Linear n-compartment catenary models: Formulas to describe tracer amount in any compartment and identification of parameters from a concentration-time curve

Resolution of kinetic equations and parameter identification are discussed for n-compartment linear catenary models with elimination allowed from any compartment. For a given input, general formulas are derived to describe the tracer amount in any compartment as a function of the model parameters. Conversely, explicit procedures are given to identify the model parameters when the concentration-time curve is known in one arbitrary compartment, the tracer being injected into the same compartment. In this inverse problem, the solution is not unique: the model transfer rate constants can only be localized in a finite set of intervals.     

A stochastic model for then-compartment irreversible system

This paper deals with a stochasticn-compartment irreversible system with a non-homogeneous Poisson input and arbitrary residence time for each of the compartments. Results relating to the number of particles present in each of the compartments as well as the total number of particles present in the system at any time are derived. Further, explicit expressions for the auto covariance function for each compartment and the cross-covariance function between any two compartments with a given time lag are obtained. As a particular case, then-compartment irreversible system is analyzed with homogeneous Poisson input and exponential residence time distribution for each of the compartments. The possible applications of the model are discussed.     

Demiarcs,creaons and genons

Useful insights into the representation of natural systems can be gained by decomposing directed graphs (digraphs) into elementary components. Arcs of digraphs can be split into male demiarcs (outarcs) which leave vertices and female demiarcs (inarcs) which enter demiarcs. Likewise, a vertex can be split into an input perceiving side called the creaon and an output generating side called the genon. Digraphs can be regarded as being hierarchically organized because each vertex in a level-1 digraph can be expanded into a level-2 digraph. In general, each vertex of a level-i digraph can be expanded into a level-(i+1) digraph. Arcs of a level-i digraph can be regarded as bundles of level-(i + 1) arcs which are split at the vertex boundary. These elementary graphical components are shown to be useful for depicting input-output systems such as organisms, ecosystems and societies.     

Tests of the Overall Hypothesis for Arbitrary Dependence Structures

Three different methods for testing an overall hypothesis which is the intersection of n elementary hypotheses are proposed and discussed. Admitted are arbitrary dependences among the tests; for each method sharp upper bounds for the error probability are obtained.     

A Statistical Model for In Vivo Neuronal Dynamics

Single neuron models have a long tradition in computational neuroscience. Detailed biophysical models such as the Hodgkin-Huxley model as well as simplified neuron models such as the class of integrate-and-fire models relate the input current to the membrane potential of the neuron. Those types of models have been extensively fitted to in vitro data where the input current is controlled. Those models are however of little use when it comes to characterize intracellular in vivo recordings since the input to the neuron is not known. Here we propose a novel single neuron model that characterizes the statistical properties of in vivo recordings. More specifically, we propose a stochastic process where the subthreshold membrane potential follows a Gaussian process and the spike emission intensity depends nonlinearly on the membrane potential as well as the spiking history. We first show that the model has a rich dynamical repertoire since it can capture arbitrary subthreshold autocovariance functions, firing-rate adaptations as well as arbitrary shapes of the action potential. We then show that this model can be efficiently fitted to data without overfitting. We finally show that this model can be used to characterize and therefore precisely compare various intracellular in vivo recordings from different animals and experimental conditions.     

Compartmental mean residence time in a mammillary model with an effect compartment after intravenous or oral administration

A general estimation of mean residence time (MRT) in an effect compartment E, associated with a linear mammillary n-compartment model is presented: elimination takes place from the central and the effect compartments. Even though no sample is available from E, the MRT of the drug in this compartment can be estimated after intravenous or oral administation. Furthermore, the effect of MRT is independent of the route of administration. Also, with no new calculation, the method provides the area under the amount-time curve in compartment E.     

A comparison of methods for determining compartmental analysis parameters

The traditional method for determining compartmental analysis parameters relies on a visual selection of data points to be used for regression of data from each cellular compartment. This method is appropriate when the compartments are kinetically discrete and are easily discernible. However, where treatment effects on compartment parameters are being evaluated, a more objective method for determining initial parameters is desirable.

Three methods were examined for determining initial isotopic contents and half-times of ⁸⁶Rb elution from cellular compartments using theoretical data with known parameters. Experimental data from roots of Douglas fir (Pseudotsuga menziesii [Mirb.] Franco) and barley (Hordeum vulgare L.) intact seedlings were also used. The three methods were a visually assisted, linear regression on data of semilog plot of isotope elution versus time, a microcomputer-assisted, linear regression on semilog plot where maximization of the square of the correlation coefficient (r²) was the criterion to determine data points needed for each regression and a mainframe computer-assisted, direct nonlinear regression on elution data using a model of the sum of three exponential decay functions. The visual method resulted in the least accurate estimates of compartmental analysis parameters. The microcomputer-assisted and nonlinear regression methods calculated the parameters equally well.

     

Effects of suppression of bone turnover on cortical and trabecular load sharing in the canine vertebral body

The relative biomechanical effects of antiresorptive treatment on cortical thickness vs. trabecular bone microarchitecture in the spine are not well understood. To address this, T-10 vertebral bodies were analyzed from skeletally mature female beagle dogs that had been treated with oral saline (n=8 control) or a high dose of oral risedronate (0.5 mg/kg/day, n=9 RIS-suppressed) for 1 year. Two linearly elastic finite element models (36-μm voxel size) were generated for each vertebral body—a whole-vertebra model and a trabecular-compartment model—and subjected to uniform compressive loading. Tissue-level material properties were kept constant to isolate the effects of changes in microstructure alone. Suppression of bone turnover resulted in increased stiffness of the whole vertebra (20.9%, p=0.02) and the trabecular compartment (26.0%, p=0.01), while the computed stiffness of the cortical shell (difference between whole-vertebra and trabecular-compartment stiffnesses, 11.7%, p=0.15) was statistically unaltered. Regression analyses indicated subtle but significant changes in the relative structural roles of the cortical shell and the trabecular compartment. Despite higher average cortical shell thickness in RIS-suppressed vertebrae (23.1%, p=0.002), the maximum load taken by the shell for a given value of shell mass fraction was lower (p=0.005) for the RIS-suppressed group. Taken together, our results suggest that—in this canine model—the overall changes in the compressive stiffness of the vertebral body due to suppression of bone turnover were attributable more to the changes in the trabecular compartment than in the cortical shell. Such biomechanical studies provide an unique insight into higher-scale effects such as the biomechanical responses of the whole vertebra.     

Interchromosomal Biased Gene Conversion, Mutation and Selection in a Multigene Family

A mathematical model of the effects of interchromosomal biased gene conversion, mutation and natural selection on a multigene family is developed and analyzed. The model assumes two allelic states at each of n loci. The effects of genetic drift are ignored. The model is developed under the assumption of no recombination, but the analysis shows that, at equilibrium, there is no linkage disequilibrium, which implies that the conclusions are valid for arbitrary recombination among loci. At equilibrium, the balance between mutation, gene conversion and selection depends on the ratio of the mutation rates to the quantity [s + g(2α - 1)/ n], where s is the increment or decrement in relative fitness with each additional copy of one of the alleles, g is the conversion rate, and α is a measure of the bias in favor of one of the alleles. When this quantity is large relative to the mutation rates, the allele that has the net advantage, combining the effects of selection and conversion, will be nearly fixed in the multigene family. A comparison of these results with those from a comparable model of intrachromosomal biased conversion shows that biased interchromosomal conversion leads to approximately the same equilibrium copy number as does intrachromosomal conversion of the same strength. Interchromosomal conversion is much more effective in causing the substitution of one allele by another. The relative frequencies of interchromosomal and intrachromosomal conversion is indicated by the extent of the linkage disequilibrium among the loci in a multigene family.     

Estimation of the Optimal Statistical Quality Control Sampling Time Intervals Using a Residual Risk Measure

Background

An open problem in clinical chemistry is the estimation of the optimal sampling time intervals for the application of statistical quality control (QC) procedures that are based on the measurement of control materials. This is a probabilistic risk assessment problem that requires reliability analysis of the analytical system, and the estimation of the risk caused by the measurement error.

Methodology/Principal Findings

Assuming that the states of the analytical system are the reliability state, the maintenance state, the critical-failure modes and their combinations, we can define risk functions based on the mean time of the states, their measurement error and the medically acceptable measurement error. Consequently, a residual risk measure rr can be defined for each sampling time interval. The rr depends on the state probability vectors of the analytical system, the state transition probability matrices before and after each application of the QC procedure and the state mean time matrices. As optimal sampling time intervals can be defined those minimizing a QC related cost measure while the rr is acceptable. I developed an algorithm that estimates the rr for any QC sampling time interval of a QC procedure applied to analytical systems with an arbitrary number of critical-failure modes, assuming any failure time and measurement error probability density function for each mode. Furthermore, given the acceptable rr, it can estimate the optimal QC sampling time intervals.

Conclusions/Significance

It is possible to rationally estimate the optimal QC sampling time intervals of an analytical system to sustain an acceptable residual risk with the minimum QC related cost. For the optimization the reliability analysis of the analytical system and the risk analysis of the measurement error are needed.     

Parameter Estimation for the Bateman Function via Moments of the Empirical Curve

The parameters of the function f(t)=c(e^?at-e^?bt) are related in a simple way to the moments ^∞ tⁿf(t)^dt(n=0, 1, 2). Using empirical values of f, the moments can be estimated by numerical integration. Therefrom estimates of the parameters are obtained by elementary algebra.     

Large and small rearrangements in the evolution of prokaryotic genomes

Relative frequencies of large and small genome rearrangements (inversions and transpositions) in the evolution of prokaryotic genomes can be evaluated using the ratio between the index S (the ratio of the number of identical pairs of neighboring genes in two genomes to the total number of genes in the sample of interest) and 1–6L/n, where L is the mean difference in intergenic distances and n is the number of genes in the sample. The S value uniformly decreases with the fixation of genome rearrangements, while the decrease rate of 1–6L/n is determined by the rearrangement size. Specifically, large inversions and transpositions lead to a dramatic decrease in the index value, while small rearrangements result in an insignificant decrease. The ratio between these indices was computed for twenty pairs of closely related species belonging to different groups of bacteria and archaea. The pairs examined strongly differed in the relative frequency of large and small rearrangements. However, computer simulation showed that the total variation can be reproduced with the same input parameters of the model. This means that the differences observed can be stochastic and can be interpreted without assuming different mechanisms and factors of genome rearrangements for different groups of prokaryotes. Relative frequencies of large and small rearrangements displayed no noticeable correlations with taxonomic position, total rate of rearrangement fixation, habitation conditions, and the abundance of transposons and repetitive sequences. It is suggested that, in some cases, phage activity increases the frequency of large genome rearrangements.     

Relationship between loading dose and infusion rate to achieve any fraction of the steady-state level in any compartment model

When a drug is infused at a constant rate K₀, the time necessary for the concentration to reach a satisfying threshold of effectiveness may be too long. To achieve this level faster, it is useful to give simultaneously a dose D, of the same drug by intravenous injection. This paper proposes the calculation, as a function of K₀ and model parameters, of the loading dose D necessary to reach, in a time T, any fraction of the asymptotic value of the amount of drug in a compartment receiving a constant rate infusion, for any n-compartment model. As an example, the expression of D for mammillary and catenary pharmacokinetic models is derived.     

An analytical solution for a class of delay-differential equations in pharmacokinetic compartment modeling

The authors obtain the analytic expression for the solution of a differential system with time lags for any n-compartment linear model with a single input, and, by convolution, for all intakes. The theoretical result is applied to the case of one-, two-, and three-compartment models and gives insight into the pharmacokinetics of drug undergoing enterohepatic circulation: the amount of drug in any compartment is expressed for all time. Statistical results, such as the mean residence time of drug, are obtained by the same calculation.     

Edge lengths of trees from sequence data

Frequently there is a need to determine lengths associated with each edge of a phylogenetic tree, as these are often used as an indication of relative time intervals. Where this tree has been constructed from sequence data of r characters for n taxa, using the maximum parsimony model, an edge length can be determined from the differences between the inferred sequences of the end vertices of that edge. These inferred sequences are often not uniquely defined; a range of possible sequences are possible at a given internal vertex. In this paper we introduce an efficient [O(r×n)] algorithm which calculates the range of lengths on any edge over all the minimal labelings and significantly reduces the number of potential cases to be considered to obtain an objective measure of edge length.     

A Novel Explanation for Observed CaMKII Dynamics in Dendritic Spines with Added EGTA or BAPTA

We present a simplified reaction network in a single well-mixed volume that captures the general features of CaMKII dynamics observed during both synaptic input and spine depolarization. Our model can also account for the greater-than-control CaMKII activation observed with added EGTA during depolarization. Calcium input currents are modeled after experimental observations, and existing models of calmodulin and CaMKII autophosphorylation are used. After calibration against CaMKII activation data in the absence of chelators, CaMKII activation dynamics due to synaptic input via n-methyl-d-aspartate receptors are qualitatively accounted for in the presence of the chelators EGTA and BAPTA without additional adjustments to the model. To account for CaMKII activation dynamics during spine depolarization with added EGTA or BAPTA, the model invokes the modulation of Ca_V2.3 (R-type) voltage-dependent calcium channel (VDCC) currents observed in the presence of EGTA or BAPTA. To our knowledge, this is a novel explanation for the increased CaMKII activation seen in dendritic spines with added EGTA, and suggests that differential modulation of VDCCs by EGTA and BAPTA offers an alternative or complementary explanation for other experimental results in which addition of EGTA or BAPTA produces different effects. Our results also show that a simplified reaction network in a single, well-mixed compartment is sufficient to account for the general features of observed CaMKII dynamics.     

Growth of cell populations with arbitrarily distributed cycle durations. I. basic model

A discrete model is proposed describing the growth of cell populations with arbitrary frequency distributions of cycle durations. The model assumes that each cell divides into two cells at the end of its cycle, and that each new cell is assigned an individual cycle duration according to a probability distribution that can be arbitrarily defined. The increase in the cell number is calculated, either from the numbers of cells at earlier time points or from the initial conditions of the population, by a recurrence formula; it is also approximated by the optimal exponential function, whose parameters are determined by the initial conditions. The appropriate average cycle duration is shown not to be the arithmetic or geometric mean, but rather the solution to a more complex equation. Age distributions are calculated and compared with those found in the literature. The results of the model calculations are compared with computer simulations and with observed data on populations of the ciliate Tetrahymena geleii.     

The origin of fibonacci phyllotaxis—an analysis of Adler's contact pressure model and Mitchison's expanding apex model

Adler's contact pressure model for Fibonacci phyllotaxis is examined theoretically. It is shown that the model, as it stands, does not account for Fibonacci phyllotaxis, since it requires, but does not provide, a mechanism for initiating new primordia with increasingly greater precision as phyllotaxis rises. Modifications are suggested which remedy this deficiency in the model; one of these modifications involves a combination of Adler's model with Mitchison's model.From a comparison of the ranges of divergence angles permitted by Adler's model against Fujita's measurements of divergence angles in plants with low phyllotaxis, it is shown that the modified contact pressure model, if based on the concept of mechanical pressures between primordia in contact, cannot account for the divergence angles found in low phyllotaxis systems. However it is shown that this deficiency can be overcome if the contact pressure effect is regarded as a chemical phenomenon, mediated by a growth inhibitor produced by the prirnordia and moving more readily in vertical directions than in other directions.Mitchison's model, which is based on the concepts of an expanding apex and primordium initiation by existing primordia, is shown to account for Fibonacci phyllotaxis only if phyllotaxis rises sufficiently slowly; to guarantee that an F_n + F_n+1 system can develop there must already be at least F_n+1 primordia present in an F_n?1 + F_n system, at least F_n primordia in an F_n?2 + F_n?1 system, and so on down to at least three primordia in a 1 + 2 system, making a total of at least F_n+3?5 primordia (where F_n = nth term of the Fibonacci series with F₁ = F₂ = 1). Adler's model, modified, requires only that F_{n + 1} primordia be present with divergence angles in the range 120–180° to guarantee that an F_n + F_{n + 1} system can develop.     

Elementary network reconstruction: A framework for the analysis of regulatory networks in biological systems

Complexity of regulatory networks arises from the high degree of interaction between network components such as DNA, RNA, proteins, and metabolites. We have developed a modeling tool, elementary network reconstruction (ENR), to characterize these networks. ENR is a knowledge-driven, steady state, deterministic, quantitative modeling approach based on linear perturbation theory. In ENR we demonstrate a novel means of expressing control mechanisms by way of dimensionless steady state gains relating input and output variables, which are purely in terms of species abundances (extensive variables). As a result of systematic enumeration of network species in n×n matrix, the two properties of linear perturbation are manifested in graphical representations: transitive property is evident in a special L-shape structure, and additive property is evident in multiple L-shape structures arriving at the same matrix cell. Upon imposing mechanistic (lowest-level) gains, network self-assembly through transitive and additive properties results in elucidation of inherent topology and explicit cataloging of higher level gains, which in turn can be used to predict perturbation results. Application of ENR to the regulatory network behind carbon catabolite repression in Escherichia coli is presented. Through incorporation of known molecular mechanisms governing transient and permanent repressions, the ENR model correctly predicts several key features of this regulatory network, including a 50% downshift in intracellular cAMP level upon exposure to glucose. Since functional genomics studies are mainly concerned with redistribution of species abundances in perturbed systems, ENR could be exploited in the system-level analysis of biological systems.     

Unidirectional fluxes in saturated single-file pores of biological and artificial membranes I. Pores containing no more than one vacancy

A theoretical treatment of isotope fluxes in filled single-file pores is given. The treatment is confined to pores permitting only one vacancy at a time. Tracer flux, unidirectional fluxes and net flux are calculated. The exponent n which is obtained by representing the ratio of unidirectional fluxes as a power of the electrochemical activity-ratio proves to be closely related with m, the number of sites per pore. The minimum and maximum values of n are m?1 and m. It is shown that measurement of n provides sufficient information to determine m if the energetic properties of sites and barriers do not change too much with varying concentrations. Unlike the unidirectional fluxes, the net fluxes yield no direct information about the number of sites. The concentration dependence of the net fluxes, however, can be used to discriminate between the two limiting cases, n = m?1 and n = m. It is interesting with regard to the instantaneous potassium currents of nerve membranes that a suitable choice of the energetic parameters of a filled pore leads to a quasi-linearity of the net flux-voltage curves, which is only slightly affected by variation of the external concentration.