A model for the regulation of apoptosis intrinsic pathway: The potential role of the transcriptional regulator E2F in the point of no return.

Apoptosis has been extensively characterized by both experimental approaches and model simulations. However, it is still not fully understood how the regulation occurs, especially in the intrinsic pathway, which can be activated by a great variety of signals. In addition, the conditions in which a point of no return could be reached remain elusive. In this work, we use differential equations models to approach these issues. Our starting point was the model for caspase activation of Legewie et al. (Legewie S, et al., PLoS Computational Biology 2006, 2(9): e120), which exhibits irreversible bistability. We added an activation module to this model, with the main events related to mitochondrial outer membrane permeabilization, which includes cytochrome C release by the mitochondria and its effects on caspase activation and respiratory chain disruption. This "Extended Legewie Model" (ELM) uses BAK as the apoptotic stimulus and active caspase 3 as a measure of apoptosis activation. Unexpectedly, in the extended model, BAK cannot trigger apoptosis activation using physiologically sound initial values of the variables, due to limitations in apoptosome concentration increase. Therefore, the next step was to find a regulatory mechanism, allowing apoptosis activation in the ELM, starting from physiological initial concentrations. For this aim, we performed a sensitivity analysis on the 61 parameters of the system, finding that those producing the most relevant changes in the qualitative behaviour were the rates of synthesis of caspase 3, caspase 9 and XIAP. Based on these results, the transcription factor E2F was included in the ELM because it directly regulates the rate of synthesis of caspase 3 and 9. Depending on the concentration of E2F, the ELM shows different qualitative behaviours. On one hand, for low E2F apoptosis is impossible and for high E2F apoptosis is inevitable. Therefore, if E2F is sufficiently increased, the point of no return is crossed. On the other hand, for intermediate values of E2F there is a bistable region where the fate of the system also depends on the concentration of BAK and other signalling species.

Constraints, Trade-offs and the Currency of Fitness.

Understanding evolutionary trajectories remains a difficult task. This is because natural evolutionary processes are simultaneously affected by various types of constraints acting at the different levels of biological organization. Of particular importance are constraints where correlated changes occur in opposite directions, called trade-offs. Here we review and classify the main evolutionary constraints and trade-offs, operating at all levels of trait hierarchy. Special attention is given to life history trade-offs and the conflict between the survival and reproduction components of fitness. Cellular mechanisms underlying fitness trade-offs are described. At the metabolic level, a linear trade-off between growth and flux variability was found, employing bacterial genome-scale metabolic reconstructions. Its analysis indicates that flux variability can be considered as the currency of fitness. This currency is used for fitness transfer between fitness components during adaptations. Finally, a discussion is made regarding the constraints which limit the increase in the amount of fitness currency during evolution, suggesting that occupancy constraints are probably the main restrictions.

Regulatory design in a simple system integrating membrane potential generation and metabolic ATP consumption. Robustness and the role of energy dissipating processes.

Bacterial physiological responses integrate energy-coupling processes at the membrane level with metabolic energy demand. The regulatory design behind these responses remains largely unexplored. Propionigenium modestum is an adequate organism to study these responses because it presents the simplest scheme known integrating membrane potential generation and metabolic ATP consumption. A hypothetical sodium leak is added to the scheme as the sole regulatory site. Allosteric regulation is assumed to be absent. Information of the rate equations is not available. However, relevant features of the patterns of responses may be obtained using Metabolic Control Analysis (MCA) and Metabolic Control Design (MCD). With these tools, we show that membrane potential disturbances can be compensated by adjusting the leak flux, without significant perturbations of ATP consumption. Perturbations of membrane potential by ATP demand are inevitable and also require compensatory changes in the leak. Numerical simulations were performed with a kinetic model exhibiting the responses for small changes obtained with MCA and MCD. A modest leak (10% of input) was assumed for the reference state. We found that disturbances in membrane potential and ATP consumption, produced by environmental perturbations of the cation concentration, may be reverted to the reference state adjusting the leak. Leak changes can also compensate for undesirable effects on membrane potential produced by changes in nutrient availability or ATP demand, in a wide range of values. The system is highly robust to parameter fluctuations. The regulatory role of energy dissipating processes and the trade-off between energetic efficiency and regulatory capacity are discussed.

In silico strategy to rationally engineer metabolite production: A case study for threonine in Escherichia coli.

Genetic engineering of metabolic pathways is a standard strategy to increase the production of metabolites of economic interest. However, such flux increases could very likely lead to undesirable changes in metabolite concentrations, producing deleterious perturbations on other cellular processes. These negative effects could be avoided by implementing a balanced increase of enzyme concentrations according to the Universal Method [Kacser and Acerenza (1993) Eur J Biochem 216:361-367]. Exact application of the method usually requires modification of many reactions, which is difficult to achieve in practice. Here, improvement of threonine production via pyruvate kinase deletion in Escherichia coli is used as a case study to demonstrate a partial application of the Universal Method, which includes performing sensitivity analysis. Our analysis predicts that manipulating a few reactions is sufficient to obtain an important increase in threonine production without major perturbations of metabolite concentrations.

Kinetic properties required for sustained or paradoxical control of metabolic fluxes under large changes in enzyme activities.

The effect that an increase in the activity of an enzyme has on its flux normally decreases with activity increase. To achieve a large increase in flux by manipulating a single step would therefore require a high initial effect that maintains or increases when the activity is increased, what has been called sustained or paradoxical control. Using metabolic control analysis for large responses, we derive conditions for sustained or paradoxical control in terms of elasticity coefficients. These are used to characterise types of rate laws contributing to this behaviour. The result that simple pathways, with normal kinetics, subject to large activity changes can lead to paradoxical control behaviour suggests that this type of pattern may be much more ubiquitous than could have, in principle, been suspected.

Modular metabolic control analysis of large responses.

Deciphering the laws that govern metabolic responses of complex systems is essential to understand physiological functioning, pathological conditions and the outcome of experimental manipulations of intact cells. To this aim, a theoretical and experimental sensitivity analysis, called modular metabolic control analysis (MMCA), was proposed. This field was previously developed under the assumptions of infinitesimal changes and/or proportionality between parameters and rates, which are usually not fulfilled in vivo. Here we develop a general MMCA for two modules, not relying on those assumptions. Control coefficients and elasticity coefficients for large changes are defined. These are subject to constraints: summation and response theorems, and relationships that allow calculating control from elasticity coefficients. We show how to determine the coefficients from top-down experiments, measuring the rates of the isolated modules as a function of the linking intermediate (there is no need to change parameters inside the modules). The novel formalism is applied to data of two experimental studies from the literature. In one of these, 40% increase in the activity of the supply module results in less than 4% increase in flux, while infinitesimal MMCA predicts more than 30% increase in flux. In addition, it is not possible to increase the flux by manipulating the activity of demand. The impossibility of increasing the flux by changing the activity of a single module is due to an abrupt decrease of the control of the modules when their corresponding activities are increased. In these cases, the infinitesimal approach can give highly erroneous predictions.

A modular modulation method for achieving increases in metabolite production.

Increasing the production of overproducing strains represents a great challenge. Here, we develop a modular modulation method to determine the key steps for genetic manipulation to increase metabolite production. The method consists of three steps: (i) modularization of the metabolic network into two modules connected by linking metabolites, (ii) change in the activity of the modules using auxiliary rates producing or consuming the linking metabolites in appropriate proportions and (iii) determination of the key modules and steps to increase production. The mathematical formulation of the method in matrix form shows that it may be applied to metabolic networks of any structure and size, with reactions showing any kind of rate laws. The results are valid for any type of conservation relationships in the metabolite concentrations or interactions between modules. The activity of the module may, in principle, be changed by any large factor. The method may be applied recursively or combined with other methods devised to perform fine searches in smaller regions. In practice, it is implemented by integrating to the producer strain heterologous reactions or synthetic pathways producing or consuming the linking metabolites. The new procedure may contribute to develop metabolic engineering into a more systematic practice.

Source and regulation of flux variability in Escherichia coli.

BACKGROUND: Metabolic responses are essential for the adaptation of microorganisms to changing environmental conditions. The repertoire of flux responses that the metabolic network can display in different external conditions may be quantified applying flux variability analysis to genome-scale metabolic reconstructions. RESULTS: A procedure is developed to classify and quantify the sources of flux variability. We apply the procedure to the latest Escherichia coli metabolic reconstruction, in glucose minimal medium, with an additional constraint to account for the mechanism coordinating carbon and nitrogen utilization mediated by α-ketoglutarate. Flux variability can be decomposed into three components: internal, external and growth variability. Unexpectedly, growth variability is the only significant component of flux variability in the physiological ranges of glucose, oxygen and ammonia uptake rates. To obtain substantial increases in metabolic flexibility, E. coli must decrease growth rate to suboptimal values. This growth-flexibility trade-off gives a straightforward interpretation to recent work showing that most overall cell-to-cell flux variability in a population of E. coli can be attained sampling a small number of enzymes most likely to constrain cell growth. Importantly, it provides an explanation for the global reorganization occurring in metabolic networks during adaptations to environmental challenges. The calculations were repeated with a pathogenic strain and an old reconstruction of the commensal strain, having less than 50% of the reactions of the latest reconstruction, obtaining the same general conclusions. CONCLUSIONS: In E. coli growing on glucose, growth variability is the only significant component of flux variability for all physiological conditions explored. Increasing flux variability requires reducing growth to suboptimal values. The growth-flexibility trade-off operates in physiological and evolutionary adaptations, and provides an explanation for the global reorganization occurring during adaptations to environmental challenges. The results obtained do not rely on the knowledge of kinetic and regulatory details of the system and are highly robust to incomplete or incorrect knowledge of the reaction network.

Modular metabolic control analysis of large responses in branched systems--application to aspartate metabolism.

Organisms subject to changing environmental conditions or experimental protocols show complex patterns of responses. The design principles behind these patterns are still poorly understood. Here, modular metabolic control analysis is developed to deal with large changes in branched pathways. Modular aggregation of the system dramatically reduces the number of explicit variables and modulation sites. Thus, the resulting number of control coefficients, which describe system responses, is small. Three properties determine the pattern for large changes in the variables: the values of infinitesimal control coefficients, the effect of large rate changes on the control coefficients and the range of rate changes preserving feasible intermediate concentrations. Importantly, this pattern gives information about the possibility of obtaining large variable changes by changing parameters inside the module, without the need to perform any parameter modulations. The framework is applied to a detailed model of Asp metabolism. The system is aggregated in one supply module, producing Thr from Asp (SM1), and two demand modules, incorporating Thr (DM2) and Ile (DM3) into protein. Their fluxes are: J(1), J(2), and J(3), respectively. The analysis shows similar high infinitesimal control coefficients of J(2) by the rates of SM1 and DM2 (C(v1)(J2) = 0.6 and C(v2)(J2) = 0.7, respectively). In addition, these coefficients present only moderate decreases when the rates of the corresponding modules are increased. However, the range of feasible rate changes in SM1 is narrow. Therefore, for large increases in J(2) to be obtained, DM2 must be modulated. Of the rich network of allosteric interactions present, only two groups of inhibitions generate the control pattern for large responses.

A strategy to calculate the patterns of nutrient consumption by microorganisms applying a two-level optimisation principle to reconstructed metabolic networks.

Bacterial responses to environmental changes rely on a complex network of biochemical reactions. The properties of the metabolic network determining these responses can be divided into two groups: the stoichiometric properties, given by the stoichiometry matrix, and the kinetic/thermodynamic properties, given by the rate equations of the reaction steps. The stoichiometry matrix represents the maximal metabolic capabilities of the organism, and the regulatory mechanisms based on the rate laws could be considered as being responsible for the administration of these capabilities. Post-genomic reconstruction of metabolic networks provides us with the stoichiometry matrix of particular strains of microorganisms, but the kinetic aspects of in vivo rate laws are still largely unknown. Therefore, the validity of predictions of cellular responses requiring detailed knowledge of the rate equations is difficult to assert. In this paper, we show that by applying optimisation criteria to the core stoichiometric network of the metabolism of Escherichia coli, and including information about reversibility/irreversibility only of the reaction steps, it is possible to calculate bacterial responses to growth media with different amounts of glucose and galactose. The target was the minimisation of the number of active reactions (subject to attaining a growth rate higher than a lower limit) and subsequent maximisation of the growth rate (subject to the number of active reactions being equal to the minimum previously calculated). Using this two-level target, we were able to obtain by calculation four fundamental behaviours found experimentally: inhibition of respiration at high glucose concentrations in aerobic conditions, turning on of respiration when glucose decreases, induction of galactose utilisation when the system is depleted of glucose and simultaneous use of glucose and galactose as carbon sources when both sugars are present in low concentrations. Preliminary results of the coarse pattern of sugar utilisation were also obtained with a genome-scale E. coli reconstructed network, yielding similar qualitative results.

A hierarchical approach to cooperativity in macromolecular and self-assembling binding systems.

The study of complex macromolecular binding systems reveals that a high number of states and processes are involved in their mechanism of action, as has become more apparent with the sophistication of the experimental techniques used. The resulting information is often difficult to interpret because of the complexity of the scheme (large size and profuse interactions, including cooperative and self-assembling interactions) and the lack of transparency that this complexity introduces into the interpretation of the indexes traditionally used to describe the binding properties. In particular, cooperative behaviour can be attributed to very different causes, such as direct chemical modification of the binding sites, conformational changes in the whole structure of the macromolecule, aggregation processes between different subunits, etc. In this paper, we propose a novel approach for the analysis of the binding properties of complex macromolecular and self-assembling systems. To quantify the binding behaviour, we use the global association quotient defined as K(c) = [occupied sites]/([free sites] L), L being the free ligand concentration. K(c) can be easily related to other measures of cooperativity (such as the Hill number or the Scatchard plot) and to the free energies involved in the binding processes at each ligand concentration. In a previous work, it was shown that K(c) could be decomposed as an average of equilibrium constants in two ways: intrinsic constants for Adair binding systems and elementary constants for the general case. In this study, we show that these two decompositions are particular cases of a more general expression, where the average is over partial association quotients, associated with subsystems from which the system is composed. We also show that if the system is split into different subsystems according to a binding hierarchy that starts from the lower, microscopic level and ends at the higher, aggregation level, the global association quotient can be decomposed following the hierarchical levels of macromolecular organisation. In this process, the partial association quotients of one level are expressed, in a recursive way, as a function of the partial quotients of the level that is immediately below, until the microscopic level is reached. As a result, the binding properties of very complex macromolecular systems can be analysed in detail, making the mechanistic explanation of their behaviour transparent. In addition, our approach provides a model-independent interpretation of the intrinsic equilibrium constants in terms of the elementary ones.

On the origins of a crowded cytoplasm.

Contemporary cells show a highly crowded macromolecular content, the processes which originated this state being largely unknown. We propose that a driving force leading to the crowded cellular state could be the increase in growth rate produced by an enhanced cytoplasmic protein concentration. Briefly, in a diluted scenario, an increase in protein concentration has two opposing effects on growth rate. The favorable effect is the increase in the activity per unit volume of the component proteins and the disadvantageous effect is the concomitant increase in the protein mass per unit volume which has to be produced. In this work we show that the first effect is quantitatively more important, resulting in an overall increase in growth rate. This result was obtained with a model of E. coli and using nonmechanistic physiological arguments. The proposed driving force operates even at low protein concentrations, where the nonspecific interactions of macromolecular crowding are not significant, and could be as ancient as the first protocells. Experimental measurement of this cytoplasmic protein concentration effect in present organisms is hindered by the prevailing nonspecific interactions, product of long-term evolution. However, chemical/biochemical systems, built up to mimic properties of living cells, could be an adequate tool to test this effect.

Necessary conditions for a minimal model of receptor to show adaptive response over a wide range of levels of stimulus.

Sensory systems respond to temporal changes in the stimulus and adapt to the new level when it persists, this pattern of response being maintained in a wide range of levels of stimulus. Here we use a simple model of adaptation developed by Segel et al. (J. Theor. Biol. 120 (1986) 151-179) and extended by Hauri and Ross (Biophys. J. 68 (1995) 708-722) to study the conditions in which it shows wide range of response. The model consists of a receptor that switches between a variable number of states, either by mass action law or by covalent modification. Using a global optimization procedure, we have optimized the adaptive response of the alternatives of the model with different number of states. We find that it is impossible to obtain a wide range of response if the receptor switches between states following mass-action laws, irrespective of the number of states. Instead, a wide range (of five orders of magnitude of ligand concentration) can be obtained if the receptor switches between several states by irreversible covalent modification, in agreement with previous models. Therefore, in this model, expenditure of energy to maintain a large number of covalent modification cycles operating outside equilibrium is necessary to achieve a wide range of response. The optimal values of the parameters present similar patterns to those reported for specific receptors, but there is no quantitative agreement. For instance, ligand affinity varies several orders of magnitude between the different states of the receptor, what is unlikely to be fulfilled by real systems. To see if the minimal model can show adaptive response and range with quantitatively plausible parameter values a sub-optimal receptor was studied, finding that adaptive response of high intensity can still be obtained in at least three orders of magnitude.

Elasticity analysis and design for large metabolic responses produced by changes in enzyme activities.

Metabolic control analysis has been extensively used to describe how the sensitivity properties of the component enzymes in a metabolic pathway (represented by the elasticity coefficients) determine the way in which metabolic variables respond (described by the control coefficients). Similarly, metabolic control design addresses the inverse problem of obtaining the sensitivity properties of the component enzymes that are required for the system to show a pre-established pattern of responses. These formalisms, including what is called elasticity analysis and design, were developed for small, strictly speaking infinitesimal, changes. Here we extend them to large metabolic responses. The new approach can be applied to simple two-step pathways or to any arbitrary metabolic system divided into two groups linked by one intermediate. General expressions that relate control and elasticity coefficients for large changes are derived. Concentration and flux connectivity relationships are obtained. The relationships for large changes indicate that the pattern of responses is not necessarily the same as the one obtained with the traditional infinitesimal approach, in some cases the patterns being qualitatively different. The general analysis is used to study the control of ketogenesis in rat liver mitochondria, starting from data available in the literature. The control profile of the pathway subject to large changes shows both quantitative and qualitative differences from the one obtained from an analysis that is performed with infinitesimal coefficients. This exemplifies the type of errors that may be introduced when drawing conclusions about large metabolic responses from results obtained with an infinitesimal treatment.

Sensitivity analysis of metabolic cascades catalyzed by bifunctional enzymes.

Covalent modification/demodification cycles are common in metabolism. When the modification and demodification steps are carried out by two independent enzymes, the degree of modification can be ultrasensitive to the total concentration of either catalyst. We recently showed that the degree of modification of a target molecule cannot exhibit ultrasensitivity to the free concentrations of effectors that decide whether a bifunctional enzyme acts as modifier or demodifier. However, here we can now demonstrate that the degree of modification of a target molecule can display ultrasensitivity to the total, rather than free, concentrations of such effectors. Our results clarify some general aspects of ultrasensitive responses to effectors, including competitive inhibitors, in mono-cyclic cascades.

Product dependence and bifunctionality compromise the ultrasensitivity of signal transduction cascades.

Covalent modification cycles are ubiquitous. Theoretical studies have suggested that they serve to increase sensitivity. However, this suggestion has not been corroborated experimentally in vivo. Here, we demonstrate that the assumptions of the theoretical studies, i.e., irreversibility and absence of product inhibition, were not trivial: when the conversion reactions are close to equilibrium or saturated by their product, "zero-order" ultrasensitivity disappears. For high sensitivities to arise, not only substrate saturation (zero-order) but also high equilibrium constants and low product saturation are required. Many covalent modification cycles are catalyzed by one bifunctional 'ambiguous' enzyme rather than by two independent proteins. This makes high substrate concentration and low product concentration for both reactions of the cycle inconsistent; such modification cycles cannot have high responses. Defining signal strength as ratios of modified (e.g., phosphorylated) over unmodified protein, signal-to-signal response sensitivity equals 1: signal strength should remain constant along a cascade of ambiguous modification cycles. We also show that the total concentration of a signalling effector protein cannot affect the signal emanating from a modification cycle catalyzed by an ambiguous enzyme if the ratio of the two forms of the effector protein is not altered. This finding may explain the experimental result that the pivotal signal transduction protein PII plus its paralogue GlnK do not control steady-state N-signal transduction in Escherichia coli. It also rationalizes the absence of strong phenotypes for many signal-transduction proteins. Emphasis on extent of modification of these proteins is perhaps more urgent than transcriptome analysis.