Chapter 2 – Orientation for the Bio-Curious  49

plants, again the forces at the base of a giant sequoia tree are in excess of 1 × 10⁷ N. For

humans, the typical body weight is several hundred newtons, so this total force can clearly be

exerted by muscles in legs. However, such macrolength scale forces are obviously distributed

throughout the cells of a tissue. For example, in muscle, the tissue may be composed of 10–​20

muscle fibers running in parallel, each of which in turn might be composed of ~10 myofibrils,

which are the equivalent cellular level units in muscle. With the additional presence of con­

nective tissue in between the fibers, the actual cellular forces are on the order of sub-​newtons.

This level of force can be compared with that required to break a covalent chemical bond

such as a carbon–​carbon bond of ~10⁻⁹ N, or a weaker noncovalent bond such as those of an

antibody binding of ~10⁻¹⁰ N.

At the lower end of the scale are forces exerted by individual molecular machines, typically

around the level of a few multiples of 10⁻¹² N. This unit is the piconewton (pN), which is often

used by biologists. The weakest biologically relevant forces in biology are due to random

thermal fluctuations of surrounding water-​solvent molecules, which is an example of the

Langevin force (or fluctuation force), depending upon the length and time scale of observa­

tion. For example, a nucleus of diameter ~10⁻⁶ m observed for a single second will experi­

ence a net Langevin force of ~10⁻¹⁴ N. Note that in biology, gravitational forces of biological

molecules are normally irrelevant (e.g., ~10⁻¹⁷ N).

One of the key attractive forces, which are essential to all organisms, is that of covalent

bonding, which allows strong chemical bonds to form between atoms, of carbon in par­

ticular. Covalent bonds involve the sharing of pairs of electrons from individual atomic

orbitals to form stable molecular orbitals. The strength of these bonds is often quantified

by the energy required to break them (as the bond force integrated over the distance of

the bond). The bond energy involving carbon atoms typically varies in a range of 50–​150

kBT energy units.

Cells and biomolecules are also affected by several weaker noncovalent forces, which can

be both attractive and repulsive. An ideal way to characterize a vector force F is through the

grad function of the respective potential energy landscape of that force, U:

(2.7)



F

U

= −∇

Electrostatic forces in the water-​solvated cellular environment involve layers of polar water

molecules and ions, in addition to an electrical double layer (EDL) (the Gouy–​Chapman

layer) composed of ions adsorbed onto molecular surfaces with a second more diffuse weakly

bound to counter charges of the first layer. The governing equation for the electrostatic poten­

tial energy is governed by Coulomb’s law, which in its simplest form describes the electric

potential Ve due to a single point of charge q at a distance r:

(2.8)

V

q

r

m

e ⁼ 4

0

πε ε

where

ε0 is the electrical permittivity in a vacuum

εm is the relative electrical permittivity in the given medium

Therefore, for example, for two charges of equal magnitude q but opposite sign, separated by

a distance d, each will experience an attractive electrostatic force Fe toward the other parallel

to the line that joins the two charges of

(2.9)

F

q

r

m

e ⁼

2

0

2

4πε ε

For dealing with multiple electrical charges in a real biological system, a method called “Ewald

summation” is employed, which treats the total electrical potential energy as the sum of

short-​range and long-​range components. The usual way to solve this often complex equation