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Figure 4.3
Biochemical inversion uses the transcription and translation cellular pro-
cesses. Ribosomal RNA translates the input mRNA into an amino acid chain, which
then folds into a three-dimensional protein structure. When the protein binds to an
operator of the gene's promoter, it prevents transcription of the gene by RNA poly-
merase (RNAp). In the absence of the repressor protein, RNAp transcribes the gene
into the output mRNA.
mRNA, representing the input signal to the inverter. In the first stage, ribo-
somal RNA translates the mRNA product
ψ
A
into the input repressor protein
φ
A
. Let
ψ
A
and
L
(translation stage) denote the steady-state mapping between
φ
A
. In general, increases in
φ
A
until an asymptotic
boundary is reached. Factors that determine this boundary include the amino
acid synthesis capabilities of the cell, the efficiency of the ribosome-binding
site, and mRNA stability. Because the cell degrades both mRNA and input
protein molecules, a continuous synthesis of the input mRNA is required for a
steady level of the input protein.
The second stage in the inverter uses cooperative binding to reduce the digital
noise. Here, the input protein monomers join to form polymers (often dimers,
occasionally tetramers), which then bind to the operator and repress the gene.
Because the cells degrades the proteins, a continuous synthesis of the input
protein is required for maintaining the repression activity. Let
ψ
A
yield linear increases in
ρ
A
denote the
strength of this repression, defined as the concentration of operator that is bound
by repressor. In steady state, the relation
C
(cooperative binding stage) between
φ
A
and
φ
A
, the amount of
repression increases only slightly as the input protein concentrations increase
because these concentrations are too low for significant dimerization. Without
dimerization, the monomeric repressor cannot bind to the DNA. Then, at higher
levels of
ρ
A
will generally be sigmoidal. For low values of
φ
A
(when the input proteins dimerize easily), cooperative binding and
dimerization result in nonlinear increases in the repression activity. Finally,
at saturating levels of the input protein when the operator is mostly bound,
the curve reaches an asymptotic boundary. Because the repressor activity is
