Calculation errors of time-varying flux control coefficients obtained from elasticity coefficients by means of summation and connectivity theorems in metabolic control analysis

This paper investigates the accuracy of a matrix method proposed by other researchers to calculate time-varying flux control coefficients (dynamic FCCs) from elasticity coefficients by means of summation and connectivity theorems in the framework of metabolic control analysis. A mathematical model for the fed-batch penicillin V fermentation process is used as a case example for discussion. Calculated results reveal that this method produces significant calculation errors because the theorems are essentially valid only in steady state, although it may provide rough time-transient behaviors of FCCs. Strictly, therefore, dynamic FCCs should be directly calculated from the differential equations for metabolite concentrations and sensitivities.     

Determination of elasticities, concentration and flux control coefficients from transient metabolite data using linlog kinetics

This paper presents a practical approach to estimate the kinetic parameters of a metabolic network from in vivo kinetics experiments. This method is based on the linlog kinetics format (Visser and Heijnen, 2003, Metab. Eng. 5(3), 164-176; Wu et al., 2004, Eur. J. Biochem. 271, 3348-3359), of which the kinetic parameters, called elasticities, are estimated by an iterative linear optimization followed by non-linear optimization, from transient metabolite concentration data which are directly obtainable from rapid pulse experiments. In this way, not only the parameters are estimated but also a full kinetic model, based on linlog kinetics, is developed. The obtained elasticities also allow immediate calculation of all control coefficients. As an in silico case study, the estimation of elasticities of a linear pathway is presented. The method is shown to be able to estimate the elasticities quite accurately and to be robust toward errors in the metabolite data originating from sampling and measurement inaccuracy. The method allows experimental redesign to get more accurate estimated parameters and accommodates various types of experimentally applied disturbances in the pathway: changes in independent metabolites, dependent metabolites or enzyme levels/activities.     

Instantaneous flux control analysis for biochemical systems.

Linear sensitivity analysis of steady-state control of enzymic systems has been extended to non-steady states yielding sensitivity coefficients which provide non-intuitive insights into the behavior of the system and the sites of metabolic control, and which are quantitative counterparts to traditional qualitative concepts. Because this information is provided in a readily understood format, these coefficients serve as convenient indices of metabolic control. This treatment was applied to a simple test system, consisting of two enzymes and one non-enzymatic reaction, which exhibits oscillatory behavior. The results indicate that oscillations in the concentrations of the intermediate metabolites are regulated almost exclusively by the second enzyme. Control of the flux through the pathway is apportioned equally among the three reactions during periods of low net flux, but it is due almost exclusively to the second enzyme during periods of high net flux.     

A simple matrix method for determining flux control coefficients in complex pathways

Flux control coefficients express in quantitative terms the extent to which the steady state flux through a metabolic pathway is controlled by a particular parameter. Enzyme flux control coefficients can be calculated using matrix algebra methods which express the control coefficients in terms of parameters which can be determined experimentally (enzyme elasticities, flux ratios, metabolite ratios). This paper describes an algorithm based on a 'constraint' matrix which enables expressions for enzyme control coefficients to be written for pathways of any complexity.     

Experimental determination of group flux control coefficients in metabolic networks

Grouping of reactions around key metabolite branch points can facilitate the study of metabolic control of complex metabolic networks. This top-down Metabolic Control Analysis is exemplified through the introduction of group (flux, as well as concentration) control coefficients whose magnitudes provide a measure of the relative impact of each reaction group on the overall network flux, as well as on the overall network stability, following enzymatic amplification. In this article, we demonstrate the application of previously developed theory to the determination of group flux control coefficients. Experimental data for the changes in metabolic fluxes obtained in response to the introduction of six different environmental perturbations are used to determine the group flux control coefficients for three reaction groups formed around the phosphoenolpyruvate/pyruvate branch point. The consistency of the obtained group flux control coefficient estimates is systematically analyzed to ensure that all necessary conditions are satisfied. The magnitudes of the determined control coefficients suggest that the control of lysine production flux in Corynebacterium glutamicum cells at a growth base state resides within the lysine biosynthetic pathway that begins with the PEP/PYR carboxylation anaplorotic pathway. Copyright 1998 John Wiley & Sons, Inc.     

Metabolic flux control analysis of branch points: an improved approach to obtain flux control coefficients from large perturbation data

An overview of published approaches for the metabolic flux control analysis of branch points revealed that often not all fundamental constraints on the flux control coefficients have been taken into account. This has led to contradictory statements in literature on the minimum number of large perturbation experiments required to estimate the complete set of flux control coefficients C(J) for a metabolic branch point. An improved calculation procedure, based on approximate Lin-log reaction kinetics, is proposed, providing explicit analytical solutions of steady state fluxes and metabolite concentrations as a function of large changes in enzyme levels. The obtained solutions allow direct calculation of elasticity ratios from experimental data and subsequently all C(J)-values from the unique relation between elasticity ratio's and flux control coefficients. This procedure ensures that the obtained C(J)-values satisfy all fundamental constraints. From these it follows that for a three enzyme branch point only one characterised or two uncharacterised large flux perturbations are sufficient to obtain all C(J)- values. The improved calculation procedure is illustrated with four experimental cases.     

Nonlinear effect of dynamic self-organization in macromolecular systems caused by photocontrolled electron flux

We use the electron-conformational interaction approach to develop a physical model which self-consistently describes the photomobilized electron transfer kinetics and structure conformational transitions in reaction centers (RCs) of purple bacteria. We consider the kinetics of electron transition from pigment onto primary acceptor and the subsequent charge recombination accounting for the change of distance between the above-mentioned cofactors. It is shown that, given natural values of RC parameters, the kinetic constant's dependence on the acting light intensity is monotone. As opposed to the previous case, similar dependencies for the chain of electron transfer between primary and secondary quinone acceptors revealed anS-like relationship. This can lead to bistability of the RC optical transmission coefficient and a fundamental dependence of charge recombination kinetics upon the prehistory of the RC's interaction with exciting radiation.     

Sensitivity summation theorems for stochastic biochemical reaction systems

We investigate how stochastic reaction processes are affected by external perturbations. We describe an extension of the deterministic metabolic control analysis (MCA) to the stochastic regime. We introduce stochastic sensitivities for mean and covariance values of reactant concentrations and reaction fluxes and show that there exist MCA-like summation theorems among these sensitivities. The summation theorems for flux variances is shown to depend on the size of the measurement time window (?) within which reaction events are counted for measuring a single flux. It is found that the degree of the ?-dependency can become significant for processes involving multi-time-scale dynamics and is estimated by introducing a new measure of time-scale separation. This ?-dependency is shown to be closely related to the power-law scaling observed in flux fluctuations in various complex networks.     

The dynamic systems approach to control and regulation of intracellular networks

Systems theory and cell biology have enjoyed a long relationship that has received renewed interest in recent years in the context of systems biology. The term 'systems' in systems biology comes from systems theory or dynamic systems theory: systems biology is defined through the application of systems- and signal-oriented approaches for an understanding of inter- and intra-cellular dynamic processes. The aim of the present text is to review the systems and control perspective of dynamic systems. The biologist's conceptual framework for representing the variables of a biochemical reaction network, and for describing their relationships, are pathway maps. A principal goal of systems biology is to turn these static maps into dynamic models, which can provide insight into the temporal evolution of biochemical reaction networks. Towards this end, we review the case for differential equation models as a 'natural' representation of causal entailment in pathways. Block-diagrams, commonly used in the engineering sciences, are introduced and compared to pathway maps. The stimulus-response representation of a molecular system is a necessary condition for an understanding of dynamic interactions among the components that make up a pathway. Using simple examples, we show how biochemical reactions are modelled in the dynamic systems framework and visualized using block-diagrams.     

Risk-aware limited lookahead control for dynamic resource provisioning in enterprise computing systems

Utility or on-demand computing, a provisioning model where a service provider makes computing infrastructure available to customers as needed, is becoming increasingly common in enterprise computing systems. Realizing this model requires making dynamic, and sometimes risky, resource provisioning and allocation decisions in an uncertain operating environment to maximize revenue while reducing operating cost. This paper develops an optimization framework wherein the resource provisioning problem is posed as one of sequential decision making under uncertainty and solved using a limited lookahead control scheme. The proposed approach accounts for the switching costs incurred during resource provisioning and explicitly encodes risk in the optimization problem. Simulations using workload traces from the Soccer World Cup 1998 web site show that a computing system managed by our controller generates up to 20% more profit than a system without dynamic control while incurring low control overhead.

Interdependence between cooperativity and control coefficients

Influence of the concentration of internal metabolites on the control coefficient (defined as fractional change in flux per fractional change in enzyme activity) and regulatory properties of a given enzyme have been studied theoretically using a cyclic model of three enzymes. This model is useful to investigate the properties of the flux control coefficient for an enzyme following different rate equations. Enzymes can have high or low values of control coefficient irrespective of the type of kinetic equation, but the results obtained show that the sensitivity of these values to substrate variations is strongly dependent on its rate equation. These results help identify which kinetic equation allows the best control of a given metabolic pathway. These results have been applied to the purine nucleotide cycle. It is demonstrated that the best control of the cycle is reached when the irreversible reaction catalyzed by AMP deaminase follows a rate law that corresponds to a rational function of 2:2 degree with respect to AMP concentration.     

Load balancing for dynamic real-time systems

Some classes of real-time systems function in environments, which cannot be modeled with static approaches. In such environments, the arrival rates of events which drive transient computations may be unknown. Also, periodic computations may be required to process varying numbers of data elements per period, but the number of data elements to be processed in an arbitrary period cannot be known at the time of system engineering, nor can an upper bound be determined for the number of data items; thus, a worst case execution time cannot be obtained for such periodics. This paper presents middleware services that support such dynamic real-time systems through load balancing. The middleware services have been implemented and employed for (1) the DynBench dynamic real-time benchmark suite and (2) an experimental Navy system. Experimental results show the effectiveness of our load balancing techniques for consistently delivering real-time quality-of-service, even in highly dynamic environments. This revised version was published online in July 2006 with corrections to the Cover Date.     

Control of the metabolic flux in a system with high enzyme concentrations and moiety-conserved cycles. The sum of the flux control coefficients can drop significantly below unity.

In a number of metabolic pathways enzyme concentrations are comparable to those of substrates. Recently it has been shown that many statements of the 'classical' metabolic control theory are violated if such a system contains a moiety-conserved cycle. For arbitrary pathways we have found: (a) the equation connecting coefficients CEiJ (obtained by varying the Ei concentration) and CviJ (obtained by varying the kicat), and (b) modified summation equations. The sum of the enzyme control coefficients (equal to unity under the 'classical' theory) appears always to be below unity in the systems considered. The relationships revealed were illustrated by a numerical example where the sum of coefficients CEiJ reached negative values. A method for experimental measurements of the above coefficients is proposed.     

Effect of bacterial density and substrate concentration on yield coefficients

Measurements were made of the yield coefficient during the aerobic metabolism of glucose by a heterogeneous bacterial mixture. Expressed in terms of carbon, the coefficient was approximately 0.48. The value did not vary with initial bacterial densities ranging from 0.4 pg to 40 micrograms of cell carbon per ml and with glucose concentrations ranging from 43 pg to 100 micrograms of carbon per ml. Under all these circumstances, about 44% of the glucose carbon was converted to CO2, and 7.4% was excreted as organic products. The significance of uncharacterized organic substrates contaminating the medium to the coefficients calculated for low glucose concentrations is discussed.