A R T I C L E S
Spies et al.
multiple data sets each with independent starting conditions (e.g.,
substrate concentration). DynaFit was chosen for the present application
due to its flexibility, ease of use, robustness, statistical features, and
scripting abilities.
randomization were as follows: k1 and k8, 5 × 104-5 × 107 M-1 s-1
;
k2, k3, k6, and k7, 50-50 000 s-1; k4 and k5, 1 × 107-1× 1010 s-1. Rate
constant sets for which any one of the four calculated steady-state
parameters (two kcat and two KM values) fell outside of the range of
50-150% of the experimental values were eliminated from further
consideration. For the pH 8.9 and 6.9 sets, this resulted in 160 and
214 parameter sets, respectively, being selected for initial global fitting.
Batch Global Fitting of Screened Rate Constant Sets. The selected
rate constant sets were fitted in batch mode using DynaFit. This requires
only an input script from the user, enabling a large number of fits to
be executed automatically. No constraints were placed on the rate
constants during the fitting procedure, and all eight rate constants were
fitted. A modification of the default settings for DynaFit was used. A
stopping criterion for the Marquardt compromise parameter (λ) was
employed (StopLambda ) 1 × 10-7), and the Levenberg-Marquardt
algorithm was set to restart once an apparent minimum had been reached
(Restarts ) 5). Also, DynaFit can adjust the initial concentrations of
the reacting species (alanine and enzyme) up to 10% from the input
values to correct for pipetting errors. This feature was used in this initial
phase of the fitting process.
Enzyme kineticists have traditionally shunned progress curve
analyses due to the difficulties in integrating the equations for complex
mechanisms and applying the resulting complicated integrated rate
equations to data analysis. This is not to say that integrated rate
equations have not been successfully employed in enzymology.35 With
the advent of programs such as DynaFit, these integration and data
analysis difficulties for progress curves are eliminated. One can collect
a large number of full progress curves that differ in, for example, initial
substrate concentration. These can be read into DynaFit and fitted to
the chemical mechanism describing the enzyme under analysis, yielding
best-fit estimates for the microscopic rate constants of the mechanism
through a straightforward procedure.
Alanine racemase is particularly amenable to this treatment, due to
the kinetic simplicity (one substrate and one product) and the definite
equilibrium constant of unity. Additionally, the chemical mechanism
of the enzyme was recently shown to be a stepwise double proton
transfer by multiple kinetic isotope effects (KIEs).7 The fits obtained
from global analyses of multiple progress curves can be confirmed by
a variety of independent measurements with alanine racemase. For
example, steady-state kinetic constants (e.g., kcat, KM) are known
aggregates of the microscopic rate constants (the fitted parameters in
the present procedure) describing the mechanism. Values for these
steady-state constants are easily obtained from initial rate experiments
and can be compared to those calculated from the fitted rate constants.
The effect of viscosity variation, which generally probes the contribu-
tions of bimolecular steps to rate limitation, offers additional validation.
Other independent measurements that can be used as checks on the
fitted rate constants include spectroscopic analysis of enzyme inter-
mediates under substrate-saturating equilibrium conditions, KIEs, and
the presence or absence of “overshoots” in progress curves conducted
in D2O.
Steady-state kinetic parameters were calculated from the fitted
microscopic rate constants and were compared to the experimental
values. Fitted rate constant sets were selected based on both the
calculated steady-state parameters (kcat’s, KM’s, and the equilibrium
constant) and the mean square value. The mean square is defined as
the sum over all data points of the square of the difference between
the predicted and observed value divided by the total number of data
points. DynaFit provides the option of weighting individual progress
curves within the global data set. The default setting (Equalize ) 1)
was used, which ensures that each progress curve contributes equally
to the global fit even though the curves do not contain identical numbers
of data points. For the pH 8.9 data set, rate constant sets were eliminated
if the values of any of the calculated steady-state kinetic parameters
fell outside of the range of 50-150% of the experimental values or
had mean square values larger than 1.02. This yielded three parameter
sets. All of these selected sets had calculated equilibrium constants
within 20% of unity. These three sets were then averaged, resulting in
the “averaged” rate constant set for pH 8.9 (Table 2). For pH 6.9, rate
constant sets were eliminated if the values of any of the calculated
steady-state kinetic parameters fell outside of the range of 50% to 150%
of the experimental and had mean square values larger than 1.7.
Refinement of Averaged Rate Constant Set. The averaged set is
likely to be near the values that give the global minimum of mean
square. Thus, the averaged rate constant set is used as initial rate
constants in an additional fit, yielding the “averaged-fitted” rate constant
set (Table 2). To examine more systematically the hypersurface around
the averaged-fitted rate constant set, the rate constants were each
randomized by 5%, to generate 70 new rate constant sets, which were
then independently globally fitted. It was found that the 10% titration
in initial substrate and enzyme concentrations had a deleterious effect
on the ability of the algorithm to optimize the 5% randomized rate
constant sets to closely similar fits, yielding mean square values that
spanned a large range. Convergence is greatly improved when the
titration feature is turned off. Furthermore, the value of any given rate
constant varies only slightly between fitted rate constant sets when the
5% randomized sets are used as initial estimates. The same procedure
was employed for the pH 6.9 data set, yielding similar results (Table
3). The mean square values spanned a large range when the titration
feature is turned on and is greatly reduced when turned off. The
“optimized” rate constant sets for the pH 6.9 and 8.9 data are those
that gave the lowest mean square values in the fits of the 5% randomized
rate constant sets using the tight convergence criteria.
The application of nonlinear regression to kinetic analyses is
straightforward when relatively simple, well-behaved equations and data
with a strong signal are employed. The difficulty increases greatly as
the number of fitted rate constants increases.36 Thus, in fitting data to
a mechanism with, for example, eight rate constants, it is a challenge
to find the set of values that gives the global minimum on the mean
square hypersurface. Here, this problem of finding the global minimum
is addressed by starting the search for the optimal set of rate constants
with the screening of quasi-randomly generated sets of rate constants.
Sets that yield values of steady-state kinetic constants not too distant
from experimentally determined values are selected for full global
analysis. The rate constants of the selected sets are used as initial
estimates in global fits to a group of experimental progress curves.
Those globally fitted sets of rate constants that are in close accord with
experimental data (e.g. equilibrium constant) are selected and averaged
together, and this averaged set of rate constants is used as initial
estimates in a further global fit.
The hypersurface about the apparent minimum is explored to ensure
that the global minimum has been reached by randomizing the fitted
rate constants by a small fraction and then globally refitting these
randomized rate constant sets. If the true minimum has been reached,
then these will reconverge on the original solution. This set of optimized
rate constants is used to construct the free energy profile for the enzyme,
which is itself validated by independent experimental data.
(B) Specific Methodology. Screening of Quasi-Randomized Sets
of Rate Constants. The screening procedure employed 1000 quasi-
randomly generated sets of rate constants. The stepwise mechanism is
defined by eight rate constants (Scheme 1). The limits on the
Energy Boundary of the Quinonoid Intermediate. The fitted
values for k4 and k5 are likely to be the least well determined, since
they are most removed from the bimolecular steps that are directly
probed by the substrate concentration dependence. Further analysis of
(35) Duggleby, R. G. Methods Enzymol. 1995, 249, 61-90.
(36) Motulsky, H. J.; Ransnas, L. A. FASEB J. 1987, 1, 365-74.
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7466 J. AM. CHEM. SOC. VOL. 126, NO. 24, 2004