A structured, segregated model for genetically modified Escherichia coli cells and its use for prediction of plasmid stability

Quorum-Sensing Synchronization of Synthetic Toggle Switches: A Design Based on Monotone Dynamical Systems Theory

Synthetic constructs in biotechnology, biocomputing, and modern gene therapy interventions are often based on plasmids or transfected circuits which implement some form of “on-off” switch. For example, the expression of a protein used for therapeutic purposes might be triggered by the recognition of a specific combination of inducers (e.g., antigens), and memory of this event should be maintained across a cell population until a specific stimulus commands a coordinated shut-off. The robustness of such a design is hampered by molecular (“intrinsic”) or environmental (“extrinsic”) noise, which may lead to spontaneous changes of state in a subset of the population and is reflected in the bimodality of protein expression, as measured for example using flow cytometry. In this context, a “majority-vote” correction circuit, which brings deviant cells back into the required state, is highly desirable, and quorum-sensing has been suggested as a way for cells to broadcast their states to the population as a whole so as to facilitate consensus. In this paper, we propose what we believe is the first such a design that has mathematically guaranteed properties of stability and auto-correction under certain conditions. Our approach is guided by concepts and theory from the field of “monotone” dynamical systems developed by M. Hirsch, H. Smith, and others. We benchmark our design by comparing it to an existing design which has been the subject of experimental and theoretical studies, illustrating its superiority in stability and self-correction of synchronization errors. Our stability analysis, based on dynamical systems theory, guarantees global convergence to steady states, ruling out unpredictable (“chaotic”) behaviors and even sustained oscillations in the limit of convergence. These results are valid no matter what are the values of parameters, and are based only on the wiring diagram. The theory is complemented by extensive computational bifurcation analysis, performed for a biochemically-detailed and biologically-relevant model that we developed. Another novel feature of our approach is that our theorems on exponential stability of steady states for homogeneous or mixed populations are valid independently of the number N of cells in the population, which is usually very large (N ≫ 1) and unknown. We prove that the exponential stability depends on relative proportions of each type of state only. While monotone systems theory has been used previously for systems biology analysis, the current work illustrates its power for synthetic biology design, and thus has wider significance well beyond the application to the important problem of coordination of toggle switches.

For the last decade, outstanding progress has been made, and considerable practical experience has accumulated, in the construction of elementary genetic circuits that perform various tasks, such as memory storage and logical operations, in response to both exogenous and endogenous stimuli. Using modern molecular “plug-and-play” technologies, various (re-)programmable cellular populations can be engineered, and they can be combined into more complex cellular systems. Among all engineered synthetic circuits, a toggle, a robust bistable switch leading to a binary response dynamics, is the simplest basic synthetic biology device, analogous to the “flip-flop” or latch in electronic design, and it plays a key role in biotechnology, biocomputing, and proposed gene therapies. However, despite many remarkable properties of the existing toggle designs, they must be tightly controlled in order to avoid spontaneous switching between different expression states (loss of long-term memory) or even the breakdown of stability through the generation of stable oscillations. To address this concrete challenge, we have developed a new design for quorum-sensing synthetic toggles, based on monotone dynamical systems theory. Our design is endowed with strong theoretical guarantees that completely exclude unpredictable chaotic behaviors in the limit of convergence, as well as undesired stable oscillations, and leads to robust consensus states.

Escherichia coli adapts to tetracycline resistance plasmid (pBR322) by mutating endogenous potassium transport: in silico hypothesis testing

Antibiotic resistance exerts a metabolic cost on bacteria and presumably a fitness disadvantage in the absence of antibiotics. However, several studies have shown that bacteria can evolve to eliminate this cost. Escherichia coli can adapt to the plasmid pBR322 carrying the tetA tetracycline-resistance gene (codes for the TetA efflux pump) by a chromosome mutation, which requires an intact tetA gene on the plasmid. The TetA pump can mediate potassium uptake. Here, the hypothesis that TetA replaces the endogenous K(+) uptake system Trk is explored using a multi-level modeling approach that explicitly resolves relevant intracellular processes (e.g., metabolism and K(+) uptake) and simulates individual bacteria in competition. The general behavior of the model is consistent with observations from the literature (e.g., growth rate and K(+) limitation). In competition experiments without tetracycline, the model correctly predicts the fitness advantage of naive susceptible over naive resistant, evolved resistant over naive resistant and evolved resistant over evolved susceptible strains. Trk takes up about 10 times the K(+) required, which costs energy. TetA takes up less K(+) , which is more efficient and leads to the evolution of the Trk mutant. The evolved Trk mutant relies on TetA to take up K(+) , and thus, carrying the plasmid is advantageous even in the absence of the antibiotic.

Structured model of genetic control via the lac promoter in Escherichia coli

A model that describes induction of protein synthesis from lac-based promoters has been developed and incorporated into the single-cell model of Escherichia coli with transcriptional and translational modifications. Unlike previous models of lac-based promoters, this model allows a priori prediction of the intracellular parameters controlling transcription from lac-based promoters with only the extracellular levels of substrate and inducer as inputs. Because of the structural detail of the model, it is possible to simulate different genetic constructions for comparison, such as Lacl(q) strains versus wild-type cells, or including lacl on a multicopy plasmid. Expression from lac to tac promoters is predicted to yield 5% and 30% of the total cellular protein, respectively, with a pBR322-type plasmid. The model predicts the experimental observation that the Lacl(q) strain is not as fully induced as the wild-type strains, even at higher inducer concentrations. Additionally, the model predicts the right order of magnitude of protein production from lac and tac promoters when mechanisms for attenuation of transcription at lower translational efficiency are considered. Finally, the model predicts that for high copy number systems ribosomes become limiting in the synthesis of plasmid-encoded proteins. (c) 1994 John Wiley & Sons, Inc.

Regulation of ribosome synthesis in Escherichia coli: effects of temperature and dilution rate changes

The effect of temperature on the synthesis of ribosome in Escherichia coli K-12 was investigated. In continuous fermentation, the total and functioning ribosome contents decreased with increasing temperature, while the non-functioning ribosome content remained unchanged. Cells contained higher amounts of functioning ribosome at lower temperatures to compensate for the decrease in translational activity. A transient study was performed to investigate the dynamic response of the cell to changes in the dilution rate. In response to the dilution rate shift-up, the cell mass decreased until the cells produced a sufficient amount of ribosomes to support the new higher growth rate. However, the response to the dilution rate shift-down resulted in an immediate increase in cell mass. This may be due to the fact that the cell already contains enough ribosomes to support a lower growth rate corresponding to the new low dilution rate. Based on the experimental results, a mathematical model was developed to describe the cell growth at transient as well as steady states. The total ribosome content was included as a variable because it affects the growth rate of the cell.

Ribosomal protein limitations in Escherichia coli under conditions of high translational activity

Details of the mechanism for ribosome synthesis have been incorporated in the single-cell Escherichia coli model, which enable us to predict the amount of protein synthesizing machinery under different environmental conditions. The predictions agree quite well with available experimental data. The model predicts that ribosomal protein limitations are important when the translational apparatus is in high demand. Ribosomal RNA synthesis is induced by an increase in translational activity, which, in turn, stimulates ribosomal protein synthesis. However, as the demand increases still more, the ribosomal protein mRNA must compete with the plasmid mRNA for ribosomes, and the efficiency of translation of ribosomal proteins is reduced. (c) 1994 John Wiley & Sons, Inc.

Modeling a minimal cell

One important aim of synthetic biology is to develop a self-replicating biological system capable of performing useful tasks. A mathematical model of a synthetic organism would greatly enhance its value by providing a platform in which proposed modifications to the system could be rapidly prototyped and tested. Such a platform would allow the explicit connection of genomic sequence information to physiological predictions. As an initial step toward this aim, a minimal cell model (MCM) has been formulated. The MCM is defined as a model of a hypothetical cell with the minimum number of genes necessary to grow and divide in an optimally supportive culture environment. It is chemically detailed in terms of genes and gene products, as well as physiologically complete in terms of bacterial cell processes (e.g., DNA replication and cell division). A mathematical framework originally developed for modeling Escherichia coli has been used to build the platform MCM. A MCM with 241 product-coding genes (those which produce protein or stable RNA products) is presented. This gene set is genomically complete in that it codes for all the functions that a minimal chemoheterotrophic bacterium would require for sustained growth and division. With this model, the hypotheses behind a minimal gene set can be tested using a chemically detailed, dynamic, whole-cell modeling approach. Furthermore, the MCM can simulate the behavior of a whole cell that depends on the cell's (1) metabolic rates and chemical state, (2) genome in terms of expression of various genes, (3) environment both in terms of direct nutrient starvation and competitive inhibition leading to starvation, and (4) genomic sequence in terms of the chromosomal locations of genes.

Investigation of subpopulation heterogeneity and plasmid stability in recombinant escherichia coli via a simple segregated model

Many microbial and cell cultures exhibit phenomena that can best be described using a segregated modeling approach. Heterogeneties are more marked in recombinant cell cultures because subpopulations, which often exhibit different growth and productivity characteristics, are more easily identified by selective markers. A simple segregated mathematical model that simulates the growth of recombinant Escherichia coli cells is developed. Subpopulations of different growth rate, plasmid replication rate, and plasmid segregation probability are explicitly considered. Results indicate that a third mechanism of plasmid instability, referred to here as a "downward selective pressure," is significant when describing plasmid loss in batch and chemostat cultures. Also, the model agrees well with experimental data from cultures under antibiotic selective pressure. Finally, model simulations of chemostat cultures reveal the importance of initial conditions on culture stability and the possible presence of nonrandom partitioning functions.

Amino acid supplementation decreases plasmid retention in Escherichia coli

The effect of amino acid supplementation on plasmid stability in Escherichia coli B/r was tested experimentally. Comparisons of experimental results to computer-predicted values were made using a detailed, structured single-cell model. The plasmid, pDW17 (a pBR322 derivative with a mutated tac promoter controlling the beta-lactamase gene), was used. In chemostat cultures, the amino acid supplemented cultures were always less stable than those grown in minimal medium. This effect was not a growth rate effect, as increasing growth rate improves stability for both cultures in minimal medium and in amino acid supplemented medium. The computer model also predicted a decrease in stability due to amino acid supplementation. The model also predicts that amino acid supplementation, combined with moderately strong plasmid-encoded protein expression, results in a depletion of low-molecular-weight organics compared with plasmid-free cells. In minimal medium the same level of plasmid-encoded protein synthesis results in a strong reduction in amino acid pools compared with plasmid-free cells. With amino acid supplementation the growth differential between plasmid-bearing and plasmid-free cells may be due to an "energy limitation," while in minimal medium the size of the growth rate differential may be due to a "building block" limitation.

Multi-scale spatio-temporal modeling: lifelines of microorganisms in bioreactors and tracking molecules in cells

Agent-based models are rigorous tools for simulating the interactions of individual entities, such as organisms or molecules within cells and assessing their effects on the dynamic behavior of the system as a whole. In context with bioprocess and biosystems engineering there are several interesting and important applications. This contribution aims at introducing this strategy with the aid of two examples characterized by striking distinctions in the scale of the individual entities and the mode of their interactions. In the first example a structured-segregated model is applied to travel along the lifelines of single cells in the environment of a three-dimensional turbulent field of a stirred bioreactor. The modeling approach is based on an Euler-Lagrange formulation of the system. The strategy permits one to account for the heterogeneity present in real reactors in both the fluid and cellular phases, respectively. The individual response of the cells to local variations in the extracellular concentrations is pictured by a dynamically structured model of the key reactions of the central metabolism. The approach permits analysis of the lifelines of individual cells in space and time.The second application of the individual modeling approach deals with dynamic modeling of signal transduction pathways in individual cells. Usually signal transduction networks are portrayed as being wired together in a spatially defined manner. Living circuitry, however, is placed in highly malleable internal architecture. Creating a homogenous bag of molecules, a well-mixed system, the dynamic behavior of which is modeled with a set of ordinary differential equations is normally not valid. The dynamics of the MAP kinase and a steroid hormone pathway serve as examples to illustrate how single molecule tracking can be linked with the stochasticity of biochemical reactions, where diffusion and reaction occur in a probabilistic manner. The problem of hindered diffusion caused by macromolecular crowding is also taken into account.

Modeling of enzyme production kinetics

Models of single cells, cell populations, and cultures can be most useful in organizing information in a comprehensive system description, as well as in optimizing and controlling actual production operations. Models discussed in this article are of various degrees of biological structure and mathematical complexity. The models are developed based on the biomass formation, substrate consumption, and product formation. the potentials asn the limitations of all the models have been reported. The parameter estimation by different methods has been discussed in this communication. These parameters will be helpful to explore the areas where future-modeling studies may be especially valuable.

Modeling the synchronization of yeast respiratory oscillations

The budding yeast Saccharomyces cerevisiae exhibits autonomous oscillations when grown aerobically in continuous culture with ethanol as the primary carbon source. A single cell model that includes the sulfate assimilation and ethanol degradation pathways recently has been developed to study these respiratory oscillations. We utilize an extended version of this single cell model to construct large cell ensembles for investigation of a proposed synchronization mechanism involving hydrogen sulfide. Ensembles with as many as 10,000 cells are used to simulate population synchronization and to compute transient number distributions from asynchronous initial cell states. Random perturbations in intracellular kinetic parameters are introduced to study the synchronization of single cells with small variations in their unsynchronized oscillation periods. The cell population model is shown to be consistent with available experimental data and to provide insights into the regulatory mechanisms responsible for the synchronization of yeast metabolic oscillations.

A modular minimal cell model: purine and pyrimidine transport and metabolism

A more complete understanding of the relationship of cell physiology to genomic structure is desirable. Because of the intrinsic complexity of biological organisms, only the simplest cells will allow complete definition of all components and their interactions. The theoretical and experimental construction of a minimal cell has been suggested as a tool to develop such an understanding. Our ultimate goal is to convert a "coarse-grain" lumped parameter computer model of Escherichia coli into a genetically and chemically detailed model of a "minimal cell." The base E. coli model has been converted into a generalized model of a heterotrophic bacterium. This coarse-grain minimal cell model is functionally complete, with growth rate, composition, division, and changes in cell morphology as natural outputs from dynamic simulations where only the initial composition of the cell and of the medium are specified. A coarse-grain model uses pseudochemical species (or modules) that are aggregates of distinct chemical species that share similar chemistry and metabolic dynamics. This model provides a framework in which these modules can be "delumped" into chemical and genetic descriptions while maintaining connectivity to all other functional elements. Here we demonstrate that a detailed description of nucleotide precursors transport and metabolism is successfully integrated into the whole-cell model. This nucleotide submodel requires fewer (12) genes than other theoretical predictions in minimal cells. The demonstration of modularity suggests the possibility of developing modules in parallel and recombining them into a fully functional chemically and genetically detailed model of a prokaryote cell.

Modelling of Mammalian cells and cell culture processes

Mammalian cell cultures represent the major source for a number of very high-value biopharmaceutical products, including monoclonal antibodies (MAbs), viral vaccines, and hormones. These products are produced in relatively small quantities due to the highly specialised culture conditions and their susceptibility to either reduced productivity or cell death as a result of slight deviations in the culture conditions. The use of mathematical relationships to characterise distinct parts of the physiological behaviour of mammalian cells and the systematic integration of this information into a coherent, predictive model, which can be used for simulation, optimisation, and control purposes would contribute to efforts to increase productivity and control product quality. Models can also aid in the understanding and elucidation of underlying mechanisms and highlight the lack of accuracy or descriptive ability in parts of the model where experimental and simulated data cannot be reconciled. This paper reviews developments in the modelling of mammalian cell cultures in the last decade and proposes a future direction - the incorporation of genomic, proteomic, and metabolomic data, taking advantage of recent developments in these disciplines and thus improving model fidelity. Furthermore, with mammalian cell technology dependent on experiments for information, model-based experiment design is formally introduced, which when applied can result in the acquisition of more informative data from fewer experiments. This represents only part of a broader framework for model building and validation, which consists of three distinct stages: theoretical model assessment, model discrimination, and model precision, which provides a systematic strategy from assessing the identifiability and distinguishability of a set of competing models to improving the parameter precision of a final validated model.

Structured model to predict intracellular amino acid shortages during recombinant protein overexpression in E. coli

Recombinant protein overexpression and the classical stringent response have been shown to induce the same proteases. Since the stringent response was the result of an intracellular amino acid shortage, it was hypothesized that the overexpression of the recombinant protein also caused an intracellular amino acid shortage. A structured non-segregated kinetic mathematical model of recombinant Escherichia coli was developed to predict intracellular amino acid shortages during recombinant protein overexpression, and thus the induction of the stringent response. Two model recombinant proteins were examined, chloramphenicol acetyl-transferase (CAT) and an 'average protein'. The model predicted an aromatic amino acid shortage during CAT overexpression, as predicted based on the CAT's amino acid content. The model also predicted a shortage of the intracellular alanine family amino acid pool during CAT overexpression. This was unexpected due to the relatively low content of alanine family amino acids in CAT compared to the average E. coli protein. The model predicted alanine, glutarate, and aspartate family amino acid shortages during recombinant 'average protein' overexpression. Additionally, the model predicted a decrease in the ribosome pool at induction for both recombinant proteins, which agrees with published experimental results. Thus, the structured kinetic model was able to predict amino acid shortages, that could potentially cause a stringent response and elevated protease activity.

Single-cell models: promise and limitations

This article describes the development of single-cell models, their uses and accomplishments, the barriers to the greater adoption, and a perspective on challenges to the biochemical engineering community where the single-cell model approach may be used advantageously. In particular, it may become an important tool in relating genomic information to cellular regulation and dynamics.

Studies on plasmid stability and LTB production by recombinant Vibrio cholerae in batch and chemostat cultures: a lesson for optimizing conditions for chemical induction

Plasmid content, its stability and the expression of B-subunit of Escherichia coli heat-labile enterotoxin (LTB) in Vibrio cholerae/r-pMMB68 system have been studied in batch as well as in chemostat cultures. Upon induction with isopropyl-beta-D-thiogalactopyranoside (IPTG), cultures secreted LTB into the extracellular milieu. Highest specific LTB production rate of 7.3 mg mg-1h-1 was achieved in batch culture induced at the late exponential growth phase. The plasmid pMMB68 was fairly stable up to 20 generations, even in the absence of selection pressure. Instability of the plasmid was accelerated in the presence of IPTG and at higher dilution rates. Maximum productivity of 2.1 mg l-1 h-1 was achieved in continuous culture, which remained constant at a range of dilution rates from 0.20 to 0.35 h-1.

Immuno- and flow cytometric analytical methods for biotechnological research and process monitoring

In this article, the applications of immunoanalysis and flow cytometry for research and process monitoring in biotechnology are discussed. Brief reviews of the two analytical methods are followed by descriptions of actual applications in various areas of biotechnology. In the case of immunoanalysis, emphasis is placed on systems for on-line bioprocess monitoring, and examples are given for a thermostable pullulanase, a mouse IgG, and antithrombin III. Although flow cytometry is not currently an on-line analytical technique, its value as an off-line method is illustrated by examples of the measurement of shear stress effects, lipid content, and sterol content.