Diffusion in Cells

Group Meeting, November 14, 2014

Wendell Smith

Motivation

Biology

  • Particles in cells do not simply diffuse
    • Movement does not obey $\left< x^2 \right> \propto t$
    • Highly non-Gaussian

not slope 1!

Motivation

Physics

  • Just how non-gaussian can hard spheres get?
    • At low density, hard spheres are perfectly gaussian
    • Closer to the glass transition, they get somewhat non-gaussian

What does non-Gaussian mean?

  • $\alpha_2 = \frac{\left< x^4 \right> }{3 \left< x^2 \right> } - 1$
    • For a Gaussian distribution, $\left< x^4 \right> = 3\left< x^2 \right>$, so $\alpha_2 = 0$

What does non-Gaussian mean?

  • A random walk (diffusion) is Gaussian at any given time:
    • $P(x, t) = \frac{1}{2 \sqrt{\pi D t}} e^{-\frac{x^2}{4 D t}}$
  • So if the particles aren't diffusing, then $\alpha_2 \neq 0$
    • For caged particles, $\alpha_2 \approx -\frac{1}{5}$

What does non-Gaussian have to do with the MSD?

  • Short answer: its what you can't see in an MSD plot
  • The MSD gives you the mean squared displacement
      It measures the width of particle displacements of time
  • $\alpha_2$ measures how "non-gaussian" the distribution is

Diffusion

$\alpha_2 = 0$

Mixed

$\alpha_2 > 0$

Caging

$\alpha_2 = -\frac{1}{5}$

$\left< x^2 \right>$ and $\left< x^4 \right>$

  • MSD has a classic shape
    • Almost linear for small $\phi$
    • Plateau region for larger $\phi$

  • $\left< x^4 \right>$ looks similar to $\left< x^2 \right>$

  • $N = 100$, monodisperse

$\left< x^2 \right>$ and $\left< x^4 \right>$

Without the time-component

  • Filled area is between
    $3\left< x^2 \right>$ and $\left< x^4 \right>$
  • Area corresponds to $\alpha_2$: $$\alpha_2 = \frac{\left< x^4 \right>}{3 \left< x^2 \right>^2 }- 1$$

$\alpha_2$

  • Goes up to $1.6$, but no higher
  • At higher densities, this is increasingly difficult to measure
  • As density increases, we expect $\alpha_2$ to remain under $1.6$

Crystallization

  • Data shown previously included crystallization
  • May switch to bidisperse

Crystallization

Characterization: Preliminary Data

  • Order parameter $Q_6$ for each simulation as a function of time
    • Each simulation is its own line
    • Dashed lines not included in previous data
    • Some simulations started from the same initial conditions