Enzyme kinetics for a two-step enzymic reaction with comparable initial enzyme-substrate ratios

Frenzen, C. L.; Maini, Philip K.

doi:10.1007/bf00276148

Cited by 31 publications

(30 citation statements)

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“…In the case of the system here, we discussed this hypothesis in Levine et al (2000Levine et al ( , 2001. Other sources include Frenzen and Maini (1988), Segel (1988) and Segel and Slemrod (1989).…”

Section: The Equations Of Mass Actionmentioning

confidence: 99%

Mathematical Modeling of Capillary Formation and Development in Tumor Angiogenesis: Penetration into the Stroma

Levine

Pamuk

Sleeman

et al. 2001

Bulletin of Mathematical Biology

235

202

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The purpose of this paper is to present a mathematical model for the tumor vascularization theory of tumor growth proposed by Judah Folkman in the early 1970s and subsequently established experimentally by him and his coworkers [Ausprunk, D. H. and J. Folkman (1977) Migration and proliferation of endothelial cells in performed and newly formed blood vessels during tumor angiogenesis, Microvasc Res., 14, 53-65; Brem, S., B. A. Preis, ScD. Langer, B. A. Brem and J. Folkman (1997) Inhibition of neovascularization by an extract derived from vitreous Am. J. Opthalmol., 84, 323-328; Folkman, J. (1976) The vascularization of tumors, Sci. Am., 234, 58-64; Gimbrone, M. A. Jr, R. S. Cotran, S. B. Leapman and J. Folkman (1974) Tumor growth and neovascularization: an experimental model using the rabbit cornea, J. Nat. Cancer Inst., 52, 413-419]. In the simplest version of this model, an avascular tumor secretes a tumor growth factor (TGF) which is transported across an extracellular matrix (ECM) to a neighboring vasculature where it stimulates endothelial cells to produce a protease that acts as a catalyst to degrade the fibronectin of the capillary wall and the ECM. The endothelial cells then move up the TGF gradient back to the tumor, proliferating and forming a new capillary network. In the model presented here, we include two mechanisms for the action of angiostatin. In the first mechanism, substantiated experimentally, the angiostatin acts as a protease inhibitor. A second mechanism for the production of protease inhibitor from angiostatin by endothelial cells is proposed to be of Michaelis-Menten type. Mathematically, this mechanism includes the former as a subcase. Our model is different from other attempts to model the process of tumor angiogenesis in that it focuses (1) on the biochemistry of the process at the level of the cell; (2) the movement of the cells is based on the theory of reinforced random walks; (3) standard transport equations for the diffusion of molecular species in porous media. One consequence of our numerical simulations is that we obtain very good computational agreement with the time of the onset of vascularization and the rate of capillary tip growth observed in rabbit cornea experiments [Ausprunk, D. H. and J. Folkman (1977) Migration and proliferation of endothelial cells in performed and newly formed blood vessels during tumor angiogenesis, Microvasc Res., 14, 73-65; Brem, S., B. A. Preis, ScD. Langer, B. A. Brem and J. Folkman (1997) Inhibition of neovascularization by an extract derived from vitreous Am. J. Opthalmol., 84, 323-328; Folkman, J. (1976) The vascularization of tumors, Sci. Am., 234, 58-64; Gimbrone, M. A. Jr, R. S. Cotran, S. B. Leapman and J. Folkman (1974) Tumor growth and neovascularization: An experimental model using the rabbit cornea. J. Nat. Cancer Inst., 52, 413-419]. Furthermore, our numerical experiments agree with the observation that the tip of a growing capillary accelerates as it approaches the tumor [Folkman, J. (1976) The vascularization of tumors, Sci. Am., 2...

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Section: The Equations Of Mass Actionmentioning

confidence: 99%

Mathematical Modeling of Capillary Formation and Development in Tumor Angiogenesis: Penetration into the Stroma

Levine

Pamuk

Sleeman

et al. 2001

Bulletin of Mathematical Biology

235

202

View full text Add to dashboard Cite

show abstract

“…To obtain the reduced model corresponding to ZDP 1 , we substitute from Eqn (20) into the first ODE in Eqn (11) …”

Section: Accurate Enzyme Kinetics Based On Zdpmentioning

confidence: 99%

“…Using the tabulated ZDP 1 manifold to reduce the PTS model As we show above, an explicit expression for a ZDP manifold, such as Eqn (20), can be used to obtain a lower-dimensional model, such as Eqn (21). When a ZDP manifold is only available in tabulated form, however, such a reduced equation cannot be written out explicitly.…”

Section: Calculation Of Zdp Manifolds For the Pts Modelmentioning

confidence: 99%

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Simplified yet highly accurate enzyme kinetics for cases of low substrate concentrations

Härdin

Zagaris

Krab³

et al. 2009

The FEBS Journal

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Much of enzyme kinetics builds on simplifications enabled by the quasi‐steady‐state approximation and is highly useful when the concentration of the enzyme is much lower than that of its substrate. However, in vivo, this condition is often violated. In the present study, we show that, under conditions of realistic yet high enzyme concentrations, the quasi‐steady‐state approximation may readily be off by more than a factor of four when predicting concentrations. We then present a novel extension of the quasi‐steady‐state approximation based on the zero‐derivative principle, which requires considerably less theoretical work than did previous such extensions. We show that the first‐order zero‐derivative principle, already describes much more accurately the true enzyme dynamics at enzyme concentrations close to the concentration of their substrates. This should be particularly relevant for enzyme kinetics where the substrate is an enzyme, such as in phosphorelay and mitogen‐activated protein kinase pathways. We illustrate this for the important example of the phosphotransferase system involved in glucose uptake, metabolism and signaling. We find that this system, with a potential complexity of nine dimensions, can be understood accurately using the first‐order zero‐derivative principle in terms of the behavior of a single variable with all other concentrations constrained to follow that behavior.

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“…However, the evidence for this is not conclusive. Segel (1988);Segel and Slemrod (1989) and Frenzen and Maini (1988) have shown that the sQSSA can provide a good approximation even when ½S 0 % ½E 0 as long as [E 0 ] is small compared to K M . The positive [S 0 ]-[E 0 ] plane can be divided into regions in which these approximations hold, but in certain circumstances there remains a region where neither holds.…”

Section: Validity Of the Quasi-steady-state Approximationmentioning

confidence: 99%

A Century of Enzyme Kinetics: Reliability of the K M and v max Estimates

Schnell¹,

Maini²

2003

Comments� on Theoretical Biology

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135

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The application of the quasi-steady-state approximation (QSSA) in biochemical kinetics allows the reduction of a complex biochemical system with an initial fast transient into a simpler system. The simplified system yields insights into the behavior of the biochemical reaction, and analytical approximations can be obtained to determine its kinetic parameters. However, this process can lead to inaccuracies due to the inappropriate application of the QSSA. Here we present a number of approximate solutions and determine in which regions of the initial enzyme and substrate concentration parameter space they are valid. In particular, this illustrates that experimentalists must be careful to use the correct approximation appropriate to the initial conditions within the parameter space.

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Enzyme kinetics for a two-step enzymic reaction with comparable initial enzyme-substrate ratios

Cited by 31 publications

References 3 publications

Mathematical Modeling of Capillary Formation and Development in Tumor Angiogenesis: Penetration into the Stroma

Mathematical Modeling of Capillary Formation and Development in Tumor Angiogenesis: Penetration into the Stroma

Simplified yet highly accurate enzyme kinetics for cases of low substrate concentrations

A Century of Enzyme Kinetics: Reliability of the K M and v max Estimates

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