Chapter 8 – Theoretical Biophysics  365

force terms, the governing equation of motion can often then be solved relatively easily, for

example, in the case of Hagen–​Poiseuille flow in a microfluidics channel (see Chapter 7) and

also fluid movement involving laminar flow around biological structures that can be mod­

eled acceptably by well-​defined geometrical shapes. The advantage here is that well-​defined

shapes have easily derived analytical formulations for drag coefficients, for example, that of

a sphere is manifested in Stokes law (see Chapter 6) or that of a cylinder that models the

effects of drag on transmembrane proteins (see Chapter 3), or that of a protein or nucleic acid

undergoing gel electrophoresis whose shape can be modeled as an ellipsoid (see Chapter 6).

The analysis of molecular combining (see Chapter 6) may also utilize cylindrical shapes as

models of biopolymers combed by the surface tension force of a retracting fluid meniscus,

though better models involve characterizing the biopolymers as WLC, which is reptating

within the confines of a virtual tube and thus treating the viscous drag forces as those on the

surface of this tube.

Fluid dynamics analysis is also relevant to swimming organisms. The most valuable par­

ameter to use in terms of determining the type of analytical method to employ to model

swimming behavior is the dimensionless Reynolds number Re (see Chapter 6). Previously, we

discussed this in the context of allowing us to determine if fluid flow was turbulent or lam­

inar, such that Re values above a rough cutoff of ~2100 (see Chapter 7). The importance for

swimming organisms is that there are two broadly distinct regimes of low Reynolds number

swimmers and high Reynolds number swimmers.

High Reynolds number hydrodynamics involve in essence relatively large, fast organisms

resulting in Re > 1. Conversely, low Reynolds number hydrodynamics involve in essence rela­

tively small, slow organisms resulting in Re < 1. For example, a human swimming in water has

a characteristic length scale of ~ 1 m, with a typical speed of ~1 m s⁻¹, indicating an Re of ~10⁶.

Low Reynolds number swimmers are typically microbial, often single-​cell organisms.

For example, E. coli bacteria have a characteristic length scale of ~1 μm but have a typical

swimming speed of a few tens of microns every second. This indicates a Re of ~10⁻⁵.

High Reynolds number hydrodynamics are more complicated to analyze since the com­

plete Navier–​Stokes equation often needs to be considered. Low Reynolds number hydro­

dynamics is far simpler. This is because the viscous drag force on the swimming organism is

far in excess of the inertial forces. Thus, when a bacterial cell stops actively swimming, then

the cell motion, barring random fluid fluctuations, stops. In other words, there is no gliding.

For example, consider the periodic back-​and-​forth motions of an oar of a rowing boat in

water, a case of high Re hydrodynamics. Such motions will propel the boat forward. However,

for low Re hydrodynamics, this simple reciprocal motion in water will result in a net swimmer

movement of zero. Thus, for bacteria and other microbes, their swimming motion needs

to be more complicated. Instead of a back-​and-​forth, bacteria rotate a helical flagellum to

propel themselves, powered by a rotary molecular machine called the flagellar motor, which

is embedded in the cell membrane. This acts in effect as a helical propeller.

Bacteria utilize this effect in bacterial chemotaxis that enables them to swim up a chem­

ical nutrient gradient to find source of food. The actual method involves a biased random

walk. For example, the small length scale of E. coli bacteria means that they cannot operate a

system of different locations of nutrient receptor on their cell surface to decide which direc­

tion to swim since there would be insufficient distance between any two receptors to discrim­

inate subtle differences of a nutrient concentration over a ~1 μm length scale. Instead, they

operate a system of alternating runs and tumbles in their swimming, such that a run involves

a bacterial cell swimming in a straight line and a tumble involving the cell stopping but then

undergoing a transient tumbling motion that then randomizes the direction of the cell body

prior to another run that will then be in a direction largely uncorrelated with the previous

run (i.e., random).

The great biological subtlety of E. coli chemotaxis is that the frequency of tumbling

increases in a low nutrient concentration gradient but decreases in a high nutrient concentra­

tion gradient using a complicated chemical cascade and adaption system that involves feed­

back between chemical receptors on the cell membrane and the flagellar motor that power

the cell swimming via diffusion-​mediated signal transduction of a response regulator protein

called CheY through the cell cytoplasm. This means that if there is an absence of food locally,

KEY BIOLOGICAL

APPLICATIONS:

REACTION–​DIFFUSION

MODELING TOOLS

Molecular mobility and turnover

analysis; Molecular motor trans­

location modeling.