Phylogeography of the green turtle, Chelonia mydas, in the Southwest Indian Ocean

Patterns of mitochondrial DNA (mtDNA) variation were used to analyse the population genetic structure of southwestern Indian Ocean green turtle (Chelonia mydas) populations. Analysis of sequence variation over 396 bp of the mtDNA control region revealed seven haplotypes among 288 individuals from 10 nesting sites in the Southwest Indian Ocean. This is the first time that Atlantic Ocean haplotypes have been recorded among any Indo-Pacific nesting populations. Previous studies indicated that the Cape of Good Hope was a major biogeographical barrier between the Atlantic and Indian Oceans because evidence for gene flow in the last 1.5 million years has yet to emerge. This study, by sampling localities adjacent to this barrier, demonstrates that recent gene flow has occurred from the Atlantic Ocean into the Indian Ocean via the Cape of Good Hope. We also found compelling genetic evidence that green turtles nesting at the rookeries of the South Mozambique Channel (SMC) and those nesting in the North Mozambique Channel (NMC) belong to separate genetic stocks. Furthermore, the SMC could be subdivided in two different genetic stocks, one in Europa and the other one in Juan de Nova. We suggest that this particular genetic pattern along the Mozambique Channel is attributable to a recent colonization from the Atlantic Ocean and is maintained by oceanic conditions in the northern and southern Mozambique Channel that influence early stages in the green turtle life cycle.

Phase relations in the system Ta-Rh-O and thermodynamic properties of TaRhO4

Phase relations in the system Ta-Rh-O were determined by analysis of quenched samples corresponding to thirteen compositions inside the ternary triangle after equilibration at 1273 K. All the Ta-Rh alloys were found to be in equilibrium with Ta2O5. Only one ternary oxide TaRhO4 was detected. Based on phase relations in the ternary system, a solid-state electrochemical cell, incorporating calcia-stabilized zirconia as the electrolyte, was designed to measure the standard Gibbs energy of formation (Delta G degrees, J mol(-1)) of TaRhO4 in the temperature range from 900 to 1300 K. For the reaction, 1/2 beta-Ta2O5 + 1/2 Rh2O3(ortho) -> TaRhO4 Delta G degrees = -42993 + 5.676T (+/- 85) The calculated decomposition temperatures of TaRhO4 are 1644 +/- 5K in pure O-2 and 1543 +/- 5K in air at a total pressure p(o) = 0.1 MPa. Thermodynamic properties of TaRhO4 at 298.15K have been evaluated from the results. The limited experimental thermodynamic data for Rh-rich alloys available in the literature are in fair accord with Miedema's model. The Gibbs energies of formation of the different phases in the binary system Ta-Rh were estimated based on these inputs, consistent with the binary phase diagram. Based on the thermodynamic information on the stability of various phases, an oxygen potential diagram for the system Ta-Rh-O at 1273K was constructed. Also presented are temperature-composition diagrams for the ternary system at constant oxygen partial pressures (po(2)/p(o) = 0.212 and 10(-6)) calculated form the basic data.

Genetic Structure of Cochliobolus sativus Populations Sampled from Roots and Leaves of Barley and Wheat in North Dakota

Common root rot (CRR) and spot blotch, caused by Cochliobolus sativus (Ito and Kurib.) Drechsl. ex Dast., are important diseases of barley (Hordeum vulgare L.) and wheat (Triticum aestivum L.) worldwide. However, the population biology of C. sativus is still poorly understood. In this study, the genetic structure of three C. sativus populations, consisting of isolates sampled respectively from barley leaves (BL), barley roots (BR) and wheat roots (WR) in North Dakota, was analysed with amplified fragment length polymorphism (AFLP) markers. A total of 127 AFLP loci were generated among 208 C. sativus isolates analysed with three primer combinations. Gene diversity (H = 0.277-0.335) were high in all three populations. Genetic variation among C. sativus individuals within population accounted for 74%, whereas 26% of the genetic variation was explained among populations. Genetic differentiation was high (empty set PT = 0.261, corrected G ''(st)= 0.39), whereas gene flow (Nm) ranged from 1.27 to 1.56 among the three populations analysed. The multilocus linkage disequilibrium (LD) ((r) over bard = 0.0760.117) was moderate in C. sativus populations. Cluster analyses indicate that C. sativus populations differentiated according to the hosts (barley and wheat) and tissues (root and leaf) although generalists also exist in North Dakota. Crop breeding may benefit from combining genes for resistance against both specialists and generalists of C. sativus.

Genetic Structure of Cochliobolus sativus Populations Sampled from Roots and Leaves of Barley and Wheat in North Dakota

Common root rot (CRR) and spot blotch, caused by Cochliobolus sativus (Ito and Kurib.) Drechsl. ex Dast., are important diseases of barley (Hordeum vulgare L.) and wheat (Triticum aestivum L.) worldwide. However, the population biology of C. sativus is still poorly understood. In this study, the genetic structure of three C. sativus populations, consisting of isolates sampled respectively from barley leaves (BL), barley roots (BR) and wheat roots (WR) in North Dakota, was analysed with amplified fragment length polymorphism (AFLP) markers. A total of 127 AFLP loci were generated among 208 C. sativus isolates analysed with three primer combinations. Gene diversity (H = 0.277-0.335) were high in all three populations. Genetic variation among C. sativus individuals within population accounted for 74%, whereas 26% of the genetic variation was explained among populations. Genetic differentiation was high (empty set PT = 0.261, corrected G ''(st)= 0.39), whereas gene flow (Nm) ranged from 1.27 to 1.56 among the three populations analysed. The multilocus linkage disequilibrium (LD) ((r) over bard = 0.0760.117) was moderate in C. sativus populations. Cluster analyses indicate that C. sativus populations differentiated according to the hosts (barley and wheat) and tissues (root and leaf) although generalists also exist in North Dakota. Crop breeding may benefit from combining genes for resistance against both specialists and generalists of C. sativus.

Conformational Choices for the Stereochemically Constrained gamma-Amino Acid Residue Gabapentin: Theoretical Studies and Correlation With Experimental Results

Gabapentin (1-aminomethylcyclohexaneacetic acid, Gpn) is an achiral, conformationally constrained gamma amino acid residue. A survey of available crystal structures of Gpn peptides reveals that the torsion angles about the C-gamma-C-beta (theta(1)) and C-beta-C-alpha(theta(2)) bonds are overwhelmingly limited to gauche, gauche (g(+)g(+)/g(-)g(-)) conformations. The Gpn residue forms C-7 and C-9 hydrogen bonds in which the donor and acceptor atoms come from the flanking peptide units. In combination with alpha amino acid residues alpha gamma and gamma alpha segments can adopt C-12 hydrogen bonded structures. The conformational choices available to the Gpn residue have been probed using energy calculations, adopting a grid search strategy. Ramachandran phi-psi maps have been constructed for fixed values of theta(1) and theta(2), corresponding to the gauche and trans conformations. The sterically allowed and energetically favorable regions of conformational space have been defined and experimental observations compared. C-7 and C-9 hydrogen bonded conformational families have been identified using a grid search approach in which theta(1) and theta(2) values are varied over a range of +/- 10 degrees about ideal values at 1 degrees intervals. The theoretical analysis together with experimental observations for 59 Gpn residues from 35 crystal structures permits definition of the limited range of conformational possibilities at this gamma amino acid residue. .

Metal-Nucleotide Interactions: Crystal Structures of Alkali (Li+, Na+, K+) and Alkaline Earth (Ca2+, Mg2+) Metal Complexes of Adenosine 2'-Monophosphate

Crystal structures of lithium, sodium, potassium, calcium and magnesium salts of adenosine 2'-monophosphate (2'-AMP) have been obtained at atomic resolution by X-ray crystallographic methods. 2'-AMP.Li belongs to the monoclinic space group P21 with a = 7.472(3)Å, b = 26.853(6) Å, c = 9.184(1)Å, b = 113.36(1)Å and Z= 4. 2'-AMP.Na and 2'-AMP.K crystallize in the trigonal space groups P31 and P3121 with a = 8.762(1)Å, c = 34.630(5)Å, Z= 6 and a = 8.931(4), Åc = 34.852(9)Å and Z= 6 respectively while 2'-AMP.Ca and 2'-AMP.Mg belong to space groups P6522 and P21 with cell parameters a = 9.487(2), c = 74.622(13), Z = 12 and a = 4.973(1), b = 10.023(2), c = 16.506(2), beta = 91.1(0) and Z = 2 respectively. All the structures were solved by direct methods and refined by full matrix least-squares to final R factors of 0.033, 0.028, 0.075, 0.069 and 0.030 for 2'-AMP.Li, 2'-AMP.Na, 2'- AMP.K, 2'-AMP.Ca and 2'-AMP.Mg, respectively. The neutral adenine bases in all the structures are in syn conformation stabilized by the O5'-N3 intramolecular hydrogen bond as in free acid and ammonium complex reported earlier. In striking contrast, the adenine base is in the anti geometry (cCN = -156.4(2)°) in 2'-AMP.Mg. Ribose moieties adopt C2'-endo puckering in 2'-AMP.Li and 2'-AMP.Ca, C2'-endo-C3'-exo twist puckering in 2'-AMP.Na and 2'-AMP.K and a C3'-endo-C2'-exo twist puckering in 2'-AMP.Mg structure. The conformation about the exocyclic C4'-C5' bond is the commonly observed gauche-gauche (g+) in all the structures except the gauche- trans (g-) conformation observed in 2'-AMP.Mg structure. Lithium ions coordinate with water, ribose and phosphate oxygens at distances 1.88 to 1.99Å. Na+ ions and K+ ions interact with phosphate and ribose oxygens directly and with N7 indirectly through a water oxygen. A distinct feature of 2'-AMP.Na and 2'-AMP.K structures is the involvement of ribose O4' in metal coordination. The calcium ion situated on a two-fold axis coordinates directly with three oxygens OW1, OW2 and O2 and their symmetry mates at distances 2.18 to 2.42Å forming an octahedron. A classic example of an exception to the existence of the O5'-N3 intramolecular hydorgen bond is the 2'-AMP.Mg strucure. Magnesium ion forms an octahedral coordination with three water and three phosphate oxygens at distances ranging from 2.02 to 2.11Å. A noteworthy feature of its coordination is the indirect link with N3 through OW3 oxygen resulting in macrochelation between the base and the phosphate group. Greater affnity of metal clays towards 5' compared to 2' and 3' nucleotides (J. Lawless, E. Edelson, and L. Manring, Am. Chem. Soc. Northwest Region Meeting, Seattle. 1978) due to macrochelation infered from solution studies (S. S. Massoud, H. Sigel, Eur. J. Biochem. 179, 451-458 (1989)) and interligand hydrogen bonding induced by metals postulated from metal-nucleotide structures in solid state (V. Swaminathan and M. Sundaralingam, CRC. Crit. Rev. Biochem. 6, 245-336 (1979)) are borne out by our structures also. The stacking patterns of adenine bases of both 2'-AMP.Na and 2'-AMP.K structures resemble the 2'-AMP.NH4 structure reported in the previous article. 2'-AMP.Li, 2'-AMP.Ca and 2'-AMP.Mg structures display base-ribose O4' stacking. An overview of interaction of monovalent and divalent cations with 2' and 5'-nucleotides has been presented.

Diverse populations of T cells with NK cell receptors accumulate in the human intestine in health and in colorectal cancer

T cells expressing NK cell receptors (NKR) display rapid MHC-unrestricted cytotoxicity and potent cytokine secretion and are thought to play roles in immunity against tumors. We have quantified and characterized NKR+ T cells freshly isolated from epithelial and lamina propria layers of duodenum and colon from 16 individuals with no evidence of gastrointestinal disease and from tumor and uninvolved tissue from 19 patients with colorectal cancer. NKR+ T cell subpopulations were differentially distributed in different intestinal compartments, and CD161+ T cells accounted for over one half of T cells at all locations tested. Most intestinal CD161+ T cells expressed alpha beta TCR and either CD4 or CD8. Significant proportions expressed HLA-DR,CD69 and Fas ligand. Upon stimulation in vitro, CD161+ T cells produced IFN-gamma and TNF-alpha but not IL-4. NKT cells expressing the Valpha24Vbeta11 TCR, which recognizes CD1d,were virtually absent from the intestine, but colonic cells produced IFN-gamma in response to the NKT cell agonist ligand alpha-galactosylceramide. NKR+ T cells were not expanded in colonic tumors compared to adjacent uninvolved tissue. The predominance, heterogeneity and differential distribution of NKR+ T cells at different intestinal locations suggests that they are central to intestinal immunity.

Two levels of MCT1 and CD147 expression in the equine red blood cells and muscle

Monocarboxylate transporters (MCTs), especially the isoforms MCT1 - MCT4, cotransport lactate and protons across the cell membranes. They are thus essential for pH regulation and homeostasis in glycolytic cells such as red blood cells (RBCs), and skeletal muscle cells during intense exercise. In 70% of the Standardbred horses the lactate transport activity (TA) in RBCs is high and transport is mediated mainly by MCTs. In the rest 30% of the Standardbreds MCT mediated transport route is not active and the TA is low. MCTs need an ancillary protein for their proper localization and functioning in the plasma membrane. The ancillary protein for MCT1 and MCT4 is a member of immunoglobulin superfamily, CD147. Here we determined the expression of MCT isoforms and CD147 in equine RBCs and gluteal muscle. We sequenced the cDNA of horse MCT1 and CD147 to achieve horse-specific antibodies and to reveal sequence variations that may affect the TA of RBCs. The amount of MCT1 and CD147 mRNA in muscle were also studied. ---- In all, 73 horses representing different breeds were used. Blood samples were drawn from the jugular vein and muscle samples were taken either from gluteal muscle using biopsy needle or during castration from expendable cremaster muscle. The TA of RBCs was studied using radiolabeled lactate and the amount of MCT isoforms and CD147 in the plasma membranes using Western blotting. The level of mRNA in muscle cells was determined using qPCR. Isoforms MCT1 and MCT2 were found in the RBCs and isoforms MCT1 and MCT4 in the muscle cells of horses. The TA of RBCs was dependent on the expression of CD147 and MCT1 in the plasma membrane. Sequence variations were found in the cDNA of both MCT1 and CD147, but they did not explain the inactivity of MCT1 mediated transport route. The single nucleotide polymorphism (SNP) Met125Val in CD147 that existed parallel with an SNP in 3´-untranslated region explained, however, attenuation in CD147 expression in Standardbreds. A single mutation Ile51Val also decreased the expression of CD147 in one Warmblood. The MCT1 and CD147 mRNA concentrations in the gluteal muscle were higher in horses with higher MCT1 and CD147 expression in RBCs and lower in horses with minor expression of CD147 and MCT1. This suggests that the bimodal distribution of TA is due to differences in transcriptional regulation that is functioning in parallel in MCT1 and CD147 gene.

Identification and characterization of a new plasmid carrying genes for degradation of 2,4-dichlorophenoxyacetate from Pseudomonas cepacia CSV90

Pseudomonas cepacia CSV90 is able to utilize 2,4-dichlorophenoxyacetate (2,4-D) and 2-methyl-4-chlorophenoxyacetate as sole sources of carbon and energy. Mutants of the strain CSV90 which had lost this ability appeared spontaneously on a nonselective medium. The wild-type strain harbored a 90-kb plasmid, pMAB1, whereas 2,4-D-negative mutants either lost the plasmid or had a 70-kb plasmid, pMAB2. The plasmid pMAB2 was found to have undergone a deletion Of a 20-kb fragment of pMAB1. The plasmid-free mutants regained the ability to degrade 2,4-D after introduction of purified pMAB1 by electroporation. Cloning in Escherichia coli of a 10-kb BamHI fragment from pMAB1, the region absent in pMAB2, resulted in the expression of the gene tfdC encoding 3,5-dichlorocatechol 1,2-dioxygenase. After subcloning, the tfdC gene was located in a 1.6-kb HindIII fragment. The nucleotide sequence of the tfdC gene and the restriction map of its contiguous region are identical to those of the well-characterized 2,4-D-degradative plasmid pJP4 of Alcaligenes eutrophus, whereas the overall restriction maps of the two plasmids are different. The N-terminal 44-amino-acid sequence of the enzyme purified from the strain CSV90 confirmed the reading frame in the DNA sequence for tfdC and indicated that the initiation codon GUG is read as methionine instead of valine.

Stretch effects extracted from inwardly and outwardly propagating spherical premixed flames

Unsteady propagation of spherical flames, both inward and outward, are studied numerically extensively for single-step reaction and for different Lewis numbers of fuel/oxidizer. The dependence of flame speed ratio (s) and flame temperature ratio are obtained for a range of Lewis numbers and stretch (kappa) values. These results of s versus kappa show that the asymptotic theory by Frankel and Sivashinsky is reasonable for outward propagation. Other theories are unsatisfactory both quantitatively and qualitatively. The stretch effects are much higher for negative stretch than for positive stretch, as also seen in the theory of Frankel and Sivashinsky. The linearity of the flame speed ratio vs stretch relationship is restricted to nondimensional stretch of +/-0.1. It is shown further that the results from cylindrical flames are identical to the spherical flame on flame speed ratio versus nondimensional stretch plot thus confirming the generality of the concept of stretch. The comparison of the variation of (ds/dkappa)kappa=0 with beta(Lc - 1) show an offset between the computed and the asymptotic results of Matalon and Matkowsky. The departure of negative stretch results from this variation is significant. Several earlier experimental results are analysed and set out in the form of s versus kappa plot. Comparison of the results with experiments seem reasonable for negative stretch. The results for positive stretch are satisfactory qualitatively for a few cases. For rich propane-air, there are qualitative differences pointing to the need for full chemistry calculations in the extraction of stretch effects.

Conjectures about phase turbulence in the complex Ginzburg-Landau equation

In the complex Ginzburg-Landau equation, we consider possible ''phase turbulent'' regimes, where asymptotic correlations are controlled by phase fluctuations rather than by topological defects. Conjecturing that the decay of such correlations is governed by the Kardar-Parisi-Zhang (KPZ) model of growing interfaces, we derive the following results: (1) A scaling ansatz implies that equal-time spatial correlations in 1d, 2d, and 3d decay like e(-Ax2 zeta), where A is a nonuniversal constant, and zeta=1/2 in 1d. (2) Temporal correlations decay as exp(-t(2 beta)h(t/L(z))), with the scaling law <(beta)over bar> = <(zeta)over bar>/z, where z = 3/2, 1.58..., and 1.66..., for d = 1,2, and 3 respectively. The scaling function h(y) approaches a constant as y --> 0, and behaves like y(2(beta-<(beta)over bar>)), for large y. If in 3d the associated KPZ model turns out to be in its weak-coupling (''smooth'') phase, then, instead of the above behavior, the CGLE exhibits rotating long-range order whose connected correlations decay like 1/x in space or 1/t(1/2) in time. (3) For system sizes, L, and times t respectively less than a crossover length, L(c), and time, t(c), correlations are governed by the free-field or Edwards-Wilkinson (EW) equation, rather than the KPZ model. In 1d, we find that L(c) is large: L(c) similar to 35,000; for L < L(c) we show numerical evidence for stretched exponential decay of temporal correlations with an exponent consistent with the EW value beta(EW)= 1/4.

Design and Characterization of Stabilized Derivatives of Human CD4D12 and CD4D1

CD4 is present on the surface of T-lymphocytes and is the primary cellular receptor for HIV-1. CD4 consists of a cytoplasmic tail, one transmembrane region, and four extracellular domains, D1-D4. A construct consisting of the first two domains of CD4 (CD4D12) is folded and binds gp120 with similar affinity as soluble 4-domain CD4 (sCD4). However, the first domain alone (CD4D1) was previously shown to be largely unfolded and had 3-fold weaker affinity for gp120 when compared to sCD4 [Sharma, D.; et al. (2005) Biochemistry 44, 16192-16202]. We now report the design and characterization of three single-site mutants of CD4D12 (G6A, L51I, and V86L) and one multisite mutant of CD4D1 (G6A/L511/L5K/F98T). G6A, L51I, and V86L are cavity-filling mutations while L5K and F98T are surface mutations which were introduced to minimize the aggregation of CD4D1 upon removal of the second domain. Two mutations, G6A and V86L in CD4D12 increased the stability and yield of the protein relative to the wild-type protein. The mutant CD4D1 (CD4D1a) with the 4 mutations was folded and more stable compared to the original CD4D1, but both bound gp120 with comparable affinity. In in vitro neutralization assays, both CD4D1a and G6A-CD4D12 were able to neutralize diverse HIV-1 viruses with similar IC(50)s as 4-domain CD4. These stabilized derivatives of human CD4 can be useful starting points for the design of other more complex viral entry inhibitors.

Experimental study of phase equilibria in the system Cu-Al-Sn

Phase equilibria in the Cu-rich corner of the ternary system Cu-Al-Sn have been re-investigated. Final equilibrium microstructures of 20 ternary alloy compositions near Cu3Al were used to refine the ternary phase diagram. The microstructures were characterized using optical microscopy (OM), x-ray diffraction (XRD), electron probe microanalysis and transmission electron microscopy. Isothermal sections at 853, 845, 833, 818, 808, 803 and 773 K have been composed. Vertical sections have been drawn at 2 and 3 at% Sn, showing beta(1) as a stable phase. Three-phase fields (alpha + beta + beta(1)) and (beta + beta(1) + gamma(1)) result from beta -> alpha + beta(1) eutectoid and beta + gamma(1) -> beta(1) peritectoid reactions forming metastable beta(1) in the binary Cu-Al. With the lowering of temperature from 853 to 818 K, these three-phase fields are shifted to lower Sn concentrations, with simultaneous shrinkage and shifting of (beta + beta(1)) two-phase field. The three-phase field (alpha + beta + gamma(1)) resulting from the binary reaction beta -> alpha + gamma(1) shifts to higher Sn contents, with associated shrinkage of the beta field, with decreasing temperature. With further reduction of temperature, a new ternary invariant reaction beta + beta(1) -> alpha + gamma(1) is observed at similar to 813 K. The beta disappears completely at 803 K, giving rise to the three-phase field (alpha + beta(1) + gamma(1)). Some general guidelines on the role of ternary additions (M) on the stability of the ordered beta(1) phase are obtained by comparing the results of this study with data in the literature on other systems in the systems group Cu-Al-M.

Aza-induced cardiomyocyte differentiation of P19 EC-cells by epigenetic co-regulation and ERK signaling

Stem cells in cell based therapy for cardiac injury is being potentially considered. However, genetic regulatory networks involved in cardiac differentiation are not clearly understood. Among stem cell differentiation models, mouse P19 embryonic carcinoma (EC) cells, are employed for studying (epi)genetic regulation of cardiomyocyte differentiation. Here, we comprehensively assessed cardiogenic differentiation potential of 5-azacytidine (Aza) on P19 EC-cells, associated gene expression profiles and the changes in DNA methylation, histone acetylation and activated-ERK signaling status during differentiation. Initial exposure of Aza to cultured EC-cells leads to an efficient (55%) differentiation to cardiomyocyte-rich embryoid bodies with a threefold (16.8%) increase in the cTnI(+) cardiomyocytes. Expression levels of cardiac-specific gene markers i.e., Isl-1, BMP-2, GATA-4, and alpha-MHC were up-regulated following Aza induction, accompanied by differential changes in their methylation status particularly that of BMP-2 and alpha-MHC. Additionally, increases in the levels of acetylated-H3 and pERK were observed during Aza-induced cardiac differentiation. These studies demonstrate that Aza is a potent cardiac inducer when treated during the initial phase of differentiation of mouse P19 EC-cells and its effect is brought about epigenetically and co-ordinatedly by hypo-methylation and histone acetylation-mediated hyper-expression of cardiogenesis-associated genes and involving activation of ERK signaling.

Intensity profile of light scattered from a rough surface

We present in this paper, approximate analytical expressions for the intensity of light scattered by a rough surface, whose elevation. xi(x,y) in the z-direction is a zero mean stationary Gaussian random variable. With (x,y) and (x',y') being two points on the surface, we have h. = 0 with a correlation, = sigma(2)g(r), where r = (x - x')(2) + ( y - y')(2)](1/2) is the distance between these two points. We consider g(r) = exp-r/l)(beta)] with 1 <= beta <= 2, showing that g(0) = 1 and g(r) -> 0 for r >> l. The intensity expression is sought to be expressed as f(v(xy)) = {1 + (c/2y)v(x)(2) + v(y)(2)]}(-y), where v(x) and v(y) are the wave vectors of scattering, as defined by the Beckmann notation. In the paper, we present expressions for c and y, in terms of sigma, l, and beta. The closed form expressions are verified to be true, for the cases beta = 1 and beta = 2, for which exact expressions are known. For other cases, i.e., beta not equal 1, 2 we present approximate expressions for the scattered intensity, in the range, v(xy) = (v(x)(2) + v(y)(2))(1/2) <= 6.0 and show that the relation for f(v(xy)), given above, expresses the scattered intensity quite accurately, thus providing a simple computational methods in situations of practical importance.