Mesenchymal Stem Cell Heterogeneity and Ageing In Vitro: A Model Approach - Computational Modeling in Tissue Engineering - page 185

Fig. 4 Mechanisms of noise driven MSC differentiation. a Modelling fluctuations of the
differentiation state a. Upper panel: Gaussian conditional probability function p(a 0 |a) for the
transition a ? a 0 for a = 0.3, 0.5 and 0.7. Lower panel: A decrease of the noise amplitude r(a)
with a results in an accumulation of cells at higher values of a. b A pedigree of a differentiated,
quiescent cell (a [ a d ) illustrating the model concept. After some time the cell regains
proliferative capacity (a s \ a \ a d ) and generates a number of progeny. Moreover, there is a non
vanishing probability of even regaining stem cell properties (a \ a s )

Stem cells (a \ a s ) and differentiated cells (a [ a d ) do not proliferate. During the
growth process cells may frequently switch between proliferative and non-proliferative
states. This will result in an effective cell growth time larger than s.
The rate of randomization R is a model parameter that can be used for fitting
experimental data. For MSCs it has been set to about 1 update per hour for a stem
cell noise-amplitude of r 0 = 0.15. Growth times are given by experimental
observed minimal cell cycles times, which is about 11 h for MSC. The choice of a s
and a d is somewhat arbitrary, as there is no clear phenotype related to it. We have
chosen a s = 0.15 and a d = 0.85. Details can be found in Krinner et al. [ 47 ].
A straightforward method for quantifying the population heterogeneity is to
calculate the normalised Shannon entropy from the distribution of the a states
(here for a histogram of n bins):

ln ð n Þ X
n

H ¼ 1 p ð a n Þ ln ð p ð a n ÞÞ

ð 3 Þ

Figure 5 shows the dependence of H on f(E) assuming a constant f(E) = f 0
between 0 and 1. H decreases with increasing f(E). However, the distribution
remains heterogeneous (close to maximum entropy) as long as f(E) gets close to 1.
These changes are much more pronounced in quiescent populations 1/s = 0
compared to proliferating populations 1/s [ 0. Thus, very efficient noise control
mechanisms are required in order to generate a homogeneous proliferating