APPLICATION OF A GENERALIZED MWC MODEL FOR THE MATHEMATICAL SIMULATION OF METABOLIC PATHWAYS REGULATED BY ALLOSTERIC ENZYMES

A functional group-guided approach to aptamers for small molecules

Aptameric receptors are important biosensor components, yet our ability to identify them depends on the target structures. We analyzed the contributions of individual functional groups on small molecules to binding within 27 target-aptamer pairs, identifying potential hindrances to receptor isolation-for example, negative cooperativity between sterically hindered functional groups. To increase the probability of aptamer isolation for important targets, such as leucine and voriconazole, for which multiple previous selection attempts failed, we designed tailored strategies focused on overcoming individual structural barriers to successful selections. This approach enables us to move beyond standardized protocols into functional group-guided searches, relying on sequences common to receptors for targets and their analogs to serve as anchors in regions of vast oligonucleotide spaces wherein useful reagents are likely to be found.

Parameter Reliability and Understanding Enzyme Function

Knowledge of the Michaelis-Menten parameters and their meaning in different circumstances is an essential prerequisite to understanding enzyme function and behaviour. The published literature contains an abundance of values reported for many enzymes. The problem concerns assessing the appropriateness and validity of such material for the purpose to which it is to be applied. This review considers the evaluation of such data with particular emphasis on the assessment of its fitness for purpose.

Metabolic Modelling as a Framework for Metabolomics Data Integration and Analysis

Metabolic networks are regulated to ensure the dynamic adaptation of biochemical reaction fluxes to maintain cell homeostasis and optimal metabolic fitness in response to endogenous and exogenous perturbations. To this end, metabolism is tightly controlled by dynamic and intricate regulatory mechanisms involving allostery, enzyme abundance and post-translational modifications. The study of the molecular entities involved in these complex mechanisms has been boosted by the advent of high-throughput technologies. The so-called omics enable the quantification of the different molecular entities at different system layers, connecting the genotype with the phenotype. Therefore, the study of the overall behavior of a metabolic network and the omics data integration and analysis must be approached from a holistic perspective. Due to the close relationship between metabolism and cellular phenotype, metabolic modelling has emerged as a valuable tool to decipher the underlying mechanisms governing cell phenotype. Constraint-based modelling and kinetic modelling are among the most widely used methods to study cell metabolism at different scales, ranging from cells to tissues and organisms. These approaches enable integrating metabolomic data, among others, to enhance model predictive capabilities. In this review, we describe the current state of the art in metabolic modelling and discuss future perspectives and current challenges in the field.

A Pycellerator Tutorial

We present a tutorial on using Pycellerator for biomolecular simulations. Models are described in human readable (and editable) text files (UTF8 or ASCII) containing collections of reactions, assignments, initial conditions, function definitions, and rate constants. These models are then converted into a Python program that can optionally solve the system, e.g., as a system of differential equations using ODEINT, or be run by another program. The input language implements an extended version of the Cellerator arrow notation, including mass action, Hill functions, S-Systems, MWC, and reactions with user-defined kinetic laws. Simple flux balance analysis is also implemented. We will demonstrate the implementation and analysis of progressively more complex models, starting from simple mass action through indexed cascades. Pycellerator can be used as a library that is integrated into other programs, run as a command line program, or in iPython notebooks. It is implemented in Python 2.7 and available under an open source GPL license.

A mechanistic kinetic description of lactate dehydrogenase elucidating cancer diagnosis and inhibitor evaluation

As a key enzyme for glycolysis, lactate dehydrogenase (LDH) remains as a topic of great interest in cancer study. Though a number of kinetic models have been applied to describe the dynamic behavior of LDH, few can reflect its actual mechanism, making it difficult to explain the observed substrate and competitor inhibitions at wide concentration ranges. A novel mechanistic kinetic model is developed based on the enzymatic processes and the interactive properties of LDH. Better kinetic simulation as well as new enzyme interactivity information and kinetic properties extracted from published articles via the novel model was presented. Case studies were presented to a comprehensive understanding of the effect of temperature, substrate, and inhibitor on LDH kinetic activities for promising application in cancer diagnosis, inhibitor evaluation, and adequate drug dosage prediction.

A probabilistic framework for the exploration of enzymatic capabilities based on feasible kinetics and control analysis

BACKGROUND

Analysis of limiting steps within enzyme-catalyzed reactions is fundamental to understand their behavior and regulation. Methods capable of unravelling control properties and exploring kinetic capabilities of enzymatic reactions would be particularly useful for protein and metabolic engineering. While single-enzyme control analysis formalism has previously been applied to well-studied enzymatic mechanisms, broader application of this formalism is limited in practice by the limited amount of kinetic data and the difficulty of describing complex allosteric mechanisms.

METHODS

To overcome these limitations, we present here a probabilistic framework enabling control analysis of previously unexplored mechanisms under uncertainty. By combining a thermodynamically consistent parameterization with an efficient Sequential Monte Carlo sampler embedded in a Bayesian setting, this framework yields insights into the capabilities of enzyme-catalyzed reactions with modest kinetic information, provided that the catalytic mechanism and a thermodynamic reference point are defined.

RESULTS

The framework was used to unravel the impact of thermodynamic affinity, substrate saturation levels and effector concentrations on the flux control and response coefficients of a diverse set of enzymatic reactions.

CONCLUSIONS

Our results highlight the importance of the metabolic context in the control analysis of isolated enzymes as well as the use of statistically sound methods for their interpretation.

GENERAL SIGNIFICANCE

This framework significantly expands our current capabilities for unravelling the control properties of general reaction kinetics with limited amount of information. This framework will be useful for both theoreticians and experimentalists in the field.

A general framework for thermodynamically consistent parameterization and efficient sampling of enzymatic reactions

Kinetic models provide the means to understand and predict the dynamic behaviour of enzymes upon different perturbations. Despite their obvious advantages, classical parameterizations require large amounts of data to fit their parameters. Particularly, enzymes displaying complex reaction and regulatory (allosteric) mechanisms require a great number of parameters and are therefore often represented by approximate formulae, thereby facilitating the fitting but ignoring many real kinetic behaviours. Here, we show that full exploration of the plausible kinetic space for any enzyme can be achieved using sampling strategies provided a thermodynamically feasible parameterization is used. To this end, we developed a General Reaction Assembly and Sampling Platform (GRASP) capable of consistently parameterizing and sampling accurate kinetic models using minimal reference data. The former integrates the generalized MWC model and the elementary reaction formalism. By formulating the appropriate thermodynamic constraints, our framework enables parameterization of any oligomeric enzyme kinetics without sacrificing complexity or using simplifying assumptions. This thermodynamically safe parameterization relies on the definition of a reference state upon which feasible parameter sets can be efficiently sampled. Uniform sampling of the kinetics space enabled dissecting enzyme catalysis and revealing the impact of thermodynamics on reaction kinetics. Our analysis distinguished three reaction elasticity regions for common biochemical reactions: a steep linear region (0> ΔGr >-2 kJ/mol), a transition region (-2> ΔGr >-20 kJ/mol) and a constant elasticity region (ΔGr <-20 kJ/mol). We also applied this framework to model more complex kinetic behaviours such as the monomeric cooperativity of the mammalian glucokinase and the ultrasensitive response of the phosphoenolpyruvate carboxylase of Escherichia coli. In both cases, our approach described appropriately not only the kinetic behaviour of these enzymes, but it also provided insights about the particular features underpinning the observed kinetics. Overall, this framework will enable systematic parameterization and sampling of enzymatic reactions.

Incorporating covalent and allosteric effects into rate equations: the case of muscle glycogen synthase

Several enzymes have been described that undergo both allosteric and covalent regulation, but, to date, there exists no succinct kinetic description that is able to account for both of these mechanisms of regulation. Muscle glycogen synthase, an enzyme implicated in the pathogenesis of several metabolic diseases, is activated by glucose 6-phosphate and inhibited by ATP and phosphorylation at multiple sites. A kinetic description of glycogen synthase could provide insight into the relative importance of these modifiers. In the present study we show, using non-linear parameter optimization with robust weight estimation, that a Monod-Wyman-Changeux model in which phosphorylation favours the inactive T conformation provides a satisfactory description of muscle glycogen synthase kinetics. The best-fit model suggests that glucose 6-phosphate and ATP compete for the same allosteric site, but that ATP also competes with the substrate UDP-glucose for the active site. The novelty of our approach lies in treating covalent modification as equivalent to allosteric modification. Using the obtained rate equation, the relationship between enzyme activity and phosphorylation state is explored and shown to agree with experimental results. The methodology we propose could also be applied to other enzymes that undergo both allosteric and covalent modification.

Time-scale separation--Michaelis and Menten's old idea, still bearing fruit

Michaelis and Menten introduced to biochemistry the idea of time-scale separation, in which part of a system is assumed to be operating sufficiently fast compared to the rest so that it may be taken to have reached a steady state. This allows, in principle, the fast components to be eliminated, resulting in a simplified description of the system's behaviour. Similar ideas have been widely used in different areas of biology, including enzyme kinetics, protein allostery, receptor pharmacology, gene regulation and post-translational modification. However, the methods used have been independent and ad hoc. In the present study, we review the use of time-scale separation as a means to simplify the description of molecular complexity and discuss recent work setting out a single framework that unifies these separate calculations. The framework offers new capabilities for mathematical analysis and helps to do justice to Michaelis and Menten's insights about individual enzymes in the context of multi-enzyme biological systems.

Cooperative binding

Molecular binding is an interaction between molecules that results in a stable association between those molecules. Cooperative binding occurs if the number of binding sites of a macromolecule that are occupied by a specific type of ligand is a nonlinear function of this ligand's concentration. This can be due, for instance, to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive (supralinear) or negative (infralinear). Cooperative binding is most often observed in proteins, but nucleic acids can also exhibit cooperative binding, for instance of transcription factors. Cooperative binding has been shown to be the mechanism underlying a large range of biochemical and physiological processes.

A linear framework for time-scale separation in nonlinear biochemical systems

Cellular physiology is implemented by formidably complex biochemical systems with highly nonlinear dynamics, presenting a challenge for both experiment and theory. Time-scale separation has been one of the few theoretical methods for distilling general principles from such complexity. It has provided essential insights in areas such as enzyme kinetics, allosteric enzymes, G-protein coupled receptors, ion channels, gene regulation and post-translational modification. In each case, internal molecular complexity has been eliminated, leading to rational algebraic expressions among the remaining components. This has yielded familiar formulas such as those of Michaelis-Menten in enzyme kinetics, Monod-Wyman-Changeux in allostery and Ackers-Johnson-Shea in gene regulation. Here we show that these calculations are all instances of a single graph-theoretic framework. Despite the biochemical nonlinearity to which it is applied, this framework is entirely linear, yet requires no approximation. We show that elimination of internal complexity is feasible when the relevant graph is strongly connected. The framework provides a new methodology with the potential to subdue combinatorial explosion at the molecular level.

ON COOPERATIVE QUASI-EQUILIBRIUM MODELS OF TRANSCRIPTIONAL REGULATION

Mechanistic models for transcriptional regulation are derived using the methods of equilibrium statistical mechanics, to model equilibrating processes that occur at a fast time scale. These processes regulate slower changes in the synthesis and expression of transcription factors that feed back and cooperatively regulate transcription, forming a gene regulation network (GRN). We rederive and extend two previous quasi-equilibrium models of transcriptional regulation, and demonstrate circumstances under which they can be approximated at each transcription complex by feed-forward artificial neural network (ANN) models. A single-level mechanistic model can be approximated by a successfully applied phenomenological model of GRNs which is based on single-layer analog-valued ANNs. A two-level hierarchical mechanistic model, with separate activation states for modules and for the whole transcription complex, can be approximated by a two-layer feed-forward ANN in several related ways. The sufficient conditions demonstrated for the ANN approximations correspond biologically to large numbers of binding sites each of which have a small effect. A further extension to the single-level and two-level models allows one-dimensional chains of overlapping and/or energetically interacting binding sites within a module. Partition functions for these models can be constructed from stylized diagrams that indicate energetic and logical interactions between binary-valued state variables. All parameters in the mechanistic models, including the two approximations, can in principle be related to experimentally measurable free energy differences, among other observables.

A scalable and integrative system for pathway bioinformatics and systems biology

MOTIVATION

Progress in systems biology depends on developing scalable informatics tools to predictively model, visualize, and flexibly store information about complex biological systems. Scalability of these tools, as well as their ability to integrate within larger frameworks of evolving tools, is critical to address the multi-scale and size complexity of biological systems.

RESULTS

Using current software technology, such as self-generation of database and object code from UML schemas, facilitates rapid updating of a scalable expert assistance system for modeling biological pathways. Distribution of key components along with connectivity to external data sources and analysis tools is achieved via a web service interface.

AVAILABILITY

All sigmoid modeling software components and supplementary information are available through: http://www.igb.uci.edu/servers/sb.html.

Computing phenomenologic Adair-Klotz constants from microscopic MWC parameters

BACKGROUND

Modellers using the MWC allosteric framework have often found it difficult to validate their models. Indeed many experiments are not conducted with the notion of alternative conformations in mind and therefore do not (or cannot) measure relevant microscopic constant and parameters. Instead, experimentalists widely use the Adair-Klotz approach in order to describe their experimental data.

RESULTS

We propose a way of computing apparent Adair-Klotz constants from microscopic association constants and allosteric parameters of a generalised concerted model with two different states (R and T), with an arbitrary number of non-equivalent ligand binding sites. We apply this framework to compute Adair-Klotz constants from existing models of calmodulin and hemoglobin, two extreme cases of the general framework.

CONCLUSION

The validation of computational models requires methods to relate model parameters to experimentally observable quantities. We provide such a method for comparing generalised MWC allosteric models to experimentally determined Adair-Klotz constants.

Amino Acid Metabolism and Fluxes

By the mid1960s, the pioneering work of Umbarger and Gerhart and Pardee had shown us that carbon flow through a biosynthetic pathway was controlled by allosteric inhibition of the first enzyme of the pathway by its end product; and, studies of the lac operon by Jacob and Monod had established that genes were controlled by an operator-repressor mechanism. During the intervening forty-plus years, knowledge and technologies have continued to explode in unanticipated ways. Today, we understand in great detail the molecular mechanisms of the many levels of metabolic and genetic regulation that control carbon flow through the amino acid biosynthetic pathways. Traditional experimental approaches are not sufficient for the integration and reconstruction of complex biological systems using data mostly generated by high-throughput experiments. Only with computational methods and adequate modeling tools will we be able to reconstruct and query these large and complicated systems. Due to complicated enzyme reaction mechanisms and the frequent lack of rate constant measurements needed for solving differential equations, most investigators have turned their attention to the development of abstract, top-down modeling tools. For example, Palsson and colleagues have used metabolic flux balance analysis (FBA) methods to simulate steady-state metabolite flux through E. coli pathways representing hundreds of enzyme steps. Recently, Yang et al. have developed a bottom-up, enzyme mechanism modeling language, kMech (kinetic mechanism), for the mathematical simulation of metabolic pathways.

An enzyme-centric approach for modelling non-linear biological complexity

Background

The current challenge of Systems Biology is to integrate high throughput data sets for simulating the complexity of biological networks, exploit the evolution of nature-designed networks that maintain the robustness of a biological system, and thereby generate novel, experimentally testable hypotheses. In order to simulate non-linear biological complexities, we have previously developed an Enzyme-Centric mechanistic modeling approach and validated it using metabolic network in E. coli. The idea is to use prior knowledge of catalytic and regulatory mechanisms of each enzyme within the metabolic network to build a dynamic model for investigating the network level regulation and thus understand the nature design principle behind the network.

Results

In this paper, we further demonstrate the application of complex enzyme catalytic and regulatory modules to simulate nonlinear network regulatory patterns vs. simple linear conversion model. We learned and validated that it is essential to incorporate prior knowledge from the literature to simulate non-linear biological complexities. The network expandability is demonstrated and validated with the complex amino acid biosynthetic network with multi-regulations. Also, we demonstrated the compatibility of mechanistic models within close species. Furthermore, the eukaryotic protein factory model for insuring steady mRNA production is simulated and the coupling of RNA transcription and splicing is validated by both mathematical simulation and experimental analysis.

Conclusion

We demonstrated the importance of modeling complex enzyme catalytic and regulatory mechanisms to further understand nonlinear network regulatory patterns. The simulations presented in this paper reveal how a living system maintains homeostasis and its robustness to continue functioning while facing environmental stresses or genetic mutations.