An Extended Filament Based Lamellipodium Model Produces Various Moving Cell Shapes in the Presence of Chemotactic Signals

The Filament Based Lamellipodium Model (FBLM) is a two-phase two-dimensional continuum model, describing the dynamcis of two interacting families of locally parallel actin filaments (C.Schmeiser and D.Oelz, How do cells move? Mathematical modeling of cytoskeleton dynamics and cell migration. Cell mechanics: from single scale-based models to multiscale modeling. Chapman and Hall, 2010). It contains accounts of the filaments' bending stiffness, of adhesion to the substrate, and of cross-links connecting the two families. An extension of the model is presented with contributions from nucleation of filaments by branching, from capping, from contraction by actin-myosin interaction, and from a pressure-like repulsion between parallel filaments due to Coulomb interaction. The effect of a chemoattractant is described by a simple signal transduction model influencing the polymerization speed. Simulations with the extended model show its potential for describing various moving cell shapes, depending on the signal transduction procedure, and for predicting transients between nonmoving and moving states as well as changes of direction

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.National Cancer Institute (U.S.) (Grant No. 5U01CA177799)National Institutes of Health (U.S.) (Ruth L. Kirschstein National Research Service Award

Multiscale computational modeling of single cell migration in 3D

La migración celular es un proceso complejo, orquestado por factores químicos y biológicos, por la microestructura y por las propiedades mecánicas de la matriz extracelular. Este fenómeno es fundamental para el desarrollo de tejidos en los organismos pluricelulares, y como seres humanos, nos acompaña durante toda la vida, desde el mismo momento de la concepción hasta la muerte. Juega un papel fundamental durante el desarrollo embrionario determinando la formación de los diferentes órganos (morfogénesis) y es clave en todos los procesos regenerativos como la renovación de la piel, la respuesta inflamatoria o la cicatrización de heridas. Sin embargo, también contribuye al desarrollo de procesos patológicos como la metástasis, el retraso mental, la osteoporosis o enfermedades vasculares entre otros. Es por ello de vital importancia el conocer los mecanismos fundamentales que controlan la migración celular con el fin de tratar de manera efectiva las diferentes patologías, así como avanzar en el trasplante de órganos y el desarrollo de tejidos artificiales. Así pues, el objetivo de esta Tesis es el desarrollo de modelos a distintas escalas y centrados en diversos aspectos de la migración, de manera que faciliten la compresión de fenómenos específicos y sirvan como guía para el diseño de experimentos. Dada la complejidad y las grandes diferencias respecto a la migración colectiva, todos los modelos y análisis de esta Tesis se centran en células individuales. En primer lugar se ha estudiado la migración tridimensional de una célula individual embebida en una matriz extracelular donde su velocidad y orientación se consideran reguladas por estímulos mecánicos. Para ello se ha desarrollado un modelo mecanosensor basado en elementos finitos y se ha analizado el comportamiento celular en función de diferentes rigideces y condiciones de contorno a escala celular. A medida que el trabajo ha progresado, los resultados del modelo unidos a nuevos avances científicos publicados en este ámbito, han reforzado la idea de que el mecansimo mecanosensor juega un papel crítico en los procesos que dirigen la migración celular. Por ello, se ha necesitado un estudio más profundo de este fenómeno para lo que se ha utilizado un modelo mucho más detallado a escala intracelular. Así pues, se ha explorado la estructura interna del citoesqueleto y su comportamiento ante cambios mecánicos en la matriz extracelular, utilizando un modelo discreto de partículas basado en dinámica Browniana con el que se ha simulado la formación de una red de actina (polimerización) entrecruzada con proteínas y motores moleculares. En concreto, se ha estudiado el comportamiento activo de estos motores y su papel como sensores de estímulos mecánicos externos (mecanosensores) de manera que los resultados obtenidos con este modelo “micro” han permitido validar las hipótesis del modelo previo. Consecuentemente, se ha revisado el modelo mecánico y se le ha añadido dependencia temporal, obteniendo un modelo continuo capaz de predecir respuestas celulares macroscópicas basadas en el comportamiento de los componentes microestructurales. En otras palabras, esta simplificación ha permitido la introducción de la respuesta macroscópica emergente obtenida del comportamiento dinámico de la microestructura, disminuyendo enormemente el coste computacional y por tanto permitiendo simulaciones a mayores escalas espacio-temporales. A continuación se han introducido las nuevas hipótesis en un modelo probabilístico de migración a escala celular basado en elementos finitos que permite al mismo tiempo el estudio de factores tanto a escala macroscópica (velocidades, trayectorias) como a escala celular (orientación, área de adhesión, tensiones celulares, desplazamientos de la matriz etc.). Adicionalmente, este modelo es sensible no sólo a la mecánica sino a las condiciones fluido-químicas del entorno, las cuales han sido analizadas igualmente mediante simulaciones por elementos finitos. Con todo esto, los modelos desarrollados todavía no incluyen una descripción detallada de procesos importantes envueltos en la migración celular como la protrusión de la membrana, la polimerización de actina en el frente celular o la formación de adhesiones focales. Por lo tanto, para completar la Tesis, se ha desarrollado un modelo continuo basado en diferencias finitas que permite el estudio del comportamiento dinámico del lamelipodio y el papel fundamental que juegan la polimerización de actina, los motores moleculares y las adhesiones focales (FAs) en el frente celular durante la migración. Cell migration is a complex process, orchestrated by biological and chemical factors, and by the microstructure and extracellular matrix (ECM) mechanical properties among others. It is essential for tissue development in multicellular organisms, and as human beings, it accompanies us throughout life, from conception to death. It plays a major role during embryonic development, defining organ formation (morphogenesis) and being crucial in all the regenerative processes such as skin renewal, inflammatory response or wound healing. However, it is also involved in several pathological processes e.g. metastasis, mental retardation, osteoporosis or vascular diseases. Therefore, understanding the fundamental mechanisms controling cell migration is vitally important to effectively treat different pathologies and to make progress in organ transplantation and tissue development. Thus, the main scope of this Thesis is the development of mathematical models at different scales and focused on different aspects of cell migration so that specific phenomena can be better understood, serving as a guide for the development of new experiments. All the models and analysis contained in this thesis are focused on single cells, firstly due to the complexity and marked differences with respect to collective cell migration, and secondly owing to the importance of individual migration in important processes such as metastatic tumor cell migration. In addition, since three- dimensional environments are physiologically more relevant, 3D approaches have been considered in most of the models here developed to better mimic in vivo conditions. Firstly, single cell migration of a cell embedded in a three-dimensional matrix was studied, regulating its velocity and polarization through mechanical clues. For this purpose, a finite element (FE) based mechanosensing model was developed, analyzing cell behavior according to different ECM rigidities and boundary conditions at the cell scale. As work advanced, results from the model together with recent findings from literature strengthened the idea that mechanosensing plays a critical role in cell motility driving processes. For this reason, a deeper understanting of this mechanism was needed, resulting in the development of a specific and more detailed model (at the intracellular scale). Hence, the cytoskeletal structure response to mechanical stimuli has been explored using a discrete particle-based Brownian dynamics model. This model was used to simulate the formation of actin networks (through actin polymerization) cross-linked with proteins (ACPs) and molecular motors. Specifically, the active role of molecular motors and their role as mechanosensors were studied, so that the results of the intracellular scale approach allowed the validation of the previous model main assumptions. As a consequence, the mechanical hypothesis were revised and a temporal dependence was incorporated, obtaining a new continuum model able to predict macroscopic cell responses based on microstructural components behavior. In other words, this simplification allowed introducing the emergent macroscopic response obtained from the active behavior of the microstructure, saving large amounts of computational time and permitting simulations at higher time and length scales. Next, the new hypotheses were incorporated into a probabilistic, FE-voxel-based cell-scale migration model, permitting simultaneously the study of macro-scale factors (velocities, trajectories) and cell-scale ones (polarization, adhesion area, cell stress, ECM displacements etc.). Additionally this model includes the effect of fluid-chemical stimuli, which was also analyzed by means of FE-simulations. With all this, the developed models still lacked a detailed description of important processes involved in cell migration such as membrane protrusion, actin polymerization or focal adhesion (FA) formation. As a result, a continuum model was designed to study the lamellipodium dynamics and the major role of actin polymerization and focal adhesions (FA) at the cell front during cell migration

Integrated Mathematical and Experimental Study of Cell Migration and Shape

Cell migration plays an essential role in many of physiological and pathological processes, including morphogenesis, inflammation, wound healing, and tumor metastasis. It is a complex process that involves multi-scale interactions between the cell and the extracellular matrix (ECM). Cells migrate through stromal ECM with native and cell-derived curvature at micron-meter scale are context-specific. How does the curvature of ECM mechanically change cell morphology and motility? Can the diverse migration behaviors from genetically identical cells be predictively using cell migrating data? We address these questions using an integrated computational and experimental approach: we developed three-dimensional biomechanical cell model and measured and analyzed a large number of cell migration images over time. Our findings suggest that 1. substrate curvature determines cell shape through contact and regulating protrusion dynamics; 2. effective cell migration is characterized with long cellular persistence time, low speed variation, spatial-temporally coordinated protrusion and contraction; 3. the cell shape variation space is low dimensional; and 4. migration behavior can be determined by a single image projected in the low dimensional cell shape variation space

Mechanical Sensing of Living Systems — From Statics to Dynamics

Living systems are fascinating sensing machines that outmatch all artificial machines. Our aim is to put a focus on the dynamics of mechanosensing in cellular systems through concepts and experimental approaches that have been developed during the past decades. By recognizing that a cellular system is not simply the intricate assembly of active and passive macromolecular actors but that it can also manifest scale-invariant and/or highly nonlinear global dynamics, biophysicists have opened a new domain of investigation of living systems. In this chapter, we review methods and techniques that have been implemented to decipher the cascade of temporal events which enable a cell to sense a mechanical stimulus and to elaborate a response to adapt or to counteract this perturbation. We mainly describe intrusive (mechanical probes) and nonintrusive (optical devices) experimental methods that have proved to be efficient for real-time characterization of stationary and nonstationary cellular dynamics. Finally, we discuss whether thermal fluctuations, which are inherent to living systems, are a source of coordination (e.g., synchronization) or randomization of the global dynamics of a cell

A Review of Mathematical Models for the Formation of\ud Vascular Networks

Mainly two mechanisms are involved in the formation of blood vasculature: vasculogenesis and angiogenesis. The former consists of the formation of a capillary-like network from either a dispersed or a monolayered population of endothelial cells, reproducible also in vitro by specific experimental assays. The latter consists of the sprouting of new vessels from an existing capillary or post-capillary venule. Similar phenomena are also involved in the formation of the lymphatic system through a process generally called lymphangiogenesis.\ud \ud A number of mathematical approaches have analysed these phenomena. This paper reviews the different modelling procedures, with a special emphasis on their ability to reproduce the biological system and to predict measured quantities which describe the overall processes. A comparison between the different methods is also made, highlighting their specific features

Atomic force microscopy-based mechanobiology

Mechanobiology emerges at the crossroads of medicine, biology, biophysics and engineering and describes how the responses of proteins, cells, tissues and organs to mechanical cues contribute to development, differentiation, physiology and disease. The grand challenge in mechanobiology is to quantify how biological systems sense, transduce, respond and apply mechanical signals. Over the past three decades, atomic force microscopy (AFM) has emerged as a key platform enabling the simultaneous morphological and mechanical characterization of living biological systems. In this Review, we survey the basic principles, advantages and limitations of the most common AFM modalities used to map the dynamic mechanical properties of complex biological samples to their morphology. We discuss how mechanical properties can be directly linked to function, which has remained a poorly addressed issue. We outline the potential of combining AFM with complementary techniques, including optical microscopy and spectroscopy of mechanosensitive fluorescent constructs, super-resolution microscopy, the patch clamp technique and the use of microstructured and fluidic devices to characterize the 3D distribution of mechanical responses within biological systems and to track their morphology and functional state.Peer ReviewedPostprint (published version

Mechanical models of proteins

textIn general, this dissertation is concerned with modeling of mechanical behavior of protein molecules. In particular, we focus on coarse-grained models, which bridge the gap in time and length scale between the atomistic simulation and biological processes. The dissertation presents three independent studies involving such models. The first study is concerned with a rigorous coarse-graining method for dynamics of linear systems. In this method, as usual, the conformational space of the original atomistic system is divided into master and slave degrees of freedom. Under the assumption that the characteristic timescales of the masters are slower than those of the slaves, the method results in Langevin-type equations of motion governed by an effective potential of mean force. In addition, coarse-graining introduces hydrodynamic-like coupling among the masters as well as non-trivial inertial effects. Application of our method to the long-timescale part of the relaxation spectra of proteins shows that such dynamic coupling is essential for reproducing their relaxation rates and modes. The second study is concerned with calibration of elastic network models based on the so-called B-factors, obtained from x-ray crystallographic measurements. We show that a proper calibration procedure must account for rigid-body motion and constraints imposed by the crystalline environment on the protein. These fundamental aspects of protein dynamics in crystals are often ignored in currently used elastic network models, leading to potentially erroneous network parameters. We develop an elastic network model that properly takes rigid-body motion and crystalline constraints into account. This model reveals that B-factors are dominated by rigid-body motion rather than deformation, and therefore B-factors are poorly suited for identifying elastic properties of protein molecules. Furthermore, it turns out that B-factors for a benchmark set of three hundred and thirty protein molecules can be well approximated by assuming that the protein molecules are rigid. The third study is concerned with the polymer mediated interaction between two planar surfaces. In particular, we consider the case where a thin polymer layer bridges two parallel plates. We consider two models of monodisperse and polydisperse for the polymer layer and obtain an analytical expression for the force-distance relationship of the two plates.Engineering Mechanic

All-scale structural analysis of biomolecules through dynamical graph partitioning

From femtosecond bond vibrations to millisecond domain motions, the dynamics of biomolecules spans a wide range of time and length scales. This hierarchy of overlapping scales links the molecular and biophysical details to key aspects of their functionality. However, the span of scales combined with their intricate coupling rapidly drives atomic simulation methods to their limits, thereby often resulting in the need for coarse-graining techniques which cannot take full account of the biochemical details. To overcome this tradeoff, a graph-theoretical framework inspired by multiscale community detection methods and stochastic processes is here introduced for the analysis of protein and DNA structures. Using biophysical force fields, we propose a general mapping of the 3D atomic coordinates onto an energy-weighted network that includes the physico-chemical details of interatomic bonds and interactions.Making use of a dynamics-based approach for community detection on networks, optimal partitionings of the structure are identified which are biochemically relevant over different scales. The structural organisation of the biomolecule is shown to be recovered bottom-up over the entire range of chemical, biochemical and biologically meaningful scales, directly from the atomic information of the structure, and without any reparameterisation. This methodology is applied and discussed in five proteins and an ensemble of DNA quadruplexes. In each case, multiple conformations associated with different states of the biomolecule or stages of the underlying catalytic reaction are analysed. Experimental observations are shown to be correctly captured, including the functional domains, regions of the protein with coherent dynamics such as rigid clusters, and the spontaneous closure of some enzymes in the absence of substrate. A computational mutational analysis tool is also derived which identifies both known and new residues with a significant impact on ligand binding. In large multimeric structures, the methodology highlights patterns of long range communication taking place between subunits. In the highly dynamic and polymorphic DNA quadruplexes, key structural features for their physical stability and signatures of their unfolding pathway are identified in the static structure.Open Acces