Sensora: Berg–Purcell Limit
Physics of Life & Earth arc, page 191 of 193. Back to The Pixelated Physics Universe
Who they are
Sensora listens to a drizzle of molecules landing on her skin, counting every one to learn how much is out there.
Sensora is a character of the Physics of Life & Earth arc: drawn in pixels, ruled by one piece of real physics, and never allowed to break it.
The physics
How precisely can a cell measure the concentration c of a chemical around it? Molecules arrive by diffusion one by one, so the count is noisy. Berg and Purcell showed in 1977 that a cell of radius a, averaging for a time T, can do no better than a fractional error of order 1/√(Dac T). Listen longer, or be bigger, to know better. Their paper also argued that a modest number of receptors spread over the surface captures most of the possible signal, and bacteria such as E. coli sense close to this physical limit.
$$ \begin{gathered} \frac{\delta c}{c} \sim \frac{1}{\sqrt{D\,a\,c\,T}} \end{gathered} $$
| Symbol | Meaning | SI unit |
|---|---|---|
| $\delta c / c$ | fractional error in the measured concentration | 1 |
| $D$ | diffusion coefficient of the molecules | m²/s |
| $a$ | cell size (radius) | m |
| $c$ | concentration | 1/m³ |
| $T$ | averaging (listening) time | s |
Honest caveat: This is a scaling law (“~”, not “=”); the careful “perfect instrument” result adds a numerical factor. Bacteria come close to the limit, not exactly to it.
Sources
- Berg & Purcell (1977) Physics of chemoreception, Biophys. J. 20, 193
- Bialek & Setayeshgar (2005) Physical limits to biochemical signaling, Proc. Natl. Acad. Sci. USA 102, 10040
Crossovers
Sensora meets Radicane (Poisson Counting Statistics). Sensora counts molecules arriving by diffusion. Radicane's √N statistics set the Berg–Purcell limit on how well a cell can know the concentration.
Sensora meets Flagella (Corkscrew Escape). Flagella swims toward food, and Sensora decides which way is better. A bacterium's chemotaxis needs both the corkscrew and the counting.
Wear the lore
Concept keywords: Berg–Purcell Limit, Sensora, Physics of Life & Earth, physics lore, Pixelated Physics