Basic Physical Scales Relevant to Cells and Molecules 

Relevant Length Scales

Cells are roughly microns in size (1 micron = 10-3 millimeters = 10-6 meters)

E. coli cells are usually around 1 x 1 x 3 microns
Yeast cells are little balls around 2 microns in diameter
Most human cells are roughly 20 microns across, with roughly nuclei about 10 microns in diameter.
Newt cells have 40 micron-diameter nuclei, and the cells are around 60 microns across.

Specialized cells in eukaryotes can be much longer (the long axons of neuron cells can be a meter or so in length), or larger (eggs of amphibians and birds are good examples)

The basic length scale used to describe molecules is a nanometer (1 nanometer = 1 nm = 10-9 meters)

Many biomolecules are made out of chemical units (e.g. nucleotides, amino acids) which are around 1 nm in size, meaning that they are a few atoms across.  Other `small' molecules such as lipids and sugars are also roughly on this length scale.  Most folded up proteins are a few nm in diameter.  So, the nanometer is really useful as a basic yardstick of biomolecules.
Another commonly used unit of length is the Angstrom (1 Angstrom = 10-10 m = 0.1 nanometer).  Atoms are roughly Angstroms in size (a hydrogen atom is about 1 A in diameter, a carbon atom is about 2 A in diameter).  You might read about Angstroms, and you should just immediately think of them as a tenth of a nanometer.

What is remarkable about biomolecules is that they can be made of many nm-size units, strung together to make long, linear polymers.

For example, the genetic DNAs in human cells are roughly 108 nucleotides in length; each nucleotide contributes a fraction of a nm, making the whole DNA a few centimeters long!

The wavelength of light is a bit shorter than one micron

Visible light ranges has a wavelength ranging from 650 nm (red) through 350 nm (violet).  As you know from your introductory physics courses, the light microscope can only be used to image details in cells down to at the very best, about 1/2 of the wavelength being used.  This means that we can't directly observe the structure of cells at scales smaller than about 100 nm.

In practice things are usually worse - usual white-light microscopes can't resolve detail smaller than about 250 nm, and heroic measures must be taken to observe with 100 nm detail.

This is important since it implies that all information about the molecular-scale operation of live cells must be gathered indirectly.

One important tool for looking at cells with higher resolution is the electron microscope, which uses electrons with sub-nm wavelengths to image at down to sub-nm resolution.   Unfortunately at present, the electron microscope only works on samples in vacuum, and therefore on dead (and usually `fixed' or cross-linked) cells.   In any case, because electrons scatter very strongly from water, the electron microscope can't be even conceivably be used at present to look into cells at the depths needed to observe their workings.

A way to image inside live cells at  10-nanometer spatial resolution would be an extremely useful tool for someone to invent.


Thermal Energy

Each microscopic degree of freedom has an energy kBT associated with it

At room temperature, molecules are jiggling around continually due to thermal motion.  In cells, everything is surrounded by water and so everything is being bumped continuously by neighboring molecules.  All of this random motion gives rise to diffusion of individual molecules, which will be one of the topics discussed in detail later.

The energy associated with a single molecular degree of freedom, e.g. the linear motion of a molecule, or the energy of stretching of a chemical bond, is a fundamental physical quantity:

    kBT = (Boltzmann's constant) x  (absolute temperature)

Here Boltzmann's constant is kB = 1.38 x 10-23 Joules/Kelvin, and is a fundamental constant determined experimentally.

Remember that room temperature (25 C) in absolute terms is around 300 Kelvin (25 + 273.1 = 298.1 K to be more precise).

So  kBT = 4.1 x 10-21 J  is the relevant thermal energy of single molecular degrees of freedom (note that there is not much change over the range from around 270 K to 330 K relevant to most living things).

So now we can roughly estimate the velocity with which a water molecule is moving in a glass of water (or in a cell) at 300 K.
Between collisions, a water molecule has kinetic energy which will be about the thermal energy:

m v
2
= kB T
 
which means that the velocity will be about
v =  æ 
ç 
è 
2 kB
ö 
÷ 
ø 
1/2  

 

æ 
ç 
è 
2 ×4 ×10-21     J 
18 ×1.6 ×10-27     kg 
ö 1/2 
÷ 
ø 
=   530    m/sec
 
Make sure you understand where the factor of 18 comes from.

Relevant Time Scales

The time between collisions of small molecules in a liquid is about a picosecond

The preceeding calculation puts us in good shape to understand a basic time scale associated with water and other small molecules - the typical time between successive collisions of a water molecule with its neighbors.   We simply have

D
D
= v 
 
 where  Dx = 0.2 nm is the rough size of a water molecule, and where v = 500 m/sec.  Rearranging and solving for the time interval gives Dt = 0.4 x 10-12 sec, about half a picosecond.

Over times longer than one collision time, molecules undergo random-walk or diffusive motion

After one collision, the direction of our water molecule will be changed in an difficult-to-predict way.  After a few collisions there will be no chance to predict its direction of motion, or even where it is.  This process is called diffusion, and we will discuss it in detail later.  Diffusive motion is totally different from the straight-line motion we just considered.  We'll see that a diffusing molecule has a relation between distance covered Dx and time interval Dt which is

(Dx)
D
= D
 
where D is the diffusion constant, with units length-squared per time, for that molecule.  For a water molecule in water we can estimate its diffusion constant very roughly as
D =  (0.2 nm)
0.4 x 10-12 sec 
= 1 x 10-3  (cm)2/sec 
 
A key point about diffusion is that the time you have to wait to go a distance Dx grows as the square of that distance.    So diffusion is increasingly slow at longer and longer distances.

Similarly, motion of larger molecules (e.g. large proteins or nucleic acids) occurs essentially by diffusion, in the absence of other forces.  Biomolecules which are flexible additionally undergo random shape changes.  We'll see that even moderately long molecules - a few thousand repeat units - can take microseconds to milliseconds to move.

Relevant time scales for us will include the time that it takes chemical reaction to occur, which if there are large energy barriers to cross, can be seconds or longer. 


Stored Energy in Cells

Cells don't rely on thermal energy or thermal motion to move molecules around.  Instead, they harvest energy from their surroundings, and then store it until it is needed.  One of the molecules used to store energy for the short term is ATP, which can be converted to ADP in an exothermic reaction which liberates about 20 kBT under cellular conditions.   Very roughly ATP -> ADP conversion steps are used to liberate energy to drive other chemical reactions in the cell.

So - cells use units of energy which are appreciably larger than single thermal excitations.  This is crucial to avoid having thermal motion simply bring the contents of a cell to thermal equilibrium - meaning death.


Forces in Cells

A force is developed when work (with a change in energy) is done over a distance:
f = DE 
Dx
 
We can use this to estimate the forces experienced by molecules during thermal fluctuations, and during biological processes.

Generally we will be worried about energies on the order of a kBT, and distance over which such energies are transferred of a nm.  So we can see that the rough forces that we will be talking about will be

f = 4x 10-21 J 
10-9 m
= 4 x 10-12 N = 4 piconewtons = 4 pN
 
For random thermal motions, we already saw that the energy of a water molecule can change by a fraction of kBT over a distance comparable to its size of 0.1 nm, so water molecules are subject to random forces that instantaneously may be a few tens of pN.

For directed motions generated by e.g. burning of ATP in the cell, the steps will be a nm or so, and the energy used per step will be a few kBT, again giving forces in the few pN range.

So at the molecular scale, we will be worried about forces of around a pN.