Covariate Adjusted Correlation Analysis with Application to FMR1 Premutation Female Carrier Data

Summary .  Motivated by molecular data on female premutation carriers of the fragile X mental retardation 1 ( FMR1 ) gene, we present a new method of covariate adjusted correlation analysis to examine the association of messenger RNA (mRNA) and number of CGG repeat expansion in the  FMR1  gene. The association between the molecular variables in female carriers needs to adjust for activation ratio (ActRatio), a measure which accounts for the protective effects of one normal X chromosome in females carriers. However, there are inherent uncertainties in the exact effects of ActRatio on the molecular measures of interest. To account for these uncertainties, we develop a flexible adjustment that accommodates both additive and multiplicative effects of ActRatio nonparametrically. The proposed adjusted correlation uses local conditional correlations, which are local method of moments estimators, to estimate the Pearson correlation between two variables adjusted for a third observable covariate. The local method of moments estimators are averaged to arrive at the final covariate adjusted correlation estimator, which is shown to be consistent. We also develop a test to check the nonparametric joint additive and multiplicative adjustment form. Simulation studies illustrate the efficacy of the proposed method. (Application to  FMR1  premutation data on 165 female carriers indicates that the association between mRNA and CGG repeat after adjusting for ActRatio is stronger.) Finally, the results provide independent support for a specific jointly additive and multiplicative adjustment form for ActRatio previously proposed in the literature.     

An error model for protein quantification

MOTIVATION: Quantitative experimental data is the critical bottleneck in the modeling of dynamic cellular processes in systems biology. Here, we present statistical approaches improving reproducibility of protein quantification by immunoprecipitation and immunoblotting. RESULTS: Based on a large data set with more than 3600 data points, we unravel that the main sources of biological variability and experimental noise are multiplicative and log-normally distributed. Therefore, we suggest a log-transformation of the data to obtain additive normally distributed noise. After this transformation, common statistical procedures can be applied to analyze the data. An error model is introduced to account for technical as well as biological variability. Elimination of these systematic errors decrease variability of measurements and allow for a more precise estimation of underlying dynamics of protein concentrations in cellular signaling. The proposed error model is relevant for simulation studies, parameter estimation and model selection, basic tools of systems biology. AVAILABILITY: Matlab and R code is available from the authors on request. The data can be downloaded from our website www.fdm.uni-freiburg.de/~ckreutz/data.     

Particle capture into the lung made simple?

Understanding the impact distribution of particles entering the human respiratory system is of primary importance as it concerns not only atmospheric pollutants or dusts of various kinds but also the efficiency of aerosol therapy and drug delivery. To model this process, current approaches consist of increasingly complex computations of the aerodynamics and particle capture phenomena, performed in geometries trying to mimic lungs in a more and more realistic manner for as many airway generations as possible. Their capture results from the complex interplay between the details of the aerodynamic streamlines and the particle drag mechanics in the resulting flow. In contrast, the present work proposes a major simplification valid for most airway generations at quiet breathing. Within this context, focusing on particle escape rather than capture reveals a simpler structure in the entire process. When gravity can be neglected, we show by computing the escape rates in various model geometries that, although still complicated, the escape process can be depicted as a multiplicative escape cascade in which each elementary step is associated with a single bifurcation. As a net result, understanding of the particle capture may not require computing particle deposition in the entire lung structure but can be abbreviated in some regions using our simpler approach of successive computations in single realistic bifurcations. Introducing gravity back into our model, we show that this multiplicative model can still be successfully applied on up to nine generations, depending on particle type and breathing conditions.     

Weak ergodicity of population evolution processes

The weak ergodic theorems of mathematical demography state that the age distribution of a closed population is asymptotically independent of the initial distribution. In this paper, we provide a new proof of the weak ergodic theorem of the multistate population model with continuous time. The main tool to attain this purpose is a theory of multiplicative processes, which was mainly developed by Garrett Birkhoff, who showed that ergodic properties generally hold for an appropriate class of multiplicative processes. First, we construct a general theory of multiplicative processes on a Banach lattice. Next, we formulate a dynamical model of a multistate population and show that its evolution operator forms a multiplicative process on the state space of the population. Subsequently, we investigate a sufficient condition that guarantees the weak ergodicity of the multiplicative process. Finally, we prove the weak and strong ergodic theorems for the multistate population and resolve the consistency problem.     

Priorities in statistics, the sensitive feet of elephants, and don’t transform data

In spite of the widespread use of statistics in plant ecology, some misunderstandings are widespread. Lájer’s warning against non-random sampling in the field is well taken, but non-randomization is probably more common than we realize in experimental work too, and a frequent cause of inexplicable “significant” results. However, in the placement of quadrats/samples, restricted randomization is always preferable to plain random. The main purpose of randomization, as R.A. Fisher made clear, is to obtain a valid estimate of the error. Random placement does not, as Fisher realized, ensure independence of samples because of spatial autocorrelation, which is present in all ecological work. If we forget this, we can end up concluding that elephants carefully select moss cushions to tread on. Although a normal distribution is often formally required, tests such as the Analysis of Variance are fairly robust against departures. Obsession with normality leads to the use of inappropriate transformations, for example a log transformation when the author had no intention of a multiplicative model. Even worse is the use of a log (x + 1) transformation, which gives answers in neither additive nor multiplicative terms, and in a way unrelated to the means presented. There are several solutions to this, including randomization tests. After all this, we should not take the arbitrary value of 0.05 too seriously. Many statisticians do not.     

Dimensions and moment arms of the hind- and forelimb muscles of common chimpanzees (Pan troglodytes).

This paper supplies quantitative data on the hind- and forelimb musculature of common chimpanzees (Pan troglodytes) and calculates maximum joint moments of force as a contribution to a better understanding of the differences between chimpanzee and human locomotion. We dissected three chimpanzees, and recorded muscle mass, fascicle length, and physiological cross-sectional area (PCSA). We also obtained flexion/extension moment arms of the major muscles about the limb joints. We find that in the hindlimb, chimpanzees possess longer fascicles in most muscles but smaller PCSAs than are predicted for humans of equal body mass, suggesting that the adaptive emphasis in chimpanzees is on joint mobility at the expense of tension production. In common chimpanzee bipedalism, both hips and knees are significantly more flexed than in humans, necessitating muscles capable of exerting larger moments at the joints for the same ground force. However, we find that when subject to the same stresses, chimpanzee hindlimb muscles provide far smaller moments at the joints than humans, particularly the quadriceps and plantar flexors. In contrast, all forelimb muscle masses, fascicle lengths, and PCSAs are smaller in humans than in chimpanzees, reflecting the use of the forelimbs in chimpanzee, but not human, locomotion. When subject to the same stresses, chimpanzee forelimb muscles provide larger moments at the joints than humans, presumably because of the demands on the forelimbs during locomotion. These differences in muscle architecture and function help to explain why chimpanzees are restricted in their ability to walk, and particularly to run bipedally.     

Estimating parameters for generalized mass action models using constraint propagation

As modern molecular biology moves towards the analysis of biological systems as opposed to their individual components, the need for appropriate mathematical and computational techniques for understanding the dynamics and structure of such systems is becoming more pressing. For example, the modeling of biochemical systems using ordinary differential equations (ODEs) based on high-throughput, time-dense profiles is becoming more common-place, which is necessitating the development of improved techniques to estimate model parameters from such data. Due to the high dimensionality of this estimation problem, straight-forward optimization strategies rarely produce correct parameter values, and hence current methods tend to utilize genetic/evolutionary algorithms to perform non-linear parameter fitting. Here, we describe a completely deterministic approach, which is based on interval analysis. This allows us to examine entire sets of parameters, and thus to exhaust the global search within a finite number of steps. In particular, we show how our method may be applied to a generic class of ODEs used for modeling biochemical systems called Generalized Mass Action Models (GMAs). In addition, we show that for GMAs our method is amenable to the technique in interval arithmetic called constraint propagation, which allows great improvement of its efficiency. To illustrate the applicability of our method we apply it to some networks of biochemical reactions appearing in the literature, showing in particular that, in addition to estimating system parameters in the absence of noise, our method may also be used to recover the topology of these networks.     

Exponential Signaling Gain at the Receptor Level Enhances Signal-to-Noise Ratio in Bacterial Chemotaxis

Cellular signaling systems show astonishing precision in their response to external stimuli despite strong fluctuations in the molecular components that determine pathway activity. To control the effects of noise on signaling most efficiently, living cells employ compensatory mechanisms that reach from simple negative feedback loops to robustly designed signaling architectures. Here, we report on a novel control mechanism that allows living cells to keep precision in their signaling characteristics – stationary pathway output, response amplitude, and relaxation time – in the presence of strong intracellular perturbations. The concept relies on the surprising fact that for systems showing perfect adaptation an exponential signal amplification at the receptor level suffices to eliminate slowly varying multiplicative noise. To show this mechanism at work in living systems, we quantified the response dynamics of the E. coli chemotaxis network after genetically perturbing the information flux between upstream and downstream signaling components. We give strong evidence that this signaling system results in dynamic invariance of the activated response regulator against multiplicative intracellular noise. We further demonstrate that for environmental conditions, for which precision in chemosensing is crucial, the invariant response behavior results in highest chemotactic efficiency. Our results resolve several puzzling features of the chemotaxis pathway that are widely conserved across prokaryotes but so far could not be attributed any functional role.     

Coherent modeling of longitudinal causal effects on binary outcomes

Analyses of biomedical studies often necessitate modeling longitudinal causal effects. The current focus on personalized medicine and effect heterogeneity makes this task even more challenging. Toward this end, structural nested mean models (SNMMs) are fundamental tools for studying heterogeneous treatment effects in longitudinal studies. However, when outcomes are binary, current methods for estimating multiplicative and additive SNMM parameters suffer from variation dependence between the causal parameters and the noncausal nuisance parameters. This leads to a series of difficulties in interpretation, estimation, and computation. These difficulties have hindered the uptake of SNMMs in biomedical practice, where binary outcomes are very common. We solve the variation dependence problem for the binary multiplicative SNMM via a reparameterization of the noncausal nuisance parameters. Our novel nuisance parameters are variation independent of the causal parameters, and hence allow for coherent modeling of heterogeneous effects from longitudinal studies with binary outcomes. Our parameterization also provides a key building block for flexible doubly robust estimation of the causal parameters. Along the way, we prove that an additive SNMM with binary outcomes does not admit a variation independent parameterization, thereby justifying the restriction to multiplicative SNMMs.     

Evidence and implications for multiplicative interactions among loci predisposing to human common disease

OBJECTIVE: The aim of this study was to utilize information on monozygotic twin concordance rates and linkage studies results for common diseases to predict the likely mode of interaction between susceptibility loci. METHODS: We calculated combinations of allele frequency and genotypic relative risk (GRR) that would generate linkage results typically observed in common human diseases. Given these single locus effects, we calculated the expected monozygotic twin concordance assuming different numbers of loci under different interaction models. RESULTS: We demonstrate that, for disorders like schizophrenia, a purely additive model of interaction among loci is not consistent with the available evidence. Instead there are likely significant multiplicative or stronger interactions. Given these interactions, we show that in a diagnostic test based on a subset of predisposing loci, the marginal increase of predictive value rises with each additional locus that is discovered. Our model was consistent with susceptibility alleles being common or rare. CONCLUSIONS: Evidence from monozygotic twin concordance rates and linkage results point to a significant degree of multiplicative interaction among loci.     

Multiplicative multi-fractal modeling of electromyography signals for discerning neuropathic conditions

In this paper, we present a new method for multi-scale analysis of electromyography signals based on an interesting fractal process known as multiplicative cascade multi-fractal. Using simulated needle electromyography signals, we show this method provides a means for discrimination of normal and neuropathic electromyography signals. We also present experimental results that show the new parameters, computed using multiplicative cascade multi-fractal modeling, are more robust than the conventional signal parameter, number of turns, in the presence of additive noise. Results of multiplicative cascade multi-fractal modeling are consistent with other multi-scale approaches; advantages and differences are high lighted.     

Experimental verification of a computational technique for determining ground reactions in human bipedal stance

We have developed a three-dimensional (3D) biomechanical model of human standing that enables us to study the mechanisms of posture and balance simultaneously in various directions in space. Since the two feet are on the ground, the system defines a kinematically closed-chain which has redundancy problems that cannot be resolved using the laws of mechanics alone. We have developed a computational (optimization) technique that avoids the problems with the closed-chain formulation thus giving users of such models the ability to make predictions of joint moments, and potentially, muscle activations using more sophisticated musculoskeletal models. This paper describes the experimental verification of the computational technique that is used to estimate the ground reaction vector acting on an unconstrained foot while the other foot is attached to the ground, thus allowing human bipedal standing to be analyzed as an open-chain system. The computational approach was verified in terms of its ability to predict lower extremity joint moments derived from inverse dynamic simulations performed on data acquired from four able-bodied volunteers standing in various postures on force platforms. Sensitivity analyses performed with model simulations indicated which ground reaction force (GRF) and center of pressure (COP) components were most critical for providing better estimates of the joint moments. Overall, the joint moments predicted by the optimization approach are strongly correlated with the joint moments computed using the experimentally measured GRF and COP (0.78 < or = r(2) < or = 0.99,median,0.96) with a best-fit that was not statistically different from a straight line with unity slope (experimental=computational results) for postures of the four subjects examined. These results indicate that this model-based technique can be relied upon to predict reasonable and consistent estimates of the joint moments using the predicted GRF and COP for most standing postures.     

Symmetry and range limits in importance indices

Recently, Mingo has analyzed the properties of I_imp, an importance index, and demonstrated that its range is not symmetrical. While agreeing with this comment, we believe that more light needs to be shed on the issue of symmetry in relation to such indices. Importance indices are calculated using three values: performance of the organism in the absence and in the presence of neighbors and maximum performance of the organism in ideal conditions. Because of this structure, importance indices can hardly ever achieve symmetry along the whole range of potential performances. We discuss the limitation of the symmetry range for different symmetry types and for both additive and multiplicative indices. We conclude that importance indices, as other interactions indices, are practical tools for interpreting ecological outcomes, especially while comparing between studies. Nevertheless, the current structure of importance indices prevents symmetry along their whole range. While the lack of “perfect” symmetry may call for the development of more sophisticated importance metrics, the current indices are still helpful for the understanding of biological systems and should not be discarded before better alternatives are well established. To prevent potential confusion, we suggest that ecologists present the relevant index symmetry range in addition to their results, thus minimizing the probability of misinterpretation.     

Development of an agar lift-DNA/DNA hybridization technique for use in visualization of the spatial distribution of Eubacteria on soil surfaces.

While microbial growth is well-understood in pure culture systems, less is known about growth in intact soil systems. The objective of this work was to develop a technique to allow visualization of the two-dimensional spatial distribution of bacterial growth on a homogenous soil surface. This technique is a two-step process wherein an agar lift is taken and analyzed using a universal gene probe. An agar lift is comprised of a thin layer of soil that is removed from a soil surface using an agar slab. The agar is incubated to allow for microbial growth, after which, colonies are transferred to a membrane for conventional bacterial colony DNA/DNA hybridization analysis. In this study, a eubacterial specific probe was used to demonstrate that growing bacterial populations on soil surfaces could be visualized. Results show that microbial growth and distribution was nonuniform across the soil surface. Spot supplementation of the soil with benzoate or glucose resulted in a localized microbial growth response. Since only growing colonies are detected, this technique should facilitate a greater understanding of the microbial distribution and its response to substrate addition in more heterogenous soil systems.     

The coupling of pathways and processes through shared components

Background

The coupling of pathways and processes through shared components is being increasingly recognised as a common theme which occurs in many cell signalling contexts, in which it plays highly non-trivial roles.

Results

In this paper we develop a basic modelling and systems framework in a general setting for understanding the coupling of processes and pathways through shared components. Our modelling framework starts with the interaction of two components with a common third component and includes production and degradation of all these components. We analyze the signal processing in our model to elucidate different aspects of the coupling. We show how different kinds of responses, including "ultrasensitive" and adaptive responses, may occur in this setting. We then build on the basic model structure and examine the effects of additional control regulation, switch-like signal processing, and spatial signalling. In the process, we identify a way in which allosteric regulation may contribute to signalling specificity, and how competitive effects may allow an enzyme to robustly coordinate and time the activation of parallel pathways.

Conclusions

We have developed and analyzed a common systems platform for examining the effects of coupling of processes through shared components. This can be the basis for subsequent expansion and understanding the many biologically observed variations on this common theme.     

The distribution of allelic effects under mutation and selection

The Price (1970, 1972) equation is applied to the problem of describing the changes in the moments of allelic effects caused by selection, mutation and recombination at loci governing a quantitative genetic character. For comparable assumptions the resulting equations are the same as those obtained by different means by Barton & Turelli (1987; Turelli & Barton, 1989). The Price equation provides a natural framework within which to examine certain kinds of non-additive allelic effects, recombination and assortative mating. The use of the Price equation is illustrated by finding the equilibrium genetic variance under multiplicative dominance and epistasis and under assortative mating at an additive locus. The limitations of the use of recursion equations for the moments of allelic effects are also discussed.     

Moment fitting for parameter inference in repeatedly and partially observed stochastic biological models

The inference of reaction rate parameters in biochemical network models from time series concentration data is a central task in computational systems biology. Under the assumption of well mixed conditions the network dynamics are typically described by the chemical master equation, the Fokker Planck equation, the linear noise approximation or the macroscopic rate equation. The inverse problem of estimating the parameters of the underlying network model can be approached in deterministic and stochastic ways, and available methods often compare individual or mean concentration traces obtained from experiments with theoretical model predictions when maximizing likelihoods, minimizing regularized least squares functionals, approximating posterior distributions or sequentially processing the data. In this article we assume that the biological reaction network can be observed at least partially and repeatedly over time such that sample moments of species molecule numbers for various time points can be calculated from the data. Based on the chemical master equation we furthermore derive closed systems of parameter dependent nonlinear ordinary differential equations that predict the time evolution of the statistical moments. For inferring the reaction rate parameters we suggest to not only compare the sample mean with the theoretical mean prediction but also to take the residual of higher order moments explicitly into account. Cost functions that involve residuals of higher order moments may form landscapes in the parameter space that have more pronounced curvatures at the minimizer and hence may weaken or even overcome parameter sloppiness and uncertainty. As a consequence both deterministic and stochastic parameter inference algorithms may be improved with respect to accuracy and efficiency. We demonstrate the potential of moment fitting for parameter inference by means of illustrative stochastic biological models from the literature and address topics for future research.