Leonard M. Sander

A Model for Strain-Induced Roughening and Coherent Island Growth

We have investigated the morphological evolution of strained films during growth. Novel Monte Carlo studies, which incorporate linear elasticity, have been performed to simulate film growth with misfit. These studies demonstrate the onset of islanding for sufficiently large misfit. We present an analytic calculation which shows that from the onset of deposition the films are energetically unstable to large-scale islanding. We argue that the kinetics ultimately determines the surface morphology. Dislocations are not necessary for surface lattice relaxation. Support for this picture is inferred from experimental results on a number of strained growth systems.

Percolation on heterogeneous networks as a model for epidemics

We consider a spatial model related to bond percolation for the spread of a disease that includes variation in the susceptibility to infection. We work on a lattice with random bond strengths and show that with strong heterogeneity, i.e. a wide range of variation of susceptibility, patchiness in the spread of the epidemic is very likely, and the criterion for epidemic outbreak depends strongly on the heterogeneity. These results are qualitatively different from those of standard models in epidemiology, but correspond to real effects. We suggest that heterogeneity in the epidemic will affect the phylogenetic distance distribution of the disease-causing organisms. We also investigate small world lattices, and show that the effects mentioned above are even stronger.

An algorithm for extracting the network geometry of three-dimensional collagen gels

The geometric structure of a biopolymer network impacts its mechanical and biological properties. In this paper, we develop an algorithm for extracting the network architecture of three‐dimensional (3d) fluorescently labeled collagen gels, building on the initial work of Wu et al., (2003) . Using artificially generated images, the network extraction algorithm is then validated for its ability to reconstruct the correct bulk properties of the network, including fiber length, persistence length, cross‐link density, and shear modulus.

Growth patterns of microscopic brain tumors.

Highly malignant brain tumors such as glioblastoma multiforme form complex growth patterns in vitro in which invasive cells organize in tenuous branches. Here, we formulate a chemotaxis model for this sort of growth. A key element controlling the pattern is homotype attraction, i.e., the tendency for invasive cells to follow pathways previously explored. We investigate this in two ways: we show that there is an intrinsic instability in the model, which leads to branch formation. We also give a discrete description for the expansion of the invasive zone, and a continuum model for the nutrient supply. The results indicate that both strong heterotype chemotaxis and strong homotype chemoattraction are required for branch formation within the invasive zone. Our model thus can give a way to assess the importance of the various processes, and a way to explore and analyze transitions between different growth regimes.

The micromechanics of three-dimensional collagen-I gels

We study the micromechanics of collagen-I gel with the goal of bridging the gap between theory and experiment in the study of biopolymer networks. Three-dimensional images of fluorescently labeled collagen are obtained by confocal microscopy, and the network geometry is extracted using a 3D network skeletonization algorithm. Each fiber is modeled as an elastic beam that resists stretching and bending, and each crosslink is modeled as torsional spring. The stress–strain curves of networks at three different densities are compared with rheology measurements. The model shows good agreement with experiment, confirming that strain stiffening of collagen can be explained entirely by geometric realignment of the network, as opposed to entropic stiffening of individual fibers. The model also suggests that at small strains, crosslink deformation is the main contributer to network stiffness, whereas at large strains, fiber stretching dominates. As this modeling effort uses networks with realistic geometries, this analysis can ultimately serve as a tool for understanding how the mechanics of fibers and crosslinks at the microscopic level produce the macroscopic properties of the network.

Lithium inhibits invasion of glioma cells; possible involvement of glycogen synthase kinase-3

Therapies targeting glioma cells that diffusely infiltrate normal brain are highly sought after. Our aim was to identify novel approaches to this problem using glioma spheroid migration assays. Lithium, a currently approved drug for the treatment of bipolar illnesses, has not been previously examined in the context of glioma migration. We found that lithium treatment potently blocked glioma cell migration in spheroid, wound-healing, and brain slice assays. The effects observed were dose dependent and reversible, and worked using every glioma cell line tested. In addition, there was little effect on cell viability at lithium concentrations that inhibit migration, showing that this is a specific effect. Lithium treatment was associated with a marked change in cell morphology, with cells retracting the long extensions at their leading edge. Examination of known targets of lithium showed that inositol monophosphatase inhibition had no effect on glioma migration, whereas inhibition of glycogen synthase kinase-3 (GSK-3) did. This suggested that the effects of lithium on glioma cell migration could possibly be mediated through GSK-3. Specific pharmacologic GSK-3 inhibitors and siRNA knockdown of GSK-3alpha or GSK-3beta isoforms both reduced cell motility. These data outline previously unidentified pathways and inhibitors that may be useful for the development of novel anti-invasive therapeutics for the treatment of brain tumors.

Extinction Times for Birth-Death Processes: Exact Results, Continuum Asymptotics, and the Failure of the Fokker--Planck Approximation

We consider extinction times for a class of birth-death processes commonly found in applications, where there is a control parameter which defines a threshold. Below the threshold, the population quickly becomes extinct; above, it persists for a long time. We give an exact expression for the mean time to extinction in the discrete case and its asymptotic expansion for large values of the population scale. We have results below the threshold, at the threshold, and above the threshold, and we observe that the Fokker--Planck approximation is valid only quite near the threshold. We compare our asymptotic results to exact numerical evaluations for the susceptible-infected-susceptible epidemic model, which is in the class that we treat. This is an interesting example of the delicate relationship between discrete and continuum treatments of the same problem.

Geography in a scale-free network model.

We offer an example of a network model with a power-law degree distribution, P(k) approximately k(-alpha), for nodes, but which nevertheless has a well-defined geography and a nonzero threshold percolation probability for alpha>2, the range of real-world contact networks. This is different from p(c)=0 for alpha<3 results for the original well-mixed scale-free networks. In our lattice-based scale-free network, individuals link to nearby neighbors on a lattice. Even considerable additional small-world links do not change our conclusion of nonzero thresholds. When applied to disease propagation, these results suggest that random immunization may be more successful in controlling human epidemics than previously suggested if there is geographical clustering.

Dynamics and pattern formation in invasive tumor growth

We study the in vitro dynamics of the malignant brain tumor glioblastoma multiforme. The growing tumor consists of a dense proliferating zone and an outer less dense invasive region. Experiments with different types of cells show qualitatively different behavior: one cell line invades in a spherically symmetric manner, but another gives rise to branches. We formulate a model for this sort of growth using two coupled reaction-diffusion equations for the cell and nutrient concentrations. When the ratio of the nutrient and cell diffusion coefficients exceeds some critical value, the plane propagating front becomes unstable with respect to transversal perturbations. The instability threshold and the full phase-plane diagram in the parameter space are determined. The results are in a qualitative agreement with experimental findings for the two types of cells.

Competing roughening mechanisms in strained heteroepitaxy: a fast kinetic Monte Carlo study.

We study the morphological evolution of strained heteroepitaxial films using kinetic Monte Carlo simulations in two dimensions. A novel Greens function approach, analogous to boundary integral methods, is used to calculate elastic energies efficiently. We observe island formation at low lattice misfit and high temperature that is consistent with the Asaro-Tiller-Grinfeld instability theory. At high misfit and low temperature, islands or pits form according to the nucleation theory of Tersoff and LeGoues.