Control in channelled pathways. A matrix method calculating the enzyme control coefficients

The usual equations expressing the enzyme control coefficients (quantitative indicators of 'global' control properties of a pathway) via the elasticity coefficients (reflecting local kinetic properties of an enzyme reaction), cannot be applied to a variety of 'non-ideal' pathways, in particular to pathways with metabolic channelling. Here we show that the relationship between the control and elasticity coefficients can be obtained by considering such a metabolic pathway as a network of elemental chemical conversions (steps). To calculate the control coefficients of enzymes one should first determine the elasticity coefficients of such elemental steps and then take their appropriate combinations. Although the method is illustrated for a channelled pathway it can be used for any non-ideal pathway including those with high enzyme concentrations where the sequestration of metabolites by enzymes cannot be neglected.     

Lipid Metabolizing Enzyme Activities Modulated by Phospholipid Substrate Lateral Distribution

Biological membranes contain many domains enriched in phospholipid lipids and there is not yet clear explanation about how these domains can control the activity of phospholipid metabolizing enzymes. Here we used the surface dilution kinetic theory to derive general equations describing how complex substrate distributions affect the activity of enzymes following either the phospholipid binding kinetic model (which assumes that the enzyme molecules directly bind the phospholipid substrate molecules), or the surface-binding kinetic model (which assumes that the enzyme molecules bind to the membrane before binding the phospholipid substrate). Our results strongly suggest that, if the enzyme follows the phospholipid binding kinetic model, any substrate redistribution would increase the enzyme activity over than observed for a homogeneous distribution of substrate. Besides, enzymes following the surface-binding model would be independent of the substrate distribution. Given that the distribution of substrate in a population of micelles (each of them a lipid domain) should follow a Poisson law, we demonstrate that the general equations give an excellent fit to experimental data of lipases acting on micelles, providing reasonable values for kinetic parameters—without invoking special effects such as cooperative phenomena. Our theory will allow a better understanding of the cellular-metabolism control in membranes, as well as a more simple analysis of the mechanisms of membrane acting enzymes.     

Defining control coefficients in non-ideal metabolic pathways

The extent to which an enzyme controls a flux has been defined as the effect on that flux of a small modulation of the activity of that enzyme divided by the magnitude of the modulation. We here show that in pathways with metabolic channelling or high enzyme concentrations and conserved moieties involving both enzymic and non-enzymic species, this definition is ambiguous; the magnitude of the corresponding flux control coefficient depends on how the enzyme activity is modulated. This is illustrated with two models of biochemically relevant pathways, one in which dynamic metabolite channelling plays a role, and one with a moiety-conserved cycle. To avoid such ambiguity, we view biochemical pathways in a more detailed manner, i.e., as a network of elemental steps. We define 'elemental control coefficients' in terms of the effect on a flux of an equal modulation of the forward and reverse rate constant of any such elemental step (which may correspond to transitions between enzyme states). This elemental control coefficient is independent of the method of modulation. We show how metabolic control analysis can proceed when formulated in terms of the elemental control coefficients and how the traditional control coefficients are related to these elemental control coefficients. An 'impact' control coefficient is defined which quantifies the effect of an activation of all elemental processes in which an enzyme is involved. It equals the sum of the corresponding elemental control coefficients. In ideal metabolic pathways this impact control coefficient reduces to the traditional flux control coefficient. Differences between the traditional control coefficients are indicative of non-ideality of a metabolic pathway, i.e. of channelling or high enzyme concentrations.     

The steady-state kinetics of isotope exchange for one substrate–one product enzymic reactions

Steady-state kinetic equations for isotope exchange are derived for a number of one substrate-one product enzymic mechanisms in which two molecules of substrate or product can be combined with an enzyme molecule at the one time (e.g. allosteric mechanisms). The usual assumption, that the radioactive material is distributed among the substrate and product components according to a first-order law, is not valid. One can recognize whether isotope-exchange kinetics of an enzyme reaction follows first-order behaviour by using various initial concentrations of the labelled substance added to a mixture.     

Dependence of control coefficient distribution on the boundaries of a metabolic system: a generalized analysis of the effects of additional input and output reactions to a linear pathway

Both experimental and theoretical studies of metabolism are likely to relate to a segment that has been isolated for analytical purposes. In practice, it will be embedded in the whole of cellular metabolism. Thus, it is necessary to consider how conclusions about the control of an isolated pathway may be modified in this wider context where the input and output metabolites are considered as variables of cellular metabolism. Here, we analyse the effect of expanding a linear metabolic pathway by adding an extra input or an extra output. In particular, we analyse the effect of the elasticities of the extra steps on control coefficients. We derive matrix algebraic relationships for obtaining flux and concentration control coefficients from expressions depending on these extra elasticities and on parameters (elasticities and control coefficients) of the original pathway. These equations can be shown in certain cases to be generalized versions of earlier rescaling relationships and to be related to top-down analysis, but also apply where the new variable metabolite of the expanded pathway is an effector of more than one step of the original pathway. We use our relationships to analyse the dependence or independence of control coefficients upon these extra elasticities for the published analyses of the pathway of mammalian serine biosynthesis (Fell & Snell, 1988) and Escherischia coli threonine biosynthesis (Chassagnole et al., 2001). The same analysis can be applied to determine whether the transport reactions of substrates and products of a pathway in and out of a cell need to be included in estimations of the control coefficients of the enzymes.     

In vitro control analysis of an enzyme system: Experimental and analytical developments

In this paper we describe a flow-through system for reconstituting parts of metabolism from purified enzymes. This involves pumping continuously into a reaction chamber, fresh enzymes and reagents so that metabolic reactions occur in the chamber. The waste products leave the chamber via the outflow so that a steady state can be setup. The system we chose consisted of a single enzyme, lactate dehydrogenase. This enzyme was chosen because it consumes NADH in the chamber which could be monitored spectrophotometrically. The aim of the work was to investigate whether a steady state could be achieved in the flow system and whether a metabolic control analysis could be done. We measured two control coefficients, C_LDH and C_pump for the enzyme flux and NADH concentration and confirmed that the summation theorem applied to this system. The advantage of a flow-through system is that the titrations necessary to estimate the control coefficients can be easily and precisely controlled; this means that accurate estimates for the control coefficients can be obtained. In the paper, we discuss some statistical aspects of the data analysis and some possible applications of the technique, including a method to determine the presence of metabolic channelling between two different enzymes.     

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.     

Analysis of substrate protection of an immobilized glucose isomerase reactor

The effect of substrate protection on enzyme deactivation was studied in a differential bed and a packed bed reactor using a commercial immobilized glucose isomerase (Swetase, Nagase Co.). Experimental data obtained from differential bed reactor were analyzed based on Briggs-Haldane kinetics in which enzyme deactivation accompanying the protection of substrate was considered. The deactivation constant of the enzyme-substrate complex was found to be about half of that of the free enzyme. The mathematical analysis describing the performance of a packed bed reactor under the considerations of the effects of substrate protection, diffusion resistance, and enzyme deactivation was studied. The system equations for the packed bed reactor were solved using an orthogonal collocation method. The presence of substrate protection and the diffusion effect within the enzyme particles resulted in an axial variation of effectiveness factor, eta(D), along the length of the packed bed. The axial distribution profile of eta(D) was found to be dependent on the operation temperature, Based on the effect of substrate protection, a better substrate feed policy could be theoretically found for promoting productivity in long-term operation. (c) 1993 John Wiley & Sons, Inc.     

Effect of error of the quasi-steady-state approximation on the estimation of Km and Vm from a single time curve

For an irreversible, one-substrate enzyme mechanism, post-transient time curves of the substrate and the product are approximately described by different equations of the steady-state type. The magnitude of error of these approximations is shown to be small either at low enzyme/substrate or at low enzyme/ Michaelis-constant ratios. The effect of error on the kinetic parameters estimated from a single time curve is evaluated. It is shown that a set of well-separated substrate con centrations (which are still high relative to the concentration of enzyme) is crucial for obtaining accurate estimates of the parameters.     

Metabolic control and its analysis. Additional relationships between elasticities and control coefficients

Existing theorems from the analysis of metabolic control have been taken and embedded in a simple matrix algebra procedure for calculating the flux control coefficients of enzymes (formerly known as sensitivities) in a metabolic pathway from their kinetic properties (their elasticities). New theorems governing the flux control coefficients of branched pathways and substrate cycles have been derived to allow the procedure to be applied to complex pathway configurations. Modifications to the elasticity terms used in the equations have been theoretically justified so that the method remains valid for pathways with conserved metabolites (for example, the adenine nucleotide pool or the intermediates of a catalytic cycle such as the tricarboxylic acid cycle) or with pools of metabolites kept very near to equilibrium by very rapid reactions. The matrix equations generated using these theorems and relationships may be solved algebraically or numerically. Algebraic solutions have been used to determine the factors responsible for the degree of amplification of flux control coefficients by substrate cycles and to show that it is possible to derive expressions for the elasticities of a group of enzymes.     

Matrix method for determining steps most rate-limiting to metabolic fluxes in biotechnological processes

The metabolic control theory developed by Kacser, Burns, Heinrich, and Rapoport is briefly outlined, extended, and transformed so as optimally to address some biotechnological questions. The extensions include (i) a new theorem that relates the control of metabolite concentrations by enzyme activities to flux ratios at branches in metabolic pathways; (ii) a new theorem that does the same for the control of the distribution of the flux over two branches; (iii) a method that expresses these controls into properties (the so-called elasticity coefficients) of the enzymes in the pathway; and (iv) a theorem that relates the effects of changes in metabolite concentrations on reaction rates to the effects of changes in enzyme properties on the same rates. Matrix equations relating the flux control and concentration control coefficients to the elasticity coefficients of enzymes in simple linear and branched pathways incorporating feedback are given, together with their general solutions and a numerical example. These equations allow one to develop rigorous criteria by which to decide the optimal strategy for the improvement of a microbial process. We show how this could be used in deciding which property of which enzyme should be changed in order to obtain the maximal concentration of a metabolite or the maximal metabolic flux.     

The matrix method of metabolic control analysis: its validity for complex pathway structures

The sensitivities of the variables of a metabolic system (such as fluxes and concentrations) to variations in enzyme concentration are expressed in metabolic control analysis as control coefficients. The matrix method is a system of writing matrix equations that generate expressions for the control coefficients in terms of the characteristics of the components (principally the enzymes). Previously, the matrix method has been considered in terms of simple pathway structures; here we justify its applicability to complex pathways, such as those with multiple branches. It is shown that this requires modification of the branch point relationship to take account of changes of flux along the limbs of the branch and of stoichiometric factors. The method of deriving the flux control coefficients with respect to different fluxes in the system is extended to cope with these circumstances.     

Theory of metabolism regulation: a complete system of equations for regulation coefficients

Basic quantitative parameters of control in a metabolic system are considered: control coefficients of enzymes with respect to metabolic fluxes and concentrations, and in the case when there are conservation laws, the response coefficients of metabolic fluxes and concentrations to changes in the conserved sums of metabolite concentrations (e. g. conserved moieties). Relationships are obtained which generalize the well known connectivity relations for the case of metabolites binding by conservation laws. Additional relationships are obtained which complement the set of connectivity relations up to the complete system of equations for determining all the control coefficients. The control coefficients are expressed through the enzyme elasticity coefficients, steady state metabolic fluxes and concentrations. Formulas are derived which express response coefficients of flux and concentrations through the enzyme control and elasticity coefficients and metabolite concentrations.     

Introducing total substrates simplifies theoretical analysis at non-negligible enzyme concentrations: pseudo first-order kinetics and the loss of zero-order ultrasensitivity

The Briggs–Haldane standard quasi-steady state approximation and the resulting rate expressions for enzyme driven biochemical reactions provide crucial theoretical insight compared to the full set of equations describing the reactions, mainly because it reduces the number of variables and equations. When the enzyme is in excess of the substrate, a significant amount of substrate can be bound in intermediate complexes, so-called substrate sequestration. The standard quasi-steady state approximation is known to fail under such conditions, a main reason being that it neglects these intermediate complexes. Introducing total substrates, i.e., the sums of substrates and intermediate complexes, provides a similar reduction of the number of variables to consider but without neglecting the contribution from intermediate complexes. The present theoretical study illustrates the usefulness of such simplifications for the understanding of biochemical reaction schemes. We show how introducing the total substrates allows a simple analytical treatment of the relevance of significant enzyme concentrations for pseudo first-order kinetics and reconciles two proposed criteria for the validity of the pseudo first-order approximation. In addition, we show how the loss of zero-order ultrasensitivity in covalent modification cycles can be analyzed, in particular that approaches such as metabolic control analysis are immediately applicable to scenarios described by the total substrates with enzyme concentrations higher than or comparable to the substrate concentrations. A simple criterion which excludes the possibility of zero-order ultrasensitivity is presented.     

The regulatory design of an allosteric feedback loop: the effect of saturation by pathway substrate

Aspects of metabolic regulation can be fruitfully studied with a combination of generic modelling, control analysis and graphical analysis using rate characteristics. This paper analyses a prototypical supply-demand system consisting of a biosynthetic subsystem subject to allosteric inhibition by its product and a demand process that consumes this product. The effect of changes in affinity of the committing supply enzyme for the pathway substrate on the regulatory properties of the supply subsystem is compared for the Monod-Wyman-Changeux and the reversible Hill allosteric enzyme models. We found that the Hill model has a distinct advantage in that the steady-state concentration at which it maintains the product is set by the half-saturating product concentration and is independent of changes in the degree of saturation for substrate. In contrast, with the Monod-Wyman-Changeux model this set point varies with affinity for substrate. Explicitly incorporating reversibility in all rate equations made it possible to distinguish between kinetic and thermodynamic aspects of regulation. Combining the supply and demand rate characteristics allows us to explore both the control distribution at steady state and the regulatory performance of the system over a wide range of demand activities.     

Effects of cell volume regulating osmolytes on glycerol 3-phosphate binding to triosephosphate isomerase

During cell volume regulation, intracellular concentration changes occur in both inorganic and organic osmolytes in order to balance the extracellular osmotic stress and maintain cell volume homeostasis. Generally, salt and urea increase the Km's of enzymes and trimethylamine N-oxide (TMAO) counteracts these effects by decreasing Km's. The hypothesis to account for these effects is that urea and salt shift the native state ensemble of the enzyme toward conformers that are substrate-binding incompetent (BI), while TMAO shifts the ensemble toward binding competent (BC) species. Km's are often complex assemblies of rate constants involving several elementary steps in catalysis, so to better understand osmolyte effects we have focused on a single elementary event, substrate binding. We test the conformational shift hypothesis by evaluating the effects of salt, urea, and TMAO on the mechanism of binding glycerol 3-phosphate, a substrate analogue, to yeast triosephosphate isomerase. Temperature-jump kinetic measurements promote a mechanism consistent with osmolyte-induced shifts in the [BI]/[BC] ratio of enzyme conformers. Importantly, salt significantly affects the binding constant through its effect on the activity coefficients of substrate, enzyme, and enzyme-substrate complex, and it is likely that TMAO and urea affect activity coefficients as well. Results indicate that the conformational shift hypothesis alone does not account for the effects of osmolytes on Km's.     

Reaction kinetics of protease with substrate phage. Kinetic model developed using stromelysin

Peptide libraries generated using phage display have been widely applied to proteolytic enzymes for substrate selection and optimization, but the reaction kinetics between the enzyme and substrate phage are not well understood. Using a quantitative ELISA assay to monitor the disappearance of substrate, we have been able to follow the course of reaction between stromelysin, a metalloprotease, and its substrate phage. We found that under the proteolytic conditions where the enzyme was present in nanomolar concentration or higher, in excess over the substrate, the proteolysis of substrate phage was a single exponential event and the observed rate linear with respect to enzyme concentration. The enzyme concentration dependence could be described by pseudo first-order kinetic equations. Our data suggest that substrate binding is slow relative to the subsequent hydrolysis step, implying that the phage display selection process enriches clones that have high binding affinity to the protease, and the selection may not discriminate those of different chemical reactivity toward the enzyme. Considering that multiple substrate molecules may be present on a single phage particle, we regard the substrate phage reaction kinetic model as empirical. The validity of the model was ascertained when we successfully applied it to determine the binding affinity of a competitive inhibitor of stromelysin.     

Lipid-induced aggregation of phospholipase A2: sucrose density gradient ultracentrifugation and crosslinking studies

The aggregation behavior of cobra venom (Naja naja naja) phospholipase A2 in the presence of lipids and Ca2+ was examined using ultracentrifugation and crosslinking techniques. Velocity sedimentation experiments were performed in sucrose gradients. The sedimentation coefficients of the cobra phospholipase A2 and various controls, including bovine serum albumin (BSA), malate dehydrogenase, carbonic anhydrase and pancreatic phospholipase A2, were calculated both in the presence and absence of ligands. The monomeric phospholipid, diheptanoylphosphatidylcholine, and the phospholipid analogue, dodecylphosphocholine (DPC), increased the sedimentation coefficient of the cobra phospholipase A2 from 2.2 S to 2.9 S, a value that is consistent with the formation of an enzyme dimer. The control proteins were unaffected by the presence of phospholipid, except for BSA, which apparently binds large amounts of DPC. Crosslinking experiments with glutaraldehyde showed that in the presence of diheptanoylphosphatidylcholine or DPC, the amount of crosslinked enzyme increased. Ca2+ had no effect on the aggregation state of the enzyme as measured by either technique. Both the ultracentrifugation data and crosslinking data are consistent with the hypothesis that the cobra venom phospholipase A2 exists as a dimer or higher-order aggregate in the presence of lipid substrate, although it is yet to be determined whether the functional subunit is a monomer, dimer or higher-order oligomer.     

Partition coefficients of substrates and products and solvent selection for biocatalysis under nearly anhydrous conditions

The effect of solvent on the activity of mushroom tyrosinase toward three substrates was studied at a constant water activity of either 0.74 or 0.86. No simple correlation was observed between enzyme activity and log P, but partition coefficients of substrate (P(s)) and product (P(p)) gave systematic relations with enzyme activity. When initial reaction rates were considered, there was a bellshaped relationship between enzyme activity and P(s) with an optimal P(s) for each substrate. This can be explained by assuming that the solvent affected the enzyme activity primarily by affecting the substrate concentration in the aqueous layer around the catalyst where the enzymic reaction occurs. When long-term reaction rates were considered, a high P(p)/P(s) ratio was consistent with preservation of enzyme activity. (c) 1994 John Wiley & Sons, Inc.     

Unusual commonality in active site structural features of substrate promiscuous and specialist enzymes

Enzyme promiscuity is the ability of (some) enzymes to perform alternate reactions or catalyze non-cognate substrate(s). The latter is referred to as substrate promiscuity, widely studied for its biotechnological applications and understanding enzyme evolution. Insights into the structural basis of substrate promiscuity would greatly benefit the design and engineering of enzymes. Previous studies on some enzymes have suggested that flexibility, hydrophobicity, and active site protonation state could play an important role in enzyme promiscuity. However, it is not known yet whether substrate promiscuous enzymes have distinctive structural characteristics compared to specialist enzymes, which are specific for a substrate. In pursuit to address this, we have systematically compared substrate/catalytic binding site structural features of substrate promiscuous with those of specialist enzymes. For this, we have carefully constructed dataset of substrate promiscuous and specialist enzymes. On careful analysis, surprisingly, we found that substrate promiscuous and specialist enzymes are similar in various binding/catalytic site structural features such as flexibility, surface area, hydrophobicity, depth, and secondary structures. Recent studies have also alluded that promiscuity is widespread among enzymes. Based on these observations, we propose that substrate promiscuity could be defined as a continuum feature that varies from narrow (specialist) to broad range of substrate preferences. Moreover, diversity of conformational states of an enzyme accessible for ligand binding may possibly regulate its substrate preferences.