The problem of the spreading of a liquid film along a solid surface: A new mathematical formulation

A new mathematical model is proposed for the spreading of a liquid film on a solid surface. The model is based on the standard lubrication approximation for gently sloping films (with the no-slip condition for the fluid at the solid surface) in the major part of the film where it is not too thin. In the remaining and relatively small regions near the contact lines it is assumed that the so-called autonomy principle holds—i.e., given the material components, the external conditions, and the velocity of the contact lines along the surface, the behavior of the fluid is identical for all films. The resulting mathematical model is formulated as a free boundary problem for the classical fourth-order equation for the film thickness. A class of self-similar solutions to this free boundary problem is considered.

Tight frames of k-plane ridgelets and the problem of representing objects that are smooth away from d-dimensional singularities in Rn

For each pair (n, k) with 1 ≤ k < n, we construct a tight frame (ρλ : λ ∈ Λ) for L2 (Rn), which we call a frame of k-plane ridgelets. The intent is to efficiently represent functions that are smooth away from singularities along k-planes in Rn. We also develop tools to help decide whether k-plane ridgelets provide the desired efficient representation. We first construct a wavelet-like tight frame on the X-ray bundle χn,k—the fiber bundle having the Grassman manifold Gn,k of k-planes in Rn for base space, and for fibers the orthocomplements of those planes. This wavelet-like tight frame is the pushout to χn,k, via the smooth local coordinates of Gn,k, of an orthonormal basis of tensor Meyer wavelets on Euclidean space Rk(n−k) × Rn−k. We then use the X-ray isometry [Solmon, D. C. (1976) J. Math. Anal. Appl. 56, 61–83] to map this tight frame isometrically to a tight frame for L2(Rn)—the k-plane ridgelets. This construction makes analysis of a function f ∈ L2(Rn) by k-plane ridgelets identical to the analysis of the k-plane X-ray transform of f by an appropriate wavelet-like system for χn,k. As wavelets are typically effective at representing point singularities, it may be expected that these new systems will be effective at representing objects whose k-plane X-ray transform has a point singularity. Objects with discontinuities across hyperplanes are of this form, for k = n − 1.

The role of MHC class I glycoproteins in the regulation of induction of cell death in immunocytes by malignant melanoma cells

A deranged expression of MHC class I glycoproteins, characteristic of a variety of malignancies, contributes to the ability of cancer to avoid destruction by T cell-mediated immunity. An abrogation of the metastatic capacity of B16 melanoma cells has been achieved by transfecting an MHC class I-encoding vector into class I-deficient B16 melanoma clones [Gorelik, E., Kim, M., Duty, L. & Galili, U. (1993) Clin. Exp. Metastasis 11, 439–452]. We report here that the deranged expression of class I molecules by B16 melanoma cells is more than a mere acquisition of the capacity to escape immune recognition. Namely, cells of the B16 melanoma prompted splenic lymphocytes to commit death after coculture. However, a class I-expressing and nonmetastatic CL8-2 clone was found to be less potent as an inducer of apoptosis than class I-deficient and metastatic BL9 and BL12 clones. Both Thy1.2+ and Thy1.2− splenocytes underwent cell death when exposed to the class I-deficient BL9 clone. A proportion of CD4+ and CD8+ cells among splenocytes exposed to the BL9 clone was lower than that observed in a coculture with cells of the CL8-2 clone. Consistently, none of the melanoma clones studied produced a ligand to the FAS receptor (FAS-L). Thus, our results provide evidence that (i) the production of FAS-L may not be the sole mechanism by which malignant cells induce apoptosis in immunocytes, and (ii) absence of MHC class I glycoproteins plays an important role in preventing the elimination of potential effector immunocytes by tumor cells.

Evolution of chlorophyll and bacteriochlorophyll: the problem of invariant sites in sequence analysis.

Competing hypotheses seek to explain the evolution of oxygenic and anoxygenic processes of photosynthesis. Since chlorophyll is less reduced and precedes bacteriochlorophyll on the modern biosynthetic pathway, it has been proposed that chlorophyll preceded bacteriochlorophyll in its evolution. However, recent analyses of nucleotide sequences that encode chlorophyll and bacteriochlorophyll biosynthetic enzymes appear to provide support for an alternative hypothesis. This is that the evolution of bacteriochlorophyll occurred earlier than the evolution of chlorophyll. Here we demonstrate that the presence of invariant sites in sequence datasets leads to inconsistency in tree building (including maximum-likelihood methods). Homologous sequences with different biological functions often share invariant sites at the same nucleotide positions. However, different constraints can also result in additional invariant sites unique to the genes, which have specific and different biological functions. Consequently, the distribution of these sites can be uneven between the different types of homologous genes. The presence of invariant sites, shared by related biosynthetic genes as well as those unique to only some of these genes, has misled the recent evolutionary analysis of oxygenic and anoxygenic photosynthetic pigments. We evaluate an alternative scheme for the evolution of chlorophyll and bacteriochlorophyll.

Desensitization of N-methyl-D-aspartate receptors: a problem of interpretation.

The phenomenon of desensitization is universal, but its mechanism is still ill-understood and controversial. A recently published study [Lin, F. & Stevens, C. F. (1994) J. Neurosci, 14, 2153-2160] attempted to cast light on the mechanism of desensitization of N-methyl-D-aspartate (NMDA) receptors, in particular the vexed question of whether the channel must open before it can desensitize. During the desensitizing preexposure to agonist in those experiments, more desensitization was produced when channel openings were observed than when no openings were observed. The conclusion that "desensitization occurs more rapidly from the open state" unfortunately was based on a stochastic fallacy, and we present here a theoretical treatment and illustration showing that the observed behavior is predicted by a simple mechanism in which desensitization can occur only from a shut state.

Types of evolutionary stability and the problem of cooperation.

The evolutionary stability of cooperation is a problem of fundamental importance for the biological and social sciences. Different claims have been made about this issue: whereas Axelrod and Hamilton's [Axelrod, R. & Hamilton, W. (1981) Science 211, 1390-1398] widely recognized conclusion is that cooperative rules such as "tit for tat" are evolutionarily stable strategies in the iterated prisoner's dilemma (IPD), Boyd and Lorberbaum [Boyd, R. & Lorberbaum, J. (1987) Nature (London) 327, 58-59] have claimed that no pure strategy is evolutionarily stable in this game. Here we explain why these claims are not contradictory by showing in what sense strategies in the IPD can and cannot be stable and by creating a conceptual framework that yields the type of evolutionary stability attainable in the IPD and in repeated games in general. Having established the relevant concept of stability, we report theorems on some basic properties of strategies that are stable in this sense. We first show that the IPD has "too many" such strategies, so that being stable does not discriminate among behavioral rules. Stable strategies differ, however, on a property that is crucial for their evolutionary survival--the size of the invasion they can resist. This property can be interpreted as a strategy's evolutionary robustness. Conditionally cooperative strategies such as tit for tat are the most robust. Cooperative behavior supported by these strategies is the most robust evolutionary equilibrium: the easiest to attain, and the hardest to disrupt.

An antibody-interleukin 2 fusion protein overcomes tumor heterogeneity by induction of a cellular immune response.

Antibody-based therapies for cancer rely on the expression of defined antigens on neoplastic cells. However, most tumors display heterogeneity in the expression of such antigens. We demonstrate here that antibody-targeted interleukin 2 delivery overcomes this problem by induction of a host immune response. Immunohistochemical analysis demonstrated that the antibody-interleukin 2 fusion protein-induced eradication of established tumors is mediated by host immune cells, particularly CD8+ T cells. Because of this cellular immune response, antibody-directed interleukin 2 therapy is capable to address established metastases displaying substantial heterogeneity in expression of the targeted antigen. This effector mechanism further enables the induction of partial regressions of large subcutaneous tumors that exceeded more than 5% of the body weight. These observations indicate that antibody-directed cytokine delivery offers an effective new tool for cancer therapy.