Measurement of protein size in concentrated solutions by small angle X‐ray scattering

By simulations on the distance distribution function (DDF) derived from small angle X‐ray scattering (SAXS) theoretical data of a dense monodisperse system, we found a quantitative mathematical correlation between the apparent size of a spherically symmetric (or nearly spherically symmetric) homogenous particle and the concentration of the solution. SAXS experiments on protein solutions of human hemoglobin and horse myoglobin validated the correlation. This gives a new method to determine, from the SAXS DDF, the size of spherically symmetric (or nearly spherically symmetric) particles of a dense monodisperse system, specifically for protein solutions with interference effects.     

Mathematical Estimation of the Level of Microbial Contamination on Spacecraft Surfaces by Volumetric Air Sampling

Microbiological sampling methods presently used for enumeration of microorganisms on spacecraft surfaces require contact with easily damaged components. Estimation of viable particles on surfaces using air sampling methods in conjunction with a mathematical model would be desirable. Parameters necessary for the mathematical model are the effect of angled surfaces on viable particle collection and the number of viable cells per viable particle. Deposition of viable particles on angled surfaces closely followed a cosine function, and the number of viable cells per viable particle was consistent with a Poisson distribution. Other parameters considered by the mathematical model included deposition rate and fractional removal per unit time. A close nonlinear correlation between volumetric air sampling and airborne fallout on surfaces was established with all fallout data points falling within the 95% confidence limits as determined by the mathematical model.     

A modular approach for modeling the cell cycle based on functional response curves

Modeling biochemical reactions by means of differential equations often results in systems with a large number of variables and parameters. As this might complicate the interpretation and generalization of the obtained results, it is often desirable to reduce the complexity of the model. One way to accomplish this is by replacing the detailed reaction mechanisms of certain modules in the model by a mathematical expression that qualitatively describes the dynamical behavior of these modules. Such an approach has been widely adopted for ultrasensitive responses, for which underlying reaction mechanisms are often replaced by a single Hill function. Also time delays are usually accounted for by using an explicit delay in delay differential equations. In contrast, however, S-shaped response curves, which by definition have multiple output values for certain input values and are often encountered in bistable systems, are not easily modeled in such an explicit way. Here, we extend the classical Hill function into a mathematical expression that can be used to describe both ultrasensitive and S-shaped responses. We show how three ubiquitous modules (ultrasensitive responses, S-shaped responses and time delays) can be combined in different configurations and explore the dynamics of these systems. As an example, we apply our strategy to set up a model of the cell cycle consisting of multiple bistable switches, which can incorporate events such as DNA damage and coupling to the circadian clock in a phenomenological way.     

Fluctuation analysis in small chemically reacting systems

A theory of noise fluctuations is developed which is applicable to systems of any size in which unimolecular or bimolecular reactions are occurring. The main difference between small and large reacting systems is that in the former the probability of finding a particle in a particular state does not obey a Gaussian distribution, but satisfies a distribution which reflects the mechanism of the chemical reaction. This difference is reflected in the main result of the theory: an autocorrelation function that is expressible as a sum of exponentials, the amplitudes of which are explicit functions of the moments of the distribution. Thus, by using small systems, the autocorrelation function,in principle, allows the elucidation of reaction mechanisms. Numerical simulations indicate that for reacting systems having ten or fewer particles, the deviation of the autocorrelation function from a single exponential should be easily detectable, and that estimates of the first four moments of the distribution should be possible. Accurate inference of the distribution, however, will require further mathematical and experimental advances.     

The alpha particulate liver glycogen. A morphometric approach to the kinetics of its synthesis and degradation.

Electron micrographs of rat hepatocytes with a glycogen content between 0.36 and 2.55% (w/w) were submitted to morphometrical analysis. From the number and size of glycogen profiles, the distribution of radius and volume of glycogen alpha particles were computed. The 7-fold difference in glycogen content was accompanied by an only 1.8-fold increase in the mean volume of the particles while their number increased by a factor of 4. On the basis of these observations, it is proposed that the population of glycogen particles can be divided in two groups. The first one is made of growing particles, still associated with glycogen synthase; they are the only particles present at low glycogen concentration and their number is limited. Application of a simple mathematical model allows to estimate their number in hepatocytes as 49 X 10(12) particles . ml-1. The second group is made of glycogen particles which have reached their maximal size and the number of which is in principle unlimited. The maximal particle size is estimated to be 0.36 X 10(-15) ml, corresponding to an average molecular weight of 178 X 10(6). The average molecular weight of glycogen, as measured from the actual size of the particles, varied from 89 X 10(6) to 161 X 10(6).     

Supramolecular photosystem II organization in grana thylakoid membranes: evidence for a structured arrangement

The distribution of photosystem (PS) II complexes in stacked grana thylakoids derived from electron microscopic images of freeze-fractured chloroplasts are examined for the first time using mathematical methods. These characterize the particle distribution in terms of a nearest neighbor distribution function and a pair correlation function. The data were compared with purely random distributions calculated by a Monte Carlo simulation. The analysis reveals that the PSII distribution in grana thylakoids does not correspond to a random protein mixture but that ordering forces lead to a structured arrangement on a supramolecular level. Neighboring photosystems are significantly more separated than would be the case in a purely random distribution. These results are explained by structural models, in which boundary lipids and light-harvesting complex (LHC) II trimers are arranged between neighboring PSII. Furthermore, the diffusion of PSII was analyzed by a Monte Carlo simulation with a protein density of 80% area occupation (determined for grana membranes). The mobility of the photosystems is severely reduced by the high protein density. From an estimate of the mean migration time of PSII from grana thylakoids to stroma lamellae, it becomes evident that this diffusion contributes significantly to the velocity of the repair cycle of photoinhibited PSII.     

Modeling Taxa-Abundance Distributions in Microbial Communities using Environmental Sequence Data

We show that inferring the taxa-abundance distribution of a microbial community from small environmental samples alone is difficult. The difficulty stems from the disparity in scale between the number of genetic sequences that can be characterized and the number of individuals in communities that microbial ecologists aspire to describe. One solution is to calibrate and validate a mathematical model of microbial community assembly using the small samples and use the model to extrapolate to the taxa-abundance distribution for the population that is deemed to constitute a community. We demonstrate this approach by using a simple neutral community assembly model in which random immigrations, births, and deaths determine the relative abundance of taxa in a community. In doing so, we further develop a neutral theory to produce a taxa-abundance distribution for large communities that are typical of microbial communities. In addition, we highlight that the sampling uncertainties conspire to make the immigration rate calibrated on the basis of small samples very much higher than the true immigration rate. This scale dependence of model parameters is not unique to neutral theories; it is a generic problem in ecology that is particularly acute in microbial ecology. We argue that to overcome this, so that microbial ecologists can characterize large microbial communities from small samples, mathematical models that encapsulate sampling effects are required.     

The transducer model for contrast detection and discrimination: formal relations, implications, and an empirical test

The transducer function mu for contrast perception describes the nonlinear mapping of stimulus contrast onto an internal response. Under a signal detection theory approach, the transducer model of contrast perception states that the internal response elicited by a stimulus of contrast c is a random variable with mean mu(c). Using this approach, we derive the formal relations between the transducer function, the threshold-versus-contrast (TvC) function, and the psychometric functions for contrast detection and discrimination in 2AFC tasks. We show that the mathematical form of the TvC function is determined only by mu, and that the psychometric functions for detection and discrimination have a common mathematical form with common parameters emanating from, and only from, the transducer function mu and the form of the distribution of the internal responses. We discuss the theoretical and practical implications of these relations, which have bearings on the tenability of certain mathematical forms for the psychometric function and on the suitability of empirical approaches to model validation. We also present the results of a comprehensive test of these relations using two alternative forms of the transducer model: a three-parameter version that renders logistic psychometric functions and a five-parameter version using Foley's variant of the Naka-Rushton equation as transducer function. Our results support the validity of the formal relations implied by the general transducer model, and the two versions that were contrasted account for our data equally well.     

Quantitation of the major proteins of the human erythrocyte membrane by amino acid analysis

The proteins of the human erythrocyte membrane have been separated by sodium dodecyl sulfate-polyacrylamide gel electrophoresis, the resulting gel cut into 2-mm sections, and the amino acid content and composition of each slice measured using a sensitive method of amino acid analysis. The distribution of proteins among bands coincides closely with that estimated using staining intensity. Composition data for the major bands agree well with those reported for the purified proteins in all cases except that of band 4.5. Using quantitative amino acid analysis and resistive particle counting the total protein content of purified membranes was found to be 3.75 X 10(-13) g/cell, which is substantially less than previous estimates based on indirect methods. These data are used to calculate the number of copies of each major protein in a single erythrocyte.     

TOWARDS A GENERAL THEORY OF GROUP SELECTION

The longstanding debate about the importance of group (multilevel) selection suffers from a lack of formal models that describe explicit selection events at multiple levels. Here, we describe a general class of models for two‐level evolutionary processes which include birth and death events at both levels. The models incorporate the state‐dependent rates at which these events occur. The models come in two closely related forms: (1) a continuous‐time Markov chain, and (2) a partial differential equation (PDE) derived from (1) by taking a limit. We argue that the mathematical structure of this PDE is the same for all models of two‐level population processes, regardless of the kinds of events featured in the model. The mathematical structure of the PDE allows for a simple and unambiguous way to distinguish between individual‐ and group‐level events in any two‐level population model. This distinction, in turn, suggests a new and intuitively appealing way to define group selection in terms of the effects of group‐level events. We illustrate our theory of group selection by applying it to models of the evolution of cooperation and the evolution of simple multicellular organisms, and then demonstrate that this kind of group selection is not mathematically equivalent to individual‐level (kin) selection.     

Binding of peripheral proteins to mixed lipid membranes: effect of lipid demixing upon binding

Binding isotherms have been determined for the association of horse heart cytochrome c with dioleoyl phosphatidylglycerol (DOPG)/dioleoyl phosphatidylcholine (DOPC) bilayer membranes over a range of lipid compositions and ionic strengths. In the absence of protein, the DOPG and DOPC lipids mix nearly ideally. The binding isotherms have been analyzed using double layer theory to account for the electrostatics, either the Van der Waals or scaled particle theory equation of state to describe the protein surface distribution, and a statistical thermodynamic formulation consistent with the mass-action law to describe the lipid distribution. Basic parameters governing the electrostatics and intrinsic binding are established from the binding to membranes composed of anionic lipid (DOPG) alone. Both the Van der Waals and scaled particle equations of state can describe the effects of protein distribution on the DOPG binding isotherms equally well, but with different values of the maximum binding stoichiometry (13 lipids/protein for Van der Waals and 8 lipids/protein for scaled particle theory). With these parameters set, it is then possible to derive the association constant, Kr, of DOPG relative to DOPC for surface association with bound cytochrome c by using the binding isotherms obtained with the mixed lipid membranes. A value of Kr (DOPG:DOPC) = 3.3-4.8, depending on the lipid stoichiometry, is determined that consistently describes the binding at different lipid compositions and different ionic strengths. Using the value of Kr obtained it is possible to derive the average in-plane lipid distribution and the enhancement in protein binding induced by lipid redistribution using the statistical thermodynamic theory.     

A residue critical for flavin binding in flavocytochrome b2 from Baker's yeast. Inactivation and labeling of flavin-free enzyme by 2-keto-3-butynoate

The reagent 2-keto-3-butynoic acid is the product formed in the reaction between the suicide reagent 2-hydroxy-3-butynoate and a number of flavoproteins. We describe in this paper the inactivation of flavin-free flavocytochrome b2 by 2-keto-3-butynoate, in a rapid reaction which introduces 0.9 mol reagent for total inactivation. The modification results in loss of affinity for flavin and affects a cysteine residue. We also describe in this paper a simple enzymatic method for preparing 2-keto-3-butynoate, as well as some properties of the reagent, in particular its stability and susceptibility to nucleophilic attack. We show that at neutral pH it is highly specific for thiol compounds. Some properties of the adduct formed with glutathione are described. These experiments should pave the way for the use of 2-keto-3-butynoate with other proteins.     

Application of template matching technique to particle detection in electron micrographs

Template matching together with the comprehensive theory of image formation in electron microscope provides an optimal (in Bayesian sense) tool for solving one of the outstanding problems in single particle analysis, i.e., automatic selection of particle views from noisy micrograph fields. The method is based on the assumption that the reference three-dimensional structure is known and that the relevant parameters of the model of the image formation process can be estimated. In the first stage of the procedure, a set of possible particle views is generated using the available reference structure. The template images are constructed as linear combinations of available particle views using a clustering technique. Next, the micrograph noise characteristic is established using an automated contrast transfer function (CTF) estimation procedure. Finally, the CTF parameters calculated are used to construct a matched filter and correlation functions corresponding to the available template images are calculated. In order to alleviate the problem of the biased caused by varying image formation conditions, a decision making strategy based on the predicted distribution of correlation coefficients is proposed. It is demonstrated that due to the inclusion of CTF considerations, the template matching method performed very well in a broad range of microscopy conditions.     

Quantifying intraclass correlations for count and time‐to‐event data

The intraclass correlation is commonly used with clustered data. It is often estimated based on fitting a model to hierarchical data and it leads, in turn, to several concepts such as reliability, heritability, inter‐rater agreement, etc. For data where linear models can be used, such measures can be defined as ratios of variance components. Matters are more difficult for non‐Gaussian outcomes. The focus here is on count and time‐to‐event outcomes where so‐called combined models are used, extending generalized linear mixed models, to describe the data. These models combine normal and gamma random effects to allow for both correlation due to data hierarchies as well as for overdispersion. Furthermore, because the models admit closed‐form expressions for the means, variances, higher moments, and even the joint marginal distribution, it is demonstrated that closed forms of intraclass correlations exist. The proposed methodology is illustrated using data from agricultural and livestock studies.     

Stochastic modelling of biochemical systems of multi-step reactions using a simplified two-variable model

Background

A fundamental issue in systems biology is how to design simplified mathematical models for describing the dynamics of complex biochemical reaction systems. Among them, a key question is how to use simplified reactions to describe the chemical events of multi-step reactions that are ubiquitous in biochemistry and biophysics. To address this issue, a widely used approach in literature is to use one-step reaction to represent the multi-step chemical events. In recent years, a number of modelling methods have been designed to improve the accuracy of the one-step reaction method, including the use of reactions with time delay. However, our recent research results suggested that there are still deviations between the dynamics of delayed reactions and that of the multi-step reactions. Therefore, more sophisticated modelling methods are needed to accurately describe the complex biological systems in an efficient way.

Results

This work designs a two-variable model to simplify chemical events of multi-step reactions. In addition to the total molecule number of a species, we first introduce a new concept regarding the location of molecules in the multi-step reactions, which is the second variable to represent the system dynamics. Then we propose a simulation algorithm to compute the probability for the firing of the last step reaction in the multi-step events. This probability function is evaluated using a deterministic model of ordinary differential equations and a stochastic model in the framework of the stochastic simulation algorithm. The efficiency of the proposed two-variable model is demonstrated by the realization of mRNA degradation process based on the experimentally measured data.

Conclusions

Numerical results suggest that the proposed new two-variable model produces predictions that match the multi-step chemical reactions very well. The successful realization of the mRNA degradation dynamics indicates that the proposed method is a promising approach to reduce the complexity of biological systems.

     

Modeling of Coulomb collisions in a kinetic description of the electron cyclotron resonance plasma heating

The electron distribution function is modeled numerically with allowance for Coulomb collisions and quasilinear effects under cyclotron resonance conditions by solving a two-dimensional kinetic equation containing the quasilinear diffusion operator and the Coulomb collision operator in the Landau form. Two simplified model collision integrals that make it possible to describe electron heating by microwave radiation are considered. The first model collision operator is obtained by introducing the parametric time dependence of the temperature of the background Maxwellian electrons into the linear collision integral. It is shown that the heating of the bulk electrons can be described in a noncontradictory way if the temperature dynamics of the background electrons is calculated from the equation of energy balance, which is governed by the amount of the microwave power absorbed by the resonant electrons with the distribution function modified due to quasilinear effects. This conclusion is confirmed in a more rigorous fashion by comparing the solutions obtained using the first model Coulomb collision integral with those obtained using the second model integral, namely, the nonlinear operator derived by averaging the distribution function of the scattering electrons over pitch angles. The time-dependent linear collision integral is used to obtain analytic solutions describing quasi-steady electron heating with allowance for the quasilinear degradation of microwave power absorption.     

Modular decomposition of metabolic systems via null-space analysis

We describe a method by which the reactions in a metabolic system may be grouped hierarchically into sets of modules to form a metabolic reaction tree. In contrast to previous approaches, the method described here takes into account the fact that, in a viable network, reactions must be capable of sustaining a steady-state flux. In order to achieve this decomposition we introduce a new concept--the reaction correlation coefficient, phi, and show that this is a logical extension of the concept of enzyme (or reaction) subsets. In addition to their application to modular decomposition, reaction correlation coefficients have a number of other interesting properties, including a convenient means for identifying disconnected subnetworks in a system and potential applications to metabolic engineering. The method computes reaction correlation coefficients from an orthonormal basis of the null-space of the stoichiometry matrix. We show that reaction correlation coefficients are uniquely defined, even though the basis of the null-space is not. Once a complete set of reaction correlation coefficients is calculated, a metabolic reaction tree can be determined through the application of standard programming techniques. Computation of the reaction correlation coefficients, and the subsequent construction of the metabolic reaction tree is readily achievable for genome-scale models using a commodity desk-top PC.     

Applications of DL_POLY to modelling of mesoscopic particulate systems

The molecular dynamics package DL_POLY has at its heart a number of versatile and efficient dynamics algorithms that can readily be adapted to extend the application of this code well beyond the time and length scales typically associated with atomistic simulations. In order to achieve this, it is necessary to substitute the appropriate interparticle potentials and forces in place of the default functional forms in DL_POLY, which are mainly suitable for molecular systems. To facilitate this, it may be required to incorporate additional factors, into the simulation, such as velocity-dependent dissipation effects (friction), rotational degrees of freedom and non-spherosymmetric forces. In this paper, we will discuss some of the practical details of implementing these changes to DL_POLY (version 2) together with applications of discrete particle dynamics methods, such as dissipative particle dynamics (DPD) and granular dynamics (GD) (also known as the discrete or distinct element method, DEM) to particle packing in composite systems and pharmaceutical powders. We also consider how well the approach of simulating particles of arbitrary shape using rigid assemblies of fused soft spheres (each individually interacting via pairwise continuous potentials) compares to true hard-body simulations of polygonal particles.