Maximum likelihood method for the analysis of time-resolved fluorescence decay curves

The usefulness of fluorescence techniques for the study of macromolecular structure and dynamics depends on the accuracy and sensitivity of the methods used for data analysis. Many methods for data analysis have been proposed and used, but little attention has been paid to the maximum likelihood method, generally known as the most powerful statistical method for parameter estimation. In this paper we study the properties and behavior of maximum likelihood estimates by using simulated fluorescence intensity decay data. We show that the maximum likelihood method provides generally more accurate estimates of lifetimes and fractions than does the standard least-squares approach especially when the lifetime ratios between individual components are small. Three novelties to the field of fluorescence decay analysis are also introduced and studied in this paper: a) discretization of the convolution integral based on the generalized integral mean value theorem: b) the likelihood ratio test as a tool to determine the number of exponential decay components in a given decay profile; and c) separability and detectability indices which provide measures on how accurately, a particular decay component can be detected. Based on the experience gained from this and from our previous study of the Padé-Laplace method, we make some recommendations on how the complex problem of deconvolution and parameter estimation of multiexponential functions might be approached in an experimental setting. Offprint requests to: F. G. Prendergast     

Fisheries bioeconomics: why is it so widely misunderstood?

Many fisheries management systems, even when based on apparently sound science, have failed to prevent severe overfishing. And even when successful in this sense, such systems have frequently resulted in a large degree of excess fishing capacity. The reason for these failures can often be found in a lack of consideration of the economic incentives affecting fishermen. Specifically, when forced to compete for a fixed total annual catch quota (TAC), fishermen are motivated to fish at high intensity, and to expand the fishing power of their vessels. Individual fishing quotas (IFQs) are being increasingly used as a method of altering economic incentives in a desirable way. IFQ systems, however, can also suffer severe shortcomings, unless substantial fees are extracted for the exclusive right to exploit a publicly owned resource. When combined with appropriate fees, or royalties, IFQs can indeed result in sustainable, profitable fisheries. There still remains the fundamental question of risk management, but this is also now beginning to be addressed. Thus there is now a strong hope for the future success of marine fisheries, at least within 200-mile coastal zones.     

In vivo temperature limitations of photosynthesis in Phaseolus vulgaris L

The purpose of this study was to evaluate the temperature response of photosynthesis in two common bean genotypes differing in crop yield when grown under warm conditions. The cultivar Nobre is sensitive to high temperatures, whereas Diplomata shows better crop yield under high temperatures. Plants were grown in a greenhouse prior to transferring to a controlled environment cabinet for the temperature treatments. In a first experiment, 30 days-old plants were subjected to a short exposure (1 day) at temperatures that varied from 9 °C to 39 °C. Diplomata had lower net CO₂ assimilation rate (A) at 15 °C and 21 °C, but higher from 27 °C to 39 °C. Photosynthetic parameters calculated from modeling the response of A to the intercellular CO₂ concentration suggested that the different temperature responses of the two genotypes are caused by different rates of diffusion of CO₂ to the assimilation site, not by differences in biochemical limitations of photosynthesis. While stomatal conductance (g_s) did not differ between the genotypes, mesophyll conductance (g_m) was slightly greater for Nobre at 15 °C, but much higher in Diplomata from 21 °C to 39 °C. In a second experiment, no difference was observed in biomass accumulation between the two genotypes after growth for 24 days under a 35/20 °C (day/night) regime. Hence, the differences in photosynthesis did not cause variation in plant growth at the vegetative stage. The differential genotypic response of g_m to temperature suggests that g_m might be an important limitation to photosynthesis in Nobre, the common bean genotype sensitive to elevated temperature. However, more studies are needed employing other methods for g_m evaluation to validate these results.     

Phylogenetic analysis using parsimony and likelihood methods

The assumptions underlying the maximum-parsimony (MP) method of phylogenetic tree reconstruction were intuitively examined by studying the way the method works. Computer simulations were performed to corroborate the intuitive examination. Parsimony appears to involve very stringent assumptions concerning the process of sequence evolution, such as constancy of substitution rates between nucleotides, constancy of rates across nucleotide sites, and equal branch lengths in the tree. For practical data analysis, the requirement of equal branch lengths means similar substitution rates among lineages (the existence of an approximate molecular clock), relatively long interior branches, and also few species in the data. However, a small amount of evolution is neither a necessary nor a sufficient requirement of the method. The difficulties involved in the application of current statistical estimation theory to tree reconstruction were discussed, and it was suggested that the approach proposed by Felsenstein (1981,J. Mol. Evol. 17: 368–376) for topology estimation, as well as its many variations and extensions, differs fundamentally from the maximum likelihood estimation of a conventional statistical parameter. Evidence was presented showing that the Felsenstein approach does not share the asymptotic efficiency of the maximum likelihood estimator of a statistical parameter. Computer simulations were performed to study the probability that MP recovers the true tree under a hierarchy of models of nucleotide substitution; its performance relative to the likelihood method was especially noted. The results appeared to support the intuitive examination of the assumptions underlying MP. When a simple model of nucleotide substitution was assumed to generate data, the probability that MP recovers the true topology could be as high as, or even higher than, that for the likelihood method. When the assumed model became more complex and realistic, e.g., when substitution rates were allowed to differ between nucleotides or across sites, the probability that MP recovers the true topology, and especially its performance relative to that of the likelihood method, generally deteriorates. As the complexity of the process of nucleotide substitution in real sequences is well recognized, the likelihood method appears preferable to parsimony. However, the development of a statistical methodology for the efficient estimation of the tree topology remains a difficult open problem.     

A maximum entropy method for inverting Laplace transforms of probability density functions

This papers presents a maximum entropy method for invertingLaplace transforms of density functions of positive random variables.The maximum entropy density is very flexible and can assumea variety of different shapes. Accurate approximations to thetrue density can be obtained even when only a few transformvalues are available. Numerical evidence is provided for gamma,lognormal, inverse Gaussian and Pareto distributions.     

Ordered multiple comparisons with the best and their applications to dose-response studies

This article considers the problem of comparing several treatments (dose levels, interventions, etc.) with the best, where the best treatment is unknown and the treatments are ordered in some sense. Order relations among treatments often occur quite naturally in practice. They may be ordered according to increasing risks, such as tolerability or safety problems with increasing dose levels in a dose-response study, for example. We tackle the problem of constructing a lower confidence bound for the smallest index of all treatments being at most marginally less effective than the (best) treatment having the largest effect. Such a bound ensures at confidence level 1 -alpha that all treatments with lower indices are relevantly less effective than the best competitor. We derive a multiple testing strategy that results in sharp confidence bounds. The proposed lower confidence bound is compared with those derived from other testing strategies. We further derive closed-form expressions for power and sample size calculations. Finally, we investigate several real data sets to illustrate various applications of our methods.     

Sustainability of exploited ecologically interdependent species

This paper examines the application of maximum sustainable yield (MSY) policy in ecosystem and indicates when the ecosystem based fisheries management approach is required for conservation purpose. To describe the possible impacts of applying global MSY policy in an ecosystem, we have considered both specialist and generalist prey–predator models with different fishing efforts. It is found that harvesting both prey and predator species in specialist prey–predator systems, to achieve global maximum sustainable total yield (MSTY) under independent efforts, will cause the extinction of the predator species. In contrast, the global MSTY may exist in a generalist prey–predator system. If global MSTY does not exist, then it is due to the extinction of predator species. Hence, the prey species never goes to extinction under independent efforts and this scenario is quite different from the one found under combined harvesting effort.     

Capture-recapture when time and behavioral response affect capture probabilities

We consider a capture-recapture model in which capture probabilities vary with time and with behavioral response. Two inference procedures are developed under the assumption that recapture probabilities bear a constant relationship to initial capture probabilities. These two procedures are the maximum likelihood method (both unconditional and conditional types are discussed) and an approach based on optimal estimating functions. The population size estimators derived from the two procedures are shown to be asymptotically equivalent when population size is large enough. The performance and relative merits of various population size estimators for finite cases are discussed. The bootstrap method is suggested for constructing a variance estimator and confidence interval. An example of the deer mouse analyzed in Otis et al. (1978, Wildlife Monographs 62, 93) is given for illustration.     

Maximum likelihood estimation of oncogenetic tree models

We present a new approach for modelling the dependences between genetic changes in human tumours. In solid tumours, data on genetic alterations are usually only available at a single point in time, allowing no direct insight into the sequential order of genetic events. In our approach, genetic tumour development and progression is assumed to follow a probabilistic tree model. We show how maximum likelihood estimation can be used to reconstruct a tree model for the dependences between genetic alterations in a given tumour type. We illustrate the use of the proposed method by applying it to cytogenetic data from 173 cases of clear cell renal cell carcinoma, arriving at a model for the karyotypic evolution of this tumour.     

The Performance of Goodness of Fit Tests for Logistic Regression with Discrete Covariates

This paper considers the distribution of previously proposed goodness of fit tests when some or all of the covariates are dichotomous variables. The simulations show that of the statistics suggested for testing fit only one appears suitable for use with discrete covariates. This statistic uses conditional maximum likelihood estimates and groups the estimated probabilities into groups of equal size or into groups based on the patterns of the covariates when these are few in number.     

A maximum likelihood approach to the detection of selection from a phylogeny

Summary A large amount of information is contained within the phylogentic relationships between species. In addition to their branching patterns it is also possible to examine other aspects of the biology of the species. The influence that deleterious selection might have is determined here. The likelihood of different phylogenies in the presence of selection is explored to determine the properties of such a likelihood surface. The calculation of likelihoods for a phylogeny in the presence and absence of selection, permits the application of a likelihood ratio test to search for selection. It is shown that even a single selected site can have a strong effect on the likelihood. The method is illustrated with an example fromDrosophila melanogaster and suggests that delerious selection may be acting on transposable elements.     

Determinate growth and the scaling of photosynthetic energy intake in the solitary coral Fungia concinna (Verrill)

For many marine invertebrates, the maximum size of an individual is influenced heavily by environmental factors and may be limited by energetic constraints. In this study, an energetic model developed originally for anemones was applied to the free-living scleractinian Fungia concinna (Verrill) from Moorea (French Polynesia) to test the hypothesis that energetic constraints limit the size of this solitary coral. The modified model assumed that photosynthesis was the primary source of metabolic energy, and that metabolic costs were represented by aerobic respiration; these sources and sinks of energy were compared using daily energy budgets that were analyzed using double logarithmic regressions of energy against coral size. With this approach, energy limitation is characterized by a scaling exponent for energetic cost (b_cost) that is larger than the scaling exponent for energy intake (b_intake). For the size range of F. concinna studied, b_intake = 0.73 ± 0.09 and b_cost = 0.46 ± 0.10, thereby demonstrating that large individuals accumulated an energetic surplus, even when the expenditure associated with host tissue and symbiont growth was included in the model. The surplus of energy that this coral acquires as it grows appears to be driven by the scaling of traits associated functionally with the scaling of respiration and photosynthesis. Specifically, tissue biomass displayed a strong positive allometry with respect to surface area (i.e., b > 1), and this constraint on surface area may be the mechanistic basis of the low scaling exponent for metabolic cost. In contrast, the capacity for autotrophy - defined indirectly as Symbiodinium population density and chlorophyll content - increased isometrically with surface area, and likely contributed to the higher scaling exponent for intake relative to cost. Our results suggest that growth in F. concinna is not limited strictly by energy, but instead maximum size must be determined by alternative physiological or ecological constraints.     

Nonparametric functional mapping of quantitative trait loci

Summary .  Functional mapping is a useful tool for mapping quantitative trait loci (QTL) that control dynamic traits. It incorporates mathematical aspects of biological processes into the mixture model-based likelihood setting for QTL mapping, thus increasing the power of QTL detection and the precision of parameter estimation. However, in many situations there is no obvious functional form and, in such cases, this strategy will not be optimal. Here we propose to use nonparametric function estimation, typically implemented with B-splines, to estimate the underlying functional form of phenotypic trajectories, and then construct a nonparametric test to find evidence of existing QTL. Using the representation of a nonparametric regression as a mixed model, the final test statistic is a likelihood ratio test. We consider two types of genetic maps: dense maps and general maps, and the power of nonparametric functional mapping is investigated through simulation studies and demonstrated by examples.     

PHYLOGENIES FROM RESTRICTION SITES: A MAXIMUM-LIKELIHOOD APPROACH

Restriction sites data can be analyzed by maximum likelihood to obtain estimates of phylogenies. The likelihood methods of Smouse and Li, who were able to compute likelihoods for up to four species under a simplified model of base change, can be extended numerically to deal with any number of species. The computational methods for doing so are outlined. The resulting algorithms are slow but take multiple gains and losses of restriction sites fully into account, unlike parsimony methods. They allow for the failure to observe potential sites that are absent from all species. Analysis of the five-species hominoid data of Ferris and coworkers confirms the pattern found by Smouse and Li with four species—that a chimpanzee-gorilla clade is favored, but not statistically significantly over other tree topologies. A large data set produced by computer simulation has also been analyzed to confirm that the method works properly. The methods used here do not allow for different rates of transitions and transversions. They can be extended to do so, but only at a cost of considerably slower computations. The present method is available in a computer program.     

A note on the estimation of allele frequencies

A new method of obtaining allele frequency estimates is described. The method may prove useful when maximum likelihood estimates are not available. No assumptions regarding the absence of alleles are required. Errors of estimates have not been obtained, but the process seems to converge to maximum likelihood.     

A Pinus cembra L. tree-ring record for late spring to late summer temperature in the Rhaetian Alps,Italy

Ongoing climate change strongly affects high-elevation environments in the European Alps, influencing the cryosphere and the biosphere and causing widespread retreat of glaciers and changes in biomes. Nevertheless, high-elevation areas often lack long meteorological series, and global datasets cannot represent local variations well. Thus, proxy data, such as tree rings, provide information on past climatic variations from these remote sites. Although maximum latewood density (MXD) chronologies provide better temperature information than those based on tree-ring width (TRW), MXD series from the European Alps are lacking. To derive high-quality temperature information for the Rhaetian Alps, Pinus cembra L. trees sampled at approximately 2000 m a.s.l. were used to build one MXD chronology spanning from 1647 to 2015. The MXD data were significantly and highly correlated with seasonal May-September mean temperatures. The MXD chronology showed a generally positive trend since the middle of the 19^th century, interrupted by short phases of climatic deterioration in the beginning of the 20^th century and in the 1970s, conforming with the temperature trends. Our results underline the potential for using Pinus cembra L. MXD to reconstruct mean temperature variations, especially during the onset and latter part of the growing season, providing additional information on parts of the growing season not inferred from TRW. Future studies on MXD for this species will increase the availability of temporal and spatial data, allowing detailed climate reconstructions.     

A concept of a general computer program for the iterative calculation of maximum likelihood estimators with application to qualitative dose response analyses

In this article we describe the construction of a general computer program for the iterative calculation of maximum likelihood estimators. The program is general in the sense that it allows the maximization of any given likelihood function. The user only has to write a subroutine LKLHD, in which the special likelihood function and their first and second derivatives will be calculated. This subroutine is an input parameter of the optimization program. This enables the user to employ one main program for the maximization of various likelihood functions. This advantage will be shown for the evaluation of qualitative dose response relationships (quantal assays: probit-, logit-analysis).