Team:Heidelberg/Modeling

From 2008.igem.org

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[[Image:schloss_altstadt.jpg|520px|left]]<br>[[Image:IGEM_Team_Heidelberg_08.jpg|375px|right]]
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__NOTOC__
-
 
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<br>
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<br>
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<br>
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<br>
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<br>
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<br>
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<br>
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<br>
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<br>
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<br>
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<br>
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<br>
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<html>
<html>
-
<style type="text/css">
 
-
#nav, #nav ul {
 
-
 
-
        position:relative
 
-
margin: 0px 0px 0px 0px; /* purpose: allow centering ul table */
 
-
padding: 0px 0px 0px 0px;                      /* es scheint, als liegen die eigentlichen border der tabelle weiter außen, und
 
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                                                        angezeigt wird nur der zellhintergrund in schwarz */
 
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-
 
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        display:table
 
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                                        /* purpose: ul doesn't stretch width 100% */
 
-
                                   
 
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}
 
-
#nav li {
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<style>
-
display: table-cell;                            /* purpose: li behaves like table-cell. legt also die zeilen nebeneinander, mit
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h1.firstHeading { display: none; }
-
                                                          block wären sie untereinander */
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position: relative; /* purpose: non-overlap li elements in ul */
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list-style: none; /* purpose: remove default html list-style */
+
-
       
+
-
}
+
-
#nav li a {
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p {text-align: justify;}
-
display: block; /* purpose: non-overlap div on a */
+
-
margin: 0 0 0 0; /* purpose: spacing main menu items */
+
-
padding: 4px 21px ;
+
-
        border:3px solid #000000;
+
-
        background-color: black;
+
-
color: #00b0e6;
+
-
text-align: center;
+
-
text-decoration: none; /* purpose: remove underline from a */
+
-
font: bold 14.4px arial;
+
-
}
+
-
#nav li a:hover {
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a:link { color: #00b0e6; text-decoration: none}
-
background-color: black;
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a:visited { color:#00b0e6; text-decoration: none}
-
color: #f29400;
+
a:hover { color:#f29400; text-decoration: none}
-
}
+
a:active { color:#f29400; text-decoration: none}
-
#nav div {
+
#bodyContent { padding: 10px auto; width: 910px; margin: auto; clear: none; }
-
position: absolute; /* Anordnung der Zellen, wenn relativ, dann alle übereinander. purpose: li of
+
-
                                                          div doesn't spread out */
+
-
display: none;
+
-
width: 7em;                                     /* breite der ausklappboxen, paris hat sie auch nicht angepasst, man könnte
+
-
                                                            sie für jede seite extra anpassen...*/
+
-
opacity: 0.8;
+
-
        filter: alpha(opacity=80);
+
-
background: black;
+
-
        border:1px solid #000000;
+
-
                               
+
-
/* wenn man hier border und farbe eingibt, kann man einen rand um die ausklappenden kästen machen, der ist allerdings falsch formatiert*/
+
-
                                                       
+
-
}
+
-
#nav span a, #nav div a {                       /* dieser teil macht die ausklappende leiste*/
+
table#team_members { text-align: justify; border: 0; }
-
position: relative;                    /* wichtig: wenn der Teil fehlt, dann wird nur ein link pro ausklapp gezeigt*/
+
table#team_members h2, table#team_members h3 { clear: both; }
-
display: block; /* purpose: a's in div have same width */
+
-
margin: 0;
+
-
padding: 5px 10px;
+
-
text-align: left;
+
-
font: 13px arial;
+
-
}
+
-
+
-
#nav span a:hover, #nav div a:hover {           /* dieser teil macht das hovern der ausklappenden leiste*/
+
-
color: #f29400;
+
-
}
+
-
#nav span div {
 
-
position: relative;
 
-
margin: 0;
 
-
border: none; /* purpose: reset border to none */
 
-
border-top: 1px solid #5970B2; /* purpose: add a seperator */
 
-
border-bottom: 1px solid #5970B2; /* purpose: add a seperator */
 
-
opacity: 1.0; /* purpose: opacity already 0.8 by #nav div */
 
-
filter: alpha(opacity=100); /* purpose: opacity already 80 by #nav div */
 
-
}
 
-
 
-
#nav span div a {
 
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text-indent: 10px;
 
-
}
 
-
 
-
#nav span div span div a {
 
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text-indent: 20px;
 
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}
 
-
#nav .expand {
+
/*-----------------------------------------------------------------------------------------------*/
-
background-image: url('https://static.igem.org/mediawiki/2008/e/ef/Icon-expand.png');
+
div.MenuBar ul li ul.DropDownMenu {
-
background-repeat: no-repeat;
+
  display: none; /* Hides all drop-down menus. */
-
background-position: 95% 50%;
+
 
 +
}
 +
/*
 +
li:hover works in IE7 and FF2.
 +
a:hover works in IE6 and FF2.
 +
a:hover breaks li:hover in FF2.
 +
*/
 +
div.MenuBar ul li:hover ul.DropDownMenu li ul.SideMenu,
 +
div.MenuBar ul li a:hover ul.DropDownMenu li a ul.SideMenu {
 +
  display: none; /* Hides all side menus. */
 +
}
 +
/*------------------------------------------------------------------------------------- Menu Bar */
 +
div.MenuBar {
 +
  font: arial, helvetica, sans-serif;
 +
  height: 30px;
 +
  width: 910px;
 +
  /*width: 100%*/
 +
  margin: 0;
 +
  border-top: 0;
 +
  border-right: 0;
 +
  border-left: 0;
 +
  padding: 0;
 +
  background: black;
 +
 
 +
}
 +
div.MenuBar ul {
 +
  font: arial, helvetica, sans-serif;
 +
  text-align: center;
 +
  list-style-type: none;
 +
  margin: 0.5em auto;
 +
  border: 0;
 +
  padding: 0;
 +
  background: black;
 +
}
 +
div.MenuBar ul li {
 +
  font: arial, helvetica, sans-serif;
 +
  display: block;
 +
  padding: 0;
 +
  margin: 0;
 +
  font-size: 1.3em;
 +
  float: left;
 +
  background: black;
 +
  text-align: center;
 +
  width: 107px;
 +
  position: relative; /* Sets the positioning context for each drop-down menu. */
}
}
-
#nav .collapse {
+
div.MenuBar ul li a {
-
background-image: url('https://static.igem.org/mediawiki/2008/c/cd/Icon-collapse.png');
+
  font: arial, helvetica, sans-serif;
-
background-repeat: no-repeat;
+
  display: block;
-
background-position: 95% 50%;
+
  background: black;
 +
  height: 22px; /* Keep height + padding-top + padding-bottom sync with the menu bar height. */
 +
  color: #ffffff;
 +
  padding-top: 4px;
 +
  padding-bottom: 4px;
 +
  padding-left: 1em; /* Sets the left space between top-level items. */
 +
  padding-right: 1em; /* Sets the right space between top-level items. */
 +
  text-decoration: none;
}
}
-
 
-
</style>
+
/*------------------------------------------------------------------------------ Drop-Down Menus */
 +
div.MenuBar ul li:hover ul.DropDownMenu,
 +
div.MenuBar ul li a:hover ul.DropDownMenu {
 +
  display: block;
 +
  width: 10em; /* Drop-down menu width.
 +
                  Use MenuTailor.css to customize. */
 +
  height: 1em;
 +
  padding: 1px; /* Sets the drop-down menu "effective border" width. */
 +
  position: absolute;
 +
  top: 23px; /* Places the drop-down menu under the menu bar.
 +
                Keep it sync with the menu bar height. */
 +
  left: 0; /* Aligns the drop-down menu to its top-level item. */
 +
  background-color: black; /* Selected item. */
 +
  color: #FFFFFF;
-
<script type="text/javascript" src="http://student.kuleuven.be/~s0173901/scripts/jquery.js"></script>
+
}
-
<script type="text/javascript">
+
div.MenuBar ul li:hover ul.DropDownMenu li a,
 +
div.MenuBar ul li a:hover ul.DropDownMenu li a {
 +
  width: 10em; /* Keep it sync with the drop-down menu width.
 +
                            Use MenuTailor.css to customize. */
 +
  height: 1em;
 +
  padding-left: 0;
 +
  padding-right: 0;
 +
  background-color: black; /* Selected item. */
 +
  color: #FFFFFF;
 +
}
 +
ul.DropDownMenu li a span {
 +
  display: block;
 +
  padding-left: 0.75em; /* Sets the left space of each drop-down menu item. */
 +
  padding-right: 0.25em; /* Sets the right space of each drop-down menu item. */
 +
  text-align: right; /* Aligns the >> symbol to the right. */
 +
}
 +
ul.DropDownMenu li a span span {
 +
  float: left; /* Aligns the text (back) to the left. */
 +
  font: 12px arial, helvetica, sans-serif; /* Required for IE55. */
 +
  height: 20px;
 +
  color: #FFFFFF;
 +
}
 +
/*----------------------------------------------------------------------------------- Side Menus */
 +
div.MenuBar ul li:hover ul.DropDownMenu li:hover ul.SideMenu,
 +
div.MenuBar ul li a:hover ul.DropDownMenu li a:hover ul.SideMenu {
 +
  display: block;
 +
  width: 11em; /* Side menu width.
 +
                  Use MenuTailor.css to customize. */
 +
  padding: 1px; /* Sets the side menu "effective border" width. */
 +
  position: absolute;
 +
  top: -1px; /* Aligns the side menu to its drop-down menu item.
 +
                Keep it sync with the side menu "effective border" width. */
 +
  left: 13em; /* Places the side menu to the right of the drop-down menu.
 +
                            Keep it sync with the drop-down menu width.
 +
                            Use MenuTailor.css to customize. */
 +
}
 +
div.MenuBar ul li:hover ul.DropDownMenu li:hover ul.SideMenu li a,
 +
div.MenuBar ul li a:hover ul.DropDownMenu li a:hover ul.SideMenu li a {
 +
  width: 11em; /* Keep it sync with the side menu width.
 +
                            Use MenuTailor.css to customize. */
 +
  font: 12px arial, helvetica, sans-serif; /* Required for IE55. */
 +
  left: 13em; /* Places the side menu to the right of the drop-down menu.
 +
                            Keep it sync with the drop-down menu width.
 +
                            Use MenuTailor.css to customize. */
 +
}
 +
div.MenuBar ul li ul.DropDownMenu li ul.SideMenu li a span {
 +
  padding-left: 1.5em; /* Sets the left space of each side menu item. */
 +
  padding-right: 0.5em; /* Sets the right space of each side menu item. */
 +
  text-align: left;
 +
  font: 12px arial, helvetica, sans-serif; /* Required for IE55. */
 +
  left: 13em; /* Places the side menu to the right of the drop-down menu.
 +
                            Keep it sync with the drop-down menu width.
 +
                            Use MenuTailor.css to customize. */
 +
}
 +
/*----------------------------------------------------------------------------- Browser Specific */
 +
* html div.MenuBar ul li a {
 +
  float: left; /* Required for IE55 and IE6.
 +
                            Breaks O9.
 +
                            Hidden (* html) from non-IE browsers. */
 +
}
 +
* html ul.DropDownMenu li a:hover {
 +
  cursor: hand; /* Required for IE55.
 +
                  Hidden (* html) from non-IE browsers. */
 +
}
 +
ul.DropDownMenu li a:hover {
 +
  cursor: pointer; /* Required for IE6 and IE7.
 +
                      Hidding it (* html) from non-IE browsers breaks IE7!
 +
}
 +
* html div.MenuBar a:hover {
 +
  text-decoration: none; /* Required for IE55 and IE6.
 +
                            Hidden (* html) from non-IE browsers. */
 +
}
 +
* html div.MenuBar ul li table,
 +
* html div.MenuBar ul li table td {
 +
  border: 0; /* Required for IE55 and IE6.
 +
                Hidden (* html) from non-IE browsers. */
 +
}
 +
/*------------------------------------------------------------------------------- Default Colors */
 +
div.MenuBar {
 +
  background-color: Menu;
 +
  border-bottom: 1px solid ButtonShadow;
 +
}
 +
div.MenuBar a {
 +
  background-color: Menu; /* Top-level unselected items. */
 +
  color: MenuText;
 +
}
 +
div.MenuBar ul li:hover a,
 +
div.MenuBar ul li a:hover {
 +
  color: #ea7f16;
 +
  background-color: Highlight; /* Top-level selected item. */
 +
  color: HighlightText;
 +
}
 +
/*...............................................................................................*/
 +
div.MenuBar ul li:hover ul.DropDownMenu,
 +
div.MenuBar ul li a:hover ul.DropDownMenu {
 +
  background-color: ButtonShadow; /* Sets the drop-down menu "effective border" color. */
 +
}
 +
div.MenuBar ul li:hover ul.DropDownMenu li a,
 +
div.MenuBar ul li a:hover ul.DropDownMenu li a {
 +
  background-color: Menu;  Drop-down menu unselected items.
 +
                            Sets the drop-down menu "effective background" color. */
 +
  color: MenuText;
 +
}
 +
div.MenuBar ul li:hover ul.DropDownMenu li:hover a,
 +
div.MenuBar ul li a:hover ul.DropDownMenu li a:hover {
 +
  background-color: Highlight; /* Drop-down menu selected item. */
 +
  color: HighlightText;
 +
}
 +
/*...............................................................................................*/
 +
div.MenuBar ul li:hover ul.DropDownMenu li:hover ul.SideMenu,
 +
div.MenuBar ul li a:hover ul.DropDownMenu li a:hover ul.SideMenu {
 +
  background-color: ButtonShadow; /* Sets the side menu "effective border" color. */
 +
}
 +
div.MenuBar ul li:hover ul.DropDownMenu li:hover ul.SideMenu li a,
 +
div.MenuBar ul li a:hover ul.DropDownMenu li a:hover ul.SideMenu li a {
 +
  background-color: Menu; /* Side menu unselected items.
 +
                            Sets the side menu "effective background" color. */
 +
  color: MenuText;
 +
}
 +
div.MenuBar ul li:hover ul.DropDownMenu li:hover ul.SideMenu li a:hover,
 +
div.MenuBar ul li a:hover ul.DropDownMenu li a:hover ul.SideMenu li a:hover {
 +
  background-color: Highlight; /* Side menu selected item. */
 +
  color: HighlightText;
 +
}
 +
/*-----------------------------------------------------------------------------------------------*/
 +
/*Script-Free 3-Level Menu 1.2 Tailor
 +
  www.CesarDaniel.info
 +
/*-------------------------------------------------------------------------------------- General */
 +
body {
 +
  background: white;
 +
  color: black;
 +
  margin: 0;
 +
  border: 0;
 +
  padding: 0;
 +
}
 +
div.MenuBar {
 +
  font: 13px arial, helvetica, sans-serif;
 +
}
 +
div.MenuBar ul {
 +
  font: 13px arial, helvetica, sans-serif; /* Required for IE55. */
 +
}
 +
/*--------------------------------------------------------------------------------------- Colors */
 +
div.MenuBar {
 +
  background-color: black; /* Selected item. */
 +
  color: #FFFFFF;
 +
  border-bottom: 1px solid ButtonShadow;
 +
}
 +
div.MenuBar a,
 +
div.MenuBar ul li:hover ul.DropDownMenu li a,
 +
div.MenuBar ul li a:hover ul.DropDownMenu li a,
 +
div.MenuBar ul li:hover ul.DropDownMenu li:hover ul.SideMenu li a,
 +
div.MenuBar ul li a:hover ul.DropDownMenu li a:hover ul.SideMenu li a {
 +
  background-color: black; /* Selected item. */
 +
  color: #FFFFFF;
 +
}
 +
div.MenuBar ul li:hover a,
 +
div.MenuBar ul li a:hover,
 +
div.MenuBar ul li:hover ul.DropDownMenu li:hover a,
 +
div.MenuBar ul li a:hover ul.DropDownMenu li a:hover,
 +
div.MenuBar ul li:hover ul.DropDownMenu li:hover ul.SideMenu li a:hover,
 +
div.MenuBar ul li a:hover ul.DropDownMenu li a:hover ul.SideMenu li a:hover {
 +
  background-color: #00b0e6; /* Selected item. */
 +
  color: #FFFFFF;
 +
}
 +
div.MenuBar ul li:hover ul.DropDownMenu,
 +
div.MenuBar ul li a:hover ul.DropDownMenu,
 +
div.MenuBar ul li:hover ul.DropDownMenu li:hover ul.SideMenu,
 +
div.MenuBar ul li a:hover ul.DropDownMenu li a:hover ul.SideMenu {
 +
  background-color: ButtonShadow; /* Sets the drop-down and side menus "effective border" color. */
 +
}
 +
/*--------------------------------------------------------------------------------------- Widths */
 +
/*
-
function ddmsie() {
+
/*
-
$("#nav ul").css('display', 'inline-block');
+
Menu Bar 1
-
$("#nav li").css('display', 'inline');
+
Drop-Down Menu #2
-
$("#nav a").css('display', 'inline-block');
+
*/
-
$("#nav a").hover(function () {$(this).css('background-color', '#d4e2ef')},
+
div.MenuBar#navi ul li:hover ul.DropDownMenu#MB1-DDM4,
-
function () {$(this).css('background-color', '#649cd7')});
+
div.MenuBar#navi ul li a:hover ul.DropDownMenu#MB1-DDM4,
-
$("#nav div a").css('display', 'block');
+
div.MenuBar#navi ul li:hover ul.DropDownMenu#MB1-DDM4 li a,
-
$("#nav div").css('left', '0');
+
div.MenuBar#navi ul li a:hover ul.DropDownMenu#MB1-DDM4 li a {
-
$("#nav div").css('top', '100%');
+
  width: 11em; /* Drop-down menu width. */
-
$("#nav span div").css('top', '0');
+
}
 +
div.MenuBar#navi ul li:hover ul.DropDownMenu#MB1-DDM5,
 +
div.MenuBar#navi ul li a:hover ul.DropDownMenu#MB1-DDM5,
 +
div.MenuBar#navi ul li:hover ul.DropDownMenu#MB1-DDM5 li a,
 +
div.MenuBar#navi ul li a:hover ul.DropDownMenu#MB1-DDM5 li a {
 +
  width: 12em; /* Drop-down menu width. */
}
}
-
function ddmozilla() {
+
/*...............................................................................................*/
-
+
/*
 +
Menu Bar 1
 +
Drop-Down Menu #2
 +
Side Menu #1
 +
*/
 +
div.MenuBar#navi ul li:hover ul.DropDownMenu li:hover ul.SideMenu#MB1-DDM2-SM1,
 +
div.MenuBar#navi ul li a:hover ul.DropDownMenu li a:hover ul.SideMenu#MB1-DDM2-SM1 {
 +
  left: 15.5em !important; /* Places the side menu to the right of the drop-down menu.
 +
                            Keep it sync with the drop-down menu width. */
 +
}
 +
div.MenuBar#navi ul li:hover ul.DropDownMenu li:hover ul.SideMenu#MB1-DDM2-SM1,
 +
div.MenuBar#navi ul li a:hover ul.DropDownMenu li a:hover ul.SideMenu#MB1-DDM2-SM1,
 +
div.MenuBar#navi ul li:hover ul.DropDownMenu li:hover ul.SideMenu#MB1-DDM2-SM1 li a,
 +
div.MenuBar#navi ul li a:hover ul.DropDownMenu li a:hover ul.SideMenu#MB1-DDM2-SM1 li a {
 +
  width: 10em; /* Side menu width. */
 +
}
 +
/*...............................................................................................*/
 +
/*
 +
Menu Bar 1
 +
Drop-Down Menu #2
 +
Side Menu #2
 +
*/
 +
div.MenuBar#navi ul li:hover ul.DropDownMenu li:hover ul.SideMenu#MB1-DDM2-SM2,
 +
div.MenuBar#navi ul li a:hover ul.DropDownMenu li a:hover ul.SideMenu#MB1-DDM2-SM2 {
 +
  left: 15.5em  !important; /* Places the side menu to the right of the drop-down menu.
 +
                            Keep it sync with the drop-down menu width. */
 +
}
 +
div.MenuBar#navi ul li:hover ul.DropDownMenu li:hover ul.SideMenu#MB1-DDM2-SM2,
 +
div.MenuBar#navi ul li a:hover ul.DropDownMenu li a:hover ul.SideMenu#MB1-DDM2-SM2,
 +
div.MenuBar#navi ul li:hover ul.DropDownMenu li:hover ul.SideMenu#MB1-DDM2-SM2 li a,
 +
div.MenuBar#navi ul li a:hover ul.DropDownMenu li a:hover ul.SideMenu#MB1-DDM2-SM2 li a {
 +
  width: 10em; /* Side menu width. */
 +
}
 +
/*...............................................................................................*/
 +
/*
 +
Menu Bar 1
 +
Drop-Down Menu #2
 +
Side Menu #3
 +
*/
 +
div.MenuBar#navi ul li:hover ul.DropDownMenu li:hover ul.SideMenu#MB1-DDM2-SM3,
 +
div.MenuBar#navi ul li a:hover ul.DropDownMenu li a:hover ul.SideMenu#MB1-DDM2-SM3 {
 +
  left: 15.5em  !important; /* Places the side menu to the right of the drop-down menu.
 +
                            Keep it sync with the drop-down menu width. */
}
}
-
+
div.MenuBar#navi ul li:hover ul.DropDownMenu li:hover ul.SideMenu#MB1-DDM2-SM3,
-
function ddnav() {
+
div.MenuBar#navi ul li a:hover ul.DropDownMenu li a:hover ul.SideMenu#MB1-DDM2-SM3,
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$("#nav li").hover(
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[model_1_heidelberg.swf]
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 +
==Background==
 +
Interaction of bacterial populations is a challenging and difficult task for modelers. Bacterial strains respond to external chemicals, secrete their own signals for communication, and even produce killing factors for other bacterial species. Many of these processes are non-linear and proceed at different time and space scales. <br>
 +
[[https://2008.igem.org/Team:Heidelberg/Modeling back]]
 +
 
 +
==Motivation==
 +
In our project we construct a system of two bacterial strains, in which one population (killer strain) chemotactically senses the chemical signal AI-2 emitted by another population (prey). The killer strain is able to swim towards the prey population using the gradient of AI-2. In the vicinity of prey, the killer strain eliminates the prey. We considered two alternative killing mechanisms: colicin secretion and lambda-phage infection. <br>
 +
[[https://2008.igem.org/Team:Heidelberg/Modeling back]]
 +
 
 +
==Our achievements==
 +
We studied the system behavior in space and time using systems of non-linear Partial Differential Equations (PDEs). Systems of PDEs are widely used in science in a variety of problems, from tornadoes prediction to quantum mechanics.
 +
Unlike ordinary differential equations (ODEs), PDEs enable simulation of spatial effects in time, taking into account diffusion effects and spatial fluxes. This is particularly important when the system is not homogeneous. In our case,
 +
the killer population swims towards the prey population, and chemical signals require a certain time to diffuse through the medium. We designed a minimal system of four non-linear reaction-diffusion PDEs to simulate the chemotactic motility of the killer population towards the prey, and the colicin killing process (Fig. 1 (a), (c)). To simulate the system, we approximated the original system of four PDEs by a system of 10,000 ODEs (method of lines), and solved it numerically using custom-written Matlab code. In a more comprehensive PDE model, we added the fifth PDE to reflect the dynamics of AI-1 (Fig. 1 (b), (d)). AI-1 is secreted by the prey cells and adsorbed by the killer cells, in which it activates the production of the colicin and lysis proteins. The killer cells induced by AI-1 lyse and release the colicin into the surrounding media, resulting in the prey elimination. The extended model allowed us to reflect every key component of the system, and to simulate the interaction of the killer and prey populations at different initial spatial distribution and various system parameters. [[https://2008.igem.org/Team:Heidelberg/Modeling back]]<br> <br>
 +
 
 +
[[Image:Colmod.jpg|center|thumb|733px|'''Figure 1''': The spatial models for chemotaxis of the killer strain to the prey and the colicin-induced killing process. <br> '''(a)''' The minimal model with four components, where AI-1 is assumed to have the same production and diffusion rates as AI-2, both described by equation four.<br>'''(b)'''The corresponding system of four partial
 +
differential equations.<br>'''(c)''' The extended model with five components, where AI-1 dynamics is modeled explicitly.<br>'''(d)'''The corresponding system of five partial differential equations.]] <br> <br>
 +
 
 +
In the case when both populations are well mixed in spatially homogeneous conditions, we modeled another possible killing mechanism – the infection by lambda-phages. Here the prey cells can become infected either by conjugation
 +
with killer cells or by free phages, which in both cases leads to lysis of the prey cells and to a release of multiple new phages, initiating a snowball effect. The assumption of spatial homogeneity allowed us to better resolve the temporal behavior of this complex system. To model the lambda-phage infection, we used a system of delay differential equations (DDEs) (Fig. 2 (a), (c)). Unlike ordinary differential equations (ODEs), DDEs allow the temporal change in concentration to depend on concentrations at earlier time points. In our system, we used this method to reflect the fact that phage maturing cannot be arbitrary fast, but takes a certain time to develop. A careful model verification enabled us to simulate the infection dynamics in detail, including the formation of conjugation complexes and the snowball killing effect. To perform a detailed mathematical analysis, we reduced the full model to a simpler ODE version (Fig. 2 (b), (d)). For the ODE system, we determined its steady states with the linear stability
 +
analysis and investigated their stability with the initial values analysis, enabling to predict different system regimes upon given parameters and initial conditions. [[https://2008.igem.org/Team:Heidelberg/Modeling back]]<br> <br> <br>
 +
 
 +
[[Image:PhageSummary.png|center|thumb|733px|'''Figure 2''': The models of ''E. Coli'' infection by lambda-phages.<br/> '''(a)''' Scheme of the simplified model. <br /> '''(b)''' The ODE system which describes the simplified model of lambda-phage infection. <br /> '''(c)''' Scheme of the full model. <br /> '''(d)''' The DDE system which describes the full model of the phage infection.]]
 +
 
 +
==Modeling Results==
 +
If you are interested in more details concerning <br><br>
 +
our modeling part of '''Spatially Resolved Modeling of Colicin-Induced Killing and Chemotactic Sensing in ''E. coli''''', please click [https://2008.igem.org/Team:Heidelberg/Modeling/Chemotaxis here] <br><br>
 +
our modeling part of '''Phage-induced infection dynamics of ''E. coli'' ''', please click [[Media:Phagedocu.pdf|here]] <br><br>
 +
'''Status reports''' of our group, please click [https://2008.igem.org/Team:Heidelberg/Notebook/team_meetings#Team_Meetings here].<br><br>
 +
To download this '''summary''' of our modeling project, please click [[Media:Summary.pdf|here]]
 +
 
 +
[[https://2008.igem.org/Team:Heidelberg/Modeling back]]

Latest revision as of 23:06, 29 October 2008



Background

Interaction of bacterial populations is a challenging and difficult task for modelers. Bacterial strains respond to external chemicals, secrete their own signals for communication, and even produce killing factors for other bacterial species. Many of these processes are non-linear and proceed at different time and space scales.
[back]

Motivation

In our project we construct a system of two bacterial strains, in which one population (killer strain) chemotactically senses the chemical signal AI-2 emitted by another population (prey). The killer strain is able to swim towards the prey population using the gradient of AI-2. In the vicinity of prey, the killer strain eliminates the prey. We considered two alternative killing mechanisms: colicin secretion and lambda-phage infection.
[back]

Our achievements

We studied the system behavior in space and time using systems of non-linear Partial Differential Equations (PDEs). Systems of PDEs are widely used in science in a variety of problems, from tornadoes prediction to quantum mechanics. Unlike ordinary differential equations (ODEs), PDEs enable simulation of spatial effects in time, taking into account diffusion effects and spatial fluxes. This is particularly important when the system is not homogeneous. In our case, the killer population swims towards the prey population, and chemical signals require a certain time to diffuse through the medium. We designed a minimal system of four non-linear reaction-diffusion PDEs to simulate the chemotactic motility of the killer population towards the prey, and the colicin killing process (Fig. 1 (a), (c)). To simulate the system, we approximated the original system of four PDEs by a system of 10,000 ODEs (method of lines), and solved it numerically using custom-written Matlab code. In a more comprehensive PDE model, we added the fifth PDE to reflect the dynamics of AI-1 (Fig. 1 (b), (d)). AI-1 is secreted by the prey cells and adsorbed by the killer cells, in which it activates the production of the colicin and lysis proteins. The killer cells induced by AI-1 lyse and release the colicin into the surrounding media, resulting in the prey elimination. The extended model allowed us to reflect every key component of the system, and to simulate the interaction of the killer and prey populations at different initial spatial distribution and various system parameters. [back]

Figure 1: The spatial models for chemotaxis of the killer strain to the prey and the colicin-induced killing process.
(a) The minimal model with four components, where AI-1 is assumed to have the same production and diffusion rates as AI-2, both described by equation four.
(b)The corresponding system of four partial differential equations.
(c) The extended model with five components, where AI-1 dynamics is modeled explicitly.
(d)The corresponding system of five partial differential equations.


In the case when both populations are well mixed in spatially homogeneous conditions, we modeled another possible killing mechanism – the infection by lambda-phages. Here the prey cells can become infected either by conjugation with killer cells or by free phages, which in both cases leads to lysis of the prey cells and to a release of multiple new phages, initiating a snowball effect. The assumption of spatial homogeneity allowed us to better resolve the temporal behavior of this complex system. To model the lambda-phage infection, we used a system of delay differential equations (DDEs) (Fig. 2 (a), (c)). Unlike ordinary differential equations (ODEs), DDEs allow the temporal change in concentration to depend on concentrations at earlier time points. In our system, we used this method to reflect the fact that phage maturing cannot be arbitrary fast, but takes a certain time to develop. A careful model verification enabled us to simulate the infection dynamics in detail, including the formation of conjugation complexes and the snowball killing effect. To perform a detailed mathematical analysis, we reduced the full model to a simpler ODE version (Fig. 2 (b), (d)). For the ODE system, we determined its steady states with the linear stability analysis and investigated their stability with the initial values analysis, enabling to predict different system regimes upon given parameters and initial conditions. [back]


Figure 2: The models of E. Coli infection by lambda-phages.
(a) Scheme of the simplified model.
(b) The ODE system which describes the simplified model of lambda-phage infection.
(c) Scheme of the full model.
(d) The DDE system which describes the full model of the phage infection.

Modeling Results

If you are interested in more details concerning

our modeling part of Spatially Resolved Modeling of Colicin-Induced Killing and Chemotactic Sensing in E. coli, please click here

our modeling part of Phage-induced infection dynamics of E. coli , please click here

Status reports of our group, please click here.

To download this summary of our modeling project, please click here

[back]

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