The effect of characteristic times on collective modes of two quorum sensing coupled identical ring oscillators

https://doi.org/10.1016/j.chaos.2021.111176Get rights and content

Highlights

  • Ring oscillator dynamics depend on heterogeneity of the variables’ timescales.

  • Two quorum sensing coupled ring oscillators create a rich variety of collective modes.

  • Numerical bifurcation analysis reveals the transitions and coexistence of modes.

  • Heterogeneity of timescales dramatically alters the structure of phase diagram.

  • Reduced heterogeneity restores dominance of steady states and in-phase limit cycle.

Abstract

There has long been interest in methods to achieve variability of collective modes in systems of coupled identical oscillators. Extensive multistability and a dynamically rich parameter space exist for two identical ring oscillators mean-field coupled by the quorum sensing mechanism when there is significant heterogeneity of the timescales of the ring elements’ variables. These findings have been shown in numerical and electrical circuit models based on the Repressilator, a ring oscillator using repressor genes as the inverter elements. Here we investigate the effect of reducing the degree of timescale heterogeneity. We find the 2D parameter map of dynamical regimes for the previously established heterogeneous timescale, using the coupling strength and the isolated oscillator strength for the parameters. The map shows a sea of anti-phase (AP) oscillation, with a large island of complex oscillations bounded by Neimark-Sacker torus bifurcation of the AP. The island is dominated by chaos produced by period doubling cascades of two prominent limit cycles (an unusual resonant cycle and a highly asymmetric cycle) and by torus destruction. Extensive regions of multistability exist, including between different oscillatory states, and, notably, coexistence of the steady state region and oscillatory states. Using numerical bifurcation analysis with timescale heterogeneity as a continuation parameter, we follow the evolution of the collective modes, reveal the appearance of new dynamical regimes, and determine changes in the multistability. We find that decreasing the heterogeneity causes: period-doubling cascades of the AP outside of the island, thereby creating new regimes of complexity; an increase in the dominance of chaos due to a reduction in the size of regions dominated by limit cycles; loss of stable oscillatory states inside steady state regions, converting those regions from multistable to sole dynamic steady state regions; and the appearance of in-phase oscillations and its sole dominance.

Introduction

The emergence of collective dynamical modes and their multistability in populations of identical oscillators have been of interest for several decades in the study of many natural phenomena. In particular, they are of principle importance for understanding the functioning of neural, genetic, and ecological networks. The mechanisms of formation of collective modes depend on the properties of the isolated oscillators and the types of coupling schemes. Synthetic genetic oscillators are a suitable platform for studying the variability of the collective modes due to the flexibility of the bacterial quorum sensing mechanism which is added to the genetic network to provide the intercellular communication [1], [2], [3], [4], [5].

Since 2000 the Repressilator [6] is a popular oscillator for designing isolated synthetic genetic constructions [7], [8], [9], [10] as well as for the investigations of collective modes in their populations [11], [12], [13]. The minimal version of a Repressilator consists of three genes (a, b, c) (nonlinear elements) whose protein products (A, B, C) repress the transcriptions of each other unidirectionally in a cyclic way (. . . A–|B–|C–|A. . .). The cooperativity of transcription repression, which is the core process of the Repressilator, is typically described by the Hill function α/(1 + xn), where x is the repressor (A, B, C) abundance and n is the degree of repression cooperativity. The dynamics of the isolated Repressilator are controlled by the value of n, which sets the steepness of repression, by the repressor production rate α, which determines the amplitudes of the limit cycles for variables (A, B, C), and by the repressor degradation rates (due to protease catalyzed breakdown of proteins), which are primarily responsible for the period of the limit cycles. Bifurcation analysis of the isolated Repressilator shows that all enumerated parameters can convert limit cycle to stable steady state via supercritical Andronov-Hopf bifurcation under small value of α, and/or n < 2, and a large difference of repressor's degradation rates (timescales). However, this simple three-dimensional limit cycle does not show transitions to complex regimes like chaos over a very large parameter space.

Recently, we investigated the dynamics of two identical three-dimensional quorum sensing (QS) coupled Repressilators using numerical and electronic circuit models [14], [15], [16], [17]. The core of bacterial QS is the production of small signal molecules (autoinducer, AI) which can, first, easily diffuse across the cell membrane and external medium and, second, work to activate/repress transcription of an intended target gene. In our model QS was implemented in a genetic oscillator as an additional element not required for the generation of the auto-oscillations. Gene b and the gene controlling autoinducer production in the cell share the same promoter while the autoinducer activates the expression of gene c. This difference between the positions of genes producing AI and accepting its activity in the limits of the 3-gene ring is important for the generation of multistability. Dynamics of AI depends on the rates of its production, degradation, and diffusion between cell and external medium where the fast averaging of AI concentration occurs. In the limit of quasi-steady state approximation for AI dynamics in medium, we use the degree of AI dilution as one of the principal bifurcation parameters (Q) taking others fixed as in previous works.

Many areas of parameters α, n, and Q were found with very rich sets of attractors. It has been shown that without any additional external stimulation, the anti-phase limit cycle (APLC) converts to a two-frequency torus, which gives rise to a family of resonant limit cycles, including a cycle with a winding number 5:5 (both oscillators have period-5 limit cycles, see LC5:5 time series in the inset in Fig. 5), which creates extended areas with chaos via period-doubling cascades. Despite using identical Repressilators, many periodic windows with inhomogeneous limit cycles have been found inside chaotic areas including the unexpected limit cycle with a 1:2 winding number (LC1:2), which is stable over a large parameter plane and coexists with other attractors including chaos. However, the investigated parameters so far do not consider the temporal structure of the isolated Repressilator's limit cycle because the set of non-uniform repressor degradation rates was fixed throughout these studies.

It has long been known that timescales of variables in the ODE system of isolated oscillators can dramatically change the development and stability of collective modes as well as the structure of phase diagram of coupled oscillators in the case of multistability. For example, in two FitzHugh-Nagumo like two-dimensional membrane oscillators coupled through the diffusion of slow variables the location of stable anti-phase limit cycle on the parameter plane and its coexistence with stable inhomogeneous steady state is controlled by the value of timescales separation [18]. Kopell and Sommers [19] demonstrated that coupled relaxation oscillators “approach synchrony using mechanisms very different from that of oscillators with a more sinusoidal waveform”. As global coupling is added to the arrays of slightly different harmonic electrochemical oscillators [20] the distribution of frequencies become narrower until synchronization is attained. However, configuring the arrays as relaxation oscillators lead to more complicated behaviors which include formation of anti-phase oscillations and clusters formation.

In three-variable ring oscillators with cyclic repression the heterogeneity of repressor's degradation times results not only in variations of the phase point velocity along limit cycle but in the significant difference of repressor's amplitudes. Our numerical estimations and the rigorous analysis of the effects of non-uniform rate constants published in [21] show that there are broad intervals of rates where the limit cycle of an isolated Repressilator is stable. In this connection it is important to continue the study of the remarkable multistability in two QS-coupled Repressilators [14], [15], [16] in the case of reducing the heterogeneity of degradation rates.

To perform this work, we choose as a starting point the previously used timescales [16] of identical rates for repressors B and C, five times faster rate for A, and present the two-parameter (Q-α)-phase diagram with the major attractors. Some features of this diagram such as the highly asymmetric LC1:2 were discussed previously [16]. Here we present new features, including the unusual coexistence of different forms of the resonant LC5:5. Next, numerical bifurcation continuations of all selected attractors as a function of repressor A timescale reveal the emergence of new types of attractors, stabilization of old ones, and dramatic complication of the (Q-α)-phase diagrams structure. These calculations prompted to what extent the difference between repressors degradation rates should be reduced to reveal and to focus on the qualitative changes in the (Q-α)-phase diagrams. Two of the diagrams are presented in detail demonstrating the modifications of multistable dynamics which, nevertheless, remain very rich. We find that reducing the timescale heterogeneity results in: creation of new regions of complex behavior due to stimulation of period-doubling cascades of the AP; increased prevalence of chaos in the complex region due to the shrinking of prominent limit cycle regions; conversion of selected regions of coexistence of oscillatory and steady states to sole dynamic steady state regions due to loss of stability of the oscillatory states; and the appearance of regions with stable in-phase limit cycles despite the dominance of repulsive coupling over the main parts of parameter plane.

Section snippets

Coupled repressilator model

We investigate the dynamics of two identical Repressilators interacting via QS coupling as used previously [22,23]. Fig. 1 shows two Repressilators located in different cells and coupled by QS mechanism via the external medium. The three genes in each ring produce mRNAs (a, b, c) which are translated to proteins (transcription factors) (A, B, C), and they impose Hill function inhibition of gene expression in a cyclic order by the preceding gene. The QS feedback is maintained by the autoinducer

Results

The results are organized to present first all the important dynamical regimes for the timescale βj= (0.5,0.1,0.1) using a 2D map in the parameter space formed by the coupling strength Q and the isolated oscillator strength α. Next, the evolution of the dynamical regimes as the timescale is changed is shown in 2D maps of Q vs timescale parameter β1. Finally, (Q-α) regime maps are shown for the timescales β1 = 0.4 and 0.3 showing how the regimes evolve and how new regimes appear.

Discussion and conclusions

The appearance and fast construction of synthetic genetic oscillators provided a new area for the study of multistability of collective modes in populations of identical oscillators implemented in living cells. The mechanisms of formation of collective modes depend on the properties of isolated oscillators as well as the types of coupling schemes. Here we considered their manifestations in the model of two identical 3-dimensional Repressilators indirectly coupled via production and diffusion of

Credit authorship contribution statement

Authors contributed equally to all sections of this work.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgement

This work was partially supported by the grant of Russian Foundation for Basic Research (project No. 19–02–00610) and by the University of North Carolina at Greensboro. We thank Alexei Kazakov and Natalya Stankevich for stimulating questions and discussions.

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