Instantaneous mutation rate in cancer initiation and progression.

BACKGROUND: Cancer is one of the leading causes for the morbidity and mortality worldwide. Although substantial studies have been conducted theoretically and experimentally in recent years, it is still a challenge to explore the mechanisms of cancer initiation and progression. The investigation for these problems is very important for the diagnosis of cancer diseases and development of treatment schemes. RESULTS: To accurately describe the process of cancer initiation, we propose a new concept of gene initial mutation rate based on our recently designed mathematical model using the non-constant mutation rate. Unlike the widely-used average gene mutation rate that depends on the number of mutations, the gene initial mutation rate can be used to describe the initiation process of a single patient. In addition, we propose the instantaneous tumour doubling time that is a continuous function of time based on the non-constant mutation rate. Our proposed concepts are supported by the clinic data of seven patients with advanced pancreatic cancer. The regression results suggest that, compared with the average mutation rate, the estimated initial mutation rate has a larger value of correlation coefficient with the patient survival time. We also provide the estimated tumour size of these seven patients over time. CONCLUSIONS: The proposed concepts can be used to describe the cancer initiation and progression for different patients more accurately. Since a quantitative understanding of cancer progression is important for clinical treatment, our proposed model and calculated results may provide insights into the development of treatment schemes and also have other clinic implications.

A mathematical model for eph/ephrin-directed segregation of intermingled cells.

Eph receptors, the largest family of receptor tyrosine kinases, control cell-cell adhesion/de-adhesion, cell morphology and cell positioning through interaction with cell surface ephrin ligands. Bi-directional signalling from the Eph and ephrin complexes on interacting cells have a significant role in controlling normal tissue development and oncogenic tissue patterning. Eph-mediated tissue patterning is based on the fine-tuned balance of adhesion and de-adhesion reactions between distinct Eph- and ephrin-expressing cell populations, and adhesion within like populations (expressing either Eph or ephrin). Here we develop a stochastic, Lagrangian model that is based on Eph/ephrin biology: incorporating independent Brownian motion to describe cell movement and a deterministic term (the drift term) to represent repulsive and adhesive interactions between neighbouring cells. Comparison between the experimental and computer simulated Eph/ephrin cell patterning events shows that the model recapitulates the dynamics of cell-cell segregation and cell cluster formation. Moreover, by modulating the term for Eph/ephrin-mediated repulsion, the model can be tuned to match the actual behaviour of cells with different levels of Eph expression or activity. Together the results of our experiments and modelling suggest that the complexity of Eph/ephrin signalling mechanisms that control cell-cell interactions can be described well by a mathematical model with a single term balancing adhesion and de-adhesion between interacting cells. This model allows reliable prediction of Eph/ephrin-dependent control of cell patterning behaviour.

A new model of time scheme for progression of colorectal cancer.

BACKGROUND: tumourigenesis can be regarded as an evolutionary process, in which the transformation of a normal cell into a tumour cell involves a number of limiting genetic and epigenetic events. To study the progression process, time schemes have been proposed for studying the process of colorectal cancer based on extensive clinical investigations. Moreover, a number of mathematical models have been designed to describe this evolutionary process. These models assumed that the mutation rate of genes is constant during different stages. However, it has been pointed that the subsequent driver mutations appear faster than the previous ones and the cumulative time to have more driver mutations grows with the growing number of gene mutations. Thus it is still a challenge to calculate the time when the first mutation occurs and to determine the influence of tumour size on the mutation rate. RESULTS: In this work we present a general framework to remedy the shortcoming of existing models. Rather than considering the information of gene mutations based on a population of patients, we for the first time determine the values of the selective advantage of cancer cells and initial mutation rate for individual patients. The averaged values of doubling time and selective advantage coefficient determined by our model are consistent with the predictions made by the published models. Our calculation showed that the values of biological parameters, such as the selective advantage coefficient, initial mutation rate and cell doubling time diversely depend on individuals. Our model has successfully predicted the values of several important parameters in cancer progression, such as the selective advantage coefficient, initial mutation rate and cell doubling time. In addition, experimental data validated our predicted initial mutation rate and cell doubling time. CONCLUSIONS: The introduced new parameter makes our proposed model more flexible to fix various types of information based on different patients in cancer progression.