scholarly journals Enzyme-sharing as a cause of multi-stationarity in signalling systems

2011 ◽  
Vol 9 (71) ◽  
pp. 1224-1232 ◽  
Author(s):  
Elisenda Feliu ◽  
Carsten Wiuf
Keyword(s):  
Steady State ◽  
Chemical Species ◽  
Steady States ◽  
Mass Action ◽  
Signalling Pathways ◽  
Necessary Condition ◽  
Full Generality ◽  
Mass Action Kinetics ◽  
Cellular Decision Making ◽  
The Stability

Multi-stationarity in biological systems is a mechanism of cellular decision-making. In particular, signalling pathways regulated by protein phosphorylation display features that facilitate a variety of responses to different biological inputs. The features that lead to multi-stationarity are of particular interest to determine, as well as the stability, properties of the steady states. In this paper, we determine conditions for the emergence of multi-stationarity in small motifs without feedback that repeatedly occur in signalling pathways. We derive an explicit mathematical relationship φ between the concentration of a chemical species at steady state and a conserved quantity of the system such as the total amount of substrate available. We show that φ determines the number of steady states and provides a necessary condition for a steady state to be stable—that is, to be biologically attainable. Further, we identify characteristics of the motifs that lead to multi-stationarity, and extend the view that multi-stationarity in signalling pathways arises from multi-site phosphorylation. Our approach relies on mass-action kinetics, and the conclusions are drawn in full generality without resorting to simulations or random generation of parameters. The approach is extensible to other systems.

The Determination of Multiple Steady States in Two Families of Biological Systems

2004 ◽  
Vol 59 (3) ◽  
pp. 136-146
Author(s):  
Guo-Syong Chuang ◽  
Pang-Yen Ho ◽  
Hsing-Ya Li
Keyword(s):  
Steady State ◽  
Transport Model ◽  
Biological Systems ◽  
Linear Equations ◽  
Steady States ◽  
Metabolic Energy ◽  
Mass Action ◽  
Multiple Steady States ◽  
Mass Action Kinetics ◽  
Subnetwork Analysis

The capacity of computational multiple steady states in two biological systems are determined by the Deficiency One Algorithm and the Subnetwork Analysis. One is a bacterial glycolysis model involving the generation of ATP, and the other one is an active membrane transport model, which is performed by pump proteins coupled to a source of metabolic energy. Mass action kinetics, is assumed and both models consist of eight coupled non-linear equations. A set of rate constants and two corresponding steady states are computed. The phenomena of bistability and hysteresis are discussed. The bifurcation of multiple steady states is also displayed. A signature of multiplicity is derived, which can be applied to mechanism identifications if steady state concentrations for some species are measured. The capacity of steady state multiplicity is extended to their families of reaction networks.

scholarly journals Training an asymmetric signal perceptron through reinforcement in an artificial chemistry

2014 ◽  
Vol 11 (93) ◽  
pp. 20131100 ◽  
Author(s):  
Peter Banda ◽  
Christof Teuscher ◽  
Darko Stefanovic
Keyword(s):  
State Of The Art ◽  
Chemical Species ◽  
Mass Action ◽  
Mass Action Kinetics ◽  
Dna Strand Displacement ◽  
Autonomous Adaptation ◽  
Biochemical Systems ◽  
Dna Strand ◽  
Logic Functions ◽  
Application Specific

State-of-the-art biochemical systems for medical applications and chemical computing are application-specific and cannot be reprogrammed or trained once fabricated. The implementation of adaptive biochemical systems that would offer flexibility through programmability and autonomous adaptation faces major challenges because of the large number of required chemical species as well as the timing-sensitive feedback loops required for learning. In this paper, we begin addressing these challenges with a novel chemical perceptron that can solve all 14 linearly separable logic functions. The system performs asymmetric chemical arithmetic, learns through reinforcement and supports both Michaelis–Menten as well as mass-action kinetics. To enable cascading of the chemical perceptrons, we introduce thresholds that amplify the outputs. The simplicity of our model makes an actual wet implementation, in particular by DNA-strand displacement, possible.

On general transformations and variational principles for the magnetohydrodynamics of ideal fluids. Part 4. Generalized isovorticity principle for three-dimensional flows

1999 ◽  
Vol 390 ◽  
pp. 127-150 ◽  
Author(s):  
V. A. VLADIMIROV ◽  
H. K. MOFFATT ◽  
K. I. ILIN
Keyword(s):  
Steady State ◽  
Variational Principles ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Steady States ◽  
Mhd Equations ◽  
Quadratic Functional ◽  
The First Variation ◽  
The Stability ◽  
And Variational Principles

The equations of magnetohydrodynamics (MHD) of an ideal fluid have two families of topological invariants: the magnetic helicity invariants and the cross-helicity invariants. It is first shown that these invariants define a natural foliation (described as isomagnetovortical, or imv for short) in the function space in which solutions {u(x, t), h(x, t)} of the MHD equations reside. A relaxation process is constructed whereby total energy (magnetic plus kinetic) decreases on an imv folium (all magnetic and cross-helicity invariants being thus conserved). The energy has a positive lower bound determined by the global cross-helicity, and it is thus shown that a steady state exists having the (arbitrarily) prescribed families of magnetic and cross-helicity invariants.The stability of such steady states is considered by an appropriate generalization of (Arnold) energy techniques. The first variation of energy on the imv folium is shown to vanish, and the second variation δ2E is constructed. It is shown that δ2E is a quadratic functional of the first-order variations δ1u, δ1h of u and h (from a steady state U(x), H(x)), and that δ2E is an invariant of the linearized MHD equations. Linear stability is then assured provided δ2E is either positive-definite or negative-definite for all imv perturbations. It is shown that the results may be equivalently obtained through consideration of the frozen-in ‘modified’ vorticity field introduced in Part 1 of this series.Finally, the general stability criterion is applied to a variety of classes of steady states {U(x), H(x)}, and new sufficient conditions for stability to three-dimensional imv perturbations are obtained.

scholarly journals A structural property for reduction of biochemical networks

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Anika Küken ◽  
Philipp Wendering ◽  
Damoun Langary ◽  
Zoran Nikoloski
Keyword(s):  
Steady State ◽  
Model Reduction ◽  
Structural Property ◽  
Large Scale ◽  
Biochemical Networks ◽  
Mass Action ◽  
Mass Action Kinetics ◽  
Reduced Models ◽  
Metabolic Phenotypes ◽  
Genome Scale

AbstractLarge-scale biochemical models are of increasing sizes due to the consideration of interacting organisms and tissues. Model reduction approaches that preserve the flux phenotypes can simplify the analysis and predictions of steady-state metabolic phenotypes. However, existing approaches either restrict functionality of reduced models or do not lead to significant decreases in the number of modelled metabolites. Here, we introduce an approach for model reduction based on the structural property of balancing of complexes that preserves the steady-state fluxes supported by the network and can be efficiently determined at genome scale. Using two large-scale mass-action kinetic models of Escherichia coli, we show that our approach results in a substantial reduction of 99% of metabolites. Applications to genome-scale metabolic models across kingdoms of life result in up to 55% and 85% reduction in the number of metabolites when arbitrary and mass-action kinetics is assumed, respectively. We also show that predictions of the specific growth rate from the reduced models match those based on the original models. Since steady-state flux phenotypes from the original model are preserved in the reduced, the approach paves the way for analysing other metabolic phenotypes in large-scale biochemical networks.

Steady-State Bifurcation for a Biological Depletion Model

2016 ◽  
Vol 26 (04) ◽  
pp. 1650066 ◽  
Author(s):  
Yan’e Wang ◽  
Jianhua Wu ◽  
Yunfeng Jia
Keyword(s):  
Steady State ◽  
Bifurcation Theory ◽  
Steady States ◽  
Two Dimensional ◽  
Implicit Function ◽  
Flux Boundary Condition ◽  
One Dimensional ◽  
Steady State Bifurcation ◽  
The Stability ◽  
Theoretical Results

A two-species biological depletion model in a bounded domain is investigated in which one species is a substrate and the other is an activator. Firstly, under the no-flux boundary condition, the asymptotic stability of constant steady-states is discussed. Secondly, by viewing the feed rate of the substrate as a parameter, the steady-state bifurcations from constant steady-states are analyzed both in one-dimensional kernel case and in two-dimensional kernel case. Finally, numerical simulations are presented to illustrate our theoretical results. The main tools adopted here include the stability theory, the bifurcation theory, the techniques of space decomposition and the implicit function theorem.

scholarly journals On the small redshift limit of steady states of the spherically symmetric Einstein–Vlasov system and their stability

2015 ◽  
Vol 159 (3) ◽  
pp. 529-546 ◽  
Author(s):  
MAHIR HADŽIĆ ◽  
GERHARD REIN
Keyword(s):  
Steady State ◽  
Steady States ◽  
Spherically Symmetric ◽  
Second Variation ◽  
Speed Of Light ◽  
Limiting Behavior ◽  
Adm Mass ◽  
The Stability ◽  
The Individual ◽  
The Second Variation

AbstractFamilies of steady states of the spherically symmetric Einstein–Vlasov system are constructed, which are parametrised by the central redshift. It is shown that as the central redshift tends to zero, the states in such a family are well approximated by a steady state of the Vlasov–Poisson system, i.e., a Newtonian limit is established where the speed of light is kept constant as it should be and the limiting behavior is analysed in terms of a parameter which is tied to the physical properties of the individual solutions. This result is then used to investigate the stability properties of the relativistic steady states with small redshift parameter in the spirit of recent work by the same authors, i.e., the second variation of the ADM mass about such a steady state is shown to be positive definite on a suitable class of states.

CRNT4SBML: a Python package for the detection of bistability in biochemical reaction networks

2020 ◽  
Vol 36 (12) ◽  
pp. 3922-3924
Author(s):  
Brandon C Reyes ◽  
Irene Otero-Muras ◽  
Michael T Shuen ◽  
Alexandre M Tartakovsky ◽  
Vladislav A Petyuk
Keyword(s):  
Bifurcation Diagram ◽  
Steady States ◽  
Mass Action ◽  
Biochemical Reaction ◽  
Reaction Networks ◽  
Biological Pathway ◽  
Biochemical Reaction Networks ◽  
Mass Action Kinetics ◽  
Bifurcation Points ◽  
Python Package

Abstract Motivation Signaling pathways capable of switching between two states are ubiquitous within living organisms. They provide the cells with the means to produce reversible or irreversible decisions. Switch-like behavior of biological systems is realized through biochemical reaction networks capable of having two or more distinct steady states, which are dependent on initial conditions. Investigation of whether a certain signaling pathway can confer bistability involves a substantial amount of hypothesis testing. The cost of direct experimental testing can be prohibitive. Therefore, constraining the hypothesis space is highly beneficial. One such methodology is based on chemical reaction network theory (CRNT), which uses computational techniques to rule out pathways that are not capable of bistability regardless of kinetic constant values and molecule concentrations. Although useful, these methods are complicated from both pure and computational mathematics perspectives. Thus, their adoption is very limited amongst biologists. Results We brought CRNT approaches closer to experimental biologists by automating all the necessary steps in CRNT4SMBL. The input is based on systems biology markup language (SBML) format, which is the community standard for biological pathway communication. The tool parses SBML and derives C-graph representations of the biological pathway with mass action kinetics. Next steps involve an efficient search for potential saddle-node bifurcation points using an optimization technique. This type of bifurcation is important as it has the potential of acting as a switching point between two steady states. Finally, if any bifurcation points are present, continuation analysis with respect to a user-defined parameter extends the steady state branches and generates a bifurcation diagram. Presence of an S-shaped bifurcation diagram indicates that the pathway acts as a bistable switch for the given optimization parameters. Availability and implementation CRNT4SBML is available via the Python Package Index. The documentation can be found at https://crnt4sbml.readthedocs.io. CRNT4SBML is licensed under the Apache Software License 2.0.

THE STABILITY OF SPATIALLY NONHOMOGENEOUS STEADY STATES IN A TUBULAR CHEMICAL REACTOR

1993 ◽  
Vol 03 (06) ◽  
pp. 1477-1486
Author(s):  
JAMES M. ROTENBERRY ◽  
ANTONMARIA A. MINZONI
Keyword(s):  
Steady State ◽  
Peclet Number ◽  
Chemical Reactor ◽  
Higher Order ◽  
Steady States ◽  
Complex Conjugate ◽  
Heat Of Reaction ◽  
Péclet Number ◽  
The Stability ◽  
Steady State Solutions

We study the axial heat and mass transfer in a highly diffusive tubular chemical reactor in which a simple reaction is occurring. The steady state solutions of the governing equations are studied using matched asymptotic expansions, the theory of dynamical systems, and by calculating the solutions numerically. In particular, the effect of varying the Peclet and Damköhler numbers (P and D) is investigated. A simple expression for the approximate location of the transition layer for large Peclet number is derived and its accuracy tested against the numerical solution. The stability of the steady states is examined by calculating the eigenvalues and eigenfunctions of the linearized equations. It is shown that a Hopf bifurcation of the CSTR model (i.e., the limit as the P approaches zero) can be continued up to order 1 in the Peclet number. Furthermore, it is shown numerically that for appropriate values of the Peclet number, the Damköhler number, and B (the heat of reaction) these Hopf bifurcations merge with the limit points of an "S–shaped" bifurcation curve in a higher order singularity controlled by the Bogdanov–Takens normal form. Consequently, there must exist a finite amplitude, nonuniform, stable periodic solution for parameter values near this singularity. The existence of higher order degeneracies is also explored. In particular, it is shown for D ≪ 1 that no value of P exists where two pairs of complex conjugate eigenvalues of the steady state solutions can cross the imaginary axis simultaneously.

scholarly journals Dynamical modelling of the heat shock response inChlamydomonas reinhardtii

2016 ◽  
Author(s):  
Stefano Magni ◽  
Antonella Succurro ◽  
Alexander Skupin ◽  
Oliver Ebenhöh
Keyword(s):  
Steady State ◽  
Heat Shock ◽  
Heat Shock Response ◽  
Model Organism ◽  
Mass Action ◽  
Large Temperature ◽  
System Response ◽  
Yield Reduction ◽  
Mass Action Kinetics ◽  
Shock Response

AbstractGlobal warming is exposing plants to more frequent heat stress, with consequent crop yield reduction. Organisms exposed to large temperature increases protect themselves typically with a heat shock response (HSR). To study the HSR in photosynthetic organisms we present here a data driven mathematical model describing the dynamics of the HSR in the model organismChlamydomonas reinhartii. Temperature variations are sensed by the accumulation of unfolded proteins, which activates the synthesis of heat shock proteins (HSP) mediated by the heat shock transcription factor HSF1. Our dynamical model employs a system of ordinary differential equations mostly based on mass-action kinetics to study the time evolution of the involved species. The signalling network is inferred from data in the literature, and the multiple experimental data-sets available are used to calibrate the model, which allows to reproduce their qualitative behaviour. With this model we show the ability of the system to adapt to temperatures higher than usual during heat shocks longer than three hours by shifting to a new steady state. We study how the steady state concentrations depend on the temperature at which the steady state is reached. We systematically investigate how the accumulation of HSPs depends on the combination of temperature and duration of the heat shock. We finally investigate the system response to a smooth variation in temperature simulating a hot day.

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