Cell population balance models can account for the phenotypic heterogeneity that characterizes isogenic cell populations. However, in order to utilize the predictive power of these models, the single-cell reaction and division rates as well as the partition probability density function are required. These functions (collectively called Intrinsic Physiological State Functions (IPSF)) depend on the cellular property (state variable) of interest such as a morphological characteristic, a protein or the DNA content. To determine the IPSF one has to solve tough inverse cell population balance problems (Ramkrishna, 1994; Ramkrishna et al., 1968; Ramkrishna and Schell, 1999). Collins and Richmond (Collins and Richmond, 1962), have developed a set of equations, which can be solved to find the IPSF under balanced growth, if the average specific growth rate as well as the distributions of the cell property of interest are known for: a) the entire cell population b) the dividing cell subpopulation and c) the newborn cell subpopulation. Although the aforementioned approach has been applied in several cases in the past, usually state variables and not the same biological system.

The current research work focuses on completely resolving the inverse cell population balance problem, in the sense of determining the three IPSF with respect to the same cellular property for the same biological system, given the three necessary experimentally determined distributions. At first, the problem has been generally addressed (no particular biological system). To this end, we have considered the set of Collins and Richmond equations extended by the additional integral equation for the partition probability density function, derived from the cell population balance model and the normalization condition for the density function. To assess the effect of heterogeneity on the extracted IPSF and our ability to effectively recover them, we have utilized simulated rather than real experimental data which have been generated by solving the forward cell population balance problem for different known IPSF. The significance of a whole set of parameters affecting the solution of the mathematical problem has been investigated. Additionally, uncertainty, which is inherent in the experimental data, has been also taken into account in recovering the IPSF. With the insight obtained from the general treatment of the problem, the theoretical methodology was then applied to a model system: an E. coli population with an IPTG-inducible genetic toggle network equipped with a gfpmut3 gene functioning as a reporter of intracellular expression levels. Real experimental data obtained for the aforementioned system have been used to determine the IPSF and assess the effect of IPTG on them. The theoretical work presented here can be used as a general framework applicable to other biological systems in order to obtain single cell information and therefore elucidate the effects and implications of cell population heterogeneity.

References

Collins JF, Richmond MH. 1962. Rate of growth of Bacillus cereus between divisions. Journal of General Microbiology

Ramkrishna D, Fredrickson AG, Tsuchiya HM. 1968. On relationships between various distribution functions in balanced unicellular growth. Bulletin of Mathematical Biophysics 30:319-323.

Ramkrishna D. 1994. Toward a self-similar theory of microbial populations. Biotechnology and Bioengineering 43:138-148.

Ramkrishna D, Schell J. 1999. On self-similar growth Journal of Biotechnology 71:255-258.