A discrete,size-structured model of phytoplankton growth in the chemostat: introduction of inhomogeneous cell division size

 We introduce inhomogeneous, substrate dependent cell division in a time discrete, nonlinear matrix model of size-structured population growth in the chemostat, first introduced by Gage et al. [8] and later analysed by Smith [13]. We show that mass conservation is verified, and conclude that our system admits one non zero globally stable equilibrium, which we express explicitly. Then we run numerical simulations of the system, and compare the predictions of the model to data related to phytoplankton growth, whose obtention we discuss. We end with the identification of several parameters of the system. Received: 9 February 2000 / Revised version: 10 October 2001 / Published online: 23 August 2002 RID="*" ID="*" Present address: Department of Mathematics and Statistics, University of Victoria, B.C., Canada. e-mail: jarino@math.uvic.ca Key words or phrases: Chemostat – Structured population models – Discrete model – Inhomogeneous division size     

Computational analysis of CFSE proliferation assay

CFSE based tracking of the lymphocyte proliferation using flow cytometry is a powerful experimental technique in immunology allowing for the tracing of labelled cell populations over time in terms of the number of divisions cells undergone. Interpretation and understanding of such population data can be greatly improved through the use of mathematical modelling. We apply a heterogenous linear compartmental model, described by a system of ordinary differential equations similar to those proposed by Kendall. This model allows division number-dependent rates of cell proliferation and death and describes the rate of changes in the numbers of cells having undergone j divisions. The experimental data set that we specifically analyze specifies the following characteristics of the kinetics of PHA-induced human T lymphocyte proliferation assay in vitroL (1) the total number of live cells, (2) the total number of dead but not disintegrated cells and (3) the number of cells divided j times. Following the maximum likelihood approach for data fitting, we estimate the model parameters which, in particular, present the CTL birth- and death rate “functions”. It is the first study of CFSE labelling data which convincingly shows that the lymphocyte proliferation and death both in vitro and in vivo are division number dependent. For the first time, the confidence in the estimated parameter values is analyzed by comparing three major methods: the technique based on the variance–covariance matrix, the profile-likelihood-based approach and the bootstrap technique. We compare results and performance of these methods with respect to their robustness and computational cost. We show that for evaluating mathematical models of differing complexity the information-theoretic approach, based upon indicators measuring the information loss for a particular model (Kullback–Leibler information), provides a consistent basis. We specifically discuss methodological and computational difficulties in parameter identification with CFSE data, e.g. the loss of confidence in the parameter estimates starting around the sixth division. Overall, our study suggests that the heterogeneity inherent in cell kinetics should be explicitly incorporated into the structure of mathematical models.      

Clonal analysis of the Arabidopsis root confirms that position, not lineage, determines cell fate

 The cellular organization of the Arabidopsis thaliana (L.) Heynh. root meristem suggests that a regular pattern of cell divisions occurs in the root tip. Deviations from this pattern of division might be expected to disrupt the organization of cells and tissues in the root. A clonal analysis of the 3-d-old primary root meristem was carried out to determine if there is variability in division patterns, and if so to discover their effect on cellular organization in the root. Clones induced in the seedling meristem largely confirmed the predicted pattern of cell divisions. However, the cellular initials that normally give rise to the different cell files in the root were shown to exhibit some instability. For example, it was calculated that a lateral root cap/epidermal initial is displaced every 13 d. Furthermore, the existence of large marked clones that included more than two adjacent cell layers suggests that intrusive growth followed by cell division may occur at low frequency, perhaps in response to local cell deaths in the meristem. These findings support the view that even in plant organs with stereotypical cell division patterns, positional information is still the key determinant of cell fate. Received: 27 August 1999 / Accepted: 4 December 1999     

Analysis of higher-order neuronal interactions based on conditional inference

 Higher-order neural interactions, i.e., interactions that cannot be reduced to interactions between pairs of cells, have received increasing attention in the context of recent attempts to understand the cooperative dynamics in cortical neural networks. Typically, likelihood-ratio tests of log-linear models are being employed for statistical inference. The parameter estimation of these models for simultaneously recorded single-neuron spiking activities is a crucial ingredient of this approach. Extending a previous investigation of a two-neuron system, we present here the general formulation of an exact test suited for the detection of positive higher-order interactions between m neurons. This procedure does not require the estimation of any interaction parameters and additionally optimizes the test power of the statistical inference. We apply the approach to a three-neuron system and show how second-order and third-order interactions can be reliably distinguished. We study the performance of the method as a function of the interaction strength. Received: 18 January 2002 / Accepted in revised form: 26 November 2002 / Published online: 13 March 2003 RID="*" ID="*" Present address: Institute for Theoretical Biology, Humboldt University, 10115 Berlin, Germany Correspondence to: R. Gütig (e-mail: r.guetig@biologie.hu-berlin.de, Tel.: +49 30 2093 9112, Fax: +49 30 2093 8801) Acknowledgements. We thank Shun-ichi Amari and Hiro Nakahara for valuable discussions on the information geometry of the exponential family of probability distributions underlying the present approach. Supported in part by the Studienstiftung des deutschen Volkes, the German-Israeli Foundation for Scientific Research and Development (GIF), the Deutsche Forschungsgemeinschaft (DFG), and the Institut für Grenzgebiete der Psychologie, Freiburg.     

Growth, ageing and death of a photoautotrophic plant cell culture

 Batch cultures of photoautotrophic cell suspensions of Chenopodiumrubrum L., growing in an inorganic medium on CO₂ under a daily balanced light–dark regime of 16 : 8 h could be maintained for approximately 100 d without subcultivation. The long-lived cultures showed an initial cell division phase of 4 weeks, followed by a stationary phase of another 4 weeks, after which ageing and progressive cell death reduced the number of living cells and the cultures usually expired after another 3–4 weeks. These developmental phases of the cell culture were characterised with respect to photosynthetic performance, dark respiration, content of phytohormones and capacity of cell division. Cell division of the majority of the cells finished in the G1- or G0-phase of the cell cycle, caused by a pronounced decline in the endogenous levels of auxin and cytokinins. Supply of these growth factors to resting cells resulted in resumption of cytokinesis, at least by some of the cells. However, responsiveness to the phytohomones declined during the stationary phase, and subcultivation was no longer possible beyond day 60 when the phases of ageing and death commenced. Ageing was characterised by a further decline in the photosynthetic capacity of the cells, by a climacteric enhancement of dark respiration, but also by a slight increase in the level of IAA and cytokinins concomitant with a decrease in ethylene. Similarities and differences between the development of batch-cultured photoautotrophic cells of C. rubrum and that of a leaf are discussed with respect to using the cell culture as a model for a leaf. Received: 30 April 1999 / Accepted: 21 August 1999     

An alternate method for estimation of cell growth kinetics from batch cultures

An alternative to estimation of cell growth kinetics via continuous culture experiments is proposed in this article. The method employed is based on batch culture experiments with very small inocula (initial cell concentrations being typically less than 5000 cells/mL). Such low initial cell concentrations result in extended exponential cell growth phase during which culture conditions remain unchanged, thereby permitting precise estimation of specific cell growth rates from batch experiments especially for fast-growing microorganisms such as Bacillus species. The effectiveness and utility of this approach are demonstrated via several experiments conducted with a wild-type strain (Bacillus subtilis TN106) and a recombinant strain (B. subtilis TN106[pAT5]). True establishment of exponential growth phase requires insignificant variance of most of the culture conditions during the initial growth phase. Satisfaction of this requirement is demonstrated for microbial systems investigated here. This approach is especially well suited for recombinant microorganisms containing segregationally unstable plasmids, since estimation of growth kinetics of these from continuous cultures is very difficult and highly unreliable due to continual reversion of recombinant ceils to plasmid-free host cells unless some selection pressure is applied at levels sufficient to keep the presence of plasmid-free cells minimal.     

In vivo diversification and migrations of chick embryo heart muscle cells: a morphometric analysis with ALV- and SNV-based non-replicative vectors

 By means of a reporter gene we previously demonstrated that non-replicative Avian Leukemia Virus- and Spleen Necrosis Virus-based retroviral vectors were preferentially expressed in the heart of avian embryos from different species. Using a computer-assisted approach, we now compare clones tagged by the two types of vectors, for volume, anatomical and subanatomical localisation, number of Hoechst-stained cell nuclei and mean cell division time during the period of heart morphogenesis, i.e. from stages 17–19 to 34 of Hamburger and Hamilton (1951). This analysis demonstrates that clones labelled by the two types of viruses display similar features and bring about new insights on the relationships between mitotic and migratory properties of the myocardial cells and histogenesis of the heart. Since only exteriormost cells were tagged with our inoculation procedure, our analysis shows that: (1) at stages 17–19, the myocardium is composed of cells with diverse potentials; some cells still retain the capacity to divide extensively and participate to different heart muscle layers, whilst most are restricted in their multiplication potential and contribute to single muscle layers; (2) about half of the clones are located deep in the heart wall, revealing extensive cell migrations from the heart surface to the ventricular trabeculae, the first migrating cells tagged being detected 20 h after viral inoculation. The presence of these cells is consistent with the finding of a large number of compact trabecular clones 5 days later suggesting that these cells divide mainly after completing migration. Our approach provides new insights as well as quantitative data on the different processes involved in heart morphogenesis, namely multiplication, migration and localisation of heart muscle cells. Received: 10 March 1996 / Accepted in revised form: 7 July 1996     

A Generalised Age- and Phase-Structured Model of Human Tumour Cell Populations Both Unperturbed and Exposed to a Range of Cancer Therapies

We develop a general mathematical model for a population of cells differentiated by their position within the cell division cycle. A system of partial differential equations governs the kinetics of cell densities in certain phases of the cell division cycle dependent on time t (hours) and an age-like variable τ (hours) describing the time since arrival in a particular phase of the cell division cycle. Transition rate functions control the transfer of cells between phases. We first obtain a theoretical solution on the infinite domain −∞ < t < ∞. We then assume that age distributions at time t=0 are known and write our solution in terms of these age distributions on t=0. In practice, of course, these age distributions are unknown. All is not lost, however, because a cell line before treatment usually lies in a state of asynchronous balanced growth where the proportion of cells in each phase of the cell cycle remain constant. We assume that an unperturbed cell line has four distinct phases and that the rate of transition between phases is constant within a short period of observation (‘short’ relative to the whole history of the tumour growth) and we show that under certain conditions, this is equivalent to exponential growth or decline. We can then gain expressions for the age distributions. So, in short, our approach is to assume that we have an unperturbed cell line on t ≤ 0, and then, at t=0 the cell line is exposed to cancer therapy. This corresponds to a change in the transition rate functions and perhaps incorporation of additional phases of the cell cycle. We discuss a number of these cancer therapies and applications of the model.     

Cell division and the stability of cellular populations

 This paper couples a general d-dimensional (d arbitrary) model for the intracellular biochemistry of a generic cell with a probabilistic division hypothesis and examines the consequence of division for stability of cell function and structure. We show rather surprisingly that cell division is capable of giving rise to a stable population of cells with respect to function and structure even if, in the absence of cell division, the underlying biochemical dynamics are unstable. In the context of a simple example, our stability condition suggests that rapid cell proliferation plays a stabilizing role for cellular populations. Received: 15 January 1996 / Revised version: 31 July 1998     

Modeling aggregation and growth processes in an algal population model: analysis and computations

 Aggregation, the formation of large particles through multiple collision of smaller ones is a highly visible phenomena in oceanic waters which can control material flux to the deep sea. Oceanic aggregates more than 1 cm in diameter have been observed and are frequently described to consist of phytoplankton cells as well as other organic matter such as fecel pellets and mucus nets from pteropods. Division of live phytoplankton cells within an aggregate can also increase the size of aggregate (assuming some daughter cells stay in the aggregate) and hence could be a significant factor in speeding up the formation process of larger aggregate. Due to the difficulty of modeling cell division within aggregates, few efforts have been made in this direction. In this paper, we propose a size structured approach that includes growth of aggregate size due to both cell division and aggregation. We first examine some basic mathematical issues associated with the development of a numerical simulation of the resulting algal aggregation model. The numerical algorithm is then used to examine the basic model behavior and present a comparison between aggregate distribution with and without division in aggregates. Results indicate that the inclusion of a growth term in aggregates, due to cell division, results in higher densities of larger aggregates; hence it has the impact to speed clearance of organic matter from the surface layer of the ocean. Received 1 July 1994; received in revised form 23 February 1996     

Robust parameter estimation during logistic modeling of batch and fed‐batch culture kinetics

Methods for robust logistic modeling of batch and fed‐batch mammalian cell cultures are presented in this study. Linearized forms of the logistic growth, logistic decline, and generalized logistic equation were derived to obtain initial estimates of the parameters by linear least squares. These initial estimates facilitated subsequent determination of refined values by nonlinear optimization using three different algorithms. Data from BHK, CHO, and hybridoma cells in batch or fed‐batch cultures at volumes ranging from 100 mL–300 L were tested with the above approach and solution convergence was obtained for all three nonlinear optimization approaches for all data sets. This result, despite the sensitivity of logistic equations to parameter variation because of their exponential nature, demonstrated that robust estimation of logistic parameters was possible by this combination of linearization followed by nonlinear optimization. The approach is relatively simple and can be implemented in a spreadsheet to robustly model mammalian cell culture batch or fed‐batch data. © 2009 American Institute of Chemical Engineers Biotechnol. Prog., 2009     

Growth at low temperature causes nitrogen limitation in the cyanobacterium Synechococcus sp. PCC 7002

The coloration of cells of the cyanobacterium Synechococcus sp. PCC 7002 changed from normal blue-green to yellow-green when cells were grown at 15° C in a medium containing nitrate as the sole nitrogen source. This change of coloration was similar to a general response to nutrient deprivation (chlorosis). For the chlorotic cells at 15° C, the total amounts of phycobiliproteins and chlorophyll a decreased, high levels of glycogen accumulated, and growth was arithmetic rather than exponential. These changes in composition and growth occurred in cells grown at low (50 μE m^–2 s^–1) as well as high (250 μE m^–2 s^–1) light intensity. After a temperature shift-up to 38° C, chlorotic cells rapidly regained their normal blue-green coloration and normal exponential growth rate within 7 h. When cells were grown at 15° C in a medium containing urea as the reduced nitrogen source, cells grew exponentially and the symptoms of chlorosis were not observed. The decrease in photosynthetic oxygen evolution activity at low temperature was much smaller than the decrease in growth rate for cells grown on nitrate as the nitrogen source. These studies demonstrate that low-temperature-induced chlorosis of Synechococcus sp. PCC 7002 is caused by nitrogen limitation and is not the result of limited photosynthetic activity or photodamage to the photosynthetic apparatus, and that nitrogen assimilation is an important aspect of the low-temperature physiology of cyanobacteria. Received: 24 April 1997 / Accepted: 5 August 1997     

On identification of Na+ channel gating schemes using moving-average filtered hidden Markov models

 Transitions between distinct kinetic states of an ion channel are described by a Markov process. Hidden Markov models (HMM) have been successfully applied in the analysis of single ion channel recordings with a small signal-to-noise ratio. However, we have recently shown that the anti-aliasing low-pass filter misleads parameter estimation. Here, we show for the case of a Na⁺ channel recording that the standard HMM do neither allow parameter estimation nor a correct identification of the gating scheme. In particular, the number of closed and open states is determined incorrectly, whereas a modified HMM considering the anti-aliasing filter (moving-average filtered HMM) is able to reproduce the characteristic properties of the time series and to perform gating scheme identification. Received: 11 February 1999 / Revised version: 18 June 1999 / Accepted: 21 June 1999     

Lysis of growing cells ofSaccharomyces cerevisiae induced by papulacandin B

Light and electron microscopy was used to study the effect of papulacandin B onSaccharomyces cerevisiae in the exponential growth phase. At 1–2 μg/mL cell division in the culture continued almost in parallel with the control, at 4 μg/mL cell proliferation was reduced and the culture contained some cells with 2–9 buds which were not separated from the mother cell by a septum, and at higher concentrations (8, 16 and 32 μg/mL) the proliferation stopped within 2 h. Cessation of proliferation was due to lysis of budding cells in the bud region including perforation of thinned cell wall (most often at the bud basis and sometimes at its apex), extrusion of cytoplasm and death of cell. Lysis was also observed in cells without visible buds. Dividing cells died without visible lysis.     

Evaluation of Multitype Mathematical Models for CFSE-Labeling Experiment Data

Carboxy-fluorescein diacetate succinimidyl ester (CFSE) labeling is an important experimental tool for measuring cell responses to extracellular signals in biomedical research. However, changes of the cell cycle (e.g., time to division) corresponding to different stimulations cannot be directly characterized from data collected in CFSE-labeling experiments. A number of independent studies have developed mathematical models as well as parameter estimation methods to better understand cell cycle kinetics based on CFSE data. However, when applying different models to the same data set, notable discrepancies in parameter estimates based on different models has become an issue of great concern. It is therefore important to compare existing models and make recommendations for practical use. For this purpose, we derived the analytic form of an age-dependent multitype branching process model. We then compared the performance of different models, namely branching process, cyton, Smith–Martin, and a linear birth–death ordinary differential equation (ODE) model via simulation studies. For fairness of model comparison, simulated data sets were generated using an agent-based simulation tool which is independent of the four models that are compared. The simulation study results suggest that the branching process model significantly outperforms the other three models over a wide range of parameter values. This model was then employed to understand the proliferation pattern of CD4+ and CD8+ T cells under polyclonal stimulation.     

Elongation and surface extension of individual cells of Escherichia coli B/r: comparison of theoretical and experimental size distributions

The way individual cells grow and divide uniquely determines the (time-invariant) cell size distribution of populations in steady-state exponential growth. In the preceding article, theoretical distributions were derived for two exponential and six linear models containing a small number of adjustable parameters but no assumptions other than that all cells obey the same growth law. The linear models differ from each other with respect to the timing of the presumptive doubling in their growth rate, the exponential models--according to whether there is or is not a part of the cell that does not contribute to the growth rate. Here we compared the size distributions predicted by each of these models with those of cell length and surface area measured by electron microscopy; the quality of the fit, as determined by the mean-square successive-differences test and the chi 2 goodness-of-fit test, was taken as a measure of the adequacy of the model. The actual data came from two slow-growing E. coli B/r cultures, an A strain (pi = 125 min) and a K strain (pi = 106 min), and a correction was introduced in each to account for the distortion caused by the finite size of the picture frame. The parameter estimates produced by the various models are quite reliable (cv less than 0.1%); we discuss them briefly and compare their values in the two strains. All the length extension models were rejected outright whereas most of the surface growth versions were not. When the same models were tested on A-strain data from a faster growing culture (tau = 21 min), those models that provided an adequate fit to the cell surface area data proved equally satisfactory in the case of cell length. These findings are evaluated and shown to be consistent with cell surface area rather than cell length being the dimension under active control. Three surface area models, all linear, are rejected--those in which doubling of the growth rate occurs with a constant probability from cell birth, at a particular cell age, and precisely at cell division. The evidence in the literature that appears to contradict this last result, rejection of the simple linear surface growth model, is shown to be faulty. The 16 original models are here reduced to five, two involving exponential surface growth and three linear, and possible reasons are presented for our inability to discriminate further at this stage.     

Calculation of Steel's cell loss factor and of the parameters of cellular kinetics for a population with an exponential growth of the cell number and cell death at G0 phase with a probability equal to 1

The method of calculation of three cell kinetics parameters (the Steel's cell loss factor phi, the proliferative pool Pc, and the mean number m of the proliferating cells after mitotic division of one cell) was shown to be the same for the exponential growth state of cell number with cell death at the G0-phase, and for the exponential growth state with cell death occurring immediately after mitosis. The value of the mean number delta of non-proliferating cells that appeared after mitotic division of one cell is different for these two models of the exponential growth state with the equal values of the other three parameters (phi, Pc, and m). A method is proposed for calculating the parameter delta on the data of the percentage of labeled cells obtained in the experiments with continuous cultivation of cells in the nutrient medium containing 3H-thymidine. The kinetics of cell line HL-60 (the experimental data of Foa et al., 1982) can be described at the first approximation, by a model of the exponential growth state with the cell death at the G0-phase, with Pc = 0.80, phi = 0.24, m = 1.61, delta = 0.39, and the life time of the non-proliferating cells tQ = 24 hours.     

Modeling Cell and Matrix Anisotropy in Fibroblast Populated Collagen Vessels

Microstructurally based models for bio-artificial tissues are needed to predict in vivo mechanical behavior and to validate assumptions for models of biologic tissues. We develop a microstructural model, based on on Zahalak et al. (2000) [Biophys 79(5):2369–2381], to describe matrix and tissue anisotropy observed in recent biaxial tests of fibroblast populated collagen vessels (FPCVs) with different cell orientations (Wagenseil et al. in Ann Biomed Eng 32(5):720–731 2004). The model includes pseudo-elastic cell behavior and pseudo-elastic, non-linear matrix behavior with recruitment of initially buckled collagen fibers. We obtained estimates of collagen matrix parameters from measurements of FPCVs treated with 2× 10⁻⁶ M Cytochalasin D and used these estimates to determine cell parameters in FPCVs activated with 5% fetal calf serum. The estimated stiffness of individual fibroblasts was 41–1,165 kPa. Parameter estimates for both cell and matrix were influenced by the non-linearity of the biaxial test data, making it difficult to obtain unique parameter values for some experiments. Additional microstructural measurements of the collagen matrix may help to more precisely determine the relative contributions of cells and matrix.     

Estimating Lymphocyte Division and Death Rates from CFSE Data

The division tracking dye, carboxyfluorescin diacetate succinimidyl ester (CFSE) is currently the most informative labeling technique for characterizing the division history of cells in the immune system. Gett and Hodgkin [Nat. Immunol. 1:239–244, 2000] have pioneered the quantitative analysis of CFSE data. We confirm and extend their data analysis approach using simple mathematical models. We employ the extended Gett and Hodgkin [Nat. Immunol. 1:239–244, 2000] method to estimate the time to first division, the fraction of cells recruited into division, the cell cycle time, and the average death rate from CFSE data on T cells stimulated under different concentrations of IL-2. The same data is also fitted with a simple mathematical model that we derived by reformulating the numerical model of Deenick et al. [J. Immunol. 170:4963–4972, 2003]. By a non-linear fitting procedure we estimate parameter values and confidence intervals to identify the parameters that are influenced by the IL-2 concentration. We obtain a significantly better fit to the data when we assume that the T cell death rate depends on the number of divisions cells have completed. We provide an outlook on future work that involves extending the Deenick et al. [J. Immunol. 170:4963–4972, 2003] model into the classical smith-martin model, and into a model with arbitrary probability distributions for death and division through subsequent divisions.     

Culturability and survival of an extreme thermophile isolated from deep-sea hydrothermal vents

The culturability of a strictly anaerobic, extremely thermophilic archaeon, Thermococcus peptonophilus (optimal growth temperature: 85° C), was studied during survival stages at various temperatures (98, 85, 70, and 4° C). Total cell number (determined by DAPI staining), active cells (rhodamine-stained cells), and culturable cells (using most-probable-number) were counted over time. The number of culturable cells decreased under each condition tested. The total number of cells significantly decreased only at temperatures close to the maximum for growth (98° C); at this temperature, the cells spontaneously lysed. Our results suggested that survival at 4° C in oxygenated waters might be a mechanism for the dispersion of extreme thermophiles in the ocean. In addition, we proved the existence of T. peptonophilus cells in several physiological states: culturable cells, active non-culturable cells, inactive non-culturable cells, and dead cells. Cell death was caused by cellular lysis. Received: 5 February 1996 / Accepted: 16 April 1996