Modeling the cell cycle: From deterministic models to hybrid systems

Computational simulation of cellular proliferation using a meshless method

BACKGROUND AND OBJECTIVE

During cell proliferation, cells grow and divide in order to obtain two new genetically identical cells. Understanding this process is crucial to comprehend other biological processes. Computational models and algorithms have emerged to study this process and several examples can be found in the literature. The objective of this work was to develop a new computational model capable of simulating cell proliferation. This model was developed using the Radial Point Interpolation Method, a meshless method that, to the knowledge of the authors, was never used to solve this type of problem. Since the efficiency of the model strongly depends on the efficiency of the meshless method itself, the optimal numbers of integration points per integration cell and of nodes for each influence-domain were investigated. Irregular nodal meshes were also used to study their influence on the algorithm.

METHODS

For the first time, an iterative discrete model solved by the Radial Point Interpolation Method based on the Galerkin weak form was used to establish the system of equations from the reaction-diffusion integro-differential equations, following a new phenomenological law proposed by the authors that describes the growth of a cell over time while dependant on oxygen and glucose availability. The discretization flexibility of the meshless method allows to explicitly follow the geometric changes of the cell until the division phase.

RESULTS

It was found that an integration scheme of 6 × 6 per integration cell and influence-domains with only seven nodes allows to predict the cellular growth and division with the best balance between the relative error and the computing cost. Also, it was observed that using irregular meshes do not influence the solution.

CONCLUSIONS

Even in a preliminary phase, the obtained results are promising, indicating that the algorithm might be a potential tool to study cell proliferation since it can predict cellular growth and division. Moreover, the Radial Point Interpolation Method seems to be a suitable method to study this type of process, even when irregular meshes are used. However, to optimize the algorithm, the integration scheme and the number of nodes inside the influence-domains must be considered.

DNA-structured mathematical model of cell-cycle progression in cyclic hypoxia

New experimental data have shown how the periodic exposure of cells to low oxygen levels (i.e., cyclic hypoxia) impacts their progress through the cell-cycle. Cyclic hypoxia has been detected in tumours and linked to poor prognosis and treatment failure. While fluctuating oxygen environments can be reproduced in vitro, the range of oxygen cycles that can be tested is limited. By contrast, mathematical models can be used to predict the response to a wide range of cyclic dynamics. Accordingly, in this paper we develop a mechanistic model of the cell-cycle that can be combined with in vitro experiments, to better understand the link between cyclic hypoxia and cell-cycle dysregulation. A distinguishing feature of our model is the inclusion of impaired DNA synthesis and cell-cycle arrest due to periodic exposure to severely low oxygen levels. Our model decomposes the cell population into five compartments and a time-dependent delay accounts for the variability in the duration of the S phase which increases in severe hypoxia due to reduced rates of DNA synthesis. We calibrate our model against experimental data and show that it recapitulates the observed cell-cycle dynamics. We use the calibrated model to investigate the response of cells to oxygen cycles not yet tested experimentally. When the re-oxygenation phase is sufficiently long, our model predicts that cyclic hypoxia simply slows cell proliferation since cells spend more time in the S phase. On the contrary, cycles with short periods of re-oxygenation are predicted to lead to inhibition of proliferation, with cells arresting from the cell-cycle in the G2 phase. While model predictions on short time scales (about a day) are fairly accurate (i.e, confidence intervals are small), the predictions become more uncertain over longer periods. Hence, we use our model to inform experimental design that can lead to improved model parameter estimates and validate model predictions.

Integrative workflows for network analysis

Due to genetic heterogeneity across patients, the identification of effective disease signatures and therapeutic targets is challenging. Addressing this challenge, we have previously developed a network-based approach, which integrates heterogeneous sources of biological information to identify disease specific core-regulatory networks. In particular, our workflow uses a multi-objective optimization function to calculate a ranking score for network components (e.g. feedback/feedforward loops) based on network properties, biomedical and high-throughput expression data. High ranked network components are merged to identify the core-regulatory network(s) that is then subjected to dynamical analysis using stimulus–response and in silico perturbation experiments for the identification of disease gene signatures and therapeutic targets. In a case study, we implemented our workflow to identify bladder and breast cancer specific core-regulatory networks underlying epithelial–mesenchymal transition from the E2F1 molecular interaction map.

In this study, we review our workflow and described how it has developed over time to understand the mechanisms underlying disease progression and prediction of signatures for clinical decision making.

A Network-Based Integrative Workflow to Unravel Mechanisms Underlying Disease Progression

Unraveling mechanisms underlying diseases has motivated the development of systems biology approaches. The key challenges for the development of mathematical models and computational tool are (1) the size of molecular networks, (2) the nonlinear nature of spatio-temporal interactions, and (3) feedback loops in the structure of interaction networks. We here propose an integrative workflow that combines structural analyses of networks, high-throughput data, and mechanistic modeling. As an illustration of the workflow, we use prostate cancer as a case study with the aim of identifying key functional components associated with primary to metastasis transitions. Analysis carried out by the workflow revealed that HOXD10, BCL2, and PGR are the most important factors affected in primary prostate samples, whereas, in the metastatic state, STAT3, JUN, and JUNB are playing a central role. The identified key elements of each network are validated using patient survival analysis. The workflow presented here allows experimentalists to use heterogeneous data sources for the identification of diagnostic and prognostic signatures.

Systems biology of cancer: a challenging expedition for clinical and quantitative biologists

A systems-biology approach to complex disease (such as cancer) is now complementing traditional experience-based approaches, which have typically been invasive and expensive. The rapid progress in biomedical knowledge is enabling the targeting of disease with therapies that are precise, proactive, preventive, and personalized. In this paper, we summarize and classify models of systems biology and model checking tools, which have been used to great success in computational biology and related fields. We demonstrate how these models and tools have been used to study some of the twelve biochemical pathways implicated in but not unique to pancreatic cancer, and conclude that the resulting mechanistic models will need to be further enhanced by various abstraction techniques to interpret phenomenological models of cancer progression.