A simple test for inactivation of an enzyme during assay

The measurement of true initial rates is not always absolutely necessary to estimate enzyme kinetic parameters

In the chapters dealing with enzyme reactions, the authors of all Biochemistry textbooks and of even more specialized texts consider that the characteristic parameters (kcat and Km) must be determined under initial or steady-state rate conditions. This implies the transformation of a very limited proportion of substrate (at most 10-20%) or a continuous recording of the product or substrate concentration vs. time. Both options can present practical difficulties. Is it possible to get around these very stringent conditions? Here we show that in the most favourable cases up  to 70% of the substrate can be converted resulting in systematic errors on the parameters (that can easily be taken account of) if the simple Henri-Michaelis-Menten equation is utilised. Alternatively, the integrated form of the same equation directly yields excellent estimates of the same parameters. Our observations should greatly facilitate the task of researchers who study systems in which measurements of the reaction progress are painstaking or when substrate concentrations close to the detection limit must be used. The general conclusion is that it is not always absolutely necessary to determine initial or steady-state rates to obtain reliable estimations of the enzyme kinetic parameters..

Dipeptide-Derived Alkynes as Potent and Selective Irreversible Inhibitors of Cysteine Cathepsins

The potential of designing irreversible alkyne-based inhibitors of cysteine cathepsins by isoelectronic replacement in reversibly acting potent peptide nitriles was explored. The synthesis of the dipeptide alkynes was developed with special emphasis on stereochemically homogeneous products obtained in the Gilbert-Seyferth homologation for C≡C bond formation. Twenty-three dipeptide alkynes and 12 analogous nitriles were synthesized and investigated for their inhibition of cathepsins B, L, S, and K. Numerous combinations of residues at positions P1 and P2 as well as terminal acyl groups allowed for the derivation of extensive structure-activity relationships, which were rationalized by computational covalent docking for selected examples. The determined inactivation constants of the alkynes at the target enzymes span a range of >3 orders of magnitude (3-10 133 M^-1 s^-1). Notably, the selectivity profiles of alkynes do not necessarily reflect those of the nitriles. Inhibitory activity at the cellular level was demonstrated for selected compounds.

On the reproducibility of enzyme reactions and kinetic modelling

Enzyme reactions are highly dependent on reaction conditions. To ensure reproducibility of enzyme reaction parameters, experiments need to be carefully designed and kinetic modeling meticulously executed. Furthermore, to enable quality control of enzyme reaction parameters, the experimental conditions, the modeling process as well as the raw data need to be reported comprehensively. By taking these steps, enzyme reaction parameters can be open and FAIR (findable, accessible, interoperable, re-usable) as well as repeatable, replicable and reproducible. This review discusses these requirements and provides a practical guide to designing initial rate experiments for the determination of enzyme reaction parameters and gives an open, FAIR and re-editable example of the kinetic modeling of an enzyme reaction. Both the guide and example are scripted with Python in Jupyter Notebooks and are publicly available (https://fairdomhub.org/investigations/483/snapshots/1). Finally, the prerequisites of automated data analysis and machine learning algorithms are briefly discussed to provide further motivation for the comprehensive, open and FAIR reporting of enzyme reaction parameters.

Biochemical Properties of a New Polysaccharide Lyase Family 25 Ulvan Lyase TsUly25B from Marine Bacterium Thalassomonas sp. LD5

Marine macroalgae, contributing much to the bioeconomy, have inspired tremendous attention as sustainable raw materials. Ulvan, as one of the main structural components of green algae cell walls, can be degraded by ulvan lyase through the β-elimination mechanism to obtain oligosaccharides exhibiting several good physiological activities. Only a few ulvan lyases have been characterized until now. This thesis explores the properties of a new polysaccharide lyase family 25 ulvan lyase TsUly25B from the marine bacterium Thalassomonas sp. LD5. Its protein molecular weight was 54.54 KDa, and it was most active under the conditions of 60 °C and pH 9.0. The Km and kcat values were 1.01 ± 0.05 mg/mL and 10.52 ± 0.28 s-1, respectively. TsUly25B was salt-tolerant and NaCl can significantly improve its thermal stability. Over 80% of activity can be preserved after being incubated at 30 °C for two days when the concentration of NaCl in the solution is above 1 M, while 60% can be preserved after incubation at 40 °C for 10 h with 2 M NaCl. TsUly25B adopted an endolytic manner to degrade ulvan polysaccharides, and the main end-products were unsaturated ulvan disaccharides and tetrasaccharides. In conclusion, our research enriches the ulvan lyase library and advances the utilization of ulvan lyases in further fundamental research as well as ulvan oligosaccharides production.

Tetracycline Derivatives Inhibit Plasmodial Cysteine Protease Falcipain-2 through Binding to a Distal Allosteric Site

Allosteric inhibitors regulate enzyme activity from remote and usually specific pockets. As they promise an avenue for less toxic and safer drugs, the identification and characterization of allosteric inhibitors has gained great academic and biomedical interest in recent years. Research on falcipain-2 (FP-2), the major papain-like cysteine hemoglobinase of Plasmodium falciparum, might benefit from this strategy to overcome the low selectivity against human cathepsins shown by active site-directed inhibitors. Encouraged by our previous finding that methacycline inhibits FP-2 noncompetitively, here we assessed other five tetracycline derivatives against this target and characterized their inhibition mechanism. As previously shown for methacycline, tetracycline derivatives inhibited FP-2 in a noncompetitive fashion, with K_i values ranging from 121 to 190 μM. A possible binding to the S' side of the FP-2 active site, similar to that described by X-ray crystallography (PDB: 6SSZ) for the noncompetitive inhibitor E-chalcone 48 (EC48), was experimentally discarded by kinetic analysis using a large peptidyl substrate spanning the whole active site. By combining lengthy molecular dynamics (MD) simulations that allowed methacycline to diffuse from solution to different FP-2 surface regions and free energy calculations, we predicted the most likely binding mode of the ligand. Of note, the proposed binding pose explains the low differences in K_i values observed for the tested tetracycline derivatives and the calculated binding free energies match the experimental values. Overall, this study has implications for the design of novel allosteric inhibitors against FP-2 and sets the basis for further optimization of the tetracycline scaffold to produce more potent and selective inhibitors.

The Moderately (D)efficient Enzyme: Catalysis-Related Damage In Vivo and Its Repair

Enzymes have in vivo life spans. Analysis of life spans, i.e., lifetime totals of catalytic turnovers, suggests that nonsurvivable collateral chemical damage from the very reactions that enzymes catalyze is a common but underdiagnosed cause of enzyme death. Analysis also implies that many enzymes are moderately deficient in that their active-site regions are not naturally as hardened against such collateral damage as they could be, leaving room for improvement by rational design or directed evolution. Enzyme life span might also be improved by engineering systems that repair otherwise fatal active-site damage, of which a handful are known and more are inferred to exist. Unfortunately, the data needed to design and execute such improvements are lacking: there are too few measurements of in vivo life span, and existing information about the extent, nature, and mechanisms of active-site damage and repair during normal enzyme operation is too scarce, anecdotal, and speculative to act on. Fortunately, advances in proteomics, metabolomics, cheminformatics, comparative genomics, and structural biochemistry now empower a systematic, data-driven approach for identifying, predicting, and validating instances of active-site damage and its repair. These capabilities would be practically useful in enzyme redesign and improvement of in-use stability and could change our thinking about which enzymes die young in vivo, and why.

In silico identification of noncompetitive inhibitors targeting an uncharacterized allosteric site of falcipain-2

Falcipain-2 (FP-2) is a Plasmodium falciparum hemoglobinase widely targeted in the search for antimalarials. FP-2 can be allosterically modulated by various noncompetitive inhibitors that have been serendipitously identified. Moreover, the crystal structures of two inhibitors bound to an allosteric site, termed site 6, of the homolog enzyme human cathepsin K (hCatK) suggest that the equivalent region in FP-2 might play a similar role. Here, we conduct the rational identification of FP-2 inhibitors through virtual screenings (VS) of compounds into several pocket-like conformations of site 6, sampled during molecular dynamics (MD) simulations of the free enzyme. Two noncompetitive inhibitors, ZINC03225317 and ZINC72290660, were confirmed using in vitro enzymatic assays and their poses into site 6 led to calculated binding free energies matching the experimental ones. Our results provide strong evidence about the allosteric inhibition of FP-2 through binding of small molecules to site 6, thus opening the way toward the discovery of new inhibitors against this enzyme.

A Novel High-Throughput FLIPR Tetra-Based Method for Capturing Highly Confluent Kinetic Data for Structure-Kinetic Relationship Guided Early Drug Discovery

Target engagement by small molecules is necessary for producing a physiological outcome. In the past, a lot of emphasis was placed on understanding the thermodynamics of such interactions to guide structure-activity relationships. It is becoming clearer, however, that understanding the kinetics of the interaction between a small-molecule inhibitor and the biological target [structure-kinetic relationship (SKR)] is critical for selection of the optimum candidate drug molecule for clinical trial. However, the acquisition of kinetic data in a high-throughput manner using traditional methods can be labor intensive, limiting the number of molecules that can be tested. As a result, in-depth kinetic studies are often carried out on only a small number of compounds, and usually at a later stage in the drug discovery process. Fundamentally, kinetic data should be used to drive key decisions much earlier in the drug discovery process, but the throughput limitations of traditional methods preclude this. A major limitation that hampers acquisition of high-throughput kinetic data is the technical challenge in collecting substantially confluent data points for accurate parameter estimation from time course analysis. Here, we describe the use of the fluorescent imaging plate reader (FLIPR), a charge-coupled device (CCD) camera technology, as a potential high-throughput tool for generating biochemical kinetic data with smaller time intervals. Subsequent to the design and optimization of the assay, we demonstrate the collection of highly confluent time-course data for various kinase protein targets with reasonable throughput to enable SKR-guided medicinal chemistry. We select kinase target 1 as a special case study with covalent inhibition, and demonstrate methods for rapid and detailed analysis of the resultant kinetic data for parameter estimation. In conclusion, this approach has the potential to enable rapid kinetic studies to be carried out on hundreds of compounds per week and drive project decisions with kinetic data at an early stage in drug discovery.

An Electrochemical Approach to Follow and Evaluate the Kinetic Catalysis of Ricin on hsDNA

International authorities classify the ricin toxin, present in castor seeds, as a potential agent for use in bioterrorism. Therefore, the detection, identification, and characterization of ricin are considered the first actions for its risk assessment during a suspected exposure, parallel to the development of therapeutic and medical countermeasures. In this study, we report the kinetic analysis of electro-oxidation of adenine released from hsDNA by the catalytic action of ricin by square wave voltammetry. The results suggest that ricin-mediated adenine release exhibited an unusual kinetic profile, with a progress curve controlled by the accumulation of the product and the values of the kinetic constants of 46.6 µM for Km and 2000 min⁻¹ for kcat, leading to a catalytic efficiency of 7.1 × 10⁵ s⁻¹ M⁻¹.

Screening and Identification of Metacaspase Inhibitors: Evaluation of Inhibition Mechanism and Trypanocidal Activity

A common strategy to identify new antiparasitic agents is the targeting of proteases, due to their essential contributions to parasite growth and development. Metacaspases (MCAs) are cysteine proteases present in fungi, protozoa, and plants. These enzymes, which are associated with crucial cellular events in trypanosomes, are absent in the human host, thus arising as attractive drug targets. To find new MCA inhibitors with trypanocidal activity, we adapted a continuous fluorescence enzymatic assay to a medium-throughput format and carried out screening of different compound collections, followed by the construction of dose-response curves for the most promising hits. We used MCA5 from Trypanosoma brucei (TbMCA5) as a model for the identification of inhibitors from the GlaxoSmithKline HAT and CHAGAS chemical boxes. We also assessed a third collection of nine compounds from the Maybridge database that had been identified by virtual screening as potential inhibitors of the cysteine peptidase falcipain-2 (clan CA) from Plasmodium falciparum Compound HTS01959 (from the Maybridge collection) was the most potent inhibitor, with a 50% inhibitory concentration (IC₅₀) of 14.39 µM; it also inhibited other MCAs from T. brucei and Trypanosoma cruzi (TbMCA2, 4.14 µM; TbMCA3, 5.04 µM; TcMCA5, 151 µM). HTS01959 behaved as a reversible, slow-binding, and noncompetitive inhibitor of TbMCA2, with a mechanism of action that included redox components. Importantly, HTS01959 displayed trypanocidal activity against bloodstream forms of T. brucei and trypomastigote forms of T. cruzi, without cytotoxic effects on Vero cells. Thus, HTS01959 is a promising starting point to develop more specific and potent chemical structures to target MCAs.

Drug discovery for enzymes

Enzymes are essential, physiological catalysts involved in all processes of life, including metabolism, cellular signaling and motility, as well as cell growth and division. They are attractive drug targets because of the presence of defined substrate-binding pockets, which can be exploited as binding sites for pharmaceutical enzyme inhibitors. Understanding the reaction mechanisms of enzymes and the molecular mode of action of enzyme inhibitors is indispensable for the discovery and development of potent, efficacious, and safe novel drugs. The combination of classical concepts of enzymology with new experimental and data analysis methods opens new routes for drug discovery.

Explicit Treatment of Non-Michaelis-Menten and Atypical Kinetics in Early Drug Discovery*

Biological systems are highly regulated. They are also highly resistant to sudden perturbations enabling them to maintain the dynamic equilibrium essential to sustain life. This robustness is conferred by regulatory mechanisms that influence the activity of enzymes/proteins within their cellular context to adapt to changing environmental conditions. However, the initial rules governing the study of enzyme kinetics were mostly tested and implemented for cytosolic enzyme systems that were easy to isolate and/or recombinantly express. Moreover, these enzymes lacked complex regulatory modalities. Now, with academic labs and pharmaceutical companies turning their attention to more-complex systems (for instance, multiprotein complexes, oligomeric assemblies, membrane proteins and post-translationally modified proteins), the initial axioms defined by Michaelis-Menten (MM) kinetics are rendered inadequate, and the development of a new kind of kinetic analysis to study these systems is required. This review strives to present an overview of enzyme kinetic mechanisms that are atypical and, oftentimes, do not conform to the classical MM kinetics. Further, it presents initial ideas on the design and analysis of experiments in early drug-discovery for such systems, to enable effective screening and characterisation of small-molecule inhibitors with desirable physiological outcomes.

Trypanosoma cruzi synthesizes proline via a Δ1-pyrroline-5-carboxylate reductase whose activity is fine-tuned by NADPH cytosolic pools

In Trypanosoma cruzi, the etiological agent of Chagas disease, the amino acid proline participates in processes related to T. cruzi survival and infection, such as ATP production, cell differentiation, host-cell invasion, and in protection against osmotic, nutritional, and thermal stresses and oxidative imbalance. However, little is known about proline biosynthesis in this parasite. Δ1-Pyrroline-5-carboxylate reductase (P5CR, EC 1.5.1.2) catalyzes the biosynthesis of proline from Δ1-pyrroline-5-carboxylate (P5C) with concomitant NADPH oxidation. Herein, we show that unlike other eukaryotes, T. cruzi biosynthesizes proline from P5C, which is produced exclusively from glutamate. We found that TcP5CR is an NADPH-dependent cytosolic enzyme with a Kmapp for P5C of 27.7 μM and with a higher expression in the insect-resident form of the parasite. High concentrations of the co-substrate NADPH partially inhibited TcP5CR activity, prompting us to analyze multiple kinetic inhibition models. The model that best explained the obtained data included a non-competitive substrate inhibition mechanism (Kiapp=45±0.7μM). Therefore, TcP5CR is a candidate as a regulatory factor of this pathway. Finally, we show that P5C can exit trypanosomatid mitochondria in conditions that do not compromise organelle integrity. These observations, together with previously reported results, lead us to propose that in T. cruzi TcP5CR participates in a redox shuttle between the mitochondria and the cytoplasm. In this model, cytoplasmic redox equivalents from NADPH pools are transferred to the mitochondria using proline as a reduced metabolite, and shuttling to fuel electrons to the respiratory chain through proline oxidation by its cognate dehydrogenase.

An NADH preferring acetoacetyl-CoA reductase is engaged in poly-3-hydroxybutyrate accumulation in Escherichia coli

Oxygen supply implies higher production cost and reduction of maximum theoretical yields. Thus, generation of fermentation products is more cost-effective. Aiming to find a key piece for the production of (poly)-3-hydroxybutyrate (PHB) as a fermentation product, here we characterize an acetoacetyl-CoA reductase, isolated from a Candidatus Accumulibacter phosphatis-enriched mixed culture, showing a (k_cat^NADH/K_M^NADH)/(k_cat^NADPH/K_M^NADPH)>500. Further kinetic analyses indicate that, at physiological concentrations, this enzyme clearly prefers NADH, presenting the strongest NADH preference so far observed among the acetoacetyl-CoA reductases. Structural and kinetic analyses indicate that residues between E37 and P41 have an important role for the observed NADH preference. Moreover, an operon was assembled combining the phaCA genes from Cupriavidus necator and the gene encoding for this NADH-preferring acetoacetyl-CoA reductase. Escherichia coli cells expressing that assembled operon showed continuous accumulation of PHB under oxygen limiting conditions and PHB titer increased when decreasing the specific oxygen consumption rate. Taken together, these results show that it is possible to generate PHB as a fermentation product in E. coli, opening opportunities for further protein/metabolic engineering strategies envisioning a more efficient anaerobic production of PHB.

Reverse transcriptase kinetics for one-step RT-PCR

Understanding reverse transcriptase (RT) activity is critical for designing fast one-step RT-PCRs. We report a stopped-flow assay that monitors SYBR Green I fluorescence to investigate RT activity in PCR conditions. We studied the influence of PCR conditions on RT activity and assessed the accuracy of cDNA synthesis predictions for one-step RT-PCR. Nucleotide incorporation increased from 26 to 89 s^-1 between 1.5 and 6 mM MgCl₂ but was largely unaffected by changes in KCl. Conversely, increasing KCl from 15 to 75 mM increased apparent rate constants for RT-oligonucleotide binding (0.010-0.026 nM^-1 s^-1) and unbinding (0.2-1.5 s^-1). All rate constants increased between 22 and 42 °C. When evaluated by PCR quantification cycle, cDNA predictions differed from experiments using RNase H+ RT (average 1.7 cycles) and RNase H- (average 4.5 cycles). Decreasing H+ RT concentrations 10 to 10⁴-fold from manufacturer recommendations improved cDNA predictions (average 0.8 cycles) and increased RT-PCR assay efficiency. RT activity assays and models can be used to aid assay design and improve the speed of RT-PCRs. RT type and concentration must be selected to promote rapid cDNA synthesis but minimize nonspecific amplification. We demonstrate 2-min one-step RT-PCR of a Zika virus target using reduced RT concentrations and extreme PCR.

Considerations when Measuring Biocatalyst Performance

As biocatalysis matures, it becomes increasingly important to establish methods with which to measure biocatalyst performance. Such measurements are important to assess immobilization strategies, different operating modes, and reactor configurations, aside from comparing protein engineered variants and benchmarking against economic targets. While conventional measurement techniques focus on a single performance metric (such as the total turnover number), here, it is argued that three metrics (achievable product concentration, productivity, and enzyme stability) are required for an accurate assessment of scalability.

In silico Guided Drug Repurposing: Discovery of New Competitive and Non-competitive Inhibitors of Falcipain-2

Malaria is among the leading causes of death worldwide. The emergence of Plasmodium falciparum resistant strains with reduced sensitivity to the first line combination therapy and suboptimal responses to insecticides used for Anopheles vector management have led to renewed interest in novel therapeutic options. Here, we report the development and validation of an ensemble of ligand-based computational models capable of identifying falcipain-2 inhibitors, and their subsequent application in the virtual screening of DrugBank and Sweetlead libraries. Among four hits submitted to enzymatic assays, two (odanacatib, an abandoned investigational treatment for osteoporosis and bone metastasis, and the antibiotic methacycline) confirmed inhibitory effects on falcipain-2, with Ki of 98.2 nM and 84.4 μM. Interestingly, Methacycline proved to be a non-competitive inhibitor (α = 1.42) of falcipain-2. The effects of both hits on falcipain-2 hemoglobinase activity and on the development of P. falciparum were also studied.

Potent and selective inhibitors for M32 metallocarboxypeptidases identified from high-throughput screening of anti-kinetoplastid chemical boxes

Enzymes of the M32 family are Zn-dependent metallocarboxypeptidases (MCPs) widely distributed among prokaryotic organisms and just a few eukaryotes including Trypanosoma brucei and Trypanosoma cruzi, the causative agents of sleeping sickness and Chagas disease, respectively. These enzymes are absent in humans and several functions have been proposed for trypanosomatid M32 MCPs. However, no synthetic inhibitors have been reported so far for these enzymes. Here, we present the identification of a set of inhibitors for TcMCP-1 and TbMCP-1 (two trypanosomatid M32 enzymes sharing 71% protein sequence identity) from the GlaxoSmithKline HAT and CHAGAS chemical boxes; two collections grouping 404 compounds with high antiparasitic potency, drug-likeness, structural diversity and scientific novelty. For this purpose, we adapted continuous fluorescent enzymatic assays to a medium-throughput format and carried out the screening of both collections, followed by the construction of dose-response curves for the most promising hits. As a result, 30 micromolar-range inhibitors were discovered for one or both enzymes. The best hit, TCMDC-143620, showed sub-micromolar affinity for TcMCP-1, inhibited TbMCP-1 in the low micromolar range and was inactive against angiotensin I-converting enzyme (ACE), a potential mammalian off-target structurally related to M32 MCPs. This is the first inhibitor reported for this family of MCPs and considering its potency and specificity, TCMDC-143620 seems to be a promissory starting point to develop more specific and potent chemical tools targeting M32 MCPs from trypanosomatid parasites.

Target-based Screening of the Chagas Box: Setting Up Enzymatic Assays to Discover Specific Inhibitors Across Bioactive Compounds

Chagas disease is a neglected tropical illness caused by the protozoan parasite Trypanosoma cruzi. The disease is endemic in Latin America with about 6 million people infected and many more being at risk. Only two drugs are available for treatment, Nifurtimox and Benznidazole, but they have a number of side effects and are not effective in all cases. This makes urgently necessary the development of new drugs, more efficient, less toxic and affordable to the poor people, who are most of the infected population. In this review we will summarize the current strategies used for drug discovery considering drug repositioning, phenotyping screenings and target-based approaches. In addition, we will describe in detail the considerations for setting up robust enzymatic assays aimed at identifying and validating small molecule inhibitors in high throughput screenings.

Visual kinetic analysis

Naked-eye comparison of reaction profiles is a powerful tool to extract mechanistic information – FAQ, video tutorial and templated examples included.

Visual kinetic analyses extract meaningful mechanistic information from experimental data using the naked-eye comparison of appropriately modified progress reaction profiles. Basic kinetic information is obtained easily and quickly from just a few experiments. Therefore, these methods are valuable tools for all chemists working in process chemistry, synthesis or catalysis with an interest in mechanistic studies. This minireview describes the visual kinetic analyses developed in the last fifteen years and provides answers to the most common queries of new users. Furthermore, a video tutorial is attached detailing the implementation of both VTNA and RPKA.

Enhancing Coupled Enzymatic Activity by Colocalization on Nanoparticle Surfaces: Kinetic Evidence for Directed Channeling of Intermediates

Multistep enzymatic cascades are becoming more prevalent in industrial settings as engineers strive to synthesize complex products and pharmaceuticals in economical, environmentally friendly ways. Previous work has shown that immobilizing enzymes on nanoparticles can enhance their activity significantly due to localized interfacial effects, and this enhancement remains in place even when that enzyme's activity is coupled to another enzyme that is still freely diffusing. Here, we investigate the effects of displaying two enzymes with coupled catalytic activity directly on the same nanoparticle surface. For this, the well-characterized enzymes pyruvate kinase (PykA) and lactate dehydrogenase (LDH) were utilized as a model system; they jointly convert phosphoenolpyruvate to lactate in two sequential steps as part of downstream glycolysis. The enzymes were expressed with terminal polyhistidine tags to facilitate their conjugation to semiconductor quantum dots (QDs) which were used here as prototypical nanoparticles. Characterization of enzyme coassembly to two different sized QDs showed a propensity to cross-link into nanoclusters consisting of primarily dimers and some trimers. Individual and joint enzyme activity in this format was extensively investigated in direct comparison to control samples lacking the QD scaffolds. We found that QD association enhances LDH activity by >50-fold and its total turnover by at least 41-fold, and that this high activation appears to be largely due to stabilization of its quarternary structure. When both enzymes are simultaneously bound to the QD surfaces, their colocalization leads to >100-fold improvements in the overall rates of coupled activity. Experimental results in conjunction with detailed kinetic simulations provide evidence that this significant improvement in coupled activity is partially attributable to a combination of enhanced enzymatic activity and stabilization of LDH. More importantly, experiments aimed at disrupting channeled processes and further kinetic modeling suggest that the bulk of the performance enhancement arises from intermediary "channeling" between the QD-colocalized enzymes. A full understanding of the underlying processes that give rise to such enhancements from coupled enzymatic activity on nanoparticle scaffolds can provide design criteria for improved biocatalytic applications.

Molecular system identification for enzyme directed evolution and design

The rational design of chemical catalysts requires methods for the measurement of free energy differences in the catalytic mechanism for any given catalyst Hamiltonian. The scope of experimental learning algorithms that can be applied to catalyst design would also be expanded by the availability of such methods. Methods for catalyst characterization typically either estimate apparent kinetic parameters that do not necessarily correspond to free energy differences in the catalytic mechanism or measure individual free energy differences that are not sufficient for establishing the relationship between the potential energy surface and catalytic activity. Moreover, in order to enhance the duty cycle of catalyst design, statistically efficient methods for the estimation of the complete set of free energy differences relevant to the catalytic activity based on high-throughput measurements are preferred. In this paper, we present a theoretical and algorithmic system identification framework for the optimal estimation of free energy differences in solution phase catalysts, with a focus on one- and two-substrate enzymes. This framework, which can be automated using programmable logic, prescribes a choice of feasible experimental measurements and manipulated input variables that identify the complete set of free energy differences relevant to the catalytic activity and minimize the uncertainty in these free energy estimates for each successive Hamiltonian design. The framework also employs decision-theoretic logic to determine when model reduction can be applied to improve the duty cycle of high-throughput catalyst design. Automation of the algorithm using fluidic control systems is proposed, and applications of the framework to the problem of enzyme design are discussed.

Mechanistic enzymology in drug discovery: a fresh perspective

Given the therapeutic and commercial success of small-molecule enzyme inhibitors, as exemplified by kinase inhibitors in oncology, a major focus of current drug-discovery and development efforts is on enzyme targets. Understanding the course of an enzyme-catalysed reaction can help to conceptualize different types of inhibitor and to inform the design of screens to identify desired mechanisms. Exploiting this information allows the thorough evaluation of diverse compounds, providing the knowledge required to efficiently optimize leads towards differentiated candidate drugs. This review highlights the rationale for conducting high-quality mechanistic enzymology studies and considers the added value in combining such studies with orthogonal biophysical methods.

One pot synthesis of GDP-mannose by a multi-enzyme cascade for enzymatic assembly of lipid-linked oligosaccharides

Glycosylation of proteins is a key function of the biosynthetic‐secretory pathway in the endoplasmic reticulum (ER) and Golgi apparatus. Glycosylated proteins play a crucial role in cell trafficking and signaling, cell‐cell adhesion, blood‐group antigenicity, and immune response. In addition, the glycosylation of proteins is an important parameter in the optimization of many glycoprotein‐based drugs such as monoclonal antibodies. In vitro glycoengineering of proteins requires glycosyltransferases as well as expensive nucleotide sugars. Here, we present a designed pathway consisting of five enzymes, glucokinase (Glk), phosphomannomutase (ManB), mannose‐1‐phosphate‐guanyltransferase (ManC), inorganic pyrophosphatase (PmPpA), and 1‐domain polyphosphate kinase 2 (1D‐Ppk2) expressed in E. coli for the cell‐free production and regeneration of GDP‐mannose from mannose and polyphosphate with catalytic amounts of GDP and ADP. It was shown that GDP‐mannose is produced at various conditions, that is pH 7–8, temperature 25–35°C and co‐factor concentrations of 5–20 mM MgCl₂. The maximum reaction rate of GDP‐mannose achieved was 2.7 μM/min at 30°C and 10 mM MgCl₂ producing 566 nmol GDP‐mannose after a reaction time of 240 min. With respect to the initial GDP concentration (0.8 mM) this is equivalent to a yield of 71%. Additionally, the cascade was coupled to purified, transmembrane‐deleted Alg1 (ALG1ΔTM), the first mannosyltransferase in the ER‐associated lipid‐linked oligosaccharide (LLO) assembly. Thereby, in a one‐pot reaction, phytanyl‐PP‐(GlcNAc)₂‐Man₁ was produced with efficient nucleotide sugar regeneration for the first time. Phytanyl‐PP‐(GlcNAc)₂‐Man₁ can serve as a substrate for the synthesis of LLO for the cell‐free in vitro glycosylation of proteins. A high‐performance anion exchange chromatography method with UV and conductivity detection (HPAEC‐UV/CD) assay was optimized and validated to determine the enzyme kinetics. The established kinetic model enabled the optimization of the GDP‐mannose regenerating cascade and can further be used to study coupling of the GDP‐mannose cascade with glycosyltransferases. Overall, the study envisages a first step towards the development of a platform for the cell‐free production of LLOs as precursors for in vitro glycoengineering of proteins.

Novel scaffolds for inhibition of Cruzipain identified from high-throughput screening of anti-kinetoplastid chemical boxes

American Trypanosomiasis or Chagas disease is a prevalent, neglected and serious debilitating illness caused by the kinetoplastid protozoan parasite Trypanosoma cruzi. The current chemotherapy is limited only to nifurtimox and benznidazole, two drugs that have poor efficacy in the chronic phase and are rather toxic. In this scenario, more efficacious and safer drugs, preferentially acting through a different mechanism of action and directed against novel targets, are particularly welcome. Cruzipain, the main papain-like cysteine peptidase of T. cruzi, is an important virulence factor and a chemotherapeutic target with excellent pre-clinical validation evidence. Here, we present the identification of new Cruzipain inhibitory scaffolds within the GlaxoSmithKline HAT (Human African Trypanosomiasis) and Chagas chemical boxes, two collections grouping 404 non-cytotoxic compounds with high antiparasitic potency, drug-likeness, structural diversity and scientific novelty. We have adapted a continuous enzymatic assay to a medium-throughput format and carried out a primary screening of both collections, followed by construction and analysis of dose-response curves of the most promising hits. Using the identified compounds as a starting point a substructure directed search against CHEMBL Database revealed plausible common scaffolds while docking experiments predicted binding poses and specific interactions between Cruzipain and the novel inhibitors.

The Multi Domain Caldicellulosiruptor bescii CelA Cellulase Excels at the Hydrolysis of Crystalline Cellulose

The crystalline nature of cellulose microfibrils is one of the key factors influencing biomass recalcitrance which is a key technical and economic barrier to overcome to make cellulosic biofuels a commercial reality. To date, all known fungal enzymes tested have great difficulty degrading highly crystalline cellulosic substrates. We have demonstrated that the CelA cellulase from Caldicellulosiruptor bescii degrades highly crystalline cellulose as well as low crystallinity substrates making it the only known cellulase to function well on highly crystalline cellulose. Unlike the secretomes of cellulolytic fungi, which typically comprise multiple, single catalytic domain enzymes for biomass degradation, some bacterial systems employ an alternative strategy that utilizes multi-catalytic domain cellulases. Additionally, CelA is extremely thermostable and highly active at elevated temperatures, unlike commercial fungal cellulases. Furthermore we have determined that the factors negatively affecting digestion of lignocellulosic materials by C. bescii enzyme cocktails containing CelA appear to be significantly different from the performance barriers affecting fungal cellulases. Here, we explore the activity and degradation mechanism of CelA on a variety of pretreated substrates to better understand how the different bulk components of biomass, such as xylan and lignin, impact its performance.

Hydrogen peroxide inhibition of bicupin oxalate oxidase

Oxalate oxidase is a manganese containing enzyme that catalyzes the oxidation of oxalate to carbon dioxide in a reaction that is coupled with the reduction of oxygen to hydrogen peroxide. Oxalate oxidase from Ceriporiopsis subvermispora (CsOxOx) is the first fungal and bicupin enzyme identified that catalyzes this reaction. Potential applications of oxalate oxidase for use in pancreatic cancer treatment, to prevent scaling in paper pulping, and in biofuel cells have highlighted the need to understand the extent of the hydrogen peroxide inhibition of the CsOxOx catalyzed oxidation of oxalate. We apply a membrane inlet mass spectrometry (MIMS) assay to directly measure initial rates of carbon dioxide formation and oxygen consumption in the presence and absence of hydrogen peroxide. This work demonstrates that hydrogen peroxide is both a reversible noncompetitive inhibitor of the CsOxOx catalyzed oxidation of oxalate and an irreversible inactivator. The build-up of the turnover-generated hydrogen peroxide product leads to the inactivation of the enzyme. The introduction of catalase to reaction mixtures protects the enzyme from inactivation allowing reactions to proceed to completion. Circular dichroism spectra indicate that no changes in global protein structure take place in the presence of hydrogen peroxide. Additionally, we show that the CsOxOx catalyzed reaction with the three carbon substrate mesoxalate consumes oxygen which is in contrast to previous proposals that it catalyzed a non-oxidative decarboxylation with this substrate.

Anaerobic Heme Degradation: ChuY Is an Anaerobilin Reductase That Exhibits Kinetic Cooperativity

Heme catabolism is an important biochemical process that many bacterial pathogens utilize to acquire iron. However, tetrapyrrole catabolites can be reactive and often require further processing for transport out of the cell or conversion to another useful cofactor. In previous work, we presented in vitro evidence of an anaerobic heme degradation pathway in Escherichia coli O157:H7. Consistent with reactions that have been reported for other radical S-adenosyl-l-methionine methyltransferases, ChuW transfers a methyl group to heme by a radical-mediated mechanism and catalyzes the β-scission of the porphyrin macrocycle. This facilitates iron release and the production of a new linear tetrapyrrole termed "anaerobilin". In this work, we describe the structure and function of ChuY, an enzyme expressed downstream from chuW within the same heme utilization operon. ChuY is structurally similar to biliverdin reductase and forms a dimeric complex in solution that reduces anaerobilin to the product we have termed anaerorubin. Steady state analysis of ChuY exhibits kinetic cooperativity that is best explained by a random addition mechanism with a kinetically preferred path for initial reduced nicotinamide adenine dinucleotide phosphate binding.

Specific Inhibition of β-Secretase Processing of the Alzheimer Disease Amyloid Precursor Protein

Development of disease-modifying therapeutics is urgently needed for treating Alzheimer disease (AD). AD is characterized by toxic β-amyloid (Aβ) peptides produced by β- and γ-secretase-mediated cleavage of the amyloid precursor protein (APP). β-secretase inhibitors reduce Aβ levels, but mechanism-based side effects arise because they also inhibit β-cleavage of non-amyloid substrates like Neuregulin. We report that β-secretase has a higher affinity for Neuregulin than it does for APP. Kinetic studies demonstrate that the affinities and catalytic efficiencies of β-secretase are higher toward non-amyloid substrates than toward APP. We show that non-amyloid substrates are processed by β-secretase in an endocytosis-independent manner. Exploiting this compartmentalization of substrates, we specifically target the endosomal β-secretase by an endosomally targeted β-secretase inhibitor, which blocked cleavage of APP but not non-amyloid substrates in many cell systems, including induced pluripotent stem cell (iPSC)-derived neurons. β-secretase inhibitors can be designed to specifically inhibit the Alzheimer process, enhancing their potential as AD therapeutics without undesired side effects.

The effect of cultivation media and washing whole-cell biocatalysts on monoamine oxidase catalyzed oxidative desymmetrization of 3-azabicyclo[3,3,0]octane

It is well known that washing whole-cells containing enzyme activities after fermentation, but prior to biocatalysis can improve their activity in the subsequent reaction. In this paper, we quantify the impact of both the fermentation media and cell washing on the performance of whole-cell biocatalysis. The results are illustrated using a recombinant monoamine oxidase (expressed in Escherichia coli, used in resting state) for the oxidative desymmetrization of 3-azabicyclo[3,3,0]octane. It was shown that the need for washing biocatalyst prior to use in a reaction is dependent upon growth medium. Unlike cells grown in LB medium, washing of the cells was essential for cells grown on TB medium. With TB media, washing the cells improved the final conversion by approximately a factor of two. Additionally, over 50-fold improvement was achieved in initial activity. A potential reason for this improvement in activity was identified to be the increase in transfer of substrates across the cell membrane as a result of cell washing.

Quantifying NAD(P)H production in the upper Entner-Doudoroff pathway from Pseudomonas putida KT2440

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The first kinetic characterization of PputG6PDH-1 is presented.

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The relative production of NADH and NADPH by PputG6PDH-1 is quantified.

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The stoichiometric matrix of in silico metabolic models for Pseudomonas putida must be modified.

Despite the lack of biochemical information, all available in silico metabolic models of Pseudomonas putida KT2440 consider NADP as the only cofactor accepted by the glucose-6-phosphate dehydrogenases. Because the Entner–Doudoroff pathway is the main glycolytic route in this bacterium, determining how much NADH and NADPH are produced in the reaction catalyzed by these enzymes is very important for the correct interpretation of metabolic flux distributions. To determine the actual cofactor preference of the glucose-6-phosphate dehydrogenase encoded by the zwf-1 gene (PputG6PDH-1), the major isoform during growth on glucose, we purified this protein and studied its kinetic properties. Based on simple kinetic principles, we estimated the in vivo relative production of NADH and NADPH during the oxidation of glucose-6-phosphate (G6P). Contrary to the general assumption, our calculations showed that the reaction catalyzed by PputG6PDH-1 yields around 1/3 mol of NADPH and 2/3 mol of NADH per mol of oxidized G6P. Additionally, we obtained data suggesting that the reaction catalyzed by the 6-phosphogluconate dehydrogenase is active during growth on glucose, and it also produces NADH. These results indicate that the stoichiometric matrix of in silico models of P. putida KT2440 must be corrected and highlight the importance of considering the physiological concentrations of the involved metabolites to estimate the actual proportion of NADH and NADPH produced by a dehydrogenase.

Catalase improves saccharification of lignocellulose by reducing lytic polysaccharide monooxygenase-associated enzyme inactivation

Objectives

Efficient enzymatic saccharification of plant cell wall material is key to industrial processing of agricultural and forestry waste such as straw and wood chips into fuels and chemicals.

Results

Saccharification assays were performed on steam-pretreated wheat straw under ambient and O₂-deprived environments and in the absence and presence of a lytic polysaccharide monooxygenase (LPMO) and catalase. A kinetic model was used to calculate catalytic rate and first-order inactivation rate constants of the cellulases from reaction progress curves. The addition of a LPMO significantly (P < 0.01, Student’s T test) enhanced the rate of glucose release from 2.8 to 6.9 h⁻¹ under ambient O₂ conditions. However, this also significantly (P < 0.01, Student’s T test) increased the rate of inactivation of the enzyme mixture, thereby reducing the performance half-life from 65 to 35 h. Decreasing O₂ levels or, strikingly, the addition of catalase significantly reduced (P < 0.01, Student’s T test) enzyme inactivation and, as a consequence, higher efficiency of the cellulolytic enzyme cocktail was achieved.

Conclusion

Oxidative inactivation of commercial cellulase mixtures is a significant factor influencing the overall saccharification efficiency and the addition of catalase can be used to protect these mixtures from inactivation.

Electronic supplementary material

The online version of this article (doi:10.1007/s10529-015-1989-8) contains supplementary material, which is available to authorized users.

Peptide Backbone Composition and Protease Susceptibility: Impact of Modification Type, Position, and Tandem Substitution

The clinical utility of peptides is limited by their rapid degradation by endogenous proteases. Modification of the peptide backbone can generate functional analogues with enhanced proteolytic stability. Existing principles for the design of such oligomers have focused primarily on effective structural mimicry. A more robust strategy would incorporate a rational approach for engineering maximal proteolytic stability with minimal unnatural residue content. We report here the systematic comparison of the proteolytic resistance imparted by four backbone modifications commonly employed in the design of protease-stable analogues of peptides with complex folding patterns. The degree of protection was quantified as a function of modification type, position, and tandem substitution in the context of a long, unstructured host sequence and a canonical serine protease. These results promise to inform ongoing work to develop biostable mimics of increasingly complex peptides and proteins.

Biocatalytic carboxylation of phenol derivatives: kinetics and thermodynamics of the biological Kolbe-Schmitt synthesis

Microbial decarboxylases, which catalyse the reversible regioselective ortho-carboxylation of phenolic derivatives in anaerobic detoxification pathways, have been studied for their reverse carboxylation activities on electron-rich aromatic substrates. Ortho-hydroxybenzoic acids are important building blocks in the chemical and pharmaceutical industries and are currently produced via the Kolbe-Schmitt process, which requires elevated pressures and temperatures (≥ 5 bar, ≥ 100 °C) and often shows incomplete regioselectivities. In order to resolve bottlenecks in view of preparative-scale applications, we studied the kinetic parameters for 2,6-dihydroxybenzoic acid decarboxylase from Rhizobium sp. in the carboxylation- and decarboxylation-direction using 1,2-dihydroxybenzene (catechol) as starting material. The catalytic properties (K(m), V(max)) are correlated with the overall thermodynamic equilibrium via the Haldane equation, according to a reversible random bi-uni mechanism. The model was subsequently verified by comparing experimental results with simulations. This study provides insights into the catalytic behaviour of a nonoxidative aromatic decarboxylase and reveals key limitations (e.g. substrate oxidation, CO2 pressure, enzyme deactivation, low turnover frequency) in view of the employment of this system as a 'green' alternative to the Kolbe-Schmitt processes.

Multiplex detection of enzymatic activity with responsive lanthanide-based luminescent probes

Multiplex analyte detection in complex dynamic systems is desirable for the investigation of cellular communication networks as well as in medical diagnostics. A family of lanthanide-based responsive luminescent probes for multiplex detection is reported. The high modularity of the probe design enabled the rapid assembly of both green and red emitters for a large variety of analytes by the simple exchange of the lanthanide or an analyte-cleavable caging group, respectively. The real-time three-color detection of up to three analytes was demonstrated, thus setting the stage for the non-invasive investigation of interconnected biological processes.

Metabolic impact of an NADH-producing glucose-6-phosphate dehydrogenase in Escherichia coli

In Escherichia coli, the oxidative branch of the pentose phosphate pathway (oxPPP) is one of the major sources of NADPH when glucose is the sole carbon nutrient. However, unbalanced NADPH production causes growth impairment as observed in a strain lacking phosphoglucoisomerase (Δpgi). In this work, we studied the metabolic response of this bacterium to the replacement of its glucose-6-phosphate dehydrogenase (G6PDH) by an NADH-producing variant. The homologous enzyme from Leuconostoc mesenteroides was studied by molecular dynamics and site-directed mutagenesis to obtain the NAD-preferring LmG6PDH(R46E,Q47E). Through homologous recombination, the zwf loci (encoding G6PDH) in the chromosomes of WT and Δpgi E. coli strains were replaced by DNA encoding LmG6PDH(R46E,Q47E). Contrary to some predictions performed with flux balance analysis, the replacements caused a substantial effect on the growth rates, increasing 59 % in the Δpgi strain, while falling 44 % in the WT. Quantitative PCR (qPCR) analysis of the zwf locus showed that the expression level of the mutant enzyme was similar to the native enzyme and the expression of genes encoding key enzymes of the central pathways also showed moderate changes among the studied strains. The phenotypic and qPCR data were integrated into in silico modelling, showing an operative G6PDH flux contributing to the NADH pool. Our results indicated that, in vivo, the generation of NADH by G6PDH is beneficial or disadvantageous for growth depending on the operation of the upper Embden-Meyerhof pathway. Interestingly, a genomic database search suggested that in bacteria lacking phosphofructokinase, the G6PDHs tend to have similar preferences for NAD and NADP. The importance of the generation of NADPH in a pathway such as the oxPPP is discussed.

Studying SIRT6 regulation using H3K56 based substrate and small molecules

SIRT6 is a modulator of chromatin structure having an important role in healthy ageing, and there is a crucial need to find specific modulators for it. Therefore, the activity of SIRT6 should be studied using a variety of methods. We examined the capability of SIRT6 to deacetylate a set of five fluorogenic substrates based on p53 and histone H3 sequences. The substrate designed around H3K56 deacetylation site exhibited the best signal-to-background ratio and was chosen for further studies. Nicotinamide is a known inhibitor for sirtuins, and it was found to be less potent inhibitor for SIRT6 than it is for SIRT1. In addition, we studied 15 other small molecule sirtuin modulators using the H3K56 based substrate. EX-527, quercetin and three pseudopeptidic compounds were found to be the most potent SIRT6 inhibitors, exhibiting over 50% deacetylation inhibition. These findings describe the first modulators of SIRT6 activity at the physiologically important H3K56 deacetylation site.

Identification of small molecules that selectively inhibit diacylglycerol lipase-α activity

Recent genetic evidence suggests that the diacylglycerol lipase (DAGL-α) isoform is the major biosynthetic enzyme for the most abundant endocannabinoid, 2-arachidonoyl-glycerol (2-AG), in the central nervous system. Revelation of its essential role in regulating retrograde synaptic plasticity and adult neurogenesis has made it an attractive therapeutic target. Therefore, it has become apparent that selective inhibition of DAGL-α enzyme activity with a small molecule could be a strategy for the development of novel therapies for the treatment of disease indications such as depression, anxiety, pain, and cognition. In this report, the authors present the identification of small-molecule inhibitor chemotypes of DAGL-α, which were selective (≥10-fold) against two other lipases, pancreatic lipase and monoacylglycerol lipase, via high-throughput screening of a diverse compound collection. Seven chemotypes of interest from a list of 185 structural clusters, which included 132 singletons, were initially selected for evaluation and characterization. Selection was based on potency, selectivity, and chemical tractability. One of the chemotypes, the glycine sulfonamide series, was prioritized as an initial lead for further medicinal chemistry optimization.

A new approach to control the enigmatic activity of aldose reductase

Aldose reductase (AR) is an NADPH-dependent reductase, which acts on a variety of hydrophilic as well as hydrophobic aldehydes. It is currently defined as the first enzyme in the so-called polyol pathway, in which glucose is transformed into sorbitol by AR and then to fructose by an NAD(+)-dependent dehydrogenase. An exaggerated flux of glucose through the polyol pathway (as can occur in diabetes) with the subsequent accumulation of sorbitol, was originally proposed as the basic event in the aethiology of secondary diabetic complications. For decades this has meant targeting the enzyme for a specific and strong inhibition. However, the ability of AR to reduce toxic alkenals and alkanals, which are products of oxidative stress, poses the question of whether AR might be better classified as a detoxifying enzyme, thus raising doubts as to the unequivocal advantages of inhibiting the enzyme. This paper provides evidence of the possibility for an effective intervention on AR activity through an intra-site differential inhibition. Examples of a new generation of aldose reductase "differential" inhibitors (ARDIs) are presented, which can preferentially inhibit the reduction of either hydrophilic or hydrophobic substrates. Some selected inhibitors are shown to preferentially inhibit enzyme activity on glucose or glyceraldehyde and 3-glutathionyl-4-hydroxy-nonanal, but are less effective in reducing 4-hydroxy-2-nonenal. We question the efficacy of D, L-glyceraldehyde, the substrate commonly used in in vitro inhibition AR studies, as an in vitro reference AR substrate when the aim of the investigation is to impair glucose reduction.

Determining enzyme kinetics for systems biology with nuclear magnetic resonance spectroscopy

Enzyme kinetics for systems biology should ideally yield information about the enzyme’s activity under in vivo conditions, including such reaction features as substrate cooperativity, reversibility and allostery, and be applicable to enzymatic reactions with multiple substrates. A large body of enzyme-kinetic data in the literature is based on the uni-substrate Michaelis-Menten equation, which makes unnatural assumptions about enzymatic reactions (e.g., irreversibility), and its application in systems biology models is therefore limited. To overcome this limitation, we have utilised NMR time-course data in a combined theoretical and experimental approach to parameterize the generic reversible Hill equation, which is capable of describing enzymatic reactions in terms of all the properties mentioned above and has fewer parameters than detailed mechanistic kinetic equations; these parameters are moreover defined operationally. Traditionally, enzyme kinetic data have been obtained from initial-rate studies, often using assays coupled to NAD(P)H-producing or NAD(P)H-consuming reactions. However, these assays are very labour-intensive, especially for detailed characterisation of multi-substrate reactions. We here present a cost-effective and relatively rapid method for obtaining enzyme-kinetic parameters from metabolite time-course data generated using NMR spectroscopy. The method requires fewer runs than traditional initial-rate studies and yields more information per experiment, as whole time-courses are analyzed and used for parameter fitting. Additionally, this approach allows real-time simultaneous quantification of all metabolites present in the assay system (including products and allosteric modifiers), which demonstrates the superiority of NMR over traditional spectrophotometric coupled enzyme assays. The methodology presented is applied to the elucidation of kinetic parameters for two coupled glycolytic enzymes from Escherichia coli (phosphoglucose isomerase and phosphofructokinase). ³¹P-NMR time-course data were collected by incubating cell extracts with substrates, products and modifiers at different initial concentrations. NMR kinetic data were subsequently processed using a custom software module written in the Python programming language, and globally fitted to appropriately modified Hill equations.