If I have a method for the measurement of kinetics parameters, how I can determine the different concentrations of substrate that I need use for obtain several initial velocities and generate the Lineweaver-Burk plot? and what kind of software is recomendded for the calculations?
You start with determining the dependence: cS on time for a given initial concentration cSo. One curve is enough. You differentiate the obtained curve numerically using the finite difference method to find the dependence of rate (not velocity): r=dcS/dt on cS. EXCEL can be used. With such a curve you can graphically find rm (upper limit) and KM which is equal to cS for rm/2. You can also use the L-B method by drawing a 1/rm plot vs 1/cS. Instead of the graphical method, you can also use the nonlinear regression method. The most important, however, is a reliable cS vs t curve.
If your v/[S] curve is hyperbolic, you need at least 12 data points, equally spaced between 0.1 Km and at least 5 Km, better 10 Km (solubility permitting), see doi:10.1006/jtbi.1996.0023. If your curve is more complex (sigmoid or several binding sites), you need more data over a larger concentration range.
As Suji George already indicated, the LWB-plot causes severe bias when used for parameter estimation, I include the plot of a simulation of this problem I once did. You can use non-linear curve fitting (Nelder-Mead simplex is more stable than the more commonly used Marquardt-Levenberg, and the only option when you need other fitting criteria than least-sum-of-squares, for an example why this can be required see doi:10.1046/j.1432-1327.1999.00643.x).
However, if your data are hyperbolic, I'd recommend the Eisenthal/Cornish-Bowden method (doi:10.1042/bj1390721, 10.1042/bj1390715) instead, it is non-parametric and hence both simple and stable when the data contain errors. I have included a completely worked-out example in the exercise-section of doi:10.1007/978-3-319-19920-7_6
Engelbert Buxbaum It is interesting what you write. Everyone has their favorite numerical methods. You probably prefer gradientless multivariable function minimization methods that are numerically stable, but very slowly converge to the minimum of the function. I prefer gradient methods, and I put the M-L algorithm first. It converges much faster than Nelder-Mead, and its numerical stability depends on a properly programmed Gaussian method. Moreover, this method can be used for any objective function (contrary to what you say), just like the N-M method.
By the way, could you explain what is "linearising regression". Regards,
Suji George You are absolutely correct that converting the M-M equation to the form required by the L-B method changes the error distribution. However, I would avoid using the term "linear transformation" here.
Mirosław Grzesik : "Linearising regression" means that curved data are transformed to a straight line, followed by linear regression. The LWB plot is a typical example, taking logarithms is another.
As to the ML and NM-methods it is my (trademark) experience, that ML takes longer, and (more importantly) gives higher sum of squares than NM run on the same problem, with the same starting parameters. This is particularly true for problems with many parameters (say, a 2D Gauss-distribution with 2 means, 2 standard deviations, the correlation between x and y and a scaling parameter for the peak height).
The biggest advantage of ML is that it returns error estimates for the parameters directly, in NM you need bootstrapping. If you include the time for bootstrapping, then NM is indeed much slower than ML.
What I particularly like about NM is its incredible flexibility. I once had a problem where the detected analyte was unstable, that is A → B → C, with B detected. I used numerical integration of the ODE system, and fed the results into NM to estimate the rate constants.
Btw, do you have a reference for the use of other fitting criteria than sum-of-squares in ML? In NM you are completely free to use any numeric function. My particular interest is minimising chi square (relative error).
Thank you very much for your answers, I need to do several enzyme inhibition reactions and I am looking for some useful information, first of all I wanted to make a diagram of Lineweaver Burk just because I read several articles where they used this diagram but with your opinions now I have more clear the ideas, best regards
Engelbert Buxbaum, Well, we will discuss it one each other.
There is "linear regression". There are minimization methods with the objective function in the form of one of fit measures, commonly called "nonlinear regression". I still do not know what is "linearizing regresion". By changing the coordinate system, a straight line can be obtained from the data in the form of a curve, but it is not a linear transformation or a linearization. Likewise, they are not a linear transformation of L-B, Hanes-Woolf or Eadie-Hofstee transforms, or taking log. So you did not convince me of the sense of using such a term.
You have your favorite method and I have mine. You had interesting experiences with NM, me with ML. Among other, I used it to solve two-point boundary value problems for ODE, to minimize G (equilibrium composition of reaction mixtures), or to determine the parameters of kinetic equations. As I wrote earlier, ML can be used for any function, whether in the form of a formula or an algorithm. I also used NM (2nd step) in conjunction with the Monte Carlo method (1st step) and the FPD method (3rd step) called MINUITL from the CERN numerical program library.
You ask for chi2. There used to be an ML routine in the CERN library with a built-in SRE (sum of relative errors) instead of SSD. But with chi2 I did not see. Regards,
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