120 research outputs found
Characteristic, completion or matching timescales? An analysis of temporary boundaries in enzyme kinetics
Scaling analysis exploiting timescale separation has been one of the most
important techniques in the quantitative analysis of nonlinear dynamical
systems in mathematical and theoretical biology. In the case of enzyme
catalyzed reactions, it is often overlooked that the characteristic timescales
used for the scaling the rate equations are not ideal for determining when
concentrations and reaction rates reach their maximum values. In this work, we
first illustrate this point by considering the classic example of the
single-enzyme, single-substrate Michaelis--Menten reaction mechanism. We then
extend this analysis to a more complicated reaction mechanism, the auxiliary
enzyme reaction, in which a substrate is converted to product in two sequential
enzyme-catalyzed reactions. In this case, depending on the ordering of the
relevant timescales, several dynamic regimes can emerge. In addition to the
characteristic timescales for these regimes, we derive matching timescales that
determine (approximately) when the transitions from initial fast transient to
steady-state kinetics occurs. The approach presented here is applicable to a
wide range of singular perturbation problems in nonlinear dynamical systems.Comment: 35 pages, 11 figure
Validity of the MichaelisâMenten equation â steadyâstate or reactant stationary assumption: that is the question
Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/102681/1/febs12564.pd
Protein Interactions and Transition Times that Influence the Pathogenesis of Protein Folding Diseases
Designing nanoparticle treatment of autoimmunity with quantitative biology
Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/141246/1/imcb201320.pd
Stochastic enzyme kinetics and the quasi-steady-state reductions: Application of the slow scale linear noise approximation \`a la Fenichel
The linear noise approximation models the random fluctuations from the mean
field model of a chemical reaction that unfolds near the thermodynamic limit.
Specifically, the fluctuations obey a linear Langevin equation up to order
, where is the size of the chemical system (usually the
volume). Under the presence of disparate timescales, the linear noise
approximation admits a quasi-steady-state reduction referred to as the slow
scale linear noise approximation. However, the slow scale linear approximation
has only been derived for fast/slow systems that are in Tikhonov standard form.
In this work, we derive the slow scale linear noise approximation directly from
Fenichel theory, without the need for a priori scaling and dimensional
analysis. In so doing, we can apply for the first time the slow scale linear
noise approximation to fast/slow systems that are not of standard form. This is
important, because often times algorithms are only computationally expensive in
parameter ranges where the system is singularly perturbed, but not in standard
form. We also comment on the breakdown of the slow scale linear noise
approximation near dynamic bifurcation points -- a topic that has remained
absent in the chemical kinetics literature, despite the presence of
bifurcations in simple biochemical reactions, such the Michaelis--Menten
reaction mechanism.Comment: 22 pages, 2 figure
The Quasi-Steady-State Approximations revisited: Timescales, small parameters, singularities, and normal forms in enzyme kinetic
In this work, we revisit the scaling analysis and commonly accepted
conditions for the validity of the standard, reverse and total
quasi-steady-state approximations through the lens of dimensional
Tikhonov-Fenichel parameters and their respective critical manifolds. By
combining Tikhonov-Fenichel parameters with scaling analysis and energy
methods, we derive improved upper bounds on the approximation error for the
standard, reverse and total quasi-steady-state approximations. Furthermore,
previous analyses suggest that the reverse quasi-steady-state approximation is
only valid when initial enzyme concentrations greatly exceed initial substrate
concentrations. However, our results indicate that this approximation can be
valid when initial enzyme and substrate concentrations are of equal magnitude.
Using energy methods, we find that the condition for the validity of the
reverse quasi-steady-state approximation is far less restrictive than was
previously assumed, and we derive a new "small" parameter that determines the
validity of this approximation. In doing so, we extend the established domain
of validity for the reverse quasi-steady-state approximation. Consequently,
this opens up the possibility of utilizing the reverse quasi-steady-state
approximation to model enzyme catalyzed reactions and estimate kinetic
parameters in enzymatic assays at much lower enzyme to substrate ratios than
was previously thought. Moreover, we show for the first time that the critical
manifold of the reverse quasi-steady-state approximation contains a singular
point where normal hyperbolicity is lost. Associated with this singularity is a
transcritical bifurcation, and the corresponding normal form of this
bifurcation is recovered through scaling analysis.Comment: 50 pages, 10 figures, 1 tabl
A multiscale mathematical model of cancer, and its use in analyzing irradiation therapies
Background: Radiotherapy outcomes are usually predicted using the Linear
Quadratic model. However, this model does not integrate complex features of
tumor growth, in particular cell cycle regulation.
Methods: In this paper, we propose a multiscale model of cancer growth based
on the genetic and molecular features of the evolution of colorectal cancer.
The model includes key genes, cellular kinetics, tissue dynamics, macroscopic
tumor evolution and radiosensitivity dependence on the cell cycle phase. We
investigate the role of gene-dependent cell cycle regulation in the response of
tumors to therapeutic irradiation protocols.
Results: Simulation results emphasize the importance of tumor tissue features
and the need to consider regulating factors such as hypoxia, as well as tumor
geometry and tissue dynamics, in predicting and improving radiotherapeutic
efficacy.
Conclusion: This model provides insight into the coupling of complex
biological processes, which leads to a better understanding of oncogenesis.
This will hopefully lead to improved irradiation therapy.Comment: 19 pages, 14, figures. Article available at
http://www.tbiomed.com/content/3/1/7 Copyright 2006 Ribba et al; licensee
BioMed Central Ltd. This is an Open Access article distributed under the
terms of the Creative Commons Attribution License
(http://creativecommons.org/licenses/by/2.0), which permits unrestricted use,
distribution, and reproduction in any medium, provided the original work is
properly cite
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