Multicellular computer simulation of morphogenesis: blastocoel roof thinning and matrix assembly in Xenopus laevis

Beyond microtubules: The cellular environment at the endoplasmic reticulum attracts proteins to the nucleus, enabling nuclear transport

All proteins are translated in the cytoplasm, yet many, including transcription factors, play vital roles in the nucleus. While previous research has concentrated on molecular motors for the transport of these proteins to the nucleus, recent observations reveal perinuclear accumulation even in the absence of an energy source, hinting at alternative mechanisms. Here, we propose that structural properties of the cellular environment, specifically the endoplasmic reticulum (ER), can promote molecular transport to the perinucleus without requiring additional energy expenditure. Specifically, physical interaction between proteins and the ER impedes their diffusion and leads to their accumulation near the nucleus. This result explains why larger proteins, more frequently interacting with the ER membrane, tend to accumulate at the perinucleus. Interestingly, such diffusion in a heterogeneous environment follows Chapman's law rather than the popular Fick's law. Our findings suggest a novel protein transport mechanism arising solely from characteristics of the intracellular environment.

Modeling the extracellular matrix in cell migration and morphogenesis: a guide for the curious biologist

The extracellular matrix (ECM) is a highly complex structure through which biochemical and mechanical signals are transmitted. In processes of cell migration, the ECM also acts as a scaffold, providing structural support to cells as well as points of potential attachment. Although the ECM is a well-studied structure, its role in many biological processes remains difficult to investigate comprehensively due to its complexity and structural variation within an organism. In tandem with experiments, mathematical models are helpful in refining and testing hypotheses, generating predictions, and exploring conditions outside the scope of experiments. Such models can be combined and calibrated with in vivo and in vitro data to identify critical cell-ECM interactions that drive developmental and homeostatic processes, or the progression of diseases. In this review, we focus on mathematical and computational models of the ECM in processes such as cell migration including cancer metastasis, and in tissue structure and morphogenesis. By highlighting the predictive power of these models, we aim to help bridge the gap between experimental and computational approaches to studying the ECM and to provide guidance on selecting an appropriate model framework to complement corresponding experimental studies.

A computational modeling approach for predicting multicell spheroid patterns based on signaling-induced differential adhesion

Physiological and pathological processes including embryogenesis and tumorigenesis rely on the ability of individual cells to work collectively to form multicell patterns. In these heterogeneous multicell systems, cell-cell signaling induces differential adhesion between cells that leads to tissue-level patterning. However, the sensitivity of pattern formation to changes in the strengths of signaling or cell adhesion processes is not well understood. Prior work has explored these issues using synthetically engineered heterogeneous multicell spheroid systems, in which cell subpopulations engage in bidirectional intercellular signaling to regulate the expression of different cadherins. While engineered cell systems provide excellent experimental tools to observe pattern formation in cell populations, computational models of these systems may be leveraged to explore more systematically how specific combinations of signaling and adhesion parameters can drive the emergence of unique patterns. We developed and validated two- and three-dimensional agent-based models (ABMs) of spheroid patterning for previously described cells engineered with a bidirectional signaling circuit that regulates N- and P-cadherin expression. Systematic exploration of model predictions, some of which were experimentally validated, revealed how cell seeding parameters, the order of signaling events, probabilities of induced cadherin expression, and homotypic adhesion strengths affect pattern formation. Unsupervised clustering was also used to map combinations of signaling and adhesion parameters to these unique spheroid patterns predicted by the ABM. Finally, we demonstrated how the model may be deployed to design new synthetic cell signaling circuits based on a desired final multicell pattern.

Blastocoel morphogenesis: A biophysics perspective

The blastocoel is a fluid-filled cavity characteristic of animal embryos at the blastula stage. Its emergence is commonly described as the result of cleavage patterning, but this historical view conceals a large diversity of mechanisms and overlooks many unsolved questions from a biophysics perspective. In this review, we describe generic mechanisms for blastocoel morphogenesis, rooted in biological literature and simple physical principles. We propose novel directions of study and emphasize the importance to study blastocoel morphogenesis as an evolutionary and physical continuum.

Convergent extension in the amphibian, Xenopus laevis

This review is a comprehensive analysis of the cell biology and biomechanics of Convergent Extension in Xenopus.

Using a continuum model to decipher the mechanics of embryonic tissue spreading from time-lapse image sequences: An approximate Bayesian computation approach

Advanced imaging techniques generate large datasets capable of describing the structure and kinematics of tissue spreading in embryonic development, wound healing, and the progression of many diseases. These datasets can be integrated with mathematical models to infer biomechanical properties of the system, typically identifying an optimal set of parameters for an individual experiment. However, these methods offer little information on the robustness of the fit and are generally ill-suited for statistical tests of multiple experiments. To overcome this limitation and enable efficient use of large datasets in a rigorous experimental design, we use the approximate Bayesian computation rejection algorithm to construct probability density distributions that estimate model parameters for a defined theoretical model and set of experimental data. Here, we demonstrate this method with a 2D Eulerian continuum mechanical model of spreading embryonic tissue. The model is tightly integrated with quantitative image analysis of different sized embryonic tissue explants spreading on extracellular matrix (ECM) and is regulated by a small set of parameters including forces on the free edge, tissue stiffness, strength of cell-ECM adhesions, and active cell shape changes. We find statistically significant trends in key parameters that vary with initial size of the explant, e.g., for larger explants cell-ECM adhesion forces are weaker and free edge forces are stronger. Furthermore, we demonstrate that estimated parameters for one explant can be used to predict the behavior of other similarly sized explants. These predictive methods can be used to guide further experiments to better understand how collective cell migration is regulated during development.

Cellular automata model for human articular chondrocytes migration, proliferation and cell death: An in vitro validation

Articular cartilage is characterized by low cell density of only one cell type, chondrocytes, and has limited self-healing properties. When articular cartilage is affected by traumatic injuries, a therapeutic strategy such as autologous chondrocyte implantation is usually proposed for its treatment. This approach requires in vitro chondrocyte expansion to yield high cell number for cell transplantation. To improve the efficiency of this procedure, it is necessary to assess cell dynamics such as migration, proliferation and cell death during culture. Computational models such as cellular automata can be used to simulate cell dynamics in order to enhance the result of cell culture procedures. This methodology has been implemented for several cell types; however, an experimental validation is required for each one. For this reason, in this research a cellular automata model, based on random-walk theory, was devised in order to predict articular chondrocyte behavior in monolayer culture during cell expansion. Results demonstrated that the cellular automata model corresponded to cell dynamics and computed-accurate quantitative results. Moreover, it was possible to observe that cell dynamics depend on weighted probabilities derived from experimental data and cell behavior varies according to the cell culture period. Thus, depending on whether cells were just seeded or proliferated exponentially, culture time probabilities differed in percentages in the CA model. Furthermore, in the experimental assessment a decreased chondrocyte proliferation was observed along with increased passage number. This approach is expected to having other uses as in enhancing articular cartilage therapies based on tissue engineering and regenerative medicine.

Mechanocellular models of epithelial morphogenesis

Embryonic epithelia achieve complex morphogenetic movements, including in-plane reshaping, bending and folding, through the coordinated action and rearrangement of individual cells. Technical advances in molecular and live-imaging studies of epithelial dynamics provide a very real opportunity to understand how cell-level processes facilitate these large-scale tissue rearrangements. However, the large datasets that we are now able to generate require careful interpretation. In combination with experimental approaches, computational modelling allows us to challenge and refine our current understanding of epithelial morphogenesis and to explore experimentally intractable questions. To this end, a variety of cell-based modelling approaches have been developed to describe cell-cell mechanical interactions, ranging from vertex and 'finite-element' models that approximate each cell geometrically by a polygon representing the cell's membrane, to immersed boundary and subcellular element models that allow for more arbitrary cell shapes. Here, we review how these models have been used to provide insights into epithelial morphogenesis and describe how such models could help future efforts to decipher the forces and mechanical and biochemical feedbacks that guide cell and tissue-level behaviour. In addition, we discuss current challenges associated with using computational models of morphogenetic processes in a quantitative and predictive way.This article is part of the themed issue 'Systems morphodynamics: understanding the development of tissue hardware'.

The Molecular Basis of Radial Intercalation during Tissue Spreading in Early Development

Radial intercalation is a fundamental process responsible for the thinning of multilayered tissues during large-scale morphogenesis; however, its molecular mechanism has remained elusive. Using amphibian epiboly, the thinning and spreading of the animal hemisphere during gastrulation, here we provide evidence that radial intercalation is driven by chemotaxis of cells toward the external layer of the tissue. This role of chemotaxis in tissue spreading and thinning is unlike its typical role associated with large-distance directional movement of cells. We identify the chemoattractant as the complement component C3a, a factor normally linked with the immune system. The mechanism is explored by computational modeling and tested in vivo, ex vivo, and in vitro. This mechanism is robust against fluctuations of chemoattractant levels and expression patterns and explains expansion during epiboly. This study provides insight into the fundamental process of radial intercalation and could be applied to a wide range of morphogenetic events.

Vertebrate Embryonic Cleavage Pattern Determination

The pattern of the earliest cell divisions in a vertebrate embryo lays the groundwork for later developmental events such as gastrulation, organogenesis, and overall body plan establishment. Understanding these early cleavage patterns and the mechanisms that create them is thus crucial for the study of vertebrate development. This chapter describes the early cleavage stages for species representing ray-finned fish, amphibians, birds, reptiles, mammals, and proto-vertebrate ascidians and summarizes current understanding of the mechanisms that govern these patterns. The nearly universal influence of cell shape on orientation and positioning of spindles and cleavage furrows and the mechanisms that mediate this influence are discussed. We discuss in particular models of aster and spindle centering and orientation in large embryonic blastomeres that rely on asymmetric internal pulling forces generated by the cleavage furrow for the previous cell cycle. Also explored are mechanisms that integrate cell division given the limited supply of cellular building blocks in the egg and several-fold changes of cell size during early development, as well as cytoskeletal specializations specific to early blastomeres including processes leading to blastomere cohesion. Finally, we discuss evolutionary conclusions beginning to emerge from the contemporary analysis of the phylogenetic distributions of cleavage patterns. In sum, this chapter seeks to summarize our current understanding of vertebrate early embryonic cleavage patterns and their control and evolution.

PROKARYO: an illustrative and interactive computational model of the lactose operon in the bacterium Escherichia coli

Background

We are creating software for agent-based simulation and visualization of bio-molecular processes in bacterial and eukaryotic cells. As a first example, we have built a 3-dimensional, interactive computer model of an Escherichia coli bacterium and its associated biomolecular processes. Our illustrative model focuses on the gene regulatory processes that control the expression of genes involved in the lactose operon. Prokaryo, our agent-based cell simulator, incorporates cellular structures, such as plasma membranes and cytoplasm, as well as elements of the molecular machinery, including RNA polymerase, messenger RNA, lactose permease, and ribosomes.

Results

The dynamics of cellular ’agents’ are defined by their rules of interaction, implemented as finite state machines. The agents are embedded within a 3-dimensional virtual environment with simulated physical and electrochemical properties. The hybrid model is driven by a combination of (1) mathematical equations (DEQs) to capture higher-scale phenomena and (2) agent-based rules to implement localized interactions among a small number of molecular elements. Consequently, our model is able to capture phenomena across multiple spatial scales, from changing concentration gradients to one-on-one molecular interactions.

We use the classic gene regulatory mechanism of the lactose operon to demonstrate our model’s resolution, visual presentation, and real-time interactivity. Our agent-based model expands on a sophisticated mathematical E. coli metabolism model, through which we highlight our model’s scientific validity.

Conclusion

We believe that through illustration and interactive exploratory learning a model system like Prokaryo can enhance the general understanding and perception of biomolecular processes. Our agent-DEQ hybrid modeling approach can also be of value to conceptualize, illustrate, and—eventually—validate cell experiments in the wet lab.

Applications of computational models to better understand microvascular remodelling: a focus on biomechanical integration across scales

Microvascular network remodelling is a common denominator for multiple pathologies and involves both angiogenesis, defined as the sprouting of new capillaries, and network patterning associated with the organization and connectivity of existing vessels. Much of what we know about microvascular remodelling at the network, cellular and molecular scales has been derived from reductionist biological experiments, yet what happens when the experiments provide incomplete (or only qualitative) information? This review will emphasize the value of applying computational approaches to advance our understanding of the underlying mechanisms and effects of microvascular remodelling. Examples of individual computational models applied to each of the scales will highlight the potential of answering specific questions that cannot be answered using typical biological experimentation alone. Looking into the future, we will also identify the needs and challenges associated with integrating computational models across scales.

Mesoderm is required for coordinated cell movements within zebrafish neural plate in vivo

Background

Morphogenesis of the zebrafish neural tube requires the coordinated movement of many cells in both time and space. A good example of this is the movement of the cells in the zebrafish neural plate as they converge towards the dorsal midline before internalizing to form a neural keel. How these cells are regulated to ensure that they move together as a coherent tissue is unknown. Previous work in other systems has suggested that the underlying mesoderm may play a role in this process but this has not been shown directly in vivo.

Results

Here we analyze the roles of subjacent mesoderm in the coordination of neural cell movements during convergence of the zebrafish neural plate and neural keel formation. Live imaging demonstrates that the normal highly coordinated movements of neural plate cells are lost in the absence of underlying mesoderm and the movements of internalization and neural tube formation are severely disrupted. Despite this, neuroepithelial polarity develops in the abnormal neural primordium but the resulting tissue architecture is very disorganized.

Conclusions

We show that the movements of cells in the zebrafish neural plate are highly coordinated during the convergence and internalization movements of neurulation. Our results demonstrate that the underlying mesoderm is required for these coordinated cell movements in the zebrafish neural plate in vivo.

The effects of cell compressibility, motility and contact inhibition on the growth of tumor cell clusters using the Cellular Potts Model

There are numerous biological examples where genes associated with migratory ability of cells also confer the cells with an increased fitness even though these genes may not have any known effect on the cell mitosis rates. Here, we provide insight into these observations by analyzing the effects of cell migration, compression, and contact inhibition on the growth of tumor cell clusters using the Cellular Potts Model (CPM) in a monolayer geometry. This is a follow-up of a previous study (Thalhauser et al. 2010) in which a Moran-type model was used to study the interaction of cell proliferation, migratory potential and death on the emergence of invasive phenotypes. Here, we extend the study to include the effects of cell size and shape. In particular, we investigate the interplay between cell motility and compressibility within the CPM and find that the CPM predicts that increased cell motility leads to smaller cells. This is an artifact in the CPM. An analysis of the CPM reveals an explicit inverse-relationship between the cell stiffness and motility parameters. We use this relationship to compensate for motility-induced changes in cell size in the CPM so that in the corrected CPM, cell size is independent of the cell motility. We find that subject to comparable levels of compression, clusters of motile cells grow faster than clusters of less motile cells, in qualitative agreement with biological observations and our previous study. Increasing compression tends to reduce growth rates. Contact inhibition penalizes clumped cells by halting their growth and gives motile cells an even greater advantage. Finally, our model predicts cell size distributions that are consistent with those observed in clusters of neuroblastoma cells cultured in low and high density conditions.

Par6b regulates the dynamics of apicobasal polarity during development of the stratified Xenopus epidermis

During early vertebrate development, epithelial cells establish and maintain apicobasal polarity, failure of which can cause developmental defects or cancer metastasis. This process has been mostly studied in simple epithelia that have only one layer of cells, but is poorly understood in stratified epithelia. In this paper we address the role of the polarity protein Partitioning defective-6 homolog beta (Par6b) in the developing stratified epidermis of Xenopus laevis. At the blastula stage, animal blastomeres divide perpendicularly to the apicobasal axis to generate partially polarized superficial cells and non-polarized deep cells. Both cell populations modify their apicobasal polarity during the gastrula stage, before differentiating into the superficial and deep layers of epidermis. Early differentiation of the epidermis is normal in Par6b-depleted embryos; however, epidermal cells dissociate and detach from embryos at the tailbud stage. Par6b-depleted epidermal cells exhibit a significant reduction in basolaterally localized E-cadherin. Examination of the apical marker Crumbs homolog 3 (Crb3) and the basolateral marker Lethal giant larvae 2 (Lgl2) after Par6b depletion reveals that Par6b cell-autonomously regulates the dynamics of apicobasal polarity in both superficial and deep epidermal layers. Par6b is required to maintain the “basolateral” state in both epidermal layers, which explains the reduction of basolateral adhesion complexes and epidermal cells shedding.

Multiscale computational models of complex biological systems

Integration of data across spatial, temporal, and functional scales is a primary focus of biomedical engineering efforts. The advent of powerful computing platforms, coupled with quantitative data from high-throughput experimental methodologies, has allowed multiscale modeling to expand as a means to more comprehensively investigate biological phenomena in experimentally relevant ways. This review aims to highlight recently published multiscale models of biological systems, using their successes to propose the best practices for future model development. We demonstrate that coupling continuous and discrete systems best captures biological information across spatial scales by selecting modeling techniques that are suited to the task. Further, we suggest how to leverage these multiscale models to gain insight into biological systems using quantitative biomedical engineering methods to analyze data in nonintuitive ways. These topics are discussed with a focus on the future of the field, current challenges encountered, and opportunities yet to be realized.

Multi-scale modeling of tissues using CompuCell3D

The study of how cells interact to produce tissue development, homeostasis, or diseases was, until recently, almost purely experimental. Now, multi-cell computer simulation methods, ranging from relatively simple cellular automata to complex immersed-boundary and finite-element mechanistic models, allow in silico study of multi-cell phenomena at the tissue scale based on biologically observed cell behaviors and interactions such as movement, adhesion, growth, death, mitosis, secretion of chemicals, chemotaxis, etc. This tutorial introduces the lattice-based Glazier-Graner-Hogeweg (GGH) Monte Carlo multi-cell modeling and the open-source GGH-based CompuCell3D simulation environment that allows rapid and intuitive modeling and simulation of cellular and multi-cellular behaviors in the context of tissue formation and subsequent dynamics. We also present a walkthrough of four biological models and their associated simulations that demonstrate the capabilities of the GGH and CompuCell3D.

Mathematical model and numerical simulation of the cell growth in scaffolds

A scaffold is a three-dimensional matrix that provides a structural base to fill tissue lesion and provides cells with a suitable environment for proliferation and differentiation. Cell-seeded scaffolds can be implanted immediately or be cultured in vitro for a period of time before implantation. To obtain uniform cell growth throughout the entire volume of the scaffolds, an optimal strategy on cell seeding into scaffolds is important. We propose an efficient and accurate numerical scheme for a mathematical model to predict the growth and distribution of cells in scaffolds. The proposed numerical algorithm is a hybrid method which uses both finite difference approximations and analytic closed-form solutions. The effects of each parameter in the mathematical model are numerically investigated. Moreover, we propose an optimization algorithm which finds the best set of model parameters that minimize a discrete l(2) error between numerical and experimental data. Using the mathematical model and its efficient and accurate numerical simulations, we could interpret experimental results and identify dominating mechanisms.

Dispersal, settling and layer formation

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

Agent-based models in translational systems biology

Effective translational methodologies for knowledge representation are needed in order to make strides against the constellation of diseases that affect the world today. These diseases are defined by their mechanistic complexity, redundancy, and nonlinearity. Translational systems biology aims to harness the power of computational simulation to streamline drug/device design, simulate clinical trials, and eventually to predict the effects of drugs on individuals. The ability of agent-based modeling to encompass multiple scales of biological process as well as spatial considerations, coupled with an intuitive modeling paradigm, suggests that this modeling framework is well suited for translational systems biology. This review describes agent-based modeling and gives examples of its translational applications in the context of acute inflammation and wound healing.

Computational modeling of epithelial tissues

There is an extensive literature on the computational modeling of epithelial tissues at all levels from subcellular to whole tissue. This review concentrates on behavior at the individual cell to whole tissue level, and particularly on organizational aspects, and provides an indication of where information from other areas, such as the modeling of angiogenesis, is relevant. The skin, and the lining of all of the body cavities (lung, gut, cervix, bladder etc) are epithelial tissues, which in a topological sense are the boundary between inside and outside the body. They are thin sheets of cells (usually of the order of 0.5 mm thick) without extracellular matrix, have a relatively simple structure, and contain few types of cells. They have important barrier, secretory and transport functions, which are essential for the maintenance of life, so homeostasis and wound healing are important aspects of the behavior of epithelial tissues. Carcinomas originate in epithelial tissues.There are essentially two approaches to modeling tissues--to start at the level of the tissue (i.e., a length scale of the order of 1 mm) and develop generalized equations for behavior (a continuum approach); or to start at the level of the cell (i.e., a length scale of the order of 10 µm) and develop tissue behavior as an emergent property of cellular behavior (an individual-based approach). As will be seen, these are not mutually exclusive approaches, and they come in a variety of flavors.

Embryo mechanics: balancing force production with elastic resistance during morphogenesis

Morphogenesis requires the spatial and temporal control of embryo mechanics, including force production and mechanical resistance to those forces, to coordinate tissue deformation and large-scale movements. Thus, biomechanical processes play a key role in directly shaping the embryo. Additional roles for embryo mechanics during development may include the patterning of positional information and to provide feedback to ensure the success of morphogenetic movements in shaping the larval body and organs. To understand the multiple roles of mechanics during development requires familiarity with engineering principles of the mechanics of structures, the viscoelastic properties of biomaterials, and the integration of force and stress within embryonic structures as morphogenesis progresses. In this chapter, we review the basic engineering principles of biomechanics as they relate to morphogenesis, introduce methods for quantifying embryo mechanics and the limitations of these methods, and outline a formalism for investigating the role of embryo mechanics in birth defects. We encourage the nascent field of embryo mechanics to adopt standard engineering terms and test methods so that studies of diverse organisms can be compared and universal biomechanical principles can be revealed.

At the biological modeling and simulation frontier

We provide a rationale for and describe examples of synthetic modeling and simulation (M&S) of biological systems. We explain how synthetic methods are distinct from familiar inductive methods. Synthetic M&S is a means to better understand the mechanisms that generate normal and disease-related phenomena observed in research, and how compounds of interest interact with them to alter phenomena. An objective is to build better, working hypotheses of plausible mechanisms. A synthetic model is an extant hypothesis: execution produces an observable mechanism and phenomena. Mobile objects representing compounds carry information enabling components to distinguish between them and react accordingly when different compounds are studied simultaneously. We argue that the familiar inductive approaches contribute to the general inefficiencies being experienced by pharmaceutical R&D, and that use of synthetic approaches accelerates and improves R&D decision-making and thus the drug development process. A reason is that synthetic models encourage and facilitate abductive scientific reasoning, a primary means of knowledge creation and creative cognition. When synthetic models are executed, we observe different aspects of knowledge in action from different perspectives. These models can be tuned to reflect differences in experimental conditions and individuals, making translational research more concrete while moving us closer to personalized medicine.

A computational approach to understand in vitro alveolar morphogenesis

Primary human alveolar type II (AT II) epithelial cells maintained in Matrigel cultures form alveolar-like cysts (ALCs) using a cytogenesis mechanism that is different from that of other studied epithelial cell types: neither proliferation nor death is involved. During ALC formation, AT II cells engage simultaneously in fundamentally different, but not fully characterized activities. Mechanisms enabling these activities and the roles they play during different process stages are virtually unknown. Identifying, characterizing, and understanding the activities and mechanisms are essential to achieving deeper insight into this fundamental feature of morphogenesis. That deeper insight is needed to answer important questions. When and how does an AT cell choose to switch from one activity to another? Why does it choose one action rather than another? We report obtaining plausible answers using a rigorous, multi-attribute modeling and simulation approach that leveraged earlier efforts by using new, agent and object-oriented capabilities. We discovered a set of cell-level operating principles that enabled in silico cells to self-organize and generate systemic cystogenesis phenomena that are quantitatively indistinguishable from those observed in vitro. Success required that the cell components be quasi-autonomous. As simulation time advances, each in silico cell autonomously updates its environment information to reclassify its condition. It then uses the axiomatic operating principles to execute just one action for each possible condition. The quasi-autonomous actions of individual in silico cells were sufficient for developing stable cyst-like structures. The results strengthen in silico to in vitro mappings at three levels: mechanisms, behaviors, and operating principles, thereby achieving a degree of validation and enabling answering the questions posed. We suggest that the in silico operating principles presented may have a biological counterpart and that a semiquantitative mapping exists between in silico causal events and in vitro causal events.

Multicell simulations of development and disease using the CompuCell3D simulation environment

Mathematical modeling and computer simulation have become crucial to biological fields from genomics to ecology. However, multicell, tissue-level simulations of development and disease have lagged behind other areas because they are mathematically more complex and lack easy-to-use software tools that allow building and running in silico experiments without requiring in-depth knowledge of programming. This tutorial introduces Glazier-Graner-Hogeweg (GGH) multicell simulations and CompuCell3D, a simulation framework that allows users to build, test, and run GGH simulations.

Essential operating principles for tumor spheroid growth

BACKGROUND

Our objective was to discover in silico axioms that are plausible representations of the operating principles realized during characteristic growth of EMT6/Ro mouse mammary tumor spheroids in culture. To reach that objective we engineered and iteratively falsified an agent-based analogue of EMT6 spheroid growth. EMT6 spheroids display consistent and predictable growth characteristics, implying that individual cell behaviors are tightly controlled and regulated. An approach to understanding how individual cell behaviors contribute to system behaviors is to discover a set of principles that enable abstract agents to exhibit closely analogous behaviors using only information available in an agent's immediate environment. We listed key attributes of EMT6 spheroid growth, which became our behavioral targets. Included were the development of a necrotic core surrounded by quiescent and proliferating cells, and growth data at two distinct levels of nutrient.

RESULTS

We then created an analogue made up of quasi-autonomous software agents and an abstract environment in which they could operate. The system was designed so that upon execution it could mimic EMT6 cells forming spheroids in culture. Each agent used an identical set of axiomatic operating principles. In sequence, we used the list of targeted attributes to falsify and revise these axioms, until the analogue exhibited behaviors and attributes that were within prespecified ranges of those targeted, thereby achieving a level of validation.

CONCLUSION

The finalized analogue required nine axioms. We posit that the validated analogue's operating principles are reasonable representations of those utilized by EMT6/Ro cells during tumor spheroid development.

A computational model of chemotaxis-based cell aggregation

We present a computational model that successfully captures the cell behaviors that play important roles in 2-D cell aggregation. A virtual cell in our model is designed as an independent, discrete unit with a set of parameters and actions. Each cell is defined by its location, size, rates of chemoattractant emission and response, age, life cycle stage, proliferation rate and number of attached cells. All cells are capable of emitting and sensing a chemoattractant chemical, moving, attaching to other cells, dividing, aging and dying. We validated and fine-tuned our in silico model by comparing simulated 24-h aggregation experiments with data derived from in vitro experiments using PC12 pheochromocytoma cells. Quantitative comparisons of the aggregate size distributions from the two experiments are produced using the Earth Mover's Distance (EMD) metric. We compared the two size distributions produced after 24 h of in vitro cell aggregation and the corresponding computer simulated process. Iteratively modifying the model's parameter values and measuring the difference between the in silico and in vitro results allow us to determine the optimal values that produce simulated aggregation outcomes closely matched to the PC12 experiments. Simulation results demonstrate the ability of the model to recreate large-scale aggregation behaviors seen in live cell experiments.

Integrating morphogenesis with underlying mechanics and cell biology

Morphogenesis integrates a wide range of cellular processes into a self-organizing, self-deforming tissue. No single molecular "magic bullet" controls morphogenesis. Wide ranging cellular processes, often without parallels in conventional cell culture systems, work together to generate the architecture and modulate forces that produce and guide shape changes in the embryo. In this review we summarize the early development of the frog Xenopus laevis from a biomechanical perspective. We describe processes operating in the embryo from whole embryo scale, the tissue-scale, to the cellular and extracellular matrix scale. We focus on describing cells, their behaviors and the unique microenvironments they traverse during gastrulation and discuss the role of tissue mechanics in these processes.

Multiscale computational analysis of Xenopus laevis morphogenesis reveals key insights of systems-level behavior

Background

Tissue morphogenesis is a complex process whereby tissue structures self-assemble by the aggregate behaviors of independently acting cells responding to both intracellular and extracellular cues in their environment. During embryonic development, morphogenesis is particularly important for organizing cells into tissues, and although key regulatory events of this process are well studied in isolation, a number of important systems-level questions remain unanswered. This is due, in part, to a lack of integrative tools that enable the coupling of biological phenomena across spatial and temporal scales. Here, we present a new computational framework that integrates intracellular signaling information with multi-cell behaviors in the context of a spatially heterogeneous tissue environment.

Results

We have developed a computational simulation of mesendoderm migration in the Xenopus laevis explant model, which is a well studied biological model of tissue morphogenesis that recapitulates many features of this process during development in humans. The simulation couples, via a JAVA interface, an ordinary differential equation-based mass action kinetics model to compute intracellular Wnt/β-catenin signaling with an agent-based model of mesendoderm migration across a fibronectin extracellular matrix substrate. The emergent cell behaviors in the simulation suggest the following properties of the system: maintaining the integrity of cell-to-cell contact signals is necessary for preventing fractionation of cells as they move, contact with the Fn substrate and the existence of a Fn gradient provides an extracellular feedback loop that governs migration speed, the incorporation of polarity signals is required for cells to migrate in the same direction, and a delicate balance of integrin and cadherin interactions is needed to reproduce experimentally observed migratory behaviors.

Conclusion

Our computational framework couples two different spatial scales in biology: intracellular with multicellular. In our simulation, events at one scale have quantitative and dynamic impact on events at the other scale. This integration enables the testing and identification of key systems-level hypotheses regarding how signaling proteins affect overall tissue-level behavior during morphogenesis in an experimentally verifiable system. Applications of this approach extend to the study of tissue patterning processes that occur during adulthood and disease, such as tumorgenesis and atherogenesis.

Simulating invasion with cellular automata: connecting cell-scale and population-scale properties

Interpretive and predictive tools are needed to assist in the understanding of cell invasion processes. Cell invasion involves cell motility and proliferation, and is central to many biological processes including developmental morphogenesis and tumor invasion. Experimental data can be collected across a wide range of scales, from the population scale to the individual cell scale. Standard continuum or discrete models used in isolation are insufficient to capture this wide range of data. We develop a discrete cellular automata model of invasion with experimentally motivated rules. The cellular automata algorithm is applied to a narrow two-dimensional lattice and simulations reveal the formation of invasion waves moving with constant speed. The simulation results are averaged in one dimension-these data are used to identify the time history of the leading edge to characterize the population-scale wave speed. This allows the relationship between the population-scale wave speed and the cell-scale parameters to be determined. This relationship is analogous to well-known continuum results for Fisher's equation. The cellular automata algorithm also produces individual cell trajectories within the invasion wave that are analogous to cell trajectories obtained with new experimental techniques. Our approach allows both the cell-scale and population-scale properties of invasion to be predicted in a way that is consistent with multiscale experimental data. Furthermore we suggest that the cellular automata algorithm can be used in conjunction with individual data to overcome limitations associated with identifying cell motility mechanisms using continuum models alone.

Coalescence of interacting cell populations

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

Multi-cell agent-based simulation of the microvasculature to study the dynamics of circulating inflammatory cell trafficking

Leukocyte trafficking through the microcirculation and into tissues is central in angiogenesis, inflammation, and the immune response. Although the literature is rich with mechanistic detail describing molecular mediators of these processes, integration of signaling events and cell behaviors within a unified spatial and temporal framework at the multi-cell tissue-level is needed to achieve a fuller understanding. We have developed a novel computational framework that combines agent-based modeling (ABM) with a network flow analysis to study monocyte homing. A microvascular network architecture derived from mouse muscle was incorporated into the ABM. Each individual cell was represented by an individual agent in the simulation. The network flow model calculates hemodynamic parameters (blood flow rates, fluid shear stress, and hydrostatic pressures) throughout the simulated microvascular network. These are incorporated into the ABM to affect monocyte transit through the network and chemokine/cytokine concentrations. In turn, simulated monocytes respond to their local mechanical and biochemical environments and make behavioral decisions based on a rule set derived from independent literature. Simulated cell behaviors give rise to emergent leukocyte rolling, adhesion, and extravasation. Molecular knockout simulations were performed to validate the model, and predictions of monocyte adhesion, rolling, and extravasation show good agreement with the independently published corresponding mouse studies.

Enabling tools for tissue engineering

Tissue engineering is a multidisciplinary field that combines engineering, physical sciences, biology, and medicine to restore or replace tissues and organs functions. In this review, enabling tools for tissue engineering are discussed in the context of four key areas or pillars: prediction, production, performance, and preservation. Prediction refers to the computational modeling where the ability to simulate cellular behavior in complex three-dimensional environments will be essential for design of tissues. Production refer imaging modalities that allow high resolution, non-invasive monitoring of the development and incorporation of tissue engineered constructs. Lastly, preservation includes biochemical tools that permit cryopreservation, vitrification, and freeze-drying of cells and tissues. Recent progress and future perspectives for development in each of these key areas are presented.

Cell proliferation drives neural crest cell invasion of the intestine

A general mathematical model of cell invasion is developed and validated with an experimental system. The model incorporates two basic cell functions: non-directed (diffusive) motility and proliferation to a carrying capacity limit. The model is used here to investigate cell proliferation and motility differences along the axis of an invasion wave. Mathematical simulations yield surprising and counterintuitive predictions. In this general scenario, cells at the invasive front are proliferative and migrate into previously unoccupied tissues while those behind the front are essentially nonproliferative and do not directly migrate into unoccupied tissues. These differences are not innate to the cells, but are a function of proximity to uninvaded tissue. Therefore, proliferation at the invading front is the critical mechanism driving apparently directed invasion. An appropriate system to experimentally validate these predictions is the directional invasion and colonization of the gut by vagal neural crest cells that establish the enteric nervous system. An assay using gut organ culture with chick-quail grafting is used for this purpose. The experimental results are entirely concordant with the mathematical predictions. We conclude that proliferation at the wavefront is a key mechanism driving the invasive process. This has important implications not just for the neural crest, but for other invasion systems such as epidermal wound healing, carcinoma invasion and other developmental cell migrations.

Radial intercalation of ciliated cells during Xenopus skin development

Cells with motile cilia cover the skin of Xenopus tadpoles in a characteristic spacing pattern. This pattern arises during early development when cells within the inner layer of ectoderm are selected out by Notch to form ciliated cell precursors (CCPs) that then radially intercalate into the outer epithelial cell layer to form ciliated cells. When Notch is inhibited and CCPs are overproduced, radial intercalation becomes limiting and the spacing of ciliated cells is maintained. To determine why this is the case, we used confocal microscopy to image intercalating cells labeled using transplantation and a transgenic approach that labels CCPs with green fluorescent protein (GFP). Our results indicate that inner cells intercalate by first wedging between the basal surface of the outer epithelium but only insert apically at the vertices where multiple outer cells make contact. When overproduced, more CCPs are able to wedge basally, but apical insertion becomes limiting. We propose that limitations imposed by the outer layer, along with restrictions on the apical insertion of CCPs, determine their pattern of radial intercalation.

Mathematical modelling of engineered tissue growth using a multiphase porous flow mixture theory

This paper outlines the framework of a porous flow mixture theory for the mathematical modelling of in vitro tissue growth, and gives an application of this theory to an aspect of tissue engineering. The problem is formulated as a set of partial differential equations governing the space and time dependence of the amounts of each component of the tissue (phase), together with the physical stresses in each component. The theory requires constitutive relations to specify the material properties of each phase, and also requires relations to specify the stresses developed due to mechanical interactions, both within each phase and between different phases. An application of the theory is given to the study of the mobility and aggregation of a population of cells seeded into an artificial polymeric scaffold. Stability analysis techniques show that the interplay of the forces between the tissue constituents results in two different regimes: either the cells form aggregates or disperse through the scaffold.