We investigated the mathematical modeling of the evolution of a Stem Cell population (most prominently Hematopoietic Stem Cells - HSCs) with the objective to construct reliable, accurate and consistent simulations of the biological processes involved at different levels of detail.

The identification of the mathematical abstractions and algorithms that match the known biological facts has been driving our effort. The choice of the reference biological system to be simulated - Stem Cells (SCs) - was made because of the challenges it poses for the construction of a flexible, multi-scale, user-friendly simulation tool capable of handling complex Hybrid Systems.

Research in Stem Cell Biology is of tremendous importance for the promise it holds for the cure of several degenerative and/or traumatic conditions. While the body of knowledge about SC's is increasing over time, many questions remain unanswered. The implementation of a "reference" simulation system (amenable to incremental refinements, updates and amendments) will provide biologists with a tool for the comparison and "validation" of their hypotheses and experimental results. Additionally, the tool will aid discovery of related results and phenomena.

The construction of a simulation tool capable of meeting the stated objectives is challenging both from a mathematical and a computer science point of view. Whilst the modeling and informatics simulation technology has developed considerably over the past years, some key questions remain regarding the modeling and simulation of systems as complex as biological ones. Three such problems pertain the efficient manipulation of hierarchical abstractions, the modeling of different time scales and spaces, and the treatment of essentially partial models of various biological processes. A related orthogonal problem pertains the accuracy of the numerical simulations.

We started out with a simple model of a three stage cell line development, where, e.g., a HSC population either self renews or differentiates into a Red Cell Progenitor population, or a White Cell Progenitor population. These in turn give rise to the final Red Blood Cells and to the final White Blood Cells.

The model is essentially a complex Birth-Death process, for which we studied both approximate deterministic, and stochastic versions. The model takes into account a simple feedback loop based on the populations' sizes and the different kinds of SCs' differentiation (either symmetric or asymmetric). Of course, the model of the feedback loop can be made more precise by modularly inserting the interaction among various signaling factors. As such, our model becomes a test-bed for a variety of Cell/Cell signaling processes.

As an important evaluation step, we implemented our initial mathematical model on a number of well known simulation platforms. In particular we concentrated on mathematical modeling tools like Matlab, Octave and Mathematica, and simulation tools used by several Electrical Engineering and Computer Science projects, e.g. SHIFT and Lambda-Shift from UC Berkeley and Charon from UPenn.

We are aware the our models need a careful parameter determination. We will proceed to complete this task by analyzing data from co-cultivation gene expression micro-array experiments.