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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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Pursuit altering the direction of a trajectory. Pursuit behaviourresults from the movement of an attractor. In this case, the direction of thetrajectory is altered since the attractor moves against the flow. Upper panelsshow (quasi-)potential surfaces, lower panels phase portraits as in Figure3C. The progress of time is shown through increasinglydark shading, and by the arrow at the bottom of the figure. (A–D)Panels show the movement of the attractors towards the origin as activationstrength (represented by αx andαy in equation 1) is decreased over time.As the attractor ‘overtakes’ the trajectory (between panelsB and C), the angle of the flow changes drastically, leadingto a reversal in the direction of the trajectory (clearly visible in D).See also Additional file 3, Supporting Movie S3.
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Figure 6: Pursuit altering the direction of a trajectory. Pursuit behaviourresults from the movement of an attractor. In this case, the direction of thetrajectory is altered since the attractor moves against the flow. Upper panelsshow (quasi-)potential surfaces, lower panels phase portraits as in Figure3C. The progress of time is shown through increasinglydark shading, and by the arrow at the bottom of the figure. (A–D)Panels show the movement of the attractors towards the origin as activationstrength (represented by αx andαy in equation 1) is decreased over time.As the attractor ‘overtakes’ the trajectory (between panelsB and C), the angle of the flow changes drastically, leadingto a reversal in the direction of the trajectory (clearly visible in D).See also Additional file 3, Supporting Movie S3.

Mentions: Transition. A transition indicates the switch of a non-autonomous systemfrom one attractor state to another. Upper panels show (quasi-)potentialsurfaces, lower panels phase portraits as in Figure 3C.The progress of time is shown through increasingly dark shading, and by thearrow at the bottom of the figure. (A) The system starts off in thebistable regime. The trajectory’s initial conditions coincide with theattractor at high x, low y (dark blue). The trajectory istherefore at steady state at the outset. (B) Changes in auto-activationthresholds a and c (equation 1) over time cause the system toundergo a subcritical pitchfork bifurcation and enter the tristable regime (seealso Figure 2). (C, D) The trajectory does notswitch attractors immediately after the bifurcation occurs. However, it doesnot remain anchored to its current attractor either. Instead, it is left behindby the moving attractor, which it starts to pursue (see also Figures 5 and 6). (E) The system entersthe monostable regime as the two bistable attractors disappear via twosimultaneous saddle-node (or fold) bifurcations. The trajectory suddenly findsitself in an alternative basin of attraction, and eventually converges to thenew, monostable attractor with low x and y. This change inbasins of attraction is represented by a change in colour of the trajectoryshown in (E). See also Additional file 1, SupportingMovie S1.


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

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

Pursuit altering the direction of a trajectory. Pursuit behaviourresults from the movement of an attractor. In this case, the direction of thetrajectory is altered since the attractor moves against the flow. Upper panelsshow (quasi-)potential surfaces, lower panels phase portraits as in Figure3C. The progress of time is shown through increasinglydark shading, and by the arrow at the bottom of the figure. (A–D)Panels show the movement of the attractors towards the origin as activationstrength (represented by αx andαy in equation 1) is decreased over time.As the attractor ‘overtakes’ the trajectory (between panelsB and C), the angle of the flow changes drastically, leadingto a reversal in the direction of the trajectory (clearly visible in D).See also Additional file 3, Supporting Movie S3.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

Figure 6: Pursuit altering the direction of a trajectory. Pursuit behaviourresults from the movement of an attractor. In this case, the direction of thetrajectory is altered since the attractor moves against the flow. Upper panelsshow (quasi-)potential surfaces, lower panels phase portraits as in Figure3C. The progress of time is shown through increasinglydark shading, and by the arrow at the bottom of the figure. (A–D)Panels show the movement of the attractors towards the origin as activationstrength (represented by αx andαy in equation 1) is decreased over time.As the attractor ‘overtakes’ the trajectory (between panelsB and C), the angle of the flow changes drastically, leadingto a reversal in the direction of the trajectory (clearly visible in D).See also Additional file 3, Supporting Movie S3.
Mentions: Transition. A transition indicates the switch of a non-autonomous systemfrom one attractor state to another. Upper panels show (quasi-)potentialsurfaces, lower panels phase portraits as in Figure 3C.The progress of time is shown through increasingly dark shading, and by thearrow at the bottom of the figure. (A) The system starts off in thebistable regime. The trajectory’s initial conditions coincide with theattractor at high x, low y (dark blue). The trajectory istherefore at steady state at the outset. (B) Changes in auto-activationthresholds a and c (equation 1) over time cause the system toundergo a subcritical pitchfork bifurcation and enter the tristable regime (seealso Figure 2). (C, D) The trajectory does notswitch attractors immediately after the bifurcation occurs. However, it doesnot remain anchored to its current attractor either. Instead, it is left behindby the moving attractor, which it starts to pursue (see also Figures 5 and 6). (E) The system entersthe monostable regime as the two bistable attractors disappear via twosimultaneous saddle-node (or fold) bifurcations. The trajectory suddenly findsitself in an alternative basin of attraction, and eventually converges to thenew, monostable attractor with low x and y. This change inbasins of attraction is represented by a change in colour of the trajectoryshown in (E). See also Additional file 1, SupportingMovie S1.

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