Periodic enzyme synthesis and oscillatory repression: why is the period of oscillation close to the cell cycle time?

Theoretical study of tryptophan operon: Application in microbial technology

A model for the tryptophan operon is formulated based on the genetic and biophysical data available on the structure of the operon and the nature of interactions between the represser and its ligands. Studies have been done, on wild-type, superrepressing, and loose-binding strains to identify conditions at which the stability of the system changes (i.e., evolves to a stable synthesis or periodic synthesis with increasing amplitude). Also, the factors that increase the yield of tryptophan are studied and predictions made, based on the results, for obtaining overproducing strains of tryptophan that can be used for the industrial production of this useful amino acid.

Self-regulation in a minimal model of chemical self-replication

In biological systems, regulation plays an important role in keeping metabolite concentrations within physiological ranges. To study the dynamical implications of self-regulation, we consider a functional form used in genetic networks and couple it to a mechanism associated with chemical self-replication. For the two-variable minimal model, we find that activation can yield chemical toggles similar to those reported for gene repression in E. coli as well as more complex dynamics.

Design of regulation and dynamics in simple biochemical pathways

Complex regulation of biochemical pathways in a cell is brought about by the interaction of simpler regulatory structures. Among the basic regulatory designs, feedback inhibition of gene expression is the most common motif in gene regulation and a ubiquitous control structure found in nature. In this work, we have studied a common structural feature (delayed feedback) in gene organisation and shown, both theoretically and experimentally, its subtle but important functional role in gene expression kinetics in a negatively auto-regulated system. Using simple deterministic and stochastic models with varying levels of realism, we present detailed theoretical representations of negatively auto-regulated transcriptional circuits with increasing delays in the establishment of feedback of repression. The models of the circuits with and without delay are studied analytically as well as numerically for variation of parameters and delay lengths. The positive invariance, boundedness of the solutions, local and global asymptotic stability of both the systems around the unique positive steady state are studied analytically. Existence of transient temporal dynamics is shown mathematically. Comparison of the two types of model circuits shows that even though the long-term dynamics is stable and not affected by delays in repression, there is interesting variation in the transient dynamical features with increasing delays. Theoretical predictions are validated through experimentally constructed gene circuits of similar designs. This combined theoretical and experimental study helps delineate the opposing effects of delay-induced instability, and the stability-enhancing property of negative feedback in the pathway behaviour, and gives rationale for the abundance of similar designs in real biochemical pathways.

Dynamic analysis of genetic control and regulation of amino acid synthesis: the tryptophan operon in Escherichia coli

A mathematical model of the tryptophan operon is analyzed to investigate the regulatory effects of feedback repression and the demand for tryptophan in the cell. In this model, feedback repression is considered to be a two-step process. First, the endproduct tryptophan combines with the inactive repressor produced by the regulatory genes to yield an active complex. This complex subsequently binds to the operator and prevents transcription of the structural genes into mRNA. The demand for tryptophan in the cell is modeled by a hyperbolic saturation function of the Michaelis-Menten type. Results are obtained for the expression of the tryptophan operon in Escherichia coli and their applicability to tryptophan production by microbial fermentation is discussed. It is shown that, depending on the strain level of the operon and the rate of utilization of tryptophan in the cell, an overproduction of tryptophan can be achieved under stable operating conditions; in other circumstances, the operon may become stable or unstable, and may lead to a periodic synthesis.

On the functional diversity of dynamical behaviour in genetic and metabolic feedback systems

Background

Feedback regulation plays crucial roles in the robust control and maintenance of many cellular systems. Negative feedbacks are found to underline both stable and unstable, often oscillatory, behaviours. We explore the dynamical characteristics of systems with single as well as coupled negative feedback loops using a combined approach of analytical and numerical techniques. Particularly, we emphasise how the loop's characterising factors (strength and cooperativity levels) affect system dynamics and how individual loops interact in the coupled-loop systems.

Results

We develop an analytical bifurcation analysis based on the stability and the Routh- Hurwitz theorem for a common negative feedback system and a variety of its variants. We demonstrate that different combinations of the feedback strengths of individual loops give rise to different dynamical behaviours. Moreover, incorporating more negative feedback loops always tend to enhance system stability. We show that two mechanisms, in addition to the lengthening of pathway, can lower the Hill coefficient to a biologically plausible level required for sustained oscillations. These include loops coupling and end-product utilisation. We find that the degradation rates solely affect the threshold Hill coefficient for sustained oscillation, while the synthesis rates have more significant roles in determining the threshold feedback strength. Unbalancing the degradation rates between the system species is found as a way to improve stability.

Conclusion

The analytical methods and insights presented in this study demonstrate that reallocation of the feedback loop may or may not make the system more stable; the specific effect is determined by the degradation rates of the newly inhibited molecular species. As the loop moves closer to the end of the pathway, the minimum Hill coefficient for oscillation is reduced. Furthermore, under general (unequal) values of the degradation rates, system extension becomes more stable only when the added species degrades slower than it is being produced; otherwise the system is more prone to oscillation. The coupling of loops significantly increases the richness of dynamical bifurcation characteristics. The likelihood of having oscillatory behaviour is directly determined by the loops' strength: stronger loops always result in smaller oscillatory regions.

Propagation of extrinsic perturbation in a negatively auto-regulated pathway

The apparent precision of the output in multi-step biochemical pathways in the face of external and intrinsic perturbations is non-obvious and conceptually difficult. Using a simple three-step negatively auto-regulated model pathway, we show that the effect of perturbation at different steps of the pathway and its transmission through the network is dependent on the context (i.e., the position) of the particular reaction step in relation to the topology of the regulatory network, stoichiometry of reactions, type of nonlinearity involved in the reactions and also on the intrinsic dynamical state of the pathway variables. We delineate the qualitative and quantitative changes in the pathway dynamics for constant ('bias') and random external perturbations acting on the pathway steps locally or globally to all steps. We show that constant perturbation induces qualitative change in dynamics, whereas random fluctuations cause significant quantitative variations in the concentrations of the different variables. Thus, the dynamic response of multi-step biochemical pathways to external perturbation depends on their biochemical, topological and dynamical features.

Synchronization in coupled cells with activator-inhibitor pathways

The functional dynamics exhibited by cell collectives are fascinating examples of robust, synchronized, collective behavior in spatially extended biological systems. To investigate the roles of local cellular dynamics and interaction strength in the spatiotemporal dynamics of cell collectives of different sizes, we study a model system consisting of a ring of coupled cells incorporating a three-step biochemical pathway of regulated activator-inhibitor reactions. The isolated individual cells display very complex dynamics as a result of the nonlinear interactions common in cellular processes. On coupling the cells to nearest neighbors, through diffusion of the pathway end product, the ring of cells yields a host of interesting and unusual dynamical features such as, suppression of chaos, phase synchronization, traveling waves, and intermittency, for varying interaction strengths and system sizes. But robust complete synchronization can be induced in these coupled cells with a small degree of random coupling among them even where regular coupling yielded only intermittent synchronization. Our studies indicate that robustness in synchronized functional dynamics in tissues and cell populations in nature can be ensured by a few transient random connections among the cells. Such connections are being discovered only recently in real cellular systems.

A systems- and signal-oriented approach to intracellular dynamics

A mathematical understanding of regulation, and, in particular, the role of feedback, has been central to the advance of the physical sciences and technology. In this article, the framework provided by systems biology is used to argue that the same can be true for molecular biology. In particular, and using basic modular methods of mathematical modelling which are standard in control theory, a set of dynamic models is developed for some illustrative cell signalling processes. These models, supported by recent experimental evidence, are used to argue that a control theoretical approach to the mechanisms of feedback in intracellular signalling is central to furthering our understanding of molecular communication. As a specific example, a MAPK (mitogen-activated protein kinase) signalling pathway is used to show how potential feedback mechanisms in the signalling process can be investigated in a simulated environment. Such ‘what if’ modelling/simulation studies have been an integral part of physical science research for many years. Using tools of control systems analysis, as embodied in the disciplines of systems biology, similar predictive modelling/simulation studies are now bearing fruit in cell signalling research.

Minimal model for complex dynamics in cellular processes

Cellular functions are controlled and coordinated by the complex circuitry of biochemical pathways regulated by genetic and metabolic feedback processes. This paper aims to show, with the help of a minimal model of a regulated biochemical pathway, that the common nonlinearities and control structures present in biomolecular interactions are capable of eliciting a variety of functional dynamics, such as homeostasis, periodic, complex, and chaotic oscillations, including transients, that are observed in various cellular processes.

Does replication-induced transcription regulate synthesis of the myriad low copy number proteins of Escherichia coli?

Over 80% of the genes in the E. coli chromosome express fewer than a hundred copies each of their protein products per cell. It is argued here that transcription of these genes is neither constitutive nor regulated by protein factors, but rather, induced by the act of replication. The utility of such replication-induced (RI) transcription to the temporal regulation of synthesis of determinate quantities of low copy number (LCN) proteins is described. It is suggested that RI transcription may be necessitated, as well as facilitated, by the folding of the bacterial chromosome into a compact nucleoid. Mechanistic aspects of the induction of transcription by replication are discussed with respect to the modulation of transcriptional initiation by negative supercoiling effects, promoter methylation status and derepression. It is shown that RI transcription offers plausible explanations for the constancy of the C period of the E. coli cell cycle and the remarkable conservation of gene order in the chromosomes of enteric bacteria. Some experimental tests of the hypothesis are proposed.

A new method for the analysis of the dynamics of the molecular genetic control systems. II. Application of the method of generalized threshold models in the investigation of concrete genetic systems

Mathematical models of the prokaryotic control systems of tryptophan biosynthesis (both normal and with cloned blocks) and arabinose catabolism have been built using the method of generalized threshold models. Kinetic curves for molecular components (mRNAs, proteins, metabolites) of the systems considered are obtained. It has been shown that the method of generalized threshold models gives a more detailed qualitative picture of the dynamics of the molecular genetic control systems in comparison with the heuristic method of threshold models. The qualitative analysis of the functioning of the following mechanisms of control of the tryptophan biosynthesis: (1) inhibition of the activity of anthranilate synthetase by tryptophan, (2) repression and (3) attenuation of transcription of the tryptophan operon on the basis of the mathematical model of the control system of the tryptophan biosynthesis demonstrates that feedback inhibition is the most operative of the considered mechanisms while repression allows the bacterium to economize intracellular resources. As regards the control system of the arabinose catabolism the results of modelling enable us to state the following. The induction by arabinose within a wide range of parameter values causes two subsystems (araBAD and transport operons) of the arabinose regulon with a low rate of arabinose utilization to pass into a stationary regime and one subsystem (araC operon) to pass into a stable periodical regime. A study of the system characterized by the effective utilization of arabinose has shown that under induction by arabinose stable oscillations with small amplitudes of the concentration of regulatory protein and oscillations with large amplitudes of the concentrations of arabinose-isomerase and transport protein may occur. The period of the oscillation depends on the mean lifetime of the "activator-DNA" complex and on the rate constant of arabinoseisomerase degradation.

State-structure models—A base for efficient control of fermentation processes

The imbedding of state-structure models into the framework of structured segregated models is described. A special type of state-structure models based on delay-differential equations is considered. Some results regarding the description of unbalanced growth processes and the development of efficient periodic control strategies are presented.

Complex behaviour of the repressible operon

The repressor-mediated repression process in bacteria is modelled using a gene-enzyme-endproduct control unit. A combined analytical-numerical study shows that the system, though stable normally, becomes unstable for super-repressing strains even at low values of the cooperativity of repression, provided demand for the endproduct saturates at large endproduct concentrations. In addition the system also shows bistability, i.e., the co-existence of a stable steady-state and a stable limit cycle. The tryptophan operon is used as a model system and the results are discussed in the light of differential regulation of gene expression in lower organisms, especially in mutant strains.

Mathematical modelling of dynamics and control in metabolic networks: VI. Dynamic bifurcations in single biochemical control loops

A dynamic stability analysis of an extended form of the Goodwin equations is presented. The Goodwin equations are extended to include Michaelis-Menten kinetics for the removal of the end-product. Inclusion of saturation kinetic behavior substantially increases the likelihood of dynamic instability in this model control loop. Oscillations are found for reaction chains of low order, as low as second order, and low degrees of co-operativity, as low as v = 2, simultaneously, thus indicating that dynamic instability in this system exists for physiologically realistic parameter values. The branches of bifurcated solutions are computed numerically and unstable Hopf bifurcations are found. Further, solution branches from stable Hopf bifurcation points are found to "fold back", i.e. have periodic limit points, producing situations where multiple stable limit cycles exist.

Why are so many biological systems periodic?

The ubiquity of oscillations in biological systems is well established. Oscillations are observed in all types of organisms from the simplest to the most complex. Periods can range from fractions of a second to months or years. From time to time, it has been suggested that many biological oscillations are the result of the breakdown of effective self-regulation. The opposite view is defended here. It is argued that most periodic behavior is not pathological but rather constitutes the normal operation for these systems. They are present because they confer positive functional advantages for the organism. The advantages fall into five general categories: temporal organization, spatial organization, prediction of repetitive events, efficiency and precision of control.

On the dynamics of controlled metabolic network and cellular behaviour

The existence of elaborate control mechanisms for the various biochemical processes inside and within living cells is responsible for the coherent behaviour observed in its spatio-temporal organisation. Stability and sensitivity are both necessary properties of living systems and these are achieved through negative and positive feedback loops as in other control systems. We have studied a three-step reaction scheme involving a negative and a positive feedback loop in the form of end-product inhibition and allosteric activation. The variety of behaviour exhibited by this system, under different conditions, includes steady state, simple limit cycle oscillations, complex oscillations and period bifurcations leading to random oscillations or chaos. The system also shows the existence of two distinct chaotic regimes under the variation of a single parameter. These results, in comparison with single biochemical control loops, show that new behaviours can be exhibited in a more complex network which are not seen in the single control loops. The results are discussed in the light of a diverse variety of cellular functions in normal and altered cells indicating the role of controlled metabolic network as the underlying basis for cellular behaviour.

Control of cell-cycle timing in early embryos of Caenorhabditis elegans

A technique has been developed for extruding either substantial amounts of cytoplasm without nuclei or individual nuclei with small amounts of cytoplasm from early embryos of C. elegans after perforating the eggshell with a laser microbeam. This technique, in conjunction with laser-induced cell fusion, has allowed the altering of nuclear/cytoplasmic ratios and the exposing of the nucleus of one cell to cytoplasm from another. Using these approaches the roles of nuclei and cytoplasm in determining the different cell-cycle periods of the several blastomere lineages in early embryos have been examined. It was found that nuclei in a common cytoplasm divide synchronously; enucleated blastomeres retain a cycling period characteristic of their lineage; cycling period is not substantially affected by changes in the ratio of nuclear to cytoplasmic volumes or the DNA content per cell; the period of a cell from one lineage can be substantially altered by introduction of cytoplasm from a cell of another lineage with a different period; and short-term effects of foreign cytoplasm on the timing of the subsequent mitosis differ depending on position of the donor cell in the cell cycle. These results are discussed in connection with models for the action of cytoplasmic factors in controlling cell-cycle timing.

Interaction of spatial diffusion and delays in models of genetic control by repression

A class of models based on the Jacob and Monod theory of genetic repression for control of biosynthetic pathways in cells is considered. Both spatial diffusion and time delays are taken into account. A method is developed for representing the effects of spatial diffusion as distributed delay terms. This method is applied to two specific models and the interaction between the diffusion and the delays is treated in detail. The destabilization of the steady-state and the bifurcation of oscillatory solutions are studied as functions of the diffusivities and the delays. The limits of very small and very large diffusivities are analyzed and comparisons with well-mixed compartment models are made.