Andrzej M. Kierzek

The effect of transcription and translation initiation frequencies on the stochastic fluctuations in prokaryotic gene expression.

The kinetics of prokaryotic gene expression has been modelled by the Monte Carlo computer simulation algorithm of Gillespie, which allowed the study of random fluctuations in the number of protein molecules during gene expression. The model, when applied to the simulation of LacZ gene expression, is in good agreement with experimental data. The influence of the frequencies of transcription and translation initiation on random fluctuations in gene expression has been studied in a number of simulations in which promoter and ribosome binding site effectiveness has been changed in the range of values reported for various prokaryotic genes. We show that the genes expressed from strong promoters produce the protein evenly, with a rate that does not vary significantly among cells. The genes with very weak promoters express the protein in “bursts” occurring at random time intervals. Therefore, if the low level of gene expression results from the low frequency of transcription initiation, huge fluctuations arise. In contrast, the protein can be produced with a low and uniform rate if the gene has a strong promoter and a slow rate of ribosome binding (a weak ribosome binding site). The implications of these findings for the expression of regulatory proteins are discussed.

STOCKS: STOChastic Kinetic Simulations of biochemical systems with Gillespie algorithm.

MOTIVATIONnThe availability of a huge amount of molecular data concerning various biochemical reactions provoked numerous attempts to study the dynamics of cellular processes by means of kinetic models and computer simulations. Biochemical processes frequently involve small numbers of molecules (e.g. a few molecules of a transcriptional regulator binding to one molecule of a DNA regulatory region). Such reactions are subject to significant stochastic fluctuations. Monte Carlo methods must be employed to study the functional consequences of the fluctuations and simulate processes that cannot be modelled by continuous fluxes of matter. This provides the motivation to develop software dedicated to Monte Carlo simulations of cellular processes with the rigorously proven Gillespie algorithm.nnnRESULTSnSTOCKS, software for the stochastic kinetic simulation of biochemical processes is presented. The program uses a rigorously derived Gillespie algorithm that has been shown to be applicable to the study of prokaryotic gene expression. Features dedicated to the study of cellular processes are implemented, such as the possibility to study a process in the range of several cell generations with the application of a simple cell division model. Taking expression of Escherichia coli beta-galactosidase as an example, it is shown that the program is able to simulate systems composed of reactions varying in several orders of magnitude by means of reaction rates and the numbers of molecules involved.nnnAVAILABILITYnThe software is available at ftp://ibbrain.ibb.waw.pl/stocksand http://www.ibb.waw.pl/stocks.nnnSUPPLEMENTARY INFORMATIONnParameters of the model of prokaryotic gene expression are available in example files of software distribution.

Bridging the Gap between Stochastic and Deterministic Regimes in the Kinetic Simulations of the Biochemical Reaction Networks

The biochemical reaction networks include elementary reactions differing by many orders of magnitude in the numbers of molecules involved. The kinetics of reactions involving small numbers of molecules can be studied by exact stochastic simulation. This approach is not practical for the simulation of metabolic processes because of the computational cost of accounting for individual molecular collisions. We present the maximal time step method, a novel approach combining the Gibson and Bruck algorithm with the Gillespie tau-leap method. This algorithm allows stochastic simulation of systems composed of both intensive metabolic reactions and regulatory processes involving small numbers of molecules. The method is applied to the simulation of glucose, lactose, and glycerol metabolism in Escherichia coli. The gene expression, signal transduction, transport, and enzymatic activities are modeled simultaneously. We show that random fluctuations in gene expression can propagate to the level of metabolic processes. In the cells switching from glucose to a mixture of lactose and glycerol, random delays in transcription initiation determine whether lactose or glycerol operon is induced. In a small fraction of cells severe decrease in metabolic activity may also occur. Both effects are epigenetically inherited by the progeny of the cell in which the random delay in transcription initiation occurred.

Models of protein crystal growth.

The growth of large and well ordered protein crystals remains the major obstacle in protein structure determination by means of X-ray crystallography. One of the reasons is that the physico-chemical aspect of protein crystallization process is not understood. This article reviews efforts towards formulation of models that could become theoretical frameworks for the interpretation of voluminous experimental data collected on protein crystal growth. Special attention is devoted to microscopic models that recognize the role of the shape of protein molecules in crystal formation.

Microscopic model of protein crystal growth

A microscopic, reversible model to study protein crystal nucleation and growth is presented. The probability of monomer attachment to the growing crystal was assumed to be proportional to the protein volume fraction and the orientational factor representing the anisotropy of protein molecules. The rate of detachment depended on the free energy of association of the given monomer in the lattice, as calculated from the buried surface area. The proposed algorithm allowed the simulation of the process of crystal growth from free monomers to complexes having 10(5) molecules, i.e. microcrystals with already formed faces. These simulations correctly reproduced the crystal morphology of the chosen model system--the tetragonal lysozyme crystal. We predicted the critical size, after which the growth rate rapidly increased to approximately 50 protein monomers. The major factors determining the protein crystallisation kinetics were the geometry of the protein molecules and the resulting number of kinetics traps on the growth pathway.

Lattice simulations of protein crystal formation.

A new algorithm is presented for the lattice simulation of protein crystal growth. The algorithm allows the calculation of the size distribution of microcrystals in the volume and timescale of experiments and within the framework of the previously-published microscopic model [A.M. Kierzek, W.M. Wolf, P. Zielenkiewicz, Biophys. J. 73 (1997) 571-580]. Simulations for the tetragonal lysozyme crystal show that there are two critical sizes in the development of ordered phase. The first one corresponds to the size of the smallest stable complex which, in the case of the tetragonal lysozyme crystal, is the particular tetramer. In a volume of 5 mul the tetramer appears in the millisecond timescale. The second critical radius of approximately 100 monomers is only reached by a few of all the smallest stable complexes formed in the solution. The model predicts that out of 10(7) tetramers which appear in solution, only eight reach the size of 100 monomers within 8 h. After exceeding the second critical radius the microcrystals grow to the size of 10(4) monomers in the minute timescale and are thus assumed to quickly lead to macroscopic crystals. The predicted number of crystals formed during 8 h of nucleation is in qualitative agreement with arrested nucleation experiments.