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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 𝜇0 𝐜𝐮𝐫𝐥 𝐁 𝑥, 𝑦, 𝑧|𝑡 = 𝐉 𝑥, 𝑦, 𝑧|𝑡 + 𝜺 𝐫 𝜀0 𝜕𝐄 𝑥, 𝑦, 𝑧|𝑡 𝜕𝑡 𝐝𝐢𝐯 𝐃 𝑥, 𝑦, 𝑧|𝑡 = 𝐝𝐢𝐯 𝜺 𝒓 𝜀0 𝐄 𝑥, 𝑦, 𝑧|𝑡 = 𝝆 𝑓 𝑥, 𝑦, 𝑧|𝑡 Bold red means dangerously oversimplified: 𝐃, 𝜺 𝐫
- 4. Applying Maxwell to Mitochondria Seems Hopeless Source: textbooks and internet
- 5. January 9, 2020 Bob Eisenberg 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
- 6. January 9, 2020 Bob Eisenberg 6 BUT …… is it Hopeless to know all charges and how they move in an IC? integrated circuit Source: textbooks and internet
- 7. January 9, 2020 Bob Eisenberg 7 Integrated Circuits are Designed with Kirchhoff’s Current Law and little else!!! Hard as that is to believe Source: textbooks and internet
- 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 𝟏𝟎−𝟏𝟎 sec
- 10. 𝑹𝑪 = charging time = 𝟏𝟎−𝟏𝟐 farads × 𝟏𝟎 𝟑 ohm = 𝟏𝟎−𝟗 𝐬𝐞𝐜 FLUXES ARE NOT CONSERVED I𝐢𝐧 𝑽 𝒐𝒖𝒕 = 𝑰𝒊𝒏 𝟏 − 𝒆 Τ−𝒕 𝑹𝑪
- 11. January 9, 2020 Bob Eisenberg 11 Maxwell and Kirchhoff DISAGREE in usual derivation of Kirchhoff using flux of charges
- 12. January 9, 2020 Bob Eisenberg 12 Must extend Kirchhoff’s Laws to general systems
- 13. January 9, 2020 Bob Eisenberg 13 Deriving Kirchhoff’s Laws from Maxwell Equations and conservation of current is Not so Easy as it seems
- 14. January 9, 2020 Bob Eisenberg 14 Using Maxwell Equations and conservation of current is NON-TRIVIAL because 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.
- 15. January 9, 2020 Bob Eisenberg 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.
- 16. January 9, 2020 Bob Eisenberg 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.
- 17. January 9, 2020 Bob Eisenberg 17 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
- 18. January 9, 2020 Bob Eisenberg 18 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. What are the PHYSICAL problems with traditional Maxwell Equations? Eisenberg, 2019. Dielectric Dilemma. arXxiv: 1901.10805.
- 19. Updating Maxwell Move 𝜺 𝒓 − 𝟏 𝜺 𝟎 terms into 𝐉 𝒐𝒕𝒉𝒆𝒓 and 𝝆 𝒐𝒕𝒉𝒆𝒓 1 𝜇0 𝐜𝐮𝐫𝐥 𝐁 𝑥, 𝑦, 𝑧|𝑡 = ሚ𝐉 𝑥, 𝑦, 𝑧|𝑡 + 𝐉 𝑜𝑡ℎ𝑒𝑟 𝑥, 𝑦, 𝑧|𝑡 + 𝜺 𝐫 𝜀0 𝜕𝐄 𝑥, 𝑦, 𝑧|𝑡 𝜕𝑡 𝐝𝐢𝐯 𝐃 𝑥, 𝑦, 𝑧|𝑡 = 𝐝𝐢𝐯 𝜺 𝒓 𝜀0 𝐄 𝑥, 𝑦, 𝑧|𝑡 = 𝝆 𝑓 𝑥, 𝑦, 𝑧|𝑡 + 𝝆 𝑜𝑡ℎ𝑒𝑟 𝑥, 𝑦, 𝑧|𝑡 *Over-simplification of a single (real number) dielectric constant 𝜺 𝒓 can easily be removed by describing explicitly the actual properties of charges 𝝆 𝒐𝒕𝒉𝒆𝒓 and currents 𝐉 𝒐𝒕𝒉𝒆𝒓 See Eisenberg 2019 Arxiv 1904.09695.
- 20. January 9, 2020 Bob Eisenberg 20 1 𝜇0 𝐜𝐮𝐫𝐥 𝐁 = ሚ𝐉 + 𝐉 𝑛𝑜𝑛−𝑖𝑑𝑒𝑎𝑙 + 𝜺 𝒓 − 1 𝜀0 𝜕𝐄 𝜕𝑡 + 𝜀0 𝜕𝐄 𝜕𝑡 Ions Non-ideal Ideal Dielectric Space Updated Derivation Conservation of Current Ions can be included and nonideal polarization, currents, charges, etc. 1 Ampere’s Law a Maxwell Equation 𝐉 𝑜𝑡ℎ𝑒𝑟 𝑥, 𝑦, 𝑧|𝑡 Bold red means dangerously oversimplified: 𝜺 𝐫
- 21. January 9, 2020 Bob Eisenberg 21 𝐝𝐢𝐯 1 𝜇0 𝐜𝐮𝐫𝐥 𝐁 = 𝐝𝐢𝐯 ሚ𝐉 + 𝐉 𝑛𝑜𝑛−𝑖𝑑𝑒𝑎𝑙 + 𝜺 𝒓 − 1 𝜀0 𝜕𝐄 𝜕𝑡 + 𝜀0 𝜕𝐄 𝜕𝑡 = 0 Ions Non-ideal Ideal Dielectric Space Updated Derivation Conservation of Current Ions can be included and nonideal polarization, currents, charges, etc. 2 Conservation of Current 𝐉 𝑜𝑡ℎ𝑒𝑟 𝑥, 𝑦, 𝑧|𝑡 Bold red means dangerously oversimplified: 𝜺 𝐫
- 22. 22 Polarization is nearly always non-ideal Eisenberg, 2019. Dielectric Dilemma. arXxiv: 1901.10805.
- 23. January 9, 2020 Bob Eisenberg 23 Conservation of Current is Universal and Exact because It is a Property of Space Τ𝝏𝑬 𝝏𝒕 in a vacuum produces B field Not a property of matter It arises from the Principle of Relativity every textbook and Dunstan, D. (2008) Phil Trans Roy Soc A: 366 1861 Charge does not vary when velocity → speed of light Mass, length, time do vary
- 24. January 9, 2020 Bob Eisenberg 24 Note Flux ሚ𝐉 + 𝐉 𝑜𝑡ℎ𝑒𝑟 of charged matter is NOT NOT NOT conserved div ሚ𝐉 + 𝐉 𝑜𝑡ℎ𝑒𝑟 ≠ 𝟎 Maxwell equations imply that Charge and Matter accumulate because 𝜀0 𝜕E 𝜕𝑡 ≠ 0 div ሚ𝐉 + 𝐉 𝑜𝑡ℎ𝑒𝑟 = 𝜀0 𝜕𝐄 𝜕𝑡 ≠ 0 In physical language, Some charge accumulates in the ‘stray’ capacitance of space independent of matter 𝐼in 𝑉𝑜𝑢𝑡 = 𝐼𝑖𝑛 1 − 𝑒 Τ−𝑡 𝑅𝐶
- 25. Conservation of Current in one-dimension Classical dielectric 𝜀 𝑟 formulation perhaps branched 𝟎 = ሚ𝐉 + 𝐉 𝑛𝑜𝑛−𝑖𝑑𝑒𝑎𝑙 + 𝜀 𝑟 − 1 𝜀0 𝜕𝐄 𝜕𝑡 + 𝜀0 𝜕𝐄 𝜕𝑡 𝐉 = − 𝜀r 𝜀0 𝜕𝐄 𝜕𝑡
- 26. January 9, 2020 Bob Eisenberg 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 𝐝𝐢𝐯 𝐉 + 𝐉 𝑛𝑜𝑛−𝑖𝑑𝑒𝑎𝑙 = − 𝜕𝛒 𝑓 𝜕𝑡
- 27. January 9, 2020 Bob Eisenberg 27 All is obvious to a fine practicing engineer “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 Alan Finkel Founder Axon Instruments, Now Chief Scientist Australian Goverment
- 28. January 9, 2020 Bob Eisenberg 28 Conservation of Current is Exact and Universal in updated Maxwell Equations Major Consequences are most easily understood in One-dimensional SERIES circuit All currents are equal No matter what the microphysics
- 29. January 9, 2020 Bob Eisenberg 29 Continuity of Current is Exact in Kelvin’s Submarine Telegraph 𝒊 𝐍𝐞𝐰𝐟𝐨𝐮𝐧𝐝𝐥𝐚𝐧𝐝 = 𝒊𝐈𝐫𝐞𝐥𝐚𝐧𝐝 How can that possibly be true? Source: internet
- 30. January 9, 2020 Bob Eisenberg 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
- 31. * i E t= * i E t= D = permittivity E Ag AgClAg AgCl Continuity of Current 𝑱… is Exact 𝑱 𝐃𝐞𝐯𝐢𝐜𝐞 𝟏 = 𝑱 𝐃𝐞𝐯𝐢𝐜𝐞 𝟐 = 𝑱 𝐃𝐞𝐯𝐢𝐜𝐞 𝟑 … no matter what carries the current, at all times and all locations! Flux is NOT continuous, current is continuous Current has very Different Physics in Different Systems
- 32. Conservation of Current is Exact even though Physics of Charge Flow ҧ𝐉 𝑥, 𝑡 Varies Profoundly Flux of Charges = ҧ𝐉 = ሚ𝐉 + 𝐉 𝑛𝑜𝑛−𝑖𝑑𝑒𝑎𝑙 + 𝜀 𝑟 − 1 𝜀0 𝜕𝐄 𝜕𝑡 𝐝𝐢𝐯 ҧ𝐉≠0 𝐝𝐢𝐯 ҧ𝐉 𝑥, 𝑡 + 𝜀0 𝜕𝐄 𝑥, 𝑡 𝜕𝑡 = 𝟎 Current Flux of Charges
- 33. Math Says Conservation of Current is Exact but Flux of Charges are not conserved 𝐝𝐢𝐯 ҧ𝐉≠0
- 34. Page 34 Electric Field takes on the Value that Conserves Current 𝐄 𝑥, 𝑡 = − 1 𝜀0 න 𝐉 𝑥, 𝑡 𝑑𝑡 Details and PROOF including quantum mechanics at https://arxiv.org/abs/1609.09175 Specifically, E moves atoms and creates the ‘ethereal’ current ε0 Τ𝜕𝑬 𝜕𝑡 So total current 𝐉 𝑥, 𝑡 + ε0 Τ𝜕𝑬 𝜕𝑡 is always conserved One-dimensional perhaps branched
- 35. Page 35 Electric Field takes on the Value that Conserves Current This is NOT mysterious Details and PROOF including quantum mechanics at https://arxiv.org/abs/1609.09175 Specifically, E moves atoms so total current 𝐉 𝑥, 𝑡 + ε0 Τ𝜕𝑬 𝜕𝑡 is always conserved One dimensional perhaps branched systems
- 36. Page 36 Τ𝝏𝑬 𝝏𝒕 creating 𝑩 field in a vacuum is Mysterious Is Τε 𝟎 𝝏𝑬 𝝏𝒕 an ethereal current?
- 37. January 9, 2020 Bob Eisenberg 37 Conservation of Current is Exact and Universal So what? Current must always be described by Continuum Equations Particle motion does NOT define Current Contradicts Intuition Current ≠ Flux of charge Flux is defined by particle motion Current is not
- 38. January 9, 2020 Bob Eisenberg 38 CurrentCurrent = 𝜺 𝟎 𝛛𝐄 𝝏𝒕 Vacuum Current
- 39. January 9, 2020 Bob Eisenberg 39 Contradicts Intuition Small Systems REQUIRE 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
- 40. January 9, 2020 Bob Eisenberg 40 Conservation of Current is Practically Important in Understanding Transporters Oxidative Phosphorylation Photosynthesis main processes in life
- 41. Bilayer Setup ‘voltage clamp’ Inhomogeneous Dirichlet Condition For voltage Classic Voltage Clamp Natural Setup: small cell, etc. Homogeneous Neumann Boundary Condition For total current Hodgkin Huxley Katz 1952 Na+ Ca++ Na+ Ca++ 𝐼 = 𝐼 𝑁𝑎 + 𝐼 𝐶𝑎 = 𝟎 𝐼 𝑁𝑎 = −𝐼 𝐶𝑎 𝐼 𝑁𝑎 𝐼 𝐶𝑎 Independent Currents
- 42. 42 Biophysical Prediction Does NOT require knowledge of all charges
- 43. January 9, 2020 Bob Eisenberg 43 With Conservation of Current mitochondria are no more difficult than large circuit problems
- 44. 44 NOT hopeless, Maxwell Predicts Important Biophysics Independent of details of the Mitochondrion if a branched One-dimensional Formulation is appropriate
- 45. January 9, 2020 Bob Eisenberg 45 Inside Channels PROFOUND SIMIPLIFICATION If we can figure out how to exploit it
- 46. January 9, 2020 Bob Eisenberg 46 Profound Implications of One Dimensional Systems for atomic view of ion channels Current is equal everywhere in a channel At all times and under all conditions that the Maxwell Equations Apply
- 47. January 9, 2020 Bob Eisenberg 47 Profound Implications of One Dimensional Systems for atomic view of ion channels Spatial Variable does NOT appear in description of current in a one dimensional channel
- 48. January 9, 2020 Bob Eisenberg 48 Spatial Variable does NOT appear in description of current in a one dimensional channel
- 49. January 9, 2020 Bob Eisenberg 49 Hopping Model is COMPLETELY INAPPROPRIATE for current that is uniform in 𝒙
- 50. January 9, 2020 Bob Eisenberg 50 Spatial Variable does appear in description of ion movement in a one dimensional channel
- 51. January 9, 2020 Bob Eisenberg 51 Spatial Variable does NOT appear in description of current in a one dimensional channel
- 52. January 9, 2020 Bob Eisenberg 52 Spatial Variable does NOT appear in description of current in a one dimensional channel Molecular Dynamics that 1) conserves current 2) has one dielectric constant 3) extends to biological time scales
- 53. January 9, 2020 Bob Eisenberg 53 Spatial Variable does NOT appear in description of current in a one dimensional channel Quasi-particle for current CONDUCTON left over is DIFFUSON + POLARON. 𝚺 = PERMION* *Elber, Chen, Rojewska, and Eisenberg. 1995. Sodium in gramicidin: An example of a permion. Biophys. J. 68:906-924.
- 54. January 9, 2020 Bob Eisenberg 54 Spatial Variable does NOT appear in description of current in a one dimensional channel Another way is to include equality of current as a constraint in time averaging schemes e.g., of Ma and Liu Ma, Li, and Liu. 2016.. arXiv:1605.04886. Ma, Li, and Liu. 2016. arXiv:1606.03625.
- 55. January 9, 2020 Bob Eisenberg 55
- 56. January 9, 2020 Bob Eisenberg 56 Paradigm Change Current does not knock on or knock off Current does not hop Current must be described by a continuum equation Current flow is equal everywhere in a one dimensional channel
- 57. January 9, 2020 Bob Eisenberg 57 Any Questions?
- 58. January 9, 2020 Bob Eisenberg 58 Extra Slides
- 59. Main Applications of Electrodynamics involve Ions that diffuse Semiconductors where Holes and Electrons diffuse and carry charge Digital Technology, Smartphones, Computers Electrolyte Solutions where ions diffuse and have bulk flow Molecular Biology, Chemistry, Nanotechnology These applications are ENORMOUSLY IMPORTANT Involving a good fraction of human life January 9, 2020 Bob Eisenberg 59
- 60. January 9, 2020 Bob Eisenberg 60 Ions holes and electrons are usually Permanent Charges Ions move by Diffusion Driven by Concentration Ions move by Convection Driven by Pressure Ions move by Migration Driven by Electric Field Ions move in Complex Dielectrics Maxwell Equations need to be Updated to deal with these realities Classical Maxwell Equations only deal with electric field
- 61. January 9, 2020 Bob Eisenberg 61 It is NOT necessary to know the charges to understand one crucial property of electrodynamics Engineers use Kirchhoff’s Current Law at 𝟏𝟎−𝟏𝟎 sec and 𝟏𝟎−𝟖 cm Kirchhoff’s Current Law is a crucial property of electrodynamics Equivalent to the Maxwell equations themselves
- 62. January 9, 2020 Bob Eisenberg 62 Polarization of Matter εr − 1 ε0 𝜺 𝒓 involves all movements of matter that do not translate mass, roughly speaking 𝜺 𝒓 − 𝟏 𝜺 𝟎 is as complex as the motion of matter itself. Seems hopeless to make a general theory What are the problems with textbook Maxwell Equations? 𝜺 𝒓 𝜺 𝟎 is a hybrid
- 63. January 9, 2020 Bob Eisenberg 63 describes Polarization of Space 𝜺 𝟎 Polarization of space 𝜀0 is uniform and as exact as Maxwell’s equations Polarization of Matter εr − 1 ε0 𝜺 𝒓 involves all movements of matter that do not translate mass, roughly speaking 𝜺 𝒓 𝜺 𝟎 is a hybrid What are the problems with textbook Maxwell Equations?
- 64. January 9, 2020 Bob Eisenberg 64 Polarization of Space 𝜺 𝟎 Polarization of space 𝜀0 is uniform and as exact as Maxwell’s equations Implies Propagation of Light Universal and Exact Conservation of Current Independent of matter Is linked to Special Relativity 𝜺 𝒓 𝜺 𝟎 is a hybrid
- 65. Applying Maxwell to Transporters 1) Sum of Currents in a Transporter is zero in ‘small cell’, mitochondrion, etc. 2) Currents are Coupled in a Transporter in a natural setting by Maxwell 3) Bilayer set up does NOT require currents to sum to zero. Bilayer setup sets voltage across transporter, currents are not controlled. 4) So transporter currents s are NOT coupled by Conservation of Current in standard bilayer setup Natural Setting Experimental Setting