# 6.021/Notes/2006-10-10

## Diffusion

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

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

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

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

Time for half the solute to diffuse x1 / 2: $t_{1/2}\approx\frac{x_{1/2}^2}{D}$

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

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

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

## Osmosis

Van't Hoff Law: π(x,t) = RTCΣ(x,t)

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

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

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

Flux: ΦV = LV((pi − πi) − (po − πo))

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

## Carrier Transport

Solution to simple symmetric 4-state carrier model:

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

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

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

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

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