Coupling geometric PDEs with physics, cell morphology, motility and pattern formation

Annual report on the 6-months research programme held at the Isaac Newton Institute for Mathematical Sciences on Coupling geometric PDEs with physics for cell morphology, motility and pattern formation from July 13th - December 18th 201

Signature of a non-harmonic potential as revealed from a consistent shape and fluctuation analysis of an adherent membrane

The interaction of fluid membranes with a scaffold, which can be a planar surface or a more complex structure, is intrinsic to a number of systems - from artificial supported bilayers and vesicles to cellular membranes. In principle, these interactions can be either discrete and protein mediated, or continuous. In the latter case, they emerge from ubiquitous intrinsic surface interaction potentials as well as nature-designed steric contributions of the fluctuating membrane or from the polymers of the glycocalyx. Despite the fact that these nonspecific potentials are omnipresent, their description has been a major challenge from experimental and theoretical points of view. Here we show that a full understanding of the implications of the continuous interactions can be achieved only by expanding the standard superposition models commonly used to treat these types of systems, beyond the usual harmonic level of description. Supported by this expanded theoretical framework, we present three independent, yet mutually consistent, experimental approaches to measure the interaction potential strength and the membrane tension. Upon explicitly taking into account the nature of shot noise as well as of finite experimental resolution, excellent agreement with the augmented theory is obtained, which finally provides a coherent view of the behavior of the membrane in a vicinity of a scaffold.Comment: 15 pages, 12 figures, accepted by Physical Review

Measuring cellular traction forces on non-planar substrates

Animal cells use traction forces to sense the mechanics and geometry of their environment. Measuring these traction forces requires a workflow combining cell experiments, image processing and force reconstruction based on elasticity theory. Such procedures have been established before mainly for planar substrates, in which case one can use the Green's function formalism. Here we introduce a worksflow to measure traction forces of cardiac myofibroblasts on non-planar elastic substrates. Soft elastic substrates with a wave-like topology were micromolded from polydimethylsiloxane (PDMS) and fluorescent marker beads were distributed homogeneously in the substrate. Using feature vector based tracking of these marker beads, we first constructed a hexahedral mesh for the substrate. We then solved the direct elastic boundary volume problem on this mesh using the finite element method (FEM). Using data simulations, we show that the traction forces can be reconstructed from the substrate deformations by solving the corresponding inverse problem with a L1-norm for the residue and a L2-norm for 0th order Tikhonov regularization. Applying this procedure to the experimental data, we find that cardiac myofibroblast cells tend to align both their shapes and their forces with the long axis of the deformable wavy substrate.Comment: 34 pages, 9 figure

Traction force microscopy with optimized regularization and automated Bayesian parameter selection for comparing cells

Adherent cells exert traction forces on to their environment, which allows them to migrate, to maintain tissue integrity, and to form complex multicellular structures. This traction can be measured in a perturbation-free manner with traction force microscopy (TFM). In TFM, traction is usually calculated via the solution of a linear system, which is complicated by undersampled input data, acquisition noise, and large condition numbers for some methods. Therefore, standard TFM algorithms either employ data filtering or regularization. However, these approaches require a manual selection of filter- or regularization parameters and consequently exhibit a substantial degree of subjectiveness. This shortcoming is particularly serious when cells in different conditions are to be compared because optimal noise suppression needs to be adapted for every situation, which invariably results in systematic errors. Here, we systematically test the performance of new methods from computer vision and Bayesian inference for solving the inverse problem in TFM. We compare two classical schemes, L1- and L2-regularization, with three previously untested schemes, namely Elastic Net regularization, Proximal Gradient Lasso, and Proximal Gradient Elastic Net. Overall, we find that Elastic Net regularization, which combines L1 and L2 regularization, outperforms all other methods with regard to accuracy of traction reconstruction. Next, we develop two methods, Bayesian L2 regularization and Advanced Bayesian L2 regularization, for automatic, optimal L2 regularization. Using artificial data and experimental data, we show that these methods enable robust reconstruction of traction without requiring a difficult selection of regularization parameters specifically for each data set. Thus, Bayesian methods can mitigate the considerable uncertainty inherent in comparing cellular traction forces

Cyclic Stress at mHz Frequencies Aligns Fibroblasts in Direction of Zero Strain

Recognition of external mechanical signals is vital for mammalian cells. Cyclic stretch, e.g. around blood vessels, is one such signal that induces cell reorientation from parallel to almost perpendicular to the direction of stretch. Here, we present quantitative analyses of both, cell and cytoskeletal reorientation of umbilical cord fibroblasts. Cyclic strain of preset amplitudes was applied at mHz frequencies. Elastomeric chambers were specifically designed and characterized to distinguish between zero strain and minimal stress directions and to allow accurate theoretical modeling. Reorientation was only induced when the applied stretch exceeded a specific amplitude, suggesting a non-linear response. However, on very soft substrates no mechanoresponse occurs even for high strain. For all stretch amplitudes, the angular distributions of reoriented cells are in very good agreement with a theory modeling stretched cells as active force dipoles. Cyclic stretch increases the number of stress fibers and the coupling to adhesions. We show that changes in cell shape follow cytoskeletal reorientation with a significant temporal delay. Our data identify the importance of environmental stiffness for cell reorientation, here in direction of zero strain. These in vitro experiments on cultured cells argue for the necessity of rather stiff environmental conditions to induce cellular reorientation in mammalian tissues

Nanoscale Topography and Poroelastic Properties of Model Tissue Breast Gland Basement Membranes

Basement membranes (BMs) are thin layers of condensed extracellular matrix proteins serving as permeability filters, cellular anchoring sites, and barriers against cancer cell invasion. It is believed that their biomechanical properties play a crucial role in determining cellular behavior and response, especially in mechanically active tissues like breast glands. Despite this, so far, relatively little attention has been dedicated to their analysis because of the difficulty of isolating and handling such thin layers of material. Here, we isolated BMs derived from MCF10A spheroids—three-dimensional breast gland model systems mimicking in vitro the most relevant phenotypic characteristics of human breast lobules—and characterized them by atomic force microscopy, enhanced resolution confocal microscopy, and scanning electron microscopy. By performing atomic force microscopy height-clamp experiments, we obtained force-relaxation curves that offered the first biomechanical data on isolated breast gland BMs to our knowledge. Based on enhanced resolution confocal microscopy and scanning electron microscopy imaging data, we modeled the system as a polymer network immersed in liquid and described it as a poroelastic material. Finite-element simulations matching the experimental force-relaxation curves allowed for the first quantification, to our knowledge, of the bulk and shear moduli of the membrane as well as its water permeability. These results represent a first step toward a deeper understanding of the mechanism of tensional homeostasis regulating mammary gland activity as well as its disruption during processes of membrane breaching and metastatic invasion

Microstructure of sheared entangled solutions of semiflexible polymers

We study the influence of finite shear deformations on the microstructure and rheology of solutions of entangled semiflexible polymers theoretically and by numerical simulations and experiments with filamentous actin. Based on the tube model of semiflexible polymers, we predict that large finite shear deformations strongly affect the average tube width and curvature, thereby exciting considerable restoring stresses. In contrast, the associated shear alignment is moderate, with little impact on the average tube parameters, and thus expected to be long-lived and detectable after cessation of shear. Similarly, topologically preserved hairpin configurations are predicted to leave a long-lived fingerprint in the shape of the distributions of tube widths and curvatures. Our numerical and experimental data support the theory

Nanometric thermal fluctuations of weakly confined biomembranes measured with microsecond time-resolution

We probe the bending fluctuations of bio-membranes using highly deflated giant unilamellar vesicles (GUVs) bound to a substrate by a weak potential arising from generic interactions. The substrate is either homogeneous, with GUVs bound only by the weak potential, or is chemically functionalized with a micro-pattern of very strong specific binders. In both cases, the weakly adhered membrane is seen to be confined at a well-defined distance above the surface while it continues to fluctuate strongly. We quantify the fluctuations of the weakly confined membrane at the substrate proximal surface as well as of the free membrane at the distal surface of the same GUV. This strategy enables us to probe in detail the damping of fluctuations in the presence of the substrate, and to independently measure the membrane tension and the strength of the generic interaction potential. Measurements were done using two complementary techniques – dynamic optical displacement spectroscopy (DODS, resolution: 20 nm, 10 μs), and dual wavelength reflection interference contrast microscopy (DW-RICM, resolution: 4 nm, 50 ms). After accounting for the spatio-temporal resolution of the techniques, an excellent agreement between the two measurements was obtained. For both weakly confined systems we explore in detail the link between fluctuations on the one hand and membrane tension and the interaction potential on the other hand

Phonon confinement and interface lattice dynamics of ultrathin high-rare earth sesquioxide films: the case of Eu₂O₃ on YSZ(001)

The spatial confinement of atoms at surfaces and interfaces significantly alters the lattice dynamics of thin films, heterostructures and multilayers. Ultrathin films with high dielectric constants (high-k) are of paramount interest for applications as gate layers in current and future integrated circuits. Here we report a lattice dynamics study of high-k Eu$_{2}$O$_{3}$ films with thicknesses of 21.3, 2.2, 1.3, and 0.8 nm deposited on YSZ(001). The Eu-partial phonon density of states (PDOS), obtained from nuclear inelastic scattering, exhibits broadening of the phonon peaks accompanied by up to a four-fold enhancement of the number of low-energy states compared to the ab initio calculated PDOS of a perfect Eu$_{2}$O$_{3}$ crystal. Our analysis demonstrates that while the former effect reflects the reduced phonon lifetimes observed in thin films due to scattering from lattice defects, the latter phenomenon arises from an ultrathin EuO layer formed between the thin Eu$_{2}$O$_{3}$ film and the YSZ(001) substrate. Thus, our work uncovers another potential source of vibrational anomalies in thin films and multilayers, which has to be cautiously considered

Structure and domain dynamics of human lactoferrin in solution and the influence of Fe(III)-ion ligand binding

BackgroundHuman lactoferrin is an iron-binding protein of the innate immune system consisting of two connected lobes, each with a binding site located in a cleft. The clefts in each lobe undergo a hinge movement from open to close when Fe3+ is present in the solution and can be bound. The binding mechanism was assumed to relate on thermal domain fluctuations of the cleft domains prior to binding. We used Small Angle Neutron Scattering and Neutron Spin Echo Spectroscopy to determine the lactoferrin structure and domain dynamics in solution.ResultsWhen Fe3+ is present in solution interparticle interactions change from repulsive to attractive in conjunction with emerging metas aggregates, which are not observed without Fe3+. The protein form factor shows the expected change due to lobe closing if Fe3+ is present. The dominating motions of internal domain dynamics with relaxation times in the 30–50 ns range show strong bending and stretching modes with a steric suppressed torsion, but are almost independent of the cleft conformation. Thermally driven cleft closing motions of relevant amplitude are not observed if the cleft is open.ConclusionThe Fe3+ binding mechanism is not related to thermal equilibrium fluctuations closing the cleft. A likely explanation may be that upon entering the cleft the iron ion first binds weakly which destabilizes and softens the hinge region and enables large fluctuations that then close the cleft resulting in the final formation of the stable iron binding site and, at the same time, stable closed conformation