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David Bernick
BIOE 497G Honors Paper
The article “Modeling Stem Cell Population Growth: Incorporating Terms for Proliferative
Heterogeneity,” by B.M. Deasy, creates and tests two models, a growth model with cell loss and a
growth model with differentiation. Stem cells have a lot of potential but not enough is known about
population expansion and the control of differentiation, we still must understand regulation by intrinsic
and extrinsic factors. A key distinction of stem cells over other proliferating cells is their affinity for a
heterogeneous state; in order to correctly create a mathematical model for growth, this must be
accounted for.
Exponential growth is often used to model populations, using the equation N= N0 2t/DT. Where
N is the number of cells at time t, N0 is the initial number of cells, and DT is the dividing time. This
equation is the same as the model used in class, X=X0 eμt. However, stems cells do not behave this way
because of non-dividing cells in the population. These cells cease to divide due to a number of factors,
including quiescence, terminal differentiation, senescence, or cell death. This can be modeled as a
growth factor of a <1, and or a death rate, μ >0. One model that is used to describe these dynamics is
the Sherley model,
N = N0 {0.5 + [1 – (2a)(t/DT)+1 / (2(1 – 2a))]}
which includes the growth fraction (mitotic fraction) a to account for non-dividing cells. This model
can be used to describe growth in “unrestricted” conditions, providing estimates for division time,
mitotic fraction, and population doubling time. The assumptions of this model include that DT is
constant, that dividing cells can create both dividing and non-dividing cells.
In order to create a growth model with cell loss we need to include three parameters, the DT,
mitotic fraction (a), and cell loss, M(t), with M being the total number of dead cells at time t. The
model equation for growth with cell loss becomes:
N = N0 {0.5 + [1 – (2a)(t/DT)+1 / (2(1 – 2a))]} – M
This equation looks and acts like the Sherley model, except for the last term, M which allows a more
accurate prediction of the growth factor (a) with conditions of cell loss. This model was tested
experimentally by plating 6-well collagen-coated plates with Muscle Derived Stem Cells (MDSC) at a
density of 225 cells/cm2 in normal growth medium. The average number of cells in each viewfield on
day 0 was ~4. Cell death was induced using the transcription inhibitor actinomycin D. Images of
population growth were acquired every 10 minutes for a 5-day period. From the video record, the live
cell count, N(t), was determined at 12-hour intervals and DTs were measured. Cell loss due to cell
death was measured by continuous viewing of the video record.
In typical in vitro stem cell growth there is a changing equilibrium between replication and
differentiation due to a variety of signals. To model this a term may be added to the Sherley model to
account for the terminally differentiated cells. The live number of cells can be split into the
proliferating cells (NProliferate) and non-dividing differentiated cells (Ndifferentiated). We can write this as
N = Nproliferate + Ndifferentiated
or,
Nproliferate + Ndifferentiated = N0 {0.5 + [1 – (2a)(t/DT)+1 / (2(1 – 2a))]} – M
In order to test this growth model the myogenic cells were examined under conditions which induce
differentiation. When myogenic cells differentiate, they fuse together to form multi-nucleated
myotubes, with the number of fused cells equal to the number of nuclei. The mono-nucleated portion is
mitotically active and consists of self-renewing MDSCs and myogenic precursor cells. We can
substitute
Nproliferate = NMononuclear , and Ndifferentiated = NFused Nuclei
Therefore,
NMononuclear + NFused Nuclei = N0 {0.5 + [1 – (2a (t/DT)+1 / (2(1 – 2a ))]} – M
To induce differentiation, MDSCs first were plated at a density of 1,000 cells/cm2 on collagen-coated
flasks in normal medium for 48 hours, and cultured for 96 hours. The average number of cells
per viewfield was 12 and growth curves were normalized to N0 = 1,000 cells/cm2.
This paper concludes with solving for more accurate mitotic fraction (a) than which could be
estimated using the Sherley model. The number of proliferating cells could be predicted by using the
mathematical model:
Nproliferate = N0 {0.5 + [1 – (2α)(t/DT)+1 / (2(1 – 2α))]} – M – NFused Nuclei
However to be able to use this equation to solve for any proliferating stem cell population we
would have to have better defined variables for M and NFused Nuclei . Since we describe a as the growth
factor, then (1– a) would be the fraction of cells that do not divide. We define cell death as cessation
of movement; cell blebbing, shrinkage, or bursting; change in phase contrast due to compromised
membrane, and finally, cell detachment from the substratum. Therefore we could then model M as
M = N0 emt(1– a)
where m is some fraction of non-dividing cells which fall into the category of cell death. Using the
same logic we can also define NFused Nuclei as
NFused Nuclei = Ndifferentiated = N0 edt(1– a)
where d is some fraction of non-dividing cells which terminally differentiates. The equation would end
up as:
Nproliferate = N0 {0.5 + [1 – (2a )(t/DT)+1 / (2(1 – 2a ))] – emt (1– a) – edt (1– a)}
It can be expected that both m and d could be modeled as non-linear, exponential functions, but
I am not sure how to give any proof of this other than a thought experiment. If you consider the options
of a stem cell, they will all eventually differentiate or die. If differentiation can be controlled as an
exponential function then we could predict, and therefore program, a sample to differentiate “on
command”. This could be very useful in stem cell therapeutics and in the creation of more natural
tissues.
Overall this was a very interesting paper to read and it has a lot of potential for the
understanding of stem cell dynamics. I really liked how they tried to determine everything
experimentally and how accurate of results were achieved. They accounted for so many factors
including the tendency toward an increase in cell life as confluency increased as well as creating the
population doubling equation using a known growth factor (a).