1Physiology 472/572 - 2009 - Quantitative modeling of biological systems Lecture 24: Muscle physiology and calcium dynamics Types of muscle • Skeletal muscle - movement • Cardiac muscle - pumping blood • Smooth muscle - blood vessels, gastrointestinal tract, etc. Sliding filament model • 'thin' filaments - chains of actin proteins: high tensile, passive • 'thick filaments' - bundles of myosin • when activated, myosin heads attach to and pull against actin filaments • each head movement moves the thick filament 5-10 nm • energy is supplied by hydrolysis of ATP to ADP, which resets head to extended position Role of calcium ions • at rest, tropomyosin lies in groove on actin filament and blocks 7 binding sites • each tropomyosin has a troponin complex at one end • troponin contains a calcium-binding protein • when calcium binds to troponin, tropomyosin moves to expose myosin-binding sites2Control of intracellular calcium • calcium is continuously pumped out of the cytoplasm and the concentration is low at rest • intracellular calcium is stored in sarcoplasmic reticulum (SR) - a mesh of closed tubes that lie close to the myofibrils • contraction can be stimulated in several ways: action potential in the heart, synapse with motor nerves in skeletal muscle, rise in vasodilator substances in smooth muscle • in each case, the stimulus leads to the release of a small amount of calcium into the cytoplasm, which is amplified by calcium-induced calcium release • this positive feedback mechanism can lead to oscillating calcium levels Two-pool model for calcium dynamics (Goldbeter et al., PNAS 87, 1461-1465, 1990) • c(t) is concentration of Ca2+ in the cytosol • cs(t) is the concentration of Ca2+ in the Ca2+-sensitive pool sfreleaseuptakesssfreleaseuptakeckJJdtdcVckJJkcrdtdcV−−=++−−= • Juptake depends only on c: nn1n1uptakecKcVJ+= • Jrelease depends on c and cs: pp3pmsm2ms2releasecKccKcVJ++=3• introduce non-dimensional variables: u = c/K1, v = cs/K2, τ = kt/V, α = K3/K1, β = V1/V2, γ = K2Vs/(K1V), δ = kfK2/V2, ε = kK2Vs/(V2V), μ = r/(kK1) • typical values α = 0.9, β = 0.13, γ = 2, δ = 0.004, ε = 0.04, μ = 0.3, m = 2, n = 2, p = 4 δvuαu1vv1uuβ v)f(u,wherev)f(u,ε1ddvv)f(u,εγuμddupppmmnn−++−+==−−=ττ • one equilibrium point, stable for these parameters • a small pulse of additional calcium into the cytoplasm causes rapid release of calcium from stores, slow reuptake and return to equilibrium • this represents behavior during muscle contraction 0 0.5 1 1.5 2 2.5 300.511.5uvu nullclinev nullclineμ = 0.30 20 30 40 500.20.40.60.811.21.41.6tPulse of
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