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  • 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)
  • [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