Mathematical models of developmental vascular remodelling: A review.

Over the past 40 years, there has been a strong focus on the development of mathematical models of angiogenesis, while developmental remodelling has received little such attention from the mathematical community. Sprouting angiogenesis can be seen as a very crude way of laying out a primitive vessel network (the raw material), while remodelling (understood as pruning of redundant vessels, diameter control, and the establishment of vessel identity and hierarchy) is the key to turning that primitive network into a functional network. This multiscale problem is of prime importance in the development of a functional vasculature. In addition, defective remodelling (either during developmental remodelling or due to a reactivation of the remodelling programme caused by an injury) is associated with a significant number of diseases. In this review, we discuss existing mathematical models of developmental remodelling and explore the important contributions that these models have made to the field of vascular development. These mathematical models are effectively used to investigate and predict vascular development and are able to reproduce experimentally observable results. Moreover, these models provide a useful means of hypothesis generation and can explain the underlying mechanisms driving the observed structural and functional network development. However, developmental vascular remodelling is still a relatively new area in mathematical biology, and many biological questions remain unanswered. In this review, we present the existing modelling paradigms and define the key challenges for the field.

Choice of spatial discretisation influences the progression of viral infection within multicellular tissues.

There has been an increasing recognition of the utility of models of the spatial dynamics of viral spread within tissues. Multicellular models, where cells are represented as discrete regions of space coupled to a virus density surface, are a popular approach to capture these dynamics. Conventionally, such models are simulated by discretising the viral surface and depending on the rate of viral diffusion and other considerations, a finer or coarser discretisation may be used. The impact that this choice may have on the behaviour of the system has not been studied. Here we demonstrate that under realistic parameter regimes - where viral diffusion is small enough to support the formation of familiar ring-shaped infection plaques - the choice of spatial discretisation of the viral surface can qualitatively change key model outcomes including the time scale of infection. Importantly, we show that the choice between implementing viral spread as a cell-scale process, or as a high-resolution converged PDE can generate distinct model outcomes, which raises important conceptual questions about the strength of assumptions underpinning the spatial structure of the model. We investigate the mechanisms driving these discretisation artefacts, the impacts they may have on model predictions, and provide guidance on the design and implementation of spatial and especially multicellular models of viral dynamics. We obtain our results using the simplest TIV construct for the viral dynamics, and therefore anticipate that the important effects we describe will also influence model predictions in more complex models of virus-cell-immune system interactions. This analysis will aid in the construction of models for robust and biologically realistic modelling and inference.

Multiscale modelling of desquamation in the interfollicular epidermis.

Maintenance of epidermal thickness is critical to the barrier function of the skin. Decreased tissue thickness, specifically in the stratum corneum (the outermost layer of the tissue), causes discomfort and inflammation, and is related to several severe diseases of the tissue. In order to maintain both stratum corneum thickness and overall tissue thickness it is necessary for the system to balance cell proliferation and cell loss. Cell proliferation in the epidermis occurs in the basal layer and causes constant upwards movement in the tissue. Cell loss occurs when dead cells at the top of the tissue are lost to the environment through a process called desquamation. Desquamation is thought to occur through a gradual reduction in adhesion between cells, due to the cleaving of adhesion proteins by enzymes, in the stratum corneum. In this paper we will investigate combining a (mass action) subcellular model of desquamation with a three dimensional (cell centre based) multicellular model of the interfollicular epidermis to better understand maintenance of epidermal thickness. Specifically, our aim is to determine if a hypothesised biological model for the degradation of cell-cell adhesion, from the literature, is sufficient to maintain a steady state tissue thickness. These investigations show the model is able to provide a consistent rate of cell loss in the multicellular model. This loss balances proliferation, and hence maintains a homeostatic tissue thickness. Moreover, we find that multiple proliferative cell populations in the basal layer can be represented by a single proliferative cell population, simplifying investigations with this model. The model is used to investigate a disorder (Netherton Syndrome) which disrupts desquamation. The model shows how biochemical changes can cause disruptions to the tissue, resulting in a reduced tissue thickness and consequently diminishing the protective role of the tissue. A hypothetical treatment result is also investigated: we compare the cases of a partially effective homogeneous treatment (where all cells partially recover) and a totally effective heterogeneous treatment (in which a proportion of the cells totally recover) with the aim to determine the difference in the response of the tissue to these different scenarios. Results show an increased benefit to corneum thickness from the heterogeneous treatment over the homogeneous treatment.

Free and Interfacial Boundaries in Individual-Based Models of Multicellular Biological systems.

Coordination of cell behaviour is key to a myriad of biological processes including tissue morphogenesis, wound healing, and tumour growth. As such, individual-based computational models, which explicitly describe inter-cellular interactions, are commonly used to model collective cell dynamics. However, when using individual-based models, it is unclear how descriptions of cell boundaries affect overall population dynamics. In order to investigate this we define three cell boundary descriptions of varying complexities for each of three widely used off-lattice individual-based models: overlapping spheres, Voronoi tessellation, and vertex models. We apply our models to multiple biological scenarios to investigate how cell boundary description can influence tissue-scale behaviour. We find that the Voronoi tessellation model is most sensitive to changes in the cell boundary description with basic models being inappropriate in many cases. The timescale of tissue evolution when using an overlapping spheres model is coupled to the boundary description. The vertex model is demonstrated to be the most stable to changes in boundary description, though still exhibits timescale sensitivity. When using individual-based computational models one should carefully consider how cell boundaries are defined. To inform future work, we provide an exploration of common individual-based models and cell boundary descriptions in frequently studied biological scenarios and discuss their benefits and disadvantages.

Maintaining the proliferative cell niche in multicellular models of epithelia.

The maintenance of the proliferative cell niche is critical to epithelial tissue morphology and function. In this paper we investigate how current modelling methods can result in the erroneous loss of proliferative cells from the proliferative cell niche. Using an established model of the inter-follicular epidermis we find there is a limit to the proliferative cell densities that can be maintained in the basal layer (the niche) if we do not include additional mechanisms to stop the loss of proliferative cells from the niche. We suggest a new methodology that enables maintenance of a desired homeostatic population of proliferative cells in the niche: a rotational force is applied to the two daughter cells during the mitotic phase of division to enforce a particular division direction. We demonstrate that this new methodology achieves this goal. This methodology reflects the regulation of the orientation of cell division.

A mathematical model of cell fate selection on a dynamic tissue.

Multicellular tissues are the building blocks of many biological systems and organs. These tissues are not static, but dynamically change over time. Even if the overall structure remains the same there is a turnover of cells within the tissue. This dynamic homeostasis is maintaned by numerous governing mechanisms which are finely tuned in such a way that the tissue remains in a homeostatic state, even across large timescales. Some of these governing mechanisms include cell motion, and cell fate selection through inter cellular signalling. However, it is not yet clear how to link these two processes, or how they may affect one another across the tissue. In this paper, we present a multicellular, multiscale model, which brings together the two phenomena of cell motility, and inter cellular signalling, to describe cell fate selection on a dynamic tissue. We find that the affinity for cellular signalling to occur greatly influences a cells ability to differentiate. We also find that our results support claims that cell differentiation is a finely tuned process within dynamic tissues at homeostasis, with excessive cell turnover rates leading to unhealthy (undifferentiated and unpatterned) tissues.

A modular framework for multiscale, multicellular, spatiotemporal modeling of acute primary viral infection and immune response in epithelial tissues and its application to drug therapy timing and effectiveness.

Simulations of tissue-specific effects of primary acute viral infections like COVID-19 are essential for understanding disease outcomes and optimizing therapies. Such simulations need to support continuous updating in response to rapid advances in understanding of infection mechanisms, and parallel development of components by multiple groups. We present an open-source platform for multiscale spatiotemporal simulation of an epithelial tissue, viral infection, cellular immune response and tissue damage, specifically designed to be modular and extensible to support continuous updating and parallel development. The base simulation of a simplified patch of epithelial tissue and immune response exhibits distinct patterns of infection dynamics from widespread infection, to recurrence, to clearance. Slower viral internalization and faster immune-cell recruitment slow infection and promote containment. Because antiviral drugs can have side effects and show reduced clinical effectiveness when given later during infection, we studied the effects on progression of treatment potency and time-of-first treatment after infection. In simulations, even a low potency therapy with a drug which reduces the replication rate of viral RNA greatly decreases the total tissue damage and virus burden when given near the beginning of infection. Many combinations of dosage and treatment time lead to stochastic outcomes, with some simulation replicas showing clearance or control (treatment success), while others show rapid infection of all epithelial cells (treatment failure). Thus, while a high potency therapy usually is less effective when given later, treatments at late times are occasionally effective. We illustrate how to extend the platform to model specific virus types (e.g., hepatitis C) and add additional cellular mechanisms (tissue recovery and variable cell susceptibility to infection), using our software modules and publicly-available software repository.

Modelling Cellular Interactions and Dynamics During Kidney Morphogenesis.

Kidney disease and renal disorders account for a significant proportion of health complications in mid-late adulthood worldwide. Many renal deficiencies are due to improper formation of the kidneys before birth, which are caused by disorders in the developmental process that arise from genetic and/or environmental factors. Mathematical modelling can help build on experimental knowledge to increase our understanding of the complexities of kidney organogenesis. In this paper, we present a discrete cell-based model of kidney development. Specifically, we model the tip of the developing ureteric tree to investigate the behaviours of cap mesenchyme cells which are required to sustain ureteric tip growth. We find that spatial regulation of the differentiation of cap mesenchyme cells through cellular signalling is sufficient to ensure robust ureteric tip development. Additionally, we find that increased adhesion interactions between cap mesenchyme cells and the ureteric tip surface can lead to a more stable tip-cap unit. Our analysis of the various processes on this scale highlights essential components for healthy kidney growth and provides insight into mechanisms to be studied further in order to replicate the process in vitro.

Modelling the effect of subcellular mutations on the migration of cells in the colorectal crypt.

BACKGROUND: Many cancers arise from mutations in cells within epithelial tissues. Mutations manifesting at the subcellular level influence the structure and function of the tissue resulting in cancer. Previous work has proposed how cell level properties can lead to mutant cell invasion, but has not incorporated detailed subcellular modelling RESULTS: We present a framework that allows the straightforward integration and simulation of SBML representations of subcellular dynamics within multiscale models of epithelial tissues. This allows us to investigate the effect of mutations in subcellular pathways on the migration of cells within the colorectal crypt. Using multiple models we find that mutations in APC, a key component in the Wnt signalling pathway, can bias neutral drift and can also cause downward invasion of mutant cells in the crypt. CONCLUSIONS: Our framework allows us to investigate how subcellular mutations, i.e. knockouts and knockdowns, affect cell-level properties and the resultant migration of cells within epithelial tissues. In the context of the colorectal crypt, we see that mutations in APC can lead directly to mutant cell invasion.

Cell morphology drives spatial patterning in microbial communities.

The clearest phenotypic characteristic of microbial cells is their shape, but we do not understand how cell shape affects the dense communities, known as biofilms, where many microbes live. Here, we use individual-based modeling to systematically vary cell shape and study its impact in simulated communities. We compete cells with different cell morphologies under a range of conditions and ask how shape affects the patterning and evolutionary fitness of cells within a community. Our models predict that cell shape will strongly influence the fate of a cell lineage: we describe a mechanism through which coccal (round) cells rise to the upper surface of a community, leading to a strong spatial structuring that can be critical for fitness. We test our predictions experimentally using strains of Escherichia coli that grow at a similar rate but differ in cell shape due to single amino acid changes in the actin homolog MreB. As predicted by our model, cell types strongly sort by shape, with round cells at the top of the colony and rod cells dominating the basal surface and edges. Our work suggests that cell morphology has a strong impact within microbial communities and may offer new ways to engineer the structure of synthetic communities.

Stochastic clonal expansion of "superstars" enhances the reserve capacity of enteric nervous system precursor cells.

We quantified cell population increase in the quail embryo enteric nervous system (ENS) from E2.5 (about 1500 cells) to E12 (about 8 million cells). We then probed ENS proliferative capacity by grafting to the chorio-allantoic membrane large (600 cells) and small (40 cells) populations of enteric neural crest (ENC) cells with aneural gut. This demonstrated that ENC cells show an extremely high capacity to regulate their proliferation while forming the ENS. Previous mathematical models and clonal label experiments revealed that a few dominant ENS "superstar" cell clones emerge but most clones are small. The model implied that "superstars" arise stochastically, but the same outcome could arise if "superstars" were pre-determined. We investigated these two modes mathematically and by grafting experiments with large and small numbers of ENCs, each including one EGFP-labelled ENC cell. The stochastic model predicts that the frequency of "superstar" detection increases as the ENC population decreases, the pre-determined model does not. Experimentally, as predicted by the stochastic model, the frequency of "superstar" detection increased with small ENC cell number. We conclude that ENS "superstar" clones achieve this status stochastically. Clonal dominance implies that clonal diversity is greatly reduced and in this case, somatic mutations may affect the phenotype. We suggest that somatic mutations coupled with loss of clonal diversity may contribute to variable penetrance and expressivity in individuals with genetically identical ENS pathologies.

Interkinetic nuclear migration and basal tethering facilitates post-mitotic daughter separation in intestinal organoids.

Homeostasis of renewing tissues requires balanced proliferation, differentiation and movement. This is particularly important in the intestinal epithelium where lineage tracing suggests that stochastic differentiation choices are intricately coupled to the position of a cell relative to a niche. To determine how position is achieved, we followed proliferating cells in intestinal organoids and discovered that the behaviour of mitotic sisters predicted long-term positioning. We found that, normally, 70% of sisters remain neighbours, while 30% lose contact and separate after cytokinesis. These post-mitotic placements predict longer term differences in positions assumed by sisters: adjacent sisters reach similar positions over time; in a pair of separating sisters, one remains close to its birthplace while the other is displaced upward. Computationally modelling crypt dynamics confirmed that post-mitotic separation leads to sisters reaching different compartments. We show that interkinetic nuclear migration, cell size and asymmetric tethering by a process extending from the basal side of cells contribute to separations. These processes are altered in adenomatous polyposis coli (Apc) mutant epithelia where separation is lost. We conclude that post-mitotic placement contributes to stochastic niche exit and, when defective, supports the clonal expansion of Apc mutant cells.

Paneth Cell-Rich Regions Separated by a Cluster of Lgr5+ Cells Initiate Crypt Fission in the Intestinal Stem Cell Niche.

The crypts of the intestinal epithelium house the stem cells that ensure the continual renewal of the epithelial cells that line the intestinal tract. Crypt number increases by a process called crypt fission, the division of a single crypt into two daughter crypts. Fission drives normal tissue growth and maintenance. Correspondingly, it becomes less frequent in adulthood. Importantly, fission is reactivated to drive adenoma growth. The mechanisms governing fission are poorly understood. However, only by knowing how normal fission operates can cancer-associated changes be elucidated. We studied normal fission in tissue in three dimensions using high-resolution imaging and used intestinal organoids to identify underlying mechanisms. We discovered that both the number and relative position of Paneth cells and Lgr5+ cells are important for fission. Furthermore, the higher stiffness and increased adhesion of Paneth cells are involved in determining the site of fission. Formation of a cluster of Lgr5+ cells between at least two Paneth-cell-rich domains establishes the site for the upward invagination that initiates fission.

Reversible host cell remodeling underpins deformability changes in malaria parasite sexual blood stages.

The sexual blood stage of the human malaria parasite Plasmodium falciparum undergoes remarkable biophysical changes as it prepares for transmission to mosquitoes. During maturation, midstage gametocytes show low deformability and sequester in the bone marrow and spleen cords, thus avoiding clearance during passage through splenic sinuses. Mature gametocytes exhibit increased deformability and reappear in the peripheral circulation, allowing uptake by mosquitoes. Here we define the reversible changes in erythrocyte membrane organization that underpin this biomechanical transformation. Atomic force microscopy reveals that the length of the spectrin cross-members and the size of the skeletal meshwork increase in developing gametocytes, then decrease in mature-stage gametocytes. These changes are accompanied by relocation of actin from the erythrocyte membrane to the Maurer's clefts. Fluorescence recovery after photobleaching reveals reversible changes in the level of coupling between the membrane skeleton and the plasma membrane. Treatment of midstage gametocytes with cytochalasin D decreases the vertical coupling and increases their filterability. A computationally efficient coarse-grained model of the erythrocyte membrane reveals that restructuring and constraining the spectrin meshwork can fully account for the observed changes in deformability.

Mechano-logical model of C. elegans germ line suggests feedback on the cell cycle.

The Caenorhabditis elegans germ line is an outstanding model system in which to study the control of cell division and differentiation. Although many of the molecules that regulate germ cell proliferation and fate decisions have been identified, how these signals interact with cellular dynamics and physical forces within the gonad remains poorly understood. We therefore developed a dynamic, 3D in silico model of the C. elegans germ line, incorporating both the mechanical interactions between cells and the decision-making processes within cells. Our model successfully reproduces key features of the germ line during development and adulthood, including a reasonable ovulation rate, correct sperm count, and appropriate organization of the germ line into stably maintained zones. The model highlights a previously overlooked way in which germ cell pressure may influence gonadogenesis, and also predicts that adult germ cells might be subject to mechanical feedback on the cell cycle akin to contact inhibition. We provide experimental data consistent with the latter hypothesis. Finally, we present cell trajectories and ancestry recorded over the course of a simulation. The novel approaches and software described here link mechanics and cellular decision-making, and are applicable to modeling other developmental and stem cell systems.

Comparing individual-based approaches to modelling the self-organization of multicellular tissues.

The coordinated behaviour of populations of cells plays a central role in tissue growth and renewal. Cells react to their microenvironment by modulating processes such as movement, growth and proliferation, and signalling. Alongside experimental studies, computational models offer a useful means by which to investigate these processes. To this end a variety of cell-based modelling approaches have been developed, ranging from lattice-based cellular automata to lattice-free models that treat cells as point-like particles or extended shapes. However, it remains unclear how these approaches compare when applied to the same biological problem, and what differences in behaviour are due to different model assumptions and abstractions. Here, we exploit the availability of an implementation of five popular cell-based modelling approaches within a consistent computational framework, Chaste (http://www.cs.ox.ac.uk/chaste). This framework allows one to easily change constitutive assumptions within these models. In each case we provide full details of all technical aspects of our model implementations. We compare model implementations using four case studies, chosen to reflect the key cellular processes of proliferation, adhesion, and short- and long-range signalling. These case studies demonstrate the applicability of each model and provide a guide for model usage.

The role of the Hes1 crosstalk hub in Notch-Wnt interactions of the intestinal crypt.

The Notch pathway plays a vital role in determining whether cells in the intestinal epithelium adopt a secretory or an absorptive phenotype. Cell fate specification is coordinated via Notch's interaction with the canonical Wnt pathway. Here, we propose a new mathematical model of the Notch and Wnt pathways, in which the Hes1 promoter acts as a hub for pathway crosstalk. Computational simulations of the model can assist in understanding how healthy intestinal tissue is maintained, and predict the likely consequences of biochemical knockouts upon cell fate selection processes. Chemical reaction network theory (CRNT) is a powerful, generalised framework which assesses the capacity of our model for monostability or multistability, by analysing properties of the underlying network structure without recourse to specific parameter values or functional forms for reaction rates. CRNT highlights the role of ß-catenin in stabilising the Notch pathway and damping oscillations, demonstrating that Wnt-mediated actions on the Hes1 promoter can induce dynamic transitions in the Notch system, from multistability to monostability. Time-dependent model simulations of cell pairs reveal the stabilising influence of Wnt upon the Notch pathway, in which ß-catenin- and Dsh-mediated action on the Hes1 promoter are key in shaping the subcellular dynamics. Where Notch-mediated transcription of Hes1 dominates, there is Notch oscillation and maintenance of fate flexibility; Wnt-mediated transcription of Hes1 favours bistability akin to cell fate selection. Cells could therefore regulate the proportion of Wnt- and Notch-mediated control of the Hes1 promoter to coordinate the timing of cell fate selection as they migrate through the intestinal epithelium and are subject to reduced Wnt stimuli. Furthermore, mutant cells characterised by hyperstimulation of the Wnt pathway may, through coupling with Notch, invert cell fate in neighbouring healthy cells, enabling an aberrant cell to maintain its neighbours in mitotically active states.

A Multicellular Model of Intestinal Crypt Buckling and Fission.

Crypt fission is an in vivo tissue deformation process that is involved in both intestinal homeostasis and colorectal tumourigenesis. Despite its importance, the mechanics underlying crypt fission are currently poorly understood. Recent experimental development of organoids, organ-like buds cultured from crypt stem cells in vitro, has shown promise in shedding light on crypt fission. Drawing inspiration from observations of organoid growth and fission in vivo, we develop a computational model of a deformable epithelial tissue layer. Results from in silico experiments show the stiffness of cells and the proportions of cell subpopulations affect the nature of deformation in the epithelial layer. In particular, we find that increasing the proportion of stiffer cells in the layer increases the likelihood of crypt fission occurring. This is in agreement with and helps explain recent experimental work.

How computational models contribute to our understanding of the germ line.

Computational models are an invaluable tool in modern biology. They provide a framework within which to summarize existing knowledge, enable competing hypotheses to be compared qualitatively and quantitatively, and to facilitate the interpretation of complex data. Moreover, models allow questions to be investigated that are difficult to approach experimentally. Theories can be tested in context, identifying the gaps in our understanding and potentially leading to new hypotheses. Models can be developed on a variety of scales and with different levels of mechanistic detail, depending on the available data, the biological questions of interest, and the available mathematical and computational tools. The goal of this review is to provide a broad picture of how modeling has been applied to reproductive biology. Specifically, we look at four uses of modeling: (i) comparing hypotheses; (ii) interpreting data; (iii) exploring experimentally challenging questions; and (iv) hypothesis evaluation and generation. We present examples of each of these applications in reproductive biology, drawing from a range of organisms-including Drosophila, Caenorhabditis elegans, mouse, and humans. We aim to describe the data and techniques used to construct each model, and to highlight the benefits of modeling to the field, as complementary to experimental work. Mol. Reprod. Dev. 83: 944-957, 2016 © 2016 Wiley Periodicals, Inc.

The free energy landscape of dimerization of a membrane protein, NanC.

Membrane proteins are frequently present in crowded environments, which favour lateral association and, on occasions, two-dimensional crystallization. To better understand the non-specific lateral association of a membrane protein we have characterized the free energy landscape for the dimerization of a bacterial outer membrane protein, NanC, in a phospholipid bilayer membrane. NanC is a member of the KdgM-family of bacterial outer membrane proteins and is responsible for sialic acid transport in E. coli. Umbrella sampling and coarse-grained molecular dynamics were employed to calculate the potentials of mean force (PMF) for a variety of restrained relative orientations of two NanC proteins as the separation of their centres of mass was varied. We found the free energy of dimerization for NanC to be in the range of -66 kJ mol(-1) to -45 kJ mol(-1). Differences in the depths of the PMFs for the various orientations are related to the shape of the proteins. This was quantified by calculating the lipid-inaccessible buried surface area of the proteins in the region around the minimum of each PMF. The depth of the potential well of the PMF was shown to depend approximately linearly on the buried surface area. We were able to resolve local minima in the restrained PMFs that would not be revealed using conventional umbrella sampling. In particular, these features reflected the local organization of the intervening lipids between the two interacting proteins. Through a comparison with the distribution of lipids around a single freely-diffusing NanC, we were able to predict the location of these restrained local minima for the orientational configuration in which they were most pronounced. Our ability to make this prediction highlights the important role that lipid organization plays in the association of two NanCs in a bilayer.