Potential for control of signaling pathways via cell size and shape

Crawling and turning in a minimal reaction-diffusion cell motility model: Coupling cell shape and biochemistry

We study a minimal model of a crawling eukaryotic cell with a chemical polarity controlled by a reaction-diffusion mechanism describing Rho GTPase dynamics. The size, shape, and speed of the cell emerge from the combination of the chemical polarity, which controls the locations where actin polymerization occurs, and the physical properties of the cell, including its membrane tension. We find in our model both highly persistent trajectories, in which the cell crawls in a straight line, and turning trajectories, where the cell transitions from crawling in a line to crawling in a circle. We discuss the controlling variables for this turning instability and argue that turning arises from a coupling between the reaction-diffusion mechanism and the shape of the cell. This emphasizes the surprising features that can arise from simple links between cell mechanics and biochemistry. Our results suggest that similar instabilities may be present in a broad class of biochemical descriptions of cell polarity.

Single-Cell Migration in Complex Microenvironments: Mechanics and Signaling Dynamics

Cells are highly dynamic and mechanical automata powered by molecular motors that respond to external cues. Intracellular signaling pathways, either chemical or mechanical, can be activated and spatially coordinated to induce polarized cell states and directional migration. Physiologically, cells navigate through complex microenvironments, typically in three-dimensional (3D) fibrillar networks. In diseases, such as metastatic cancer, they invade across physiological barriers and remodel their local environments through force, matrix degradation, synthesis, and reorganization. Important external factors such as dimensionality, confinement, topographical cues, stiffness, and flow impact the behavior of migrating cells and can each regulate motility. Here, we review recent progress in our understanding of single-cell migration in complex microenvironments.

Real-time prediction of cell division timing in developing zebrafish embryo

Combination of live-imaging and live-manipulation of developing embryos in vivo provides a useful tool to study developmental processes. Identification and selection of target cells for an in vivo live-manipulation are generally performed by experience- and knowledge-based decision-making of the observer. Computer-assisted live-prediction method would be an additional approach to facilitate the identification and selection of the appropriate target cells. Herein we report such a method using developing zebrafish embryos. We choose V2 neural progenitor cells in developing zebrafish embryo as their successive shape changes can be visualized in real-time in vivo. We developed a relatively simple mathematical method of describing cellular geometry of V2 cells to predict cell division-timing based on their successively changing shapes in vivo. Using quantitatively measured 4D live-imaging data, features of V2 cell-shape at each time point prior to division were extracted and a statistical model capturing the successive changes of the V2 cell-shape was developed. By applying sequential Bayesian inference method to the model, we successfully predicted division-timing of randomly selected individual V2 cells while the cell behavior was being live-imaged. This system could assist pre-selecting target cells desirable for real-time manipulation–thus, presenting a new opportunity for in vivo experimental systems.

Memory of cell shape biases stochastic fate decision-making despite mitotic rounding

Cell shape influences function, and the current model suggests that such shape effect is transient. However, cells dynamically change their shapes, thus, the critical question is whether shape information remains influential on future cell function even after the original shape is lost. We address this question by integrating experimental and computational approaches. Quantitative live imaging of asymmetric cell-fate decision-making and their live shape manipulation demonstrates that cellular eccentricity of progenitor cell indeed biases stochastic fate decisions of daughter cells despite mitotic rounding. Modelling and simulation indicates that polarized localization of Delta protein instructs by the progenitor eccentricity is an origin of the bias. Simulation with varying parameters predicts that diffusion rate and abundance of Delta molecules quantitatively influence the bias. These predictions are experimentally validated by physical and genetic methods, showing that cells exploit a mechanism reported herein to influence their future fates based on their past shape despite dynamic shape changes.

Cell shape influences function but during mitotic cell rounding the original shape is lost. Here the authors show that the cellular eccentricity of progenitor cell biases stochastic fate-decisions using a combination of quantitative live imaging, genetic manipulations and computational simulations.

Effects of 3D geometries on cellular gradient sensing and polarization

During cell migration, cells become polarized, change their shape, and move in response to various internal and external cues. Cell polarization is defined through the spatio-temporal organization of molecules such as PI3K or small GTPases, and is determined by intracellular signaling networks. It results in directional forces through actin polymerization and myosin contractions. Many existing mathematical models of cell polarization are formulated in terms of reaction-diffusion systems of interacting molecules, and are often defined in one or two spatial dimensions. In this paper, we introduce a 3D reaction-diffusion model of interacting molecules in a single cell, and find that cell geometry has an important role affecting the capability of a cell to polarize, or change polarization when an external signal changes direction. Our results suggest a geometrical argument why more roundish cells can repolarize more effectively than cells which are elongated along the direction of the original stimulus, and thus enable roundish cells to turn faster, as has been observed in experiments. On the other hand, elongated cells preferentially polarize along their main axis even when a gradient stimulus appears from another direction. Furthermore, our 3D model can accurately capture the effect of binding and unbinding of important regulators of cell polarization to and from the cell membrane. This spatial separation of membrane and cytosol, not possible to capture in 1D or 2D models, leads to marked differences of our model from comparable lower-dimensional models.

On the importance of modelling the internal spatial dynamics of biological cells

Spatial effects such as cell shape have very often been considered negligible in models of cellular pathways, and many existing simulation infrastructures do not take such effects into consideration. Recent experimental results are reversing this judgement by showing that very small spatial variations can make a big difference in the fate of a cell. This is particularly the case when considering eukaryotic cells, which have a complex physical structure and many subtle control mechanisms, but bacteria are also interesting for the huge variation in shape both between species and in different phases of their lifecycle. In this work we perform simulations that measure the effect of three common bacterial shapes on the behaviour of model cellular pathways. To perform these experiments we develop ReDi-Cell, a highly scalable GPGPU cell simulation infrastructure for the modelling of cellular pathways in spatially detailed environments. ReDi-Cell is validated against known-good simulations, prior to its use in new work. We then use ReDi-Cell to conduct novel experiments that demonstrate the effect that three common bacterial shapes (Cocci, Bacilli and Spirilli) have on the behaviour of model cellular pathways. Pathway wavefront shape, pathway concentration gradients, and chemical species distribution are measured in the three different shapes. We also quantify the impact of internal cellular clutter on the same pathways. Through this work we show that variations in the shape or configuration of these common cell shapes alter model cell behaviour.

A computational method for the coupled solution of reaction-diffusion equations on evolving domains and manifolds: Application to a model of cell migration and chemotaxis

In this paper, we devise a moving mesh finite element method for the approximate solution of coupled bulk-surface reaction-diffusion equations on an evolving two dimensional domain. Fundamental to the success of the method is the robust generation of bulk and surface meshes. For this purpose, we use a novel moving mesh partial differential equation (MMPDE) approach. The developed method is applied to model problems with known analytical solutions; these experiments indicate second-order spatial and temporal accuracy. Coupled bulk-surface problems occur frequently in many areas; in particular, in the modelling of eukaryotic cell migration and chemotaxis. We apply the method to a model of the two-way interaction of a migrating cell in a chemotactic field, where the bulk region corresponds to the extracellular region and the surface to the cell membrane.

Nanofiber scaffolds influence organelle structure and function in bone marrow stromal cells

Recent work demonstrates that osteoprogenitor cell culture on nanofiber scaffolds can promote differentiation. This response may be driven by changes in cell morphology caused by the three-dimensional (3D) structure of nanofibers. We hypothesized that nanofiber effects on cell behavior may be mediated by changes in organelle structure and function. To test this hypothesis, human bone marrow stromal cells (hBMSCs) were cultured on poly(ε-caprolactone) (PCL) nanofibers scaffolds and on PCL flat spuncoat films. After 1 day-culture, hBMSCs were stained for actin, nucleus, mitochondria, and peroxisomes, and then imaged using 3D confocal microscopy. Imaging revealed that the hBMSC cell body (actin) and peroxisomal volume were reduced during culture on nanofibers. In addition, the nucleus and peroxisomes occupied a larger fraction of cell volume during culture on nanofibers than on films, suggesting enhancement of the nuclear and peroxisomal functional capacity. Organelles adopted morphologies with greater 3D-character on nanofibers, where the Z-Depth (a measure of cell thickness) was increased. Comparisons of organelle positions indicated that the nucleus, mitochondria, and peroxisomes were closer to the cell center (actin) for nanofibers, suggesting that nanofiber culture induced active organelle positioning. The smaller cell volume and more centralized organelle positioning would reduce the energy cost of inter-organelle vesicular transport during culture on nanofibers. Finally, hBMSC bioassay measurements (DNA, peroxidase, bioreductive potential, lactate, and adenosine triphosphate (ATP)) indicated that peroxidase activity may be enhanced during nanofiber culture. These results demonstrate that culture of hBMSCs on nanofibers caused changes in organelle structure and positioning, which may affect organelle functional capacity and transport. Published 2016. This article is a U.S. Government work and is in the public domain in the USA. J Biomed Mater Res Part B: Appl Biomater, 2016. © 2016 Wiley Periodicals, Inc. J Biomed Mater Res Part B: Appl Biomater, 105B: 989-1001, 2017.

Bistability of a coupled Aurora B kinase-phosphatase system in cell division

Aurora B kinase, a key regulator of cell division, localizes to specific cellular locations, but the regulatory mechanisms responsible for phosphorylation of substrates located remotely from kinase enrichment sites are unclear. Here, we provide evidence that this activity at a distance depends on both sites of high kinase concentration and the bistability of a coupled kinase-phosphatase system. We reconstitute this bistable behavior and hysteresis using purified components to reveal co-existence of distinct high and low Aurora B activity states, sustained by a two-component kinase autoactivation mechanism. Furthermore, we demonstrate these non-linear regimes in live cells using a FRET-based phosphorylation sensor, and provide a mechanistic theoretical model for spatial regulation of Aurora B phosphorylation. We propose that bistability of an Aurora B-phosphatase system underlies formation of spatial phosphorylation patterns, which are generated and spread from sites of kinase autoactivation, thereby regulating cell division.

Dendritic spine geometry can localize GTPase signaling in neurons

Dendritic spines are the postsynaptic terminals of most excitatory synapses in the mammalian brain. Learning and memory are associated with long-lasting structural remodeling of dendritic spines through an actin-mediated process regulated by the Rho-family GTPases RhoA, Rac, and Cdc42. These GTPases undergo sustained activation after synaptic stimulation, but whereas Rho activity can spread from the stimulated spine, Cdc42 activity remains localized to the stimulated spine. Because Cdc42 itself diffuses rapidly in and out of the spine, the basis for the retention of Cdc42 activity in the stimulated spine long after synaptic stimulation has ceased is unclear. Here we model the spread of Cdc42 activation at dendritic spines by means of reaction-diffusion equations solved on spine-like geometries. Excitable behavior arising from positive feedback in Cdc42 activation leads to spreading waves of Cdc42 activity. However, because of the very narrow neck of the dendritic spine, wave propagation is halted through a phenomenon we term geometrical wave-pinning. We show that this can account for the localization of Cdc42 activity in the stimulated spine, and, of interest, retention is enhanced by high diffusivity of Cdc42. Our findings are broadly applicable to other instances of signaling in extreme geometries, including filopodia and primary cilia.

Early Signaling in Primary T Cells Activated by Antigen Presenting Cells Is Associated with a Deep and Transient Lamellal Actin Network

Cellular signaling transduction critically depends on molecular interactions that are in turn governed by dynamic subcellular distributions of the signaling system components. Comprehensive insight into signal transduction requires an understanding of such distributions and cellular structures driving them. To investigate the activation of primary murine T cells by antigen presenting cells (APC) we have imaged more than 60 signaling intermediates during T cell stimulation with microscopy across resolution limits. A substantial number of signaling intermediates associated with a transient, wide, and actin-associated lamellum extending from an interdigitated T cell:APC interface several micrometers into the T cell, as characterized in detail here. By mapping the more than 60 signaling intermediates onto the spatiotemporal features of cell biological structures, the lamellum and other ones previously described, we also define distinct spatial and temporal characteristics of T cell signal initiation, amplification, and core signaling in the activation of primary T cells by APCs. These characteristics differ substantially from ones seen when T cells are activated using common reductionist approaches.

Aggregation-fragmentation model of robust concentration gradient formation

Concentration gradients of signaling molecules are essential for patterning during development and they have been observed in both unicellular and multicellular systems. In subcellular systems, clustering of the signaling molecule has been observed. We develop a theoretical model of cluster-mediated concentration gradient formation based on the Becker-Döring equations of aggregation-fragmentation processes. We show that such a mechanism produces robust concentration gradients on realistic time and spatial scales so long as the process of clustering does not significantly stabilize the signaling molecule. Finally, we demonstrate that such a model is applicable to the pom1p subcellular gradient in fission yeast.

Methodologies for the modeling and simulation of biochemical networks, illustrated for signal transduction pathways: a primer

Biochemical networks depict the chemical interactions that take place among elements of living cells. They aim to elucidate how cellular behavior and functional properties of the cell emerge from the relationships between its components, i.e. molecules. Biochemical networks are largely characterized by dynamic behavior, and exhibit high degrees of complexity. Hence, the interest in such networks is growing and they have been the target of several recent modeling efforts. Signal transduction pathways (STPs) constitute a class of biochemical networks that receive, process, and respond to stimuli from the environment, as well as stimuli that are internal to the organism. An STP consists of a chain of intracellular signaling processes that ultimately result in generating different cellular responses. This primer presents the methodologies used for the modeling and simulation of biochemical networks, illustrated for STPs. These methodologies range from qualitative to quantitative, and include structural as well as dynamic analysis techniques. We describe the different methodologies, outline their underlying assumptions, and provide an assessment of their advantages and disadvantages. Moreover, publicly and/or commercially available implementations of these methodologies are listed as appropriate. In particular, this primer aims to provide a clear introduction and comprehensive coverage of biochemical modeling and simulation methodologies for the non-expert, with specific focus on relevant literature of STPs.

Scaffold biomaterials for nano-pathophysiology

This review is intended to provide an overview of tissue engineering strategies using scaffold biomaterials to develop a vascularized tissue engineered construct for nano-pathophysiology. Two primary topics are discussed. The first is the biological or synthetic microenvironments that regulate cell behaviors in pathological conditions and tissue regeneration. Second is the use of scaffold biomaterials with angiogenic factors and/or cells to realize vascularized tissue engineered constructs for nano-pathophysiology. These topics are significantly overlapped in terms of three-dimensional (3-D) geometry of cells and blood vessels. Therefore, this review focuses on neovascularization of 3-D scaffold biomaterials induced by angiogenic factors and/or cells. The novel strategy of this approach in nano-pathophysiology is to utilize the vascularized tissue engineered construct as a tissue model to predict the distribution and subsequent therapeutic efficacy of a drug delivery system with different physicochemical and biological properties.

The actin-driven spatiotemporal organization of T-cell signaling at the system scale

T cells are activated through interaction with antigen-presenting cells (APCs). During activation, receptors and signaling intermediates accumulate in diverse spatiotemporal distributions. These distributions control the probability of signaling interactions and thus govern information flow through the signaling system. Spatiotemporally resolved system-scale investigation of signaling can extract the regulatory information thus encoded, allowing unique insight into the control of T-cell function. Substantial technical challenges exist, and these are briefly discussed herein. While much of the work assessing T-cell spatiotemporal organization uses planar APC substitutes, we focus here on B-cell APCs with often stark differences. Spatiotemporal signaling distributions are driven by cell biologically distinct structures, a large protein assembly at the interface center, a large invagination, the actin-supported interface periphery as extended by smaller individual lamella, and a newly discovered whole-interface actin-driven lamellum. The more than 60 elements of T-cell activation studied to date are dynamically distributed between these structures, generating a complex organization of the signaling system. Signal initiation and core signaling prefer the interface center, while signal amplification is localized in the transient lamellum. Actin dynamics control signaling distributions through regulation of the underlying structures and drive a highly undulating T-cell/APC interface that imposes substantial constraints on T-cell organization. We suggest that the regulation of actin dynamics, by controlling signaling distributions and membrane topology, is an important rheostat of T-cell signaling.

Sonic hedgehog signaling regulates mode of cell division of early cerebral cortex progenitors and increases astrogliogenesis

The morphogen Sonic Hedgehog (SHH) plays a critical role in the development of different tissues. In the central nervous system, SHH is well known to contribute to the patterning of the spinal cord and separation of the brain hemispheres. In addition, it has recently been shown that SHH signaling also contributes to the patterning of the telencephalon and establishment of adult neurogenic niches. In this work, we investigated whether SHH signaling influences the behavior of neural progenitors isolated from the dorsal telencephalon, which generate excitatory neurons and macroglial cells in vitro. We observed that SHH increases proliferation of cortical progenitors and generation of astrocytes, whereas blocking SHH signaling with cyclopamine has opposite effects. In both cases, generation of neurons did not seem to be affected. However, cell survival was broadly affected by blockade of SHH signaling. SHH effects were related to three different cell phenomena: mode of cell division, cell cycle length and cell growth. Together, our data in vitro demonstrate that SHH signaling controls cell behaviors that are important for proliferation of cerebral cortex progenitors, as well as differentiation and survival of neurons and astroglial cells.

Periodic migration in a physical model of cells on micropatterns

We extend a model for the morphology and dynamics of a crawling eukaryotic cell to describe cells on micropatterned substrates. This model couples cell morphology, adhesion, and cytoskeletal flow in response to active stresses induced by actin and myosin. We propose that protrusive stresses are only generated where the cell adheres, leading to the cell's effective confinement to the pattern. Consistent with experimental results, simulated cells exhibit a broad range of behaviors, including steady motion, turning, bipedal motion, and periodic migration, in which the cell crawls persistently in one direction before reversing periodically. We show that periodic motion emerges naturally from the coupling of cell polarization to cell shape by reducing the model to a simplified one-dimensional form that can be understood analytically.

Functional morphometric analysis in cellular behaviors: shape and size matter

Cellular morphogenesis in response to biophysical and topographical cues provides insights into cytoskeletal status, biointerface communications, and phenotypic adaptations in an incessant signaling feedback that governs cellular fate. Morphometric characterization is an important element in the study of the dynamic cellular behaviors, in their interactive response to environmental influence exerted by culture system. They collectively serve to reflect cellular proliferation, migration, and differentiation, which may serve as prognostic indices for clinical and pathological diagnosis. Various parameters are proposed to categorize morphological adaptations in relation to cellular function. In this review, the underlying principles, assumptions, and limitations of morphological characterizations are discussed. The significance, challenges, and implications of quantitative morphometric characterization of cell shapes and sizes in determining cellular functions are discussed.

Excitable behavior in amoeboid chemotaxis

Chemotaxis, the directed motion of cells in response to chemical gradients, is a fundamental process. Eukaryotic cells detect spatial differences in chemoattractant receptor occupancy with high precision and use these differences to bias the location of actin-rich protrusions to guide their movement. Research into chemotaxis has benefitted greatly from a systems biology approach that combines novel experimental and computational tools to pose and test hypotheses. Recently, one such hypothesis has been postulated proposing that chemotaxis in eukaryotic cells is mediated by locally biasing the activity of an underlying excitable system. The excitable system hypothesis can account for a number of cellular behaviors related to chemotaxis, including the stochastic nature of the movement of unstimulated cells, the directional bias imposed by chemoattractant gradients, and the observed spatial and temporal distribution of signaling and cytoskeleton proteins.

Cortical geometry may influence placement of interface between Par protein domains in early Caenorhabditis elegans embryos

During polarization, proteins and other polarity determinants segregate to the opposite ends of the cell (the poles) creating biochemically and dynamically distinct regions. Embryos of the nematode worm Caenorhabditis elegans (C. elegans) polarize shortly after fertilization, creating distinct regions of Par protein family members. These regions are maintained through to first cleavage when the embryo divides along the plane specified by the interface between regions, creating daughter cells with different protein content. In wild type single cell embryos the interface between these Par protein regions is reliably positioned at approximately 60% egg length, however, it is not known what mechanisms are responsible for specifying the position of the interface. In this investigation, we use two mathematical models to investigate the movement and positioning of the interface: a biologically based reaction-diffusion model of Par protein dynamics, and the analytically tractable perturbed Allen-Cahn equation. When we numerically simulate the models on a static 2D domain with constant thickness, both models exhibit a persistently moving interface that specifies the boundary between distinct regions. When we modify the simulation domain geometry, movement halts and the interface is stably positioned where the domain thickness increases. Using asymptotic analysis with the perturbed Allen-Cahn equation, we show that interface movement depends explicitly on domain geometry. Using a combination of analytic and numeric techniques, we demonstrate that domain geometry, a historically overlooked aspect of cellular simulations, may play a significant role in spatial protein patterning during polarization.

A comparison of computational models for eukaryotic cell shape and motility

Eukaryotic cell motility involves complex interactions of signalling molecules, cytoskeleton, cell membrane, and mechanics interacting in space and time. Collectively, these components are used by the cell to interpret and respond to external stimuli, leading to polarization, protrusion, adhesion formation, and myosin-facilitated retraction. When these processes are choreographed correctly, shape change and motility results. A wealth of experimental data have identified numerous molecular constituents involved in these processes, but the complexity of their interactions and spatial organization make this a challenging problem to understand. This has motivated theoretical and computational approaches with simplified caricatures of cell structure and behaviour, each aiming to gain better understanding of certain kinds of cells and/or repertoire of behaviour. Reaction–diffusion (RD) equations as well as equations of viscoelastic flows have been used to describe the motility machinery. In this review, we describe some of the recent computational models for cell motility, concentrating on simulations of cell shape changes (mainly in two but also three dimensions). The problem is challenging not only due to the difficulty of abstracting and simplifying biological complexity but also because computing RD or fluid flow equations in deforming regions, known as a “free-boundary” problem, is an extremely challenging problem in applied mathematics. Here we describe the distinct approaches, comparing their strengths and weaknesses, and the kinds of biological questions that they have been able to address.

The mysterious human epidermal cell cycle, or an oncogene-induced differentiation checkpoint

Fifteen years ago, we reported that proto-oncogene MYC promoted differentiation of human epidermal stem cells, a finding that was surprising to the MYC and the skin research communities. MYC was one of the first human oncogenes identified, and it had been strongly associated with proliferation. However, it was later shown that MYC could induce apoptosis under low survival conditions. Currently, the notion that MYC promotes epidermal differentiation is widely accepted, but the cell cycle mechanisms that elicit this function remain unresolved. We have recently reported that keratinocytes respond to cell cycle deregulation and DNA damage by triggering terminal differentiation. This mechanism might constitute a homeostatic protection face to cell cycle insults. Here, I discuss recent and not-so-recent evidence suggesting the existence of a largely unexplored oncogene-induced differentiation response (OID) analogous to oncogene-induced apoptosis (OIA) or senescence (OIS). In addition, I propose a model for the role of the cell cycle in skin homeostasis maintenance and for the dual role of MYC in differentiation.

Chondrogenesis on sulfonate-coated hydrogels is regulated by their mechanical properties

Many studies have demonstrated that sulfur-containing acidic groups induce chondrogenesis in vitro and in vivo. Recently, it is increasingly clear that mechanical properties of cell substrates largely influence cell differentiation. Thus, the present study investigated how mechanical properties of sulfonate-coated hydrogels influences chondrogenesis of mesenchymal stem cells (MSCs). Sulfonate-coated polyacrylamide gels (S-PAAm gels) which have the elastic modulus, E, of about 1, 15 and 150 kPa, were used in this study. MSCs cultured on the high stiffness S-PAAm gels (E=∼150 kPa) spread out with strong expression of stress fibers, while MSCs cultured on the low stiffness S-PAAm gels (E=∼1 kPa) had round shapes with less stress fibers but more cortical actins. Importantly, even in the absence of differentiation supplements, the lower stiffness S-PAAm gels led to the higher mRNA levels of chondrogenic markers such as Col2a1, Agc and Sox9 and the lower mRNA levels of an undifferentiation marker Sca1, indicating that the mechanical properties of S-PAAm gels strongly influence chondrogenesis. Blebbistatin which blocks myosin II-mediated mechanical sensing suppressed chondrogenesis induced by the low stiffness S-PAAm gels. The present study demonstrates that the soft S-PAAm gels effectively drive MSC chondrogenesis even in the absence of soluble differentiation factors and thus suggests that sulfonate-containing hydrogels with low stiffness could be a powerful tool for cartilage regeneration.

Effects of compartment size on the kinetics of intracompartmental multimeric protein synthesis

The cell contents are encapsulated within a compartment, the volume of which is a fundamental physical parameter that may affect intracompartmental reactions. However, there have been few studies to elucidate whether and how volume changes alone can affect the reaction kinetics. It is difficult to address these questions in vivo, because forced cell volume changes, e.g., by osmotic inflation/deflation, globally alters the internal state. Here, we prepared artificial cell-like compartments with different volumes but with identical constituents, which is not possible with living cells, and synthesized two tetrameric enzymes, β-glucuronidase (GUS) and β-galactosidase (GAL), by cell-free protein synthesis. Tetrameric GUS but not GAL was synthesized more quickly in smaller compartments. The difference between the two was dependent on the rate-limiting step and the reaction order. The observed acceleration mechanism would be applicable to living cells as multimeric protein synthesis in a microcompartment is ubiquitous in vivo.

Spatial aspects in the SMAD signaling pathway

Among other approaches, differential equations are used for a deterministic quantitative description of time-dependent biological processes. For intracellular systems, such as signaling pathways, most existing models are based on ordinary differential equations. These models describe temporal processes, while they neglect spatial aspects. We present a model for the SMAD signaling pathway, which gives a temporal and spatial description on the basis of reaction diffusion equations to answer the question whether cell geometry plays a role in signaling. In this article we simulate the ordinary differential equations as well as partial differential equations of parabolic type with suile numerical methods, the latter on different cell geometries. In addition to manual construction of idealized cells, we also construct meshes from microscopy images of real cells. The main focus of the paper is to compare the results of the model without and with spatial aspects to answer the addressed question. The results show that diffusion in the model can lead to significant intracellular gradients of signaling molecules and changes the level of response to the signal transduced by the signaling pathway. In particular, the extent of these observations depends on the geometry of the cell.

Geometry sensing by self-organized protein patterns

In the living cell, proteins are able to organize space much larger than their dimensions. In return, changes of intracellular space can influence biochemical reactions, allowing cells to sense their size and shape. Despite the possibility to reconstitute protein self-organization with only a few purified components, we still lack knowledge of how geometrical boundaries affect spatiotemporal protein patterns. Following a minimal systems approach, we used purified proteins and photolithographically patterned membranes to study the influence of spatial confinement on the self-organization of the Min system, a spatial regulator of bacterial cytokinesis, in vitro. We found that the emerging protein pattern responds even to the lateral, two-dimensional geometry of the membrane such that, as in the three-dimensional cell, Min protein waves travel along the longest axis of the membrane patch. This shows that for spatial sensing the Min system does not need to be enclosed in a three-dimensional compartment. Using a computational model we quantitatively analyzed our experimental findings and identified persistent binding of MinE to the membrane as requirement for the Min system to sense geometry. Our results give insight into the interplay between geometrical confinement and biochemical patterns emerging from a nonlinear reaction-diffusion system.

Deconstructing the third dimension: how 3D culture microenvironments alter cellular cues

Much of our understanding of the biological mechanisms that underlie cellular functions, such as migration, differentiation and force-sensing has been garnered from studying cells cultured on two-dimensional (2D) glass or plastic surfaces. However, more recently the cell biology field has come to appreciate the dissimilarity between these flat surfaces and the topographically complex, three-dimensional (3D) extracellular environments in which cells routinely operate in vivo. This has spurred substantial efforts towards the development of in vitro 3D biomimetic environments and has encouraged much cross-disciplinary work among biologists, material scientists and tissue engineers. As we move towards more-physiological culture systems for studying fundamental cellular processes, it is crucial to define exactly which factors are operative in 3D microenvironments. Thus, the focus of this Commentary will be on identifying and describing the fundamental features of 3D cell culture systems that influence cell structure, adhesion, mechanotransduction and signaling in response to soluble factors, which - in turn - regulate overall cellular function in ways that depart dramatically from traditional 2D culture formats. Additionally, we will describe experimental scenarios in which 3D culture is particularly relevant, highlight recent advances in materials engineering for studying cell biology, and discuss examples where studying cells in a 3D context provided insights that would not have been observed in traditional 2D systems.

Modelling cell polarization driven by synthetic spatially graded Rac activation

The small GTPase Rac is known to be an important regulator of cell polarization, cytoskeletal reorganization, and motility of mammalian cells. In recent microfluidic experiments, HeLa cells endowed with appropriate constructs were subjected to gradients of the small molecule rapamycin leading to synthetic membrane recruitment of a Rac activator and direct graded activation of membrane-associated Rac. Rac activation could thus be triggered independent of upstream signaling mechanisms otherwise responsible for transducing activating gradient signals. The response of the cells to such stimulation depended on exceeding a threshold of activated Rac. Here we develop a minimal reaction-diffusion model for the GTPase network alone and for GTPase-phosphoinositide crosstalk that is consistent with experimental observations for the polarization of the cells. The modeling suggests that mutual inhibition is a more likely mode of cell polarization than positive feedback of Rac onto its own activation. We use a new analytical tool, Local Perturbation Analysis, to approximate the partial differential equations by ordinary differential equations for local and global variables. This method helps to analyze the parameter space and behaviour of the proposed models. The models and experiments suggest that (1) spatially uniform stimulation serves to sensitize a cell to applied gradients. (2) Feedback between phosphoinositides and Rho GTPases sensitizes a cell. (3) Cell lengthening/flattening accompanying polarization can increase the sensitivity of a cell and stabilize an otherwise unstable polarization.

Influence of the nuclear membrane, active transport, and cell shape on the Hes1 and p53-Mdm2 pathways: insights from spatio-temporal modelling

There are many intracellular signalling pathways where the spatial distribution of the molecular species cannot be neglected. These pathways often contain negative feedback loops and can exhibit oscillatory dynamics in space and time. Two such pathways are those involving Hes1 and p53-Mdm2, both of which are implicated in cancer. In this paper we further develop the partial differential equation (PDE) models of Sturrock et al. (J. Theor. Biol., 273:15-31, 2011) which were used to study these dynamics. We extend these PDE models by including a nuclear membrane and active transport, assuming that proteins are convected in the cytoplasm towards the nucleus in order to model transport along microtubules. We also account for Mdm2 inhibition of p53 transcriptional activity. Through numerical simulations we find ranges of values for the model parameters such that sustained oscillatory dynamics occur, consistent with available experimental measurements. We also find that our model extensions act to broaden the parameter ranges that yield oscillations. Hence oscillatory behaviour is made more robust by the inclusion of both the nuclear membrane and active transport. In order to bridge the gap between in vivo and in silico experiments, we investigate more realistic cell geometries by using an imported image of a real cell as our computational domain. For the extended p53-Mdm2 model, we consider the effect of microtubule-disrupting drugs and proteasome inhibitor drugs, obtaining results that are in agreement with experimental studies.

How to build a robust intracellular concentration gradient

Concentration gradients of morphogens are critical regulators of patterning in developmental biology. Increasingly, intracellular concentration gradients have also been found to orchestrate spatial organization, but inside single cells, where they regulate processes such as cell division, polarity and mitotic spindle dynamics. Here, we discuss recent progress in understanding how such intracellular gradients can be built robustly. We focus particularly on the Pom1p gradient in fission yeast, elucidating how various buffering mechanisms operate to ensure precise gradient formation. In this case, a systems-level understanding of the entire mechanism of precise gradient construction is now within reach, with important implications for gradients in both intracellular and developmental contexts.

Endocytosis and signaling: cell logistics shape the eukaryotic cell plan

Our understanding of endocytosis has evolved remarkably in little more than a decade. This is the result not only of advances in our knowledge of its molecular and biological workings, but also of a true paradigm shift in our understanding of what really constitutes endocytosis and of its role in homeostasis. Although endocytosis was initially discovered and studied as a relatively simple process to transport molecules across the plasma membrane, it was subsequently found to be inextricably linked with almost all aspects of cellular signaling. This led to the notion that endocytosis is actually the master organizer of cellular signaling, providing the cell with understandable messages that have been resolved in space and time. In essence, endocytosis provides the communications and supply routes (the logistics) of the cell. Although this may seem revolutionary, it is still likely to be only a small part of the entire story. A wealth of new evidence is uncovering the surprisingly pervasive nature of endocytosis in essentially all aspects of cellular regulation. In addition, many newly discovered functions of endocytic proteins are not immediately interpretable within the classical view of endocytosis. A possible framework, to rationalize all this new knowledge, requires us to "upgrade" our vision of endocytosis. By combining the analysis of biochemical, biological, and evolutionary evidence, we propose herein that endocytosis constitutes one of the major enabling conditions that in the history of life permitted the development of a higher level of organization, leading to the actuation of the eukaryotic cell plan.

Noise reduction in the intracellular pom1p gradient by a dynamic clustering mechanism

Chemical gradients can generate pattern formation in biological systems. In the fission yeast Schizosaccharomyces pombe, a cortical gradient of pom1p (a DYRK-type protein kinase) functions to position sites of cytokinesis and cell polarity and to control cell length. Here, using quantitative imaging, fluorescence correlation spectroscopy, and mathematical modeling, we study how its gradient distribution is formed. Pom1p gradients exhibit large cell-to-cell variability, as well as dynamic fluctuations in each individual gradient. Our data lead to a two-state model for gradient formation in which pom1p molecules associate with the plasma membrane at cell tips and then diffuse on the membrane while aggregating into and fragmenting from clusters, before disassociating from the membrane. In contrast to a classical one-component gradient, this two-state gradient buffers against cell-to-cell variations in protein concentration. This buffering mechanism, together with time averaging to reduce intrinsic noise, allows the pom1p gradient to specify positional information in a robust manner.

Dimensional and temporal controls of three-dimensional cell migration by zyxin and binding partners

Spontaneous molecular oscillations are ubiquitous in biology. But to our knowledge, periodic cell migratory patterns have not been observed. Here we report the highly regular, periodic migration of cells along rectilinear tracks generated inside three-dimensional matrices, with each excursion encompassing several cell lengths, a phenotype that does not occur on conventional substrates. Short hairpin RNA depletion shows that these one-dimensional oscillations are uniquely controlled by zyxin and binding partners α-actinin and p130Cas, but not vasodilator-stimulated phosphoprotein and cysteine-rich protein 1. Oscillations are recapitulated for cells migrating along one-dimensional micropatterns, but not on two-dimensional compliant substrates. These results indicate that although two-dimensional motility can be well described by speed and persistence, three-dimensional motility requires two additional parameters, the dimensionality of the cell paths in the matrix and the temporal control of cell movements along these paths. These results also suggest that the zyxin/α-actinin/p130Cas module may ensure that motile cells in a three-dimensional matrix explore the largest space possible in minimum time.

How cells integrate complex stimuli: the effect of feedback from phosphoinositides and cell shape on cell polarization and motility

To regulate shape changes, motility and chemotaxis in eukaryotic cells, signal transduction pathways channel extracellular stimuli to the reorganization of the actin cytoskeleton. The complexity of such networks makes it difficult to understand the roles of individual components, let alone their interactions and multiple feedbacks within a given layer and between layers of signalling. Even more challenging is the question of if and how the shape of the cell affects and is affected by this internal spatiotemporal reorganization. Here we build on our previous 2D cell motility model where signalling from the Rho family GTPases (Cdc42, Rac, and Rho) was shown to organize the cell polarization, actin reorganization, shape change, and motility in simple gradients. We extend this work in two ways: First, we investigate the effects of the feedback between the phosphoinositides (PIs) , and Rho family GTPases. We show how that feedback increases heights and breadths of zones of Cdc42 activity, facilitating global communication between competing cell “fronts”. This hastens the commitment to a single lamellipodium initiated in response to multiple, complex, or rapidly changing stimuli. Second, we show how cell shape feeds back on internal distribution of GTPases. Constraints on chemical isocline curvature imposed by boundary conditions results in the fact that dynamic cell shape leads to faster biochemical redistribution when the cell is repolarized. Cells with frozen cytoskeleton, and static shapes, consequently respond more slowly to reorienting stimuli than cells with dynamic shape changes, the degree of the shape-induced effects being proportional to the extent of cell deformation. We explain these concepts in the context of several in silico experiments using our 2D computational cell model.

Single cells, such as amoeba and white blood cells, change shape and move in response to environmental stimuli. Their behaviour is a consequence of the intracellular properties balanced by external forces. The internal regulation is modulated by several proteins that interact with one another and with membrane lipids. We examine, through in silico experiments using a computational model of a moving cell, the interactions of an important class of such proteins (Rho GTPases) and lipids (phosphoinositides, PIs), their spatial redistribution, and how they affect and are affected by cell shape. Certain GTPases promote the assembly of the actin cytoskeleton. This then leads to the formation of a cell protrusion, the leading edge. The feedback between PIs and GTPases facilitates global communication across the cell, ensuring that multiple, complex, or rapidly changing stimuli can be resolved into a single decision for positioning the leading edge. Interestingly, the cell shape itself affects the intracellular biochemistry, resulting from interactions between the curvature of the chemical fronts and the cell edge. Cells with static shapes consequently respond more slowly to reorienting stimuli than cells with dynamic shape changes. This potential to respond more rapidly to external stimuli depends on the degree of cellular shape deformation.

CROSSTALK FACILITATES SPATIAL SIGNAL PROPAGATION THROUGH MAPK CASCADES

In intracellular mitogen-activated protein kinase (MAPK) cascades, it has been shown that signals can be propagated across the cell cytosol in the form of phosphoprotein waves arising from the bistable response of MAPK to active MAPK kinase. Without such a bistable response, however, they can not propagate into distant cell compartments, although a long positive feedback endows a mechanistically-distinct bistable response of MAPK to extracellular signal. Here we provide a compensate means that uses crosstalk between parallel identical pathways of MAPK cascades. For a spherical cell, we find that both unidirectional and bidirectional crosstalk with enhancement of phosphorylation can facilitate phosphoprotein signal propagation from the plasma membrane to the periphery of cell nucleus. Moreover, different shallow spatial gradients of biphosphorylated MAPK occur in the cytosol under different strengths of pathway interactions. These results suggest that crosstalk would be utilized by living organisms for spatial information transfer and cellular decision-making processing.

Regulation of the MEX-5 gradient by a spatially segregated kinase/phosphatase cycle

Protein concentration gradients encode spatial information across cells and tissues and often depend on spatially localized protein synthesis. Here, we report that a different mechanism underlies the MEX-5 gradient. MEX-5 is an RNA-binding protein that becomes distributed in a cytoplasmic gradient along the anterior-to-posterior axis of the one-cell C. elegans embryo. We demonstrate that the MEX-5 gradient is a direct consequence of an underlying gradient in MEX-5 diffusivity. The MEX-5 diffusion gradient arises when the PAR-1 kinase stimulates the release of MEX-5 from slow-diffusive, RNA-containing complexes in the posterior cytoplasm. PAR-1 directly phosphorylates MEX-5 and is antagonized by the spatially uniform phosphatase PP2A. Mathematical modeling and in vivo observations demonstrate that spatially segregated phosphorylation and dephosphorylation reactions are sufficient to generate stable protein concentration gradients in the cytoplasm. The principles demonstrated here apply to any spatially segregated modification cycle that affects protein diffusion and do not require protein synthesis or degradation.

B cell activation triggered by the formation of the small receptor cluster: a computational study

We proposed a spatially extended model of early events of B cell receptors (BCR) activation, which is based on mutual kinase-receptor interactions that are characteristic for the immune receptors and the Src family kinases. These interactions lead to the positive feedback which, together with two nonlinearities resulting from the double phosphorylation of receptors and Michaelis-Menten dephosphorylation kinetics, are responsible for the system bistability. We demonstrated that B cell can be activated by a formation of a tiny cluster of receptors or displacement of the nucleus. The receptors and Src kinases are activated, first locally, in the locus of the receptor cluster or the region where the cytoplasm is the thinnest. Then the traveling wave of activation propagates until activity spreads over the whole cell membrane. In the models in which we assume that the kinases are free to diffuse in the cytoplasm, we found that the fraction of aggregated receptors, capable to initiate B cell activation decreases with the decreasing thickness of cytoplasm and decreasing kinase diffusion. When kinases are restricted to the cell membrane - which is the case for most of the Src family kinases - even a cluster consisting of a tiny fraction of total receptors becomes activatory. Interestingly, the system remains insensitive to the modest changes of total receptor level. The model provides a plausible mechanism of B cells activation due to the formation of small receptors clusters collocalized by binding of polyvalent antigens or arising during the immune synapse formation.

B cells are activated in response to binding of appropriate ligands, which induces the aggregation of B cell receptors. The formation of even small clusters containing less than 1% of all the receptors is sufficient for activation. This observation led us to a model in which the receptor cluster serves only as a switch that turns on the activation process involving also the remaining receptors. The idea of the model exploits the fact the Src kinase - BCR system is bistable, and thus its local activation may start the propagation of a traveling wave, which spreads activation over the entire membrane. We found that the minimal size of the activatory cluster decreases with the thickness of the cytoplasm and kinase diffusion coefficient. It is particularly small when kinases are restricted to the membrane. These findings are consistent with the properties of B cells, which prior to activation have extremely thin cytoplasmic layer and in which Src family kinases (interacting with the receptors) are tethered to the membrane.

Engineered approaches to the stem cell microenvironment for cardiac tissue regeneration

Micro- and nanoscale engineering approaches in medicine have the potential to recreate physiologically relevant stem cell microenvironments to enhance our understanding of stem cell behaviour and bring stem cell therapy closer to fruition. The realization of such advancements will impact a number of therapeutic applications, the most immediate of which may be the repair of heart tissue. Despite profound advances in creating physiologically relevant in vivo stem cell niches through the control of biochemical regulatory factors, further synergism of innovative techniques promise to elucidate the impact of a number of physical cues such as stem cell differentiation into cardiac cells, the electromechanical coupling among these cells, and the formation of bioengineered cardiac tissue grafts. This review examines the recent physiologically relevant micro- and nanoengineering efforts that have been made to address these factors. In Sections II and III, we introduce the traditional focuses of stem cell derived cardiac tissue: differentiation directed by transcription factors and structural cues within the stem cell niche. However, the majority of this review, Sections IV-VII, endeavours to highlight innovative and unconventional microscale engineering techniques that have employed topographic, biomaterial, microfluidic, mechanical, electrical, and optical stimulation for stem cell based cardiac tissue engineering.

Spatio-temporal modelling of the NF-κB intracellular signalling pathway: the roles of diffusion, active transport, and cell geometry

The nuclear factor kappa B (NF-κB) intracellular signalling pathway is central to many stressful, inflammatory, and innate immune responses. NF-κB proteins themselves are transcription factors for hundreds of genes. Experiments have shown that the NF-κB pathway can exhibit oscillatory dynamics-a negative feedback loop causes oscillatory nuclear-cytoplasmic translocation of NF-κB. Given that cell size and shape are known to influence intracellular signal transduction, we consider a spatio-temporal model of partial differential equations for the NF-κB pathway, where we model molecular movement by diffusion and, for several key species including NF-κB, by active transport as well. Through numerical simulations we find values for model parameters such that sustained oscillatory dynamics occur. Our spatial profiles and animations bear a striking resemblance to experimental images and movie clips employing fluorescent fusion proteins. We discover that oscillations in nuclear NF-κB may occur when active transport is across the nuclear membrane only, or when no species are subject to active transport. However, when active transport is across the nuclear membrane and NF-κB is additionally actively transported through the cytoplasm, oscillations are lost. Hence transport mechanisms in a cell will influence its response to activation of its NF-κB pathway. We also demonstrate that sustained oscillations in nuclear NF-κB are somewhat robust to changes in the shape of the cell, or the shape, location, and size of its nucleus, or the location of ribosomes. Yet if the cell is particularly flat or the nucleus sufficiently small, then oscillations are lost. Thus the geometry of a cell may partly determine its response to NF-κB activation. The NF-κB pathway is known to be constitutively active in several human cancers. Our spatially explicit modelling approach will allow us, in future work, to investigate targeted drug therapy of tumours.

Modeling of spatially-restricted intracellular signaling

Understanding the signaling capabilities of a cell presents a major challenge, not only due to the number of molecules involved, but also because of the complex network connectivity of intracellular signaling. Recently, the proliferation of quantitative imaging techniques has led to the discovery of the vast spatial organization of intracellular signaling. Computational modeling has emerged as a powerful tool for understanding how inhomogeneous signaling originates and is maintained. This article covers the current imaging techniques used to obtain quantitative spatial data and the mathematical approaches used to model spatial cell biology. Modeling-derived hypotheses have been experimentally tested and the integration of modeling and imaging approaches has led to non-intuitive mechanistic insights.

Precision of sensing cell length via concentration gradients

Unicellular organisms are typically found to have a characteristic cell size. To achieve a homeostatic distribution of cell sizes over many generations requires that cell length is actively sensed and regulated. However, the mechanisms by which cell size is controlled remain poorly understood. Recent experiments in fission yeast have shown that cell length is controlled in part by polar gradients of the protein Pom1 together with localized measurement of concentration at midcell. Dilution as the cell grows leads to a reduction in the midcell protein concentration, which lifts a block on mitosis. Here we analyze the precision of this mechanism for length sensing in the presence of inevitable intrinsic noise in the processes leading to formation and measurement of this gradient. We find that the use of concentration gradients allows for more robust length sensing than a comparable spatially uniform system, and allows for reliable length determination even if the average protein concentration throughout the cell remains constant as the cell grows. Optimal values for the gradient decay length and receptor dissociation constant emerge from maximizing sensitivity while minimizing the impact of density fluctuations.

The role of substratum compliance of hydrogels on vascular endothelial cell behavior

Cardiovascular disease (CVD) remains a leading cause of death both within the United States (US) as well as globally. In 2006 alone, over one-third of all deaths in the US were attributable to CVD. The high prevalence, mortality, morbidity, and socioeconomic impact of CVD has motivated a significant research effort; however, there remain significant knowledge gaps regarding disease onset and progression as well as pressing needs for improved therapeutic approaches. One critical area of research that has received limited attention is the role of biophysical cues on the modulation of endothelial cell behaviors; specifically, the impact of local compliance, or the stiffness, of the surrounding vascular endothelial extracellular matrix. In this study, the impact of substratum compliance on the modulation of cell behaviors of several human primary endothelial cell types, representing different anatomic sites and differentiation states in vivo, were investigated. Substrates used within our studies span the range of compliance that has been reported for the vascular endothelial basement membrane. Differences in substratum compliance had a profound impact on cell attachment, spreading, elongation, proliferation, and migration. In addition, each cell population responded differentially to changes in substratum compliance, documenting endothelial heterogeneity in the response to biophysical cues. These results demonstrate the importance of incorporating substratum compliance in the design of in vitro experiments as well as future prosthetic design. Alterations in vascular substratum compliance directly influence endothelial cell behavior and may participate in the onset and/or progression of CVDs.

A comparison of mathematical models for polarization of single eukaryotic cells in response to guided cues

Polarization, a primary step in the response of an individual eukaryotic cell to a spatial stimulus, has attracted numerous theoretical treatments complementing experimental studies in a variety of cell types. While the phenomenon itself is universal, details differ across cell types, and across classes of models that have been proposed. Most models address how symmetry breaking leads to polarization, some in abstract settings, others based on specific biochemistry. Here, we compare polarization in response to a stimulus (e.g., a chemoattractant) in cells typically used in experiments (yeast, amoebae, leukocytes, keratocytes, fibroblasts, and neurons), and, in parallel, responses of several prototypical models to typical stimulation protocols. We find that the diversity of cell behaviors is reflected by a diversity of models, and that some, but not all models, can account for amplification of stimulus, maintenance of polarity, adaptation, sensitivity to new signals, and robustness.

Getting cells and tissues into shape

Cells of all shapes and sizes are able to calculate the location of their middles in order to divide into two during mitosis. Minc et al. (2011) and Gibson et al. (2011) now show that simple mechanical models accurately predict cleavage-plane positioning, and that geometrical interactions between neighboring cells are sufficient to generate ordered patterns of mitosis in growing epithelia.

Model of cell signal transduction in a three-dimensional domain

Intracellular signalling molecules form pathways inside the cell. These pathways carry a signal to target proteins which results in cellular responses. We consider a spherical cell with two internal compartments containing localized activating enzymes where as deactivating enzymes are spread uniformly through out the cytosol. Two diffusible signalling molecules are activated at the compartments and later deactivated in the cytosol due to deactivating enzymes. The two signalling molecules are a single link in a cascade reaction and form a self regulated dynamical system involving positive and negative feedback. Using matched asymptotic expansions we obtain approximate solutions of the steady state diffusion equation with a linear decay rate. We obtain three-dimensional concentration profiles for the signalling molecules. We also investigate an extension of the above system which has multiple cascade reactions occurring between multiple signalling molecules. Numerically, we show that the speed of the signal is an increasing function of the number of links in the cascade.