Predicted steady-state cell size distributions for various growth models

The question of how an individual bacterial cell grows during its life cycle remains controversial. In 1962 Collins and Richmond derived a very general expression relating the size distributions of newborn, dividing and extant cells in steady-state growth and their growth rate; it represents the most powerful framework currently available for the analysis of bacterial growth kinetics. The Collins-Richmond equation is in effect a statement of the conservation of cell numbers for populations in steady-state exponential growth. It has usually been used to calculate the growth rate from a measured cell size distribution under various assumptions regarding the dividing and newborn cell distributions, but can also be applied in reverse--to compute the theoretical cell size distribution from a specified growth law. This has the advantage that it is not limited to models in which growth rate is a deterministic function of cell size, such as in simple exponential or linear growth, but permits evaluation of far more sophisticated hypotheses. Here we employed this reverse approach to obtain theoretical cell size distributions for two exponential and six linear growth models. The former differ as to whether there exists in each cell a minimal size that does not contribute to growth, the latter as to when the presumptive doubling of the growth rate takes place: in the linear age models, it is taken to occur at a particular cell age, at a fixed time prior to division, or at division itself; in the linear size models, the growth rate is considered to double with a constant probability from cell birth, with a constant probability but only after the cell has reached a minimal size, or after the minimal size has been attained but with a probability that increases linearly with cell size. Each model contains a small number of adjustable parameters but no assumptions other than that all cells obey the same growth law. In the present article, the various growth laws are described and rigorous mathematical expressions developed to predict the size distribution of extant cells in steady-state exponential growth; in the following paper, these predictions are tested against high-quality experimental data.     

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.     

Cell cycle progression

In this paper we consider cell cycle models for which the transition operator for the evolution of birth mass density is a simple, linear dynamical system with a stochastic perturbation. The convolution model for a birth mass distribution is presented. Density functions of birth mass and tail probabilities in n-th generation are calculated by a saddle-point approximation method. With these probabilities, representing the probability of exceeding an acceptable mass value, we have more control over pathological growth. A computer simulation is presented for cell proliferation in the age-dependent cell cycle model. The simulation takes into account the fact that the age-dependent model with a linear growth is a simple linear dynamical system with an additive stochastic perturbation. The simulated data as well as the experimental data (generation times for mouse L) are fitted by the proposed convolution model.     

A mathematical model for the batch reactor kinetics of algae growth

A mathematical model based on the Einstein law of photochemical equivalence is proposed to describe the batch growth of unicellular algae. The model was applied in an integrated form to cell concentration versus growth time data taken over an extended range of cell concentrations which include both the regions of “exponential” and “linear” growth. It is shown that a certain function of cell concentration contained in the integrated form of the model is linearly dependent on the growth time over both the “exponential” and “linear” growth regions.     

A new kinetic model of protein adsorption on suspended anion-exchange resin particles.

The kinetics of adsorption of bovine serum albumin on an anion-exchange resin were measured in a batch system using a flow cell and ultraviolet absorbance, as a function of initial liquid-phase protein concentration and solid-to-liquid phase ratio. A new mathematical model for adsorption kinetics is presented that fits the experimental data to give a highly linear relationship with time, following a short transient period. Numerical integration of the differential form of the new composite nonlinear (CNL) kinetic model, containing three independent parameters, is shown to describe the dynamics of batch adsorption much better than alternative lumped parameter models. Although the new model is phenomenological rather than mechanistic, its principal parameter is shown to be a direct linear function of a physically measurable quantity. This study demonstrates that the model can accurately simulate protein concentration-time profiles using parameter estimates derived from correlations over a wide range of initial protein concentrations and phase ratios. The new CNL model is shown to be considerably superior to the Langmuir and solid-film linear kinetic models in this regard, having the additional advantage that an equilibrium isotherm for the system is not required.     

Experimental testing of dynamic energy budget models

1. Dynamic energy budget (DEB) models describing the allocation of assimilate to the competing processes of growth, reproduction and maintenance in individual organisms have been applied to a variety of species with some success. There are two contrasting model formulations based on dynamic allocation rules that have been widely used (net production and net assimilation formulations). However, the predictions of these two classes of DEB models are not easily distinguished on the basis of simple growth and fecundity data.
2. It is shown that different assumptions incorporated in the rules determining allocation to growth and reproduction in two classes of commonly applied DEB models predict qualitatively distinct patterns for an easily measured variable, cumulative reproduction by the time an individual reaches an arbitrary size.
3. A comparison with experimental data from Daphnia pulex reveals that, in their simplest form, neither model predicts the observed qualitative pattern of reproduction, despite the fact that both formulations capture basic growth features.
4. An examination of more elaborate versions of the two models, in which the allocation rules are modified to account for brief periods of starvation experienced in the laboratory cultures, reveals that a version of the net production model can predict the qualitative pattern seen for cumulative eggs as a function of mass in D. pulex . The analysis leads to new predictions which can be easily tested with further laboratory experiments.     

Kinetic models of C3H mouse mammary tumor growth: implications regarding tumor cell loss

Three models of tumor cell loss are described. The effects of cell loss on other cellular kinetic parameters are evaluated, and experiments which may distinguish among the models are discussed. Each model is based on a different cell-loss mechanism, and equations for the cell-cycle, cell-frequency distribution, the growth of both the proliferating and non-proliferating cell population, the growth fraction (GF), and the relative rate of volumetric growth, (dV/dt)/V, are derived. The following types of data are simulated for each model: the pulse labelling index, the mitotic index, and the labeling index as a function of time after a single or a series of 3H-TdR injections. The relative volumetric growth rate has the same mathematical form for each model. The PLM curves predicted by each model for the tumor lines studied (S102F and Slow) are not appreciably different. The predicted initial labeling index and mitotic index may differ significantly among the models depending upon the tumor line. The most striking difference among the models lies in the predictions regarding the labeling index as a function of time after a single or after a series of 3H-TdR injections. These types of labeling experiments should be valuable for distinguishing the different cell-loss mechanisms in solid tumors.     

Connection between stochastic and deterministic modelling of microbial growth

We present in this paper various links between individual and population cell growth. Deterministic models of the lag and subsequent growth of a bacterial population and their connection with stochastic models for the lag and subsequent generation times of individual cells are analysed. We derived the individual lag time distribution inherent in population growth models, which shows that the Baranyi model allows a wide range of shapes for individual lag time distribution. We demonstrate that individual cell lag time distributions cannot be retrieved from population growth data. We also present the results of our investigation on the effect of the mean and variance of the individual lag time and the initial cell number on the mean and variance of the population lag time. These relationships are analysed theoretically, and their consequence for predictive microbiology research is discussed.     

Kinetic Models of C3h Mouse Mammary Tumor Growth: Implications Regarding Tumor Cell Loss

Three models of tumor cell loss are described. the effects of cell loss on other cellular kinetic parameters are evaluated, and experiments which may distinguish among the models are discussed. Each model is based on a different cell-loss mechanism, and equations for the cell-cycle, cell-frequency distribution, the growth of both the proliferating and non-proliferating cell population, the growth fraction (GF), and the relative rate of volumetric growth, (dV/dt)/V, are derived. The following types of data are simulated for each model: the pulse labelling index, the mitotic index, and the labeling index as a function of time after a single or a series of ³H-TdR injections. the relative volumetric growth rate has the same mathematical form for each model. the PLM curves predicted by each model for the tumor lines studied (S102F and Slow) are not appreciably different. the predicted initial labeling index and mitotic index may differ significantly among the models depending upon the tumor line. the most striking difference among the models lies in the predictions regarding the labeling index as a function of time after a single or after a series of ³H-TdR injections. These types of labeling experiments should be valuable for distinguishing the different cell-loss mechanisms in solid tumors.     

Kinetics of protein and DNA synthesis studied by mathematical modelling of flow cytometric protein and DNA histograms

Abstract. Mathematical models for histograms of cellular protein content as measured by flow cytometry were developed, based on theoretical protein distributions. These were derived from the age distribution of cells and the accumulation function for cellular protein content as a function of age within the cell cycle. A model assuming an exponential age distribution and an exponential protein. accumulation function was found to give the best representation of protein histograms of exponentially growing NHIK 3025 cells. This is in good agreement with the known kinetic behaviour of such cells. By the combined use of the protein histogram model and a similar model for DNA content, and assuming linear DNA accumulation during S, the fraction of cells in S, as a function of cellular protein content, was simulated. This function showed good agreement with values of the [³H]TdR labelling index scored in cells sorted by flow cytometry from 5-channel intervals of the protein histogram. The protein and DNA histogram models were combined into a two-dimensional model for correlated protein/DNA measurements. Comparison between simulated data and experimentally derived two-dimensional protein/DNA histograms gave further support to the cell kinetic assumptions underlying the models, but also identified some minor deviations which could not be recognized in the analysis of the one-dimensional histograms.     

A case study of linear versus non-linear modelling

Theories for the facilitation of neurotransmitter release are discussed in a case study of the properties of linear and non-linear models for a phenomenon whose time course can be represented by a sum of decaying exponentials. Particular attention is paid to the effects of a "key factor" on the slopes and amplitudes of the exponentials that can be derived from semilog plots of the data. It is shown that the presence of such effects can give strong evidence for the inappropriateness of linear models. A non-linear model is demonstrated to be capable of describing the changes with extracellular Ca concentration of straight line segments that fit data in semilog plots of facilitation as a function of time. The conclusion is reached that even if data seems to be representable by several independently alterable exponentials one must be cautious in drawing inferences concerning the number, linearity, or independence of the underlying processes.     

Cell Growth and Division: I. A Mathematical Model with Applications to Cell Volume Distributions in Mammalian Suspension Cultures

A mathematical model is formulated for the development of a population of cells in which the individual members may grow and divide or die. A given cell is characterized by its age and volume, and these parameters are assumed to determine the rate of volume growth and the probability per unit time of division or death. The initial value problem is formulated, and it is shown that if cell growth rate is proportional to cell volume, then the volume distribution will not converge to a time-invariant shape without an added dispersive mechanism. Mathematical simplications which are possible for the special case of populations in the exponential phase or in the steady state are considered in some detail. Experimental volume distributions of mammalian cells in exponentially growing suspension cultures are analyzed, and growth rates and division probabilities are deduced. It is concluded that the cell volume growth rate is approximately proportional to cell volume and that the division probability increases with volume above a critical threshold. The effects on volume distribution of division into daughter cells of unequal volumes are examined in computer models.     

State vector models of the cell cycle. III: Continuous time cell cycle models

In this paper the elements of the matrix of the Hahn cell-cycle model are identified with the infinitesimal transition probabilities of a Markov process, and as a limiting process a differential equation analogue is derived. The probability density function of the discrete time model is derived and used to obtain the density function for transit times of the continuous time model. It is shown that the mean transit time remains constant and that the variances of the discrete and continuous time models are the same to the order of the time increment. Finally, it is shown how to derive the Takahashi model from the continuous time Hahn model.     

The matrix representation of pharmacokinetic models.

An equation is developed from the matrix of rate constants which describes the behaviour of linear pharmacokinetic models for any initial condition as a function of time. This general matrix equation is then used to derive analogous expressions for drug distribution after a period of infusion, at the steady state, or during a multiple constant-dosage regimen. Matrix expressions are also derived for areas under drug concentration curves for any compartment after single doses or during multiple dosing. General matrix equations are shown to yield loading dosage schedules to achieve plateau concentrations throughout any open system.It is suggested that matrix methods have advantages over previously used mathematical techniques in pharmacokinetics in the simplicity of the algebraic expressions, and their ease of manipulation. An algebraic example of an open two-compartment model is worked to indicate the applicability of the general expressions.     

Optimal control of tumor size used to maximize survival time when cells are resistant to chemotherapy.

The high failure rates encountered in the chemotherapy of some cancers suggest that drug resistance is a common phenomenon. In the current study, the tumor burden during therapy is used to slow the growth of the drug-resistant cells, thereby maximizing the survival time of the host. Three types of tumor growth model are investigated--Gompertz, logistic, and exponential. For each model, feedback controls are constructed that specify the optimal tumor mass as a function of the size of the resistant subpopulation. For exponential and logistic tumor growth, the tumor burden during therapy is shown to have little impact upon survival time. When the tumor is in Gompertz growth, therapies maintaining a large tumor burden double and sometimes triple the survival time under aggressive therapies. Aggressive therapies aim for a rapid reduction in the sensitive cell subpopulation. These conclusions are not dependent upon the values of the model constants that determine the mass of resistant cells. Since treatments maintaining a high tumor burden are optimal for Gompertz tumor growth and close to optimal for exponential and logistic tumor growth, it may no longer be necessary to know the growth characteristics of a tumor to schedule anticancer drugs.     

On the distribution of state values of reproducing cells

Characterizing a cell state by measuring the degree of gene expression as well as its noise has gathered much attention. The distribution of such state values (e.g., abundances of some proteins) over cells has been measured, and is not only a result of intracellular process, but is also influenced by the growth in cell number that depends on the state. By incorporating the growth-death process into the standard Fokker-Planck equation, a nonlinear temporal evolution equation of distribution is derived and then solved by means of eigenfunction expansions. This general formalism is applied to the linear relaxation case. First, when the growth rate of a cell increases linearly with the state value x, the shift of the average x due to the growth effect is shown to be proportional to the variance of x and the relaxation time, similar to the biological fluctuation-response relationship. Second, when there is a threshold value of x for growth, the existence of a critical growth rate, represented again by the variance and the relaxation time, is demonstrated. The relevance of the results to the analysis of biological data on the distribution of cell states, as obtained for example by flow cytometry, is discussed.     

Modes of Growth in Mammalian Cells

The increase of cell volume as a function of time was studied throughout the generation cycle in synchronous cultures of Chinese hamster cells using a Coulter aperture and a multichannel analyzer calibrated against known cell volumes. The experimental results were compared to a mathematical model of cell volume increase which considered the effect of the distribution of individual cell generation times on the progress of the population. Several modes of volume increase, including linear and exponential, were considered. The mean volume vs. time curve was rounded at the ends of the cycle even when linear growth was assumed. The experimental results show that cell volume increased in a smooth fashion as a function of time, with no discontinuities in rate detectable at periods when cells may have been undergoing metabolic shifts as, for example, through the phases associated with DNA synthesis, G₁, S, G₂. A statistical test on the comparison of the modal cell volume vs. time data to the predictions of linear and exponential growth models accepted both hypotheses within the resolution of these experiments. However, exponential growth was favored over linear growth in one cell line. Volume dispersion was almost constant with time in both sublines which is also consistent with exponential growth. Limitations of the electronic technique of volume measurement and indications for future experiments are discussed.     

The distributions of cell size and generation time in a model of the cell cycle incorporating size control and random transitions

A deterministic/probabilistic model of the cell division cycle is analysed mathematically and compared to experimental data and to other models of the cell cycle. The model posits a random-exiting phase of the cell cycle and a minimum-size requirement for entry into the random-exiting phase. By design, the model predicts exponential "beta-curves", which are characteristic of sister cell generation times. We show that the model predicts "alpha-curves" with exponential tails and hyperbolic-sine-like shoulders, and that these curves fit observed generation-time data excellently. We also calculate correlation coefficients for sister cells and for mother-daughter pairs. These correlation coefficients are more negative than is generally observed, which is characteristic of all size-control models and is generally attributed to some unknown positive correlation in growth rates of related cells. Next we compare theoretical size distributions with observed distributions, and we calculate the dependence of average cell mass on specific growth rate and show that this dependence agrees with a well-known relation in bacteria. In the discussion we argue that unequal division is probably not the source of stochastic fluctuations in deterministic size-control models, transition-probability models with no feedback from cell size cannot account for the rapidity with which the new, stable size distribution is established after perturbation, and Kubitschek's rate-normal model is not consistent with exponential beta-curves.     

Evolution of life history strategies for an asexual annual plant model

At any given moment in time a plant is partitioning total growth mass into its various component parts such as leaves, roots, reproductive material, etc. The view is taken that the plant has evolved a life history strategy to control this partitioning process. This paper illustrates the utility of optimal control theory for use in determining life history strategies which maximize fitness for a given asexual plant model. The optimal control methods are first used on a model previously analyzed by Professor Dan Cohen, who used a different method. His results of a change from 100% vegetable growth to 100% reproductive growth at a fixed switching time is again obtained. This 100% switching result is shown to be more generally applicable by using a qualitatively described model. However the switching time in general is shown to be a function of both leaf mass and time remaining to the end of the growing season. The allocation to toxin production is also considered. It is shown that under this model an inequality between system parameters must be satisfied before the plant should allocate growth to toxin production. Although the particular model explored here may rarely be realistic in nature, these same methods of optimal control theory can be applied in a similar fashion to many other proposed models of plant resource allocation.