A mathematical model of the methionine cycle

Building on the work of Martinov et al. (2000), a mathematical model is developed for the methionine cycle. A large amount of information is available about the enzymes that catalyse individual reaction steps in the cycle, from methionine to S-adenosylmethionine to S-adenosylhomocysteine to homocysteine, and the removal of mass from the cycle by the conversion of homocysteine to cystathionine. Nevertheless, the behavior of the cycle is very complicated since many substrates alter the activities of the enzymes in the reactions that produce them, and some can also alter the activities of other enzymes in the cycle. The model consists of four differential equations, based on known reaction kinetics, that can be solved to give the time course of the concentrations of the four main substrates in the cycle under various circumstances. We show that the behavior of the model in response to genetic abnormalities and dietary deficiencies is similar to the changes seen in a wide variety of experimental studies. We conduct computational "experiments" that give understanding of the regulatory behavior of the methionine cycle under normal conditions and the behavior in the presence of genetic variation and dietary deficiencies.     

Optimization and mathematical modeling of the transtubular bioreactor for the production of monoclonal antibodies from a hybridoma cell line

This report describes the use of a transtubular bioreactor to study the relative effects of diffusion versus perfusion of medium on antibody production by a hybridoma cell line. The study was performed with a high-density cell culture maintained in a serum-free, low-protein medium for 77 days. It was determined that the reactor possessed a macro-mixing pattern residence time distribution similar to a continuous stirred tank reactor (CSTR). However, due to the arrangement of the medium lines in the reactor, the flow patterns for nutrient distribution consist of largely independent medium path lengths ranging from short to long. When operated with cyclic, reversing, transtubular medium flow, some regions of the reactor (with short residence times) are more accessible to medium than others (with long residence times). From this standpoint, the reactor can be divided into three regions: a captive volume, which consists of medium primarily delivered via diffusion; a lapped volume, which provides nutrients through unilateral convection; and a swept volume, which operates through bilateral convection. The relative sizes of these three volumes were modified experimentally by changing the period over which the direction of medium flow was reversed from 15 min (larger captive volume) to 9 h (larger swept volume). The results suggest that antibody concentration increases as the size of the diffusion-limited (captive) volume is increased to a maximum at around 30 min with a sharp decrease thereafter. As reflected by changes in measured consumption of glucose and production of lactate, no significant difference in cellular metabolism occurred as the reactor was moved between these different states. These results indicate that the mode of operation of the transtubular bioreactor may influence antibody productivity under serum-free, low-protein conditions with minimal effects on cellular metabolism.     

A mathematical model of the Calvin photosynthesis cycle

1. A mathematical model is presented for photosynthetic carbohydrate formation in C3 plants under conditions of light and carbon dioxide saturation. The model considers reactions of the Calvin cycle with triose phosphate export and starch production as main output processes, and treats concentrations of NADPH, NAD+, CO2, and H+ as fixed parameters of the system. Using equilibrium approximations for all reaction steps close to equilibrium steady-state and transient-state relationships are derived which may be used for calculation of reaction fluxes and concentrations of the 13 carbohydrate cycle intermediates, glucose 6-phosphate, glucose 1-phosphate, ATP, ADP, and inorganic (ortho)phosphate. 2. Predictions of the model were examined with the assumption that photosynthate export from the chloroplast occurs to a medium containing orthophosphate as the only exchangeable metabolite. The results indicate that the Calvin cycle may operate in a single dynamically stable steady state when the external concentration of orthophosphate does not exceed 1.9 mM. At higher concentrations of the external metabolite, the reaction system exhibits overload breakdown; the excessive rate of photosynthate export deprives the system of cycle intermediates such that the cycle activity progressively approaches zero. 3. Reactant concentrations calculated for the stable steady state that may obtain are in satisfactory agreement with those observed experimentally, and the model accounts with surprising accuracy for experimentally observed effects of external orthophosphate on the steady-state cycle activity and rate of starch production. 4. Control analyses are reported which show that most of the non-equilibrium enzymes in the system have a strong regulatory influence on the steady-state level of all of the cycle intermediates. Substrate concentration control coefficients for cycle enzymes may be positive, such that an increase in activity of an enzyme may raise the steady-state concentration of the substrate is consumes. 5. Under optimal external conditions (0.15-0.5 mM orthophosphate), reaction flux in the Calvin cycle is controlled mainly by ATP synthetase and sedoheptulose bisphosphatase; the cycle activity approaches the maximum velocity that can be supported by the latter enzyme. At lower concentrations of external orthophosphate the cycle activity is controlled almost exclusively by the phosphate translocator.(ABSTRACT TRUNCATED AT 400 WORDS)     

A mathematical model for analysis of the cell cycle in cell lines derived from human tumors

The growth of human cancers is characterised by long and variable cell cycle times that are controlled by stochastic events prior to DNA replication and cell division. Treatment with radiotherapy or chemotherapy induces a complex chain of events involving reversible cell cycle arrest and cell death. In this paper we have developed a mathematical model that has the potential to describe the growth of human tumour cells and their responses to therapy. We have used the model to predict the response of cells to mitotic arrest, and have compared the results to experimental data using a human melanoma cell line exposed to the anticancer drug paclitaxel. Cells were analysed for DNA content at multiple time points by flow cytometry. An excellent correspondence was obtained between predicted and experimental data. We discuss possible extensions to the model to describe the behaviour of cell populations in vivo.     

Effect of NH3 on the cell growth of a hybridoma cell line

Ammonia often has been reported to inhibit cell growth. The aqueous ammonia equilibrium between the un-ionized form (NH₃) and the ammonium ion (NH₄ ⁺) depends on the pH of the solution. Extensive studies in batch and continuous cultivation by varying pH and total ammonia concentration were carried out to investigate whether a kinetic model describing growth inhibition by ammonia has to be based on the total ammonia concentration, or the concentration of NH₃. A significant relationship between the specific growth rate and death rate, respectively, and the NH₃ concentration but not the total ammonia concentration, was detected. An adaptation of the cells to high ammonia levels was not observed. Based on these results a new kinetic model for ammonia mediated growth inhibition is suggested. For high density cultivation it is recommended to control the pH at the lower limit of the growth optimum to keep the NH₃ level low.     

A cell cycle model for somitogenesis: mathematical formulation and numerical simulation

After many years of research, the mechanisms that generate a periodic pattern of repeated elements (somites) along the length of the embryonic body axis is still one of the major unresolved problems in developmental biology. Here we present a mathematical formulation of the cell cycle model for somitogenesis proposed in Development105 (1989), 119-130. Somite precursor cells in the node are asynchronous, and therefore, as a population, generate continuously pre-somite cells which enter the segmental plate. The model makes the hypothesis that there exists a time window within the cell cycle, making up one-seventh of the cycle, which gates the pre-somite cells so that they make somites discretely, seven per cycle. We show that the model can indeed account for the spatiotemporal patterning of somite formation during normal development as well as the periodic abnormalities produced by heat shock treatment. We also relate the model to recent molecular data on the process of somite formation.     

A mathematical model of the sleep/wake cycle

We present a biologically-based mathematical model that accounts for several features of the human sleep/wake cycle. These features include the timing of sleep and wakefulness under normal and sleep-deprived conditions, ultradian rhythms, more frequent switching between sleep and wakefulness due to the loss of orexin and the circadian dependence of several sleep measures. The model demonstrates how these features depend on interactions between a circadian pacemaker and a sleep homeostat and provides a biological basis for the two-process model for sleep regulation. The model is based on previous “flip–flop” conceptual models for sleep/wake and REM/NREM and we explore whether the neuronal components in these flip–flop models, with the inclusion of a sleep-homeostatic process and the circadian pacemaker, are sufficient to account for the features of the sleep/wake cycle listed above. The model is minimal in the sense that, besides the sleep homeostat and constant cortical drives, the model includes only those nuclei described in the flip–flop models. Each of the cell groups is modeled by at most two differential equations for the evolution of the total population activity, and the synaptic connections are consistent with those described in the flip–flop models. A detailed analysis of the model leads to an understanding of the mathematical mechanisms, as well as insights into the biological mechanisms, underlying sleep/wake dynamics.     

A complex mathematical model of the human menstrual cycle

Despite the fact that more than 100 million women worldwide use birth control pills and that half of the world's population is concerned, the menstrual cycle has so far received comparatively little attention in the field of mathematical modeling. The term menstrual cycle comprises the processes of the control system in the female body that, under healthy circumstances, lead to ovulation at regular intervals, thus making reproduction possible. If this is not the case or ovulation is not desired, the question arises how this control system can be influenced, for example, by hormonal treatments. In order to be able to cover a vast range of external manipulations, the mathematical model must comprise the main components where the processes belonging to the menstrual cycle occur, as well as their interrelations. A system of differential equations serves as the mathematical model, describing the dynamics of hormones, enzymes, receptors, and follicular phases. Since the processes take place in different parts of the body and influence each other with a certain delay, passing over to delay differential equations is deemed a reasonable step. The pulsatile release of the gonadotropin-releasing hormone (GnRH) is controlled by a complex neural network. We choose to model the pulse time points of this GnRH pulse generator by a stochastic process. Focus in this paper is on the model development. This rather elaborate mathematical model is the basis for a detailed analysis and could be helpful for possible drug design.     

Parameter estimation for a mathematical model of the cell cycle in frog eggs.

Parameter values for a kinetic model of the nuclear replication-division cycle in frog eggs are estimated by fitting solutions of the kinetic equations (nonlinear ordinary differential equations) to a suite of experimental observations. A set of optimal parameter values is found by minimizing an objective function defined as the orthogonal distance between the data and the model. The differential equations are solved by LSODAR and the objective function is minimized by ODRPACK. The optimal parameter values are close to the guesstimates of the modelers who first studied this problem. These tools are sufficiently general to attack more complicated problems, where guesstimation is impractical or unreliable.     

Cell cycle kinetics of the accumulation of heavy and light chain immunoglobulin proteins in a mouse hybridoma cell line

Rates of accumulation of immunoglobulin proteins have been determined using flow cytometry and population balance equations for exponentially growing murine hybridoma cells in the individual G₁, S and G₂+M cell cycle phases. A producer cell line that secretes monoclonal antibodies, and a nonproducer clone that synthesizes only -light chains were analyzed. The pattern for the kinetics of total intracellular antibody accumulation during the cell cycle is very similar to the previously described pattern for total protein accumulation (Kromenaker & Srienc 1991). The relative mean rate of heavy chain accumulation during the S phase was approximately half the relative mean rate of light chain accumulation during this cell cycle phase. This indicates an unbalanced synthesis of heavy and light chains that becomes most pronounced during this cell cycle phase. The nonproducer cells have on average an intracellular light chain content that is 42% lower than that of the producer cells. The nonproducer cells in the G₁ phase with low light chain content did not have a significantly higher rate of light chain accumulation relative to other G₁ phase nonproducer cells. This is in sharp contrast to what was observed for the G₁ phase producer cells. In addition, although the relative mean rate of accumulation of light chain was negative for G₂+M phase nonproducer cells, the magnitude of this relative mean rate was less than half that observed for the producer cells in this cell cycle phase. This suggests that the mechanisms that regulate the transport of fully assembled antibody molecules through the secretion pathway differ from those which regulate the secretion of free light chains. The results reported here indicate that there is a distinct pattern for the cell cycle dynamics of antibody synthesis and secretion in hybridomas. These results are consistent with a model for the dynamics of secretion which suggests that the rate of accumulation of secreted proteins will be greatest for newborn cells due to an interruption of the secretion pathway during mitosis.     

A temperature-sensitive cell cycle mutant of the BHK cell line

A temperature-sensitive growth mutant derived from the BHK 21 cell Line, ts AF8, was found to have greatly reduced DNA synthesis at the nonpermissive temperature. This reduction is mainly due to a decrease in the frequency of cells synthesizing DNA. Upon shift up, ts AF8 becomes blocked in the G1 phase of the cell cycle. The cells acquire elevated cAMP levels and a unimodal distribution of DNA content, equivalent to that of G1 cells at the permissive temperature, Ts AF8 cells blocked at the G1/S boundary with hydroxyurea will enter S when shifted to the nonpermissive temperature. On the other hand, ts AF8 cells arrested m G1 by serum deprivation and shifted to the nonpermissive temperature at the moment of serum addition do not enter S, while those synchronized by isoleucine deprivation and shifted at the time of isoleucine addition will enter S. These data suggest that the cycle arrest point of the ts AF8 mutation is located in G1 between the blocks induced by serum starvation and isoleucine deprivation. The reduction in DNA synthesis caused by the ts AF8 mutation is not reversed by infection or transformation with Polyoma virus. Mitochondrial DNA continues to be synthesized at wild-type levels at the nonpermissive temperature.     

A new parental cell line for human x human hybridoma production

The potential of a new HAT-sensitive human lymphoblastoid cell line TK6 TG^r.P1. as a fusion partner was assessed, by comparison with the established human parental cell line UC729.6. Both of these cell lines were fused with the peripheral blood mononuclear cells of a patient with B-chronic lymphocytic leukaemia. The hybridomas generated in these fusion experiments were analysed by the fluorescence activated cell sorter and karyotyping. An anti-idiotype ELISA assay detected the presence of the patient's characteristic idiotype bearing immunoglobulin in the supernatant of a number of the hybridoma cell lines generated in both fusions.     

Family tree analysis of a transformed cell line and the transition probability model for the cell cycle

Variation in intermitotic time between individual cells in culture can be ascribed to the occurrence of random transitions in the cell cycle. We have analysed a family tree of mouse neuroblastoma cells, and observed that variation in difference in intermitotic time between sister cells is smaller than between cousin cells, and this difference is again smaller than between second-cousin and unrelated cells. This observation is incompatible with all transition probability models presented so far. We propose a model for the cell cycle with a single random transition, but with the additional assumption that the (two) system parameters may show variability within the population such that the closer cells are in their relation to each other, the closer their values of the system parameters will be. This model describes correctly the behaviour of the family tree of the cell line and in addition is able to explain why differences in intermitotic time between sister cells are exponentially distributed, while intermitotic times themselves are more or less normally distributed. Methods have been described to quantify the various system parameters.     

A deterministic model of the cell cycle

The variability of the duration of the cell cycle is explained by the phenomenon of sensitive dependence upon initial conditions; as may occur in deterministic non-linear systems. Chaotic dynamics of a system is the result of this sensitive dependence. First a deterministic system is formulated that is equivalent to the Smith-Martin transition probability model of the cell cycle. Next the model is extended to a dynamic process that ranges over the cell generations. A deterministic non-linear relationship between the cycle time of the mother and daughter cell is established. It clarifies the variability of mother-daughter correlation for the different cell types. The model is fitted to two different cell cultures; it shows that the graph of the non-linear relation has the same shape for different cell types.     

Dichloroacetate increases cell and antibody yields in batch cultures of a hybridoma cell line

We have studied the effect of the pyruvate dehydrogenase (PDH) activator, dichloroacetate (DCA), on the growth, metabolism, and productivity of the PQXB (1/2) hybridoma cell line. In control batch cultures, cessation of growth and the onset of decline phase coincided with the time at which the media became exhausted of glutamine. Supplementation of the media with DCA (1 mM) extended the growth phase of this cell line by approximately 20 h without affecting its growth rate. This prolonged period of growth resulted in an increased maximum cell density (16%) and final antibody yield (55%). Repeat experiments showed these effects to be reproducible, with the increases in antibody yield being between 50 and 60%. DCA did not affect the specific rates of glucose utilization and lactate production. However, it decreased the specific glutamine consumption rate. This characteristic of DCA action appeared, at least in part, to provide an explanation for the extended growth phase exhibited by DCA-treated cultures, since it delayed the time at which the media became depleted of glutamine. The consumption and production kinetics for various nutrients and their metabolites in both control and DCA-treated cultures suggested that: (1) glutamine catabolism proceeded by a pathway involving conversion to glutamate by glutaminase followed by subsequent transamination by alanine aminotransferase, and (2) DCA decreased the specific glutamine consumption rate by directly or indirectly inhibiting the transamination. It is expected that the routine inclusion of DCA in media used for hybridoma cultivation will be valuable for enhancement of monoclonal antibody (Mab) yields on a laboratory scale. (c) 1996 John Wiley & Sons, Inc.