 𝒙𝒚; 𝑷 𝒛𝒛 = cross, self-power spectral density = estimator of
  • 16.
    Page 16 Structural Complexity socharacteristic of life, so daunting to mathematicians is the Hierarchy of Devices
  • 17.
    Page 17 Biology ismade of Devices and they are Multiscale Structural Complexity of Life is the Hierarchy of Devices
  • 18.
    18 One Cell contains many Devices StructuralComplexity of Life is the Hierarchy of Devices
  • 19.
    19 Real Biological System ANerve Cell is a Hierarchy of Devices Cell Body, Dendrites, Axon, Terminals Example: Bob, I would not put it that way….. Andrew Huxley with his mentor Alan Hodgkin
  • 20.
    20 Nerve Signal isthe propagating ‘Action Potential’, a waveform in x and t Purely Local Theory is Impossible because the phenomena involves centimeter scales, as well as Angstroms. All atom Molecular Dynamics is impossible because ~1023 atoms are involved Atomic Properties of the Voltage Sensor are coupled to the macroscopic electric field a centimeter away. The electric field controls the sensor; the sensor controls the electric field. That is how propagation works!
  • 21.
    Page 21 Channels areSource of Signal
  • 22.
  • 23.
    Page 23 Start withVoltage Sensor
  • 24.
    24 Vargas, E., Yarov-Yarovoy,V., Khalili-Araghi, F., Catterall, W. A., Klein, M. L., Tarek, M., Lindahl, E., Schulten, K., Perozo, E., Bezanilla, F. & Roux, B. An emerging consensus on voltage-dependent gating from computational modeling and molecular dynamics simulations. The Journal of General Physiology 140, 587-594 (2012). Emerging Consensus …. Voltage Sensor Structure (NOT conduction channel)
  • 25.
    25 Vargas, E., Yarov-Yarovoy,V., Khalili-Araghi, F., Catterall, W. A., Klein, M. L., Tarek, M., Lindahl, E., Schulten, K., Perozo, E., Bezanilla, F. & Roux, B. An emerging consensus on voltage-dependent gating from computational modeling and molecular dynamics simulations. The Journal of General Physiology 140, 587-594 (2012). Emerging Consensus …. Voltage Sensor Structure (NOT conduction channel)
  • 26.
  • 27.
    27 Physiologists§ Mistakenly call aSaturating distribution ‘Boltzmann’ e.g., Bezanilla, Villalba-Galea J. Gen. Physiol (2013) 142: 575 𝑸 𝑽 = 𝑸 𝒎𝒂𝒙 𝟏 + 𝒆𝒙𝒑 −𝑸 𝒎𝒂𝒙 𝑽 − 𝑽 𝟏 𝟐 𝒌 𝑩 𝑻 Physicists: Saturation Fermi distribution Boltzmann* distribution does NOT saturate. Boltzmann is exponential, like 𝒆𝒙𝒑 −𝑽 𝒌 𝑩 𝑻 . *Boltzmann (1904) Lectures on Gas Theory, Berkeley § p.503 of Hodgkin and Huxley. 1952. ‘Quantitative description ...’ J. Physiol. 117:500-544. Bezanilla. How membrane proteins sense voltage Nature Rev Mol Cell Biol (2008) 9, 323 Fermi Distribution not Boltzmann Arginines Internal Dissolved Ions 𝑲+, 𝒐𝒓𝒈𝒂𝒏𝒊𝒄𝒔− External Dissolved Ions 𝑵𝒂+ 𝑪𝒍− Dissolved Ions 𝑲+, 𝒐𝒓𝒈𝒂𝒏𝒊𝒄𝒔− External Dissolved Ions 𝑵𝒂+ 𝑪𝒍− Voltage Clamp = Test Potential Voltage Clamp =0 Holding Potential Current 𝐈 𝐦 to maintain test potential Holding Potenti al Test Potentials Holding Potenti al Test Potentials 𝐈 𝐦
  • 28.
    Perhaps the first ConsistentModel of a Protein Machine made by a conformation change 28 Francisco Bezanilla Chun Liu 柳 春 Allen Tzyy-Leng Horng 洪子倫 Bob Eisenberg
  • 29.
    Figure 1. Geometricconfiguration of the reduced mechanical model. Voltage sensor works by charge injection through a fluid dielectric of side chains, attachments of arginines to the S4 segment. intracellular extracellular Voltage Sensor works by Charge Injection through a fluid dielectric of side chains, not yet fully known. More work needed!
  • 30.
  • 31.
    31 𝐸 = 𝑘𝐵 𝑇 𝑐𝑖 𝑙𝑜𝑔𝑐𝑖 − 𝜀0 𝜀𝑟 2𝑎𝑙𝑙 𝑖 ∇𝜙 2 + 𝑧𝑖 𝑒 𝑎𝑙𝑙 𝑖 𝑐𝑖 𝜙 + ( 𝑉𝑖 + 𝑉𝑏 ) 𝑎𝑟𝑔𝑖𝑛𝑖𝑛𝑒𝑠 𝑐𝑖 𝑉 + 𝑔𝑖𝑗 2 𝑐𝑖 𝑐𝑗 𝑎𝑟𝑔𝑖𝑛𝑖𝑛𝑒𝑠 𝑖,𝑗 𝑑𝑉, Variational Formulation EnVarA because ‘Everything’ interacts with ‘Everything Else’ through the electric field and often through steric interactions of Pauli exclusion principle Poisson Equation and Transport equation are DERIVED, not assumed from variations like 𝛿𝐸 𝛿𝜙 = 0, so cross terms are always consistent, i.e., satisfy all equations with one set of parameters.
  • 32.
    30 Defining Laws Charge CreatesElectric Field − 1 𝐴 𝑑 𝑑𝑧 Γ𝐴 𝑑𝜙 𝑑𝑧 = 𝑖=1 𝑁 𝑧𝑖 𝑐𝑖 , 𝑖 = Na, Cl, 1, 2, 3, 4 Transport of Mass 𝜕𝑐 𝑖 𝜕𝑡 + 1 𝐴 𝜕 𝜕𝑧 𝐴𝐽𝑖 = 0, 𝑖 =Na, Cl, 1,2,3,4
  • 33.
    33 𝑽𝒊 𝒛, 𝒕= 𝑲(𝒛 − 𝒛𝒊 + 𝒁 𝑺𝟒(𝒕) ) 𝟐 , where K is the spring constant, zi is the fixed anchoring position of the spring for each arginine ci on S4, 𝑍 𝑆4(𝑡) is the center-of-mass z position of S4 by treating S4 as a rigid body. 𝑍 𝑆4(𝑡) follows the motion of equation based on spring-mass system: Conformation Change of Arginines is Described by an Elastic System
  • 34.
    Current Carried byArginines note cross terms 𝟏= − 𝟏 𝑪 𝟏 𝒛 + 𝒛 𝒂𝒓𝒈 𝑪 𝟏 𝒛 + 𝑪 𝟏 𝑽 𝟏 𝒛 + 𝑽 𝒛 + 𝒈𝑪 𝟏 𝑪 𝟐 𝒛 + 𝑪 𝟑 𝒛 + 𝑪 𝟒 𝒛 𝟐 = − 𝟐 𝑪 𝟐 𝒛 + 𝒛 𝒂𝒓𝒈 𝑪 𝟐 𝒛 + 𝑪 𝟐 𝑽 𝟐 𝒛 + 𝑽 𝒛 + 𝒈𝑪 𝟐 𝑪 𝟏 𝒛 + 𝑪 𝟑 𝒛 + 𝑪 𝟒 𝒛 𝟑 = − 𝟑 𝑪 𝟑 𝒛 + 𝒛 𝒂𝒓𝒈 𝑪 𝟑 𝒛 + 𝑪 𝟑 𝑽 𝟑 𝒛 + 𝑽 𝒛 + 𝒈𝑪 𝟑 𝑪 𝟏 𝒛 + 𝑪 𝟐 𝒛 + 𝑪 𝟒 𝒛 𝟒= − 𝟒 𝑪 𝟒 𝒛 + 𝒛 𝒂𝒓𝒈 𝑪 𝟒 𝒛 + 𝑪 𝟒 𝑽 𝟒 𝒛 + 𝑽 𝒛 + 𝒈𝑪 𝟒 𝑪 𝟏 𝒛 + 𝑪 𝟐 𝒛 + 𝑪 𝟑 𝒛
  • 35.
    Page 35 Figure 9.(a) Time courses of subtracted gating current [A1] with voltage rising from -90mV to V mV at t=10, holds on till t=150, and drops back to -90mV, where V=-62, -50, … -8 mV. (b) τ2 versus V compared with experiment [7]. Fitting Data
  • 36.
    Page 36 Figure 3.(a) QV curve and comparison with [7]. Steady-state distributions for Na, Cl and arginines at (b) V=-90mV, (c) V=-48mV, (d) V=-8mV. Fitting Data
  • 37.
    37 Ions Electric Field Current is Conserved OUTPUT Currentincluding 𝜺 𝟎 𝑬 𝒕 INPUT Voltage Clamp t Conservation of Current is an Important Constraint. Rate Models of Chemical Kinetics; Molecular Dynamics do not conserve current A different talk
  • 38.
    Page 38 Nerve Signaling isa Hierarchy of Devices
  • 39.
    Page 39 Channels areSource of Signal
  • 40.
    40 Classical cable theoryof transmission lines, telegrapher’s equations, Kelvin, Hodgkin, Noble, including 3D-cable theory, ~10 papers, e.g., Eisenberg and Johnson. 1970. Three dimensional electrical field problems in physiology. Prog. Biophys. Mol. Biol. 20:1-65 Barcilon, Cole, Eisenberg. 1971. Singular Perturbation … SIAM J. Appl. Math. 21:339-354. Another talk, another time! Channels + lipid capacitance Poisson Equation
  • 41.
    How do ionsmove through channels? >75 papers since 1986 41
  • 42.
    42 Working Hypothesis bio-speak: Crucial BiologicalAdaptation is Crowded Ions and Side Chains Biology occurs in concentrated >0.3 M mixtures of spherical charges NOT IDEAL AT ALL Poisson Boltzmann does NOT fit data!! 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
  • 43.
    43 Solutions are ExtraordinarilyConcentrated >10M Solid NaCl is 37M where they are most important, DNA, enzyme active sites, channels and electrodes of batteries and electrochemical cells
  • 44.
    Active Sites ofProteins are Very Charged 7 charges ~ 20M net charge Selectivity Filters and Gates of Ion Channels are Active Sites = 1.2×1022 cm-3 - + + + + + - - - 4 Å K+ Na+ Ca2+ Hard Spheres 44 Figure adapted from Tilman Schirmer liquid Water is 55 M solid NaCl is 37 M OmpF Porin Physical basis of function K+ Na+ Induced Fit of Side Chains Ions are Crowded
  • 45.
    Crowded Active Sites in573 Enzymes 45 Enzyme Type Catalytic Active Site Density (Molar) Protein Acid (positive) Basic (negative) | Total | Elsewhere Total (n = 573) 10.6 8.3 18.9 2.8 EC1 Oxidoreductases (n = 98) 7.5 4.6 12.1 2.8 EC2 Transferases (n = 126) 9.5 7.2 16.6 3.1 EC3 Hydrolases (n = 214) 12.1 10.7 22.8 2.7 EC4 Lyases (n = 72) 11.2 7.3 18.5 2.8 EC5 Isomerases (n = 43) 12.6 9.5 22.1 2.9 EC6 Ligases (n = 20) 9.7 8.3 18.0 3.0 Jimenez-Morales, Liang, Eisenberg
  • 46.
    Charge-Space Competition 46 More than35 papers are available at ftp://ftp.rush.edu/users/molebio/Bob_Eisenberg/reprints Monte Carlo Methods Dezső Boda Wolfgang NonnerDoug Henderson Bob Eisenberg
  • 47.
    47 Mutants of ompFPorin Atomic Scale Macro Scale 30 60 -30 30 60 0 pA mV LECE (-7e) LECE-MTSES- (-8e) LECE-GLUT- (-8e)ECa ECl WT (-1e) Calcium selective Experiments have ‘engineered’ channels (5 papers) including Two Synthetic Calcium Channels As density of permanent charge increases, channel becomes calcium selective Erev  ECa in 0.1M 1.0 M CaCl2 ; pH 8.0 Unselective Natural ‘wild’ Type built by Henk Miedema, Wim Meijberg of BioMade Corp. Groningen, Netherlands Miedema et al, Biophys J 87: 3137–3147 (2004); 90:1202-1211 (2006); 91:4392-4400 (2006) MUTANT ─ Compound Glutathione derivatives Designed by Theory || Evidence RyR (start)
  • 48.
    48 Ca Channel log (Concentration/M) 0.5 -6-4 -2 Na+ 0 1 Ca2+ Charge -3e Occupancy(number) E E E A Monte Carlo simulations of Boda, et al Same Parameters pH 8 Mutation Same Parameters Mutation EEEE has full biological selectivity in similar simulations Na Channel Concentration/M pH =8 Na+ Ca2+ 0.004 0 0.002 0.05 0.10 Charge -1e D E K A
  • 49.
    49 Mutants of ompFPorin Atomic Scale Macro Scale 30 60 -30 30 60 0 pA mV LECE (-7e) LECE-MTSES- (-8e) LECE-GLUT- (-8e)ECa ECl WT (-1e) Calcium selective Experiments have ‘engineered’ channels (5 papers) including Two Synthetic Calcium Channels As density of permanent charge increases, channel becomes calcium selective Erev  ECa in 0.1M 1.0 M CaCl2 ; pH 8.0 Unselective Natural ‘wild’ Type built by Henk Miedema, Wim Meijberg of BioMade Corp. Groningen, Netherlands Miedema et al, Biophys J 87: 3137–3147 (2004); 90:1202-1211 (2006); 91:4392-4400 (2006) MUTANT ─ Compound Glutathione derivatives Designed by Theory || Evidence RyR (start)
  • 50.
    Poisson Fermi Approachto Ion Channels 50 . 劉晉良 Jinn-Liang Liu discovered role of SATURATION Bob Eisenberg helped with applications
  • 51.
    Motivation Natural Description ofCrowded Charge is a Fermi Distribution designed to describe saturation Simulating saturation by interatomic repulsion (Lennard Jones) is a significant mathematical challenge to be side-stepped if possible 51
  • 52.
    Motivation Largest Effect of Crowded Charge is Saturation important inchannels, DNA, enzymes, and electrochemical devices in general Saturation cannot be described at all by classical Poisson Boltzmann approach and is described in a (wildly) uncalibrated way by present day Molecular Dynamics 52
  • 53.
    53 Liu, Eisenberg. 2018.Journal of chemical physics 148:054501 also arXiv:1801.03470 . Calibration in Bulk Solution: New Result
  • 54.
    54 Individual activity coefficientsof 2:1 electrolytes. Comparison of PF results with experimental data [26] on i = Pos2+ (cation) and Neg− (anion) activity coefficients γi in various [PosNeg2] from 0 to 1.5 M. Calibration in Bulk Solution: New Result
  • 55.
    Na Channel 55 Signature ofCardiac Calcium Channel CaV1.n Anomalous* Mole Fraction (non-equilibrium) Liu & Eisenberg (2015) Physical Review E 92: 012711 *Anomalous because CALCIUM CHANNEL IS A SODIUM CHANNEL at [CaCl2]  10-3.4 Ca2+ is conducted for [Ca2+] > 10-3.4, but Na+ is conducted for [Ca2+] <10-3. Ca Channel
  • 56.
    56 Three Dimensional Theory Comparisonwith Experiments Gramicidin A
  • 57.
    ‘Law’ of MassAction including Interactions From Bob Eisenberg p. 1-6, in this issue Variational Approach EnVarA 1 2- 0E      6 7 8 6 7 8 r r x u Conservative Dissipative Chun Liu 柳 春
  • 58.
    58 Energetic Variational Approach allows accuratecomputation of Flow and Interactions in Complex Fluids like Liquid Crystals Engineering needs Calibrated Theories and Simulations Engineering Devices almost always use flow Classical theories and Molecular Dynamics have difficulties with flow, interactions, and complex fluids
  • 59.
    59 Energetic Variational Approach EnVarA ChunLiu, Rolf Ryham, and Yunkyong Hyon Mathematicians and Modelers: two different ‘partial’ variations written in one framework, using a ‘pullback’ of the action integral } } 1 2 0 E       '' Dissipative'Force'Conservative Force r r x u Action Integral, after pullback Rayleigh Dissipation FunctionAction Integral, after pullback Rayleigh Dissipation Function Field Theory of Ionic Solutions: Liu, Ryham, Hyon, Eisenberg Allows boundary conditions and flow Deals Consistently with Interactions of Components 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 Composite Variational Principle Shorthand for Euler Lagrange process with respect to r x Shorthand for Euler Lagrange process with respect to r u
  • 60.
    2 , = , =, i i i B i i j j B i i n p j n p D c c k T z e c d y dx k T c                    = Dissipative r r% 1 4 4 4 4 4 4 4 4 4 2 4 4 4 4 4 4 4 4 4 3 , = = , , 0 , , = 1 log 2 2 i B i i i i i j j i n p i n p i j n p c k T c c z ec c d y dx d dt                        Conservative r r% 6 4 4 4 4 4 4 4 4 4 4 44 7 4 4 4 4 4 4 4 4 4 4 4 48 Hard Sphere Terms Permanent Charge of proteintime ci number density; thermal energy; Di diffusion coefficient; n negative; p positive; zi valence; ε dielectric constantBk T Number Density Thermal Energy valence proton charge Dissipation Principle Conservative Energy dissipates into Friction = , 0 2 1 22 i i i n p z ec              Note that with suitable boundary conditions 60
  • 61.
    61 PNP (Poisson NernstPlanck) for Spheres Eisenberg, Hyon, and Liu 12 , 14 12 , 14 12 ( ) ( ) = ( ) | | 6 ( ) ( ) ( ) , | | n n n nn n n n n n B n p n p p a a x yc c D c z e c y dy t k T x y a a x y c y dy x y                                r r r r r r r r r r r r Nernst Planck Diffusion Equation for number density cn of negative n ions; positive ions are analogous Non-equilibrium variational field theory EnVarA Coupling Parameters Ion Radii =1 or( ) = N i i i z ec i n p               0ρ Poisson Equation Permanent Charge of Protein Number Densities Diffusion Coefficient Dielectric Coefficient valence proton charge Thermal Energy
  • 62.
    62 Semiconductor PNP Equations ForPoint Charges       i i i i d J D x A x x dx    Poisson’s Equation Drift-diffusion & Continuity Equation          0 i i i d d x A x e x e z x A x dx dx            Ρ 0idJ dx         ex * x x x ln xi i i iz e kT              1 2 3 Finite Size Special Chemistry Chemical Potential Thermal Energy Valence Proton charge Permanent Charge of Protein Doping of Semiconductor Cross sectional Area Flux Diffusion Coefficient Number Densities Dielectric Coefficient valence proton charge Not in Semiconductor ( )i x
  • 63.
    All we haveto do is Solve them! with Boundary Conditions defining Charge Carriers ions, holes, quasi-electrons Geometry 63
  • 64.
    64 Mathematical Solutions can besimple and easy to recognize! f b k k L R
  • 65.
    Page 65    out inJ J k f k b kJ l k C l k C    6 44 7 4 48 6 44 7 4 48 Unidirectional Efflux Unidirectional Infflux eft ightL R f b k k L R *but potential profile must be computed by solving electrostatics problem. Complex coupled nonlinearities are in the electric field!!! Solution is a Chemical Reaction* exactly
  • 66.
    66 Solution* of PNPEquation *MATHEMATICS This solution was actually DERIVED from several conditional probability measures by ANALYTICAL integrations No approximations, no numerics 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 Simple formulas are available for the probabilities   {                       Unidirectional Efflux Unidirectional I ConditionalDiffusion ChannelSource ProbabilityVelocity LengthConcentration 6 4 4 44 7 4 4 4 48 1 4 2 4 31 2 3 1 4 4 2 4 4 3 k k k k k D D CR LJ R L RC L Prob Prob Rate Constant nfflux 6 4 4 44 7 4 4 4 48
  • 67.
    Page 67 Please donot be deceived by the eventual simplicity of Results. This took >2 years! Solution was actually DERIVED 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 on-line textbook of USA Biophysical Society, DeFelice editor. Eisenberg, B. (2011) Chemical Physics Letters 511: 1-6
  • 68.
    All we haveto do is Solve them! Don’t Despair Semiconductor Technology has Already Done That! 68
  • 69.
    Semiconductor Devices PNP equationsdescribe 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 formulas are possible Weishi Liu University of Kansas
  • 70.
    Device converts Inputto Output by a simple ‘law’ 70 Device is ROBUST and TRANSFERRABLE because it uses POWER and has complexity! Dotted lines outline: current mirrors (red); differential amplifiers (blue); class A gain stage (magenta); voltage level shifter (green); output stage (cyan). Circuit Diagram of common 741 op-amp: Twenty transistors needed to make linear robust device INPUT Vin(t) OUTPUT Vout (t) Power Supply Dirichlet Boundary Condition independent of time and everything else Power Supply
  • 71.
    Integrated Circuit A Hierarchyof Devices and Structures Technology as of ~2014 IBM Power8 71 Too small to see!
  • 72.