A mathematical theory of stochastic microlensing. II. Random images, shear, and the Kac-Rice formula

Continuing our development of a mathematical theory of stochastic microlensing, we study the random shear and expected number of random lensed images of different types. In particular, we characterize the first three leading terms in the asymptotic expression of the joint probability density function (pdf) of the random shear tensor due to point masses in the limit of an infinite number of stars. Up to this order, the pdf depends on the magnitude of the shear tensor, the optical depth, and the mean number of stars through a combination of radial position and the star's mass. As a consequence, the pdf's of the shear components are seen to converge, in the limit of an infinite number of stars, to shifted Cauchy distributions, which shows that the shear components have heavy tails in that limit. The asymptotic pdf of the shear magnitude in the limit of an infinite number of stars is also presented. All the results on the random microlensing shear are given for a general point in the lens plane. Extending to the general random distributions (not necessarily uniform) of the lenses, we employ the Kac-Rice formula and Morse theory to deduce general formulas for the expected total number of images and the expected number of saddle images. We further generalize these results by considering random sources defined on a countable compact covering of the light source plane. This is done to introduce the notion of global expected number of positive parity images due to a general lensing map. Applying the result to microlensing, we calculate the asymptotic global expected number of minimum images in the limit of an infinite number of stars, where the stars are uniformly distributed. This global expectation is bounded, while the global expected number of images and the global expected number of saddle images diverge as the order of the number of stars. © 2009 American Institute of Physics.

Display of cell surface sites for fibronectin assembly is modulated by cell adherence to (1)F3 and C-terminal modules of fibronectin.

BACKGROUND: Fibronectin-null cells assemble soluble fibronectin shortly after adherence to a substrate coated with intact fibronectin but not when adherent to the cell-binding domain of fibronectin (modules (7)F3-(10)F3). Interactions of adherent cells with regions of adsorbed fibronectin other than modules (7)F3-(10)F3, therefore, are required for early display of the cell surface sites that initiate and direct fibronectin assembly. METHODOLOGY/PRINCIPAL FINDINGS: To identify these regions, coatings of proteolytically derived or recombinant pieces of fibronectin containing modules in addition to (7)F3-(10)F3 were tested for effects on fibronectin assembly by adherent fibronectin-null fibroblasts. Pieces as large as one comprising modules (2)F3-(14)F3, which include the heparin-binding and cell adhesion domains, were not effective in supporting fibronectin assembly. Addition of module (1)F3 or the C-terminal modules to modules (2)F3-(14)F3 resulted in some activity, and addition of both (1)F3 and the C-terminal modules resulted in a construct, (1)F3-C, that best mimicked the activity of a coating of intact fibronectin. Constructs (1)F3-C V0, (1)F3-C V64, and (1)F3-C Delta(V(15)F3(10)F1) were all able to support fibronectin assembly, suggesting that (1)F3 through (11)F1 and/or (12)F1 were important for activity. Coatings in which the active parts of (1)F3-C were present in different proteins were much less active than intact (1)F3-C. CONCLUSIONS: These results suggest that (1)F3 acts together with C-terminal modules to induce display of fibronectin assembly sites on adherent cells.

The Haemophilus influenzae HMW1C protein is a glycosyltransferase that transfers hexose residues to asparagine sites in the HMW1 adhesin.

The Haemophilus influenzae HMW1 adhesin is a high-molecular weight protein that is secreted by the bacterial two-partner secretion pathway and mediates adherence to respiratory epithelium, an essential early step in the pathogenesis of H. influenzae disease. In recent work, we discovered that HMW1 is a glycoprotein and undergoes N-linked glycosylation at multiple asparagine residues with simple hexose units rather than N-acetylated hexose units, revealing an unusual N-glycosidic linkage and suggesting a new glycosyltransferase activity. Glycosylation protects HMW1 against premature degradation during the process of secretion and facilitates HMW1 tethering to the bacterial surface, a prerequisite for HMW1-mediated adherence. In the current study, we establish that the enzyme responsible for glycosylation of HMW1 is a protein called HMW1C, which is encoded by the hmw1 gene cluster and shares homology with a group of bacterial proteins that are generally associated with two-partner secretion systems. In addition, we demonstrate that HMW1C is capable of transferring glucose and galactose to HMW1 and is also able to generate hexose-hexose bonds. Our results define a new family of bacterial glycosyltransferases.

Role of DNA binding sites and slow unbinding kinetics in titration-based oscillators.

Genetic oscillators, such as circadian clocks, are constantly perturbed by molecular noise arising from the small number of molecules involved in gene regulation. One of the strongest sources of stochasticity is the binary noise that arises from the binding of a regulatory protein to a promoter in the chromosomal DNA. In this study, we focus on two minimal oscillators based on activator titration and repressor titration to understand the key parameters that are important for oscillations and for overcoming binary noise. We show that the rate of unbinding from the DNA, despite traditionally being considered a fast parameter, needs to be slow to broaden the space of oscillatory solutions. The addition of multiple, independent DNA binding sites further expands the oscillatory parameter space for the repressor-titration oscillator and lengthens the period of both oscillators. This effect is a combination of increased effective delay of the unbinding kinetics due to multiple binding sites and increased promoter ultrasensitivity that is specific for repression. We then use stochastic simulation to show that multiple binding sites increase the coherence of oscillations by mitigating the binary noise. Slow values of DNA unbinding rate are also effective in alleviating molecular noise due to the increased distance from the bifurcation point. Our work demonstrates how the number of DNA binding sites and slow unbinding kinetics, which are often omitted in biophysical models of gene circuits, can have a significant impact on the temporal and stochastic dynamics of genetic oscillators.