New infinite families of exact sums of squares formulas, Jacobi elliptic functions, and Ramanujan’s tau function

In this paper, we give two infinite families of explicit exact formulas that generalize Jacobi’s (1829) 4 and 8 squares identities to 4n2 or 4n(n + 1) squares, respectively, without using cusp forms. Our 24 squares identity leads to a different formula for Ramanujan’s tau function τ(n), when n is odd. These results arise in the setting of Jacobi elliptic functions, Jacobi continued fractions, Hankel or Turánian determinants, Fourier series, Lambert series, inclusion/exclusion, Laplace expansion formula for determinants, and Schur functions. We have also obtained many additional infinite families of identities in this same setting that are analogous to the η-function identities in appendix I of Macdonald’s work [Macdonald, I. G. (1972) Invent. Math. 15, 91–143]. A special case of our methods yields a proof of the two conjectured [Kac, V. G. and Wakimoto, M. (1994) in Progress in Mathematics, eds. Brylinski, J.-L., Brylinski, R., Guillemin, V. & Kac, V. (Birkhäuser Boston, Boston, MA), Vol. 123, pp. 415–456] identities involving representing a positive integer by sums of 4n2 or 4n(n + 1) triangular numbers, respectively. Our 16 and 24 squares identities were originally obtained via multiple basic hypergeometric series, Gustafson’s Cℓ nonterminating 6φ5 summation theorem, and Andrews’ basic hypergeometric series proof of Jacobi’s 4 and 8 squares identities. We have (elsewhere) applied symmetry and Schur function techniques to this original approach to prove the existence of similar infinite families of sums of squares identities for n2 or n(n + 1) squares, respectively. Our sums of more than 8 squares identities are not the same as the formulas of Mathews (1895), Glaisher (1907), Ramanujan (1916), Mordell (1917, 1919), Hardy (1918, 1920), Kac and Wakimoto, and many others.

Histone H1 overexpressed to high level in tobacco affects certain developmental programs but has limited effect on basal cellular functions.

Histone H1, a major structural component of chromatin fiber, is believed to act as a general repressor of transcription. To investigate in vivo the role of this protein in transcription regulation during development of a multicellular organism, we made transgenic tobacco plants that overexpress the gene for Arabidopsis histone H1. In all plants that overexpressed H1 the total H1-to-DNA ratio in chromatin increased 2.3-2.8 times compared with the physiological level. This was accompanied by 50-100% decrease of native tobacco H1. The phenotypic changes in H1-overexpressing plants ranged from mild to severe perturbations in morphological appearance and flowering. No correlation was observed between the extent of phenotypic change and the variation in the amount of overexpressed H1 or the presence or absence of the native tobacco H1. However, the severe phenotypic changes were correlated with early occurrence during plant growth of cells with abnormally heterochromatinized nuclei. Such cells occurred considerably later in plants with milder changes. Surprisingly, the ability of cells with highly heterochromatinized nuclei to fulfill basic physiological functions, including differentiation, was not markedly hampered. The results support the suggestion that chromatin structural changes dependent on H1 stoichiometry and on the profile of major H1 variants have limited regulatory effect on the activity of genes that control basal cellular functions. However, the H1-mediated chromatin changes can be of much greater importance for the regulation of genes involved in control of specific developmental programs.

Selectively enhanced contextual fear conditioning in mice lacking the transcriptional regulator CCAAT/enhancer binding protein δ

CCAAT/enhancer binding protein δ (C/EBPδ) is a transcriptional regulator implicated in the hepatic acute phase response and in adipogenic and myeloid cell differentiation. We found that C/EBPδ is widely expressed in the peripheral and central nervous systems, including neurons of the hippocampal formation, indicating a role in neural functions. To examine the role of C/EBPδ in vivo, we generated mice with a targeted deletion of the C/EBPδ gene. This mutation does not interfere with normal embryonic and postnatal development. Performance in a battery of behavioral tests indicates that basic neurological functions are normal. Furthermore, performance in a Morris water maze task suggests that C/EBPδ mutant mice have normal spatial learning. However, in the contextual and auditory-cue-conditioned fear task, C/EBPδ null mice displayed significantly more conditioned freezing to the test context than did wild-type controls, but equivalent conditioning to the auditory cue. These data demonstrate a selectively enhanced contextual fear response in mice carrying a targeted genomic mutation and implicate C/EBPδ in the regulation of a specific type of learning and memory.

Targeted disruption of the Rad51 gene leads to lethality in embryonic mice.

The mouse Rad51 gene is a mammalian homologue of the Escherichia coli recA and yeast RAD51 genes, both of which are involved in homologous recombination and DNA repair. To elucidate the physiological role of RAD51 protein, the gene was targeted in embryonic stem (ES) cells. Mice heterozygous for the Rad51 null mutation were intercrossed and their offspring were genotyped. There were no homozygous (Rad51-/-) pups among 148 neonates examined but a few Rad51-/- embryos were identified when examined during the early stages of embryonic development. Doubly knocked-out ES cells were not detected under conditions of selective growth. These results are interpreted to mean that RAD51 protein plays an essential role in the proliferation of cell. The homozygous Rad51 null mutation can be categorized in cell-autonomous defects. Pre-implantational lethal mutations that disrupt basic molecular functions will thus interfere with cell viability.

Shared functions in vivo of a glycosyl-phosphatidylinositol-linked aspartyl protease, Mkc7, and the proprotein processing protease Kex2 in yeast.

The MKC7 gene was isolated as a multicopy suppressor of the cold-sensitive growth phenotype of a yeast kex2 mutant, which lacks the protease that cleaves pro-alpha-factor and other secretory proproteins at pairs of basic residues in a late Golgi compartment in yeast. MKC7 encodes an aspartyl protease most closely related to product of the YAP3 gene, a previously isolated multicopy suppressor of the pro-alpha-factor processing defect of a kex2 null. Multicopy MKC7 suppressed the alpha-specific mating defect of a kex2 null as well as multicopy YAP3 did, but multicopy YAP3 was a relatively weak suppressor of kex2 cold sensitivity. Overexpression of MKC7 resulted in production of a membrane-associated proteolytic activity that cleaved an internally quenched fluorogenic peptide substrate on the carboxyl side of a Lys-Arg site. Treatment with phosphatidylinositol-specific phospholipase C shifted Mkc7 activity from the detergent to the aqueous phase in a Triton X-114 phase separation, indicating that membrane attachment of Mkc7 is mediated by a glycosyl-phosphatidylinositol anchor. Although disruption of MKC7 or YAP3 alone resulted in no observable phenotype, mkc7 yap3 double disruptants exhibited impaired growth at 37 degrees C. Disruption of MKC7 and YAP3 in a kex2 null mutant resulted in profound temperature sensitivity and more generalized cold sensitivity. The synergism of mkc7, yap3, and kex2 null mutations argues that Mkc7 and Yap3 are authentic processing enzymes whose functions overlap those of Kex2 in vivo.

Myelin basic protein kinase activity in tomato leaves is induced systemically by wounding and increases in response to systemin and oligosaccharide elicitors

In response to wounding, a 48-kDa myelin basic protein (MBP) kinase is activated within 2 min, both locally and systemically, in leaves of young tomato plants. The activating signal is able to pass through a steam girdle on the stem, indicating that it moves through the xylem and does not require intact phloem tissue. A 48-kDa MBP kinase is also activated by the 18-amino acid polypeptide systemin, a potent wound signal for the synthesis of systemic wound response proteins (swrps). The kinase activation by systemin is strongly inhibited by a systemin analog having a Thr-17 → Ala-17 substitution, which is a powerful antagonist of systemin activation of swrp genes. A 48-kDa MBP kinase activity also increases in response to polygalacturonic acid and chitosan but not in response to jasmonic acid or phytodienoic acid. In def1, a mutant tomato line having a defective octadecanoid pathway, the 48-kDa MBP kinase is activated by wounding and systemin as in the wild-type plants. This indicates that MBP kinase functions between the perception of primary signals and the DEF1 gene product. In response to wounding, the MBP kinase is phosphorylated on phosphotyrosine residues, indicating a relationship to the mitogen-activated protein kinase family of protein kinases.

Nuclear Activities of Basic Fibroblast Growth Factor: Potentiation of Low-Serum Growth Mediated by Natural or Chimeric Nuclear Localization Signals

Human basic fibroblast growth factor (FGF-2) occurs in four isoforms: a low molecular weight (LMW FGF-2, 18 kDa) and three high molecular weight (HMW FGF-2, 22, 22.5, and 24 kDa) forms. LMW FGF-2 is primarily cytoplasmic and functions in an autocrine manner, whereas HMW FGF-2s are nuclear and exert activities through an intracrine, perhaps nuclear, pathway. Selective overexpression of HMW FGF-2 forms in fibroblasts promotes growth in low serum, whereas overexpression of LMW FGF-2 does not. The HMW FGF-2 forms have two functional domains: an amino-terminal extension and a common 18-kDa amino acid sequence. To investigate the role of these regions in the intracrine signaling of HMW FGF-2, we produced stable transfectants of NIH 3T3 fibroblasts overexpressing either individual HMW FGF-2 forms or artificially nuclear-targeted LMW FGF-2. All of these forms of FGF-2 localize to the nucleus/nucleolus and induce growth in low serum. The nuclear forms of FGF-2 trigger a mitogenic stimulus under serum starvation conditions and do not specifically protect the cells from apoptosis. These data indicate the existence of a specific role for nuclear FGF-2 and suggest that LMW FGF-2 represents the biological messenger in both the autocrine/paracrine and intracrine FGF-2 pathways.

Generalized flows, intrinsic stochasticity, and turbulent transport

The study of passive scalar transport in a turbulent velocity field leads naturally to the notion of generalized flows, which are families of probability distributions on the space of solutions to the associated ordinary differential equations which no longer satisfy the uniqueness theorem for ordinary differential equations. Two most natural regularizations of this problem, namely the regularization via adding small molecular diffusion and the regularization via smoothing out the velocity field, are considered. White-in-time random velocity fields are used as an example to examine the variety of phenomena that take place when the velocity field is not spatially regular. Three different regimes, characterized by their degrees of compressibility, are isolated in the parameter space. In the regime of intermediate compressibility, the two different regularizations give rise to two different scaling behaviors for the structure functions of the passive scalar. Physically, this means that the scaling depends on Prandtl number. In the other two regimes, the two different regularizations give rise to the same generalized flows even though the sense of convergence can be very different. The “one force, one solution” principle is established for the scalar field in the weakly compressible regime, and for the difference of the scalar in the strongly compressible regime, which is the regime of inverse cascade. Existence and uniqueness of an invariant measure are also proved in these regimes when the transport equation is suitably forced. Finally incomplete self similarity in the sense of Barenblatt and Chorin is established.

LXRα functions as a cAMP-responsive transcriptional regulator of gene expression

LXRα is a member of a nuclear receptor superfamily that regulates transcription. LXRα forms a heterodimer with RXRα, another member of this family, to regulate the expression of cholesterol 7α-hydroxylase by means of binding to the DR4-type cis-element. Here, we describe a function for LXRα as a cAMP-responsive regulator of renin and c-myc gene transcriptions by the interaction with a specific cis-acting DNA element, CNRE (an overlapping cAMP response element and a negative response element). Our previous studies showed that renin gene expression is regulated by cAMP, at least partly, through the CNRE sequence in its 5′-flanking region. This sequence is also found in c-myc and several other genes. Based on our cloning results using the yeast one-hybrid system, we discovered that the mouse homologue of human LXRα binds to the CNRE and demonstrated that it binds as a monomer. To define the function of LXRα on gene expression, we transfected the renin-producing renal As4.1 cells with LXRα expression plasmid. Overexpression of LXRα in As4.1 cells confers cAMP inducibility to reporter constructs containing the renin CNRE. After stable transfection of LXRα, As4.1 cells show a cAMP-inducible up-regulation of renin mRNA expression. In parallel experiments, we demonstrated that LXRα can also bind to the homologous CNRE in the c-myc promoter. cAMP promotes transcription through c-myc/CNRE:LXRα interaction in LXRα transiently transfected cells and increases c-myc mRNA expression in stably transfected cells. Identification of LXRα as a cAMP-responsive nuclear modulator of renin and c-myc expression not only has cardiovascular significance but may have generalized implication in the regulation of gene transcription.

Purification and Characterization of a Galactose-Rich Basic Glycoprotein in Tobacco1

We found a galactose-rich basic glycoprotein (GBGP) in the cell walls of cultured tobacco (Nicotiana tabacum) cells. GBGP and extensin were isolated as the major components of basic, salt-extracted cell wall glycoproteins. GBGP and extensin were separated by gel filtration in 6 m guanidine hydrochloride as 49- and 90-kD peaks, respectively, and further purified with reverse-phase chromatography. The protein moiety of GBGP constitutes about one-half of the molecule (w/w) and contains lysine (16%), proline (12%), hydroxyproline (10%), tyrosine (4%), alanine (7%), leucine (6%), and cystine (1.4%). Galactose accounted for 72% of the sugar moiety, arabinose content was low (17%), and a significant amount of mannose (7%) was found. No immunological cross-reaction was detected between GBGP and extensin. The antibody against native GBGP with sugar chains reacted with other glycoproteins on the gel blots, whereas the antibodies against deglycosylated GBGP and native extensin were highly specific. Immunolocalization analysis in tobacco stems showed that GBGP is specific to parenchyma tissue and that extensin localizes in the epidermis. This tissue-specific and exclusive distribution suggests important functions of these basic glycoproteins.