Cell migration and proliferation on homogeneous and non-homogeneous domains : modelling on the scale of individuals and populations

Cell migration is a behaviour critical to many key biological effects, including wound healing, cancerous cell invasion and morphogenesis, the development of an organism from an embryo. However, given that each of these situations is distinctly different and cells are extremely complicated biological objects, interest lies in more basic experiments which seek to remove conflating factors and present a less complex environment within which cell migration can be experimentally examined. These include in vitro studies like the scratch assay or circle migration assay, and ex vivo studies like the colonisation of the hindgut by neural crest cells. The reduced complexity of these experiments also makes them much more enticing as problems to mathematically model, like done here. The primary goal of the mathematical models used in this thesis is to shed light on which cellular behaviours work to generate the travelling waves of invasion observed in these experiments, and to explore how variations in these behaviours can potentially predict differences in this invasive pattern which are experimentally observed when cell types or chemical environment are changed. Relevant literature has already identified the difficulty of distinguishing between these behaviours when using traditional mathematical biology techniques operating on a macroscopic scale, and so here a sophisticated individual-cell-level model, an extension of the Cellular Potts Model (CPM), is been constructed and used to model a scratch assay experiment. This model includes a novel mechanism for dealing with cell proliferations that allowed for the differing properties of quiescent and proliferative cells to be implemented into their behaviour. This model is considered both for its predictive power and used to make comparisons with the travelling waves which result in more traditional macroscopic simulations. These comparisons demonstrate a surprising amount of agreement between the two modelling frameworks, and suggest further novel modifications to the CPM that would allow it to better model cell migration. Considerations of the model’s behaviour are used to argue that the dominant effect governing cell migration (random motility or signal-driven taxis) likely depends on the sort of invasion demonstrated by cells, as easily seen by microscopic photography. Additionally, a scratch assay simulated on a non-homogeneous domain consisting of a ’fast’ and ’slow’ region is also used to further differentiate between these different potential cell motility behaviours. A heterogeneous domain is a novel situation which has not been considered mathematically in this context, nor has it been constructed experimentally to the best of the candidate’s knowledge. Thus this problem serves as a thought experiment used to test the conclusions arising from the simulations on homogeneous domains, and to suggest what might be observed should this non-homogeneous assay situation be experimentally realised. Non-intuitive cell invasion patterns are predicted for diffusely-invading cells which respond to a cell-consumed signal or nutrient, contrasted with rather expected behaviour in the case of random-motility-driven invasion. The potential experimental observation of these behaviours is demonstrated by the individual-cell-level model used in this thesis, which does agree with the PDE model in predicting these unexpected invasion patterns. In the interest of examining such a case of a non-homogeneous domain experimentally, some brief suggestion is made as to how this could be achieved.

Mathematical modelling of soft callus formation in early murine bone repair

Fracture healing is a complicated coupling of many processes. Yet despite the apparent complexity, fracture repair is usually effective. There is, however, no comprehensive mathematical model addressing the multiple interactions of cells, cytokines and oxygen that includes extra-cellular matrix production and that results in the formation of the early stage soft callus. This thesis develops a one dimensional continuum transport model in the context of early fracture healing. Although fracture healing is a complex interplay of many local factors, critical components are identified and used to construct an hypothesis about regulation of the evolution of early callus formation. Multiple cell lines, cellular differentiation, oxygen levels and cytokine concentrations are examined as factors affecting this model of early bone repair. The model presumes diffusive and chemotactic cell migration mechanisms. It is proposed that the initial signalling regime and oxygen availability arising as consequences of bone fracture, are sufficient to determine the quantity and quality of early soft callus formation. Readily available software and purpose written algorithms have been used to obtain numerical solutions representative of various initial conditions. These numerical distributions of cellular populations reflect available histology obtained from murine osteotomies. The behaviour of the numerical system in response to differing initial conditions can be described by alternative in vivo healing pathways. An experimental basis, as illustrated in murine fracture histology, has been utilised to validate the mathematical model outcomes. The model developed in this thesis has potential for future extension, to incorporate processes leading to woven bone deposition, while maintaining the characteristics that regulate early callus formation.

A geometric construction of travelling wave solutions to the Keller–Segel model

We study a version of the Keller–Segel model for bacterial chemotaxis, for which exact travelling wave solutions are explicitly known in the zero attractant diffusion limit. Using geometric singular perturbation theory, we construct travelling wave solutions in the small diffusion case that converge to these exact solutions in the singular limit.

Distinguishing between mechanisms of cell aggregation using pair-correlation functions

Many cell types form clumps or aggregates when cultured in vitro through a variety of mechanisms including rapid cell proliferation, chemotaxis, or direct cell-to-cell contact. In this paper we develop an agent-based model to explore the formation of aggregates in cultures where cells are initially distributed uniformly, at random, on a two-dimensional substrate. Our model includes unbiased random cell motion, together with two mechanisms which can produce cell aggregates: (i) rapid cell proliferation, and (ii) a biased cell motility mechanism where cells can sense other cells within a finite range, and will tend to move towards areas with higher numbers of cells. We then introduce a pair-correlation function which allows us to quantify aspects of the spatial patterns produced by our agent-based model. In particular, these pair-correlation functions are able to detect differences between domains populated uniformly at random (i.e. at the exclusion complete spatial randomness (ECSR) state) and those where the proliferation and biased motion rules have been employed - even when such differences are not obvious to the naked eye. The pair-correlation function can also detect the emergence of a characteristic inter-aggregate distance which occurs when the biased motion mechanism is dominant, and is not observed when cell proliferation is the main mechanism of aggregate formation. This suggests that applying the pair-correlation function to experimental images of cell aggregates may provide information about the mechanism associated with observed aggregates. As a proof of concept, we perform such analysis for images of cancer cell aggregates, which are known to be associated with rapid proliferation. The results of our analysis are consistent with the predictions of the proliferation-based simulations, which supports the potential usefulness of pair correlation functions for providing insight into the mechanisms of aggregate formation.

A finite volume method with linearisation in time for the solution of advection-reaction-diffusion systems

The numerical solution in one space dimension of advection--reaction--diffusion systems with nonlinear source terms may invoke a high computational cost when the presently available methods are used. Numerous examples of finite volume schemes with high order spatial discretisations together with various techniques for the approximation of the advection term can be found in the literature. Almost all such techniques result in a nonlinear system of equations as a consequence of the finite volume discretisation especially when there are nonlinear source terms in the associated partial differential equation models. This work introduces a new technique that avoids having such nonlinear systems of equations generated by the spatial discretisation process when nonlinear source terms in the model equations can be expanded in positive powers of the dependent function of interest. The basis of this method is a new linearisation technique for the temporal integration of the nonlinear source terms as a supplementation of a more typical finite volume method. The resulting linear system of equations is shown to be both accurate and significantly faster than methods that necessitate the use of solvers for nonlinear system of equations.

Epidermal growth factor promotes MDA-MB-231 breast cancer cell migration through a phosphatidylinositol 3'-kinase and phospholipase C-dependent mechanism

Epidermal growth factor receptor (EGFR) levels predict a poor outcome in human breast cancer and are most commonly associated with proliferative effects of epidermal growth factor (EGF), with little emphasis placed on motogenic responses to EGF. We found that MDA-MB-231 human breast cancer cells elicited a potent chemotactic response despite their complete lack of a proliferative response to EGF. Antagonists of EGFR ligation, the EGFR kinase, phosphatidylinositol 3'-kinase, and phospholipase C, but not the mitogen- activated protein kinases (extracellular signal-regulated protein kinase 1 and 2), blocked MDA-MB-231 chemotaxis. These findings suggest that EGF may influence human breast cancer progression via migratory pathways, the signaling for which appears to be dissociated, at least in part, from the proliferative pathways.

Hepatocyte growth factor stimulates invasion across reconstituted basement membranes by a new human small intestinal cell line

Hepatocyte growth factor/scatter factor (HGF/SF) is a protein growth factor whose pleiotropic effects on epithelial cells include the stimulation of motility, mitosis and tubulogenesis. These responses are mediated by the cell surface tyrosine kinase receptor c-met. Because both the cytokine and receptor are found in the gastrointestinal tract, we have studied the effects of HGF/SF on transformed gut epithelial cells which express c-met. Here we describe the response of a new transformed human jejunal epithelioid cell line (HIE-7) to HGF/SF. Morphologically HIE-7 cells are immature. Their epithelial lineage was confirmed by reactivity with the epithelial specific antibodies AE1/AE3, Cam 5.2, Ber-EP4 and anti-EMA and is consistent with their expression of c-met mRNA and protein. In addition, electron microscopic analysis revealed the presence of primitive junctions and rudimentary microvilli, but features of polarization were absent. When grown on reconstituted basement membranes, HIE-7 cells formed closely associated multicellular cord-like structures adjacent to acellular spaces. However, the cells did not mature structurally, form lumen-like structures or express disaccharidase mRNA, even in the presence of recombinant HGF (rHGF). On the other hand, rHGF induced HIE-7 cells to scatter and stimulated their rapid migration in a modified wound assay. To determine whether the motogenic effect caused by rHGF is associated with HIE-7 cell invasiveness across reconstituted basement membranes, a Boyden chamber chemoinvasion assay was performed. rHGF stimulated a 10-fold increase in the number of HIE-7 cells that crossed the basement membrane barrier, while only stimulating a small increase in chemotaxis across a collagen IV matrix, suggesting that the cytokine activates matrix penetration by these cells. rHGF also stimulated the invasion of basement membranes by an undifferentiated rat intestinal cell line (IEC-6) and by two human colon cancer cell lines which are poorly differentiated (DLD-1 and SW 948). In contrast, two moderately well differentiated colon cancer cell lines (Caco-2 and HT-29) did not manifest an invasive response when exposed to rHGF. These results suggest that HGF/SF may play a significant role in the invasive behavior of anaplastic and poorly differentiated gut epithelial tumors.

Invasive activity and chemotactic response to growth factors by Kaposi's sarcoma cells

Kaposi's sarcoma (KS) is a relatively low grade neoplasm, classically occurring in the skin of elderly men. A more virulent and invasive form of Kaposi's sarcoma has been described in patients with acquired immune deficiency syndrome (AIDS). The origin and identification of the tumor cells in these lesions is controversial. Here we have studied the behavior of cells derived from KS lesions in an in vitro assay which measures the ability of cells to invade through a reconstituted basement membrane. In agreement with previous work, KS cells obtained under selective culture conditions were invasive showing activity comparable to that of malignant tumor cells. Normal fibroblasts, smooth muscle cells, and endothelial cells did not demonstrate invasive behavior under the same experimental conditions. To characterize further the nature of the KS cells we tested the chemotactic response of cells from the most invasive line to a variety of growth factors and compared their response to those of fibroblasts, smooth muscle, and endothelial cells. These studies suggest that normal cells respond to a unique repertoire of chemotactic factors. The chemotactic response of the KS cells most closely resembled that of smooth muscle cells and was quite distinct from endothelial cells. These results indicate that the KS-derived cultures contain invasive cells with a smooth muscle cell-like phenotype.

Invasive phenotype of MCF10A cells overexpressing c-Ha-ras and c-erbB-2 oncogenes

Infection with erbB-2 (E) of Ha-ras (H) oncogene-transfected cells has been previously shown to cooperatively induce anchorage-independent growth of the MCF10A human mammary epithelial cell line in vitro, but not to induce nude mouse tumorigenicity. Here we show that oncogene-transformed MCF10A are able to halt in the lungs of nude mice, a sign of organ colonization potential. We have therefore studied the transformants for in vitro migratory and invasive properties known to correlate with the metastatic potential of human mammary carcinoma cells in nude mice. MCF10A transfected with Ha-ras, infected with a recombinant retroviral vector containing the human c-erB-2 proto-oncogene (MCF10A-HE cells), show a higher invasive index than either the single transfectant (MCF10A-H) or MCF10A-erB-2(MCF10A-E) cells in the Boyden chamber chemotaxis and chemoinvasion assays. The MCF10A-HE cells also adopted an invasive stellate growth pattern when plated or embedded in Matrigel, in contrast to the spherical colonies formed by the single transformants MCF10A-H, MCF10A-E, and the parental cells. Dot-blot analysis of gelatinase A and TIMP-2 mRNA levels revealed increasing gelatinase A mRNA levels (HE > E > H > MCF10A) and reduced TIMP-2 expression in both single and double transformants. Furthermore, MCF10A-HE cells show more MMP-2 activity than parental MCF10A cells or the single transformants. CD44 analysis revealed differential isoform banding for the MCF10A-HE cells compared to parental cells, MCF10A-H and MCF10A-E, accompanied by increased binding of hyaluronan by the double transformants. Our results indicate that erB-2 and Ha-ras co-expression can induce a more aggressive phenotype in vitro, representative of the malignancy of mammary carcinomas.

Meta-analysis of microarray data identifies GAS6 expression as an independent predictor of poor survival in ovarian cancer

Seeking new biomarkers for epithelial ovarian cancer, the fifth most common cause of death from all cancers in women and the leading cause of death from gynaecological malignancies, we performed a meta-analysis of three independent studies and compared the results in regard to clinicopathological parameters. This analysis revealed that GAS6 was highly expressed in ovarian cancer and therefore was selected as our candidate of choice. GAS6 encodes a secreted protein involved in physiological processes including cell proliferation, chemotaxis, and cell survival. We performed immunohistochemistry on various ovarian cancer tissues and found that GAS6 expression was elevated in tumour tissue samples compared to healthy control samples (P < 0.0001). In addition, GAS6 expression was also higher in tumours from patients with residual disease compared to those without. Our data propose GAS6 as an independent predictor of poor survival, suggesting GAS6, both on the mRNA and on the protein level, as a potential biomarker for ovarian cancer. In clinical practice, the staining of a tumour biopsy for GAS6 may be useful to assess cancer prognosis and/or to monitor disease progression.

Conformational Spread as a Mechanism for Cooperativity in the Bacterial Flagellar Switch

The bacterial flagellar switch that controls the direction of flagellar rotation during Chemotaxis has a highly cooperative response. This has previously been understood in terms of the classic two-state, concerted model of allosteric regulation. Here, we used high-resolution optical microscopy to observe switching of single motors and uncover the stochastic multistate nature of the switch. Our observations are in detailed quantitative agreement with a recent general model of allosteric cooperativity that exhibits conformational spread-the stochastic growth and shrinkage of domains of adjacent subunits sharing a particular conformational state. We expect that conformational spread will be important in explaining cooperativity in other large signaling complexes.

Numerical computation of an Evans function for travelling waves

We demonstrate a geometrically inspired technique for computing Evans functions for the linearised operators about travelling waves. Using the examples of the F-KPP equation and a Keller–Segel model of bacterial chemotaxis, we produce an Evans function which is computable through several orders of magnitude in the spectral parameter and show how such a function can naturally be extended into the continuous spectrum. In both examples, we use this function to numerically verify the absence of eigenvalues in a large region of the right half of the spectral plane. We also include a new proof of spectral stability in the appropriate weighted space of travelling waves of speed c≥sqrt(2δ) in the F-KPP equation.

Infection and cellular defense dynamics in a novel 17beta-estradiol murine model of chronic human group B streptococcus genital tract colonization reveal a role for hemolysin in persistence and neutrophil accumulation

Genital tract carriage of group B streptococcus (GBS) is prevalent among adult women; however, the dynamics of chronic GBS genital tract carriage, including how GBS persists in this immunologically active host niche long term, are not well defined. To our knowledge, in this study, we report the first animal model of chronic GBS genital tract colonization using female mice synchronized into estrus by delivery of 17β-estradiol prior to intravaginal challenge with wild-type GBS 874391. Cervicovaginal swabs, which were used to measure bacterial persistence, showed that GBS colonized the vaginal mucosa of mice at high numbers (106–107 CFU/swab) for at least 90 d. Cellular and histological analyses showed that chronic GBS colonization of the murine genital tract caused significant lymphocyte and PMN cell infiltrates, which were localized to the vaginal mucosal surface. Long-term colonization was independent of regular hormone cycling. Immunological analyses of 23 soluble proteins related to chemotaxis and inflammation showed that the host response to GBS in the genital tract comprised markers of innate immune activation including cytokines such as GM-CSF and TNF-α. A nonhemolytic isogenic mutant of GBS 874391, Δcyle9, was impaired for colonization and was associated with amplified local PMN responses. Induction of DNA neutrophil extracellular traps, which was observed in GBS-infected human PMNs in vitro in a hemolysin-dependent manner, appeared to be part of this response. Overall, this study defines key infection dynamics in a novel murine model of chronic GBS genital tract colonization and establishes previously unknown cellular and soluble defense responses to GBS in the female genital tract.

Synthetic peptides as chemoattractants for bull spermatozoa structure activity correlations

The ability of various synthetic peptide analogs of. Formyl-Met-Leu-Phe to induce chemotaxis in bull sperm is compared using an inverted capillary assay. The formyl group is essential for chemotactic activity and corresponding t-butyloxycarbonyl tripeptides are inactive. Sequence analogs, Formyl-Met-Phe-Leu, Formyl-Leu-Met-Phe and Formyl-Leu-Phe-Met are active. Replacement of Met and Leu by Pro does not diminish activity. Formyl-Met-Leu-Phe-NH2 is active suggesting that electrostatic interactions involving the carboxyl group may be unimportant in receptor interactions. The studies establish the importance of an amino terminal formyl group and a sequence of at least three hydrophobic residues, for inducing sperm chemotaxis.

Bimodal distribution of motility and cell fate in Dictyostelium discoideum

Pre-starvation amoebae of Dictyostelium discoideum exhibit random movements. Starved cells aggregate by directed movements (chemotaxis) towards cyclic AMP and differentiate into live spores or dead stalk cells. Many differences between presumptive spore and stalk cells precede differentiation. We have examined whether cell motility-related factors are also among them. Cell speeds and localisation of motility-related signalling molecules were monitored by live cell imaging and immunostaining (a) in nutrient medium during growth, (b) immediately following transfer to starvation medium and (c) in nutrient medium that was re-introduced after a brief period of starvation. Cells moved randomly under all three conditions but mean speeds increased following transfer from nutrient medium to starvation medium; the transition occurred within 15 min. The distribution of speeds in starvation medium was bimodal: about 20% of the cells moved significantly faster than the remaining 80%. The motility-related molecules F-actin, PTEN and PI3 kinase were distributed differently in slow and fast cells. Among starved cells, the calcium content of slower cells was lower than that of the faster cells. All differences reverted within 15 min after restoration of the nutrient medium. The slow/fast distinction was missing in Polysphondylium pallidum, a cellular slime mould that lacks the presumptive stalk and spore cell classes, and in the trishanku (triA(center dot)) mutant of D. discoideum, in which the classes exist but are unstable. The transition from growth to starvation triggers a spontaneous and reversible switch in the distribution of D. discoideum cell speeds. Cells whose calcium content is relatively low (known to be presumptive spore cells) move slower than those whose calcium levels are higher (known to be presumptive stalk cells). Slow and fast cells show different distributions of motility-related proteins. The switch is indicative of a bistable mechanism underlying cell motility.