A mathematical model of metabolic insulin signaling pathways

We develop a mathematical model that explicitly represents many of the known signaling components mediating translocation of the insulin-responsive glucose transporter GLUT4 to gain insight into the complexities of metabolic insulin signaling pathways. A novel mechanistic model of postreceptor events including phosphorylation of insulin receptor substrate-1, activation of phosphatidylinositol 3-kinase, and subsequent activation of downstream kinases Akt and protein kinase C-zeta is coupled with previously validated subsystem models of insulin receptor binding, receptor recycling, and GLUT4 translocation. A system of differential equations is defined by the structure of the model. Rate constants and model parameters are constrained by published experimental data. Model simulations of insulin dose-response experiments agree with published experimental data and also generate expected qualitative behaviors such as sequential signal amplification and increased sensitivity of downstream components. We examined the consequences of incorporating feedback pathways as well as representing pathological conditions, such as increased levels of protein tyrosine phosphatases, to illustrate the utility of our model for exploring molecular mechanisms. We conclude that mathematical modeling of signal transduction pathways is a useful approach for gaining insight into the complexities of metabolic insulin signaling.     

Conditions for effective allosteric feedforward and feedback in metabolic pathways

Whether an allosteric feedback or feedforward modifier actually has an effect on the steady-state properties of a metabolic pathway depends not only on the allosteric modifier effect itself, but also on the control properties of the affected allosteric enzyme in the pathway of which it is part. Different modification mechanisms are analysed: mixed inhibition, allosteric inhibition and activation of the reversible Monod-Wyman-Changeux and reversible Hill models. In conclusion, it is shown that, whereas a modifier effect on substrate and product binding (specific effects) can be an effective negative feedback mechanism, it is much less effective as a positive feedforward mechanism. The prediction is that catalytic effects that change the apparent limiting velocity would be more effective in feedforward activation.     

A model for the allosteric regulation of pH-sensitive enzymes.

1. In an enzyme that has two independent binding sites for a ligand, any inhibitor that binds solely to the free enzyme will give rise to positive co-operativity. 2. A model is considered for the allosteric control of enzymes by effectors in which their effects are mediated by ligand-induced perturbations of the ionization constants of a group or groups involved in the binding of substrate to the active site. 3. The model described offers a plausible explanation for the observation that the sigmoidal initial-rate curves reported for some regulatory enzymes are not expressed at all pH values where the enzyme is catalytically active.     

Description of the kinetics of allosteric polymeric enzymes with 2 ligands based on the generalized Ising model

The two-ligand model for allosteric regulation of complex polymeric enzymes is offered. Two types of cooperative interactions are observed--the cooperative transformations inside the promoter and interactions of promoters themselves--along with the direct interaction of two ligands inside each subunit. The solution of the model is given and its kinetic properties are discussed.     

An enzyme mechanism language for the mathematical modeling of metabolic pathways

MOTIVATION: As a first step toward the elucidation of the systems biology of complex biological systems, it was our goal to mathematically model common enzyme catalytic and regulatory mechanisms that repeatedly appear in biological processes such as signal transduction and metabolic pathways. RESULTS: We describe kMech, a Cellerator language extension that describes a suite of enzyme mechanisms. Each enzyme mechanism is parsed by kMech into a set of fundamental association-dissociation reactions that are translated by Cellerator into ordinary differential equations that are numerically solved by Mathematica. In addition, we present methods that use commonly available kinetic measurements to estimate rate constants required to solve these differential equations.     

Kinetic cooperativity in the concerted model for allosteric enzymes.

The cooperativity of enzyme-substrate interactions is investigated in the concerted allosteric model of Monod, Wyman and Changeux. The general case of K-V systems is considered, in which the two protomer conformational states R and T postulated in the theory differ in catalytic and binding properties. An expression for the Hill coefficient nH defined with respect to the asymptotic velocity V infinity to is analyzed in conditions which exclude substrate inhibition. Kinetic cooperativity is always positive (nH greater than 1) in the case of a dimer enzyme, and in the case of an inactive T state. Slight kinetic negative cooperativity (nH less than 1) occurs under restrictive conditions for larger numbers of protomers when the substrate binds significantly to the less active state of the enzyme, but the phenomenon remains negligible for trimers and tetramers. These conclusions differ from those obtained [A. Goldbeter, J. Mol.Biol.90 (1974) 185] with the Hill coefficient based on the absolute maximum velocity, which may exceed the experimental value V infinity to in K-V systems. The results extend those of Paulus and DeRiel [J. Mol. Biol. 97 (1975) 667] and support the view that in most cases, negative cooperativity is not compatible with a mechanism based on a concerted and conservative allosteric transition. The Hill coefficients for binding and catalysis are compared in K-V systems.     

Time dependent model for hemoglobin and allosteric enzymes. II

The time-dependent theory developed in Part I is specialized to treat tetrameric hemoglobin, and the results of the theory for dimeric-and tetrameric hemoglobin are compared with data on the kinetics of the reactions of hemoglobin with carbon monoxide and oxygen at various salt concentrations for the case of large concentration of ligand relative to that of hemoglobin. The fit of the theoretical results to the data suggests that hemoglobin at a 2 M salt concentration is predominantly dimeric and that the tetramer should be taken as the functional unit to explain the kinetics of the reactions of normal hemoglobin. A relationship is established between the time-dependent theory arid Adair's Intermediate Compound Hypothesis (I.C.H.) for hemoglobin, as brought to its present state by Gibson and Roughton. A generalization (G.I.C.H.) of the I.C.H. is presented and is shown to be equivalent to the time-dependent theory in the limit of infinite ligand concentration. The I.C.H. is shown to be an excellent approximation to the centralized theory (G.I.C.H.) in this limit.     

A markov classification model for metabolic pathways

Background  

This paper considers the problem of identifying pathways through metabolic networks that relate to a specific biological response. Our proposed model, HME3M, first identifies frequently traversed network paths using a Markov mixture model. Then by employing a hierarchical mixture of experts, separate classifiers are built using information specific to each path and combined into an ensemble prediction for the response.     

A generalized mathematical model of biological oscillators

Biological rhythms such as circadian rhythms, biochemical rhythms and neural oscillators are based on the mathematical model of the theory of harmonic oscillators. These are solutions of certain second-order differential equations. They can also be viewed as spherical harmonics on the circle in the two-dimensional Euclidean space. The spherical harmonics on (n-1)-spheres and, more generally, the Stiefel harmonics can represent oscillatory phenomena, and we expect that they can serve as models for more complex biological rhythms.     

温室茄子（Solanum melongena L.）光合数学模型与光合生化模型模拟分析

针对植物光合与内外环境因子间的关系以及光合“午睡”现象中的气孔限制与非气孔限制问题,以温室茄子‘茄杂一号’为试材,对叶室温光组合方式下测定的净光合速率Pn对胞间CO2浓度Ci响应曲线,和人工增施CO2处理下测定的Pn日变化进程,进行了光合数学模型和Farquhar、von Caemmerer和Berry的光合生化动力学模型（简称为FvCB模型）模拟分析。采用美国思爱迪生态仪器有限公司的CI-301PS光合作用测定仪进行净光合速率（Pn）、光合有效辐射（PAR）、气温（Ta）、叶温（Tl）、环境二氧化碳浓度（Ca）、胞间二氧化碳浓度（Ci）和空气相对湿度（Hr）参数测定。其结果表明,无论是Pn对Ci的响应曲线还是光合日进程中,数学模型对Pn的拟合度明显优于为FvCB模型。因此,通过数学模型可以解析出光合日进程受单一环境因子（PAR、Ta、Ca、Hr）及其复合环境因子的综合影响。然而,FvCB模型模拟结果显示出,温光组合下受Rubisco（即RuBP羧化/加氧酶）数量与活性及动力学特性限制的羧化速率Ac、受RuBP（1,5-二磷酸核酮糖）再生限制的羧化速率Aj以及受TPU（磷酸丙糖）可利用量限制的羧化速率Ap对Ci响应的主控作用呈现交替变化趋势。其交替变化转折点胞间二氧化碳浓度Cicj在强光高温组合中较高,而在弱光低温组合中较低;同时还发现,Cicj和Cijp受叶温的影响强于光照。光合日进程中的FvCB模型模拟分析揭示出,早晨和傍晚弱光下为Aj限制时段;晴天上午和中午前后的充足日照下为Ac限制时段。多云和阴天下Aj的限制时段延长。增施CO2会延长Aj的限制时段,同时相应缩短Ac的限制时段;冬季2次增施CO2的出现了Ap限制时段。     

Chromosomal assignments for porcine genes encoding enzymes in hepatic metabolic pathways

Increasing the number of mapped genes will facilitate (1) the identification of potential candidate genes for a trait of interest within quantitative trait loci regions and (2) comparative mapping. The metabolic activities of the liver are essential for providing fuel to peripheral organs, for regulation of amino acid, carbohydrate and lipid metabolism and for homoeostasis of vitamins, minerals and electrolytes. We aimed to identify and map genes coding for enzymes active in the liver by somatic cell genetics in order to contribute to the improvement of the porcine gene map. We mapped 28 genes of hepatic metabolic pathways including six genes whose locations could be confirmed and 22 new assignments. Localization information in human was available for all but one gene. In total 24 genes were assigned to in the expected chromosomal regions on the basis of the currently available information on the comparative human and pig map while for four genes our results suggest a new correspondence or extended regions of conservation between porcine and human chromosomes.     

Pollen wall development: the associated enzymes and metabolic pathways

Pollen grains are surrounded by a sculpted wall, which protects male gametophytes from various environmental stresses and microbial attacks, and also facilitates pollination. Pollen wall development requires lipid and polysaccharide metabolism, and some key genes and proteins that participate in these processes have recently been identified. Here, we summarise the genes and describe their functions during pollen wall development via several metabolic pathways. A working model involving substances and catalytic enzyme reactions that occur during pollen development is also presented. This model provides information on the complete process of pollen wall development with respect to metabolic pathways.     

Toward a detailed description of the pathways of allosteric communication in the GroEL chaperonin through atomistic simulation

GroEL, along with its coprotein GroES, is essential for ensuring the correct folding of unfolded or newly synthesized proteins in bacteria. GroEL is a complex, allosteric molecule, composed of two heptameric rings stacked back to back, that undergoes large structural changes during its reaction cycle. These structural changes are driven by the cooperative binding and subsequent hydrolysis of ATP, by GroEL. Despite numerous previous studies, the precise mechanisms of allosteric communication and the associated structural changes remain elusive. In this paper, we describe a series of all-atom, unbiased, molecular dynamics simulations over relatively long (50-100 ns) time scales of a single, isolated GroEL subunit and also a heptameric GroEL ring, in the presence and absence of ATP. Combined with results from a distance restraint-biased simulation of the single ring, the atomistic details of the earliest stages of ATP-driven structural changes within this complex molecule are illuminated. Our results are in broad agreement with previous modeling studies of isolated subunits and with a coarse-grained, forcing simulation of the single ring. These are the first reported all-atom simulations of the GroEL single-ring complex and provide a unique insight into the role of charged residues K80, K277, R284, R285, and E388 at the subunit interface in transmission of the allosteric signal. These simulations also demonstrate the feasibility of performing all-atom simulations of very large systems on sufficiently long time scales on typical high performance computing facilities to show the origins of the earliest events in biologically relevant processes.     

The metabolic pathway collection from EMP: the enzymes and metabolic pathways database.

The Enzymes and Metabolic Pathways database (EMP) is an encoding of the contents of over 10 000 original publications on the topics of enzymology and metabolism. This large body of information has been transformed into a queryable database. An extraction of over 1800 pictorial representations of metabolic pathways from this collection is freely available on the World Wide Web. We believe that this collection will play an important role in the interpretation of genetic sequence data, as well as offering a meaningful framework for the integration of many other forms of biological data.     

The determination of positive and negative co-operativity with allosteric enzymes and the interpretation of sigmoid curves and non-linear double reciprocal plots for the MWC and KNF models

(i) It is proved that only four independent constants can ever be obtained by extrapolation procedures applied to non-hyperbolic steady-state or binding data, (ii) Analysis of the algebraic graphs $\text{y}\text{x}$, $\text{(1/y)}\text{(1/x)}$, $\text{y}\text{(}\text{y}\text{x}\text{)}$ and ($\text{x}\text{y}\text{)/x}$ is shown to require a knowledge of the sign of six curve shape determinants. In each case, the sign is a necessary and sufficient condition for a specific curve shape feature, (iii) The precise graphical effect of positive and negative co-operativity then requires the definition of two reference curves, the osculating hyperbola at zero substrate concentration, OH(0), and the osculating hyperbola at infinite substrate concentration OH(∞). These are better first order approximations than the Hill equation, (iv) Rules for determining unambiguously the sign of initial, final and overall co-operativity coefficients by inspection of non-hyperbolic binding curves are then possible, (v) These rules require that saturation data for:

$\text{y=}\text{∑}\text{i=1}\text{n}\text{a}{}_{\mathrm{i}}\text{x}{}^{\mathrm{i}}\text{∑}\text{i=0}\text{n}\text{β}{}_{\mathrm{i}}\text{x}{}^{\mathrm{i}}$

be fitted by computer for low concentrations to the hyperbola:

$\text{OH}\text{(o)=}\text{(}\text{-a}{}_1{}^2\text{ψ}{}_{11}{}^{20}\text{)x}\text{[(}\text{-a}{}_1\text{β}{}_0\text{ψ}{}_{11}{}^{20}\text{)+x]}$

while regression of high substrate concentration data is to:

$\text{OH}\text{(∞)=}\text{(}\text{a}{}_{\mathrm{n}}\text{β}{}_{\mathrm{n}}\text{)x}\text{[(}\text{φ}{}_{\mathrm{n,n-1}}\text{a}{}_{\mathrm{n}}\text{β}{}_{\mathrm{n}}\text{)+x]}$

. Comparisons of the best fit pseudo-kinetic constants then gives the type of co-operativity present in an unambiguous way with no assumptions as to molecular mechanism, (vi) These rules are then applied to the MWC and KNF allosteric models of ligand binding and the constraints necessary for specific curve shape effects are given, (vii) The graphical expression of positive or negative final co-operativity depends only on events at high substrate concentration but overall and initial co-operativities produce specific geometric effects depending upon the difference between behaviour of saturation data at both extremes of concentration, (viii) This apparent anomaly is explained by a discussion of the relationships between the osculating hyperbolae, the theoretical parent hyperbola and the Hill plot asymptotes.     

Time-dependent model for hemoglobin and allosteric enzymes. I. General formulation

A previous equilibrium model is generalized to study time-dependent behavior of hemoglobin and allosteric enzymes. An exact solution for two interacting subunits (e.g., diheme) is given, and a general method for solving the resulting set of differential equations is outlined. At half saturation (equilibrium) concentration, the model takes a particularly simple form which suggests an experiment to determine the number of subunits of an allosteric enzyme, or in particular to distinguish diheme from ordinary hemoglobin. The relation between the present model and other kinetic models is also discussed.     

Benzoxazinoid biosynthesis,a model for evolution of secondary metabolic pathways in plants

Benzoxazinoids are secondary metabolites that are effective in defence and allelopathy. They are synthesised in two subfamilies of the Poaceae and sporadically found in single species of the dicots. The biosynthesis is fully elucidated in maize; here the genes encoding the enzymes of the pathway are in physical proximity. This “biosynthetic cluster” might facilitate coordinated gene regulation. Data from Zea mays, Triticum aestivum and Hordeum lechleri suggest that the pathway is of monophyletic origin in the Poaceae. The branchpoint from the primary metabolism (Bx1 gene) can be traced back to duplication and functionalisation of the alpha-subunit of tryptophan synthase (TSA). Modification of the intermediates by consecutive hydroxylation is catalysed by members of a cytochrome P450 enzyme subfamily (Bx2–Bx5). Glucosylation by an UDP-glucosyltransferase (UGT, Bx8, Bx9) is essential for the reduction of autotoxicity of the benzoxazinoids. In some species 2,4-dihydroxy-1,4-benzoxazin-3-one-glucoside (DIBOA-glc) is further modified by the 2-oxoglutarate-dependent dioxygenase BX6 and the O-methyltransferase BX7. In the dicots Aphelandra squarrosa, Consolida orientalis, and Lamium galeobdolon, benzoxazinoid biosynthesis is analogously organised: The branchpoint is established by a homolog of TSA, P450 enzymes catalyse hydroxylations and at least the first hydroxylation reaction is identical in dicots and Poaceae, the toxic aglucon is glucosylated by an UGT. Functionally, TSA and BX1 are indole-glycerolphosphate lyases (IGLs). Igl genes seem to be generally duplicated in angiosperms. Modelling and biochemical characterisation of IGLs reveal that the catalytic properties of the enzyme can easily be modified by mutation. Independent evolution can be assumed for the BX1 function in dicots and Poaceae.