Transient dynamics around unstable periodic orbits in the generalized repressilator model

Abstract

We study the temporal dynamics of the generalized repressilator, a network of coupled repressing genes arranged in a directed ring topology, and give analytical conditions for the emergence of a finite sequence of unstable periodic orbits that lead to reachable long-lived oscillating transients. Such transients dominate the finite time horizon dynamics that is relevant in confined, noisy environments such as bacterial cells (see our previous work [Strelkowa and Barahona, J. R. Soc. Interface 7, 1071 (2010)]), and are therefore of interest for bioengineering and synthetic biology. We show that the family of unstable orbits possesses spatial symmetries and can also be understood in terms of traveling wave solutions of kink-like topological defects. The long-lived oscillatory transients correspond to the propagation of quasistable two-kink configurations that unravel over a long time. We also assess the similarities between the generalized repressilator model and other unidirectionally coupled electronic systems, such as magnetic flux gates, which have been implemented experimentally. Bacterial cells are noisy environments that evolve over a limited time horizon. Under these conditions, the relevant dynamics is not dictated exclusively by the long-term attractors of the system but can be dominated by reachable long-lived transients with qualitatively different observable dynamics. Such transients are an important feature of naturally occurring biological networks, in particular in developmental biology, and can also be exploited for forward design of genetic elements in Synthetic Biology. Here, we investigate the dynamics of unidirectional rings of genetic repressors and show that their observable dynamics is dominated by oscillatory transients around unstable periodic orbits with strong temporal symmetries in the discrete ring lattice. The oscillatory transients around such orbits can be seen to correspond to the propagation of groups of kink-like excitations against the background of a dimerized ("updown") solution, a picture with strong parallelisms with classic discrete lattice models, 1-3 some of which have been implemented experimentally with electronic elements.

Transient dynamics and their control in time-delay autonomous Boolean ring networks

“…Long transient times were also reported in a model for inhibitory ring GRNs that does not incorporate time delays, but instead includes an intermediate step in gene expression, which gives an effective delay time [68].…”

Stochastic oscillatory dynamics of generalized repressilators

“…The oscillations and their quality are simultaneously affected by changes in the parameter set (c i ) and the system size Ω. A reasonable experimental strategy to sweep the bifurcation parameter c 1 studied in our earlier work [10] is to change the gene copy number, d 0 , since they are linearly related. The gene copy number d 0 can be manipulated experimentally and used as a "biological knob" to induce oscillations in synthetic repressilators implemented in living bacterial populations.…”

“…where J is the drift matrix equal to the deterministic Jacobian (see [10]) and σ is the diffusion matrix. The stationary stochastic power spectrum of this linear SDE is calculated with the well-known formula for multi-variable linear SDEs ( [18], Chap.…”

“…The generalized repressilator model, a genetic circuit of coupled repressing genes arranged in the form of a directed ring, has been extensively explored in the deterministic setting using nonlinear complexity analysis [8,9,7,10] and the special case of three genes has already been implemented in living bacterial cells using synthetic biology tools [11]. The model shows qualitatively different steady state characteristics if the ring consists of even number of repressors compared to rings with odd number of repressors [8].…”

“…We explore the impact of low copy number noise on the onset and quality of oscillations in the generalized repressilator model with odd-number of elements. In our previous work [Strelkowa & Barahona, 2011] we applied deterministic complexity analysis and provided analytical conditions for the emergence of stable limit cycles via Hopf Bifurcations in oddnumbered rings. Here, we extend this analysis to the stochastic description of the model and study the influence of a biochemical design 'knob' -the gene copy number -on the onset and quality of the oscillations.…”