6.021/Notes/2006-10-10

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Diffusion

Fick's 1st law: [math]\phi(x,t)=-D\frac{\partial c(x,t)}{\partial x}[/math]

Continuity: [math]-\frac{\partial\phi}{\partial x} = \frac{\partial c}{\partial t}[/math]

Diffusion Equation: [math]\frac{\partial c}{\partial t} = D\frac{\partial^2 c}{\partial x^2}[/math]

Solution of diffusion equation to impulse stimulus is Gaussian: [math]c(x,t)=\frac{n_0}{\sqrt{4\pi Dt}}e^{-x^2/(4Dt)}[/math]

Time for half the solute to diffuse [math]x_{1/2}[/math]: [math]t_{1/2}\approx\frac{x_{1/2}^2}{D}[/math]

Fick's law for membranes: [math]\phi_n(t)=P_n(c_n^i(t)-c_n^o(t))[/math]; [math]P_n=\frac{D_nk_n}{d}[/math]

Membrane steady state time constant: [math]\tau_{ss}=\frac{d^2}{\pi^2 D}[/math]

Solution for dissolve and diffuse: [math]c_n^i(t)=c_n^i(\infty)+(c_n^i(0)-c_n^i(\infty))e^{-t/\tau_{eq}}[/math]; [math]c_n^i(\infty) = \frac{N_n}{V_i+V_o}, \tau_{eq}=\frac{1}{AP(1/V_i+1/V_o)}[/math]

Osmosis

Van't Hoff Law: [math]\pi(x,t)= RTC_\Sigma(x,t)[/math]

Darcy's Law: [math]\Phi_V(x,t)= -\kappa\frac{\partial p}{\partial x}[/math]

Continuity: [math]-\rho_m\frac{\partial \Phi_V}{\partial x} = 0[/math]

Hydraulic conductivity: [math]L_V = \frac{\kappa}{d}[/math]

Flux: [math]\Phi_V = L_V((p^i-\pi^i)-(p^o-\pi^o))[/math]

Cells: [math]\frac{dV^i}{dt} = -A(t)\Phi_V[/math] with solution [math]v_c(\infty) = v_c' + \frac{N^i_\Sigma}{C^o_\Sigma}[/math]

Carrier Transport

Solution to simple symmetric 4-state carrier model:

[math]\mathfrak{N}^i_{ES}=\frac{\beta}{\alpha+\beta}\frac{c^i_s}{c^i_s+K}\mathfrak{N}_{ET}[/math]

[math]\mathfrak{N}^i_{E}=\frac{\beta}{\alpha+\beta}\frac{K}{c^i_s+K}\mathfrak{N}_{ET}[/math]

[math]\mathfrak{N}^o_{ES}=\frac{\alpha}{\alpha+\beta}\frac{c^o_s}{c^o_s+K}\mathfrak{N}_{ET}[/math]

[math]\mathfrak{N}^o_{E}=\frac{\alpha}{\alpha+\beta}\frac{K}{c^o_s+K}\mathfrak{N}_{ET}[/math]

[math]\phi_s=(\phi_s)_{max}(\frac{c^i_s}{c^i_s+K}-\frac{c^o_s}{c^o_s+K})[/math]; [math](\phi_s)_{max}=\frac{\alpha\beta}{\alpha+\beta}\mathfrak{N}_{ET}[/math]