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From Maxwell to Mitochondria

Applying the Maxwell equations to mitochondria seems a hopeless task: there is so much complexity. But computers and their chips are nearly as complicated. Design of circuits is done every day by uncounted numbers of engineers and scientists, thanks to Kirchoff's Current Law, which is a conservation law in one dimensional (branched) systems of devices. Kirchoff's current law conserves flux, not current in its usual derivation. But Maxwell's equations do not conserve flux; they conserve total current. Total current J equals flux plus displacement current J eps_0 partial E/partial t . Maxwell's definition of current allows circuit laws to be applied to complex systems of devices, over a wide range of times and conditions. Channels and enzymes are devices because they localize current flow. Channels and enzymes can be analyzed by the methods of circuit theory, for that reason.<br><br>

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From Maxwell to Mitochondria

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  1. From Maxwell to Mitochondria Robert (‘Bob’) Eisenberg Dept of Applied Mathematics Illiois Institute of Technology Dept of Physiology and Biophysics Rush University Chicago IL USA ResearchGate DOI: 10.13140/RG.2.2.22234.29129 January 7 2020

  2. From Maxwell to Mitochondria Lecture Series on Mathematical Biology ICMS, National Tsin Hua University Hsinchu Taiwan

  3. Applying Maxwell Textbook formulation 1 ?0 ?? ?,?,?|? ?? ???? ? ?,?,?|? = ? ?,?,?|? + ???0 ??? ? ?,?,?|? = ??? ???0? ?,?,?|? = ???,?,?|? Bold red means dangerously oversimplified: ?,??

  4. Applying Maxwell to Mitochondria Seems Hopeless Source: textbooks and internet

  5. Hopeless, if one must “… exhibit in every case all the charges, whatever their origin” p.10-4 of Feynman, Leighton, and Sands (1963) Electromagnetism and Matter January 9, 2020 Bob Eisenberg 5

  6. BUT …… is it Hopeless to know all charges and how they move in an IC? integrated circuit Source: textbooks and internet January 9, 2020 Bob Eisenberg 6

  7. Integrated Circuits are Designed with Kirchhoff’s Current Law and little else!!! Hard as that is to believe Source: textbooks and internet January 9, 2020 Bob Eisenberg 7

  8. Derivation of Kirchhoff’s Law is at DC Derivation of Kirchhoff’s law is about fluxes References and Proofs in Eisenberg. 2019. Kirchhoff's Law can be Exact. arXiv: 1905.13574 Eisenberg, Gold, Song, and Huang. 2018. What Current Flows Through a Resistor? arXiv:1805.04814

  9. Derivation of Kirchhoff’s law is about fluxes BUT FLUXES ARE NOT CONSERVED according to experiment or Maxwell equations NOT AT ALL at ??−??sec

  10. I?? Τ −? ?? ????= ???? − ? ?? = charging time = ??−??farads × ???ohm = ??−???? FLUXES ARE NOT CONSERVED

  11. Maxwell and Kirchhoff DISAGREE DISAGREE in usual derivation of Kirchhoff using flux of charges January 9, 2020 Bob Eisenberg 11

  12. Must extend Kirchhoff’s Laws to general systems January 9, 2020 Bob Eisenberg 12

  13. Deriving Kirchhoff’s Laws from Maxwell Equations and conservation of current is Not so Easy as it seems January 9, 2020 Bob Eisenberg 13

  14. Using Maxwell Equations and conservation of current is NON-TRIVIAL because Traditional Maxwell Equations Cannot be Used Traditional Maxwell Equations Cannot be Used Maxwell’s equations do not deal with Diffusion Convection Complex materials Complicated dielectric properties Dielectrics are almost NEVER ideal Eisenberg, 2019. Dielectric Dilemma. arXxiv: 1901.10805. January 9, 2020 Bob Eisenberg 14

  15. Traditional Form of Maxwell Equations Cannot be Used Maxwell’s equations do not deal with Diffusion Convection Complex materials Complicated dielectric properties Eisenberg, 2019. Dielectric Dilemma. arXxiv: 1901.10805. January 9, 2020 Bob Eisenberg 15

  16. It is necessary to update Maxwell’s Equations to see how Kirchhoff’s law & Conservation of Current apply to Complex Liquids and Complex Biological Systems Eisenberg, R. S. 2019. Updating Maxwell with Electrons, Charge, and More Realistic Polarization. arXiv 1904.09695 . Eisenberg, Oriols, and Ferry. 2017. Dynamics of Current, Charge, and Mass. Molecular Based Mathematical Biology 5:78-115 and arXiv preprint 1708.07400. January 9, 2020 Bob Eisenberg 16

  17. It is necessary to update Maxwell’s Equations It is necessary to update Maxwell’s Equations https://arxiv.org/abs/1904.09695 Not just my opinion This is the opinion* of Nobel Prize winners in Physics, Richard Feynman (quantum electrodynamics) and Edward Purcell (nuclear magnetic resonance) *p. 10-7 of Feynman, Leighton, and Sands. 1963. Mainly Electromagnetism and Matter *p. 506 of Purcell and Morin. 2013. Electricity and Magnetism January 9, 2020 Bob Eisenberg 17

  18. What are the PHYSICAL problems with traditional Maxwell Equations? Maxwell’s equations do not deal with Diffusion Convection Complex materials Complicated dielectric properties Indeed, Maxwell’s original equations do not include ions or electrons or their movement! Textbook treatments do not deal with other forces like diffusion or convection at all. Eisenberg, 2019. Dielectric Dilemma. arXxiv: 1901.10805. January 9, 2020 Bob Eisenberg 18

  19. Updating Maxwell Move ??− ? ??terms into ??????and ?????? 1 ?0 ?? ?,?,?|? ?? ???? ? ?,?,?|? =ሚ? ?,?,?|? + ???ℎ???,?,?|? + ???0 ??? ? ?,?,?|? = ??? ???0? ?,?,?|? = ???,?,?|? + ???ℎ???,?,?|? *Over-simplification of a single (real number)dielectric constant ??can easily be removed by describing explicitly the actual properties of charges ??????and currents ?????? See Eisenberg 2019 Arxiv 1904.09695.

  20. Updated Derivation Conservation of Current Ions can be included and nonideal polarization, currents, charges, etc. 1 Space ???ℎ???,?,?|? 1 ?0 ?? ??+ ?0 ?? ?? ???? ? =ሚ? + ????−?????+ ??− 1 ?0 Ampere’s Law a Maxwell Equation Ions Ideal Dielectric Non-ideal Bold red means dangerously oversimplified:?? January 9, 2020 Bob Eisenberg 20

  21. Updated Derivation Conservation of Current Ions can be included and nonideal polarization, currents, charges, etc. 2 Space ???ℎ???,?,?|? 1 ?0 ?? ??+ ?0 ?? ?? ???? ? = ??? ሚ? + ????−?????+ ??− 1 ?0 ??? = 0 Conservation Conservation of of Current Current Ions Ideal Dielectric Non-ideal Bold red means dangerously oversimplified:?? January 9, 2020 Bob Eisenberg 21

  22. Polarization is nearly always non Polarization is nearly always non- -ideal ideal Eisenberg, 2019. Dielectric Dilemma. arXxiv: 1901.10805. 22

  23. Conservation of Current Conservation of Current is Universal and Exact Universal and Exact because because It is a Property of Space It is a Property of Space Τ ?? ?? in a vacuum produces B field Not a property of matter Not a property of matter It arises from the Principle of It arises from the Principle of Relativity Relativity every textbook and Dunstan, D. (2008) Phil Trans Roy Soc A: 366 1861 Charge does not vary when velocity → Mass, length, time do vary → speed of light January 9, 2020 Bob Eisenberg 23

  24. Note Flux ሚ? + ???ℎ??of charged matter is NOT NOT NOT conserved div ሚ? + ???ℎ?? ≠ ? Maxwell equations imply that Charge and Matter accumulate because ?0 ?E ??≠ 0 ?? ??≠ 0 div ሚ? + ???ℎ?? = ?0 In physical language, Some charge accumulates in the ‘stray’ capacitance of space independent of matter Τ −? ?? ?in ????= ???1 − ? Bob Eisenberg January 9, 2020 24

  25. Conservation of Current in one-dimension Classical dielectric ??formulation perhaps branched ?? ??+ ?0 ?? ?? ? =ሚ? + ????−?????+ ??− 1 ?0 ?? ?? ? = − ?r?0

  26. Continuity Equation = −??? ??? ? + ????−????? ?? seems to be equivalent to Conservation of Current mathematically But it is NOT equivalent to Conservation of Current in practice Conservation of Current does NOT require knowledge of charges Continuity Equation requires knowledge of charges and their motion 26 January 9, 2020 Bob Eisenberg

  27. All is obvious to a fine practicing engineer Alan Finkel Founder Axon Instruments, Now Chief Scientist Australian Goverment “Bob, why do you need all that math? Everyone knows how to use Kirchhoff. Everyone knows you have to include the displacement current. No one would try to keep track of all the charges” Paraphrase of email exchange January 9, 2020 Bob Eisenberg 27

  28. Conservation of Current is Exact and Universal in updated Maxwell Equations Major Consequences are most easily understood in One One- -dimensional SERIES circuit dimensional SERIES circuit All currents are equal All currents are equal No matter what the microphysics No matter what the microphysics January 9, 2020 Bob Eisenberg 28

  29. Continuity of Current is Exact in Continuity of Current is Exact in Kelvin’s Submarine Telegraph Kelvin’s Submarine Telegraph ?????????????= ???????? How can that possibly be true? How can that possibly be true? Source: internet January 9, 2020 Bob Eisenberg 29

  30. All currents are equal no matter what the microphysics in one-dimensional series circuit over any length scale, including trans-atlantic or less chauvinistically trans-oceanic January 9, 2020 Bob Eisenberg 30

  31. Current has Current has very very Different Physics Different Physics in Different Systems in Different Systems Ag AgCl Ag AgCl =  =  * * i E t i E t D = permittivity   E Continuity of Current Continuity of Current ?…is Exact ??????? ?= ??????? ?= ??????? ?… … is Exact no matter what carries the current, at all times and all locations! Flux is NOT continuous, current is continuous

  32. Conservation of Current is Exact Conservation of Current is Exact Flux of Charges Flux of Charges = ҧ? =ሚ? + ????−?????+ ??− 1 ?0 ?? ?? Flux of Charges ?? ?,? ?? ??? ҧ? ?,? + ?0 = ? Current even though Physics of Charge Flow Physics of Charge Flow ҧ? ?,? Varies Profoundly Varies Profoundly ??? ҧ?≠0

  33. Math Says Math Says Conservation of Current is Exact but Flux of Charges are not conserved ??? ҧ?≠0

  34. Electric Field takes on the Electric Field takes on the Value that Conserves Current Value that Conserves Current One-dimensional perhaps branched ? ?,? = −1 ?0න? ?,? ?? Specifically, E moves atoms and creates the ‘ethereal’ current ε0 So total current ? ?,? + ε0 Τ ?? ?? ?? ?? is always conserved Τ Details and PROOF including quantum mechanics at https://arxiv.org/abs/1609.09175 Page 34

  35. Electric Field takes on the Value that Conserves Current This is NOT mysterious This is NOT mysterious Specifically, E E moves atoms moves atoms so total current ? ?,? + ε0 ?? ?? is always conserved Τ One dimensional perhaps branched systems Details and PROOF including quantum mechanics at https://arxiv.org/abs/1609.09175 Page 35

  36. ?? ?? creating ? field in a vacuum Τ is is Mysterious Mysterious Is ε ε??? ?? an ethereal current? Τ Page 36

  37. Conservation of Current is Exact and Universal So what? Current must always be described Current must always be described by Continuum Equations by Continuum Equations Particle motion does NOT define Current Contradicts Intuition Contradicts Intuition Current Current ≠ Flux is defined by particle motion Flux is defined by particle motion Current is not Current is not ≠ Flux of charge Flux of charge January 9, 2020 Bob Eisenberg 37

  38. Vacuum Vacuum Current Current Current ?? ?? = ?? 38 January 9, 2020 Bob Eisenberg

  39. Contradicts Intuition Small Systems Small Systems REQUIRE REQUIRE Continuum Description Continuum Description of Electric Current Current does NOT flow by hopping Current is independent of location in series systems Particle Motion can be hopping, not current January 9, 2020 Bob Eisenberg 39

  40. Conservation of Current is Practically Important in Understanding Transporters Oxidative Phosphorylation Photosynthesis main processes in life January 9, 2020 Bob Eisenberg 40

  41. Natural Setup: small cell, etc. Homogeneous Neumann Boundary Condition For total current Hodgkin Huxley Katz 1952 Bilayer Setup ‘voltage clamp’ Inhomogeneous Dirichlet Condition For voltage Classic Voltage Clamp Na+ Ca++ Ca++ Na+ ??? ??? ? = ???+ ???= ? ???= −??? Independent Independent Currents

  42. Biophysical Prediction Does NOT require knowledge of all charges 42

  43. With Conservation of Current With Conservation of Current mitochondria are no more difficult than large circuit problems January 9, 2020 Bob Eisenberg 43

  44. NOT hopeless, Maxwell Predicts Important Biophysics Independent Independent of details of the Mitochondrion if a branched One One- -dimensional Formulation dimensional Formulation is appropriate 44

  45. Inside Channels PROFOUND SIMIPLIFICATION If we can figure out how to exploit it January 9, 2020 Bob Eisenberg 45

  46. Profound Implications of One Profound Implications of One Dimensional Systems for atomic Dimensional Systems for atomic view of ion channels view of ion channels Current is equal everywhere in a channel At all times and under all conditions that the Maxwell Equations Apply January 9, 2020 Bob Eisenberg 46

  47. Profound Implications of One Dimensional Systems for atomic Profound Implications of One Dimensional Systems for atomic view of ion channels view of ion channels Spatial Variable does NOT appear Spatial Variable does NOT appear in description of current in a one in description of current in a one dimensional channel dimensional channel January 9, 2020 Bob Eisenberg 47

  48. Spatial Variable does NOT appear Spatial Variable does NOT appear in description of current in a one in description of current in a one dimensional channel dimensional channel January 9, 2020 Bob Eisenberg 48

  49. Hopping Model is Hopping Model is COMPLETELY INAPPROPRIATE COMPLETELY INAPPROPRIATE for current that is uniform in ? January 9, 2020 Bob Eisenberg 49

  50. Spatial Variable Spatial Variable does description of ion movement in a description of ion movement in a one dimensional channel one dimensional channel does appear in appear in January 9, 2020 Bob Eisenberg 50

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