This chapter describes a practical procedure to dissect metabolic systems, simplify them, and use or derive enzyme rate equations in order to build a mathematical model of a metabolic system and run simulations. We first deal with a simple example, modeling a single enzyme that follows Michaelis-Menten kinetics and operates in the middle of an unbranched metabolic pathway. Next we describe the rules that can be followed to isolate sub-systems from their environment to simulate their behavior. Finally we use examples to show how to derive suitable rate equations, simpler than those needed for mechanistic studies, though adequate to describe the behavior over the physiological range of conditions.Many of the general characteristics of kinetic models will be obvious to readers familiar with the theory of metabolic control analysis (Cornish-Bowden, Fundamentals of Enzyme Kinetics, Wiley-Blackwell, Weinheim, 327-380, 2012), but here we shall not assume such knowledge, as the chapter is directed toward practical application rather than theory.
Victor Henri's great contribution to the understanding of enzyme kinetics and mechanism is not always given the credit that it deserves. In addition, his earlier work in experimental psychology is totally unknown to biochemists, and his later work in spectroscopy and photobiology almost equally so. Applying great rigour to his analysis he succeeded in obtaining a model of enzyme action that explained all of the observations available to him, and he showed why the considerable amount of work done in the preceding decade had not led to understanding. His view was that only physical chemistry could explain the behaviour of enzymes, and that models should be judged in accordance with their capacity not only to explain previously known facts but also to predict new observations against which they could be tested. The kinetic equation usually attributed to Michaelis and Menten was in reality due to him. His thesis of 1903 is now available in English.
Biochemical information has been crucial for the development of evolutionary biology. On the one hand, the sequence information now appearing is producing a huge increase in the amount of data available for phylogenetic analysis; on the other hand, and perhaps more fundamentally, it allows understanding of the mechanisms that make evolution possible. Less well recognized, but just as important, understanding evolutionary biology is essential for understanding many details of biochemistry that would otherwise be mysterious, such as why the structures of NAD and other coenzymes are far more complicated than their functions would seem to require. Courses of biochemistry should thus pay attention to the essential role of evolution in selecting the molecules of life.
Fructose-1,6-bisphosphatase, a major enzyme of gluconeogenesis, is inhibited by AMP, Fru-2,6-P2 and by high concentrations of its substrate Fru-1,6-P2. The mechanism that produces substrate inhibition continues to be obscure.
Methods and equations for analysing the kinetics of enzyme-catalysed reactions were developed at the beginning of the 20th century in two centres in particular; in Paris, by Victor Henri, and, in Berlin, by Leonor Michaelis and Maud Menten. Henri made a detailed analysis of the work in this area that had preceded him, and arrived at a correct equation for the initial rate of reaction. However, his approach was open to the important objection that he took no account of the hydrogen-ion concentration (a subject largely undeveloped in his time). In addition, although he wrote down an expression for the initial rate of reaction and described the hyperbolic form of its dependence on the substrate concentration, he did not appreciate the great advantages that would come from analysis in terms of initial rates rather than time courses. Michaelis and Menten not only placed Henris analysis on a firm experimental foundation, but also defined the experimental protocol that remains standard today. Here, we review this development, and discuss other scientific contributions of these individuals. The three parts have different authors, as indicated, and do not necessarily agree on all details, in particular about the relative importance of the contributions of Michaelis and Menten on the one hand and of Henri on the other. Rather than force the review into an unrealistic consensus, we consider it appropriate to leave the disagreements visible.
The paper that introduced biochemists to the idea of allosteric feedback inhibition [Monod J, Changeux J-P & Jacob F (1963) J Mol Biol 6, 306-329] is now 50 years old, and the two papers on models for enzyme cooperativity that followed it [Monod J, Wyman J & Changeux J-P (1965) J Mol Biol 12, 88-118; Koshland DE, Némethy G & Filmer D (1966) Biochemistry 5, 365-385] are almost as old. All of these papers continue to be heavily cited today - more in the 21st century than they were in the last two decades of the 20th. This is because they continue to be central for understanding enzyme regulation, and increasingly important in the age of systems biology.
The equation commonly called the Michaelis-Menten equation is sometimes attributed to other authors. However, although Victor Henri had derived the equation from the correct mechanism, and Adrian Brown before him had proposed the idea of enzyme saturation, it was Leonor Michaelis and Maud Menten who showed that this mechanism could also be deduced on the basis of an experimental approach that paid proper attention to pH and spontaneous changes in the product after formation in the enzyme-catalysed reaction. By using initial rates of reaction they avoided the complications due to substrate depletion, product accumulation and progressive inactivation of the enzyme that had made attempts to analyse complete time courses very difficult. Their methodology has remained the standard approach to steady-state enzyme kinetics ever since.
Chemical equations are normally written in terms of specific ionic and elemental species and balance atoms of elements and electric charge. However, in a biochemical context it is usually better to write them with ionic reactants expressed as totals of species in equilibrium with each other. This implies that atoms of elements assumed to be at fixed concentrations, such as hydrogen at a specified pH, should not be balanced in a biochemical equation used for thermodynamic analysis. However, both kinds of equations are needed in biochemistry. The apparent equilibrium constant K for a biochemical reaction is written in terms of such sums of species and can be used to calculate standard transformed Gibbs energies of reaction ?(r)G°. This property for a biochemical reaction can be calculated from the standard transformed Gibbs energies of formation ?(f)G(i)° of reactants, which can be calculated from the standard Gibbs energies of formation of species ?(f)G(j)° and measured apparent equilibrium constants of enzyme-catalyzed reactions. Tables of ?(r)G° of reactions and ?(f)G(i)° of reactants as functions of pH and temperature are available on the web, as are functions for calculating these properties. Biochemical thermodynamics is also important in enzyme kinetics because apparent equilibrium constant K can be calculated from experimentally determined kinetic parameters when initial velocities have been determined for both forward and reverse reactions. Specific recommendations are made for reporting experimental results in the literature.
The nature of life has been a topic of interest from the earliest of times, and efforts to explain it in mechanistic terms date at least from the 18th century. However, the impressive development of molecular biology since the 1950s has tended to have the question put on one side while biologists explore mechanisms in greater and greater detail, with the result that studies of life as such have been confined to a rather small group of researchers who have ignored one anothers work almost completely, often using quite different terminology to present very similar ideas. Central among these ideas is that of closure, which implies that all of the catalysts needed for an organism to stay alive must be produced by the organism itself, relying on nothing apart from food (and hence chemical energy) from outside. The theories that embody this idea to a greater or less degree are known by a variety of names, including (M,R) systems, autopoiesis, the chemoton, the hypercycle, symbiosis, autocatalytic sets, sysers and RAF sets. These are not all the same, but they are not completely different either, and in this review we examine their similarities and differences, with the aim of working towards the formulation of a unified theory of life.
The specificity of an enzyme obeying the Michaelis?Menten equation is normally measured by comparing the kcat/Km for different substrates, but this is inappropriate for enzymes with a Hill coefficient h different from 1. The obvious alternative of generalizing Km in the expression as K0.5, the substrate concentration for half-saturation, is better, but it is not entirely satisfactory either, and here we show that kcat/K0.5(h) gives satisfactory results for analyzing the kinetic behavior of metabolic pathways. The importance of using kcat/K0.5(h) increases with the value of h, but even when h is small, it makes an appreciable difference, as illustrated for the mammalian hexokinases. Reinterpretation of data for the specificity of these enzymes in terms of the proposed definition indicates that hexokinase D, often believed highly specific for glucose, and accordingly called “glucokinase”, actually has the lowest preference for glucose over fructose of the four isoenzymes found in mammals.
A living organism must not only organize itself from within; it must also maintain its organization in the face of changes in its environment and degradation of its components. We show here that a simple (M,R)-system consisting of three interlocking catalytic cycles, with every catalyst produced by the system itself, can both establish a non-trivial steady state and maintain this despite continuous loss of the catalysts by irreversible degradation. As long as at least one catalyst is present at a sufficient concentration in the initial state, the others can be produced and maintained. The system shows bistability, because if the amount of catalyst in the initial state is insufficient to reach the non-trivial steady state the system collapses to a trivial steady state in which all fluxes are zero. It is also robust, because if one catalyst is catastrophically lost when the system is in steady state it can recreate the same state. There are three elementary flux modes, but none of them is an enzyme-maintaining mode, the entire network being necessary to maintain the two catalysts.
In a previous paper, we pointed out that the capability to synthesize glycine from serine is constrained by the stoichiometry of the glycine hydroxymethyltransferase reaction, which limits the amount of glycine produced to be no more than equimolar with the amount of C 1 units produced. This constraint predicts a shortage of available glycine if there are no adequate compensating processes. Here, we test this prediction by comparing all reported fl uxes for the production and consumption of glycine in a human adult. Detailed assessment of all possible sources of glycine shows that synthesis from serine accounts for more than 85% of the total, and that the amount of glycine available from synthesis, about 3 g/day, together with that available from the diet, in the range 1.5-3.0 g/day, may fall significantly short of the amount needed for all metabolic uses, including collagen synthesis by about 10 g per day for a 70 kg human. This result supports earlier suggestions in the literature that glycine is a semi-essential amino acid and that it should be taken as a nutritional supplement to guarantee a healthy metabolism.
The major insight in Robert Rosens view of a living organism as an (M,R)-system was the realization that an organism must be "closed to efficient causation", which means that the catalysts needed for its operation must be generated internally. This aspect is not controversial, but there has been confusion and misunderstanding about the logic Rosen used to achieve this closure. In addition, his corollary that an organism is not a mechanism and cannot have simulable models has led to much argument, most of it mathematical in nature and difficult to appreciate. Here we examine some of the mathematical arguments and clarify the conditions for closure.
The aspartate-derived amino-acid pathway from plants is well suited for analysing the function of the allosteric network of interactions in branched pathways. For this purpose, a detailed kinetic model of the system in the plant model Arabidopsis was constructed on the basis of in vitro kinetic measurements. The data, assembled into a mathematical model, reproduce in vivo measurements and also provide non-intuitive predictions. A crucial result is the identification of allosteric interactions whose function is not to couple demand and supply but to maintain a high independence between fluxes in competing pathways. In addition, the model shows that enzyme isoforms are not functionally redundant, because they contribute unequally to the flux and its regulation. Another result is the identification of the threonine concentration as the most sensitive variable in the system, suggesting a regulatory role for threonine at a higher level of integration.
In this work we attempt to find out the extent to which realistic prebiotic compartments, such as fatty acid vesicles, would constrain the chemical network dynamics that could have sustained a minimal form of metabolism. We combine experimental and simulation results to establish the conditions under which a reaction network with a catalytically closed organization (more specifically, an (M,R-system) would overcome the potential problem of self-suffocation that arises from the limited accessibility of nutrients to its internal reaction domain. The relationship between the permeability of the membrane, the lifetime of the key catalysts and their efficiency (reaction rate enhancement) turns out to be critical. In particular, we show how permeability values constrain the characteristic time scale of the bounded protometabolic processes. From this concrete and illustrative example we finally extend the discussion to a wider evolutionary context.
A fundamental landmark in the emergence and maintenance of the first proto-biological systems must have been the formation of a "closed" metabolic organization, and this paper describes a stochastic analysis of a simple model of a system that is closed to efficient causation. Although it shows an absorbing barrier corresponding to the trivial solution that implies collapse and extinction, for certain values of the kinetic parameters it can also show a "coexistence state" in which there are non-null populations of its intermediates, which corresponds approximately to a non-trivial deterministic stable steady state. Depending on the initial conditions, fluctuations can drive the system either to the self-maintaining regime or to extinction, with different probabilities. Different lines of equal probability have been obtained and compared with the deterministic results, and the average time for reaching these states (characteristic time) has been estimated. The system shows strong dependence on volume size, and there is a critical volume below which it collapses very rapidly. The characteristic time is also affected by the volume, with faster responses for lower system volumes. All these results are discussed in the context of the origin of living organization.
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