Chapter 9 – Emerging Biophysics Techniques  399

creatures. But a key feature of bacterial chemotaxis is that different proteins in the pathway

can be monitored using fluorescent protein labeling strategies (see Chapter 7) coupled with

advanced single-​molecule localization microscopy techniques (see Chapter 4) to monitor

the spatial distribution of each component in real time as a function of perturbation in the

external chemoattractant concentration, which enables systems biophysics models of the

whole-​cell level process of bacterial chemotaxis to be developed.

9.2.2  MOLECULAR NETWORKS

Many complex systems, both biological and nonbiological, can be represented as networks,

and these share several common features. Here, the components of the system feature as

nodes, while the interactions between the components are manifested as edges that link nodes.

There also exist motifs in networks, which are commonly occurring subdomain patterns

found across many different networks, for example, including feedback loops. Modularity

is therefore also common to these motifs, in that a network can be seen to be composed

of different modular units of motifs. Also, many networks in biology tend to be scale-​free

networks. This means that their degree of distribution follows a power law such that the

fraction P(k) of nodes in the network having k connections to other nodes satisfies

(9.1)

P k

Ak

( ) =

−γ

where

γ is usually in the range ~2–​3

A is a normalization constant ensuring that the sum of all P values is exactly 1

Molecular networks allow regulation of processes at the cost of some redundancy in the

system, but also impart robustness to noise. The most detailed type of molecular network

involves metabolic reactions, since these involve not only reactions, substrates, and products,

but also enzymes that catalyze the reactions.

The Barabási–​Albert model is an algorithm for generating random scale-​free networks.

It operates by generating new edges at each node in an initial system by a method of prob­

abilistic attachment. It is valuable here in the context of creating a synthetic, controlled net­

work that has scale-​free properties but which is a more reduced version of real, complex

biological network. Thus, it can be used to develop general analytical methods for investi­

gating scale-​free network properties. Of key importance here is the robust identification of

genuine nodes in a real network. There are several node clustering algorithms available, for

example, the k-​means algorithm alluded to briefly previously in the context of identifying

different Förster resonance energy transfer (FRET) states in fluorescence microscopy (see

Chapter 4) and to clustering of images of the same subclass in principal component analysis

(PCA) (see Chapter 8).

The general k-​means clustering algorithm functions to output k mean clusters from a data

set of n points, such that k < n. It is structured as follows:

1 Initialize by randomly generating k initial clusters, each with k associated mean values,

from the data set where k is usually relatively small compared to n.

2 k clusters are created by associating each data point with the nearest mean from

a cluster. This can often be represented visually using partitions between the data

points on a Voronoi diagram. This is mathematically equivalent to assigning each data

point to the cluster whose mean value results in the minimum within-​cluster sum of

squares value.

3 After partitioning, the data points then calculate the new centroid value from each of

the k clusters.

4 Iterate steps 2 and 3 until convergence. At this stage, rejection/​acceptance criteria

can also be applied on putative clusters (e.g., to insist that to be within a given cluster