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Classification of transient behaviours in a time-dependent toggle switch model.

Verd B, Crombach A, Jaeger J - BMC Syst Biol (2014)

Bottom Line: We describe these in detail, and illustrate the usefulness of our classification scheme by providing a number of examples that demonstrate how it can be employed to gain specific mechanistic insights into the dynamics of gene regulation.The practical aim of our proposed classification scheme is to make the analysis of explicitly time-dependent transient behaviour tractable, and to encourage the wider use of non-autonomous models in systems biology.Our method is applicable to a large class of biological processes.

View Article: PubMed Central - HTML - PubMed

Affiliation: EMBL/CRG Research Unit in Systems Biology, Centre for Genomic Regulation (CRG), Barcelona, Spain. yogi.jaeger@crg.eu.

ABSTRACT

Background: Waddington's epigenetic landscape is an intuitive metaphor for the developmental and evolutionary potential of biological regulatory processes. It emphasises time-dependence and transient behaviour. Nowadays, we can derive this landscape by modelling a specific regulatory network as a dynamical system and calculating its so-called potential surface. In this sense, potential surfaces are the mathematical equivalent of the Waddingtonian landscape metaphor. In order to fully capture the time-dependent (non-autonomous) transient behaviour of biological processes, we must be able to characterise potential landscapes and how they change over time. However, currently available mathematical tools focus on the asymptotic (steady-state) behaviour of autonomous dynamical systems, which restricts how biological systems are studied.

Results: We present a pragmatic first step towards a methodology for dealing with transient behaviours in non-autonomous systems. We propose a classification scheme for different kinds of such dynamics based on the simulation of a simple genetic toggle-switch model with time-variable parameters. For this low-dimensional system, we can calculate and explicitly visualise numerical approximations to the potential landscape. Focussing on transient dynamics in non-autonomous systems reveals a range of interesting and biologically relevant behaviours that would be missed in steady-state analyses of autonomous systems. Our simulation-based approach allows us to identify four qualitatively different kinds of dynamics: transitions, pursuits, and two kinds of captures. We describe these in detail, and illustrate the usefulness of our classification scheme by providing a number of examples that demonstrate how it can be employed to gain specific mechanistic insights into the dynamics of gene regulation.

Conclusions: The practical aim of our proposed classification scheme is to make the analysis of explicitly time-dependent transient behaviour tractable, and to encourage the wider use of non-autonomous models in systems biology. Our method is applicable to a large class of biological processes.

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Dynamical regimes of the toggle switch model. The toggle switch model canexhibit three different dynamical regimes depending on parameter values.(A) In the monostable regime, the phase portrait has one attractor pointonly (represented by the blue dot on the quasi-potential landscape). At thisattractor, both products of X and Y are present at lowconcentrations. (B) In the bistable regime, which gives the toggle switchits name, there are two attractor points (shown in different shades of blue) andone saddle (red) on a separatrix (black line), which separates the two basins ofattraction. The attractors correspond to high x, low y (darkblue), or low x, high y (light blue). The two factors nevercoexist when equilibrium is reached in this regime. (C) In the tristableregime, both bistable switch attractors and the steady state at low co-existingconcentrations are present (shown in different shades of blue). In addition, thereare two separatrices with associated saddle points (red). These regimes convertinto each other as follows (double-headed black arrows indicate reversibility ofbifurcations): the monostable attractor is converted into two bistable attractorsand a saddle point through a supercritical pitchfork bifurcation; the saddle inthe bistable regime is converted into an attractor and two additional saddles inthe tristable regime through a subcritical pitchfork bifurcation; the bistableattractors and their saddles collide and annihilate in two simultaneoussaddle-node (or fold) bifurcations to turn the tristable regime into a monostableone. Graph axes as in Figure 1B, Panel 4.
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Figure 2: Dynamical regimes of the toggle switch model. The toggle switch model canexhibit three different dynamical regimes depending on parameter values.(A) In the monostable regime, the phase portrait has one attractor pointonly (represented by the blue dot on the quasi-potential landscape). At thisattractor, both products of X and Y are present at lowconcentrations. (B) In the bistable regime, which gives the toggle switchits name, there are two attractor points (shown in different shades of blue) andone saddle (red) on a separatrix (black line), which separates the two basins ofattraction. The attractors correspond to high x, low y (darkblue), or low x, high y (light blue). The two factors nevercoexist when equilibrium is reached in this regime. (C) In the tristableregime, both bistable switch attractors and the steady state at low co-existingconcentrations are present (shown in different shades of blue). In addition, thereare two separatrices with associated saddle points (red). These regimes convertinto each other as follows (double-headed black arrows indicate reversibility ofbifurcations): the monostable attractor is converted into two bistable attractorsand a saddle point through a supercritical pitchfork bifurcation; the saddle inthe bistable regime is converted into an attractor and two additional saddles inthe tristable regime through a subcritical pitchfork bifurcation; the bistableattractors and their saddles collide and annihilate in two simultaneoussaddle-node (or fold) bifurcations to turn the tristable regime into a monostableone. Graph axes as in Figure 1B, Panel 4.

Mentions: The toggle switch model (1) exhibits different dynamical regimes depending on the valuesof its parameters (Figure 2A–C). Its name derives from thefact that it can exhibit bistability over a wide range of parameters. When in thisbistable region of parameter space, the underlying phase portrait has two attractingstates and one saddle point (Figure 2B). All phase portraitsassociated with parameters in the bistable range are topologically equivalent to eachother, meaning that they can be mapped onto each other by a continuous deformation ofphase space called a homeomorphism [34].


Classification of transient behaviours in a time-dependent toggle switch model.

Verd B, Crombach A, Jaeger J - BMC Syst Biol (2014)

Dynamical regimes of the toggle switch model. The toggle switch model canexhibit three different dynamical regimes depending on parameter values.(A) In the monostable regime, the phase portrait has one attractor pointonly (represented by the blue dot on the quasi-potential landscape). At thisattractor, both products of X and Y are present at lowconcentrations. (B) In the bistable regime, which gives the toggle switchits name, there are two attractor points (shown in different shades of blue) andone saddle (red) on a separatrix (black line), which separates the two basins ofattraction. The attractors correspond to high x, low y (darkblue), or low x, high y (light blue). The two factors nevercoexist when equilibrium is reached in this regime. (C) In the tristableregime, both bistable switch attractors and the steady state at low co-existingconcentrations are present (shown in different shades of blue). In addition, thereare two separatrices with associated saddle points (red). These regimes convertinto each other as follows (double-headed black arrows indicate reversibility ofbifurcations): the monostable attractor is converted into two bistable attractorsand a saddle point through a supercritical pitchfork bifurcation; the saddle inthe bistable regime is converted into an attractor and two additional saddles inthe tristable regime through a subcritical pitchfork bifurcation; the bistableattractors and their saddles collide and annihilate in two simultaneoussaddle-node (or fold) bifurcations to turn the tristable regime into a monostableone. Graph axes as in Figure 1B, Panel 4.
© Copyright Policy - open-access
Related In: Results  -  Collection

License 1 - License 2
Show All Figures
getmorefigures.php?uid=PMC4109741&req=5

Figure 2: Dynamical regimes of the toggle switch model. The toggle switch model canexhibit three different dynamical regimes depending on parameter values.(A) In the monostable regime, the phase portrait has one attractor pointonly (represented by the blue dot on the quasi-potential landscape). At thisattractor, both products of X and Y are present at lowconcentrations. (B) In the bistable regime, which gives the toggle switchits name, there are two attractor points (shown in different shades of blue) andone saddle (red) on a separatrix (black line), which separates the two basins ofattraction. The attractors correspond to high x, low y (darkblue), or low x, high y (light blue). The two factors nevercoexist when equilibrium is reached in this regime. (C) In the tristableregime, both bistable switch attractors and the steady state at low co-existingconcentrations are present (shown in different shades of blue). In addition, thereare two separatrices with associated saddle points (red). These regimes convertinto each other as follows (double-headed black arrows indicate reversibility ofbifurcations): the monostable attractor is converted into two bistable attractorsand a saddle point through a supercritical pitchfork bifurcation; the saddle inthe bistable regime is converted into an attractor and two additional saddles inthe tristable regime through a subcritical pitchfork bifurcation; the bistableattractors and their saddles collide and annihilate in two simultaneoussaddle-node (or fold) bifurcations to turn the tristable regime into a monostableone. Graph axes as in Figure 1B, Panel 4.
Mentions: The toggle switch model (1) exhibits different dynamical regimes depending on the valuesof its parameters (Figure 2A–C). Its name derives from thefact that it can exhibit bistability over a wide range of parameters. When in thisbistable region of parameter space, the underlying phase portrait has two attractingstates and one saddle point (Figure 2B). All phase portraitsassociated with parameters in the bistable range are topologically equivalent to eachother, meaning that they can be mapped onto each other by a continuous deformation ofphase space called a homeomorphism [34].

Bottom Line: We describe these in detail, and illustrate the usefulness of our classification scheme by providing a number of examples that demonstrate how it can be employed to gain specific mechanistic insights into the dynamics of gene regulation.The practical aim of our proposed classification scheme is to make the analysis of explicitly time-dependent transient behaviour tractable, and to encourage the wider use of non-autonomous models in systems biology.Our method is applicable to a large class of biological processes.

View Article: PubMed Central - HTML - PubMed

Affiliation: EMBL/CRG Research Unit in Systems Biology, Centre for Genomic Regulation (CRG), Barcelona, Spain. yogi.jaeger@crg.eu.

ABSTRACT

Background: Waddington's epigenetic landscape is an intuitive metaphor for the developmental and evolutionary potential of biological regulatory processes. It emphasises time-dependence and transient behaviour. Nowadays, we can derive this landscape by modelling a specific regulatory network as a dynamical system and calculating its so-called potential surface. In this sense, potential surfaces are the mathematical equivalent of the Waddingtonian landscape metaphor. In order to fully capture the time-dependent (non-autonomous) transient behaviour of biological processes, we must be able to characterise potential landscapes and how they change over time. However, currently available mathematical tools focus on the asymptotic (steady-state) behaviour of autonomous dynamical systems, which restricts how biological systems are studied.

Results: We present a pragmatic first step towards a methodology for dealing with transient behaviours in non-autonomous systems. We propose a classification scheme for different kinds of such dynamics based on the simulation of a simple genetic toggle-switch model with time-variable parameters. For this low-dimensional system, we can calculate and explicitly visualise numerical approximations to the potential landscape. Focussing on transient dynamics in non-autonomous systems reveals a range of interesting and biologically relevant behaviours that would be missed in steady-state analyses of autonomous systems. Our simulation-based approach allows us to identify four qualitatively different kinds of dynamics: transitions, pursuits, and two kinds of captures. We describe these in detail, and illustrate the usefulness of our classification scheme by providing a number of examples that demonstrate how it can be employed to gain specific mechanistic insights into the dynamics of gene regulation.

Conclusions: The practical aim of our proposed classification scheme is to make the analysis of explicitly time-dependent transient behaviour tractable, and to encourage the wider use of non-autonomous models in systems biology. Our method is applicable to a large class of biological processes.

Show MeSH
Related in: MedlinePlus