Physics¶

A note on units¶

The model’s numerical work is done in a nanometer, gram, second unit-space to, well, match the scale of the system. This means that we use conversion factors to get from mks units to our nmgs system. Our unit of force is piconewtons (\(1 g*nm^2/s^2\)). Conversion factors for common units are:

Joules (\(1 kg\,m^2/s^2\)) multiply by \(10^{21}\) to get it in \(pN\,nm\)
Poise (\(1 g/cm\,s\)) multiply by \(10^{-7}\) to get it in \(g/nm\,s\)

Actin diffusion¶

On first glance at a model visualization, it appears that actin is trucking along at a prodigious rate when unbound. I was initially concerned that I had overestimated the diffusive constants for actin filaments. Let’s check to see that they are within reason.

We find the diffusive coefficient for actin by treating it as a cylinder diffusing about its long axis. From page 107 of Howard, 2001 we get that this will produce a drag coefficient of

\[\gamma = \frac{2 \pi \eta L}{\ln(L/2r) - 0.20}\]

We start with \(\eta\) as the viscosity of water, but multiply it by a factor of 3.2 to account for the increased crowding of cytoplasm. \(L\) will vary from filament to filament, but let’s take a short example length of 14 g-actin pairs or approximately \(40 nm\).

From this we can calculate the diffusion constant of our sample actin as

\[D = \frac{k T}{ \gamma }\]

Now, I’m always a bit paranoid about dimension checking, so let’s explicitly calculate the diffusion constant taking units into account. The misu package will help here. Here is a summary of our units

Variable	Value	Units	Source
\(k\)	\(1.38E-23\)	\(J/K\)	1
\(T\)	\(277\)	\(K\)	choice
\(\eta\)	\(0.00365\)	\(Pa\,s\)	2, 3
\(L\)	\(40\)	\(nm\)	choice
\(r\)	\(3\)	\(nm\)	4

import misu as m
import numpy as np

k = 1.38E-23 * m.J * m.K**-1
T = 288 * m.K
eta = 0.00356 * m.Pa_s
L = 40 * m.nanometre
r = 3 * m.nanometre

D = (k * T)/((2 * np.pi * eta * L)/(np.log(L/(2*r)) - 0.20))
print("D is %0.1f microns**2/sec"%(D >> (m.micron**2/m.s)))

D is 7.5 microns**2/sec

That compares very reasonably with an observed diffusion constant of \(5.2 \mu m ^2/s\) for \(40 nm\) long actin fragments in yeast cytosol. We can be reassured that, while our actin diffusion appears to be quite quick, it matches well to observed values.