Murray PJ, Walter A, Fletcher AG, Edwards CM, Tindall MJ, Maini PK (2011) Comparing a discrete and continuum model of the intestinal crypt, PHYSICAL BIOLOGY 8 (2) ARTN 026011
van Leeuwen IMM, Edwards CM, Ilyas M, Byrne HM (2007) Towards a multiscale model of colorectal cancer, World Journal of Gastroenterology 13 (9) pp. 1399-1407
Colorectal cancer (CRC) is one of the best characterised cancers, with extensive data documenting the sequential gene mutations that underlie its development. Complementary datasets are also being generated describing changes in protein and RNA expression, tumour biology and clinical outcome. Both the quantity and the variety of information are inexorably increasing and there is now an accompanying need to integrate these highly disparate datasets. In this article we aim to explain why we believe that mathematical modelling represents a natural tool or language with which to integrate these data and, in so doing, to provide insight into CRC. © 2007 The WJG Press. All rights reserved.
Breward C, Dellar PJ, Edwards CM, Kaouri K, Richardson G, Wilson S (2002) Mathematical modelling of pipe-flow and extrusion of composite materials,
This paper deals with violent discontinuities in shallow water flows with large Froude number F. On a horizontal base, the paradigm problem is that of the impact of two fluid layers in situations where the flow can be modelled as two smooth regions joined by a singularity in the flow field. Within the framework of shallow water theory, we show that, over a certain time-scale, this discontinuity may be described by a delta shock, which is a weak solution of the underlying conservation laws in which the depth and mass and momentum fluxes have both delta function and step function components. We also make some conjectures about how this model evolves from the traditional model for jet impacts in which a spout is emitted. For flows on a sloping base, we show that for flow with an aspect ratio of O(F?2) on a base with an O(1) or larger slope, the governing equations admit a new type of discontinuous solution that is also modelled as a delta shock. The physical manifestation of this discontinuity is a small ?tube? of fluid bounding the flow. The delta-shock conditions for this flow are derived and solved for a point source on an inclined plane. This latter delta-shock framework also sheds light on the evolution of the layer impact on a horizontal base
Murray PJ, Edwards CM, Tindall MJ, Maini PK (2009) From a discrete to a continuum model of cell dynamics in one dimension, PHYSICAL REVIEW E 80 (3) ARTN 031912
In syncytial embryos nuclei undergo cycles of division and rearrangement within a common cytoplasm. It is presently unclear to what degree and how the nuclear array maintains positional order in the face of rapid cell divisions. Here we establish a quantitative assay, based on image processing, for analysing the dynamics of the nuclear array. By tracking nuclear trajectories in Drosophila melanogaster embryos, we are able to define and evaluate local and time-dependent measures for the level of geometrical order in the array. We find that after division, order is re-established in a biphasic manner, indicating the competition of different ordering processes. Using mutants and drug injections, we show that the order of the nuclear array depends on cytoskeletal networks organised by centrosomes. While both f-actin and microtubules are required for re-establishing order after mitosis, only f-actin is required to maintain the stability of this arrangement. Furthermore, f-actin function relies on myosin-independent non-contractile filaments that suppress individual nuclear mobility, whereas microtubules promote mobility and attract adjacent nuclei. Actin caps are shown to act to prevent nuclear incorporation into adjacent microtubule baskets. Our data demonstrate that two principal ordering mechanisms thus simultaneously contribute: (1) a passive crowding mechanism in which nuclei and actin caps act as spacers and (2) an active self-organisation mechanism based on a microtubule network.
Syncytial embryos develop through cycles of nuclear division and rearrangement within a common cytoplasm. A paradigm example is Drosophila melanogaster in which nuclei form an ordered array in the embryo surface over cell cycles 10-13. This ordering process is assumed to be essential for subsequent cellularisation. Using quantitative tissue analysis, it has previously been shown that the regrowth of actin and microtubule networks after nuclear division generates reordering forces that counteract its disordering effect (Kanesaki et al., 2011). We present here an individual-based computer simulation modelling the nuclear dynamics. In contrast to similar modelling approaches e.g. epithelial monolayers or tumour spheroids, we focus not on the spatial dependence, but rather on the time-dependence of the interaction laws. We show that appropriate phenomenological inter-nuclear force laws reproduce the experimentally observed dynamics provided that the cytoskeletal network regrows sufficiently quickly after mitosis. Then repulsive forces provided by the actin system are necessary and sufficient to regain the observed level of order in the system, after the strong disruption resulting from cytoskeletal network disassembly and spindle formation. We also observe little mixing of nuclei through cell cycles. Our study highlights the importance of the dynamics of cytoskeletal forces during this critical phase of syncytial development and emphasises the need for real-time experimental data at high temporal resolution. © 2014 Elsevier Ltd.
This paper presents a biomechanical model for the small pits, called crypts, that line the colon. A continuum approach is adopted, with the crypt epithelium modelled as a growing beam attached to the underlying lamina by cell bonds, which generate tension within the layer. These cell attachments are assumed to be viscoelastic thus allowing for cell progression along the crypt. It is shown that any combination of: an increase in net proliferation (i.e. cell production minus apoptosis), an enlargement of the proliferative compartment, an increase in the strength of the cellular attachment to the underlying lamina, or a change in the rate of cell growth or cell bonding may generate buckling of the tissue. These changes can all be generated by an activating mutation of the Wnt cascade, which is generally accepted to be the first genetic change in colorectal cancer, with subsequent deformation, budding, and crypt fission an observed feature of the adenomatous crypt.
Breward C, Dyson R, Edwards CM, Metcalfe P, Please CP, Zyskin M (2004) Modelling of melt on spinning wheels,
Murray PJ, Edwards CM, Tindall MJ, Maini PK (2012) Classifying general nonlinear force laws in cell-based models via the continuum limit, PHYSICAL REVIEW E 85 (2) ARTN 021921
Johnston MD, Edwards CM, Bodmer WF, Maini PK, Chapman SJ (2007) Mathematical modeling of cell population dynamics in the colonic crypt and in colorectal cancer, Proceedings of the National Academy of Sciences of the United States of America 104 (10) pp. 4008-4013
Johnston MD, Maini PK, Chapman SJ, Edwards CM, Bodmer WF (2010) On the proportion of cancer stem cells in a tumour, JOURNAL OF THEORETICAL BIOLOGY 266 (4) pp. 708-711
Johnston MD, Edwards CM, Bodmer WF, Maini PK, Chapman SJ (2007) Examples of mathematical modeling: Tales from the crypt, Cell Cycle 6 (17) pp. 2106-2112
Mathematical modeling is being increasingly recognized within the biomedical sciences as an important tool that can aid the understanding of biological systems. The heavily regulated cell renewal cycle in the colonic crypt provides a good example of how modeling can be used to find out key features of the system kinetics, and help to explain both the breakdown of homeostasis and the initiation of tumorigenesis. We use the cell population model by Johnston et al.5 to illustrate the power of mathematical modeling by considering two key questions about the cell population dynamics in the colonic crypt. We ask: how can a model describe both homeostasis and unregulated growth in tumorigenesis; and to which parameters in the system is the model most sensitive? In order to address these questions, we discuss what type of modeling approach is most appropriate in the crypt. We use the model to argue why tumorigenesis is observed to occur in stages with long lag phases between periods of rapid growth, and we identify the key parameters. ©2007 Landes Bioscience.
Van Leeuwen IMM, Byrne HM, Johnston MD, Edwards CM, Chapman SJ, Bodmer WF, Maini PK (2007) Modelling multiscale aspects of colorectal cancer, AIP Conference Proceedings 971 pp. 3-7
Colorectal cancer (CRC) is responsible for nearly half a million deaths annually world-wide . We present a series of mathematical models describing the dynamics of the intestinal epithelium and the kinetics of the molecular pathway most commonly mutated in CRC, the Wnt signalling network. We also discuss how we are coupling such models to build a multiscale model of normal and aberrant guts. This will enable us to combine disparate experimental and clinical data, to investigate interactions between phenomena taking place at different levels of organisation and, eventually, to test the efficacy of new drugs on the system as a whole. © 2008 American Institute of Physics.
Epithelial cell layers on soft elastic substrates or pillar arrays are commonly used as model systems for investigating the role of force in tissue growth, maintenance and repair. Here we show analytically that the experimentally observed localization of traction forces to the periphery of the cell layers
does not necessarily imply increased local cell activity, but follows naturally from the elastic problem of a finite-sized contractile layer coupled to an elastic foundation. For homogeneous contractility, the force localization is determined by one dimensionless parameter interpolating between linear and
exponential force profiles for the extreme cases of very soft and very stiff substrates, respectively. If contractility is sufficiently increased at the periphery, outward directed displacements can occur at intermediate positions. We also show that anisotropic extracellular stiffness can lead to force
localization in the stiffer direction, as observed experimentally.
Schwarz US, Dunlop CM (2012) Developmental biology: A growing role for computer simulations, Current Biology 22 (11)
Keeping cells separated in well-defined domains is essential for development. A new computational-experimental study elucidates the physical mechanisms that establish and maintain the dorsal-ventral compartment boundary in the Drosophila wing disc and demonstrates the increasing value of computer simulations in developmental biology. © 2012 Elsevier Ltd.
Edwards CM, Ovendon N, Rottschäfer V (2003) Some cracking ideas on egg incubation,
There is increasing experimental interest in mechanotransduction in
multi-cellular tissues as opposed to single cells. This is driven by a growing awareness of
the importance of physiologically relevant three-dimensional culture and of cell-cell and
cell-gel interactions in directing growth and development. The paradigm biophysical
technique for investigating tissue level mechanobiology in this context is to grow
model tissues in artificial gels with well-defined mechanical properties. These studies
often indicate that the stiµness of the encapsulating gel can significantly alter cellular
behaviours. We demonstrate here potential mechanisms linking tissue growth with
stiµness-mediated mechanotransduction. We show how tissue growth in gel systems
generates points at which there is a significant qualitative change in the cellular stress
and strain experienced. We show analytically how these potential switching points
depend on the mechanical properties of the constraining gel and predict when they
will occur. Significantly, we identify distinct mechanisms that act separately in each of
the stress and strain fields at diµerent times. These observations suggest growth as a
potential physical mechanism coupling gel stiµness with cellular mechanotransduction
in three-dimensional tissues. We additionally show that non-proliferating areas, in
the case that the constraining gel is soft compared with the tissue, will expand and
contract passively as a result of growth. Central compartment size is thus seen to not
be a reliable indicator on its own for growth initiation or active behaviour.
Maintenance and activation of the limited supply of primordial follicles in the ovary are
important determinants of reproductive lifespan. Currently, the molecular programme that
maintains the primordial phenotype and the early events associated with follicle activation
are not well defined. Here we have systematically analysed these events using microscopy
and detailed image analysis. Using the immature mouse ovary as a model, we demonstrate
that the onset of granulosa cell (GC) proliferation results in increased packing density on the
oocyte surface and consequent GC cuboidalisation. These events precede oocyte growth
and nuclear translocation of FOXO3a, a transcription factor important in follicle activation.
Immunolabelling of the TGFb signalling mediators and transcription factors, SMAD2/3,
revealed a striking expression pattern specific to GCs of small follicles. SMAD2/3 was
expressed in the nuclei of primordial GCs but was mostly excluded in early growing follicles.
In activated follicles, GC nuclei lacking SMAD2/3 generally expressed Ki67. These findings
suggest that the first phenotypic changes during follicle activation are observed in GCs, and
that TGFb signalling is fundamental for regulating GC arrest and the onset of proliferation.