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 $$x_{1/2}$$: $$t_{1/2}\approx\frac{x_{1/2}^2}{D}$$

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: $$\pi(x,t)= RTC_\Sigma(x,t)$$

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: $$\Phi_V = L_V((p^i-\pi^i)-(p^o-\pi^o))$$

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:

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

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

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

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

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