New plots for old kinetics

In which basic enzyme kinetics are revisited, systematically incorrect lab statistics are bemoaned, and a little-known elegant 40 year old solution to estimating enzyme kinetic parameters is explored.

Michaelis-Menten enzyme kinetics.  The backbone of everyone's first approximation to real kinetics.  Basic biochemistry.  Everyone knows everything there is to know about Michaelis-Menten/Briggs-Haldane kinetics, don't they?  Don't they?  Well, not really.  At least, I assume that I still don't.  Until fairly recently I was unaware of a paper from the early 1970s that presented, at least to me, a fresh perspective on measuring enzyme kinetics...

We all know the Michaelis-Menten equation:

v is the measured initial rate of reaction, [S] is the concentration of substrate, V_max is the limiting rate (or maximum velocity) of the reaction, and K_m is the Michaelis constant (whose precise meaning differs depending on whether you derive the model like Michaelis-Menten from a rate-limiting step assumption, or like Briggs-Haldane by assuming steady-state).  In a lab experiment to investigate the kinetics of an enzyme, we would vary the substrate concentration [S] and measure the initial rate of reaction v, in order to be able to calculate both V_max and K_m.

Unfortunately, in its standard form, this equation describes a rectangular hyperbola, and fitting experimental results to this in the lab graphically in the days before computers were common was a bit tricky.  These days it's straightforward enough to fit nonlinear equations on your desktop, or even on your phone, but to get past this problem before computing power was so ubiquitous, linearised forms of the Michaelis-Menten equation were used, so that least-squares linear regression (or, more usually, your eye and a ruler) could be employed.

These linear transformations are pretty venerable now, having been introduced by Woolf in the early part of the 20th Century:

The first of these transformations is the well-known Lineweaver-Burk plot of 1/v against 1/[S].  The second is the Hanes-Woolf plot of [S]/v against [S].  They're all problematic, and were improved upon in several ways by a 1974 paper by Eisenthal and Cornish-Bowden that sadly doesn't seem to have gained much traction in the intervening four decades.

The three linear transformations given above have issues in that they distort the experimental data.  Statistically-speaking, this causes problems when fitting the linear regression to the linearised forms to estimate K_m and V_max, in that linear regression assumes that the errors in the data to be fitted are normally-distributed.  Even if this is the case for the experimental measurement, once a reciprocal is taken, the error structure is distorted.  This can be easily seen with some example plots.

In the first plot, a Michaelis-Menten kinetic system is shown in a plot of v against [S].  The typical rectangular hyperbola curve produced by this system is shown in blue, and some simulated experimental datapoints, taken at regular intervals of [S], are shown as crosses.  The experimental data has a normally-distributed error in v, with mean of the 'true' value of v, and standard deviation of 2.5% of that value.  Note that errors scale proportionally with measured initial rate.

The 'real' curve passes more or less through the centre of the simulated points, as we might expect.  Drawing a curve through the simulated points by hand would be a pain, but at least we can see that, by spacing the substrate concentrations evenly, we span a fair proportion of the curve, and computational nonlinear fitting would probably give us a good estimate.  But, where there is no computer to hand, we have to use other approaches.

The Lineweaver-Burk plot of 1/v against 1/[S] was the dominant linear form used in the lab when I was an undergraduate.  It produces linear output, but suffers from two problems that are easily seen when plotted with simulated data:

Firstly, our nicely-spaced experimental datapoints are now stretched and compressed.  The 1/[S] axis in particular now crams most of our experimental datapoints into the left-hand portion of the graph.  The lonely points off to the right have high leverage on the fit as a result, so errors in these values would have disproportionately large influence on the result.  In order to space our substrate concentrations evenly for a Lineweaver-Burk plot, we'd have to think carefully about substrate concentrations in the experiment.  A second problem though is that the 'real' line no longer passes through the centre of the experimental points, but instead hangs just below them.  This indicates that our error structure is no longer symmetrical about the 'real value', and a standard least-squares linear regression on this graph will systematically fit an incorrect line.

The 'exotic' alternative linear fit of my undergraduate days was the Hanes-Woolf plot.  This avoids one problem of the Lineweaver-Burk plot by making the x-axis be [S], so we could step our substrate concentrations evenly, and not have to think too hard beforehand.

However, the y-axis still uses a reciprocal, and biases the error structure, so the corresponding least-squares regression again systematically estimates the wrong fitted line.  Moreover, both axes involve our measurement of [S], and so are not independent.  This makes it difficult to assess goodness of fit.

In 1974, Eisenthal and Cornish-Bowden published the 'Direct Line' method which, instead of linearising the Michaelis-Menten equation, uses the simple mathematical trick of converting the Michaelis-Menten equation into a parametric equation:

Parametric equations like these are alternative representations of the system in a different parameter space.  In this case, we can say that K_m goes on the x-axis, and V_max on the y-axis to give "K_mV_max space".  When we do this, then for every experimental observation we make of v and [S], the values of K_m and V_max that could satisfy the equation form a straight line, and this line should pass through the point (K_m, V_max) that corresponds to their values in the real system.  If we make several measurements of v and [S], then the straight lines from each measurement should pass through this point, and no other.

Looking at the equation, if we put V_max=0, then -K_m/[S] = 1, and our measured value of [S] must equal -K_m.  Likewise, if we put K_m=0, then our measured value of v must equal V_max.  So for each measured combination of v and [S] we draw a line that passes through (-[S], 0) and (0, v).  Where these lines intersect is then the point (K_m, V_max).

In the graph above, the same experimental data as seen in the basic Michaelis-Menten plot, the Lineweaver-Burk plot, and the Hanes-Woolf plot are shown.  Each of the coloured lines corresponds to a single experimental observation of [S] and v, and the actual values of K_m and V_max for the system are shown by the black cross.

This approach makes no assumptions about the form of the experimental error when making measurements.  Experimental error shows itself by offsetting of the points of intersection for the lines in the graph, representing our uncertainty in the estimates of K_m and V_max.  The authors recommend that the final estimate is made by taking the median predicted values for each of K_m and V_max.  These predicted values can be obtained straightforwardly by solving simultaneous equations for each pair of (v, [S]) observations in turn, which is arguably less onerous than carrying out least-squares linear regression by hand.

Of course, these days we can just solve the problem of estimating kinetic parameters directly by nonlinear fitting, even for much more complex kinetic models than Michaelis-Menten.  This is nice, and very welcome, but doesn't allow us these little digressions into parameter space, and sadly sidelines elegant and insightful solutions to practical problems, such as this.