A mathematical model for the growth of a single cell of E. coli on a glucose/glutamine/ammonium medium

Computer-Aided Whole-Cell Design: Taking a Holistic Approach by Integrating Synthetic With Systems Biology

Computer-aided design (CAD) for synthetic biology promises to accelerate the rational and robust engineering of biological systems. It requires both detailed and quantitative mathematical and experimental models of the processes to (re)design biology, and software and tools for genetic engineering and DNA assembly. Ultimately, the increased precision in the design phase will have a dramatic impact on the production of designer cells and organisms with bespoke functions and increased modularity. CAD strategies require quantitative models of cells that can capture multiscale processes and link genotypes to phenotypes. Here, we present a perspective on how whole-cell, multiscale models could transform design-build-test-learn cycles in synthetic biology. We show how these models could significantly aid in the design and learn phases while reducing experimental testing by presenting case studies spanning from genome minimization to cell-free systems. We also discuss several challenges for the realization of our vision. The possibility to describe and build whole-cells in silico offers an opportunity to develop increasingly automatized, precise and accessible CAD tools and strategies.

Metabolic ensemble modeling for strain engineers

Previous mathematical modeling efforts have made significant contributions to the development of systems biology for predicting biological behavior quantitatively. However, dynamic metabolic model construction remains challenging due to uncertainties in mechanistic structures and parameters. In addition, parameter estimation and model validation often require designated experiments conducted only for purpose of modeling. Such difficulties have hampered the progress of modeling in biology and biotechnology. To circumvent these problems, ensemble approaches have been used to account for uncertainties in model structure and parameters. Specifically, this review focuses on approaches that utilize readily available fermentation data for parameter screening and model validation. Time course data for metabolite measurements, if available, can further calibrate the model. The basis for this approach is explained in non-mathematical terms accessible to experimentalists. Information gained from such an approach has been shown to be useful in designing Escherichia coli strains for metabolic engineering and synthetic biology.

Mathematical modeling of induced foreign protein production by recombinant bacteria

A new generalized mathematical model for recombinant bacteria which includes inducer effects on cell growth and foreign protein production is developed. The model equation set was applied to a host-vector system, Escherichi coli D1210 and plasmid pSD8. Batch experiments were designed and performed in shake flasks to verify the model. A parameter estimation method was developed and proven to be efficient. Although simple, the model can effectively describe the dynamics of the production of foreign protein in recombinant bacteria and can be used for optimization and control studies to maximize foreign protein production.

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.

Characterizing the metabolic phenotype: a phenotype phase plane analysis

Genome-scale metabolic maps can be reconstructed from annotated genome sequence data, biochemical literature, bioinformatic analysis, and strain-specific information. Flux-balance analysis has been useful for qualitative and quantitative analysis of metabolic reconstructions. In the past, FBA has typically been performed in one growth condition at a time, thus giving a limited view of the metabolic capabilities of a metabolic network. We have broadened the use of FBA to map the optimal metabolic flux distribution onto a single plane, which is defined by the availability of two key substrates. A finite number of qualitatively distinct patterns of metabolic pathway utilization were identified in this plane, dividing it into discrete phases. The characteristics of these distinct phases are interpreted using ratios of shadow prices in the form of isoclines. The isoclines can be used to classify the state of the metabolic network. This methodology gives rise to a "phase plane" analysis of the metabolic genotype-phenotype relation relevant for a range of growth conditions. Phenotype phase planes (PhPPs) were generated for Escherichia coli growth on two carbon sources (acetate and glucose) at all levels of oxygenation, and the resulting optimal metabolic phenotypes were studied. Supplementary information can be downloaded from our website (http://epicurus.che.udel.edu).

A mathematical model of the cell cycle of a hybridoma cell line

A one-dimensional age-based population balance model of the cell cycle is proposed for a mouse-mouse hybridoma cell line (mm321) producing immunoglobulin G antibody to paraquat. It includes the four conventional cell cycle phases, however, G1 is divided into two parts (G1a and G1b). Two additional phases have been added, a non-cycling state G1', and a pre-death phase D. The duration of these additional phases is determined by cumulative glutamine content and ammonia concentration, respectively. It is assumed that glutamine is only consumed during G1 and antibody is only produced during G1b and S, the kinetics are assumed to be zero-order. Glucose is consumed throughout the cell cycle at a rate that is dependent upon its prevalent concentration. Ammonia and lactate are produced in direct proportion to glutamine and glucose consumption, respectively. Parameters in the model have been determined from experimental data or from fitting the model to post-synchronisation data. The model thus fitted has been used to successfully predict this cell lines behaviour in conventional batch culture at different initial glutamine concentrations, and in chemostat culture at steady-state and in response to a glutamine pulse. The model predicts viable cell, glutamine, glucose and lactate kinetics well, but there are some discrepancies in the prediction for ammonia and antibody. Overall, the results obtained support the assumptions made in the model relating to the regulation of cell cycle progression. It is concluded that this approach has the potential to be exploited with other cell lines and used in a model-based control scheme.

The Escherichia coli MG1655 in silico metabolic genotype: its definition, characteristics, and capabilities

The Escherichia coli MG1655 genome has been completely sequenced. The annotated sequence, biochemical information, and other information were used to reconstruct the E. coli metabolic map. The stoichiometric coefficients for each metabolic enzyme in the E. coli metabolic map were assembled to construct a genome-specific stoichiometric matrix. The E. coli stoichiometric matrix was used to define the system's characteristics and the capabilities of E. coli metabolism. The effects of gene deletions in the central metabolic pathways on the ability of the in silico metabolic network to support growth were assessed, and the in silico predictions were compared with experimental observations. It was shown that based on stoichiometric and capacity constraints the in silico analysis was able to qualitatively predict the growth potential of mutant strains in 86% of the cases examined. Herein, it is demonstrated that the synthesis of in silico metabolic genotypes based on genomic, biochemical, and strain-specific information is possible, and that systems analysis methods are available to analyze and interpret the metabolic phenotype.

Cellular growth in biofilms

In this paper we develop a macroscopic evolutionary equation for the growth of the cellular phase starting from a microscopic description of mass transport and a simple structured model for product formation. The methods of continuum mechanics and volume averaging are used to develop the macroscopic representation by carefully considering the fluxes of chemical species that pertain to cell growth. The concept of structured models is extended to include the transport of reacting chemical species at the microscopic scale. The resulting macroscopic growth model is similar in form to previously published models for the transport of a single substrate and electron donor and for the production of cellular mass and exopolymer. The method of volume averaging indicates under what conditions the developed growth model is valid and provides an explicit connection between the relevant microscopic model parameters and their corresponding macroscopic counterparts.

Characterization of stress and protein turnover from protein overexpression in fed-batch E. coli cultures

A structured kinetic model that accounts for proteolytic degradation due to recombinant protein overexpression is introduced and its performance evaluated by comparison with previously reported fed-batch experimental data. This mathematical model contains an additional pool for a generic key precursor (in our case phenylalanine), an improved IPTG transport term, a phenylalanine transport term, and a variable protein turnover expression that accounts for proteolytic activity. The model predictions concerning proteolytic activity, glucose level, and cell growth are in very good agreement with an amino acid depletion hypothesis. Cultures exposed to greater stress showed higher and/or longer proteolysis, whereas less overall proteolytic activity was observed when the effect of induction was somewhat ameliorated.

Mathematical models of metabolic pathways

There have been recent advances in metabolic flux analysis. In particular, the marriage of traditional flux balancing with NMR isotopomer distribution analysis holds great promise for the detailed quantification of physiology. Nevertheless, flux analysis yields only static snap-shots of metabolism. To robustly predict the time evolution of metabolic networks, dynamic mathematical models, especially those that contain a description of both gene expression as well as enzyme activity, must be utilized. When mechanistic control and regulatory information is not available, heuristic-based methods, such as the cybernetic framework, can be employed to describe the action of these control mechanisms. In the 'high-information' future, as more biological information becomes available, such heuristic-based approaches can be replaced by mechanistic mass-action representations of physiology that stem directly from genetic sequence.

The relation of proton motive force, adenylate energy charge and phosphorylation potential to the specific growth rate and efficiency of energy transduction in Bacillus licheniformis under aerobic growth conditions

The magnitude of the proton motive force (delta p) and its constituents, the electrical (delta psi) and chemical potential (-Z delta pH), were established for chemostat cultures of a protease-producing, relaxed (rel-) variant and a not protease-producing, stringent (rel+) variant of an industrial strain of Bacillus licheniformis (respectively referred to as the A- and the B-type). For both types, an inverse relation of delta p with the specific growth rate mu was found. The calculated intracellular pH (pHin) was not constant but inversely related to mu. This change in pHin might be related to regulatory functions of metabolism but a regulatory role for pHin itself could not be envisaged. Measurement of the adenylate energy charge (EC) showed a direct relation with mu for glucose-limited chemostat cultures; in nitrogen-limited chemostat cultures, the EC showed an approximately constant value at low mu and an increased value at higher mu. For both limitations, the ATP/ADP ratio was directly related to mu. The phosphorylation potential (delta G'p) was invariant with mu. From the values for delta G'p and delta p, a variable -->H+/ATP-stoichiometry was inferred: -->H+/ATP = 1.83 +/- 0.52 mu, so that at a given -->H+/O-ratio of four (4), the apparent P/O-ratio (inferred from regression analysis) showed a decline of 2.16 to 1.87 for mu = 0 to mu max (we discuss how more than half of this decline will be independent of any change in internal cell-volume). We propose that the constancy of delta G'p and the decrease in the efficiency of energy-conservation (P/O-value) with increasing mu are a way in which the cells try to cope with an apparent less than perfect coordination between anabolism and catabolism to keep up the highest possible mu with a minimum loss of growth-efficiency. Protease production in nitrogen-limited cultures as compared to glucose-limited cultures, and the difference between the A- and B-type, could not be explained by a different energy-status of the cells.

Quantification of control of microbial metabolism by substrates and enzymes

The control of substrates or enzymes on metabolic processes can be expressed in quantitative terms. Most of the experimental material found in the literature, however, has been obtained under non-standardized conditions, precluding definite conclusions concerning the magnitude of control. A number of representative examples is discussed and it is concluded that a quantitative analysis of the factors that control metabolism is essential for understanding the microbial behaviour.

A structured model for vegetative growth and sporulation in Bacillus thuringiensis

A mathematical model has been developed for the delta-endotoxin producing Bacillus thuringiensis. The structure of the model involves the processes taking place during vegetative growth, those leading to the initiation of sporulation under conditions of carbon and/or nitrogen limitation, and the sporulation events. The key features in the model are the pools of compounds, such as PRPP, IMP, ADP/ATP, GDP/GTP, pyrimidine nucleotides, NAD/NADH2, amino acids, nucleic acids, cell wall, and vegetative and sporulation proteins. These, along with sigma-factors that control the nature of RNA-polymerase during the different phases, effectively stimulate the vegetative growth and sporulation. The initiation of sporulation is controlled by the intracellular concentration of GTP. Results of simulation of vegetative growth, initiation of sporulation, spore protein formation, and production of delta-endotoxin under C- or N-limitation are presented.