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John J. Tyson - One of the best experts on this subject based on the ideXlab platform.
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Review Article All Cyclic AMP waves during aggregation of Dictyostelium amoebae
2016Co-Authors: John J. Tyson, J. D. MurrayAbstract:During the aggregation phase of their life cycle, Dictyo-stelium discoideum amoebae communicate with each other by traveling waves of cyclic AMP. These waves are generated by an interplay between random diffusion of cyclic AMP in the extracellular milieu and the signal-reception/signal-relaying capabilities of individual amoebae. Kinetic properties of the enzymes, transport proteins and cell-surface receptor proteins involved in the cyclic AMP signaling system have been painstakingly worked out over the past fifteen years in many labora-tories. Recently Martiel & Goldbeter (1987) incorpor-ated this biochemical information into a unified math-ematical model of communication among Dictyostelium amoebae. Numerical simulations of the mathematical model, carried out by Tyson et al. (1989), agree in quantitative detail with experimental observations of cyclic AMP traveling waves in Dictyostelium cultures. Such mathematical modeling and numerical experimen-tation provide a necessary link between detailed studies of the molecular control mechanism and experimental observations of the intact developmental system
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Printed in Great Britain © The Company of Biologists Limited 1989 Cyclic AMP waves during aggregation of Dictyostelium amoebae
2008Co-Authors: John J. Tyson, J. D. MurrayAbstract:During the aggregation phase of their life cycle, Dictyostelium discoideum amoebae communicate with each other by traveling waves of cyclic AMP. These waves are generated by an interplay between random diffusion of cyclic AMP in the extracellular milieu and the signalreception/signal-relaying capabilities of individual amoebae. Kinetic properties of the enzymes, transport proteins and cell-surface receptor proteins involved in the cyclic AMP signaling system have been painstakingly worked out over the past fifteen years in many laboratories. Recently Martiel & Goldbeter (1987) incorporated this biochemical information into a unified mathematical model of communication among Dictyostelium Biological background When amoebae of the myxomycete Dictyostelium discoideum are left to starve on an agar surface, they begin signaling to each other with a chemical messenger, cyclic adenosine 3',5'-monophosphate (cAMP) (Bonner, 1969). Individual cells receive the signal by binding extracellular cAMP to a membrane receptor (Newell, 1986), and this binding stimulates the synthesis of cAMP from ATP by adenylate cyclase within the cell. Newly synthesized cAMP is transported to the extracellular medium and in this fashion the chemical signal is amplified. Amplification of the signal is limited by the fact that, on prolonged exposure to cAMP, the membrane receptor becomes desensitized (Devreotes & Sherring, 1985), i.e. it no longer stimulates adenylate cyclase activity. In the absence of cAMP synthesis, the concentration of cAMP decreases by the action of phosphodiesterase, which hydrolyzes cAMP to 5'-AMP. In a field of signaling amoebae spread over an agar surface, pulses of extracellular cAMP travel across the field in the form of either expanding concentric circular waves or rotating spiral waves. As the waves pass periodically through the field of independent amoebal cells, they stimulate a chemotactic movement of the amoebae towards the center of the pattern (the origin of the circles or the pivot point of the spiral). Eventuall
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Stochastic simulation of enzyme-catalyzed reactions with disparate timescales.
Biophysical journal, 2008Co-Authors: Debashis Barik, Mark Paul, William T. Baumann, Yang Cao, John J. TysonAbstract:Many physiological characteristics of living cells are regulated by protein interaction networks. Because the total numbers of these protein species can be small, molecular noise can have significant effects on the dynamical properties of a regulatory network. Computing these stochastic effects is made difficult by the large timescale separations typical of protein interactions (e.g., complex formation may occur in fractions of a second, whereas catalytic conversions may take minutes). Exact stochastic simulation may be very inefficient under these circumstances, and methods for speeding up the simulation without sacrificing accuracy have been widely studied. We show that the "total quasi-steady-state approximation" for enzyme-catalyzed reactions provides a useful framework for efficient and accurate stochastic simulations. The method is applied to three examples: a simple enzyme-catalyzed reaction where enzyme and substrate have comparable abundances, a Goldbeter-Koshland switch, where a kinase and phosphatase regulate the phosphorylation state of a common substrate, and coupled Goldbeter-Koshland switches that exhibit bistability. Simulations based on the total quasi-steady-state approximation accurately capture the steady-state probability distributions of all components of these reaction networks. In many respects, the approximation also faithfully reproduces time-dependent aspects of the fluctuations. The method is accurate even under conditions of poor timescale separation.
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Modeling networks of coupled enzymatic reactions using the total quasi-steady state approximation
PLoS Computational Biology, 2007Co-Authors: Andrea Ciliberto, Fabrizio Capuani, John J. TysonAbstract:In metabolic networks, metabolites are usually present in great excess over the enzymes that catalyze their interconversion, and describing the rates of these reactions by using the Michaelis-Menten rate law is perfectly valid. This rate law assumes that the concentration of enzyme-substrate complex (C) is much less than the free substrate concentration (S0). However, in protein interaction networks, the enzymes and substrates are all proteins in comparable concentrations, and neglecting C with respect to S0 is not valid. Borghans, DeBoer, and Segel developed an alternative description of enzyme kinetics that is valid when C is comparable to S0. We extend this description, which Borghans et al. call the total quasi-steady state approximation, to networks of coupled enzymatic reactions. First, we analyze an isolated Goldbeter-Koshland switch when enzymes and substrates are present in comparable concentrations. Then, on the basis of a real example of the molecular network governing cell cycle progression, we couple two and three Goldbeter-Koshland switches together to study the effects of feedback in networks of protein kinases and phosphatases. Our analysis shows that the total quasi-steady state approximation provides an excellent kinetic formalism for protein interaction networks, because (1) it unveils the modular structure of the enzymatic reactions, (2) it suggests a simple algorithm to formulate correct kinetic equations, and (3) contrary to classical Michaelis-Menten kinetics, it succeeds in faithfully reproducing the dynamics of the network both qualitatively and quantitatively.
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Modeling networks of coupled enzymatic reactions using the total quasi-steady state approximation, PLoS
2007Co-Authors: Andrea Ciliberto, Fabrizio Capuani, John J. TysonAbstract:In metabolic networks, metabolites are usually present in great excess over the enzymes that catalyze their interconversion, and describing the rates of these reactions by using the Michaelis–Menten rate law is perfectly valid. This rate law assumes that the concentration of enzyme–substrate complex (C) is much less than the free substrate concentration (S 0). However, in protein interaction networks, the enzymes and substrates are all proteins in comparable concentrations, and neglecting C with respect to S0 is not valid. Borghans, DeBoer, and Segel developed an alternative description of enzyme kinetics that is valid when C is comparable to S 0. We extend this description, which Borghans et al. call the total quasi-steady state approximation, to networks of coupled enzymatic reactions. First, we analyze an isolated Goldbeter–Koshland switch when enzymes and substrates are present in comparable concentrations. Then, on the basis of a real example of the molecular network governing cell cycle progression, we couple two and three Goldbeter–Koshland switches together to study the effects of feedback in networks of protein kinases and phosphatases. Our analysis shows that the total quasi-steady state approximation provides an excellent kinetic formalism for protein interaction networks, because (1) it unveils the modular structure of the enzymatic reactions, (2) it suggests a simple algorithm to formulate correct kinetic equations, and (3) contrary to classical Michaelis–Menten kinetics, it succeeds in faithfully reproducing the dynamics of the network both qualitatively and quantitatively. Citation: Ciliberto A, Capuani F, Tyson JJ (2007) Modeling networks of coupled enzymatic reactions using the total quasi-steady state approximation. PLoS Comput Biol 3(3): e45. doi:10.1371/journal.pcbi.003004
Albert Goldbeter - One of the best experts on this subject based on the ideXlab platform.
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effect of positive feedback loops on the robustness of oscillations in the network of cyclin dependent kinases driving the mammalian cell cycle
FEBS Journal, 2012Co-Authors: Claude Gerard, Didier Gonze, Albert GoldbeterAbstract:The transitions between the G(1), S, G(2) and M phases of the mammalian cell cycle are driven by a network of cyclin-dependent kinases (Cdks), whose sequential activation is regulated by intertwined negative and positive feedback loops. We previously proposed a detailed computational model for the Cdk network, and showed that this network is capable of temporal self-organization in the form of sustained oscillations, which govern ordered progression through the successive phases of the cell cycle [Gerard and Goldbeter (2009) Proc Natl Acad Sci USA 106, 21643-21648]. We subsequently proposed a skeleton model for the cell cycle that retains the core regulatory mechanisms of the detailed model [Gerard and Goldbeter (2011) Interface Focus 1, 24-35]. Here we extend this skeleton model by incorporating Cdk regulation through phosphorylation/dephosphorylation and by including the positive feedback loops that underlie the dynamics of the G(1)/S and G(2)/M transitions via phosphatase Cdc25 and via phosphatase Cdc25 and kinase Wee1, respectively. We determine the effects of these positive feedback loops and ultrasensitivity in phosphorylation/dephosphorylation on the dynamics of the Cdk network. The multiplicity of positive feedback loops as well as the existence of ultrasensitivity promote the occurrence of bistability and increase the amplitude of the oscillations in the various cyclin/Cdk complexes. By resorting to stochastic simulations, we further show that the presence of multiple, redundant positive feedback loops in the G(2)/M transition of the cell cycle markedly enhances the robustness of the Cdk oscillations with respect to molecular noise.
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The Cell Cycle is a Limit Cycle
'EDP Sciences', 2012Co-Authors: C. Gérard, Albert GoldbeterAbstract:Progression along the successive phases of the mammalian cell cycle is driven by a network of cyclin-dependent kinases (Cdks). This network is regulated by a variety of negative and positive feedback loops. We previously proposed a detailed, 39-variable model for the Cdk network and showed that it is capable of temporal self-organization in the form of sustained oscillations, which correspond to the repetitive, transient, sequential activation of the cyclin- Cdk complexes that govern the successive phases of the cell cycle [Gérard and Goldbeter (2009) Proc Natl Acad Sci 106, 21643-8]. Here we compare the dynamical behavior of three models of different complexity for the Cdk network driving the mammalian cell cycle. The first is the detailed model that counts 39 variables and is based on Michaelis-Menten kinetics for the enzymatic steps. From this detailed model, we build a version based only on mass-action kinetics, which counts 80 variables. In this version we do not need to assume that enzymes are present in much smaller amounts that their substrates, which is not necessarily the case in the cell cycle. We show that these two versions of the model for the Cdk network yield similar results. In particular they predict sustained oscillations of the limit cycle type. We show that the model for the Cdk network can be reduced to a version containing only 5 variables, which is more amenable to stochastic simulations. This skeleton version retains the dynamic properties of the more complex versions of the model for the Cdk network in regard to Cdk oscillations. The regulatory wiring of the Cdk network therefore governs its dynamic behavior, regardless of the degree of molecular detail. We discuss the relative advantages of each version of the model, all of which support the view that the mammalian cell cycle behaves as a limit cycle oscillator
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Edited by:
2012Co-Authors: Claude Gerard, Albert Goldbeter, Matteo Barberis HumboldtAbstract:We recently proposed a detailed model describing the dynamics of the network of cyclin-dependent kinases (Cdks) driving the mammalian cell cycle (Gérard and Goldbeter, 2009). The model contains four modules, each centered around one cyclin/Cdk complex. Cyclin D/Cdk4–6 and cyclin E/Cdk2 promote progression in G1 and elicit the G1/S transition, respectively; cyclin A/Cdk2 ensures progression in S and the transition S/G2, while the activity of cyclin B/Cdk1 brings about the G2/M transition. This model shows that in the presence of sufficient amounts of growth factor the Cdk network is capable of temporal self-organization in the form of sustained oscillations, which correspond to the ordered, sequential activation of the various cyclin/Cdk complexes that control the successive phases of the cell cycle. The results suggest that the switch from cellular quiescence to cell proliferation corresponds to the transition from a stable steady state to sustained oscillations in the Cdk network. The transition depends on a finely tuned balance between factors that promote or hinder progression in the cell cycle. We show that the transition from quiescence to proliferation can occur in multiple ways that alter this balance. By resorting to bifurcation diagrams, we analyze the mechanism of oscillations in the Cdk network. Finally, we show that the complexity of the detailed model can be greatly reduced, without losing its key dynamical properties, by considering a skeleton model for the Cdk network. Using such a skeleton model for the mammalian cell cycle we show that positive feedback (PF) loops enhance the amplitude and the robustness of Cdk oscillations with respect to molecular noise. We compare the relative merits of the detailed and skeleton versions of the model for the Cdk network driving the mammalian cell cycle
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from simple to complex patterns of oscillatory behavior in a model for the mammalian cell cycle containing multiple oscillatory circuits
Chaos, 2010Co-Authors: Claude Gerard, Albert GoldbeterAbstract:We previously proposed an integrated computational model for the network of cyclin-dependent kinases (Cdks) that controls the dynamics of the mammalian cell cycle [C. Gerard and A. Goldbeter, "Temporal self-organization of the cyclin/Cdk network driving the mammalian cell cycle," Proc. Natl. Acad. Sci. U.S.A. 106, 21643 (2009)]. The model contains four Cdk modules regulated by reversible phosphorylation, Cdk inhibitors, protein synthesis or degradation, and the balance between antagonistic effects of the tumor suppressor pRB and the transcription factor E2F. Increasing the level of a growth factor above a critical threshold triggers the transition from a quiescent, stable steady state to self-sustained oscillations in the Cdk network. These oscillations correspond to the repetitive, transient activation of cyclin D/Cdk4-6 in G1, cyclin E/Cdk2 at the G1/S transition, cyclin A/Cdk2 in S and at the S/G2 transition, and cyclin B/Cdk1 at the G2/M transition. This periodic, ordered activation of the various cyclin/Cdk complexes can be associated with cell proliferation. The multiplicity of feedback loops within the Cdk network is such that it contains at least four distinct circuits capable of producing oscillations. The tight coupling of these oscillatory circuits generally results in simple periodic behavior associated with repetitive cycles of mitosis or with endoreplication. The latter corresponds to multiple passages through the phase of DNA replication without mitosis. We show here that, as a result of the interaction between the multiple oscillatory circuits, particularly when attenuating the strength of the oscillatory module involving cyclin B/Cdk1, the model for the Cdk network can also produce complex periodic oscillations, quasiperiodic oscillations, and chaos. Numerical simulations based on limited explorations in parameter space nevertheless suggest that these complex modes of oscillatory behavior remain less common than the evolution to simple periodic oscillations of the limit cycle type, holding with the view that simple periodic oscillations in the Cdk network correspond to its physiological mode of dynamic behavior.
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the mitotic oscillator temporal self organization in a phosphorylation dephosphorylation enzymatic cascade
Deutsche Bunsen-Gesellschaft für physikalische Chemie. Joint meeting and international Bunsen discussion meeting, 1994Co-Authors: Pierrecharles Romond, J M Guilmot, Albert GoldbeterAbstract:The conditions for temporal self-organization in the form of sustained oscillations are determined in a minimal cascade model previously proposed (A. Goldbeter, Proc. Natl. Acad. Sci. USA 88, 9107–9111 (1991)) for the mitotic oscillator driving the embryonic cell division cycle. The model is based on a phosphorylation-dephosphorylation cascade involving cyclin and cdc2 kinase. In the first cycle of the cascade, cdc2 kinase is activated through dephosphorylation triggered by the accumulation of cyclin, while in a second cycle the activation of a cyclin protease is brought about through phosphorylation by cdc2 kinase. The fact that cyclin promotes the activation of cdc2 kinase while the latter enzyme triggers cyclin degradation introduces a negative feedback loop which is at the core of the periodic operation of the cascade. We analyze the mechanism of oscillatory behavior by constructing stability diagrams as a function of some of the main parameters of the model. Investigated in turn are the roles of negative feedback and of phosphorylation-dephosphorylation thresholds. Such thresholds arise from the phenomenon of zero-order ultrasensitivity associated with the kinetics of covalent modification cycles. An extension of the minimal model allows one to address the effect of additional phosphorylation-dephosphorylation cycles and the possible role of autocatalysis by cdc2 kinase in the generation of periodic behavior. The analysis shows that the existence of thresholds as well as an increase in the number of cycles in the cascade favor the occurrence of sustained oscillations. The results further indicate that negative and positive feedback may both contribute to the repetitive activation of cdc2 kinase.
Eberhard Bodenschatz - One of the best experts on this subject based on the ideXlab platform.
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convective instability and boundary driven oscillations in a reaction diffusion advection model
Chaos, 2017Co-Authors: Estefania Vidalhenriquez, Eberhard Bodenschatz, Vladimir S Zykov, Azam GholamiAbstract:In a reaction-diffusion-advection system, with a convectively unstable regime, a perturbation creates a wave train that is advected downstream and eventually leaves the system. We show that the convective instability coexists with a local absolute instability when a fixed boundary condition upstream is imposed. This boundary induced instability acts as a continuous wave source, creating a local periodic excitation near the boundary, which initiates waves travelling both up and downstream. To confirm this, we performed analytical analysis and numerical simulations of a modified Martiel-Goldbeter reaction-diffusion model with the addition of an advection term. We provide a quantitative description of the wave packet appearing in the convectively unstable regime, which we found to be in excellent agreement with the numerical simulations. We characterize this new instability and show that in the limit of high advection speed, it is suppressed. This type of instability can be expected for reaction-diffusion sy...
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convective instability and boundary driven oscillations in a reaction diffusion advection model
arXiv: Pattern Formation and Solitons, 2017Co-Authors: Estefania Vidalhenriquez, Eberhard Bodenschatz, Vladimir S Zykov, Azam GholamiAbstract:In a reaction-diffusion-advection system, with a convectively unstable regime, a perturbation creates a wave train that is advected downstream and eventually leaves the system. We show that the convective instability coexists with a local absolute instability when a fixed boundary condition upstream is imposed. This boundary induced instability acts as a continuous wave source, creating a local periodic excitation near the boundary, which initiates waves traveling both up and downstream. To confirm this, we performed analytical analysis and numerical simulations of a modified Martiel-Goldbeter reaction-diffusion model with the addition of an advection term. We provide a quantitative description of the wave packet appearing in the convectively unstable regime, which we found to be in excellent agreement with the numerical simulations. We characterize this new instability and show that in the limit of high advection speed, it is suppressed. This type of instability can be expected for reaction-diffusion systems that present both a convective instability and an excitable regime. In particular, it can be relevant to understand the signaling mechanism of the social amoeba Dictyostelium discoideum that may experience fluid flows in its natural habitat.
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flow driven two dimensional waves in colonies of dictyostelium discoideum
New Journal of Physics, 2015Co-Authors: Azam Gholami, Oliver Steinbock, Vladimir S Zykov, Eberhard BodenschatzAbstract:Dictyostelium discoideum (D.d.) is a valuable model organism to study self-organization and pattern formation in biology. Recently we reported flow-driven waves in experiments with uniformly distributed populations of signaling amobae, D.d., and carried out a theoretical study in a one-dimensional model. In this work, we perform two-dimensional numerical simulations using the well-known Martiel–Golbeter model to study the effect of the flow profile and intrinsic noise on the flow-driven waves. We show that, in the presence of flow, a persistence noise due to spontaneous cell firing events can lead to sustained structures that fill the whole length of the system. We also show that external periodic stimuli of cyclic adenosine monophosphate can induce 1:1 and 2:1 entrainments which are in agreement with our experimental observations.
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flow driven instabilities during pattern formation of dictyostelium discoideum
New Journal of Physics, 2015Co-Authors: Azam Gholami, Oliver Steinbock, Vladimir S Zykov, Eberhard BodenschatzAbstract:The slime mold Dictyostelium discoideum is a well known model system for the study of biological pattern formation. In the natural environment, aggregating populations of starving Dictyostelium discoideum cells may experience fluid flows that can profoundly change the underlying wave generation process. Here we study the effect of advection on the pattern formation in a colony of homogeneously distributed Dictyostelium discoideum cells described by the standard Martiel–Goldbeter model. The external flow advects the signaling molecule cyclic adenosine monophosphate (cAMP) downstream, while the chemotactic cells attached to the solid substrate are not transported with the flow. The evolution of small perturbations in cAMP concentrations is studied analytically in the linear regime and by corresponding numerical simulations. We show that flow can significantly influence the dynamics of the system and lead to a flow-driven instability that initiate downstream traveling cAMP waves. We also show that boundary conditions have a significant effect on the observed patterns and can lead to a new kind of instability.
Andrea Ciliberto - One of the best experts on this subject based on the ideXlab platform.
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Modeling networks of coupled enzymatic reactions using the total quasi-steady state approximation
PLoS Computational Biology, 2007Co-Authors: Andrea Ciliberto, Fabrizio Capuani, John J. TysonAbstract:In metabolic networks, metabolites are usually present in great excess over the enzymes that catalyze their interconversion, and describing the rates of these reactions by using the Michaelis-Menten rate law is perfectly valid. This rate law assumes that the concentration of enzyme-substrate complex (C) is much less than the free substrate concentration (S0). However, in protein interaction networks, the enzymes and substrates are all proteins in comparable concentrations, and neglecting C with respect to S0 is not valid. Borghans, DeBoer, and Segel developed an alternative description of enzyme kinetics that is valid when C is comparable to S0. We extend this description, which Borghans et al. call the total quasi-steady state approximation, to networks of coupled enzymatic reactions. First, we analyze an isolated Goldbeter-Koshland switch when enzymes and substrates are present in comparable concentrations. Then, on the basis of a real example of the molecular network governing cell cycle progression, we couple two and three Goldbeter-Koshland switches together to study the effects of feedback in networks of protein kinases and phosphatases. Our analysis shows that the total quasi-steady state approximation provides an excellent kinetic formalism for protein interaction networks, because (1) it unveils the modular structure of the enzymatic reactions, (2) it suggests a simple algorithm to formulate correct kinetic equations, and (3) contrary to classical Michaelis-Menten kinetics, it succeeds in faithfully reproducing the dynamics of the network both qualitatively and quantitatively.
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Modeling networks of coupled enzymatic reactions using the total quasi-steady state approximation, PLoS
2007Co-Authors: Andrea Ciliberto, Fabrizio Capuani, John J. TysonAbstract:In metabolic networks, metabolites are usually present in great excess over the enzymes that catalyze their interconversion, and describing the rates of these reactions by using the Michaelis–Menten rate law is perfectly valid. This rate law assumes that the concentration of enzyme–substrate complex (C) is much less than the free substrate concentration (S 0). However, in protein interaction networks, the enzymes and substrates are all proteins in comparable concentrations, and neglecting C with respect to S0 is not valid. Borghans, DeBoer, and Segel developed an alternative description of enzyme kinetics that is valid when C is comparable to S 0. We extend this description, which Borghans et al. call the total quasi-steady state approximation, to networks of coupled enzymatic reactions. First, we analyze an isolated Goldbeter–Koshland switch when enzymes and substrates are present in comparable concentrations. Then, on the basis of a real example of the molecular network governing cell cycle progression, we couple two and three Goldbeter–Koshland switches together to study the effects of feedback in networks of protein kinases and phosphatases. Our analysis shows that the total quasi-steady state approximation provides an excellent kinetic formalism for protein interaction networks, because (1) it unveils the modular structure of the enzymatic reactions, (2) it suggests a simple algorithm to formulate correct kinetic equations, and (3) contrary to classical Michaelis–Menten kinetics, it succeeds in faithfully reproducing the dynamics of the network both qualitatively and quantitatively. Citation: Ciliberto A, Capuani F, Tyson JJ (2007) Modeling networks of coupled enzymatic reactions using the total quasi-steady state approximation. PLoS Comput Biol 3(3): e45. doi:10.1371/journal.pcbi.003004
Roger B Sidje - One of the best experts on this subject based on the ideXlab platform.
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stochastic chemical kinetics and the total quasi steady state assumption application to the stochastic simulation algorithm and chemical master equation
Journal of Chemical Physics, 2008Co-Authors: Shev Macnamara, Alberto Maria Bersani, Kevin Burrage, Roger B SidjeAbstract:Recently the application of the quasi-steady-state approximation QSSA to the stochastic simulation algorithm SSA was suggested for the purpose of speeding up stochastic simulations of chemical systems that involve both relatively fast and slow chemical reactions Rao and Arkin, J.Chem. Phys. 118, 4999 2003 and further work has led to the nested and slow-scale SSA. Improved numerical efficiency is obtained by respecting the vastly different time scales characterizing the system and then by advancing only the slow reactions exactly, based on a suitable approximation to the fast reactions. We considerably extend these works by applying the QSSA to numerical methods for the direct solution of the chemical master equation CME and, in particular, to the finite state projection algorithm Munsky and Khammash, J. Chem. Phys. 124, 044104 2006, in conjunction with Krylov methods. In addition, we point out some important connections to the literature on the deterministic total QSSA tQSSA and place the stochastic analogue of the QSSA within the more general framework of aggregation of Markov processes. We demonstrate the new methods on four examples: Michaelis–Menten enzyme kinetics, double phosphorylation, the Goldbeter–Koshland switch, and the mitogen activated protein kinase cascade. Overall, we report dramatic improvements by applying the tQSSA to the CME solver.