6.021/Notes/2006-10-11

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Nernst-Planck Equation: J_n = -z_nFD_n\frac{\partial c_n}{\partial x}-u_nz_n^2F^2c_n\frac{\partial\psi}{\partial x}

Continuity: \frac{\partial J_n}{\partial x} = -z_nF\frac{\partial c_n}{\partial t}

Poisson's Equation: \frac{\partial^2 \psi}{\partial x^2} = -\frac{1}{\epsilon}\sum_n z_nFc_n(x,t)

Flux through membranes

  • Assume membrane in steady state as before
  • concentrations of charge charge can't change so current is constant
  • Four inputs: voltage on inside and outside, concentration on inside and outside
  • Jn = Gn(VmVn)
  • G_n = \frac{1}{\int_0^d{\frac{dx}{u_nz_n^2F^2c_n(x)}}} (electrical conductivity)
    • always greater than zero, means transport will always go down electrochemical gradient (lose energy)
    • Not really constant (depends on concentration) but in real cells, will seldom see much change in concentrations so we will assume Gn is constant.
  • Vm = ψ(0) − ψ(d) (potential difference across membrane)
  • V_n=\frac{RT}{z_nF}{\rm ln}\frac{c^o_n}{c^i_n} (Nernst equilibrium potential)
    • this constant is part of the model and not directly measurable (not physical)
    • is electrical representation of chemical phenomenon
    • But can indirectly measure this by changing Vm. The Nernst potential is the same potential that when applied externally to the membrane causes no current.
    • \frac{RT}{F}\approx 26mV at room temperature
    • \frac{RT}{F}{\rm ln(10)}\approx 60mV, so can use V_n \approx \frac{60mV}{z_n}{\rm log}\frac{c^o_n}{c^i_n}
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