Split-Plot Microarray Experiments

This article focuses on microarray experiments with two or more factors in which treatment combinations of the factors corresponding to the samples paired together onto arrays are not completely random. A main effect of one (or more) factor(s) is confounded with arrays (the experimental blocks). This is called a split-plot microarray experiment. We utilise an analysis of variance (ANOVA) model to assess differentially expressed genes for between-array and within-array comparisons that are generic under a split-plot microarray experiment. Instead of standard t- or F-test statistics that rely on mean square errors of the ANOVA model, we use a robust method, referred to as 'a pooled percentile estimator', to identify genes that are differentially expressed across different treatment conditions. We illustrate the design and analysis of split-plot microarray experiments based on a case application described by Jin et al. A brief discussion of power and sample size for split-plot microarray experiments is also presented.     

Split-plot designs for robotic serial dilution assays

This article explores effective implementation of split-plot designs in serial dilution bioassay using robots. We show that the shortest path for a robot to fill plate wells for a split-plot design is equivalent to the shortest common supersequence problem in combinatorics. We develop an algorithm for finding the shortest common supersequence, provide an R implementation, and explore the distribution of the number of steps required to implement split-plot designs for bioassay through simulation. We also show how to construct collections of split plots that can be filled in a minimal number of steps, thereby demonstrating that split-plot designs can be implemented with nearly the same effort as strip-plot designs. Finally, we provide guidelines for modeling data that result from these designs.     

Nested analysis of variance with autocorrelated errors

In this paper we consider the problem where there is a randomized experimental design with several successive time measurements on each experimental unit. One approach to the analysis of such data is to treat time as the subplot treatment and to use a split-plot analysis of variance. Alternatively, the problem may be considered in a more general multivariate framework. Here we recognize the time-induced correlations and apply an autoregressive time series modelling approach. Estimation and testing are addressed. Two examples are presented to illustrate the practicality of our procedure. Some extensions are also considered briefly.     

Statistical Methods for Using Nonlinear Models to Compare Silage Treatments

Statistical methods are described which can be used to compare treatments where the response is described by a nonlinear model. The nonlinear analysis of covariance is described for a one-way treatment structure and a two-way treatment structure. Methods for constructing confidence bands about a model and about the difference of two models are described. Examples involving the dynamics of silage fermentation are used to demonstrate the methods.     

耳蜗滤波器设计

提出了一种非线性耳蜗滤波器组的设计方法,设计的耳蜗滤波器组结构简单,易于实现,能模拟因输入信号电平的不同而引起的耳蜗滤波器的非线性特性,而且其特征参数能随着更新实验结果的出现灵活地加以修正。文中给出了设计细节以及具体的数字实现方法。     

Scheffy's Confidence Intervals in the Models of Analysis of Covariance

Scheffe's confidence intervals for linear functions of some subvectors of a vector of parameters are presented. The considered subvectors are such that covariance matrices of their estimators are known non-negative definite matrices multiplied by unknown positive constants. This property is characteristic of the least squares estimators of vectors of main and interaction effects in the analysis of covariance models of the following experimental designs: split-block, split-plot, completely randomized two-factor design and randomized complete block design. The formulas for confidence intervals for linear functions of vectors of main or interaction effects in the designs mentioned above are given in the paper. The practical example is given as an illustration.     

A method for the identification of in-vivo segmental stiffness properties of the spine

A numerical algorithm is used to estimate in-vivo segmental stiffness properties of individual spine segments based upon existing load-displacement data. A static nonlinear finite element model stimulates a pathological spine and corrective instrumentation system. A systematic procedure for establishing the model's stiffness parameters is described, in the form of a nonlinear constrained optimal design problem. The numerical method is demonstrated using as an example a case of adolescent idiopathic scoliosis requiring corrective surgery.     

Signal propagation in nonlinear stochastic gene regulatory networks

The ability to build genetic circuits with a reproducible response to external stimuli depends on the experimental and theoretical methods used in the process. A theoretical formalism that describes the response of a nonlinear stochastic genetic network to the external stimuli (input signals), is proposed. Two applications are studied in detail: the design of a logic pulse and the interference of three signal generators in the E2F1 regulatory element. The gene interactions are presented using molecular diagrams that have a precise mathematical structure and retain the biological meaning of the processes.     

Search for critical loading condition of the spine–A meta analysis of a nonlinear viscoelastic finite element model

The relative vulnerability of spinal motion segments to different loading combinations remains unknown. The meta-analysis described here using the results of a validated L2–L3 nonlinear viscoelastic finite element model was designed to investigate the critical loading and its effect on the internal mechanics of the human lumbar spine. A Box-Behnken experimental design was used to design the magnitude of seven independent variables associated with loads, rotations and velocity of motion. Subsequently, an optimization method was used to find the primary and secondary variables that influence spine mechanical output related to facet forces, disc pressure, ligament forces, annulus matrix compressive/shear stresses and anulus fibers strain. The mechanical responses with respect to the two most-relevant variables were then regressed linearly using the response surface quadratic model. Axial force and sagittal rotation were identified as the most-relevant variables for mechanical responses. The procedure developed can be used to find the critical loading for finite element models with multi input variables. The derived meta-models can be used to predict the risk associated with various loading parameters and in setting safer load limits.     

Nonlinear transient computation as a potential "kernel trick" in cortical processing

Evidence has been found for the presence of chaotic dynamics at all levels of the mammalian brain. This has led to some searching questions about the potential role that nonlinear dynamics may have in neural information processing. We propose that chaos equips the brain with the equivalent of a kernel trick for solving hard nonlinear problems. The approach presented, which is described as nonlinear transient computation, uses the dynamics of a well known chaotic attractor. The paper provides experimental results to show that this approach can be used to solve some challenging pattern recognition tasks. The paper also offers evidence to suggest that the efficacy of nonlinear transient computation for nonlinear pattern classification is dependent only on the generic properties of chaotic attractors and is not sensitive to the particular dynamics of specific sub-regions of chaotic phase space. If, as this work suggests, nonlinear transient computation is independent of the particulars of any given chaotic attractor, then it could be offered as a possible explanation of how the chaotic dynamics that have been observed in brain structures contribute to neural information processing tasks.     

Search for critical loading condition of the spine--a meta analysis of a nonlinear viscoelastic finite element model

The relative vulnerability of spinal motion segments to different loading combinations remains unknown. The meta-analysis described here using the results of a validated L2-L3 nonlinear viscoelastic finite element model was designed to investigate the critical loading and its effect on the internal mechanics of the human lumbar spine. A Box-Behnken experimental design was used to design the magnitude of seven independent variables associated with loads, rotations and velocity of motion. Subsequently, an optimization method was used to find the primary and secondary variables that influence spine mechanical output related to facet forces, disc pressure, ligament forces, annulus matrix compressive/shear stresses and anulus fibers strain. The mechanical responses with respect to the two most-relevant variables were then regressed linearly using the response surface quadratic model. Axial force and sagittal rotation were identified as the most-relevant variables for mechanical responses. The procedure developed can be used to find the critical loading for finite element models with multi input variables. The derived meta-models can be used to predict the risk associated with various loading parameters and in setting safer load limits.     

Bayesian experimental design for nonlinear mixed-effects models with application to HIV dynamics

Bayesian experimental design is investigated for Bayesian analysis of nonlinear mixed-effects models. Existence of the posterior risk for parameter estimation is shown. When the same prior distribution is used for both design and inference, existence of the preposterior risk for design is also proven. If the prior distribution used in design is different from that used for inference, sufficient conditions are established for existence of the preposterior risk for design. A case study of design for an experiment in population HIV dynamics is provided.     

模拟氮沉降和降雨变化对贝加尔针茅草原土壤细菌群落结构的影响

研究氮沉降和降雨变化对土壤细菌群落结构的影响,对未来预测多个气候变化因子对草地生态系统影响的交互作用具有重要意义。以施氮和灌溉分别模拟氮沉降和降雨增加,采用高通量测序技术,研究8个氮添加水平(0、15、30、50、100、150、200、300kg N hm-2a-1)和2个水分添加水平(不灌溉、模拟夏季增雨100 mm灌溉)对土壤细菌群落结构的影响。结果表明,氮素和水分输入增加后,土壤细菌群落组成、丰度均显著变化(P0.05)。在群落中占主导的细菌门类有疣微菌门Verrucomicrobia(30.61%—48.51%)、变形菌门Proteobacteria(21.37%—29.97%)、酸杆菌门Acidobacteria(9.54%—20.67%)和拟杆菌门Bacteroidetes(4.96%—9.74%)。在常规降雨和水分添加两种条件下,随着氮添加水平的增加,占主导的细菌门类(相对丰度1%)表现出不同的变化趋势。疣微菌门相对丰度在常规降雨N100—N300条件下显著降低,但在氮素和水分同时添加条件下随氮添加水平升高而逐渐升高,在N200—N300时显著升高。变形菌门和拟杆菌门相对丰度在常规降雨高氮添加条件下呈升高趋势,但在水分添加时却无明显变化。酸杆菌门相对丰度在常规降雨高氮添加条件下升高,但在水分添加后呈明显下降趋势。放线菌门Actinobacteria相对丰度在常规降雨N100—N300条件下显著升高,但在水分添加后高氮添加时显著降低。厚壁菌门Firmicutes相对丰度在常规降雨条件下无显著变化,但在水分和高氮添加条件下降低。浮霉菌门Planctomycetes相对丰度在两种不同的水分添加条件下均呈先升高后降低的趋势。氮素和水分添加对土壤细菌群落结构的变化存在明显的互作效应(P0.0001)。在不同氮素和水分输入条件下共有19个土壤细菌门类相对丰度有显著差异。土壤细菌群落结构的变化主要来自于疣微菌门和酸杆菌门的相对丰度变化,两者可作为土壤细菌群落结构变化的指示种。综上,氮素和水分添加显著改变了土壤细菌群落结构,氮素和水分对土壤细菌不同门类相对丰度变化存在明显的互作效应。     

Application of iterative robust model‐based optimal experimental design for the calibration of biocatalytic models

The aim of model calibration is to estimate unique parameter values from available experimental data, here applied to a biocatalytic process. The traditional approach of first gathering data followed by performing a model calibration is inefficient, since the information gathered during experimentation is not actively used to optimize the experimental design. By applying an iterative robust model‐based optimal experimental design, the limited amount of data collected is used to design additional informative experiments. The algorithm is used here to calibrate the initial reaction rate of an ω‐transaminase catalyzed reaction in a more accurate way. The parameter confidence region estimated from the Fisher Information Matrix is compared with the likelihood confidence region, which is not only more accurate but also a computationally more expensive method. As a result, an important deviation between both approaches is found, confirming that linearization methods should be applied with care for nonlinear models. © 2017 American Institute of Chemical Engineers Biotechnol. Prog., 33:1278–1293, 2017     

Genetic expectations of quantitative trait loci main and interaction effects obtained with the triple testcross design and their relevance for the analysis of heterosis

Interpretation of experimental results from quantitative trait loci (QTL) mapping studies on the predominant type of gene action can be severely affected by the choice of statistical model, experimental design, and provision of epistasis. In this study, we derive quantitative genetic expectations of (i) QTL effects obtained from one-dimensional genome scans with the triple testcross (TTC) design and (ii) pairwise interactions between marker loci using two-way analyses of variance (ANOVA) under the F(2)- and the F(infinity)-metric model. The theoretical results show that genetic expectations of QTL effects estimated with the TTC design are complex, comprising both main and epistatic effects, and that genetic expectations of two-way marker interactions are not straightforward extensions of effects estimated in one-dimensional scans. We also demonstrate that the TTC design can partially overcome the limitations of the design III in separating QTL main effects and their epistatic interactions in the analysis of heterosis and that dominance x additive epistatic interactions of individual QTL with the genetic background can be estimated with a one-dimensional genome scan. Furthermore, we present genetic expectations of variance components for the analysis of TTC progeny tested in a split-plot design, assuming digenic epistasis and arbitrary linkage.     

Optimal Experimental Design and Accuracy of Parameter Estimation for Nonlinear Regression Models Used in Long-term Selection

An experimental design problem is considered for the analysis of long-term selection experiments with nonlinear regression models. For a 3-parametric exponential regression function whose parameters have also a reasonable biological interpretation approximate formulas for the determination of the necessary number of observations at each generation are constructed in such a way that the half expected length of an (1 — α)-confidence interval for a chosen parameter is not greater than a given value. In this sense the accuracy of the parameter estimators can be described.     

Nonlinear regression methods in design of experiments and mathematical modelling. Applications to the analysis of the steady-state kinetics of glutathione reductase.

A branching reaction pathway involving a ping pong and a sequential loop has been proposed for glutathione reductase (Biochem. Biophys. Res. Commun. 53 (1973) 1151). In the present investigation nonlinear regression methods have been applied in the fitting of rate equations to experimental data to test the validity of the model proposed and to discriminate between alternative mathematical models (cf. FEBS Lett. 26 (1972) 252). In the best rate law, some of the parameters were numerically redundant. Therefore, a feature-wise analysis of the rate equation was carried out by varying one substrate concentration at a time. The overall strategy used was a cyclic procedure involving: experimentation - analysis of data - modelling - design of experiments - new experimentation etc. Consideration was given to the experimental error structure and to the importance of weighting in the regression analysis. In the design of experiments for discrimination between rival models, a previously defined discrimination function was used. The results of the analysis support the branching reaction scheme proposed for glutathione reductase.     

Transformations of count data for tests of interaction in factorial and split-plot experiments

In applied entomological experiments, when the response is a count-type variable, certain transformation remedies such as the square root, logarithm (log), or rank transformation are often used to normalize data before analysis of variance. In this study, we examine the usefulness of these transformations by reanalyzing field-collected data from a split-plot experiment and by performing a more comprehensive simulation study of factorial and split-plot experiments. For field-collected data, significant interactions were dependent upon the type of transformation. For the simulation study, Poisson distributed errors were used for a 2 by 2 factorial arrangement, in both randomized complete block and split-plot settings. Various sizes of main effects were induced, and type I error rates and powers of the tests for interaction were examined for the raw response values, log-, square root-, and rank-transformed responses. The aligned rank transformation also was investigated because it has been shown to perform well in testing interactions in factorial arrangements. We found that for testing interactions, the untransformed response and the aligned rank response performed best (preserved nominal type I error rates), whereas the other transformations had inflated error rates when main effects were present. No evaluations of the tests for main effects or simple effects have been conducted. Potentially these transformations will still be necessary when performing these tests.     

Ab initio prediction of the three-dimensional structure of a de novo designed protein: a double-blind case study

Ab initio structure prediction and de novo protein design are two problems at the forefront of research in the fields of structural biology and chemistry. The goal of ab initio structure prediction of proteins is to correctly characterize the 3D structure of a protein using only the amino acid sequence as input. De novo protein design involves the production of novel protein sequences that adopt a desired fold. In this work, the results of a double-blind study are presented in which a new ab initio method was successfully used to predict the 3D structure of a protein designed through an experimental approach using binary patterned combinatorial libraries of de novo sequences. The predicted structure, which was produced before the experimental structure was known and without consideration of the design goals, and the final NMR analysis both characterize this protein as a 4-helix bundle. The similarity of these structures is evidenced by both small RMSD values between the coordinates of the two structures and a detailed analysis of the helical packing.     

Modeling and observer design for recombinant Escherichia coli strain

A mathematical model for recombinant bacteria which includes foreign protein production is developed. The experimental system consists of an Escherichia Coli strain and plasmid pIT34 containing genes for bioluminescence and production of a protein, β-galactosidase. This recombinant strain is constructed to facilitate on-line estimation and control in a complex bioprocess. Several batch experiments are designed and performed to validate the developed model. The design of a model structure, the identification of the model parameters and the estimation problem are three parts of a joint design problem. A nonlinear observer is designed and an experimental evaluation is performed on a batch fermentation process to estimate the substrate consumption.