Meaning and mechanism
The surface of the cell membrane is a boundary for exchanging with the outside, and the internal volume is a space that needs to be supplied. The two quantities do not increase at the same rate according to size.
If you increase the length of one side of a cube by a factor of two, its surface area becomes four times larger, and its volume becomes eight times larger. The surface area corresponding to one unit of volume is halved.
Relationships and conditions
The surface area to volume ratio [SA/V] of a cube with side length a is 6/a. The representative distance for one-directional diffusion is √(2Dt)), so the time required to move the same distance doubled is four times as long.
Real cells utilize compartmentalization, membrane wrinkles, and active transport and internal flow. It is not possible to determine the maximum size of all cells using only simple size models.
What to observe in the simulation
Smaller cubes have a larger surface area/volume and reach a larger volume ratio with the same depth of diffusion. The penetration rate is 1−((a−2d)/a)³ and is 100% if a≤2d.
It is a model that penetrates an agar cube uniformly from six sides with a constant diffusion coefficient. A sharp boundary and d=√(2Dt) are approximate. The actual concentration distribution, cell membrane transport, metabolism, and spherical cell geometry are omitted. The color area ratio of the cross section and the three-dimensional penetration volume ratio are different.
Key relationships and symbols
d = √(2Dt) · SA/V = 6/a
| Symbol | Meaning | Unit |
|---|---|---|
| a | Side length of the model cell | m |
| D | Diffusion coefficient | m²/s |
| SA/V | Surface-area-to-volume ratio | m⁻¹ |
Worked example
What is the penetration volume ratio when a=2 cm and penetration depth d=0.5 cm?
The remaining central side is 2−2×0.5=1 cm. The uninfiltrated volume ratio is 1³/2³=1/8, so the infiltration ratio is 87.5%.
Common misconceptions
- The surface area does not necessarily become better than the volume in terms of exchange conditions simply because it has increased.