Using prior knowledge in the determination of macromolecular size-distributions by analytical ultracentrifugation

Analytical ultracentrifugation has reemerged as a widely used tool for the study of ensembles of biological macromolecules to understand, for example, their size-distribution and interactions in free solution. Such information can be obtained from the mathematical analysis of the concentration and signal gradients across the solution column and their evolution in time generated as a result of the gravitational force. In sedimentation velocity analytical ultracentrifugation, this analysis is frequently conducted using high resolution, diffusion-deconvoluted sedimentation coefficient distributions. They are based on Fredholm integral equations, which are ill-posed unless stabilized by regularization. In many fields, maximum entropy and Tikhonov-Phillips regularization are well-established and powerful approaches that calculate the most parsimonious distribution consistent with the data and prior knowledge, in accordance with Occam's razor. In the implementations available in analytical ultracentrifugation, to date, the basic assumption implied is that all sedimentation coefficients are equally likely and that the information retrieved should be condensed to the least amount possible. Frequently, however, more detailed distributions would be warranted by specific detailed prior knowledge on the macromolecular ensemble under study, such as the expectation of the sample to be monodisperse or paucidisperse or the expectation for the migration to establish a bimodal sedimentation pattern based on Gilbert-Jenkins' theory for the migration of chemically reacting systems. So far, such prior knowledge has remained largely unused in the calculation of the sedimentation coefficient or molecular weight distributions or was only applied as constraints. In the present paper, we examine how prior expectations can be built directly into the computational data analysis, conservatively in a way that honors the complete information of the experimental data, whether or not consistent with the prior expectation. Consistent with analogous results in other fields, we find that the use of available prior knowledge can have a dramatic effect on the resulting molecular weight, sedimentation coefficient, and size-and-shape distributions and can significantly increase both their sensitivity and their resolution. Further, the use of multiple alternative prior information allows us to probe the range of possible interpretations consistent with the data.     

Iterative algorithms for processing experimental data

The need to solve linear and nonlinear integral equations arise, e.g., in recovering plasma parameters from the data of multichannel diagnostics. The paper presents an iterative method for solving integral equations with a singularity at the upper limit of integration. The method consists in constructing successive approximations and calculating the integral by quadrature formulas in each integration interval. An example of application of the iterative algorithm to numerically solve an integral equation similar to those arising in recovering the plasma density profile from reflectometry data is presented.     

Dynamic formation of oriented patches in chondrocyte cell cultures

Growth factors have a significant impact not only on the growth dynamics but also on the phenotype of chondrocytes (Barbero et al. in J. Cell. Phys. 204:830–838, 2005). In particular, as chondrocytes approach confluence, the cells tend to align and form coherent patches. Starting from a mathematical model for fibroblast populations at equilibrium (Mogilner et al. in Physica D 89:346–367, 1996), a dynamic continuum model with logistic growth is developed. Both linear stability analysis and numerical solutions of the time-dependent nonlinear integro-partial differential equation are used to identify the key parameters that lead to pattern formation in the model. The numerical results are compared quantitatively to experimental data by extracting statistical information on orientation, density and patch size through Gabor filters.     

Population dynamics with age- and time-dependent birth and death rates

A continous, deterministic mathematical model is used to predict population distributions by age at any time, given the initial distribution and the variation of birth and death rates with age and time. Solutions are obtained on a computer using a semi-discretization algorithm in which time derivatives in the partial differential equations are replaced by finite-difference expressions. The resulting sets of ordinary differential equations are solved by a predictor-corrector method. Graphical results are shown for some examples.     

Strategic Treatment Interruptions During Imatinib Treatment of Chronic Myelogenous Leukemia

Although imatinib is an effective treatment for chronic myelogenous leukemia (CML), and nearly all patients treated with imatinib attain some form of remission, imatinib does not completely eliminate leukemia. Moreover, if the imatinib treatment is stopped, most patients eventually relapse (Cortes et al. in Clin. Cancer Res. 11:3425–3432, 2005). In Kim et al. (PLoS Comput. Biol. 4(6):e1000095, 2008), the authors presented a mathematical model for the dynamics of CML under imatinib treatment that incorporates the anti-leukemia immune response. We use the mathematical model in Kim et al. (PLoS Comput. Biol. 4(6):e1000095, 2008) to study and numerically simulate strategic treatment interruptions as a potential therapeutic strategy for CML patients. We present the results of numerous simulated treatment programs in which imatinib treatment is temporarily stopped to stimulate and leverage the anti-leukemia immune response to combat CML. The simulations presented in this paper imply that treatment programs that involve strategic treatment interruptions may prevent leukemia from relapsing and may prevent remission for significantly longer than continuous imatinib treatment. Moreover, in many cases, strategic treatment interruptions may completely eliminate leukemic cells from the body. Thus, strategic treatment interruptions may be a feasible clinical approach to enhancing the effects of imatinib treatment for CML. We study the effects of both the timing and the duration of the treatment interruption on the results of the treatment. We also present a sensitivity analysis of the results to the parameters in the mathematical model.     

A Two-Step Penalized Regression Method with Networked Predictors

Penalized regression incorporating prior dependency structure of predictors can be effective in high-dimensional data analysis (Li and Li in Bioinformatics, 24:1175–1118, 2008). Pan et al. (Biometrics, 66:474–484, 2010) proposed a penalized regression method for better outcome prediction and variable selection by smoothing parameters over a given predictor network, which can be applied to analysis of microarray data with a given gene network. In this paper, we develop two modifications to their method for further performance enhancement. First, we employ convex programming and show its improved performance over an approximate optimization algorithm implemented in their original proposal. Second, we perform bias reduction after initial variable selection through a new penalty, leading to better parameter estimates and outcome prediction. Simulations have demonstrated substantial performance improvement of the proposed modifications over the original method.     

Adding Adhesion to a Chemical Signaling Model for Somite Formation

Somites are condensations of mesodermal cells that form along the two sides of the neural tube during early vertebrate development. They are one of the first instances of a periodic pattern, and give rise to repeated structures such as the vertebrae. A number of theories for the mechanisms underpinning somite formation have been proposed. For example, in the “clock and wavefront” model (Cooke and Zeeman in J. Theor. Biol. 58:455–476, 1976), a cellular oscillator coupled to a determination wave progressing along the anterior-posterior axis serves to group cells into a presumptive somite. More recently, a chemical signaling model has been developed and analyzed by Maini and coworkers (Collier et al. in J. Theor. Biol. 207:305–316, 2000; Schnell et al. in C. R. Biol. 325:179–189, 2002; McInerney et al. in Math. Med. Biol. 21:85–113, 2004), with equations for two chemical regulators with entrained dynamics. One of the chemicals is identified as a somitic factor, which is assumed to translate into a pattern of cellular aggregations via its effect on cell–cell adhesion. Here, the authors propose an extension to this model that includes an explicit equation for an adhesive cell population. They represent cell adhesion via an integral over the sensing region of the cell, based on a model developed previously for adhesion driven cell sorting (Armstrong et al. in J. Theor. Biol. 243:98–113, 2006). The expanded model is able to reproduce the observed pattern of cellular aggregates, but only under certain parameter restrictions. This provides a fuller understanding of the conditions required for the chemical model to be applicable. Moreover, a further extension of the model to include separate subpopulations of cells is able to reproduce the observed differentiation of the somite into separate anterior and posterior halves. N.J. Armstrong was supported by a Doctoral Training Account Studentship from EPSRC. K.J. Painter and J.A. Sherratt were supported in part by Integrative Cancer Biology Program Grant CA113004 from the US National Institute of Health and in part by BBSRC grant BB/D019621/1 for the Centre for Systems Biology at Edinburgh.     

Optimal biocatalyst loading in a fixed bed

The optimal distribution of biocatalyst in a fixed bed operating at steady state was determined to minimize the length of the bed for a fixed conversion of 95%. The distribution in terms of the biocatalyst loading on an inert support depends upon the axial distance from the bed entrance (continuous solution) as well as a set of dimensionless parameters that reflect the bed geometry, the bulk flow, reaction kinetics and diffusional characteristics. A mathematical model of the system with the following features was used to obtain the results: (1) convective mass transfer and dispersion in the bulk phase; (2) mass transfer from the bulk phase to the surface of the catalyst particle; and (3) simultaneous diffusion and chemical reaction in the catalyst particle with Michaelis–Menton kinetics and a reliable diffusion model (Zhao and DeLancey in Biotechnol Bioeng 64:434–441, 1999, 2000). The solution to the mathematical model was obtained with Mathematica utilizing the Runge Kutta 4–5 method. The dimensionless length resulting from the continuous solution was compared with the optimal length restricted to a uniform or constant cell loading across the entire bed. The maximum difference in the dimensionless length between the continuous and uniform solutions was determined to be 6.5%. The model was applied to published conversion data for the continuous production of ethanol that included cell loading (Taylor et al. in Biotechnol Prog 15:740–751, 2002). The data indicated a minimum production cost at a catalyst loading within 10% of the optimum predicted by the mathematical model. The production rate versus cell loading in most cases displayed a sufficiently broad optimum that the same (non-optimal) rate could be obtained at a significantly smaller loading such as a rate at 80% loading being equal to the rate at 20% loading. These results are particularly important because of the renewed interest in ethanol production (Novozymes and BBI International, Fuel ethanol: a technological evolution, 2004).     

Mathematical Model for Two Germline Stem Cells Competing for Niche Occupancy

In the Drosophila germline stem cell ovary niche, two stem cells compete with each other for niche occupancy to maintain stem cell quality by ensuring that differentiated stem cells are rapidly pushed out the niche and replenished by normal ones (Jin et al. in Cell Stem Cell 2:39–49, 2008). To gain a deeper understanding of this biological phenomenon, we have derived a mathematical model for explaining the physical interactions between two stem cells. The model is a system of two nonlinear first order and one second order differential equations coupled with E-cadherins expression levels. The model can explain the dynamics of the competition process of two germline stem cells and may help to reveal missing information obtained from experimental results. The model predicts several qualitative features in the competition process, which may help to design rational experiments for a better understanding of the stem cell competition process.     

TGF-Mediated Dynamics in a System of Many Coupled Nephrons

This paper presents a mathematical model of a system of many coupled nephrons branching from a common cortical radial artery, and accompanying analysis of that system. This modeling effort is a first step in understanding how coupling magnifies the tendency of nephrons to oscillate owing to tubuloglomerular feedback. Central to the present work is the single nephron integral model (as in Pitman et al., The IMA Volumes in Mathematics and Its Applications, vol. 129, pp. 345–364, 2002 and in Zaritski, Ph.D. Dissertation, 1999) which is a simplification of the single nephron PDE model of Layton et al. (Am. J. Physiol. 261, F904–F919, 1991). A second principal idea used in the present model is a coupling of model nephrons, generalizing the work of Pitman et al. (Bull. Math. Biol. 66, 1463–1492, 2004) who proposed a model of two coupled nephrons. In this study, we couple nephrons through a nearest neighbor interaction. Speaking generally, our results suggest that a series of similar nephrons coupled to their nearest neighbors are more prone to be found in an oscillatory mode, relative to a single nephron with the same properties. More specifically, we show analytically that, for N coupled identical nephrons, the region supporting oscillatory solutions in the time delay–gain parameter plane increases with N. Numerical simulations suggest that, if N nephrons have gains and time delays that do not differ by much, the system is, again, more prone to oscillate, relative to a single nephron, and the oscillations tend to be approximately synchronous and in-phase. We examine the effect of parameters on bifurcation. We also examine alternative models of coupling; this analysis allows us to conclude that the increased propensity of coupled nephrons to oscillate is a robust finding, true for several models of nephron interaction.     

Determination of Interspin Distance Distributions by cw-ESR Is a Single Linear Inverse Problem

Cw-ESR distance measurement method is extremely valuable for studying the dynamics-function relationship of biomolecules. However, extracting distance distributions from experiments has been a highly technique-demanding procedure. It has never been conclusively identified, to our knowledge, that the problems involved in the analysis are ill posed and are best solved using Tikhonov regularization. We treat the problems from a novel point of view. First of all, we identify the equations involved and uncover that they are actually two linear first-kind Fredholm integral equations. They can be combined into one single linear inverse problem and solved in a Tikhonov regularization procedure. The improvement with our new treatment is significant. Our approach is a direct and reliable mathematical method capable of providing an unambiguous solution to the ill-posed problem. It need not perform nonlinear least-squares fitting to infer a solution from noise-contaminated data and, accordingly, substantially reduces the computation time and the difficulty of analysis. Numerical tests and experimental data of polyproline II peptides with variant spin-labeled sites are provided to demonstrate our approach. The high resolution of the distance distributions obtainable with our new approach enables a detailed insight into the flexibility of dynamic structure and the identification of conformational species in solution state.     

On relationships between various distribution functions in balanced unicellular growth

The relationships between various size distributions in balanced exponential growth of a batch culture of microorganisms are presented. Starting from the partial differential integral equations (Eakmanet al., 1966; Fredricksonet al., 1967) derived for the growth of a microbial culture expressions are obtained for the growth rate of organisms of specific size and size range. These expressions were first obtained by Collins and Richmond (1962) by an entirely different method. Also derived are equations which link probability functions, which are basic to the growth of a microbial culture, with other size distributions that can be estimated experimentally.     

Perceptual learning with perceptions

In this work we present an approach to understand neuronal mechanisms underlying perceptual learning. Experimental results achieved with stimulus patterns of coherently moving dots are considered to build a simple neuronal model. The design of the model is made transparent and underlying behavioral assumptions made explicit. The key aspect of the suggested neuronal model is the learning algorithm used: We evaluated an implementation of Hebbian learning and are thus able to provide a straight-forward model capable to explain the neuronal dynamics underlying perceptual learning. Moreover, the simulation results suggest a very simple explanation for the aspect of “sub-threshold” learning (Watanabe et al. in Nature 413:844–884, 2001) as well as the relearning of motion discrimination after damage to primary visual cortex as recently reported (Huxlin et al. in J Neurosci 29:3981–3991, 2009) and at least indicate that perceptual learning might only occur when accompanied by conscious percepts.     

Phosphorescence lifetime analysis with a quadratic programming algorithm for determining quencher distributions in heterogeneous systems.

A new method for analysis of phosphorescence lifetime distributions in heterogeneous systems has been developed. This method is based on decomposition of the data vector to a linearly independent set of exponentials and uses quadratic programming principles for x2 minimization. Solution of the resulting algorithm requires a finite number of calculations (it is not iterative) and is computationally fast and robust. The algorithm has been tested on various simulated decays and for analysis of phosphorescence measurements of experimental systems with descrete distributions of lifetimes. Critical analysis of the effect of signal-to-noise on the resolving capability of the algorithm is presented. This technique is recommended for resolution of the distributions of quencher concentration in heterogeneous samples, of which oxygen distributions in tissue is an important example. Phosphors of practical importance for biological oxygen measurements: Pd-meso-tetra (4-carboxyphenyl) porphyrin (PdTCPP) and Pd-meso-porphyrin (PdMP) have been used to provide experimental test of the algorithm.     

Characterization of oxylipins and dioxygenase genes in the asexual fungus Aspergillus niger

Background  

Aspergillus niger is an ascomycetous fungus that is known to reproduce through asexual spores, only. Interestingly, recent genome analysis of A. niger has revealed the presence of a full complement of functional genes related to sexual reproduction [1]. An example of such genes are the dioxygenase genes which in Aspergillus nidulans, have been shown to be connected to oxylipin production and regulation of both sexual and asexual sporulation [2–4]. Nevertheless, the presence of sex related genes alone does not confirm sexual sporulation in A. niger.     

Causal isolation robustness analysis: the combinatorial strategy of circadian clock research

This paper distinguishes between causal isolation robustness analysis and independent determination robustness analysis and suggests that the triangulation of the results of different epistemic means or activities serves different functions in them. Circadian clock research is presented as a case of causal isolation robustness analysis: in this field researchers made use of the notion of robustness to isolate the assumed mechanism behind the circadian rhythm. However, in contrast to the earlier philosophical case studies on causal isolation robustness analysis (Weisberg and Reisman in Philos Sci 75:106–131, 2008; Kuorikoski et al. in Br J Philos Sci 61:541–567, 2010), robustness analysis in the circadian clock research did not remain in the level of mathematical modeling, but it combined it with experimentation on model organisms and a new type of model, a synthetic model.     

Validation-based model selection for 13C metabolic flux analysis with uncertain measurement errors

Accurate measurements of metabolic fluxes in living cells are central to metabolism research and metabolic engineering. The gold standard method is model-based metabolic flux analysis (MFA), where fluxes are estimated indirectly from mass isotopomer data with the use of a mathematical model of the metabolic network. A critical step in MFA is model selection: choosing what compartments, metabolites, and reactions to include in the metabolic network model. Model selection is often done informally during the modelling process, based on the same data that is used for model fitting (estimation data). This can lead to either overly complex models (overfitting) or too simple ones (underfitting), in both cases resulting in poor flux estimates. Here, we propose a method for model selection based on independent validation data. We demonstrate in simulation studies that this method consistently chooses the correct model in a way that is independent on errors in measurement uncertainty. This independence is beneficial, since estimating the true magnitude of these errors can be difficult. In contrast, commonly used model selection methods based on the χ²-test choose different model structures depending on the believed measurement uncertainty; this can lead to errors in flux estimates, especially when the magnitude of the error is substantially off. We present a new approach for quantification of prediction uncertainty of mass isotopomer distributions in other labelling experiments, to check for problems with too much or too little novelty in the validation data. Finally, in an isotope tracing study on human mammary epithelial cells, the validation-based model selection method identified pyruvate carboxylase as a key model component. Our results argue that validation-based model selection should be an integral part of MFA model development.