6.021/Notes/2006-09-13
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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
- solute outside dissolves into membrane
- solute diffuses through membrane
- solute dissolves into cytoplasm
- concentration of solute in cell increases
- Assume dissolving is faster than diffusion (assume dissolving is instant)
- [math]\displaystyle{ c_n^i }[/math]: concentration inside of solute
- [math]\displaystyle{ c_n^o }[/math]: 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 [math]\displaystyle{ k_{oil:water}=\frac{c_n^{oil}}{c_n^{water}} }[/math] (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 [math]\displaystyle{ t=d^2/D }[/math] but this is overestimate as don't need that many particles to cross membrane. For midway is [math]\displaystyle{ t=d^2/(4D) }[/math]
- Exact solution: [math]\displaystyle{ \tau_{ss}=\frac{d^2}{\pi^2 D} }[/math] (steady state time constant for membrane)
Solute enters cell
- [math]\displaystyle{ \phi_n(t)=P_n(c_n^i(t)-c_n^o(t)), P_n=\frac{D_nk_n}{d} }[/math]
- [math]\displaystyle{ P_n }[/math]: 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
- [math]\displaystyle{ c_n^i(t)V_i + c_n^o(t)V_o = N_n }[/math] (total amount of solute is conserved)
- [math]\displaystyle{ \phi_n(t)=P_n(c_n^i(t)-c_n^o(t)) }[/math]
- Solution to equations: [math]\displaystyle{ c_n^i(t)=c_n^i(\infty)+(c_n^i(0)-c_n^i(\infty))e^{-t/\tau_{eq}} }[/math]
- [math]\displaystyle{ c_n^i(\infty) = \frac{N_n}{V_i+V_o}, \tau_{eq}=\frac{1}{AP(1/V_i+1/V_o)} }[/math]
Check assumption dissolving is fast
- 2 time constants
- [math]\displaystyle{ \tau_{ss}=\frac{d^2}{\pi^2 D}, \tau_{eq}=\frac{1}{AP(1/V_i+1/V_o)} }[/math]
- For cell [math]\displaystyle{ V_o\gg V_i }[/math] so [math]\displaystyle{ \tau_{eq}=\frac{V_i}{AP} }[/math]
- Assume spherical cell, [math]\displaystyle{ r=10\mu{\rm m}, d=10{\rm nm}, k=1 }[/math]
- [math]\displaystyle{ \frac{\tau_{eq}}{\tau_{ss}}=\frac{V_i}{AP}\frac{\pi^2D}{d^2} \approx 3\frac{r}{d} \approx 10^3 }[/math]
- Assumption that [math]\displaystyle{ \tau_{eq} \gg \tau_{ss} }[/math] is ok