  Plant hormone auxin has critical roles in plant growth, dependent on its
heterogeneous distribution in plant tissues. Exactly how auxin transport and
developmental processes such as growth coordinate to achieve the precise
patterns of auxin observed experimentally is not well understood. Here we use
mathematical modelling to examine the interplay between auxin dynamics and
growth and their contribution to formation of patterns in auxin distribution in
plant tissues. Mathematical models describing the auxin-related signalling
pathway, PIN and AUX1 dynamics, auxin transport, and cell growth in plant
tissues are derived. A key assumption of our models is the regulation of PIN
proteins by the auxin-responsive ARF-Aux/IAA signalling pathway, with
upregulation of PIN biosynthesis by ARFs. Models are analysed and solved
numerically to examine the long-time behaviour and auxin distribution. Changes
in auxin-related signalling processes are shown to be able to trigger
transition between passage and spot type patterns in auxin distribution. The
model was also shown to be able to generate isolated cells with oscillatory
dynamics in levels of components of the auxin signalling pathway which could
explain oscillations in levels of ARF targets that have been observed
experimentally. Cell growth was shown to have influence on PIN polarisation and
determination of auxin distribution patterns. Numerical simulation results
indicate that auxin-related signalling processes can explain the different
patterns in auxin distributions observed in plant tissues, whereas the
interplay between auxin transport and growth can explain the `reverse-fountain'
pattern in auxin distribution observed at plant root tips.
