Analysis of tracer experiments in non-conservative steady-state systems

A method is developed for finding the transfer and localization rates and the volumes ofN compartment steady-state biological systems from experimental results. It is shown that a complete solution for certain systems in which the rates and volumes remain constant and in which there is access to all compartments can be obtained by using a single radioactive tracer. The information obtainable from experiments wherein some compartments are not accessible is analyzed for mammillary and catenary systems. Conservative systems are handled as special cases in which the localization is zero while anisotropic membranes separating compartments are shown to introduce no additional mathematical difficulty whenever all compartments are accessible. The limitations on the use of this method of multi-compartment tracer analysis are briefly discussed. Research supported by the Atomic Energy Commission, Contract AT (30-1)-1551.     

Optimal association with partly missing key vectors

A new association scheme which can still recall appropriate data when some key elements are missing (blank) is presented. The traditional associative memory models are designed to deal with complete (memorized) keys, but in the real world, key elements are often missing due to error, equipment failure, observation difficulty, etc. The traditional models, in this case, can not have an optimal association except for special cases. When an incomplete key containing blanks is given, we wish to get the same data, as nearly as possible, as would be obtained with the complete key. In this paper, the optimal associative memory model which operates with partly missing keys is proposed. The model is constructed on the basis of the theory of the pseudoinverse of matrices. Even from the incomplete keys which contain a large percentage of blanks, the model recalls the appropriate data optimally under the MSE criterion. From the results of computer simulations, we can show that the model has the expected ability.     

Analysis of tracer experiments: VII. General multi-compartment systems imbedded in non-homogeneous inaccessible media

An integral equation analysis of generaln compartment steady state systems imbedded in static media of arbitrary complexity has been developed. A set of initial entry functions can be found which serve to determine a corresponding set of partitioned initial entry functions. The partitioned functions, in turn, can be used to predict the probabilities and time courses of various transport histories and to determine all steady state rates of flow between measured compartments. The method is quite general, being completely applicable, for example, to closed systems, to cyclic systems and to systems in which relatively rapid (but finite) exchange between compartments occurs.     

Measuring the stability of partly folded proteins using TMAO

Standard methods for measuring free energy of protein unfolding by chemical denaturation require complete folding at low concentrations of denaturant so that a native baseline can be observed. Alternatively, proteins that are completely unfolded in the absence of denaturant can be folded by addition of the osmolyte trimethylamine N-oxide (TMAO), and the unfolding free energy can then be calculated through analysis of the refolding transition. However, neither chemical denaturation nor osmolyte-induced refolding alone is sufficient to yield accurate thermodynamic unfolding parameters for partly folded proteins, because neither method produces both native and denatured baselines in a single transition. Here we combine urea denaturation and TMAO stabilization as a means to bring about baseline-resolved structural transitions in partly folded proteins. For Barnase and the Notch ankyrin domain, which both show two-state equilibrium unfolding, we found that DeltaG degrees for unfolding depends linearly on TMAO concentration, and that the sensitivity of DeltaG degrees to urea (the m-value) is TMAO independent. This second observation confirms that urea and TMAO exert independent effects on stability over the range of cosolvent concentrations required to bring about baseline-resolved structural transitions. Thermodynamic parameters calculated using a global fit that assumes additive, linear dependence of DeltaG degrees on each cosolvent are similar to those obtained by standard urea-induced unfolding in the absence of TMAO. Finally, we demonstrate the applicability of this method to measurement of the free energy of unfolding of a partly folded protein, a fragment of the full-length Notch ankyrin domain.     

Singular solutions of the diffusion equation of population genetics

The forward diffusion equation for gene frequency dynamics is solved subject to the condition that the total probability is conserved at all times. This can lead to solutions developing singular spikes (Dirac delta functions) at the gene frequencies 0 and 1. When such spikes appear in solutions they signal gene loss or gene fixation, with the "weight" associated with the spikes corresponding to the probability of loss or fixation. The forward diffusion equation is thus solved for all gene frequencies, namely the absorbing frequencies of 0 and 1 along with the continuous range of gene frequencies on the interval (0,1) that excludes the frequencies of 0 and 1. Previously, the probabilities of the absorbing frequencies of 0 and 1 were found by appeal to the backward diffusion equation, while those in the continuous range (0,1) were found from the forward diffusion equation. Our unified approach does not require two separate equations for a complete dynamical treatment of all gene frequencies within a diffusion approximation framework. For cases involving mutation, migration and selection, it is shown that a property of the deterministic part of gene frequency dynamics determines when fixation and loss can occur. It is also shown how solution of the forward equation, at long times, leads to the standard result for the fixation probability.     

Segregation of partly melted DNA molecules

Segregation of partly melted DNA molecules is a convenient and efficient method to isolate DNA fragments associated with CpG islands. The method stands on the observation that the electrophoretic mobility of partly melted DNA fragments in a denaturing gradient gel is low and that they persist in the gel so long as the remaining helical part is sufficiently resistant to strand dissociation and dissociates slowly. Such features are observed in DNA fragments derived from CpG islands. These DNA fragments are preferentially retained in a denaturing gradient gel after prolonged electric field exposure, permitting the enrichment of DNA fragments derived from CpG islands. The principle and practical application of this method are reviewed.     

DeepCME: A deep learning framework for computing solution statistics of the chemical master equation

Stochastic models of biomolecular reaction networks are commonly employed in systems and synthetic biology to study the effects of stochastic fluctuations emanating from reactions involving species with low copy-numbers. For such models, the Kolmogorov’s forward equation is called the chemical master equation (CME), and it is a fundamental system of linear ordinary differential equations (ODEs) that describes the evolution of the probability distribution of the random state-vector representing the copy-numbers of all the reacting species. The size of this system is given by the number of states that are accessible by the chemical system, and for most examples of interest this number is either very large or infinite. Moreover, approximations that reduce the size of the system by retaining only a finite number of important chemical states (e.g. those with non-negligible probability) result in high-dimensional ODE systems, even when the number of reacting species is small. Consequently, accurate numerical solution of the CME is very challenging, despite the linear nature of the underlying ODEs. One often resorts to estimating the solutions via computationally intensive stochastic simulations. The goal of the present paper is to develop a novel deep-learning approach for computing solution statistics of high-dimensional CMEs by reformulating the stochastic dynamics using Kolmogorov’s backward equation. The proposed method leverages superior approximation properties of Deep Neural Networks (DNNs) to reliably estimate expectations under the CME solution for several user-defined functions of the state-vector. This method is algorithmically based on reinforcement learning and it only requires a moderate number of stochastic simulations (in comparison to typical simulation-based approaches) to train the “policy function”. This allows not just the numerical approximation of various expectations for the CME solution but also of its sensitivities with respect to all the reaction network parameters (e.g. rate constants). We provide four examples to illustrate our methodology and provide several directions for future research.     

Computer model of the metabolism of phenylalanine in normal subjects and in patients with phenylketonuria

The techniques of systems analysis have been applied to the metabolism of phenylalanine in healthy subjects and in phenylketonuria to derive a computer model. The model consists of seven compartments and four biological functions. Enzyme reactions are assumed to obey Michaelis-Menten kinetics. The computer program is written in FORTRAN. A biological validation of the model has been performed. The computer model has potential value for estimating concentrations of key metabolites in human organs which are not accessible to direct analysis from measurement of metabolites in blood and urine and phenylalanine intake. Such estimates would be helpful in clinical decisions concerning modification or termination of dietary treatment in phenylketonuria.     

The analysis of enzyme progress curves by numerical differentiation, including competitive product inhibition and enzyme reactivation

A new method for analyzing steady-state enzyme kinetic data is presented. The technique, which is based on the numerical differentiation of the complete reaction curve, has several advantages over initial velocity and integrated Michaelis-Menten equation methods. The differentiated data are fit to the differential equation describing the appropriate kinetic scheme. This approach is particularly valuable in cases of strong competitive product inhibition and of changing concentrations of active enzyme. The method assumes a reversible reaction and is applicable to a very wide variety of steady-state kinetic schemes. A particular advantage of this approach over integrated methods is that it is independent of [S0] and hence of errors in [S0]. The combination of complete progress curve and computer analysis makes this approach very efficient with respect to both time and materials. Running on an IBM PC XT or equivalent microcomputer with an 8087 coprocessor, the analyses are very fast, the complete process usually being complete in a minute or two. The utility of the technique is demonstrated by application to both simulated and real data. We show that the differentiation of the progress curve for the ribonuclease-catalyzed hydrolysis of 2',3'-cyclic cytidine monophosphate reveals strong product inhibition by 3'-CMP, and this product inhibition accounts for the large discrepancies reported in the literature for the value of Km for this substrate. The method was also applied to determine the rate of reactivation of beta-lactamase which had been reversibly inactivated by cloxacillin. Since large numbers of data points are required for the numerical differentiation the method has become practical only with the advent of computer-acquired data systems.     

Calculation of physiological acid-base parameters in multicompartment systems with application to human blood.

A general formalism for calculating parameters describing physiological acid-base balance in single compartments is extended to multicompartment systems and demonstrated for the multicompartment example of human whole blood. Expressions for total titratable base, strong ion difference, change in total titratable base, change in strong ion difference, and change in Van Slyke standard bicarbonate are derived, giving calculated values in agreement with experimental data. The equations for multicompartment systems are found to have the same mathematical interrelationships as those for single compartments, and the relationship of the present formalism to the traditional form of the Van Slyke equation is also demonstrated. The multicompartment model brings the strong ion difference theory to the same quantitative level as the base excess method.     

Simplified calculation of cell water content during freezing and thawing in nonideal solutions of cryoprotective agents and its possible application to the study of “solution effects” injury

A simplified equation has been derived which reduces the time and complexity of calculating subzero cell water content during freezing and thawing as compared to calculation by means of the Mazur equation. The simplified equation also allows inclusion of the effects of nonideality of glycerol and dimethyl sulfoxide aqueous electrolyte solutions. Furthermore, a very simple, iterative method of solving the simplified equation has been shown to give results which are equivalent to those obtained using the far more difficult and involved Runge-Kutta technique. It is hoped that these simplifications will make calculation of cell water content accessible to more cryobiologists. In addition, possible applications of such calculations to mechanistic issues in the area of “solution effects” injury are discussed.     

Generalized log-rank tests for partly interval-censored failure time data

In this paper, we consider incomplete survival data: partly interval-censored failure time data where observed data include both exact and interval-censored observations on the survival time of interest. We present a class of generalized log-rank tests for this type of survival data and establish their asymptotic properties. The method is evaluated using simulation studies and illustrated by a set of real data from a diabetes study.     

A Heuristic Search Approach Using Approximate Solutions to Petri Net State Equations for Scheduling Flexible Manufacturing Systems

This paper proposes a new heuristic search approach based on an analytic theory of the Petri net state equations for scheduling flexible manufacturing systems (FMSs) with the goal of minimizing makespan. The proposed method models an FMS using a timed Petri net and exploits approximate solutions of the net's state equation to predict the total cost (makespan) from the initial state through the current state to the goal. That is, the heuristic function considers global information provided by the state equation. This makes the method possible to obtain solutions better than those obtained using prior works (Lee and DiCesare, 1994a, 1994b) that consider only the current status or limited global information. In addition, to reduce memory requirement and thus to increase the efficiency of handling larger systems, the proposed scheduling algorithm contains a procedure to reduce the searched state space.     

A partly parametric additive risk model

Aalen's additive risk model allows the influence of each covariateto vary separately over time. Although allowing greater flexibilityof temporal structure than a Cox model, Aalen's model is morelimited in the number of covariates it can handle. We introducea partly parametric version of Aalen's model in which the influenceof only a few covariates varies nonparametrically over time,and that of the remaining covariates is constant. Efficientprocedures for fitting this new model are developed and studied.The approach is applied to data from the Medical Research Council'smyelomatosis trials.     

Complete Numerical Solution of the Diffusion Equation of Random Genetic Drift

A numerical method is presented to solve the diffusion equation for the random genetic drift that occurs at a single unlinked locus with two alleles. The method was designed to conserve probability, and the resulting numerical solution represents a probability distribution whose total probability is unity. We describe solutions of the diffusion equation whose total probability is unity as complete. Thus the numerical method introduced in this work produces complete solutions, and such solutions have the property that whenever fixation and loss can occur, they are automatically included within the solution. This feature demonstrates that the diffusion approximation can describe not only internal allele frequencies, but also the boundary frequencies zero and one. The numerical approach presented here constitutes a single inclusive framework from which to perform calculations for random genetic drift. It has a straightforward implementation, allowing it to be applied to a wide variety of problems, including those with time-dependent parameters, such as changing population sizes. As tests and illustrations of the numerical method, it is used to determine: (i) the probability density and time-dependent probability of fixation for a neutral locus in a population of constant size; (ii) the probability of fixation in the presence of selection; and (iii) the probability of fixation in the presence of selection and demographic change, the latter in the form of a changing population size.     

Oxidative quenching of spruce thermomechanical pulp fiber autofluorescence monitored in real time by confocal laser scanning microscopy-implications for lignin autofluorescence

Confocal laser scanning microscopy (CLSM) was used to monitor real-time lignin autofluorescence intensities from different cell wall compartments of spruce fibers during oxidation by laccase-2,2'-azinobis-3-ethylbenzthiazoline-6-sulfonate (ABTS) treatment. CLSM data revealed an instant emission quenching from all cell wall compartments, including those not physically accessible to enzyme or ABTS, followed by an additional decrease simultaneously in all cell wall compartments over a 45 min time course. The importance of site-to-site excitation energy transfer in lignin efficient over long distance is suggested.     

Model diagnostic tests for selecting informative correlation structure in correlated data

In the generalized method of moments approach to longitudinaldata analysis, unbiased estimating functions can be constructedto incorporate both the marginal mean and the correlation structureof the data. Increasing the number of parameters in the correlationstructure corresponds to increasing the number of estimatingfunctions. Thus, building a correlation model is equivalentto selecting estimating functions. This paper proposes a chi-squaredtest to choose informative unbiased estimating functions. Weshow that this methodology is useful for identifying which sourceof correlation it is important to incorporate when there aremultiple possible sources of correlation. This method can alsobe applied to determine the optimal working correlation forthe generalized estimating equation approach.     

Specific accumulation of 17 alpha-hydroxyprogesterone in microsomal membranes during the process of cytochrome P-450(C-17)-catalysed androgen biosynthesis. A dynamic study of intermediate formation and turnover.

A complete dynamic analysis of cytochrome P-450(C-17)-catalysed androgen biosynthesis from a single dose of progesterone and 17 alpha-hydroxyprogesterone in a double-label double-substrate experiment was performed in order to elucidate the controversial intermediacy of 17 alpha-hydroxyprogesterone. Label distribution within the steroid fractions as well as in the membrane and buffer compartments yields direct evidence that the endogenously formed 17 alpha-hydroxyprogesterone (which is in an 'intermediate state') accumulates to a higher degree in microsomal membranes than does the exogenously added 17 alpha-hydroxyprogesterone (which is in a 'substrate state') under certain conditions. It is also demonstrated that endogenously formed 17 alpha-hydroxyprogesterone may partly leave the membrane compartment (in terms of a 'leakage' or 'overflow' phenomenon) and is then able to equilibrate with the pool of exogenously added 17 alpha-hydroxyprogesterone. Since only the label distribution in the membrane-associated (but not always in the aqueous) 17 alpha-hydroxyprogesterone pool corresponds to the label distribution in the androgen fraction, it is concluded that only the membrane-associated 17 alpha-hydroxyprogesterone pool is directly accessible to cytochrome P-450(C-17)-catalysed conversion into androgens.