Symbolic Versus Numerical Computation and Visualization of Parameter Regions for Multistationarity of Biological Networks

We investigate models of the mitogenactivated protein kinases (MAPK) network, with the aim of determining where in parameter space there exist multiple positive steady states. We build on recent progress which combines various symbolic computation methods for mixed systems of equalities and inequalities. We demonstrate that those techniques benefit tremendously from a newly implemented graph theoretical symbolic preprocessing method. We compare computation times and quality of results of numerical continuation methods with our symbolic approach before and after the application of our preprocessing.Comment: Accepted into Proc. CASC 201

Switches, Excitable Responses and Oscillations in the Ring1B/Bmi1 Ubiquitination System

In an active, self-ubiquitinated state, the Ring1B ligase monoubiquitinates histone H2A playing a critical role in Polycomb-mediated gene silencing. Following ubiquitination by external ligases, Ring1B is targeted for proteosomal degradation. Using biochemical data and computational modeling, we show that the Ring1B ligase can exhibit abrupt switches, overshoot transitions and self-perpetuating oscillations between its distinct ubiquitination and activity states. These different Ring1B states display canonical or multiply branched, atypical polyubiquitin chains and involve association with the Polycomb-group protein Bmi1. Bistable switches and oscillations may lead to all-or-none histone H2A monoubiquitination rates and result in discrete periods of gene (in)activity. Switches, overshoots and oscillations in Ring1B catalytic activity and proteosomal degradation are controlled by the abundances of Bmi1 and Ring1B, and the activities and abundances of external ligases and deubiquitinases, such as E6-AP and USP7

Signaling Cascades Modulate the Speed of Signal Propagation through Space

Cells are not mixed bags of signaling molecules. As a consequence, signals must travel from their origin to distal locations. Much is understood about the purely diffusive propagation of signals through space. Many signals, however, propagate via signaling cascades. Here, we show that, depending on their kinetics, cascades speed up or slow down the propagation of signals through space, relative to pure diffusion.We modeled simple cascades operating under different limits of Michaelis-Menten kinetics using deterministic reaction-diffusion equations. Cascades operating far from enzyme saturation speed up signal propagation; the second mobile species moves more quickly than the first through space, on average. The enhanced speed is due to more efficient serial activation of a downstream signaling module (by the signaling molecule immediately upstream in the cascade) at points distal from the signaling origin, compared to locations closer to the source. Conversely, cascades operating under saturated kinetics, which exhibit zero-order ultrasensitivity, can slow down signals, ultimately localizing them to regions around the origin.Signal speed modulation may be a fundamental function of cascades, affecting the ability of signals to penetrate within a cell, to cross-react with other signals, and to activate distant targets. In particular, enhanced speeds provide a way to increase signal penetration into a cell without needing to flood the cell with large numbers of active signaling molecules; conversely, diminished speeds in zero-order ultrasensitive cascades facilitate strong, but localized, signaling

The interplay of intrinsic and extrinsic bounded noises in genetic networks

After being considered as a nuisance to be filtered out, it became recently clear that biochemical noise plays a complex role, often fully functional, for a genetic network. The influence of intrinsic and extrinsic noises on genetic networks has intensively been investigated in last ten years, though contributions on the co-presence of both are sparse. Extrinsic noise is usually modeled as an unbounded white or colored gaussian stochastic process, even though realistic stochastic perturbations are clearly bounded. In this paper we consider Gillespie-like stochastic models of nonlinear networks, i.e. the intrinsic noise, where the model jump rates are affected by colored bounded extrinsic noises synthesized by a suitable biochemical state-dependent Langevin system. These systems are described by a master equation, and a simulation algorithm to analyze them is derived. This new modeling paradigm should enlarge the class of systems amenable at modeling. We investigated the influence of both amplitude and autocorrelation time of a extrinsic Sine-Wiener noise on: $(i)$ the Michaelis-Menten approximation of noisy enzymatic reactions, which we show to be applicable also in co-presence of both intrinsic and extrinsic noise, $(ii)$ a model of enzymatic futile cycle and $(iii)$ a genetic toggle switch. In $(ii)$ and $(iii)$ we show that the presence of a bounded extrinsic noise induces qualitative modifications in the probability densities of the involved chemicals, where new modes emerge, thus suggesting the possibile functional role of bounded noises

Unlimited multistability in multisite phosphorylation systems

Reversible phosphorylation on serine, threonine and tyrosine is the most widely studied posttranslational modification of proteins (1, 2). The number of phosphorylated sites on a protein (n) shows a significant increase from prokaryotes, with n less than or equal to 7 sites, to eukaryotes, with examples having n greater than or equal to 150 sites (3). Multisite phosphorylation has many roles (4, 5) and site conservation indicates that increasing numbers of sites cannot be due merely to promiscuous phosphorylation. A substrate with n sites has an exponential number (2^n) of phospho-forms and individual phospho-forms may have distinct biological effects (6, 7). The distribution of these phospho-forms and how this distribution is regulated have remained unknown. Here we show that, when kinase and phosphatase act in opposition on a multisite substrate, the system can exhibit distinct stable phospho-form distributions at steady state and that the maximum number of such distributions increases with n. Whereas some stable distributions are focused on a single phospho-form, others are more diffuse, giving the phospho-proteome the potential to behave as a fluid regulatory network able to encode information and flexibly respond to varying demands. Such plasticity may underlie complex information processing in eukaryotic cells (8) and suggests a functional advantage in having many sites. Our results follow from the unusual geometry of the steady-state phospho-form concentrations, which we show to constitute a rational algebraic curve, irrespective of n. We thereby reduce the complexity of calculating steady states from simulating 3 times 2^n differential equations to solving two algebraic equations, while treating parameters symbolically. We anticipate that these methods can be extended to systems with multiple substrates and multiple enzymes catalysing different modifications, as found in posttranslational modification 'codes' (9) such as the histone code (10, 11). Whereas simulations struggle with exponentially increasing molecular complexity, mathematical methods of the kind developed here can provide a new language in which to articulate the principles of cellular information processing (12)

EhMAPK, the Mitogen-Activated Protein Kinase from Entamoeba histolytica Is Associated with Cell Survival

Mitogen Activated Protein Kinases (MAPKs) are a class of serine/threonine kinases that regulate a number of different cellular activities including cell proliferation, differentiation, survival and even death. The pathogen Entamoeba histolytica possess a single homologue of a typical MAPK gene (EhMAPK) whose identification was previously reported by us but its functional implications remained unexplored. EhMAPK, the only mitogen-activated protein kinase from the parasitic protist Entamoeba histolytica with Threonine-X-Tyrosine (TXY) phosphorylation motif was cloned, expressed in E. coli and functionally characterized under different stress conditions. The expression profile of EhMAPK at the protein and mRNA level remained similar among untreated, heat shocked and hydrogen peroxide-treated samples in all cases of dose and time. But a significant difference was obtained in the phosphorylation status of the protein in response to different stresses. Heat shock at 43°C or 0.5 mM H2O2 treatment enhanced the phosphorylation status of EhMAPK and augmented the kinase activity of the protein whereas 2.0 mM H2O2 treatment induced dephosphorylation of EhMAPK and loss of kinase activity. 2.0 mM H2O2 treatment reduced parasite viability significantly but heat shock and 0.5 mM H2O2 treatment failed to adversely affect E. histolytica viability. Therefore, a distinct possibility that activation of EhMAPK is associated with stress survival in E. histolytica is seen. Our study also gives a glimpse of the regulatory mechanism of the protein under in vivo conditions. Since the parasite genome lacks any typical homologue of mammalian MEK, the dual specificity kinases which are the upstream activators of MAPK, indications of the existence of some alternate regulatory mechanisms of the EhMAPK activity is perceived. These may include the autophosphorylation activity of the protein itself in combination with some upstream phosphatases which are not yet identified

Diffusive coupling can discriminate between similar reaction mechanisms in an allosteric enzyme system

<p>Abstract</p> <p>Background</p> <p>A central question for the understanding of biological reaction networks is how a particular dynamic behavior, such as bistability or oscillations, is realized at the molecular level. So far this question has been mainly addressed in well-mixed reaction systems which are conveniently described by ordinary differential equations. However, much less is known about how molecular details of a reaction mechanism can affect the dynamics in diffusively coupled systems because the resulting partial differential equations are much more difficult to analyze.</p> <p>Results</p> <p>Motivated by recent experiments we compare two closely related mechanisms for the product activation of allosteric enzymes with respect to their ability to induce different types of reaction-diffusion waves and stationary Turing patterns. The analysis is facilitated by mapping each model to an associated complex Ginzburg-Landau equation. We show that a sequential activation mechanism, as implemented in the model of Monod, Wyman and Changeux (MWC), can generate inward rotating spiral waves which were recently observed as glycolytic activity waves in yeast extracts. In contrast, in the limiting case of a simple Hill activation, the formation of inward propagating waves is suppressed by a Turing instability. The occurrence of this unusual wave dynamics is not related to the magnitude of the enzyme cooperativity (as it is true for the occurrence of oscillations), but to the sensitivity with respect to changes of the activator concentration. Also, the MWC mechanism generates wave patterns that are more stable against long wave length perturbations.</p> <p>Conclusions</p> <p>This analysis demonstrates that amplitude equations, which describe the spatio-temporal dynamics near an instability, represent a valuable tool to investigate the molecular effects of reaction mechanisms on pattern formation in spatially extended systems. Using this approach we have shown that the occurrence of inward rotating spiral waves in glycolysis can be explained in terms of an MWC, but not with a Hill mechanism for the activation of the allosteric enzyme phosphofructokinase. Our results also highlight the importance of enzyme oligomerization for a possible experimental generation of Turing patterns in biological systems.</p

DBSolve Optimum: a software package for kinetic modeling which allows dynamic visualization of simulation results

<p>Abstract</p> <p>Background</p> <p>Systems biology research and applications require creation, validation, extensive usage of mathematical models and visualization of simulation results by end-users. Our goal is to develop novel method for visualization of simulation results and implement it in simulation software package equipped with the sophisticated mathematical and computational techniques for model development, verification and parameter fitting.</p> <p>Results</p> <p>We present mathematical simulation workbench DBSolve Optimum which is significantly improved and extended successor of well known simulation software DBSolve5. Concept of "dynamic visualization" of simulation results has been developed and implemented in DBSolve Optimum. In framework of the concept graphical objects representing metabolite concentrations and reactions change their volume and shape in accordance to simulation results. This technique is applied to visualize both kinetic response of the model and dependence of its steady state on parameter. The use of the dynamic visualization is illustrated with kinetic model of the Krebs cycle.</p> <p>Conclusion</p> <p>DBSolve Optimum is a user friendly simulation software package that enables to simplify the construction, verification, analysis and visualization of kinetic models. Dynamic visualization tool implemented in the software allows user to animate simulation results and, thereby, present them in more comprehensible mode. DBSolve Optimum and built-in dynamic visualization module is free for both academic and commercial use. It can be downloaded directly from <url>http://www.insysbio.ru</url>.</p

Modelling molecular interaction pathways using a two-stage identification algorithm

In systems biology, molecular interactions are typically modelled using white-box methods, usually based on mass action kinetics. Unfortunately, problems with dimensionality can arise when the number of molecular species in the system is very large, which makes the system modelling and behavior simulation extremely difficult or computationally too expensive. As an alternative, this paper investigates the identification of two molecular interaction pathways using a black-box approach. This type of method creates a simple linear-in-the-parameters model using regression of data, where the output of the model at any time is a function of previous system states of interest. One of the main objectives in building black-box models is to produce an optimal sparse nonlinear one to effectively represent the system behavior. In this paper, it is achieved by applying an efficient iterative approach, where the terms in the regression model are selected and refined using a forward and backward subset selection algorithm. The method is applied to model identification for the MAPK signal transduction pathway and the Brusselator using noisy data of different sizes. Simulation results confirm the efficacy of the black-box modelling method which offers an alternative to the computationally expensive conventional approach

Monitoring dynamics of defects and single Fe atoms in N-functionalized few-layer graphene by in situ temperature programmed scanning transmission electron microscopy

In this study, we aim to contribute an understanding of the pathway of formation of Fe species during top-down synthesis of dispersed Fe on N-functionalized few layer graphene. We use X-ray absorption spectroscopy to determine the electronic structure and coordination geometry of the Fe species and in situ high angle annular dark field scanning transmission electron microscopy combined with atomic resolved electron energy loss spectroscopy to localize these, identify their chemical configuration and monitor their dynamics during thermal annealing. We show the high mobility of peripheral Fe atoms, first diffusing rapidly at the trims of the graphene layers and at temperatures as high as 573 K, diffusing from the edge planes towards in-plane locations of the graphene layers forming three-, four-coordinated metal sites and more complexes polynuclear Fe species. This process occurs via bond breaking which partially reduces the extension of the graphene domains. However, the vast majority of Fe is segregated as a metal phase. This dynamic interconversion depends on the structural details of the surrounding graphitic environment in which these are formed as well as the Fe loading. N species appear stabilizing isolated and polynuclear Fe species even at temperatures as high as 873 K. The significance of our results lies on the fact that single Fe atoms in graphene are highly mobile and therefore a structural description of the active sites as such is insufficient and more complex species might be more relevant, especially in the case of multielectron transfer reaction. Here we provide the experimental evidence on the formation of these polynuclear Fe-N sites and their structural characteristics