Why do cells divide?

Geometry Level 3

Why do cells divide? Turns out it is simply a matter of mathematics. A cell can only survive if the rate of change of materials with the surrounding medium is "big" so it can absorb nutrients and excrete waste fast enough. The rate of change, of course, depends on the surface area of the cell. The amount of nutrients needed for survival and waste produced depend on the volume of the cell. Now, if we think of a cell as a sphere, as it gets bigger the volume will increase faster than the surface area. Eventually so much waste will be produced and not enough nutrients will be absorbed and the cell must decide what to do: either it stops growing or it divides.

Suppose a spherical cell must divide. When it does, no nutrient should be lost, and therefore the "daughter-cells" must add up to the same volume as the "mother-cell". Suppose also that when a cell devides, it devides into two identical cells. The ratio of surface area to the volume of the daughter-cells can be written as

$x \frac{A}{V},$

where $A$ and $V$ are the surface area and volume of the mother-cell, respectively. Find $x$ .

The answer is 1.26.

1 solution

A Former Brilliant Member
May 4, 2020

Let the radius of the mother cell be $R$ and of each daughter cell be $r$ .

Then $R^3=2r^3\implies R=2^{\frac{1}{3}}r, \dfrac{A}{V}=\dfrac{3}{R}$ .

The ratio of area to volume of each daughter cell is

$\dfrac{3}{r}=\dfrac{3}{R}\times \dfrac{R}{r}=2^{\frac{1}{3}}\dfrac{A}{V}$ .

Hence, $x=2^{\frac{1}{3}}\approx \boxed {1.2599210498949}$ .

Why do cells divide?

The answer is 1.26.

1 solution

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