6.021/Notes/2006-10-04

Ion Transport

 * Major constituents of cells
 * important functions
 * charge is substrate for neural communication
 * every charged particle (in principle) affects every other ion
 * more complicated than other mechanisms

Mechanisms
2 distinct mechanisms of diffusion and drift

Diffusion
Given by Fick's law $$\phi_n = -D_n\frac{\partial c_n}{\partial x}$$

Drift

 * Effect of electrical forces on montion of charged particles.
 * Electric Field (vector field) $$E(x,t)$$
 * force on particle $$f_p = QE(x,t) = z_neE(x,t)$$ where $$z_n$$ is valence and $$e\approx 1.6\cdot 10^{-19}$$ C.
 * Motions of small particles in water are viscosity dominated (Stokes 1855)

$$v\propto f_p = u_p f_p = u_n f$$ where $$u_p$$ is mechanical mobility in units of velocity/force, $$u_n$$ is the molar mechanical mobility and $$f$$ becomes the force on a mole of particle.

For charged particles: $$v=u_nz_neN_AE(x,t)=u_nz_nFE(x,t)$$ ($$F=eN_A$$ which is Faradya's number) = charge/mole about 96500 C/mol.

$$D_n=u_nRT$$: Einstein's relation

Flux due to drift: $$\phi_n = \lim_{A\rightarrow 0 ; \Delta t\rightarrow 0}\frac{c_n(x,t)A\Delta x}{A\Delta t} = vc_n(x,t)$$

$$\phi_n = c_n(x,t)u_nz_nFE(x,t) = -c_nu_nz_nF\frac{\partial\psi}{\partial x}$$ where $$E=-\frac{\partial\psi}{\partial x}$$ (electric field depends on the potential gradient)

The flux of ions is the current density given by $$J_n = z_nF\phi_n$$ This is in units of current/area and is easier to measure than flux.

Combined transport
Combining diffusion and drift to get Nernst-Planck Equation:

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

Note that this is really just a combination of Fick's and Ohm's Laws.

Continuity: (needed to solve equations just like in other transport mechanisms)

$$\frac{\partial\phi_n}{\partial x} = -\frac{\partial c_n}{\partial t}$$ or equivalently

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

Unlike diffusion, also need one more equation for $$\psi$$ but this electric potential depends on all particles.

From Gauss' law: $$\frac{\partial E}{\partial x} = \frac{1}{\epsilon}\rho(x,t) = \frac{1}{\epsilon}\sum_n z_nFc_n(x,t)$$ where $$\epsilon$$ is the permitivity and $$\rho$$ is the charge density.

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

Electroneutrality

 * In a solution with some charge, after some time, all charges go to the edges away from each other.
 * $$\tau_r$$ is the relaxation time and is on the order of nanoseconds for physiological salines
 * Similarly, in space, a region around the charge is formed that negates the charge. This is known as the Debye layer and has a thickness of around a nanometer.
 * Thus for times much greater than the relaxation time and distances much greater than the Debye distance, we can assume electroneutrality of the solution.