Mathematica Functions to study
Gene Induction

A GeneInduction Package Tutorial

Jean Peccoud
TIMC-IMAG
Faculté de médecine de Grenoble
Institut Albert Bonniot
F-38706 La Tronche
France
Jean.Peccoud@imag.fr

Overview

Scope

Since the synthesis of proteins can be viewed, at the cellular level, as a catalysis with a single molecule of catalyser i.e. the gene, the number of protein molecules is subjected to random fluctuations. The gene induction phenomenon has been modelled by a birth-and-death process in random environment which has been studied in:
Peccoud,J.; Ycart, B. (1995) Markovian Modelling of Gene Product Synthesis. Theoretical Population Biology (in press).
The GeneInduction package provides the reader of this paper the tools to compute himself either numerically or symbolically the results reported in the paper.

Content

iap.ps: postscript file of the Theor. Pop. Biol. paper.
iap.tex:	The TeX file of the Theor. Pop. Biol. paper.
GeneInduction.m: provides function to compute the generating function, the mean, and the variance of the protein number.
GITutorial.ma: this file.

Naming Conventions

The functions defined in the package are designed to compute the generating functions, the mean value, and the variance of the protein number noted P in the paper. Thus P is used as a prefix in every function names. Some functions are designed to give numerical results. Their names begin with N. Although the symbolic function, such as PMean, can return a numerical result when called with numercal arguments, it is recommended to use its numerical counterpart, NPMean, when a numerical result is expected since the consistency of the arguments is then checked carefully.

<<GeneInduction`

Names["GeneInduction`*"]

The other symbols are the model parameters. They will be explained in a following section.

Symbols and Functions description

Symbols and model parameters

Since a single set of parameters is used by all the functions defined in the package, they have been given a usage message to help the package user. See the Theor. Pop. Biol. paper for more details.

?lambda

?mu

?nu

?delta

?t

?z

Functions

Generating Function

The PGeneratingFunction and NPGeneratingFunction are used to compute respectively  symbolic and numercal values of the generating function of the distribution of probability of the protein number.

?PGeneratingFunction

The last argument of NPGeneratingFunction, delta is optional. Its default value is set to 1

?NPGeneratingFunction

Mean Value

PMean and NPMean return respectively the symbolic and numerical mean value of the protein number.

?PMean

?NPMean

Again the last argument is optional with 1 as a default value.

Variance

PVariance and NPVariance return respectively the symbolic and numerical variance of the protein number.

?PVariance

?NPVariance

Once again the last argument is optional with 1 as a default value.

Messages

Three error messages are returned when the expression evaluation is not possible.

When delta=0 and t=Infinity the system does not have any stationnary distribution. The functions cannot be evaluated but PMean and NPMean.

Transient values of the generating function and the variance have not been integrated in the package when delta is positive. Thus, a message is return when the expression is not available.

This message is the default message indicating that the function has not been correctly called.

Possible uses of the package

This material is intended to assist readers of the paper in evaluating the values of the results. Symbolic functions could be used by specialists of probability to manipulate them. Biologists can be interested in studying the way the protein number is affected by the different parameters of the model. As a biologist myself, I would be extremely happy to have contacts with biologists interested in an experimental validation of this theoretical work. I could provide more tools to analyse experiment results. So, if you have any interest in this question, please, feel free to contact me.