Kinetic Monte Carlo and cellular particle dynamics simulations of multicellular systems

Computational methods for biofabrication in tissue engineering and regenerative medicine - a literature review

This literature review rigorously examines the growing scientific interest in computational methods for Tissue Engineering and Regenerative Medicine biofabrication, a leading-edge area in biomedical innovation, emphasizing the need for accurate, multi-stage, and multi-component biofabrication process models. The paper presents a comprehensive bibliometric and contextual analysis, followed by a literature review, to shed light on the vast potential of computational methods in this domain. It reveals that most existing methods focus on single biofabrication process stages and components, and there is a significant gap in approaches that utilize accurate models encompassing both biological and technological aspects. This analysis underscores the indispensable role of these methods in understanding and effectively manipulating complex biological systems and the necessity for developing computational methods that span multiple stages and components. The review concludes that such comprehensive computational methods are essential for developing innovative and efficient Tissue Engineering and Regenerative Medicine biofabrication solutions, driving forward advancements in this dynamic and evolving field.

Multi-scale nature of the tissue surface tension: Theoretical consideration on tissue model systems

Tissue surface tension is one of the key parameters that govern tissue rearrangement, shaping, and segregation within various compartments during organogenesis, wound healing, and cancer diseases. Deeper insight into the relationship between tissue surface tension and cell residual stress accumulation caused by collective cell migration can help us to understand the multi-scale nature of cell rearrangement with pronounced oscillatory trend. Oscillatory change of cell velocity that caused strain and generated cell residual stress were discussed in the context of mechanical waves. The tissue surface tension also showed oscillatory behaviour. The main goal of this theoretical consideration is to emphasize an inter-relation between various scenarios of cell rearrangement and tissue surface tension by distinguishing liquid-like and solid-like surfaces. This complex phenomenon is discussed in the context of an artificial tissue model system, namely cell aggregate rounding after uni-axial compression between parallel plates. Experimentally obtained oscillatory changes in the cell aggregate shape during the aggregate rounding, which is accompanied by oscillatory decrease in the aggregate surface area, points to oscillatory changes in the tissue surface tension. Besides long-time oscillations, cell surface tension can perform short time relaxation cycles. This behaviour of the tissue surface tension distinguishes living matter from other soft matter systems. This complex phenomenon is discussed based on dilatational viscoelasticity and thermodynamic approach.

Mathematical and computational models in spheroid-based biofabrication

Ubiquitous in embryonic development, tissue fusion is of interest to tissue engineers who use tissue spheroids or organoids as building blocks of three-dimensional (3D) multicellular constructs. This review presents mathematical models and computer simulations of the fusion of tissue spheroids. The motivation of this study stems from the need to predict the post-printing evolution of 3D bioprinted constructs. First, we provide a brief overview of differential adhesion, the main morphogenetic mechanism involved in post-printing structure formation. It will be shown that clusters of cohesive cells behave as an incompressible viscous fluid on the time scale of hours. The discussion turns then to mathematical models based on the continuum hydrodynamics of highly viscous liquids and on statistical mechanics. Next, we analyze the validity and practical use of computational models of multicellular self-assembly in live constructs created by tissue spheroid bioprinting. Finally, we discuss the perspectives of the field as machine learning starts to reshape experimental design, and modular robotic workstations tend to alleviate the burden of repetitive tasks in biofabrication. STATEMENT OF SIGNIFICANCE: Bioprinted constructs are living systems, which evolve via morphogenetic mechanisms known from developmental biology. This review presents mathematical and computational tools devised for modeling post-printing structure formation. They help achieving a desirable outcome without expensive optimization experiments. While previous reviews mainly focused on assumptions, technical details, strengths, and limitations of computational models of multicellular self-assembly, this article discusses their validity and practical use in biofabrication. It also presents an overview of mathematical models that proved to be useful in the evaluation of experimental data on tissue spheroid fusion, and in the calibration of computational models. Finally, the perspectives of the field are discussed in the advent of robotic biofabrication platforms and bioprinting process optimization by machine learning.

Arrested coalescence of multicellular aggregates

Multicellular aggregates are known to exhibit liquid-like properties. The fusion process of two cell aggregates is commonly studied as the coalescence of two viscous drops. However, tissues are complex materials and can exhibit viscoelastic behaviour. It is known that elastic effects can prevent the complete fusion of two drops, a phenomenon known as arrested coalescence. Here we study this phenomenon in stem cell aggregates and provide a theoretical framework which agrees with the experiments. In addition, agent-based simulations show that active cell fluctuations can control a solid-to-fluid phase transition, revealing that arrested coalescence can be found in the vicinity of an unjamming transition. By analysing the dynamics of the fusion process and combining it with nanoindentation measurements, we obtain the effective viscosity, shear modulus and surface tension of the aggregates. More generally, our work provides a simple, fast and inexpensive method to characterize the mechanical properties of viscoelastic materials.

Fusion Dynamics of Hybrid Cell-Microparticle Aggregates: A Jelly Pearl Model

We study the fusion of homogeneous cell aggregates and of hybrid aggregates combining cells and microparticles. In all cases, we find that the contact area does not vary linearly over time, as observed for liquid drops, but rather it follows a power law in t^2/3. This result is interpreted by generalizing the fusion model of soft viscoelastic solid balls to viscoelastic liquid balls, akin to jelly pearls. We also explore the asymmetric fusion between a homogeneous aggregate and a hybrid aggregate. This latter experiment allows the determination of the self-diffusion coefficient of the cells in a tissue by following the spatial distribution of internalized particles in the cells.

Continuum Theory of Active Phase Separation in Cellular Aggregates

Dense cellular aggregates are common in biology, ranging from bacterial biofilms to organoids, cell spheroids, and tumors. Their dynamics, driven by intercellular forces, is intrinsically out of equilibrium. Motivated by bacterial colonies as a model system, we present a continuum theory to study dense, active, cellular aggregates. We describe the process of aggregate formation as an active phase separation phenomenon, while the merging of aggregates is rationalized as a coalescence of viscoelastic droplets where the key timescales are linked to the turnover of the active force. Our theory provides a general framework for studying the rheology and nonequilibrium dynamics of dense cellular aggregates.

Cell spheroid fusion: beyond liquid drops model

Biological self-assembly is crucial in the processes of development, tissue regeneration, and maturation of bioprinted tissue-engineered constructions. The cell aggregates-spheroids-have become widely used model objects in the study of this phenomenon. Existing approaches describe the fusion of cell aggregates by analogy with the coalescence of liquid droplets and ignore the complex structural properties of spheroids. Here, we analyzed the fusion process in connection with structure and mechanical properties of the spheroids from human somatic cells of different phenotypes: mesenchymal stem cells from the limbal eye stroma and epithelial cells from retinal pigment epithelium. A nanoindentation protocol was applied for the mechanical measurements. We found a discrepancy with the liquid drop fusion model: the fusion was faster for spheroids from epithelial cells with lower apparent surface tension than for mesenchymal spheroids with higher surface tension. This discrepancy might be caused by biophysical processes such as extracellular matrix remodeling in the case of mesenchymal spheroids and different modes of cell migration. The obtained results will contribute to the development of more realistic models for spheroid fusion that would further provide a helpful tool for constructing cell aggregates with required properties both for fundamental studies and tissue reparation.

A lattice kinetic Monte-Carlo method for simulating chromosomal dynamics and other (non-)equilibrium bio-assemblies

Biological assemblies in living cells such as chromosomes constitute large many-body systems that operate in a fluctuating, out-of-equilibrium environment. Since a brute-force simulation of that many degrees of freedom is currently computationally unfeasible, it is necessary to perform coarse-grained stochastic simulations. Here, we develop all tools necessary to write a lattice kinetic Monte-Carlo (LKMC) algorithm capable of performing such simulations. We discuss the validity and limits of this approach by testing the results of the simulation method in simple settings. Importantly, we illustrate how at large external forces Metropolis-Hastings kinetics violate the fluctuation-dissipation and steady-state fluctuation theorems and discuss better alternatives. Although this simulation framework is rather general, we demonstrate our approach using a DNA polymer with interacting SMC condensin loop-extruding enzymes. Specifically, we show that the scaling behavior of the loop-size distributions that we obtain in our LKMC simulations of this SMC-DNA system is consistent with that reported in other studies using Brownian dynamics simulations and analytic approaches. Moreover, we find that the irreversible dynamics of these enzymes under certain conditions result in frozen, sterically jammed polymer configurations, highlighting a potential pitfall of this approach.

Synthetic development: building mammalian multicellular structures with artificial genetic programs

Synthetic biology efforts began in simple single-cell systems, which were relatively easy to manipulate genetically (Cameron et al., 2014). The field grew exponentially in the last two decades, and one of the latest frontiers are synthetic developmental programs for multicellular mammalian systems (Black et al., 2017; Wieland and Fussenegger, 2012) to genetically control features such as patterning or morphogenesis. These programs rely on engineered cell-cell communications, multicellular gene regulatory networks and effector genes. Here, we contextualize the first of these synthetic developmental programs, examine molecular and computational tools that can be used to generate next generation versions, and present the general logic that underpins these approaches. These advances are exciting as they represent a novel way to address both control and understanding in the field of developmental biology and tissue development (Elowitz and Lim, 2010; Velazquez et al., 2018; White et al., 2018; Morsut, 2017). This field is just at the beginning, and it promises to be of major interest in the upcoming years of biomedical research.

Jamming state transition and collective cell migration

Jamming state transition has been used in literature to describe migrating-to-resting cell state transition during collective cell migration without proper rheological confirmation. Yield stress often has been used as an indicator of a jamming state. Yield stress points to the liquid-to-solid state transition, but not a priori to jamming state transition. Various solid states such as elastic solid and viscoelastic solids can be considered in the context of their ability to relax. The relaxation time for (1) an elastic solid tends to zero, (2) Kelvin-Voigt viscoelastic solid is finite, and (3) jamming state tends to infinity.

In order to clarify the meaning of jamming state from the rheological standpoint we formulated the constitutive model of this state based on following conditions (1) migration of the system constituents is much damped such that the diffusion coefficient tends to zero, (2) relaxation time tends to infinity, (3) storage and loss moduli satisfy the condition G^′(ω)/G^"(ω) = const > 1. Jamming state represents the non-linear viscoelastic solid state. The main characteristic of this state is that the system cannot relax.

Jamming state transition of multicellular systems caused by collective cell migration is discussed on a model system such as cell aggregate rounding after uni-axial compression between parallel plates based on the data from the literature. Cell aggregate rounding occurs via successive relaxation cycles. Every cycle corresponds to a different scenario of cell migration. Three scenarios were established depending on the magnitude of mechanical and biochemical perturbations (1) ordered scenario with reduced perturbations corresponds to the case that most of the cells migrate, (2) disordered scenario corresponds to the case that some cell groups migrate while the others (at the same time) stay in resting state (corresponds to medium perturbations), and (3) highly suppressed cell migration under large perturbations corresponds to the viscoelastic solid under jamming state. If cells reach the jamming state in one cycle, they are able to overcome this undesirable state and start migrating again in the next cycle by achieving the first or second scenarios again.

Functional Epithelium Remodeling in Response to Applied Stress under In Vitro Conditions

Mathematical modeling is often used in tissue engineering in order to overcome one of its major challenges: transformation of complex biological and rheological behaviors of cells and tissue in a mathematically predictive and physically manipulative engineering process. The successive accomplishment of this task will greatly help in quantifying and optimizing clinical application of the tissue engineering products. One of the problems emerging in this area is the relation between resting and migrating cell groups, as well as between different configurations of migrating cells and viscoelasticity. A deeper comprehension of the relation between various configurations of migrating cells and viscoelasticity at the supracellular level represents the prerequisite for optimization of the performance of the artificial epithelium. Since resting and migrating cell groups have a considerable difference in stiffness, a change in their mutual volume ratio and distribution may affect the viscoelasticity of multicellular surfaces. If those cell groups are treated as different phases, then an analogous model may be applied to represent such systems. In this work, a two-step Eyring model is developed in order to demonstrate the main mechanical and biochemical factors that influence configurations of migrating cells. This model could be also used for considering the long-time cell rearrangement under various types of applied stress. The results of this theoretical analysis point out the cause-consequence relationship between the configuration of migrating cells and rheological behavior of multicellular surfaces. Configuration of migrating cells is influenced by mechanical and biochemical perturbations, difficult to measure experimentally, which lead to uncorrelated motility. Uncorrelated motility results in (1) decrease of the volume fraction of migrating cells, (2) change of their configuration, and (3) softening of multicellular surfaces.

Molecular Motors Govern Liquidlike Ordering and Fusion Dynamics of Bacterial Colonies

Bacteria can adjust the structure of colonies and biofilms to enhance their survival rate under external stress. Here, we explore the link between bacterial interaction forces and colony structure. We show that the activity of extracellular pilus motors enhances local ordering and accelerates fusion dynamics of bacterial colonies. The radial distribution function of mature colonies shows local fluidlike order. The degree and dynamics of ordering are dependent on motor activity. At a larger scale, the fusion dynamics of two colonies shows liquidlike behavior whereby motor activity strongly affects surface tension and viscosity.

3D bioprinting of tissues and organs for regenerative medicine

3D bioprinting is a pioneering technology that enables fabrication of biomimetic, multiscale, multi-cellular tissues with highly complex tissue microenvironment, intricate cytoarchitecture, structure-function hierarchy, and tissue-specific compositional and mechanical heterogeneity. Given the huge demand for organ transplantation, coupled with limited organ donors, bioprinting is a potential technology that could solve this crisis of organ shortage by fabrication of fully-functional whole organs. Though organ bioprinting is a far-fetched goal, there has been a considerable and commendable progress in the field of bioprinting that could be used as transplantable tissues in regenerative medicine. This paper presents a first-time review of 3D bioprinting in regenerative medicine, where the current status and contemporary issues of 3D bioprinting pertaining to the eleven organ systems of the human body including skeletal, muscular, nervous, lymphatic, endocrine, reproductive, integumentary, respiratory, digestive, urinary, and circulatory systems were critically reviewed. The implications of 3D bioprinting in drug discovery, development, and delivery systems are also briefly discussed, in terms of in vitro drug testing models, and personalized medicine. While there is a substantial progress in the field of bioprinting in the recent past, there is still a long way to go to fully realize the translational potential of this technology. Computational studies for study of tissue growth or tissue fusion post-printing, improving the scalability of this technology to fabricate human-scale tissues, development of hybrid systems with integration of different bioprinting modalities, formulation of new bioinks with tuneable mechanical and rheological properties, mechanobiological studies on cell-bioink interaction, 4D bioprinting with smart (stimuli-responsive) hydrogels, and addressing the ethical, social, and regulatory issues concerning bioprinting are potential futuristic focus areas that would aid in successful clinical translation of this technology.

Jamming transitions in cancer

The traditional picture of tissues, where they are treated as liquids defined by properties such as surface tension or viscosity has been redefined during the last few decades by the more fundamental question: under which conditions do tissues display liquid-like or solid-like behaviour? As a result, basic concepts arising from the treatment of tissues as solid matter, such as cellular jamming and glassy tissues, have shifted into the current focus of biophysical research. Here, we review recent works examining the phase states of tissue with an emphasis on jamming transitions in cancer. When metastasis occurs, cells gain the ability to leave the primary tumour and infiltrate other parts of the body. Recent studies have shown that a linkage between an unjamming transition and tumour progression indeed exists, which could be of importance when designing surgery and treatment approaches for cancer patients.

'Printability' of Candidate Biomaterials for Extrusion Based 3D Printing: State-of-the-Art

Regenerative medicine has been highlighted as one of the UK's 8 'Great Technologies' with the potential to revolutionize patient care in the 21st Century. Over the last decade, the concept of '3D bioprinting' has emerged, which allows the precise deposition of cell laden bioinks with the aim of engineering complex, functional tissues. For 3D printing to be used clinically, there is the need to produce advanced functional biomaterials, a new generation of bioinks with suitable cell culture and high shape/print fidelity, to match or exceed the physical, chemical and biological properties of human tissue. With the rapid increase in knowledge associated with biomaterials, cell-scaffold interactions and the ability to biofunctionalize/decorate bioinks with cell recognition sequences, it is important to keep in mind the 'printability' of these novel materials. In this illustrated review, we define and refine the concept of 'printability' and review seminal and contemporary studies to highlight the current 'state of play' in the field with a focus on bioink composition and concentration, manipulation of nozzle parameters and rheological properties.

Physical models of collective cell motility: from cell to tissue

In this article, we review physics-based models of collective cell motility. We discuss a range of techniques at different scales, ranging from models that represent cells as simple self-propelled particles to phase field models that can represent a cell's shape and dynamics in great detail. We also extensively review the ways in which cells within a tissue choose their direction, the statistics of cell motion, and some simple examples of how cell-cell signaling can interact with collective cell motility. This review also covers in more detail selected recent works on collective cell motion of small numbers of cells on micropatterns, in wound healing, and the chemotaxis of clusters of cells.

Design and Printing Strategies in 3D Bioprinting of Cell-Hydrogels: A Review

Bioprinting is an emerging technology that allows the assembling of both living and non-living biological materials into an ideal complex layout for further tissue maturation. Bioprinting aims to produce engineered tissue or organ in a mechanized, organized, and optimized manner. Various biomaterials and techniques have been utilized to bioprint biological constructs in different shapes, sizes and resolutions. There is a need to systematically discuss and analyze the reported strategies employed to fabricate these constructs. We identified and discussed important design factors in bioprinting, namely shape and resolution, material heterogeneity, and cellular-material remodeling dynamism. Each design factors are represented by the corresponding process capabilities and printing parameters. The process-design map will inspire future biomaterials research in these aspects. Design considerations such as data processing, bio-ink formulation and process selection are discussed. Various printing and crosslinking strategies, with relevant applications, are also systematically reviewed. We categorized them into 5 general bioprinting strategies, including direct bioprinting, in-process crosslinking, post-process crosslinking, indirect bioprinting and hybrid bioprinting. The opportunities and outlook in 3D bioprinting are highlighted. This review article will serve as a framework to advance computer-aided design in bioprinting technologies.

Lateral assembly of N-cadherin drives tissue integrity by stabilizing adherens junctions

Cadherin interactions ensure the correct registry and anchorage of cells during tissue formation. Along the plasma membrane, cadherins form inter-junctional lattices via cis- and trans-dimerization. While structural studies have provided models for cadherin interactions, the molecular nature of cadherin binding in vivo remains unexplored. We undertook a multi-disciplinary approach combining live cell imaging of three-dimensional cell assemblies (spheroids) with a computational model to study the dynamics of N-cadherin interactions. Using a loss-of-function strategy, we demonstrate that each N-cadherin interface plays a distinct role in spheroid formation. We found that cis-dimerization is not a prerequisite for trans-interactions, but rather modulates trans-interfaces to ensure tissue stability. Using a model of N-cadherin junction dynamics, we show that the absence of cis-interactions results in low junction stability and loss of tissue integrity. By quantifying the binding and unbinding dynamics of the N-cadherin binding interfaces, we determined that mutating either interface results in a 10-fold increase in the dissociation constant. These findings provide new quantitative information on the steps driving cadherin intercellular adhesion and demonstrate the role of cis-interactions in junction stability.

Predictive modeling of post bioprinting structure formation

Cellular particle dynamics (CPD) is an effective computational method to describe the shape evolution and biomechanical relaxation processes in systems composed of micro tissues such as multicellular aggregates. Therefore, CPD is a useful tool to predict the outcome of postprinting structure formation in bioprinting. The predictive power of CPD has been demonstrated for multicellular systems composed of identical volume-conserving spherical and cylindrical bioink units. Experiments and computer simulations were related through an independently developed theoretical formalism based on continuum mechanics. Here we generalize the CPD formalism to (i) include non-identical bioink particles often used in specific bioprinting applications, (ii) describe the more realistic experimental situation in which during the post-printing structure formation via the fusion of spherical bioink units the volume of the system decreases, and (iii) directly connect CPD simulations to the corresponding experiments without the need of the intermediate continuum theory inherently based on simplifying assumptions.

In-silico analysis on biofabricating vascular networks using kinetic Monte Carlo simulations

We present a computational modeling approach to study the fusion of multicellular aggregate systems in a novel scaffold-less biofabrication process, known as 'bioprinting'. In this novel technology, live multicellular aggregates are used as fundamental building blocks to make tissues or organs (collectively known as the bio-constructs,) via the layer-by-layer deposition technique or other methods; the printed bio-constructs embedded in maturogens, consisting of nutrient-rich bio-compatible hydrogels, are then placed in bioreactors to undergo the cellular aggregate fusion process to form the desired functional bio-structures. Our approach reported here is an agent-based modeling method, which uses the kinetic Monte Carlo (KMC) algorithm to evolve the cellular system on a lattice. In this method, the cells and the hydrogel media, in which cells are embedded, are coarse-grained to material's points on a three-dimensional (3D) lattice, where the cell-cell and cell-medium interactions are quantified by adhesion and cohesion energies. In a multicellular aggregate system with a fixed number of cells and fixed amount of hydrogel media, where the effect of cell differentiation, proliferation and death are tactically neglected, the interaction energy is primarily dictated by the interfacial energy between cell and cell as well as between cell and medium particles on the lattice, respectively, based on the differential adhesion hypothesis. By using the transition state theory to track the time evolution of the multicellular system while minimizing the interfacial energy, KMC is shown to be an efficient time-dependent simulation tool to study the evolution of the multicellular aggregate system. In this study, numerical experiments are presented to simulate fusion and cell sorting during the biofabrication process of vascular networks, in which the bio-constructs are fabricated via engineering designs. The results predict the feasibility of fabricating the vascular structures via the bioprinting technology and demonstrate the morphological development process during cellular aggregate fusion in various engineering designed structures. The study also reveals that cell sorting will perhaps not significantly impact the final fabricated products, should the maturation process be well-controlled in bioprinting.

Partial mixing phase of binary cells in finite systems

We study the self-organization of binary cell mixtures in finite cubic lattices. Depending on the relative attractions between cell types, the binary mixture model generates four distinct cellular associations: complete sorting, shell-core sorting, partial mixing, and complete mixing of heterotypic cells. At the boundaries between these four phases, the cellular associations show large variations, representing phase transitions. We find that the partial mixing phase is highly tolerant to thermal fluctuations. Interestingly, human pancreatic islets, the micro-organs for glucose homeostasis, adapt the partial mixing phase consisting of α and β cells.

Emergent complexity of the cytoskeleton: from single filaments to tissue

Despite their overwhelming complexity, living cells display a high degree of internal mechanical and functional organization which can largely be attributed to the intracellular biopolymer scaffold, the cytoskeleton. Being a very complex system far from thermodynamic equilibrium, the cytoskeleton's ability to organize is at the same time challenging and fascinating. The extensive amounts of frequently interacting cellular building blocks and their inherent multifunctionality permits highly adaptive behavior and obstructs a purely reductionist approach. Nevertheless (and despite the field's relative novelty), the physics approach has already proved to be extremely successful in revealing very fundamental concepts of cytoskeleton organization and behavior. This review aims at introducing the physics of the cytoskeleton ranging from single biopolymer filaments to multicellular organisms. Throughout this wide range of phenomena, the focus is set on the intertwined nature of the different physical scales (levels of complexity) that give rise to numerous emergent properties by means of self-organization or self-assembly.