# 6.021/Notes/2006-09-13

## Diffusion

• Non-linear relationship between space and time is non-intuitive

### Diffusion applied to cells

• Membrane diffusion
• The membrane is around 10nm whereas the cell is about 10μm
• Can treat as 1D diffusion (diffusion across membrane ignoring other dimension)
• Reference direction for flux: positive is out of cell

### Dissolve and Diffuse model

1. solute outside dissolves into membrane
2. solute diffuses through membrane
3. solute dissolves into cytoplasm
4. concentration of solute in cell increases
• Assume dissolving is faster than diffusion (assume dissolving is instant)
• $\displaystyle{ c_n^i }$: concentration inside of solute
• $\displaystyle{ c_n^o }$: concentration outside of solute

### Dissolve model

• Membrane is like oil, cytoplasm and outside bath is like water
• Some solutes like oil, some like water
• Find relative solubilities of solute n in oil and water
• Partition coefficient $\displaystyle{ k_{oil:water}=\frac{c_n^{oil}}{c_n^{water}} }$ (at equilibrium)

### Diffusion in membrane

• Difficult to solve analytically but numerically easy
• From point of view of membrane, both inside and outside baths are constant
• If wait long enough (reach equilibrium), the concentration will become flat in membrane
• But short term, will be a straight line
• How long to straight line?
• Membrane width d
• Can estimate it. For half of particles to cross membrane is $\displaystyle{ t=d^2/D }$ but this is overestimate as don't need that many particles to cross membrane. For midway is $\displaystyle{ t=d^2/(4D) }$
• Exact solution: $\displaystyle{ \tau_{ss}=\frac{d^2}{\pi^2 D} }$ (steady state time constant for membrane)

### Solute enters cell

• $\displaystyle{ \phi_n(t)=P_n(c_n^i(t)-c_n^o(t)), P_n=\frac{D_nk_n}{d} }$
• $\displaystyle{ P_n }$: permeability of membrane to solute n
• concentration in cell changes: 2 compartment diffusion
• assume volumes constant, baths are well-stirred, membrane is thin (ignore solute in membrane), and membranes always in steady state
• $\displaystyle{ c_n^i(t)V_i + c_n^o(t)V_o = N_n }$ (total amount of solute is conserved)
• $\displaystyle{ \phi_n(t)=P_n(c_n^i(t)-c_n^o(t)) }$
• Solution to equations: $\displaystyle{ c_n^i(t)=c_n^i(\infty)+(c_n^i(0)-c_n^i(\infty))e^{-t/\tau_{eq}} }$
• $\displaystyle{ c_n^i(\infty) = \frac{N_n}{V_i+V_o}, \tau_{eq}=\frac{1}{AP(1/V_i+1/V_o)} }$

### Check assumption dissolving is fast

• 2 time constants
• $\displaystyle{ \tau_{ss}=\frac{d^2}{\pi^2 D}, \tau_{eq}=\frac{1}{AP(1/V_i+1/V_o)} }$
• For cell $\displaystyle{ V_o\gg V_i }$ so $\displaystyle{ \tau_{eq}=\frac{V_i}{AP} }$
• Assume spherical cell, $\displaystyle{ r=10\mu{\rm m}, d=10{\rm nm}, k=1 }$
• $\displaystyle{ \frac{\tau_{eq}}{\tau_{ss}}=\frac{V_i}{AP}\frac{\pi^2D}{d^2} \approx 3\frac{r}{d} \approx 10^3 }$
• Assumption that $\displaystyle{ \tau_{eq} \gg \tau_{ss} }$ is ok