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2.5  Physical Quantities in Biology

is to use the particle mesh Ewald method (PME method), which treats the short-​range and

long-​range components separately in real space and Fourier space (see Chapter 8).

Van der Waals forces (dispersion-​steric repulsion), as discussed, are short-​range steric

repulsive potential energy with 1/​r¹² (distance r) dependence fundamentally due to the Pauli

exclusion principle in quantum mechanics. The exclusion principle disallows the overlap of

electron orbitals. When we combine this short-​range repulsive component with a longer-​

range attractive component from interactions with nonbonding electrons inducing electrical

dipoles, a 1/​r⁶ dependence, we get the so-​called “Lennard–​Jones potential,” which is also

referred to as the L-​J, 6-​12, and 12-​6 potential) UL-​J:

(2.10)

U

A

r

B

r

L J

−⁼

−

12

6

Here, A and B are the constants of the particular biological system.

Hydrogen (or H–​) bonding, already referred to, is a short-​range force operating over

~0.2–​0.3 nm. These are absolutely essential to forming the higher-​order structures of

many different biological molecules. The typical energy required to break an H-​bond

is ~5kBT.

Hydrophobic forces are largely entropic based resulting from the tendency of nonpolar

molecules to pool together to exclude polar water molecules. There is no simple law to

describe hydrophobic forces, but they are the strongest at 10–​20 nm distances, and so are

generally perceived as long range. Hydrophobic bonds are very important in stabilizing the

structural core of globular protein molecules.

Finally, there are Helfrich forces. These result from the thermal fluctuations of cell

membranes due to random collisions of solvent water molecules. They are a source entropic

force, manifest as short-​range repulsion.

2.5.2  LENGTH, AREA, AND VOLUME

At the high end of the biological length scale, for single organisms at least, is a few tens of

meters (e.g., the length of the largest animal is the blue whale at ~30 m, the largest plant is

the giant sequoia tree at almost 90 m in height). Colonies of multiple organisms, and whole

ecosystems, can clearly be much larger still. At the low end of the scale are single biological

molecules, which are typically characterized by a few nanometers (unit nm or 10⁻⁹ m; i.e., 1

m/​1000 million) barring exceptions such as filamentous biopolymers, like DNA, which can

be much longer. Crystallographers also use a unit called the “Angstrom” (Å), equal to 10⁻¹⁰

m, since this is the length scale of the hydrogen atom diameter, and so typical covalent and

hydrogen bond lengths will be a few Angstroms.

Surface area features investigated in biology often involve cell membranes, and since

the length scale of cell diameter is typically a few microns (μm), the μm² area unit is not

uncommon. For volume, biochemistry in general refers to liter units (L) of 10⁻³ m³, but typical

quantities for biochemical assays often involve volumes in the range 1–​1000 μL (microliters),

though more bulk assays potentially use several milliliters (mL).

2.5.3  ENERGY AND TEMPERATURE

Molecular scale (pN) forces integrated over with nanometer spatial displacements result in

an energy scale of a few piconewton nanometers. The piconewton nanometer unit (pN nm)

equals 10⁻²¹ J.

Organisms have a high temperature, from a physics perspective, since quantum energy

transitions are small relative to classical levels, and so the equipartition theorem, that each

independent quadratic term, or degree of freedom, in the energy equation for a molecule in

an ensemble at absolute temperature T has an average energy kBT/​2. A water molecule has