Degradation rate uniformity determines success of oscillations in repressive feedback regulatory networks

Dynamical modules in metabolism, cell and developmental biology

Modularity is an essential feature of any adaptive complex system. Phenotypic traits are modules in the sense that they have a distinguishable structure or function, which can vary (quasi-)independently from its context. Since all phenotypic traits are the product of some underlying regulatory dynamics, the generative processes that constitute the genotype-phenotype map must also be functionally modular. Traditionally, modular processes have been identified as structural modules in regulatory networks. However, structure only constrains, but does not determine, the dynamics of a process. Here, we propose an alternative approach that decomposes the behaviour of a complex regulatory system into elementary activity-functions. Modular activities can occur in networks that show no structural modularity, making dynamical modularity more widely applicable than structural decomposition. Furthermore, the behaviour of a regulatory system closely mirrors its functional contribution to the outcome of a process, which makes dynamical modularity particularly suited for functional decomposition. We illustrate our approach with numerous examples from the study of metabolism, cellular processes, as well as development and pattern formation. We argue that dynamical modules provide a shared conceptual foundation for developmental and evolutionary biology, and serve as the foundation for a new account of process homology, which is presented in a separate contribution by DiFrisco and Jaeger to this focus issue.

Emergence of multistability and strongly asymmetric collective modes in two quorum sensing coupled identical ring oscillators

The simplest ring oscillator is made from three strongly nonlinear elements repressing each other unidirectionally, resulting in the emergence of a limit cycle. A popular implementation of this scheme uses repressor genes in bacteria, creating the synthetic genetic oscillator known as the Repressilator. We consider the main collective modes produced when two identical Repressilators are mean-field-coupled via the quorum-sensing mechanism. In-phase and anti-phase oscillations of the coupled oscillators emerge from two Andronov-Hopf bifurcations of the homogeneous steady state. Using the rate of the repressor's production and the value of coupling strength as the bifurcation parameters, we performed one-parameter continuations of limit cycles and two-parameter continuations of their bifurcations to show how bifurcations of the in-phase and anti-phase oscillations influence the dynamical behaviors for this system. Pitchfork bifurcation of the unstable in-phase cycle leads to the creation of novel inhomogeneous limit cycles with very different amplitudes, in contrast to the well-known asymmetrical limit cycles arising from oscillation death. The Neimark-Sacker bifurcation of the anti-phase cycle determines the border of an island in two-parameter space containing almost all the interesting regimes including the set of resonant limit cycles, the area with stable inhomogeneous cycle, and very large areas with chaotic regimes resulting from torus destruction and period doubling of resonant cycles and inhomogeneous cycles. We discuss the structure of the chaos skeleton to show the role of inhomogeneous cycles in its formation. Many regions of multistability and transitions between regimes are presented. These results provide new insights into the coupling-dependent mechanisms of multistability and collective regime symmetry breaking in populations of identical multidimensional oscillators.

Tractable nonlinear memory functions as a tool to capture and explain dynamical behaviors

Mathematical approaches from dynamical systems theory are used in a range of fields. This includes biology where they are used to describe processes such as protein-protein interaction and gene regulatory networks. As such networks increase in size and complexity, detailed dynamical models become cumbersome, making them difficult to explore and decipher. This necessitates the application of simplifying and coarse graining techniques to derive explanatory insight. Here we demonstrate that Zwanzig-Mori projection methods can be used to arbitrarily reduce the dimensionality of dynamical networks while retaining their dynamical properties. We show that a systematic expansion around the quasi-steady-state approximation allows an explicit solution for memory functions without prior knowledge of the dynamics. The approach not only preserves the same steady states but also replicates the transients of the original system. The method correctly predicts the dynamics of multistable systems as well as networks producing sustained and damped oscillations. Applying the approach to a gene regulatory network from the vertebrate neural tube, a well-characterized developmental transcriptional network, identifies features of the regulatory network responsible for its characteristic transient behavior. Taken together, our analysis shows that this method is broadly applicable to multistable dynamical systems and offers a powerful and efficient approach for understanding their behavior.

Design Principles for Compartmentalization and Spatial Organization of Synthetic Genetic Circuits

Compartmentalization is a hallmark of cellular systems and an ingredient actively exploited in evolution. It is also being engineered and exploited in synthetic biology, in multiple ways. While these have demonstrated important experimental capabilities, understanding design principles underpinning compartmentalization of genetic circuits has been elusive. We develop a systems framework to elucidate the interplay between the nature of the genetic circuit, the spatial organization of compartments, and their operational state (well-mixed or otherwise). In so doing, we reveal a number of unexpected features associated with compartmentalizing synthetic and template-based circuits. These include (i) the consequences of distributing circuits including trade-offs and how they may be circumvented, (ii) hidden constraints in realizing a distributed circuit, and (iii) appealing new features of compartmentalized circuits. We build on this to examine exemplar applications, which consolidate and extend the design principles we have obtained. Our insights, which emerge from the most basic and general considerations of compartmentalizing genetic circuits, are relevant in a broad range of settings.

Modularity, criticality, and evolvability of a developmental gene regulatory network

The existence of discrete phenotypic traits suggests that the complex regulatory processes which produce them are functionally modular. These processes are usually represented by networks. Only modular networks can be partitioned into intelligible subcircuits able to evolve relatively independently. Traditionally, functional modularity is approximated by detection of modularity in network structure. However, the correlation between structure and function is loose. Many regulatory networks exhibit modular behaviour without structural modularity. Here we partition an experimentally tractable regulatory network-the gap gene system of dipteran insects-using an alternative approach. We show that this system, although not structurally modular, is composed of dynamical modules driving different aspects of whole-network behaviour. All these subcircuits share the same regulatory structure, but differ in components and sensitivity to regulatory interactions. Some subcircuits are in a state of criticality, while others are not, which explains the observed differential evolvability of the various expression features in the system.

Oscillations in well-mixed, deterministic feedback systems: Beyond ring oscillators

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I present a way of breaking down regulatory networks to find Hopf bifurcations. This helps find optimal conditions for oscillations in dynamical systems models of these networks.

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In a model of negative auto-regulation of a gene by its dimeric protein, it is optimal for the monomer to degrade faster than the mRNA and the mRNA to degrade faster than the dimer.

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Adding a weak positive feedback loop to a repressilator increases the probability of oscillations.

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The optimal degradation rate of species in the sub-loop is higher than that of species outside it.

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The opposite is true for a negative feedback sub-loop or a very strong positive feedback sub-loop.

A ring oscillator is a system in which one species regulates the next, which regulates the next and so on until the last species regulates the first. In addition, the number of the regulations which are negative, and so result in a reduction in the regulated species, is odd, making the overall feedback in the loop negative. In ring oscillators, the probability of oscillations is maximised if the degradation rates of the species are equal. When there is more than one loop in the regulatory network, the dynamics can be more complicated. Here, a systematic way of organising the characteristic equation of ODE models of regulatory networks is provided. This facilitates the identification of Hopf bifurcations. It is shown that the probability of oscillations in non-ring systems is maximised for unequal degradation rates. For example, when there is a ring and a second ring employing a subset of the genes in the first ring, then the probability of oscillations is maximised when the species in the sub-ring degrade more slowly than those outside, for a negative feedback subring. When the sub-ring forms a positive feedback loop, the optimal degradation rates are larger for the species in the sub-ring, provided the positive feedback is not too strong. By contrast, optimal degradation rates are smaller for the species in the sub-ring, when the positive feedback is very strong. Adding a positive feedback loop to a repressilator increases the probability of oscillations, provided the positive feedback is not too strong, whereas adding a negative feedback loop decreases the probability of oscillations. The work is illustrated with numerical simulations of example systems: an autoregulatory gene model in which transcription is downregulated by the protein dimer and three-species and four-species gene regulatory network examples.

Revisiting a synthetic intracellular regulatory network that exhibits oscillations

In 2000, Elowitz and Leibler introduced the repressilator-a synthetic gene circuit with three genes that cyclically repress transcription of the next gene-as well as a corresponding mathematical model. Experimental data and model simulations exhibited oscillations in the protein concentrations across generations. Müller et al. (J Math Biol 53(6):905-937, 2006) generalized the model to an arbitrary number of genes and analyzed the resulting dynamics. Their new model arose from five key assumptions, two of which are restrictive given current biological knowledge. Accordingly, we propose a new repressilator system that allows for general functions to model transcription, degradation, and translation. We prove that, with an odd number of genes, the new model has a unique steady state and the system converges to this steady state or to a periodic orbit. We also give a necessary and sufficient condition for stability of steady states when the number of genes is even and conjecture a condition for stability for an odd number. Finally, we derive a new rate function describing transcription that arises under more reasonable biological assumptions than the widely used single-step binding assumption. With this new transcription-rate function, we compare the model's amplitude and period with that of a model with the conventional transcription-rate function. Taken together, our results enhance our understanding of genetic regulation by repression.

Comparison between Effects of Retroactivity and Resource Competition upon Change in Downstream Reporter Genes of Synthetic Genetic Circuits

Reporter genes have contributed to advancements in molecular biology. Binding of an upstream regulatory protein to a downstream reporter promoter allows quantification of the activity of the upstream protein produced from the corresponding gene. In studies of synthetic biology, analyses of reporter gene activities ensure control of the cell with synthetic genetic circuits, as achieved using a combination of in silico and in vivo experiments. However, unexpected effects of downstream reporter genes on upstream regulatory genes may interfere with in vivo observations. This phenomenon is termed as retroactivity. Using in silico and in vivo experiments, we found that a different copy number of regulatory protein-binding sites in a downstream gene altered the upstream dynamics, suggesting retroactivity of reporters in this synthetic genetic oscillator. Furthermore, by separating the two sources of retroactivity (titration of the component and competition for degradation), we showed that, in the dual-feedback oscillator, the level of the fluorescent protein reporter competing for degradation with the circuits' components is important for the stability of the oscillations. Altogether, our results indicate that the selection of reporter promoters using a combination of in silico and in vivo experiments is essential for the advanced design of genetic circuits.