141 research outputs found
Model reduction of biochemical reactions networks by tropical analysis methods
We discuss a method of approximate model reduction for networks of
biochemical reactions. This method can be applied to networks with polynomial
or rational reaction rates and whose parameters are given by their orders of
magnitude. In order to obtain reduced models we solve the problem of tropical
equilibration that is a system of equations in max-plus algebra. In the case of
networks with nonlinear fast cycles we have to solve the problem of tropical
equilibration at least twice, once for the initial system and a second time for
an extended system obtained by adding to the initial system the differential
equations satisfied by the conservation laws of the fast subsystem. The two
steps can be reiterated until the fast subsystem has no conservation laws
different from the ones of the full model. Our method can be used for formal
model reduction in computational systems biology
Tropical geometries and dynamics of biochemical networks. Application to hybrid cell cycle models
We use the Litvinov-Maslov correspondence principle to reduce and hybridize
networks of biochemical reactions. We apply this method to a cell cycle
oscillator model. The reduced and hybridized model can be used as a hybrid
model for the cell cycle. We also propose a practical recipe for detecting
quasi-equilibrium QE reactions and quasi-steady state QSS species in
biochemical models with rational rate functions and use this recipe for model
reduction. Interestingly, the QE/QSS invariant manifold of the smooth model and
the reduced dynamics along this manifold can be put into correspondence to the
tropical variety of the hybridization and to sliding modes along this variety,
respectivelyComment: conference SASB 2011, to be published in Electronic Notes in
Theoretical Computer Scienc
Tropicalization and tropical equilibration of chemical reactions
Systems biology uses large networks of biochemical reactions to model the
functioning of biological cells from the molecular to the cellular scale. The
dynamics of dissipative reaction networks with many well separated time scales
can be described as a sequence of successive equilibrations of different
subsets of variables of the system. Polynomial systems with separation are
equilibrated when at least two monomials, of opposite signs, have the same
order of magnitude and dominate the others. These equilibrations and the
corresponding truncated dynamics, obtained by eliminating the dominated terms,
find a natural formulation in tropical analysis and can be used for model
reduction.Comment: 13 pages, 1 figure, workshop Tropical-12, Moskow, August 26-31, 2012;
in press Contemporary Mathematic
Flexible and robust patterning by centralized gene networks
We consider networks with two types of nodes. The v-nodes, called centers,
are hyperconnected and interact one to another via many u-nodes, called
satellites. This centralized architecture, widespread in gene networks, realize
a bow-tie scheme and possesses interesting properties. Namely, this
organization creates feedback loops that are capable to generate any prescribed
patterning dynamics, chaotic or periodic, and create a number of equilibrium
states. We show that activation or silencing of a node can sharply switch the
network attractor, even if the activated or silenced node is weakly connected.
We distinguish between two dynamically different situations, "power of center"
(PC) when satellite response is fast and "satellite power" (SP) when center
response is fast. Using a simple network example we show that a centralized
network is more robust with respect to time dependent perturbations, in the PC
relative to the SP case. In theoretical molecular biology, this class of models
can be used to reveal a non-trivial relation between the architecture of
protein-DNA and protein-protein interaction networks and controllability of
space-time dynamics of cellular processes.Comment: 23 pages, Fundamenta Informaticae, in pres
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
Reduction of dynamical biochemical reaction networks in computational biology
Biochemical networks are used in computational biology, to model the static
and dynamical details of systems involved in cell signaling, metabolism, and
regulation of gene expression. Parametric and structural uncertainty, as well
as combinatorial explosion are strong obstacles against analyzing the dynamics
of large models of this type. Multi-scaleness is another property of these
networks, that can be used to get past some of these obstacles. Networks with
many well separated time scales, can be reduced to simpler networks, in a way
that depends only on the orders of magnitude and not on the exact values of the
kinetic parameters. The main idea used for such robust simplifications of
networks is the concept of dominance among model elements, allowing
hierarchical organization of these elements according to their effects on the
network dynamics. This concept finds a natural formulation in tropical
geometry. We revisit, in the light of these new ideas, the main approaches to
model reduction of reaction networks, such as quasi-steady state and
quasi-equilibrium approximations, and provide practical recipes for model
reduction of linear and nonlinear networks. We also discuss the application of
model reduction to backward pruning machine learning techniques
Protein synthesis driven by dynamical stochastic transcription
In this manuscript we propose a mathematical framework to couple
transcription and translation in which mRNA production is described by a set of
master equations while the dynamics of protein density is governed by a random
differential equation. The coupling between the two processes is given by a
stochastic perturbation whose statistics satisfies the master equations. In
this approach, from the knowledge of the analytical time dependent distribution
of mRNA number, we are able to calculate the dynamics of the probability
density of the protein population.Comment: 20 pages, 3 figure
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