A modeling and simulation approach to the study of metabolic control analysis

Authors

  • Carlos Rodríguez-Caso,

    1. Departamento de Biología Molecular y Bioquímica, Facultad de Ciencias, Universidad de Málaga, E-29071 Málaga, Spain
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  • Francisca Sánchez-Jiménez,

    1. Departamento de Biología Molecular y Bioquímica, Facultad de Ciencias, Universidad de Málaga, E-29071 Málaga, Spain
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  • Miguel Ángel Medina

    Corresponding author
    1. Departamento de Biología Molecular y Bioquímica, Facultad de Ciencias, Universidad de Málaga, E-29071 Málaga, Spain
    • Departamento de Biología Molecular y Bioquímica, Facultad de Ciencias, Universidad de Málaga, E-29071 Málaga, Spain Tel.: 34-95-2137132; Fax: 34-95-2132000
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  • This work was supported by funds from the Andalusian Government (to the CVI-267 Group) and from the Spanish Ministry for Science and Technology.

Abstract

Metabolic control analysis has contributed to the rapid advance in our understanding of metabolic regulation. However, up to now this topic has not been covered properly in biochemistry courses. This work reports the development and implementation of a practical lesson on metabolic control analysis using modeling and simulation.

The abbreviation used is: MCA, metabolic control analysis.

An understanding of metabolism is essential for the understanding of life. However, on many occasions students look on the study of metabolism as a boring task; they are almost reduced to learning a number of metabolic pathways by heart. On the other hand, the central role of metabolic regulation is not always stressed conveniently in biochemistry courses. To study regulation requires a knowledge of changes of metabolites and enzymes with time, that is, of kinetics. Traditionally, kinetics has been taught in biochemistry courses in terms of enzyme steady-state kinetics. Obviously, this is a detailed study of the local properties of the individual enzymes [1]. However, control emerges as a systemic property of metabolic networks [2], thus requiring a systemic approach to its study, including sensitivity analysis. Among the different theories built to make this approach possible, metabolic control analysis (MCA)1 has become a standard approach [3, 4].

Despite the fact that MCA has contributed to the rapid advance in our understanding of metabolic regulation at the research level in recent years, paradoxically this topic is completely absent from most general biochemistry textbooks, with the remarkable exception of the text by Voet and Voet [5]. Fortunately, some special books devoted to metabolic control [6, 7] and the inclusion of MCA in some recent enzymology books [8, 9] are contributing to change this situation.

In our department, MCA has been included in the syllabus of a course devoted to metabolic biochemistry (formerly called Metabolic Regulation) for 20 years. Our own experience as teachers is that, in a significant proportion of students, the study of MCA induces what has been called “numerophobia” [10], as is the case for the study of enzyme kinetics. In our experience, this is mainly because of a “scenic panic” against the non-linear equations of MCA because of an insufficient mathematical background. This “numerophobia” frequently causes repulse and/or an inadequate assimilation of the central concepts of MCA. To avoid this problem, we think that a practical approach could contribute to facilitate the assimilation of MCA. Several bench practical approaches to the study of MCA have been developed previously and applied successfully [11, 12]. However, more than often the non-linear equations of MCA have no analytical solutions; modeling and simulation can be particularly useful here.

Herein, we communicate our teaching experience in the development and implementation of a practical lesson on MCA using modeling and simulation.

OBJECTIVES

  • To promote an easy connection between enzymology and metabolism

  • To build a model of a metabolic pathway from the kinetic parameters of the enzymes involved and some data on the concentrations of substrates and modulators

  • To show clearly that there is a close relationship between the local behavior of any enzyme and the systemic responses of the metabolic pathways

  • To get, by simulation, an easy, fast, and inexpensive method to obtain the required data to apply MCA theory

  • To use concepts that are not always clearly assimilated by students

  • To facilitate the learning of the different coefficients and theorems of MCA

STRATEGY

First, students are taught how to easily design a model of a metabolic pathway from available kinetic parameters by using programs that allow such a design in an intuitive manner, without the need of intense computer knowledge.

In a second stage, students are taught how to simulate the model by changing the parameters required to apply MCA. Here, students have to identify accurately when a steady state is reached, a required condition to apply MCA.

Finally, students obtain the numerical data required to apply MCA from the simulations. Several coefficients can be calculated, and the different theorems of MCA can be verified.

RESULTS AND DISCUSSION

In our Metabolic Biochemistry course, we have developed a practical protocol to be carried out in a 2–3-h session in the computer laboratory. Students, grouped in pairs, were asked to study a virtual linear four-step pathway, with the last step being irreversible (Scheme 1). The initial values of parameters are given in Table I

The model was built and designed as a compartment system by using the program STELLA (13). This is an intuitive program that allows the easy design of metabolic pathways in a graphical manner. Assuming Michaelian behavior of all of the involved enzymes, functions relating to the compartments were defined by using the Michaelis-Menten equation, the total velocity equation, and the Haldane relationship (9) (Scheme 2). Starting with the initial values given in Table I, the behavior of this model system was simulated by using the program MADONNA (14). Once steady-state conditions were reached, numerical data were acquired to allow for the determination of different coefficients of MCA. In our practical session, we suggested calculating flux control coefficients for all the enzymes. The values obtained by our students are shown in Table II. From these data, the summation theorem of flux control coefficients can be confirmed immediately. Sequential simulations after changes in the external metabolite (X1) over a couple orders of magnitude allowed calculating the response coefficient for X1. The values obtained by our students were in the range of 0.25 to 0.30. We have observed that, at least with the programs used, it is compulsory to use as many decimal values as possible to obtain accurate results (otherwise, if we round values to the second or third decimal value, the truncation error increases exponentially over the steps in the simulation, giving rise to final erroneous values).

Although we have used Stella and Madonna, the procedure can be carried out using other programs. We recommend using GEPASY (15) because of its easy use, its popularity, and because it can be downloaded free of charge from the World Wide Web.

The approach described here can be used to model any kind of metabolic pathway, either real or virtual and linear or branched. This training is actually useful for a proper assimilation of the concepts of MCA, as we have confirmed with our students by a triple approach, which is described as follows.

  • According to our own records and notes on students' performance, the final degree of assimilation of MCA by those students who carried out the modeling and simulation practical session was higher than that of students in previous years who did not carry out this practical session.

  • Students had to solve numerical problems on MCA both before and after the practical session. The scores obtained in the exercises carried out after the practice were consistently higher than those obtained by the same students in those exercises carried out before the practical session.

  • Taking into account the results obtained in a survey completed by students after training, the students felt they had reached a better comprehension of MCA. Students agreed that the practical session indeed contributed to clarifying their doubts.

In conclusion, we are confident that this kind of approach can facilitate to students the study and proper assimilation of MCA, a systemic theory of control that we hope to be introduced in biochemistry teaching in a degree similar to the extension reached by metabolic regulation at research level.

Figure Scheme 1..

Virtual pathway consisting of four enzymes (E1, E2, and E3 showing reversible kinetics and E4 being reversible) and four metabolites (X1 is considered a parameter, and S2, S3, and S4 are variables).

Figure Scheme 2..

Equations used in the model. For enzymes 1 to 3, total velocity equations containing the Haldane relationship are considered; for enzyme 4 a simple Michaelis-Menten equation is considered.

Table 1. Initial condition values for enzyme kinetics parameters and metabolite concentrations
Kmath imageSj means the affinity constant of the enzyme Ei for the metabolite Sj.
Metabolitesm)Activitiesm min−1) KM for E1m)KM for E2m)KM for E3m)KM for E4m)
X1100V10.05Keq150KX1M1100KS2M2500KS3M3104KS4M4250
S20V2100Keq210KS2M1100KS3M2250KS4M320  
S30V3250Keq350        
S40V4100Keq4         
Table 2. Control flux coefficients for initial conditions
 Enzyme 1Enzyme 2Enzyme 3Enzyme 4Summation
Control flux coefficient(for initial conditions)0.320.070.150.521.06

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