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Physical/Mathematical Background Physics/Math Background Bioengineering 6003 Cellular Electrophysiology & Biophysics The Basics • Fields, Flows, and Circuits – Electric field, potential field – current, conductors – Ohms law, IV curves, dynamic circuit analysis • Sources, Sinks, and Vector Calculus – Current monopoles, dipoles – Volume conductor fields – Div, grad, curl and all that Physics/Math Background Bioengineering 6003 Cellular Electrophysiology & Biophysics Coulomb’s Law +qt -q1 +q2 Physics/Math Background Bioengineering 6003 Cellular Electrophysiology & Biophysics Superposition of Electric Field +qt -q1 +q2 Physics/Math Background Bioengineering 6003 Cellular Electrophysiology & Biophysics Current and Ohm’s Law • Different forms of Ohm’s Law • Without potential difference there is no current! • Essentials of Ohm’s Law: – linear relationship between current and voltage – not universal, especially not in living systems What do we mean by non-linearity in this context? Physics/Math Background Bioengineering 6003 Cellular Electrophysiology & Biophysics Conductors & Resistors • Conductors - - – Electrons free to move – Current flow in response to electric field – In static state, no net charge (E=0) - • Resistors – Electrons less free to move – Create potential differences – Depend on material properties - E!0 - E=0 !V = 0 - - V Note: Electric field is the (negative) gradient of potential, E = -!V Physics/Math Background Bioengineering 6003 Cellular Electrophysiology & Biophysics Capacitance -Q - Total charge is a function of electric field, which depends on V and spacing. - +Q + + V + - • Dielectric – Charges not free to move, just shift – E!0 inside, opposes applied E – Result is increased Q for the same v and increased C Does anything change when the plates move? + + -Q - + + - + - + - - + +Q + ? V Increasing spacing decreases the E field, which means less charge and reduced capacitance Q = CV Physics/Math Background Bioengineering 6003 Cellular Electrophysiology & Biophysics Membrane Equivalent Circuit Channel Lipid Bilayer Charged Polar Head Rm Cm + " Em Physics/Math Background Bioengineering 6003 Cellular Electrophysiology & Biophysics Current and Ohm’s Law • Without potential difference there is no current! • Without conductance, there is no current. • Ohm’s law: – linear relationship between current and voltage – not universal, especially not in living systems 1 I = V = GV R v(0) jx v(x) x 0 L Physics/Math Background Bioengineering 6003 Cellular Electrophysiology & Biophysics Current-Voltage (I-V) i A ?? v V Physics/Math Background Bioengineering 6003 Cellular Electrophysiology & Biophysics Equivalent circuits 1 3g i 2g g g v Physics/Math Background Bioengineering 6003 Cellular Electrophysiology & Biophysics Equivalent circuits 2 i V1 V2 V1 V2 v v g g Driving force I1=(v+V1) g Physics/Math Background I2=(v-V2) g Bioengineering 6003 Cellular Electrophysiology & Biophysics I-V Curve Examples I Rectifying current K-current Rectifying current + Nernst potential Positive Nernst potential V Vrev Vrev Na-current Physics/Math Background Bioengineering 6003 Cellular Electrophysiology & Biophysics Circuit Analysis • Conservation of charge: currents sum at nodes • Conservation of energy: sum of voltages = 0 v1 i1 i2 i3 i1 + i2 + i3 = 0 v4 v2 v4 = v1+ v2+ v3 v3 Physics/Math Background Bioengineering 6003 Cellular Electrophysiology & Biophysics Voltage Divider i R1 i=vB/(R1+R2) i=v2/R2 vB v2=vB R2/(R1+R2) R2 v2 i Examples of voltage dividers in EP measurements? Physics/Math Background Bioengineering 6003 Cellular Electrophysiology & Biophysics Electrical Profile of a Cell v E Outside + - + - + + + - + - + Outside - - + - - - - Inside Physics/Math Background + + Bioengineering 6003 Cellular Electrophysiology & Biophysics Equilibrium Potential a) Membrane is impermeable b) Membrane becomes permeable to potassium only (semipermeable) c) Equilibrium established when electrostatic and chemical gradients balance. Physics/Math Background Bioengineering 6003 Cellular Electrophysiology & Biophysics Nernst Equillibrium Electrical and Chemical work At equilibrium there will be no net movement of X Physics/Math Background Bioengineering 6003 Cellular Electrophysiology & Biophysics Example Nernst Potentials Nernst Potential (mV) Ion External Physics/Math Background Internal Bioengineering 6003 Cellular Electrophysiology & Biophysics Diffusion • Driven by thermodynamic random motion • Similar linear relationship as Ohm’s Law • where D = rate of diffusion • C = concentration • P = permeability Physics/Math Background Bioengineering 6003 Cellular Electrophysiology & Biophysics
