text: This work investigates the emergence of oscillations in one of the simplest cellular signaling networks exhibiting oscillations, namely, the dual-site phosphorylation and dephosphorylation network (futile cycle), in which the mechanism for phosphorylation is processive while the one for dephosphorylation is distributive (or vice-versa). The fact that this network yields oscillations was shown recently by Suwanmajo and Krishnan. Our results, which significantly extend their analyses, are as follows. First, in the three-dimensional space of total amounts, the border between systems with a stable versus unstable steady state is a surface defined by the vanishing of a single Hurwitz determinant. Second, this surface consists generically of simple Hopf bifurcations. Next, simulations suggest that when the steady state is unstable, oscillations are the norm. Finally, the emergence of oscillations via a Hopf bifurcation is enabled by the catalytic and association constants of the distributive part of the mechanism: if these rate constants satisfy two inequalities, then the system generically admits a Hopf bifurcation. Our proofs are enabled by the Routh-Hurwitz criterion, a Hopf-bifurcation criterion due to Yang, and a monomial parametrization of steady states.
  @cite_37: Post-translational modification of proteins plays a central role in cellular regulation but its study has been hampered by the exponential increase in substrate modification forms (“modforms”) with increasing numbers of sites. We consider here biochemical networks arising from post-translational modification under mass-action kinetics, allowing for multiple substrates, having different types of modification (phosphorylation, methylation, acetylation, etc.) on multiple sites, acted upon by multiple forward and reverse enzymes (in total number L), using general enzymatic mechanisms. These assumptions are substantially more general than in previous studies. We show that the steady-state modform concentrations constitute an algebraic variety that can be parameterised by rational functions of the L free enzyme concentrations, with coefficients which are rational functions of the rate constants. The parameterisation allows steady states to be calculated by solving L algebraic equations, a dramatic reduction compared to simulating an exponentially large number of differential equations. This complexity collapse enables analysis in contexts that were previously intractable and leads to biological predictions that we review. Our results lay a foundation for the systems biology of post-translational modification and suggest deeper connections between biochemical networks and algebraic geometry.
  @cite_22: 
  @cite_15: 
  @cite_10: Prompted by the recent growing evidence of oscillatory behavior involving MAPK cascades we present a systematic approach of analyzing models and elucidating the nature of biochemical oscillations based on reaction network theory. In particular, we formulate a minimal biochemically consistent mass action subnetwork of the Huang-Ferrell model of the MAPK signalling that provides an oscillatory response when a parameter controlling the activation of the top-tier kinase is varied. Such dynamics are either intertwined with or separated from the earlier found bistable hysteretic behavior in this model. Using the theory of stability of stoichiometric networks, we reduce the original MAPK model, convert kinetic to convex parameters and examine those properties of the minimal subnetwork that underlie the oscillatory dynamics. We also use the methods of classification of chemical oscillatory networks to explain the rhythmic behavior in physicochemical terms, i.e., we identify of the role of individual biochemical species in positive and negative feedback loops and describe their coordinated action leading to oscillations. Our approach provides an insight into dynamics without the necessity of knowing rate coefficients and thus is useful prior the statistical evaluation of parameters.
  @cite_39: Dual phosphorylation of proteins is a principal component of intracellular signalling. Bistability is considered an important property of such systems and its origin is not yet completely understood. Theoretical studies have established parameter values for multistationarity and bistability for many types of proteins. However, up to now no formal criterion linking multistationarity and bistability to the parameter values characterizing dual phosphorylation has been established. Deciding whether an unclassified protein has the capacity for bistability, therefore requires careful numerical studies. Here, we present two general algebraic conditions in the form of inequalities. The first employs the catalytic constants, and if satisfied guarantees multistationarity (and hence the potential for bistability). The second involves the catalytic and Michaelis constants, and if satisfied guarantees uniqueness of steady states (and hence absence of bistability). Our method also allows for the direct computation of the total concentration values such that multistationarity occurs. Applying our results yields insights into the emergence of bistability in the ERK–MEK–MKP system that previously required a delicate numerical effort. Our algebraic conditions present a practical way to determine the capacity for bistability and hence will be a useful tool for examining the origin of bistability in many models containing dual phosphorylation.
  @cite_19: We present conditions which guarantee a parametrization of the set of positive equilibria of a generalized mass-action system. Our main results state that (i) if the underlying generalized chemical reaction network has an effective deficiency of zero, then the set of positive equilibria coincides with the parametrized set of complex-balanced equilibria and (ii) if the network is weakly reversible and has a kinetic deficiency of zero, then the equilibrium set is nonempty and has a positive, typically rational, parametrization. Via the method of network translation, we apply our results to classical mass-action systems studied in the biochemical literature, including the EnvZ-OmpR and shuttled WNT signaling pathways. A parametrization of the set of positive equilibria of a (generalized) mass-action system is often a prerequisite for the study of multistationarity and allows an easy check for the occurrence of absolute concentration robustness (ACR), as we demonstrate for the EnvZ-OmpR pathway.
  @cite_27: Recent proteomic data indicate that a majority of the phosphorylated proteins in a eucaryotic cell contain multiple sites of phosphorylation. In many signaling events, a single kinase phosphorylates multiple sites on a target protein. Processive phosphorylation occurs when a protein kinase binds once to a substrate and phosphorylates all of the available sites before dissociating. In this review, we discuss examples of processive phosphorylation by serine threonine kinases and tyrosine kinases. We describe current experimental approaches for distinguishing processive from non-processive phosphorylation. Finally, we contrast the biological situations that are suited to regulation by processive and non-processive phosphorylation.
  @cite_23: We give necessary and sufficient conditions in terms of sign vectors for the injectivity of families of polynomial maps with arbitrary real exponents defined on the positive orthant. Our work relates and extends existing injectivity conditions expressed in terms of Jacobian matrices and determinants. In the context of chemical reaction networks with power-law kinetics, our results can be used to preclude as well as to guarantee multiple positive steady states. In the context of real algebraic geometry, our work recognizes a prior result of Craciun, Garcia-Puente, and Sottile, together with work of two of the authors, as the first partial multivariate generalization of the classical Descartes' rule, which bounds the number of positive real roots of a univariate real polynomial in terms of the number of sign variations of its coefficients.
  @cite_2: Abstract 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 results showed that these networks can have switch-like behavior, bistability and oscillations. In this work, we consider three representative ERK networks, one with a negative feedback loop, which present a binomial steady state ideal under mass-action kinetics. We therefore apply the theoretical result present in [27] to find a set of rate constants that allow two significantly different stable steady states in the same stoichiometric compatibility class for each network. Our approach makes it possible to study certain aspects of the system, such as multistationarity, without relying on simulation, since we do not assume a priori any constant but the topology of the network. As the performed analysis is general it could be applied to many other important biochemical networks.
  @cite_31: Bifurcation theory is one of the most widely used approaches for analysis of dynamical behaviour of chemical and biochemical reaction networks. Some of the interesting qualitative behaviour that are analyzed are oscillations and bistability (a situation where a system has at least two coexisting stable equilibria). Both phenomena have been identified as central features of many biological and biochemical systems. This paper, using the theory of stoichiometric network analysis (SNA) and notions from algebraic geometry, presents sufficient conditions for a reaction network to display bifurcations associated with these phenomena. The advantage of these conditions is that they impose fewer algebraic conditions on model parameters than conditions associated with standard bifurcation theorems. To derive the new conditions, a coordinate transformation will be made that will guarantee the existence of branches of positive equilibria in the system. This is particularly useful in mathematical biology, where only positive variable values are considered to be meaningful. The first part of the paper will be an extended introduction to SNA and algebraic geometry-related methods which are used in the coordinate transformation and set up of the theorems. In the second part of the paper we will focus on the derivation of bifurcation conditions using SNA and algebraic geometry. Conditions will be derived for three bifurcations: the saddle-node bifurcation, a simple branching point, both linked to bistability, and a simple Hopf bifurcation. The latter is linked to oscillatory behaviour. The conditions derived are sufficient and they extend earlier results from stoichiometric network analysis as can be found in (Aguda and Clarke in J Chem Phys 87:3461-3470, 1987; Clarke and Jiang in J Chem Phys 99:4464-4476, 1993; in J Symb Comput 40:1361-1382, 2005). In these papers some necessary conditions for two of these bifurcations were given. A set of examples will illustrate that algebraic conditions arising from given sufficient bifurcation conditions are not more difficult to interpret nor harder to calculate than those arising from necessary bifurcation conditions. Hence an increasing amount of information is gained at no extra computational cost. The theory can also be used in a second step for a systematic bifurcation analysis of larger reaction networks.
  @cite_13: Mathematical modelling has become an established tool for studying the dynamics of biological systems. Current applications range from building models that reproduce quantitative data to identifying systems with predefined qualitative features, such as switching behaviour, bistability or oscillations. Mathematically, the latter question amounts to identifying parameter values associated with a given qualitative feature. We introduce a procedure to partition the parameter space of a parameterized system of ordinary differential equations into regions for which the system has a unique or multiple equilibria. The procedure is based on the computation of the Brouwer degree, and it creates a multivariate polynomial with parameter depending coefficients. The signs of the coefficients determine parameter regions with and without multistationarity. A particular strength of the procedure is the avoidance of numerical analysis and parameter sampling. The procedure consists of a number of steps. Each of these steps might be addressed algorithmically using various computer programs and available software, or manually. We demonstrate our procedure on several models of gene transcription and cell signalling, and show that in many cases we obtain a complete partitioning of the parameter space with respect to multistationarity.
  @cite_20: Many biochemical and industrial applications involve complicated networks of simultaneously occurring chemical reactions. Under the assumption of mass action kinetics, the dynamics of these chemical reaction networks are governed by systems of polynomial ordinary differential equations. The steady states of these mass action systems have been analyzed via a variety of techniques, including stoichiometric network analysis, deficiency theory, and algebraic techniques (e.g., Grobner bases). In this paper, we present a novel method for characterizing the steady states of mass action systems. Our method explicitly links a network’s capacity to permit a particular class of steady states, called toric steady states, to topological properties of a generalized network called a translated chemical reaction network. These networks share their reaction vectors with their source network but are permitted to have different complex stoichiometries and different network topologies. We apply the results to examples drawn from the biochemical literature.
summary: Our work joins a growing number of works that harness steady-state parametrizations. Such results include criteria for when such parametrizations exist @cite_19 @cite_37 and methods for using them to determine whether a network is multistationary @cite_20 @cite_23 @cite_27 @cite_2 . Going further, steady-state parametrizations can also be used to find a witness to multistationarity or even the precise parameter regions that yield multistationarity @cite_13 @cite_39 . In this work, we use a steady-state parametrization in a novel way: to study oscillations via Hopf bifurcations. (Our approach is similar in spirit to using Clarke's convex parameters together with a Hopf-bifurcation criterion @cite_31 @cite_15 @cite_22 @cite_10 ).