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Mathematics in Molecular and Cellular Biology

Mathematics in Molecular and Cellular Biology. Many thanks to Zhenli for inviting me!!. Multiscale Models of Nerve and Muscle. also called Physiology of Nerve and Muscle. From Structure to Function By Fundamental Physical Laws. From Anatomy to Physiology using

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Mathematics in Molecular and Cellular Biology

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  1. Mathematics in Molecular and Cellular Biology Many thanks to Zhenli for inviting me!!

  2. Multiscale Models of Nerve and Muscle also called Physiology of Nerve and Muscle

  3. From Structure to Function By Fundamental Physical Laws

  4. FromAnatomy toPhysiology using Biophysics & Biochemistry From Structure to Function using Fundamental Physical Laws

  5. Multiscale Analysis is also called Physiology

  6. Multiscale MATHEMATICAL Analysishas rarely been possible until now

  7. Multiscale MATHEMATICAL Analysishas rarely been possible until now Except forNerve Cells Hodgkin, Huxley and Katz

  8. Multiscale Mathematical Analysis is not available for any other tissue or cell,although many are working to change this! MultiScale Analysis of Nerve Functionis more complete than of ANY other cell/tissuein Biology

  9. Multiscale Analysis is Structural. (Almost) all Structures are known and can be described on all scales. MultiScale Analysis of Nerve Function

  10. Multiscale Analysis is physical, as well as mathematical. Physical variables and equations can be used in almost all steps. Description is needed only in one important case, namely gating. MultiScale Analysis of Nerve Function

  11. Gating is the process that opens and closes ion channels. Gating is widely thought to be a conformational change. Conformational change of what? Driven by whatphysics? How (physically) does conformation change function? Gating

  12. Conformational change of what? Driven by whatphysics? How (physically) does conformation change function? Gating Answers to these questions are needed if phrase ‘conformational change’ is to be more than vague description* *Opinion of Bob Eisenberg, not generally shared

  13. Verbal Models Are Popular with Biologists but Inadequate

  14. Multiscale Analysis is Physical, as well as Mathematical. Physical Analysis can only be done once Structures are Known Function is Known Fundamental Physics is Known

  15. Physical Analysis can only be done once Structures are Known Function is Known Fundamental Physics is Known Structures and Fundamental Physics of Gating are not known

  16. FromAnatomy toPhysiology using Biophysics & Biochemistry From Structure to Function using Fundamental Physical Laws

  17. PHYSIOLOGYof NerveandSkeletal Muscle

  18. Background Skeletal Cardiac Web Hyperlink EC Coupling Michael Fill, Ph.D., Dept. Physiology, Rush University

  19. What is the information signal of a nerve? How does that signal move down a nerve fiber? What are the molecular mechanisms involved? Completely solved in outline Mathematics available Many IMPORTANT problems unsolvedOptimization of function Neuronal Conduction

  20. Neuronal Signal

  21. Voltage Signals are Found Throughout Life

  22. Channels Determine Electrical Properties

  23. Resting Potential Set by K Channels

  24. Potential Determined by the Ion with the greatest Conductance

  25. Potential Determined by the Ion with the greatest Conductance

  26. Action Potential Gating Animation Action Potential

  27. Propagation

  28. Propagation Action Potential Propagation

  29. Myelinated Nerve Propagation Action Potential Propagation

  30. Na Channel

  31. K channel

  32. Reminder: the Action Potential

  33. Na & K channels Gating Animation Action Potential

  34. Ca Channels

  35. Ach Channel

  36. Single Channel Traces First Latency 1 2 3 4 5 Number Time Ensemble of 10 Traces Activation Rate Current Time Inactivation Rate Ensemble of 500 Traces Current Time Non-Stationary Single Channel Analysis Michael Fill, Ph.D., Dept. Physiology, Rush University

  37. Non-Stationary RyR2 Channel Recording pCa 5 O C 250 ms pCa 7 (+ caged-Ca) 20 pA pCa 7  6 Michael Fill, Ph.D. Dept. Physiology Rush University Flash Burst

  38. More?

  39. 100% 75% 56% NUMBER TIME, ms General Single Channel Theory “GATING” = Process by which a channel opens & closes [ C  O ] Transitions between open & closed states are “STOCHASTIC”. This means that the transition between the O and C states occurs randomly. Suppose, a channel is closed and that the probability of it opening in the next 1 ms is 0.25 (i.e. 25%). The exact instant the channel will open can not be precisely predicted. Channel “gating” is a “MARKOVIAN” process. This means that the probability of transition is always constant. It does not depend on what has happened in the past. In other words, the system does not have a “memory”. The 25% probability of a closed channel opening (see above) will be the same regardless of whether the channel was closed for a ms, µs or ns. “DWELL TIMES” are exponentially distributed. If opening probability is 0.25 in 1 ms (25%), then 75% of channels will still be open after 1 ms. About 56% (75% of 75%) would be open after 2 ms. About 42% (75% of 56%) would be open after 3 ms and so on.

  40. O1 I C3 O2 C1 C2 C4 C5 O3 C1 C2 C3 C4 C5 C1 C2 O O1 O2 O3 “BURSTING BEHAVIOR” when openings are grouped together. Bursting can be a consequence of multiple closed states. For example, when the channel is in C2 it will show bursting behavior if probability of transition to C2 is less than probability of transition to the 0 state. General Single Channel Theory (cont.) Most channels have multiple open & closed states. RyR2 Channel KCa Channel “FIRST LATENCY” is the time to first opening following a stimulus. Most channels activate in response to some sort of stimulus. First latency is the average time it takes the channel to respond to a abrupt step-like stimulus. If a channel has one closed state, the first latency is a simple monotonic function. If a channel has multiple closed states, then first latency will be a more complex function. Michael Fill, Ph.D., Dept. Physiology, Rush University

  41. TRANS CIS 20 mM CsCH3SO3 200 mM CsCH3SO3 pCa 7, pH 7.4 pCa 7, pH 7.4 1 mM EGTA 1 mM EGTA Septa Incorporating Channelsin Planar Lipid Bilayers Experimental Conditions Cup  O  C 100 – 150 µm Channel Enriched Liposomes Phosphatidylethanolamine & Phosphatidylcholine (7:3) Michael Fill, Ph.D., Dept. Physiology, Rush University

  42. Steady-State Single Channel Analysis Time Current Dwell Times Current Amplitude Closed Threshold Idealize Open Open Current Closed Pool & Plot Open/Closed Times Current-Voltage Relationship pA pS mV Count Count Time Time Michael Fill, Ph.D., Dept. Physiology, Rush University

  43. Single Channel Traces First Latency 1 2 3 4 5 Number Time Ensemble of 10 Traces Activation Rate Current Time Inactivation Rate Ensemble of 500 Traces Current Time Non-Stationary Single Channel Analysis Michael Fill, Ph.D., Dept. Physiology, Rush University

  44. Non-Stationary RyR2 Channel Recording pCa 5 O C 250 ms pCa 7 (+ caged-Ca) 20 pA pCa 7  6 Michael Fill, Ph.D. Dept. Physiology Rush University Flash Burst

  45. Michael Fill, Ph.D., Dept. Physiology, Rush University RyR Channel Ca2+ Activation & Deactivation

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