5,227 research outputs found
Tikhonov-Fenichel reduction for parameterized critical manifolds with applications to chemical reaction networks
We derive a reduction formula for singularly perturbed ordinary differential
equations (in the sense of Tikhonov and Fenichel) with a known parameterization
of the critical manifold. No a priori assumptions concerning separation of slow
and fast variables are made, or necessary.We apply the theoretical results to
chemical reaction networks with mass action kinetics admitting slow and fast
reactions. For some relevant classes of such systems there exist canonical
parameterizations of the variety of stationary points, hence the theory is
applicable in a natural manner. In particular we obtain a closed form
expression for the reduced system when the fast subsystem admits complex
balanced steady states
Analysis of Reaction Network Systems Using Tropical Geometry
We discuss a novel analysis method for reaction network systems with
polynomial or rational rate functions. This method is based on computing
tropical equilibrations defined by the equality of at least two dominant
monomials of opposite signs in the differential equations of each dynamic
variable. In algebraic geometry, the tropical equilibration problem is
tantamount to finding tropical prevarieties, that are finite intersections of
tropical hypersurfaces. Tropical equilibrations with the same set of dominant
monomials define a branch or equivalence class. Minimal branches are
particularly interesting as they describe the simplest states of the reaction
network. We provide a method to compute the number of minimal branches and to
find representative tropical equilibrations for each branch.Comment: Proceedings Computer Algebra in Scientific Computing CASC 201
A geometric method for model reduction of biochemical networks with polynomial rate functions
Model reduction of biochemical networks relies on the knowledge of slow and
fast variables. We provide a geometric method, based on the Newton polytope, to
identify slow variables of a biochemical network with polynomial rate
functions. The gist of the method is the notion of tropical equilibration that
provides approximate descriptions of slow invariant manifolds. Compared to
extant numerical algorithms such as the intrinsic low dimensional manifold
method, our approach is symbolic and utilizes orders of magnitude instead of
precise values of the model parameters. Application of this method to a large
collection of biochemical network models supports the idea that the number of
dynamical variables in minimal models of cell physiology can be small, in spite
of the large number of molecular regulatory actors
Asymptotology of Chemical Reaction Networks
The concept of the limiting step is extended to the asymptotology of
multiscale reaction networks. Complete theory for linear networks with well
separated reaction rate constants is developed. We present algorithms for
explicit approximations of eigenvalues and eigenvectors of kinetic matrix.
Accuracy of estimates is proven. Performance of the algorithms is demonstrated
on simple examples. Application of algorithms to nonlinear systems is
discussed.Comment: 23 pages, 8 figures, 84 refs, Corrected Journal Versio
Dynamic optimization of metabolic networks coupled with gene expression
The regulation of metabolic activity by tuning enzyme expression levels is
crucial to sustain cellular growth in changing environments. Metabolic networks
are often studied at steady state using constraint-based models and
optimization techniques. However, metabolic adaptations driven by changes in
gene expression cannot be analyzed by steady state models, as these do not
account for temporal changes in biomass composition. Here we present a dynamic
optimization framework that integrates the metabolic network with the dynamics
of biomass production and composition, explicitly taking into account enzyme
production costs and enzymatic capacity. In contrast to the established dynamic
flux balance analysis, our approach allows predicting dynamic changes in both
the metabolic fluxes and the biomass composition during metabolic adaptations.
We applied our algorithm in two case studies: a minimal nutrient uptake
network, and an abstraction of core metabolic processes in bacteria. In the
minimal model, we show that the optimized uptake rates reproduce the empirical
Monod growth for bacterial cultures. For the network of core metabolic
processes, the dynamic optimization algorithm predicted commonly observed
metabolic adaptations, such as a diauxic switch with a preference ranking for
different nutrients, re-utilization of waste products after depletion of the
original substrate, and metabolic adaptation to an impending nutrient
depletion. These examples illustrate how dynamic adaptations of enzyme
expression can be predicted solely from an optimization principle
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