https://burrito.bio/essays/what-limits-a-cells-size

Biology is a Burrito *

A View of the Cell

Why Are Cells Small?

Two physical constraints -- surface area and diffusion -- explain why
cells are so small.

By Niko McCarty, Fellow at Astera Institute

A human body is built from 30 trillion cells -- excluding microbes --
that each arise from a lone, fertilized egg. These cells come in a
multiplicity of shapes and sizes, with internal volumes spanning five
orders of magnitude. The smallest human cell, a sperm, has a volume
of just 30 um3, whereas an oocyte has a volume of 4,000,000 um3,
making it the largest cell in the human body.^1

What accounts for this huge range? A simplistic answer is that
evolution has made each cell the size best suited to its function.
Maybe sperm are small because the body needs to make many of them,
and tiny cells cost less energy to make. (Sperm consist of little
more than DNA and a few mitochondria, which are necessary for
providing energy to spin their whip-like tails.) By contrast, an
oocyte needs massive reserves of mitochondria and nutrients to
support early embryonic growth. In short, every cell is as large or
small as it needs to be -- within reason.

But we can derive far more satisfying answers from physics.

The first major limit on a cell's size is its surface area-to-volume
ratio. Assuming that a cell is roughly spherical in shape, its
internal volume grows proportionally to the cube of its radius,
whereas its surface area grows proportionally to the square of that
radius. In other words, a cell's volume grows much faster than its
surface area.

This ratio has big consequences for cell survival. The cell's
membrane funnels nutrients into the cell and secretes waste. It's
also where the energy in a prokaryotic cell -- like E. coli -- gets
made. If the interior grows too large relative to the membrane, the
cell will not be able to produce enough energy or excrete waste
quickly enough to maintain all the 'stuff' inside, and metabolism
will slow down.

A second constraint is diffusion, or the tendency for molecules to
migrate from areas of high concentration to areas of lower
concentration. This migration dictates how quickly enzymes find
substrates, or how signaling molecules reach receptors, and how often
ribosomes collide with messenger RNAs. Inside a cell, nearly
everything happens by chance encounters amongst molecules! As a
cell's volume grows, though, the chance that these encounters will
happen decreases (assuming the total numbers of molecules stay
constant).^2

A molecule's diffusion rate changes based on various factors. The
cytoplasm is extremely crowded, for example, and so molecules spend
lots of time ricocheting off obstacles, delaying their arrival at a
distant location. Every protein in a cell collides with about 10
billion water molecules per second on average. The vast majority of
proteins in a bacterium have diffusion coefficients of only 5 to 10
um^2 per second (a measure of how quickly molecules spread through
space). Some molecules also aggregate or stick to charged surfaces,
further slowing their movement.^3 In general, large molecules diffuse
slower than small ones.

Metabolites in E. coli can diffuse from one side of the cell to the
other in milliseconds, which means collisions -- and cellular outcomes
-- happen quickly. A typical protein takes just 0.01 seconds to
traverse a bacterium's diameter (about 1 micrometer), but the same
protein would take around four minutes to move one millimeter and
more than six hours to move one centimeter. This is, in part, why
cells are so tiny.

With these constraints in mind, we can begin to speculate as to why
various cells are shaped the way they are.

Red blood cells are tiny and shaped like biconcave discs to aid with
diffusion; by abandoning a spherical shape and evolving more toward a
'donut,' they increase their surface area without compromising their
compact volume. This, in turn, enhances their ability to exchange
oxygen with cells in the body. Their small size (just 8 micrometers
across) also helps them move through narrow capillaries.

Simulating Diffusion

Cells must make tradeoffs between many different constraints, from
energy demands to volume to surface area and nutrient import. It can
feel overwhelming to track all these things at once! Fortunately, we
don't even need to appeal to complicated mathematics to wrap our head
around these constraints, and why cells make the decisions they do.

Everything in biology runs on collisions between molecules. The
"magic" of life cannot happen without these collisions. So let's
imagine the simplest possible scenario: a cell with one molecule and
one target. Hit the "Run" button to see how long it takes for the two
to collide.

Of course, real cells are not empty! Every molecule in every cell
collides with billions of water molecules per second, not to mention
all the RNA, DNA, and proteins floating around. Below is the same
simulation, but now there are barriers between the molecule and its
target. Run the simulation to see how this increases the time it
takes for the two to collide.

A bigger cell can do more things than a small one. It can carry more
proteins, which means it can execute more functions and do more
complex behaviors. But this bigger size also comes with tradeoffs. If
the cell grows too large, its surface area won't be able to keep up;
it may not be able to import the nutrients (and export the waste)
needed to maintain the larger volume. As the cell grows, the average
distance between a molecule and its target also expands, meaning
collisions take longer.

A large cell can speed up "collision" times by increasing the number
of molecules! More molecules bouncing around the cell means there's a
higher chance that one of them will find the target quickly, such
that the magic of life can proceed. But this is not a perfect
solution, since making more molecules requires that the cell burns
more energy, which demands more nutrients, which is limited by
surface area! So always remember: A cell must carefully balance
tradeoffs between its volume, diffusion, surface area, and energy.

In contrast, oocytes can grow so large, in part, because they are
less metabolically active than other types of human cells -- and thus
don't depend so much on random collisions. They stockpile nutrients
during oogenesis to wait out fertilization. Eukaryotic cells also
grow large, in general, because they've evolved compartmentalization;
by modularizing specific functions into organelles, they bring
molecules closer together to help get the job done.

Cell sizes are not fixed, however, even within a single species.
Cells often swell as they increase their production of proteins and
metabolites in preparation for division. This is in line with
biology's only rule: namely, there are exceptions to every rule!

Case in point: a giant bacterium called Thiomargarita magnifica can
extend about one centimeter in length, so large that it can be seen
by the naked eye. It does so by breaking the surface area-to-volume
rule, filling between 65-80 percent of its internal volume with an
empty vacuole. In other words, it pushes most of its molecules to the
cell periphery, thus shortening diffusion distances.^4

Thiomargarita magnifica, a giant bacterium visible to the naked eye
Thiomargarita magnifica is a bacterial species that can extend about
one centimeter in length, several orders of magnitude more than E.
coli. These microbes are visible to the naked eye. Credit: Jean-Marie
Volland Bubble algae, or Valonia ventricosa, a giant single-celled
organism Bubble algae (aka Valonia ventricosa). Credit: Trident's
Cove

Despite their variety, these architectures still hinge on molecules
bumping into each other, guided by the immutable laws of physics. Or,
as D'Arcy Wentworth Thompson mused in On Growth and Form (1917), "The
form of an object is a 'diagram of forces.'" Cells bear witness to
both internal and external forces; they are constrained by diffusion
and shaped by the delicate trade-off between volume and surface area.

1 In other words, the egg is more than 100,000-times larger than the
sperm that swims to it. But it's difficult to wrap our heads around
such large numbers, and so it's often helpful to rescale entities
into everyday objects that we can more easily imagine. If a sperm
cell were 'blown up' to the size of a glass marble, for example, then
an oocyte, scaled in the same way, would be roughly the size of a
modern refrigerator. 2 Obviously, this assumption is not true. As
cells grow, they make more molecules to keep collision times low. But
it costs energy to make more molecules, and this is a tradeoff I
return to below. 3 Several experiments have shown that molecules move
more slowly near the densely-packed nucleoid, for example, and that
these location-dependent diffusion rates are affected by things like
charges between proteins. 4 So-called 'bubble algae,' or sea grapes,
do something similar. They are the largest single-celled organism, up
to 10,000-times larger than an average microbe (and visible to the
naked eye!) The bubble algae achieve this by filling up 95% of their
internal volume with a vacuole. The vacuole is filled with big sugar
chains, which the cell uses to repair damage on its cell extremities.
Each cell also has dozens or hundreds of nuclei, pressed tightly
against the walls.

Notes

 1. | In other words, the egg is more than 100,000-times larger than
    the sperm that swims to it. But it's difficult to wrap our heads
    around such large numbers, and so it's often helpful to rescale
    entities into everyday objects that we can more easily imagine.
    If a sperm cell were 'blown up' to the size of a glass marble,
    for example, then an oocyte, scaled in the same way, would be
    roughly the size of a modern refrigerator.
 2. | Obviously, this assumption is not true. As cells grow, they
    make more molecules to keep collision times low. But it costs
    energy to make more molecules, and this is a tradeoff I return to
    below.
 3. | Several experiments have shown that molecules move more slowly
    near the densely-packed nucleoid, for example, and that these
    location-dependent diffusion rates are affected by things like
    charges between proteins.
 4. | So-called 'bubble algae,' or sea grapes, do something similar.
    They are the largest single-celled organism, up to 10,000-times
    larger than an average microbe (and visible to the naked eye!)
    The bubble algae achieve this by filling up 95% of their internal
    volume with a vacuole. The vacuole is filled with big sugar
    chains, which the cell uses to repair damage on its cell
    extremities. Each cell also has dozens or hundreds of nuclei,
    pressed tightly against the walls.

Feedback

Found an error or have a question? I update these essays based on
your feedback. If you send a comment that improves the essay, I'll
publish your name on this website and also in a forthcoming, physical
book. Please write to me at nsmccarty3@gmail.com.

This essay was improved by:

-- A., Ben Andrew, Ashish Uppala

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Part of Biology is a Burrito (& Other Essays) * Written by Niko
McCarty