Dynamic Bayesian sensitivity analysis of a myocardial metabolic model

Dynamic compartmentalized metabolic models are identified by a large number of parameters, several of which are either non-physical or extremely difficult to measure. Typically, the available data and prior information is insufficient to fully identify the system. Since the models are used to predict the behavior of unobserved quantities, it is important to understand how sensitive the output of the system is to perturbations in the poorly identifiable parameters. Classically, it is the goal of sensitivity analysis to asses how much the output changes as a function of the parameters. In the case of dynamic models, the output is a function of time and therefore its sensitivity is a time dependent function. If the output is a differentiable function of the parameters, the sensitivity at one time instance can be computed from its partial derivatives with respect to the parameters. The time course of these partial derivatives describes how the sensitivity varies in time.When the model is not uniquely identifiable, or if the solution of the parameter identification problem is known only approximately, we may have not one, but a distribution of possible parameter values. This is always the case when the parameter identification problem is solved in a statistical framework. In that setting, the proper way to perform sensitivity analysis is to not rely on the values of the sensitivity functions corresponding to a single model, but to consider the distributed nature of the sensitivity functions, inherited from the distribution of the vector of the model parameters.In this paper we propose a methodology for analyzing the sensitivity of dynamic metabolic models which takes into account the variability of the sensitivity over time and across a sample. More specifically, we draw a representative sample from the posterior density of the vector of model parameters, viewed as a random variable. To interpret the output of this doubly varying sensitivity analysis, we propose visualization modalities particularly effective at displaying simultaneously variations over time and across a sample. We perform an analysis of the sensitivity of the concentrations of lactate and glycogen in cytosol, and of ATP, ADP, NAD⁺ and NADH in cytosol and mitochondria, to the parameters identifying a three compartment model for myocardial metabolism during ischemia.     

Potential for anaerobic bioremediation of BTEX in petroleum-contaminated aquifers

Laboratory incubations of aquifer material or enrichments derived from aquifer material as well as geochemical data have suggested that, under the appropriate conditions, BTEX components of petroleum (benzene, toluene, ethylbenzene and xylene) can be degraded in the absence of molecular oxygen with either Fe(III), sulfate, or nitrate serving as the electron acceptor. BTEX degradation under methanogenic conditions has also been observed. However, especially for benzene, the BTEX contaminant of greatest concern, anaerobic degradation is often difficult to establish and maintain in laboratory incubations. Although studies to date have suggested that naturally occurring anaerobic BTEX degradation has the potential to remove significant quantities of BTEX from petroleum-contaminated aquifers, and mechanisms for stimulating anaerobic BTEX degradation in laboratory incubations have been developed, further study of the organisms involved in this metabolism and the factors controlling their distribution and activity are required before it will be possible to design rational strategies for accelerating anaerobic BTEX degradation in contaminated aquifers. Received 21 November 1995/ Accepted in revised form 20 February 1996     

The design of long-term effective uranium bioremediation strategy using a community metabolic model

Acetate amendment at uranium contaminated sites in Rifle, CO. leads to an initial bloom of Geobacter accompanied by the removal of U(VI) from the groundwater, followed by an increase of sulfate‐reducing bacteria (SRBs) which are poor reducers of U(VI). One of the challenges associated with bioremediation is the decay in Geobacter abundance, which has been attributed to the depletion of bio‐accessible Fe(III), motivating the investigation of simultaneous amendments of acetate and Fe(III) as an alternative bioremediation strategy. In order to understand the community metabolism of Geobacter and SRBs during artificial substrate amendment, we have created a genome‐scale dynamic community model of Geobacter and SRBs using the previously described Dynamic Multi‐species Metabolic Modeling framework. Optimization techniques are used to determine the optimal acetate and Fe(III) addition profile. Field‐scale simulation of acetate addition accurately predicted the in situ data. The simulations suggest that batch amendment of Fe(III) along with continuous acetate addition is insufficient to promote long‐term bioremediation, while continuous amendment of Fe(III) along with continuous acetate addition is sufficient to promote long‐term bioremediation. By computationally minimizing the acetate and Fe(III) addition rates as well as the difference between the predicted and target uranium concentration, we showed that it is possible to maintain the uranium concentration below the environmental safety standard while minimizing the cost of chemical additions. These simulations show that simultaneous addition of acetate and Fe(III) has the potential to be an effective uranium bioremediation strategy. They also show that computational modeling of microbial community is an important tool to design effective strategies for practical applications in environmental biotechnology. Biotechnol. Bioeng. 2012; 109: 2475–2483. © 2012 Wiley Periodicals, Inc.     

Optimization-based metabolic control analysis

In this work, a novel optimization-based metabolic control analysis (OMCA) method is introduced for reducing data requirement for metabolic control analysis (MCA). It is postulated that using the optimal control approach, the fluxes in a metabolic network are correlated to metabolite concentrations and enzyme activities as a state-feedback control system that is optimal with respect to a homeostasis objective. It is then shown that the optimal feedback gains are directly related to the elasticity coefficients (ECs) of MCA. This approach requires determination of the relative "importance" of metabolites and fluxes for the system, which is possible with significantly reduced experimental data, as compared with typical MCA requirements. The OMCA approach is applied to a top-down control model of glycolysis in hepatocytes. It is statistically demonstrated that the OMCA model is capable of predicting the ECs observed experimentally with few exceptions. Further, an OMCA-based model reconciliation study shows that the modification of four assumed stoichiometric coefficients in the model can explain most of the discrepancies, with the exception of elasticities with respect to the NADH/NAD ratio.     

Alleviation of metal and BTEX inhibition on BTEX degradation using PVA-immobilized degrader: kinetic model of BTEX degradation

Alleviation of metal inhibition on BTEX degradation using PVA-immobilized degrader (Mycobacterium sp. CHXY119) was investigated. When BTEX of 29 mg L^?1 [B:T:E:X = 1:1:1:1 (mg)] was used, more than 99 % of BTEX was simultaneously degraded by the free cells within 170 h. In contrast, BTEX of 114–172 mg L^?1 seriously inhibited degradation. High concentrations of metals (Mn²⁺: 15, Ni²⁺: 10, and Zn²⁺: 10 mg L^?1) also strongly inhibited BTEX degradation by the free cells at BTEX of 29 mg L^?1. Immobilization of degraders alleviated the inhibition of BTEX and heavy metals at high concentrations. A modified non-competitive inhibition model well described the BTEX degradation by the free and immobilized cells in the absence and presence of metal ions (R ² = 0.92–0.99). The above results provide valuable information on treatment of metal-BTEX co-contaminated wastewater by the immobilized degrader.     

Topological analysis of metabolic control

A topological approach is presented for the analysis of control and regulation in metabolic pathways. In this approach, the control structure of a metabolic pathway is represented by a weighted directed graph. From an inspection of the topology of the graph, the control coefficients of the enzymes are evaluated in a heuristic manner in terms of the enzyme elasticities. The major advantage of the topological approach is that it provides a visual framework for (1) calculating the control coefficients of the enzymes, (2) analyzing the cause-effect relationships of the individual enzymes, (3) assessing the relative importance of the enzymes in metabolic regulation, and (4) simplifying the structure of a given pathway, from a regulatory viewpoint. Results are obtained for (a) an unbranched pathway in the absence of feedback the feedforward regulation and (b) an unbranched pathway with feedback inhibition. Our formulation is based on the metabolic control theory of Kacser and Burns (1973) and Heinrich and Rapoport (1974).     

Investigating biological control over soil carbon temperature sensitivity

Understanding the temperature sensitivity of soil respiration is critical for predicting the response of ecosystems to climate change, yet the microbial communities responsible are rarely considered explicitly in studies or models. In this study, we assessed total microbial community composition, quantified bacterial respiration temperature response, and investigated the temperature dependence of bacterial carbon substrate utilization in tropical, temperate, and taiga soils (from Puerto Rico, California, and Alaska). Microbial community composition was characterized using phospholipid fatty acid analysis. Bacterial community respiration on a standardized set of substrates was ascertained using the BiOLOG^™ substrate utilization assay incubated at four temperatures: 4, 12, 28, and 40 °C. First, we found that microbial communities from the three latitudes were compositionally distinct and that the bacterial component of the three communities had markedly different respiration temperature–response curves corresponding with their experienced temperature regimes. We use these data to highlight limitations of widely used temperature–response equations and investigate temperature-dependent patterns of substrate utilization. We found that temperature response, in terms of both respiration rates and substrate use, varied for these bacterial communities independent of substrate quality or quantity interactions such as labile depletion. In contrast to the common assumption of heterotrophic microbial ubiquity, we found that bacterial community differences from these diverse systems appeared to determine both rates of respiration and patterns of carbon substrate usage. We suggest that microbial community composition-specific responses to changing climate may be important in predicting the long-term role of ecosystems in atmospheric CO₂ dynamics.     

Design of large metabolic responses. Constraints and sensitivity analysis

Metabolic control analysis (Kacser & Burns (1973). Symp. Soc. Exp. Biol.27, 65-104; Heinrich & Rapoport (1974). Eur. J. Biochem.42, 89-95) has been extensively used to describe the response of metabolic concentrations and fluxes to small (infinitesimal) changes in enzyme concentrations and effectors. Similarly, metabolic control design (Acerenza (1993). J. theor. Biol.165, 63-85) has been proposed to design small metabolic responses. These approaches have the limitation that they were not devised to deal with large (non-infinitesimal) responses. Here we develop a strategy to design large changes in the metabolic variables. The only assumption made is that, for all the parameter values under consideration, the system has a unique stable steady state. The procedure renders the kinetic parameters of the rate equations that when embedded in the metabolic network produce the pattern of large changes in the steady-state variables that we aim to design. Structural and kinetic constraints impose restrictions on the type of responses that could be designed. We show that these conditions can be transformed into the language of mean-sensitivity coefficients and, as a consequence, a sensitivity analysis of large metabolic responses can be performed after the system has been designed. The mean-sensitivity coefficients fulfil conservation and summation relationships that in the limit reduce to the well-known theorems for infinitesimal changes. Finally, it is shown that the same procedure that was used to design metabolic responses and analyse their sensitivity properties can also be used to determine the values of kinetic parameters of the rate laws operating "in situ".     

An integrated metabolic model for the aerobic and denitrifying biological phosphorus removal

In this work, an integrated metabolic model for biological phosphorus removal is presented. Using a previously proposed mathematical model it was shown to be possible to describe the two known biological phosphorus removal processes, under aerobic and denitrifying conditions, with the same biochemical reactions, where only the difference in electron acceptor (oxygen and nitrate) is taken into account. Though, apart from the ATP/NADH ratio, the stoichiometry in those models is identical, different kinetic parameters were found. Therefore, a new kinetic structure is proposed that adequately describes phosphorus removal under denitrifying and aerobic conditions with the same kinetic equations and parameters. The ATP/NADH ratio (delta) is the only model parameter that is different for aerobic and denitrifying growth. With the new model, simulations of anaerobic/aerobic and anaerobic/denitrifying sequencing batch reactors (A(2) SBR and A/O SBR) were made for verification of the model. Not only short-term behavior, but also steady state, was simulated. The results showed very good agreement between model predictions and experimental results for a wide range of dynamic conditions and sludge retention times. Sensitivity analysis shows the influence of the model parameters and the feed substrate concentrations on both systems. (c) 1997 John Wiley & Sons, Inc. Biotechnol Bioeng 54: 434-450, 1997.     

Enhancement of the microbial community biomass and diversity during air sparging bioremediation of a soil highly contaminated with kerosene and BTEX

In order to obtain insights in complexity shifts taking place in natural microbial communities under strong selective pressure, soils from a former air force base in the Czech Republic, highly contaminated with jet fuel and at different stages of a bioremediation air sparging treatment, were analyzed. By tracking phospholipid fatty acids and 16S rRNA genes, a detailed monitoring of the changes in quantities and composition of the microbial communities developed at different stages of the bioventing treatment progress was performed. Depending on the length of the air sparging treatment that led to a significant reduction in the contamination level, we observed a clear shift in the soil microbial community being dominated by Pseudomonads under the harsh conditions of high aromatic contamination to a status of low aromatic concentrations, increased biomass content, and a complex composition with diverse bacterial taxonomical branches. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users. The online version of an erratum to this article can be found at http://dx.doi.org/. An erratum to this article can be found at     

Increasing the flux in metabolic pathways: A metabolic control analysis perspective

The problems of engineering increased flux in metabolic pathways are analyzed in terms of the understanding provided by metabolic control analysis. Over-expression of a single enzyme is unlikely to be effective unless it is known to have a high flux control coefficient, which can be used as an approximate predictive tool. This is likely to rule out enzymes subject to feedback inhibition, because it transfers control downstream from the inhibited enzyme to the enzymes utilizing the feedback metabolite. Although abolishing feedback inhibition can restore flux control to an enzyme, it is also likely to cause large increases in the concentrations of metabolic intermediates. Simultaneous and coordinated over-expression of most of the enzymes in a pathway can, in principle, produce substantial flux increases without changes in metabolite levels, though technically it may be difficult to achieve. It is, however, closer to the method used by cells to change flux levels, where coordinated changes in the level of activity of pathway enzymes are the norm. Another option is to increase the demand for the pathway product, perhaps by increasing its rate of excretion or removal. Copyright 1998 John Wiley & Sons, Inc.     

Integration of temporal analysis and control analysis of metabolic systems.

A theory is developed that integrates approaches to the analysis of pathway transient response and metabolic control analysis. A Temporal Control Coefficient is defined that is a measure of the system's transient response to modulation of enzyme activity or concentration. The approach allows for the analysis of the establishment of a steady state from rest, of the system's 'agility' of response to minor perturbations of a pre-existing steady state and of the macroscopic transition between steady states. In the last-mentioned case it is shown that, like the transient time itself, the control of transient response retains the property of independence from the mechanism of the transition. In consequence, the Temporal Control Coefficient can be defined in terms of the control properties of the initial and final states alone without reference to the mechanism of transition. A summation property is shown to apply to the Temporal Control Coefficients in each case. Connectivity relationships between elasticities and Temporal Control Coefficients are also established.     

The mathematics of metabolic control analysis revisited

In this minireview, several different approaches to derivation of the theorems and relationships of Metabolic Control Analysis (MCA) are discussed and an alternative approach is presented. This new approach consists of solving the steady-state mass balances for the intracellular metabolites using linearized kinetics. The application of linearized kinetics reflects the fact that MCA is based on linearization of the system equations in a reference steady state. Our derivation is valid for metabolic networks of arbitrary complexity, including those containing conserved moieties and branches. The value of our approach is its simplicity: the derivation is straightforward and therefore easy to follow. It can serve as a compact introduction to the mathematical basis of MCA.     

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 metabolic model for biological phosphorus removal by denitrifying organisms

A metabolic model for biological phosphorus removal under denitrifying conditions has been established. The model is based on previous work with aerobic phosphorus removal. The form of the kinetic equations used is the same as for the aerobic model. The main difference is the value of P/NADH(2) ratio in the electron transport phosphorylation with nitrate (delta(N)). This value was determined independently from batch tests with an enriched culture of denitrifying phosphorus-removing bacteria. The measured delta(N) was approximately 1.0 mol ATP/mol NADH(2). This indicates that the energy production efficiency with nitrate compared to oxygen is approximately 40% lower. These batch tests were also used to identify a proper set of kinetic parameters. The obtained model was subsequently applied for the simulation of cyclic behavior in an anaerobic-anoxic sequencing batch reactor at different biomass retention times. The simulation results showed that the metabolic model can be used successfully for the denitrifying dephosphatation process. The obtained kinetic parameters for denitrifying enrichment cultures, however, deviated from those obtained for the aerobic enrichment cultures. (c) 1996 John Wiley & Sons, Inc.