Michaelis-Menten Formula Research Articles

Commutativity of spatiochromatic covariance matrices in natural image statistics

Statistic of natural images is a growing field of research both in vision and image processing. On the vision research side, fine statistical details about object distribution in real-world scenes help understanding the human visual system behavior. On the image processing side, by using the information gathered from statistics of natural scenes, we can obtain reliable priors and insights that can be used in many models. In has been rigorously proven in [ 16 ] that, if second order stationarity and commutativity of spatiochromatic covariance matrices hold true for natural scenes, then the codification of spatial and chromatic information by the human visual system can be separated through a tensor product. Spatial features are coded via local and oriented Fourier basis elements, while color features are coded via a triad given by an achromatic channel followed by two color opponent channels. In this paper, we will show that, while stationarity is guaranteed, commutativity is not. However, we shall see that commutativity of spatiochromatic covariance matrices can be approached if the database of images used to model visual scenes is modified accordingly to a suitable transformation that describes the response of retinal photoreceptors to light absorption: the Michaelis-Menten formula. A thorough investigation of the effects of a parameter of this formula will be performed and its influence on commutativity of covariance matrices will be detailed.

Structural conditions on complex networks for the Michaelis–Menten input–output response

The Michaelis-Menten (MM) fundamental formula describes how the rate of enzyme catalysis depends on substrate concentration. The familiar hyperbolic relationship was derived by timescale separation for a network of three reactions. The same formula has subsequently been found to describe steady-state input-output responses in many biological contexts, including single-molecule enzyme kinetics, gene regulation, transcription, translation, and force generation. Previous attempts to explain its ubiquity have been limited to networks with regular structure or simplifying parametric assumptions. Here, we exploit the graph-based linear framework for timescale separation to derive general structural conditions under which the MM formula arises. The conditions require a partition of the graph into two parts, akin to a "coarse graining" into the original MM graph, and constraints on where and how the input variable occurs. Other features of the graph, including the numerical values of parameters, can remain arbitrary, thereby explaining the formula's ubiquity. For systems at thermodynamic equilibrium, we derive a necessary and sufficient condition. For systems away from thermodynamic equilibrium, especially those with irreversible reactions, distinct structural conditions arise and a general characterization remains open. Nevertheless, our results accommodate, in much greater generality, all examples known to us in the literature.

A Generalized Michaelis-Menten Equation in Protein Synthesis: Effects of Mis-Charged Cognate tRNA and Mis-Reading of Codon.

The sequence of amino acid monomers in the primary structure of a protein is decided by the corresponding sequence of codons (triplets of nucleic acid monomers) on the template messenger RNA (mRNA). The polymerization of a protein, by incorporation of the successive amino acid monomers, is carried out by a molecular machine called ribosome. We develop a stochastic kinetic model that captures the possibilities of mis-reading of mRNA codon and prior mis-charging of a tRNA. By a combination of analytical and numerical methods, we obtain the distribution of the times taken for incorporation of the successive amino acids in the growing protein in this mathematical model. The corresponding exact analytical expression for the average rate of elongation of a nascent protein is a 'biologically motivated' generalization of the Michaelis-Menten formula for the average rate of enzymatic reactions. This generalized Michaelis-Menten-like formula (and the exact analytical expressions for a few other quantities) that we report here display the interplay of four different branched pathways corresponding to selection of four different types of tRNA.

Response-guided dosing for rheumatoid arthritis

ABSTRACTRheumatoid arthritis (RA) is an auto-immune disease with an unknown cause. Many patients receiving traditional methotrexate treatment continue to exhibit joint damage and are then treated with biologics. Biologic treatment is difficult owing to the uncertainty in dose-response, high cost, side effects, and intravenous administration. Recent clinical trials have therefore attempted response-guided dosing (RGD), where the hope is to adapt biologic doses over the treatment course based on each patient’s observed evolution of the 28-joint disease activity score (DAS28).We provide a stochastic dynamic programming (DP) framework to facilitate RGD. We present a concrete formulation where the DAS28 response is modeled using a stochastic Michaelis-Menten formula. The goal is to balance the DAS28 attained at the end of the course with the weighted total dose administered. We perform numerical experiments using data from the OPTION trial and observe that the optimal dosing policy gives higher doses in worse DAS28 scores. We present sensitivity analyses to provide further insights into this monotone dosing policy.This basic formulation is extended to a general stochastic DP for RGD. This is also applicable to other diseases and conditions such as hepatitis C, hyperlipidemia, hypertension, and AIDS. The DP allows for an arbitrary dose-response function, and balances the disutility of doses with the disutility of the disease condition reached. We prove that, when the decision-maker is risk-averse and the dose-response is supermodular and convex, there exists an optimal policy that gives higher doses in worse disease conditions. We provide several examples where these conditions are met.

Response-Guided Dosing for Rheumatoid Arthritis

Rheumatoid arthritis (RA) is an auto-immune disease with an unknown cause. Many patients receiving traditional methotrexate treatment continue to exhibit progressive joint damage and are then often treated with biologics. Biologic treatment is difficult owing to the uncertainty in dose-response, high cost, side effects, and intravenous administration. Recent clinical trials have therefore attempted response-guided dosing (RGD), where the hope is to adapt biologic doses over the treatment course based on each individual patient's observed evolution of the 28-joint disease activity score (DAS28). We provide a rigorous, stochastic dynamic programming (DP) framework to facilitate RGD. We first present a concrete formulation where the DAS28 response is modeled using a stochastic Michaelis-Menten formula. The goal is to balance the DAS28 attained at the end of the course with the weighted total dose administered. We perform numerical experiments using data from the OPTION trial and observe that the optimal dosing policy has a monotone structure -- it gives higher doses in worse DAS28 scores. Our sensitivity analyses bring forth the intuitive trend that optimal doses decrease with increasing biologic efficacy and with increasing aversion to dose. We also provide a so-called efficient frontier that is obtained by varying dose weights; this makes explicit the trade-off between DAS28 scores and total doses administered, and it could be used as a decision-making tool in practice. The sensitivity analyses also reveal that a more aggressive treatment strategy, that administers a larger incremental dose over two given states, is optimal when there is more uncertainty in dose-response. Our basic formulation is then extended to a general stochastic DP for RGD. This is applicable mot only to RA, but also to other diseases such as hepatitis C, cholesterol, hypertension, and AIDS, where the measured disease condition could correspond to viral loads, LDL cholesterol levels, blood pressure, or CD4 counts. The DP allows for an arbitrary dose-response function, and balances the disutility of doses with the disutility of the disease condition reached. We prove that, when the decision-maker is risk-averse and the dose-response is supermodular and convex, there exists an optimal policy that gives higher doses in worse disease conditions. We provide several examples where these conditions are met, and also show by counterexamples that such a monotone policy may not be optimal when convexity is violated.

Kinetic Analysis on Rotating Biological Contactor Treatment of Organic Matter

This paper adopts a 3-stage rotating biological contactor (RBC), studies kinetic analysis on RBC treatment of organic matter .Water temperature 25 °C and hydraulic retention time (HRT) 5h , COD concentration of influent varies from about 400 to 700mg / L, the average COD removal rate is 85% . According to Monod and Michaelis-Menten formula, organic matter removal kinetics equation in the first stage of RBC can be catched. When DO≥2mg/L, organic matter removal kinetic equation: U1=7.02S1/( S1+640.70), DO<2mg/L, organic matter removal kinetic equation: U2=35.97S2/( S2+4848.92) .Where U is substrate removal rate of unit area of disc, S is the organic matter concentration of substrate. Some reference values of the equation are very useful for practical application of RBC.

Study on biofiltration capacity and kinetics of nutrient uptake by Gracilaria cervicornis (Turner) J. Agardh (Rhodophyta, Gracilariaceae)

The absorption efficiency and kinetic parameters (Vmax, Ks and Vmax:Ks) of the seaweed Gracilaria cervicornis for the nutrients NH4+, NO3- and PO4(3-) were evaluated. Absorption efficiency was measured by monitoring nutrient concentrations for 5 h in culture media with initial concentrations of 5, 10, 20 and 30µM. Kinetic parameters were determined by using the Michaelis-Menten formula. Absorption efficiencies for this algae were greater in treatments with lower concentrations, as evidenced by a reduction of 85.3, 97.5 and 81.2% for NH4+, NO3- and PO4(3-), respectively. Kinetic parameters show that G. cervicornis exhibits greater ability to take up high concentrations of NH4+ (Vmax=158.5µM g dw-1 h-1) and low concentrations of PO4(3-) (Ks=5µM and Vmax:Ks=10.3). These results suggest that this algal species has good absorption capacity for the nutrients tested and may be a promising candidate as a bioremediator of eutrophized environments.

Legitimacy of the stochastic Michaelis–Menten approximation

Michaelis-Menten kinetics are commonly used to represent enzyme-catalysed reactions in biochemical models. The Michaelis-Menten approximation has been thoroughly studied in the context of traditional differential equation models. The presence of small concentrations in biochemical systems, however, encourages the conversion to a discrete stochastic representation. It is shown that the Michaelis-Menten approximation is applicable in discrete stochastic models and that the validity conditions are the same as in the deterministic regime. The authors then compare the Michaelis-Menten approximation to a procedure called the slow-scale stochastic simulation algorithm (ssSSA). The theory underlying the ssSSA implies a formula that seems in some cases to be different from the well-known Michaelis-Menten formula. Here those differences are examined, and some special cases of the stochastic formulas are confirmed using a first-passage time analysis. This exercise serves to place the conventional Michaelis-Menten formula in a broader rigorous theoretical framework.

Numerical integration methods for large-scale biophysical simulations

Simulations of biophysical systems inevitably include steps that correspond to time integrations of ordinary differential equations. These equations are often related to enzyme action in the synthesis and destruction of molecular species, and in the regulation of transport of molecules into and out of the cell or cellular compartments. Enzyme action is almost invariably modeled with the quasi-steady-state Michaelis–Menten formula or its close relative, the Hill formula: this description leads to systems of equations that may be stiff and hard to integrate, and poses unusual computational challenges in simulations where a smooth evolution is interrupted by the discrete events that mark the cells' lives, like initiation and termination of DNA synthesis or mitosis. These discrete events lead to abrupt parameter changes and to variable system size. This is the case of a numerical model (Virtual Biophysics Lab – VBL) that we are developing to simulate the growth of three-dimensional tumor cell aggregates (spheroids). The underlying cellular events have characteristic timescales that span approximately 12 orders of magnitude, and thus the program must be robust and stable. Moreover the program must be able to manage a very large number of equations (of the order of 107–108), and finally it must be able to accept frequent modifications of the underlying theoretical model. Here we study the applicability of known integration methods to this context – quite unusual from the point of view of the standard theory of differential equations, but extremely relevant to biophysics – and we describe the results of numerical tests in situations similar to those found in actual simulations.

ON REACTION DYNAMICS AT DOPAMINE SYNAPSES

Dopamine neurons play a key role in normal and pathological cognitive processes as well as in the effects of certain drugs of addiction. Models of the synapses of such neurons include transporter mechanisms and reaction dynamics. This article focuses attention on the fundamental reaction that converts tyrosine to DOPA, which involves a cofactor. The Michaelis-Menten formula for the rate of an enzymatic reaction is modified by the presence of cofactors, which may be either essential or nonessential. In the essential case, the reaction rate is found to depend on the relative magnitudes of the concentrations of the primary enzyme and the cofactor. The case of a nonessential cofactor is more complex and it is shown for the first time how this leads to reaction rate formulas that depend explicitly on the concentrations of the enzyme and cofactor. The extended Michaelis-Menten formulas are applied to the aforementioned reaction with tyrosine hydroxylase as enzyme and biopterin as cofactor. The results are useful in constructing accurate models of dopamine synapses.

Effect of Polyethers for Photo and Thermal Decomposition of Photosensitive Diazo Compounds

Inclusion compound formation of polyethylene glycol (PEG) with p-substituted benzene diazo compounds (SBD) in aqueous solutions was decided from activation energy (Ea) of thermal decomposition, dissociation constant Kd by Lineweaver-Burk plot modified with Michaelis-Menten formula and degradation temperature Dt, and compared with constitution of complex by WinMOPAC calculation. Thermal decomposition rate of p-chlorobenzene diazo compound (ClBD) with PEG was accelerated as with βcyclodextrine (CD) which include the benzene ring of the diazo compound. Kd of ClBD with PEG was larger than that with CD. PEG is not a ring, but a linear molecule. Dt of SBD with PEG in solid was higher than that without PEG300. Effect of molecular weight of PEG for Dt was, in a decreasing order, no additives - PEG300

Gel catalysts that switch on and off.

We report development of a polymer gel with a catalytic activity that can be switched on and off when the solvent composition is changed. The gel consists of two species of monomers. The major component, N-isopropylacrylamide, makes the gel swell and shrink in response to a change in composition of ethanol/water mixtures. The minor component, vinylimidazole, which is capable of catalysis, is copolymerized into the gel network. The reaction rate for catalytic hydrolysis of p-nitrophenyl caprylate was small when the gel was swollen. In contrast, when the gel was shrunken, the reaction rate increased 5 times. The activity changes discontinuously as a function of solvent composition, thus the catalysis can be switched on and off by an infinitesimal change in solvent composition. The kinetics of catalysis by the gel in the shrunken state is well described by the Michaelis-Menten formula, indicating that the absorption of the substrate by the hydrophobic environment created by the N-isopropylacrylamide polymer in the shrunken gel is responsible for enhancement of catalytic activity. In the swollen state, the rate vs. active site concentration is linear, indicating that the substrate absorption is not a primary factor determining the kinetics. Catalytic activity of the gel is studied for substrates with various alkyl chain lengths; of those studied the switching effect is most pronounced for p-nitrophenyl caprylate.

Cytochrome Oxidase Activity of Individual Mitochondrion as Quantified by Platinum -Diaminobenzidine Reaction with Energy Dispersive X-ray Analyzer.

The platinum diaminobenzidine (Pt-DAB) reaction was characterized by the oxidative polymerization of DAB with a concurrent incorporation of Pt atoms into the polymerization product in the presence of cytochrome oxidase activity. It conformed to the Michaelis-Menten's formula showing its enzymatic nature. It was specific to the oxidase as proved by the suppressive effects due to some inhibitors and heat. It permitted the quantification of the oxidase activity of an individual mitochondrion and its constituent part by energy dispersive X-ray analysis (EDAX) owing to the incorporated Ptatoms. To elucidate the gradient in the oxidase activity along the radial axis of a liver lobule, the local activity difference between the hepatocytes in the peripheral region (PE) adjacent to the triad and those in the central region (CN) close to the central vein was quantified at individual mitochondrial level. The mitochondria in the PE hepatocytes were found 1.5 times stronger in activity per area of the reactive site than those in the CN hepatocytes.

Potential of Datura innoxia cell suspension cultures for glucosylating hydroquinone

The efficiency of glucosylation of hydroquinone by Datura innoxia cell suspension cultures was investigated. The yield of arbutin was 2.4 g/l medium when 10 mM of hydroquinone was added to a suspension culture that was then incubated for 24 h, but the yield decreased at a higher concentration. This decrease, which could not be overcome by changing the growth phase or increasing the cell density used, could be avoided by the repeated addition of a low concentration of hydroquinone over 3 days. This increased the yield of arbutin to 4.2 g/l at the usual cell density and to 7.1 g/l at a high density. The kinetics of this reaction were explained by the Michaelis-Menten formula. The theoretical maximum velocity of the arbutin-forming reaction was estimated as 0.77 mg/h/g. The velocity increased linearly up to a cell density of 300 g/l under standard aeration, the theoretical maximum yield of arbutin being calculated to be 5.5 g/l/day.

Cobalt acetylacetonate-diaminobenzidine reaction for microspectrophotometry of cytochrome oxidase.

The cobalt acetylacetonate-diaminobenzidine (CAA-DAB) reaction for cytochrome oxidase has been examined. The medium consists of 2.5mM CAA, 2.5mM DAB·4HCl, 0.25mg/ml cytochrome c and 0.1mg/ml catalase dissolved in 0.1M cacodylate buffer of pH 7.4. Hepatocytes were stained deep blue in granular form due to mitochondrial reaction in rat liver specimens incubated in this medium at 37°C for 10min. The reaction was inhibited by KCN, and by 10min preheating at 80°C. It conformed to Michaelis-Menten's formula and was more sensitive than the color-unmodified DAB reaction processed in the complete medium minus CAA. The CAA-DAB reaction gave the reaction product whose absorption peak locates at 600nm. The microspectrophotometry, carried out by scanning the specimens at 600nm, gave the following results; 1) the adaptability of Beer's law was proved, 2) a decreasing gradient in cytochrome oxidase activity was found along the radial axis of the hepatic lobule from the periphery to the center, 3) using a standard step wedge as a reference, cytochrome oxidase activity was expressed in terms of DAB equivalent units.

A microspectrophotometric quantification of cytochrome oxidase activity by means of color-modified diaminobenzidine reaction.

A new microspectrophotometric method for quantifying cytochrome oxidase activity has been developed based on a color-modified diaminobenzidine (DAB) reaction. The reaction consists of pretreating tissue sections or cell smears with 10mM cobalt acetate dissolved in 0.1M tris-HCl buffer of pH 7.4 for 10min at 37°C, washing and then treating with the DAB reagent for cytochrome oxidase for 15min at 37°C. Mitochondria turn reddish blue in the stained sections or smears. This reaction is enzymatic and conforms to Michaelis-Menten's formula. The microspectrophotometric quantification is effected by scanning the selected cytoplasmic area at 575nm or by measuring the cytoplasmic plug at two wavelengths, 575 and 665nm. As an example, the monomodal distribution in cytochrome oxidase activity was demonstrated among isolated rat liver cells by the two wavelength method.

Light-shade adaptation by the oceanic dinoflagellates Pyrocystis noctiluca and P. fusiformis

Species-specific rates of photosynthetic carbon uptake (P), chlorophyll a content and P versus irradiance (P-I), have been measured for cells of Pyrocystis noctiluca and P. fusiformis isolated from natural populations collected in the euphotic zone within and below the surface mixed layer in the Sargasso Sea. These same measurements and the assay for ribulose bis-phosphate carboxylase (RuBP-Case), have been made for cultures of P. noctiluca in a 12 h L: 12 h D photoperiod at 9 different constant or at changing light intensities. In nature chl a cell-1 was constant throughout the euphotic zone. The photosynthetic capacity (Pmax), of cells captured below the surface mixed layer was lower by a factor of 10 compared with cells collected from the surface mixed layer. The Pmax for P. noctiluca collected and incubated within the surface mixed layer was the same as for cell cultures grown under high light, non nutrient-limiting conditions, suggesting that photosynthesis in the natural system was not nutrient limited. In laboratory cultures under constant low light intensities, chl a cell-1 increased by a factor of 5 while both Pmax and RuBPCase activity decreased by a factor of ca 4 compared with high light intensities. In changing light intensities both Pmax and RuBPCase activities were decreased by factors of 4 during low light intervals while chl a cell-1 approached a constant intermediate value. The change in chl a cell-1 in response to prolonged exposure to constant low light intensities was first order with a rate constant of 0.33 d-1. For all irradiance conditions in culture, the P-I dependence could be described by the simple Michaelis-Menten formula. The ratio of Pmax to KI, (the light intensity where P=Pmax/2) was a constant with a Coefficient of Variation of 12%: The constancy of this ratio, the parallel changes in RuBPCase activity with Pmax and the constant chl a cell-1 in the Sargasso Sea imply that for P. noctiluca and presumably P. fusiformis in nature, a dark enzymatic step rather than changes in photosynthetic pigment concentrations may regulate the photosynthetic capacity in the changing photic environment.

Cation uptake by Douglas-fir seedlings grown in solution culture

Douglas-fir (Pseudotsugamenziesii (Mirb.) Franco) seedlings were grown in nutrient solutions and cation uptake was measured by disappearance from solution. Although seedlings were uniform in size and weight, a range of uptake rates was observed. Potassium and calcium uptake rates were determined over a range of cation concentrations. Using the Michaelis–Menten formula, kinetic constants were calculated for potassium (Km = 0.017 mM) and for calcium (Km = 0.043 mM). These values were similar to those reported for other plant species. Temperature optima were determined (20 °C for potassium and 26 °C for calcium), indicating that cation uptake was an active metabolic process, sensitive to temperature. Low levels of calcium were required for both potassium and magnesium uptake; at higher levels of calcium, magnesium uptake was reduced.

A further study of 17 beta-hydroxysteroid oxidoreductase in the human testis: mechanism of in vitro activation.

The mechanism and specificity of product activation of 17 beta-hydroxysteroid oxidoreductase for interconversion of androstenedione and testosterone were investigated using cell-free homogenates of human testes as an enzyme preparation. The reactions of oxidation and reduction catalyzed by the enzyme followed the Michaelis-Menten formula Addition of testosterone to the incubation medium resulted in increased Vmax and decreased the Km of reductase function of the enzyme. Androstenedione added to the incubation medium increased both Vmax and Km of the enzyme for the oxidase function. The apparent Km of the reductase function for [3H] androstenedione as a substrate in the presence of a fixed concentration of [14C] testosterone was found identical with the apparent Kd for the androstenedione as an effector (activator) for the oxidase function of the enzyme calculated from the simultaneous increase of [14C] androstnedione produced from the fixed concentration of [14C] testosterone. Similarly, the apparent Km of the oxidase function for [3H] testosterone as a substrate in the presence of a fixed concentration of [14C] androstenedione was identical to the apparent Kd for the testosterone as an effector (activator) for the reductase function of the enzyme calculated from the simultaneous increase of [14C] testosterone produced from the fixed concentration of [14C] androstenedione. None of the free 17-oxo and 17 beta-hydroxy steroids examined showed substrate activation or product inhibition of the enzyme except for androstenedione which showed product inhibition at high concentration. The present findings in combination with our previous data suggest the presence of two sites on the 17 beta-hydroxysteroid oxidoreductase of the human testis, one specific for androstenedione and the other specific for testosterone, each serving as an active site for the substrate and also as an activation site for the other active site.

Substrate specificity of endo-polyguluronide lyases from Pseudomonas sp. on the basis of their kinetic properties.

Two endo-alginate lyases [EC 4.2.2.3] differing in their mode of degradation of substrates and practically free of polymannuronide lyase activity were partially purified from Pseudomonas sp. cells. Their substrate specificities were investigated for two different kinds of alginate fragments; a polyguluronide (SG) and a polyuronide consisting of mannuronic (M) and guluronic (G) acid residues (SMG). The effects of various salts and some organic compounds such as EDTA and p-chloromercuribenzoate on the degradation of the two substrates were similar. High concentrations of the substrates similarly inhibited the action ofthe lyases, giving a bell-shaped plot. A polymannuronide alginate fragment (SM) which was a substrate for polymannuronide lyase but was not attacked by these guluronide lyases also inhibited the degradation of SG and SMG. The overall degradation velocities of a mixture of SG and SMG by both lyases coincided with those calculated from the Michaelis-Menten formula. Based on the above results, it was concluded that SG and SMG are attacked by the same endo-polyguluronide lyase.