Introduction to Physiology IV - Calcium Dynamics

Introduction to Physiology IV - Calcium Dynamics J. P. Keener Mathematics Department 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

Muscle Calcium Dynamics p.3/26

Muscle Calcium Dynamics p.4/26

Phototransduction Calcium Dynamics p.5/26

Phototransduction Calcium Dynamics p.6/26

Taste Calcium Dynamics p.7/26

Synaptic Transmission Calcium Dynamics p.8/26

Synaptic Transmission Calcium Dynamics p.9/26

Calcium Oscillations A 0.4 nm 0.6nM 0.9nM VP CCh 10mM B A) Hepatocytes B) Rat parotid gland C GnRH 1nM 10s D 20s C) Gonadotropes D) Hamster eggs 20s 30s 1min E, F) Insulinoma Cells E 25mM F 200mM CCh 50s 50s Calcium Dynamics p.10/26

Calcium Handling Calcium Dynamics p.11/26

IPR Calcium Handling J L JSL Extracellular Space v dc dt = J IP R J SERCA + J L J SL (Intracellular Space) Cytosol J IPR JSERCA Sarcoplasmic Reticulum (Calcium stores) Calcium Dynamics p.12/26

IPR Calcium Handling J L JSL Extracellular Space v dc dt = J IP R J SERCA + J L J SL (Intracellular Space) Cytosol Sarcoplasmic Reticulum J IPR JSERCA (Calcium stores) with J IP R IP 3 Receptor - calcium regulated calcium channel, J SERCA Sarco- and Endoplasmic Reticulum Calcium ATPase, Calcium Dynamics p.12/26

Calcium Handling group I k 1 p S 000 S 100 k -1 k -5 k -1 S 110 S 010 k 5 c k 1 p k 5 c k -5 S 110 k -2 k 2 c k -4 k 4 c k -4 k 4 c k -2 k 2 c k -2 k 2 c k 3 p k 5 c k 3 p k 5 c S 111 k -3 S 011 S 001 k -5 group II k -3 S 101 S 111 k -5 Calcium Dynamics p.13/26

the Imagine IP3 Receptors 10 ++ Ca ++ ++ Ca ++ Ca Ca S S 11 01 00 S S Calcium Dynamics p.14/26

IP 3 Receptors Flux through IP 3 receptor is diffusive, J IP R = g max P o (c c sr ) where P o = S10 3 = m3 h 3 is the open probability, and dm dh dt = φ m (c)(1 m) ψ m (c)m, dt = φ h(c)(1 h) ψ(c)h. Furthermore, m is a fast variable, so is in qss, m = m (c). Ca ++ S 10 00 "fast" S ++ Ca "slow" "slow" Ca ++ S 01 Ca ++ "fast" S 11 Consequently, (h is reminiscent of HH h)... Calcium Dynamics p.15/26

Calcium Dynamics dc dt = (g maxp o + J er )(c e c) J SERCA, dh dt = φ h(c)(1 h) ψ h (c)h, 1 0.8 0.6 where J SERCA = V max c 2 K 2 s +c 2, P o = h 3 f(c) h 0.4 0.2 0 0 5 10 15 20 25 30 35 40 45 50 1 0.8 0.6 0.4 time (s) 0.2 0 10 2 10 1 10 0 c (µ M) Calcium Dynamics p.16/26

Bifurcation Diagram A 0.8 Stable oscillation c (µm) 0.6 0.4 c max unstable oscillation 0.2 HB unstable steady state HB stable 0 5 p 10 15 c (µm) B 0.5 0.4 0.3 0.2 0.1 0.0 p=10 c (µm) C 0.6 0.4 0.2 0.0 p=1 65 70 75 80 85 time (s) 90 95 50 100 time (s) 150 Calcium Dynamics p.17/26

RYR Calcium Handling Calcium Dynamics p.18/26

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 diadic cleft which excites CICR which causes muscles to contract. Calcium Dynamics p.19/26

EC Calcium Handling J L JNCX Extracellular Space v dc dt = J RY R J SERCA + J L J NCX Cytosol (Intracellular Space) J RYR JSERCA Sarcoplasmic Reticulum (Calcium stores) Calcium Dynamics p.20/26

EC Calcium Handling J L JNCX Extracellular Space v dc dt = J RY R J SERCA + J L J NCX Cytosol (Intracellular Space) Sarcoplasmic Reticulum J RYR JSERCA (Calcium stores) with J RY R Ryanodine Receptor - calcium regulated calcium channel, J SERCA Sarco- and Endoplasmic Reticulum Calcium ATPase, Calcium Dynamics p.20/26

Serious Problems There are (at least) three problems with this (and all similar) models: A J ICa J RyR B Graded response 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

Bursting A) Pancreatic β-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 relasing cell Calcium Dynamics p.22/26

Pancreatic β cells C m dv dt τ n (V ) dn dt dc dt = I Ca (V ) = n (V ) n, = f( k 1 I Ca (V ) k c c), ( ḡ K n 4 + ḡk,cac ) (V V K ) ḡ L (V V L ), K d + c where I Ca = ḡ Ca m 3 (V )h (V )(V V Ca ). Calcium Dynamics p.23/26

Fast Phase Plane n A B n V Low Ca ++ High Ca ++ V Calcium Dynamics p.24/26

Bifurcation Diagram A V HB max V osc HC dc/dt=0 min V osc SN V ss c hb c hc c B V dc/dt=0 V ss c Calcium Dynamics p.25/26

Bursting Oscillations -20 V (mv) -30-40 -50 0.8 0.7 c (µm) 0.6 0.5 0.4 0 5 10 Time (s) 15 20 Calcium Dynamics p.26/26