6.021/Notes/2006-10-10

Diffusion

Fick's 1st law: $\displaystyle{ \phi(x,t)=-D\frac{\partial c(x,t)}{\partial x} }$

Continuity: $\displaystyle{ -\frac{\partial\phi}{\partial x} = \frac{\partial c}{\partial t} }$

Diffusion Equation: $\displaystyle{ \frac{\partial c}{\partial t} = D\frac{\partial^2 c}{\partial x^2} }$

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

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

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

Membrane steady state time constant: $\displaystyle{ \tau_{ss}=\frac{d^2}{\pi^2 D} }$

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

Osmosis

Van't Hoff Law: $\displaystyle{ \pi(x,t)= RTC_\Sigma(x,t) }$

Darcy's Law: $\displaystyle{ \Phi_V(x,t)= -\kappa\frac{\partial p}{\partial x} }$

Continuity: $\displaystyle{ -\rho_m\frac{\partial \Phi_V}{\partial x} = 0 }$

Hydraulic conductivity: $\displaystyle{ L_V = \frac{\kappa}{d} }$

Flux: $\displaystyle{ \Phi_V = L_V((p^i-\pi^i)-(p^o-\pi^o)) }$

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

Carrier Transport

Solution to simple symmetric 4-state carrier model:

$\displaystyle{ \mathfrak{N}^i_{ES}=\frac{\beta}{\alpha+\beta}\frac{c^i_s}{c^i_s+K}\mathfrak{N}_{ET} }$

$\displaystyle{ \mathfrak{N}^i_{E}=\frac{\beta}{\alpha+\beta}\frac{K}{c^i_s+K}\mathfrak{N}_{ET} }$

$\displaystyle{ \mathfrak{N}^o_{ES}=\frac{\alpha}{\alpha+\beta}\frac{c^o_s}{c^o_s+K}\mathfrak{N}_{ET} }$

$\displaystyle{ \mathfrak{N}^o_{E}=\frac{\alpha}{\alpha+\beta}\frac{K}{c^o_s+K}\mathfrak{N}_{ET} }$

$\displaystyle{ \phi_s=(\phi_s)_{max}(\frac{c^i_s}{c^i_s+K}-\frac{c^o_s}{c^o_s+K}) }$; $\displaystyle{ (\phi_s)_{max}=\frac{\alpha\beta}{\alpha+\beta}\mathfrak{N}_{ET} }$