Branching stochastic processes with immigration in analysis of renewing cell populations

This paper considers the utility of a new class of stochastic branching processes with non-homogeneous immigration in modeling complex renewing cell systems. Such systems typically include the population of stem cells that provides an inexhaustible supply of cells necessary for maintaining the cellular composition of a tissue. A stem cell may be induced to transform (differentiate) into a progenitor cell. Progenitor cells retain the ability to proliferate and their function is believed to provide a quick proliferative response to an increased demand for cells in the population. There may be several sub-types of progenitor cells. Terminally differentiated cells do not divide under normal conditions; they are responsible for maintaining tissue-specific functions. Recent advancements in experimental techniques offer considerable scope for quantitative studies of in vivo cell kinetics based on stochastic modeling of renewing cell populations. However, no ready-made theory is currently available to take full advantage of these advancements. This paper introduces such a theory with a special focus on its feasibility in biological applications.     

Evolution of linkage disequilibrium of the founders in exponentially growing populations

Evolution of linkage disequilibrium of the founders in exponentially growing populations was studied using a time-inhomogeneous It? process model. The model is an extension of the diffusion approximation of the Wright-Fisher model. As a measure of linkage disequilibrium, the squared standard linkage deviation, which is defined by a ratio of the moments, was considered. A system of ordinary differential equations that these moments obey was obtained. This system can be solved numerically. By simulations, it was shown that the squared standard linkage deviation gives a good approximation of the expectation of the squared correlation coefficient of gamete frequencies. In addition, a perturbative solution was obtained when the growth rate is not large. By using the perturbation, an asymptotic formula for the squared standard linkage deviation after a large number of generations was obtained. According to the formula, the squared standard linkage deviation tends to be 1/(4Nc), where N is the current size of the population and c is the recombination fraction between two loci. It is dependent on neither the initial effective size, the growth rate, nor the mutation rate. In exponentially growing populations, linkage disequilibrium will be asymptotically the same as that in a constant size population, the effective size of which is the current effective size.     

A Fokker-Planck equation for growing cell populations

Closed form solutions are obtained for a Fokker-Planck model for cell growth as a function of maturation velocity and degree of maturation. For reproduction rules where daughter cells inherit their parent's maturation velocity the complete solution is derived in terms of Airy functions. For more complicated reproduction rules partial results are obtained. Emphasis is given to the relationship of these problems to time dependent linear transport theory.     

Parameter inference for stochastic biochemical models from perturbation experiments parallelised at the single cell level

Understanding and characterising biochemical processes inside single cells requires experimental platforms that allow one to perturb and observe the dynamics of such processes as well as computational methods to build and parameterise models from the collected data. Recent progress with experimental platforms and optogenetics has made it possible to expose each cell in an experiment to an individualised input and automatically record cellular responses over days with fine time resolution. However, methods to infer parameters of stochastic kinetic models from single-cell longitudinal data have generally been developed under the assumption that experimental data is sparse and that responses of cells to at most a few different input perturbations can be observed. Here, we investigate and compare different approaches for calculating parameter likelihoods of single-cell longitudinal data based on approximations of the chemical master equation (CME) with a particular focus on coupling the linear noise approximation (LNA) or moment closure methods to a Kalman filter. We show that, as long as cells are measured sufficiently frequently, coupling the LNA to a Kalman filter allows one to accurately approximate likelihoods and to infer model parameters from data even in cases where the LNA provides poor approximations of the CME. Furthermore, the computational cost of filtering-based iterative likelihood evaluation scales advantageously in the number of measurement times and different input perturbations and is thus ideally suited for data obtained from modern experimental platforms. To demonstrate the practical usefulness of these results, we perform an experiment in which single cells, equipped with an optogenetic gene expression system, are exposed to various different light-input sequences and measured at several hundred time points and use parameter inference based on iterative likelihood evaluation to parameterise a stochastic model of the system.     

Probability of resistance evolution for exponentially growing virus in the host

Chemotherapy for tumor and pathogenic virus often faces an emergence of resistant mutants, which may lead to medication failure. Here we study the risk of resistance to evolve in a virus population which grows exponentially. We assume that infected cells experience a "proliferation event" of virus at a random time and that the number of newly infected cells from an infected cell follows a Poisson distribution. Virus starts from a single infected cell and the virus infection is detected when the number of infected cells reaches a detection size. Initially virus is sensitive to a drug but later acquires resistance by mutations. We ask the probability that one or more cells infected with drug-resistant virus exist at the time of detection. We derive a formula for the probability of resistance and confirm its accuracy by direct computer simulations. The probability of resistance increases with detection size and mutation rate but decreases with the population growth rate of sensitive virus. The risk of resistance is smaller when more cells are newly infected by viral particles from a single infected cell if the viral growth rate is the same.     

Rate of DNA synthesis in exponentially growing cell lines in the presence and absence of antimetabolites

Experimental tumor cell lines were used to show that in the presence of 5-fluorodeoxyuridine (FdUrd), the rate of DNA synthesis remains unaltered as long as a saturating concentration of thymidine is present. This unimpeded rate of DNA synthesis in combination with FdUrd-blocked de novo thymidylate synthesis makes it possible to accurately measure the total rate of increase of DNA using tritiated thymidine of known specific activity. The observed amount of incorporated tritiated thymidine is in excellent agreement with the calculated theoretical maximal incorporation in cultures with exponentially increasing DNA and cell number.     

Timing the changes of cyclin E cell content in G1 in exponentially growing cells

We present a method for measuring the content of immunocytochemically detected proteins in individual cells progressing through G(1) phase and its application in the analysis of cyclin E levels. The sequence of G(1) events is tracked under unaltered cycling conditions, in a cell line in the phase of balanced growth in vitro, to avoid the pitfalls of synchronization. Cells were pulse-labeled with BrdUrd and analyzed sequentially by multiparameter flow cytometry, focusing on the subpopulation of labeled cells progressively entering G(1). We use the time-from-birth ("age") of individual cells to track their position inside G(1). Using the average content of cyclin E in the whole population of G(1) cells as the internal reference for each sample, we analyzed the time course of the frequency histograms of cyclin E content within BrdUrd-labeled G(1) cells by exploiting the properties of the age distributions of asynchronous populations. This way we could calculate the average cyclin E content of cells in each age cohort. Cyclin E values were low until age 3 h, after which they rose gradually, reaching six times the value of newborn cells at the end of G(1).     

A radiative transfer modeling approach for accurate interpretation of PAM fluorometry experiments in suspended algal cultures

The results of a numerical study on the simulation of pulse amplitude modulated (PAM) fluorometry within dense suspensions of photosynthetic microorganisms are presented. The Monte Carlo method was used to solve the radiative transfer equation in an algae‐filled cuvette, taking into account absorption, anisotropic scattering, and fluorescence, as well as Fresnel reflections at interfaces. This method was used to simulate the transport of excitation and fluorescence light in a common laboratory fluorometer. In this fluorometer, detected fluorescence originates from a multitude of locations within the algal suspension, which can be exposed to very different fluence rates. The fluorescence‐weighted fluence rate is reported, which is the local fluence rate of actinic light, averaged over all locations from which detected fluorescence originated. A methodology is reported for recovering the fluorescence‐weighted fluence rate as a function of the transmittance of measuring light and actinic light through the sample, which are easily measured with common laboratory fluorometers. The fluorescence‐weighted fluence rate can in turn be used as a correction factor for recovering intrinsic physiological parameters, such as the functional cross section of Photosystem II, from apparent (experimental) values. © 2016 American Institute of Chemical Engineers Biotechnol. Prog., 32:1601–1615, 2016     

Protein expression in relation to the cell cycle of exponentially growing human prostatic epithelial cells

This report concerns the study of the relationship between protein expression and the cell cycle in exponentially proliferating benign and malignant human prostate epithelial cells in short-term cultures. Multiparameter flow cytometric measurements were performed to correlate the expression of prostate-specific acid phosphatase, epithelial membrane antigen and epitectin with cell cycle progression. The expression of the three proteins was heterogeneous in G1 cells. The early post-mitotic cells exhibited the lowest levels when compared with late G1 cells, wherein the expression was many times greater. There was no further increase as the cells progressed through S and G2 + M. These findings, corroborating prior observations in other systems, suggest the possibility that the levels of the proteins studied increase during the G1 phase of the cell cycle and drop during or immediately after cytokinesis. As an alternate explanation, the heterogeneity of protein expression characteristic of G1 cells may be due, at least in part, to an asymmetric apportionment of cell constituents at mitosis.     

A visual analytics approach for models of heterogeneous cell populations

In recent years, cell population models have become increasingly common. In contrast to classic single cell models, population models allow for the study of cell-to-cell variability, a crucial phenomenon in most populations of primary cells, cancer cells, and stem cells. Unfortunately, tools for in-depth analysis of population models are still missing. This problem originates from the complexity of population models. Particularly important are methods to determine the source of heterogeneity (e.g., genetics or epigenetic differences) and to select potential (bio-)markers. We propose an analysis based on visual analytics to tackle this problem. Our approach combines parallel-coordinates plots, used for a visual assessment of the high-dimensional dependencies, and nonlinear support vector machines, for the quantification of effects. The method can be employed to study qualitative and quantitative differences among cells. To illustrate the different components, we perform a case study using the proapoptotic signal transduction pathway involved in cellular apoptosis.     

A kinetic approach to the determination of the S phase pool size of thymidine triphosphate in exponentially growing mouse L cells

A mathematical model has been constructed to describe the accumulation of radioactivity in the DNA of mouse L cells growing exponentially in the presence of [³H]thymidine. The model depends on three parameters: (1) the rate of transformation of exogenous thymidine into dTTP; (2) the rate of synthesis of DNA; and (3) the pool size of dTTP. From experiments in which cells are labeled over short and long periods, respectively, data may be obtained by which the parameters may be estimated. The results show that the size of the dTTP pool estimated in this way agrees with the total amount of dTTP in the cell as estimated by an enzymatic assay; thus all the dTTP in the cell serves as precursor in DNA synthesis. In addition, experiments in which satellite DNA has been separated from bulk DNA show that these two species are made from the same precursor pool of dTTP.     

DNA distribution of mast cell populations in growing rats

The proliferation of rat peritoneal mast cells was examined under normal conditions in vivo. DNA content of individual mast cells was measured by cytofluorometry after staining with the bibenzimidazole dye Hoechst 33258. Diploid non mast cells from each rat were used as a biological standard, which resulted in small long-term variations in the method. The proportion of mast cells in the S + G2 region of the DNA distribution was about 4% for young rats (24 days old, body-weights about 60 g). It decreased in relation to body-weight, and was less than 1% for 105-day-old rats weighing 400 g. During the same growth period the total number of mast cells in the peritoneal cavity increased about 8-fold. The total number of proliferating cells, about 30,000, remained constant throughout the observation period. No evidence of polyploidization or accumulation in G2 of mast cell nuclei was found. It is concluded that peritoneal mast cells increase in number by mitotic proliferation of differentiated cells.     

DNA distribution of mast cell populations in growing rats

Summary The proliferation of rat peritoneal mast cells was examined under normal conditions in vivo. DNA content of individual mast cells was measured by cytofluorometry after staining with the bibenzimidazole dye Hoechst 33258. Diploid non mast cells from each rat were used as a biological standard, which resulted in small long-term variations in the method. The proportion of mast cells in the S+G2 region of the DNA distribution was about 4% for young rats (24 days old, body-weights about 60 g). It decreased in relation to body-weight, and was less than 1% for 105-day-old rats weighing 400 g. During the same growth period the total number of mast cells in the peritoneal cavity increased about 8-fold. The total number of proliferating cells, about 30,000, remained constant throughout the observation period. No evidence of polyploidization or accumulation in G2 of mast cell nuclei was found. It is concluded that peritoneal mast cells increase in number by mitotic proliferation of differentiated cells.Supported by grants from the Swedish Medical Research Council, Project No. 2235     

Signatures of neutral evolution in exponentially growing tumors: A theoretical perspective

Recent work of Sottoriva, Graham, and collaborators have led to the controversial claim that exponentially growing tumors have a site frequency spectrum that follows the 1/f law consistent with neutral evolution. This conclusion has been criticized based on data quality issues, statistical considerations, and simulation results. Here, we use rigorous mathematical arguments to investigate the site frequency spectrum in the two-type model of clonal evolution. If the fitnesses of the two types are λ₀ < λ₁, then the site frequency spectrum is c/f^α where α = λ₀/λ₁. This is due to the advantageous mutations that produce the founders of the type 1 population. Mutations within the growing type 0 and type 1 populations follow the 1/f law. Our results show that, in contrast to published criticisms, neutral evolution in an exponentially growing tumor can be distinguished from the two-type model using the site frequency spectrum.     

A nonparametric approach to the discrimination of two cell populations

An approach to the statistical testing of differences between two cell populations the elements of which are characterized by multivariate data is described. The approach is based on the Fisher discriminant and the Kruskal-Wallis test. The method, which is sufficient but not necessary, makes no assumptions about the normality of the data or about the equality of the covariance matrices for each population.