General statistics of stochastic process of gene expression in eukaryotic cells.

Thousands of genes are expressed at such very low levels (< or =1 copy per cell) that global gene expression analysis of rarer transcripts remains problematic. Ambiguity in identification of rarer transcripts creates considerable uncertainty in fundamental questions such as the total number of genes expressed in an organism and the biological significance of rarer transcripts. Knowing the distribution of the true number of genes expressed at each level and the corresponding gene expression level probability function (GELPF) could help resolve these uncertainties. We found that all observed large-scale gene expression data sets in yeast, mouse, and human cells follow a Pareto-like distribution model skewed by many low-abundance transcripts. A novel stochastic model of the gene expression process predicts the universality of the GELPF both across different cell types within a multicellular organism and across different organisms. This model allows us to predict the frequency distribution of all gene expression levels within a single cell and to estimate the number of expressed genes in a single cell and in a population of cells. A random "basal" transcription mechanism for protein-coding genes in all or almost all eukaryotic cell types is predicted. This fundamental mechanism might enhance the expression of rarely expressed genes and, thus, provide a basic level of phenotypic diversity, adaptability, and random monoallelic expression in cell populations.

Lumped parameter estimation for the embryonic chick vascular system: a time-domain approach using MLAB.

We have evaluated several lumped parameter analog models for the early chick embryonic vascular system that may be used to infer loading characteristics of the developing heart. We measured dorsal aortic pressure and flow simultaneously with a servo-null pressure system and a pulsed Doppler velocimeter. Four different analog circuit models were chosen for comparisons. We formulated the time-domain differential equations specifying the relations between pressure and flow in the models, and then estimated the lumped parameters that produced the best fit. The MLAB mathematical modeling software was used for solving differential equations, and for minimizing the difference between model-predicted values and experimental data. The traditional three-element Windkessel model with an added inductance term was most often the best-fitting model. This is compatible with the previous study using a frequency-domain approach. The procedures developed for the current study are adaptable for the study of a variety of nonlinear models, and distributed parameter models for mammalian cardiovascular development with mechanically, pharmacologically, or genetically altered conditions.

Mlab--a mathematical modeling tool.

An interactive interpreter called Mlab is described. One uses Mlab by typing commands. In this sense, Mlab is a programming language. It has various mathematical and graphical facilities which make it a useful tool for mathematical modeling. The curve fitting capabilities of Mlab are augmented with differential-equation-handling and matrix-manipulation capabilities which provide a powerful and civilized facility for curve fitting. Many people are engaged in this activity, and, in general, they use programs which are neither sufficiently general nor easy to use. (Some conventional programming is usually required, for example.) Mlab purports to be easier than alternate approaches. The nature of Mlab is discussed with accompanying examples. The main example is the use of curve fitting to determine molecular weight from ultracentrifuge data. This example was chosen because it exhibits a special feature of Mlab, namely the root operator, which appears in the definition of the model function.

Simulation studies of self-associating systems; discrimination between specific and isodesmic associations.

The possibilities of discriminating between definite and indefinite (isodesmic) modes of self-association are explored by fitting the different models to stimulated data, using non-linear least-squares curve-fitting to determine the fitting parameters for real and impostor models. It was found that over an extensive range of values for the equilibrium constant of a non-ideal isodesmic generating model, only a non-ideal monomer-dimer-tetramer-octamer was a successful impostor model. Some criteria for rejecting inappropriate models are discussed.

Kinetics of gonadotropin binding by receptors of the rat testis. Analysis by a nonlinear curve-fitting method.

The kinetics of the reaction between human chorionic gonadotropin (hCG) and specific gonadotropin receptors in the rat testis were determined at 24 and 37 degrees, over a wide range of hormone concentrations. Hormone concentrations were corrected for the binding activity of the (-125I)hCG tracer preparations. Analysis of the experimental data was performed with an interactive nonlinear curve fitting program, based upon the second-order chemical kinetic differential equation. The mean values for the association rate constant (k1) were 4.7 x 10-7 M-1 min-1 at 24 degrees, and 11.0 x 10-7 M-1 min-1 at 37 degrees. At both temperatures, the values of kl were independent of hormone concentration. Initial dissociation rates were consistent with first order kinetics, with dissociation rate constant (k2) of 1.7 x 10 minus -3 and 4.6 x 10 minus -3 min minus -1 at 24 and 37 degrees, respectively. When studied over longer periods at 24 degrees, the dissociation process appeared to be multiexponential. The kinetics of degradation of (-125I)hCG and receptors were determined at both temperatures, and a mathematical model was developed by modification of the second-order chemical kinetic differential equation to take these factors into account. The application of such a model to hCG kinetic binding data demonstrated that reactant degradation had little significant effect on the derivation of the association rate constant (k1), but caused significant overestimation of the dissociation rate constant (k2) values derived from association experiments. The model was also applied by computer simulation to a theoretical analysis of the effects of degradation of free hormone and receptor sites upon kinetic and steadystate binding data. By this method, the initial velocities of hormone binding were shown to be less affected by degradation than the steady-state levels of hormone-receptor complex. Also, reactant degradation in simulated steady-state experiments caused an underestimate of the apparent equilibrium association constant, but had relatively less effect on the determination of binding site concentration.