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 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}

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: Dn = unRT

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

Jn = Gn(VmVn)

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

Gm = Gn
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})

Ion channels

I = γ(VmVn)

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

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

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

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

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

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