Mathematical analysis of the asymptotic behavior of the Leslie population matrix model

LetL be a Leslie population matrix. Leslie (1945) and others have shown that the matrixL has a leading positive eigenvalueλ ₀ and that in general: (1) $$\mathop {\lim }\limits_{t \to \infty } \frac{{L^t X}}{{\lambda _0^t }} = \gamma X_{\lambda _0 } $$ whereX _{λ 0} is an eigenvector corresponding toλ ₀,X is any initial population vector, and γ is a scalar quantity detormined byX. In this article we generalize (1) exhaustively by removing the mild restrictions on the fertility rates which most writers impose. The result is an oscillatory limit of a kind first noted by Bernardelli (1941) and Lewis (1942) and described by Bernardelli as “population waves”. We calculate in terms ofλ ₀ and the entries of the matrixL the values of this oscillatory limit as well as its time-independent average over one period. This calculation includes as its leading special case the result of (1), confirming incidentally that γ is nonzero. To stabilize a population, the matrixL must be adjusted so thatλ ₀=1. The limits calculated for the oscillatory and non-oscillatory cases then have maximum significance since they represent the limiting population vectors. We discuss a simple scheme for accomplishing stanbilization which yields as a byproduct an easily accessible scalar measure ofL's tendency to promote population growth. The reciprocal of this measure is the familiar net reproduction rate.     

Random Leslie matrices in population dynamics

We generalize the concept of the population growth rate when a Leslie matrix has random elements (correlated or not), i.e., characterizing the disorder in the vital parameters. In general, we present a perturbative formalism to deal with linear non-negative random matrix difference equations, then the non-trivial effective eigenvalue of which defines the long-time asymptotic dynamics of the mean-value population vector state is presented as the effective growth rate. This effective eigenvalue is calculated from the smallest positive root of a secular polynomial. Analytical (exact and perturbative calculations) results are presented for several models of disorder. In particular, a 3 × 3 numerical example is applied to study the effective growth rate characterizing the long-time dynamics of a biological population model. The present analysis is a perturbative method for finding the effective growth rate in cases when the vital parameters may have negative covariances across populations.     

Metabolizable amino acids and energy requirements of Nellore and crossbred Angus × Nellore bulls fed rations of different crude protein concentrations

Growth rate of cattle depends on their genetic makeup and nutrient intake. Moreover, increased growth rate may lead to increased amino acid (AA) requirements. Therefore, we evaluated the AA content of the empty body and estimated the net AA and energy requirements of purebred and crossbred beef bulls fed rations of different dietary CP concentrations. We performed a comparative slaughter experiment with 24 Nellore and 24 Angus × Nellore (A × N) bulls (8 months; initial shrunk BW: Nellore = 208.0 ± 12.78 kg; A × N = 221.9 ± 14.16 kg). Eight bulls (four Nellore and four A × N) were designated as the reference group, eight bulls (four Nellore and four A × N) were fed to maintenance level and 32 bulls (16 Nellore and 16 A × N) were fed ad libitum. The 32 bulls fed ad libitum were distributed using a completely randomized design in a 2 × 3 factorial scheme with two genetic groups (Nellore or A × N) and three dietary CP contents (100, 120 or 140 g CP/kg DM), being four groups with five bulls and two groups with six bulls. The experimental period lasted for 224 days. There were no interactions (P ≥ 0.056) between the dietary CP contents and genetic groups for any of the response variables. The dietary CP contents did not affect (P ≥ 0.062) the AA content in the empty body (g/kg empty BW [EBW]), with exception for Tryptophan (P = 0.027, linear effect). The dietary CP contents did not affect (P ≥ 0.051) AA content in the empty body (g/100 g of CP), with exception for Alanine (P = 0.013) that responded quadratically to dietary CP increase. The equations to estimate the net Lysine (Lys) and Methionine (Met) requirements (g/100 g of CP) were: Lys = 5.1 × EBW^0.0594 and Met = 1.7 × EBW^0.0255. Metabolizable Lys and Met to metabolizable energy (ME) ratios decreased as bulls EBW increased. Also, the metabolizable protein to ME ratio decreased as bulls EBW increased. In conclusion, the present study provides useful information regarding net and metabolizable requirements of AA of purebred and crossbred beef bulls. In the future, after the validation of the equations, these results can be used to calculate the AA requirements for growth of purebred and crossbred beef bulls. Nevertheless, it is important to highlight that the small sample size was one limitation of this present experiment.     

Aggregation of Leslie matrix models with application to ten diverse animal species

Some grouping is necessary when constructing a Leslie matrix model because it involves discretizing a continuous process of births and deaths. The level of grouping is determined by the number of age classes and frequency of sampling. It is largely unknown what is lost or gained by using fewer age classes, and I address this question using aggregation theory. I derive an aggregator for a Leslie matrix model using weighted least squares, determine what properties an aggregated matrix inherits from the original matrix, evaluate aggregation error, and measure the influence of aggregation on asymptotic and transient behaviors. To gauge transient dynamics, I employ reactivity of the standardized Leslie matrix. I apply the aggregator to 10 Leslie models developed for animal populations drawn from a diverse set of species. Several properties are inherited by the aggregated matrix: (a) it is a Leslie matrix; (b) it is irreducible whenever the original matrix is irreducible; (c) it is primitive whenever the original matrix is primitive; and (d) its stable population growth rate and stable age distribution are consistent with those of the original matrix if the least squares weights are equal to the original stable age distribution. In the application, depending on the population modeled, when the least squares weights do not follow the stable age distribution, the stable population growth rate of the aggregated matrix may or may not be approximately consistent with that of the original matrix. Transient behavior is lost with high aggregation.     

Effects of grain type and conditioning temperature during pelleting on growth performance,ruminal fermentation,meat quality and blood metabolites of fattening lambs

Ruminants can tolerate moderate concentrations of dietary tannin, making it feasible to replace corn with sorghum in ruminant diets; however, conditioning temperature of pelleted total mixed ration (PTMR) greatly affects nutrient digestibility. The objective was to determine effects of grain type and conditioning temperature during pelleting on growth performance, ruminal fermentation, meat quality and blood metabolites of fattening lambs. This was a 2 × 3 factorial study, with corn and sorghum and three conditioning temperatures (65, 75 and 85 °C) in a randomized complete design, with 36 lambs (120 ± 10.2 d and 24.9 ± 3.3 kg) grouped by weight and randomly allocated. The resulting six PTMRs were referred to as 65-S, 75-S and 85-S for sorghum-based diets, and 65-C, 75-C and 85-C for corn-based diets, for low, medium and high pelleting temperatures, respectively. There was no grain type × conditioning temperature (Grain × Temp) interaction on growth performance and apparent nutrient digestibility. Furthermore, grain type did not affect DM intake (DMI), average daily gain (ADG) or feed conversion ratio (FCR) of fattening lambs. Pelleting at 75 °C improved ADG (P < 0.03) and FCR (P < 0.02) of fattening lambs compared to other temperatures. There was a Grain × Temp interaction (P < 0.01) on ruminal pH (lowest in lambs fed 75-S). There tended (P = 0.07) to be a Grain × Temp interaction for total volatile fatty acid (VFA), and there were Grain × Temp interactions for molar proportions of acetate (P < 0.04), butyrate (P < 0.03) and branch-chained VFA (P < 0.01). Lambs fed sorghum-based PTMR had greater molar proportion of propionate (P < 0.03) and lower acetate to propionate ratio (A:P, P < 0.04). Lambs fed sorghum-based PTMR had higher plasma concentrations of urea nitrogen (N) (P < 0.03), glucose (P < 0.01) and alkaline phosphatase (P < 0.05), whereas other blood metabolites were not affected by treatments. There were Grain × Temp (P < 0.03) interactions for color coordinates of longissimus and mid-gluteal muscle. Lambs fed sorghum-based PTMR had lower (P < 0.01) dressing percentage and meat quality than those fed corn-based PTMR. We concluded that sorghum can replace corn in lamb diets without compromising growth performance and feed efficiency; furthermore, feeding sorghum vs corn improved rumen fermentation, with reduced A:P ratio and enhanced N and glucose utilization. Finally, pelleting at 75 °C increased feeding value of either sorghum- or corn-based PTMR for fattening lambs.     

Yearly variation in recruitment and its effect on population dynamics inYoldia notabilis (Mollusca: Bivalvia), analyzed using projection matrix model

Long-term variation in recruitment was estimated by constructing projection matrices for a marine bivalve, Yoldia notabilis, at two stations in Otsuchi Bay, northeastern Japan, and the effects of its variation on population dynamics were examined using a simple matrix model. The matrix model was developed from the Leslie matrix, in which the population growth rate λ was expressed as a function of recruitment rate r₀. The equilibrium recruitment rate r_s, or the recruitment rate required to maintain population at constant size (λ=1), was expressed by the reciprocal of the reproductive value of a newly recruited individual. The estimates of r_s for the field population were lower at the shallower station than at the deeper station, reflecting higher survivorship and fecundity. Past recruitment rate estimated both by the field samplings for 3 years and by the back-calculation from the current age structure for over 10 years showed large yearly variation, ranging between 0 and 58.6×10⁻⁴. The estimates were larger than r_s, and hence, large enough to increase population size (λ>1) only in approximately one-third of the estimated years. This suggests that the population has been maintained by occasional successful recruitment occurring once every few years.     

Vector and scalar fields in modeling of spatial variations of growth rates within plant organs

The paper confronts a question having broad implication: if it is assumed that growth is symplastic and that the anticlinal and periclinal orientations of newly formed cell walls are maintained during growth, what can then be inferred about the field of the displacement velocity vectors, V, in a growing organ? An answer is approached by describing the V field by means of a scalar field, G, such that V = μ grad G, where μ is a simple function of G. The G field can be fully specified if the pattern of displacement lines and magnitudes of V along one displacement line are known. This may, for example, be along the axis or along a meridional line on the surface. The V field allows calculations of linear, areal and volume growth rate patterns in space as well as expansions in time.     

Differences in daily stem size variation and growth in two hybrid eucalypt clones

Understanding daily stem size variation is important as the net increment of a forest stand is ultimately determined by the accumulation of daily increment events. In this study, measurements of stem size at high spatial and temporal resolution were made using two commercial hybrid Eucalyptus clones [E. grandis × urophylla (GU) and E. grandis × camaldulensis (GC)] over a period of more than 3.5 years in order to better understand how daily stem growth is effected by variations in environmental conditions. It was evident that GU had fewer days on which net growth occurred than GC. However, when growth did occur, GU grew for longer each day and at a higher rate than GC. Thus, it still had an overall larger net stem increment during the study period. The GU clone had a markedly intermittent pattern of growth, such that growth essentially ceased under drought conditions, but responded rapidly when water became available. This confirms other findings that E. grandis × urophylla is more susceptible to drought stress than E. grandis × camaldulensis, but emphasizes that a strategy of “rapid response” when environmental conditions become temporarily non-limiting is a good one in terms of net increment at sites such as in this study.

Energy and protein requirements of Holstein × Gyr crossbred heifers

Nutrient requirements in cattle are dependent on physiological stage, breed and environmental conditions. In Holstein × Gyr crossbred dairy heifers, the lack of data remains a limiting factor for estimating energy and protein requirements. Thus, we aimed to estimate the energy and protein requirements of Holstein × Gyr crossbred heifers raised under tropical conditions. Twenty-two crossbred (½ Holstein × ½ Gyr) heifers with an average initial BW of 102.2 ± 3.4 kg and 3 to 4 months of age were used. To estimate requirements, the comparative slaughter technique was used: four animals were assigned to the reference group, slaughtered at the beginning of the experiment to estimate the initial empty BW (EBW) and composition of the animals that remained in the experiment. The remaining animals were randomized into three treatments based on targeted rates of BW gain: high (1.0 kg/day), low (0.5 kg/day) and close to maintenance (0.1 kg/day). At the end of the experiment, all animals were slaughtered to determine EBW, empty body gain (EBG) and body energy and protein contents. The linear regression parameters were estimated using PROC MIXED of SAS (version 9.4). Estimates of the parameters of non-linear regressions were adjusted through PROC NLIN of SAS using the Gauss–Newton method for parameter fit. The net requirements of energy for maintenance (NE_m) and metabolizable energy for maintenance (ME_m) were 0.303 and 0.469 MJ/EBW^0.75 per day, respectively. The efficiency of use of ME_m was 64.5%. The estimated equation to predict the net energy requirement for gain (NE_g) was: NE_g (MJ/day) = 0.299 × EBW^0.75 × EBG^0.601. The efficiency of use of ME for gain (k_g) was 30.7%. The requirement of metabolizable protein for maintenance was 3.52 g/EBW^0.75 per day. The equation to predict net protein requirement for gain (NP_g) was: NP_g (g/day) = 243.65 × EBW^−0.091 × EBG. The efficiency of use of metabolizable protein for gain (k) was 50.8%. We observed noteworthy differences when comparing to ME and protein requirements of Holstein × Gyr crossbred heifers with other systems. In addition, we also observed differences in estimates for NE_m, NE_g, NP_g, k_g and k. Therefore, we propose that the equations generated in the present study should be used to estimate energy and protein requirements for Holstein × Gyr crossbred dairy heifers raised in tropical conditions in the post-weaning phase up to 185 kg of BW.     

Lymphocytic Thyroiditis is Associated with Increased Number of Benign Cervical Nodes and Fewer Central Neck Compartment Metastatic Lymph Nodes in Patients with Differentiated Thyroid Cancer

Objective: Whether or not autoimmune thyroid disease influences the progression of differentiated thyroid cancer (DTC) remains controversial. Findings of previous studies are influenced by lead time bias and/or procedure bias selection. These biases can be reduced by studying a single-institution patient population that underwent a similar extent of surgical resection.Methods: From a cohort of 660 patients with DTC who underwent thyroidectomy, we retrospectively studied 357 patients who underwent total thyroidectomy and central compartment node dissection (CCND) for DTC between 2003 and 2013.Results: Forty-one percent (140/345) of study patients had lymphocytic thyroiditis (LT), and 30% (91/301) had serum positive for thyroglobulin antibody (TgAb). LT was reported in 78% of the TgAb-positive cases. Sixty percent (213/357) of cases had metastatic thyroid carcinoma in 1 or more neck lymph nodes (55% [198/357] central compartment, and 22% [77/356] lateral compartment). Patients with LT had fewer metastatic cervical lymph nodes than those with no LT (2.7 ± 4.7 vs 3.5 ± 4.8, respectively, P = .0285). Patients with positive TgAb and thyroiditis had a larger number of benign cervical lymph nodes removed than those with negative TgAb or no LT. No significant difference was observed in age, tumor size, multifocality, extrathyroidal extension, vascular invasion, or frequency of cervical lymph node metastasis between TgAb-negative and -positive cases or between cases with and without LT.Conclusion: Lymphocytic thyroiditis is associated with fewer central neck compartment metastatic lymph nodes and a larger number of excised reactive benign cervical lymph nodes. Whether this association indicates a protective role of thyroid autoimmunity in lymph node spreading remains unclear.Abbreviations:CCND = central compartment node dissectionDTC = differentiated thyroid cancerHT = Hashimoto thyroiditisLT = lymphocytic thyroiditisTgAb = thyroglobulin antibodyTPO = thyroid peroxidase     

Comparison of nonlinear models to describe the feather growth and development curve in yellow-feathered chickens

Feathers play a critical role in thermoregulation and directly influence poultry production. Poor feathering adversely affects living appearance and carcass quality, thus reducing profits. However, producers tend to ignore the importance of feather development and do not know the laws of feather growth and development. The objective of this study was to fit growth curves to describe the growth and development of feathers in yellow-feathered broilers during the embryonic and posthatching periods using different nonlinear functions (Gompertz, logistic and Bertalanffy). Feather mass and length were determined during the embryonic development and posthatching stages to identify which growth model most accurately described the feather growth pattern. The results showed that chick embryos began to grow feathers at approximately embryonic (E) day 10, and the feathers grew rapidly from E13 to E17. There was little change from E17 to the day of hatching (DOH). During the embryonic period, the Gompertz function (Y = 798.48e^{−203 431exp(−0.87t)}, Akaike’s information criterion (AIC) = −0.950 × 10³, Bayesian information criterion (BIC) = −0.711 × 10³ and mean square error (MSE) = 559.308) provided the best fit for the feather growth curve compared with the other two functions. After hatching, feather mass and length changed little from the DOH to day (D) 14, increased rapidly from D21 to D91 and then grew slowly after D91. The first stage of feather molting occurred from 2 to 3 weeks of age when the down feathers were mostly shed and replaced with juvenile feathers, and the second stage occurred at approximately 13 to 15 weeks of age. The three nonlinear functions could overall fit the feather growth curve well, but the Bertalanffy model (Y = 116.88 × (1−0.86e^−0.02t)³, AIC = 1.065 × 10⁵, BIC = 1.077 × 10⁵ and MSE = 11.308) showed the highest degree of fit among the models. Therefore, the Gompertz model exhibited the best goodness of fit for the feather growth curve during the embryonic development, while the Bertalanffy model was the most suitable model due to its accurate ability to predict the growth and development of feathers during the growth period, which is an important commercial characteristic of yellow-feathered chickens.     

Machine Learning to Detect Alzheimer’s Disease from Circulating Non-coding RNAs

Blood-borne small non-coding (sncRNAs) are among the prominent candidates for blood-based diagnostic tests. Often, high-throughput approaches are applied to discover biomarker signatures. These have to be validated in larger cohorts and evaluated by adequate statistical learning approaches. Previously, we published high-throughput sequencing based microRNA (miRNA) signatures in Alzheimer’s disease (AD) patients in the United States (US) and Germany. Here, we determined abundance levels of 21 known circulating miRNAs in 465 individuals encompassing AD patients and controls by RT-qPCR. We computed models to assess the relation between miRNA expression and phenotypes, gender, age, or disease severity (Mini-Mental State Examination; MMSE). Of the 21 miRNAs, expression levels of 20 miRNAs were consistently de-regulated in the US and German cohorts. 18 miRNAs were significantly correlated with neurodegeneration (Benjamini-Hochberg adjusted P < 0.05) with highest significance for miR-532-5p (Benjamini-Hochberg adjusted P = 4.8 × 10⁻³⁰). Machine learning models reached an area under the curve (AUC) value of 87.6% in differentiating AD patients from controls. Further, ten miRNAs were significantly correlated with MMSE, in particular miR-26a/26b-5p (adjusted P = 0.0002). Interestingly, the miRNAs with lower abundance in AD were enriched in monocytes and T-helper cells, while those up-regulated in AD were enriched in serum, exosomes, cytotoxic t-cells, and B-cells. Our study represents the next important step in translational research for a miRNA-based AD test.     

Compartmental models and their application : Keith Godfrey,Academic Press,London, 1983, xiv+293 pp., $50.00

For the Leslie-matrix population projection model, a simple auxiliary quantity, termed the single-year reduced growth factor, is introduced and used to deduce bounds for the growth factor of the model, i.e. the dominant eigenvalue of the Leslie matrix involved. Extension to the case of internal survival in the age groups and reformulation of parameter sensitivity are briefly discussed.     

Medical Treatment Landscape for Active Acromegaly in A Pituitary Center in Israel

Objective: To evaluate current real-life experience with medical treatment for active acromegaly in a large cohort.Methods: Data on demographic parameters, blood tests, imaging studies, and treatments were extracted from the medical records.Results: The cohort included 87 patients (43 male) with active acromegaly. The mean age at diagnosis was 40.2 ± 11.4 years, and the mean duration of follow-up was 7.9 ± 5.8 years. Seventy patients presented with a macroadenoma. Mean baseline insulin growth factor 1 (IGF-1) (n = 67) was 3.2 ± 1.9 × upper limit of normal (ULN). Surgery and radiotherapy were performed in 75 and 10 patients, respectively. Currently, 38 subjects receive somatostatin analogues, pegvisomant as a monotherapy is given to 8 patients, pasireotide is given to 17 patients, cabegoline to 4 patients, estrogen to 2 females, and SSAs combined with pegvisomant to 10 patients. Eight patients are not being actively treated, including 4 following radiotherapy. Good biochemical control (IGF-1 <1.3 × ULN) was achieved in 76 patients (87%), and 11 patients (13%) are currently uncontrolled (IGF-1 >1.3 × ULN). Seventy-eight percent of controlled patients are being given 1 medication; 11% are on combination therapy; 4 patients are well controlled after radiotherapy and 2 are partially controlled without any treatment. The main adverse effects of treatment were diabetes mellitus in 7 patients (on pasireotide) and symptomatic cholelithiasis in 5 patients.Conclusion: Active acromegaly can be controlled medically in most patients, with a low rate of adverse effects. This study displays the characteristic variety of treatment options available for active acromegaly.     

Population Genetics of SARS-CoV-2: Disentangling Effects of Sampling Bias and Infection Clusters

A novel RNA virus, the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), is responsible for the ongoing outbreak of coronavirus disease 2019 (COVID-19). Population genetic analysis could be useful for investigating the origin and evolutionary dynamics of COVID-19. However, due to extensive sampling bias and existence of infection clusters during the epidemic spread, direct applications of existing approaches can lead to biased parameter estimations and data misinterpretation. In this study, we first present robust estimator for the time to the most recent common ancestor (TMRCA) and the mutation rate, and then apply the approach to analyze 12,909 genomic sequences of SARS-CoV-2. The mutation rate is inferred to be 8.69 × 10⁻⁴ per site per year with a 95% confidence interval (CI) of [8.61 × 10⁻⁴, 8.77 × 10⁻⁴], and the TMRCA of the samples inferred to be Nov 28, 2019 with a 95% CI of [Oct 20, 2019, Dec 9, 2019]. The results indicate that COVID-19 might originate earlier than and outside of Wuhan Seafood Market. We further demonstrate that genetic polymorphism patterns, including the enrichment of specific haplotypes and the temporal allele frequency trajectories generated from infection clusters, are similar to those caused by evolutionary forces such as natural selection. Our results show that population genetic methods need to be developed to efficiently detangle the effects of sampling bias and infection clusters to gain insights into the evolutionary mechanism of SARS-CoV-2. Software for implementing VirusMuT can be downloaded at https://bigd.big.ac.cn/biocode/tools/BT007081.     

Prodrugs of fumarate esters for the treatment of psoriasis and multiple sclerosis—a computational approach

Density functional theory (DFT) calculations at B3LYP/6-31 G (d,p) and B3LYP/6-311?+?G(d,p) levels for the substituted pyridine-catalyzed isomerization of monomethyl maleate revealed that isomerization proceeds via four steps, with the rate-limiting step being proton transfer from the substituted pyridinium ion to the C=C double bond in INT1. In addition, it was found that the isomerization rate (maleate to fumarate) is solvent dependent. Polar solvents, such as water, tend to accelerate the isomerization rate, whereas apolar solvents, such as chloroform, act to slow down the reaction. A linear correlation was obtained between the isomerization activation energy and the dielectric constant of the solvent. Furthermore, linearity was achieved when the activation energy was plotted against the pK _a value of the catalyst. Substituted-pyridine derivatives with high pK _a values were able to catalyze isomerization more efficiently than those with low pK _a values. The calculated relative rates for prodrugs 1–6 were: 1 (406.7), 2 (7.6?×?10⁶), 3 (1.0), 4 (20.7), 5 (13.5) and 6 (2.2?×?10³). This result indicates that isomerizations of prodrugs 1 and 3–5 are expected to be slow and that of prodrugs 2 and 6 are expected to be relatively fast. Hence, prodrugs 2 and 3–5 have the potential to be utilized as prodrugs for the slow release of monomethylfumarate in the treatment of psoriasis and multiple sclerosis.

Population growth with stochastic fluctuations in the life table

Monte Carlo simulations with the Leslie matrix and similar population models show that as the variance in survivorship or fecundity increases, the expected population growth rate decreases. This is attributed to Jensen's inequality with the observation that the rate of increase is a concave function of age-specific survivorship and fertility rates. Applications of this observation are advised for demographic studies, population simulation, optimal harvest strategies, and natural selection for variance in fertility and survivorship rates.     

Pituitary Gigantism - Experience of a Single Center from Western India

Objective: Limited data are available on pituitary gigantism, as it is a rare disorder. This study was carried out to assess the clinical, hormonal, and radiologic profiles and management outcomes of patients with pituitary gigantism.Methods: We conduced a retrospective analysis of 14 patients with pituitary gigantism who presented to a single tertiary care institute from 1990 to 2014.Results: Thirteen patients were male, and 1 was female. The mean age at diagnosis was 21.9 ± 6.1 years, with a mean lag period of 6.5 ± 5.6 years. The mean height SD score at the time of diagnosis was 3.2 ± 0.6. Symptoms of tumor mass effect were the chief presenting complaint in the majority (50%) of patients, while 2 patients were asymptomatic. Six patients had hyperprolactinemia. At presentation, the nadir PGGH (postglucose GH) and insulin-like growth factor (IGF 1)-ULN (× upper limit of normal) were 63.2 ± 94.9 ng/mL and 1.98 ± 0.5, respectively. All (except 1 with mild pituitary hyperplasia) had pituitary macroadenoma. Six patients had invasive pituitary adenoma. Transsphenoidal surgery (TSS) was the primary modality of treatment in 13/14 patients, and it achieved remission in 4/13 (30.76%) patients without recurrence over a median follow-up of 7 years. Post-TSS radiotherapy (RT) achieved remission in 3/5 (60%) patients over a median follow-up of 3.5 years. None of the patients received medical management at any point of time.Conclusion: Gigantism is more common in males, and remission can be achieved in the majority of the patients with the help of multimodality treatment (TSS and RT).Abbreviations: GH = growth hormone GHRH = growth hormone-releasing hormone IGF 1 = insulin-like growth factor 1 MEN 1 = multiple endocrine neoplasia type 1 PGGH = postglucose growth hormone RT = radiotherapy TSS = transsphenoidal surgery     

Ligation of EcoRI endonuclease-generated DNA fragments into linear and circular structures.

Double-stranded DNA fragments terminated at their 5′-ends by the singlestranded sequence pA-A-T-T-, generated by digestion of DNA with EcoRI restriction endonuclease, were ligated with Escherichia coli polynucleotide ligase under various conditions of temperature, concentration and time. The linear and circular products of ligation were separated by electrophoresis in agarose gel and quantitated by densitometry. The rate of ligation of (EcoRI-cleaved) simian virus (SV40) DNA at a concentration of 100 μg/ml increased from 0 °C to 5 °C to 10 °C (6-fold increase overall); raising the temperature to 15 °C did not further increase the rate of ligation. At the appropriate DNA concentrations, the predominant products of ligation are either linear concatemers that are integral multimers of the starting DNA fragment, or covalently closed circular structures of the monomeric DNA fragment. Ligating a mixture of two different length DNA fragments gives rise to all of the possible expected recombinant molecules.Linear or circular products of ligation were predicted by consideration of the total concentration of DNA termini, i, and the local concentration of one terminus in the neighborhood of the other on the same DNA molecule, j. The parameter j is a function of the length of a DNA molecule, providing this length is greater than the random coil segment of DNA. Experimentally it was found that circular structures are formed in significant amounts only under conditions when the value of j is several times greater than that of i. When j = i, equal amounts of linear and circular products would be expected, but most of the molecules were ligated into linear concatemers. No circular structure of a DNA fragment whose contour length l (6 × 10⁻² μm) is smaller than the random coil segment value b (7·17 × 10⁻² μm) was observed, while circular structures of the dimer of the same molecule (12 × 10⁻² μm) were detected.