Improved analysis of bacterial CGH data beyond the log-ratio paradigm

Background

While the theory of enzyme kinetics is fundamental to analyzing and simulating biochemical systems, the derivation of rate equations for complex mechanisms for enzyme-catalyzed reactions is cumbersome and error prone. Therefore, a number of algorithms and related computer programs have been developed to assist in such derivations. Yet although a number of algorithms, programs, and software packages are reported in the literature, one or more significant limitation is associated with each of these tools. Furthermore, none is freely available for download and use by the community.

Results

We have implemented an algorithm based on the schematic method of King and Altman (KA) that employs the topological theory of linear graphs for systematic generation of valid reaction patterns in a GUI-based stand-alone computer program called KAPattern. The underlying algorithm allows for the assumption steady-state, rapid equilibrium-binding, and/or irreversibility for individual steps in catalytic mechanisms. The program can automatically generate MathML and MATLAB output files that users can easily incorporate into simulation programs.

Conclusion

A computer program, called KAPattern, for generating rate equations for complex enzyme system is a freely available and can be accessed at http://www.biocoda.org.     

Activation of wheat chloroplast sedoheptulose bisphosphatase: a continuous spectrophotometric assay

A new continuous spectrophotometric assay for sedoheptulose 1,7-bisphosphatase, applied to studies of the activation and steady-state kinetics of the wheat enzyme, is described. The assay enzyme sequence couples the formation of sedoheptulose 7-phosphate to the oxidation of NADH. The recycling of the reaction substrate enables measurements to be made at essentially constant substrate concentrations. Activation of wheat chloroplast sedoheptulose 1,7-bisphosphatase required a reducing agent and could be described by a first-order rate constant. The rate of activation was greatly increased in the presence of Mg²⁺ and sedoheptulose 1,7-bisphosphate. The K_m of the activated enzyme for sedoheptulose 1,7-bisphosphate. and its S_0.5 for Mg²⁺ were found to be 13.3 μm and 1.6 mm respectively. A high recovery method for purifying wheat chloroplast sedoheptulose 1,7-bisphosphatase is also detailed.     

Periodic operation of well mixed enzyme reactors with steady state multiplicities using relay feedback

Periodic operation of a well mixed enzyme reactor with input and output multiplicities is theoretically analyzed. The system is an isothermal continuously stirred tank enzyme reactor with a non ideal mixing represented by a Cholettes’ model and a reaction kinetics given by k ₁ c/(k ₁ c+1)². The system exhibits input multiplicities and output multiplicities. Some of the input steady-states correspond to unstable steady-states. The periodic operation in the inlet feed concentration cannot be done by open loop oscillation method. Relay feedback method is essential for the periodic operation of such systems. The improvement in the average conversion is calculated for the two unstable input steady-states. Higher average conversion is obtained at the largest value of the feed concentration. The limitation of relay feedback method for the stable input steady-state is also brought out.     

Growth Kinetics of Thiobacillus ferrooxidans Isolated from Arsenic Mine Drainage

Thiobacillus ferrooxidans is found in many Alaskan and Canadian drainages contaminated by metals dissolved from placer and lode gold mines. We have examined the iron-limited growth and iron oxidation kinetics of a T. ferrooxidans isolate, AK1, by using batch and continuous cultures. Strain AK1 is an arsenic-tolerant isolate obtained from placer gold mine drainage containing large amounts of dissolved arsenic. The steady-state growth kinetics are described with equations modified for threshold ferrous iron concentrations. The maximal specific growth rate (μ_max) for isolate AK1 at 22.5°C was 0.070 h⁻¹, and the ferrous iron concentration at which the half-maximal growth rate occurred (K_μ) was 0.78 mM. Cell yields varied inversely with growth rate. The iron oxidation kinetics of this organism were dependent on biomass. We found no evidence of ferric inhibition of ferrous iron oxidation for ferrous iron concentrations between 9.0 and 23.3 mM. A supplement to the ferrous medium of 2.67 mM sodium arsenite did not result in an increased steady-state biomass, nor did it appear to affect the steady-state growth kinetics observed in continuous cultures.     

Kinetic Studies of β-Galactosidase Induction

The kinetics of β-galactosidase induction in E. coli ML 3 have been studied. Following addition of inducer, the rate of enzyme synthesis accelerates from the uninduced to a steady-state rate. At saturating concentration of inducer the time constant (T_c) for this process is 2.5 to 3 minutes. With decreasing inducer concentration (I), increasing time constants are observed. I/I + K′ approximates I/T_c. The steady-state rate of β-galactosidase synthesis is approximated by I²/I² + K². K′ and K have been estimated for IPTG and TMG. The kinetics of β-galactosidase production after the removal of inducer by dilution or after the addition of glucose have been investigated. A transition time of 2.5 to 3 minutes is observed before enzyme synthesis slows or stops. These results are consistent with the hypothesis that the enzyme-forming unit is unstable.     

Control of CO2 fixation during the induction period. The role of thiol-mediated enzyme activation in the alga,Dunaliella

(1) Illumination of the unicellular green alga, Dunaliella, produced a 2–3-fold enhancement of ATPase activity in subsequently lysed algae. Using the inhibitor, tentoxin, it was shown that this light-induced activity, but not the light-independent activity, was attributable to the chloroplast coupling factor, CF₁. (1) A 4–5-fold increase in fructose-1,6-bisphosphatase activity was measured in Dunaliella lysed subsequent to illumination. (3) Experiments with methyl viologen demonstrated that both light-induced CF₁-ATPase and fructose-1,6-bisphosphatase activities were due to thiol-modulation of the enzymes by the algal thioredoxin system. (4) The light-induced increase in fructose-1,6-bisphosphatase activity could be simulated by incubation of intact algae in the dark with dithiothreitol. This thiol-induced increase in enzyme activity was accompanied by a decrease in the induction period of CO₂-dependent O₂ evolution upon subsequent measurement. (5) The kinetics of induction of both enzyme activities were very similar to the kinetics of induction of CO₂-dependent O₂ evolution in Dunaliella. As the light intensity was increased to 180 W · m² the steady-state enzyme activities increased in parallel with the rate of CO₂-dependent O₂ evolution. (6) The results are consistent with the imposition of a kinetic restraint on CO₂ fixation by the extent of enzyme activation under certain conditions in Dunaliella.     

Fitting of enzyme kinetic data without prior knowledge of weights.

A method is described for fitting equations to enzyme kinetic data that requires minimal assumptions about the error structure of the data. The dependence of the variances on the velocities is not assumed, but is deduced from internal evidence in the data. The effect of very bad observations ('outliers') is mitigated by decreasing the weight of observations that give large deviations from the fitted equation. The method works well in a wide range of circumstances when applied to the Michaelis-Menten equation, but it is not limited to this equation. It can be applied to most of the equations in common use for the analysis of steady-state enzyme kinetics. It has been implemented as a computer program that can fit a wide variety of equations with two, three or four parameters and two or three variables.     

Effects of isoleucine 135 side chain length on the cofactor donor-acceptor distance within F420H2:NADP+ oxidoreductase: A kinetic analysis

F₄₂₀H₂:NADP⁺ Oxidoreductase (Fno) catalyzes the reversible reduction of NADP⁺ to NADPH by transferring a hydride from the reduced F₄₂₀ cofactor. Here, we have employed binding studies, steady-state and pre steady-state kinetic methods upon wtFno and isoleucine 135 (I135) Fno variants in order to study the effects of side chain length on the donor-acceptor distance between NADP⁺ and the F₄₂₀ precursor, FO. The conserved I135 residue of Fno was converted to a valine, alanine and glycine, thereby shortening the side chain length. The steady-state kinetic analysis of wtFno and the variants showed classic Michaelis-Menten kinetics with varying FO concentrations. The data revealed a decreased k_cat as side chain length decreased, with varying FO concentrations. The steady-state plots revealed non-Michaelis-Menten kinetic behavior when NADPH was varied. The double reciprocal plot of the varying NADPH concentrations displays a downward concave shape, while the NADPH binding curves gave Hill coefficients of less than 1. These data suggest that negative cooperativity occurs between the two identical monomers. The pre steady-state Abs₄₂₀ versus time trace revealed biphasic kinetics, with a fast phase (hydride transfer) and a slow phase. The fast phase displayed an increased rate constant as side chain length decreased. The rate constant for the second phase, remained ~2 s^?1 for each variant. Our data suggest that I135 plays a key role in sustaining the donor-acceptor distance between the two cofactors, thereby regulating the rate at which the hydride is transferred from FOH₂ to NADP⁺. Therefore, Fno is a dynamic enzyme that regulates NADPH production.     

The quaternary structure of pyruvate kinase type 1 from Escherichia coli at low nanomolar concentrations

Pyruvate kinase (PK) is the key control point of glycolysis—the biochemical pathway central to energy metabolism and the production of precursors used in biosynthesis. PK type 1 from Escherichia coli (Ec-PK1) is activated by both fructose-1,6-bisphosphate (FBP) and its substrate, phosphoenol pyruvate (PEP). To date, it has not been possible to determine whether the enzyme is tetrameric at the low concentrations (i.e. low nM range) used to study the steady-state kinetics, or assess whether its allosteric effectors alter the oligomeric state of the enzyme at these concentrations. Employing the new technique of analytical ultracentrifugation with fluorescence detection we have, for the first time, shown that the K_D^4–2 for Ec-PK1 is in the subnanomolar range, well below the concentrations used in kinetic studies. In addition, we show that, unlike some other PK isoenzymes, the modulation of oligomeric state by the allosteric effectors FBP and PEP does not occur at a concentration of 10 nM or above.     

The monomer -- dimer association of rabbit lver aryl sulfatase A and its relationship to the anomalous kinetics.

The polymerization of aryl sulfatase A (aryl sulfate sulfohydrolase, EC 3.1.6.1) has been studied by frontal gel chromatography on Sephadex G-200 and Bio-Gel A-5m under various conditions of pH, ionic strength, and temperature. The aryl sulfatase A molecule exists as a monomer and as a dimer at pH 7.5 and pH 4.5, respectively. The extent of dissociation is markedly pH-, protein concentration-, and ionic strength-dependent. Only a small effect of temperature was observed. The enthalpy change (ΔH^o) for the dissociation was ?2.5 ± 1 kcal/mol at pH 5.5–5.6, and the entropy change for dissociation of the enzyme dimer to two monomeric units was ?47 cal mol^?1 deg^?1. Sulfate ion has little effect on the extent of dissociation of the enzyme at pH 5.6. The present studies suggest that the dissociation of rabbit liver aryl sulfatase A is regulated by the ionization of amino acid residues whose apparent pK is between pH 5 and 6. The driving force for the association of the subunits of the enzyme is primarily ionic and/or ionic/hydrogen bond formation. The small enthalpy change and the fact that dissociation is strongly favored by an increase in the ionic strength suggest that hydrophobic interactions play only a minor role in stabilizing the dimeric quaternary structure relative to the monomeric state. The monomeric form of the enzyme exhibits the anomalous kinetics often observed with sulfatase A but the dimer does not show anomalous kinetics. Since aryl sulfatase A is probably in the dimeric form in the lysosome, the anomalous kinetics of the enzyme are unlikely to be of physiological importance in the intact lysosome.     

Kinetic analysis of active membrane transport systems: equations for net velocity and isotope exchange.

Equations for the initial net velocity and for isotope exchange velocities in active membrane transport systems are presented. The equations are expressed entirely in terms of kinetic constants which are experimentally determinable from appropriate reciprocal plots, and replots of slope and intercept (for net velocity) or 1/Δslope and 1/Δintercept (for isotope exchange). The equations and plots are equally applicable to a soluble iso uni uni enzyme system.The effect of pH on sulfate transport by an ATP-sulfurylase negative mutant of Penicillium notatum was analyzed assuming that ³⁵SO₄²⁻ and H⁺ are cosubstrates of the transport system. The kinetics are consistent with an ordered addition to the carrier of one H⁺ ion followed by ³⁵SO₄²⁻, with H⁺ in equilibrium with the carrier and the carrier-H⁺ complex.     

Δ6-desaturase: improved methodology and analysis of the kinetics in a multi-enzyme system

A new method of assay for the A6-desaturation of linoleic acid was developed. This method, which uses HPLC for separation of the fatty acid substrate and product, exhibited a lower coefficient of variation (0.3%) than the reported TLC method (3.5%) [l], and avoided the step of methylation of the saponified fatty acid substrate and product. Using this new method of assay, the kinetics of the Δ6-desaturase in a multi-enzyme system were analysed. A number of factors that could have striking effects on desaturase kinetics were investigated, including the effect of (i) endogenous microsomal linoleic acid on total substrate concentration, and (ii) the pre-reaction catalysed by acyl-CoA synthetase and competing reactions catalysed by lysophospholipid acyltransferase and acyl-CoA hydrolase. Endogenous free linoleate in the hepatic microsomes was found to be 2.9 ± l.0 AM (0.5 mg microsomal protein/ml), which was comparable to added substrate concentrations (l.8 to 7.9 μM). The kinetics of the Δ6-desaturase were dissected from the kinetics of the above mentioned pre-reaction and competing reactions through a combination of experimental approaches and computer modeling. From computer modeling, a K_m and V_max of l.5 μM and 0.063 nmol/min were calculated for the Δ6-desaturase, compared to K_m and V_max of 10.7 μM and 0.08 nmol/min calculated directly from data uncorrected for endogenous substrate. It was concluded that lysophospholipid acyltransferase, acyl-CoA synthetase and endogenous linoleic acid significantly affect the kinetic measurements of hepatic microsomal Δ6-desaturase. These results have implications for kinetic analyses of all desaturases in microsomal systems.     

Characterization of recombinant human acetyl-CoA carboxylase-2 steady-state kinetics

Acetyl-CoA carboxylase (ACC) catalyzes the carboxylation of acetyl-CoA to form malonyl-CoA, a key metabolite in the fatty acid synthetic and oxidation pathways. The present study describes the steady-state kinetic analysis of a purified recombinant human form of the enzyme, namely ACC2, using a novel LC/MS/MS assay to directly measure malonyl-CoA formation. Four dimensional matrices, in which bicarbonate (HCO₃^?), ATP, acetyl-CoA, and citrate were varied, and global data fitting to appropriate steady-state equations were used to generate kinetic constants. Product inhibition studies support the notion that the enzyme proceeds through a hybrid (two-site) random Ter Ter mechanism, one that likely involves a two-step reaction at the biotin carboxylase domain. Citrate, a known activator of animal forms of ACC, activates both by increasing k_cat and k_cat/K_M for ATP and acetyl-CoA.     

Assay of both activities of the bifunctional tRNA-modifying enzyme MnmC reveals a kinetic basis for selective full modification of cmnm5s2U to mnm5s2U

Transfer RNA (tRNA) contains a number of complex ‘hypermodified’ nucleosides that are essential for a number of genetic processes. Intermediate forms of these nucleosides are rarely found in tRNA despite the fact that modification is not generally a complete process. We propose that the modification machinery is tuned into an efficient ‘assembly line’ that performs the modification steps at similar, or sequentially increasing, rates to avoid build-up of possibly deleterious intermediates. To investigate this concept, we measured steady-state kinetics for the final two steps of the biosynthesis of the mnm⁵s²U nucleoside in Escherichia coli tRNA^Glu, which are both catalysed by the bifunctional MnmC enzyme. High-performance liquid chromatography-based assays using selectively under-modified tRNA substrates gave a K_m value of 600 nM and k_cat 0.34 s⁻¹ for the first step, and K_m 70 nM and k_cat 0.31 s⁻¹ for the second step. These values show that the second reaction occurs faster than the first reaction, or at a similar rate at very high substrate concentrations. This result indicates that the enzyme is kinetically tuned to produce fully modified mnm⁵(s²)U while avoiding build-up of the nm⁵(s²)U intermediate. The assay method developed here represents a general approach for the comparative analysis of tRNA-modifying enzymes.     

ZONE BEHAVIOR OF ENZYMES : ILLUSTRATED BY THE EFFECT OF DISSOCIATION CONSTANT AND DILUTION ON THE SYSTEM CHOLINESTERASE-PHYSOSTIGMINE

1. The kinetics of the reversible combination of one enzyme center with one molecule of a substrate or inhibitor is treated as a true bimolecular instead of a pseudomonomolecular reaction. The general equations describing such a reaction are presented and analyzed algebraically and graphically. 2. A new term, "specific concentration," is introduced to denote the concentration of reactants in units equal to the dissociation constant. Its use makes the kinetic equations universally applicable to all reversible systems of the given type. 3. It is shown that such a system exhibits three "zones" of behavior. Each zone is characterized and shown to exhibit significant differences in the function relating the concentrations of the components of the system at equilibrium. The zone boundaries are rigorously defined in terms of the specific enzyme concentration, for the mathematical error tolerable with a given experimental accuracy; and approximate boundaries for practical use are proposed. 4. The classical treatment of enzyme kinetics is shown to be a limiting case valid only for low specific enzyme concentrations (zone A) and to be inapplicable in a number of systems whose dissociation constants are very small or whose molar enzyme concentrations are very great, and in which, therefore, the specific enzyme concentrations are large. See Table I for a summary of zone differences. 5. In an enzyme system containing substrate or inhibitor, dilution before determination of reaction velocities is shown to be a crucial operation, entailing large changes in the fraction of enzyme in the form of a complex. The changes in fractional activity or inhibition with dilution are shown to be a function of specific enzyme concentration, the dilution factor, and the fraction of enzyme initially in the form of complex. Equations are given permitting the calculation of the state of the system at any concentration. The errors introduced into physiological work by failure to take the dilution effect into account are pointed out. 6. Experimental data are presented showing that the system composed of serum cholinesterase and physostigmine behaves as predicted by the dilution effect equations. 7. Two other conclusions of practical pharmacological importance are drawn from the theory of zone behavior: (a) The finding that a biological response is a linear function of the dose of a drug does not necessarily mean that the reaction is irreversible, but only that if reversible, the reactant with which the drug combines has a high specific concentration. (b) If a tissue enzyme has a high specific concentration, all reversible inhibitors will be equally potent in combining with it, regardless of their relative potency in dilute systems; provided only that their dissociation constants are within certain broad limits. 8. It is shown how the type of analysis here applied to bimolecular reactions can be applied in toto to systems of the type E + nX ⇋ EX_n, where n molecules of substrate or inhibitor unite with one enzyme center. The zone boundaries and the magnitude of the dilution effect change with n, but the general characteristics of the zones are the same for all values of n. 9. Since the analysis is based only on mass law assumptions, it is applicable to any system that is formally analogous to the one here treated.     

Kinetics of Growth and Amylase Production of Saccharomycopsis fibuligera on Potato Processing Wastewater

The kinetics of growth and amylase production of Saccharomycopsis fibuligera were studied in a chemostat on a synthetic potato processing blancher water. Dilution rates (D) from 0.101 to 0.480 h⁻¹ were examined. A mathematical model based on the Monod equation was developed. The yield of cell mass from carbohydrates was constant and equal to 0.84. The maximum specific growth rate and the Monod constant were determined to be 0.596 h⁻¹ and 0.226 mg/ml, respectively. An equation for the steady-state starch concentrations was empirically derived. The steady-state noncarbohydrate carbon levels rose linearly with D. Reducing sugars were the growth-limiting substrate, and their steady-state levels conformed to Monod kinetics. The yield of amylase from the cell mass (Y_z) declined as D rose and was described by the equation Y_z = (−8.005D + 4.076). The model predicted that the maximum production of cell mass should occur at D = 0.35 h⁻¹ and the maximum production of amylase should occur at D = 0.22 h⁻¹. The mathematical model presented agreed with the experimental results in its prediction of the steady-state level of reducing sugar, starch, cell mass, and amylase concentrations as well as the productivity of amylase.     

Purification and characterization of a mutant ATP phosphoribosyltransferase hypersensitive to histidine feedback inhibition.

A mutant form of ATP phosphoribosyltranferase (EC 2.4.2.17), hisG1708c, which results in abnormally slow growth of Salmonella typhimurium at 20 °C was purified to homogeneity and kinetic and chemical behavior were characterized. Initial velocity steady-state substrate kinetics of wild-type and mutant enzymes at 37 °C were consistent with sequential kinetics and demonstrated that standard assay concentrations of substrates were sufficient to substantially saturate both enzymes. Nearly time-independent inhibition by histidine at 37 °C could be obtained only after incubation in the presence of product and histidine. Studies at 37 °C showed that the mutant enzyme is 24 times more sensitive to histidine than the wild type in a negatively cooperative manner instead of the positively cooperative manner seen for wild type. Pure mutant enzyme exhibits two major electrophoretic species of native enzyme. Although one less cysteine is titratable in native mutant enzyme, the amino acid compositions of mutant and wild-type enzymes are similar. Histidine produces an ultraviolet difference spectrum in mutant enzyme closely resembling that produced in wild type. Binding of histidyl-tRNA to mutant enzyme is substantially inhibited by histidine. It is concluded that the hisG1708c mutation alters some conformational processes coupled to the histidine binding site while not affecting others.     

Investigation of the enzymatic activity of the Na,K-ATPase via isothermal titration microcalorimetry

Isothermal titration microcalorimetry (ITC) is shown here to be a sensitive and accurate method for assaying the steady-state enzyme activity of the Na⁺,K⁺-ATPase. Single ATP injection experiments yield an apparent enthalpy change for the ATP hydrolysis reaction catalyzed by the enzyme of −51 (± 1) kJ mol⁻¹. This value is independent of the amount of ADP accumulated in the sample cell, which indicates that under the experimental conditions studied here (saturating Na⁺ and K⁺ concentrations) ADP does not inhibit enzyme activity by reversal of the phosphorylation reaction and resynthesizing ATP. Multiple ATP injection titration experiments in which varying concentrations of ADP were initially included in the sample cell could be adequately explained by a Michaelis-Menten kinetic model incorporating noncompetitive inhibition. This suggests that ADP inhibits the enzyme by binding to more than one enzyme intermediate and inhibiting forward reactions of the enzyme. Values of K_m and K_I obtained for the fits agree with literature values obtained by other methods. Because ITC is a direct method of continually monitoring enzyme activity, it is a valuable supplement to less direct or noncontinuous methods such as colorimetric, enzyme-coupled or radioactivity-based assays.     

Determination of diadenosine tetraphosphate (Ap4A) levels in subpicomole quantities by a phosphodiesterase luciferin-luciferase coupled assay: Application as a specific assay for diadenosine tetraphosphatase

A coupled enzymatic assay for diadenosine 5′, 5?-P¹, P⁴-tetraphosphate(Ap₄A) is described. Luciferin-luciferase produces light by consuming the ATP that is liberated by the action of snake venom phosphodiesterase on Ap₄A. The procedure is linear with Ap₄A levels ranging from 0.02 to 2 pmol. The pool size of Ap₄A in human leukemic cells was determined by acid extraction of the cells followed by initial fractionation of the extract on a DEAE-cellulose column and application of the phosphodiesterase luciferin-luciferase coupled assay. The method was also used to follow the purification of a diadenosine tetraphosphate-degrading enzyme (diadenosine tetraphosphatase, Ap₄Aase) from mouse ascites tumor cells. The partially purified enzyme had a K_m of 2.8 μm for Ap₄A when applying the coupled enzymatic assay for the determination of initial rate kinetics.