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Abstract
An understanding of cellular biochemistry and its control mechanisms demands an appreciation of the kinetics of systems containing many enzymes and substrates. Such systems are too complex for their kinetics to be represented by simple equations of the Michaelis-Menten type, and progress has only been made by the use of computers. Several workers1,2 have described digital computer solutions of differential equations representing the concentrations of each reactant. These solutions, however, require much computer time because of the nature of the mathematics of enzyme kinetics. This communication describes a stochastic method of calculating the kinetic behaviour of biochemical systems, which is more efficient than the methods formerly used.
... The simulation applications shown in this section are stochastic approaches (Kibby, 1969). They either implement proprietary algorithms of their own (MCell and StochSim) or make use of simulation algorithms from literature (BioNetS, Copasi, and SmartCell). ...
This work discusses the definition and implementation of a hybrid four-dimensional (4D) Cell Simulator - 4DiCeS. Despite the great variety of software packages available for modeling, simulation, and analysis of data, there is no application that features a complete and variable integration of different simulation methods in a 3D environment. The key improvement, which 4DiCeS has over other existing systems, is the ability not to be fixed on either deterministic or stochastic modeling and simulation approaches. A system of specialized interfaces therefore was designed to allow the bonding of interchangeable algorithmic modules of various types. The internal representation of the model is designed for the easy exchangeability of data. Furthermore, the integration of cell model file format standards is permitted by exchangeable plug-ins. The implemented system includes an Application Programming Interface (API) for writing individual plug-ins to utilize different simulation algorithms. This facilitates the implementation of tailored programs and specific algorithms that can be developed for data mining as well as visualization. The resulting 4DiCeS framework presented in this work describes a concept for the integration of heterogenous data into an easy-to-use software. Applications for such a modeling and simulation tool that allows for both 3D visualization and concurrent algorithms are imperative. The applications lie in the areas of electrical excitable cells, circadian rhythm, cell cycle, cellular motility, membrane transporters, metabolic pathways, and signal transduction networks. In comparison to 2D methods, the use of a 3D geometry provides considerably more significant data. The possibility of simulating different compartments of a cell with diverse algorithms can reduce computational effort dramatically. Consequently, this allows for both improved modeling and more information output on the spatio-temporal behavior of a system. Thus it is a scientific challenge to integrate concurrently running algorithms in combination with 3D visualization.
... For the purposes of this simulation-optimization methodology, the strategy that is used for the simulation part is not really important, so long as it is capable of taking values of the model parameters as inputs and providing the (calculated) values of the variables as outputs. Simulation based on differential equations, arguably the most popular method, is quite adequate, but so are other methods such as Monte Carlo (Kibby, 1969) or cellular automata (Ermentrout and Edelstein-Keshet, 1993). In the implementation described below, we have used our own simulation package Gepasi (Mendes, 1993(Mendes, , 1997, which is based on differential equations and is written in the C++ language. ...
The simulation of biochemical kinetic systems is a powerful approach that can be used for: (i) checking the consistency of a postulated model with a set of experimental measurements, (ii) answering 'what if?' questions and (iii) exploring possible behaviours of a model. Here we describe a generic approach to combine numerical optimization methods with biochemical kinetic simulations, which is suitable for use in the rational design of improved metabolic pathways with industrial significance (metabolic engineering) and for solving the inverse problem of metabolic pathways, i.e. the estimation of parameters from measured variables.
We discuss the suitability of various optimization methods, focusing especially on their ability or otherwise to find global optima. We recommend that a suite of diverse optimization methods should be available in simulation software as no single one performs best for all problems. We describe how we have implemented such a simulation-optimization strategy in the biochemical kinetics simulator Gepasi and present examples of its application.
The new version of Gepasi (3.20), incorporating the methodology described here, is available on the Internet at http://gepasi.dbs.aber.ac.uk/softw/Gepasi. html.
prm@aber.ac.uk
This chapter discusses, with real examples, the rate laws of first-order and second-order elementary reactions and shows how complex reactions can be treated as sequences of parallel, consecutive and reversible reactions. Reaction mechanisms can be complicated. The method of Laplace transforms is presented in detail, and with examples, to show how linear differential equations can be solved analytically. Numerical methods such as the Matrix, Runge–Kutta, Markov chain and Monte Carlo methods are presented in detail and illustrated with examples. It is also shown that complex kinetic schemes can be simplified with some chemical knowledge of the system, and this is illustrated with the isolation method, the pre-equilibrium approximation, the steady-state approximation and the identification of the rate-determining step of a chemical mechanism. Finally, global and target analysis of large and complex time-resolved spectroscopic data are discussed and exemplified.
In a scheme (Fig. 1) the assumed connections between the elements responsible for the regulation of the female sexual functions are summarized.
We consider the Darvey, Ninham and Staff model for reversible chemical reactions, in the case where the ratio of the rate constants is either very large or very small. It is shown that the distribution of the number of molecules at equilibrium may sometimes be closely approximated by the Poisson distribution, and on other occasions, by a distribution with much smaller tails than the Poisson. The second type of approximating distribution is termed a Bessel distribution, and its properties are studied.
Genomics has revolutionized research in biological sciences. The reductionist approach of single-molecule analysis is being slowly, but steadily, replaced by global views of the entire cellular machinery. This started with genetics, which turned to determining complete DNA sequences of organisms and is resulting in global comparisons of these sequences to infer their evolutionary past. With the availability of the first complete genome sequences, however, it became obvious that we know very little about how cells work. The problem is that a large number of genes in any genome have functions that are yet unknown, as judged by the lack of obvious phenotype of the corresponding mutants. These “orphan” genes are also not similar at the DNA or protein sequence levels to other genes of known function. The first complete eukaryotic genome, of the yeast Saccharomyces cerevisiae (Goffeau et al., 1996), perhaps the species for which we know most biochemistry, revealed a staggering 40% of genes to which no function could be assigned. Based on this, Oliver called for a systematic approach to the discovery of gene function (Oliver, 1996), which since became known as functional genomics.
The molecular weight distribution of A2-B2 type condensation polymers in the presence of capping monomer C has been derived with statistical calculation and Monte Carlo simulation methods. The Monte Carlo simulation result agrees with that of statistical calculation. The number distribution function and weight distribution function of seven types of molecules existing in A2-B2-C system have been obtained. The effect of reactivity of capping monomer C on these distributions are discussed.
We consider the Darvey, Ninham and Staff model for reversible chemical reactions, in the case where the ratio of the rate constants is either very large or very small. It is shown that the distribution of the number of molecules at equilibrium may sometimes be closely approximated by the Poisson distribution, and on other occasions, by a distribution with much smaller tails than the Poisson. The second type of approximating distribution is termed a Bessel distribution, and its properties are studied.
Die Gewinnung mathematischer Modelle mit Hilfe elektronischer Rechenanlagen wurde an Hand einiger Beispiele aus dem Bereich der organischen Chemie erläutert. Ausgehend von den beiden grundlegenden mathematischen Modelltypen, den analytischen und den Regressionsmodellen, wurden die Computersimulation der analytisch formulierbaren kinetischen Modelle, die Regressionsanalyse sowie die Monte-Carlo-Methode dargelegt. Besonders hervorgehoben wurden die Methoden der Regressionsanalyse. Die große Bedeutung der linearen Regressionsmodelle wird am Beispiel der LFE-Beziehungen demonstriert.
Chemical Kinetics is a journey into the world of atoms and molecules, guided by an explanation of how molecular structures change with time. Time-dependent molecular phenomena connoted by a new rational (ISM, intersecting-state model, interacting-state model). Chemical kinetics is more about structure and reactivity than number and calculations, although the reader is led calculation of rate constants for several kind of reactions: atom transfers, proton transfers, electron transfers, catalysis, substitution reactions, catalysis,energy transfer.
A relationship between the frequency of the occurrence of starquakes on pulsars and the shear strength of the solid crust of these objects is derived assuming various mechanisms of damping of their rates of rotation. Tentative conclusions concerning the shear strength, the age, and the composition of the Crab and the Vela pulsars are made.
A new stochastic model named CORUB has been used to do a phenomenological description of the photopolymerization of furfuryl methacrylate, with the following methodology. Statistical analysis of the model's adequacy shows a good correlation between experimental data and those modelled stochastically. This model was developed using two stochastic variables that represent the random nature of the intermolecular encounters upon which the Collision Theory is sustained. The amounts of random numbers generated simulate time in a new way, based on the standard physical concept, with satisfactory results. Also, calculating the number of times that each individual step of the reaction mechanism takes place, a new possibility for carrying out the sensitivity analysis is presented. In addition to which, the analogy assumed between the frequency of the random number generated and the effective intermolecular collision allows us to model, exclusively and accurately enough, the effect of temperature as well as to estimate the apparent activation energy precisely. The algorithm of CORUB model was coded in Turbo Pascal 6.0, resulting in the CHEMOD-X program, which constitutes a software specifically elaborated as an integrative part of this work, allowing the model to be used.
THE outer layers of neutron stars form a solid crust with a calculable rigidity (shear modulus) very soon after the stars are born. Subsequent changes in stellar shape from oblate toward spherical, as the neutron star angular velocity decreases, will induce stresses in the crust until the maximum shear strain which the solid can support is reached. Beyond this yield point there will be a sudden relaxation of the stress, and a very slight change in stellar shape and moment of inertia. The calculated accompanying jump in angular velocity is close to that which has been observed in a pulsar.
Stochastic models for a number of reversible chemical reactions involving one or two second‐order steps are examined and exact solutions which involve standard orthogonal polynomials are found for the equilibrium states. The expected values and variances of the numbers of molecules present in the systems at equilibrium are expressed in terms of these polynomials. Comparison of the stochastic mean value and the classical deterministic expression, shows that they are equivalent for most systems of physical interest, that is when large numbers of molecules are present. The relative fluctuation is found to be O (N−☒), where N is of the order of the number of molecules present in the system, in agreement with the results of equilibrium statistical mechanics.
Methods of simulation of biochemical experiments, by the numerical integration of the differential equations arising from models in terms of reaction networks, are discussed. Hand calculations are used to illustrate the type of integration method required. It is found that the optimum method depends both on the structure of the model and the current values of its parameters. A method which is optimum for the early stages of reaction is described. It is a suitable basis for a method which optimises itself as the integration proceeds, and this development is discussed.