ENZO: a web tool for derivation and evaluation of kinetic models of enzyme catalyzed reactions

Abstract

We describe a web tool ENZO (Enzyme Kinetics), a graphical interface for building kinetic models of enzyme catalyzed reactions. ENZO automatically generates the corresponding differential equations from a stipulated enzyme reaction scheme. These differential equations are processed by a numerical solver and a regression algorithm which fits the coefficients of differential equations to experimentally observed time course curves. ENZO allows rapid evaluation of rival reaction schemes and can be used for routine tests in enzyme kinetics. It is freely available as a web tool, at http://enzo.cmm.ki.si.

Figure 1. Michaelis-Menten reaction scheme.

E is the free enzyme, S the substrate, ES the Michaelis complex and P the product; k₀ is a second order and k₁ and k₂ are first order rate constants, respectively. The differential equations were automatically generated from the drawn reaction scheme by ENZO.

Figure 2. ENZO URL:

http://enzo.cmm.ki.si . The ENZO web page provides a short introduction and links to a quick guide, examples and ENZO tool.

Figure 3. Enzyme titration reaction scheme.

E is the Torpedo californica acetylcholinesterase enzyme, I the m-(N,N,N-trimethylammonio) trifluoroacetophenone (TMTFA) inhibitor, and EI their complex; k₀ is a second order association rate constant. The reaction was drawn using the Reaction Scheme tab of ENZO.

Figure 4. Converged results of parameter fitting for enzyme active site titration experiment.

Initial concentrations of enzyme E and inhibitor I for progress curve files tfk1.dat, tfk2.dat, tfk3.dat (Experimental Data panel shows tfk1.dat) are fitted in the interval of [0, 10²⁰]; the initial concentration of EI is zero and fixed; the checkbox “E” is checked under Measured Species, which signifies that E is the measured quantity and the progress curves below represent the time course of its residual activity. The respective units of the residual activity in the Y-axis are OD/min and the units of time in X-axis are seconds. Fitted rate constant k₀ and initial values of E and I at three different concentrations of I are displayed under the Evaluated Parameters in the upper right corner panel, the experimental progress curves are blue and the fitted curves are red as shown in Time Course of the Reaction chart at the bottom panel of the screen. The arrows mark the difference between the inital value and the plateau.

Figure 5. Normalization curve for the determination of active site concentration from enzyme activity.

X-axis represents the concentration of added TMTFA corresponding to actual enzyme concentrations after 300 times delution. Y-axis represents the difference between the initial enzyme activity and the activity at plateau caused by the presence of experimental TMTFA concentration. The concentration of the substrate in all activity determinations was 0.5 mM.

Figure 6. Reaction scheme for autoactivation of procathepsin B.

E stands for free active enzyme, S for enzyme precursor and ES is the complex. Note the stoichiometry of the reaction in the differential equations.

Figure 7. Autoactivation of procathepsin B.

The data originally presented in were used. The total amount of protein was determined by Pace et al. and used as a fixed value for each substrate/precursor concentration. The initial values of ES is zero and E is fitted in the interval [0, 1]. The initial value of rate constant k₀ is set to diffusion rate value of 10⁸ M⁻¹ min⁻¹ and fixed, while initial values of k₁ and k₂ are 200 min⁻¹ and 7.2 min⁻¹ respectively, and fitted in the interval of [0, 10²⁰]. The sum of free active enzyme (E) and the instantaneusly dissociated complex (ES) is the measured species. Y-axis represents the concentration of cathepsin B in molar concentration and the X-axis represents time in minutes. The final estimated values of rate constants and initial concentrations of active enzyme portion for each individual curve are displayed under the Evaluated Parameters.

Figure 8. Cholinesterase reaction with a substrate.

The reaction scheme and the corresponding differential equations were created by ENZO. Substrate (butyrylthiocholine) bound to the peripheral anionic site is denoted by “S” on the left of the label name (e.g., SE, SES, SEA, SEAS). When bound to the catalytic anionic site, the “S” is placed on the right of the name (e.g., ES, SES, EAS SEAS). Covalent acyl-enzyme is represented by EA and P is the first product (thiocholine) released upon enzyme acylation. The acyl group is denoted by A.

Figure 9. Converged results of parameter fitting for inhibition of cholinesterase.

Fitted rate constants k₁, k₃, k₄, k₅, k₁₀, k₁₁, k₁₂ are displayed under the Evaluated Parameters. k₀ is fixed at 8.8·10⁶ s⁻¹. ENZO allows, if necessary, for constraining several rate constants, thus simplifying the fitting and speeding up the evaluation process (k₁₄ = k₈ = k₆ = k₀, k₁₅ = k₉ = k₇ = k₂ = k₁, k₁₃ = k₃, k₁₀ = 2k₄, k₁₁ = 50k₅, where k₁₄, k₈, k₆, k₀ are second order while others are first order rate constants). For all progress curves, the initial concentrations of SES, SE, EAS, ES, SEAS, P, ES, SEA are fixed to zero and not fitted. E is fixed to 2.6 nM as independently determined by active site titration. For the progress curves chst1.dat to chst14.dat, S is fixed to 2 µM, 5 µM, 10 µM, 20 µM, 4.35 µM, 7.63 µM, 0.16 mM, 0.5 mM, 1 mM, 2 mM, 5 mM, 10 mM, 20 mM and 5 mM respectively. P is the measured species. Y-axis shows the product in molar concentration and X-axis shows the time in seconds.