Once a taxonomy has been used

in documenting treatment, a logical step would be to use the same information in billing, in the way the CPT is currently used by physicians. The detail could be used to justify procedures and/or quantities billed, or as accounting for time use and charges in situations where there is no direct link between the fee collected and the intensity of services rendered (eg, under capitation). Stem Cell Compound Library cell line An additional advantage of classification-aided documentation is that medical record abstracting becomes faster and much more unambiguous because all therapists administering the same treatment designate it with the same standardized nomenclature. Clinicians participating in the previously mentioned SCI PBE study have suggested that the point of care form (which is a series of menus loaded on a personal digital assistant) that they completed for each session might (in a somewhat simplified format) be an eminent way of documenting treatment sessions (Julie Gassaway, Alectinib concentration oral communication, June 29, 2010). Currently, treatment documentation is primarily focused on information needed to obtain third party payment, and it is done with freestyle notes that show tremendous variation from one clinician to the next. The improvement of communication within and between disciplines represented on the

rehabilitation team would be a potential byproduct of documentation based on an interdisciplinary treatment taxonomy (see Mintken et al113). If professionals can agree with the theory that generated the RTT as to what intervention(s) are appropriate for specific patient problems/deficits and given feasible treatment goals, a typology might be used in a more or less prescriptive mode. the Many professional groups are currently developing clinical practice guidelines; the most prominent example in the field of rehabilitation medicine is the effort by the Consortium for Spinal Cord Medicine.114 With the availability of an RTT, the optimal course of treatment

for patients with a given set of problems or diagnoses might be described using clearly defined doses and timing of a series of defined interventions. Similarly, the development of clinical paths (pathways) now used in many rehabilitation programs115, 116, 117 and 118 would benefit from a standard nomenclature that provides detail about treatments (characteristics, quantity, or intensity) in a comprehensible manner. In a closely related application, an RTT might be used in quality assurance to describe the treatment actually delivered and compare this to the ideal.119 Routine rehabilitation program evaluation would find an RTT (in simplified form) very useful for evaluating whether all patients received the minimum treatment program promised by the facility.

The correlation coefficients between the hourly data series were 0.93 at Kõiguste and 0.91 at Matsi (960 and 1440 value pairs respectively). The RMSE was slightly better at Matsi (0.135 vs. 0.167 m). The RMSE, standardized with the variability range, which illustrates the prognostic value of the calibration, was 8.9% at Matsi and 9.4% at Kõiguste. These sets of comparative Z-VAD-FMK purchase statistics were only marginally worse than in our previous experiences at the Harilaid Peninsula (Suursaar & Kullas 2009) and at Letipea (Suursaar 2010). The main reason for this was the absence of storm wave conditions and the relatively

smaller range of values used in the calibration procedure. Strictly speaking, the calibration is fully valid for the conditions and variability range during the calibration. We can assume that on moving back from the calibration time, the results may get gradually worse. Validation is possible when additional data sets of the same type are available outside the calibration periods. At Matsi, 60 days of relatively calm summer measurements were used in the calibration. But at Kõiguste in the autumn, apart from 40 days of calibration, some 30 days were left for validation just before sea-ice began to affect waves by shortening fetch distances. check details The validation results were very good (Figure 5d), r was 0.89 and the RMSE was equal to 0.197. Also, the validation (verification) we performed

earlier at Letipea (see Figure 3 in Suursaar 2010) showed remarkably good agreement between measurements and calculations. Depending largely on morphometry, coastline and bottom many topography, the current

velocity components and sea surface height at every single point in the model domain possess a specific way of reacting to wind forcing. By choosing the points (of measurements) at Kõiguste and Matsi and applying the same methodology as Suursaar & Kullas (2006), the reaction of currents to wind direction can be investigated (Figure 6 and Figure 7). At Matsi, the strongest currents appeared in wind directions 150 and 330 degrees, and in 70 and 250 degrees at Kõiguste. However, when choosing a neighbouring, or just a different point, the result would be somewhat different as well (see Figure 6b). In stationary or persistent wind forcing conditions, downwind flows prevail near the coasts of medium-size oval basins and large lakes, whereas compensatory flows against the wind evolve along the deeper middle section of a sub-basin (Csanady 1973). For instance at Pärnu Bay, our simulations revealed two well-defined basin-scale flow regimes with cyclonic and anticyclonic circulation cells (Suursaar & Kullas 2006). The two wind directions which switch between the two regimes were approximately 120 and 300 degrees. Determined by the size and coastline, similar patterns were also found in simulations with stationary and uniform winds blowing from different directions in the northern Gulf of Riga (Figure 6 and Figure 7).

In this study, we tested different protocols to recover DNA from molar and pre-molar teeth of cadavers in bad decomposition stages with different post-mortem intervals. We were able to obtain DNA profiles from the questioned samples and to compare them with reference samples. No significant differences

were observed in the total quantity of DNA obtained in the procedures with distinct incubation times showing that short cell lysis time can be used in urgent genetic identification with good quality results. The use of concentration column (Microcon™-100) resulted in an increased amount of DNA when compared to isopropanol. However, the lower concentration of DNA obtained by precipitation with isopropanol seemed to have been compensated by the higher purity, since the measurement by optic density fraction was higher and because no significant www.selleckchem.com/products/ldk378.html differences in the number of amplified loci BTK inhibitor were found between these protocols. Isopropanol was, in fact, very effective in DNA precipitation, as it has already been reported, 22 besides being low-priced. Six samples had an apparent good amount of total DNA but resulted in poor autosomal profiles. It was probably

due to unsatisfactory quality of DNA by degradation or reduced DNA quantity by microorganism DNA contamination.23 To verify this, specific human DNA quantification by Real Time PCR analysis would be necessary.24 We compared the DNA amount and the DNA profiles with the time elapsed between death and laboratory procedures, but the increase of post-mortem interval did not interfere in any of these variables. In conclusion, our work showed molar or pre-molar teeth as good candidates to obtain satisfactory

DNA profiles suggesting the high potential Cediranib (AZD2171) of tooth samples as a source to DNA typing independently of the decomposed corpse’s time or laboratory procedures. This study was financed by SENASP-IGP-RS. The study is part of the Master’s Degree thesis of the first author who had a fellowship from PUCRS, Brazil. We declare that we have no conflict of interest. This project was approved by the Research Ethics Committee of the Pontifical Catholic University of Rio Grande do Sul (PUCRS-code #1107/05; Tel.: +55 51 33203345), and the consent or assent to take part in this study was obtained from Forensic Laboratory of Instituto-Geral de Perícias of Rio Grande do Sul, Brazil. We thank E. Eizirik for his suggestions and J.Z. Bolsi for technical assistance. This study was financed by SENASP-IGP-RS. The study is part of the Master’s Degree thesis of the first author who had a fellowship from PUCRS, Brazil. “
“Periodontitis is a chronic inflammatory disease of destructive and non-reversible action, that, if not treated, can cause tooth mobility leading to subsequent tooth loss.

In fact, the second-best BLASTX hit, after BgP, is to a Eubacterium acidaminophilum FdhC (CAC39240.1) that has been characterized experimentally ( Graentzdoerffer et al., 2003). Formate could serve as an electron donor, carbon substrate, or both. A possible formate dehydrogenase gamma subunit gene (01341_2381) is found in a cluster with other ORFs

variously annotated as formate, thiosulfate, CH5424802 cost and tetrathionite reductase component genes; it is doubtful whether their in vivo roles can be deduced from the sequences alone. Phosphotransferase systems for carbohydrate uptake typically consist of one or two membrane (EIIC/EIID) and one or two cytosolic (EIIA/EIIB) components specific for a given carbohydrate, and two more general cytoplasmic components (EI and HPr), which may be in various combinations of fused and separate proteins (reviewed in Deutscher et al. (2006)). EI is a phosphoenolpyruvate:protein phosphotransferase, and HPr is a phosphocarrier transferring phosphate groups from EI to EIIA. In Gram-negative bacteria, phosphate groups are transferred in a cascade from phosphoenolpyruvate (PEP) to the membrane PTS components, and thence to a periplasmic carbohydrate

molecule, concomitant with its uptake. The phosphorylated carbohydrates are typically fed into the glycolysis pathway. PTS genes are also involved in transcriptional regulation of carbohydrate metabolism. 5-Fluoracil in vitro Only two sets

Selleck ZD1839 of putative PTS-related genes have been annotated in the BOGUAY genome. One is related to ascorbate uptake systems, and includes possible EIIA (ulaC, 00136_0633), EI (ptsI, 00136_0635) and HPr (hprK, 00136_0634) genes; the other is related to regulatory systems that are thought to coordinate nitrogen and carbon uptake, and includes putative EIIA (ptsN, 00726_1444) and HPr (hprK, 00726_1445) genes. No membrane-protein genes have been identified for either of these potential PTS systems, however; they may have strictly regulatory (or other) functions, or novel membrane components. The BOGUAY genome encodes a complete glycolytic pathway, and apparently two types of energy-generating electron transport pathways. In addition to the common oxidative phosphorylation pathway, in several possible variants, it possesses two different genes for most components of a putative Rnf complex, a potentially energy-generating ion pump whose detailed function is not yet well understood. This suggests that the BOGUAY strain may be able to access a range of electron donors and acceptors. Details are discussed immediately below. All glycolysis genes seem to be present in the BOGUAY genome (Table S6), with energy-conserving pyrophosphate-consuming enzymes apparently preferred to those hydrolyzing ATP. There are two possible pyrophosphate-dependent 6-phosphofructokinases (PFKs; Fig.

The data obtained are in qualitative agreement with the results of the previous field studies by Lehr et al. (1984a) and Elliot (1986). An example of the dependence of L in the downwind speed direction and film area S on time is presented in Figure 4. Data obtained at various wind speeds are shown in this figure by the symbols (°) – 1.6 m s− 1 – 3.3 m s− 1, (△) – 7.8 m s− 1, (⋄) – 11.7 m s− 1. The origin of the coordinates in Figure 4 corresponds to the moment when the vegetable oil was first spilt. As follows from Figure 4 the values of L at a fixed time point grow when the wind speed increases. STI571 order The same tendency is observed for areas of SF GDC-0941 research buy ( Figure 4b). We did not find an explicit dependence of the film slick axis l on wind speed. The values of the ratio L/l describing slick elongation at various times in the wind speed range from 9 to 11.7 m s− 1, from 6 to 9 m s− 1 and < 3.3 m s− 1 are shown in Figure 5 by the symbols (+), (⋄) and (°) respectively. The solid line shows the value of L/l = 1. As can be seen from Figure 5 at U < 3.6 m s− 1 the values of L/l change from 0.9 to 1.1. Thus under calm wind conditions SF is circular in shape. Film slick elongation

increases with a strengthening wind and at U ∼ 12 ms− 1 values of L/l are ∼ 18. Let us define the rate of semi-major axis growth in the downwind and upwind directions as udsp = ∂Ld/∂t and uupsp = ∂Lup/∂t respectively. Wind speed dependences of spreading rates in the downwind and upwind direction are presented in Figure 6 and denoted by the symbols (°) and (+) accordingly. Values of uupsp and udsp were calculated using all the data of each measurement and thus represent average values. Spreading rates at weak wind speeds varied from 0.01 to 0.02 m s− 1.

There is an increase of values of usp for moderate and strong winds. According to the results of the experiments, the observed spreading rate of the semi-major axis of the film at U = 12 m s− 1 is ∼ 4 times higher than the value typical of U = 1.6 m s− 1 – 3.6 m s− 1 (see Figure 6). Now we consider the growth of the surface film size under various wave conditions. The dependence of udsp on Hs is presented in Figure 7, where the symbols correspond Endonuclease to measurements at various inverse wave ages α = U/Cp (Cp – wave phase velocity of the spectral peak): (°) – α = 0.9–1.3; (+) – α = 2–3. The case denoted by (•) relates to calm wind conditions and to the presence of a swell. As follows from Figure 7, no explicit dependence of the SF spreading rate on Hs for the whole set of points is observed. In contrast, the tendency of udsp to increase with increasing wave height for the obtained data set is visible when α = 0.9–1.3 and α = 2–3. At the same time spreading rates for the case denoted by (•) measured at Hs = 0.62 m and U = 1.

equation(2) v0[E]=kcat[S]Km+[S] equation(3) [E]v0=Kmkcat1[S]+1kcat Plotting 1/[vo] versus 1/[S] gives a linear line with a slope check details of Km/kcat (the reciprocal of the second-order rate constant of the enzyme, kcat/Km), a y-intercept of 1/kcat and an x-intercept of −1/Km. While these values can easily be determined without the aid of a computer, they are heavily dependent on the precision of rates determined

at the lowest substrate concentrations as illustrated in Figure 1. This is problematic since the precision of the measurement is lowest at low substrate concentrations due the slower rates and correspondingly small signal changes in the kinetic assay employed. As can be seen in Figure 1, small changes in the rates determined at low substrate concentrations can dramatically affect both the slope and intercepts of the Lineweaver–Burk plot and thus the kinetic parameters and associated KIEs determined using this method. This sensitivity is overcome when plotting the untransformed data and using the non-linear Michaelis–Menten equation (Eq. (2)) to determine the kinetic parameters. Similar uncertainties arise when using alternate methods of plotting enzyme kinetic data and should therefore be avoided. When isotope effects are measured for a multi-dimensional model the data should be fit globally to equations describing the kinetic mechanism under

study. Common and general examples can be expressed as y=F[xi], where E7080 supplier xi is more than a single variable, such as multi-substrate enzymes (y=F[S1,S2,…[Si]]), temperature and pressure (y=F[P1,P2,…[Pi]]), etc. The kinetic parameters obtained from these fits should be used to calculate both the isotope effects on the relevant parameters and their associated errors. The relevant equations used to fit the data should be reported as well as the software package used for the analysis, the regression

method, and the specific methods for errors assessment, incorporation, selleck kinase inhibitor statistical weighing, and propagation. In graphic presentation of the data, the individual curves should be plotted using the kinetic parameters obtained from the global fitting, rather than a single dimensional fit for a specific set of variables (e.g., concentration of inhibitor in Figure 2). In addition, the statistical confidence of the global fit should be reported either in a table or the figure legend. It is important to use global fits of the data to determine a KIE, since the values obtained from fitting to a model of lower dimension (e.g., fitting to individual curves measured under different conditions) may not represent meaningful and general parameters ( Cook, 1991, Cook and Cleland, 2007, Cleland, 1963 and Kohen and Limbach, 2006). Furthermore, the plots presented should be fit using the parameters obtained from the global fits to allow for a visual inspection of the quality of the data.

“
“Current Opinion in Genetics & Development 2013, 23:53–62 This review comes from a themed issue on Cancer genomics Edited by Nahum Sonenberg and Nissim Hay For a complete overview see the Issue and the Editorial Available online 11th Jan 2013 0959-437X/$ – see front matter, © 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.gde.2012.12.005

SRT1720 cost Target of rapamycin (TOR) is a conserved serine/threonine kinase that regulates cell growth, aging and metabolism, from yeast to human [1, 2, 3, 4 and 5]. TOR is found in two structurally and functionally distinct complexes termed TOR complex 1 (TORC1) and TORC2 (Figure 1 and Figure 2). The immunosuppressive macrolide rapamycin inhibits TORC1 activity. In metazoans, TORC1 controls growth-related processes such as ribosome biogenesis, protein synthesis, transcription, nutrient uptake and autophagy in response to nutrients, growth factors, and cellular energy status. The best-characterized substrates of TORC1 are 4E-BP and S6K via which mammalian TORC1 (mTORC1) controls protein synthesis. The core components of mTORC1 are mTOR, raptor and mLST8. mTORC2 is activated

by growth factors alone, via PI3K-dependent ribosome association [6•• and 7••]. The commonly described substrates of TORC2 are AGC kinase family members such as Akt, SGK, and PKCα in mammals [8]. The core components of mTORC2 are mTOR, rictor, mSIN1 and mLST8. mTOR plays a particularly important role in metabolic organs — such as the liver, muscle, and adipose tissue — to Cabozantinib cell line regulate whole body energy homeostasis. Thus, deregulation of mTOR signaling leads to metabolic disorders, such as obesity and type 2 diabetes, and cancer, that is, some of the most common causes of death in Western society. Furthermore, consistent with its role as a nutrient and growth factor sensor, decreased

mTOR signaling reduces aging and thereby extends lifespan. Importantly, aging is a major risk factor for the development of cancer and metabolic disorders. Org 27569 Thus, mTOR underlies both aging and age-related diseases, suggesting that insight in mTOR signaling may provide a means to counter both aging and age-related disease by a single ‘treatment’. In other words, an understanding of mTOR signaling may allow one to collectively ‘treat’ age-related diseases by delaying aging. Here, we review the major recent findings on mTOR signaling in different metabolic organs and how this may affect aging and age-related disease. Aging is defined as an accumulation of cellular damage over time, promoting disease and death. Genetic or pharmacological inhibition of TORC1 signaling extends lifespan in yeast, worms, flies and mice [9, 10•, 11, 12, 13••, 14, 15, 16, 17, 18 and 19]. Importantly, rapamycin delays the onset of age-related disease and extends lifespan even in old mice [13•• and 15].

Groundwater seepage in the study area has been the subject of several recent studies (Pempkowiak et al., 2010, Szymczycha et al., 2012, Szymczycha et al., 2013 and Kotwicki et al., 2013). It has been established that the groundwater outflow varies seasonally from 3.6 to 21.3 L d− 1 m− 2. Groundwater rates were lower in February and May (2010) and higher in September and November (2009) and correlated well with the average monthly precipitation characteristic of the area (Korzeniewski 2003). The average concentrations of nutrients were calculated at 60.6 ± 5.9 μmol L− 1 PD-1/PD-L1 inhibitor clinical trial (PO4) and 119.4 ± 42 μmol L− 1 (NH4 + NO2 + NO3). SGD at the study site is apparently a major factor behind the abundance of

biota there ( Kotwicki et al. 2013). The seepage rate in the study site is influenced by several factors, including sea level, wave action, precipitation, sea bottom relief and dynamics. Storm surges seem to be the most significant factor influencing the groundwater seepage rate and the residence time of pore

water in the study area ( Szymczycha et al. 2012). The additional study sites were situated along the Polish coast at Międzyzdroje (M), Kołobrzeg (K), Łeba (Ł) and Władysławowo (W). These locations were selected in accordance with literature reports indicating areas that were expected to be impacted by groundwater (Kryza & Kryza 2005). This additional sampling campaign was carried out in order to investigate DIC and DOC concentrations in seeping water Androgen Receptor signaling pathway Antagonists collected at locations other than the main study area – the Bay of Puck. Assessment of SGD into the Baltic Sea was the aim of several research studies and projects. Piekarek-Jankowska (1994) estimated the groundwater seepage to the Bay of Puck to be 3500 m3 h− 1. Kryza & Kryza (2006) calculated that the volume of SGD to the Polish coastal zone of the Baltic Sea was equal to some 16 570 m3 h− 1. Kozerski (2007) estimated the rate of

SGD to the Gulf of Gdańsk including the Bay of Puck to be 6700 m3 h− 1. Peltonen Molecular motor (2002) estimated that the total volume of SGD entering the Baltic Sea was 4.4 km3 yr− 1 accounting for some 1% of the total river run-off volume. It was estimated that around 75% of the groundwater discharge enters the Baltic along its southern coast (Peltonen 2002). Uścinowicz (2011) concluded that SGD in the Bay of Puck/Gulf of Gdańsk exceeds by far the SGDs in other regions of the Baltic. Thus the study area can be regarded as representing the most important southern Baltic Sea groundwater seepage area. This study is a continuation of earlier investigations by Pempkowiak et al. (2010), Szymczycha et al. (2012) and Szymczycha et al. (2013). Five sampling campaigns were carried out during the following periods: 31.08–3.09.2009, 2–6.11.2009, 28.02–1.03.2010, 5–7.05.2010 and 10–17.07.2013. The study area in the Bay of Puck (H) covers about 9200 m2 and is shown on Figure 1.

Although we do not have an explanation for this overestimate, future studies should endeavour to use in situ pellets. If this is not possible and culture pellets are used, their characteristics should resemble those of in situ ones (e.g. similar algae for feeding, similar textures, C:N ratios, lipids contents, etc.). Interestingly, it was recently FK506 solubility dmso argued that degradation by bacteria and protozooplankton of culture (Rhodomonas sp.) faecal pellets of C. finmarchicus incubated in cold waters takes three days to be significant (32% after 3 days) ( Svensen et al. 2012).

In the present study, FP-CSD by bacteria and protozooplankton was measured over the first two days of incubation (about 10.1% d− 1 for culture pellets, Figure 2). Assuming a constant carbon to volume ratio, the degradation would thus be about 30% in 72 h, which is comparable to the results of Svensen et al. (2012). Those authors used a different method, where microscopic measurements UK-371804 were performed in order to estimate faecal pellet volume changes and

therefore degradation. It seems, however, that this microscopic method did not make it possible to determine statistical differences in volume during the first two days, in contrast to the respiration method used in the present study. The use of micro-respiration chambers may therefore be more sensitive. In addition, the method used in the present study is less subjective and less time-consuming than the microscopic method. Despite the limited data set, the novel 4-Aminobutyrate aminotransferase use of micro-respiration chambers for faecal pellet

FP-CSD in the present study highlights the importance of bacteria from the pellet matrix, free-living bacteria and protozooplankton for faecal pellet degradation. Bacteria and protozooplankton play an important role in faecal pellet degradation at the chl a max compared to deeper water, and it most likely an important factor in areas where primary production is high, as the abundance of bacteria and protozooplankton is correlated with primary production. Few studies have addressed the importance of protozooplankton in Arctic areas, but this knowledge will be crucial for our understanding of the role of protozooplankton for the vertical flux of faecal pellets, which have been underestimated in the past owing to the low temperatures. In addition, the comparison between in situ and culture pellets addresses the importance of using in situ pellets if we wish to extrapolate results to natural field conditions. The results obtained from the experiments with culture pellets should be treated with caution, as they may overestimate the degradation rates. Special thanks go to the chief scientist of the Conflux cruise, M. Reigstad. A ‘thank you’ also goes to S. Øygarde, C. Svensen and C. Wexels Riser for their help in the field, in the lab and for the carbon analysis. We are grateful to T. Tamelander for the bacteria data. This manuscript has benefited from the valuable comments and suggestions from C. Lalande, C.

The wind effects are directly related to the pressure distribution over an area. However, as shown by tide gauge records, true sea level surges and falls can be several times higher than the values resulting from the action of tangential wind stress upon a fluid surface (Wiśniewski & Holec 1983). Suursaar et al. (2003) pointed out that the highest surge events on the west Estonian coast are associated with deep depressions producing strong SW and W winds in suitably oriented bays such as Pärnu Bay. An example is the mid-latitude

depression Gudrun, which occurred in January 2005 and caused the heaviest storm surge along the coasts of the Gulf of Riga (Suursaar et al. 2006). The sea level at Pärnu was 2.75 m higher selleck screening library than the mean level there. In the Gulf of Finland, record increases in sea level were measured

as well, e.g. at Helsinki (1.51 m). Skriptunov & Gorelits (2001) PLX 4720 showed that significant wind-induced variations in the water level near the River Neva as well as their magnitude and duration result from the wind regime and the morphology of the near-mouth offshore zone. Averkiev and Klevanny, 2007 and Averkiev and Klevanny, 2010 analysed the effects of atmospheric pressure as well as wind direction and speed on the sea level in the Gulf of Finland. They showed the low pressure system trajectory to be potentially important in generating storm surges particularly damaging for St. Petersburg (Russia). The problem of sea surface deformation by concentric, mesoscale, fast-moving deep low-pressure systems was addressed by Lisowski, 1960, Lisowski, 1961 and Lisowski, 1963, Wiśniewski, 1996, Wiśniewski, 1997 and Wiśniewski, 2003, Wiśniewski & Holec (1983), Wiśniewski & Kowalewska-Kalkowska (2007) and Wiśniewski and Wolski, 2009a and Wiśniewski and Wolski, 2011. It seems, however, that this not factor has been generally underestimated, even downright ignored, in the literature, a situation that has been detrimental to attempts at explaining mechanisms

of such extreme phenomena as coastal floods or low sea levels that adversely affect navigation safety, stability of hydraulic engineering structures, etc. It is true that a lowered atmospheric pressure system (a tropical cyclone or a concentric low pressure system) overlies a water cushion, moving together with the pressure system at the sea surface. Wave height depends on the pressure decrease in the centre of the system. A pressure drop of Δp = 1 hPa results in a static sea level rise of ΔHs = 1 cm under a stationary low ( Figure 1a, formula  (3)). When the depression moves over the sea surface, the latter becomes dynamically deformed (ΔHd). The sea level deformation shows positive wave elevations in the centre and negative elevations on the flanks of the deformation ( Figure 1b, formula  (4)).