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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 geometry 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 iscaused to move by a break in the symmetry of the repressive interactionsbetween X and Y. No new attractor states or separatrices arecreated. Therefore, the change in landscape topography is purely geometricaland does not affect the topology of phase space. Upper panels show(quasi-)potential surfaces, lower panels phase portraits as in Figure 3C. The progress of time is shown through increasingly darkshading, and by the arrow at the bottom of the figure. (A) The systemstarts off in the bistable regime and the initial conditions place thetrajectory in the basin of the low x, high y attractor (lightblue). (B) Changing the threshold of one of the repressive interactionsonly (b in equation 1) shrinks the basin of attraction of the lowx, high y attractor (light blue) causing the separatrix tomove towards the upper left corner of the phase portrait. (C) Theshifting separatrix catches up with the converging trajectory, recruiting itinto the basin of the high x, low y attractor (dark blue). Acapture event has taken place without any preceding bifurcation. This change inbasins of attraction is represented by the colour coding of the trajectory onthe phase portrait. (D) After the capture, the system converges to itsnew attractor (dark blue). Interestingly, the geometry of this capture has madethe trajectory loop over itself. Such self-crossing trajectories are neverobserved in autonomous dynamical systems. The subsequent saddle-nodebifurcation by which the saddle (red) and the light blue attractor annihilateeach other does not affect the trajectory any further. See also Additional file5, Supporting Movie S5.
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Figure 8: Capture due to a change in the geometry 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 iscaused to move by a break in the symmetry of the repressive interactionsbetween X and Y. No new attractor states or separatrices arecreated. Therefore, the change in landscape topography is purely geometricaland does not affect the topology of phase space. Upper panels show(quasi-)potential surfaces, lower panels phase portraits as in Figure 3C. The progress of time is shown through increasingly darkshading, and by the arrow at the bottom of the figure. (A) The systemstarts off in the bistable regime and the initial conditions place thetrajectory in the basin of the low x, high y attractor (lightblue). (B) Changing the threshold of one of the repressive interactionsonly (b in equation 1) shrinks the basin of attraction of the lowx, high y attractor (light blue) causing the separatrix tomove towards the upper left corner of the phase portrait. (C) Theshifting separatrix catches up with the converging trajectory, recruiting itinto the basin of the high x, low y attractor (dark blue). Acapture event has taken place without any preceding bifurcation. This change inbasins of attraction is represented by the colour coding of the trajectory onthe phase portrait. (D) After the capture, the system converges to itsnew attractor (dark blue). Interestingly, the geometry of this capture has madethe trajectory loop over itself. Such self-crossing trajectories are neverobserved in autonomous dynamical systems. The subsequent saddle-nodebifurcation by which the saddle (red) and the light blue attractor annihilateeach other does not affect the trajectory any further. See also Additional file5, Supporting Movie S5.

Mentions: In the second scenario, we consider a situation where the capture event is notpreceded by a bifurcation. In this case, a pre-existing separatrix is moving throughphase space due to parameter changes. This movement of the separatrix reconfiguresthe geometry of the basins of attraction without changing the topology of the phaseportrait. We simulate this event in our toggle switch model by introducing asymmetricchanges in the values of the thresholds determining the two mutually repressingregulatory interactions (i.e. b≠d in 1) (Figure 8, see also Additional file 5,Supporting Movie S5). This shifts the position of the separatrix in the bistableregime towards one of the two attractors without creating or annihilating any steadystates.


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 geometry 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 iscaused to move by a break in the symmetry of the repressive interactionsbetween X and Y. No new attractor states or separatrices arecreated. Therefore, the change in landscape topography is purely geometricaland does not affect the topology of phase space. Upper panels show(quasi-)potential surfaces, lower panels phase portraits as in Figure 3C. The progress of time is shown through increasingly darkshading, and by the arrow at the bottom of the figure. (A) The systemstarts off in the bistable regime and the initial conditions place thetrajectory in the basin of the low x, high y attractor (lightblue). (B) Changing the threshold of one of the repressive interactionsonly (b in equation 1) shrinks the basin of attraction of the lowx, high y attractor (light blue) causing the separatrix tomove towards the upper left corner of the phase portrait. (C) Theshifting separatrix catches up with the converging trajectory, recruiting itinto the basin of the high x, low y attractor (dark blue). Acapture event has taken place without any preceding bifurcation. This change inbasins of attraction is represented by the colour coding of the trajectory onthe phase portrait. (D) After the capture, the system converges to itsnew attractor (dark blue). Interestingly, the geometry of this capture has madethe trajectory loop over itself. Such self-crossing trajectories are neverobserved in autonomous dynamical systems. The subsequent saddle-nodebifurcation by which the saddle (red) and the light blue attractor annihilateeach other does not affect the trajectory any further. See also Additional file5, Supporting Movie S5.
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Figure 8: Capture due to a change in the geometry 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 iscaused to move by a break in the symmetry of the repressive interactionsbetween X and Y. No new attractor states or separatrices arecreated. Therefore, the change in landscape topography is purely geometricaland does not affect the topology of phase space. Upper panels show(quasi-)potential surfaces, lower panels phase portraits as in Figure 3C. The progress of time is shown through increasingly darkshading, and by the arrow at the bottom of the figure. (A) The systemstarts off in the bistable regime and the initial conditions place thetrajectory in the basin of the low x, high y attractor (lightblue). (B) Changing the threshold of one of the repressive interactionsonly (b in equation 1) shrinks the basin of attraction of the lowx, high y attractor (light blue) causing the separatrix tomove towards the upper left corner of the phase portrait. (C) Theshifting separatrix catches up with the converging trajectory, recruiting itinto the basin of the high x, low y attractor (dark blue). Acapture event has taken place without any preceding bifurcation. This change inbasins of attraction is represented by the colour coding of the trajectory onthe phase portrait. (D) After the capture, the system converges to itsnew attractor (dark blue). Interestingly, the geometry of this capture has madethe trajectory loop over itself. Such self-crossing trajectories are neverobserved in autonomous dynamical systems. The subsequent saddle-nodebifurcation by which the saddle (red) and the light blue attractor annihilateeach other does not affect the trajectory any further. See also Additional file5, Supporting Movie S5.
Mentions: In the second scenario, we consider a situation where the capture event is notpreceded by a bifurcation. In this case, a pre-existing separatrix is moving throughphase space due to parameter changes. This movement of the separatrix reconfiguresthe geometry of the basins of attraction without changing the topology of the phaseportrait. We simulate this event in our toggle switch model by introducing asymmetricchanges in the values of the thresholds determining the two mutually repressingregulatory interactions (i.e. b≠d in 1) (Figure 8, see also Additional file 5,Supporting Movie S5). This shifts the position of the separatrix in the bistableregime towards one of the two attractors without creating or annihilating any steadystates.

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