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 [math]\displaystyle{ \phi_n = -D_n\frac{\partial c_n}{\partial x} }[/math]

Drift

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

Forces	Size scale	Time scale
inertial (F=ma)	[math]\displaystyle{ radius^3 }[/math]	acceleration
viscosity	[math]\displaystyle{ radius }[/math]	velocity

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

For charged particles: [math]\displaystyle{ v=u_nz_neN_AE(x,t)=u_nz_nFE(x,t) }[/math] ([math]\displaystyle{ F=eN_A }[/math] which is Faradya's number) = charge/mole about 96500 C/mol.

[math]\displaystyle{ D_n=u_nRT }[/math]: Einstein's relation

Flux due to drift: [math]\displaystyle{ \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) }[/math]

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

The flux of ions is the current density given by [math]\displaystyle{ J_n = z_nF\phi_n }[/math] 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:

[math]\displaystyle{ J_n = -z_nFD_n\frac{\partial c_n}{\partial x}-u_nz_n^2F^2c_n\frac{\partial\psi}{\partial x} }[/math]

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)

[math]\displaystyle{ \frac{\partial\phi_n}{\partial x} = -\frac{\partial c_n}{\partial t} }[/math] or equivalently

[math]\displaystyle{ \frac{\partial J_n}{\partial x} = -z_nF\frac{\partial c_n}{\partial t} }[/math]

Unlike diffusion, also need one more equation for [math]\displaystyle{ \psi }[/math] but this electric potential depends on all particles.

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

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

Electroneutrality

• In a solution with some charge, after some time, all charges go to the edges away from each other.
• [math]\displaystyle{ \tau_r }[/math] 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.