Analysis of Travelling Waves Associated with the Modelling of Aerosolised Skin Grafts

Mathematical Models for Cell Migration with Real-Time Cell Cycle Dynamics

The fluorescent ubiquitination-based cell cycle indicator, also known as FUCCI, allows the visualization of the G1 and S/G2/M cell cycle phases of individual cells. FUCCI consists of two fluorescent probes, so that cells in the G1 phase fluoresce red and cells in the S/G2/M phase fluoresce green. FUCCI reveals real-time information about cell cycle dynamics of individual cells, and can be used to explore how the cell cycle relates to the location of individual cells, local cell density, and different cellular microenvironments. In particular, FUCCI is used in experimental studies examining cell migration, such as malignant invasion and wound healing. Here we present, to our knowledge, new mathematical models that can describe cell migration and cell cycle dynamics as indicated by FUCCI. The fundamental model describes the two cell cycle phases, G1 and S/G2/M, which FUCCI directly labels. The extended model includes a third phase, early S, which FUCCI indirectly labels. We present experimental data from scratch assays using FUCCI-transduced melanoma cells, and show that the predictions of spatial and temporal patterns of cell density in the experiments can be described by the fundamental model. We obtain numerical solutions of both the fundamental and extended models, which can take the form of traveling waves. These solutions are mathematically interesting because they are a combination of moving wavefronts and moving pulses. We derive and confirm a simple analytical expression for the minimum wave speed, as well as exploring how the wave speed depends on the spatial decay rate of the initial condition.

INVESTIGATION OF A STRUCTURED FISHER'S EQUATION WITH APPLICATIONS IN BIOCHEMISTRY

Recent biological research has sought to understand how biochemical signaling pathways, such as the mitogen-activated protein kinase (MAPK) family, influence the migration of a population of cells during wound healing. Fisher's Equation has been used extensively to model experimental wound healing assays due to its simple nature and known traveling wave solutions. This partial differential equation with independent variables of time and space cannot account for the effects of biochemical activity on wound healing, however. To this end, we derive a structured Fisher's Equation with independent variables of time, space, and biochemical pathway activity level and prove the existence of a self-similar traveling wave solution to this equation. We exhibit that these methods also apply to a general structured reaction-diffusion equation and a chemotaxis equation. We also consider a more complicated model with different phenotypes based on MAPK activation and numerically investigate how various temporal patterns of biochemical activity can lead to increased and decreased rates of population migration.

Exact Solutions of Coupled Multispecies Linear Reaction-Diffusion Equations on a Uniformly Growing Domain

Embryonic development involves diffusion and proliferation of cells, as well as diffusion and reaction of molecules, within growing tissues. Mathematical models of these processes often involve reaction–diffusion equations on growing domains that have been primarily studied using approximate numerical solutions. Recently, we have shown how to obtain an exact solution to a single, uncoupled, linear reaction–diffusion equation on a growing domain, 0 < x < L(t), where L(t) is the domain length. The present work is an extension of our previous study, and we illustrate how to solve a system of coupled reaction–diffusion equations on a growing domain. This system of equations can be used to study the spatial and temporal distributions of different generations of cells within a population that diffuses and proliferates within a growing tissue. The exact solution is obtained by applying an uncoupling transformation, and the uncoupled equations are solved separately before applying the inverse uncoupling transformation to give the coupled solution. We present several example calculations to illustrate different types of behaviour. The first example calculation corresponds to a situation where the initially–confined population diffuses sufficiently slowly that it is unable to reach the moving boundary at x = L(t). In contrast, the second example calculation corresponds to a situation where the initially–confined population is able to overcome the domain growth and reach the moving boundary at x = L(t). In its basic format, the uncoupling transformation at first appears to be restricted to deal only with the case where each generation of cells has a distinct proliferation rate. However, we also demonstrate how the uncoupling transformation can be used when each generation has the same proliferation rate by evaluating the exact solutions as an appropriate limit.

Dispersal, settling and layer formation

Motivated by examples in developmental biology and ecology, we develop a model for convection-dominated invasion of a spatial region by initially motile agents which are able to settle permanently. The motion of the motile agents and their rate of settling are affected by the local concentration of settled agents. The model can be formulated as a nonlinear partial differential equation for the time-integrated local concentration of the motile agents, from which the instantaneous density of settled agents and its long-time limit can be extracted. In the limit of zero diffusivity, the partial differential equation is of first order; for application-relevant initial and boundary-value problems, shocks arise in the time-integrated motile agent density, leading to delta-function components in the motile agent density. Furthermore, there are simple solutions for a model of successive layer formation. In addition some analytic results for a one-dimensional system with non-zero diffusivity can also be obtained. A case study, both with and without diffusion, is examined numerically. Some important predictions of the model are insensitive to the specific settling law used and the model offers insight into biological processes involving layered growth or overlapping generations of colonization.

Coalescence of interacting cell populations

We analyse the coalescence of invasive cell populations by studying both the temporal and steady behaviour of a system of coupled reaction-diffusion equations. This problem is relevant to recent experimental observations of the dynamics of opposingly directed invasion waves of cells. Two cell types, u and v, are considered with the cell motility governed by linear or nonlinear diffusion. The cells proliferate logistically so that the long-term total cell density, u+v approaches a carrying capacity. The steady-state solutions for u and v are denoted u(s) and v(s). The steady solutions are spatially invariant and satisfy u(s)+v(s)=1. However, this expression is underdetermined so the relative proportion of each cell type u(s) and v(s) cannot be determined a priori. Various properties of this model are studied, such as how the relative proportion of u(s) and v(s) depends on the relative motility and relative proliferation rates. The model is analysed using a combination of numerical simulations and a comparison principle. This investigation unearths some novel outcomes regarding the role of overcrowding and cell death in this type of cell migration assay. These observations have relevance to experimental design and interpretation regarding the identification and parameterisation of mechanisms involved in cell invasion.

Travelling Waves of Attached and Detached Cells in a Wound-Healing Cell Migration Assay

During a wound-healing cell migration assay experiment, cells are observed to detach and undergo mitosis before reattaching as a pair of cells on the substrate. During experiments with mice 3T3 fibroblasts, cell detachment can be confined to the wavefront region or it can occur over the whole wave profile. A multi-species continuum approach to modelling a wound-healing assay is taken to account for this phenomenon. The first cell population is composed of attached motile cells, while the second population is composed of detached immotile cells undergoing mitosis and returning to the migrating population after successful division. The first model describes cell division occurring only in the wavefront region, while a second model describes cell division over the whole of the scrape wound. The first model reverts to the Fisher equation when the reattachment rate relative to the detachment rate is large, while the second case does not revert to the Fisher equation in any limit. The models yield travelling wave solutions. The minimum wave speed is slower than the minimum Fisher wave speed and its dependence on the cell detachment and reattachment rate parameters is analysed. Approximate travelling wave profiles of the two cell populations are determined asymptotically under certain parameter regimes.

Mathematical modelling of aerosolised skin grafts incorporating keratinocyte clonal subtypes

Severe burns can be very traumatic for the patient, and while burns caused by industrial or domestic accidents are common, there are also increasing numbers of burns associated with terrorism. A novel technique to assist in the healing process is to spray skin cells, keratinocytes, that are cultured from the patient's own tissue, directly onto the burn site. This process involves taking some undamaged skin from the patient, allowing the skin cells to proliferate rapidly in the laboratory over a period of 5-10 days, harvesting and separating the cells and then spraying them onto the burn. This paper deals with keratinocytes that have been cultured in vitro for a short period of time (early passage cultured cells). The spraying process has yet to be optimised with respect to the seeding density required for fastest re-epithelisation and thus there is a need for this process to be modelled. In this paper, we review some of the skin biology and develop a mathematical model of the growth patterns of cell colonies after they have been applied using a aerosolised technique. The model allows us to predict coverage over time and can be used as a decision support tool for clinicians.