Team:IBB Pune/Modeling

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<p><span style="font-weight:bold; font-size:200%; color:#6600FF;">Modelling</span></p></html>
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<p><span style="font-weight:bold; font-size:200%; color:#006600;">Modelling</span></p></html>
<p>The aim of the modelling aspect of our project is to develop simulations that would enable us to predict the behavior of the export tag with the Snowdrift Game as the backdrop. </p>
<p>The aim of the modelling aspect of our project is to develop simulations that would enable us to predict the behavior of the export tag with the Snowdrift Game as the backdrop. </p>
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<html>
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<p><span style="font-weight:bold; font-size:200%; color:#F07211;">
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<p><span style="font-weight:bold; font-size:200%; color:#006600;">
<br>
<br>
Project 1- A Model Of The Snowdrift Game</span>
Project 1- A Model Of The Snowdrift Game</span>
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This model is for a Single iteration of the experiment i.e. the culture is initiated in shaker flask and allowed to grow. In order to find ESS or stable equilibria, we will have to repeatedly subculture the cells under similar conditions, by replacing the medium or supplying fresh medium. In such a setup, the input for the next cycle of experiments will be the output of the previous cycle of experiments.  
This model is for a Single iteration of the experiment i.e. the culture is initiated in shaker flask and allowed to grow. In order to find ESS or stable equilibria, we will have to repeatedly subculture the cells under similar conditions, by replacing the medium or supplying fresh medium. In such a setup, the input for the next cycle of experiments will be the output of the previous cycle of experiments.  
   
   
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===Assumptions===
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<span style="font-weight:bold; font-size:125%; color:#006600;"><p>Assumptions</p></span>
''At time t=0;''
''At time t=0;''
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*There are 'k' co-operators and 'N-k' defectors
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*There are ''''k'''' co-operators and ''''N-k'''' defectors
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*Medium contains 'L' mg/ml of lactose, glucose conc. (g) = 0
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*Medium contains ''''L'''' mg/ml of lactose, glucose conc. (g) = 0
*Culture is well mixed.
*Culture is well mixed.
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*Extracellular Enzyme conc (Ec) = 0 units/ml
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*Extracellular Enzyme conc ('''Ec''') = 0 units/ml
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<span style="font-weight:bold; font-size:150%; color:#006600;"><p>Artificial assumptions</p></span>
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'''Artificial assumptions'''
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*<p>Glucose is consumed by all cells. Galactose is also consumed at the same rate  Gc mg/cell/min/ml.
*<p>Glucose is consumed by all cells. Galactose is also consumed at the same rate  Gc mg/cell/min/ml.
*The metabolic benefit due to glucose and galactose is same.So effectively each lactose molecule gives rise to 2 glucose molecules</p>
*The metabolic benefit due to glucose and galactose is same.So effectively each lactose molecule gives rise to 2 glucose molecules</p>
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*<p>Rate of degradation of enzyme is zero</p>
*<p>Rate of degradation of enzyme is zero</p>
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<span style="font-weight:bold; font-size:125%; color:#006600;"><p>Model</p></span>
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===Model===
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In this Model,
In this Model,
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1. Each co-operator secretes ''''B' units of enzyme/ min and pays cost of 'c' / min'''
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1. Each co-operator secretes ''''B'''' units of enzyme/ min and pays cost of ''''c''''/min
2. These molecules diffuse freely through the medium
2. These molecules diffuse freely through the medium
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3. They convert Lactose to Glucose + Galactose given by
3. They convert Lactose to Glucose + Galactose given by
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4. Rate of conversion of lactose to glucose is given by (k2)*(Ec)*(L)  
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g mg/ml/min
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g = (k2)*(Ec)*(L) mg/ml/min                       .... (1)
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5. Absorbed glucose confers growth rate advantage of '''r'''  
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4. Absorbed glucose confers growth rate advantage of '''r'''  
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Advantage conferred by Glucose r= (R)*(g)*(Gc)  
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r = (R)*(g)*(Gc)                                      .... (2)
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After each min, the population of co-operators and defectors is updated depending upon the growth
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After each min, the population of co-operators and defectors is updated depending upon their respective growth rates.
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rates.
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<p>
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D(t) = D(t-1)+ r * D(t-1)                             .... (3)    Defector population (t)
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D(t) = D(t-1)+r*D(t-1) Defector population
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k(t) = k(t-1) + (r-c) * k(t-1)                        .... (4)    Co-operator population (t)
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</p><p>
 
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k(t) = k(t-1) + (r-c)*k(t-1) Co-operator population</p>
 
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===Equations===
 
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<p>
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the Lactose that remains at time 't'
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The Lactose that remains at time ''''t'''' is given by:
<br>
<br>
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L=L-L*Ec*k2
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L = L - L * Ec * k2                                   .... (5)
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</p>
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<p>
 
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The net Glucose present in the medium:
 
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</p>
 
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<p>
 
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G=G+((2*L*Ec*k2)-(N*Gc)
 
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</p>
 
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===Results===
 
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[[Image:Pic1.GIF |center|400px|thumbnail|the lactose and glc levels ]]
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The net Glucose present in the medium at time ''''t'''' is given by:
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<br><br><br><br><br>
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<br>
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G = G +(( 2* L* Ec* k2)-( N * Gc)                    .... (6)
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<span style="font-weight:bold; font-size:125%; color:#006600;"><p>Results</p></span>
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[[Image:Pic1.GIF |center|400px|thumbnail|Plot of lactose and glucose levels at k2=1, Gc = 0.0001 mg/cell/min, R=0.99, c=0.001  ]]
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<br>
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[[Image:Pic2.GIF |center|400px|thumbnail|the population distribution ]]
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[[Image:Pic2.GIF |center|400px|thumbnail|The population distribution at k2=1, Gc = 0.0001 mg/cell/min, R=0.99, c=0.001]]
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<br>
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[[Image:Pic3.gif |center|400px|thumbnail|the Cooperator to defector ratio ]]
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[[Image:Pic3.gif |center|400px|thumbnail|The Cooperator to defector ratio at k2=1, Gc = 0.0001 mg/cell/min, R=0.99, c=0.001 ]]

Latest revision as of 00:03, 22 October 2009





Modelling

The aim of the modelling aspect of our project is to develop simulations that would enable us to predict the behavior of the export tag with the Snowdrift Game as the backdrop.


Project 1- A Model Of The Snowdrift Game



Model



This is a Model of the Snowdrift game. This model assumes use of two different strains as co-operators and defectors.

Co-operators constitutively secrete/produce b-gal. Defectors cannot secrete/produce B-gal.

This model is for a Single iteration of the experiment i.e. the culture is initiated in shaker flask and allowed to grow. In order to find ESS or stable equilibria, we will have to repeatedly subculture the cells under similar conditions, by replacing the medium or supplying fresh medium. In such a setup, the input for the next cycle of experiments will be the output of the previous cycle of experiments.

Assumptions

At time t=0;

  • There are 'k' co-operators and 'N-k' defectors
  • Medium contains 'L' mg/ml of lactose, glucose conc. (g) = 0
  • Culture is well mixed.
  • Extracellular Enzyme conc (Ec) = 0 units/ml

Artificial assumptions

  • Glucose is consumed by all cells. Galactose is also consumed at the same rate Gc mg/cell/min/ml.

  • The metabolic benefit due to glucose and galactose is same.So effectively each lactose molecule gives rise to 2 glucose molecules

  • There is no intracellular lactose metabolism (only extracellular).

  • There is no lag in enzyme production and secretion.

  • Rate of degradation of enzyme is zero

Model

In this Model,

1. Each co-operator secretes 'B' units of enzyme/ min and pays cost of 'c'/min

2. These molecules diffuse freely through the medium

3. They convert Lactose to Glucose + Galactose given by 


g = (k2)*(Ec)*(L) mg/ml/min	                       .... (1)

4. Absorbed glucose confers growth rate advantage of r 

r = (R)*(g)*(Gc)                                      .... (2)

After each min, the population of co-operators and defectors is updated depending upon their respective growth rates. 

D(t) = D(t-1)+ r * D(t-1)                             .... (3)     Defector population (t)	
k(t) = k(t-1) + (r-c) * k(t-1)                        .... (4)     Co-operator population (t)



The Lactose that remains at time 't' is given by: 

L = L - L * Ec * k2                                   .... (5)


The net Glucose present in the medium at time 't' is given by: 

G = G +(( 2* L* Ec* k2)-( N * Gc)                     .... (6)

Results
Plot of lactose and glucose levels at k2=1, Gc = 0.0001 mg/cell/min, R=0.99, c=0.001


The population distribution at k2=1, Gc = 0.0001 mg/cell/min, R=0.99, c=0.001


The Cooperator to defector ratio at k2=1, Gc = 0.0001 mg/cell/min, R=0.99, c=0.001
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