Mathematics of cell growth : a thesis presented in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Albany, New Zealand
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Date
2014
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Massey University
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Abstract
We present a model that describes growth, division and death of cells structured
by size. Here, size can be interpreted as DNA content or physical
size. The model is an extension of that studied by Hall and Wake [24] and
incorporates the symmetric as well as the asymmetric division of cells.
We first consider the case of symmetric cell division. This leads to an
initial boundary value problem that involves a first-order linear PDE with a
functional term. We study the separable solution to this problem which plays
an important role in the long term behaviour of solutions. We also derive a
solution to the problem for arbitrary initial cell distributions. The method
employed exploits the hyperbolic character of the underlying differential operator,
and the advanced nature of the functional argument to reduce the
problem to a sequence of simple Cauchy problems. The existence of solutions
for arbitrary initial distributions is established along with uniqueness.
The asymptotic relationship with the separable solution is established, and
because the solution is known explicitly, higher order terms in the asymptotics
can be obtained. Adding variability to the growth rate of cells leads
to a modified Fokker-Planck equation with a functional term. We find the
steady size distribution solution to this equation. We also obtain a constructive
existence and uniqueness theorem for this equation with an arbitrary
initial size-distribution and with a no-flux condition.
We then proceed to study the binary asymmetric division of cells. This
leads to an initial boundary value problem that involves a first-order linear
PDE with two functional terms. We find and prove the unimodality of the
steady size distribution solution to this equation. The existence of higher
eigenfunctions is also discussed. Adding stochasticity to the growth rate of
cells yields a second-order functional differential equation with two non-local
terms.
These problems, being a particular kind of functional differential equations
exhibit unusual characteristics. Although the associated boundary
value problems are well-posed, the spectral problems that arise by separating
the variables, cannot be easily shown to have a complete set of eigenfunctions
or the usual orthogonality properties.
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Keywords
Cells, Growth, Cell growth, Mathematical models, Research Subject Categories::MATHEMATICS::Applied mathematics