6.021/Notes/Equations

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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: $π(x, t) = R T C Σ (x, t)$

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: $Φ V = L V ((p i - π i) - (p o - π o))$

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:

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

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

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

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

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

Electrodiffusion

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

Einstein's relation: $D n = u n R T$

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

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

Membranes

$J n = G n (V m - V n)$

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

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

Cells

G_m =	∑	G_n
	n

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

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

Core conductor model

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

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

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

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

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

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

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

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

Hodgkin-Huxley

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

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

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

Cable model

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

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

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

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

$v_m = V_m - V_m^o$

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

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