Download Calculating enzyme kinetic parameters from protein structures

Bringing Together Biomolecular Simulation and Experimental Studies
Calculating enzyme kinetic parameters from
protein structures
Matthias Stein*1 , Razif R. Gabdoulline*† and Rebecca C. Wade*
*Molecular and Cellular Modeling Group, EML Research gGmbH, Schloss-Wolfsbrunnenweg 33, 69118 Heidelberg, Germany, and †BIOMS, University of
Heidelberg, Im Neuenheimer Feld 368, 69120 Heidelberg, Germany
Abstract
Enzyme kinetic parameters can differ between different species and isoenzymes for the same catalysed
reaction. Computational approaches to calculate enzymatic kinetic parameters from the three-dimensional
structures of proteins will be reviewed briefly here. Enzyme kinetic parameters may be derived by modelling
and simulating the rate-determining process. An alternative, approximate, but more computationally
efficient approach is the comparison of molecular interaction fields for experimentally characterized
enzymes and those for which parameters should be determined. A correlation between differences in
interaction fields and experimentally determined kinetic parameters can be used to determine parameters
for orthologous enzymes from other species. The estimation of enzymatic kinetic parameters is an important
step in setting up mathematical models of biochemical pathways in systems biology.
Introduction
In systems biology, the fluxes and kinetics of substrates,
metabolites and products in a biomolecular network
are quantified. The network can involve metabolic and/or
signalling pathways. The network may be described by a set of
coupled deterministic or stochastic differential equations that
may be solved by numerical integration. Setting up this set of
coupled differential equations requires the input of molecular
concentrations or particle numbers and a detailed set of
enzymatic kinetic parameters that depend on environmental
conditions such as temperature, pH and ionic strength. In
enzyme catalysis, enzyme–substrate association and turnover
are often described by Michaelis–Menten-like kinetics:
k1
k2
E + S ES −→ E + P
k−1
(1)
where E is the enzyme, S is the substrate and P is the
product. Here, kcat or the catalytic activity is given by k2
and K m ≈ k−1 /k1 (when k2 k−1 ) and can be interpreted as
a substrate binding affinity. The ratio kcat /K m is a measure of
the catalytic efficiency of the enzyme (see Figure 1).
When constructing models of biochemical networks,
it is often found that a necessary enzymatic kinetic parameter
is missing or available only under different experimental
conditions or for a different organism from that required.
Here, we will review computational approaches to calculate
these missing enzymatic kinetic parameters from threedimensional protein structures. Mathematical fitting of
Key words: computational approach, enzymatic kinetic parameters, enzyme–substrate
association, molecular interaction field, systems biology, three-dimensional structure of protein.
Abbreviations used: KIE, kinetic isotope effect; MM, molecular mechanical; QM, quantum
mechanical; PIPSA, protein interaction property similarity analysis; qPIPSA, quantitative PIPSA.
1
To whom correspondence should be addressed (email [email protected]).
Biochem. Soc. Trans. (2008) 36, 51–54; doi:10.1042/BST0360051
experimental data or parameter estimation based on network
models will not be considered here.
Different steps of enzymatic catalysis
need different modelling approaches
Enzymes lower the transition state barrier and thus accelerate
the reaction compared with the uncatalysed reaction in
solution. The formation of an initial encounter complex is
the first step in enzyme catalysis (affecting k1 and thus K m )
and can be investigated by modelling the enzyme–substrate
association process. Enzymatic turnover may be modelled
for the substrate bound in the active site of the enzyme.
Recent progress has delivered insights into the degree
of transition state stabilization by enzymes, e.g. the
preferential binding of the transition state by a preformed
enzyme pocket. The appropriate choice of computational
tool to calculate enzyme kinetic parameters depends critically
on the time scale of the slowest and thus rate-limiting
step. Depending on the underlying chemistry, very fast
processes, such as proton transfer (10−12 s) or side-chain
reorientation (10−11 –10−10 s), may be rate-limiting. On the
other hand, slower processes such as hinge bending at domain
interfaces (10−11 –10−7 s) or other conformational changes
may be critical [1] (for a review of simulation techniques,
see [2]).
Modelling the transition state
When chemical bond breaking and formation processes
are rate-determining, accurate modelling of the free energy
differences between the enzyme–substrate complex and
the transition state [E–S]# can yield the barrier to catalysis
and thus kcat (see Figure 1). Modern transition state theory
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Figure 1 Schematic diagram of the factors determining the rate
of an enzymatic reaction
The enzyme (E) and the substrate (S) form an enzyme–substrate
complex (E–S). Upon converting the substrate, the enzyme–
substrate complex passes over a transition state ([E–S]# ) to yield
the product (P) plus the recovered enzyme. Sufficient conformational
sampling around the transition state is required to achieve a statistically
significant determination of the transition-state free energy changes.
In transition-state theory, the changes in free energy [G# (T)] can
be used to determine the kinetics [k(T)] of over-the-barrier motion.
For light nuclei, QM corrections to transition state theory [among
others, the insertion of a transmission coefficient γ (T) to account for
through-the-barrier tunnelling motion of protons, hydrogen radicals and
hydrides] may explain the KIEs of some reaction rates. T is temperature,
β = k B T, and k B is Boltzmann’s constant.
of enzymatic reactions can be found in [3]. Once the
transition state has been localized, sufficient sampling of
the configurational space around the transition state is critical
for obtaining kinetic parameters. Usually, a VB (valence bond)
description for educts, products and intermediates along a
simplified reaction co-ordinate is used. Along this reaction
co-ordinate, VTST (variational transition state theory) can
be employed to calculate the free energies of activation and
thus to calculate rate constants. For a review of possible
potential energy functions to model the transition state,
see [4].
QM (quantum mechanical) tunnelling effects become
important when light nuclei (protons/hydrides) are involved.
Then the reaction displays a KIE (kinetic isotope effect)
and the reaction rate is lower when deuterated substrates or
solvents are used. Through-barrier tunnelling rather than
over-the-barrier motion (see Figure 1) can be dominating.
Modelling of the rate-determining proton transfer steps
requires a QM treatment of the bond breaking/forming
process plus consideration of the effect of the surrounding
protein atoms on the barrier height by hybrid QM/MM
(molecular mechanical) models.
An alternative to the static QM treatment of the active site
is the consideration of Newtonian atomic motion plus explicit
integration of the time-dependent Schrödinger equation in a
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and, recently, the extension to electronic motion in a hybrid
Car–Parrinello QM/MM approach (see [5]). If sufficient
sampling is carried out, accurate free energy barriers in
enzymatic catalysis can be obtained. A comprehensive
review on QM methods to calculate enzymatic rate constants
can be found in [6].
Modelling diffusional enzyme–substrate
association
The enzyme–substrate association step can be responsible
for a large modulation of enzymatic activity. When the
enzyme–substrate interaction is strong and catalytic
turnover is fast, the transition state barrier for the enzymatic
substrate-to-product conversion is not rate-determining
and kcat /K m is close to the diffusion limit [7]. In this case,
the formation of an initial enzyme–substrate diffusional
association complex becomes limiting. Indications for this
are the dependence of the rate constants on ionic strength
or viscosity of the solvent. Brownian dynamics simulation
can be used to simulate the diffusional association of
substrate and enzyme [8]. In these simulations, the solvent
is considered implicitly by solving the Poisson–Boltzmann
equation to model electrostatic interactions and its dynamic
effects are modelled by stochastic terms. Examples of
Brownian dynamic simulation that yield quantitative results
in agreement with experimental findings include enzyme–
substrate interactions (e.g. acetylcholinesterase, superoxide
dismutase and triose phosphate isomerase), enzyme–
inhibitor binding (e.g. barnase–barstar), antibody–antigen
binding and the association of electron transfer proteins [9].
Molecular interaction fields
Electrostatic potential has been recognized as one of the
major determinants of enzymatic catalysis both in enzyme–
substrate association and in transition state stabilization. The
comparative analysis of the electrostatic potentials of a large
number of enzymes by qPIPSA [quantitative PIPSA (protein
interaction property analysis)] [10] allows differences in
the molecular interaction field to be related directly to
differences in enzymatic kinetic parameters between species
or between mutants and wild-type (see Figure 2). For
deriving such a correlation, a reference set of experimentally
well-characterized enzymes is required.
It has been shown that results from PIPSA are comparable
with those from Brownian dynamics simulation but are
obtained at a fraction of the computing time [11]. From
qPIPSA, estimates of enzymatic K m and kcat /K m values
can be made based on a comparison of the electrostatic
potentials of enzymes around the active site or any region
responsible for specific enzyme kinetics. The computational
efficiency allows the functional and kinetic characterization
of a large number of enzymes, the detection of experimental
outliers and the characterization of the enzymes of an entire
metabolic pathway [10,12,13].
Bringing Together Biomolecular Simulation and Experimental Studies
Figure 2 Scheme showing a comparative approach for estimating enzymatic kinetic parameters based on an analysis of molecular
interaction fields
The molecular interaction fields of enzymes that catalyse the same reaction and possess similar three-dimensional structures
are calculated and compared. These molecular interaction fields can be, for example, the electrostatic potential or a
hydrophobic field. Differences in molecular interaction fields may be correlated with differences in enzymatic kinetic
parameters, such as K m and k cat /K m , for a given set of experimentally well-characterized enzymes. Such a correlation
can be used to determine enzyme kinetic constants for enzymes from other species or enzyme mutants [10].
Conclusions
References
Molecular simulations play an important role in elucidating
enzymatic reaction mechanisms. Progress in computing
algorithms and hardware has enabled the calculation of
enzymatic kinetic parameters at various levels of theory.
The direct applicability of simulated kinetic parameters
in systems biology is still hampered by the required high
accuracy of computed kinetic parameters. One example can
be found in [14].
In systems biology, the focus shifts from investigating
single enzymes to protein families, enzyme classes or entire
pathways. Thus computationally efficient simulations are
necessary to deal with a large number of biomolecules.
Coarse-graining and comparative approaches are general
strategies to accomplish this task [15]. The availability of
more and more accurate experimental data and computational
approaches will bring molecular and biochemical network
simulations closer together, thus enabling insights into more
complex biological phenomena.
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Financial support from the German Federal Ministry for Research
(BMBF) ‘HepatoSys’ project (grant numbers 0313076 and
0313078C), the Klaus Tschira Foundation and the Center for
Modelling and Simulation in the Biosciences (BIOMS; Heidelberg,
Germany) is gratefully acknowledged.
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kinetic parameters in systems biology by comparing molecular
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Characterizations; Proceedings of the 2nd International Beilstein
Workshop on ESCEC (Hicks, M.G. and Kettner, C., eds), pp. 237–253,
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Received 14 September 2007
doi:10.1042/BST0360051