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

Breaking Out of Poverty Traps:Internal Migration and Interregional Convergence in Russia

We study barriers to labor mobility using panel data on gross region-to-region migration flows in Russia in 1996–2010. Using both parametric and semiparametric methods and controlling for region-to-region pairwise fixed effects, we find a non-monotonic relationship between income and migration. In richer regions, higher incomes result in lower migration outflows. However, in the poorest regions, an increase in incomes results in higher emigration. This is consistent with the presence of geographical poverty traps: potential migrants want to leave the poor regions but cannot afford to move. We also show that economic growth and financial development have allowed most Russian regions to grow out of poverty traps bringing down interregional differentials of wages, incomes and unemployment rates

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Algorithm for identification of piecewise smooth hybrid systems; application to eukaryotic cel