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notebook.tex
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% Default to the notebook output style
% Inherit from the specified cell style.
\documentclass[11pt]{article}
\usepackage[T1]{fontenc}
% Nicer default font (+ math font) than Computer Modern for most use cases
\usepackage{mathpazo}
% Basic figure setup, for now with no caption control since it's done
% automatically by Pandoc (which extracts  syntax from Markdown).
\usepackage{graphicx}
% We will generate all images so they have a width \maxwidth. This means
% that they will get their normal width if they fit onto the page, but
% are scaled down if they would overflow the margins.
\makeatletter
\def\maxwidth{\ifdim\Gin@nat@width>\linewidth\linewidth
\else\Gin@nat@width\fi}
\makeatother
\let\Oldincludegraphics\includegraphics
% Set max figure width to be 80% of text width, for now hardcoded.
\renewcommand{\includegraphics}[1]{\Oldincludegraphics[width=.8\maxwidth]{#1}}
% Ensure that by default, figures have no caption (until we provide a
% proper Figure object with a Caption API and a way to capture that
% in the conversion process - todo).
\usepackage{caption}
\DeclareCaptionLabelFormat{nolabel}{}
\captionsetup{labelformat=nolabel}
\usepackage{adjustbox} % Used to constrain images to a maximum size
\usepackage{xcolor} % Allow colors to be defined
\usepackage{enumerate} % Needed for markdown enumerations to work
\usepackage{geometry} % Used to adjust the document margins
\usepackage{amsmath} % Equations
\usepackage{amssymb} % Equations
\usepackage{textcomp} % defines textquotesingle
% Hack from http://tex.stackexchange.com/a/47451/13684:
\AtBeginDocument{%
\def\PYZsq{\textquotesingle}% Upright quotes in Pygmentized code
}
\usepackage{upquote} % Upright quotes for verbatim code
\usepackage{eurosym} % defines \euro
\usepackage[mathletters]{ucs} % Extended unicode (utf-8) support
\usepackage[utf8x]{inputenc} % Allow utf-8 characters in the tex document
\usepackage{fancyvrb} % verbatim replacement that allows latex
\usepackage{grffile} % extends the file name processing of package graphics
% to support a larger range
% The hyperref package gives us a pdf with properly built
% internal navigation ('pdf bookmarks' for the table of contents,
% internal cross-reference links, web links for URLs, etc.)
\usepackage{hyperref}
\usepackage{longtable} % longtable support required by pandoc >1.10
\usepackage{booktabs} % table support for pandoc > 1.12.2
\usepackage[inline]{enumitem} % IRkernel/repr support (it uses the enumerate* environment)
\usepackage[normalem]{ulem} % ulem is needed to support strikethroughs (\sout)
% normalem makes italics be italics, not underlines
% Colors for the hyperref package
\definecolor{urlcolor}{rgb}{0,.145,.698}
\definecolor{linkcolor}{rgb}{.71,0.21,0.01}
\definecolor{citecolor}{rgb}{.12,.54,.11}
% ANSI colors
\definecolor{ansi-black}{HTML}{3E424D}
\definecolor{ansi-black-intense}{HTML}{282C36}
\definecolor{ansi-red}{HTML}{E75C58}
\definecolor{ansi-red-intense}{HTML}{B22B31}
\definecolor{ansi-green}{HTML}{00A250}
\definecolor{ansi-green-intense}{HTML}{007427}
\definecolor{ansi-yellow}{HTML}{DDB62B}
\definecolor{ansi-yellow-intense}{HTML}{B27D12}
\definecolor{ansi-blue}{HTML}{208FFB}
\definecolor{ansi-blue-intense}{HTML}{0065CA}
\definecolor{ansi-magenta}{HTML}{D160C4}
\definecolor{ansi-magenta-intense}{HTML}{A03196}
\definecolor{ansi-cyan}{HTML}{60C6C8}
\definecolor{ansi-cyan-intense}{HTML}{258F8F}
\definecolor{ansi-white}{HTML}{C5C1B4}
\definecolor{ansi-white-intense}{HTML}{A1A6B2}
% commands and environments needed by pandoc snippets
% extracted from the output of `pandoc -s`
\providecommand{\tightlist}{%
\setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}}
\DefineVerbatimEnvironment{Highlighting}{Verbatim}{commandchars=\\\{\}}
% Add ',fontsize=\small' for more characters per line
\newenvironment{Shaded}{}{}
\newcommand{\KeywordTok}[1]{\textcolor[rgb]{0.00,0.44,0.13}{\textbf{{#1}}}}
\newcommand{\DataTypeTok}[1]{\textcolor[rgb]{0.56,0.13,0.00}{{#1}}}
\newcommand{\DecValTok}[1]{\textcolor[rgb]{0.25,0.63,0.44}{{#1}}}
\newcommand{\BaseNTok}[1]{\textcolor[rgb]{0.25,0.63,0.44}{{#1}}}
\newcommand{\FloatTok}[1]{\textcolor[rgb]{0.25,0.63,0.44}{{#1}}}
\newcommand{\CharTok}[1]{\textcolor[rgb]{0.25,0.44,0.63}{{#1}}}
\newcommand{\StringTok}[1]{\textcolor[rgb]{0.25,0.44,0.63}{{#1}}}
\newcommand{\CommentTok}[1]{\textcolor[rgb]{0.38,0.63,0.69}{\textit{{#1}}}}
\newcommand{\OtherTok}[1]{\textcolor[rgb]{0.00,0.44,0.13}{{#1}}}
\newcommand{\AlertTok}[1]{\textcolor[rgb]{1.00,0.00,0.00}{\textbf{{#1}}}}
\newcommand{\FunctionTok}[1]{\textcolor[rgb]{0.02,0.16,0.49}{{#1}}}
\newcommand{\RegionMarkerTok}[1]{{#1}}
\newcommand{\ErrorTok}[1]{\textcolor[rgb]{1.00,0.00,0.00}{\textbf{{#1}}}}
\newcommand{\NormalTok}[1]{{#1}}
% Additional commands for more recent versions of Pandoc
\newcommand{\ConstantTok}[1]{\textcolor[rgb]{0.53,0.00,0.00}{{#1}}}
\newcommand{\SpecialCharTok}[1]{\textcolor[rgb]{0.25,0.44,0.63}{{#1}}}
\newcommand{\VerbatimStringTok}[1]{\textcolor[rgb]{0.25,0.44,0.63}{{#1}}}
\newcommand{\SpecialStringTok}[1]{\textcolor[rgb]{0.73,0.40,0.53}{{#1}}}
\newcommand{\ImportTok}[1]{{#1}}
\newcommand{\DocumentationTok}[1]{\textcolor[rgb]{0.73,0.13,0.13}{\textit{{#1}}}}
\newcommand{\AnnotationTok}[1]{\textcolor[rgb]{0.38,0.63,0.69}{\textbf{\textit{{#1}}}}}
\newcommand{\CommentVarTok}[1]{\textcolor[rgb]{0.38,0.63,0.69}{\textbf{\textit{{#1}}}}}
\newcommand{\VariableTok}[1]{\textcolor[rgb]{0.10,0.09,0.49}{{#1}}}
\newcommand{\ControlFlowTok}[1]{\textcolor[rgb]{0.00,0.44,0.13}{\textbf{{#1}}}}
\newcommand{\OperatorTok}[1]{\textcolor[rgb]{0.40,0.40,0.40}{{#1}}}
\newcommand{\BuiltInTok}[1]{{#1}}
\newcommand{\ExtensionTok}[1]{{#1}}
\newcommand{\PreprocessorTok}[1]{\textcolor[rgb]{0.74,0.48,0.00}{{#1}}}
\newcommand{\AttributeTok}[1]{\textcolor[rgb]{0.49,0.56,0.16}{{#1}}}
\newcommand{\InformationTok}[1]{\textcolor[rgb]{0.38,0.63,0.69}{\textbf{\textit{{#1}}}}}
\newcommand{\WarningTok}[1]{\textcolor[rgb]{0.38,0.63,0.69}{\textbf{\textit{{#1}}}}}
% Define a nice break command that doesn't care if a line doesn't already
% exist.
\def\br{\hspace*{\fill} \\* }
% Math Jax compatability definitions
\def\gt{>}
\def\lt{<}
% Document parameters
\title{Genome\_regulatory\_circuits}
% Pygments definitions
\makeatletter
\def\PY@reset{\let\PY@it=\relax \let\PY@bf=\relax%
\let\PY@ul=\relax \let\PY@tc=\relax%
\let\PY@bc=\relax \let\PY@ff=\relax}
\def\PY@tok#1{\csname PY@tok@#1\endcsname}
\def\PY@toks#1+{\ifx\relax#1\empty\else%
\PY@tok{#1}\expandafter\PY@toks\fi}
\def\PY@do#1{\PY@bc{\PY@tc{\PY@ul{%
\PY@it{\PY@bf{\PY@ff{#1}}}}}}}
\def\PY#1#2{\PY@reset\PY@toks#1+\relax+\PY@do{#2}}
\expandafter\def\csname PY@tok@w\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.73,0.73,0.73}{##1}}}
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\expandafter\def\csname PY@tok@sa\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}}
\expandafter\def\csname PY@tok@sb\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}}
\expandafter\def\csname PY@tok@sc\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}}
\expandafter\def\csname PY@tok@dl\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}}
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\expandafter\def\csname PY@tok@mb\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.40,0.40,0.40}{##1}}}
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\expandafter\def\csname PY@tok@mh\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.40,0.40,0.40}{##1}}}
\expandafter\def\csname PY@tok@mi\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.40,0.40,0.40}{##1}}}
\expandafter\def\csname PY@tok@il\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.40,0.40,0.40}{##1}}}
\expandafter\def\csname PY@tok@mo\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.40,0.40,0.40}{##1}}}
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\expandafter\def\csname PY@tok@cm\endcsname{\let\PY@it=\textit\def\PY@tc##1{\textcolor[rgb]{0.25,0.50,0.50}{##1}}}
\expandafter\def\csname PY@tok@cpf\endcsname{\let\PY@it=\textit\def\PY@tc##1{\textcolor[rgb]{0.25,0.50,0.50}{##1}}}
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\expandafter\def\csname PY@tok@cs\endcsname{\let\PY@it=\textit\def\PY@tc##1{\textcolor[rgb]{0.25,0.50,0.50}{##1}}}
\def\PYZbs{\char`\\}
\def\PYZus{\char`\_}
\def\PYZob{\char`\{}
\def\PYZcb{\char`\}}
\def\PYZca{\char`\^}
\def\PYZam{\char`\&}
\def\PYZlt{\char`\<}
\def\PYZgt{\char`\>}
\def\PYZsh{\char`\#}
\def\PYZpc{\char`\%}
\def\PYZdl{\char`\$}
\def\PYZhy{\char`\-}
\def\PYZsq{\char`\'}
\def\PYZdq{\char`\"}
\def\PYZti{\char`\~}
% for compatibility with earlier versions
\def\PYZat{@}
\def\PYZlb{[}
\def\PYZrb{]}
\makeatother
% Exact colors from NB
\definecolor{incolor}{rgb}{0.0, 0.0, 0.5}
\definecolor{outcolor}{rgb}{0.545, 0.0, 0.0}
% Prevent overflowing lines due to hard-to-break entities
\sloppy
% Setup hyperref package
\hypersetup{
breaklinks=true, % so long urls are correctly broken across lines
colorlinks=true,
urlcolor=urlcolor,
linkcolor=linkcolor,
citecolor=citecolor,
}
% Slightly bigger margins than the latex defaults
\geometry{verbose,tmargin=1in,bmargin=1in,lmargin=1in,rmargin=1in}
\begin{document}
\maketitle
\#
Computational Systems Biology
\#\#
Gene Regulatory Circuits Exercises
\#\#\#\#
Ferran Pegenaute, [email protected]
\subsubsection{Table of Content}\label{table-of-content}
\begin{itemize}
\tightlist
\item
Section \ref{exercise-1}
\item
Section \ref{exercise-2}
\item
Section \ref{exercise-3}
\begin{itemize}
\tightlist
\item
Section \ref{change-ak}
\item
Section \ref{change-as}
\item
Section \ref{change-bs}
\end{itemize}
\end{itemize}
\subsubsection{Exercise 1 }\label{exercise-1}
\textbf{Consider a gene whose expression is affected by a direct
positive feedback, with Hill coefficient 2, maximum expression rate 1
nM/s, activation threshold 100 nM, and degradation rate 0.001 s-1.
Integrate the differential equation that represents the dynamics of the
concentration of the expressed protein, and determine the threshold
value of the initial condition that separates the basins of attraction
of the two coexisting equilibrium states of the system.}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}8}]:} \PY{k+kn}{import} \PY{n+nn}{numpy} \PY{k}{as} \PY{n+nn}{np}
\PY{k+kn}{from} \PY{n+nn}{scipy}\PY{n+nn}{.}\PY{n+nn}{integrate} \PY{k}{import} \PY{n}{odeint}
\PY{k+kn}{import} \PY{n+nn}{matplotlib}\PY{n+nn}{.}\PY{n+nn}{pyplot} \PY{k}{as} \PY{n+nn}{plt}
\PY{k+kn}{from} \PY{n+nn}{IPython}\PY{n+nn}{.}\PY{n+nn}{display} \PY{k}{import} \PY{n}{Image}
\end{Verbatim}
First, store the values given in the exercise in the corresponding
variables:
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}9}]:} \PY{c+c1}{\PYZsh{} Hill coefficient}
\PY{n}{n}\PY{o}{=}\PY{l+m+mi}{2}
\PY{c+c1}{\PYZsh{} Maximum expression rate}
\PY{n}{b}\PY{o}{=}\PY{l+m+mi}{1}
\PY{c+c1}{\PYZsh{} Activation threshold}
\PY{n}{K} \PY{o}{=} \PY{l+m+mi}{100}
\PY{c+c1}{\PYZsh{} Degradation rate}
\PY{n}{y}\PY{o}{=}\PY{l+m+mf}{0.001}
\end{Verbatim}
Let's define the functions that return dx/dt:
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}14}]:} \PY{k}{def} \PY{n+nf}{dx\PYZus{}dt}\PY{p}{(}\PY{n}{x}\PY{p}{,}\PY{n}{t}\PY{p}{)}\PY{p}{:}
\PY{n}{dxdt} \PY{o}{=} \PY{p}{(}\PY{p}{(}\PY{n}{b}\PY{o}{*}\PY{n}{x}\PY{o}{*}\PY{o}{*}\PY{n}{n}\PY{p}{)}\PY{o}{/}\PY{p}{(}\PY{n}{K}\PY{o}{*}\PY{o}{*}\PY{n}{n}\PY{o}{+} \PY{n}{x}\PY{o}{*}\PY{o}{*}\PY{n}{n}\PY{p}{)}\PY{p}{)}\PY{o}{\PYZhy{}}\PY{n}{y}\PY{o}{*}\PY{n}{x}
\PY{k}{return} \PY{n}{dxdt}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}11}]:} \PY{c+c1}{\PYZsh{}time state}
\PY{n}{t} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{linspace}\PY{p}{(}\PY{l+m+mi}{0}\PY{p}{,} \PY{l+m+mi}{15000}\PY{p}{,} \PY{l+m+mi}{1000}\PY{p}{)}
\PY{n}{x0}\PY{o}{=}\PY{p}{[}\PY{l+m+mi}{10}\PY{p}{,}\PY{l+m+mi}{50}\PY{p}{,}\PY{l+m+mi}{1000}\PY{p}{]}
\PY{k}{for} \PY{n}{x} \PY{o+ow}{in} \PY{n}{x0}\PY{p}{:}
\PY{n}{y0} \PY{o}{=} \PY{n}{odeint}\PY{p}{(}\PY{n}{dx\PYZus{}dt}\PY{p}{,} \PY{n}{x}\PY{p}{,} \PY{n}{t}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{plot}\PY{p}{(}\PY{n}{t}\PY{p}{,} \PY{n}{y0}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{xlabel}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{time}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{ylabel}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{x(t)}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{legend}\PY{p}{(}\PY{p}{[}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{x= }\PY{l+s+si}{\PYZpc{}.1f}\PY{l+s+s2}{\PYZdq{}} \PY{o}{\PYZpc{}}\PY{k}{x}])
\PY{n}{plt}\PY{o}{.}\PY{n}{title}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{Positive feedback model for x = }\PY{l+s+si}{\PYZpc{}.1f}\PY{l+s+s1}{\PYZsq{}}\PY{o}{\PYZpc{}}\PY{k}{x})
\PY{n}{plt}\PY{o}{.}\PY{n}{show}\PY{p}{(}\PY{p}{)}
\PY{n}{len\PYZus{}y0}\PY{o}{=}\PY{n+nb}{len}\PY{p}{(}\PY{n}{y0}\PY{p}{)}
\PY{n+nb}{print}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{The function tents to the value: }\PY{l+s+s2}{\PYZdq{}}\PY{p}{,} \PY{n+nb}{round}\PY{p}{(}\PY{n+nb}{float}\PY{p}{(}\PY{n}{y0}\PY{p}{[}\PY{n}{len\PYZus{}y0}\PY{o}{\PYZhy{}}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{)}\PY{p}{,}\PY{l+m+mi}{4}\PY{p}{)}\PY{p}{)}
\PY{n}{y1}\PY{o}{=}\PY{n+nb}{round}\PY{p}{(}\PY{n+nb}{float}\PY{p}{(}\PY{n}{y0}\PY{p}{[}\PY{n}{len\PYZus{}y0}\PY{o}{\PYZhy{}}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{)}\PY{p}{,}\PY{l+m+mi}{3}\PY{p}{)}
\end{Verbatim}
\begin{center}
\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_9_0.png}
\end{center}
{ \hspace*{\fill} \\}
\begin{Verbatim}[commandchars=\\\{\}]
The function tents to the value: 0.0003
\end{Verbatim}
\begin{center}
\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_9_2.png}
\end{center}
{ \hspace*{\fill} \\}
\begin{Verbatim}[commandchars=\\\{\}]
The function tents to the value: 989.8974
\end{Verbatim}
\begin{center}
\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_9_4.png}
\end{center}
{ \hspace*{\fill} \\}
\begin{Verbatim}[commandchars=\\\{\}]
The function tents to the value: 989.898
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}12}]:} \PY{n+nb}{print}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{The threshold value for the initial condition is: }\PY{l+s+s2}{\PYZdq{}}\PY{p}{,} \PY{n+nb}{round}\PY{p}{(}\PY{p}{(}\PY{l+m+mi}{1000}\PY{o}{\PYZhy{}}\PY{n}{y1}\PY{p}{)}\PY{p}{,}\PY{l+m+mi}{4}\PY{p}{)}\PY{p}{)}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
The threshold values of the initial condition is: 10.102
\end{Verbatim}
By doing the integration of the differential equation that represent the
concentration of the expresse protein, we found that the two equilibrium
states of the systems are found at protein concentration of 0.003 nM and
of 989.898 nM.
The \textbf{threshold value is 10.102} approximately, as we can see on
the above graphs obtained by changing the initial condition of the
protein concentration. We have use the values 10, 50 and 1000 nM
respectively
Observing the three plots with different possible initial conditions
tend towards two different stable equilibriums of the system. The
initial concentration 10, as it is lower than the threshold tends to a
stable state of 0 nM and both the concentration 50 nM and 1000 nM tend
to the value 989.898 nM, the former increasing the concentration while
the latter decreasing it.
\subsubsection{Exercise 2 }\label{exercise-2}
\textbf{Consider the following activator-repressor model discussed in
class (slide titled ``From oscillations to pulses''):}
\textbf{Simulate this model for the following parameter values: a2=0.025
nM/s, b1=15 nM/s, b2=0.8 nM/s, d1=d2=5·10-5 s-1, g=2.5·10-7 nM-1s-1,
K1=3000 nM, K2=750 nM, and n=m=2. Vary a1 as shown in the slides.
Reproduce the dynamics obtained there.}
\begin{itemize}
\tightlist
\item
Fist we will define the function of the differential equation
\end{itemize}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}5}]:} \PY{n}{K1} \PY{o}{=} \PY{l+m+mi}{3000}
\PY{n}{K2} \PY{o}{=} \PY{l+m+mi}{750}
\PY{n}{g} \PY{o}{=} \PY{l+m+mf}{2.5}\PY{o}{*}\PY{l+m+mi}{10}\PY{o}{*}\PY{o}{*}\PY{p}{(}\PY{o}{\PYZhy{}}\PY{l+m+mi}{7}\PY{p}{)}
\PY{n}{b1} \PY{o}{=} \PY{l+m+mi}{15}
\PY{n}{b2} \PY{o}{=} \PY{l+m+mf}{0.8}
\PY{n}{d1} \PY{o}{=} \PY{n}{d2} \PY{o}{=} \PY{l+m+mi}{5}\PY{o}{*}\PY{l+m+mi}{10}\PY{o}{*}\PY{o}{*}\PY{p}{(}\PY{o}{\PYZhy{}}\PY{l+m+mi}{5}\PY{p}{)}
\PY{n}{n} \PY{o}{=} \PY{n}{m} \PY{o}{=} \PY{l+m+mi}{2}
\PY{n}{y} \PY{o}{=} \PY{l+m+mf}{0.001}
\PY{n}{x} \PY{o}{=} \PY{l+m+mi}{3}
\PY{n}{a2} \PY{o}{=} \PY{l+m+mf}{0.025}
\PY{c+c1}{\PYZsh{} function that returns dx/dt and dy/dt}
\PY{k}{def} \PY{n+nf}{dx\PYZus{}dy}\PY{p}{(}\PY{n}{y}\PY{p}{,} \PY{n}{t}\PY{p}{,}\PY{n}{a1}\PY{p}{)}\PY{p}{:}
\PY{l+s+sd}{\PYZdq{}\PYZdq{}\PYZdq{}}
\PY{l+s+sd}{ ODE exercise 2}
\PY{l+s+sd}{ \PYZdq{}\PYZdq{}\PYZdq{}}
\PY{n}{x}\PY{p}{,} \PY{n}{y} \PY{o}{=} \PY{n}{y}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}\PY{p}{,} \PY{n}{y}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]}
\PY{n}{dxdt} \PY{o}{=} \PY{p}{(}\PY{n}{a1}\PY{o}{+}\PY{p}{(}\PY{p}{(}\PY{n}{b1}\PY{o}{*}\PY{n}{x}\PY{o}{*}\PY{o}{*}\PY{n}{n}\PY{p}{)}\PY{o}{/}\PY{p}{(}\PY{n}{K1}\PY{o}{*}\PY{o}{*}\PY{n}{n} \PY{o}{+} \PY{n}{x}\PY{o}{*}\PY{o}{*}\PY{n}{n}\PY{p}{)}\PY{p}{)}\PY{p}{)}\PY{o}{\PYZhy{}}\PY{n}{g}\PY{o}{*}\PY{n}{x}\PY{o}{*}\PY{n}{y}\PY{o}{\PYZhy{}}\PY{n}{d1}\PY{o}{*}\PY{n}{x}
\PY{n}{dydt} \PY{o}{=} \PY{p}{(}\PY{n}{a2}\PY{o}{+}\PY{p}{(}\PY{p}{(}\PY{n}{b2}\PY{o}{*}\PY{n}{x}\PY{o}{*}\PY{o}{*}\PY{n}{m}\PY{p}{)}\PY{o}{/}\PY{p}{(}\PY{n}{K2}\PY{o}{*}\PY{o}{*}\PY{n}{m} \PY{o}{+} \PY{n}{x}\PY{o}{*}\PY{o}{*}\PY{n}{m}\PY{p}{)}\PY{p}{)}\PY{p}{)}\PY{o}{\PYZhy{}}\PY{n}{d2}\PY{o}{*}\PY{n}{y}
\PY{k}{return} \PY{p}{[}\PY{n}{dxdt}\PY{p}{,} \PY{n}{dydt}\PY{p}{]}
\end{Verbatim}
\begin{itemize}
\tightlist
\item
We will set the initial state.
\end{itemize}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}6}]:} \PY{c+c1}{\PYZsh{} initial state:}
\PY{n}{y0} \PY{o}{=} \PY{p}{[}\PY{l+m+mi}{1}\PY{p}{,} \PY{l+m+mi}{2}\PY{p}{]}
\PY{c+c1}{\PYZsh{} time coodinate to solve the ODE for}
\PY{n}{t} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{linspace}\PY{p}{(}\PY{l+m+mi}{0}\PY{p}{,} \PY{l+m+mi}{1000000}\PY{p}{,}\PY{l+m+mi}{1000}\PY{p}{)}
\end{Verbatim}
\begin{itemize}
\tightlist
\item
Finally we will plot the dinamic of the
\end{itemize}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}7}]:} \PY{n}{a1}\PY{o}{=}\PY{p}{[}\PY{l+m+mf}{0.005}\PY{p}{,}\PY{l+m+mf}{0.007}\PY{p}{,}\PY{l+m+mf}{0.01}\PY{p}{]}
\PY{k}{for} \PY{n}{a} \PY{o+ow}{in} \PY{n}{a1}\PY{p}{:}
\PY{n}{y1} \PY{o}{=} \PY{n}{odeint}\PY{p}{(}\PY{n}{dx\PYZus{}dy}\PY{p}{,} \PY{n}{y0}\PY{p}{,} \PY{n}{t}\PY{p}{,} \PY{n}{args}\PY{o}{=}\PY{p}{(}\PY{n}{a}\PY{p}{,}\PY{p}{)}\PY{p}{)}
\PY{n}{dx}\PY{o}{=} \PY{n}{y1}\PY{p}{[}\PY{p}{:}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{]}
\PY{n}{dyy}\PY{o}{=}\PY{n}{y1}\PY{p}{[}\PY{p}{:}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{]}
\PY{n}{plt}\PY{o}{.}\PY{n}{plot}\PY{p}{(}\PY{n}{t}\PY{p}{,} \PY{n}{dx}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{plot}\PY{p}{(}\PY{n}{t}\PY{p}{,} \PY{n}{dyy}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{xlabel}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{time (s)}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{ylabel}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{x(t) and y(t)}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{legend}\PY{p}{(}\PY{p}{[}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{dxdt}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{dydt}\PY{l+s+s2}{\PYZdq{}}\PY{p}{]}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{title}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{Positive feedback model α1= }\PY{l+s+si}{\PYZpc{}.3f}\PY{l+s+s1}{\PYZsq{}}\PY{o}{\PYZpc{}}\PY{k}{a})
\PY{n}{plt}\PY{o}{.}\PY{n}{show}\PY{p}{(}\PY{p}{)}
\PY{c+c1}{\PYZsh{}oscillation}
\PY{n}{plt}\PY{o}{.}\PY{n}{plot}\PY{p}{(}\PY{n}{dx}\PY{p}{,}\PY{n}{dyy}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{xlabel}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{x}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{ylabel}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{y}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{legend}\PY{p}{(}\PY{p}{[}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{dxdy}\PY{l+s+s2}{\PYZdq{}}\PY{p}{]}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{title}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{Oscillation when α1= }\PY{l+s+si}{\PYZpc{}.3f}\PY{l+s+s1}{ }\PY{l+s+s1}{\PYZsq{}}\PY{o}{\PYZpc{}}\PY{k}{a})
\PY{n}{plt}\PY{o}{.}\PY{n}{show}\PY{p}{(}\PY{p}{)}
\end{Verbatim}
\begin{center}
\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_19_0.png}
\end{center}
{ \hspace*{\fill} \\}
\begin{center}
\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_19_1.png}
\end{center}
{ \hspace*{\fill} \\}
\begin{center}
\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_19_2.png}
\end{center}
{ \hspace*{\fill} \\}
\begin{center}
\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_19_3.png}
\end{center}
{ \hspace*{\fill} \\}
\begin{center}
\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_19_4.png}
\end{center}
{ \hspace*{\fill} \\}
\begin{center}
\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_19_5.png}
\end{center}
{ \hspace*{\fill} \\}
As in the above plots, we use the same time frame and kept all the
values the same except the a1 value which was modified in order to see
how it affects the behaviour of the system. we could notice that If a1
value increases from 0.005 to 0.01 and maintaining the same time frame,
we get a more unstable behavior, where the number of pulses increase. In
other words, when a1 get values higher than 0.005 the system is less
clear and there are more number of oscillations.
\subsubsection{Exercise 3. }\label{exercise-3.}
\textbf{The following model describes genetic competence in B.
subtilis:}
\textbf{Simulate this model for the parameter values given in the table
below:}
\textbf{Next, vary the parameters 𝛼k, 𝛼s and 𝛽s (one at a time) and
study the response of the system in the different situations. Reproduce
the different dynamical regimes studied in class.}
The K and S represent the concentrations of ComK and ComS in the cell,
respectively. Their basal expression is represented by the parameters: *
αk: Basal expression rate of ComK * αs: Basal expression rate of ComS,
maximum rate of regulated expression:
\begin{itemize}
\tightlist
\item
βk: Saturating expression rate of ComK positive feedback
\item
βs: Unrepressed expression rate of ComS
\end{itemize}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}8}]:} \PY{n}{bk} \PY{o}{=} \PY{l+m+mf}{7.5}
\PY{n}{dk} \PY{o}{=} \PY{n}{ds} \PY{o}{=} \PY{l+m+mi}{1}\PY{o}{*}\PY{l+m+mi}{10}\PY{o}{*}\PY{o}{*}\PY{p}{(}\PY{o}{\PYZhy{}}\PY{l+m+mi}{4}\PY{p}{)}
\PY{n}{kk}\PY{o}{=}\PY{l+m+mi}{5000}
\PY{n}{ks}\PY{o}{=} \PY{l+m+mi}{833}
\PY{n}{yk} \PY{o}{=} \PY{n}{ys} \PY{o}{=} \PY{l+m+mf}{0.001}
\PY{n}{Tk}\PY{o}{=}\PY{l+m+mi}{25000}
\PY{n}{Ts}\PY{o}{=}\PY{l+m+mi}{20}
\PY{n}{n} \PY{o}{=} \PY{l+m+mi}{2}
\PY{n}{p}\PY{o}{=}\PY{l+m+mi}{5}
\PY{n}{ak} \PY{o}{=} \PY{l+m+mf}{0.0875}
\PY{n}{a\PYZus{}s} \PY{o}{=} \PY{l+m+mf}{0.0004}
\PY{k}{def} \PY{n+nf}{dK\PYZus{}dS}\PY{p}{(}\PY{n}{y}\PY{p}{,} \PY{n}{t}\PY{p}{,}\PY{n}{a\PYZus{}k}\PY{p}{,}\PY{n}{a\PYZus{}s}\PY{p}{,}\PY{n}{bs}\PY{p}{)}\PY{p}{:}
\PY{l+s+sd}{\PYZdq{}\PYZdq{}\PYZdq{}}
\PY{l+s+sd}{ function that returns the ODE: dK/dt and dS/st}
\PY{l+s+sd}{ \PYZdq{}\PYZdq{}\PYZdq{}}
\PY{n}{K}\PY{p}{,} \PY{n}{S} \PY{o}{=} \PY{n}{y}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}\PY{p}{,} \PY{n}{y}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]}
\PY{n}{dKdt} \PY{o}{=} \PY{p}{(}\PY{n}{a\PYZus{}k}\PY{o}{+}\PY{p}{(}\PY{p}{(}\PY{n}{bk}\PY{o}{*}\PY{n}{K}\PY{o}{*}\PY{o}{*}\PY{n}{n}\PY{p}{)}\PY{o}{/}\PY{p}{(}\PY{n}{kk}\PY{o}{*}\PY{o}{*}\PY{n}{n} \PY{o}{+} \PY{n}{K}\PY{o}{*}\PY{o}{*}\PY{n}{n}\PY{p}{)}\PY{p}{)}\PY{p}{)}\PY{o}{\PYZhy{}}\PY{p}{(}\PY{p}{(}\PY{n}{yk}\PY{o}{*}\PY{n}{K}\PY{p}{)}\PY{o}{/}\PY{p}{(}\PY{l+m+mi}{1}\PY{o}{+}\PY{p}{(}\PY{n}{K}\PY{o}{/}\PY{n}{Tk}\PY{p}{)}\PY{o}{+}\PY{p}{(}\PY{n}{S}\PY{o}{/}\PY{n}{Ts}\PY{p}{)}\PY{p}{)}\PY{p}{)}\PY{o}{\PYZhy{}}\PY{n}{dk}\PY{o}{*}\PY{n}{K}
\PY{n}{dSdt} \PY{o}{=} \PY{p}{(}\PY{n}{a\PYZus{}s}\PY{o}{+}\PY{p}{(}\PY{n}{bs}\PY{o}{/}\PY{p}{(}\PY{l+m+mi}{1} \PY{o}{+} \PY{p}{(}\PY{n}{K}\PY{o}{/}\PY{n}{ks}\PY{p}{)}\PY{o}{*}\PY{o}{*}\PY{n}{p}\PY{p}{)}\PY{p}{)}\PY{p}{)}\PY{o}{\PYZhy{}}\PY{p}{(}\PY{p}{(}\PY{n}{ys}\PY{o}{*}\PY{n}{S}\PY{p}{)}\PY{o}{/}\PY{p}{(}\PY{l+m+mi}{1}\PY{o}{+}\PY{p}{(}\PY{n}{K}\PY{o}{/}\PY{n}{Tk}\PY{p}{)}\PY{o}{+}\PY{p}{(}\PY{n}{S}\PY{o}{/}\PY{n}{Ts}\PY{p}{)}\PY{p}{)}\PY{p}{)}\PY{o}{\PYZhy{}}\PY{n}{ds}\PY{o}{*}\PY{n}{S}
\PY{k}{return} \PY{p}{[}\PY{n}{dKdt}\PY{p}{,} \PY{n}{dSdt}\PY{p}{]}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}9}]:} \PY{c+c1}{\PYZsh{} initial state:}
\PY{n}{y0} \PY{o}{=} \PY{p}{[}\PY{l+m+mi}{1}\PY{p}{,} \PY{l+m+mf}{0.1}\PY{p}{]}
\PY{c+c1}{\PYZsh{} time coodinate to solve the ODE for}
\PY{n}{t} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{linspace}\PY{p}{(}\PY{l+m+mi}{0}\PY{p}{,} \PY{l+m+mi}{1000000}\PY{p}{,}\PY{l+m+mi}{1000}\PY{p}{)}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}10}]:} \PY{c+c1}{\PYZsh{} solve the ODE problem }
\PY{n}{a\PYZus{}k} \PY{o}{=} \PY{l+m+mf}{0.00875}
\PY{n}{a\PYZus{}s} \PY{o}{=} \PY{l+m+mf}{0.0004}
\PY{n}{bs} \PY{o}{=} \PY{l+m+mf}{0.06}
\PY{n}{y1} \PY{o}{=} \PY{n}{odeint}\PY{p}{(}\PY{n}{dK\PYZus{}dS}\PY{p}{,} \PY{n}{y0}\PY{p}{,} \PY{n}{t}\PY{p}{,}\PY{n}{args}\PY{o}{=}\PY{p}{(}\PY{n}{a\PYZus{}k}\PY{p}{,}\PY{n}{a\PYZus{}s}\PY{p}{,}\PY{n}{bs}\PY{p}{)}\PY{p}{)}
\PY{n}{dK}\PY{o}{=} \PY{n}{y1}\PY{p}{[}\PY{p}{:}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{]}
\PY{n}{dS}\PY{o}{=}\PY{n}{y1}\PY{p}{[}\PY{p}{:}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{]}
\PY{n}{plt}\PY{o}{.}\PY{n}{plot}\PY{p}{(}\PY{n}{t}\PY{p}{,} \PY{n}{dK}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{plot}\PY{p}{(}\PY{n}{t}\PY{p}{,} \PY{n}{dS}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{xlabel}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{time(s)}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{ylabel}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{Concentration (mM)}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{legend}\PY{p}{(}\PY{p}{[}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{ComK}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{ComS}\PY{l+s+s2}{\PYZdq{}}\PY{p}{]}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{title}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{Activador\PYZhy{}repressor model for ak = 0.00875, as = 0.0004, βs=0.06}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{show}\PY{p}{(}\PY{p}{)}
\PY{n}{len\PYZus{}yS}\PY{o}{=}\PY{n+nb}{len}\PY{p}{(}\PY{n}{dS}\PY{p}{)}
\PY{n}{len\PYZus{}yK}\PY{o}{=}\PY{n+nb}{len}\PY{p}{(}\PY{n}{dK}\PY{p}{)}
\PY{n+nb}{print}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{[ComS] tents to the value: }\PY{l+s+s2}{\PYZdq{}}\PY{p}{,} \PY{n+nb}{round}\PY{p}{(}\PY{n+nb}{float}\PY{p}{(}\PY{n}{dS}\PY{p}{[}\PY{n}{len\PYZus{}yS}\PY{o}{\PYZhy{}}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{)}\PY{p}{,}\PY{l+m+mi}{3}\PY{p}{)}\PY{p}{)}
\PY{n+nb}{print}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{[ComK] tents to the value: }\PY{l+s+s2}{\PYZdq{}}\PY{p}{,} \PY{n+nb}{round}\PY{p}{(}\PY{n+nb}{float}\PY{p}{(}\PY{n}{dK}\PY{p}{[}\PY{n}{len\PYZus{}yK}\PY{o}{\PYZhy{}}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{)}\PY{p}{,}\PY{l+m+mi}{3}\PY{p}{)}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{plot}\PY{p}{(}\PY{n}{dK}\PY{p}{,}\PY{n}{dS}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{xlabel}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{comK}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{ylabel}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{comS}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{title}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{Oscillation when αk = 0.00875, αs = 0.0004, βs=0.06}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{show}\PY{p}{(}\PY{p}{)}
\end{Verbatim}
\begin{center}
\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_27_0.png}
\end{center}
{ \hspace*{\fill} \\}
\begin{Verbatim}[commandchars=\\\{\}]
[ComS] tents to the value: 413.255
[ComK] tents to the value: 69.892
\end{Verbatim}
\begin{center}
\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_27_2.png}
\end{center}
{ \hspace*{\fill} \\}
\subsubsection{Changing the basal expression rate of ComK (αk) value.
}\label{changing-the-basal-expression-rate-of-comk-ux3b1k-value.}
\paragraph{When ak= 0.000875, 0.0875, 0.875, 2 and
10}\label{when-ak-0.000875-0.0875-0.875-2-and-10}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}11}]:} \PY{n}{a\PYZus{}k} \PY{o}{=} \PY{p}{[}\PY{l+m+mf}{0.000875}\PY{p}{,}\PY{l+m+mf}{0.0875}\PY{p}{,}\PY{l+m+mf}{0.5}\PY{p}{,}\PY{l+m+mf}{0.875}\PY{p}{,}\PY{l+m+mi}{2}\PY{p}{,}\PY{l+m+mi}{50}\PY{p}{]}
\PY{k}{for} \PY{n}{ak} \PY{o+ow}{in} \PY{n}{a\PYZus{}k}\PY{p}{:}
\PY{n}{a\PYZus{}s} \PY{o}{=} \PY{l+m+mf}{0.0004}
\PY{n}{bs} \PY{o}{=} \PY{l+m+mf}{0.06}
\PY{n}{y1} \PY{o}{=} \PY{n}{odeint}\PY{p}{(}\PY{n}{dK\PYZus{}dS}\PY{p}{,} \PY{n}{y0}\PY{p}{,} \PY{n}{t}\PY{p}{,} \PY{n}{args}\PY{o}{=}\PY{p}{(}\PY{n}{ak}\PY{p}{,} \PY{n}{a\PYZus{}s}\PY{p}{,}\PY{n}{bs}\PY{p}{)}\PY{p}{)}
\PY{n}{dK} \PY{o}{=} \PY{n}{y1}\PY{p}{[}\PY{p}{:}\PY{p}{,} \PY{l+m+mi}{0}\PY{p}{]}
\PY{n}{dS} \PY{o}{=} \PY{n}{y1}\PY{p}{[}\PY{p}{:}\PY{p}{,} \PY{l+m+mi}{1}\PY{p}{]}
\PY{n}{plt}\PY{o}{.}\PY{n}{plot}\PY{p}{(}\PY{n}{t}\PY{p}{,} \PY{n}{dK}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{plot}\PY{p}{(}\PY{n}{t}\PY{p}{,} \PY{n}{dS}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{xlabel}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{time(s)}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{ylabel}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{Concentration (mM)}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{legend}\PY{p}{(}\PY{p}{[}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{ComK}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{ComS}\PY{l+s+s2}{\PYZdq{}}\PY{p}{]}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{title}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{Activador\PYZhy{}repressor model for αk = }\PY{l+s+si}{\PYZpc{}.4f}\PY{l+s+s1}{\PYZsq{}}\PY{o}{\PYZpc{}}\PY{k}{ak})
\PY{n}{plt}\PY{o}{.}\PY{n}{show}\PY{p}{(}\PY{p}{)}
\PY{n}{len\PYZus{}yS}\PY{o}{=}\PY{n+nb}{len}\PY{p}{(}\PY{n}{dS}\PY{p}{)}
\PY{n}{len\PYZus{}yK}\PY{o}{=}\PY{n+nb}{len}\PY{p}{(}\PY{n}{dK}\PY{p}{)}
\PY{n+nb}{print}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{When αk = }\PY{l+s+si}{\PYZpc{}.4f}\PY{l+s+s2}{\PYZdq{}} \PY{o}{\PYZpc{}}\PY{k}{ak})
\PY{n+nb}{print}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{[ComS] tents to the value: }\PY{l+s+s2}{\PYZdq{}}\PY{p}{,} \PY{n+nb}{round}\PY{p}{(}\PY{n+nb}{float}\PY{p}{(}\PY{n}{dS}\PY{p}{[}\PY{n}{len\PYZus{}yS}\PY{o}{\PYZhy{}}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{)}\PY{p}{,}\PY{l+m+mi}{3}\PY{p}{)}\PY{p}{)}
\PY{n+nb}{print}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{[ComK] tents to the value: }\PY{l+s+s2}{\PYZdq{}}\PY{p}{,} \PY{n+nb}{round}\PY{p}{(}\PY{n+nb}{float}\PY{p}{(}\PY{n}{dK}\PY{p}{[}\PY{n}{len\PYZus{}yK}\PY{o}{\PYZhy{}}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{)}\PY{p}{,}\PY{l+m+mi}{3}\PY{p}{)}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{plot}\PY{p}{(}\PY{n}{dK}\PY{p}{,} \PY{n}{dS}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{xlabel}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{ComK}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{ylabel}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{ComS}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{title}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{ ComK vs ComS when αk = }\PY{l+s+si}{\PYZpc{}.4f}\PY{l+s+s1}{\PYZsq{}} \PY{o}{\PYZpc{}} \PY{n}{ak}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{show}\PY{p}{(}\PY{p}{)}
\end{Verbatim}
\begin{center}
\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_30_0.png}
\end{center}
{ \hspace*{\fill} \\}
\begin{Verbatim}[commandchars=\\\{\}]
When αk = 0.0009
[ComS] tents to the value: 413.235
[ComK] tents to the value: 6.062
\end{Verbatim}
\begin{center}
\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_30_2.png}
\end{center}
{ \hspace*{\fill} \\}
\begin{center}
\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_30_3.png}
\end{center}
{ \hspace*{\fill} \\}
\begin{Verbatim}[commandchars=\\\{\}]
When αk = 0.0875
[ComS] tents to the value: 57.738
[ComK] tents to the value: 966.31
\end{Verbatim}
\begin{center}
\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_30_5.png}
\end{center}
{ \hspace*{\fill} \\}
\begin{center}
\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_30_6.png}
\end{center}
{ \hspace*{\fill} \\}
\begin{Verbatim}[commandchars=\\\{\}]
When αk = 0.5000
[ComS] tents to the value: 9.642
[ComK] tents to the value: 1254.614
\end{Verbatim}
\begin{center}
\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_30_8.png}
\end{center}
{ \hspace*{\fill} \\}
\begin{center}
\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_30_9.png}
\end{center}
{ \hspace*{\fill} \\}
\begin{Verbatim}[commandchars=\\\{\}]
When αk = 0.8750
[ComS] tents to the value: 0.448
[ComK] tents to the value: 5548.956
\end{Verbatim}
\begin{center}
\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_30_11.png}
\end{center}
{ \hspace*{\fill} \\}
\begin{center}
\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_30_12.png}
\end{center}
{ \hspace*{\fill} \\}
\begin{Verbatim}[commandchars=\\\{\}]
When αk = 2.0000
[ComS] tents to the value: 0.499
[ComK] tents to the value: 9950.873
\end{Verbatim}
\begin{center}
\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_30_14.png}
\end{center}
{ \hspace*{\fill} \\}
\begin{center}
\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_30_15.png}
\end{center}
{ \hspace*{\fill} \\}
\begin{Verbatim}[commandchars=\\\{\}]
When αk = 50.0000
[ComS] tents to the value: 2.392
[ComK] tents to the value: 343804.813
\end{Verbatim}
\begin{center}
\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_30_17.png}
\end{center}
{ \hspace*{\fill} \\}
In this system we have the combination of possitive and negative
feedback, ComK and ComS respectivelly.
Observing the plot above, we can notice that when the basal expression
rate of ComK gets lower the concentration of ComS does not change
whereas the concentration of ComK gets lower as well. For instance,
when: αk = 0.00875 and αk = 0.0009 the concentration of ComS is 413.235
mM. This may be the reason that there are not oscillation behavior of
the system since the ComS is supressing the expression of ComK in a
negative feedback. That stable fixed point corresponds to the vegetative
state.
However when the basal expression rate of ComK gets higher values ( αk =
0.0875 to 0.875) we can notice that the concentration of ComS decrease
significantly (from 57.728 to 0.448) while the concentration on ComK
gets much higher (from 966.518 to 5548.956 mM). These may be because the
ComS is not expressed enough to inhibit the expression of ComK. Also, we
can see that within these values the system is unstable. The system
exhibits limit cycle oscillations between a mid-ComK and a high-ComK
level. In this case oscillation of the system is clearer to see. In
contrast, with lower values of αk there are not oscillation, we can see
that the curve tends to the 400 mM and there not a recirculation of the
reaction. But when αk get higher values than 0.5 the oscillation
dissapear and the system reach a higher concentration of ComK but it
stays in a stable state whitout concentration changes in ComS and ComK.
\subsubsection{Changing the basal expression rate of ComS (αs) value
}\label{changing-the-basal-expression-rate-of-coms-ux3b1s-value}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}12}]:} \PY{c+c1}{\PYZsh{} solve the ODE problem }
\PY{n}{a\PYZus{}s1} \PY{o}{=} \PY{p}{[}\PY{l+m+mf}{0.004}\PY{p}{,}\PY{l+m+mf}{0.05}\PY{p}{,}\PY{l+m+mf}{0.85}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{,}\PY{l+m+mi}{2}\PY{p}{,}\PY{l+m+mi}{5}\PY{p}{,}\PY{l+m+mi}{10}\PY{p}{]}
\PY{k}{for} \PY{n}{a\PYZus{}s} \PY{o+ow}{in} \PY{n}{a\PYZus{}s1}\PY{p}{:}
\PY{n}{a\PYZus{}k} \PY{o}{=} \PY{l+m+mf}{0.00875}
\PY{n}{bs} \PY{o}{=} \PY{l+m+mf}{0.06}
\PY{n}{y1} \PY{o}{=} \PY{n}{odeint}\PY{p}{(}\PY{n}{dK\PYZus{}dS}\PY{p}{,} \PY{n}{y0}\PY{p}{,} \PY{n}{t}\PY{p}{,} \PY{n}{args}\PY{o}{=}\PY{p}{(}\PY{n}{a\PYZus{}k}\PY{p}{,} \PY{n}{a\PYZus{}s}\PY{p}{,}\PY{n}{bs}\PY{p}{)}\PY{p}{)}
\PY{n}{dK} \PY{o}{=} \PY{n}{y1}\PY{p}{[}\PY{p}{:}\PY{p}{,} \PY{l+m+mi}{0}\PY{p}{]}
\PY{n}{dS} \PY{o}{=} \PY{n}{y1}\PY{p}{[}\PY{p}{:}\PY{p}{,} \PY{l+m+mi}{1}\PY{p}{]}
\PY{n}{plt}\PY{o}{.}\PY{n}{plot}\PY{p}{(}\PY{n}{t}\PY{p}{,} \PY{n}{dK}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{plot}\PY{p}{(}\PY{n}{t}\PY{p}{,} \PY{n}{dS}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{xlabel}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{time(s)}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{ylabel}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{Concentration (mM)}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{legend}\PY{p}{(}\PY{p}{[}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{ComK}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{ComS}\PY{l+s+s2}{\PYZdq{}}\PY{p}{]}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{title}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{Activador\PYZhy{}repressor model for αs =}\PY{l+s+si}{\PYZpc{}.4f}\PY{l+s+s1}{\PYZsq{}}\PY{o}{\PYZpc{}}\PY{k}{a\PYZus{}s})
\PY{n}{plt}\PY{o}{.}\PY{n}{show}\PY{p}{(}\PY{p}{)}
\PY{n}{len\PYZus{}yS}\PY{o}{=}\PY{n+nb}{len}\PY{p}{(}\PY{n}{dS}\PY{p}{)}
\PY{n}{len\PYZus{}yK}\PY{o}{=}\PY{n+nb}{len}\PY{p}{(}\PY{n}{dK}\PY{p}{)}
\PY{n+nb}{print}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{When αs = }\PY{l+s+si}{\PYZpc{}.4f}\PY{l+s+s2}{\PYZdq{}} \PY{o}{\PYZpc{}}\PY{k}{a\PYZus{}s})
\PY{n+nb}{print}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{[ComS] tents to the value: }\PY{l+s+s2}{\PYZdq{}}\PY{p}{,} \PY{n+nb}{round}\PY{p}{(}\PY{n+nb}{float}\PY{p}{(}\PY{n}{dS}\PY{p}{[}\PY{n}{len\PYZus{}yS}\PY{o}{\PYZhy{}}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{)}\PY{p}{,}\PY{l+m+mi}{3}\PY{p}{)}\PY{p}{)}
\PY{n+nb}{print}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{[ComK] tents to the value: }\PY{l+s+s2}{\PYZdq{}}\PY{p}{,} \PY{n+nb}{round}\PY{p}{(}\PY{n+nb}{float}\PY{p}{(}\PY{n}{dK}\PY{p}{[}\PY{n}{len\PYZus{}yK}\PY{o}{\PYZhy{}}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{)}\PY{p}{,}\PY{l+m+mi}{3}\PY{p}{)}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{plot}\PY{p}{(}\PY{n}{dK}\PY{p}{,} \PY{n}{dS}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{xlabel}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{ComK}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{ylabel}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{ComS}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{title}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{ComK vs ComS αs =}\PY{l+s+si}{\PYZpc{}.4f}\PY{l+s+s1}{\PYZsq{}} \PY{o}{\PYZpc{}} \PY{n}{a\PYZus{}s}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{show}\PY{p}{(}\PY{p}{)}
\end{Verbatim}
\begin{center}
\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_33_0.png}
\end{center}
{ \hspace*{\fill} \\}
\begin{Verbatim}[commandchars=\\\{\}]
When αs = 0.0040
[ComS] tents to the value: 448.558
[ComK] tents to the value: 72.322
\end{Verbatim}
\begin{center}
\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_33_2.png}
\end{center}
{ \hspace*{\fill} \\}
\begin{center}
\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_33_3.png}
\end{center}
{ \hspace*{\fill} \\}
\begin{Verbatim}[commandchars=\\\{\}]
When αs = 0.0500
[ComS] tents to the value: 904.333
[ComK] tents to the value: 93.482
\end{Verbatim}
\begin{center}
\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_33_5.png}
\end{center}
{ \hspace*{\fill} \\}
\begin{center}
\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_33_6.png}
\end{center}
{ \hspace*{\fill} \\}
\begin{Verbatim}[commandchars=\\\{\}]
When αs = 0.8500
[ComS] tents to the value: 8899.643
[ComK] tents to the value: 229.975
\end{Verbatim}
\begin{center}
\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_33_8.png}
\end{center}
{ \hspace*{\fill} \\}
\begin{center}
\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_33_9.png}
\end{center}
{ \hspace*{\fill} \\}
\begin{Verbatim}[commandchars=\\\{\}]
When αs = 1.0000
[ComS] tents to the value: 9801.591
[ComK] tents to the value: 73256.832
\end{Verbatim}
\begin{center}
\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_33_11.png}
\end{center}
{ \hspace*{\fill} \\}
\begin{center}
\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_33_12.png}
\end{center}
{ \hspace*{\fill} \\}
\begin{Verbatim}[commandchars=\\\{\}]
When αs = 2.0000
[ComS] tents to the value: 19800.797
[ComK] tents to the value: 74002.184
\end{Verbatim}
\begin{center}
\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_33_14.png}
\end{center}
{ \hspace*{\fill} \\}
\begin{center}
\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_33_15.png}
\end{center}
{ \hspace*{\fill} \\}
\begin{Verbatim}[commandchars=\\\{\}]
When αs = 5.0000
[ComS] tents to the value: 49800.319
[ComK] tents to the value: 74452.236
\end{Verbatim}
\begin{center}
\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_33_17.png}
\end{center}
{ \hspace*{\fill} \\}
\begin{center}
\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_33_18.png}
\end{center}
{ \hspace*{\fill} \\}
\begin{Verbatim}[commandchars=\\\{\}]
When αs = 10.0000
[ComS] tents to the value: 99800.16
[ComK] tents to the value: 74602.729
\end{Verbatim}
\begin{center}
\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_33_20.png}
\end{center}
{ \hspace*{\fill} \\}
Observing the behaviour of the system when we change the value of the
basal expression rate of ComS, we can notice that raising the value form
0.0004 to 0.8 the concentration of comS increase significantly faster
than the concentration of ComK. Once these concentration get its maximum