Nanorheology of viscoelastic shells: applications to viral capsids

Geometrically induced localization of flexural waves on thin warped physical membranes

We consider the propagation of flexural waves across a nearly flat, thin membrane, whose stress-free state is curved. The stress-free configuration is specified by a quenched height field, whose Fourier components are drawn from a Gaussian distribution with power-law variance. Gaussian curvature couples the in-plane stretching to out-of-plane bending. Integrating out the faster stretching modes yields a wave equation for undulations in the presence of an effective random potential, determined purely by geometry. We show that at long times and lengths, the undulation intensity obeys a diffusion equation. The diffusion coefficient is found to be frequency dependent and sensitive to the quenched height field distribution. Finally, we consider the effect of coherent backscattering corrections, yielding a weak localization correction that decreases the diffusion coefficient proportional to the logarithm of the system size, and induces a localization transition at large amplitude of the quenched height field. The localization transition is confirmed via a self-consistent extension to the strong disorder regime.

Studying nucleic envelope and plasma membrane mechanics of eukaryotic cells using confocal reflectance interferometric microscopy

Mechanical stress on eukaryotic nucleus has been implicated in a diverse range of diseases including muscular dystrophy and cancer metastasis. Today, there are very few non-perturbative methods to quantify nuclear mechanical properties. Interferometric microscopy, also known as quantitative phase microscopy (QPM), is a powerful tool for studying red blood cell biomechanics. The existing QPM tools, however, have not been utilized to study biomechanics of complex eukaryotic cells either due to lack of depth sectioning, limited phase measurement sensitivity, or both. Here, we present depth-resolved confocal reflectance interferometric microscopy as the next generation QPM to study nuclear and plasma membrane biomechanics. The proposed system features multiple confocal scanning foci, affording 1.5 micron depth-resolution and millisecond frame rate. Furthermore, a near common-path interferometer enables quantifying nanometer-scale membrane fluctuations with better than 200 picometers sensitivity. Our results present accurate quantification of nucleic envelope and plasma membrane fluctuations in embryonic stem cells.

Biomechanical studies of eukaryotic cells have been limited due to low sensitivity and axial resolution in interferometric imaging. Here, the authors present depth-resolved confocal reflectance interferometric microscopy with high sensitivity and temporal resolution, which enables quantification of nucleic envelope and plasma membrane fluctuations.

Geometric localization of thermal fluctuations in red blood cells

Cell shapes are related to their biological function. More generally, membrane morphology plays a role in the segregation and activity of transmembrane proteins. Here we show geometric implications regarding how cellular mechanics plays a role in localizing thermal fluctuations on the membrane. We show theoretically that certain geometric features of curved shells control the spatial distribution of membrane undulations. We experimentally verify this theory using discocyte red blood cells and find that geometry alone is sufficient to account for the observed spatial distribution of fluctuations. Our results, based on statistical physics and membrane elasticity, have general implications for the use of membrane shape to control thermal undulations in a variety of nanostructured materials ranging from cell membranes to graphene sheets.

The thermal fluctuations of membranes and nanoscale shells affect their mechanical characteristics. Whereas these fluctuations are well understood for flat membranes, curved shells show anomalous behavior due to the geometric coupling between in-plane elasticity and out-of-plane bending. Using conventional shallow shell theory in combination with equilibrium statistical physics we theoretically demonstrate that thermalized shells containing regions of negative Gaussian curvature naturally develop anomalously large fluctuations. Moreover, the existence of special curves, “singular lines,” leads to a breakdown of linear membrane theory. As a result, these geometric curves effectively partition the cell into regions whose fluctuations are only weakly coupled. We validate these predictions using high-resolution microscopy of human red blood cells (RBCs) as a case study. Our observations show geometry-dependent localization of thermal fluctuations consistent with our theoretical modeling, demonstrating the efficacy in combining shell theory with equilibrium statistical physics for describing the thermalized morphology of cellular membranes.

Cellular normoxic biophysical markers of hydroxyurea treatment in sickle cell disease

Hydroxyurea (HU) has been used clinically to reduce the frequency of painful crisis and the need for blood transfusion in sickle cell disease (SCD) patients. However, the mechanisms underlying such beneficial effects of HU treatment are still not fully understood. Studies have indicated a weak correlation between clinical outcome and molecular markers, and the scientific quest to develop companion biophysical markers have mostly targeted studies of blood properties under hypoxia. Using a common-path interferometric technique, we measure biomechanical and morphological properties of individual red blood cells in SCD patients as a function of cell density, and investigate the correlation of these biophysical properties with drug intake as well as other clinically measured parameters. Our results show that patient-specific HU effects on the cellular biophysical properties are detectable at normoxia, and that these properties are strongly correlated with the clinically measured mean cellular volume rather than fetal hemoglobin level.

Reflection and refraction of flexural waves at geometric boundaries

We present a theory of flexural wave propagation on elastic shells having nontrivial geometry and develop an analogy to geometric optics. The transport of momentum within the shell itself is anisotropic due to the curvature, and as such complex classical effects such as birefringence are generically found. We determine the equations of reflection and refraction of such waves at boundaries between different local geometries, showing that waves are totally internally reflected, especially at boundaries between regions of positive and negative Gaussian curvature. We verify these effects by using finite element simulations and discuss the ramifications of these effects for the statistical mechanics of thin curved materials.

Optical measurement of biomechanical properties of individual erythrocytes from a sickle cell patient

Sickle cell disease (SCD) is characterized by the abnormal deformation of red blood cells (RBCs) in the deoxygenated condition, as their elongated shape leads to compromised circulation. The pathophysiology of SCD is influenced by both the biomechanical properties of RBCs and their hemodynamic properties in the microvasculature. A major challenge in the study of SCD involves accurate characterization of the biomechanical properties of individual RBCs with minimum sample perturbation. Here we report the biomechanical properties of individual RBCs from a SCD patient using a non-invasive laser interferometric technique. We optically measure the dynamic membrane fluctuations of RBCs. The measurements are analyzed with a previously validated membrane model to retrieve key mechanical properties of the cells: bending modulus; shear modulus; area expansion modulus; and cytoplasmic viscosity. We find that high cytoplasmic viscosity at ambient oxygen concentration is principally responsible for the significantly decreased dynamic membrane fluctuations in RBCs with SCD, and that the mechanical properties of the membrane cortex of irreversibly sickled cells (ISCs) are different from those of the other types of RBCs in SCD.

Measurement of the nonlinear elasticity of red blood cell membranes

The membranes of human red blood cells (RBCs) are a composite of a fluid lipid bilayer and a triangular network of semiflexible filaments (spectrin). We perform cellular microrheology using the dynamic membrane fluctuations of the RBCs to extract the elastic moduli of this composite membrane. By applying known osmotic stresses, we measure the changes in the elastic constants under imposed strain and thereby determine the nonlinear elastic properties of the membrane. We find that the elastic nonlinearities of the shear modulus in tensed RBC membranes can be well understood in terms of a simple wormlike chain model. Our results show that the elasticity of the spectrin network can mostly account for the area compression modulus at physiological osmolality, suggesting that the lipid bilayer has significant excess area. As the cell swells, the elastic contribution from the now tensed lipid membrane becomes dominant.

Measurement of red blood cell mechanics during morphological changes

The human red blood cell (RBC) membrane, a fluid lipid bilayer tethered to an elastic 2D spectrin network, provides the principal control of the cell's morphology and mechanics. These properties, in turn, influence the ability of RBCs to transport oxygen in circulation. Current mechanical measurements of RBCs rely on external loads. Here we apply a noncontact optical interferometric technique to quantify the thermal fluctuations of RBC membranes with 3 nm accuracy over a broad range of spatial and temporal frequencies. Combining this technique with a new mathematical model describing RBC membrane undulations, we measure the mechanical changes of RBCs as they undergo a transition from the normal discoid shape to the abnormal echinocyte and spherical shapes. These measurements indicate that, coincident with this morphological transition, there is a significant increase in the membrane's shear, area, and bending moduli. This mechanical transition can alter cell circulation and impede oxygen delivery.