Model-free model elimination: A new step in the model-free dynamic analysis of NMR relaxation data

Model-free analysis is a technique commonly used within the field of NMR spectroscopy to extract atomic resolution, interpretable dynamic information on multiple timescales from the R ₁, R ₂, and steady state NOE. Model-free approaches employ two disparate areas of data analysis, the discipline of mathematical optimisation, specifically the minimisation of a χ² function, and the statistical field of model selection. By searching through a large number of model-free minimisations, which were setup using synthetic relaxation data whereby the true underlying dynamics is known, certain model-free models have been identified to, at times, fail. This has been characterised as either the internal correlation times, τ _e , τ _f , or τ _s , or the global correlation time parameter, local τ _m , heading towards infinity, the result being that the final parameter values are far from the true values. In a number of cases the minimised χ² value of the failed model is significantly lower than that of all other models and, hence, will be the model which is chosen by model selection techniques. If these models are not removed prior to model selection the final model-free results could be far from the truth. By implementing a series of empirical rules involving inequalities these models can be specifically isolated and removed. Model-free analysis should therefore consist of three distinct steps: model-free minimisation, model-free model elimination, and finally model-free model selection. Failure has also been identified to affect the individual Monte Carlo simulations used within error analysis. Each simulation involves an independent randomised relaxation data set and model-free minimisation, thus simulations suffer from exactly the same types of failure as model-free models. Therefore, to prevent these outliers from causing a significant overestimation of the errors the failed Monte Carlo simulations need to be culled prior to calculating the parameter standard deviations.     

Conformational and functional properties of hemoglobin in perturbed solvent: Kinetics of O2 release

We studied the kinetics of O₂ release by oxyhemoglobin caused by sodium dithionite, in the presence and in the absence of organic cosolvents (monohydric alcohols and formamide) at 10°C. This study was performed by using standard stopped-flow techniques coupled with microprocessor-based data acquisition. We have fitted the experimental data to a mathematical expression obtained on the basis of a two-state model that takes into account the kinetic heterogeneity between α- and β-chains and the presence of αβ-dimers in oxyhemoglobin solutions. Results indicate that the cosolvents mainly affect the allosteric parameter L, i.e., the T ? R conformational equilibrium of hemoglobin, leaving the intrinsic deoxygenation rates of both R and T states almost unaltered. The L values obtained in the present work are in excellent agreement with analogous values previously estimated from oxygen equilibrium measurements.     

Genetic noise control via protein oligomerization

Background

Biochemical equilibria are usually modeled iteratively: given one or a few fitted models, if there is a lack of fit or over fitting, a new model with additional or fewer parameters is then fitted, and the process is repeated. The problem with this approach is that different analysts can propose and select different models and thus extract different binding parameter estimates from the same data. An alternative is to first generate a comprehensive standardized list of plausible models, and to then fit them exhaustively, or semi-exhaustively.

Results

A framework is presented in which equilibriums are modeled as pairs (g, h) where g = 0 maps total reactant concentrations (system inputs) into free reactant concentrations (system states) which h then maps into expected values of measurements (system outputs). By letting dissociation constants K _d be either freely estimated, infinity, zero, or equal to other K _d , and by letting undamaged protein fractions be either freely estimated or 1, many g models are formed. A standard space of g models for ligand-induced protein dimerization equilibria is given. Coupled to an h model, the resulting (g, h) were fitted to dTTP induced R1 dimerization data (R1 is the large subunit of ribonucleotide reductase). Models with the fewest parameters were fitted first. Thereafter, upon fitting a batch, the next batch of models (with one more parameter) was fitted only if the current batch yielded a model that was better (based on the Akaike Information Criterion) than the best model in the previous batch (with one less parameter). Within batches models were fitted in parallel. This semi-exhaustive approach yielded the same best models as an exhaustive model space fit, but in approximately one-fifth the time.

Conclusion

Comprehensive model space based biochemical equilibrium model selection methods are realizable. Their significance to systems biology as mappings of data into mathematical models warrants their development.     

ANALYSIS OF THE KINETICS OF GRANULOSA CELL POPULATIONS IN THE MOUSE OVARY

The kinetics of granulosa cell populations in two types of follicles in ovaries of 28-day-old Bagg mice are investigated. The analysis includes estimations of mean values and standard deviations of the transit times (T_G1, T_S, T_G2 and T_C), the doubling time T_D, and the proliferative fraction p. First the percentage of labelled mitosis curve (PLM-curve) and the continuous labelling curve (CL-curve) are estimated. Then a hypothesis concerning the cell kinetics of the granulosa cells in the two follicle types is set up. The normal distribution is chosen to simulate the probability density functions of the transit times. On the basis of the hypothesis mathematical expressions for the PLM- and CL-curves are worked out. By fitting the calculated PLM-curve to the experimental one it is possible to estimate mean values and standard deviations of T_G1, T_S>, and T_G2. As a test of the hypothesis the CL-curve is calculated by means of the estimated parameter values and compared to the experimental one. The calculated PLM- and CL-curves are found to be in good agreement with the experimental data as far as both follicle types are concerned. It is concluded that the method is a useful procedure. The choice of a normal distribution does not imply a significant limitation of the method in these investigations. Moreover it is concluded that the hypothesis is plausible. This means, e.g., that the proliferative fraction is unity in the two follicle types and that there is no cell loss from the cell systems.     

On the relationship of anthranilic derivatives structure and the FXR (Farnesoid X receptor) agonist activity

Farnesoid X receptor (FXR) is a nuclear receptor related to lipid and glucose homeostasis and is considered an important molecular target to treatment of metabolic diseases as diabetes, dyslipidemia, and liver cancer. Nowadays, there are several FXR agonists reported in the literature and some of it in clinical trials for liver disorders. Herein, a compound series was employed to generate QSAR models to better understand the structural basis for FXR activation by anthranilic acid derivatives (AADs). Furthermore, here we evaluate the inclusion of the standard deviation (SD) of EC₅₀ values in QSAR models quality. Comparison between the use of experimental variance plus average values in model construction with the standard method of model generation that considers only the average values was performed. 2D and 3D QSAR models based on the AAD data set including SD values showed similar molecular interpretation maps and quality (Q²_LOO, Q²_(F2), and Q²_(F3)), when compared to models based only on average values. SD-based models revealed more accurate predictions for the set of test compounds, with lower mean absolute error indices as well as more residuals near zero. Additionally, the visual interpretation of different QSAR approaches agrees with experimental data, highlighting key elements for understanding the biological activity of AADs. The approach using standard deviation values may offer new possibilities for generating more accurate QSAR models based on available experimental data.     

Modelling of Ca2+-Activated Chloride Current in Tracheal Smooth Muscle Cells

Stimulation of airway myocytes by contractile agents such as acetylcholine (ACh) activates a Ca²⁺-activated Cl^– current (I_ClCa) which may play a key role in calcium homeostasis of airway myocytes and hence in airway reactivity. The aim of the present study was to model I_ClCa in airway smooth muscle cells using a computerised model previously designed for simulation of cardiac myocyte functioning. Modelling was based on a simple resistor-battery permeation model combined with multiple binding site activation by calcium. In order to validate the model, a combination of equations, used to mimic [Ca²⁺]_i response to ACh stimulation, were incorporated into the model. The results indicate that the model developed in this article accounts for experimental recordings and electrophysiological characteristics of this current in airway smooth muscle cells, with parameter values consistent with those calculated from experimental data. Such a model may thus be used to predict I_ClCa functioning, though additional experimental data from airway myocytes would be useful to more accurately determine some parameter values of the model.     

Distributed parameter identification for a label-structured cell population dynamics model using CFSE histogram time-series data

In this work we address the problem of the robust identification of unknown parameters of a cell population dynamics model from experimental data on the kinetics of cells labelled with a fluorescence marker defining the division age of the cell. The model is formulated by a first order hyperbolic PDE for the distribution of cells with respect to the structure variable x (or z) being the intensity level (or the log₁₀-transformed intensity level) of the marker. The parameters of the model are the rate functions of cell division, death, label decay and the label dilution factor. We develop a computational approach to the identification of the model parameters with a particular focus on the cell birth rate α(z) as a function of the marker intensity, assuming the other model parameters are scalars to be estimated. To solve the inverse problem numerically, we parameterize α(z) and apply a maximum likelihood approach. The parametrization is based on cubic Hermite splines defined on a coarse mesh with either equally spaced a priori fixed nodes or nodes to be determined in the parameter estimation procedure. Ill-posedness of the inverse problem is indicated by multiple minima. To treat the ill-posed problem, we apply Tikhonov regularization with the regularization parameter determined by the discrepancy principle. We show that the solution of the regularized parameter estimation problem is consistent with the data set with an accuracy within the noise level in the measurements.      

Storage and growth of neuroblastoma cells immobilized in calcium-alginate beads

Summary Mouse neuroblastoma cells (N18) were immobilized in calcium-alginate gel beads. Under standard culture conditions (37° C; 5% CO₂), cell growth was observed inside the beads. The number of cells increased threefold during 7 days of culture with cell division and differentiation visualized by electron microscopy. Cell properties maintained after short-term storage (2–3 days at 4° C) included: (i) properties of voltage-dependent ionic channels tested by patch-clamp electrophysiological techniques; (ii) expression of cell-adhesion membrane proteins tested by immunohistochemistry (iii) morphological differentiation obtained by depletion of foetal calf serum in culture medium. The advantages of such an immobilization technique as applied to neurone cells are discussed. Offprint requests to: M. Simonneau     

Prenatal viral infection leads to pyramidal cell atrophy and macrocephaly in adulthood: implications for genesis of autism and schizophrenia

We investigated the role of maternal exposure to human influenza virus (H1N1) in C57BL/6 mice on Day 9 of pregnancy on pyramidal and nonpyramidal cell density, pyramidal nuclear area, and overall brain size in Day 0 neonates and 14-week-old progeny and compared them to sham-infected cohorts. Pyramidal cell density increased significantly (p < 0.0038) by 170% in Day 0 infected mice vs. controls. Nonpyramidal cell density decreased by 33% in Day 0 infected progeny vs. controls albeit, nonsignificantly. Pyramidal cell nuclear size decreased significantly (p < 0.0465) by 29% in exposed newborn mice vs. controls. Fourteen-week-old exposed mice continued to show significant increases in both pyramidal and nonpyramidal cell density values vs. controls respectively (p < 0.0085 E1 (exposed group 1), p < 0.0279 E2 (exposed group 2) pyramidal cell density; p < 0.0092 E1, p < 0.0252 E2, nonpyramidal cell density). By the same token, pyramidal cell nuclear size exhibited 37–43% reductions when compared to control values; these were statistically significant vs. controls (p < 0.04 E1, p < 0.0259 E2). Brain and ventricular area measurements in adult exposed mice also showed significant increases and decreases respectively vs. controls. Ventricular brain ratios exhibited 38–50% decreases in exposed mice vs. controls. While the rate of pyramidal cell proliferation per unit area decreased from birth to adulthood in both control and exposed groups, nonpyramidal cell growth rate increased only in the exposed adult mice. These data show for the first time that prenatal exposure of pregnant mice on Day 9 of pregnancy to a sublethal intranasal administration of influenza virus has both short-term and long-lasting deleterious effects on developing brain structure in the progeny as evident by altered pyramidal and nonpyramidal cell density values; atrophy of pyramidal cells despite normal cell proliferation rate and final enlargement of brain. Moreover, abnormal corticogenesis is associated with development of abnormal behavior in the exposed adult mice.     

Modelling the flow of cytometric data obtained from unperturbed human tumour cell lines: Parameter fitting and comparison

In this paper we firstly present three alternative formulations of a mathematical model for human tumour cell lines unperturbed by cancer therapy. The model counts the number density of cells in each phase of the cell cycle over time where cells are differentiated by their DNA content. Data are available from the Auckland Cancer Society Research Centre, Auckland, New Zealand, in the form of DNA histograms or profiles from 11 different human tumour cell lines (i.e. in vitro) unperturbed by cancer therapy. We then apply one (computationally fast) formulation of the model and discover that although in general different combinations of parameter values give rise to very different DNA profiles it is possible that different combinations of parameter values give rise to virtually identical profiles. Experimental estimates of the rate of transition from the G ₁-phase (growth) to the S-phase (DNA synthesis) enable us to uniquely determine other model parameters of interest that give the least square error between the model and data. We finally apply our model to each of the 11 different cell lines and compare cell cycle phase transit times. Although the DNA histograms of each of the cell lines have similar shapes these cell lines have different combinations of transit times to each other, which could explain why they often react very differently when exposed to anti-cancer therapies during laboratory experiments. An understanding of the in vitro situation may give an insight into why some human cancer patients do not respond to cancer therapy. An erratum to this article is available at .     

Morphological alterations in newly born dentate gyrus granule cells that emerge after status epilepticus contribute to make them less excitable

Computer simulations of external current stimulations of dentate gyrus granule cells of rats with Status Epilepticus induced by pilocarpine and control rats were used to evaluate whether morphological differences alone between these cells have an impact on their electrophysiological behavior. The cell models were constructed using morphological information from tridimensional reconstructions with Neurolucida software. To evaluate the effect of morphology differences alone, ion channel conductances, densities and distributions over the dendritic trees of dentate gyrus granule cells were the same for all models. External simulated currents were injected in randomly chosen dendrites belonging to one of three different areas of dentate gyrus granule cell molecular layer: inner molecular layer, medial molecular layer and outer molecular layer. Somatic membrane potentials were recorded to determine firing frequencies and inter-spike intervals. The results show that morphologically altered granule cells from pilocarpine-induced epileptic rats are less excitable than control cells, especially when they are stimulated in the inner molecular layer, which is the target area for mossy fibers that sprout after pilocarpine-induced cell degeneration. This suggests that morphological alterations may act as a protective mechanism to allow dentate gyrus granule cells to cope with the increase of stimulation caused by mossy fiber sprouting.     

Richards's equation and nonlinear mixed models applied to avian growth: why use them?

Postnatal growth is an important life‐history trait that varies widely across avian species, and several equations with a sigmoidal shape have been used to model it. Classical three‐parameter models have an inflection point fixed at a percentage of the upper asymptote which could be an unrealistic assumption generating biased fits. The Richards model emerged as an interesting alternative because it includes an extra parameter that determines the location of the inflection point which can move freely along the growth curve. Recently, nonlinear mixed models (NLMM) have been used in modeling avian growth because these models can deal with a lack of independence among data as typically occurs with multiple measurements on the same individual or on groups of related individuals. Here, we evaluated the usefulness of von Bertalanffy, Gompertz, logistic, U₄ and Richards's equations modeling chick growth in the imperial shag Phalacrocorax atriceps. We modelled growth in commonly used morphological traits, including body mass, bill length, head length and tarsus length, and compared the performance of models by using NLMM. Estimated adult size, age at maximum growth and maximum growth rates markedly differed across models. Overall, the most consistent performance in estimated adult size was obtained by the Richards model that showed deviations from mean adult size within 5%. Based on AICc values, the Richards equation was the best model for all traits analyzed. For tarsus length, both Richards and U₄ models provided indistinguishable fits because the relative inflection value estimated from the Richards model was very close to that assumed by the U₄ model. Our results highlight the bias incurred by three‐parameter models when the assumed inflection placement deviates from that derived from data. Thus, the application of the Richards equation using the NLMM framework represents a flexible and powerful tool for the analysis of avian growth.     

Making the most of survey data: Incorporating age uncertainty when fitting growth parameters

Individual growth is an important parameter and is linked to a number of other biological processes. It is commonly modeled using the von Bertalanffy growth function (VBGF), which is regularly fitted to age data where the ages of the animals are not known exactly but are binned into yearly age groups, such as fish survey data. Current methods of fitting the VBGF to these data treat all the binned ages as the actual ages. We present a new VBGF model that combines data from multiple surveys and allows the actual age of an animal to be inferred. By fitting to survey data for Atlantic herring (Clupea harengus) and Atlantic cod (Gadus morhua), we compare our model with two other ways of combining data from multiple surveys but where the ages are as reported in the survey data. We use the fitted parameters as inputs into a yield‐per‐recruit model to see what would happen to advice given to management. We found that each of the ways of combining the data leads to different parameter estimates for the VBGF and advice for policymakers. Our model fitted to the data better than either of the other models and also reduced the uncertainty in the parameter estimates and models used to inform management. Our model is a robust way of fitting the VBGF and can be used to combine data from multiple sources. The model is general enough to fit other growth curves for any taxon when the age of individuals is binned into groups.     

Integration of Process-based Soil Respiration Models with Whole-Ecosystem CO2 Measurements

We integrated soil models with an established ecosystem process model (SIPNET, simplified photosynthesis and evapotranspiration model) to investigate the influence of soil processes on modelled values of soil CO₂ fluxes (R _Soil). Model parameters were determined from literature values and a data assimilation routine that used a 7-year record of the net ecosystem exchange of CO₂ and environmental variables collected at a high-elevation subalpine forest (the Niwot Ridge AmeriFlux site). These soil models were subsequently evaluated in how they estimated the seasonal contribution of R _Soil to total ecosystem respiration (TER) and the seasonal contribution of root respiration (R _Root) to R _Soil. Additionally, these soil models were compared to data assimilation output of linear models of soil heterotrophic respiration. Explicit modelling of root dynamics led to better agreement with literature values of the contribution of R _Soil to TER. Estimates of R _Soil/TER when root dynamics were considered ranged from 0.3 to 0.6; without modelling root biomass dynamics these values were 0.1–0.3. Hence, we conclude that modelling of root biomass dynamics is critically important to model the R _Soil/TER ratio correctly. When soil heterotrophic respiration was dependent on linear functions of temperature and moisture independent of soil carbon pool size, worse model-data fits were produced. Adding additional complexity to the soil pool marginally improved the model-data fit from the base model, but issues remained. The soil models were not successful in modelling R _Root/R _Soil. This is partially attributable to estimated turnover parameters of soil carbon pools not agreeing with expected values from literature and being poorly constrained by the parameter estimation routine. We conclude that net ecosystem exchange of CO₂ alone cannot constrain specific rhizospheric and microbial components of soil respiration. Reasons for this include inability of the data assimilation routine to constrain soil parameters using ecosystem CO₂ flux measurements and not considering the effect of other resource limitations (for example, nitrogen) on the microbe biomass. Future data assimilation studies with these models should include ecosystem-scale measurements of R _Soil in the parameter estimation routine and experimentally determine soil model parameters not constrained by the parameter estimation routine.     

Growth and mortality of larval Myctophum affine (Myctophidae,Teleostei)

The growth and mortality rates of Myctophum affine larvae were analysed based on samples collected during the austral summer and winter of 2002 from south‐eastern Brazilian waters. The larvae ranged in size from 2·75 to 14·00 mm standard length (L_S). Daily increment counts from 82 sagittal otoliths showed that the age of M. affine ranged from 2 to 28 days. Three models were applied to estimate the growth rate: linear regression, exponential model and Laird–Gompertz model. The exponential model best fitted the data, and L₀ values from exponential and Laird–Gompertz models were close to the smallest larva reported in the literature (c. 2·5 mm L_S). The average growth rate (0·33 mm day^?1) was intermediate among lanternfishes. The mortality rate (12%) during the larval period was below average compared with other marine fish species but similar to some epipelagic fishes that occur in the area.     

Reconciling the optimal and empirical approaches to modelling stomatal conductance

Models of vegetation function are widely used to predict the effects of climate change on carbon, water and nutrient cycles of terrestrial ecosystems, and their feedbacks to climate. Stomatal conductance, the process that governs plant water use and carbon uptake, is fundamental to such models. In this paper, we reconcile two long‐standing theories of stomatal conductance. The empirical approach, which is most commonly used in vegetation models, is phenomenological, based on experimental observations of stomatal behaviour in response to environmental conditions. The optimal approach is based on the theoretical argument that stomata should act to minimize the amount of water used per unit carbon gained. We reconcile these two approaches by showing that the theory of optimal stomatal conductance can be used to derive a model of stomatal conductance that is closely analogous to the empirical models. Consequently, we obtain a unified stomatal model which has a similar form to existing empirical models, but which now provides a theoretical interpretation for model parameter values. The key model parameter, g₁, is predicted to increase with growth temperature and with the marginal water cost of carbon gain. The new model is fitted to a range of datasets ranging from tropical to boreal trees. The parameter g₁ is shown to vary with growth temperature, as predicted, and also with plant functional type. The model is shown to correctly capture responses of stomatal conductance to changing atmospheric CO₂, and thus can be used to test for stomatal acclimation to elevated CO₂. The reconciliation of the optimal and empirical approaches to modelling stomatal conductance is important for global change biology because it provides a simple theoretical framework for analyzing, and simulating, the coupling between carbon and water cycles under environmental change.     

Molecular dynamics simulation of human interleukin-4: comparison with NMR data and effect of pH,counterions and force field on tertiary structure stability

The human protein interleukin-4 (IL-4) has been simulated at two different pH values 2 and 6, with different amounts of counterions present in the aqueous solution, and with two different force-field parameter sets using molecular dynamics simulation with the aim of validation of force field and simulation set-up by comparison to experimental nuclear magnetic resonance data, such as proton–proton nuclear Overhauser effect (NOE) distance bounds, ³ J(HN,HC_α) coupling constants and backbone N–H order parameters. Thirteen simulations varying in the length from 3 to 7 ns are compared.

At pH 6 both force-field parameter sets used do largely reproduce the NOE's and order parameters, the GROMOS 45A3 set slightly better than the GROMOS 53A6 set. ³ J values predicted from the simulation agree less well with experimental values. At pH 2 the protein unfolds, unless counterions are explicitly present in the system, but even then the agreement with experiment is worse than at pH 6. When simulating a highly charged protein, such as IL-4 at pH 2, the inclusion of counterions in the simulation seems mandatory.     

A kinetic model of Escherichia coli core metabolism satisfying multiple sets of mutant flux data

In contrast to stoichiometric-based models, the development of large-scale kinetic models of metabolism has been hindered by the challenge of identifying kinetic parameter values and kinetic rate laws applicable to a wide range of environmental and/or genetic perturbations. The recently introduced ensemble modeling (EM) procedure provides a promising remedy to address these challenges by decomposing metabolic reactions into elementary reaction steps and incorporating all phenotypic observations, upon perturbation, in its model parameterization scheme. Here, we present a kinetic model of Escherichia coli core metabolism that satisfies the fluxomic data for wild-type and seven mutant strains by making use of the EM concepts. This model encompasses 138 reactions, 93 metabolites and 60 substrate-level regulatory interactions accounting for glycolysis/gluconeogenesis, pentose phosphate pathway, TCA cycle, major pyruvate metabolism, anaplerotic reactions and a number of reactions in other parts of the metabolism. Parameterization is performed using a formal optimization approach that minimizes the discrepancies between model predictions and flux measurements. The predicted fluxes by the model are within the uncertainty range of experimental flux data for 78% of the reactions (with measured fluxes) for both the wild-type and seven mutant strains. The remaining flux predictions are mostly within three standard deviations of reported ranges. Converting the EM-based parameters into a Michaelis–Menten equivalent formalism revealed that 35% of K_m and 77% of k_cat parameters are within uncertainty range of the literature-reported values. The predicted metabolite concentrations by the model are also within uncertainty ranges of metabolomic data for 68% of the metabolites. A leave-one-out cross-validation test to evaluate the flux prediction performance of the model showed that metabolic fluxes for the mutants located in the proximity of mutations used for training the model can be predicted more accurately. The constructed model and the parameterization procedure presented in this study pave the way for the construction of larger-scale kinetic models with more narrowly distributed parameter values as new metabolomic/fluxomic data sets are becoming available for E. coli and other organisms.     

Deriving physical connectivity from neuronal morphology

 A model is presented that allows prediction of the probability for the formation of appositions between the axons and dendrites of any two neurons based only on their morphological statistics and relative separation. Statistics of axonal and dendritic morphologies of single neurons are obtained from 3D reconstructions of biocytin-filled cells, and a statistical representation of the same cell type is obtained by averaging across neurons according to the model. A simple mathematical formulation is applied to the axonal and dendritic statistical representations to yield the probability for close appositions. The model is validated by a mathematical proof and by comparison of predicted appositions made by layer 5 pyramidal neurons in the rat somatosensory cortex with real anatomical data. The model could be useful for studying microcircuit connectivity and for designing artificial neural networks. Received: 11 February 2002 / Accepted: 5 November 2002 / Published online: 20 February 2003 Correspondence to: H. Markram (e-mail: Henry.Markram@epfl.ch Tel.: +41-21-6939537, Fax: +41-21-6935350) Acknowledgements. This study was supported by the National Alliance for Autism Research, the Minerva Foundation, the US Navy, the Ebner Center for Biomedical Research, and the Edith Blum Foundation.