In distributive phosphorylation catalytic constants enable non-trivial dynamics.

Ordered distributive double phosphorylation is a recurrent motif in intracellular signaling and control. It is either sequential (where the site phosphorylated last is dephosphorylated first) or cyclic (where the site phosphorylated first is dephosphorylated first). Sequential distributive double phosphorylation has been extensively studied and an inequality involving only the catalytic constants of kinase and phosphatase is known to be sufficient for multistationarity. As multistationarity is necessary for bistability it has been argued that these constants enable bistability. Here we show for cyclic distributive double phosphorylation that if its catalytic constants satisfy an analogous inequality, then Hopf bifurcations and hence sustained oscillations can occur. Hence we argue that in distributive double phosphorylation (sequential or distributive) the catalytic constants enable non-trivial dynamics. In fact, if the rate constant values in a network of cyclic distributive double phosphorylation satisfy this inequality, then a network of sequential distributive double phosphorylation with the same rate constant values will show multistationarity-albeit for different values of the total concentrations. For cyclic distributive double phosphorylation we further describe a procedure to generate rate constant values where Hopf bifurcations and hence sustained oscillations can occur. This may, for example, allow for an efficient sampling of oscillatory regions in parameter space. Our analysis is greatly simplified by the fact that it is possible to reduce the network of cyclic distributive double phosphorylation to what we call a network with a single extreme ray. We summarize key properties of these networks.

Dynamics of ERK regulation in the processive limit.

We consider a model of extracellular signal-regulated kinase regulation by dual-site phosphorylation and dephosphorylation, which exhibits bistability and oscillations, but loses these properties in the limit in which the mechanisms underlying phosphorylation and dephosphorylation become processive. Our results suggest that anywhere along the way to becoming processive, the model remains bistable and oscillatory. More precisely, in simplified versions of the model, precursors to bistability and oscillations (specifically, multistationarity and Hopf bifurcations, respectively) exist at all "processivity levels". Finally, we investigate whether bistability and oscillations can exist together.

On the existence of Hopf bifurcations in the sequential and distributive double phosphorylation cycle.

Protein phosphorylation cycles are important mechanisms of the post translational modification of a protein and as such an integral part of intracellular signaling and control. We consider the sequential phosphorylation and dephosphorylation of a protein at two binding sites. While it is known that proteins where phosphorylation is processive and dephosphorylation is distributive admit oscillations (for some value of the rate constants and total concentrations) it is not known whether or not this is the case if both phosphorylation and dephosphorylation are distributive. We study simplified mass action models of sequential and distributive phosphorylation and show that for each of those there do not exist rate constants and total concentrations where a Hopf bifurcation occurs. To arrive at this result we use convex parameters to parametrize the steady state and Hurwitz matrices.

Multistationarity in the Space of Total Concentrations for Systems that Admit a Monomial Parametrization.

We apply tools from real algebraic geometry to the problem of multistationarity of chemical reaction networks. A particular focus is on the case of reaction networks whose steady states admit a monomial parametrization. For such systems, we show that in the space of total concentrations multistationarity is scale invariant: If there is multistationarity for some value of the total concentrations, then there is multistationarity on the entire ray containing this value (possibly for different rate constants)-and vice versa. Moreover, for these networks it is possible to decide about multistationarity independent of the rate constants by formulating semi-algebraic conditions that involve only concentration variables. These conditions can easily be extended to include total concentrations. Hence, quantifier elimination may give new insights into multistationarity regions in the space of total concentrations. To demonstrate this, we show that for the distributive phosphorylation of a protein at two binding sites multistationarity is only possible if the total concentration of the substrate is larger than either the total concentration of the kinase or the total concentration of the phosphatase. This result is enabled by the chamber decomposition of the space of total concentrations from polyhedral geometry. Together with the corresponding sufficiency result of Bihan et al., this yields a characterization of multistationarity up to lower-dimensional regions.

Emergence of Oscillations in a Mixed-Mechanism Phosphorylation System.

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.

Dynamics of Posttranslational Modification Systems: Recent Progress and Future Directions.

Posttranslational modification of proteins is important for signal transduction, and hence significant effort has gone toward understanding how posttranslational modification networks process information. This involves, on the theory side, analyzing the dynamical systems arising from such networks. Which networks are, for instance, bistable? Which networks admit sustained oscillations? Which parameter values enable such behaviors? In this Biophysical Perspective, we highlight recent progress in this area and point out some important future directions. Along the way, we summarize several techniques for analyzing general networks, such as eliminating variables to obtain steady-state parameterizations, and harnessing results on how incorporating intermediates affects dynamics.

Identifying parameter regions for multistationarity.

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.

Challenges in horizontal model integration.

BACKGROUND: Systems Biology has motivated dynamic models of important intracellular processes at the pathway level, for example, in signal transduction and cell cycle control. To answer important biomedical questions, however, one has to go beyond the study of isolated pathways towards the joint study of interacting signaling pathways or the joint study of signal transduction and cell cycle control. Thereby the reuse of established models is preferable, as it will generally reduce the modeling effort and increase the acceptance of the combined model in the field. RESULTS: Obtaining a combined model can be challenging, especially if the submodels are large and/or come from different working groups (as is generally the case, when models stored in established repositories are used). To support this task, we describe a semi-automatic workflow based on established software tools. In particular, two frequent challenges are described: identification of the overlap and subsequent (re)parameterization of the integrated model. CONCLUSIONS: The reparameterization step is crucial, if the goal is to obtain a model that can reproduce the data explained by the individual models. For demonstration purposes we apply our workflow to integrate two signaling pathways (EGF and NGF) from the BioModels Database.

A global convergence result for processive multisite phosphorylation systems.

Multisite phosphorylation plays an important role in intracellular signaling. There has been much recent work aimed at understanding the dynamics of such systems when the phosphorylation/dephosphorylation mechanism is distributive, that is, when the binding of a substrate and an enzyme molecule results in the addition or removal of a single phosphate group and repeated binding therefore is required for multisite phosphorylation. In particular, such systems admit bistability. Here, we analyze a different class of multisite systems, in which the binding of a substrate and an enzyme molecule results in the addition or removal of phosphate groups at all phosphorylation sites, that is, we consider systems in which the mechanism is processive, rather than distributive. We show that in contrast to distributive systems, processive systems modeled with mass-action kinetics do not admit bistability and, moreover, exhibit rigid dynamics: each invariant set contains a unique equilibrium, which is a global attractor. Additionally, we obtain a monomial parametrization of the steady states. Our proofs rely on a technique of Johnston for using "translated" networks to study systems with "toric steady states," recently given sign conditions for the injectivity of polynomial maps, and a result from monotone systems theory due to Angeli and Sontag.

N-Site phosphorylation systems with 2n-1 steady states.

Multisite protein phosphorylation plays a prominent role in intracellular processes like signal transduction, cell-cycle control and nuclear signal integration. Many proteins are phosphorylated in a sequential and distributive way at more than one phosphorylation site. Mathematical models of n-site sequential distributive phosphorylation are therefore studied frequently. In particular, in Wang and Sontag (J Math Biol 57:29­52, 2008), it is shown that models of n-site sequential distributive phosphorylation admit at most 2n - 1 steady states.Wang and Sontag furthermore conjecture that for odd n, there are at most n and that, for even n, there are at most n + 1 steady states. This, however, is not true: building on earlier work in Holstein et al. (BullMath Biol 75(11):2028­2058, 2013), we present a scalar determining equation for multistationarity which will lead to parameter values where a 3-site system has 5 steady states and parameter values where a 4-site system has 7 steady states. Our results therefore are counterexamples to the conjecture of Wang and Sontag.We furthermore study the inherent geometric properties of multistationarity in n-site sequential distributive phosphorylation: the complete vector of steady state ratios is determined by the steady state ratios of free enzymes and unphosphorylated protein and there exists a linear relationship between steady state ratios of phosphorylated protein.

Catalytic constants enable the emergence of bistability in dual phosphorylation.

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.

Multistationarity in sequential distributed multisite phosphorylation networks.

Multisite phosphorylation networks are encountered in many intracellular processes like signal transduction, cell-cycle control, or nuclear signal integration. In this contribution, networks describing the phosphorylation and dephosphorylation of a protein at n sites in a sequential distributive mechanism are considered. Multistationarity (i.e., the existence of at least two positive steady state solutions of the associated polynomial dynamical system) has been analyzed and established in several contributions. It is, for example, known that there exist values for the rate constants where multistationarity occurs. However, nothing else is known about these rate constants. Here, we present a sign condition that is necessary and sufficient for multistationarity in n-site sequential, distributive phosphorylation. We express this sign condition in terms of linear systems, and show that solutions of these systems define rate constants where multistationarity is possible. We then present, for n≥2, a collection of feasible linear systems, and hence give a new and independent proof that multistationarity is possible for n≥2. Moreover, our results allow to explicitly obtain values for the rate constants where multistationarity is possible. Hence, we believe that, for the first time, a systematic exploration of the region in parameter space where multistationarity occurs has become possible. One consequence of our work is that, for any pair of steady states, the ratio of the steady state concentrations of kinase-substrate complexes equals that of phosphatase-substrate complexes.

Reciprocal enzyme regulation as a source of bistability in covalent modification cycles.

Covalent modification cycles (CMCs) are the building blocks of many regulatory networks in biological systems. Under proper kinetic conditions such mono-cyclic enzyme systems can show a higher sensitivity to effectors than enzymes subject to direct allosteric regulation. Using methods from reaction network theory it has been argued that CMCs can potentially exhibit multiple steady states if the converter enzymes are regulated in a reciprocal manner, but the underlying mechanism as well as the kinetic requirements for the emergence of such a behavior remained unclear. Here, we reinvestigate CMCs with reciprocal regulation of the converter enzymes for two common regulatory mechanisms: allosteric regulation and covalent modification. To analyze the steady state behavior of the corresponding mass-action equations, we derive reduced models by means of a quasi-steady state approximation (QSSA). We also derive reduced models using the total QSSA which often better reproduces the transient dynamics of enzyme-catalyzed reaction systems. Through a steady state analysis of the reduced models we show that the occurrence of bistability can be associated with the presence of a double negative feedback loop. We also derive constraints for the model parameters which might help to evaluate the potential significance of the mechanisms described here for the generation of bistability in natural systems. In particular, our results support the view of a possible bistable response in the metabolic PFK1/F1,6BPase cycle as observed experimentally in rat liver extracts, and it suggests an alternative view on the origin of bistability in the Cdk1-Wee1-Cdc25 system.

The Process-Interaction-Model: a common representation of rule-based and logical models allows studying signal transduction on different levels of detail.

BACKGROUND: Signaling systems typically involve large, structured molecules each consisting of a large number of subunits called molecule domains. In modeling such systems these domains can be considered as the main players. In order to handle the resulting combinatorial complexity, rule-based modeling has been established as the tool of choice. In contrast to the detailed quantitative rule-based modeling, qualitative modeling approaches like logical modeling rely solely on the network structure and are particularly useful for analyzing structural and functional properties of signaling systems. RESULTS: We introduce the Process-Interaction-Model (PIM) concept. It defines a common representation (or basis) of rule-based models and site-specific logical models, and, furthermore, includes methods to derive models of both types from a given PIM. A PIM is based on directed graphs with nodes representing processes like post-translational modifications or binding processes and edges representing the interactions among processes. The applicability of the concept has been demonstrated by applying it to a model describing EGF insulin crosstalk. A prototypic implementation of the PIM concept has been integrated in the modeling software ProMoT. CONCLUSIONS: The PIM concept provides a common basis for two modeling formalisms tailored to the study of signaling systems: a quantitative (rule-based) and a qualitative (logical) modeling formalism. Every PIM is a compact specification of a rule-based model and facilitates the systematic set-up of a rule-based model, while at the same time facilitating the automatic generation of a site-specific logical model. Consequently, modifications can be made on the underlying basis and then be propagated into the different model specifications - ensuring consistency of all models, regardless of the modeling formalism. This facilitates the analysis of a system on different levels of detail as it guarantees the application of established simulation and analysis methods to consistent descriptions (rule-based and logical) of a particular signaling system.

Multistationarity in mass action networks with applications to ERK activation.

Ordinary Differential Equations (ODEs) are an important tool in many areas of Quantitative Biology. For many ODE systems multistationarity (i.e. the existence of at least two positive steady states) is a desired feature. In general establishing multistationarity is a difficult task as realistic biological models are large in terms of states and (unknown) parameters and in most cases poorly parameterized (because of noisy measurement data of few components, a very small number of data points and only a limited number of repetitions). For mass action networks establishing multistationarity hence is equivalent to establishing the existence of at least two positive solutions of a large polynomial system with unknown coefficients. For mass action networks with certain structural properties, expressed in terms of the stoichiometric matrix and the reaction rate-exponent matrix, we present necessary and sufficient conditions for multistationarity that take the form of linear inequality systems. Solutions of these inequality systems define pairs of steady states and parameter values. We also present a sufficient condition to identify networks where the aforementioned conditions hold. To show the applicability of our results we analyse an ODE system that is defined by the mass action network describing the extracellular signal-regulated kinase (ERK) cascade (i.e. ERK-activation).

Chemical reaction systems with toric steady states.

Mass-action chemical reaction systems are frequently used in computational biology. The corresponding polynomial dynamical systems are often large (consisting of tens or even hundreds of ordinary differential equations) and poorly parameterized (due to noisy measurement data and a small number of data points and repetitions). Therefore, it is often difficult to establish the existence of (positive) steady states or to determine whether more complicated phenomena such as multistationarity exist. If, however, the steady state ideal of the system is a binomial ideal, then we show that these questions can be answered easily. The focus of this work is on systems with this property, and we say that such systems have toric steady states. Our main result gives sufficient conditions for a chemical reaction system to have toric steady states. Furthermore, we analyze the capacity of such a system to exhibit positive steady states and multistationarity. Examples of systems with toric steady states include weakly-reversible zero-deficiency chemical reaction systems. An important application of our work concerns the networks that describe the multisite phosphorylation of a protein by a kinase/phosphatase pair in a sequential and distributive mechanism.

Multistationarity in the activation of a MAPK: parametrizing the relevant region in parameter space.

Mathematical models of biochemical reaction networks in the form of ordinary differential equations can exhibit all sorts of complex dynamical behaviour. It is for example known, that even a single layer of a MAPK cascade can exhibit bistability (i.e. there exist multiple (positive) steady state solutions). It is almost a common-place that bistability or some other form of multistationarity are observed in many biochemical reaction networks, especially if the focus is on signal transduction or cell cycle regulation. However, multistationarity is only exhibited if the parameter vector is located in an appropriate region of parameter space. To find these regions, for example by using numerical tools like bifurcation analysis, is a non-trivial task as it amounts to searching the whole parameter space. In this paper we show that for a model of a single layer of a MAPK cascade it is possible to derive analytical descriptions of these regions, if mass action kinetics are used. Moreover, our results give rise to a straightforward explanation for the 'robust yet fragile' behaviour in the activation of a MAPK.

Subnetwork analysis reveals dynamic features of complex (bio)chemical networks.

In analyzing and mathematical modeling of complex (bio)chemical reaction networks, formal methods that connect network structure and dynamic behavior are needed because often, quantitative knowledge of the networks is very limited. This applies to many important processes in cell biology. Chemical reaction network theory allows for the classification of the potential network behavior-for instance, with respect to the existence of multiple steady states-but is computationally limited to small systems. Here, we show that by analyzing subnetworks termed elementary flux modes, the applicability of the theory can be extended to more complex networks. For an example network inspired by cell cycle control in budding yeast, the approach allows for model discrimination, identification of key mechanisms for multistationarity, and robustness analysis. The presented methods will be helpful in modeling and analyzing other complex reaction networks.