introduction to Physiology iV - Calcium Dynamics J. P. Keener Mathematics Department University of Utah Introduction to Physiology IV - Calcium Dynamics - p. 1/26 Introduction Previous lectures emphasized the role of sodium and potassium in control of membrane size and potential; Calcium is equally important in almost every cell type; Calcium controls secretion, cell movement, muscular, contraction, cell differentiation, ciliary beating, etc. Calcium is important in both excitable and inexcitable cells. Calcium Dynamics - p.2/26 Imagine the «§ Mathematical Biology I University of Utah Mathematical Biology University of Utah Muscle Mvi.r.lnni / ín-innituftn# Trďiivuru Tftrminil rn;prnn O1 ířiiCylgm Tait,iiz:flh i>l Mlňňíli.MíllIC ■ulN.ii'jrp 1 DepolariraTHjr> _^ .....-- v..........m ■ SR HlfifuLMHi? Calcium Dynamics - p.4/26 Mathematical Biology University of Utah Phototransduction Rad In datknsts IdlJttllYB rtiodrjp&fi (opsin arid retinal) "Disks TrnnaduCin (Gprttfain) levels hěQH WltfrnbcaiiE pqtenliaf in Tome reteaae d iteuntjlransmahw Rod in light: fhodapsin bleaching AfthqilwlntinB, Acrtvůfoa Nöuri'lrJfl Splitt« nolanae itedwaee In proportion to amount of light Calcium Dynamics - p.5/26 Mathematical Biology University of Utah Phototransduction 5'GMP I l&UTSIDEl Guanylate cyclase GTP i [Tran sduc in) «Rhrt-P ^ inhibits- Rh -r Photon ***** Na 1K+ rf1 1K' 4Na' Td U% purrp n innsríBj?r*řH Calcium Dynamics - p.6/26 Taste :ium Dynamics - p.7/26 Synaptic Transmission University of Utah Calcium Dynamics - p.8/26 Synaptic Transmission University of Utah Calcium and synapses: II ► Release of Neuralransmitter : (JHSIMUHII Pnnynaphc Calcium Dynamics - p.9/26 Mathematical Biology University of Utah Calcium Oscillations 0.4 nM 0.6nM 0.9nM A) Hepatocytes B) Rat parotid gland C) Gonadotropes D) Hamster eggs E, F) Insulinoma Cells CCh10mM 20s 10s GnRH 1nM i 30s 20s 25mM 200mM V I_I 50s 1min ■ CCh 50s Calcium Dynamics - p. 10/26 Calcium Handling P3 Receptor pathway O u i sid c the Calcium Dynamics - p. 11/26 Possibilities Mathematical Biology University of Utah IPR Calcium Handling V1E = JlPR ~ JSERCA + Jl — JSL Extracellular Space Cytosol (Intracellular Space) ' IPR JSERCA Sarcoplasmic Reticulum (Calcium stores) Calcium Dynamics - p. 12/26 Mathematical Biology University of Utah IPR Calcium Handling dc U dt JlPR - JSERCA + Jl — JSL Extracellular Space Cytosol (Intracellular Space) ' IPR JSERCA Sarcoplasmic Reticulum (Calcium stores) with Jipr IP3 Receptor - calcium regulated calcium channel Calcium Dynamics - p. 12/26 Mathematical Biology University of Utah IPR Calcium Handling dc U dt JlPR - JSERCA \ + Jl - JSL Extracellular Space Cytosol (Intracellular Space) ' IPR JSERCA Sarcoplasmic Reticulum (Calcium stores) with Jipr IP3 Receptor - calcium regulated calcium channel, Jserca Sarco- and Endoplasmic Reticulum Calcium ATPase: Calcium Dynamics - p. 12/26 Mathematical Biology University of Utah IPR Calcium Handling dc U dt JlPR — JS ERCA + Jl - JSL irh Extracellular Space Cytosol (Intracellular Space) ' IPR JSERCA Sarcoplasmic Reticulum (Calcium stores) with Jipr IP3 Receptor - calcium regulated calcium channel, Jserca Sarco- and Endoplasmic Reticulum Calcium ATPase: JL L-calcium leak, Calcium Dynamics - p. 12/26 Mathematical Biology University of Utah IPR Calcium Handling dc U dt JlPR — JS ERCA + Jl — JSL irh Extracellular Space Cytosol (Intracellular Space) ' IPR JSERCA Sarcoplasmic Reticulum (Calcium stores) with Jipr IP3 Receptor - calcium regulated calcium channel, Jserca Sarco- and Endoplasmic Reticulum Calcium ATPase: JL L-calcium leak, Jsl SarcoLemnal pump (ATPase) . Calcium Dynamics - p. 12/26 Mathematical Biology University of Utah IPR Calcium Handling dc U dt JlPR — JS ERCA + Jl — JSL irh Extracellular Space Cytosol (Intracellular Space) ' IPR JSERCA Sarcoplasmic Reticulum (Calcium stores) with Jipr IP3 Receptor - calcium regulated calcium channel, Jserca Sarco- and Endoplasmic Reticulum Calcium ATPase: JL L-calcium leak, Jsl SarcoLemnal pump (ATPase) . Challenge: Determine the flux terms. Calcium Dynamics - p. 12/26 imagine the Possibilities Mathematical Biology University of Utah Calcium Handling 0.15 c o ■Š 010 CO l_ u_ c O 0.05 0.00 -8.0 /c-ip 2.0 \iU s s 110 "►°010 — -7.0 -6.0 logio[Ca2+] group I k5c kyp -~ s, -5.0 k5c »000 '100 »110 1 i i *-1 - l *-5 /C2C /c.4 kůp k_4 /C4C /c_2 /(2C /c_2 *3P k5c 1 ^3P 111 011 001 »101 - »111 k-5 k-3 group II Calcium Dynamics - p. 13/26 /P3 Receptors University of Utah Calcium Dynamics - p. 14/26 Mathematical Biology University of Utah /P3 Receptors Flux through IP3 receptor is diffusive, JlPR Qmax-Po^C Csr) where PQ = Sf0 = m3h3 is the open probability, and dm 0m(c)(l - m) - ipm(c)m, dh Furthermore, m is a fast variable, so is in qss, m h{c){\ - h) - ^{c)K -- moo(c). Ca ++ ++ Ca „o »10 "fast" t o o ++ Ca ++ Ca »01 t Sil "fast" Consequently, (h is reminiscent of HH h).... Calcium Dynamics - p. 15/26 Mathematical Biology University of Utah Calcium Dynamics dc — (dmaxPo + Jer)(Ce ~ c) ~ JsERCA, dh = (j)h{c){l - h) - iph(c)h. where Js erc a — Vmax k2+c2 Po = hsf(c) Calcium Dynamics - p. 16/26 Bifurcation Diagram University of Utah Calcium Dynamics - p. 17/26 RYR Calcium Handling Ryanodine Receptor pathway Oiit_si the Calcium Dynamics - p. 18/26 Mathematical Biology University of Utah Excitation-Contraction Coupling Cardiac cells are interesting because they contain TWO excitable systems that are interconnected • The sodium-potassium electrical action potential, that stimulates an inward calcium flux • which excites CICR • which causes muscles to contract. Calcium Dynamics - p. 19/26 Mathematical Biology University of Utah EC Calcium Handling JRYR — JSERCA + Jl — JnCX JNCX Extracellular Space Cytosol (Intracellular Space) ' RYR JSERCA Sarcoplasmic Reticulum (Calcium stores) Calcium Dynamics - p.20/26 University of Utah dc U dt Jryr\— Jserca + J l — Jncx JNCX # Extracellular Space 4=4 Cytosol (Intracellular Space) ' RYR JSERCA Sarcoplasmic Reticulum (Calcium stores) with Jryr Ryanodine Receptor - calcium regulated calcium channel Calcium Dynamics - p.20/26 Mathematical Biology University of Utah EC Calcium Handling Jryr\ — \Jserca\+ Jl — Jncx JNCX Extracellular Space Cytosol (Intracellular Space) ' RYR JSERCA Sarcoplasmic Reticulum (Calcium stores) with Jryr Ryanodine Receptor - calcium regulated calcium channel Jserca Sarco- and Endoplasmic Reticulum Calcium ATPase, Calcium Dynamics - p.20/26 Mathematical Biology University of Utah EC Calcium Handling Jryr jserca + Jl — Jncx JNCX ifh Extracellular Space Cytosol (Intracellular Space) ' RYR JSERCA Sarcoplasmic Reticulum (Calcium stores) with Jryr Ryanodine Receptor - calcium regulated calcium channel Jserca Sarco- and Endoplasmic Reticulum Calcium ATPase, jl L-type voltage regulated calcium channel, Calcium Dynamics - p.20/26 Mathematical Biology University of Utah EC Calcium Handling Jryr jserca + Jl Jncx JNCX ifh Extracellular Space Cytosol (Intracellular Space) ' RYR JSERCA Sarcoplasmic Reticulum (Calcium stores) with Jryr Ryanodine Receptor - calcium regulated calcium channel Jserca Sarco- and Endoplasmic Reticulum Calcium ATPase, jl L-type voltage regulated calcium channel, Jncx sodium(Na++)- Calcium exchanger. Calcium Dynamics - p.20/26 Mathematical Biology University of Utah EC Calcium Handling Jryr jserca + Jl Jncx JNCX ifh Extracellular Space Cytosol (Intracellular Space) ' RYR JSERCA Sarcoplasmic Reticulum (Calcium stores) with Jryr Ryanodine Receptor - calcium regulated calcium channel Jserca Sarco- and Endoplasmic Reticulum Calcium ATPase, jl L-type voltage regulated calcium channel, Jncx sodium(Na++)- Calcium exchanger. Challenge: Determine the flux terms. Calcium Dynamics - p.20/26 Mathematical Biology University of Utah Serious Problems There are (at least) three problems with this (and all similar) models: Graded response -40 0 40 80 "40 0 40 80 membrane potential (mV) Calcium is not spatially homogenious; channels are controlled by local calcium concentration. Thus, whole cell models are inappropriate - geometry mattters. Channel openings are not deterministic and numbers are not large. Stochastic modeling is needed. Calcium Dynamics - p.21/26 Mathematical Biology University of Utah Bursting • A) Pancreatic /3-cell B) Rat midbrain C) Cat Thalamocortical relay neuron D) Guinea pig olivary neuron E Aplysia R15 neuroon F) Cat thalamic reticular neuron G) Sepia giant axon H) Rat thalamic reticular neuron • I) Mouse neocortical pyramidal neuron • J) Pituitary gonadotropin re-lasing cell B. c. 4s _|20mV 200 ms -AUA -60 mV -1.5 nA j -68 mV _|20mV u jl1111 _120 mV 8s Ii kiu -50 msec 200 ms 40 _ 1 mV nA 20 -20 U > E -60 -100C 10 15 t(s) Calcium Dynamics - p.22/26 University of Utah c. m dV ~dt dn dt dc ~dt -20 -40 I -60 -80 -J_I_I_I_I_I 0 1 2 3 4 Time (min) 5 6 Ic*(V) - (gKn4 + P^-) (V - VK) - 9l(V - VL) \ Kd + cJ Tn{V)-rr = noo(y)-n. = f(-hIc&{V)-kcc), where JCa = öcam^(F)/i00(y)(F - Vca). Calcium Dynamics - p.23/26 Possibilities Mathematical Biology University of Utah Bifurcation Diagram v max V, chc dc/dt=0 B V A ÍÍ 7 T^1---^-- dc/dt=0 Calcium Dynamics - p.25/26 Possibilities Mathematical Biology University of Utah Bursting Oscillations -30 -\ ^ -40 i 0.6 Calcium Dynamics - p.26/26