L3: Random Walks in Biological Systems

mmodels the path of a molecule as it undergoes successive, random steps.

• Steps are independent and random in direction and magnitude.

• underpin diffusion processes in cells.

One Dimensional Random Walk

number line: ← -\delta — 0 — \delta —>

t=0, x(0)=0

v_x

1) once left/right

one step: \delta = ± v_x\tau

• \tau = duration of step

2) P_→=P_←=1/2

successive steps are independent (decrease collisions, decrease concentration, increase reaction vessel)

3) particles do not interact

one particle’s movement: x_i(n) = x_i(n-1) ± \delta

Average position of a molecule: mean squared displacement (MSD):

<x(n)> = 1/N \Sigma^N_i=1 (x_i(n))

= 1/N \Sigma^N_i=1 (x_i(n-1) ± \delta)

= 1/N \Sigma^N_i=1 (x_i(n-1))

= <x(n-1)>

= <x(n+1)>

= 0

• deltas will usually cancel out - average zero displacement

• mean position does not change from step to step

spread: root mean squared displacement (RMSD):

x_i(n) = x_i(n-1) ± \delta

x_i²(n) = x_i²(n-1) ± 2x_i(n-1)\delta + \delta²

<x_i²(n)> = 1/N \Sigma^N_i=1 (x_i²(n-1) ± 2x_i(n-1)\delta + \delta²)

= <x_i²(n-1)> + \delta²

<x_i²(0)> = 0 (by definition)

<x_i²(1)> = <x_i²(n-1)> + \delta² = <x_i²(0)> + \delta² = \delta²

<x_i²(2)> = <x_i²(n-1)> + \delta² = <x_i²(1)> + \delta² = \delta² + \delta² = 2\delta²

^…

<x_i²(n)> = n\delta²

t = n/\tau

n = t/\tau

<x_i²(t)> = (t/\tau)\delta²

= (\delta²/\tau)t

\delta²/2\tau = D (diffusion constant)

<x_i²(t)> = 2Dt

One-dimensional random walk equation

compare: x = vt

• deterministic

  • random walk is always changing = unpredictable

• describes one particle instead of spread

<x_i²(t)>^1/2 = (2Dt)^1/2

lysozyme - D = 10^-5 cm²/s
bacterial cell - 1 nm = 10^-4 cm

<x_i²(t)> = 2Dt

(10^-4 cm)² = 2(10^-5 cm²/s)t

t = (10^-4 cm)²/(2(10^-5 cm²/s))

= 5x10^-4 s

t = (1 cm)²/(2(10^-5 cm²/s))

= 5x10^-4 s

≈ 14 hr

Higher Dimensional Random walk

3D:

<r²> = 2nDt

where n is the number of dimensions

Diffusion

why do things diffuse? concentration gradients

J_{x(_L→R)} = -1/2 [(N(x+\delta) - (N(x))/A\tau

• A = area of wall

= -1/2 (\delta²/\tau\delta)[(N(x+\delta)/A\delta - (N(x)/A\delta)

= - D 1/\delta [C(x+\delta) - C(x)]

as \delta → 0

= -D ∂C/∂x Fick’s First Law

as \tau → 0

= 1/\tau [C(x+\delta) - C(x)]

= ∂C/∂t

as \delta → 0 and \tau → 0

∂C/∂t = D ∂²C/∂x² Fick’s Second Law

Key Takeaways

microscopic view - individual molecule stake random steps in a fluid

macroscopic view - collective behavior of many particles obey’s Fick’s Law

Brownian motion - concentration-driven diffusion

• signalling molecules

• drug delivery

• neurotransmitter diffusion

• fluorescence microscopy and protein diffusion