Team:USTC-China/modeling

From 2012.igem.org

(Difference between revisions)
(reference)
 
(51 intermediate revisions not shown)
Line 3: Line 3:
<link rel="stylesheet" href="https://2012.igem.org/Team:USTC-China/basic.css?action=raw&ctype=text/css" type="text/css">
<link rel="stylesheet" href="https://2012.igem.org/Team:USTC-China/basic.css?action=raw&ctype=text/css" type="text/css">
-
<link rel="stylesheet" href="https://2012.igem.org/Team:USTC-China/home.css?action=raw&ctype=text/css" type="text/css" media="screen">
+
<link rel="stylesheet" href="https://2012.igem.org/Team:USTC-China/background.css?action=raw&ctype=text/css" type="text/css" media="screen">
 +
<link href='http://fonts.googleapis.com/css?family=Capriola' rel='stylesheet' type='text/css'>
 +
<link href='http://fonts.googleapis.com/css?family=Dosis' rel='stylesheet' type='text/css'>
 +
<link href='http://fonts.googleapis.com/css?family=Life+Savers' rel='stylesheet' type='text/css'>
 +
 
<script type="text/javascript" src="https://2012.igem.org/Team:USTC-China/jquery-1.2.3.js?action=raw&ctype=text/javascript"></script>
<script type="text/javascript" src="https://2012.igem.org/Team:USTC-China/jquery-1.2.3.js?action=raw&ctype=text/javascript"></script>
<script type="text/javascript" src="https://2012.igem.org/Team:USTC-China/jquery.easing.1.3.js?action=raw&ctype=text/javascript"></script>
<script type="text/javascript" src="https://2012.igem.org/Team:USTC-China/jquery.easing.1.3.js?action=raw&ctype=text/javascript"></script>
Line 16: Line 20:
});
});
</script>
</script>
-
<script type="text/javascript">
+
 
-
function goTopEx(){
+
-
        var obj=document.getElementById("goTopBtn");
+
-
        function getScrollTop(){
+
-
                return document.documentElement.scrollTop;
+
-
            }
+
-
        function setScrollTop(value){
+
-
                document.documentElement.scrollTop=value;
+
-
            }   
+
-
        window.onscroll=function(){getScrollTop()>0?obj.style.display="":obj.style.display="none";}
+
-
        obj.onclick=function(){
+
-
            var goTop=setInterval(scrollMove,10);
+
-
            function scrollMove(){
+
-
                    setScrollTop(getScrollTop()/1.1);
+
-
                    if(getScrollTop()<1)clearInterval(goTop);
+
-
                }
+
-
        }
+
-
    }
+
-
</script>
+
     <!--this is the top menu-->
     <!--this is the top menu-->
Line 97: Line 83:
<div id="centercontent">
<div id="centercontent">
-
<div id="modeling">
+
<div class="stage">
-
<p style="width:975px;height:800px;">write your content here<p>
+
 
 +
<div id="menu" class="menuword">
 +
 
 +
<h1.5><a href="#0">I. Mathematical models</a></h1.5>
 +
<ol>
 +
<li><a href="#1"> Toggle Switch without designed genetic circuit</a></li>
 +
<li><a href="#2"> Toggle Switch with designed genetic circuit</a></li>
 +
<li><a href="#3"> Result</a></li>
 +
<li><a href="#5"> Value Table</a></li>
 +
</ol>
 +
<h1.5><a href="#4">II. Computer simulations</a></h1.5>
 +
</br>
 +
<h1.5><a href="#6">III. Reference</a></h1.5>
 +
 
</div>
</div>
 +
</br>
 +
<h2><a name="0">I. Mathematical models</a></h2>
 +
<p>The function of the anticro is to repress the expression of cro. That means the anticro can strongly prevent the lambda phage from turning into lytic life cycle. It is the toggle switch in the genome of lambda phage that control the decision of lysogenic or lytic life cycle and it is the target of our anticro. So, we study the steady state property of two system, without and with our designed genetic circuit separately. Finally, we give the phase graphs to show the steady state property visibly and draw the stream diagram to simulate the fate of each point on the graph.
 +
</p>
 +
<p>We study the steady state property of two system, without and with our designed genetic circuit separately.</p>
 +
<h2><a name="1">Toggle Switch without designed genetic circuit</a></h2>
 +
<p>The chemical reactions are:</p>
 +
<img src="https://static.igem.org/mediawiki/2012/7/73/Ustc-igem_m1.jpg">
 +
<p>According to the Law of mass action, we can easily write down the equation approximations to reduce these equations.</p></br>
 +
<p>First of all, we do quasi-steady state approximation for dimerization reactions</p>
 +
<img src="https://static.igem.org/mediawiki/2012/a/a1/Ustc-igem_modeling2.jpg">
 +
<p>where <img src="https://static.igem.org/mediawiki/2012/6/62/11.jpg"> and <img src="https://static.igem.org/mediawiki/2012/8/8d/12.jpg"> are the total concentration of <img src="https://static.igem.org/mediawiki/2012/9/94/13.jpg"> and <img src="https://static.igem.org/mediawiki/2012/e/ed/14.jpg">.</p>
 +
<p>Besides, we decide to take the degradation into account in the very end step. So, considering the conservation of
 +
binding site</P>
 +
<img src="https://static.igem.org/mediawiki/2012/2/2a/Ustc-igem_m3.jpg">
 +
<p>Where R is the total binding site number, we get</p>
 +
<img src="https://static.igem.org/mediawiki/2012/d/d8/Ustcm4.jpg">
 +
<p>The above equations give us the expression of <img src="https://static.igem.org/mediawiki/2012/d/db/15.jpg">.
 +
The final step is to write down the evolution equations for <img src="https://static.igem.org/mediawiki/2012/9/94/13.jpg"> and <img src="https://static.igem.org/mediawiki/2012/e/ed/14.jpg">.</br>
 +
<img src="https://static.igem.org/mediawiki/2012/2/22/Ustcm5.jpg">
 +
<p>where</p>
 +
<img src="https://static.igem.org/mediawiki/2012/c/c9/Ustcm6.jpg">
-
</div><!--the centercontent-->
+
<h2><a name="2">Toggle Switch with designed genetic circuit</a></h2>
 +
<p>There is a difference if we add our designed genetic circuit:</p>
 +
<img src="https://static.igem.org/mediawiki/2012/6/67/Ustcm7.jpg">
 +
<p>where <img src="https://static.igem.org/mediawiki/2012/8/85/16.jpg"> refers to <img src="https://static.igem.org/mediawiki/2012/6/64/17.jpg">.</p>
 +
<p>Three additional equation should be taken into account</p>
 +
<img src="https://static.igem.org/mediawiki/2012/6/6a/Ustcm8.jpg">
 +
<p>The above gives us the expression for <img src="https://static.igem.org/mediawiki/2012/a/a2/18.jpg">, which is a function of <img src="https://static.igem.org/mediawiki/2012/3/3a/Popsprm.png">. It is then straightforward to derive <img src="https://static.igem.org/mediawiki/2012/c/ca/20.jpg"> from the following</p>
 +
<img src="https://static.igem.org/mediawiki/2012/1/1e/Ustcm9.jpg">
 +
<p>Once obtained <img src="https://static.igem.org/mediawiki/2012/c/ca/20.jpg">, substitute it into the following to get the final equation</p>
 +
<img src="https://static.igem.org/mediawiki/2012/a/ae/10.jpg">
-
<div id="nav_5" class="nav_5style" style="background-color:transparent;position:fixed; left:10px; top:120px; width:200px; height:578px;">
+
<h2><a name="3">Result</a></h2>
-
<p class="level0"><a href="./Home_files/Home.htm">Home</a></p>
+
<p>We plot the stream diagram for <img src="https://static.igem.org/mediawiki/2012/6/62/11.jpg"> and <img src="https://static.igem.org/mediawiki/2012/8/8d/12.jpg">:</p>
 +
<img src="https://static.igem.org/mediawiki/2012/5/54/Pic1.jpg">
 +
</br></br></br>
 +
<img src="https://static.igem.org/mediawiki/2012/3/37/Pic2.jpg">
 +
<p>We can see the apparent two steady states on this phase graph. The stream diagram shows the changing trend of each point.</p>
 +
<p>After adding our designed circuit, the stream flow diagram is</p>
 +
<img src="https://static.igem.org/mediawiki/2012/b/bb/Pic3.jpg">
 +
</br></br></br>
 +
<img src="https://static.igem.org/mediawiki/2012/2/2a/Pic4.jpg">
 +
<p>This result demonstrates the efficiency of the anticro of repressing the lytic life cycle.</p>
 +
<h2><a name="5">Value Table</a></h2>
 +
<img src="https://static.igem.org/mediawiki/2012/a/ad/24.jpg">
 +
 
 +
<h2><a name="4">II. Computer simulations</a></h2>
 +
<p>In order to show the visible process of our engineered bacteria defending against the bacteriophage, we write a JAVA program to simulate the evolution of the colony which is invaded by the bacteriophage. The program interface is showed below:</p>
 +
<img src="https://static.igem.org/mediawiki/2012/e/e1/QQ%E6%88%AA%E5%9B%BE%E6%9C%AA%E5%91%BD%E5%90%8D.jpg" width="775">
 +
 
 +
 
 +
<p>Here is the description of our JAVA program: (if you want to skip the description and operate the program directly, please <a href="http://home.ustc.edu.cn/~xjdu/igem2012_USTC_China.htm">click here</a>.)</p>
 +
 
 +
<p>The <font color="blue">blue</font> box: presents the lysogenies</p>
 +
<p>The <font color="red">red</font> box: presents the phage in the host is in lytic life cycle</p>
 +
<p>The <font color="grey">grey </font>box: presents the normal bacterium</p>
 +
<p>The amount of this three kinds of bacteria have been set as three changeable parameters in our program. The other three changeable parameters are:</p>
 +
<p><strong>P1</strong>: The newly infected cells turn into lytic life cycle at possibility of P1 and turn into lysogenic life cycle at possibility 1-P1.</p>
 +
<p><strong>P2</strong>: The lysogen suicides at possibility of P2.</p>
 +
<p><strong>P3</strong>: At the grids marked as competence states, the lysogenic marks win the occupancy of these grids in the competence at possibility of P3. Thus, the marks represent competence states transform into lysogenic marks at possibility of P3 and transform into normal marks at possibility of 1-P3.</p>
 +
 
 +
<p>Model: </p>
 +
<p>Grid model of solid LB medium</p>
 +
<p>1. The cell can only reproduce new cells when there are empty grids around it.</p>
 +
<p>2. The newly assembled and emitted phages will only infect the cells in the grids around its host.</p>
 +
<p>3. The step length of time for this JAVA applet to operate is the time of a cell cycle.</p>
 +
<p>4. The basic level of expression of the lysis gene and the quorum sensing among cells are ignored.</p>
 +
<p>5. We adopt the Monte-Carlo simulation to deal with the process of competence among cells</p>
 +
 
 +
<p>Description of the algorithm</p>
 +
<p>The process of infection: </p>
 +
<p>1. When the host is lysed by the newly reproduced phages, the cells in the grids around it will be infected by the emitted phages and the grid the host occupied will return to be empty.</p>
 +
<p>2. The newly infected cells turn into lytic life cycle at possibility of P1 and turn into lysogenic life cycle at possibility 1-P1.</p>
 +
<p>The process of growth of the cell:</p>
 +
<p>1. The normal cell can only reproduce new cells when there are empty grids around it.</p>
 +
<p>2. The lysogen suicides at possibility of P2 and reproduce a new lysogen at empty grid around it.</p>
 +
<p>3. If a empty grid abut both a normal cell and a lysogen and both the two cell will reproduce new cell, the offspring of the lysogen win the empty grid at possibility of P3.</p>
 +
 
 +
<p>Procedures of the algorithm</p>
 +
<p>1. Arrange three kinds of cells(grey: the normal cell, red: the host in which the phage is at lytic life cycle, blue: the lysogen) at the screen according to the parameters at random.</p>
 +
<p>2. The red cell is lysed and releases the grid it occupied. All the cells in the grids abut the lysed host will be infected.</p>
 +
<p>3. Scan the screen again. The newly infected cells turn into lytic life cycle at possibility of P1 and turn into lysogenic life cycle at possibility 1-P1.</p>
 +
<p>4. The normal cell sets marks represent the reproduction of normal cells (called the normal marks) in the grids around it.</p>
 +
<p>5. The lysogen suicides at possibility of P2. Otherwise, it sets marks represent the reproduction of lysogens (called the lysogenic marks) in the grids around it. If some of these grids around the lysogen have been marked as reproduction of normal cells, then the lysogen marks these grids as competence states.</p>
 +
<p>6. At the grids marked as competence states, the lysogenic marks win the occupancy of these grids in the competence at possibility of P3. Thus, the marks represent competence states transform into lysogenic marks at possibility of P3 and transform into normal marks at possibility of 1-P3.</p>
 +
<p>7. Those grids with normal mark are occupied by the newly reproduced normal cells and those grids with lysogenic mark are occupied by the lysogenies.</p>
 +
 
 +
<p>Here is  our JAVA program:  <a href="http://home.ustc.edu.cn/~xjdu/igem2012_USTC_China.htm">click here</a>.</p>
 +
<h2><a name="6">Reference</a></h2>
 +
<ul>
 +
<li>Timothy S. Gardner*, Charles R. Cantor* & James J. Collins*: Construction of a genetic toggle switch in Escherichia coli. NATURE, VOL 403, 20 JANUARY 2000</li>
 +
 
 +
<li>Moises Santillan*, Michael C. Mackey: Why the Lysogenic State of Phage λ Is So Stable: A Mathematical Modeling Approach. Biophysical Journal volume 86, issue 1, Jan 2004, page 75-84</li>
 +
 
 +
<li>Paul J. Darling, Jo M. Holt and Gary K. Ackers: Coupled Energetics of l cro Repressor Self-assembly and Site-specific DNA Operator Binding II: Cooperative Interactions of cro Dimers. J. Mol. Biol. (2000) 302, 625-638</li>
 +
 
 +
<li>Chunbo Lou, Xili Liu, Ming Ni: Synthesizing a novel genetic sequential logic circuit: a push-on push-off switch. Molecular Systems Biology 6:350</li>
 +
</br>
 +
</br>
 +
</div>
 +
</div><!--the centercontent-->
 +
 
 +
<div id="nav_5" class="nav_5style" style="position:fixed; left:5px; top:120px; width:110px; height:430px;background-color: #DFFFDF; -moz-border-radius: 10px;  -webkit-border-radius: 10px;border-radius: 10px; border: 1px solid DDD; padding:8px;margin:0px;border:none;">
 +
<p class="level0"><a href="https://2012.igem.org/Team:USTC-China">HOME</a></p>
-
<p class="level0"><a href="https://2012.igem.org/Team:USTC-China/project">Project</a></p>
 
-
<p class="level1"><a href="https://2012.igem.org/Team:USTC-China/project">Background</a></p>
+
<p class="level0"><a href="https://2012.igem.org/Team:USTC-China/background">PROJECT</a></p>
-
<p class="level1"><a href="https://2012.igem.org/Team:USTC-China/project">Methods</a></p>
+
<p class="level1"><a href="https://2012.igem.org/Team:USTC-China/background">Background</a></p>
-
<p class="level0"><a href="https://2012.igem.org/Team:USTC-China/project">Achievements</a></p>
+
<p class="level1"><a href="https://2012.igem.org/Team:USTC-China/methods">Methods</a></p>
-
<p class="level1"><a href="https://2012.igem.org/Team:USTC-China/project">Results</a></p>
+
<p class="level0"><a href="https://2012.igem.org/Team:USTC-China/results">ACHIEVEMENTS</a></p>
-
<p class="level1"><a href="https://2012.igem.org/Team:USTC-China/project">Parts</a></p>
+
<p class="level1"><a href="https://2012.igem.org/Team:USTC-China/results">Results</a></p>
-
<p class="level1"><a href="https://2012.igem.org/Team:USTC-China/project">Future Work</a></p>
+
<p class="level1"><a href="https://2012.igem.org/Team:USTC-China/parts">Parts</a></p>
-
<p class="level1"><a href="https://2012.igem.org/Team:USTC-China/project">Judging Criteria</a></p>
+
<p class="level1"><a href="https://2012.igem.org/Team:USTC-China/futurework">Future Work</a></p>
-
<p class="level1"><a href="https://2012.igem.org/Team:USTC-China/project">Asia Jamboree</a></p>
+
<p class="level1"><a href="https://2012.igem.org/Team:USTC-China/judgingcriteria">Judging Criteria</a></p>
-
<p class="level0"><a href="https://2012.igem.org/Team:USTC-China/project">Modeling</a></p>
+
<p class="level1"><a href="https://2012.igem.org/Team:USTC-China/asiajamboree">Asia Jamboree</a></p>
-
<p class="level0"><a href="https://2012.igem.org/Team:USTC-China/project">Outreach</a></p>
+
<p class="level0"><a href="https://2012.igem.org/Team:USTC-China/modeling">MODELING</a></p>
-
<p class="level1"><a href="https://2012.igem.org/Team:USTC-China/project">Activities</a></p>
+
<p class="level0"><a href="https://2012.igem.org/Team:USTC-China/outreach">OUTREACH</a></p>
-
<p class="level1"><a href="https://2012.igem.org/Team:USTC-China/project">Food Safety</a></p>
+
<p class="level1"><a href="https://2012.igem.org/Team:USTC-China/foodsafety">Food Safety</a></p>
-
<p class="level1"><a href="https://2012.igem.org/Team:USTC-China/project">Open Day</a></p>
+
<p class="level1"><a href="https://2012.igem.org/Team:USTC-China/openday">Open Day</a></p>
-
<p class="level1"><a href="https://2012.igem.org/Team:USTC-China/project">Outreach</a></p>
+
<p class="level1"><a href="https://2012.igem.org/Team:USTC-China/outreach">Outreach</a></p>
-
<p class="level1"><a href="https://2012.igem.org/Team:USTC-China/project">Safety</a></p>
+
<p class="level1"><a href="https://2012.igem.org/Team:USTC-China/safety">Safety</a></p>
-
<p class="level0"><a href="https://2012.igem.org/Team:USTC-China/project">Notes</a></p>
+
<p class="level0"><a href="https://2012.igem.org/Team:USTC-China/protocols">NOTES</a></p>
-
<p class="level1"><a href="https://2012.igem.org/Team:USTC-China/project">Protocols</a></p>
+
<p class="level1"><a href="https://2012.igem.org/Team:USTC-China/protocols">Protocols</a></p>
-
<p class="level1"><a href="https://2012.igem.org/Team:USTC-China/project">Group meetings</a></p>
+
<p class="level1"><a href="https://2012.igem.org/Team:USTC-China/groupmeetings">Group meetings</a></p>
-
<p class="level0"><a href="https://2012.igem.org/Team:USTC-China/project">Team</a></p>
+
<p class="level0"><a href="https://2012.igem.org/Team:USTC-China/team">TEAM</a></p>
Line 155: Line 254:
<!--the top menu-->
<!--the top menu-->
-
<div style="display: none" id="goTopBtn"><img border=0 src="https://static.igem.org/mediawiki/igem.org/b/b5/Shang.gif" ></div>
+
<div class="returnTop" show="true" style="visibility: visible; zoom: 1; opacity: 1; ">
-
<script  type="text/javascript">goTopEx();</script>
+
<a href="#top">backtotop</a></div>

Latest revision as of 13:08, 26 September 2012

MODELING


I. Mathematical models

The function of the anticro is to repress the expression of cro. That means the anticro can strongly prevent the lambda phage from turning into lytic life cycle. It is the toggle switch in the genome of lambda phage that control the decision of lysogenic or lytic life cycle and it is the target of our anticro. So, we study the steady state property of two system, without and with our designed genetic circuit separately. Finally, we give the phase graphs to show the steady state property visibly and draw the stream diagram to simulate the fate of each point on the graph.

We study the steady state property of two system, without and with our designed genetic circuit separately.

Toggle Switch without designed genetic circuit

The chemical reactions are:

According to the Law of mass action, we can easily write down the equation approximations to reduce these equations.


First of all, we do quasi-steady state approximation for dimerization reactions

where and are the total concentration of and .

Besides, we decide to take the degradation into account in the very end step. So, considering the conservation of binding site

Where R is the total binding site number, we get

The above equations give us the expression of . The final step is to write down the evolution equations for and .

where

Toggle Switch with designed genetic circuit

There is a difference if we add our designed genetic circuit:

where refers to .

Three additional equation should be taken into account

The above gives us the expression for , which is a function of . It is then straightforward to derive from the following

Once obtained , substitute it into the following to get the final equation

Result

We plot the stream diagram for and :




We can see the apparent two steady states on this phase graph. The stream diagram shows the changing trend of each point.

After adding our designed circuit, the stream flow diagram is




This result demonstrates the efficiency of the anticro of repressing the lytic life cycle.

Value Table

II. Computer simulations

In order to show the visible process of our engineered bacteria defending against the bacteriophage, we write a JAVA program to simulate the evolution of the colony which is invaded by the bacteriophage. The program interface is showed below:

Here is the description of our JAVA program: (if you want to skip the description and operate the program directly, please click here.)

The blue box: presents the lysogenies

The red box: presents the phage in the host is in lytic life cycle

The grey box: presents the normal bacterium

The amount of this three kinds of bacteria have been set as three changeable parameters in our program. The other three changeable parameters are:

P1: The newly infected cells turn into lytic life cycle at possibility of P1 and turn into lysogenic life cycle at possibility 1-P1.

P2: The lysogen suicides at possibility of P2.

P3: At the grids marked as competence states, the lysogenic marks win the occupancy of these grids in the competence at possibility of P3. Thus, the marks represent competence states transform into lysogenic marks at possibility of P3 and transform into normal marks at possibility of 1-P3.

Model:

Grid model of solid LB medium

1. The cell can only reproduce new cells when there are empty grids around it.

2. The newly assembled and emitted phages will only infect the cells in the grids around its host.

3. The step length of time for this JAVA applet to operate is the time of a cell cycle.

4. The basic level of expression of the lysis gene and the quorum sensing among cells are ignored.

5. We adopt the Monte-Carlo simulation to deal with the process of competence among cells

Description of the algorithm

The process of infection:

1. When the host is lysed by the newly reproduced phages, the cells in the grids around it will be infected by the emitted phages and the grid the host occupied will return to be empty.

2. The newly infected cells turn into lytic life cycle at possibility of P1 and turn into lysogenic life cycle at possibility 1-P1.

The process of growth of the cell:

1. The normal cell can only reproduce new cells when there are empty grids around it.

2. The lysogen suicides at possibility of P2 and reproduce a new lysogen at empty grid around it.

3. If a empty grid abut both a normal cell and a lysogen and both the two cell will reproduce new cell, the offspring of the lysogen win the empty grid at possibility of P3.

Procedures of the algorithm

1. Arrange three kinds of cells(grey: the normal cell, red: the host in which the phage is at lytic life cycle, blue: the lysogen) at the screen according to the parameters at random.

2. The red cell is lysed and releases the grid it occupied. All the cells in the grids abut the lysed host will be infected.

3. Scan the screen again. The newly infected cells turn into lytic life cycle at possibility of P1 and turn into lysogenic life cycle at possibility 1-P1.

4. The normal cell sets marks represent the reproduction of normal cells (called the normal marks) in the grids around it.

5. The lysogen suicides at possibility of P2. Otherwise, it sets marks represent the reproduction of lysogens (called the lysogenic marks) in the grids around it. If some of these grids around the lysogen have been marked as reproduction of normal cells, then the lysogen marks these grids as competence states.

6. At the grids marked as competence states, the lysogenic marks win the occupancy of these grids in the competence at possibility of P3. Thus, the marks represent competence states transform into lysogenic marks at possibility of P3 and transform into normal marks at possibility of 1-P3.

7. Those grids with normal mark are occupied by the newly reproduced normal cells and those grids with lysogenic mark are occupied by the lysogenies.

Here is our JAVA program: click here.

Reference

  • Timothy S. Gardner*, Charles R. Cantor* & James J. Collins*: Construction of a genetic toggle switch in Escherichia coli. NATURE, VOL 403, 20 JANUARY 2000
  • Moises Santillan*, Michael C. Mackey: Why the Lysogenic State of Phage λ Is So Stable: A Mathematical Modeling Approach. Biophysical Journal volume 86, issue 1, Jan 2004, page 75-84
  • Paul J. Darling, Jo M. Holt and Gary K. Ackers: Coupled Energetics of l cro Repressor Self-assembly and Site-specific DNA Operator Binding II: Cooperative Interactions of cro Dimers. J. Mol. Biol. (2000) 302, 625-638
  • Chunbo Lou, Xili Liu, Ming Ni: Synthesizing a novel genetic sequential logic circuit: a push-on push-off switch. Molecular Systems Biology 6:350


backtotop

Retrieved from "http://2012.igem.org/Team:USTC-China/modeling"