The fractal structure of glycogen: A clever solution to optimize cell metabolism.

Fractal objects are complex structures built with a simple procedure involving very little information. This has an obvious interest for living beings, because they are splendid examples of optimization to achieve the most efficient structure for a number of goals by means of the most economic way. The lung alveolar structure, the capillary network, and the structure of several parts of higher plant organization, such as ears, spikes, umbels, etc., are supposed to be fractals, and, in fact, mathematical functions based on fractal geometry algorithms can be developed to simulate them. However, the statement that a given biological structure is fractal should imply that the iterative process of its construction has a real biological meaning, i.e., that its construction in nature is achieved by means of a single genetic, enzymatic, or biophysical mechanism successively repeated; thus, such an iterative process should not be just an abstract mathematical tool to reproduce that object. This property has not been proven at present for any biological structure, because the mechanisms that build the objects mentioned above are unknown in detail. In this work, we present results that show that the glycogen molecule could be the first known real biological fractal structure.

Generalization of the theory of transition times in metabolic pathways: a geometrical approach.

Cell metabolism is able to respond to changes in both internal parameters and boundary constraints. The time any system variable takes to make this response has relevant implications for understanding the evolutionary optimization of metabolism as well as for biotechnological applications. This work is focused on estimating the magnitude of the average time taken by any observable of the system to reach a new state when either a perturbation or a persistent variation occurs. With this aim, a new variable, called characteristic time, based on geometric considerations, is introduced. It is stressed that this new definition is completely general, being useful for evaluating the response time, even in complex transitions involving periodic behavior. It is shown that, in some particular situations, this magnitude coincides with previously defined transition times but differs drastically in others. Finally, to illustrate the applicability of this approach, a model of a reaction mediated by an allosteric enzyme is analyzed.

Activity and metabolic roles of the pentose phosphate cycle in several rat tissues.

Activity of the pentose-phosphate pathway in several rat tissues was investigated, developing a new method that gives the activity of each phase (oxidative and non-oxidative) as well as the whole pathway separately. Our results demonstrate that this method is easy to carry out and that it has not the problems of indirect determinations of the previous ones. The activities of the oxidative and non-oxidative phases assayed separately gives us new information on the design of the pathway in the different tissues, from which several conclusions about the physiological role of this pathway can be derived. In all cases the activity of the oxidative phase was much higher than the non-oxidative one, and the global activity of the whole pathway was the same as the activity of the non-oxidative phase. The highest activity was found in lactating mammary gland and adipose tissue. Lung and liver showed to have a moderately high activity. Brain, kidney, skeletal muscle, and intestinal mucosa showed to have also a significant activity although less than other tissues. The switch in the mammary gland from the non-lactating state to the lactating one causes a very high increase of activity of 22 times, remaining the same ratio between the activity of the two phases.

Physical constraints in the synthesis of glycogen that influence its structural homogeneity: a two-dimensional approach.

Several aspects of glycogen optimization as an efficient fuel storage molecule have been studied in previous works: the chain length and the branching degree. These results demonstrated that the values of these variables in the cellular molecule are those that optimize the structure-function relationship. In the present work we show that structural homogeneity of the glycogen molecule is also an optimized variable that plays an important role in its metabolic function. This problem was studied by means of a two-dimensional approach, which allowed us to simplify the very complicated structure of glycogen. Our results demonstrate that there is a molecular size limit that guarantees the structural homogeneity, beyond which the structure of the molecule degenerates, as many chains do not grow. This strongly suggests that such a size limit is precisely what the molecule possesses in the cell.

How did glycogen structure evolve to satisfy the requirement for rapid mobilization of glucose? A problem of physical constraints in structure building.

Optimization of molecular design in cellular metabolism is a necessary condition for guaranteeing a good structure-function relationship. We have studied this feature in the design of glycogen by means of the mathematical model previously presented that describes glycogen structure and its optimization function [Meléndez-Hevia et al. (1993), Biochem J 295: 477-483]. Our results demonstrate that the structure of cellular glycogen is in good agreement with these principles. Because the stored glucose in glycogen must be ready to be used at any phase of its synthesis or degradation, the full optimization of glycogen structure must also imply the optimization of every intermediate stage in its formation. This case can be viewed as a molecular instance of the eye problem, a classical paradigm of natural selection which states that every step in the evolutionary formation of a functional structure must be functional. The glycogen molecule has a highly optimized structure for its metabolic function, but the optimization of the full molecule has meaning and can be understood only by taking into account the optimization of each intermediate stage in its formation.

Network organization of cell metabolism: monosaccharide interconversion.

The structural properties of carbohydrate metabolism are being studied. The present contribution focuses mainly on those processes involving the transfer of carbon fragments among sugars. It is shown how enzymatic activities fix the way the system self-organizes stoichiometrically at the steady state. It is proven that there exists a specific correspondence between the set of all possible enzymic activities, the activity set, and the set of stoichiometrically compatible flux distributions through the pathway. On the one hand, there are enzymic activities that do not allow a stoichiometrically feasible coupling at the steady state of the reactions involved in the conversion. On the other hand, there are enzymic activities that are related to one or more flux distributions at the steady state (i.e. with one or several rate vectors respectively). For this latter group, it can be demonstrated that the structure of the system depends on other non-structural factors, such as boundary constraints and the kinetic parameters. As a consequence, it is suggested that this kind of metabolic process must be viewed as a complex reaction network instead of a sequential number of steps. Some implications of these derivations are illustrated for the particular conversion of CO2 --> C3. General remarks are also discussed within the framework of network models of cell metabolism.

Theoretical approaches to the evolutionary optimization of glycolysis--chemical analysis.

In the first part of this work [Heinrich, R., Montero, F., Klipp, E., Waddell, T. G. & Melendez-Hevia, E. (1997) Eur. J. Biochem. 243, 191-201] the kinetic and thermodynamic constraints under which an optimal glycolysis must be designed have been analysed. In this second part, we present a chemical analysis of the glycolytic pathway in order to determine if its design is chemically optimized according the possibilities that a glycolytic design can have. Our results demonstrate that glycolysis in modern-day cells (from glucose to lactate) has an optimized design for maximizing the flux of ATP production, and a thermodynamic profile which guarantees a high kinetic efficiency. We also discuss some cases of paleometabolism for this pathway as alternative metabolic pathways, less optimized, that exist in some bacteria. Our analysis relates mainly to metabolism designed under constant chemical affinity (substrates and products of the pathway constant), where the target of optimization can be the flux of ATP production. We also discuss the case of an externally imposed input flux, whose target of optimization is the stoichiometric yield of ATP.

Theoretical approaches to the evolutionary optimization of glycolysis: thermodynamic and kinetic constraints.

It is analyzed whether the structural design of contemporary glycolysis can be explained theoretically on the basis of optimization principles originating from natural selection during evolution. Particular attention is paid to the problem of how the kinetic and thermodynamic properties of the glycolytic pathway are related to its stoichiometry with respect to the number and location of ATP-coupling sites. The mathematical analysis of a minimal model of unbranched energy-converting pathways shows that the requirement of high ATP-production rate favours a structural design that includes not only ATP-producing reactions (P-sites) but also ATP-consuming reactions (C-sites). It is demonstrated that, at fixed overall thermodynamic properties of a chain, the ATP-production rate may be enhanced by kinetic optimization. The ATP-production rate is increased if the C-sites are concentrated at the beginning and all the P-sites at the end of the pathway. An optimum is attained, which is characterized by numbers of coupling sites corresponding to those found in glycolysis. Various extensions of the minimal model are considered, which allow the effects of internal feedback-regulations, variable enzyme concentrations, and the symmetric branching of glycolysis at the aldolase step to be considered.

The metabolic productivity of the cell factory.

It is widely accepted that some performance function has been optimized during the evolution of metabolic pathways. One can study the nature of such a function by analogy with the industrial manufacturing world, in which there have been efforts over recent decades to optimize production chains, and in which it is now accepted that fluxes are not the only important system variables that determine process efficiency, because inventory turnover must also be considered. Inspired by the parallels between living cells and manufacturing factories, we propose that fluxes and transit time may have simultaneously been major targets of natural selection in the optimization of the design, structure and kinetic parameters of metabolic pathways. Accordingly we define the ratio of flux to transit time as a performance index of productivity in metabolic systems: it measures the efficiency with which stocks are administered, and facilitates comparison of a pathway in different steady states or in different tissues or organisms. For a linear chain of two enzymes, at a fixed total equilibrium constant, we have analysed the variation of flux, transit time and productivity index as functions of the equilibrium constants of the two steps. The results show that only the productivity index has a maximum, which represents a good compromise in optimizing flux and transit time. We have extended control analysis to the productivity index and derived the summation theorem that applies to it. For linear chains of different length with maximum productivity index values, the distribution of control coefficients with regard to the three parameters has a characteristic profile independent of the length of the chain. Finally, this control profile changes when other variables are optimized, and we compare the theoretical results with the control profile of the first steps of glycolysis in rat liver.

Molecular bureaucracy: who controls the delays? Transient times in branched pathways and their control.

Analysis of metabolic control has until now been mainly confined to systems at steady state. This includes studies of the control of "transition time", which is actually a steady-state transit time that does not refer to the transient state. In this paper we examine the control of the transition state of a metabolic pathway in the approach to a stable steady state, showing that the time needed to attain it can be decreased or increased in different branches. Our analysis only applies to branched pathways, and we discuss why similar deviations cannot occur in unbranched pathways. In systems with several branches the acceleration of some branches during the transient phase, so that they reach their steady states more quickly, occurs at the expense of others, which are thus delayed. We present theorems that describe properties of the transient variables and their control.

The puzzle of the Krebs citric acid cycle: assembling the pieces of chemically feasible reactions, and opportunism in the design of metabolic pathways during evolution.

The evolutionary origin of the Krebs citric acid cycle has been for a long time a model case in the understanding of the origin and evolution of metabolic pathways: How can the emergence of such a complex pathway be explained? A number of speculative studies have been carried out that have reached the conclusion that the Krebs cycle evolved from pathways for amino acid biosynthesis, but many important questions remain open: Why and how did the full pathway emerge from there? Are other alternative routes for the same purpose possible? Are they better or worse? Have they had any opportunity to be developed in cellular metabolism evolution? We have analyzed the Krebs cycle as a problem of chemical design to oxidize acetate yielding reduction equivalents to the respiratory chain to make ATP. Our analysis demonstrates that although there are several different chemical solutions to this problem, the design of this metabolic pathway as it occurs in living cells is the best chemical solution: It has the least possible number of steps and it also has the greatest ATP yielding. Study of the evolutionary possibilities of each one-taking the available material to build new pathways-demonstrates that the emergence of the Krebs cycle has been a typical case of opportunism in molecular evolution. Our analysis proves, therefore, that the role of opportunism in evolution has converted a problem of several possible chemical solutions into a single-solution problem, with the actual Krebs cycle demonstrated to be the best possible chemical design. Our results also allow us to derive the rules under which metabolic pathways emerged during the origin of life.

Control analysis of transit time for free and enzyme-bound metabolites: physiological and evolutionary significance of metabolic response times.

Control analysis of transit time, defined as tau = delta/J, has previously been considered with the constraint of low enzyme concentrations compared with free pools of metabolites [Meléndez-Hevia, Torres, Sicilia and Kacser (1990) Biochem. J. 265, 195-202]. One of the conclusions was that the sum of the control coefficients of the transition time with respect to enzyme concentration was -1. Here we demonstrate that, if the enzyme-bound pools are taken into consideration (which would be important at high enzyme concentrations and high affinities), the sum lies between 0 and -1. The transition time between two steady states, which are frequent physiological events, is mainly governed by time constants involved in changing the enzyme concentrations. Some physiological and evolutionary aspects are discussed.

Optimization of molecular design in the evolution of metabolism: the glycogen molecule.

The animal glycogen molecule has to be designed in accordance with its metabolic function as a very effective fuel store allowing quick release of large amounts of glucose. In addition, the design should account for a high capacity of glucose storage in the least possible space. We have studied the optimization of these variables by means of a mathematical model of the glycogen molecule. Our results demonstrate that the structure is optimized to maximize (a) the total glucose stored in the smallest possible volume, (b) the proportion of it that can be directly released by phosphorylase before any debranching occurs, and (c) the number of non-reducing ends (points of attack for phosphorylase), which maximizes the speed of fuel release. The optimization of these four variables is achieved with appropriate values for two key parameters in glycogen design: the degree of branching and the length of the chains. The optimal values of these two parameters are precisely those found in cellular glycogen.

Control analysis of rat liver glycolysis under different glucose concentrations. The substrate approach and the role of glucokinase.

Control Analysis has been carried out in the first steps of a rat liver glycolytic system. Attention has been focused on the effect of several glucose concentrations on the control, particularly regarding the role of glucokinase. From kinetic studies of the whole metabolic system we have obtained information on the flux variation under different glucose concentrations. This information together with the kinetics of glucokinase has allowed us to calculate Flux Control and Elasticity Coefficients for glucokinase and the Response Coefficient of the system with respect to glucose. The changes in of the value of Flux Control Coefficients demonstrates that in conditions of low glucose concentration, glucokinase is the main enzyme in controlling the flux through the pathway, but at high glucose concentration the control moves to phosphofructokinase. Next, we have compared our results with those obtained with the shortening and titration method, previously described (Torres, N.V., Mateo, F., Meléndez-Hevia, E. and Kacser, H., (1986) Biochem. J. 234, 169-174; Torres, N.V. and Meléndez-Hevia, E. 1991. Molec. Cell. Biochem. 101, 1-10). Furthermore, from knowledge of the enzyme kinetics of the system we have been able to build a model of the pathway that allows us computer similation of its behavior and calculation of the Flux Control Coefficient profile at different glucose concentrations. By the three methods the results correlate, supporting the use of the pathway substrate as external modulator of the metabolic system as a tool for practical application of Control Analysis.

Transition time control analysis of a glycolytic system under different glucose concentrations. Control of transition time versus control of flux.

Control and Response Coefficients of transition time have been determined in a rat liver glycolytic system under different glucose concentrations. Results have been compared with the Flux Control and Flux Response Coefficients measured in the same conditions, showing that transition time and flux are different responses of the system, subject to different regulation and control. Control Coefficients of flux and transition time show a very different profile in each condition of glucose concentration assayed. Ratio of Flux Control coefficients of glucokinase over phosphofructokinase at 5 and 20 mM glucose concentration changes from 3.2 to 0.5, while the same ratio in the case of Transition Time Control Coefficients moves from 0.6 to 0.93. Moreover, the absolute values of Transition Time Control Coefficients in glycolytic conditions are one order of magnitude bigger than in gluconeogenic conditions. Values of Response Coefficients also show that the transition time has a bigger sensitivity to changes in glucose concentration than the flux in all conditions assayed, but particularly in glycolytic ones.

Analysis and characterization of transition states in metabolic systems. Transition times and the passivity of the output flux.

In this paper we study the transitions between steady states in metabolic systems. In order to deal with this task we divided the total metabolite concentration at steady state, sigma, into two new fractions, delta (the Output Transition Time) and tau beta (Input Transition Time), which are related with the course of output and input mass to the system respectively. We show the equivalence time between these terms and the Total Transition Time, tau T, previously defined [Easterby (1986) Biochem. J. 233, 871-875]. Next, we define a new magnitude, the Output Passivity of a transition, rho, which quantifies a new aspect of the transition phase that we call the passivity of the output progress curve. With these magnitudes, all of them being experimentally accessible, several features of the transient state can be measured. We apply the present analysis to (a) the case of coupled enzyme assays, which allows us to reach conclusions about the progress curves in these particular transitions and the equivalence between tau sigma and tau delta, and (b) some experimental results that allow us to discuss the biological significance of the Output Passivity in the transition between steady states in metabolic systems.

Detailed protocol and critical view for the analysis of control in metabolic systems by shortening and enzyme titration.

In this paper we give a general description of the 'shortening and enzyme titration method'. This method allows us to determine the Flux Control Coefficients of the different steps of a metabolic pathway in an in vitro experimental system. The system submitted to study is shortened in vitro by means of auxiliary enzymes, and the shortened pathway is titrated with extraneous enzymes. In this way we can modulate the activity of every enzyme of the system and thus every Flux Control Coefficient can be obtained. We criticize its different features in order to comment on the possibilities of its application to different types of systems. Our conclusion is that the method has a general applicability provided: a) that a correct definition of the metabolic pathway is given; b) that the system occurs in only one subcellular fraction and, c) that the dilution of the system by a given factor drives to the same reduction in every enzyme activity of the system.

Control analysis of transition times. Extension of analysis and matrix method.

The present theoretical basis of Control Analysis is extended with the definition of Transition Time Response Coefficients. Some new relationships between local and global coefficients defined in Control Analysis are presented. These relationships are in the form of matrix products constructed in a priori form. The use of these straightforward relationships is shown in an exemplary application corresponding to an experimental system consisting of the glycolytic degradation from glucose to glyceraldehyde-3-phosphate.