GSAC February 21, 2006 Dynamics of the Electric Field Mechanism Elizabeth Doman 1 the heart... • • transports blood to and from the body and lungs right and left component... ◮ each consisting of atrium and ventricle • blood travels... ◮ into the right atrium... ◮ pumped by the right ventricle, out of the heart, to the lungs ◮ into the left artrium... ◮ pumped, by the left ventricle, out of the heart, to the body 2 propagation... ⋆ myocardial cells are excitable and contractile • transmembrane potential can undergo an action potential ⋆ ions (Na+ , K+ , Ca2+ ) move across the cell membrane ⋆ many models of single cell action potentials • • intracellular Ca2+ dynamics cause the cell to contract ← Nessy action potentials jump from cell to cell ⇒ propagation of excitation ⋆ conduction system ⋆ ⋆ cardiac action potential ⋆ 3 propagation... • gap junction channels are the primary mechanism ◮ they allow for the intracellular spread of current ◮ propagation fails under conditions of gap junctional uncoupling • • is there a second mechanism to modulate propagation? Sperelakis, 1959 ◮ proposed an electric field interaction between cells ◮ recent experimental findings... ⋆ high density of Na Chs in cell junctions ⋆ propagation in Cx43 deficient mouse hearts + ... the electric field mechanism might exist 4 Electric Field Mechanism... • the idea... ◮ ◮ ◮ ◮ ◮ ◮ the juctional cleft is very narrow ∼ 35 − 250 Å high radial resistance to the movement of ions high radial resistance to the flow of current drastic changes in ionic cleft concentrations possibility of a separate electric field in the cleft consider the cleft as a separate domain junctional cleft cell 2 cell 1 pre-junctional membrane post-junctional membrane direction of propagation 5 Electric Field Mechanism... 1 0 Iapp 1 0 Im1 Ir12 1 0 • 1 0 Ir23 Im2 1 0 Ij12 V1 1 0 1 0 Ij21 Vj12 1 0 Im3 1 0 Ij23 V2 1 0 Ij32 Vj23 Ij34 V3 non-junctional membrane current: Imn = d Am Cm dt Vn + Iion (Vn ) • junctional membrane current: d (Vn − Vjn,n+1 ) + I˜ion (Vn − Vjn,n+1 ) Ijn,n+1 = Aj Cj dt • radial cleft current: Irn,n+1 = 1 rjc Vjn,n+1 6 Electric Field Mechanism... • balancing the currents at the nodes... ◮ cell 1: − C1 m ◮ m m • d d d αǫ dt V1 − 2αǫ dt Vj12 + αǫ dt V2 = “ ” αI˜ion (V − Vj ) + αI˜ion (V − Vj ) + sVj 1 12 2 12 12 cell 2: − C1 ◮ Iapp Cm A junction 1-2: − C1 ◮ d d V1 − αǫ dt Vj12 = (1 + αǫ) dt “ ” Iion (V1 ) + αI˜ion (V1 − Vj12 ) + d d d −αǫ dt Vj12 + (1 + 2αǫ) dt V2 − αǫ dt Vj23 = “ ” ˜ ˜ αIion (V2 − Vj12 ) + Iion (V2 ) + αIion (V2 − Vj23 ) etc... a system of coupled ODEs important parameters... ◮ ◮ ◮ surface area ratio: α = Aj /Am capacitance ratio: ǫ = Cj /Cm junctional cleft conductance: s = 1/Am Cm rjc 7 Electric Field Mechanism... • single cell membrane dynamics: Mitchell-Schaeffer ◮ a generic cardiac ionic model with two currents ◮ we care about excitation only ⇒ ignore recovery Iion (V ) = GN a m2 (V )(V − VN a ) + GK (V − VK ) I˜ion (V ) = βGN a m2 (V )(V − VN a ) + GK (V − VK ) Dimensionalized Modified Mitchell−Schaeffer 300 non−junctional junctional 250 200 150 100 50 0 −120 −100 −80 −60 −40 −20 0 20 40 60 V (mV) 8 Electric Field Mechanism... • numerical simulation... Propagation via Electric Field Mechanism using Dimensionalized Modified Mitchell−Schaeffer 50 α = 1/20 ǫ = 1/10 β=6 s = 1.3 potential (mV) ◮ parameters: 0 V1 V2 V3 V4 V5 Vj12 Vj23 Vj34 Vj45 −50 −100 0 1 2 3 4 5 6 time (ms) 9 analysis? • continuous approximation? traveling wave solutions? ◮ not yet... • can we characterize propagation failure? ◮ what are the important parameters? • • • ǫ → capacitance ratio s → junctional resistance β → junctional sodium conductance ◮ what are the underlying dynamics? ... notice: cleft potentials appear to be fast variables 10 dynamics... • nondimensionalize... ◮ Vn → φn and t → τ , such that 0 < φn < 1 and 0 < τ < 1 • two cell system... ◮ ◮ ◮ • • d dτ φ1 d + αǫ dτ (φ1 − φj ) = −iion (φ1 ) − αĩion (φ1 − φj ) + iapp d d (φ1 − φj ) + αǫ dτ (φ2 − φj ) = −αĩion (φ1 − φj ) − αĩion (φ2 − φj ) + σφj αǫ dτ d dτ φ2 d + αǫ dτ (φ2 − φj ) = −iion (φ2 ) − αĩion (φ2 − φj ) notice: the variable αǫ is very small ∼ 10−2 quasi-steady state approximation: ◮ letting αǫ → 0, we assume the junctional potentials are in qss ◮ not the same as taking dτd (φ1 − φj ) = 0 and dτd (φ2 − φj ) = 0 ◮ rather... φj is changing while restricted to some manifold 11 quasi-steady state approximation... • letting αǫ → 0, ◮ ◮ • d dτ φ1 d dτ φ2 = −iion (φ1 ) − αĩion (φ1 − φj ) + iapp = −iion (φ2 ) − αĩion (φ2 − φj ) φj changes while restricted to some manifold, ◮ αĩion (φ1 − φj ) + αĩion (φ2 − φj ) = σφj • • • remember, ĩion (φ) is a cubic the quasi-steady state, φ∗j , depends on φ1 and φ2 let’s characterize this quasi-steady state φ∗j 12 dynamics... • the quasi-steady state, φ∗j , will be intersections of ... ◮ f1 (φj ) = σφj ◮ f2 (φj ) = αĩion (φ1 − φj ) + αĩion (φ2 − φj ) φ*j as intersection of f1 and f2 with β=3 , σ=2.39 φ*j as intersection of f1 and f2 with β=3 , σ=2.39 3 φ*j as intersection of f1 and f2 with β=3 , σ=2.39 3 φ1=0 , φ2=0 ⇒ φ*j =0 2 3 φ1=0.07 , φ2=0 ⇒ φ*j =−0.01 2 1 1 1 0 0 0 −1 −1 −1 −2 −2 −2 −3 −1 −0.8 −0.6 −0.4 −0.2 0 φj 0.2 0.4 0.6 0.8 1 −3 −1 −0.8 −0.6 φ*j as intersection of f1 and f2 with β=3 , σ=2.39 −0.4 −0.2 0 φj 0.2 0.4 0.6 0.8 1 −3 −1 3 φ1=0.71 , φ2=0.16 ⇒ φ*j =−0.27 2 0 0 0 −1 −1 −1 −2 −2 −2 −0.4 −0.2 0 φj 0.2 0.4 0.6 0.8 1 −3 −1 −0.2 0 φj 0.2 0.4 0.6 0.8 1 −0.8 −0.6 −0.4 −0.2 0 φj 0.2 0.4 0.6 0.8 1 φ1=0.85 , φ2=0.85 ⇒ φ*j =−0.09 2 1 −0.6 −0.4 3 φ1=0.83 , φ2=0.48 ⇒ φ*j =−0.2 1 −0.8 −0.6 φ*j as intersection of f1 and f2 with β=3 , σ=2.39 1 −3 −1 −0.8 φ*j as intersection of f1 and f2 with β=3 , σ=2.39 3 2 φ1=0.16 , φ2=0.01 ⇒ φ*j =−0.72 2 0.8 1 −3 −1 −0.8 −0.6 −0.4 −0.2 0 φj 0.2 0.4 0.6 13 conclusions... • conditions for successful propagation... ◮ ǫ must be small enough ⋆ quasi-steady state approximation must be valid ⇒ junctional membranes must have little or no capacitance ◮ σ must be small enough ⋆ the slope of the line must not be too steep ⇒ radial cleft resistance must be high ◮ β must be large enough ⋆ the amplitude of the cubic must be large enough ⇒ the junctional membranes must be highly excitable • otherwise... propagation will fail 14