Fits the Michaelis-Menten equation for substrate concentration versus time data (M.C. Hannah).
|What to print (
||What to plot (
||Window in which to plot the graphs; default 1|
||Title for the graphs; default
||Title for the times axis; if this is unset, the identifier of the
||Title for the concentrations axis; if this is unset, the identifier of the
||Title for the rates axis; if this is unset, the identifier of the
||Weights for the observations, to use in the fit, if required; default * i.e. all observations with weight one|
||Times at which substrate concentration data were measured|
||Substrate concentration data|
||Variate with four values defining initial step lengths for the parameters S0, Vmax, Km and K1 (in that order)|
||Variate containing initial values for the parameters, similarly to
||Saves the residuals from each fit|
||Saves the fitted concentration values|
||Saves the parameter estimates|
||Saves the standard errors of the estimates|
||Saves the variance-covariance matrix of the estimates|
||Saves reaction rates, calculated from the observed concentrations|
||Saves fitted reaction rates|
The Michaelis-Menten equation, for biochemical reaction rate v, versus substrate concentration S
v(t) = dS(t) / dt = Vmax S(t) / ( Km + S(t) )
can be fitted in Genstat using
FITCURVE [CURVE=ldl; CONSTANT=omit]
with v as the response variate, and 1/S as the explanatory variate. However, in practice, data are available only for substrate concentration S at time t, and not for the reaction rate v. Instead of attempting to derive rate data, it is better statistically to fit S(t) to the directly observed concentration data. The solution to the above differential equation, S(t), has a characteristic hockey-stick shape where the response decreases linearly initially, and then curves to become horizontal as it approaches the x-axis. However, no closed form expression for S(t) exists. The procedure thus uses Golicnik’s (2010) method to fit the model.
So, the procedure fits the curve S(t) to observed concentration versus time data, obtaining parameter estimates for Vmax and Km. It can also estimate the initial concentration S0, and an additive constant K1 representing the concentration of non-reactive substrate (i.e. a lower asymptote). This generalized Michaelis-Menten curve is given by
v(t) = dS(t) / dt = Vmax ( S(t) – K1 ) / ( Km + S(t) – K1 )
The substrate concentration data and the corresponding time values must be supplied, in variates, using the
TIMES parameters. Weights can be supplied using the
You can supply initial values for the parameters, in a variate, using the
INITIAL parameter. The variate should have four values, corresponding to the parameters S0, Vmax, Km and K1 (in that order). If
INITIAL is unset, or if any of the values in the variate is missing, the procedure finds its own starting values for those not supplied. The
STEPLENGTHS parameter can supply step lengths, again in a variate. You can fix a parameters at a specific value by specifying that value as the initial value, and defining a step length of zero. When doing this, it is usually simplest to fill the positions of the other, non-fixed, parameters with missing values, in both the
Printed output is controlled by the
FITNONLINEAR directive (which is used to fit the model). The default is to print a description of the model, the analysis summary and the estimated parameters.
PLOT option controls the graphs that are plotted, with settings
||to plot the curve fitted to the concentrations, and|
||to plot the estimated reaction rates against the concentrations, and against time.|
WINDOW option specifies the window to use for the graphs (default 1). The
TITLE option can specify an overall title, and the
TRATES options can specify titles for the axes for times, concentrations and rates, respectively.
You can save the fitted concentrations using the
FITTEDVALUES parameter, and the residuals from the fit using the
RESIDUALS parameter. The parameter estimates, their standard errors and variance-covariance matrix can be saved using the
VCOVARIANCE parameters. You can also save “observed” reaction rates (calculated from the observed concentrations) with the
OBSRATES parameter, and fitted reaction rated with the
You can use the post-regression directives,
RKEEP etc., in the usual way to display or save additional output. You can also use an associated procedure,
MMPREDICT, to predict S(t) and v(t) for a new time vector, given the parameter values estimated by
The procedure uses Golicnik’s (2010) method to fit the model.
The data variates must not be restricted.
Golicnik, M. 2010. Explicit reformulations of time-dependent solution for a Michaelis-Menten enzyme reaction model. Analytical Biochemistry, 406, 94-96.
Commands for: Regression analysis.
CAPTION 'MICHAELISMENTEN example'; STYLE=meta " Read in concentration and time data." READ Concentration 25.89 26.12 24.43 24.13 23.74 23.48 23.33 * 21.82 20.94 19.13 17.77 15.11 13.23 10.24 7.85 7.57 6.08 4.53 3.40 3.35 3.26 2.72 2.67 2.00 1.74 : READ Time 0.00 0.60 4.70 5.00 5.50 6.00 6.70 7.50 9.90 12.40 15.30 19.40 25.30 30.10 37.10 43.40 45.30 48.70 54.50 60.60 62.20 63.60 64.80 66.90 72.60 81.10 : " Fit standard Michaelis-Menten model with asymptote, K1, fixed at zero." MICHAELISMENTEN [PLOT=concentration,rate] TIME=Time;\ CONCENTRATION=Concentration; INITIAL=!(3(*),0); STEP=!(3(*),0) RCHECK " Fit generalized Michaelis-Menten model with asymptote, K1, estimated." MICHAELISMENTEN [PLOT=concentration,rate] CONCENTRATION=Concentration; TIME=Time RCHECK " Predict the curves at new times using companion procedure MMPREDICT." RKEEP ESTIMATES=final VARIATE [VALUES=0...90] newTimes MMPREDICT [PLOT=concentration,rate] PARAMETER=final; TIME=newTimes;\ CONCENTRATIONS=predConc; RATES=predRate PRINT newTimes,predConc,predRate; DECIMALS=0,4,4