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Register a new userWeakly reversible mass-action systems with infinitely many positive steady states
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We show that weakly reversible mass-action systems can have a continuum of positive steady states, coming from the zeroes of a multivariate polynomial. Moreover, the same is true of systems whose underlying reaction network is reversible and has a single connected component. In our construction, we relate operations on the reaction network to the multivariate polynomial occurring as a common factor in the system of differential equations.
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We present a model for continuous cell culture coupling intra-cellular metabolism to extracellular variables describing the state of the bioreactor, taking into account the growth capacity of the cell and the impact of toxic byproduct accumulation. We provide a method to determine the steady states of this system that is tractable for metabolic networks of arbitrary complexity. We demonstrate our approach in a toy model first, and then in a genome-scale metabolic network of the Chinese hamster ovary cell line, obtaining results that are in qualitative agreement with experimental observations. More importantly, we derive a number of consequences from the model that are independent of parameter values. First, that the ratio between cell density and dilution rate is an ideal control parameter to fix a steady state with desired metabolic properties invariant across perfusion systems. This conclusion is robust even in the presence of multi-stability, which is explained in our model by the negative feedback loop on cell growth due to toxic byproduct accumulation. Moreover, a complex landscape of steady states in continuous cell culture emerges from our simulations, including multiple metabolic switches, which also explain why cell-line and media benchmarks carried out in batch culture cannot be extrapolated to perfusion. On the other hand, we predict invariance laws between continuous cell cultures with different parameters. A practical consequence is that the chemostat is an ideal experimental model for large-scale high-density perfusion cultures, where the complex landscape of metabolic transitions is faithfully reproduced. Thus, in order to actually reflect the expected behavior in perfusion, performance benchmarks of cell-lines and culture media should be carried out in a chemostat.
Mass-action kinetics and its generalizations appear in mathematical models of (bio-)chemical reaction networks, population dynamics, and epidemiology. The dynamical systems arising from directed graphs are generally non-linear and difficult to analyze. One approach to studying them is to find conditions on the network which either imply or preclude certain dynamical properties. For example, a vertex-balanced steady state for a generalized mass-action system is a state where the net flux through every vertex of the graph is zero. In particular, such steady states admit a monomial parametrization. The problem of existence and uniqueness of vertex-balanced steady states can be reformulated in two different ways, one of which is related to Birchs theorem in statistics, and the other one to the bijectivity of generalized polynomial maps, similar to maps appearing in geometric modelling. We present a generalization of Birchs theorem, by providing a sufficient condition for the existence and uniqueness of vertex-balanced steady states.
The aim of this work is to make available to the community a large collection of mass-action reaction networks of a given size for further research. The set is limited to what can be computed on a modern multi-core desktop in reasonable time (< 20 days). We have currently generated over 47 million unique reaction networks. All currently generated sets of networks are available and as new sets are completed they will also be made available. Also provided are programs for translating them into different formats, along with documentation and examples. Source code and binaries for all the programs are included. These can be downloaded from (http://www.sys-bio.org/networkenumeration). This library of networks will allow for thorough studies of the reaction network space. Additionally, these methods serve as an example for future work on enumerating other types of biological networks, such as genetic regulatory networks and mass-action networks that include regulation.
In the early 1980s, S. T. Yau conjectured that any compact Riemannian three-manifold admits an infinite number of closed immersed minimal surfaces. We use min-max theory for the area functional to prove this conjecture in the positive Ricci curvature setting. More precisely, we show that every compact Riemannian manifold with positive Ricci curvature and dimension at most seven contains infinitely many smooth, closed, embedded minimal hypersurfaces. In the last section we mention some open problems related with the geometry of these minimal hypersurfaces.
The paper deals with an eigenvalue problems possessing infinitely many positive and negative eigenvalues. Inequalities for the smallest positive and the largest negative eigenvalues, which have the same properties as the fundamental frequency, are derived. The main question is whether or not the classical isoperimetric inequalities for the fundamental frequency of membranes hold in this case. The arguments are based on the harmonic transplantation for the global results and the shape derivatives (domain variations) for nearly circular domain.
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