Cellular control models with linked positive and negative feedback and delays. II. Linear analysis and local stability

An analysis of local behavior is made of two nonlinear models which incorporate both an induction or positive feedback control mechanism and a repression or negative feedback control mechanism. The systems of differential equations with delays are linearized about their equilibria. The related characteristic equations which are exponential polynomials are studied to determine the local stability of the models. Computer studies are included to show the range of stability for different parameter values, and the biological significance is discussed briefly.     

Comparison of Deterministic and Stochastic Models of the lac Operon Genetic Network

The lac operon has been a paradigm for genetic regulation with positive feedback, and several modeling studies have described its dynamics at various levels of detail. However, it has not yet been analyzed how stochasticity can enrich the system's behavior, creating effects that are not observed in the deterministic case. To address this problem we use a comparative approach. We develop a reaction network for the dynamics of the lac operon genetic switch and derive corresponding deterministic and stochastic models that incorporate biological details. We then analyze the effects of key biomolecular mechanisms, such as promoter strength and binding affinities, on the behavior of the models. No assumptions or approximations are made when building the models other than those utilized in the reaction network. Thus, we are able to carry out a meaningful comparison between the predictions of the two models to demonstrate genuine effects of stochasticity. Such a comparison reveals that in the presence of stochasticity, certain biomolecular mechanisms can profoundly influence the region where the system exhibits bistability, a key characteristic of the lac operon dynamics. For these cases, the temporal asymptotic behavior of the deterministic model remains unchanged, indicating a role of stochasticity in modulating the behavior of the system.     

Sustained oscillations and threshold phenomena in an operon control circuit

A genetic regulatory model involving a positive feedback (via induction) and a negative feedback (via catabolite repression) is analyzed and applied to the problem of the lac operon regulation in E. coli. Damped and sustained oscillations of the limit cycle type are found along with threshold phenomena corresponding to multiple limit cycles or to multiple steady states, for values of the parameters compatible with experimental data. A comparison With the observations of Knorre and Goodwin is outlined.     

Analytical approximations for the amplitude and period of a relaxation oscillator

Background  

Analysis and design of complex systems benefit from mathematically tractable models, which are often derived by approximating a nonlinear system with an effective equivalent linear system. Biological oscillators with coupled positive and negative feedback loops, termed hysteresis or relaxation oscillators, are an important class of nonlinear systems and have been the subject of comprehensive computational studies. Analytical approximations have identified criteria for sustained oscillations, but have not linked the observed period and phase to compact formulas involving underlying molecular parameters.     

Role of feedback inhibition in stabilizing the classical operon

The Goodwin equations for a repressible operon (Goodwin, 1965) are modified (1) to describe a time lag between genetic regulation and appearance of functional enzyme, (2) to describe consumption of endproduct in protein synthesis, and (3) to describe feedback inhibition of enzyme activity. The stability of the modified equations is determined by a method outlined in the appendix which treats a class of negative feedback systems with time delays. With parameters estimated from experimental data on the tryptophan operon of Escherichia coli, we conclude that the operon becomes unstable as normal feedback inhibition is lost. Numerical solution of the modified equations shows that an example with a partial loss of feedback inhibition can have a period of oscillation less than the cell generation time, and the numerical solutions are shown to be in qualitative agreement with experiments showing oscillations in tryptophan operon expression.     

Genetic Analysis of the Maltose A Region in Escherichia coli

The genetic map of the maltose A locus of Escherichia coli contains at least three closely linked genes, malT, malP, and malQ. The order of these genes is established by deletion mapping. MalP and malQ, the presumed structural genes for maltodextrin phosphorylase and amylomaltase, belong to the same operon. MalT may be a regulator gene involved in the positive control of this operon.     

A Critical Quantity for Noise Attenuation in Feedback Systems

Feedback modules, which appear ubiquitously in biological regulations, are often subject to disturbances from the input, leading to fluctuations in the output. Thus, the question becomes how a feedback system can produce a faithful response with a noisy input. We employed multiple time scale analysis, Fluctuation Dissipation Theorem, linear stability, and numerical simulations to investigate a module with one positive feedback loop driven by an external stimulus, and we obtained a critical quantity in noise attenuation, termed as “signed activation time”. We then studied the signed activation time for a system of two positive feedback loops, a system of one positive feedback loop and one negative feedback loop, and six other existing biological models consisting of multiple components along with positive and negative feedback loops. An inverse relationship is found between the noise amplification rate and the signed activation time, defined as the difference between the deactivation and activation time scales of the noise-free system, normalized by the frequency of noises presented in the input. Thus, the combination of fast activation and slow deactivation provides the best noise attenuation, and it can be attained in a single positive feedback loop system. An additional positive feedback loop often leads to a marked decrease in activation time, decrease or slight increase of deactivation time and allows larger kinetic rate variations for slow deactivation and fast activation. On the other hand, a negative feedback loop may increase the activation and deactivation times. The negative relationship between the noise amplification rate and the signed activation time also holds for the six other biological models with multiple components and feedback loops. This principle may be applicable to other feedback systems.     

Ribitol and D-arabitol catabolism in Escherichia coli.

In Escherichia coli C, the catabolism of the pentitols ribitol and D-arabitol proceeds through separate, inducible operons, each consisting of a dehydrogenase and a kinase. The ribitol operon is induced in response to ribulose, and the D-arabitol operon is induced in response to D-arabitol. Each operon is under negative control. The genes of the ribitol and D-arabitol operons are very closely linked and lie in a mirror image arrangement, rtlB-rtlA-rtlC-atlC-atlA-atlB, between metG and his on the E. coli chromosome.     

Protein symmetry and the co-operative ligand-binding behaviour predicted by allosteric Koshland models

It is shown that the nearest-neighbour interaction two-conformation allosteric models of Koshland, Nemethy & Filmer (1966) predict binding curves with a centre of symmetry when the protein is also symmetrical and induced-fit is assumed. When nonexclusive binding to both conformations is assumed, the models predict that the family of homotropic binding curves obtained by varying the heterotropic ligand has a centre of symmetry. It is argued that the symmetry or asymmetry of binding curves is the main experimentally verifiable prediction of allosteric models insofar as they are models of interaction between protein subunits.Symmetry in a binding curve greatly simplifies the analysis of cooperative behaviour. The co-operative features possible with a symmetric binding curve for a four-site protein are analysed. The sign of co-operativity may either be uniform or change twice as saturation increases; the conditions for the various possibilities are given. For example, in terms of the intrinsic binding constants per site A^∥₁, A2, etc. the necessary and sufficient condition for positive macroscopic co-operativity over the whole symmetric binding curve is A^∥₁≤ A2, A ^∥₁ ≤ A3 which should be contrasted with the obvious A^∥₁ ≤ Al, AZ ≤A^∥₃ (positive microscopic co-operativity) which is only a sufficient but not a necessary condition. A symmetric curve may have one or three, but no more, extrema of the “Hill coefficient” h. For three extrema a change of sign of microscopic (but not necessarily macroscopic) co-operativity is necessary but not sufficient. In the case where there are off-centre maxima of h, then h < 2 everywhere on the curve.The Koshland models predict qualitative and quantitative restrictions on the forms of binding curves additional to that of symmetry. In tetrameric induced fit models, negative co-operativity in the mid-region of the curve and positive co-operativity in the outside regions is possible, but not the opposite, and three extrema of h are possible with uniform positive but not with uniform negative co-operativity.Thus by recognising the importance of symmetry it has been possible to describe and categorise all the co-operativity behaviour possible with the most plausible Koshland tetrameric models. Several experimental examples of probable non-exclusive binding to proteins and enzymes are discussed, and it is shown how the symmetry point of view illuminates their interpretation.     

Distinctive topologies of partner-switching signaling networks correlate with their physiological roles

Regulatory networks controlling bacterial gene expression often evolve from common origins and share homologous proteins and similar network motifs. However, when functioning in different physiological contexts, these motifs may be re-arranged with different topologies that significantly affect network performance. Here we analyze two related signaling networks in the bacterium Bacillus subtilis in order to assess the consequences of their different topologies, with the aim of formulating design principles applicable to other systems. These two networks control the activities of the general stress response factor sigma(B) and the first sporulation-specific factor sigma(F). Both networks have at their core a "partner-switching" mechanism, in which an anti-sigma factor forms alternate complexes either with the sigma factor, holding it inactive, or with an anti-anti-sigma factor, thereby freeing sigma. However, clear differences in network structure are apparent: the anti-sigma factor for sigma(F) forms a long-lived, "dead-end" complex with its anti-anti-sigma factor and ADP, whereas the genes encoding sigma(B) and its network partners lie in a sigma(B)-controlled operon, resulting in positive and negative feedback loops. We constructed mathematical models of both networks and examined which features were critical for the performance of each design. The sigma(F) model predicts that the self-enhancing formation of the dead-end complex transforms the network into a largely irreversible hysteretic switch; the simulations reported here also demonstrate that hysteresis and slow turn off kinetics are the only two system properties associated with this complex formation. By contrast, the sigma(B) model predicts that the positive and negative feedback loops produce graded, reversible behavior with high regulatory capacity and fast response time. Our models demonstrate how alterations in network design result in different system properties that correlate with regulatory demands. These design principles agree with the known or suspected roles of similar networks in diverse bacteria.     

A kinetic model for interaction of regulatory proteins and RNA polymerase with the control region of the lac operon of Escherichia coli.

The proposed model deals with kinetic aspects of the interaction of repressor, CRP and RNA polymerase with the control region of the lactose operon and is formulated as a system of linear differential equations. Several variants of the model are considered. They differ in the assumed mechanisms which limit expression of the operon (due to diffusion of the molecules of polymerase to the promoter and/or due to a specific interaction of polymerase and promoter) and in the existence or non-existence of an indirect interaction between the molecules of repressor and CRP, when they are bound to the control region. An analysis of the model provides a unified interpretation for several phenomena connected with regulation of the lactose operon, in particular, for the dependence of expression on concentrations of regulatory proteins and for different patterns of expression in vivo and in vitro for a class of promoter mutations.