Enzyme catalyzed reactions: From experiment to computational mechanism reconstruction

Abstract

The traditional experimental practice in enzyme kinetics involves the measurement of substrate or product concentrations as a function of time. Advances in computing have produced novel approaches for modeling enzyme catalyzed reactions from time course data. One example of such an approach is the selection of appropriate chemical reactions that best fit the data. A common limitation of this approach resides in the number of chemical species considered. The number of possible chemical reactions grows exponentially with the number of chemical species, which makes difficult to select reactions that uniquely describe the data and diminishes the efficiency of the methods. In addition, a method’s performance is also dependent on several quantitative and qualitative properties of the time course data, of which we know very little. This information is important to experimentalists as it could allow them to setup their experiments in ways that optimize the network reconstruction. We have previously described a method for inferring reaction mechanisms and kinetic rate parameters from time course data. Here, we address the limitations in the number of chemical reactions by allowing the introduction of information about chemical interactions. We also address the unknown properties of the input data by determining experimental data properties that maximize our method’s performance. We investigate the following properties: initial substrate–enzyme concentration ratios; initial substrate–enzyme concentration variation ranges; number of data points; number of different experiments (time courses); and noise. We test the method using data generated in silico from the Michaelis–Menten and the Hartley–Kilby reaction mechanisms. Our results demonstrate the importance of experimental design for time course assays that has not been considered in experimental protocols. These considerations can have far reaching implications for the computational mechanism reconstruction process.

Introduction

Enzyme kinetics is the study of the rates and mechanisms of enzyme catalyzed reactions. The traditional experimental practice to study kinetics requires the measurement of concentrations as a function of time. Data obtained from these measurements is known as time course data and it can be employed to estimate the initial reaction rates. With this initial rate data, enzymologists estimate the rate parameters or study underlying reaction mechanisms. This is accomplished using theoretical models based either on the steady-state approximation or rapid-equilibrium approximation (Cornish-Bowden, 1995b, Schnell and Maini, 2003, Segel, 1975).

Advances in computing have brought novel tools for the estimation of kinetic parameters and modeling of enzyme catalyzed reactions (reviewed in detail by Garfinkel et al., 1970). A number of well-known programs were developed to numerically model enzyme catalyzed reactions. Among these are KINSIM (Barshop et al., 1983), FITSIM (Zimmerle and Frieden, 1989), Leonora (Cornish-Bowden, 1995a), DYNAFIT (Kuzmič, 1996) and Gepasi (Mendes and Kell, 1998). Most of these programs are used to investigate the behavior of kinetic equations over a range of parameter values, or estimate the parameters through robust fitting algorithms to existing models. However, there is little available literature on the computational determination of enzyme reaction mechanisms (Arkin and Ross, 1995, Fine and Hwang, 1979, Pi and Leary, 2004, Walsh et al., 1978, Wang et al., 2006, Yeung et al., 2002).

In a previous paper, we developed a novel computational method for both inferring reaction mechanism and estimating the kinetic parameters of biochemical pathways: Method to Infer Kinetics And Network Architecture (MIKANA). Given a set of time course curves, the program identifies elementary chemical reaction steps that constitute the enzyme catalyzed reaction using a global non-linear modeling technique. In order to reconstruct the biochemical pathway, MIKANA selects appropriate chemical reactions from a dictionary of elementary chemical reaction schemes. This process is aided by a cost function namely the Information Criterion (IC), that penalizes the use of an excessive number of reactions to reconstruct the pathway.

In the previous work, we illustrated the effectiveness of our method by successfully identifying the topology (wiring diagram) of the glycolytic pathway of a bacterium from a C¹³ NMR time course curves (Srividhya et al., 2007). The data consisted of seven chemical species concentration values over time. In order to obtain the topology, the selection process consisted only of unimolecular elementary reactions. The number of elementary reactions grows exponentially with the number of chemical species, which increases when considering bimolecular reactions. In the presence of a high number of potential reactions, MIKANA fails to identify reactions that contribute the most in explaining the experimental data (Crampin et al., 2004). Note that more than one set of elementary reactions can fit the data and therefore explain the observations. This is known as the mechanism distinguishability problem. Moreover, a vast search space prohibits exhaustive search methods as we have to impose limits on computational time. The performance of the method is also dependent in several quantitative and qualitative properties of the data provided by the experiments. These properties include initial chemical species concentrations, number of time points and noise. Understanding how MIKANA behaves using data with different properties helps to identify characteristics of the data that maximize the method’s accuracy. This information is of importance to experimentalists, as it allows them to setup their experiments in ways that optimize the application of the network reconstruction model to the data they collect and provide.

In this paper, we address the above limitations by allowing the introduction of information about chemical interactions and determining properties of the data that maximize MIKANA’s performance. The information about chemical interactions consists of reactions known to be absent in the mechanism under study. It may also include identities of known chemical species such as enzymes or substrates. The chemical species interaction information is used to reduce the number of possible interactions in the dictionary of chemical interactions. To address the importance of introducing this information, we generate data in silico from two reaction mechanisms: the Michaelis–Menten (MM) reaction mechanism; and the Hartley–Kilby (HK) reaction mechanism. We compare the performance of our method in the presence and in the absence of information about chemical interactions.

Using data generated in silico from the reaction mechanisms stated above, we determined the patterns of the experimental data that maximize MIKANA’s performance. We tested for: different initial substrate–enzyme concentration ratio $(s_0/e_0)$; different initial concentration ranges of substrate ($s_0$) and enzyme ($e_0$); different number of data points; different number of time courses (experiments); and different data noise levels. This analysis can have far reaching implications in the field of experimental enzyme kinetics. The next subsection describes MIKANA. In Section 2 we detail the methodology employed in this work, followed by results in Section 3. The last section presents discussion and conclusions.

MIKANA infers biochemical pathway mechanisms from time course data using a global non-linear modeling technique to identify the elementary reaction steps which constitute the pathway. A number of different steps are taken to arrive at a reaction mechanism. The sequence of steps involved in the method is: construction of a model design matrix; construction of a derivative matrix; model selection module; and reconstruction of the ordinary differential equations (ODE’s). The final output consists of the predicted reaction steps and the reconstructed ODE model for the pathway (see Fig. 1).

The model design matrix is a matrix with columns representing unscaled velocities corresponding to all possible elementary reaction steps involving the different species in the pathway. Therefore, the first step of our method consists of the construction of a complete dictionary of chemically feasible elementary reaction steps for a given number of species. The logic used to generate these reactions is based on mass action kinetics. We have also restricted the reactions to be only unimolecular or bimolecular elementary reactions. Time course of the chemical species is used to calculate the derivative matrix. The data points are interpolated if the time series contains few time points. All the reactions from the complete dictionary are used to form the initial model. A sensitivity analysis is used to determine the reaction that will least damage the model fit to the data. This way, reactions are discarded iteratively, until a cost function is minimized and a mechanism is identified. Differential rate equations are then reconstructed from the reactions and the coefficients are inferred. Finally, the ODE reconstructor gives the differential equations as the final output of the algorithm (Srividhya et al., 2007).

We built a website interface for kinetic software. It can be found at http://cheminfo.informatics.indiana.edu:8080/biofw/biofw. The MIKANA software package is available for download at this address. We are also currently working on an user interface for MIKANA.