Mathematical Models of Cell Motility

Cell motility is an essential biological action in the creation, operation and maintenance of our bodies. Developing mathematical models elucidating cell motility will greatly advance our understanding of this fundamental biological process. With accurate models it is possible to explore many permutations of the same event and concisely investigate their outcome. While great advancements have been made in experimental studies of cell motility, it now has somewhat fallen on mathematical models to taking a leading role in future developments. The obvious reason for this is the complexity of cell motility. Employing the processing power of today’s computers will give researches the ability to run complex biophysical and biochemical scenarios, without the inherent difficulty and time associated with in vitro investigations. Before any great advancement can be made, the basics of cell motility will have to be well-defined. Without this, complicated mathematical models will be hindered by their inherent conjecture. This review will look at current mathematical investigations of cell motility, explore the reasoning behind such work and conclude with how best to advance this interesting and challenging research area.     

Mathematical Models of the Kinetics of the Cultivation of Microorganisms

Various equations of mathematical models for the kinetics of the development of various biological processes were obtained on the basis of the generalized differential equation of biomass growth. Aerobic periodic cultivation of the yeast Saccharomyces cerevisiae was carried out to provide a comparative evaluation of advantages and disadvantages of four types of mathematical models. It is shown that the exponential model is a particular solution to the generalized differential equation. The developed mathematical model can be used to predict the course of biological processes in time and can serve as a tool for a computational experiment in order to clarify the dependence of the rate of a biological process on changes in certain parameters that affect the development of cells.     

Evidence For Cell Generation Controlled Proliferation In the Small Intestinal Crypt

Abstract. In this paper we discuss the hypothesis that cell proliferation is controlled by the number of generations after leaving an 'eternal' stem cell. the theory is based on a simulation of the kinetic behaviour of cells in the intestinal crypts. There is evidence of three, four and five generations of cells which are allowed to enter mitosis in the lower and upper part of the normal intestinal tract, and in some disease states, respectively. We suggest an internal proliferation control: some kind of knowledge that cells carry from generation to generation. It is an open question what sets and changes the generation counter: internal genetic information or external influences such as growth factors or chalones.
The geometric shape of the epithelial tissue in the intestinal tract can be understood as the steady state of a highly dynamic process. Age and death are determined from the beginning; cell-cell interaction or communication is not necessary and can be neglected. Our theory will be illustrated using the intestinal crypts as they are easily accessible, of a simple structure and completely described in the literature.     

Mathematical Model For Human Myeloma Relating Growth Kinetics and Drug Resistance

Abstract. We present a computer-based mathematical model that can simulate characteristic features of the clinical time course of human myeloma. It asserts that therapy resistance in myeloma cells is an inherited trait associated with the longer inter-mitotic times of some cells and that the strength of this trait affects tumour growth characteristics. These kinetic differences within the malignant cell clone may also influence therapeutic efficacy. In the model, the same total therapy, administered in different time-dose fractions, could be 'curative' or 'minimally effective' depending on kinetic properties. For example, as others have shown, in myeloma pulsed intermittent therapy is often more effective than low dose continuous therapy. According to our model this finding is compatible with a high coefficient of inheritability of resistance from one cell generation to the next. the model also suggests that if there are subclones of varying resistance, a therapy must have some effect on each of them if it is to be employed in a curative fashion. While many aspects of the model are not yet clinically testable, exploration of its concepts might increase knowledge about fundamental neoplastic mechanisms.     

Methylmercury Effects On Cell Cycle Kinetics

Abstract. Methylmercury (MeHg) effects on cell cycle kinetics were investigated to help identify its mechanisms of action. Flow cytometric analysis of normal human fibroblasts grown in vitro in the presence of BrdU allowed quantitation of the proportion of cells in G₁, S, G₂ and the next G₁ phase. This technique provides a rapid and easily performed method of characterizing phase lengths and transition rates for the complete cell cycle. After first exposure to MeHg the cell cycle time was lengthened due to a prolonged G₁. At 3, μm MeHg the G₁ phase length was 25% longer than the control. the G₁/S transition rate was also decreased in a dose-related manner. Confluent cells exposed to MeHg and replated with MeHg respond in the same way as cells which have not been exposed to MeHg before replating. Cells exposed for long times to MeHg lost a detectable G₁ effect, and instead showed an increase in the G₂ percentage, which was directly related to MeHg concentration and length of exposure. After 8 days at 5 μM MeHg, 45% of the population was in G₂. the G₂ accumulation was reversible up to 3 days, but at 6 days the cells remained in G₂ when the MeHg was removed. Cell counts and viability indicated that there was not a selective loss of cells from the MeHg. MeHg has multiple effects on the cell cycle which include a lengthened G₁ and decreased transition probability after short term exposure of cycling cells, and a G₂ accumulation after a longer term exposure. There were no detectable S phase effects. It appears that mitosis (the G₂ accumulation) and probably synthesis of some macromolecules in G₁ (the lengthened G₁ and lowered transition probability) are particularly susceptible to MeHg.     

Quantitative Decomposition of Dynamics of Mathematical Cell Models: Method and Application to Ventricular Myocyte Models

Mathematical cell models are effective tools to understand cellular physiological functions precisely. For detailed analysis of model dynamics in order to investigate how much each component affects cellular behaviour, mathematical approaches are essential. This article presents a numerical analysis technique, which is applicable to any complicated cell model formulated as a system of ordinary differential equations, to quantitatively evaluate contributions of respective model components to the model dynamics in the intact situation. The present technique employs a novel mathematical index for decomposed dynamics with respect to each differential variable, along with a concept named instantaneous equilibrium point, which represents the trend of a model variable at some instant. This article also illustrates applications of the method to comprehensive myocardial cell models for analysing insights into the mechanisms of action potential generation and calcium transient. The analysis results exhibit quantitative contributions of individual channel gating mechanisms and ion exchanger activities to membrane repolarization and of calcium fluxes and buffers to raising and descending of the cytosolic calcium level. These analyses quantitatively explicate principle of the model, which leads to a better understanding of cellular dynamics.     

A Mathematical Model for the Effects of HER2 Overexpression on Cell Proliferation in Breast Cancer

We present a mathematical model to study the effects of HER2 over-expression on cell proliferation in breast cancer. The model illustrates the proliferative behavior of cells as a function of HER2 and EGFR receptors numbers, and the growth factor EGF. This mathematical model comprises kinetic equations describing the cell surface binding of EGF growth factor to EGFR and HER2 receptors, coupled to a model for the dependence of cell proliferation rate on growth factor receptors binding. The simulation results from this model predict: (1) a growth advantage associated with excess HER2 receptors; (2) that HER2-over-expression is an insufficient parameter to predict the proliferation response of cancer cells to epidermal growth factors; and (3) the EGFR receptor expression level in HER2-over-expressing cells plays a key role in mediating the proliferation response to receptor-ligand signaling. This mathematical model also elucidates the interaction and roles of other model parameters in determining cell proliferation rate of HER2-over-expressing cells.     

细胞命运抉择的数学模型:活性氧与p53的关系及其意义(英文)

生物体内的细胞生活在复杂的环境中。在生物体内,活性氧是普遍存在的。生物体内的活性氧可以诱导DNA损伤,最终破坏基因组稳定性。其中,对基因组损伤最严重的是DNA双链断裂损伤。肿瘤抑制因子p53是细胞内介导DNA损伤反应的重要因子。p53可以修复损伤DNA,保护轻度受损细胞。而当细胞受到严重损伤时,p53能够诱发细胞凋亡,从而维持机体稳态。p53的动力学对于细胞的反应性具有重要影响,然而对这方面却缺少系统的认识。因此在本文中,我们主要关注运用数学模型方法研究p53脉冲的动力学性质,从而揭示细胞内潜在的生死选择机制。     

Aspects of Statistics In Studies of Cell Proliferation Iii. Alternative Models Give Different Results

Alternative models for the analysis of the spatial distribution of cells in an intestinal crypt cross-section are discussed. It is shown that different, though reasonable, models lead to different conclusions.     

Systemic Cancer Progression and Tumor Dormancy: Mathematical Models Meet Single Cell Genomics

Metastatic progression is thought to result from genetically advanced ?fully-malignant“ tumor cells. Within the concept the prevailing view holds that such cells disseminate mostly from large tumors and are capable of growing into metastases once they arrive at a distant site. Support for this scenario comes from numerous mouse models in which transplanted tumor cells grow into metastases within days or weeks. However, the assumption of such fully-malignant disseminating cells in human cancer is misleading and is neither supported by mathematical modeling of survival data from cancer patients nor by ex-vivo genomic data from disseminated cancer cells. For example, in breast cancer the growth of metastases is highly homogeneous and takes on average six years, the number of disseminated tumor cells before diagnosis of metastasis is similar for different tumor stages, and the genomic aberrations of disseminated cancer cells do rarely correspond to those in the primary tumor. Since these facts question conventional concepts of metastatic progression we provide a model of cancer progression in which time considerations and direct ex-vivo data form a starting point. In the proposed model tumor dormancy is a characteristic of almost all migrated tumor cells and metastatic growth is a rare, stochastic, evolutionary process of selection and mutation of cells that often disseminate shortly after transformation at the primary site.     

Effect of Chemical Ablation of Myenteric Neurones On Intestinal Cell Proliferation

Abstract. The duodenum or descending colon of male Wistar rats (average weight 60 g) was treated by a serosal application of a 0.2% solution of benzalkonium chloride (BAC) for 30 min. Control animals were treated with 0.9% (physiological) saline. the rats were allocated to four groups: Group DC ( N = 8) in which the duodenum was treated with physiological saline; Group DB ( N = 8) in which the duodenum was treated with BAC; Group CC ( N = 7) in which the descending colon was treated with physiological saline and Group CB ( N = 7) in which the descending colon was treated with BAC. After treatment, the animals were followed up for 5 months. At the end of the experiment, the animals were injected intraperitoneally with vincristine sulphate before sacrifice. Three segments were removed from the duodenum and descending colon for neuronal counting, catecholamine and serotonin measurements and morphokinetic studies of the epithelium. the following results were obtained: (1) there was a significant reduction in neurone number in the myenteric plexus of segments treated with BAC; (2) in the denervated intestinal segments, catecholamine levels were unchanged whereas serotonin levels were increased; (3) epithelial hyperplasia was observed in the denervated duodenum and descending colon; and (4) crypt cell production rate in the duodenum was similar in groups DC and DB but was significantly increased in the descending colon in group CB as compared with controls (CC). the present findings indicate that selective myenteric neuronal denervation caused by benzalkonium chloride plays a causative role in the hyperplasia and crypt cell production rate of the intestinal epithelium (duodenum and descending colon). These changes are probably induced by functional imbalance by the surviving neuronal elements in the gut, implicating neurotransmitters such as acetylcholine, noradrenaline, serotonin, somatostatin and vasoactive intestinal peptide.     

Cell Proliferation in the Presence of Telomerase

Background

Telomerase, which is active early in development and later in stem and germline cells, is also active in the majority of human cancers. One of the known functions of telomerase is to extend the ends of linear chromosomes, countering their gradual shortening at each cell division due to the end replication problem and postreplication processing. Telomerase concentration levels vary between different cell types as well as between different tumors. In addition variable telomerase concentrations will exist in different cells in the same tumor when telomerase inhibitors are used, because of limitations of drug delivery in tissue. Telomerase extends short telomeres more frequently than long telomeres and the relation between the extension frequency and the telomere length is nonlinear.

Methodolgy/Principal Findings

Here, the biological data of the nonlinear telomerase-telomere dynamics is incorporated in a mathematical theory to relate the proliferative potential of a cell to the telomerase concentration in that cell. The main result of the paper is that the proliferative capacity of a cell grows exponentially with the telomerase concentration.

Conclusions/Significance

The theory presented here suggests that long term telomerase inhibition in every cancer progenitor or cancer stem cell is needed for successful telomere targeted cancer treatment. This theory also can be used to plan and asses the results of clinical trials targeting telomerase.     

Explicit Kinetic Heterogeneity: Mathematical Models for Interpretation of Deuterium Labeling of Heterogeneous Cell Populations

Estimation of division and death rates of lymphocytes in different conditions is vital for quantitative understanding of the immune system. Deuterium, in the form of deuterated glucose or heavy water, can be used to measure rates of proliferation and death of lymphocytes in vivo. Inferring these rates from labeling and delabeling curves has been subject to considerable debate with different groups suggesting different mathematical models for that purpose. We show that the three most common models, which are based on quite different biological assumptions, actually predict mathematically identical labeling curves with one parameter for the exponential up and down slope, and one parameter defining the maximum labeling level. By extending these previous models, we here propose a novel approach for the analysis of data from deuterium labeling experiments. We construct a model of “kinetic heterogeneity” in which the total cell population consists of many sub-populations with different rates of cell turnover. In this model, for a given distribution of the rates of turnover, the predicted fraction of labeled DNA accumulated and lost can be calculated. Our model reproduces several previously made experimental observations, such as a negative correlation between the length of the labeling period and the rate at which labeled DNA is lost after label cessation. We demonstrate the reliability of the new explicit kinetic heterogeneity model by applying it to artificially generated datasets, and illustrate its usefulness by fitting experimental data. In contrast to previous models, the explicit kinetic heterogeneity model 1) provides a novel way of interpreting labeling data; 2) allows for a non-exponential loss of labeled cells during delabeling, and 3) can be used to describe data with variable labeling length.