Modeling operon dynamics: the tryptophan and lactose operons as paradigms

Boolean Models of the Transport, Synthesis, and Metabolism of Tryptophan in Escherichia coli

The tryptophan (trp) operon in Escherichia coli codes for the proteins responsible for the synthesis of the amino acid tryptophan from chorismic acid, and has been one of the most well-studied gene networks since its discovery in the 1960s. The tryptophanase (tna) operon codes for proteins needed to transport and metabolize it. Both of these have been modeled individually with delay differential equations under the assumption of mass-action kinetics. Recent work has provided strong evidence for bistable behavior of the tna operon. The authors of Orozco-Gómez et al. (Sci Rep 9(1):5451, 2019) identified a medium range of tryptophan in which the system has two stable steady-states, and they reproduced these experimentally. In this paper, we will show how a Boolean model can capture this bistability. We will also develop and analyze a Boolean model of the trp operon. Finally, we will combine these two to create a single Boolean model of the transport, synthesis, and metabolism of tryptophan. In this amalgamated model, the bistability disappears, presumably reflecting the ability of the trp operon to produce tryptophan and drive the system toward homeostasis. All of these models have longer attractors that we call "artifacts of synchrony", which disappear in the asynchronous automata. This curiously matches the behavior of a recent Boolean model of the arabinose operon in E. coli, and we discuss some open-ended questions that arise along these lines.

Differential roles of transcriptional and translational negative autoregulations in protein dynamics

Cells continuously respond to stimuli to function properly by employing a wide variety of regulatory mechanisms that often involve protein up or down regulations. This study focuses on dynamics of a protein with negative autoregulations in E. coli, and assumes that the input signal up-regulates the protein, and then the protein down-regulates its own production via 2 distinct types of mechanisms. The mathematical models describe the dynamics of mRNA and protein for 3 scenarios: (i) a simplistic model with no regulation, (ii) a model with transcriptional negative autoregulation, and (iii) a model with translational negative autoregulation. Our analysis shows that the negative autoregulation models produce faster responses and quicker return times to the input signals compared to the model with no regulation, while the transcriptional autoregulation model is the only model capable of producing oscillatory dynamics. The stochastic simulations predict that the transcriptional autoregulation model is the noisiest followed by the simplistic model, and the translational autoregulation model has the least noise. The noise level depends on the strength of inhibition. Furthermore, the transcriptional autoregulation model filters out the noise in the input signal for longer periods of time, and this time increases as the strength of the feedback gets stronger.

In silico Determination of Some Conditions Leading to Glycolytic Oscillations and Their Interference With Some Other Processes in E. coli Cells

Autonomous oscillations of species levels in the glycolysis express the self-control of this essential cellular pathway belonging to the central carbon metabolism (CCM), and this phenomenon takes place in a large number of bacteria. Oscillations of glycolytic intermediates in living cells occur according to the environmental conditions and to the cell characteristics, especially the adenosine triphosphate (ATP) recovery system. Determining the conditions that lead to the occurrence and maintenance of the glycolytic oscillations can present immediate practical applications. Such a model-based analysis allows in silico (model-based) design of genetically modified microorganisms (GMO) with certain characteristics of interest for the biosynthesis industry, medicine, etc. Based on our kinetic model validated in previous works, this paper aims to in silico identify operating parameters and cell factors leading to the occurrence of stable glycolytic oscillations in the Escherichia coli cells. As long as most of the glycolytic intermediates are involved in various cellular metabolic pathways belonging to the CCM, evaluation of the dynamics and average level of its intermediates is of high importance for further applicative analyses. As an example, by using a lumped kinetic model for tryptophan (TRP) synthesis from literature, and its own kinetic model for the oscillatory glycolysis, this paper highlights the influence of glycolytic oscillations on the oscillatory TRP synthesis through the PEP (phosphoenolpyruvate) glycolytic node shared by the two oscillatory processes. The numerical analysis allows further TRP production maximization in a fed-batch bioreactor (FBR).

Tryptophan Biochemistry: Structural, Nutritional, Metabolic, and Medical Aspects in Humans

L-Tryptophan is the unique protein amino acid (AA) bearing an indole ring: its biotransformation in living organisms contributes either to keeping this chemical group in cells and tissues or to breaking it, by generating in both cases a variety of bioactive molecules. Investigations on the biology of Trp highlight the pleiotropic effects of its small derivatives on homeostasis processes. In addition to protein turn-over, in humans the pathways of Trp indole derivatives cover the synthesis of the neurotransmitter/hormone serotonin (5-HT), the pineal gland melatonin (MLT), and the trace amine tryptamine. The breakdown of the Trp indole ring defines instead the "kynurenine shunt" which produces cell-response adapters as L-kynurenine, kynurenic and quinolinic acids, or the coenzyme nicotinamide adenine dinucleotide (NAD(+)). This review aims therefore at tracing a "map" of the main molecular effectors in human tryptophan (Trp) research, starting from the chemistry of this AA, dealing then with its biosphere distribution and nutritional value for humans, also focusing on some proteins responsible for its tissue-dependent uptake and biotransformation. We will thus underscore the role of Trp biochemistry in the pathogenesis of human complex diseases/syndromes primarily involving the gut, neuroimmunoendocrine/stress responses, and the CNS, supporting the use of -Omics approaches in this field.

Mechanistically consistent reduced models of synthetic gene networks

Designing genetic networks with desired functionalities requires an accurate mathematical framework that accounts for the essential mechanistic details of the system. Here, we formulate a time-delay model of protein translation and mRNA degradation by systematically reducing a detailed mechanistic model that explicitly accounts for the ribosomal dynamics and the cleaving of mRNA by endonucleases. We exploit various technical and conceptual advantages that our time-delay model offers over the mechanistic model to probe the behavior of a self-repressing gene over wide regions of parameter space. We show that a heuristic time-delay model of protein synthesis of a commonly used form yields a notably different prediction for the parameter region where sustained oscillations occur. This suggests that such heuristics can lead to erroneous results. The functional forms that arise from our systematic reduction can be used for every system that involves transcription and translation and they could replace the commonly used heuristic time-delay models for these processes. The results from our analysis have important implications for the design of synthetic gene networks and stress that such design must be guided by a combination of heuristic models and mechanistic models that include all relevant details of the process.

Mathematical modeling of gene expression: a guide for the perplexed biologist

The detailed analysis of transcriptional networks holds a key for understanding central biological processes, and interest in this field has exploded due to new large-scale data acquisition techniques. Mathematical modeling can provide essential insights, but the diversity of modeling approaches can be a daunting prospect to investigators new to this area. For those interested in beginning a transcriptional mathematical modeling project, we provide here an overview of major types of models and their applications to transcriptional networks. In this discussion of recent literature on thermodynamic, Boolean, and differential equation models, we focus on considerations critical for choosing and validating a modeling approach that will be useful for quantitative understanding of biological systems.

Deterministic and stochastic simulation and analysis of biochemical reaction networks the lactose operon example

A brief introduction to mathematical modeling of biochemical regulatory reaction networks is presented. Both deterministic and stochastic modeling techniques are covered with examples from enzyme kinetics, coupled reaction networks with oscillatory dynamics and bistability. The Yildirim-Mackey model for lactose operon is used as an example to discuss and show how deterministic and stochastic methods can be used to investigate various aspects of this bacterial circuit.

The origins of time-delay in template biopolymerization processes

Time-delays are common in many physical and biological systems and they give rise to complex dynamic phenomena. The elementary processes involved in template biopolymerization, such as mRNA and protein synthesis, introduce significant time delays. However, there is not currently a systematic mapping between the individual mechanistic parameters and the time delays in these networks. We present here the development of mathematical, time-delay models for protein translation, based on PDE models, which in turn are derived through systematic approximations of first-principles mechanistic models. Theoretical analysis suggests that the key features that determine the time-delays and the agreement between the time-delay and the mechanistic models are ribosome density and distribution, i.e., the number of ribosomes on the mRNA chain relative to their maximum and their distribution along the mRNA chain. Based on analytical considerations and on computational studies, we show that the steady-state and dynamic responses of the time-delay models are in excellent agreement with the detailed mechanistic models, under physiological conditions that correspond to uniform ribosome distribution and for ribosome density up to 70%. The methodology presented here can be used for the development of reduced time-delay models of mRNA synthesis and large genetic networks. The good agreement between the time-delay and the mechanistic models will allow us to use the reduced model and advanced computational methods from nonlinear dynamics in order to perform studies that are not practical using the large-scale mechanistic models.

Stochastic approach to molecular interactions and computational theory of metabolic and genetic regulations

The underlying molecular mechanisms of metabolic and genetic regulations are computationally identical and can be described by a finite state Markov process. We establish a common computational model for both regulations based on the stationary distribution of the Markov process with the aim of establishing a unified, quantitative model of general biological regulations. Various existing results regarding intracellular regulations are derived including the classical Michaelis-Menten equation and its generalization to more complex allosteric enzymes in a systematic way. The notion of probability flow is introduced to distinguish the equilibrium stationary distribution from the non-equilibrium one; it plays a crucial role in the analysis of stationary state equations. A graphical criterion to guarantee the existence of an equilibrium stationary distribution is derived, which turns out to be identical to the classical Wegscheider condition. Simple graphical methods to compute the equilibrium and non-equilibrium stationary distributions are derived based crucially on the probability flow, which dramatically simplifies the classical methods still used in enzymology.

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.

Topology and static response of interaction networks in molecular biology

We introduce a mathematical framework describing static response of networks occurring in molecular biology. This formalism has many similarities with the Laplace-Kirchhoff equations for electrical networks. We introduce the concept of graph boundary and we show how the response of the biological networks to external perturbations can be related to the Dirichlet or Neumann problems for the corresponding equations on the interaction graph. Solutions to these two problems are given in terms of path moduli (measuring path rigidity with respect to the propagation of interaction along the graph). Path moduli are related to loop products in the interaction graph via generalized Mason-Coates formulae. We apply our results to two specific biological examples: the lactose operon and the genetic regulation of lipogenesis. Our applications show consistency with experimental results and in the case of lipogenesis check some hypothesis on the behaviour of hepatic fatty acids on fasting.