BE6003/Physiol 6003 Cellular Electrophysiology and Biophysics Modeling of Ion Channels II CVRTI Frank B. Sachse, University of UtahCellular Electrophysiology and Biophysics - Page 2 CVRTI Overview • Modeling of Mutations • Modeling of Drug Effects • Model Parameterization • Solution Methods • Software Tools • Molecular Modeling • Background • Homology • Drug Binding Group work Group work Group workCellular Electrophysiology and Biophysics - Page 3 CVRTI Applications of Ion Channel Models in System Biology • Approach: Ion channel models are integrated into cell models • Many recent cell models are based on Markov ion channel models • Application: Studies of complex electrophysiological interactions in cells • Mutations: Wild type models are modified with/extended to models of mutations • Drug effects: “Normal” models are modified with/extended to “drug” models Noble et al, 1998: Mathematical description of ionic currents and concentrations, transmembrane voltage, and conductivities of guinea-pig ventricular myocytes Myoplasma Sarcoplasmic reticulum IbCa ICa,L,Ca,ds INaCa INaCa,ds IUp Ip,Na Ib,Na ICa,L,Na INa,stretch INaK IK1 Ib,K Irel Itr INa ICa,L,Ca ICa,L,K IK IK,stretch ICa,stretch Troponin Itrop IK,AChCellular Electrophysiology and Biophysics - Page 4 CVRTI Markov Models for WT and 1795insD Cardiac Na Channels (Clancy and Rudy. Circulation 2002;105:1208-1213) Wild-type Na channel 1795insD Na channel: Insertion of amino acid aspartic acid at location 1795 Background mode (normal) Burst mode (pathological)Cellular Electrophysiology and Biophysics - Page 5 CVRTI Genetic Disease: Timothy Syndrome (Splawski et al, Proc Natl Acad Sci USA, 2005)Cellular Electrophysiology and Biophysics - Page 6 CVRTI Topology of Ion Channel Protein CaV1.2 G402S Glycine Serine G406R Glycine Arginine “De novo” mutationsCellular Electrophysiology and Biophysics - Page 7 CVRTI Modeling of Timothy Syndrome Channel Modeling Steady state inactivation is reduced in mutated channels versus wild type (WT) Model of mutated channels created with numerical optimization Prediction of course of transmembrane voltage in myocyte Changes dependent on % of channels with mutation Significant increase of action potential duration (and intracellular calcium concentrations)Cellular Electrophysiology and Biophysics - Page 8 CVRTI Slow Inward Rectifying Potassium Current IKs KCNQ1 KCNE1 Mutations • S140G Serine Glycine found in family with hereditary atrial fibrillation (Chen et al., Science, 2003) • V141M Valine Methionine found in new born child with atrial fibrillation and short QT syndrome “de novo” (Kong et al., Cardiovasc Res, 2005) * Location of Mutation KCNQ1 Genetic Disease: Mutation of KCNQ1 S4: Voltage sensing subunitCellular Electrophysiology and Biophysics - Page 9 CVRTI Model-Based Prediction of Ion Current WT KCNQ1 + KCNE1 KCNQ1 (S140G or V141M) + KCNE1 50 % WT / 50 % mutation gain of function! (Kong et al., Cardiovasc Res, 2005) constitutively openCellular Electrophysiology and Biophysics - Page 10 CVRTI Group Work Predict the effect of mutation S140G or V141M on action potential duration in myocytes!Cellular Electrophysiology and Biophysics - Page 11 CVRTI Modeling of Drug Effects 1 0 T+R TR € k1 € k−1 € k1 € k−11:1 binding € T + R TRT : ToxinR : Receptork1: forward rate constant [s-1M−1]k−1: backward rate constant [s-1]Dissociation constant [M] : Kd=k−1k1=T[ ]R[ ]TR[ ]Fractional occupancy : y =TR[ ]TR[ ]+ R[ ]=11+ Kd/ T[ ]Cellular Electrophysiology and Biophysics - Page 12 CVRTI Markovian Modeling of Drug Effects: Example • RPR: Activator of hErg1 channel current • Whole cell voltage clamp data measured in oocytesCellular Electrophysiology and Biophysics - Page 13 CVRTI Markovian Modeling of Drug Effects: Example Control 3 µM 10 µM 30µM hERG hERG/RPR 1s (Perry et al. Proc Natl Acad Sci U S A, 2007) Model includes two compartments connected by transitions between the open state O and inactivated state I:Cellular Electrophysiology and Biophysics - Page 14 CVRTI Model Parameterization by Optimization Approaches Commonly, model parameterization does not directly apply measurement data, but is based on numerical optimization procedures to minimize an error function. Error function can be defined as the difference between features extracted from experimental and simulated data: Numerical approaches: • Steep descent • Conjugate gradient • Levenberg–Marquardt • Stochastic approaches, e.g. particle swarm • ... € E =fm,i− fe,i2fe,i2⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ 2i=1...n∑fe: Experimental datafm: Model data…2: Euclidean normCellular Electrophysiology and Biophysics - Page 15 CVRTI Hodgkin-Huxley Channel Model: 1st Order ODE 1st order ODE € Iion= Gion,maxf Vm− Eion( )dfdt= αf1− f( )− βffαf≡ αfVm( ): Rate coefficientβf≡ βfVm( ): Rate coefficientf : Gating variableGion,max: Maximal conductivity for ionEion: Nernst voltageVm: Transmembrane voltage“1-f” “f” α βCellular Electrophysiology and Biophysics - Page 16 CVRTI Analytical Solution of 1st Order ODE € dfdt= αf1− f( )− βff ← → ⎯ dfdt=f∞− fτfTime constant : τf=1αf+ βfSteady − state value : f∞=αfαf+ βfResponse to step : f t( )= f∞− f∞− f0( )e−tτfCellular Electrophysiology and Biophysics - Page 17 CVRTI Markov Models: System of 1st Order ODEs € ddtP = QPwith the N states P =p1pN⎛ ⎝ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ and the matrix Q =q11 q1N  qN1 qNN⎛ ⎝ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ N-dimensional system of 1st order ODEsCellular Electrophysiology and Biophysics - Page 18 CVRTI Analytical Solution of Markov Model € P t( )= P 0( )eQteQt= 1+ Qt +Qt( )22!+Qt( )33!+…P t( )= P∞+ P 0( )Aii=2k∑e−tτi(Colquhoun and Hawkes, chap. 20, Single-Channel Recording, eds. Sakmann and Neher)Cellular Electrophysiology and Biophysics - Page 19 CVRTI Numerical Solution of Systems of ODEs • “Numerical integration” • Necessary for solving ODEs, which do not have analytical solutions • ODEs of n-th order can be reduced to set of 1st order ODEs 2nd