Stability of cyclic gene models for systems involving repression

Operon dynamics with state dependent transcription and/or translation delays

Transcription and translation retrieve and operationalize gene encoded information in cells. These processes are not instantaneous and incur significant delays. In this paper we study Goodwin models of both inducible and repressible operons with state-dependent delays. The paper provides justification and derivation of the model, detailed analysis of the appropriate setting of the corresponding dynamical system, and extensive numerical analysis of its dynamics. Comparison with constant delay models shows significant differences in dynamics that include existence of stable periodic orbits in inducible systems and multistability in repressible systems. A combination of parameter space exploration, numerics, analysis of steady state linearization and bifurcation theory indicates the likely presence of Shilnikov-type homoclinic bifurcations in the repressible operon model.

Implications of network topology on stability

In analogy to chemical reaction networks, I demonstrate the utility of expressing the governing equations of an arbitrary dynamical system (interaction network) as sums of real functions (generalized reactions) multiplied by real scalars (generalized stoichiometries) for analysis of its stability. The reaction stoichiometries and first derivatives define the network's "influence topology", a signed directed bipartite graph. Parameter reduction of the influence topology permits simplified expression of the principal minors (sums of products of non-overlapping bipartite cycles) and Hurwitz determinants (sums of products of the principal minors or the bipartite cycles directly) for assessing the network's steady state stability. Visualization of the Hurwitz determinants over the reduced parameters defines the network's stability phase space, delimiting the range of its dynamics (specifically, the possible numbers of unstable roots at each steady state solution). Any further explicit algebraic specification of the network will project onto this stability phase space. Stability analysis via this hierarchical approach is demonstrated on classical networks from multiple fields.

Theoretical foundations for traditional and generalized sensitivity functions for nonlinear delay differential equations

In this paper we present new results for differentiability of delay systems with respect to initial conditions and delays. After motivating our results with a wide range of delay examples arising in biology applications, we further note the need for sensitivity functions (both traditional and generalized sensitivity functions), especially in control and estimation problems. We summarize general existence and uniqueness results before turning to our main results on differentiation with respect to delays, etc. Finally we discuss use of our results in the context of estimation problems.

A comparative analysis of synthetic genetic oscillators

Synthetic biology is a rapidly expanding discipline at the interface between engineering and biology. Much research in this area has focused on gene regulatory networks that function as biological switches and oscillators. Here we review the state of the art in the design and construction of oscillators, comparing the features of each of the main networks published to date, the models used for in silico design and validation and, where available, relevant experimental data. Trends are apparent in the ways that network topology constrains oscillator characteristics and dynamics. Also, noise and time delay within the network can both have constructive and destructive roles in generating oscillations, and stochastic coherence is commonplace. This review can be used to inform future work to design and implement new types of synthetic oscillators or to incorporate existing oscillators into new designs.

Control of Streptococcus pyogenes virulence: modeling of the CovR/S signal transduction system

The CovR/S system in Streptococcus pyogenes (Group A Streptococcus, or GAS), a two-component signal transduction/transcription regulation system, controls the expression of major virulence factors. The presence of a negative feedback loop distinguishes the CovR/S system from the majority of bacterial two-component systems. We developed a deterministic model of the CovR/S system consisting of eight delay differential equations. Computational experiments showed that the system possessed a unique stable steady state. The dynamical behavior of the system showed a tendency for oscillations becoming more pronounced for longer but still biochemically realistic delays resulting from reductions in the rates of translation elongation. We have devised an efficient procedure for computing the system's steady state. Further, we have shown that the signal-response curves are hyperbolic for the default parameter values. However, in experiments with randomized parameters we demonstrated that sigmoidality of signal-response curves, implying a response threshold, is not only possible, but seems to be rather typical for CovR/S-like systems even when binding of the CovR response regulator protein to a promoter is non-cooperative. We used sensitivity analysis to simplify the model in order to make it analytically tractable. The existence and uniqueness of the steady state and hyperbolicity of signal-response curves for the majority of the variables was proved for the simplified model. Also, we found that provided CovS was active, the system was insensitive to changes in the concentration of any other phosphoryl donor such as acetyl phosphate.

A generalized model of the repressilator

The repressilator is a regulatory cycle of n genes where each gene represses its successor in the cycle: [see text]. The system is modelled by ODEs for an arbitrary number of identical genes and arbitrarily strong repressor binding. A detailed mathematical analysis of the dynamical behavior is provided for two model systems: (i) a repressilator with leaky transcription and single-step cooperative repressor binding, and (ii) a repressilator with auto-activation and cooperative regulator binding. Genes are assumed to be present in constant amounts, transcription and translation are modelled by single-step kinetics, and mRNAs as well as proteins are assumed to be degraded by first order reactions. Several dynamical patterns are observed: multiple steady states, periodic and aperiodic oscillations corresponding to limit cycles and heteroclinic cycles, respectively. The results of computer simulations are complemented by a detailed and complete stability analysis of all equilibria and of the heteroclinic cycle.

Non-linear qualitative signal processing for biological systems: application to the algal growth in bioreactors

We present in this paper a qualitative method to validate and monitor the structure of a non-linear model with respect to experimental data, under some hypotheses. This method is broadly independent of the analytical formulation of the model, and depends only on the qualitative structure (the signs of the Jacobian matrix). The temporal sequences of the extrema of a filtered experimental signal are compared with the transitions allowed by a graph. In particular, we show that the usual moving average of the outputs follows this transition graph. We apply this method to compare models of algal growth in a bioreactor with experimental data.

Oscillations and multiple steady states in a cyclic gene model with repression

In this paper we study the cyclic gene model with repression considered by H. T. Banks and J. M. Mahaffy. Roughly, the model describes a biochemical feedback loop consisting of an integer number G of single gene reaction sequences in series. The model leads to a system of functional differential equations. We show that there is a qualitative difference in the dynamics between even and odd G if the feedback repression is sufficiently large. For even G, multiple stable steady states can coexist while for odd G, periodic orbits exist.

Cellular control models with linked positive and negative feedback and delays. I. The models

Basic techniques from biochemical kinetics are used to develop models for a cellular control system with linked positive and negative feedback. The models are represented by a system of nonlinear differential equations with delays. The lac operon provides an example of a control system where the transcription of the operon is controlled by induction or positive feedback control and catabolite repression or negative feedback control. These processes are linked through the metabolism of lactose.