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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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Waddington’s epigenetic landscape and potential surfaces.(A)Two different views of Waddington’s epigenetic landscape taken from“The Strategy of the Genes” published in 1957 [3]. The top panel shows a top view of the landscape. The path that theball will follow represents the developmental trajectory (chreode) of a givensystem. Valleys indicate alternative differentiation pathways, branch points implydevelopmental decisions. The bottom panel shows the view from below the landscape.It illustrates how genes remodel the surface by pulling on it through ropes.Waddington used this sketch to show how the landscape’s topography changesduring development and evolution. (B) Panel 1: Diagrammatic representationof the toggle switch network used in the simulations. Activating interactions areindicated by arrows, repressing ones by T-bar connectors. See Model and methodssection for detailed parameter descriptions. Panel 2: Mathematical formulation ofthe toggle switch model. x and y indicate concentrations of theprotein products of genes X and Y. Ordinary differentialequations define the rate of change in protein concentrations( and ). Sigmoid functions with fixed Hill coefficients of4 are used to represent auto-activation and mutual repression. Decay and externalactivation are taken to be linear. Parameters as in Panel 1. Panel 3: Phaseportrait for a constant set of parameter values of the toggle switch model in thebistable regime. X- and Y-axes represent protein concentrationsx and y. We use this example to illustrate relevant featuresof phase space: arrows indicate flow, blue points mark the position of stablesteady states (attractors), the red point shows an unstable steady state (saddle)lying on the separatrix that divides the two basins of attraction (grey line). Seethe main text for detailed descriptions of the highlighted features. Panel 4:Quasi-potential landscape associated to the the phase portrait shown in Panel 3.The steepness of the quasi-potential surface correlates with the flow at eachcorresponding point on the phase portrait. Attractors, saddle, and separatrix areindicated as in Panel 3. See main text for details.
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Figure 1: Waddington’s epigenetic landscape and potential surfaces.(A)Two different views of Waddington’s epigenetic landscape taken from“The Strategy of the Genes” published in 1957 [3]. The top panel shows a top view of the landscape. The path that theball will follow represents the developmental trajectory (chreode) of a givensystem. Valleys indicate alternative differentiation pathways, branch points implydevelopmental decisions. The bottom panel shows the view from below the landscape.It illustrates how genes remodel the surface by pulling on it through ropes.Waddington used this sketch to show how the landscape’s topography changesduring development and evolution. (B) Panel 1: Diagrammatic representationof the toggle switch network used in the simulations. Activating interactions areindicated by arrows, repressing ones by T-bar connectors. See Model and methodssection for detailed parameter descriptions. Panel 2: Mathematical formulation ofthe toggle switch model. x and y indicate concentrations of theprotein products of genes X and Y. Ordinary differentialequations define the rate of change in protein concentrations( and ). Sigmoid functions with fixed Hill coefficients of4 are used to represent auto-activation and mutual repression. Decay and externalactivation are taken to be linear. Parameters as in Panel 1. Panel 3: Phaseportrait for a constant set of parameter values of the toggle switch model in thebistable regime. X- and Y-axes represent protein concentrationsx and y. We use this example to illustrate relevant featuresof phase space: arrows indicate flow, blue points mark the position of stablesteady states (attractors), the red point shows an unstable steady state (saddle)lying on the separatrix that divides the two basins of attraction (grey line). Seethe main text for detailed descriptions of the highlighted features. Panel 4:Quasi-potential landscape associated to the the phase portrait shown in Panel 3.The steepness of the quasi-potential surface correlates with the flow at eachcorresponding point on the phase portrait. Attractors, saddle, and separatrix areindicated as in Panel 3. See main text for details.

Mentions: In Waddington’s epigenetic landscape, the current state of a developing system isindicated by a ball on an undulated surface (Figure 1A, top panel) [3,5,6]. The topography of this landscape determines the developmental potential orrepertoire of the system. The top-most edge of the surface shown in Figure 1A (top panel) represents the initial state of the system given by,for example, a particular set of initial protein concentrations in a cell. Valleys inthe landscape symbolise the various differentiation pathways that are available. Thelandscape’s topography—together with the initial state—determine adevelopmental trajectory that follows a particular valley. The structure of thelandscape is such that, if the system is slightly perturbed, the sloping valley wallswill cause it to correct and readjust its trajectory. This behavour is called‘homeorhesis’—the maintenance of a dynamic trajectory—in analogyto the more static concept of homeostasis—the maintenance of a (steady) state ofthe system [3]. The wider and deeper a valley is, the more canalised the developmentaltrajectory. Waddington named such canalised trajectories ‘chreodes’.


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

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

Waddington’s epigenetic landscape and potential surfaces.(A)Two different views of Waddington’s epigenetic landscape taken from“The Strategy of the Genes” published in 1957 [3]. The top panel shows a top view of the landscape. The path that theball will follow represents the developmental trajectory (chreode) of a givensystem. Valleys indicate alternative differentiation pathways, branch points implydevelopmental decisions. The bottom panel shows the view from below the landscape.It illustrates how genes remodel the surface by pulling on it through ropes.Waddington used this sketch to show how the landscape’s topography changesduring development and evolution. (B) Panel 1: Diagrammatic representationof the toggle switch network used in the simulations. Activating interactions areindicated by arrows, repressing ones by T-bar connectors. See Model and methodssection for detailed parameter descriptions. Panel 2: Mathematical formulation ofthe toggle switch model. x and y indicate concentrations of theprotein products of genes X and Y. Ordinary differentialequations define the rate of change in protein concentrations( and ). Sigmoid functions with fixed Hill coefficients of4 are used to represent auto-activation and mutual repression. Decay and externalactivation are taken to be linear. Parameters as in Panel 1. Panel 3: Phaseportrait for a constant set of parameter values of the toggle switch model in thebistable regime. X- and Y-axes represent protein concentrationsx and y. We use this example to illustrate relevant featuresof phase space: arrows indicate flow, blue points mark the position of stablesteady states (attractors), the red point shows an unstable steady state (saddle)lying on the separatrix that divides the two basins of attraction (grey line). Seethe main text for detailed descriptions of the highlighted features. Panel 4:Quasi-potential landscape associated to the the phase portrait shown in Panel 3.The steepness of the quasi-potential surface correlates with the flow at eachcorresponding point on the phase portrait. Attractors, saddle, and separatrix areindicated as in Panel 3. See main text for details.
© Copyright Policy - open-access
Related In: Results  -  Collection

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Figure 1: Waddington’s epigenetic landscape and potential surfaces.(A)Two different views of Waddington’s epigenetic landscape taken from“The Strategy of the Genes” published in 1957 [3]. The top panel shows a top view of the landscape. The path that theball will follow represents the developmental trajectory (chreode) of a givensystem. Valleys indicate alternative differentiation pathways, branch points implydevelopmental decisions. The bottom panel shows the view from below the landscape.It illustrates how genes remodel the surface by pulling on it through ropes.Waddington used this sketch to show how the landscape’s topography changesduring development and evolution. (B) Panel 1: Diagrammatic representationof the toggle switch network used in the simulations. Activating interactions areindicated by arrows, repressing ones by T-bar connectors. See Model and methodssection for detailed parameter descriptions. Panel 2: Mathematical formulation ofthe toggle switch model. x and y indicate concentrations of theprotein products of genes X and Y. Ordinary differentialequations define the rate of change in protein concentrations( and ). Sigmoid functions with fixed Hill coefficients of4 are used to represent auto-activation and mutual repression. Decay and externalactivation are taken to be linear. Parameters as in Panel 1. Panel 3: Phaseportrait for a constant set of parameter values of the toggle switch model in thebistable regime. X- and Y-axes represent protein concentrationsx and y. We use this example to illustrate relevant featuresof phase space: arrows indicate flow, blue points mark the position of stablesteady states (attractors), the red point shows an unstable steady state (saddle)lying on the separatrix that divides the two basins of attraction (grey line). Seethe main text for detailed descriptions of the highlighted features. Panel 4:Quasi-potential landscape associated to the the phase portrait shown in Panel 3.The steepness of the quasi-potential surface correlates with the flow at eachcorresponding point on the phase portrait. Attractors, saddle, and separatrix areindicated as in Panel 3. See main text for details.
Mentions: In Waddington’s epigenetic landscape, the current state of a developing system isindicated by a ball on an undulated surface (Figure 1A, top panel) [3,5,6]. The topography of this landscape determines the developmental potential orrepertoire of the system. The top-most edge of the surface shown in Figure 1A (top panel) represents the initial state of the system given by,for example, a particular set of initial protein concentrations in a cell. Valleys inthe landscape symbolise the various differentiation pathways that are available. Thelandscape’s topography—together with the initial state—determine adevelopmental trajectory that follows a particular valley. The structure of thelandscape is such that, if the system is slightly perturbed, the sloping valley wallswill cause it to correct and readjust its trajectory. This behavour is called‘homeorhesis’—the maintenance of a dynamic trajectory—in analogyto the more static concept of homeostasis—the maintenance of a (steady) state ofthe system [3]. The wider and deeper a valley is, the more canalised the developmentaltrajectory. Waddington named such canalised trajectories ‘chreodes’.

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