Non-linear PCA: a missing data approach

MOTIVATION: Visualizing and analysing the potential non-linear structure of a dataset is becoming an important task in molecular biology. This is even more challenging when the data have missing values. RESULTS: Here, we propose an inverse model that performs non-linear principal component analysis (NLPCA) from incomplete datasets. Missing values are ignored while optimizing the model, but can be estimated afterwards. Results are shown for both artificial and experimental datasets. In contrast to linear methods, non-linear methods were able to give better missing value estimations for non-linear structured data.Application: We applied this technique to a time course of metabolite data from a cold stress experiment on the model plant Arabidopsis thaliana, and could approximate the mapping function from any time point to the metabolite responses. Thus, the inverse NLPCA provides greatly improved information for better understanding the complex response to cold stress. CONTACT: scholz@mpimp-golm.mpg.de.     

Confidence interval about the response at the stationary point of a response surface, with an application to preclinical cancer therapy

A method of calculating a confidence interval about the response at the stationary point of a response surface is presented. We show that the technique can also be used to calculate a confidence region about the location of the stationary point. The procedure is applied, via Cox's proportional-hazards model, to the analysis of survival data from a preclinical cancer chemotherapy experiment involving the combination of two drugs. It is seen that the results can be useful in determining the existence of a therapeutic synergism.     

Effect of boundaries on the response of a neural network.

The effect an abrupt boundary has upon the dynamical response of a neural network is investigated. The retina of the Limulus eye is used as a model system for studying this effect. A theoretical technique is presented for the quantitative prediction of the manner in which this neural network responds in the vicinity of its boundary. Corresponding experimental measurements of the response to moving stimuli by single optic neurons located near retinal boundaries are presented. Theory and experiment show detailed quantitative agreement.     

Interplay between order-parameter and system parameter dynamics: considerations on perceptual-cognitive-behavioral mode-mode transitions exhibiting positive and negative hysteresis and on response times

A mathematical model is presented for the emergence of perceptual-cognitive-behavioral modes in psychophysical experiments in which participants are confronted with two alternatives. The model is based on the theory of self-organization and, in particular, the order parameter concept such that the emergence of a mode is conceptualized as an instability leading to the emergence of an appropriately defined order parameter. The order parameter model is merged with a second model that describes adaptation in terms of a system parameter dynamics. It is shown that the two-component model predicts hysteretic mode-mode transitions when control parameters are increased or decreased beyond critical values. The two-component model can account for both positive and negative hysteresis effects due to the interaction between order parameter and system parameter dynamics. Moreover, the model-based analysis reveals that response time curves look rather flat when response times are relatively decoupled from the mode-mode transition phenomenon. In general, response time curves exhibit a peaked close to the mode-mode transition point. In this context, the possibility is discussed that such peaked response time curves belong to the class of critical phenomena of self-organizing systems. In order to illustrate the relevance of peaked response time curves for future research and research reported in the past, results from a perceptual judgment experiment are reported, in which participants judged their ability to stand on a tilted slope for various angles of inclination. Response time curves were found that exhibited a peak around the mode-mode-transition points between “yes” and “no” responses.     

Analysis of the Influence of Environmental Parameters on Clostridium botulinum Time-to-Toxicity by Using Three Modeling Approaches

This study used the technique of waiting time modeling to analyze the combined effects of temperature, pH, carbohydrate, protein, and lipid on the time-to-toxicity of Clostridium botulinum 56A. Waiting time models can be used whenever the time to the occurrence of some event is the variable of interest. In the case of the time-to-toxicity data, the variable is the time from the beginning of an experiment until a tube is identified as positive. The statistical analysis used the SAS procedure LIFEREG and included determination of the form of the response surface, identification of the error distribution, and simplification of the response surface. We found that increasing the macromolecule concentration decreased the probability of toxin formation. The probability of toxin formation also decreased at lower temperatures and at pHs further from the optimum. The waiting time modeling approach to developing models for botulinal toxin formation compared favorably with other approaches but had one specific advantage. Waiting time models have the inherent advantage that safety concerns regarding predictions are automatically quantified in the analysis by formally identifying a distribution of times-to-toxicity. The use of this time-to-toxicity distribution permits a customizable margin of safety (e.g., one in a million) not possible with other approaches.     

Oscillator model reduction preserving the phase response: application to the circadian clock

Mathematical model reduction is a long-standing technique used both to gain insight into model subprocesses and to reduce the computational costs of simulation and analysis. A reduced model must retain essential features of the full model, which, traditionally, have been the trajectories of certain state variables. For biological clocks, timing, or phase, characteristics must be preserved. A key performance criterion for a clock is the ability to adjust its phase correctly in response to external signals. We present a novel model reduction technique that removes components from a single-oscillator clock model and discover that four feedback loops are redundant with respect to its phase response behavior. Using a coupled multioscillator model of a circadian clock, we demonstrate that by preserving the phase response behavior of a single oscillator, we preserve timing behavior at the multioscillator level.     

A systems-theory approach to the analysis of multiexponential fluorescence decay.

A mathematical model of the fluorescence decay experiment based on linear systems theory is presented. The model suggests an experimental technique that increases the probability of correctly determining the decay constants of a multicomponent system. The use of moment methods for data analysis improves accuracy by combining information obtained from several discrete experiments. Examples are presented to show that the analysis of a three component system composed of known standards is improved as the number of experimental determinations is increased from one to four. The discrete measurements are made by changing the excitation and emission wavelengths.     

An in-vivo measurement and analysis of viscoelastic properties of the spinal cord of cats

An in-vivo experimental technique was employed to determine the linear and nonlinear characteristics of viscoelastic properties of the spinal cord of anesthetized cats. The stress relaxation and recovery curves were reproducible in a group of cat experiments. The data of linear viscoelastic properties were used to develop a power law model with Boltzmann's convolution integral. The model was capable of predicting a prolonged stress relaxation and recovery curve. For larger deformation, the results were quantified using a nonlinear analysis of viscoelastic response of the spinal cord under the uniaxial experiment.     

Multiplication noise in the human visual system at threshold: 2. Probit estimation of parameters

A mathematical technique is described that relates detection model parameters to stimulus magnitude and experimental probability of detection. The normalizing transform is used to make the response statistics approximately Gaussian. Conventional probit analysis is then applied. From measurements at M stimulus levels, a system of M equations is solved and estimates of M unknown parameters of the detection model are obtained. The technique is applied to a threshold vision model based on additive and multiplicative Poisson noise. Results are obtained for the parameter estimates for individual subjects, and for the standard deviation of the estimates, for various values of the stimulus energy and number of trials. A frequency-of-seeing experiment is performed using a point-source stimulus that randomly assumes 3 energy levels with 200 trials per level. With a central efficiency of 50%, the estimated ocular quantum efficiency for our four subjects lies between 12% and 23%, the average dark count at the retina lies between 8 and 36 counts, and the threshold count for our (low falsereport rate) data lies between 11 and 32. The theoretical results reduce to those obtained by Barlow (J. Physiol. London 160, 155–168, 1962), in the absence of dark light and multiplication noise.This work was supported by the National Science Foundation     

Cumulant analysis in fluorescence fluctuation spectroscopy

A novel technique for the analysis of fluorescence fluctuation experiments is introduced. Fluorescence cumulant analysis (FCA) exploits the factorial cumulants of the photon counts and resolves heterogeneous samples based on differences in brightness. A simple analytical model connects the cumulants of the photon counts with the brightness epsilon and the number of molecules N in the optical observation volume for each fluorescent species. To provide the tools for a rigorous error analysis of FCA, expressions for the variance of factorial cumulants are developed and tested. We compare theory with experiment by analyzing dye mixtures and simple fluorophore solutions with FCA. A comparison of FCA with photon-counting histogram (PCH) analysis, a related technique, shows that both methods give identical results within experimental uncertainty. Both FCA and PCH are restricted to data sampling times that are short compared to the diffusion time of molecules through the observation volume of the instrument. But FCA theory, in contrast to PCH, can be extended to treat arbitrary sampling times. Here, we derive analytical expressions for the second factorial cumulant as a function of the sampling time and demonstrate that the theory successfully models fluorescence fluctuation data.     

Analyzing optima in the exploration of multiple response surfaces

A method is presented for testing the equality of some or all (constrained or unconstrained) optima in a response surface analysis. An estimator of a common location of stationary points is obtained by a standard multivariate testing procedure and a confidence region associated with the common optimum is derived. The procedure is illustrated by the estimation of a common optimum in a multiple response experiment. The multivariate approach facilitates a more efficient estimation of the optimum than the usual univariate response surface analysis and provides additional tests of the response surface model.     

Response surface method was used to optimize the ultrasonic assisted extraction of flavonoids from Crinum asiaticum

In this study, ultrasound-assisted extraction of flavonoid from Crinum asiaticum was studied through response surface methodology (RSM) to gain the best extraction process of flavonoid and enhance the extraction rate of flavonoid. In the following RSM experiment, we selected the corresponding data of every factor as the center point through the single-factor experiments, then the experimental data was subjected to multiple regression analysis. According to the statistical analysis results, the results were consistent with the polynomial regression model, the determination coefficient (R2) was 0.9769. The best conditions for maximum flavonoid yield were 60% ethanol concentration, 64 °C for extraction temperature, 1:28 (v/w) solid-to-liquid ratio with extraction time for 47 min. The best response of flavonoid yield was 1.63972%. The predicted results for best reaction conditions were in good agreement with experiment values. Ultrasound-assisted extraction method can enhance the extraction rate of flavonoid significantly. It is a powerful tool to extract of important phytochemicals from nature plant.     

The analysis of nonlinear synaptic transmission

In order to characterize synaptic transmission at a unitary facilitating synapse in the lobster cardiac ganglion, a new nonlinear systems analysis technique for discrete-input systems was developed and applied. From the output of the postsynaptic cell in response to randomly occurring presynaptic nerve impulses, a set of kernels, analogous to Wiener kernels, was computed. The kernels up to third order served to characterize, with reasonable accuracy, the input-output properties of the synapse. A mathematical model of the synapse was also tested with a random impulse train and model predictions were compared with experimental synaptic output. Although the model proved to be even more accurate overall than the kernel characterization, there were slight but consistent errors in the model's performance. These were also reflected as differences between model and experimental kernels. It is concluded that a random train analysis provides a comprehensive and objective comparison between model and experiment and automatically provides an arbitrarily accurate characterization of a system's input-output behavior, even in complicated cases where other approaches are impractical.     

Individual differences in metabolomics: individualised responses and between-metabolite relationships

Many metabolomics studies aim to find ‘biomarkers’: sets of molecules that are consistently elevated or decreased upon experimental manipulation. Biological effects, however, often manifest themselves along a continuum of individual differences between the biological replicates in the experiment. Such differences are overlooked or even diminished by methods in standard use for metabolomics, although they may contain a wealth of information on the experiment. Properly understanding individual differences is crucial for generating knowledge in fields like personalised medicine, evolution and ecology. We propose to use simultaneous component analysis with individual differences constraints (SCA-IND), a data analysis method from psychology that focuses on these differences. This method constructs axes along the natural biochemical differences between biological replicates, comparable to principal components. The model may shed light on changes in the individual differences between experimental groups, but also on whether these differences correspond to, e.g., responders and non-responders or to distinct chemotypes. Moreover, SCA-IND reveals the individuals that respond most to a manipulation and are best suited for further experimentation. The method is illustrated by the analysis of individual differences in the metabolic response of cabbage plants to herbivory. The model reveals individual differences in the response to shoot herbivory, where two ‘response chemotypes’ may be identified. In the response to root herbivory the model shows that individual plants differ strongly in response dynamics. Thereby SCA-IND provides a hitherto unavailable view on the chemical diversity of the induced plant response, that greatly increases understanding of the system.     

Rank products: a simple, yet powerful, new method to detect differentially regulated genes in replicated microarray experiments

One of the main objectives in the analysis of microarray experiments is the identification of genes that are differentially expressed under two experimental conditions. This task is complicated by the noisiness of the data and the large number of genes that are examined simultaneously. Here, we present a novel technique for identifying differentially expressed genes that does not originate from a sophisticated statistical model but rather from an analysis of biological reasoning. The new technique, which is based on calculating rank products (RP) from replicate experiments, is fast and simple. At the same time, it provides a straightforward and statistically stringent way to determine the significance level for each gene and allows for the flexible control of the false-detection rate and familywise error rate in the multiple testing situation of a microarray experiment. We use the RP technique on three biological data sets and show that in each case it performs more reliably and consistently than the non-parametric t-test variant implemented in Tusher et al.'s significance analysis of microarrays (SAM). We also show that the RP results are reliable in highly noisy data. An analysis of the physiological function of the identified genes indicates that the RP approach is powerful for identifying biologically relevant expression changes. In addition, using RP can lead to a sharp reduction in the number of replicate experiments needed to obtain reproducible results.     

Tissue microarrays: bridging the gap between research and the clinic

Tissue microarrays are a high-throughput method for the investigation of biomarkers in multiple tissue specimens at once. This technique allows for the analysis of up to 500 tissue samples in a single experiment using immunohistochemistry and in situ hybridization. Recently, cell lines and xenografts have been reduced to a tissue microarray format and are being applied to preclinical drug development. In clinical research, tissue microarrays are applied at multiple levels: comprehensive analysis of samples in the context of a clinical trial or across a population. Tissue microarrays play a central role in translational research, facilitating the discovery of molecules that have potential roles in the diagnosis, prognosis and prediction of response to therapy.     

Monitoring the inhibitive effect of the static magnetic field on the activity of lysozyme with acoustic wave impedance analysis technique

The inhibitive effect of static magnetic field on the activity of lysozyme was studied using acoustic wave impedance analysis technique. Equivalent circuit parameters of piezoelectric quartz crystal (PQC) were obtained and discussed. The results showed that the activity of lysozyme was inhibited due to the effect of static magnetic field and the inhibitive effect becomes greater with an increase in magnetization time or magnetic field intensity. According to the response characteristics of motional resistance change (deltaR1), which is related to the change in the bacterial number, a quantitative response model reflecting the activity of lysozyme was theoretically derived. By fitting deltaR1 versus time curves under a specific magnetic field intensity but different magnetic time to the model, the relationship between K1 reflecting the activity of lysozyme and magnetic time t(m) was established. Based on the relationship, a new impedance response model that indicates the inhibitive influence of the magnetization time on the activity of lysozyme was derived as follows: deltaR1 = R0((K4(exp[K0exp(-0.26t(m))]t - 1) + 1)1/2 - 1). Similarly, another response model that indicates the effect of magnetic field intensity was derived as follows: deltaR1 = R0((K4(exp(K0 exp(- 5.17B)t) - 1) + 1)1/2 - 1).     

Mathematical modeling and stochastic H(infinity) identification of the dynamics of the MF-influenced oxidation of hexane

This paper presents a mathematical model explicitly reflecting the magnetic-field-induced transitions in a biologically significant process: n-hexane oxidation. The range of the magnetic field strength is found (0.05-0.3 T) with the trend indicating significant magnetic-field-induced change in the rates of reactions involving hexane (up to 50% at 0.2 T). The equations describing the effects of the magnetic field on the photoinduced free radical reaction of oxidation involving a lipid-modeling substance, hexane, are obtained on the basis of chemical kinetics and data from a batch experiment. The magnetic-field-induced changes in n-hexane oxidation are validated using the identification technique based on the real time input-output data in a separately conducted flow-through experiment.     

Development and evaluation of the herd dynamic milk model with focus on the individual cow component

The herd dynamic milk (HDM) model is a dynamic model capable of simulating the performance of individual dairy animals (from birth to death), with a daily time step. Within this study, the HDM model is described and evaluated in relation to milk production, body condition score (BCS) and BCS change throughout lactation by comparing model simulations against data from published experimental studies. The model’s response to variation in genetic potential, herbage allowance and concentrate supplementation was tested in a sensitivity analysis. Data from experiments in Ireland and France over a 3-year period (2009–11) were used to complete the evaluation. The aim of the Irish experiment was to determine the impact of different stocking rates (SRs) (SR1: 3.28 cow/ha, SR2: 2.51 cow/ha) on key physical, biological and economic performance. The aim of the French experiment was to evaluate over a prolonged time period, the ability of two breeds of dairy cows (Holstein and Normande) to produce and to reproduce under two feeding strategies (high level and low level) in the context of compact calving. The model evaluation was conducted at the herd level with separate evaluations for the primiparous and multiparous cows. The evaluation included the two extreme SRs for the Irish experiment, and an evaluation at the overall herd and individual animal level for the different breeds and feeding levels for the French data. The comparison of simulation and experimental data for all scenarios resulted in a relative prediction error, which was consistently <15% across experiments for weekly milk production and BCS. In relation to BCS, the highest root mean square error was 0.27 points of BCS, which arose for Holstein cows in the low feeding group in late lactation. The model responded in a realistic fashion to variation in genetic potential for milk production, herbage allowance and concentrate supplementation.     

Tissue microarrays: bridging the gap between research and the clinic

Tissue microarrays are a high-throughput method for the investigation of biomarkers in multiple tissue specimens at once. This technique allows for the analysis of up to 500 tissue samples in a single experiment using immunohistochemistry and in situ hybridization. Recently, cell lines and xenografts have been reduced to a tissue microarray format and are being applied to preclinical drug development. In clinical research, tissue microarrays are applied at multiple levels: comprehensive analysis of samples in the context of a clinical trial or across a population. Tissue microarrays play a central role in translational research, facilitating the discovery of molecules that have potential roles in the diagnosis, prognosis and prediction of response to therapy.