Chapter 2 – Orientation for the Bio-Curious  53

The concentration of protons [H^+​] is one of the most important measures in biology,

though in practice, as discussed, a proton in a liquid aqueous environment, such as inside a

cell, is generally coupled through hydrogen bonding to a water molecule as the hydronium/​

hydroxonium ion H3O^+​. As discussed, the normal biological representation of proton con­

centration is as −log10[H^+​], referred to as the pH, with neutral pH =​ 7, acids <7, and bases (or

alkalis) >7 assuming [H^+​] is measured in M units.

2.5.6  MOBILITY

The wide speed range of different biological features obviously reflects the broad time scale of

the process in life. At the molecular end of the biology length scale, a key speed measure is that

of the translocation of molecular machines, which is typically in the range of a few microns

per second, μm s⁻¹. In the cellular regime, there are motile cells, such as self-​propelling bac­

teria that swim an order of magnitude faster. And then at the whole organism scale, there are

speeds of more like meters per second, m s⁻¹.

To characterize the net flow of matter due to largely random motions, we talk about

diffusion, which has a dimensionality not the same as that of speed, which is [L]/​[T], but

rather of [L]²/​[T], and so conceptually more equivalent to rate at which an “area” is explored

in a given time. For purely random-​walk behavior of particles, we say that they are exhibiting

Brownian diffusion (or normal diffusion). The effective Brownian diffusion coefficient of a

biomolecule, D, assuming free and unrestricted diffusion, relates to the variation of effective

frictional drag coefficient, γ, through the Stokes–​Einstein relation of

(2.11)

D

k T

=

B

γ

For the simple case of a sphere of radius r diffusing in a fluid of viscosity η (sometimes

referred to more fully as the “dynamic” viscosity, to distinguish it from the “kinematic”

viscosity, which is the dynamic viscosity divided by the fluid density), γ is given by 6πηr.

This is often a good approximation for a globular-​like protein diffusing in the cytoplasm,

however different biomolecules in different environments need to be approximated with

different shape factors (for example, integrated membrane proteins in a lipid bilayer will

typically rotate rapidly over a microsecond timescale or faster perpendicular to the plane

of the membrane, and so the effective shape when averaged over a timescale of millisecond

or more, which is appropriate for typical light microscopy sampling, is closer to a cylinder

perpendicular to the membrane).

In the watery part of the cell, such as the cytoplasm, values of D of a few μ²m s⁻¹ are typical,

whereas for a molecule integrated into phospholipid bilayer membranes (30% of all proteins

come into this category), the local viscosity is higher by a factor of 100–​1000, with resultant

values of D smaller by this factor. The theoretical mean squared displacement 〈R²〉, after a

time t of a freely diffusing particle in n-​dimensional space (e.g., in the cytoplasm n =​ 3, in the

cell membrane n =​ 2, for a molecular motor diffusing on track, for example, a kinesin mol­

ecule on a stiff microtubule filament track, n =​ 1) is given by

(2.12)

〈

〉=

R

nDt

2

2

But note, in reality, blurring within experimental time sample windows as well as detection

precision error leads to a correction for experimental measurements, which involve single

particle tracking, and also note that in general there can be several other more complex

modes of diffusion inside living cells due to the structural heterogeneity of the intracellular

environment.