Glial cell line-derived neurotrophic factor induces barrier function of endothelial cells forming the blood-brain barrier

Since a deep involvement of astrocytes, a kind of glial cells, in differentiation of the blood-brain barrier (BBB) has been suggested, we examined the relation of glial cell line-derived neurotrophic factor (GDNF) to the BBB. First, immunohistochemical examination of the cerebral cortex of rats revealed that glial cell line-derived neurotrophic factor receptor (GFRalpha1) was preferentially expressed on the cell membranes of capillary endothelial cells. Second, to elucidate the effects of GDNF on the BBB, capillary endothelial cells isolated from the porcine cerebral cortex were cultured and then changes in tight junction function of the endothelial cells were examined after addition of GDNF, in terms of transendothelial electrical resistance (TER) and permeability. GDNF at concentrations of 0.1 and 1 ng/ml significantly activated the barrier function of the endothelial cells in the presence of cAMP. Since GDNF is secreted from astrocytes sheathing capillary endothelial cells in the brain cortex, our results strongly suggest that GDNF enhances the barrier function of tight junctions of the BBB on the one hand, and also supports the survival of neurons on the other hand.

Sld2, which interacts with Dpb11 in Saccharomyces cerevisiae, is required for chromosomal DNA replication

The DPB11 gene, which genetically interacts with DNA polymerase II (epsilon), one of three replicative DNA polymerases, is required for DNA replication and the S phase checkpoint in Saccharomyces cerevisiae. To identify factors interacting with Dbp11, we have isolated sld (synthetically lethal with dpb11-1) mutations which fall into six complementation groups (sld1 to -6). In this study, we characterized SLD2, encoding an essential 52-kDa protein. High-copy SLD2 suppressed the thermosensitive growth defect caused by dpb11-1. Conversely, high-copy DPB11 suppressed the temperature-sensitive growth defect caused by sld2-6. The interaction between Sld2 and Dpb11 was detected in a two-hybrid assay. This interaction was evident at 25 degreesC but not at 34 degreesC when Sld2-6 or Dpb11-1 replaced its wild-type protein. No interaction between Sld2-6 and Dpb11-1 could be detected even at 25 degreesC. Immunoprecipitation experiments confirmed that Dpb11 physically interacts with Sld2. sld2-6 cells were defective in DNA replication at the restrictive temperature, as were dpb11-1 cells. Further, in dpb11-1 and sld2-6 cells, the bubble-shaped replication intermediates formed in the region of the autonomously replicating sequence reduced quickly after a temperature shift. These results strongly suggest the involvement of the Dpb11-Sld2 complex in a step close to the initiation of DNA replication.

Expression of GFRalpha-1, receptor for GDNF, in rat brain capillary during postnatal development of the BBB

It is well known that the blood-brain barrier (BBB) matures at approximately 2 wk after birth in the rat. Recently, we showed that glial cell line-derived neurotrophic factor (GDNF) enhances the barrier function of porcine endothelial cells forming the BBB in culture. In the present study, we examined the relation between permeability of the BBB, using Evans blue as a tracer, and expression of the GDNF family receptor (GFRalpha-1) during postnatal development of the BBB. Morphometric analysis showed that exudation of Evans blue from capillaries of the cerebral cortex progressively decreased until postnatal day 21. Inversely, immunohistochemical examinations showed expression of GFRalpha-1 in the capillaries at postnatal day 3 and expression that reached the same levels as observed in adult rats by postnatal day 10. However, c-ret, which is thought to mediate a signal evoked by binding of GDNF to GFRalpha-1, was not expressed in the capillaries of the brain cortex in 3-mo-old rats. On the other hand, the tight junction proteins occludin and ZO-1 appeared to be fully expressed at birth. The reciprocal relation between GFRalpha-1 expression and the permeability of the BBB strongly suggests active participation of GDNF in postnatal development of the BBB, although the mechanism(s) involved is still veiled.

Large intramedullary lipoma of the cervical cord and posterior fossa. Case report

A large intramedullary lipoma of the cervical cord extending into the posterior fossa is reported in a 7-year-old boy. Magnetic resonance imaging was very useful for delineation of the anatomy of the lipoma as an aid in planning the operation.

Right-handed penises of the earwig Labidura riparia (Insecta, Dermaptera, Labiduridae): evolutionary relationships between structural and behavioral asymmetries

The number of penises vary in the insect suborder Forficulina (order Dermaptera; earwigs). Males of the families Diplatyidae, Pigidicranidae, Anisolabididae, Apachyidae, and Labiduridae have two penises (right and left), while those of the Spongipohridae, Chelisochidae, and Forficulidae have a single penis. The proposed phylogenetic relationships among these families suggest that the single-penis families evolved from an ancestor possessing two penises. To date, examinations of double-penis earwig species have found that only a single penis is used per single copulation. These diversities in structural and behavioral aspects of genitalia raises the following intriguing questions: How are the two penises used? Why did a penis degenerate in several earwig families, and which one was lost? To address these questions, structural and behavioral asymmetries were examined in detail for a representative species Labidura riparia (Labiduridae). Although there was no detectable morphological differentiation between the right and left penises, male L. riparia predominantly used the right one for insemination. This significant "right-handedness" developed without any experience of mating and was also manifested in the resting postures of the two penises when not engaged in copulation. However, surgical ablation of the right penis did not influence the insemination capacity of males. In wild-caught males, only about 10% were left-handed; within this group, abnormalities were frequently observed in the right penis. These lines of evidence indicate that the left penis is merely a spare intromittent organ, which most L. riparia males are likely never to use. Additional observations of five species of single-penis families revealed that the left penis degenerated in the common ancestor of this group. Considering the proposed sister relationship between the Labiduridae and the single-penis families, it is possible that such behavioral asymmetries in penis' use, as observed in L. riparia, are parental to the evolutionary degeneration of the infrequently used left penis.

Lack of promoting effects of the electromagnetic near-field used for cellular phones (929.2 MHz) on rat liver carcinogenesis in a medium-term liver bioassay

The possible cancer promotion potential of local exposure to a pulse modulated 929.2 MHz electromagnetic near-field on chemically-initiated rat liver carcinogenesis was investigated employing a medium-term bioassay. A 929.2-MHz electromagnetic near-field of time division multiple access (TDMA) signal for PDC (Personal Digital Cellular, Japanese cellular telephone standard) system was directed to rats through a quarter-wavelength monopole antenna. Maximum local specific absorption rates (SARs) on temporal average were 7.2-6.6 W/kg within the whole body and 2.0-1.7 W/kg within the liver, which was the target organ. The whole-body average SARs on temporal average were 0.80-0.58 W/kg. Temporal peak SARs had three times these values due to the duty ratio of the PDC signal. Exposure was for 90 min a day, 5 days a week, over 6 weeks. The exposure apparatus was specially designed for this experiment, to allow exposure of the lateral mid-section of the rat body to the electromagnetic near-field. Male F344 rats, 6 week-old, were initially (at week 0) given a single dose of diethylnitrosamine (DEN, 200 mg/kg body wt, i.p.). At 2 weeks later, exposure (48 rats) or sham-exposure (48 rats) was started. The exposure of electromagnetic near-fields was performed using the exposure apparatus mentioned above. At week 3, all rats were subjected to a 2/3 partial hepatectomy. At week 8 (i.e. after 6 weeks exposure or sham-exposure), the experiment was terminated and all rats were killed. Carcinogenic potential was scored by comparing the numbers and areas of the induced glutathione S-transferase placental form (GST-P) positive foci in the livers of the exposed and sham-exposed rats. A further group of 24 animals, given only DEN and partial hepatectomy, served as the controls. The numbers (no./cm2) of GST-P positive foci were 4.61 +/- 1.77, 5.21 +/- 1.92 (P < 0.05, versus control) and 4.09 +/- 1.47 and the areas (mm2/cm2) were 0.30 +/- 0.16, 0.36 +/- 0.21 and 0.28 +/- 0.15, for the exposed, sham-exposed and control groups, respectively. There were no significant differences between the exposed and sham-exposed groups. These findings clearly indicated that local body exposure to a 929.2-MHz field, modulated in a PDC waveform, has no significant effect on rat liver carcinogenesis under the experimental conditions employed.

The 1.5 GHz electromagnetic near-field used for cellular phones does not promote rat liver carcinogenesis in a medium-term liver bioassay

We have recently established that local exposure to a 929.2 MHz electromagnetic near-field, used for cellular phones, does not promote rat liver carcinogenesis in a medium-term bioassay system. In the present study, a 1.439 GHz electromagnetic near-field (EMF), another microwave band employed for cellular phones in Japan, was similarly investigated. Time division multiple access (TDMA) signals for the Personal Digital Cellular (PDC) Japanese cellular telephone standard system were directed to rats through a quarter-wavelength monopole antenna. Numerical dosimetry showed that the peak SARs within the liver were 1.91-0.937 W/kg, while the whole-body average specific absorption rates (SARs) were 0.680-0.453 W/kg, when the time-averaged antenna radiation power was 0.33 W. Exposure was for 90 min a day, 5 days a week, over 6 weeks, to male F344 rats given a single dose of diethylnitrosamine (200 mg/kg, i.p.) 2 weeks previously. At week 3, all rats were subjected to a two-thirds partial hepatectomy. At week 8, the experiment was terminated and the animals were killed. Carcinogenic potential was scored by comparing the numbers and areas of the induced glutathione S-transferase placental form (GST-P)-positive foci in the livers of exposed (48) and sham-exposed rats (48). Despite increased serum levels of corticosterone, adrenocorticotropic hormone (ACTH) and melatonin, the numbers and the areas of GST-P-positive foci were not significantly altered by the exposure. These findings clearly indicated that local body exposure to a 1.439 GHz EMF, as in the case of a 929.2 MHz field, has no promoting effect on rat liver carcinogenesis in the present model.

Effects of empagliflozin on progression of chronic kidney disease: a prespecified secondary analysis from the empa-kidney trial

BACKGROUND

Sodium-glucose co-transporter-2 (SGLT2) inhibitors reduce progression of chronic kidney disease and the risk of cardiovascular morbidity and mortality in a wide range of patients. However, their effects on kidney disease progression in some patients with chronic kidney disease are unclear because few clinical kidney outcomes occurred among such patients in the completed trials. In particular, some guidelines stratify their level of recommendation about who should be treated with SGLT2 inhibitors based on diabetes status and albuminuria. We aimed to assess the effects of empagliflozin on progression of chronic kidney disease both overall and among specific types of participants in the EMPA-KIDNEY trial.

METHODS

EMPA-KIDNEY, a randomised, controlled, phase 3 trial, was conducted at 241 centres in eight countries (Canada, China, Germany, Italy, Japan, Malaysia, the UK, and the USA), and included individuals aged 18 years or older with an estimated glomerular filtration rate (eGFR) of 20 to less than 45 mL/min per 1·73 m2, or with an eGFR of 45 to less than 90 mL/min per 1·73 m2 with a urinary albumin-to-creatinine ratio (uACR) of 200 mg/g or higher. We explored the effects of 10 mg oral empagliflozin once daily versus placebo on the annualised rate of change in estimated glomerular filtration rate (eGFR slope), a tertiary outcome. We studied the acute slope (from randomisation to 2 months) and chronic slope (from 2 months onwards) separately, using shared parameter models to estimate the latter. Analyses were done in all randomly assigned participants by intention to treat. EMPA-KIDNEY is registered at ClinicalTrials.gov, NCT03594110.

FINDINGS

Between May 15, 2019, and April 16, 2021, 6609 participants were randomly assigned and then followed up for a median of 2·0 years (IQR 1·5-2·4). Prespecified subgroups of eGFR included 2282 (34·5%) participants with an eGFR of less than 30 mL/min per 1·73 m2, 2928 (44·3%) with an eGFR of 30 to less than 45 mL/min per 1·73 m2, and 1399 (21·2%) with an eGFR 45 mL/min per 1·73 m2 or higher. Prespecified subgroups of uACR included 1328 (20·1%) with a uACR of less than 30 mg/g, 1864 (28·2%) with a uACR of 30 to 300 mg/g, and 3417 (51·7%) with a uACR of more than 300 mg/g. Overall, allocation to empagliflozin caused an acute 2·12 mL/min per 1·73 m2 (95% CI 1·83-2·41) reduction in eGFR, equivalent to a 6% (5-6) dip in the first 2 months. After this, it halved the chronic slope from -2·75 to -1·37 mL/min per 1·73 m2 per year (relative difference 50%, 95% CI 42-58). The absolute and relative benefits of empagliflozin on the magnitude of the chronic slope varied significantly depending on diabetes status and baseline levels of eGFR and uACR. In particular, the absolute difference in chronic slopes was lower in patients with lower baseline uACR, but because this group progressed more slowly than those with higher uACR, this translated to a larger relative difference in chronic slopes in this group (86% [36-136] reduction in the chronic slope among those with baseline uACR <30 mg/g compared with a 29% [19-38] reduction for those with baseline uACR ≥2000 mg/g; ptrend<0·0001).

INTERPRETATION

Empagliflozin slowed the rate of progression of chronic kidney disease among all types of participant in the EMPA-KIDNEY trial, including those with little albuminuria. Albuminuria alone should not be used to determine whether to treat with an SGLT2 inhibitor.

FUNDING

Boehringer Ingelheim and Eli Lilly.

Local mate competition with lethal male combat: effects of competitive asymmetry and information availability on a sex ratio game

We constructed a sex allocation model for local mate competition considering the asymmetry of competitive abilities among sons. This model assumes two females of a parasitoid wasp oviposit on the same host in sequential order. The evolutionarily stable strategy will be in either Stackelberg or Nash equilibrium, depending on whether the females can recognize their opponent's sex ratio or not, respectively. The Nash equilibrium predicts the second female produce more males than the first. If the second female is able to know and respond to the strategy of the first (a Stackelberg equilibrium), the first will decide an optimal sex ratio assuming that the second reply to it. Under such an assumption, our model predicts that not producing sons is adaptive for the second female when the sons she produces have low competitive ability. Males of parasitoid wasps Melittobia spp. are engaged in lethal male-male combat, indicating large asymmetry in mating success among sons. If females have the ability to recognize their opponent's sex ratio, our model suggests that the severe lethal male-male combat may be one factor explaining their extremely female-biased sex ratio that is unexplainable by pre-existent models.

Isolation, properties and a possible function of a water-soluble chlorophyll a/b-protein from brussels sprouts

A water-soluble Chl a/b-protein (CP673) was isolated and purified from Brussels sprouts (Brassica oleracea L. var. gemmifera DC). The protein had a molecular mass of 78 kDa and an isoelectric point of 4.7, consisted of three or four subunits of 22 kDa and was extremely heat-stable. Although CP673 contained about one Chl a per protein, the blue and red absorption bands of Chl a that consisted of three or four Chl a forms with different absorption maxima suggested that there are several different modes or sites of binding for Chl a. Chl a/b ratio of larger than 10 also indicated that Chl b is present only in a small fraction of CP673. The heterogeneity of CP673 in terms of composition and binding of Chl suggests that Chl is not an intrinsic component of the Chl-protein. Homology search showed that the N-terminal amino acid sequence of CP673 is highly homologous with that of a 22 kDa protein that accumulates in water-stressed leaves of two Brassicaceae plants, rapeseed and radish, but not with those of the light-harvesting Chl a/b-proteins of photosynthesis. A possible function of the water-soluble Chl-protein was discussed.

Teratoma in a feline unilateral cryptorchid testis

A 14- x 12- x 6-cm (approximately 600 g) cryptorchid testis was surgically removed with the omentum from the posterior portion of the abdominal cavity of a 2-year-old male American Shorthair cat. At the time of a previous orchiectomy, a unilateral testis had been discovered in the scrotum. The cut surface of the cryptorchid testis consisted of solid areas with many cysts containing fluid and spongy soft areas. Histologically, the cryptorchid testis was composed of neuroectodermal components and diffuse immature glial tissues, cystic, tubular, and papillary epithelial tissues, and immature cartilaginous tissues. A metastatic mass in the omentum had histologic structures similar to those of the cryptorchid testis. The present case was diagnosed as teratoma in a feline unilateral cryptorchid testis.

A "spare" compensates for the risk of destruction of the elongated penis of earwigs (Insecta: Dermaptera)

Male animals in several groups have multiple intromittent organs that outnumber the corresponding female gonopore. In Dermaptéra (earwigs), males of the family Anisolabididae have paired, elongated male intromittent organs (virgae), while females have a single sperm-storage organ (spermatheca). Several authors have assumed that one of the paired virgae is non-functional, because it points in the "wrong" direction. We investigated the mating success of handicapped males of Euborellia plebeja in which one of their paired virgae was removed experimentally. These handicapped males succeeded in inseminating a mate. Males with genital damage are found in the field, suggesting that the "spare" functions under natural conditions. Based on phylogenetic information on earwigs, we discuss possible evolutionary scenarios for this genital peculiarity.

In vivo function of immune inhibitory molecule B7-H4 in alloimmune responses

B7 ligands deliver both costimulatory and coinhibitory signals to the CD28 family of receptors on T lymphocytes, the balance between which determines the ultimate immune response. Although B7-H4, a recently discovered member of the B7 family, is known to negatively regulate T cell immunity in autoimmunity and cancer, its role in solid organ allograft rejection and tolerance has not been established. Targeting the B7-H4 molecule by a blocking antibody or use of B7-H4(-/-) mice as recipients of fully MHC-mismatched cardiac allografts did not affect graft survival. However, B7-H4 blockade resulted in accelerated allograft rejection in CD28-deficient recipients. B7-1/B7-2-double-deficient recipients are truly independent of CD28/CTLA-4:B7 signals and usually accept MHC-mismatched heart allografts. Blockade of B7-H4 in these mice also precipitated rejection, demonstrating regulatory function of this molecule independent of an intact CD28/CTLA-4:B7 costimulatory pathway. Accelerated allograft rejection was always accompanied by increased frequencies of alloreactive IFN-γ-, IL-4- and Granzyme B-producing splenocytes. Finally, intact recipient, but not donor, B7-H4 is essential for prolongation of allograft survival by blocking CD28/CTLA4:B7 pathway using CTLA4-Ig. These data are the first to provide evidence of the regulatory effects of B7-H4 in alloimmune responses in a murine model of solid organ transplantation.

Effects of light on the photosynthetic apparatus and a novel type of degradation of the photosystem I peripheral antenna complexes under darkness

A novel type of degradation of photosystem I peripheral antenna complexes has been observed in rice leaves under darkness in the present study. Photosynthesis, chlorophyll content, the chlorophyll a/b ratio, and relative amounts of ribulose-1,5-bisphosphate carboxylase/oxygenase decrease during dark treatment. The levels of photosystem II reaction-center complex and cytochrome f on the basis of units of chlorophyll also decline rapidly under darkness. In contrast, the levels of photosystem I reaction-center complex remain stable under darkness for six days. Low-temperature fluorescence emission spectra ascribed to photosystem I antennae clearly show a blue shift. A similar shift is also observed in the photosystem I complexes resolved with dodecyl maltoside-polyacrylamide gel electrophoresis. Moreover, polypeptide analysis of the thylakoids and photosystem I complexes isolated from the green gels shows that some polypeptides originating from photosystem I peripheral antenna complexes disappear during the dark treatment. A curve-fitting method also displays remarkable changes in the chlorophyll components between the light and dark treatments. It is likely that these results indicate the disconnection/disassembly of the photosystem I antenna as well as the photosystem II complexes induced by dark treatment. Moreover, these findings also imply the existence of different degradation mechanisms for the photosystem I and II complexes.

Endothelin-1 induces contraction of bile canaliculi in isolated rat hepatocytes

To examine the effects of endothelin-1(ET-1) on bile canalicular contractions, rat hepatocytes were isolated by a collagenase perfusion technique and cultured on p-n-p-vinylbenzenyl-D-lactonamide-coated dishes under serum-free conditions. The frequency of bile canalicular contractions was expressed by the percentage of the number of bile canaliculi(BC) that contracted compared to the total number of BC observed for 10 min. The treatment of the hepatocytes with ET-1 increased the frequency of contraction of the BC in a dose-dependent manner up to 2nM, control: 10.4%; 0.5nM ET-1:31.6%; 1nM ET-1:54.0%; 2nM ET-1:58.0%; 4nM ET-1:58.9%. The contraction, once it occurred, lasted for rest of the observation period. We also observed a transient increase in the concentration of intracellular Ca in cultured hepatocytes upon the addition of ET-1 to the medium. These results suggest that ET-1 plays an important role in the mortility of BC in vivo.

Mouse popliteal lymph node assay for assessment of allergic and autoimmunity-inducing potentials of low-molecular-weight drugs

In the present collaborative study, popliteal lymph node (PLN) responses to penicillin G (an allergenic chemical), D-penicillamine (an autoimmunity-inducing chemical), and barbital (a negative reference chemical) were investigated in three different mouse strains by ten pharmaceutical companies. Two inbred mouse strains (BALB/c and A/J) and one outbred strain (ICR) were subcutaneously injected with saline solutions containing penicillin G (1.25, 2.5 and 5 mg/mouse), D-penicillamine (0.5, 1 and 2 mg/mouse), or barbital (2 mg/mouse) into one hind footpad and saline only was injected into the contralateral footpad. PLN cellularity indices were determined on day 7. In the three strains tested, the penicillin G and D-penicillamine injections resulted in approximately dose-dependent responses. In contrast, barbital failed to generate a significant PLN reaction. In the typical data from one of the participating laboratories, the PLN responses of A/J, BALB/c, and ICR to penicillin G were high, intermediate and low, respectively, while their PLN responses to D-penicillamine were all high. Some variation in PLN cellularity indices was observed among the participating laboratories, but reproducibility of the popliteal lymph node assay (PLNA) evaluation was partly confirmed. Although the appropriate selection of mouse strains and drug dosage levels has to be considered, these results suggest that the PLNA may be an appropriate screening system for prediction of the allergic or autoimmunity-inducing potentials of low-molecular-weight drugs.

Venous malformation in the posterior fossa: guidelines for treatment

Venous malformations in the posterior fossa are relatively rare. Although the introduction of CT and MRI has made them easier to detect, their treatment is still controversial. Based on our experience with six patients and a review of the literature, we have tried to establish guidelines for their treatment. Since they have a benign natural course and may provide venous drainage in the posterior fossa, venous malformation found incidentally, unruptured venous malformation with nonhaemorrhagic complications, and those accompanied by small intracerebellar haematoma of less than 2 cm in diameter due to their rupture, should be treated conservatively unless they are associated with a coexistent malformation. Venous malformation with intracerebellar haematoma larger than 3 cm or reexpansion of the haematoma due to rebleeding should be treated surgically by evacuation of the haematoma. Resection of venous malformations in the posterior fossa should be restricted to cases in which the malformation is small and does not serve as a functional venous drainage route. Radiation therapy, including the "gamma-knife", may be a treatment of choice in the future.

Possible atavisms of genitalia in two species of earwig (Dermaptera), Proreus simulans (Chelisochidae) and Euborellia plebeja (Anisolabididae)

Male and female genitalia generally show a rapid evolutionary rate, which raises the problems related to homologization and the determination of the polarities of evolutionary changes. In earwigs (Dermaptera), multiple or branched female sperm-storage organs (spermathecae) have been reported for members of the Karschiellidae, Pygidicranidae, and Diplatyidae, collectively termed the "basal" Dermaptera. Whether the complicated spermathecae represent a plesiomorphy or an apomorphy has not been resolved. Here I report the occurrence of multiple or branched spermathecae in gamma-irradiated samples of two earwig species, Euborellia plebeja (Dohrn, 1863) (Anisolabididae) and Proreus simulans (Stål, 1860) (Chelisochidae), which belong to the "higher" Dermaptera (Apachyidae, Labiduridae, Anisolabididae, Spongiphoridae, Chelisochidae, and Forficulidae). Females belonging to the higher Dermaptera normally have a single-unbranched spermatheca. I discuss examples of possible atavisms in relation to the evolutionary pathways of spermathecal morphology. Possible atavisms in the number of male organs for sperm transfer (virgae) are also reported.

Epstein-Barr virus-positive primary brain lymphoma complicating immunoblastic lymphadenopathy

A 54-year-old man developed primary brain lymphoma after immunosuppressive treatment for immunoblastic lymphadenopathy. The lymphoma cells were positive for the Epstein-Barr virus genome and had the surface phenotype of early B cells and a normal karyotype.

Inverse bifurcation problem, singular Wiener-Hopf equations, and mathematical models in ecology

A single-species population dynamics with dispersal in a spatially heterogeneous environment is modeled by a nonlinear reaction-diffusion equation with a potential term. To each nonlinear kinetics there corresponds a bifurcation curve that describes the relation between the growth rate and the central density of a steady-state population distribution. Our main concern is an inverse problem for this correspondence. The existence of nonlinear kinetics realizing a prescribed bifurcation curve is established. It is shown that the freedom of such kinetics is of degree finite and even, depending only on the heterogeneity of the environment, and conversely that any nonnegative even integer occurs as the degree of freedom in some environments. A discussion is also made on under what kind of environment the degree is equal to zero or is positive. The mathematical analysis involves the development of a general theory for singular multiplicative Wiener-Hopf integral equations.