A simple computer program with statistical tests for the analysis of enzyme kinetics.

A simple computer program that calculates the kinetic parameters of enzyme reactions is described. Parameters are determined by nonlinear, least-squares regression using either Marquardt-Levenberg or Gauss-Newton algorithms to find the minimum sum of squares. Three types of enzyme reactions can be analyzed: single substrate reactions (Michaelis-Menten and sigmoidal kinetics), enzyme activation at a fixed substrate value or enzyme inhibition at a fixed substrate value. The user can monitor goodness of fit through nonparametric statistical tests (performed automatically by the computer) and through visual examination of the pattern of residuals. The program is unique in providing equations for activator and inhibition analysis as well as in enabling the user to fix some of the parameters before regression analysis. The simplicity of the program makes it extremely useful for quickly determining kinetic parameters during the data-gathering process.     

A microscopic model of enzyme kinetics.

Many in vivo enzymatic processes, such as those of the tissue factor pathway of blood coagulation, occur in environments with facilitated substrate delivery or enzymes bound to cellular or lipid surfaces, which are quite different from the ideal fluid environment for which the Michaelis-Menten equation was derived. To describe the kinetics of such reactions, we propose a microscopic model that focuses on the kinetics of a single-enzyme molecule. This model provides the foundation for macroscopic models of the system kinetics of reactions occurring in both ideal and nonideal environments. For ideal reaction systems, the corresponding macroscopic models thus derived are consistent with the Michaelis-Menten equation. It is shown that the apparent Km is in fact a function of the mechanism of substrate delivery and should be interpreted as the substrate level at which the enzyme vacancy time equals the residence time of ES-complexes; it is suggested that our microscopic model parameters characterize more accurately an enzyme and its catalytic efficiency than does the classical Km. This model can also be incorporated into computer simulations of more complex reactions as an alternative to explicit analytical formulation of a macroscopic model.     

Metal search: A computer program that helps design tetrahedral metal-binding sites

We describe a computer program (Metal Search) that helps design tetrahedrally coordinated metal binding sites in proteins of known structure. The program takes as input the backbone coordinates of a protein and outputs lists of four residues that might form tetrahedral sites if wild-type amino acids were replaced by cysteine or histidine. The program also outputs the side chain dihedral angles of the amino acids and the coordinates of the predicted metal ion. The only function evaluated by Metal Search is the ability of side chains to meet simple geometric criteria for formation of a tetrahedral site, but these criteria are sufficient to produce a manageably small list that can then be evaluated by other means. The program has been used in the introduction of zinc binding sites in the designed four-helix bundle protein α 4 and in the B1 domain of streptococcal protein G, and in both cases the tetrahedral coordination of a bound metal ion has been confirmed¹ (Klemba, M., Gardner, K. H., Marino, S., Clarke, N. D., and Regan, L., Nature: Structural Biology 2:368–373, 1995). © 1995 Wiley-Liss, Inc.     

A computer program for modeling the kinetics of gene expression

Enzyme induction may be modeled on the basis of four, quantifiable processes that control the rates at which specific gene products accumulate and decay. These processes include synthesis of functional mRNA, translation and degradation of mRNA, and degradation of the protein product. We present a simple computer program that permits mathematical simulation of gene expression on the basis of experimentally determined rates of synthesis and degradation. The program was implemented as a spreadsheet using Microsoft Excel for Macintosh and MS-DOS operating systems and also was adapted for HyperCard on the Macintosh. It contains a formula to account for growth of tissue or cell populations. The program predicts amounts of individual mRNAs and proteins (or enzyme activities) in cells as a function of time after a stimulus alters their rates of synthesis or degradation.     

A computationally tractable birth-death model that combines phylogenetic and epidemiological data

Inferring the dynamics of pathogen transmission during an outbreak is an important problem in infectious disease epidemiology. In mathematical epidemiology, estimates are often informed by time series of confirmed cases, while in phylodynamics genetic sequences of the pathogen, sampled through time, are the primary data source. Each type of data provides different, and potentially complementary, insight. Recent studies have recognised that combining data sources can improve estimates of the transmission rate and the number of infected individuals. However, inference methods are typically highly specialised and field-specific and are either computationally prohibitive or require intensive simulation, limiting their real-time utility. We present a novel birth-death phylogenetic model and derive a tractable analytic approximation of its likelihood, the computational complexity of which is linear in the size of the dataset. This approach combines epidemiological and phylodynamic data to produce estimates of key parameters of transmission dynamics and the unobserved prevalence. Using simulated data, we show (a) that the approximation agrees well with existing methods, (b) validate the claim of linear complexity and (c) explore robustness to model misspecification. This approximation facilitates inference on large datasets, which is increasingly important as large genomic sequence datasets become commonplace.     

Stieltjes integration and differential geometry: A model for enzyme recognition,discrimination, and catalysis

A model for enzymic catalysis is presented using the mathematical theories of differential geometry and Stieltjes integration. The Stieltjesintegrator is a complex-valued function of bounded variation which represents the curvature and torsion, hence the conformation, of the backbone of an enzyme molecule. Theintegrand is a complex-valued continuous function which describes the shape of the surface of a substrate molecule. We postulate that enzyme-substrate interactions correspond to evaluations of Stieltjes integrals, and that observables of enzymic catalysis correspond to projections. Results from the mathematical theory of the Stieltjes integral are discussed together with their biological interpretations. We contrast the difference between structural and functional proteins, and construct analogues of enzyme cofactors, modifications, and regulation. Various techniques of locating the active site on enzymes are also given. We construct a total variation metric, which is particularly useful for detecting similarities among proteins. An examination on the many different modes of convergence of mathematical functions representing biological molecules leads to a mathematical statement of the fundamental dogma of molecular biology, that ‘structure implies function’. Similar arguments also result in the converse statement ‘function dictates structure’, which is a basic premise of relational biology. Stepped-helical approximations of the backbone space curves of enzymes provide a concrete computational tool with which to calculate the Stieltjes integrals that model enzymic catalysis, by replacing the integral with a finite series. The duality between enzymes and substrates (that they aremeters ‘observing’ one another) is shown to be a consequence of the mathematical duality of Banach spaces. The Stieltjes integrals of enzyme-substrate interactions are hence shown to be bounded bilinear functionals. The mechanism of enzymic catalysis, the transformation from substrate to product, is also formulated in the Stieltjes integration context via the mathematical theory of adjoints. The paper closes with suggestions for generalizations, prospects for future studies, and a review of the correspondence between mathematical and biological concepts.     

Conditions when statistical tests for model discrimination have high power. Some examples from pharmacokinetics, ligand binding, transient and steady-state enzyme kinetics

The situation where data pairs xi, yi are actually generated by a true model, f(delta, x), but erroneously fitted by a deficient model, g(phi, x), is explored. A function, Q(delta), is described which is the average squared distance between f(delta, x) and the best-fit false model, g(phi, x). For a range of x covering 5-95% 'saturation' of f(delta, x), Q(delta) is calculated numerically for sums of exponentials, binding functions and rational functions. In each case, the region of delta when the second model in the series can be reliably differentiated from the first by statistical tests is described.     

Efficient and accurate experimental design for enzyme kinetics: Bayesian studies reveal a systematic approach

In areas such as drug development, clinical diagnosis and biotechnology research, acquiring details about the kinetic parameters of enzymes is crucial. The correct design of an experiment is critical to collecting data suitable for analysis, modelling and deriving the correct information. As classical design methods are not targeted to the more complex kinetics being frequently studied, attention is needed to estimate parameters of such models with low variance. We demonstrate that a Bayesian approach (the use of prior knowledge) can produce major gains quantifiable in terms of information, productivity and accuracy of each experiment. Developing the use of Bayesian Utility functions, we have used a systematic method to identify the optimum experimental designs for a number of kinetic model data sets. This has enabled the identification of trends between kinetic model types, sets of design rules and the key conclusion that such designs should be based on some prior knowledge of K(M) and/or the kinetic model. We suggest an optimal and iterative method for selecting features of the design such as the substrate range, number of measurements and choice of intermediate points. The final design collects data suitable for accurate modelling and analysis and minimises the error in the parameters estimated.     

A computer program that draws pedigree charts for inbred strains of animals

We produced a computer program that draws pedigree charts for inbred strains of animals such as mice or rats. This program is composed of four subprograms, which are (1) inputting the data, (2) drawing pedigree charts, (3) listing the data which have been input, and (4) backup of the system and the data. Pedigree charts and lists of data can be displayed on a TV screen and printed out on the papers. Using this program, we drew the pedigree charts of the inbred strains of rats which we are maintaining by brother-sister inbreeding in our institute and found that there were three sublines in one of the strains, WKAH/Hkm, because of unsuitable maintenance. This program is very convenient to draw the pedigree charts and useful for checking the maintenance of inbred strains or the strains of animal models of human diseases.     

A computer program for selecting animals for control and experimental groups in biochemical studies

The standard method for selecting experimental animals for controland experimental groups is by a randomization process in whicheach animal is placed into one of the groups without lookingat any of its individual differences. An alternative, activematching process is proposed by which three to five animalsare chosen to form ‘experimental units’ that areclosely matched with respect to average weight and with respectto distribution of weights (standard distribution values). Severalsuch units are then randomly assigned to each control or treatmentgroup and, at the end of the experiment, the animals withineach unit are pooled prior to analysis. The merits of this approachand a computer algorithm for making the selections are described. ; accepted on March 10, 1986     

A Macintosh computer program for designing DNA sequences that code for specific peptides and proteins

A computer program (PINCERS) is described for use in the design of synthetic genes and mixed-probe DNA sequences. A protein sequence is reverse translated with generation of synonymous codons at each position producing a degenerate sequence. In order to locate potential restriction enzyme sites, the degenerate sequence is searched with a library of restriction enzymes for sites that utilize any combination of synonymous codons. These sites are indicated in a map so that they may be incorporated into the synthetic gene sequence. The program allows the user to select the appropriate codon usage table for the organism of interest and then to set a threshold usage frequency below which codons are not generated. PINCERS may also be used to assist in planning the synthesis of mixed-probe DNA sequences for cross-hybridization experiments. It can identify regions of specified length with the protein sequence that have the least overall degeneracy, thereby minimizing the number of probes to be synthesized and, therefore, maximizing the concentration of a given probe sequence.     

A minimal model of non-hyperbolic enzyme and receptor kinetics

A simplified version of P.W. Kühl's Recovery Model [Biochem. J. 298 (1994) 171-180] has been developed in which the duration of the recovery phase of receptor or enzyme (macro)molecule was assumed to be a random value distributed exponentially like other model parameters. The model has been shown to retain all the properties of the original Recovery Model except for its ability to yield asymmetric dose-response curves (if plotted in semi-logarithmic scale). Due to its simplicity, the present model is applicable for routine fitting to experimental data. In enzyme kinetics, the model yields a diversity of non-hyperbolic dose-response curves both with higher and lower steepness than that of Henri-type ones. In receptor kinetics, the diversity of dose-response curves is further increased due to virtually no restraints being imposed on the efficacies of any state of the macromolecule.