Statistical error in isothermal titration calorimetry: variance function estimation from generalized least squares

The method of generalized least squares (GLS) is used to assess the variance function for isothermal titration calorimetry (ITC) data collected for the 1:1 complexation of Ba(2+) with 18-crown-6 ether. In the GLS method, the least squares (LS) residuals from the data fit are themselves fitted to a variance function, with iterative adjustment of the weighting function in the data analysis to produce consistency. The data are treated in a pooled fashion, providing 321 fitted residuals from 35 data sets in the final analysis. Heteroscedasticity (nonconstant variance) is clearly indicated. Data error terms proportional to q(i) and q(i)/v are well defined statistically, where q(i) is the heat from the ith injection of titrant and v is the injected volume. The statistical significance of the variance function parameters is confirmed through Monte Carlo calculations that mimic the actual data set. For the data in question, which fall mostly in the range of q(i)=100-2000 microcal, the contributions to the data variance from the terms in q(i)(2) typically exceed the background constant term for q(i)>300 microcal and v<10 microl. Conversely, this means that in reactions with q(i) much less than this, heteroscedasticity is not a significant problem. Accordingly, in such cases the standard unweighted fitting procedures provide reliable results for the key parameters, K and DeltaH(degrees) and their statistical errors. These results also support an important earlier finding: in most ITC work on 1:1 binding processes, the optimal number of injections is 7-10, which is a factor of 3 smaller than the current norm. For high-q reactions, where weighting is needed for optimal LS analysis, tips are given for using the weighting option in the commercial software commonly employed to process ITC data.     

Two novel two-dimensional cluster polymers {[NMe4]2[MOS3Cu3(μ2-I)3]} (M=Mo,W): synthesis, crystal structure and interesting optical alternation from self-defocusing to self-focusing

Two novel cluster polymers {[NMe₄]₂[MoOS₃Cu₃(μ₂-I)₃]} 1 and {[NMe₄]₂[WOS₃Cu₃(μ₂-I)₃]}_n2 have been prepared. Bithiometalates MO₂S₂²⁻ (M=Mo,W), CuI, Me₄NBr and 4,4^′-trimethylene-dipyridine(tdp) reacted in solid state under low-heating temperature resulting in these two clusters. X-ray diffraction analysis reveals that the two polymers have the same two-dimensional network structures, which are formed from nest-shaped units along crystallographic b and c directions. I atoms take on a μ₂-fashion in bridging all the units. An alternative view along crystallographic a direction shows that they are layered features. Their third-order non-linear optical (NLO) properties were determined by Z-scan techniques. Both compounds possess very strong NLO refractive effect with n₂ value of −5.02 × 10⁻¹⁷ m²W⁻¹ for 1 and 1.20 × 10⁻¹⁶ m²W⁻¹ for 2, respectively. Different from routine work, there is an alternation from cluster polymer 1-2 in their optical properties: 1 shows self-defocusing behavior, while 2 gives self-focusing effect. Effective third-order NLO susceptibilities χ⁽³⁾ were calculated to be 4.90 × 10⁻¹¹ esu 1 and 1.48 × 10⁻⁹ esu 2. The corresponding hyperpolarizabilities γ are 3.39 × 10⁻²⁸ and 1.02 × 10⁻²⁶ esu, respectively.     

Anticipating epileptic seizures: from mathematics to clinical applications

The study of dynamical changes in the neural activity preceding an epileptic seizure allows the characterization of a preictal state several minutes prior to seizure onset. This opens new perspectives for studying the mechanisms of ictogenesis as well as for possible therapeutic interventions that represent a major breakthrough. In this review we present and discuss the results from our group in this domain using nonlinear analysis of brain signals, as well as its limitation and open questions.     

A framework for whole-cell mathematical modeling

The default framework for modeling biochemical processes is that of a constant-volume reactor operating under steady-state conditions. This is satisfactory for many applications, but not for modeling growth and division of cells. In this study, a whole-cell modeling framework is developed that assumes expanding volumes and a cell-division cycle. A spherical newborn cell is designed to grow in volume during the growth phase of the cycle. After 80% of the cycle period, the cell begins to divide by constricting about its equator, ultimately affording two spherical cells with total volume equal to twice that of the original. The cell is partitioned into two regions or volumes, namely the cytoplasm (Vcyt) and membrane (Vmem), with molecular components present in each. Both volumes change during the cell cycle; Vcyt changes in response to osmotic pressure changes as nutrients enter the cell from the environment, while Vmem changes in response to this osmotic pressure effect such that membrane thickness remains invariant. The two volumes change at different rates; in most cases, this imposes periodic or oscillatory behavior on all components within the cell. Since the framework itself rather than a particular set of reactions and components is responsible for this behavior, it should be possible to model various biochemical processes within it, affording stable periodic solutions without requiring that the biochemical process itself generates oscillations as an inherent feature. Given that these processes naturally occur in growing and dividing cells, it is reasonable to conclude that the dynamics of component concentrations will be more realistic than when modeled within constant-volume and/or steady-state frameworks. This approach is illustrated using a symbolic whole cell model.     

Non-photochemical quenching kinetics during the dark to light transition in relation to the formation of antheraxanthin and zeaxanthin

Nonlinear regression analysis (NLR) is applied to quantify the dynamic response of non-photochemical fluorescence quenching (NPQ) of Trifolium repens cv. Regal upon dark to light transition. Commonly, only steady-state levels of NPQ are evaluated, ignoring transient kinetics. Experimental NPQ kinetics are fitted best with a sum of two functions: a sigmoidal Hill function plus a transient logarithmic normal function. It is shown that not only steady-state level of NPQ, but also the speed at which steady state is reached, increased with light intensity. The question is raised which biological processes cause the induction of the components of NPQ kinetics. The NPQ kinetics are found to resemble the kinetics of antheraxanthin and zeaxanthin formation during a dark to light transition. Furthermore, both molecules are known to induce NPQ. The hypothesis is put forward that a transient phase of NPQ (0-2 min after transition) is dependent upon concentrations of antheraxanthin, whereas the saturating phase corresponds with the production of zeaxanthin. A mathematical model, based on the presented hypothesis, predicts the effect of increasing light intensity on concentrations of antheraxanthin and zeaxanthin which correspond with experimental results. Implications of the hypothesis are discussed as well as the role of NLR in evaluating chlorophyll a fluorescence kinetics.     

A fibril reinforced nonhomogeneous poroelastic model for articular cartilage: inhomogeneous response in unconfined compression

The depth dependence of material properties of articular cartilage, known as the zonal differences, is incorporated into a nonlinear fibril-reinforced poroelastic model developed previously in order to explore the significance of material heterogeneity in the mechanical behavior of cartilage. The material variations proposed are based on extensive observations. The collagen fibrils are modeled as a distinct constituent which reinforces the other two constituents representing proteoglycans and water. The Young's modulus and Poisson's ratio of the drained nonfibrillar matrix are so determined that the aggregate compressive modulus for confined geometry fits the experimental data. Three nonlinear factors are considered, i.e. the effect of finite deformation, the dependence of permeability on dilatation and the fibril stiffening with its tensile strain. Solutions are extracted using a finite element procedure to simulate unconfined compression tests. The features of the model are then demonstrated with an emphasis on the results obtainable only with a nonhomogeneous model, showing reasonable agreement with experiments. The model suggests mechanical behaviors significantly different from those revealed by homogeneous models: not only the depth variations of the strains which are expected by qualitative analyses, but also, for instance, the relaxation-time dependence of the axial strain which is normally not expected in a relaxation test. Therefore, such a nonhomogeneous model is necessary for better understanding of the mechanical behavior of cartilage.     

Estimating data transformations in nonlinear mixed effects models

A routine practice in the analysis of repeated measurement data is to represent individual responses by a mixed effects model on some transformed scale. For example, for pharmacokinetic, growth, and other data, both the response and the regression model are typically transformed to achieve approximate within-individual normality and constant variance on the new scale; however, the choice of transformation is often made subjectively or by default, with adoption of a standard choice such as the log. We propose a mixed effects framework based on the transform-both-sides model, where the transformation is represented by a monotone parametric function and is estimated from the data. For this model, we describe a practical fitting strategy based on approximation of the marginal likelihood. Inference is complicated by the fact that estimation of the transformation requires modification of the usual standard errors for estimators of fixed effects; however, we show that, under conditions relevant to common applications, this complication is asymptotically negligible, allowing straightforward implementation via standard software.     

Statistical evaluation of the regulatory guidelines for use of furosemide in race horses

The pharmacokinetic behavior of furosemide concentrations in performance horses is of great interest to the equine industry and equine researchers. Specifically, such information is useful for the evaluation of the existing guidelines in several racing jurisdictions and for the possible development of new guidelines for varying time periods after administration. We studied several approaches within the framework of nonlinear mixed effects models to increase the accuracy of evaluating these guidelines. Theoretical properties of the proposed methods were examined and the variances of the resulting estimators compared. Their numerical performances were further evaluated through simulations. Finally, we applied these methods to a furosemide concentration profile data set and used our findings to address certain important practical issues.     

Semiparametric maximum likelihood for measurement error model regression

This paper presents an EM algorithm for semiparametric likelihood analysis of linear, generalized linear, and nonlinear regression models with measurement errors in explanatory variables. A structural model is used in which probability distributions are specified for (a) the response and (b) the measurement error. A distribution is also assumed for the true explanatory variable but is left unspecified and is estimated by nonparametric maximum likelihood. For various types of extra information about the measurement error distribution, the proposed algorithm makes use of available routines that would be appropriate for likelihood analysis of (a) and (b) if the true x were available. Simulations suggest that the semiparametric maximum likelihood estimator retains a high degree of efficiency relative to the structural maximum likelihood estimator based on correct distributional assumptions and can outperform maximum likelihood based on an incorrect distributional assumption. The approach is illustrated on three examples with a variety of structures and types of extra information about the measurement error distribution.