Biomedical Engineering Reference
In-Depth Information
cells should still be able to profit from any potential favourable change in the
environment, thus facilitating a ''hedge-betting'' approach. Thus it should be
possible that quorum sensing is also used to control behaviour at a level some-
where between the whole population and the single cell.
Quorum sensing can therefore serve to regulate at a variety of levels. Multiscale
modelling is consequently required to describe quorum sensing accurately: sub-
cellular gene regulation informs behaviour of an individual cell which influences
increasing numbers of cells, potentially through to the whole population pheno-
type. At each of these levels, signal is relayed back to the intracellular processes,
linking each level together. Unlike the majority of previous mathematical models
of quorum sensing, we shall focus on heterogeneity of the phenotype within a
population of cells, rather than the population as a whole becoming fully quorate,
investigating effects associated with both discrete and continuous models. Many
quorum-sensing systems have been shown mathematically to be bistable, enabling
switch-like behaviour to arise between phenotypes (i.e. between downregulated
and upregulated behaviour). This bi- (or indeed multi-) stability is dependent upon
the number and nature of feedback loops within the system [ 26 ] and quorum-
sensing-induced transcription being significantly faster than the basal rate [ 7 ]. For
the case study proposed in Sect. 4 of this chapter, we consider a system with the
appropriate conditions to be bistable.
More broadly, our focus here is on the spatio-temporal modelling of autoin-
ductive (positive-feedback) processes such as those central to quorum sensing and,
specifically, on investigating effects that go beyond those that can be captured by
the classical [i.e. partial-differential equation (PDE)] models for such phenomena.
In keeping with this focus, we investigate in Sects. 2 and 3 two model problems
associated with the most explicit mathematical abstraction of a positive feedback
process leading to upregulation, namely blow-up behaviour, investigating firstly
the effects of delays associated with gene expression and secondly the effects of
discreteness on cell-cell communication. The latter is complemented by an
exploration in Sect. 4 of the influence of discreteness on wave propagation and
pinning in a more mechanistic, and hence significantly more complicated, discrete
model for a specific quorum-sensing system.
The ostensibly simple model problems we explore first seek to illustrate the
implications of a number of key effects: positive feedback (reflected by the source
terms in the systems studied below), nonlinearity (describing effects associated
with cooperativity in gene regulation: we focus on the simplest (quadratic) pos-
sibility, as applicable when, for example, regulation is governed by protein di-
merisation), discreteness (associated with the population being made up of
individuals or distinct compartments: it should be stressed that, while the discrete
models below take the form of finite-difference approximations to PDEs, here it is
the discrete problem that is to be regarded as the 'true' model and its continuum
limit as the approximation, rather than vice versa) and delays (which are inevitably
present in autoregulation for a variety of reasons). Our analysis will apply ideas
from (multiscale) matched asymptotic expansions and similarity methods, as well
as numerical simulations.
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