C/C++ Interactive Guide/c-cplusplus-interactive-guide.iso/c_ref/csource4/209_01/ldhfitr.doc

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LDHFITR.DOC       VERS:- 01.00 DATE:- 09/26/86 TIME:- 10:02:47 PM


description of computer reduction of data from kinetic
    measurements for the enzyme lactate dehydrogenase
information on how to run program LDHFIT







        By J. A. Rupley, Tucson, Arizona


         NOTES ON DATA REDUCTION BY COMPUTER
         AND INFORMATION ON RUNNING THE PROGRAM LDHFIT


        INTRODUCTION:

         In order to obtain conclusions from quantitative measurements,
        there must be some form of data reduction. This can be as simple
        as a comparison by eye of two curves drawn through the data. If,
        however, the data set is large and complex, for example with more
        than one independent variable, and if the questions posed are
        detailed or involve a complicated nonlinear model, then visual or
        graphical methods are less satisfactory than a computer-based
        analysis. Procedures of the latter type are now widely used.

         This laboratory is a short introduction to data reduction by use
        of a computer. The intent is to show that a sophisticated
        computer program can be handled easily, that its use saves time
        and effort, that it can treat a more complicated model than can
        be treated graphically, and that it produces information such as
        estimates of uncertainties in the parameters that is difficult or
        impossible to obtain from graphical methods.

         The data to be analyzed are initial rate measurements made
        on the lactate dehydrogenase catalyzed reaction of pyruvate with
        NADH, in the presence and absence of lactate as inhibitor. The
        results of the computer fit are the following:  (1) values of the
        kinetic constants V, KmA, KmB, KmAB, KmQ/KmPQ, and KBInhib. The
        first five constants are those that can be evaluated by the
        standard graphical methods of primary and secondary reciprocal
        plots.  The constant KBInhib is the dissociation constant for the
        dead-end complex LDH-NADH-lactate, which is included in the
        mechanism fit by the computer program but cannot be included in
        the mechanism on which the graphical methods are based.  (2)
        Estimates of the standard deviations of the kinetic constants.
        These are needed for an understanding of the reliability and
        significance of the values calculated for the kinetic constants.
        (3) A list of the coordinates of points suitable for construction
        of the lines of the reciprocal plots of the standard graphical
        methods.














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        By J. A. Rupley, Tucson, Arizona

        THEORY:

        A. REMARKS ON FITTING OF A MODEL TO DATA

         In a typical data reduction, a particular model to be tested is
        fit to a set of data points under some criterion for best fit.
        The ith data point of a set of N data points consists of a single
        value for the dependent variable Yobserved(i) measured for
        corresponding single values for the one or more independent
        variables Xobserved(i). The commonly-used least squares criterion
        for quality of fit is the minimum value of the sum of the squares
        of the deviations between the observed values of Y and the values
        of Y calculated according to the model being tested.

         Working from the model to be fit to the data, one develops an
        equation relating, for each of the N data points, the dependent
        variable Y to the independent variables X and to a set of M
        variable parameters p:

             Ymodel(i) = F(Xobserved(i); p(j), j=1,M)        eq. 1

        For example, if the model predicts a linear relationship between
        Y and a single independent variable X :

             Ymodel(i) = p(1) + p(2) * Xobserved(i)          eq. 2

        The constants p(1) and p(2) of equation (2) are the Y axis
        intercept and the slope, respectively, and of course are the same
        for all data points (for all pairs of values Y(i) and X(i)).

         The fitting of a model to data consists of finding the values of
        the M variable parameters p that give the best agreement between
        the N pairs of values of Ymodel(i) and Yobserved(i). Best
        agreement can be defined as the minimum value of the least
        squares function y:


                  N
             y = SUM (Yobserved(i) - Ymodel(i))^2 * W(i)     eq. 3
                 i=1


        The factor W(i) of equation (3) is the normalized reciprocal
        variance (the statistical weighting) of the ith data point, and
        it can be set at unity if the data points are all of equal
        estimated uncertainty.

         Combining equations (1) and (3), one sees that the least squares
        function y of equation (3) is a function of the full set of N
        data points and a set of M variable parameters:

          y = f(Yobserved(i), Xobserved(i), i=1,N; p(j), j=1,M)   eq. 4



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         The fitting problem therefore consists of finding the minimum
        value of the least squares function y, which for a given set of
        data depends only on the M variable parameters p (the data points
        Yobserved(i)---Xobserved(i) in equation (4) are constant in the
        fitting). There are several methods commonly used to find the
        minimum of y and thus evaluate the best fit values of the
        parameters p. The more useful of these can handle nonlinear model
        functions F (equation (1)) of arbitrary mathematical form. The
        rate law for lactate dehydrogenase is an example of a nonlinear
        model function.

         In the simplex method used here, one constructs an M dimensional
        polyhedron with M + 1 vertices (the simplex). Each dimension of
        the simplex corresponds to a variable parameter of equation (4).
        Each vertex of the simplex is a point in the M dimensional space,
        which is called "parameter space" or "factor space." The M
        coordinates of each vertex are values of the M parameters. Thus
        each vertex of the simplex has an associated value of the least
        squares function y. The starting simplex is constructed to be so
        large as to include within it the point corresponding to the
        minimum value of y. This minimum point has as its coordinates of
        the best fit values of the parameters.

         The minimization process shrinks the simplex about the minimum
        point, even though the coordinates of the minimum are not known
        beforehand, until the vertices of the simplex are so close
        together and so nearly equal that an exit test is satisfied. The

        exit test is set so that a desired level of accuracy is obtained.
        The values of the M parameters averaged over all the vertices, ie
        the parameter values for the centroid of the simplex, serve as
        reliable estimates of the best fit parameter values (those for
        the least squares function minimum), because the minimum point is
        known to be inside the shrunken simplex and thus near the
        centroid.

         We generally want to estimate the uncertainties in the parameter
        values obtained for a model fit to a particular set of data
        points. Standard deviations of the parameters are calculated by
        the program used here.

         There are likely to be large uncertainties in the parameters if
        there are few data points or if there are large deviations
        between Ymodel and Yobserved. As a rule, one should have 5 to 10
        times as many data points as parameters.

         The first try at estimating uncertainties of the parameters can
        fail. The calculation involves matrix inversion, the use of
        differences between nearly equal large numbers, and the
        approximation of a complex surface by a simple quadratic
        function. It may be necessary to change certain test values and
        then to repeat the calculation of the standard deviations. In
        particular, if a parameter is close to a bound, so that expansion
        of the simplex in that dimension is not possible, then that
        parameter should be fixed in the quadratic fit.


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         All fitting methods can fail. We will not discuss problems with
        bounds, local minima, ill-behaved functions, poor quality data,
        physically unreasonable best fits, etc. References given in
        subsection C below discuss fitting methods in more detail.

         For additional discussion of fitting by use of the simplex
        method, see "Notes on the fitting program", and the article by
        Nelder and Mead (1965).


        B. THE FUNCTION FIT TO THE DATA

         In the computer program you will use, the function fit to the
        data, which you obtained in the kinetic experiment with lactate
        dehydrogenase, is for an ordered ternary-complex pathway with

        dead-end complexes (EAP and EQB). The following scheme describes
        the pathway:

                            EAP
                             |   KPInhib = ea * p/eap
                             |
                  ----------EA-----------
           k1 * a | k-1             k-2 | k2 * b
                  |                     |
                  |                     |
                  E               EAB <---> EPQ                 eq. 5
                  |                     |
                  |                     |
               k4 | k-4 * q     k-3 * p | k3
                  ----------EQ-----------
                             |
                             |   KBInhib = eq * b/eqb
                            EQB

         This pathway is more complicated than the one, without dead- end
        complexes, on which are based the standard graphical methods of
        analysis of two-substrate--two-product reactions:

                  -----------EA----------
                  |                     |
                  |                     |
                  E               EAB <---> EPQ                 eq. 6
                  |                     |
                  |                     |
                  -----------EQ----------

         For the direction of reaction (pyruvate reduction by NADH) and
        the product inhibitor (lactate) studied in these measurements,
        the rate law for the pathway of equation (5) is:






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             vo  =  V / [ 1  +  KmQ/KmPQ * p  +  KmA/a       eq. 7

                       +  KmB * (1 + KmQ/KmPQ * p) * (1 + 1/KPInhib * p)

                       +  KmAB/(a * b) * (1 + KmQ/KmPQ * p)


                       +  k3/(k3 + k4) * 1/KBInhib * b ]

         The presence in the pathway of equation (5) of the dead end
        complexes EAP and EQB leads to a significantly more complicated
        rate law than is found for the simpler pathway of equation (6);
        compare equation (7) with the following equation, for the "bare"
        compulsory order pathway without dead-end complexes (the pathway
        of equation (6)):

             vo  =  V / [ 1  +  KmQ/KmPQ * p  +  KmA/a       eq. 8

                       +  KmB * (1 + KmQ/KmPQ * p)

                       +  KmAB/(a * b) * (1 + KmQ/KmPQ * p) ]

         The computer program used to reduce the kinetic data fits
        equation (7) to measurements of the initial rate v(0) as a
        function of the concentrations of NADH, pyruvate, and lactate.

         The parameters that are evaluated in the fitting are listed in
        Table I and are those of equation (7).

         Several points should be noted regarding equations (7) and (8):
        (1) The last term of equation (7), containing the equilibrium
        constant KBInhib, probably can be neglected under initial rate
        conditions with q=0; in the fitting program this is recognized by
        setting the last term at a small value, 1E-10, which removes it
        from the equation.  (2) With this change, equations (7) and (8)
        are identical except for the appearance in one term of equation
        (7) of a factor containing the equilibrium constant KPInhib,
        which is for dissociation of the dead-end complex EAP defined in
        the pathway of equation (5).


















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        By J.A. Rupley, Tucson, Arizona

        REFERENCES:

        A simplex method for function minimization.
        J.A. Nelder and R. Mead (1965. Computer J. 7, 308.

        Digital computer user's handbook.
        M. Klerer and G.A. Korn (1967). Mcgraw-Hill, New York.

        Data analysis in biochemistry and biophysics.
        M.E. Magar (1972). Academic, New York.

        The solution of the general least squares problem with special
        reference to high-speed computers.
        R.H. Moore and R.K. Ziegler (1960). Los Alamos Scientific
        Laboratory Report LA-2367.








































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        By J.A. Rupley, Tucson, Arizona

        TABLE I:

                                 EQUATION (7)
        FITTING                  RATE LAW
        PARAMETERS               PARAMETERS

        1                        V

        2                        KmA  =  KmNADH

        3                        KmB  =  KmPyruvate

        4                        KmAB =  KmNADH-Pyruvate

        5                        KmQ/KmPQ = KmNAD/KmLactate-NAD

        6                        1/KPInhib = 1/KInhibLactate

        7                        k3/(k3 + k4) * 1/KInhibPyruvate
                                      parm(7) approx. equal to 0 at t=0


        INDEPENDENT VARIABLES:

        a = [NADH]     b = [Pyruvate]    p = [Lactate]    q = [NAD]
                                                          q = 0 at t=0


        DEPENDENT VARIABLE:

        vo = initial rate of conversion of pyruvate to lactate
























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        By J. A. Rupley, Tucson, Arizona

        PROCEDURES:

        A. SUMMARY OF STEPS IN THE DATA REDUCTION

         (1) Startup. Create a new file with your data, following the
        template of the DATA SHEET. Setup the title and the starting
        ranges of the parameters, from which the starting simplex is
        constructed. Be sure to save the setup on disk.

         (2) Minimization. The minimum of the least squares function is
        located by an iterative method, in which the simplex, a
        polyhedron in parameter space, is shrunk about the point
        representing the minimum. The starting simplex is large,
        sufficiently so to certainly include the minimum point. After
        shrinking, all the vertices of the simplex are so close together,
        and thus so close to the minimum point, that the average of the
        vertices, the centroid, is a good estimate of the minimum point.
        The minimization can take more than two hours. These results
        should be saved on a disk file, so if the fitting fails, it is
        possible to quickly return to this point.

         (3) Quadratic fit and standard deviations. The least squares
        function about the minimum is approximated by a quadratic
        surface, the shape of which gives estimates of the standard
        deviations of the parameters. This step takes about 10 minutes.

        There is considerable output, the last of which is a list of the
        standard deviations.

         The fitting program can be set to cycle between a set of
        minimization iterations and the quadratic approximation. This
        option can produce quicker convergence.


        B. FAILURE OF THE FITTING

         Fitting can lead to physically unreasonable values of the
        parameters, for example for a kinetics experiment a set of
        parameters for which V is unreasonably large or small, or a set
        with standard deviations such that one or more of the principal
        parameters (V, KmA, KmB, KmAB) have no significance.

         The best fit may be unacceptable. To judge whether a fit is
        acceptable, compare the root mean square error (RMS error) with
        the expected uncertainty in your independent variable, the meas-
        ured rates.

         The most likely causes of failure are:

         (1) A few data points (flyers) that are widely wrong. The
        results should always be examined for flyers, which we will
        somewhat arbitrarily define as measured values differing from
        calculated values by more than 3 standard deviations: ABS(Ymodel


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        - Yobserved) > 3 x RMS ERROR. If there are flyers, delete them
        and repeat the calculation, or correct the values by
        recalculation of vo from your experimental data.

         (2) Trapping of the fit in a local minimum in parameter space
        not near the true minimum, or a broad and ill-defined minimum
        region. To test for trapping or a broad minimum, start the
        fitting with a different simplex. For example, set tighter
        starting ranges on one or more of the principal parameters (V,
        KmA, KmB, KmAB). Alternatively, set one principal parameter, e.g.
        V, at a reasonable value.

         Generally, to test for trapping, one repeats the fitting with a
        more expanded starting simplex, not with a tighter one as
        suggested above. However, assuming the tighter ranges include
        physically reasonable values of the parameters, the approach

        above, using a restricted simplex, has the advantage of testing
        whether a set of physically reasonable values can give an
        acceptable fit of the data. To judge whether a fit is acceptable,
        even though it may not be as good as a physically unreasonable
        best fit, compare the root mean square error (RMS error) with the
        expected uncertainty in your dependent variable, the measured
        rates.

         (3) Systematic error in the measurements or poor data. Look for
        systematic bias in the extraction of the slope. Look for errors
        in calculating the value of vo.

         One can plot the measured values with the calculated lines, ie
        construct the reciprocal plots. It should then be clear what the
        problem is. Correction, if it is possible, generally requires
        examination of the original measurements, ie the tracings of
        transmittance versus time. A point to be made is that the
        computer fitting, because it is unbiased, is more likely than the
        experimenter to detect systematic error or poor data.





















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        By J. A. Rupley, Tucson, Arizona


        INSTRUCTIONS FOR FITTING KINETIC DATA FOR LACTATE DEHYDROGENASE
        WITH THE PROGRAM LDHFIT.COM (C-LANGUAGE):



        A. CREATE THE DATA FILE

         By use of a word processing program, modify the template file
        LDHDATA, incorporating your vo values and so constructing your
        own data file, eg TUE-C.DAT.


        B. RUN THE FITTING PROGRAM

         Run the program LDHFIT with your data file (eg TUE-C.DAT) to
        produce the results file (eg TUE-C.PRT)

         A>LDHFIT TUE-C.DAT TUE-C.PRT

         It takes about 50 minutes for 100 iterations. 200 to 400
        iterations are commonly required for convergence. The fitting
        will cycle between sessions of Nelder-Mead minimization (30
        iterations) and quadratic fitting.

         You should watch the fitting. Let the program run freely until
        the test value is < 10-4. Then:  (1) See if a parameter needs to
        be eliminated. If the calculation of -pmin- fails at this stage
        in the fitting, probably the fitting will not converge. If one of
        the parameter values listed under -pmin- in the summary of the
        quadratic fit results is negative, then this parameter should be
        fixed at 1.e-10. This can be done by use of a program option (see
        below).  (2) Examine the data listing for flyers: (Yobserved(i) -
        Ycalculated(i)) > 3 * rms error. If these are present, you should
        delete them from the data set, or set their weight to zero. This
        is done by reworking the data file, with a word processor, then
        rerunning LDHFIT. You will get faster convergence by setting the
        starting simplex in the data file at an intermediate point in the
        fitting.

         To exercise options, during the minimization type:  CTRL-X

         After some output, you will be asked to select an option:
             display of the data and then quadratic fit (<CR>)
             fix of a parameter (CTRL-F)
             exit from the program to operating system (CTRL-C)
             return to the fitting (CTRL-X)








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        B. AFTER THE FITTING CONVERGES

         When the simplex has shrunk to where it satisfies the exit test,
        the minimization pauses and expects input from the operator:

             respond with <RETURN> when asked if you want to  display data;

             respond with <RETURN> when asked at the next pause if you
        want to carry out quadratic fit;

             at the next pause, record the number of the iteration, and
        exit by typing CTRL-C;


        D. CONDENSE THE RESULTS FILE

         The results file (eg TUE-C.PRT) will be too long to print.
        Select the information from the last data display and quadratic
        fitting by use of a word processor or a special search program.


        E. FAILURE OF THE FIT

         Repeat the fit after checking your data and possibly resetting
        the starting parameter values.
































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