6.021/Notes/Equations

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:

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

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

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

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

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

Electrodiffusion

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

Einstein's relation: $\displaystyle{ D_n=u_nRT }$

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

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

Membranes

$\displaystyle{ J_n = G_n (V_m-V_n) }$

$\displaystyle{ G_n = \frac{1}{\int_0^d{\frac{dx}{u_nz_n^2F^2c_n(x)}}} }$ (electrical conductivity)

Nernst potential: $\displaystyle{ V_n=\frac{RT}{z_nF}{\rm ln}\frac{c^o_n}{c^i_n} \approx \frac{60 {\rm mV}}{z_n}{\rm log}\frac{c^o_n}{c^i_n} }$

Cells

$\displaystyle{ G_m=\sum_n G_n }$

Resting membrane potential: $\displaystyle{ V_m^o = \sum_n \frac{G_n}{G_m}V_n }$

Resting potential with active pumps: $\displaystyle{ V_m^o = \sum_n \frac{G_n}{G_m}V_n - \frac{1}{G_m}\sum_n J_n^a }$

Core conductor model

$\displaystyle{ \frac{\partial I_i(z,x)}{\partial z}=-K_m(z,t) }$

$\displaystyle{ \frac{\partial I_o(z,x)}{\partial z}=K_m(z,t)-K_e(z,t) }$

$\displaystyle{ \frac{\partial V_i(z,t)}{\partial z}=-r_iI_i(z,t) }$

$\displaystyle{ \frac{\partial V_o(z,t)}{\partial z}=-r_oI_o(z,t) }$

THE core conductor equation: $\displaystyle{ \frac{\partial^2 V_m(z,t)}{\partial z^2}=(r_o+r_i)K_m(z,t)-r_oK_e(z,t) }$

wave equation: $\displaystyle{ \frac{\partial^2 V_m(z,t)}{\partial z^2}=\frac{1}{\nu^2}\frac{\partial^2 V_m(z,t)}{\partial t^2} }$

$\displaystyle{ \nu = \sqrt{\frac{C}{(r_o+r_i)2\pi a}} }$, $\displaystyle{ \nu = \sqrt{\frac{Ca}{2\rho}} \propto \sqrt{a} }$

$\displaystyle{ v_o(z)=-\frac{r_o}{r_o+r_i}(v_m(z) - V_m^o) }$

Hodgkin-Huxley

$\displaystyle{ G_{K}(V_m,t) = \overline{G_K} n^4(V_m,t) }$, $\displaystyle{ G_{Na}(V_m,t) = \overline{G_{Na}} m^3(V_m,t)h(V_m,t) }$

$\displaystyle{ n(V_m,t) + \tau_n(V_m)\frac{dn(V_m,t)}{dt} = n_\infty(V_m) }$, $\displaystyle{ m(V_m,t) + \tau_m(V_m)\frac{dm(V_m,t)}{dt} = m_\infty(V_m) }$, $\displaystyle{ h(V_m,t) + \tau_h(V_m)\frac{dh(V_m,t)}{dt} = h_\infty(V_m) }$

$\displaystyle{ x_\infty=\frac{\alpha_x}{\alpha_x+\beta_x}, \tau_x=\frac{1}{\alpha_x+\beta_x} }$

Cable model

$\displaystyle{ J_m = C_m\frac{dV_m}{dt}+G_m(V_m-V_m^o) }$

Cable Equation: $\displaystyle{ v_m+\tau_m\frac{\partial v_m}{\partial t}-\lambda_c^2\frac{\partial^2v_m}{\partial z^2}=r_o\lambda_c^2K_e }$

$\displaystyle{ \tau_m=\frac{c_m}{g_m} }$

$\displaystyle{ \lambda_c = \frac{1}{\sqrt{g_m(r_o+r_i)}} }$

$\displaystyle{ v_m = V_m - V_m^o }$

Steady state solution of cable equation to impulse stimulus: $\displaystyle{ v_m(z) = \frac{r_o\lambda_c}{2}I_e e^{-|z|/\lambda_c} }$

Dynamics: $\displaystyle{ v_m(z,t)=w(z,t) e^{-t/\tau_m} }$ where $\displaystyle{ \frac{\partial w}{\partial t} = \frac{\lambda_c^2}{\tau_m} \frac{\partial^2 w}{\partial z^2} }$ (Diffusion equation with $\displaystyle{ D=\frac{\lambda_c^2}{\tau_m} }$)

Ion channels

$\displaystyle{ I = \gamma (V_m-V_n) }$

$\displaystyle{ E[\tilde{s}(t)] = x }$, $\displaystyle{ E[\tilde{g}(t)]=\gamma x=g }$, $\displaystyle{ E[\tilde{i}(t)]= g(V_m - V_n) }$

$\displaystyle{ G = \frac{N}{A} g }$, $\displaystyle{ J = \frac{N}{A} g(V_m-V_n) }$

$\displaystyle{ x(t) = x_\infty+(x(0)-x_\infty)e^{-t/\tau_x}, \tau_x=\frac{1}{\alpha+\beta}, x_\infty=\frac{\alpha}{\alpha+\beta} }$

$\displaystyle{ \tilde{i}_g = \frac{d}{dt}\tilde{q}_g }$

$\displaystyle{ i_g = E[\tilde{i}_g] = Q\frac{dx}{dt} }$