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.

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.

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.

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.

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.

Calcium channel activation and self-biting in mice

The L type calcium channel agonist (±)Bay K 8644 has been reported to cause characteristic motor abnormalities in adult mice. The current study shows that administration of this drug can also cause the unusual phenomenon of self-injurious biting, particularly when given to young mice. Self-biting is provoked by injecting small quantities of (±)Bay K 8644 directly into the lateral ventricle of the brain, suggesting a central effect of the drug. Similar behaviors can be provoked by administration of another L type calcium channel agonist, FPL 64176. The self-biting provoked by (±)Bay K 8644 can be inhibited by pretreating the mice with dihydropyridine L type calcium channel antagonists such as nifedipine, nimodipine, or nitrendipine. However, self-biting is not inhibited by nondihydropyridine antagonists including diltiazem, flunarizine, or verapamil. The known actions of (±)Bay K 8644 as an L type calcium channel agonist, the reproduction of similar behavior with another L type calcium channel agonist, and the protection afforded by certain L type calcium channel antagonists implicate calcium channels in the mediation of the self-biting behavior. This phenomenon provides a model for studying the neurobiology of this unusual behavior.

Intron self-complementarity enforces exon inclusion in a yeast pre-mRNA

Skipping of internal exons during removal of introns from pre-mRNA must be avoided for proper expression of most eukaryotic genes. Despite significant understanding of the mechanics of intron removal, mechanisms that ensure inclusion of internal exons in multi-intron pre-mRNAs remain mysterious. Using a natural two-intron yeast gene, we have identified distinct RNA–RNA complementarities within each intron that prevent exon skipping and ensure inclusion of internal exons. We show that these complementarities are positioned to act as intron identity elements, bringing together only the appropriate 5′ splice sites and branchpoints. Destroying either intron self-complementarity allows exon skipping to occur, and restoring the complementarity using compensatory mutations rescues exon inclusion, indicating that the elements act through formation of RNA secondary structure. Introducing new pairing potential between regions near the 5′ splice site of intron 1 and the branchpoint of intron 2 dramatically enhances exon skipping. Similar elements identified in single intron yeast genes contribute to splicing efficiency. Our results illustrate how intron secondary structure serves to coordinate splice site pairing and enforce exon inclusion. We suggest that similar elements in vertebrate genes could assist in the splicing of very large introns and in the evolution of alternative splicing.

Ubiquitously Expressed Dynamin-II Has a Higher Intrinsic GTPase Activity and a Greater Propensity for Self-assembly Than Neuronal Dynamin-I

To begin to understand mechanistic differences in endocytosis in neurons and nonneuronal cells, we have compared the biochemical properties of the ubiquitously expressed dynamin-II isoform with those of neuron-specific dynamin-I. Like dynamin-I, dynamin-II is specifically localized to and highly concentrated in coated pits on the plasma membrane and can assemble in vitro into rings and helical arrays. As expected, the two closely related isoforms share a similar mechanism for GTP hydrolysis: both are stimulated in vitro by self-assembly and by interaction with microtubules or the SH3 domain-containing protein, grb2. Deletion of the C-terminal proline/arginine-rich domain from either isoform abrogates self-assembly and assembly-dependent increases in GTP hydrolysis. However, dynamin-II exhibits a ∼threefold higher rate of intrinsic GTP hydrolysis and higher affinity for GTP than dynamin-I. Strikingly, the stimulated GTPase activity of dynamin-II can be >40-fold higher than dynamin-I, due principally to its greater propensity for self-assembly and the increased resistance of assembled dynamin-II to GTP-triggered disassembly. These results are consistent with the hypothesis that self-assembly is a major regulator of dynamin GTPase activity and that the intrinsic rate of GTP hydrolysis reflects a dynamic, GTP-dependent equilibrium of assembly and disassembly.

Involvement of a cytosine side chain in proton transfer in the rate-determining step of ribozyme self-cleavage

Ribozymes of hepatitis delta virus have been proposed to use an active-site cytosine as an acid-base catalyst in the self-cleavage reaction. In this study, we have examined the role of cytosine in more detail with the antigenomic ribozyme. Evidence that proton transfer in the rate-determining step involved cytosine 76 (C76) was obtained from examining cleavage activity of the wild-type and imidazole buffer-rescued C76-deleted (C76Δ) ribozymes in D2O and H2O. In both reactions, a similar kinetic isotope effect and shift in the apparent pKa indicate that the buffer is functionally substituting for the side chain in proton transfer. Proton inventory of the wild-type reaction supported a mechanism of a single proton transfer at the transition state. This proton transfer step was further characterized by exogenous base rescue of a C76Δ mutant with cytosine and imidazole analogues. For the imidazole analogues that rescued activity, the apparent pKa of the rescue reaction, measured under kcat/KM conditions, correlated with the pKa of the base. From these data a Brønsted coefficient (β) of 0.51 was determined for the base-rescued reaction of C76Δ. This value is consistent with that expected for proton transfer in the transition state. Together, these data provide strong support for a mechanism where an RNA side chain participates directly in general acid or general base catalysis of the wild-type ribozyme to facilitate RNA cleavage.

Nanohedra: Using symmetry to design self assembling protein cages, layers, crystals, and filaments

A general strategy is described for designing proteins that self assemble into large symmetrical nanomaterials, including molecular cages, filaments, layers, and porous materials. In this strategy, one molecule of protein A, which naturally forms a self-assembling oligomer, An, is fused rigidly to one molecule of protein B, which forms another self-assembling oligomer, Bm. The result is a fusion protein, A-B, which self assembles with other identical copies of itself into a designed nanohedral particle or material, (A-B)p. The strategy is demonstrated through the design, production, and characterization of two fusion proteins: a 49-kDa protein designed to assemble into a cage approximately 15 nm across, and a 44-kDa protein designed to assemble into long filaments approximately 4 nm wide. The strategy opens a way to create a wide variety of potentially useful protein-based materials, some of which share similar features with natural biological assemblies.

Self-assembled monolayers from a designed combinatorial library of de novo β-sheet proteins

A variety of naturally occurring biomaterials owe their unusual structural and mechanical properties to layers of β-sheet proteins laminated between layers of inorganic mineral. To explore the possibility of fabricating novel two-dimensional protein layers, we studied the self-assembly properties of de novo proteins from a designed combinatorial library. Each protein in the library has a distinct 63 amino acid sequence, yet they all share an identical binary pattern of polar and nonpolar residues, which was designed to favor the formation of six-stranded amphiphilic β-sheets. Characterization of proteins isolated from the library demonstrates that (i) they self assemble into monolayers at an air/water interface; (ii) the monolayers are dominated by β-sheet secondary structure, as shown by both circular dichroism and infrared spectroscopies; and (iii) the measured areas (500- 600 Å2) of individual protein molecules in the monolayers match those expected for proteins folded into amphiphilic β-sheets. The finding that similar structures are formed by distinctly different protein sequences suggests that assembly into β-sheet monolayers can be encoded by binary patterning of polar and nonpolar amino acids. Moreover, because the designed binary pattern is compatible with a wide variety of different sequences, it may be possible to fabricate β-sheet monolayers by using combinations of side chains that are explicitly designed to favor particular applications of novel biomaterials.

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.

Similar antigenic surfaces, rather than sequence homology, dictate T-cell epitope molecular mimicry.

Molecular mimicry, normally defined by the level of primary-sequence similarities between self and foreign antigens, has been considered a key element in the pathogenesis of autoimmunity. Here we describe an example of molecular mimicry between two overlapping peptides within a single self-antigen, both of which are recognized by the same human self-reactive T-cell clone. Two intervening peptides did not stimulate the T-cell clone, even though they share nine amino acids with the stimulatory peptides. Molecular modeling of major histocompatibility complex class II-peptide complexes suggests that both of the recognized peptides generate similar antigenic surfaces, although these are composed of different sets of amino acids. The molecular modeling of a peptide shifted one residue from the stimulatory peptide, which was recognized in the context of the same HLA molecule by another T-cell clone, generated a completely different antigenic surface. Functional studies using truncated peptides confirmed that the anchor residues of the two "mimicking" epitopes in the HLA groove differ. Our results show, for two natural epitopes, how molecular mimicry can occur and suggest that studies of potential antigenic surfaces, rather than sequence similarity, are necessary for analyzing suspected peptide mimicry.

A secondary-structure model for the self-cleaving region of Neurospora VS RNA.

Neurospora VS RNA performs an RNA-mediated self-cleavage reaction whose products contain 2',3'-cyclic phosphate and 5'-hydroxyl termini. This reaction is similar to those of hammerhead, hairpin, and hepatitis delta virus ribozymes; however, VS RNA is not similar in sequence to these other self-cleaving motifs. Here we propose a model for the secondary structure of the self-cleaving region of VS RNA, supported by site-directed mutagenesis and chemical modification structure probing data. The secondary structure of VS RNA is distinct from those of the other naturally occurring RNA self-cleaving domains. In addition to a unique secondary structure, several Mg-dependent interactions occur during the folding of VS RNA into its active tertiary conformation.