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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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Numerical approximation of non-autonomous trajectories.(A) Toggleswitch network. Red arrows representing auto-activation indicatetime-dependence of threshold parameters ax anday (see equation 1). (B) Values ofauto-activation thresholds ax anday are altered simultaneously and linearlyover time. The graph shows the step-wise approximation of a continuous change,in this case, an increase in parameter values. Step size is taken as small ascomputational efficiency allows. (C) During every time step, parameterscan be considered constant, and the phase portrait and (quasi-)potentiallandscape are calculated for the current set of parameter values. Trajectoriesare then integrated over the duration of the time step using the previous endpoint as the current initial condition. The result is mapped onto the potentialsurface. The four panels in (C) show examples of potential landscapes (upperpanels) calculated based on sets of parameter values at time points indicatedby dashed arrows from (B). Important events altering the geometry of thetrajectory are indicated. Lower panels show the corresponding instantaneousphase portraits with the integrated progression of the trajectory across timesteps. See Model and methods for details.
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Figure 3: Numerical approximation of non-autonomous trajectories.(A) Toggleswitch network. Red arrows representing auto-activation indicatetime-dependence of threshold parameters ax anday (see equation 1). (B) Values ofauto-activation thresholds ax anday are altered simultaneously and linearlyover time. The graph shows the step-wise approximation of a continuous change,in this case, an increase in parameter values. Step size is taken as small ascomputational efficiency allows. (C) During every time step, parameterscan be considered constant, and the phase portrait and (quasi-)potentiallandscape are calculated for the current set of parameter values. Trajectoriesare then integrated over the duration of the time step using the previous endpoint as the current initial condition. The result is mapped onto the potentialsurface. The four panels in (C) show examples of potential landscapes (upperpanels) calculated based on sets of parameter values at time points indicatedby dashed arrows from (B). Important events altering the geometry of thetrajectory are indicated. Lower panels show the corresponding instantaneousphase portraits with the integrated progression of the trajectory across timesteps. See Model and methods for details.

Mentions: As we have argued in the Background Section, we cannot generally assume thatparameter values remain constant over time when modelling biological processes. Wetake a step-wise approximation approach to the change in parameter values to addressthis problem (Figure 3). We chose a time increment (step size)as small as possible. Parameter values are kept constant for the duration of eachtime step. As a consequence, the associated phase portrait will also remain constantduring this time interval, and is visualised for each step by calculating aquasi-potential landscape as described in the previous section (Figure 3C, top row).


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

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

Numerical approximation of non-autonomous trajectories.(A) Toggleswitch network. Red arrows representing auto-activation indicatetime-dependence of threshold parameters ax anday (see equation 1). (B) Values ofauto-activation thresholds ax anday are altered simultaneously and linearlyover time. The graph shows the step-wise approximation of a continuous change,in this case, an increase in parameter values. Step size is taken as small ascomputational efficiency allows. (C) During every time step, parameterscan be considered constant, and the phase portrait and (quasi-)potentiallandscape are calculated for the current set of parameter values. Trajectoriesare then integrated over the duration of the time step using the previous endpoint as the current initial condition. The result is mapped onto the potentialsurface. The four panels in (C) show examples of potential landscapes (upperpanels) calculated based on sets of parameter values at time points indicatedby dashed arrows from (B). Important events altering the geometry of thetrajectory are indicated. Lower panels show the corresponding instantaneousphase portraits with the integrated progression of the trajectory across timesteps. See Model and methods for details.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

Figure 3: Numerical approximation of non-autonomous trajectories.(A) Toggleswitch network. Red arrows representing auto-activation indicatetime-dependence of threshold parameters ax anday (see equation 1). (B) Values ofauto-activation thresholds ax anday are altered simultaneously and linearlyover time. The graph shows the step-wise approximation of a continuous change,in this case, an increase in parameter values. Step size is taken as small ascomputational efficiency allows. (C) During every time step, parameterscan be considered constant, and the phase portrait and (quasi-)potentiallandscape are calculated for the current set of parameter values. Trajectoriesare then integrated over the duration of the time step using the previous endpoint as the current initial condition. The result is mapped onto the potentialsurface. The four panels in (C) show examples of potential landscapes (upperpanels) calculated based on sets of parameter values at time points indicatedby dashed arrows from (B). Important events altering the geometry of thetrajectory are indicated. Lower panels show the corresponding instantaneousphase portraits with the integrated progression of the trajectory across timesteps. See Model and methods for details.
Mentions: As we have argued in the Background Section, we cannot generally assume thatparameter values remain constant over time when modelling biological processes. Wetake a step-wise approximation approach to the change in parameter values to addressthis problem (Figure 3). We chose a time increment (step size)as small as possible. Parameter values are kept constant for the duration of eachtime step. As a consequence, the associated phase portrait will also remain constantduring this time interval, and is visualised for each step by calculating aquasi-potential landscape as described in the previous section (Figure 3C, top row).

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