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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.

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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There are also practical reasons to consider transient dynamics explicitly. For developmental processes that consist of continuous transitions between patterns rather than the production of a final output, it is impossible to decide a priori whether the system is representing a non-autonomous succession of steady states (see below), or whether its behaviour is truly transient (i. e. far from steady state). In the case of gap domain shifts, we have evidence for the latter [35], although there is no reason to assume that the two situations need to be mutually exclusive. The gap gene model analysed by Manu and colleagues [35] exhibits boundary shifts that are caused by trajectories following a canalising unstable manifold. Assuming steady state dynamics would collapse the trajectories representing these shifts into a single attractor point at the final configuration of gene expression. Any such analysis would miss the relevant underlying features of phase space (the transient manifold), and therefore fail to provide a proper characterisation and explanation for the observed gene expression dynamics (the shift in domain position over time).

Still, it is rare to find studies based on explicitly non-autonomous models in the literature, and most authors avoid the challenge of dealing with dynamical systems where the parameters representing external cues are time-dependent. This is the case in the study of gap domain shifts by Manu and colleagues [35, 89], where maternal morphogen gradients providing regulatory input to the system were assumed to reach steady state before gap gene boundary positioning was analysed. Such simplifications can be risky, especially when describing biological phenomena where the time scales of change in parameters and state variables are of similar order. In such cases, time scales should not be separated, nor quasi-steady states considered since it is easy for dynamical properties and behaviours of the system to be missed or misinterpreted under these conditions.

To develop our methodology for analysing transient behaviour in non-autonomous dynamical systems, we use a simple toggle switch model (see [22], and references therein) with time-dependent parameters. We consider two interacting genes X and Y (Figure 1B, panel 1). Concentrations of the corresponding protein products are labelled x and y. X and Y mutually repress each other, are linearly activated by external signals and can auto-activate themselves (Figure 1B, panel 1). Protein products decay linearly dependent on their concentration. The mathematical formulation of our toggle switch model is thus given by

Dynamical regimes of the toggle switch model. The toggle switch model can exhibit three different dynamical regimes depending on parameter values. (A) In the monostable regime, the phase portrait has one attractor point only (represented by the blue dot on the quasi-potential landscape). At this attractor, both products of X and Y are present at low concentrations. (B) In the bistable regime, which gives the toggle switch its name, there are two attractor points (shown in different shades of blue) and one saddle (red) on a separatrix (black line), which separates the two basins of attraction. The attractors correspond to high x, low y (dark blue), or low x, high y (light blue). The two factors never coexist when equilibrium is reached in this regime. (C) In the tristable regime, both bistable switch attractors and the steady state at low co-existing concentrations are present (shown in different shades of blue). In addition, there are two separatrices with associated saddle points (red). These regimes convert into each other as follows (double-headed black arrows indicate reversibility of bifurcations): the monostable attractor is converted into two bistable attractors and a saddle point through a supercritical pitchfork bifurcation; the saddle in the bistable regime is converted into an attractor and two additional saddles in the tristable regime through a subcritical pitchfork bifurcation; the bistable attractors and their saddles collide and annihilate in two simultaneous saddle-node (or fold) bifurcations to turn the tristable regime into a monostable one. Graph axes as in Figure 1B, Panel 4.

The toggle switch model has two other dynamical regimes: monostable and tristable. Phase portraits associated with parameters in the monostable range have only one attractor point (Figure 2A), while those in the tristable range have three attractor states and two saddle points (Figure 2C). Again, phase portraits within each regime are topologically equivalent to each other. While phase space can be geometrically deformed within each regime (through movements of attractors or separatrices), its topology only changes when one regime transitions into another through different types of bifurcations [32, 34] (Figure 2). The transition from monostable to bistable is known to be governed by a supracritical pitchfork, the transition from bistable to tristable involves a subcritical pitchfork bifurcation, and the transition from tristable to monostable takes place through two simultaneous saddle-node bifurcations involving the two attractors labelled in darker blue in Figure 2C.

Condition (3) will not always be met. In particular, dynamical systems representing gene interaction networks are not in general gradient systems, and therefore an associated potential function and landscape may not exist. In such cases, we can still take advantage of the visualisation power of potential landscapes by approximating the true potential using a numerical method. The numerical approximation method we adopt for our study was developed by Bhattacharya and colleagues [38] using a toggle switch model very similar to the one used here. This allows us to calculate a quasi-potential landscape for any specific set of fixed parameter values.

Finally, interpolation of all the normalised trajectories results in a continuous quasi-potential landscape. Bhattacharya et al.[38] validated this approach by demonstrating that the quasi-potential values of the steady states were inversely correlated with their probability of occurrence using a stochastic version of the toggle switch dynamical system.

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

Numerical approximation of non-autonomous trajectories. (A) Toggle switch network. Red arrows representing auto-activation indicate time-dependence of threshold parameters a x and a y (see equation 1). (B) Values of auto-activation thresholds a x and a y are altered simultaneously and linearly over 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 as computational efficiency allows. (C) During every time step, parameters can be considered constant, and the phase portrait and (quasi-)potential landscape are calculated for the current set of parameter values. Trajectories are then integrated over the duration of the time step using the previous end point as the current initial condition. The result is mapped onto the potential surface. The four panels in (C) show examples of potential landscapes (upper panels) calculated based on sets of parameter values at time points indicated by dashed arrows from (B). Important events altering the geometry of the trajectory are indicated. Lower panels show the corresponding instantaneous phase portraits with the integrated progression of the trajectory across time steps. See Model and methods for details.

The smaller the time increments considered, the better we are able to approximate continuous changes in parameter values, and the consequent changes to the associated phase portrait and quasi-potential landscape. Such accurate approximation allows us to faithfully reproduce non-autonomous trajectories produced by models with continuously time-variable parameters. This is done by integrating trajectories using constant parameters during each time step, and then using the resulting end position in phase space as the initial condition for the next time step. The resulting integrated trajectories can then be visualised by mapping them from the underlying phase plane onto the associated quasi-potential landscape as described above. This allows us to track and analyse in detail how changes in the phase portrait and quasi-potential landscape shape the trajectories as the values of the parameters are changing. ff782bc1db