(*^
::[ Information =
"This is a Mathematica Notebook file. It contains ASCII text, and can be
transferred by email, ftp, or other text-file transfer utility. It should
be read or edited using a copy of Mathematica or MathReader. If you
received this as email, use your mail application or copy/paste to save
everything from the line containing (*^ down to the line containing ^*)
into a plain text file. On some systems you may have to give the file a
name ending with ".ma" to allow Mathematica to recognize it as a Notebook.
The line below identifies what version of Mathematica created this file,
but it can be opened using any other version as well.";
FrontEndVersion = "Macintosh Mathematica Notebook Front End Version 2.2";
MacintoshStandardFontEncoding;
fontset = title, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, e8, 24, "Times";
fontset = subtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, e6, 18, "Times";
fontset = subsubtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, italic, e6, 14, "Times";
fontset = section, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, grayBox, M22, bold, a20, 18, "Times";
fontset = subsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, blackBox, M19, bold, a15, 14, "Times";
fontset = subsubsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, whiteBox, M18, bold, a12, 12, "Times";
fontset = text, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times";
fontset = smalltext, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 10, "Times";
fontset = input, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeInput, M42, N23, bold, L-5, 12, "Courier";
fontset = output, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-5, 12, "Courier";
fontset = message, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, R65535, L-5, 12, "Courier";
fontset = print, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-5, 12, "Courier";
fontset = info, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, B65535, L-5, 12, "Courier";
fontset = postscript, PostScript, formatAsPostScript, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeGraphics, M7, l34, w282, h287, 12, "Courier";
fontset = name, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, italic, 10, "Geneva";
fontset = header, inactive, noKeepOnOnePage, preserveAspect, M7, 12, "Times";
fontset = leftheader, inactive, L2, 12, "Times";
fontset = footer, inactive, noKeepOnOnePage, preserveAspect, center, M7, 12, "Times";
fontset = leftfooter, inactive, L2, 12, "Times";
fontset = help, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 10, "Times";
fontset = clipboard, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times";
fontset = completions, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times";
fontset = special1, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times";
fontset = special2, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times";
fontset = special3, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times";
fontset = special4, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times";
fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times";
paletteColors = 128; automaticGrouping; currentKernel;
]
:[font = title; inactive; preserveAspect; startGroup]
Mathematica Functions to study
Gene Induction
;[s]
2:0,1;11,0;46,-1;
2:1,25,18,Times,1,24,0,0,0;1,25,18,Times,3,24,0,0,0;
:[font = subtitle; inactive; preserveAspect]
A GeneInduction Package Tutorial
:[font = subsubtitle; inactive; preserveAspect]
Jean Peccoud
TIMC-IMAG
FacultŽ de mŽdecine de Grenoble
Institut Albert Bonniot
F-38706 La Tronche
France
Jean.Peccoud@imag.fr
:[font = section; inactive; preserveAspect; startGroup]
Overview
:[font = subsection; inactive; preserveAspect; startGroup]
Scope
:[font = text; inactive; preserveAspect; endGroup]
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.
;[s]
3:0,0;331,1;449,0;609,-1;
2:2,13,9,Times,0,12,0,0,0;1,13,9,Times,2,12,0,0,0;
:[font = subsection; inactive; preserveAspect; startGroup]
Content
:[font = text; inactive; preserveAspect; endGroup]
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.
:[font = subsection; inactive; Cclosed; preserveAspect; startGroup]
Naming Conventions
:[font = text; inactive; preserveAspect]
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.
;[s]
5:0,0;337,1;342,0;463,1;469,0;573,-1;
2:3,13,9,Times,0,12,0,0,0;2,13,10,Courier,1,12,0,0,0;
:[font = input; preserveAspect]
<A.
:[font = input; preserveAspect; startGroup]
?mu
:[font = print; inactive; preserveAspect; endGroup]
mu is the transition rate of A->I.
:[font = input; preserveAspect; startGroup]
?nu
:[font = print; inactive; preserveAspect; endGroup]
nu is the transition rate of A->A + P.
:[font = input; preserveAspect; startGroup]
?delta
:[font = print; inactive; preserveAspect; endGroup]
delta is the transition rate of P->¯.
:[font = input; preserveAspect; startGroup]
?t
:[font = print; inactive; preserveAspect; endGroup]
t is the time after the beginning of induction; t can be
set Infinity in order to have the asymptotic behavior.
:[font = input; preserveAspect; startGroup]
?z
:[font = print; inactive; preserveAspect]
z is the variable of the Generating function.
:[font = message; inactive; preserveAspect; endGroup; endGroup]
Warning: since these symbol names are not very specific
to this application they may interfer with symbols previously defined
during the session.
:[font = subsection; inactive; Cclosed; preserveAspect; startGroup]
Functions
:[font = subsubsection; inactive; preserveAspect; startGroup]
Generating Function
:[font = text; inactive; preserveAspect]
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.
;[s]
5:0,0;4,1;23,0;28,1;48,0;197,-1;
2:3,13,9,Times,0,12,0,0,0;2,13,10,Courier,1,12,0,0,0;
:[font = input; preserveAspect; startGroup]
?PGeneratingFunction
:[font = print; inactive; preserveAspect; endGroup]
PGeneratingFunction[z, t, lambda, mu, nu,delta] is the generating function
of the distribution of probability at time t of the gene product P
synthetized according to the parameters lambda, mu, nu, and delta.
:[font = text; inactive; preserveAspect]
The last argument of NPGeneratingFunction, delta is optional. Its default value is set to 1
;[s]
3:0,0;21,1;48,0;92,-1;
2:2,13,9,Times,0,12,0,0,0;1,13,10,Courier,1,12,0,0,0;
:[font = input; preserveAspect; startGroup]
?NPGeneratingFunction
:[font = print; inactive; preserveAspect; endGroup; endGroup]
NPGeneratingFunction[z, t, lambda, mu, nu,delta] gives a numerical
approximation of the generating function of the distribution of
probability at time t of the gene product P synthetized according to the
numerical parameters lambda, mu, nu, and delta which is an optional
argument the default value of which is 1.
:[font = subsubsection; inactive; preserveAspect; startGroup]
Mean Value
:[font = text; inactive; preserveAspect]
PMean and NPMean return respectively the symbolic and numerical mean value of the protein number.
;[s]
4:0,1;5,0;10,1;16,0;98,-1;
2:2,13,9,Times,0,12,0,0,0;2,13,10,Courier,1,12,0,0,0;
:[font = input; preserveAspect; startGroup]
?PMean
:[font = print; inactive; preserveAspect; endGroup]
PMean[t, lambda, mu, nu, delta] returns the mean of the probability
distribution at time t of the gene product synthetized according to the
parameters lambda, mu, nu, and delta.
:[font = input; preserveAspect; startGroup]
?NPMean
:[font = print; inactive; preserveAspect; endGroup]
NPMean[t, lambda, mu, nu, delta] gives a numerical approximation of the mean
value of the distribution of probability at time t of the gene product P
synthetized according to the numerical parameters lambda, mu, nu, and
delta which is an optional argument the default value of which is 1.
:[font = text; inactive; preserveAspect; endGroup]
Again the last argument is optional with 1 as a default value.
:[font = subsubsection; inactive; preserveAspect; startGroup]
Variance
:[font = text; inactive; preserveAspect]
PVariance and NPVariance return respectively the symbolic and numerical variance of the protein number.
;[s]
4:0,1;9,0;14,1;24,0;104,-1;
2:2,13,9,Times,0,12,0,0,0;2,13,10,Courier,1,12,0,0,0;
:[font = input; preserveAspect; startGroup]
?PVariance
:[font = print; inactive; preserveAspect; endGroup]
PVariance[t, lambda, mu, nu, delta]returns the variance of the probability
distribution at time t of the gene product synthetized according to the
parameters lambda, mu, nu, and delta.
:[font = input; preserveAspect; startGroup]
?NPVariance
:[font = print; inactive; preserveAspect; endGroup]
NPVariance[t, lambda, mu, nu, delta] gives a numerical approximation of the
variance of the distribution of probability at time t of the gene
product P synthetized according to the numerical parameters lambda, mu,
nu, and delta which is an optional argument the default value of which
is 1.
:[font = text; inactive; preserveAspect; endGroup; endGroup]
Once again the last argument is optional with 1 as a default value.
:[font = subsection; inactive; Cclosed; preserveAspect; startGroup]
Messages
:[font = text; inactive; preserveAspect]
Three error messages are returned when the expression evaluation is not possible.
:[font = message; inactive; preserveAspect]
PGeneratingFunction::infi:
When delta==0 the process does not have any stationnary distribution.
:[font = text; inactive; preserveAspect]
When delta=0 and t=Infinity the system does not have any stationnary distribution. The functions cannot be evaluated but PMean and NPMean.
;[s]
5:0,0;121,1;126,0;131,1;137,0;139,-1;
2:3,13,9,Times,0,12,0,0,0;2,13,10,Courier,1,12,0,0,0;
:[font = message; inactive; preserveAspect]
PGeneratingFunction::notavail:
PGeneratingFunction is not available with this set of arguments.
:[font = text; inactive; preserveAspect]
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.
:[font = message; inactive; preserveAspect]
PGeneratingFunction::badarg:
PGeneratingFunction called with an invalid set of parameters.
:[font = text; inactive; preserveAspect; endGroup; endGroup]
This message is the default message indicating that the function has not been correctly called.
:[font = section; inactive; preserveAspect; startGroup]
Possible uses of the package
:[font = text; inactive; preserveAspect; endGroup; endGroup]
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.
^*)