We address several questions in reduced versus extended networks via the elimination or addition of intermediate complexes in the framework of chemical reaction networks with mass-action kinetics. We clarify and extend advances in the literature concerning multistationarity in this context. We establish general results about MESSI systems, which we use to compute the circuits of multistationarity for significant biochemical networks.
Biochemical mechanisms with mass action kinetics are often modeled by systems of polynomial differential equations (DE). Determining directly if the DE system has multiple equilibria (multistationarity) is difficult for realistic systems, since they are large, nonlinear and contain many unknown parameters. Mass action biochemical mechanisms can be represented by a directed bipartite graph with species and reaction nodes. Graph-theoretic methods can then be used to assess the potential of a given biochemical mechanism for multistationarity by identifying structures in the bipartite graph referr…
Given a real sparse polynomial system, we present a general framework to find explicit coefficients for which the system has more than one positive solution, based on the recent article by Bihan, Santos and Spaenlehauer. We apply this approach to find explicit reaction rate constants and total conservation constants in biochemical reaction networks for which the associated dynamical system is multistationary.
Many biological, physical, and social interactions have a particular dependence on where they take place. In living cells, protein movement between the nucleus and cytoplasm affects cellular response (i.e., proteins must be present in the nucleus to regulate their target genes). Here we use recent developments from dynamical systems and chemical reaction network theory to identify and characterize the key-role of the spatial organization of eukaryotic cells in cellular information processing. In particular the existence of distinct compartments plays a pivotal role in whether a system is capab…
For reaction networks arising in systems biology, the capacity for two or more steady states, that is, multistationarity, is an important property that underlies biochemical switches. Another property receiving much attention recently is absolute concentration robustness (ACR), which means that some species concentration is the same at all positive steady states. In this work, we investigate the prevalence of each property while paying close attention to when the properties occur together. Specifically, we consider a stochastic block framework for generating random networks, and prove edge-pro…
Multisite phosphorylation is a signaling mechanism well known to give rise to multiple steady states, a property termed multistationarity. When phosphorylation occurs in a sequential and distributive manner, we obtain a family of networks indexed by the number of phosphorylation sites $n$. This work addresses the problem of understanding the parameter region where this family of networks displays multistationarity, by focusing on the projection of this region onto the set of kinetic parameters. The problem is phrased in the context of real algebraic geometry and reduced to studying whether a p…
Mitogen-activated protein kinase (MAPK) signaling pathways play an essential role in the transduction of environmental stimuli to the nucleus, thereby regulating a variety of cellular processes, including cell proliferation, differentiation and programmed cell death. The components of the MAPK extracellular activated protein kinase (ERK) cascade represent attractive targets for cancer therapy as their aberrant activation is a frequent event among highly prevalent human cancers. MAPK networks are a model for computational simulation, mostly using Ordinary and Partial Differential Equations. Key…
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 …
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…
We apply a Gaussian variational approximation to model reduction in large biochemical networks of unary and binary reactions. We focus on a small subset of variables (subnetwork) of interest, e.g. because they are accessible experimentally, embedded in a larger network (bulk). The key goal is to write dynamical equations reduced to the subnetwork but still retaining the effects of the bulk. As a result, the subnetwork-reduced dynamics contains a memory term and an extrinsic noise term with non-trivial temporal correlations. We first derive expressions for this memory and noise in the linearize…
The application of the quasi-steady-state approximation to the Michaelis-Menten reaction embedded in large open chemical reaction networks is a popular model reduction technique in deterministic and stochastic simulations of biochemical reactions inside cells. It is frequently assumed that the predictions of the reduced master equations obtained using the stochastic quasi-steady-state approach are in very good agreement with the predictions of the full master equations, provided the conditions for the validity of the deterministic quasi-steady-state approximation are fulfilled. We here use the…
We develop an approach aimed at optimizing the parameters of a network of biochemical logic gates for reduction of the "analog" noise buildup. Experiments for three coupled enzymatic AND gates are reported, illustrating our procedure. Specifically, starch - one of the controlled network inputs - is converted to maltose by beta-amylase. With the use of phosphate (another controlled input), maltose phosphorylase then produces glucose. Finally, nicotinamide adenine dinucleotide (NAD+) - the third controlled input - is reduced under the action of glucose dehydrogenase to yield the optically detect…
Biochemical molecules interact through modification and binding reactions, giving raise to a combinatorial number of possible biochemical species. The time-dependent evolution of concentrations of the species is commonly described by a system of coupled ordinary differential equations (ODEs). However, the analysis of such high-dimensional, non-linear system of equations is often computationally expensive and even prohibitive in practice. The major challenge towards reducing such models is providing the guarantees as to how the solution of the reduced model relates to that of the original model…
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…
The quasi-steady state approximation and time-scale separation are commonly applied methods to simplify models of biochemical reaction networks based on ordinary differential equations (ODEs). The concentrations of the "fast" species are assumed effectively to be at steady state with respect to the "slow" species. Under this assumption the steady state equations can be used to eliminate the "fast" variables and a new ODE system with only the slow species can be obtained. We interpret a reduced system obtained by time-scale separation as the ODE system arising from a unique reaction network, by…
The Michaelis-Menten equation has played a central role in our understanding of biochemical processes. It has long been understood how this equation approximates the dynamics of irreversible enzymatic reactions. However, a similar approximation in the case of networks, where the product of one reaction can act as an enzyme in another, has not been fully developed. Here we rigorously derive such an approximation in a class of coupled enzymatic networks where the individual interactions are of Michaelis-Menten type. We show that the sufficient conditions for the validity of the total quasi stead…
Biochemical computing attempts to process information with biomolecules and biological objects. In this work we review our results on analysis and optimization of single biochemical logic gates based on enzymatic reactions, and a network of three gates, for reduction of the "analog" noise buildup. For a single gate, optimization is achieved by analyzing the enzymatic reactions within a framework of kinetic equations. We demonstrate that using co-substrates with much smaller affinities than the primary substrate, a negligible increase in the noise output from the logic gate is obtained as compa…
We studied in silico effects of structural stress in Mycoplasma pneumoniae, a genome-reduced model bacterial organism, by tracking the damage propagating on its metabolic network after a deleterious perturbation. First, we analyzed failure cascades spreading from individual reactions and pairs of reactions and compared the results to those in Staphylococcus aureus and Escherichia coli. To alert to the potential damage caused by the failure of individual reactions, we propose a generic predictor based on local information that identifies target reactions for structural vulnerability. With respe…
