LSM2106 Chap.6 Enzyme Kinetics and Enzyme Inhibitors
Enzyme Kinetics and Enzyme Inhibitors
The rate that can be attributed to a substrate concentration
The sixth lecture covers kinetics and inhibitors, and the third practical measures them. An assay produces a curve rather than a number, and the curve bends for three reasons: substrate is consumed, product accumulates and can inhibit, and the enzyme loses activity over minutes.
All three are negligible at the instant the reaction begins and none is negligible four minutes later, so the initial velocity is the only rate attributable to the substrate concentration that was prepared. The practical obtains it by fitting each progress curve with a second-order polynomial forced through the origin and taking the linear coefficient, which is the gradient of the tangent at time zero.
Forcing the intercept through zero encodes a fact rather than a preference: at time zero no product exists.
Plotting those velocities against substrate concentration gives a rectangular hyperbola described by two constants. The maximum velocity is a capacity and scales with the amount of enzyme present, so it cannot be compared between preparations without normalising.
The Michaelis constant is the substrate concentration at which the enzyme runs at half that maximum, is independent of enzyme concentration, and is therefore comparable.
A low value is usually described as high apparent affinity, and the word apparent matters, since the constant collects several rate constants and equals a true binding constant only when catalysis is slow relative to substrate release.
Why the practical asks for a second plot
Reading both constants off the hyperbola means estimating an asymptote by eye.
Taking reciprocals of both sides converts the relation into a straight line whose vertical intercept is the reciprocal of the maximum velocity, whose horizontal intercept is the negative reciprocal of the Michaelis constant, and whose slope is the ratio of the two. Both constants become intercepts on a fitted line.
The price is that reciprocals magnify error at low substrate concentrations, and those points carry the most leverage, so a single careless reading at the lowest concentration moves both constants.
Inhibition is where the linearisation earns its place.
Running the assay with and without inhibitor and comparing the two lines identifies the mechanism: a competitive inhibitor raises the apparent Michaelis constant and leaves the maximum velocity untouched, so the lines share a vertical intercept; a non-competitive inhibitor lowers the maximum velocity and leaves the constant alone, so they share a horizontal intercept.
Both conclusions can be rebuilt from the mechanism rather than memorised.
What this chapter covers
- 01
Why a progress curve bends, and why only the slope at the origin is usable
- 02
Fitting a quadratic through the origin and reading the linear coefficient
- 03
The saturation relation and the operational meaning of the Michaelis constant
- 04
Maximum velocity as a capacity that depends on enzyme amount
- 05
Converting absorbance per minute into moles per minute
- 06
The double-reciprocal form and its two intercepts
- 07
Competitive and non-competitive inhibition read from the intercepts
- 08
Error leverage at low substrate concentration
Kinetic constants from two measured velocities
- 2Obtain the absorption coefficient from the standard reading.
- 2Tabulate the two pairs of reciprocals.
- 3Fit the line and extract both constants.
- 2Convert the maximum velocity into moles per minute.
Key terms
- Initial velocity
- The rate of an enzyme-catalysed reaction at the instant it begins, obtained as the gradient of the tangent to the progress curve at time zero.
- Progress curve
- A plot of product formed against time for a single reaction, whose curvature comes from substrate depletion, product inhibition and loss of enzyme activity.
- Michaelis constant
- The substrate concentration giving half the maximum velocity. It is independent of how much enzyme is present, so it can be compared between preparations.
- Maximum velocity
- The rate approached as substrate becomes saturating. It is proportional to the amount of active enzyme present and is an asymptote rather than an attainable value.
- Double-reciprocal plot
- A plot of the reciprocal of velocity against the reciprocal of substrate concentration, which linearises the saturation relation and turns both constants into intercepts.
- Competitive inhibition
- Inhibition by a compound binding the same site as the substrate, raising the apparent Michaelis constant while leaving the maximum velocity unchanged.
- Non-competitive inhibition
- Inhibition by a compound binding away from the substrate site, lowering the maximum velocity while leaving the apparent Michaelis constant unchanged.
- Absorption coefficient
- The proportionality between absorbance and concentration at a given wavelength and path length, needed to convert a rate in absorbance per minute into a chemical rate.
Enzyme Kinetics and Enzyme Inhibitors FAQ
Why not just divide the total absorbance change by the total time?
Because the curve is steepest at the start, so an average over the whole run underestimates every velocity. Worse, the underestimate is larger at high substrate where the curve bends earliest, so the bias varies systematically with the variable being plotted against. Both constants come out too low and the resulting plot still looks perfectly straight.
Is the Michaelis constant the same thing as a binding constant?
Not in general. It is built from several rate constants including the catalytic step, and it reduces to a dissociation constant only when catalysis is slow compared with the release of substrate. Describing it as apparent affinity is safe; calling it the affinity without qualification overstates what a kinetic measurement can show.
How do I tell competitive from non-competitive inhibition experimentally?
Repeat the assay across a range of substrate concentrations with and without the inhibitor and compare the two double-reciprocal lines. A competitive inhibitor is progressively outcompeted, so the lines converge on a shared vertical intercept. A non-competitive inhibitor still acts at saturation, so the vertical intercepts differ while the horizontal intercept stays put.
Why does the practical force the fitted curve through the origin?
Because at time zero no product has formed, which is a known fact about the system rather than something the data should be allowed to estimate. Leaving the intercept free lets it absorb part of the slope, and the slope is the quantity being measured, so the fit would quietly take away what the experiment was designed to obtain.
Exam move
Practise the full chain once with your own numbers: gradient at the origin, absorbance per minute to moles per litre per minute, then to moles per minute, then reciprocals, then both constants. Carry the units on every line, because errors in this topic are almost always a lost volume or a lost path length rather than a misunderstanding. Then rebuild the inhibition table from the mechanism rather than reciting it.
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