Abstract

The division tracking dye, carboxyfluorescin diacetate succinimidyl ester (CFSE) is currently the most informative
labeling technique for characterizing the division history of cells in the immune system. Gett and Hodgkin have proposed to normalize CFSE data by the 2-fold expansion that is associated with
each division, and have argued that the mean
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of the normalized data increases linearly with time, t, with a slope
reflecting the division rate p. We develop a number of mathematical models for the clonal expansion of quiescent
cells after stimulation and show, within the context of these models, under which conditions this approach is valid.
We compare three means of the distribution of cells over the CFSE profile at time t: the mean, µ(t), the mean of the
normalized distribution, µ₂(t), and the mean of the normalized distribution excluding nondivided cells, µ₂(t).
In the simplest models, which deal with homogeneous populations of cells with constant division and death rates,
the normalized frequency distribution of the cells over the respective division numbers is a Poisson distribution
with mean µ₂(t) = pt, where p is the division rate. The fact that in the data these distributions seem Gaussian is
therefore insufficient to establish that the times at which cells are recruited into the first division have a Gaussian
variation because the Poisson distribution approaches the Gaussian distribution for large pt. Excluding nondivided
cells complicates the data analysis because µ₂(t) ≠ pt, and only approaches a slope p after an initial transient.
In models where the first division of the quiescent cells takes longer than later divisions, all three means have an
initial transient before they approach an asymptotic regime, which is the expected µ(t)=2pt and µ₂(t)= µ₂(t)=pt.
Such a transient markedly complicates the data analysis.After the same initial transients, the normalized cell numbers
tend to decrease at a rate e⁻dt , where d is the death rate.
Nonlinear parameter fitting of CFSE data obtained from Gett and Hodgkin to ordinary differential equation
(ODE) models with first-order terms for cell proliferation and death gave poor fits to the data.The Smith–Martin
model with an explicit time delay for the deterministic phase of the cell cycle performed much better. Nevertheless,
the insights gained from analysis of the ODEs proved useful as we showed by generating virtual CFSE data with a simulation model, where cell cycle times were drawn from various distributions, and then computing the various
mean division numbers.
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