Chapter 9 – Emerging Biophysics Techniques  405

include oscillators (e.g., periodic fluctuations in time of the protein output of an expressed

gene), pulse generators, latches, and time-​delayed responses.

One of the simplest biological circuits is that found in many gene expression systems in

which the protein product from the gene expression activates further expression from the

promoter of the gene in question. In this simple case, the concentration C of an expressed

protein can be modeled by the following rate equation:

(9.4)

C

kC

Rp

=

−

where

k is the effective activation rate

R is the concentration of repressor complexes (e.g., a relevant TF)

p is the probability per unit time that a given promoter of a gene is occupied

The value of p can be modeled as being proportional to a Boltzmann factor exp(−ΔGp/​kBT)

where Gp is the free energy change involved in binding a repressor molecule to the promoter

of the gene. The solution to this rate equation is essentially a sigmoidal response in terms of

rate of expression versus levels of expressed protein. In other words, the output expression

rate switches from low to high over a small range of protein levels, thus acting in effect as a

binary switch, which is controlled by the concentration of that particular protein in the cell

(see Worked Case Example 9.1).

Ultimately, natural biological circuits have two core features in common. First, they are

optimized to maximize robustness. By this, we mean that the output of a biological circuit

is relatively insensitive to changes in biochemical parameters from cell to cell in a popula­

tion. In order to ensure this, gene circuits also share the feature of involving feedback, that

is, some communication between the level of output response and the level of input signal

equating in effect to a gain function of the circuit, which depends on the output. Artificial

biological circuits need to follow these same core design principles. Many of the components

of biological circuits can be monitored and characterized using a wide range of biophysical

techniques already discussed previously in this book.

FIGURE 9.2  Biological circuits. (a) Equivalent protein circuit for bacterial chemotaxis, A, B,

W, Y, and Z are chemotaxis proteins CheA, CheB, CheW, CheY, and CheZ, respectively, with P

indicating phosphorylation. (b) A common method of cellular signal transduction following

detection of a ligand by a receptor, typically in the cell membrane, involves a cascade of serial

phosphorylation steps of tyrosine kinase enzymes (depicted here by example with some gen­

etic kinases α, β, and γ), which ultimately activates a transcription factor, which in turn can then

result in regulation of a specific gene or set of genes.