Computational models reveal a passive mechanism for cell migration in the crypt

An adaptive numerical method for multi-cellular simulations of tissue development and maintenance

In recent years, multi-cellular models, where cells are represented as individual interacting entities, are becoming ever popular. This has led to a proliferation of novel methods and simulation tools. The first aim of this paper is to review the numerical methods utilised by multi-cellular modelling tools and to demonstrate which numerical methods are appropriate for simulations of tissue and organ development, maintenance, and disease. The second aim is to introduce an adaptive time-stepping algorithm and to demonstrate it's efficiency and accuracy. We focus on off-lattice, mechanics based, models where cell movement is defined by a series of first order ordinary differential equations, derived by assuming over-damped motion and balancing forces. We see that many numerical methods have been used, ranging from simple Forward Euler approaches through to higher order single-step methods like Runge-Kutta 4 and multi-step methods like Adams-Bashforth 2. Through a series of exemplar multi-cellular simulations, we see that if: care is taken to have events (births deaths and re-meshing/re-arrangements) occur on common time-steps; and boundaries are imposed on all sub-steps of numerical methods or implemented using forces, then all numerical methods can converge with the correct order. We introduce an adaptive time-stepping method and demonstrate that the best compromise between L∞ error and run-time is to use Runge-Kutta 4 with an increased time-step and moderate adaptivity. We see that a judicious choice of numerical method can speed the simulation up by a factor of 10-60 from the Forward Euler methods seen in Osborne et al. (2017), and a further speed up by a factor of 4 can be achieved by using an adaptive time-step.

The ECM and tissue architecture are major determinants of early invasion mediated by E-cadherin dysfunction

Germline mutations of E-cadherin cause Hereditary Diffuse Gastric Cancer (HDGC), a highly invasive cancer syndrome characterised by the occurrence of diffuse-type gastric carcinoma and lobular breast cancer. In this disease, E-cadherin-defective cells are detected invading the adjacent stroma since very early stages. Although E-cadherin loss is well established as a triggering event, other determinants of the invasive process persist largely unknown. Herein, we develop an experimental strategy that comprises in vitro extrusion assays using E-cadherin mutants associated to HDGC, as well as mathematical models epitomising epithelial dynamics and its interaction with the extracellular matrix (ECM). In vitro, we verify that E-cadherin dysfunctional cells detach from the epithelial monolayer and extrude basally into the ECM. Through phase-field modelling we demonstrate that, aside from loss of cell-cell adhesion, increased ECM attachment further raises basal extrusion efficiency. Importantly, by combining phase-field and vertex model simulations, we show that the cylindrical structure of gastric glands strongly promotes the cell's invasive ability. Moreover, we validate our findings using a dissipative particle dynamics simulation of epithelial extrusion. Overall, we provide the first evidence that cancer cell invasion is the outcome of defective cell-cell linkages, abnormal interplay with the ECM, and a favourable 3D tissue structure.

Modelling realistic 3D deformations of simple epithelia in dynamic homeostasis

The maintenance of tissue and organ structures during dynamic homeostasis is often not well understood. In order for a system to be stable, cell renewal, cell migration and cell death must be finely balanced. Moreover, a tissue's shape must remain relatively unchanged. Simple epithelial tissues occur in various structures throughout the body, such as the endothelium, mesothelium, linings of the lungs, saliva and thyroid glands, and gastrointestinal tract. Despite the prevalence of simple epithelial tissues, there are few models which accurately describe how these tissues maintain a stable structure. Here, we present a novel, 3D, deformable, multilayer, cell-centre model of a simple epithelium. Cell movement is governed by the minimisation of a bending potential across the epithelium, cell-cell adhesion, and viscous effects. We show that the model is capable of maintaining a consistent tissue structure while undergoing self renewal. We also demonstrate the model's robustness under tissue renewal, cell migration and cell removal. The model presented here is a valuable advancement towards the modelling of tissues and organs with complex and generalised structures.

ChemChaste: Simulating spatially inhomogeneous biochemical reaction-diffusion systems for modeling cell-environment feedbacks

BACKGROUND

Spatial organization plays an important role in the function of many biological systems, from cell fate specification in animal development to multistep metabolic conversions in microbial communities. The study of such systems benefits from the use of spatially explicit computational models that combine a discrete description of cells with a continuum description of one or more chemicals diffusing within a surrounding bulk medium. These models allow the in silico testing and refinement of mechanistic hypotheses. However, most existing models of this type do not account for concurrent bulk and intracellular biochemical reactions and their possible coupling.

CONCLUSIONS

Here, we describe ChemChaste, an extension for the open-source C++ computational biology library Chaste. ChemChaste enables the spatial simulation of both multicellular and bulk biochemistry by expanding on Chaste's existing capabilities. In particular, ChemChaste enables (i) simulation of an arbitrary number of spatially diffusing chemicals, (ii) spatially heterogeneous chemical diffusion coefficients, and (iii) inclusion of both bulk and intracellular biochemical reactions and their coupling. ChemChaste also introduces a file-based interface that allows users to define the parameters relating to these functional features without the need to interact directly with Chaste's core C++ code. We describe ChemChaste and demonstrate its functionality using a selection of chemical and biochemical exemplars, with a focus on demonstrating increased ability in modeling bulk chemical reactions and their coupling with intracellular reactions.

AVAILABILITY AND IMPLEMENTATION

ChemChaste version 1.0 is a free, open-source C++ library, available via GitHub at https://github.com/OSS-Lab/ChemChaste under the BSD license, on the Zenodo archive at zendodo doi, as well as on BioTools (biotools:chemchaste) and SciCrunch (RRID:SCR022208) databases.

A rigid body framework for multicellular modeling

Off-lattice models are a well-established approach in multicellular modeling, where cells are represented as points that are free to move in space. The representation of cells as point objects is useful in a wide range of settings, particularly when large populations are involved; however, a purely point-based representation is not naturally equipped to deal with objects that have length, such as cell boundaries or external membranes. Here we introduce an off-lattice modeling framework that exploits rigid body mechanics to represent objects using a collection of conjoined one-dimensional edges in a viscosity-dominated system. This framework can be used to represent cells as free moving polygons, to allow epithelial layers to smoothly interact with themselves, to model rod-shaped cells such as bacteria and to robustly represent membranes. We demonstrate that this approach offers solutions to the problems that limit the scope of current off-lattice multicellular models.

Intestinal villus structure contributes to even shedding of epithelial cells

In the intestinal epithelium, proliferated epithelial cells ascend the crypts and villi and shed at the villus tips into the gut lumen. In this study, we theoretically investigate the roles of the villi on cell turnover. We present a stochastic model that focuses on the duration over which cells migrate the shortest paths between the crypt orifices and the villus tips, where shedding cells are randomly chosen from among those older than the shortest-path cell migration times. By extending the length of the shortest path to delay cell shedding, the finger-like shape of the villus would tightly regulate shedding-cell ages compared with flat surfaces and shorter projections; the villus allows epithelial cells to shed at around the same age, which limits them from shedding early or staying in the epithelium for long periods. Computational simulations of cell dynamics agreed well with the predictions. We also examine various mechanical conditions of cells and confirm that coordinated collective cell migration supports the predictions. These results suggest the important roles of the villi in homeostatic maintenance of the small intestine, and we discuss the applicability of our approach to other tissues with collective cell movement.

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.

Evolution of cooperation in an epithelium

Cooperation is prevalent in nature, not only in the context of social interactions within the animal kingdom but also on the cellular level. In cancer, for example, tumour cells can cooperate by producing growth factors. The evolution of cooperation has traditionally been studied for well-mixed populations under the framework of evolutionary game theory, and more recently for structured populations using evolutionary graph theory (EGT). The population structures arising due to cellular arrangement in tissues, however, are dynamic and thus cannot be accurately represented by either of these frameworks. In this work, we compare the conditions for cooperative success in an epithelium modelled using EGT, to those in a mechanical model of an epithelium—the Voronoi tessellation (VT) model. Crucially, in this latter model, cells are able to move, and birth and death are not spatially coupled. We calculate fixation probabilities in the VT model through simulation and an approximate analytic technique and show that this leads to stronger promotion of cooperation in comparison with the EGT model.

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.

Regulators of Intestinal Epithelial Migration in Sepsis

The gut is a continuously renewing organ, with cell proliferation, migration, and death occurring rapidly under basal conditions. As the impact of critical illness on cell movement from crypt base to villus tip is poorly understood, the purpose of this study was to determine how sepsis alters enterocyte migration. Wild-type, transgenic, and knockout mice were injected with 5-bromo-2'deoxyuridine (BrdU) to label cells in S-phase before and after the onset of cecal ligation and puncture and were sacrificed at predetermined endpoints to determine distance proliferating cells migrated up the crypt-villus unit. Enterocyte migration rate was decreased from 24 to 96 h after sepsis. BrdU was not detectable on villi 6 days after sham laparotomy, meaning all cells had migrated the length of the gut and been exfoliated into its lumen. However, BrdU positive cells were detectable on villi 10 days after sepsis. Multiple components of gut integrity altered enterocyte migration. Sepsis decreased crypt proliferation, which further slowed enterocyte transit as mice injected with BrdU after the onset of sepsis (decreased proliferation) had slower migration than mice injected with BrdU before the onset of sepsis (normal proliferation). Decreasing intestinal apoptosis via gut-specific overexpression of Bcl-2 prevented sepsis-induced slowing of enterocyte migration. In contrast, worsened intestinal hyperpermeability by genetic deletion of JAM-A increased enterocyte migration. Sepsis therefore significantly slows enterocyte migration, and intestinal proliferation, apoptosis and permeability all affect migration time, which can potentially be targeted both genetically and pharmacologically.

Chronic TNFα-driven injury delays cell migration to villi in the intestinal epithelium

The intestinal epithelium is a single layer of cells which provides the first line of defence of the intestinal mucosa to bacterial infection. Cohesion of this physical barrier is supported by renewal of epithelial stem cells, residing in invaginations called crypts, and by crypt cell migration onto protrusions called villi; dysregulation of such mechanisms may render the gut susceptible to chronic inflammation. The impact that excessive or misplaced epithelial cell death may have on villus cell migration is currently unknown. We integrated cell-tracking methods with computational models to determine how epithelial homeostasis is affected by acute and chronic TNFα-driven epithelial cell death. Parameter inference reveals that acute inflammatory cell death has a transient effect on epithelial cell dynamics, whereas cell death caused by chronic elevated TNFα causes a delay in the accumulation of labelled cells onto the villus compared to the control. Such a delay may be reproduced by using a cell-based model to simulate the dynamics of each cell in a crypt-villus geometry, showing that a prolonged increase in cell death slows the migration of cells from the crypt to the villus. This investigation highlights which injuries (acute or chronic) may be regenerated and which cause disruption of healthy epithelial homeostasis.

Towards personalized computational oncology: from spatial models of tumour spheroids, to organoids, to tissues

A main goal of mathematical and computational oncology is to develop quantitative tools to determine the most effective therapies for each individual patient. This involves predicting the right drug to be administered at the right time and at the right dose. Such an approach is known as precision medicine. Mathematical modelling can play an invaluable role in the development of such therapeutic strategies, since it allows for relatively fast, efficient and inexpensive simulations of a large number of treatment schedules in order to find the most effective. This review is a survey of mathematical models that explicitly take into account the spatial architecture of three-dimensional tumours and address tumour development, progression and response to treatments. In particular, we discuss models of epithelial acini, multicellular spheroids, normal and tumour spheroids and organoids, and multi-component tissues. Our intent is to showcase how these in silico models can be applied to patient-specific data to assess which therapeutic strategies will be the most efficient. We also present the concept of virtual clinical trials that integrate standard-of-care patient data, medical imaging, organ-on-chip experiments and computational models to determine personalized medical treatment strategies.

Antrodin C, an NADPH Dependent Metabolism, Encourages Crosstalk between Autophagy and Apoptosis in Lung Carcinoma Cells by Use of an AMPK Inhibition-Independent Blockade of the Akt/mTOR Pathway

The current study aims to explore the possible anti-lung carcinoma activity of ADC as well as the underlying mechanisms by which ADC exerts its actions in NSCLC. Findings showed that ADC potently inhibited the viability of SPCA-1, induced apoptosis triggered by ROS, and arrested the cell cycle at the G2/M phase via a P53 signaling pathway. Interestingly, phenomena such as autophagosomes accumulation, conversion of the LC3-I to LC3-II, etc., indicated that autophagy could be activated by ADC. The blockage of autophagy-augmented ADC induced inhibition of cell proliferation, while autophagy activation restored cell death, indicating that autophagy had a protective effect against cell death which was induced by ADC treatment. Meanwhile, ADC treatment suppressed both the Akt/mTOR and AMPK signaling pathways. The joint action of both ADC and the autophagy inhibitor significantly increased the death of SPCA-1. An in vitro phase I metabolic stability assay showed that ADC was highly metabolized in SD rat liver microsomes and moderately metabolized in human liver microsomes, which will assist in predicting the outcomes of clinical pharmacokinetics and toxicity studies. These findings imply that blocking the Akt/mTOR signaling pathway, which was independent of AMPK inhibition, could activate ADC-induced protective autophagy in non-small-cell lung cancer cells.

Asymmetric cell division-dominant neutral drift model for normal intestinal stem cell homeostasis

The normal intestinal epithelium is continuously regenerated at a rapid rate from actively cycling Lgr5-expressing intestinal stem cells (ISCs) that reside at the crypt base. Recent mathematical modeling based on several lineage-tracing studies in mice shows that the symmetric cell division-dominant neutral drift model fits well with the observed in vivo growth of ISC clones and suggests that symmetric divisions are central to ISC homeostasis. However, other studies suggest a critical role for asymmetric cell division in the maintenance of ISC homeostasis in vivo. Here, we show that the stochastic branching and Moran process models with both a symmetric and asymmetric division mode not only simulate the stochastic growth of the ISC clone in silico but also closely fit the in vivo stem cell dynamics observed in lineage-tracing studies. In addition, the proposed model with highest probability for asymmetric division is more consistent with in vivo observations reported here and by others. Our in vivo studies of mitotic spindle orientations and lineage-traced progeny pairs indicate that asymmetric cell division is a dominant mode used by ISCs under normal homeostasis. Therefore, we propose the asymmetric cell division-dominant neutral drift model for normal ISC homeostasis. NEW & NOTEWORTHY The prevailing mathematical model suggests that intestinal stem cells (ISCs) divide symmetrically. The present study provides evidence that asymmetric cell division is the major contributor to ISC maintenance and thus proposes an asymmetric cell division-dominant neutral drift model. Consistent with this model, in vivo studies of mitotic spindle orientation and lineage-traced progeny pairs indicate that asymmetric cell division is the dominant mode used by ISCs under normal homeostasis.

Truncated Adenomatous Polyposis Coli Mutation Induces Asef-Activated Golgi Fragmentation

Adenomatous polyposis coli (APC) is a key molecule to maintain cellular homeostasis in colonic epithelium by regulating cell-cell adhesion, cell polarity, and cell migration through activating the APC-stimulated guanine nucleotide-exchange factor (Asef). The APC-activated Asef stimulates the small GTPase, which leads to decreased cell-cell adherence and cell polarity, and enhanced cell migration. In colorectal cancers, while truncated APC constitutively activates Asef and promotes cancer initiation and progression, regulation of Asef by full-length APC is still unclear. Here, we report the autoinhibition mechanism of full-length APC. We found that the armadillo repeats in full-length APC interact with the APC residues 1362 to 1540 (APC-2,3 repeats), and this interaction competes off and inhibits Asef. Deletion of APC-2,3 repeats permits Asef interactions leading to downstream signaling events, including the induction of Golgi fragmentation through the activation of the Asef-ROCK-MLC2. Truncated APC also disrupts protein trafficking and cholesterol homeostasis by inhibition of SREBP2 activity in a Golgi fragmentation-dependent manner. Our study thus uncovers the autoinhibition mechanism of full-length APC and a novel gain of function of truncated APC in regulating Golgi structure, as well as cholesterol homeostasis, which provides a potential target for pharmaceutical intervention against colon cancers.

Host-microbe interaction in the gastrointestinal tract

The gastrointestinal tract is a highly complex organ in which multiple dynamic physiological processes are tightly coordinated while interacting with a dense and extremely diverse microbial population. From establishment in early life, through to host-microbe symbiosis in adulthood, the gut microbiota plays a vital role in our development and health. The effect of the microbiota on gut development and physiology is highlighted by anatomical and functional changes in germ-free mice, affecting the gut epithelium, immune system and enteric nervous system. Microbial colonisation promotes competent innate and acquired mucosal immune systems, epithelial renewal, barrier integrity, and mucosal vascularisation and innervation. Interacting or shared signalling pathways across different physiological systems of the gut could explain how all these changes are coordinated during postnatal colonisation, or after the introduction of microbiota into germ-free models. The application of cell-based in-vitro experimental systems and mathematical modelling can shed light on the molecular and signalling pathways which regulate the development and maintenance of homeostasis in the gut and beyond.

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.

Deciphering the Complex Signaling Systems That Regulate Intestinal Epithelial Cell Death Processes and Shedding

Intestinal epithelial cells play a fundamental role in maintaining homeostasis. Shedding of intestinal cells in a controlled manner is critical to maintenance of barrier function. Barrier function is maintained during this shedding process by a redistribution of tight junctional proteins to facilitate closure of the gap left by the shedding cell. However, despite the obvious importance of epithelial cell shedding to gut health, a central question is how the extrusion of epithelial cells is achieved, enabling barrier integrity to be maintained in the healthy gut and restored during inflammation remains largely unanswered. Recent studies have provided evidence that excessive epithelial cell shedding and loss of epithelial barrier integrity is triggered by exposure to lipopolysaccharide or tumor necrosis factor alpha. Subsequent studies have provided evidence of the involvement of specific cellular components and signaling mechanisms as well as the functionality of microbiota that can be either detrimental or beneficial for intestinal barrier integrity. This review will focus on the evidence and decipher how the signaling systems through which the mucosal immune system and microbiota can regulate epithelial cell shedding and how these mechanisms interact to preserve the viability of the epithelium.

A hierarchical Bayesian model for understanding the spatiotemporal dynamics of the intestinal epithelium

Our work addresses two key challenges, one biological and one methodological. First, we aim to understand how proliferation and cell migration rates in the intestinal epithelium are related under healthy, damaged (Ara-C treated) and recovering conditions, and how these relations can be used to identify mechanisms of repair and regeneration. We analyse new data, presented in more detail in a companion paper, in which BrdU/IdU cell-labelling experiments were performed under these respective conditions. Second, in considering how to more rigorously process these data and interpret them using mathematical models, we use a probabilistic, hierarchical approach. This provides a best-practice approach for systematically modelling and understanding the uncertainties that can otherwise undermine the generation of reliable conclusions-uncertainties in experimental measurement and treatment, difficult-to-compare mathematical models of underlying mechanisms, and unknown or unobserved parameters. Both spatially discrete and continuous mechanistic models are considered and related via hierarchical conditional probability assumptions. We perform model checks on both in-sample and out-of-sample datasets and use them to show how to test possible model improvements and assess the robustness of our conclusions. We conclude, for the present set of experiments, that a primarily proliferation-driven model suffices to predict labelled cell dynamics over most time-scales.

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.

Cell proliferation within small intestinal crypts is the principal driving force for cell migration on villi

The functional integrity of the intestinal epithelial barrier relies on tight coordination of cell proliferation and migration, with failure to regulate these processes resulting in disease. It is not known whether cell proliferation is sufficient to drive epithelial cell migration during homoeostatic turnover of the epithelium. Nor is it known precisely how villus cell migration is affected when proliferation is perturbed. Some reports suggest that proliferation and migration may not be related while other studies support a direct relationship. We used established cell-tracking methods based on thymine analog cell labeling and developed tailored mathematical models to quantify cell proliferation and migration under normal conditions and when proliferation is reduced and when it is temporarily halted. We found that epithelial cell migration velocities along the villi are coupled to cell proliferation rates within the crypts in all conditions. Furthermore, halting and resuming proliferation results in the synchronized response of cell migration on the villi. We conclude that cell proliferation within the crypt is the primary force that drives cell migration along the villus. This methodology can be applied to interrogate intestinal epithelial dynamics and characterize situations in which processes involved in cell turnover become uncoupled, including pharmacological treatments and disease models.—Parker, A., Maclaren, O. J., Fletcher, A. G., Muraro, D., Kreuzaler, P. A., Byrne, H. M., Maini, P. K., Watson, A. J. M., Pin, C. Cell proliferation within small intestinal crypts is the principal driving force for cell migration on villi.

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 promises of quantitative systems pharmacology modelling for drug development

Recent growth in annual new therapeutic entity (NTE) approvals by the U.S. Food and Drug Administration (FDA) suggests a positive trend in current research and development (R&D) output. Prior to this, the cost of each NTE was considered to be rising exponentially, with compound failure occurring mainly in clinical phases. Quantitative systems pharmacology (QSP) modelling, as an additional tool in the drug discovery arsenal, aims to further reduce NTE costs and improve drug development success. Through in silico mathematical modelling, QSP can simulate drug activity as perturbations in biological systems and thus understand the fundamental interactions which drive disease pathology, compound pharmacology and patient response. Here we review QSP, pharmacometrics and systems biology models with respect to the diseases covered as well as their clinical relevance and applications. Overall, the majority of modelling focus was aligned with the priority of drug-discovery and clinical trials. However, a few clinically important disease categories, such as Immune System Diseases and Respiratory Tract Diseases, were poorly covered by computational models. This suggests a possible disconnect between clinical and modelling agendas. As a standard element of the drug discovery pipeline the uptake of QSP might help to increase the efficiency of drug development across all therapeutic indications.

Enabling Anyone to Translate Clinically Relevant Ideas to Therapies

How do we inspire new ideas that could lead to potential treatments for rare or neglected diseases, and allow for serendipity that could help to catalyze them? How many potentially good ideas are lost because they are never tested? What if those ideas could have lead to new therapeutic approaches and major healthcare advances? If a clinician or anyone for that matter, has a new idea they want to test to develop a molecule or therapeutic that they could translate to the clinic, how would they do it without a laboratory or funding? These are not idle theoretical questions but addressing them could have potentially huge economic implications for nations. If we fail to capture the diversity of ideas and test them we may also lose out on the next blockbuster treatments. Many of those involved in the process of ideation may be discouraged and simply not know where to go. We try to address these questions and describe how there are options to raising funding, how even small scale investments can foster preclinical or clinical translation, and how there are several approaches to outsourcing the experiments, whether to collaborators or commercial enterprises. While these are not new or far from complete solutions, they are first steps that can be taken by virtually anyone while we work on other solutions to build a more concrete structure for the "idea-hypothesis testing-proof of concept-translation-breakthrough pathway".

Plasticity of Cell Migration In Vivo and In Silico

Cell migration results from stepwise mechanical and chemical interactions between cells and their extracellular environment. Mechanistic principles that determine single-cell and collective migration modes and their interconversions depend upon the polarization, adhesion, deformability, contractility, and proteolytic ability of cells. Cellular determinants of cell migration respond to extracellular cues, including tissue composition, topography, alignment, and tissue-associated growth factors and cytokines. Both cellular determinants and tissue determinants are interdependent; undergo reciprocal adjustment; and jointly impact cell decision making, navigation, and migration outcome in complex environments. We here review the variability, decision making, and adaptation of cell migration approached by live-cell, in vivo, and in silico strategies, with a focus on cell movements in morphogenesis, repair, immune surveillance, and cancer metastasis.

The role of cell location and spatial gradients in the evolutionary dynamics of colon and intestinal crypts

BACKGROUND

Colon and intestinal crypts serve as an important model system for adult stem cell proliferation and differentiation. We develop a spatial stochastic model to study the rate of somatic evolution in a normal crypt, focusing on the production of two-hit mutants that inactivate a tumor suppressor gene. We investigate the effect of cell division pattern along the crypt on mutant production, assuming that the division rate of each cell depends on its location.

RESULTS

We find that higher probability of division at the bottom of the crypt, where the stem cells are located, leads to a higher rate of double-hit mutant production. The optimal case for delaying mutations occurs when most of the cell divisions happen at the top of the crypt. We further consider an optimization problem where the "evolutionary" penalty for double-hit mutant generation is complemented with a "functional" penalty that assures that fully differentiated cells at the top of the crypt cannot divide.

CONCLUSION

The trade-off between the two types of objectives leads to the selection of an intermediate division pattern, where the cells in the middle of the crypt divide with the highest rate. This matches the pattern of cell divisions obtained experimentally in murine crypts.

REVIEWERS

This article was reviewed by David Axelrod (nominated by an Editorial Board member, Marek Kimmel), Yang Kuang and Anna Marciniak-Czochra. For the full reviews, please go to the Reviewers' comments section.

Promotion of Intestinal Epithelial Cell Turnover by Commensal Bacteria: Role of Short-Chain Fatty Acids

The life span of intestinal epithelial cells (IECs) is short (3–5 days), and its regulation is thought to be important for homeostasis of the intestinal epithelium. We have now investigated the role of commensal bacteria in regulation of IEC turnover in the small intestine. The proliferative activity of IECs in intestinal crypts as well as the migration of these cells along the crypt-villus axis were markedly attenuated both in germ-free mice and in specific pathogen–free (SPF) mice treated with a mixture of antibiotics, with antibiotics selective for Gram-positive bacteria being most effective in this regard. Oral administration of chloroform-treated feces of SPF mice to germ-free mice resulted in a marked increase in IEC turnover, suggesting that spore-forming Gram-positive bacteria contribute to this effect. Oral administration of short-chain fatty acids (SCFAs) as bacterial fermentation products also restored the turnover of IECs in antibiotic-treated SPF mice as well as promoted the development of intestinal organoids in vitro. Antibiotic treatment reduced the phosphorylation levels of ERK, ribosomal protein S6, and STAT3 in IECs of SPF mice. Our results thus suggest that Gram-positive commensal bacteria are a major determinant of IEC turnover, and that their stimulatory effect is mediated by SCFAs.

Combined changes in Wnt signaling response and contact inhibition induce altered proliferation in radiation-treated intestinal crypts

Wnt concentration gradients operate in many tissues. Modeling of proliferation in control and irradiated intestinal crypts shows that the Wnt concentrations that cells experience when they are born set their proliferative fate and cell cycle duration. The simulations also predict the initial proportion of cells damaged by tumor-promoting radiation.

Curative intervention is possible if colorectal cancer is identified early, underscoring the need to detect the earliest stages of malignant transformation. A candidate biomarker is the expanded proliferative zone observed in crypts before adenoma formation, also found in irradiated crypts. However, the underlying driving mechanism for this is not known. Wnt signaling is a key regulator of proliferation, and elevated Wnt signaling is implicated in cancer. Nonetheless, how cells differentiate Wnt signals of varying strengths is not understood. We use computational modeling to compare alternative hypotheses about how Wnt signaling and contact inhibition affect proliferation. Direct comparison of simulations with published experimental data revealed that the model that best reproduces proliferation patterns in normal crypts stipulates that proliferative fate and cell cycle duration are set by the Wnt stimulus experienced at birth. The model also showed that the broadened proliferation zone induced by tumorigenic radiation can be attributed to cells responding to lower Wnt concentrations and dividing at smaller volumes. Application of the model to data from irradiated crypts after an extended recovery period permitted deductions about the extent of the initial insult. Application of computational modeling to experimental data revealed how mechanisms that control cell dynamics are altered at the earliest stages of carcinogenesis.

Multiscale Model of Colorectal Cancer Using the Cellular Potts Framework

Colorectal cancer (CRC) is one of the major causes of death in the developed world and forms a canonical example of tumorigenesis. CRC arises from a string of mutations of individual cells in the colorectal crypt, making it particularly suited for multiscale multicellular modeling, where mutations of individual cells can be clearly represented and their effects readily tracked. In this paper, we present a multicellular model of the onset of colorectal cancer, utilizing the cellular Potts model (CPM). We use the model to investigate how, through the modification of their mechanical properties, mutant cells colonize the crypt. Moreover, we study the influence of mutations on the shape of cells in the crypt, suggesting possible cell- and tissue-level indicators for identifying early-stage cancerous crypts. Crucially, we discuss the effect that the motility parameters of the model (key factors in the behavior of the CPM) have on the distribution of cells within a homeostatic crypt, resulting in an optimal parameter regime that accurately reflects biological assumptions. In summary, the key results of this paper are 1) how to couple the CPM with processes occurring on other spatial scales, using the example of the crypt to motivate suitable motility parameters; 2) modeling mutant cells with the CPM; 3) and investigating how mutations influence the shape of cells in the crypt.

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.

The Interplay between Wnt Mediated Expansion and Negative Regulation of Growth Promotes Robust Intestinal Crypt Structure and Homeostasis

The epithelium of the small intestinal crypt, which has a vital role in protecting the underlying tissue from the harsh intestinal environment, is completely renewed every 4–5 days by a small pool of stem cells at the base of each crypt. How is this renewal controlled and homeostasis maintained, particularly given the rapid nature of this process? Here, based on the recent observations from in vitro “mini gut” studies, we use a hybrid stochastic model of the crypt to investigate how exogenous niche signaling (from Wnt and BMP) combines with auto-regulation to promote homeostasis. This model builds on the sub-cellular element method to account for the three-dimensional structure of the crypt, external regulation by Wnt and BMP, internal regulation by Notch signaling, as well as regulation by internally generated diffusible signals. Results show that Paneth cell derived Wnt signals, which have been observed experimentally to sustain crypts in cultured organs, have a dramatically different influence on niche dynamics than does mesenchyme derived Wnt. While this signaling can indeed act as a redundant backup to the exogenous gradient, it introduces a positive feedback that destabilizes the niche and causes its uncontrolled expansion. We find that in this setting, BMP has a critical role in constraining this expansion, consistent with observations that its removal leads to crypt fission. Further results also point to a new hypothesis for the role of Ephrin mediated motility of Paneth cells, specifically that it is required to constrain niche expansion and maintain the crypt’s spatial structure. Combined, these provide an alternative view of crypt homeostasis where the niche is in a constant state of expansion and the spatial structure of the crypt arises as a balance between this expansion and the action of various sources of negative regulation that hold it in check.

The small intestinal epithelium, like our skin, is constantly being renewed. In the intestine however, this epithelium is exposed to the harsh digestive environment, necessitating much more rapid renewal. Remarkably, the entire epithelium is renewed every 4–5 days. This raises the question, how can the size and structure of this tissue be maintained given this pace. Motivated by recent experimental observations, we construct a three-dimensional, hybrid stochastic model to investigate the mechanisms responsible for homeostasis of this tissue. We find that there are redundant signals created by both the epithelium itself and surrounding tissues that act in parallel to maintain epithelial structure. This redundancy comes at a price however: it introduces the possibility of explosive stem cell population growth. Additional results suggest that other signals along with choreographed motion of cells are responsible for repressing this expansion. Taken together, our results provide a novel hypothesis for how robust but fast renewal of the crypt is achieved: as a balance between expansion, which drives fast renewal and repression, which holds that expansion in check to maintain the crypt’s structure.

A three-dimensional computational model of collagen network mechanics

Extracellular matrix (ECM) strongly influences cellular behaviors, including cell proliferation, adhesion, and particularly migration. In cancer, the rigidity of the stromal collagen environment is thought to control tumor aggressiveness, and collagen alignment has been linked to tumor cell invasion. While the mechanical properties of collagen at both the single fiber scale and the bulk gel scale are quite well studied, how the fiber network responds to local stress or deformation, both structurally and mechanically, is poorly understood. This intermediate scale knowledge is important to understanding cell-ECM interactions and is the focus of this study. We have developed a three-dimensional elastic collagen fiber network model (bead-and-spring model) and studied fiber network behaviors for various biophysical conditions: collagen density, crosslinker strength, crosslinker density, and fiber orientation (random vs. prealigned). We found the best-fit crosslinker parameter values using shear simulation tests in a small strain region. Using this calibrated collagen model, we simulated both shear and tensile tests in a large linear strain region for different network geometry conditions. The results suggest that network geometry is a key determinant of the mechanical properties of the fiber network. We further demonstrated how the fiber network structure and mechanics evolves with a local formation, mimicking the effect of pulling by a pseudopod during cell migration. Our computational fiber network model is a step toward a full biomechanical model of cellular behaviors in various ECM conditions.