Self-similar intermediate asymptotics for a degenerate parabolic filtration-absorption equation

The equation ∂tu = u∂xx2u − (c − 1)(∂xu)2 is known in literature as a qualitative mathematical model of some biological phenomena. Here this equation is derived as a model of the groundwater flow in a water-absorbing fissurized porous rock; therefore, we refer to this equation as a filtration-absorption equation. A family of self-similar solutions to this equation is constructed. Numerical investigation of the evolution of non-self-similar solutions to the Cauchy problems having compactly supported initial conditions is performed. Numerical experiments indicate that the self-similar solutions obtained represent intermediate asymptotics of a wider class of solutions when the influence of details of the initial conditions disappears but the solution is still far from the ultimate state: identical zero. An open problem caused by the nonuniqueness of the solution of the Cauchy problem is discussed.

Direct observation of the enhancement of noncooperative protein self-assembly by macromolecular crowding: Indefinite linear self-association of bacterial cell division protein FtsZ

Recent measurements of sedimentation equilibrium and sedimentation velocity have shown that the bacterial cell division protein FtsZ self-associates to form indefinitely long rod-like linear aggregates in the presence of GDP and Mg2+. In the present study, the newly developed technique of non-ideal tracer sedimentation equilibrium was used to measure the effect of high concentrations—up to 150 g/liter—of each of two inert “crowder” proteins, cyanmethemoglobin or BSA, on the thermodynamic activity and state of association of dilute FtsZ under conditions inhibiting (−Mg2+) and promoting (+Mg2+) FtsZ self-association. Analysis of equilibrium gradients of both FtsZ and crowder proteins indicates that, under the conditions of the present experiment, FtsZ interacts with each of the two crowder proteins essentially entirely via steric repulsion, which may be accounted for quantitatively by a simple model in which hemoglobin, albumin, and monomeric FtsZ are modeled as effective spherical hard particles, and each oligomeric species of FtsZ is modeled as an effective hard spherocylinder. The functional dependence of the sedimentation of FtsZ on the concentrations of FtsZ and either crowder indicates that, in the presence of high concentrations of crowder, both the weight-average degree of FtsZ self-association and the range of FtsZ oligomer sizes present in significant abundance are increased substantially.

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.

Spectral bifurcations in dispersive wave turbulence

Dispersive wave turbulence is studied numerically for a class of one-dimensional nonlinear wave equations. Both deterministic and random (white noise in time) forcings are studied. Four distinct stable spectra are observed—the direct and inverse cascades of weak turbulence (WT) theory, thermal equilibrium, and a fourth spectrum (MMT; Majda, McLaughlin, Tabak). Each spectrum can describe long-time behavior, and each can be only metastable (with quite diverse lifetimes)—depending on details of nonlinearity, forcing, and dissipation. Cases of a long-live MMT transient state dcaying to a state with WT spectra, and vice-versa, are displayed. In the case of freely decaying turbulence, without forcing, both cascades of weak turbulence are observed. These WT states constitute the clearest and most striking numerical observations of WT spectra to date—over four decades of energy, and three decades of spatial, scales. Numerical experiments that study details of the composition, coexistence, and transition between spectra are then discussed, including: (i) for deterministic forcing, sharp distinctions between focusing and defocusing nonlinearities, including the role of long wavelength instabilities, localized coherent structures, and chaotic behavior; (ii) the role of energy growth in time to monitor the selection of MMT or WT spectra; (iii) a second manifestation of the MMT spectrum as it describes a self-similar evolution of the wave, without temporal averaging; (iv) coherent structures and the evolution of the direct and inverse cascades; and (v) nonlocality (in k-space) in the transferral process.

A family of membrane-embedded metalloproteases involved in regulated proteolysis of membrane-associated transcription factors

We present evidence that the sporulation protein SpoIVFB of Bacillus subtilis is a member of a newly recognized family of metalloproteases that have catalytic centers adjacent to or within the membrane. SpoIVFB is required for converting the membrane-associated precursor protein, pro-σK, to the mature and active transcription factor σK by proteolytic removal of an N-terminal extension of 20 amino acids. SpoIVFB and other family members share the conserved sequence HEXXH, a hallmark of metalloproteases, as well as a second conserved motif NPDG, which is unique to the family. Both motifs, which are expected to form the catalytic center of the protease, overlap hydrophobic segments that are predicted to be separate transmembrane domains. The only other characterized member of this family of membrane-embedded metalloproteases is the mammalian Site-2 protease (S2P), which is required for the intramembrane cleavage of the eukaryotic transcription factor sterol regulatory element binding protein (SREBP). We report that amino acid substitutions in the two conserved motifs of SpoIVFB impair pro-σK processing and σK-directed gene expression during sporulation. These results and those from a similar analysis of S2P support the interpretation that both proteins are founding members of a family of metalloproteases involved in the activation of membrane-associated transcription factors. Thus, the pathways that govern the activation of the prokaryotic transcription factor pro-σK and the mammalian transcription factor SREBP not only are analogous but also use processing enzymes with strikingly homologous features.

Predicting species diversity in tropical forests

A fundamental question in ecology is how many species occur within a given area. Despite the complexity and diversity of different ecosystems, there exists a surprisingly simple, approximate answer: the number of species is proportional to the size of the area raised to some exponent. The exponent often turns out to be roughly 1/4. This power law can be derived from assumptions about the relative abundances of species or from notions of self-similarity. Here we analyze the largest existing data set of location-mapped species: over one million, individually identified trees from five tropical forests on three continents. Although the power law is a reasonable, zeroth-order approximation of our data, we find consistent deviations from it on all spatial scales. Furthermore, tropical forests are not self-similar at areas ≤50 hectares. We develop an extended model of the species-area relationship, which enables us to predict large-scale species diversity from small-scale data samples more accurately than any other available method.

Symmetries in geology and geophysics

Symmetries have played an important role in a variety of problems in geology and geophysics. A large fraction of studies in mineralogy are devoted to the symmetry properties of crystals. In this paper, however, the emphasis will be on scale-invariant (fractal) symmetries. The earth’s topography is an example of both statistically self-similar and self-affine fractals. Landforms are also associated with drainage networks, which are statistical fractal trees. A universal feature of drainage networks and other growth networks is side branching. Deterministic space-filling networks with side-branching symmetries are illustrated. It is shown that naturally occurring drainage networks have symmetries similar to diffusion-limited aggregation clusters.

Sequence-specific recognition of cytosine C5 and adenine N6 DNA methyltransferases requires different deformations of DNA.

DNA methyltransferases modify specific cytosines and adenines within 2-6 bp recognition sequences. We used scanning force microscopy and gel shift analysis to show that M.HhaI, a cytosine C-5 DNA methyltransferase, causes only a 2 degree bend upon binding its recognition site. Our results are consistent with prior crystallographic analysis showing that the enzyme stabilizes an extrahelical base while leaving the DNA duplex otherwise unperturbed. In contrast, similar analysis of M.EcoRI, an adenine N6 DNA methyltransferase, shows an average bend angle of approximately 52 degrees. This distortion of DNA conformation by M.EcoRI is shown to be important for sequence-specific binding.

Slip complexity in earthquake fault models.

We summarize studies of earthquake fault models that give rise to slip complexities like those in natural earthquakes. For models of smooth faults between elastically deformable continua, it is critical that the friction laws involve a characteristic distance for slip weakening or evolution of surface state. That results in a finite nucleation size, or coherent slip patch size, h*. Models of smooth faults, using numerical cell size properly small compared to h*, show periodic response or complex and apparently chaotic histories of large events but have not been found to show small event complexity like the self-similar (power law) Gutenberg-Richter frequency-size statistics. This conclusion is supported in the present paper by fully inertial elastodynamic modeling of earthquake sequences. In contrast, some models of locally heterogeneous faults with quasi-independent fault segments, represented approximately by simulations with cell size larger than h* so that the model becomes "inherently discrete," do show small event complexity of the Gutenberg-Richter type. Models based on classical friction laws without a weakening length scale or for which the numerical procedure imposes an abrupt strength drop at the onset of slip have h* = 0 and hence always fall into the inherently discrete class. We suggest that the small-event complexity that some such models show will not survive regularization of the constitutive description, by inclusion of an appropriate length scale leading to a finite h*, and a corresponding reduction of numerical grid size.

Knowledge-based analysis of microarray gene expression data by using support vector machines

We introduce a method of functionally classifying genes by using gene expression data from DNA microarray hybridization experiments. The method is based on the theory of support vector machines (SVMs). SVMs are considered a supervised computer learning method because they exploit prior knowledge of gene function to identify unknown genes of similar function from expression data. SVMs avoid several problems associated with unsupervised clustering methods, such as hierarchical clustering and self-organizing maps. SVMs have many mathematical features that make them attractive for gene expression analysis, including their flexibility in choosing a similarity function, sparseness of solution when dealing with large data sets, the ability to handle large feature spaces, and the ability to identify outliers. We test several SVMs that use different similarity metrics, as well as some other supervised learning methods, and find that the SVMs best identify sets of genes with a common function using expression data. Finally, we use SVMs to predict functional roles for uncharacterized yeast ORFs based on their expression data.

The paradox of alloreactivity and self MHC restriction: Quantitative analysis and statistics

Although 1–24% of T cells are alloreactive, i.e., respond to MHC molecules encoded by a foreign haplotype, it is generally believed that T cells cannot recognize foreign peptides binding foreign MHC molecules. We show using a quantitative model that, if T cell selection and activation are affinity-driven, then an alloreactivity of 1–24% is incompatible with the textbook notion that self MHC restriction is absolute. If an average of 1% of clones are alloreactive, then according to our model, at most 20-fold more clones should, on average, be activated by antigens presented on self MHC than by antigens presented on foreign MHC. This ratio is at best 5 if alloreactivity is 5%. These results describe average properties of the murine immune system, but not the outcome of individual experiments. Using supercomputer technology, we simulated 100,000 MHC restriction experiments. Although the average restriction ratio was 7.1, restriction was absolute in 10% of the simulated experiments, greater than 100, although not absolute, in 29%, and below 6 in 24%. This extreme variability agrees with experimental estimates. Our analysis suggests that alloreactivity and average self MHC restriction both cannot be high, but that a low average restriction level is compatible with high levels in a significant number of experiments.

Income distribution, socioeconomic status, and self rated health in the United States: multilevel analysis

Objective: To determine the effect of inequalities in income within a state on self rated health status while controlling for individual characteristics such as socioeconomic status.

Structural Analysis and Molecular Model of a Self-Incompatibility RNase from Wild Tomato1

Self-incompatibility RNases (S-RNases) are an allelic series of style glycoproteins associated with rejection of self-pollen in solanaceous plants. The nucleotide sequences of S-RNase alleles from several genera have been determined, but the structure of the gene products has only been described for those from Nicotiana alata. We report on the N-glycan structures and the disulfide bonding of the S3-RNase from wild tomato (Lycopersicon peruvianum) and use this and other information to construct a model of this molecule. The S3-RNase has a single N-glycosylation site (Asn-28) to which one of three N-glycans is attached. S3-RNase has seven Cys residues; six are involved in disulfide linkages (Cys-16-Cys-21, Cys-46-Cys-91, and Cys-166-Cys-177), and one has a free thiol group (Cys-150). The disulfide-bonding pattern is consistent with that observed in RNase Rh, a related RNase for which radiographic-crystallographic information is available. A molecular model of the S3-RNase shows that four of the most variable regions of the S-RNases are clustered on one surface of the molecule. This is discussed in the context of recent experiments that set out to determine the regions of the S-RNase important for recognition during the self-incompatibility response.

Self-renewal of primitive human hematopoietic cells (long-term-culture-initiating cells) in vitro and their expansion in defined medium.

A major goal of experimental and clinical hematology is the identification of mechanisms and conditions that support the expansion of transplantable hematopoietic stem cells. In normal marrow, such cells appear to be identical to (or represent a subset of) a population referred to as long-term-culture-initiating cells (LTC-ICs) so-named because of their ability to produce colony-forming cell (CFC) progeny for > or = 5 weeks when cocultured with stromal fibroblasts. Some expansion of LTC-ICs in vitro has recently been described, but identification of the factors required and whether LTC-IC self-renewal divisions are involved have remained unresolved issues. To address these issues, we examined the maintenance and/or generation of LTC-ICs from single CD34+ CD38- cells cultured for variable periods under different culture conditions. Analysis of the progeny obtained from cultures containing a feeder layer of murine fibroblasts engineered to produce steel factor, interleukin (IL)-3, and granulocyte colony-stimulating factor showed that approximately 20% of the input LTC-ICs (representing approximately 2% of the original CD34+ CD38- cells) executed self-renewal divisions within a 6-week period. Incubation of the same CD34+ CD38- starting populations as single cells in a defined (serum free) liquid medium supplemented with Flt-3 ligand, steel factor, IL-3, IL-6, granulocyte colony-stimulating factor, and nerve growth factor resulted in the proliferation of initial cells to produce clones of from 4 to 1000 cells within 10 days, approximately 40% of which included > or = 1 LTC-IC. In contrast, in similar cultures containing methylcellulose, input LTC-ICs appeared to persist but not divide. Overall the LTC-IC expansion in the liquid cultures was 30-fold in the first 10 days and 50-fold by the end of another 1-3 weeks. Documentation of human LTC-IC self-renewal in vitro and identification of defined conditions that permit their extensive and rapid amplification should facilitate analysis of the molecular mechanisms underlying these processes and their exploitation for a variety of therapeutic applications.