Mathematical investigation of normal and abnormal wound healing dynamics: local and non-local models.

The movement of cells during (normal and abnormal) wound healing is the result of biomechanical interactions that combine cell responses with growth factors as well as cell-cell and cell-matrix interactions (adhesion and remodelling). It is known that cells can communicate and interact locally and non-locally with other cells inside the tissues through mechanical forces that act locally and at a distance, as well as through long non-conventional cell protrusions. In this study, we consider a non-local partial differential equation model for the interactions between fibroblasts, macrophages and the extracellular matrix (ECM) via a growth factor (TGF-$ \beta $) in the context of wound healing. For the non-local interactions, we consider two types of kernels (i.e., a Gaussian kernel and a cone-shaped kernel), two types of cell-ECM adhesion functions (i.e., adhesion only to higher-density ECM vs. adhesion to higher-/lower-density ECM) and two types of cell proliferation terms (i.e., with and without decay due to overcrowding). We investigate numerically the dynamics of this non-local model, as well as the dynamics of the localised versions of this model (i.e., those obtained when the cell perception radius decreases to 0). The results suggest the following: (ⅰ) local models explain normal wound healing and non-local models could also explain abnormal wound healing (although the results are parameter-dependent); (ⅱ) the models can explain two types of wound healing, i.e., by primary intention, when the wound margins come together from the side, and by secondary intention when the wound heals from the bottom up.

Modelling Keloids Dynamics: A Brief Review and New Mathematical Perspectives.

Keloids are fibroproliferative disorders described by excessive growth of fibrotic tissue, which also invades adjacent areas (beyond the original wound borders). Since these disorders are specific to humans (no other animal species naturally develop keloid-like tissue), experimental in vivo/in vitro research has not led to significant advances in this field. One possible approach could be to combine in vitro human models with calibrated in silico mathematical approaches (i.e., models and simulations) to generate new testable biological hypotheses related to biological mechanisms and improved treatments. Because these combined approaches do not really exist for keloid disorders, in this brief review we start by summarising the biology of these disorders, then present various types of mathematical and computational approaches used for related disorders (i.e., wound healing and solid tumours), followed by a discussion of the very few mathematical and computational models published so far to study various inflammatory and mechanical aspects of keloids. We conclude this review by discussing some open problems and mathematical opportunities offered in the context of keloid disorders by such combined in vitro/in silico approaches, and the need for multi-disciplinary research to enable clinical progress.

Investigation into the role of macrophages heterogeneity on solid tumour aggregations.

Macrophages are one of the most important immune cell populations that can be found inside solid tumours. For a long time, it was thought that these cells have an anti-tumour role, but relatively recent research has shown that they can have both anti-tumour and pro-tumour roles as determined by their phenotypes. Due to the heterogeneity and plasticity of macrophage population, with cells changing their phenotypes in response to the tumour microenvironment, it is difficult to fully understand their role inside the solid tumours. Here we consider a mathematical modelling and computational approach to investigate the change in macrophages phenotypes (either determined by the tumour itself, or by external interventions) on overall tumour growth/control/decay. To this end we consider two simple models: one focusing on two extreme phenotypes (the M1 anti-tumour cells, and the M2 pro-tumour cells), and one considering a macrophage population structured by a continuous phenotype variable. We investigate their asymptotic dynamics (through steady-state analysis), as well as their transient behaviours (through numerical simulations). We show that while a re-polarisation of the phenotype of macrophages, as considered by many recent experimental studies, can lead to tumour control, for tumour elimination it is required that macrophages are fully functional (i.e., the rate at which they kill tumour cells is high). We also show that a mixed macrophage's phenotype can keep the tumour under control in a state of dormancy. Moreover, an increase in this mixed phenotype can cause a delay in tumour reduction (accompanied by a larger tumour reduction), as well as a delay in tumour relapse.

Investigating Macrophages Plasticity Following Tumour-Immune Interactions During Oncolytic Therapies.

Over the last few years, oncolytic virus therapy has been recognised as a promising approach in cancer treatment, due to the potential of these viruses to induce systemic anti-tumour immunity and selectively killing tumour cells. However, the effectiveness of these viruses depends significantly on their interactions with the host immune responses, both innate (e.g., macrophages, which accumulate in high numbers inside solid tumours) and adaptive (e.g., [Formula: see text] T cells). In this article, we consider a mathematical approach to investigate the possible outcomes of the complex interactions between two extreme types of macrophages (M1 and M2 cells), effector [Formula: see text] T cells and an oncolytic Vesicular Stomatitis Virus (VSV), on the growth/elimination of B16F10 melanoma. We discuss, in terms of VSV, [Formula: see text] and macrophages levels, two different types of immune responses which could ensure tumour control and eventual elimination. We show that both innate and adaptive anti-tumour immune responses, as well as the oncolytic virus, could be very important in delaying tumour relapse and eventually eliminating the tumour. Overall this study supports the use mathematical modelling to increase our understanding of the complex immune interaction following oncolytic virotherapies. However, the complexity of the model combined with a lack of sufficient data for model parametrisation has an impact on the possibility of making quantitative predictions.

Modelling the collective response of heterogeneous cell populations to stationary gradients and chemical signal relay.

The directed motion of cell aggregates toward a chemical source occurs in many relevant biological processes. Understanding the mechanisms that control this complex behavior is of great relevance for our understanding of developmental biological processes and many diseases. In this paper, we consider a self-propelled particle model for the movement of heterogeneous subpopulations of chemically interacting cells towards an imposed stable chemical gradient. Our simulations show explicitly how self-organisation of cell populations (which could lead to engulfment or complete cell segregation) can arise from the heterogeneity of chemotactic responses alone. This new result complements current theoretical and experimental studies that emphasise the role of differential cell-cell adhesion on self-organisation and spatial structure of cellular aggregates. We also investigate how the speed of individual cell aggregations increases with the chemotactic sensitivity of the cells, and decreases with the number of cells inside the aggregates.

Modelling cell movement, cell differentiation, cell sorting and proportion regulation in Dictyostelium discoideum aggregations.

Understanding the mechanisms that control tissue morphogenesis and homeostasis is a central goal not only in developmental biology but also has great relevance for our understanding of various diseases, including cancer. A model organism that is widely used to study the control of tissue morphogenesis and proportioning is the Dictyostelium discoideum. While there are mathematical models describing the role of chemotactic cell motility in the Dictyostelium assembly and morphogenesis of multicellular tissues, as well as models addressing possible mechanisms of proportion regulation, there are no models incorporating both these key aspects of development. In this paper, we introduce a 1D hyperbolic model to investigate the role of two morphogens, DIF and cAMP, on cell movement, cell sorting, cell-type differentiation and proportioning in Dictyostelium discoideum. First, we use the non-spatial version of the model to study cell-type transdifferentiation. We perform a steady-state analysis of it and show that, depending on the shape of the differentiation rate functions, multiple steady-state solutions may occur. Then we incorporate spatial dynamics into the model, and investigate the transdifferentiation and spatial positioning of cells inside the newly formed structures, following the removal of prestalk or prespore regions of a Dictyostelium slug. We show that in isolated prespore fragments, a tipped mound-like aggregate can be formed after a transdifferentiation from prespore to prestalk cells and following the sorting of prestalk cells to the centre of the aggregate. For isolated prestalk fragments, we show the formation of a slug-like structure containing the usual anterior-posterior pattern of prestalk and prespore cells.

Weakly nonlinear analysis of a hyperbolic model for animal group formation.

We consider an one-dimensional nonlocal hyperbolic model for group formation with application to self-organizing collectives of animals in homogeneous environments. Previous studies have shown that this model displays at least four complex spatial and spatiotemporal group patterns. Here, we use weakly nonlinear analysis to better understand the mechanisms involved in the formation of two of these patterns, namely stationary pulses and traveling trains. We show that both patterns arise through subcritical bifurcations from spatially homogeneous steady states. We then use these results to investigate the effect of two social interactions (attraction and alignment) on the structure of stationary and moving animal groups. While attraction makes the groups more compact, alignment has a dual effect, depending on whether the groups are stationary or moving. More precisely, increasing alignment makes the stationary groups compact, and the moving groups more elongated. Also, the results show the existence of a threshold for the total group density, above which, coordinated behaviors described by stationary and moving groups persist for a long time.

Modeling group formation and activity patterns in self-organizing collectives of individuals.

We construct and analyze a nonlocal continuum model for group formation with application to self-organizing collectives of animals in homogeneous environments. The model consists of a hyperbolic system of conservation laws, describing individual movement as a correlated random walk. The turning rates depend on three types of social forces: attraction toward other organisms, repulsion from them, and a tendency to align with neighbors. Linear analysis is used to study the role of the social interaction forces and their ranges in group formation. We demonstrate that the model can generate a wide range of patterns, including stationary pulses, traveling pulses, traveling trains, and a new type of solution that we call zigzag pulses. Moreover, numerical simulations suggest that all three social forces are required to account for the complex patterns observed in biological systems. We then use the model to study the transitions between daily animal activities that can be described by these different patterns.

Complex spatial group patterns result from different animal communication mechanisms.

We present previously undescribed spatial group patterns that emerge in a one-dimensional hyperbolic model for animal group formation and movement. The patterns result from the assumption that the interactions governing movement depend not only on distance between conspecifics, but also on how individuals receive information about their neighbors and the amount of information received. Some of these patterns are classical, such as stationary pulses, traveling waves, ripples, or traveling trains. However, most of the patterns have not been reported previously. We call these patterns zigzag pulses, semizigzag pulses, breathers, traveling breathers, and feathers.

The MyoD family of myogenic factors is regulated by electrical activity: isolation and characterization of a mouse Myf-5 cDNA.

A full-length cDNA coding for a homolog of the human Myf-5 was isolated from a BC3H-1 mouse library and characterized. The clone codes for a protein of 255 amino acids that is 89%, 88% and 68% identical to the human, bovine and Xenopus myf-5, respectively. The mouse Myf-5 cDNA (mmyf-5), as well as sequences coding for MyoD, myogenin and Mrf-4, were used to probe Northern blots to analyze the effects of innervation on the expression of the MyoD family of myogenic factors. Mouse myf-5, MyoD and myogenin mRNAs levels were found to decline in hind limb muscles of mice between embryonic day 15 (E15) and the first postnatal week, a period that coincides with innervation. In contrast, Mrf-4 transcripts increase during this period and reach steady-state levels by 1-week after birth. To distinguish if the changes in myogenic factor expression are due to a developmental program or to innervation, mRNA levels were analyzed at different times after muscle denervation. Mmyf-5 transcripts begin to accumulate 2 days postdenervation; after 1 week levels are 7-fold higher than in innervated muscle. Mrf-4, MyoD and myogenin transcripts begin to accumulate as soon as 8h after denervation, and attain levels that are 8-, 15- and 40-fold higher than found in innervated skeletal muscle, respectively. The accumulation of these three mRNAs precedes the increase of nicotinic acetylcholine receptor alpha subunit transcripts, a gene that is transcriptionally regulated by MyoD-related factors in vitro. Using extracellular electrodes to directly stimulate in situ the soleus muscle of rats, we found that 'electrical activity' per se, in absence of the nerve, represses the increases of myogenic factor mRNAs associated with denervation.

Myogenin and MyoD join a family of skeletal muscle genes regulated by electrical activity.

Myogenin and MyoD are proteins that bind to the regulatory regions of a battery of skeletal muscle genes and can activate their transcription during muscle differentiation. We have recently found that both proteins interact with the enhancer of the nicotinic acetylcholine receptor (nAChR) alpha subunit, a gene that is regulated by innervation. This observation prompted us to study if myogenin and MyoD transcript levels are also regulated by skeletal muscle innervation. Using Northern blot analysis, we found that MyoD and myogenin mRNA levels begin to decline at embryonic day 17 and attain adult levels in muscle of newborn and 3-week-old mice, respectively. In contrast, nAChR mRNAs are highest in newborn and 1-week-old mouse muscle and decline thereafter to reach adult levels in 3-week-old mice. To determine if the down-regulation of myogenin and MyoD mRNA levels during development is due to innervation, we quantitated message levels in adult calf muscles after denervation. We found that in denervated muscle myogenin and MyoD mRNAs reach levels that are approximately 40- and 15-fold higher than those found in innervated muscle. Myogenin mRNAs begin to accumulate rapidly between 8 and 16 hr after denervation, and MyoD transcripts levels begin to increase sharply between 16 hr and 1 day after denervation. The increases in myogenin and MyoD mRNA levels precede the rapid accumulation of nAChR alpha-subunit transcripts; receptor mRNAs begin to accumulate significantly after 1 day of denervation. The effects of denervation are specific because skeletal alpha-actin mRNA levels are not affected by denervation. In addition, we found that the repression of myogenin and MyoD expression by innervation is due, at least in part, to "electrical activity." Direct stimulation of soleus muscle with extracellular electrodes repressed the increase of myogenin and MyoD transcripts after denervation by 4- to 3-fold, respectively. In view of these results, it is interesting to speculate that myogenin and/or MyoD may regulate a repertoire of skeletal muscle genes that are down-regulated by electrical activity.