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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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Capture due to a change in the topology of the phase portrait. A captureresults from a trajectory being recruited into a new basin of attraction due tothe movement of a separatrix. In this example, the relevant separatrix iscreated and caused to move by a preceding bifurcation event, which leads to theappearance of a new attractor state, resulting in a change of phase spacetopology. Upper panels show (quasi-)potential surfaces, lower panels phaseportraits as in Figure 3C. The progress of time is shownthrough increasingly dark shading, and by the arrow at the bottom of thefigure. (A) The system starts off in the bistable regime and the initialconditions place the trajectory in the basin of the high x, lowy attractor (dark blue). (B) Changes in the values of theauto-activation thresholds (a and c, see equation 1) causethe system to undergo a subcritical pitchfork bifurcation and enter thetristable regime (see also Figure 2). At the time of thebifurcation, the trajectory is still attracted towards the dark blue attractor.(C) As auto-activation thresholds are further increased, the twoseparatrices surrounding the new attractor state (shown in light blue) separatefrom each other, enlarging the corresponding basin of attraction. A captureevent takes place as the separatrix ‘overtakes’ the trajectory,recruiting it into the new basin of attraction. The system will now convergetowards the light blue attractor. This change in basins of attraction isrepresented by the colour coding of the trajectory on the phase portrait.(D) As auto-activation thresholds are further increased, the systemwill transition from the tristable into the monostable regime (see also Figures2 and 4). This causes the darkblue attractors and their basins to disappear altogether, but does notinfluence the direction of the trajectory anymore, which will eventuallyconverge to the light blue attractor at low x and y. See alsoAdditional file 4, Supporting Movie S4.
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Figure 7: Capture due to a change in the topology of the phase portrait. A captureresults from a trajectory being recruited into a new basin of attraction due tothe movement of a separatrix. In this example, the relevant separatrix iscreated and caused to move by a preceding bifurcation event, which leads to theappearance of a new attractor state, resulting in a change of phase spacetopology. Upper panels show (quasi-)potential surfaces, lower panels phaseportraits as in Figure 3C. The progress of time is shownthrough increasingly dark shading, and by the arrow at the bottom of thefigure. (A) The system starts off in the bistable regime and the initialconditions place the trajectory in the basin of the high x, lowy attractor (dark blue). (B) Changes in the values of theauto-activation thresholds (a and c, see equation 1) causethe system to undergo a subcritical pitchfork bifurcation and enter thetristable regime (see also Figure 2). At the time of thebifurcation, the trajectory is still attracted towards the dark blue attractor.(C) As auto-activation thresholds are further increased, the twoseparatrices surrounding the new attractor state (shown in light blue) separatefrom each other, enlarging the corresponding basin of attraction. A captureevent takes place as the separatrix ‘overtakes’ the trajectory,recruiting it into the new basin of attraction. The system will now convergetowards the light blue attractor. This change in basins of attraction isrepresented by the colour coding of the trajectory on the phase portrait.(D) As auto-activation thresholds are further increased, the systemwill transition from the tristable into the monostable regime (see also Figures2 and 4). This causes the darkblue attractors and their basins to disappear altogether, but does notinfluence the direction of the trajectory anymore, which will eventuallyconverge to the light blue attractor at low x and y. See alsoAdditional file 4, Supporting Movie S4.

Mentions: In the first situation, the trajectory gets captured after a bifurcation event haslead to the creation of a new attractor state. This increases the number of attractorbasins and, in this way, introduces new separatrices into the phase portrait. Wesimulate this situation using the bistable-to-tristable transition caused by anincrease in the auto-activation threshold as described above (Figure 7, see also Additional file 4, Supporting MovieS4). In this example, a subcritical pitchfork bifurcation creates a new attractor andtwo associated saddles from a pre-existing saddle point (Figure 7A,B). This results in a change in phase space topology. What used to be asingle separatrix now ‘opens up’, giving rise to two different forkedseparatrices (Figure 7B,C). Further parameter changes thencause the new separatrices to move outward through phase space, catching up with, andovertaking, trajectories as they recruit points into the newly created and expandingbasin of attraction (Figure 7B–D).


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

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

Capture due to a change in the topology of the phase portrait. A captureresults from a trajectory being recruited into a new basin of attraction due tothe movement of a separatrix. In this example, the relevant separatrix iscreated and caused to move by a preceding bifurcation event, which leads to theappearance of a new attractor state, resulting in a change of phase spacetopology. Upper panels show (quasi-)potential surfaces, lower panels phaseportraits as in Figure 3C. The progress of time is shownthrough increasingly dark shading, and by the arrow at the bottom of thefigure. (A) The system starts off in the bistable regime and the initialconditions place the trajectory in the basin of the high x, lowy attractor (dark blue). (B) Changes in the values of theauto-activation thresholds (a and c, see equation 1) causethe system to undergo a subcritical pitchfork bifurcation and enter thetristable regime (see also Figure 2). At the time of thebifurcation, the trajectory is still attracted towards the dark blue attractor.(C) As auto-activation thresholds are further increased, the twoseparatrices surrounding the new attractor state (shown in light blue) separatefrom each other, enlarging the corresponding basin of attraction. A captureevent takes place as the separatrix ‘overtakes’ the trajectory,recruiting it into the new basin of attraction. The system will now convergetowards the light blue attractor. This change in basins of attraction isrepresented by the colour coding of the trajectory on the phase portrait.(D) As auto-activation thresholds are further increased, the systemwill transition from the tristable into the monostable regime (see also Figures2 and 4). This causes the darkblue attractors and their basins to disappear altogether, but does notinfluence the direction of the trajectory anymore, which will eventuallyconverge to the light blue attractor at low x and y. See alsoAdditional file 4, Supporting Movie S4.
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Figure 7: Capture due to a change in the topology of the phase portrait. A captureresults from a trajectory being recruited into a new basin of attraction due tothe movement of a separatrix. In this example, the relevant separatrix iscreated and caused to move by a preceding bifurcation event, which leads to theappearance of a new attractor state, resulting in a change of phase spacetopology. Upper panels show (quasi-)potential surfaces, lower panels phaseportraits as in Figure 3C. The progress of time is shownthrough increasingly dark shading, and by the arrow at the bottom of thefigure. (A) The system starts off in the bistable regime and the initialconditions place the trajectory in the basin of the high x, lowy attractor (dark blue). (B) Changes in the values of theauto-activation thresholds (a and c, see equation 1) causethe system to undergo a subcritical pitchfork bifurcation and enter thetristable regime (see also Figure 2). At the time of thebifurcation, the trajectory is still attracted towards the dark blue attractor.(C) As auto-activation thresholds are further increased, the twoseparatrices surrounding the new attractor state (shown in light blue) separatefrom each other, enlarging the corresponding basin of attraction. A captureevent takes place as the separatrix ‘overtakes’ the trajectory,recruiting it into the new basin of attraction. The system will now convergetowards the light blue attractor. This change in basins of attraction isrepresented by the colour coding of the trajectory on the phase portrait.(D) As auto-activation thresholds are further increased, the systemwill transition from the tristable into the monostable regime (see also Figures2 and 4). This causes the darkblue attractors and their basins to disappear altogether, but does notinfluence the direction of the trajectory anymore, which will eventuallyconverge to the light blue attractor at low x and y. See alsoAdditional file 4, Supporting Movie S4.
Mentions: In the first situation, the trajectory gets captured after a bifurcation event haslead to the creation of a new attractor state. This increases the number of attractorbasins and, in this way, introduces new separatrices into the phase portrait. Wesimulate this situation using the bistable-to-tristable transition caused by anincrease in the auto-activation threshold as described above (Figure 7, see also Additional file 4, Supporting MovieS4). In this example, a subcritical pitchfork bifurcation creates a new attractor andtwo associated saddles from a pre-existing saddle point (Figure 7A,B). This results in a change in phase space topology. What used to be asingle separatrix now ‘opens up’, giving rise to two different forkedseparatrices (Figure 7B,C). Further parameter changes thencause the new separatrices to move outward through phase space, catching up with, andovertaking, trajectories as they recruit points into the newly created and expandingbasin of attraction (Figure 7B–D).

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