A stochastic model for adhesion-mediated cell random motility and haptotaxis

Phenotypic switching mechanisms determine the structure of cell migration into extracellular matrix under the 'go-or-grow' hypothesis

A fundamental feature of collective cell migration is phenotypic heterogeneity which, for example, influences tumour progression and relapse. While current mathematical models often consider discrete phenotypic structuring of the cell population, in-line with the 'go-or-grow' hypothesis (Hatzikirou et al., 2012; Stepien et al., 2018), they regularly overlook the role that the environment may play in determining the cells' phenotype during migration. Comparing a previously studied volume-filling model for a homogeneous population of generalist cells that can proliferate, move and degrade extracellular matrix (ECM) (Crossley et al., 2023) to a novel model for a heterogeneous population comprising two distinct sub-populations of specialist cells that can either move and degrade ECM or proliferate, this study explores how different hypothetical phenotypic switching mechanisms affect the speed and structure of the invading cell populations. Through a continuum model derived from its individual-based counterpart, insights into the influence of the ECM and the impact of phenotypic switching on migrating cell populations emerge. Notably, specialist cell populations that cannot switch phenotype show reduced invasiveness compared to generalist cell populations, while implementing different forms of switching significantly alters the structure of migrating cell fronts. This key result suggests that the structure of an invading cell population could be used to infer the underlying mechanisms governing phenotypic switching.

Reassessing the Standard Chemotaxis Framework for Understanding Biased Migration in Helicobacter pylori

Helicobacter pylori infections are a major cause of peptic ulcers and gastric cancers. The development of robust inflammation in response to these flagellated, motile bacteria is correlated with poor prognosis. Chemotaxis plays a crucial role in H. pylori colonization, enabling the bacteria to swim toward favorable chemical environments. Unlike the model species of bacterial chemotaxis, Escherichia coli, H. pylori cells possess polar flagella. They run forward by rotating their flagella counterclockwise, whereas backward runs are achieved by rotating their flagella clockwise. We delve into the implications of certain features of the canonical model of chemotaxis on our understanding of biased migration in polarly flagellated bacteria such as H. pylori. In particular, we predict how the translational displacement of H. pylori cells during a backward run could give rise to chemotaxis errors within the canonical framework. Also, H. pylori lack key chemotaxis enzymes found in E. coli, without which sensitive detection of ligands with a wide dynamic range seems unlikely. Despite these problems, H. pylori exhibit robust ability to migrate toward urea-rich sources. We emphasize various unresolved questions regarding the biophysical mechanisms of chemotaxis in H. pylori, shedding light on potential directions for future research. Understanding the intricacies of biased migration in H. pylori could offer valuable insights into how pathogens breach various protective barriers in the human host.

CRIPTO-based micro-heterogeneity of mouse muscle satellite cells enables adaptive response to regenerative microenvironment

Skeletal muscle repair relies on heterogeneous populations of satellite cells (SCs). The mechanisms that regulate SC homeostasis and state transition during activation are currently unknown. Here, we investigated the emerging role of non-genetic micro-heterogeneity, i.e., intrinsic cell-to-cell variability of a population, in this process. We demonstrate that micro-heterogeneity of the membrane protein CRIPTO in mouse-activated SCs (ASCs) identifies metastable cell states that allow a rapid response of the population to environmental changes. Mechanistically, CRIPTO micro-heterogeneity is generated and maintained through a process of intracellular trafficking coupled with active shedding of CRIPTO from the plasma membrane. Irreversible perturbation of CRIPTO micro-heterogeneity affects the balance of proliferation, self-renewal, and myogenic commitment in ASCs, resulting in increased self-renewal in vivo. Our findings demonstrate that CRIPTO micro-heterogeneity regulates the adaptative response of ASCs to microenvironmental changes, providing insights into the role of intrinsic heterogeneity in preserving stem cell population diversity during tissue repair.

Semi-autonomous wound invasion via matrix-deposited, haptotactic cues

Proper wound healing relies on invasion of fibroblasts via directed migration. While the related experimental and mathematical modeling literature has mainly focused on cell migration directed by soluble cues (chemotaxis), there is ample evidence that fibroblast migration is also directed by insoluble, matrix-bound cues (haptotaxis). Furthermore, numerous studies indicate that fibronectin (FN), a haptotactic ligand for fibroblasts, is present and dynamic in the provisional matrix throughout the proliferative phase of wound healing. In the present work, we show the plausibility of a hypothesis that fibroblasts themselves form and maintain haptotactic gradients in a semi-autonomous fashion. As a precursor to this, we examine the positive control scenario where FN is pre-deposited in the wound matrix, and fibroblasts maintain haptotaxis by removing FN at an appropriate rate. After developing conceptual and quantitative understanding of this scenario, we consider two cases in which fibroblasts activate the latent form of a matrix-loaded cytokine, TGFβ, which upregulates the fibroblasts' own secretion of FN. In the first of these, the latent cytokine is pre-patterned and released by the fibroblasts. In the second, fibroblasts in the wound produce the latent TGFβ, with the presence of the wound providing the only instruction. In all cases, wound invasion is more effective than a negative control model with haptotaxis disabled; however, there is a trade-off between the degree of fibroblast autonomy and the rate of invasion.

Assessing the roles of collagen fiber morphology and matrix stiffness on ovarian cancer cell migration dynamics using multiphoton fabricated orthogonal image-based models

Ovarian cancer remains the deadliest of the gynecological cancers, where this arises from poor screening and imaging tools that can detect early disease, and also limited understanding of the structural and functional aspects of the tumor microenvironment. To gain insight into the underlying cellular dynamics, we have used multiphoton excited fabrication to create Second Harmonic Generation (SHG) image-based orthogonal models from collagen/GelMA that represent both the collagen matrix morphology and stiffness (∼2-8 kPa) of normal ovarian stroma and high grade serous ovarian cancers (HGSOC). These scaffolds are used to study migration/cytoskeletal dynamics of normal (IOSE) and ovarian cancer (OVCA433) cell lines. We found that the highly aligned fiber morphology of HGSOC promotes aspects of motility (motility coefficient, motility, and focal adhesion expression) through a contact guidance mechanism and that stiffer matrix further promotes these same processes through a mechanosensitive mechanism, where these trends were similar for both normal and cancer cells. However, cell specific differences were found on these orthogonal models relative to those providing only morphology, showing the importance of presenting both morphology and stiffness cues. Moreover, we found increased cadherin expression and decreased cell alignment only for cancer cells on scaffolds of intermediate modulus suggesting different stiffness-dependent mechanotransduction mechanisms are engaged. This overall approach affords decoupling the roles of matrix morphology, stiffness and cell genotype and affords hypothesis testing of the factors giving rise to disease progression and metastasis. Further, more established fabrication techniques cannot simultaneously reproduce both the 3D collagen fiber morphology and stiffness. STATEMENT OF SIGNIFICANCE: Ovarian cancer metastasizes when lesions are small, where cells exfoliate from the surface of the ovary and reattach at distal sites in the peritoneum. The adhesion/migration dynamics are not well understood and there is a need for new 3D in vitro models of the extracellular matrix to study the biology. Here we use multiphoton excited crosslinking to fabricate ECM orthogonal models that represent the collagen morphology and stiffness in human ovarian tissues. These are then used to study ovarian cancer cell migration dynamics and we found that contact guidance and a mechanosensitive response and cell genotype all combine to affect the behavior. These models provide insight into disease etiology and progression not readily possible by other fabrication methods.

Bacterial Proprioception: Can a Bacterium Sense Its Movement?

The evolution of the bacterial flagellum gave rise to motility and repurposing of a signaling network, now termed the chemotaxis network, enabled biasing of cell movements. This made it possible for the bacterium to seek out favorable chemical environments. To enable chemotaxis, the chemotaxis network sensitively detects extracellular chemical stimuli and appropriately modulates flagellar functions. Additionally, the flagellar motor itself is capable of detecting mechanical stimuli and adapts its structure and function in response, likely triggering a transition from planktonic to surface-associated lifestyles. Recent work has shown a link between the flagellar motor's response to mechanical stimuli and the chemotactic output. Here, we elaborate on this link and discuss how it likely helps the cell sense and adapt to changes in its swimming speeds in different environments. We discuss the mechanism whereby the motor precisely tunes its chemotaxis output under different mechanical loads, analogous to proprioception in higher order organisms. We speculate on the roles bacterial proprioception might play in a variety of phenomena including the transition to surface-associated lifestyles such as swarming and biofilms.

In silico framework to inform the design of repair constructs for peripheral nerve injury repair

Peripheral nerve injuries affect millions of people per year and cause loss of sensation and muscle control alongside chronic pain. The most severe injuries are treated through a nerve autograft; however, donor site morbidity and poor outcomes mean alternatives are required. One option is to engineer nerve replacement tissues to provide a supportive microenvironment to encourage nerve regeneration as an alternative to nerve grafts. Currently, progress is hampered due to a lack of consensus on how to arrange materials and cells in space to maximize rate of regeneration. This is compounded by a reliance on experimental testing, which precludes extensive investigations of multiple parameters due to time and cost limitations. Here, a computational framework is proposed to simulate the growth of repairing neurites, captured using a random walk approach and parameterized against literature data. The framework is applied to a specific scenario where the engineered tissue comprises a collagen hydrogel with embedded biomaterial fibres. The size and number of fibres are optimized to maximize neurite regrowth, and the robustness of model predictions is tested through sensitivity analyses. The approach provides an in silico tool to inform the design of engineered replacement tissues, with the opportunity for further development to multi-cue environments.

BIO-LGCA: A cellular automaton modelling class for analysing collective cell migration

Collective dynamics in multicellular systems such as biological organs and tissues plays a key role in biological development, regeneration, and pathological conditions. Collective tissue dynamics—understood as population behaviour arising from the interplay of the constituting discrete cells—can be studied with on- and off-lattice agent-based models. However, classical on-lattice agent-based models, also known as cellular automata, fail to replicate key aspects of collective migration, which is a central instance of collective behaviour in multicellular systems. To overcome drawbacks of classical on-lattice models, we introduce an on-lattice, agent-based modelling class for collective cell migration, which we call biological lattice-gas cellular automaton (BIO-LGCA). The BIO-LGCA is characterised by synchronous time updates, and the explicit consideration of individual cell velocities. While rules in classical cellular automata are typically chosen ad hoc, rules for cell-cell and cell-environment interactions in the BIO-LGCA can also be derived from experimental cell migration data or biophysical laws for individual cell migration. We introduce elementary BIO-LGCA models of fundamental cell interactions, which may be combined in a modular fashion to model complex multicellular phenomena. We exemplify the mathematical mean-field analysis of specific BIO-LGCA models, which allows to explain collective behaviour. The first example predicts the formation of clusters in adhesively interacting cells. The second example is based on a novel BIO-LGCA combining adhesive interactions and alignment. For this model, our analysis clarifies the nature of the recently discovered invasion plasticity of breast cancer cells in heterogeneous environments.

Pattern formation during embryonic development and pathological tissue dynamics, such as cancer invasion, emerge from individual intercellular interactions. In order to study the impact of single cell dynamics and cell-cell interactions on tissue behaviour, one needs to develop space-time-dependent on- or off-lattice agent-based models (ABMs), which consider the behaviour of individual cells. However, classical on-lattice agent-based models also known as cellular automata fail to replicate key aspects of collective migration, which is a central instance of collective behaviour in multicellular systems. Here, we present the rule- and lattice-based BIO-LGCA modelling class which allows for (i) rigorous derivation of rules from biophysical laws and/or experimental data, (ii) mathematical analysis of collective migration, and (iii) computationally efficient simulations.

In Silico Mathematical Modelling for Glioblastoma: A Critical Review and a Patient-Specific Case

Glioblastoma extensively infiltrates the brain; despite surgery and aggressive therapies, the prognosis is poor. A multidisciplinary approach combining mathematical, clinical and radiological data has the potential to foster our understanding of glioblastoma evolution in every single patient, with the aim of tailoring therapeutic weapons. In particular, the ultimate goal of biomathematics for cancer is the identification of the most suitable theoretical models and simulation tools, both to describe the biological complexity of carcinogenesis and to predict tumor evolution. In this report, we describe the results of a critical review about different mathematical models in neuro-oncology with their clinical implications. A comprehensive literature search and review for English-language articles concerning mathematical modelling in glioblastoma has been conducted. The review explored the different proposed models, classifying them and indicating the significative advances of each one. Furthermore, we present a specific case of a glioblastoma patient in which our recently proposed innovative mechanical model has been applied. The results of the mathematical models have the potential to provide a relevant benefit for clinicians and, more importantly, they might drive progress towards improving tumor control and patient's prognosis. Further prospective comparative trials, however, are still necessary to prove the impact of mathematical neuro-oncology in clinical practice.

Role of neutrophil extracellular traps in regulation of lung cancer invasion and metastasis: Structural insights from a computational model

Lung cancer is one of the leading causes of cancer-related deaths worldwide and is characterized by hijacking immune system for active growth and aggressive metastasis. Neutrophils, which in their original form should establish immune activities to the tumor as a first line of defense, are undermined by tumor cells to promote tumor invasion in several ways. In this study, we investigate the mutual interactions between the tumor cells and the neutrophils that facilitate tumor invasion by developing a mathematical model that involves taxis-reaction-diffusion equations for the critical components in the interaction. These include the densities of tumor and neutrophils, and the concentrations of signaling molecules and structure such as neutrophil extracellular traps (NETs). We apply the mathematical model to a Boyden invasion assay used in the experiments to demonstrate that the tumor-associated neutrophils can enhance tumor cell invasion by secreting the neutrophil elastase. We show that the model can both reproduce the major experimental observation on NET-mediated cancer invasion and make several important predictions to guide future experiments with the goal of the development of new anti-tumor strategies. Moreover, using this model, we investigate the fundamental mechanism of NET-mediated invasion of cancer cells and the impact of internal and external heterogeneity on the migration patterning of tumour cells and their response to different treatment schedules.

Computational models of migration modes improve our understanding of metastasis

Tumor cells migrate through changing microenvironments of diseased and healthy tissue, making their migration particularly challenging to describe. To better understand this process, computational models have been developed for both the ameboid and mesenchymal modes of cell migration. Here, we review various approaches that have been used to account for the physical environment's effect on cell migration in computational models, with a focus on their application to understanding cancer metastasis and the related phenomenon of durotaxis. We then discuss how mesenchymal migration models typically simulate complex cell–extracellular matrix (ECM) interactions, while ameboid migration models use a cell-focused approach that largely ignores ECM when not acting as a physical barrier. This approach greatly simplifies or ignores the mechanosensing ability of ameboid migrating cells and should be reevaluated in future models. We conclude by describing future model elements that have not been included to date but would enhance model accuracy.

Nonlocal and local models for taxis in cell migration: a rigorous limit procedure

A rigorous limit procedure is presented which links nonlocal models involving adhesion or nonlocal chemotaxis to their local counterparts featuring haptotaxis and classical chemotaxis, respectively. It relies on a novel reformulation of the involved nonlocalities in terms of integral operators applied directly to the gradients of signal-dependent quantities. The proposed approach handles both model types in a unified way and extends the previous mathematical framework to settings that allow for general solution-dependent coefficient functions. The previous forms of nonlocal operators are compared with the new ones introduced in this paper and the advantages of the latter are highlighted by concrete examples. Numerical simulations in 1D provide an illustration of some of the theoretical findings.

A Stochastic Modelling Framework for Single Cell Migration: Coupling Contractility and Focal Adhesions

The interaction of the actin cytoskeleton with cell–substrate adhesions is necessary for cell migration. While the trajectories of motile cells have a stochastic character, investigations of cell motility mechanisms rarely elaborate on the origins of the observed randomness. Here, guided by a few fundamental attributes of cell motility, I construct a minimal stochastic cell migration model from ground-up. The resulting model couples a deterministic actomyosin contractility mechanism with stochastic cell–substrate adhesion kinetics, and yields a well-defined piecewise deterministic process. Numerical simulations reproduce several experimentally observed results, including anomalous diffusion, tactic migration and contact guidance. This work provides a basis for the development of cell–cell collision and population migration models.

Role of Collagen Fiber Morphology on Ovarian Cancer Cell Migration Using Image-Based Models of the Extracellular Matrix

Remodeling of the extracellular matrix (ECM) is an important part in the development and progression of many epithelial cancers. However, the biological significance of collagen alterations in ovarian cancer has not been well established. Here we investigated the role of collagen fiber morphology on cancer cell migration using tissue engineered scaffolds based on high-resolution Second-Harmonic Generation (SHG) images of ovarian tumors. The collagen-based scaffolds are fabricated by multiphoton excited (MPE) polymerization, which is a freeform 3D method affording submicron resolution feature sizes (~0.5 µm). This capability allows the replication of the collagen fiber architecture, where we constructed models representing normal stroma, high-risk tissue, benign tumors, and high-grade tumors. These were seeded with normal and ovarian cancer cell lines to investigate the separate roles of the cell type and matrix morphology on migration dynamics. The primary finding is that key cell–matrix interactions such as motility, cell spreading, f-actin alignment, focal adhesion, and cadherin expression are mainly determined by the collagen fiber morphology to a larger extent than the initial cell type. Moreover, we found these aspects were all enhanced for cells on the highly aligned, high-grade tumor model. Conversely, the weakest corresponding responses were observed on the more random mesh-like normal stromal matrix, with the partially aligned benign tumor and high-risk models demonstrating intermediate behavior. These results are all consistent with a contact guidance mechanism. These models cannot be synthesized by other conventional fabrication methods, and we suggest this approach will enable a variety of studies in cancer biology.

A mechanical toy model linking cell-substrate adhesion to multiple cellular migratory responses

During cell migration, forces applied to a cell from its environment influence the motion. When the cell is placed on a substrate, such a force is provided by the cell-substrate adhesion. Modulation of adhesivity, often performed by the modulation of the substrate stiffness, tends to cause common responses for cell spreading, cell speed, persistence, and random motility coefficient. Although the reasons for the response of cell spreading and cell speed have been suggested, other responses are not well understood. In this study, we develop a simple toy model for cell migration driven by the relation of two forces: the adhesive force and the plasma membrane tension. The simplicity of the model allows us to perform the calculation not only numerically but also analytically, and the analysis provides formulas directly relating the adhesivity to cell spreading, persistence, and the random motility coefficient. Accordingly, the results offer a unified picture on the causal relations between those multiple cellular responses. In addition, cellular properties that would influence the migratory behavior are suggested.

Migration dynamics of ovarian epithelial cells on micro-fabricated image-based models of normal and malignant stroma

A profound remodeling of the collagen in the extracellular matrix (ECM) occurs in human ovarian cancer but it is unknown how this affects migration dynamics and ultimately tumor growth. Here, we investigate the influence of collagen morphology on ovarian cell migration through the use of second harmonic generation (SHG) image-based models of ovarian tumors. The scaffolds are fabricated by multiphoton excited (MPE) polymerization, where the process is akin to 3D printing except it achieves much greater resolution (∼0.5 µm) and utilizes collagen and collagen analogs. We used this technique to create scaffolds with complex 3D submicron features representing the collagen fiber morphology in normal stroma, high risk stroma, benign tumors, and high grade ovarian tumors. We found the highly aligned malignant stromal structure promoted enhanced motility and also increased cell and f-Actin alignment relative to the other tissues. However, using models based on fiber crimping characteristics, we found cells seeded on linear fibers based on normal stromal models yielded the highest degree of alignment but least motility. These results show that both the fiber properties themselves and as well as their overall alignment govern the resulting migration dynamics. These models cannot be synthesized by other conventional fabrication methods and we suggest the MPE image-based fabrication method will enable a variety of studies in cancer biology. STATEMENT OF SIGNIFICANCE: The extracellular matrix collagen in ovarian cancer is highly remodeled but the consequences on cell function remain unknown. It is important to understand the operative cell matrix interactions, as this could lead to better prognostics and better prediction of therapeutic efficacy. We probe migration dynamics using high resolution (∼0.5 µm) multiphoton excited fabrication to synthesize scaffolds whose designs are derived directly from Second Harmonic Generation microscope images of the collagen in normal ovarian tissues as well as benign and malignant tumors. Collectively our results show the importance of the matrix morphology (fiber shape and alignment) on driving cell motility, cell shape and f-Actin alignment. These collagen-based models have complex fiber morphology and cannot be created by conventional fabrication technologies.

Modeling Cell Migration Mechanics

Cell migration is the physical movement of cells and is responsible for the extensive cellular invasion and metastasis that occur in high-grade tumors. Motivated by decades of direct observation of cell migration via light microscopy, theoretical models have emerged to capture various aspects of the fundamental physical phenomena underlying cell migration. Yet, the motility mechanisms actually used by tumor cells during invasion are still poorly understood, as is the role of cellular interactions with the extracellular environment. In this chapter, we review key physical principles of cytoskeletal self-assembly and force generation, membrane tension, biological adhesion, hydrostatic and osmotic pressures, and their integration in mathematical models of cell migration. With the goal of modeling-driven cancer therapy, we provide examples to guide oncologists and physical scientists in developing next-generation models to predict disease progression and treatment.

A stochastic algorithm for accurately predicting path persistence of cells migrating in 3D matrix environments

Cell mobility plays a critical role in immune response, wound healing, and the rate of cancer metastasis and tumor progression. Mobility within a three-dimensional (3D) matrix environment can be characterized by the average velocity of cell migration and the persistence length of the path it follows. Computational models that aim to predict cell migration within such 3D environments need to be able predict both of these properties as a function of the various cellular and extra-cellular factors that influence the migration process. A large number of models have been developed to predict the velocity of cell migration driven by cellular protrusions in 3D environments. However, prediction of the persistence of a cell's path is a more tedious matter, as it requires simulating cells for a long time while they migrate through the model extra-cellular matrix (ECM). This can be a computationally expensive process, and only recently have there been attempts to quantify cell persistence as a function of key cellular or matrix properties. Here, we propose a new stochastic algorithm that can simulate and analyze 3D cell migration occurring over days with a computation time of minutes, opening new possibilities of testing and predicting long-term cell migration behavior as a function of a large variety of cell and matrix properties. In this model, the matrix elements are generated as needed and stochastically based on the biophysical and biochemical properties of the ECM the cell migrates through. This approach significantly reduces the computational resources required to track and calculate cell matrix interactions. Using this algorithm, we predict the effect of various cellular and matrix properties such as cell polarity, cell mechanoactivity, matrix fiber density, matrix stiffness, fiber alignment, and fiber binding site density on path persistence of cellular migration and the mean squared displacement of cells over long periods of time.

Guidance and Self-Sorting of Active Swimmers: 3D Periodic Arrays Increase Persistence Length of Human Sperm Selecting for the Fittest

Male infertility is a reproductive disease, and existing clinical solutions for this condition often involve long and cumbersome sperm sorting methods, including preprocessing and centrifugation-based steps. These methods also fall short when sorting for sperm free of reactive oxygen species, DNA damage, and epigenetic aberrations. Although several microfluidic platforms exist, they suffer from structural complexities, i.e., pumps or chemoattractants, setting insurmountable barriers to clinical adoption. Inspired by the natural filter-like capabilities of the female reproductive tract for sperm selection, a model-driven design, featuring pillar arrays that efficiently and noninvasively isolate highly motile and morphologically normal sperm, with lower epigenetic global methylation, from raw semen, is presented. The Simple Periodic ARray for Trapping And isolatioN (SPARTAN) created here modulates the directional persistence of sperm, increasing the spatial separation between progressive and nonprogressive motile sperm populations within an unprecedentedly short 10 min assay time. With over 99% motility of sorted sperm, a 5-fold improvement in morphology, 3-fold increase in nuclear maturity, and 2-4-fold enhancement in DNA integrity, SPARTAN offers to standardize sperm selection while eliminating operator-to-operator variations, centrifugation, and flow. SPARTAN can also be applied in other areas, including conservation ecology, breeding of farm animals, and design of flagellar microrobots for diagnostics.

Extracting cellular automaton rules from physical Langevin equation models for single and collective cell migration

Cellular automata (CA) are discrete time, space, and state models which are extensively used for modeling biological phenomena. CA are "on-lattice" models with low computational demands. In particular, lattice-gas cellular automata (LGCA) have been introduced as models of single and collective cell migration. The interaction rule dictates the behavior of a cellular automaton model and is critical to the model's biological relevance. The LGCA model's interaction rule has been typically chosen phenomenologically. In this paper, we introduce a method to obtain lattice-gas cellular automaton interaction rules from physically-motivated "off-lattice" Langevin equation models for migrating cells. In particular, we consider Langevin equations related to single cell movement (movement of cells independent of each other) and collective cell migration (movement influenced by cell-cell interactions). As examples of collective cell migration, two different alignment mechanisms are studied: polar and nematic alignment. Both kinds of alignment have been observed in biological systems such as swarms of amoebae and myxobacteria. Polar alignment causes cells to align their velocities parallel to each other, whereas nematic alignment drives cells to align either parallel or antiparallel to each other. Under appropriate assumptions, we have derived the LGCA transition probability rule from the steady-state distribution of the off-lattice Fokker-Planck equation. Comparing alignment order parameters between the original Langevin model and the derived LGCA for both mechanisms, we found different areas of agreement in the parameter space. Finally, we discuss potential reasons for model disagreement and propose extensions to the CA rule derivation methodology.

Mechanical models for cell movement – locomotion, translocation, migration

This paper provides a detailed stochastic analysis of leucocyte cell movement based on the dynamics of a rigid body. The cell's behavior is studied in two relevant anisotropic environments displaying adhesion mediated movement (haptotaxis) and stimulus mediated movement (chemotaxis). This behavior is modeled by diffusion processes on three successively longer time scales, termed locomotion, translocation, and migration.

Mechanical models for cell movement – locomotion, translocation, migration

This paper provides a detailed stochastic analysis of leucocyte cell movement based on the dynamics of a rigid body. The cell's behavior is studied in two relevant anisotropic environments displaying adhesion mediated movement (haptotaxis) and stimulus mediated movement (chemotaxis). This behavior is modeled by diffusion processes on three successively longer time scales, termed locomotion, translocation, and migration.

Programming cancer through phase-functionalized silicon based biomaterials

Applications of biomaterials in cancer therapy has been limited to drug delivery systems and markers in radiation therapy. In this article, we introduce the concept of phase-functionalization of silicon to preferentially select cancer cell populations for survival in a catalyst and additive free approach. Silicon is phase-functionalized by the interaction of ultrafast laser pulses, resulting in the formation of rare phases of SiO2 in conjunction with differing silicon crystal lattices. The degree of phase-functionalization is programmed to dictate the degree of repulsion of cancer cells. Unstable phases of silicon oxides are synthesized during phase-functionalization and remain stable at ambient conditions. This change in phase of silicon as well as formation of oxides contributes to changes in surface chemistry as well as surface energy. These material properties elicit in precise control of migration, cytoskeleton shape, direction and population. To the best of our knowledge, phase-functionalized silicon without any changes in topology or additive layers and its applications in cancer therapy has not been reported before. This unique programmable phase-functionalized silicon has the potential to change current trends in cancer research and generate focus on biomaterials as cancer repelling or potentially cancer killing surfaces.

Effect of adhesion and chemokine presentation on T-lymphocyte haptokinesis

Motility is critical for the function of T-lymphocytes. Motility in T-lymphocytes is driven by the occupancy of chemokine receptors by chemokines, and modulated by adhesive interactions. However, it is not well understood how the combination of adhesion and chemokine binding affects T-lymphocyte migration. We used microcontact printing on polymeric substrates to measure how lymphocyte migration is quantitatively controlled by adhesion and chemokine ligation. Focusing only on random motion, we found that T-lymphocytes exhibit biphasic motility in response to the substrate concentration of either ICAM-1 or VCAM-1, and generally display more active motion on ICAM-1 surfaces. Furthermore, we examined how the combination of the homeostatic chemokines CCL19 and CCL21 contribute to motility. By themselves, CCL19 and CCL21, ligands for CCR7, elicit biphasic motility, but their combination synergistically increases CCR7 mediated chemokinesis on ICAM-1. By presenting CCL21 with ICAM-1 on the surface with soluble CCL19, we observed random motion that is greater than what is observed with soluble chemokines alone. These data suggest that ICAM-1 has a greater contribution to motility than VCAM-1 and that both adhesive interactions and chemokine ligation work in concert to control T-lymphocyte motility.

A mathematical model for eph/ephrin-directed segregation of intermingled cells

Eph receptors, the largest family of receptor tyrosine kinases, control cell-cell adhesion/de-adhesion, cell morphology and cell positioning through interaction with cell surface ephrin ligands. Bi-directional signalling from the Eph and ephrin complexes on interacting cells have a significant role in controlling normal tissue development and oncogenic tissue patterning. Eph-mediated tissue patterning is based on the fine-tuned balance of adhesion and de-adhesion reactions between distinct Eph- and ephrin-expressing cell populations, and adhesion within like populations (expressing either Eph or ephrin). Here we develop a stochastic, Lagrangian model that is based on Eph/ephrin biology: incorporating independent Brownian motion to describe cell movement and a deterministic term (the drift term) to represent repulsive and adhesive interactions between neighbouring cells. Comparison between the experimental and computer simulated Eph/ephrin cell patterning events shows that the model recapitulates the dynamics of cell-cell segregation and cell cluster formation. Moreover, by modulating the term for Eph/ephrin-mediated repulsion, the model can be tuned to match the actual behaviour of cells with different levels of Eph expression or activity. Together the results of our experiments and modelling suggest that the complexity of Eph/ephrin signalling mechanisms that control cell-cell interactions can be described well by a mathematical model with a single term balancing adhesion and de-adhesion between interacting cells. This model allows reliable prediction of Eph/ephrin-dependent control of cell patterning behaviour.

Effect of Cdc42 domains on filopodia sensing, cell orientation, and haptotaxis

Filopodia are sensors which, along with microtubules, regulate the persistence of locomotion. To determine whether protrusions were involved in sensing adhesion, epithelial cells were cultured on platinum and tantalum gradients. Protrusions were defined by an unbiased statistical method of classification as factors 4 (filopodia), 5 (mass distribution), and 7 (nascent neurites). When the prevalence of protrusions was measured in zones of high (H), middle (M), and low (L) adhesiveness, the main differences were in factor 4. Its values were highest at H and declined at M and L regardless of the gradient composition. The significance of the differences was enhanced when T (top/adhesive end) and B (bottom/nonadhesive end) sides of cells were analyzed separately. Since information about sidedness increased the statistical power of the test, this result suggested that cells pointed more filopodia toward the adhesive end. Trends occurred in factors 5 and 7 only when conditions allowed for a marked trend in factor 4. The data showed that gradient sensing is proportional to the prevalence of filopodia, and filopodia are the only protrusions engaged in comparing adhesiveness across a cell. The probability (P) of the significance of a trend was then used to determine how cells sense the gradient. Binding peptides (BPs) were introduced representing sequences critical for Cdc42 docking on a specific partner. BPs for IQGAP (IQ(calmodulin-binding domain)-containing GTPase-activating protein) and ACK (Cdc42-associated kinase) reduced factor 4 values and prevented cell orientation on the gradient. Micrographs showed attenuated or stubby filopodia. These effectors may be implicated in gradient sensing. Another IQGAP BP increased filopodia prevalence and enhanced orientation on the gradient (P<0.00015). A Wiskott-Aldrich syndrome protein (WASP) BP had no effect. When sensing and orientation were abolished, they both failed at the level of filopodia, indicating that filopodia are both sensors and implementers of signals transduced by adhesion.

Artificial tribotactic microscopic walkers: walking based on friction gradients

Friction, the resistive force between two surfaces sliding past each other, is at the core of a wide diversity of locomotion schemes. While such schemes are well described for homogeneous environments, locomotion based on friction in inhomogeneous environments has not received much attention. Here we introduce and demonstrate the concept of tribotaxis, a motion that is guided by gradients in the friction coefficient. Our system is composed of microwalkers that undergo an effective frictional interaction with biological receptors on the substrate, which is regulated by the density of such receptors. When actuated stochastically, microwalkers migrate to regions of higher friction, much like a chemotactic cell migrates to regions of higher chemoattractant concentration. Simulations and theory based on biased random walks are in excellent agreement with experiments. We foresee important implications for tribotaxis in artificial and natural locomotion in biological environments.

Selection of functional human sperm with higher DNA integrity and fewer reactive oxygen species

Fertilization and reproduction are central to the survival and propagation of a species. Couples who cannot reproduce naturally have to undergo in vitro clinical procedures. An integral part of these clinical procedures includes isolation of healthy sperm from raw semen. Existing sperm sorting methods are not efficient and isolate sperm having high DNA fragmentation and reactive oxygen species (ROS), and suffer from multiple manual steps and variations between operators. Inspired by in vivo natural sperm sorting mechanisms where vaginal mucus becomes less viscous to form microchannels to guide sperm towards egg, a chip is presented that efficiently sorts healthy, motile and morphologically normal sperm without centrifugation. Higher percentage of sorted sperm show significantly lesser ROS and DNA fragmentation than the conventional swim-up method. The presented chip is an easy-to-use high-throughput sperm sorter that provides standardized sperm sorting assay with less reliance on operators's skills, facilitating reliable operational steps.

Ordered, random, monotonic and non-monotonic digital nanodot gradients

Cell navigation is directed by inhomogeneous distributions of extracellular cues. It is well known that noise plays a key role in biology and is present in naturally occurring gradients at the micro- and nanoscale, yet it has not been studied with gradients in vitro. Here, we introduce novel algorithms to produce ordered and random gradients of discrete nanodots--called digital nanodot gradients (DNGs)--according to monotonic and non-monotonic density functions. The algorithms generate continuous DNGs, with dot spacing changing in two dimensions along the gradient direction according to arbitrary mathematical functions, with densities ranging from 0.02% to 44.44%. The random gradient algorithm compensates for random nanodot overlap, and the randomness and spatial homogeneity of the DNGs were confirmed with Ripley's K function. An array of 100 DNGs, each 400×400 µm2, comprising a total of 57 million 200×200 nm2 dots was designed and patterned into silicon using electron-beam lithography, then patterned as fluorescently labeled IgGs on glass using lift-off nanocontact printing. DNGs will facilitate the study of the effects of noise and randomness at the micro- and nanoscales on cell migration and growth.

Exhaustion of racing sperm in nature-mimicking microfluidic channels during sorting

Fertilization is central to the survival and propagation of a species, however, the precise mechanisms that regulate the sperm's journey to the egg are not well understood. In nature, the sperm has to swim through the cervical mucus, akin to a microfluidic channel. Inspired by this, a simple, cost-effective microfluidic channel is designed on the same scale. The experimental results are supported by a computational model incorporating the exhaustion time of sperm.

PKCδ localization at the membrane increases matrix traction force dependent on PLCγ1/EGFR signaling

Introduction

During wound healing, fibroblasts initially migrate into the wound bed and later contract the matrix. Relevant mediators of transcellular contractility revealed by systems analyses are protein kinase c delta/myosin light chain-2 (PKCδ/MLC-2). PKCδ is activated by growth factor-driven PLCγ1 hydrolysis of phosphoinositide bisphosphate (PIP₂) hydrolysis when it becomes tranlocated to the membrane. This leads to MLC-2 phosphorylation that regulates myosin for contractility. Furthermore, PKCδ n-terminus mediates PKCδ localization to the membrane in relative proximity to PLCγ1 activity. However, the role this localization and the relationship to its activation and signaling of force is not well understood. Therefore, we investigated whether the membrane localization of PKCδ mediates the transcellular contractility of fibroblasts.

Methods

To determine PKCδ activation in targeted membrane locations in mouse fibroblast cells (NR6-WT), two PKCδ constructs were generated; PKCδ-CaaX with farnesylation moiety targeting PKCδ to the membrane and PKCδ-SaaX a non-targeting control.

Results

Increased mean cell force was observed before and during EGF stimulation in fibroblasts expressing membrane-targeted PKCδ (PKCδ-CaaX) when analyzed with 2D cell traction force and 3D compaction of collagen matrix. This effect was reduced in cells deficient in EGFR/PLCy1 signaling. In cells expressing non-membrane targeted PKCδ (PKCδ-SaaX), the cell force exerted outside the ECM (extracellular matrix) was less, but cell motility/speed/persistence was increased after EGF stimulation. Change in cell motility and increased force exertion was also preceded by change in cell morphology. Organization of actin stress fibers was also decreased as a result of increasing membrane targeting of PKCδ.

Conclusion

From these results membrane tethering of PKCδ leads to increased force exertion on ECM. Furthermore, our data show PLCγ1 regulation of PKCδ, at least in part, drives transcellular contractility in fibroblasts.

Master equation-based analysis of a motor-clutch model for cell traction force

Microenvironmental mechanics play an important role in determining the morphology, traction, migration, proliferation, and differentiation of cells. A stochastic motor-clutch model has been proposed to describe this stiffness sensitivity. In this work, we present a master equation-based ordinary differential equation (ODE) description of the motor-clutch model, from which we derive an analytical expression to for a cell's optimum stiffness (i.e. the stiffness at which the traction force is maximal). This analytical expression provides insight into the requirements for stiffness sensing by establishing fundamental relationships between the key controlling cell-specific parameters. We find that the fundamental controlling parameters are the numbers of motors and clutches (constrained to be nearly equal), and the time scale of the on-off kinetics of the clutches (constrained to favor clutch binding over clutch unbinding). Both the ODE solution and the analytical expression show good agreement with Monte Carlo motor-clutch output, and reduce computation time by several orders of magnitude, which potentially enables long time scale behaviors (hours-days) to be studied computationally in an efficient manner. The ODE solution and the analytical expression may be incorporated into larger scale models of cellular behavior to bridge the gap from molecular time scales to cellular and tissue time scales.

Quantification and regulation of cell migration

Cell migration is essential for many physiological and pathological processes that include embryonic development, the immune response, wound healing, angiogenesis, and cancer metastasis. It is also important for emerging tissue engineering applications such as tissue reconstitution and the colonization of biomedical implants. By summarizing results from recent experimental and theoretical studies, this review outlines the role played by growth factors or substrate-adhesion molecules in modulating cell motility and shows that cell motility can be an important factor in determining the rates of tissue formation. The application of cell motility assays and the use of theoretical models for analyzing cell migration and proliferation are also discussed.

Model for in vivo progression of tumors based on co-evolving cell population and vasculature

With countless biological details emerging from cancer experiments, there is a growing need for minimal mathematical models which simultaneously advance our understanding of single tumors and metastasis, provide patient-personalized predictions, whilst avoiding excessive hard-to-measure input parameters which complicate simulation, analysis and interpretation. Here we present a model built around a co-evolving resource network and cell population, yielding good agreement with primary tumors in a murine mammary cell line EMT6-HER2 model in BALB/c mice and with clinical metastasis data. Seeding data about the tumor and its vasculature from in vivo images, our model predicts corridors of future tumor growth behavior and intervention response. A scaling relation enables the estimation of a tumor's most likely evolution and pinpoints specific target sites to control growth. Our findings suggest that the clinically separate phenomena of individual tumor growth and metastasis can be viewed as mathematical copies of each other differentiated only by network structure.

Subcellular spatial segregation of integrin subtypes by patterned multicomponent surfaces

While it is well known that individual integrins are critical mediators of cell behavior, recent work has shown that when multiple types of integrins simultaneously engage the ECM, cell functions are enhanced. However, it is not known how integrins spatially coordinate to regulate cell adhesion because no reliable method exists to segregate integrins on the cell membrane. Here, we use a microcontact printing-based strategy to pattern multiple ECMs that bind distinct integrins in order to study how integrins might interact. In our technique, proteins are first adsorbed uniformly to a poly(dimethyl siloxane) stamp, and then selectively "de-inked." Our strategy overcomes several inherent limitations of conventional microcontact printing, including stamp collapse and limited functionality of the surface patterns. We show that integrins spatially segregate on surfaces patterned with multiple ECMs, as expected. Interestingly, despite spatial segregation of distinct integrins, cells could form adhesions and migrate across multicomponent surfaces as well as they do on single component surfaces. Together, our data indicate that although cells can segregate individual integrins on the cell surface to mediate ECM-specific binding, integrins function cooperatively to guide cell adhesion and migration.

Plasma-assisted surface modification of organic biopolymers to prevent bacterial attachment

Despite many synthetic biomaterials having physical properties that are comparable or even superior to those of natural body tissues, they frequently fail due to the adverse physiological reactions they cause within the human body, such as infection and inflammation. The surface modification of biomaterials is an economical and effective method by which biocompatibility and biofunctionality can be achieved while preserving the favorable bulk characteristics of the biomaterial, such as strength and inertness. Amongst the numerous surface modification techniques available, plasma surface modification affords device manufacturers a flexible and environmentally friendly process that enables tailoring of the surface morphology, structure, composition, and properties of the material to a specific need. There are a vast range of possible applications of plasma modification in biomaterial applications, however, the focus of this review paper is on processes that can be used to develop surface morphologies and chemical structures for the prevention of adhesion and proliferation of pathogenic bacteria on the surfaces of in-dwelling medical devices. As such, the fundamental principles of bacterial cell attachment and biofilm formation are also discussed. Functional organic plasma polymerised coatings are also discussed for their potential as biosensitive interfaces, connecting inorganic/metallic electronic devices with their physiological environments.

A numerical model for durotaxis

Cell migration is a phenomenon that is involved in several physiological processes. In the absence of external guiding factors it shares analogies with Brownian motion. The presence of biochemical or biophysical cues, on the other hand, can influence cell migration transforming it in a biased random movement. Recent studies have shown that different cell types are able to recognise the mechanical properties of the substratum over which they move and that these properties direct the motion through a process called durotaxis. In this work a 2D mathematical model for the description of this phenomenon is presented. The model is based on the Langevin equation that has been modified to take into account the local mechanical properties of the substratum perceived by the cells. Numerical simulations of the model provide individual cell tracks, whose characteristics can be compared with experimental observations directly. The present model is solved for two important cases: an isotropic substratum, to check that random motility is recovered as a subcase, and a biphasic substratum, to investigate durotaxis. The degree of agreement is satisfactory in both cases. The model can be a useful tool for quantifying relevant parameters of cell migration as a function of the substratum mechanical properties.

Computational modelling of biomaterial surface interactions with blood platelets and osteoblastic cells for the prediction of contact osteogenesis

Surface microroughness can induce contact osteogenesis (bone formation initiated at the implant surface) around oral implants, which may result from different mechanisms, such as blood platelet-biomaterial interactions and/or interaction with (pre-)osteoblast cells. We have developed a computational model of implant endosseous healing that takes into account these interactions. We hypothesized that the initial attachment and growth factor release from activated platelets is crucial in achieving contact osteogenesis. In order to investigate this, a computational model was applied to an animal experiment [7] that looked at the effect of surface microroughness on endosseous healing. Surface-specific model parameters were implemented based on in vitro data (Lincks et al. Biomaterials 1998;19:2219-32). The predicted spatio-temporal patterns of bone formation correlated with the histological data. It was found that contact osteogenesis could not be predicted if only the osteogenic response of cells was up-regulated by surface microroughness. This could only be achieved if platelet-biomaterial interactions were sufficiently up-regulated as well. These results confirmed our hypothesis and demonstrate the added value of the computational model to study the importance of surface-mediated events for peri-implant endosseous healing.

Actomyosin tension exerted on the nucleus through nesprin-1 connections influences endothelial cell adhesion, migration, and cyclic strain-induced reorientation

Endothelial cell polarization and directional migration is required for angiogenesis. Polarization and motility requires not only local cytoskeletal remodeling but also the motion of intracellular organelles such as the nucleus. However, the physiological significance of nuclear positioning in the endothelial cell has remained largely unexplored. Here, we show that siRNA knockdown of nesprin-1, a protein present in the linker of nucleus to cytoskeleton complex, abolished the reorientation of endothelial cells in response to cyclic strain. Confocal imaging revealed that the nuclear height is substantially increased in nesprin-1 depleted cells, similar to myosin inhibited cells. Nesprin-1 depletion increased the number of focal adhesions and substrate traction while decreasing the speed of cell migration; however, there was no detectable change in nonmuscle myosin II activity in nesprin-1 deficient cells. Together, these results are consistent with a model in which the nucleus balances a portion of the actomyosin tension in the cell. In the absence of nesprin-1, actomyosin tension is balanced by the substrate, leading to abnormal adhesion, migration, and cyclic strain-induced reorientation.

On the modelling of biological patterns with mechanochemical models: Insights from analysis and computation

The diversity of biological form is generated by a relatively small number of underlying mechanisms. Consequently, mathematical and computational modelling can, and does, provide insight into how cellular level interactions ultimately give rise to higher level structure. Given cells respond to mechanical stimuli, it is therefore important to consider the effects of these responses within biological self-organisation models. Here, we consider the self-organisation properties of a mechanochemical model previously developed by three of the authors in Acta Biomater. 4, 613-621 (2008), which is capable of reproducing the behaviour of a population of cells cultured on an elastic substrate in response to a variety of stimuli. In particular, we examine the conditions under which stable spatial patterns can emerge with this model, focusing on the influence of mechanical stimuli and the interplay of non-local phenomena. To this end, we have performed a linear stability analysis and numerical simulations based on a mixed finite element formulation, which have allowed us to study the dynamical behaviour of the system in terms of the qualitative shape of the dispersion relation. We show that the consideration of mechanotaxis, namely changes in migration speeds and directions in response to mechanical stimuli alters the conditions for pattern formation in a singular manner. Furthermore without non-local effects, responses to mechanical stimuli are observed to result in dispersion relations with positive growth rates at arbitrarily large wavenumbers, in turn yielding heterogeneity at the cellular level in model predictions. This highlights the sensitivity and necessity of non-local effects in mechanically influenced biological pattern formation models and the ultimate failure of the continuum approximation in their absence.

Stochastic modelling of biased cell migration and collagen matrix modification

Matrix dynamics plays a crucial role in several physiological and pathological processes. In this paper we develop a model framework, which describes the temporal fibre network evolution depending on the influence of migrating fibroblasts. The cells are regarded as discrete objects in the plane, whose velocities are determined by a generalised Langevin equation. For its solution we verify existence and uniqueness. The courses of the trajectories are affected by two external impulses, chemotaxis and contact guidance, respectively. The extracellular matrix is described by a continuous vector field which contains both information on density and orientation of the fibrous material. Modelling dynamic interaction between the discrete and the continuum variables is an essential point of this paper. In particular, the smoothing of the fluctuating paths plays a key role. Besides a detailed description of the formulated equations, we also supply the condensed pseudo code of the algorithm. We investigate several examples and present results both from artificial and real data.

Multiscale modelling and nonlinear simulation of vascular tumour growth

In this article, we present a new multiscale mathematical model for solid tumour growth which couples an improved model of tumour invasion with a model of tumour-induced angiogenesis. We perform nonlinear simulations of the multi-scale model that demonstrate the importance of the coupling between the development and remodeling of the vascular network, the blood flow through the network and the tumour progression. Consistent with clinical observations, the hydrostatic stress generated by tumour cell proliferation shuts down large portions of the vascular network dramatically affecting the flow, the subsequent network remodeling, the delivery of nutrients to the tumour and the subsequent tumour progression. In addition, extracellular matrix degradation by tumour cells is seen to have a dramatic affect on both the development of the vascular network and the growth response of the tumour. In particular, the newly developing vessels tend to encapsulate, rather than penetrate, the tumour and are thus less effective in delivering nutrients.

Protrusion of a Virtual Model Lamellipodium by Actin Polymerization: A Coarse-grained Langevin Dynamics Model

We report the development of a coarse-grained Langevin dynamics model of a lamellipodium featuring growing F-actin filaments in order to study the effect of stiffness of the F-actin filament, the G-actin monomer concentration, and the number of polymerization sites on lamellipodium protrusion. The virtual lamellipodium is modeled as a low-aspect-ratio doubly capped cylinder formed by triangulated particles on its surface. It is assumed that F-actin filaments are firmly attached to a lamellipodium surface where polymerization sites are located, and actin polymerization takes place by connecting a G-actin particle to a polymerization site and to the first particle of a growing F-actin filament. It is found that there is an optimal number of polymerization sites for rapid lamellipodium protrusion. The maximum speed of lamellipodium protrusion is related to competition between the number of polymerization sites and the number of available G-actin particles, and the degree of pulling and holding of the lamellipodium surface by non-polymerizing actin filaments. The lamellipodium protrusion by actin polymerization displays saltatory motion exhibiting pseudo-thermal equilibrium: the lamellipodium speed distribution is Maxwellian in two dimensions but the lamellipodium motion is biased so that the lamellipodium speed in the direction of the lamellipodium motion is much larger than that normal to the lamellipodium motion.