= Trypsin colorimetric assay
A <trypsin colorimetric assay> follows <enzyme> activity through <absorbance> at $405\,\mathrm{nm}$. <Trypsin> cleaves an <amide> bond in the synthetic <arginine> derivative <BAPNA>, releasing <p-Nitroaniline>.<Enzymatic Assay of APROTININ - Sigma Quality Control Test Procedure>[Enzymatic Assay of APROTININ, Sigma quality-control procedure, p. 1: Principle and Conditions]{ref}
The inspected Sigma procedure prepares fresh <trypsin> in cold $1\,\mathrm{mM}$ <hydrochloric acid> and fully dissolves $1\,\mathrm{mg/ml}$ <BAPNA>. Its <buffer solution> stock contains $200\,\mathrm{mM}$ <triethanolamine> and $20\,\mathrm{mM}$ <calcium chloride>, at <pH> $7.8$. Three $3.00\,\mathrm{ml}$ <cuvettes> compare <trypsin>, <trypsin> plus the <enzyme inhibitor> <aprotinin>, and a blank replacing <trypsin> with its acid solvent. <Sodium chloride> solution replaces <aprotinin> in the uninhibited control. A thermostatted instrument for <spectrophotometry> holds $25\,{}^\circ\mathrm C$ and a $1\,\mathrm{cm}$ light path. After equilibration, adding <BAPNA> starts approximately five minutes of <absorbance> recording. The uninhibited sample checks activity; the blank checks background; the inhibited sample tests suppression.<Enzymatic Assay of APROTININ - Sigma Quality Control Test Procedure>[Enzymatic Assay of APROTININ, pp. 1–2: Reagents A–F, cuvette table and Procedure]{ref}
\Image[/trypsin-colorimetric-readout-model.png]
{title=<Trypsin colorimetric assay>: controlled <absorbance> readout and an original numerical model}
{description=Original apparatus schematic and constructed straight-line traces. The traces illustrate blank subtraction and inhibition arithmetic; they are not measurements from Sigma's procedure. U is uninhibited, I inhibited, and B blank.}
Let $s_U$, $s_I$ and $s_B$ denote the measured <absorbance> <slopes> per minute over the same linear time interval. Blank subtraction gives $s_U-s_B$ and $s_I-s_B$. If <p-Nitroaniline> accounts for the changing <absorbance>, the <Beer-Lambert law> gives
$$
A_{405}=\varepsilon l c,\qquad
\frac{dc}{dt}=\frac{s_U-s_B}{\varepsilon l}.
$$
Here $c$ is product <molar concentration> and $l$ is the geometric light path through the <chemical solution>. Sigma uses $\varepsilon=9.96\,\mathrm{mM^{-1}cm^{-1}}$.<Enzymatic Assay of APROTININ - Sigma Quality Control Test Procedure>[Enzymatic Assay of APROTININ, p. 2: Calculations, millimolar extinction coefficient at 405 nm]{ref}
Choose original illustrative <slopes> $s_U=0.1036$, $s_I=0.0538$ and $s_B=0.0040\,\mathrm{min^{-1}}$. The corrected uninhibited <slope> is $0.0996\,\mathrm{min^{-1}}$, giving $dc/dt=0.0100\,\mathrm{mM\,min^{-1}}=10.0\,\mathrm{\mu M\,min^{-1}}$ at $l=1\,\mathrm{cm}$. In $3.00\,\mathrm{ml}$, the product <amount of substance> increases at $0.0300\,\mathrm{\mu mol\,min^{-1}}$. The corrected inhibited <slope> is $0.0498\,\mathrm{min^{-1}}$, half as large. The relative suppression is therefore
$$
100\left(1-\frac{s_I-s_B}{s_U-s_B}\right)
=100\frac{s_U-s_I}{s_U-s_B}=50\%.
$$
The background cancels in the numerator, but remains essential in the denominator. Using uncorrected <slopes> would instead give about $48.07\%$. Differentiating $A_{405}=\varepsilon lc$ with fixed $\varepsilon$ and $l$ explains the rate conversion; <dimensional analysis> checks it independently, since $\mathrm{min^{-1}}/(\mathrm{mM^{-1}cm^{-1}}\,\mathrm{cm})=\mathrm{mM\,min^{-1}}$.
The inference is a <reaction rate> on <BAPNA> under the stated conditions, rather than a direct count of <trypsin> molecules or proof of every cut in a <protein>. Cloudiness, other absorbing <solutes>, or different product <protonation> can compromise the <Beer-Lambert law> interpretation. A matched blank and a linear interval are therefore part of the measurement's meaning. A coloured <enzyme inhibitor> may require its own matched blank; a changed <substrate> or <pH> requires a fresh calibration. The constructed constant <slopes> above describe only that interval, not indefinite product accumulation.
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