Equivalence of truncated count mixture distributions and mixtures of truncated count distributions

This article is about modeling count data with zero truncation. A parametric count density family is considered. The truncated mixture of densities from this family is different from the mixture of truncated densities from the same family. Whereas the former model is more natural to formulate and to interpret, the latter model is theoretically easier to treat. It is shown that for any mixing distribution leading to a truncated mixture, a (usually different) mixing distribution can be found so that the associated mixture of truncated densities equals the truncated mixture, and vice versa. This implies that the likelihood surfaces for both situations agree, and in this sense both models are equivalent. Zero-truncated count data models are used frequently in the capture-recapture setting to estimate population size, and it can be shown that the two Horvitz-Thompson estimators, associated with the two models, agree. In particular, it is possible to achieve strong results for mixtures of truncated Poisson densities, including reliable, global construction of the unique NPMLE (nonparametric maximum likelihood estimator) of the mixing distribution, implying a unique estimator for the population size. The benefit of these results lies in the fact that it is valid to work with the mixture of truncated count densities, which is less appealing for the practitioner but theoretically easier. Mixtures of truncated count densities form a convex linear model, for which a developed theory exists, including global maximum likelihood theory as well as algorithmic approaches. Once the problem has been solved in this class, it might readily be transformed back to the original problem by means of an explicitly given mapping. Applications of these ideas are given, particularly in the case of the truncated Poisson family.     

Theoretical distribution of truncation lengths in incremental truncation libraries

Incremental truncation is a method for constructing libraries of every one base pair truncation of a segment of DNA. Incremental truncation libraries can be created using a time-dependent nuclease method or through the incorporation of alpha-phosphothioate dNTPs by PCR or by primer extension (THIO(pcr) truncation and THIO(extension) truncation, respectively). Libraries created by the fusion of two truncation libraries, known as ITCHY libraries, can be created using the above methods or by the incremental truncation-like method SHIPREC. Knowing and being able to tailor the distribution of truncations in incremental truncation, ITCHY and SHIPREC libraries would be beneficial for their use in protein engineering and other applications. However, the experimental determination of the distributions would require extensive, cost-prohibitive, DNA sequencing to obtain statistically relevant data. Instead, a theoretical prediction of the distributions was developed. Time-dependent incremental truncation libraries had the most uniform distribution of truncation lengths, but were biased against longer truncations. Essentially uniform distribution over the desired truncation range (from zero to N(max) base pairs) required that truncations be prepared up to at least 1.2-1.5 N(max). THIO(pcr) and THIO(extension) truncation libraries had a very nonuniform distribution of truncation lengths with a bias against longer truncations. Such nonuniformity could be significantly diminished by decreasing the incorporation rate of alphaS-dNTPs but at the expense of having a large fraction of the DNA truncated beyond the desired range or completely degraded. ITCHY libraries created using time-dependent truncation had the most uniform distribution of possible fusions and had the highest fraction of the library being parental-length fusions. However, the distribution of parental-length fusions was biased against fusions near the beginning/ends of genes unless the truncation libraries are prepared with a uniform distribution up to N(max). In contrast, SHIPREC libraries and THIO(pcr) ITCHY libraries, by the very nature of the nonuniform distributions of the truncated DNA, are ensured of having a uniform distribution of fusion points in parental-length fusions. This comes at the expense of having a smaller fraction of the library being parental-length fusions; however, this limitation can be overcome by performing size selection on the library.     

A first principles derivation of animal group size distributions

Several empirical studies have shown that the animal group size distribution of many species can be well fit by power laws with exponential truncation. A striking empirical result due to Niwa is that the exponent in these power laws is one and the truncation is determined by the average group size experienced by an individual. This distribution is known as the logarithmic distribution. In this paper we provide first principles derivation of the logarithmic distribution and other truncated power laws using a site-based merge and split framework. In particular, we investigate two such models. Firstly, we look at a model in which groups merge whenever they meet but split with a constant probability per time step. This generates a distribution similar, but not identical to the logarithmic distribution. Secondly, we propose a model, based on preferential attachment, that produces the logarithmic distribution exactly. Our derivation helps explain why logarithmic distributions are so widely observed in nature. The derivation also allows us to link splitting and joining behavior to the exponent and truncation parameters in power laws.     

An adaptive hierarchical test procedure for selecting safe and efficient treatments

We consider the situation where during a multiple treatment (dose) control comparison high doses are truncated because of lack of safety and low doses are truncated because of lack of efficacy, e.g., by decisions of a data safety monitoring committee in multiple interim looks. We investigate the properties of a hierarchical test procedure for the efficacy outcome in the set of doses carried on until the end of the trial, starting with the highest selected dose group to be compared with the placebo at the full level alpha. Left truncation, i.e., dropping doses in a sequence starting with the lowest dose, does not inflate the type I error rate. It is shown that right truncation does not inflate the type I error if efficacy and toxicity are positively related and dose selection is based on monotone functions of the safety data. A positive relation is given e.g. in the case where the efficacy and toxicity data are normally distributed with a positive pairwise correlation. A positive relation also applies if the probability for an adverse event is increasing with a normally distributed efficacy outcome. The properties of such truncation procedures are investigated by simulations. There is a conflict between achieving a small number of unsafely treated patients and a high power to detect safe and efficient doses. We also investigated a procedure to increase power where a reallocation of the sample size to the truncated treatments and the control remaining at the following stages is performed.     

Goodness of Fit of Probability Distributions for Sightings as Species Approach Extinction

Estimating the probability that a species is extinct and the timing of extinctions is useful in biological fields ranging from paleoecology to conservation biology. Various statistical methods have been introduced to infer the time of extinction and extinction probability from a series of individual sightings. There is little evidence, however, as to which of these models provide adequate fit to actual sighting records. We use L-moment diagrams and probability plot correlation coefficient (PPCC) hypothesis tests to evaluate the goodness of fit of various probabilistic models to sighting data collected for a set of North American and Hawaiian bird populations that have either gone extinct, or are suspected of having gone extinct, during the past 150 years. For our data, the uniform, truncated exponential, and generalized Pareto models performed moderately well, but the Weibull model performed poorly. Of the acceptable models, the uniform distribution performed best based on PPCC goodness of fit comparisons and sequential Bonferroni-type tests. Further analyses using field significance tests suggest that although the uniform distribution is the best of those considered, additional work remains to evaluate the truncated exponential model more fully. The methods we present here provide a framework for evaluating subsequent models.     

Nonparametric estimation of the cumulative incidences of competing risks under double truncation

Registry data typically report incident cases within a certain calendar time interval. Such interval sampling induces double truncation on the incidence times, which may result in an observational bias. In this paper, we introduce nonparametric estimation for the cumulative incidences of competing risks when the incidence time is doubly truncated. Two different estimators are proposed depending on whether the truncation limits are independent of the competing events or not. The asymptotic properties of the estimators are established, and their finite sample performance is investigated through simulations. For illustration purposes, the estimators are applied to childhood cancer registry data, where the target population is peculiarly defined conditional on future cancer development. Then, in our application, the cumulative incidences inform on the distribution by age of the different types of cancer.     

Testing goodness of fit of a uniform truncation model

Several goodness-of-fit tests of a lifetime distribution have been suggested in the literature; many take into account censoring and/or truncation of event times. In some contexts, a goodness-of-fit test for the truncation distribution is of interest. In particular, better estimates of the lifetime distribution can be obtained when knowledge of the truncation law is exploited. In cross-sectional sampling, for example, there are theoretical justifications for the assumption of a uniform truncation distribution, and several studies have used it to improve the efficiency of their survival estimates. The duality of lifetime and truncation in the absence of censoring enables methods for testing goodness of fit of the lifetime distribution to be used for testing goodness of fit of the truncation distribution. However, under random censoring, this duality does not hold and different tests are required. In this article, we introduce several goodness-of-fit tests for the truncation distribution and investigate their performance in the presence of censored event times using simulation. We demonstrate the use of our tests on two data sets.     

Maternal weight correction for alpha-fetoprotein: mathematical truncations revisited

The objective of this study was to determine if maternal weight correction as implemented 20 years ago for alpha-fetoprotein (AFP) testing is still appropriate, and what should be the weight cutoff for correction. AFP measurements from 44,951 biochemical screening cases from a single laboratory were divided by maternal weight, and separated into high >/=2.5 multiples of the median (MOM), and low <0.4 MOM. Without weight correction, AFP values are still skewed to lower levels of higher maternal weight (1.1% >/= 2.5 MOM). Using a truncated range to 200 lbs. (for women greater than 200), produces more uniform results than using an upper cutoff of 250 for truncation (2.7% vs 5.0% >/= 2.5 MOM). The same relationship is seen for "lows." Without correction, higher weight lowers AFP, correction to 250 lbs. overcompensates, and 200 lbs. gives the most even distribution, and, therefore, optimizes the distribution of highs.     

Emergence patterns in male butterflies: A hypothesis and a test

A game theoretical model is advanced to explain the emergence time schedule of male butterflies under temporal “apostatic” selection, so that males emerging on different days enjoy equal fitness in evolutionary equilibrium. The model predicts not only the position of the peak date but also the shape of the male emergence curve for any given female emergence schedule. Where the female emergence curve is smooth with one peak, a flight season can be divided into an earlier phase, when some males emerge every day, and a later phase in which no male emerges. The male emergence curve is truncated at the boundary of the phases. The position of the truncation point is determined by the difference between pre- and postemergence mortality. Preemergence mortality also determines the rate coefficient of the decrease in sex ratio through the season. The model is applied to a well-studied population of the butterfly Euphydryas editha. The male presence curve fits well, but no clear truncation exists in male emergence, and some males emerge earlier than predicted. Reasons for deviations are discussed.     

The Linear Discriminant Function: Sampling from the Truncated Normal Distribution

A study is made of the behaviour of the linear discriminant function in the classification of an observation when sampling from a truncated normal distribution. It is shown that the truncation proves ‘beneficial’ in that it reduces the error rates.     

Increased cross-bridge cycling kinetics after exchange of C-terminal truncated troponin I in skinned rat cardiac muscle

The precise mechanism of cardiac troponin I (cTnI) proteolysis in myocardial stunning is not fully understood. Accordingly, we determined the effect of cTnI C terminus truncation on chemo-mechanical transduction in isolated skinned rat trabeculae. Recombinant troponin complex (cTn), containing either mouse cTnI-(1-193) or human cTnI-(1-192) was exchanged into skinned cardiac trabeculae; Western blot analysis confirmed that 60-70% of the endogenous cTn was replaced by recombinant Tn. Incorporation of truncated cTnI induced significant reductions ( approximately 50%) in maximum force and cooperative activation as well as increases ( approximately 50%) in myofilament Ca(2+) sensitivity and tension cost. Similar results were obtained with either mouse or human truncated cTn. Presence of truncated cTnI increased maximum actin-activated S1 ATPase activity as well as its Ca(2+) sensitivity in vitro. Partial exchange (50%) for truncated cTnI resulted in similar reductions in maximum force and cooperativity; tension cost was increased in proportion to truncated cTnI content. In vitro, to determine the molecular mechanism responsible for the enhanced myofilament Ca(2+) sensitivity, we measured Ca(2+) binding to cTn as reported using a fluorescent probe. Incorporation of truncated cTnI did not affect Ca(2+) binding affinity to cTn alone. However, when cTn was incorporated into thin filaments, cTnI truncation induced a significant increase in Ca(2+) binding affinity to cTn. We conclude that cTnI truncation induces depressed myofilament function. Decreased cardiac function after ischemia/reperfusion injury may directly result, in part, from proteolytic degradation of cTnI, resulting in alterations in cross-bridge cycling kinetics.     

Role of caspase-3 in tau truncation at D421 is restricted in transgenic mouse models for tauopathies

Truncated tau is widely detected in Alzheimer's disease brain, and caspase-3 has been considered as a major executioner for tau truncation at aspartate421 (D421), according to its capability of cleaving recombinant tau in vitro . Here we investigated the relationship between D421 truncated tau and caspase-3 in two transgenic mouse models for tauopathies. In adult transgenic mice, activated caspase-3 could not be detected in neurons containing truncated tau, with the exception of a few glia-like cells or neurons in postnatal mice. Caspase-3 expression exhibited a dramatic decrease at the early development stage, and kept at constantly low levels during adult stages in both wild type and transgenic mice. On the other hand, co-incubating brain homogenates from adult tau transgenic mice and ethanol-treated postnatal mice promoted tau truncation at D421, which was mildly reduced by caspase inhibitor, but completely suppressed by phosphatase inhibitor, indicating that hyperphosphorylated tau becomes a poor substrate for truncation at D421. Taken together, our study shows that insufficient caspase-3 expression and hyperphosphorylated status of tau in the adult transgenic mouse brain restrict caspase-3 as an efficient enzyme for tau truncation in vivo . Clearly, there is a caspase-3 independent mechanism responsible for tau truncation at D421 in these models.     

The self-perpetuating tau truncation circle

Pathological truncations of human brain proteins represent the common feature of many neurodegenerative disorders including AD (Alzheimer's disease), Parkinson's disease and Huntington's disease. Protein truncations significantly change the structure and function of these proteins and thus can engender their pathological metamorphosis. We have shown previously that truncated forms of tau protein are contained in the core of the paired helical filaments that represent the main constituent of neurofibrillary pathology. Recently, we have identified truncated tau species of a different molecular signature. We have found that tau truncation is not produced by a random process, but rather by highly specific proteolytic cleavage and/or non-enzymatic fragmentation. In order to characterize the pathophysiology of AD-specific truncated tau species, we have used a transgenic rat model for AD expressing human truncated tau. Expression of the tau protein induces the formation of novel truncated tau species that originate from both transgenic human tau and endogenous rat tau proteins. Moreover, these truncated tau proteins are found exclusively in the misfolded fraction of tau, suggesting that they actively participate in the tau misfolding process. These findings corroborate further the idea that the appearance of truncated tau species starts a self-perpetuating cycle of further tau protein truncation leading to and accelerating tau misfolding and formation of neurofibrillary pathology.     

Posttranslational Maturation of the Prohormone Convertase SPC3 In Vitro

Abstract: The subtilisin-like prohormone convertase SPC3 is likely to play a role in the biosynthesis of a variety of biologically active peptides. SPC3 undergoes a series of posttranslational processing events during its biosynthesis. Multiple forms have been identified that show varying degrees of truncation at the carboxyl terminus. In this study we show that the 86-kDa form of recombinant SPC3 with an intact carboxyl terminus can undergo rapid carboxyl-terminus truncation to produce a 64-kDa form. We have defined the optimal conditions for carboxyl-terminus truncation in vitro. The carboxyl-terminus truncation reaction was less calcium sensitive, active over a broader pH range, and showed differences in inhibitor sensitivity compared with the enzymatic activities of full-length and truncated forms of SPC3 toward a fluorescent peptide substrate. Increases in enzymatic activity of 86-kDa SPC3 were also measured over a time frame consistent with conversion to the 64-kDa form. However, similar specific activities for both forms of the enzyme suggest such activity increases may not be due to carboxyl-terminus truncation. The different enzymatic properties of the major molecular forms of SPC3 highlight the importance of understanding the molecular events regulating carboxyl-terminal processing of this endoprotease.     

Tuning Plasmonic Properties of Truncated Gold Nanospheres by Coating

The influence of TiO₂ coating on resonant properties of gold nanoisland films deposited on silica substrates was studied numerically and in experiments. The model describing plasmonic properties of a metal truncated nanosphere placed on a substrate and covered by a thin dielectric layer has been developed. The model allows calculating a particle polarizability spectrum and, respectively, its surface plasmon resonance (SPR) wavelength for any given cover thickness, particle radius and truncation parameter, and dielectric functions of the particle, the substrate, the coating layer, and the surrounding medium. Dependence of the SPR position calculated for truncated gold nanospheres has coincided with the measured one for the gold nanoisland films covered with titania of different thicknesses. In the experiments, gold films with thickness of 5 nm were deposited on a silica glass substrate, annealed at 500 °C to form nanoislands of 20 nm in diameter, and covered with amorphous titania layers using atomic layer deposition technique. The resulting structures were characterized with scanning electron microscopy and optical absorption spectroscopy. The measured dependence of the SPR position on titania film thickness corresponded to the one calculated for truncated sphere-shaped nanoparticles with the truncation angle of ~50°. We demonstrated the possibility of tuning the SPR position within ~100 nm range by depositing to 30 nm thick titania layer.

     

Truncation of the cytoplasmic domain of the simian immunodeficiency virus envelope glycoprotein alters the conformation of the external domain.

We previously reported that truncation of the cytoplasmic domain of the macaque simian immunodeficiency virus SIVmac239 envelope glycoprotein enhanced its ability to induce cell fusion in a variety of cell lines. In the present study, we examined the expression of the full-length and truncated SIVmac239 envelope glycoprotein complex on cell surfaces. Using a membrane-impermeable reagent to biotinylate proteins on cell surfaces followed by immunoprecipitation, we found that under conditions in which the full-length TM protein could not be detected on the surfaces of CD4-positive or CD4-negative cell lines, the truncated TM protein was detected efficiently. In contrast, using a membrane-impermeable iodination reagent to label proteins on cell surfaces, we could detect both the full-length and truncated TM proteins. No difference between the full-length and truncated proteins was observed in the detection of the SU proteins in the biotinylation assay. Additionally, we used an assay in which SIV-specific antibodies are prebound to the native envelope proteins expressed on the cell surface and then the proteins are immunoprecipitated. Using this assay, we could not detect the truncated or full-length TM protein on the cell surface, whereas we could detect the SU subunits of both proteins. We also observed that the truncated TM protein formed more stable sodium dodecyl sulfate-resistant oligomers than the full-length TM protein did. These results indicate that truncation of the cytoplasmic domain of the SIVmac239 envelope glycoprotein affects the conformation of the external domain of the TM protein on the cell surface, even though the two proteins have no differences in the amino acid sequences of their external domains. This altered conformation could play a role in the enhanced fusion activity of the truncated SIV glycoprotein.     

A power equation for the sprint in speed skating.

An analysis of the start of the 500 m speed skating races during the 1988 Olympic Winter Games showed a remarkably high correlation between the acceleration of the skater in the first second of the sprint and the final time (r = -0.75). In this study a power equation is used to explain this high coefficient of correlation. The performance in speed skating is determined by the capability of external power production by the speed skater. This power is necessary to overcome the air and ice friction and to increase the kinetic energy of the skater. Numerical values of the power dissipated to air and ice friction, both dependent on speed, are obtained from ice friction and wind tunnel experiments. Using aerobic and anaerobic power production as measured during supra maximal bicycle tests of international-level speed skaters, a model of the kinetics of power production is obtained. Simulation of power production and power dissipation yields values of speed and acceleration and, finally, the performance time of the sprint during speed skating. The mean split time at 100 m and the final time at 500 m in these races, derived from simulation, were 10.57 s (+/- 0.31) and 37.82 s (+/- 0.96), respectively. The coefficient of correlation between the simulated 500 m times and the actual 500 m times was 0.90. From the results of this study it can be concluded that the distribution of the available anaerobic energy is an important factor in the short lasting events. For the same amount of anaerobic energy the better sprinters appear to be able to liberate considerably more energy at the onset of the race than skaters of lower performance level.     

Evidence for a recent mutation giving rise to a truncated copy of a cysteine proteinase gene in Leishmania pifanoi

We have previously identified and characterized two amastigote-specific cysteine proteinases of Leishmania pifanoi. The slightly different isoforms of the more abundant proteinase are coded by a gene family of approximately 20 gene copies, that contain a C-terminal extension characteristic of cysteine proteinases of trypanosomatids. In this gene family, we have detected a copy that codes for a truncated form of this proteinase, lacking the C-terminal extension. Interestingly, when the deletion of a nucleotide that creates a stop codon causing this truncation is disregarded, the translated sequence gives rise to a divergent C-terminal extension that has many conserved amino acids when compared to Leishmania and Trypanosome, suggesting that a recent mutation led to the truncation.