Fits the Michaelis-Menten equation for substrate concentration versus time data (M.C. Hannah).

### Options

`PRINT` = string tokens |
What to print (`model` , `deviance` , `summary` , `estimates` , `correlations` , `fittedvalues` , `monitoring` ); default `mode` , `summ` , `esti` |
---|---|

`PLOT` = string tokens |
What to plot (`concentration` , `rate` ); default `conc` |

`WINDOW` = scalar |
Window in which to plot the graphs; default 1 |

`TITLE` = text |
Title for the graphs; default `'Michaelis-Menten process'` |

`TTIMES` = text |
Title for the times axis; if this is unset, the identifier of the `TIMES` variate is used |

`TCONCENTRATIONS` = text |
Title for the concentrations axis; if this is unset, the identifier of the `CONCENTRATIONS` variate is used if available, otherwise `'Concentration'` |

`TRATES` = text |
Title for the rates axis; if this is unset, the identifier of the `RATES` variate is used if available, otherwise `'Rate'` |

`WEIGHTS` = variate |
Weights for the observations, to use in the fit, if required; default * i.e. all observations with weight one |

### Parameters

`TIMES` = variates |
Times at which substrate concentration data were measured |
---|---|

`CONCENTRATIONS` = variates |
Substrate concentration data |

`STEPLENGTHS` = variates |
Variate with four values defining initial step lengths for the parameters S_{0}, V, _{max}K and _{m}K_{1} (in that order) |

`INITIAL` = variates |
Variate containing initial values for the parameters, similarly to `STEPLENGTHS` |

`RESIDUALS` = variates |
Saves the residuals from each fit |

`FITTEDVALUES` = variates |
Saves the fitted concentration values |

`ESTIMATES` = variates |
Saves the parameter estimates |

`SE` = variates |
Saves the standard errors of the estimates |

`VCOVARIANCE` = symmetric matrix |
Saves the variance-covariance matrix of the estimates |

`OBSRATES` = variates |
Saves reaction rates, calculated from the observed concentrations |

`FITRATE` = variates |
Saves fitted reaction rates |

### Description

The Michaelis-Menten equation, for biochemical reaction rate *v*, versus substrate concentration *S*

*v*(*t*) = *dS*(*t*) / *dt* = *V _{max} S*(

*t*) / (

*K*+

_{m}*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 *V _{max}* and

*K*. It can also estimate the initial concentration

_{m}*S*

_{0}, and an additive constant

*K*

_{1}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* = *V _{max}* (

*S*(

*t*) –

*K*

_{1}) / (

*K*+

_{m}*S*(

*t*) –

*K*

_{1})

The substrate concentration data and the corresponding time values must be supplied, in variates, using the `CONCENTRATIONS`

and `TIMES`

parameters. Weights can be supplied using the `WEIGHTS`

option.

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 *S*_{0}, *V _{max}*,

*K*and

_{m}*K*

_{1}(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 `INITIAL`

and `STEPLENGTHS`

variates.Printed output is controlled by the `PRINT`

option. The settings all operate as in 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.

The `PLOT`

option controls the graphs that are plotted, with settings

`concentration` |
to plot the curve fitted to the concentrations, and |
---|---|

`rate` |
to plot the estimated reaction rates against the concentrations, and against time. |

By default, `PLOT=concentration`

.

The `WINDOW`

option specifies the window to use for the graphs (default 1). The `TITLE`

option can specify an overall title, and the `TTIMES`

, `TCONCENTRATIONS`

and `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 `ESTIMATES`

, `SE`

and `VCOVARIANCE`

parameters. You can also save “observed” reaction rates (calculated from the observed concentrations) with the `OBSRATES`

parameter, and fitted reaction rated with the `FITRATES`

parameter.

You can use the post-regression directives, `RCHECK`

, `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 `MICHAELISMENTEN`

.

Options: `PRINT`

, `PLOT`

, `WINDOW`

, `TITLE`

, `TTIMES`

, `TCONCENTRATIONS`

, `TRATES`

, `WEIGHTS`

.

Parameters: `TIMES`

, `CONCENTRATIONS`

, `STEPLENGTHS`

, `INITIAL`

, `RESIDUALS`

, `FITTEDVALUES`

, `ESTIMATES`

, `SE`

, `VCOVARIANCE`

, `OBSRATES`

, `FITRATES`

.

### Method

The procedure uses Golicnik’s (2010) method to fit the model.

### Action with `RESTRICT`

The data variates must not be restricted.

### Reference

Golicnik, M. 2010. Explicit reformulations of time-dependent solution for a Michaelis-Menten enzyme reaction model. *Analytical Biochemistry*, 406, 94-96.

### See also

Directives: `FITCURVE`

, `FITNONLINEAR`

.

Procedure: `MMPREDICT`

.

Commands for: Regression analysis.

### Example

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