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{{Imperial/StartPage2}}
{{Imperial/StartPage2}}
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{{Imperial/Box2|Growth Curve|
Part of our dry lab team concentrated on modelling the growth curve of ''B. subtilis''. This is important to characterise the chassis; particularly, the growth of ''subtilis'' is a vital parameter for planning experiments for future projects. Characterisation increases the predictability of the growth of  B. subtilis by determining, for example, its growth rate and the duration of its distinctive growth phases. In order to model the growth of ''B. subtilis'', the process was broken down into three main steps, where a separate submodel is produced in MATLAB for each step. Each submodel is an ODE model, which can be simulated using MATLAB. The variables in each submodel can be adjusted according to the boundary conditions (from experimental results).


= Growth Curves =
In the final step, a combination of Submodels 1 and 2 are superposed with Submodel 3, resulting in a more complex model which enhances the accuracy of illustrating bacterial growth.  For more details about the submodels, please click on the following links:


== Why model the growth curve? == 


The aim of modelling the growth curve is to characterise the ''B. subtilis'' chassis used in the project.  Characterisation increases the predictability of the growth of '' B. subtilis'' by determining, for example, its growth rate and the duration of its distinctive growth phases.
'''[[Tutorial for Growth Curve]]''' (More details about each submodel)


== How to model the growth curve? ==
'''[[Media:Modelling_Growth_Curve.pdf]]''' (The tutorial questions)


In order to model the growth of ''B. subtilis'', the process was broken down into three main steps, where a separate submodel is produced in MATLAB for each step. Each submodel is an ODE model, which can be simulated using MATLAB. The variables in each submodel can be adjusted according to boundary conditions (from experimental results).
The graph on the right shows the log graph used to determine the growth rate.
|[[Image:Log_graph.tif|300px|Log-Graph used to determine the growth rate]]}}


In the final step, a combination of Submodels 1 and 2 are superposed with Submodel 3, resulting in a more complex model, which illustrates bacterial growth. For more details about the submodels, please click on the following link: [[Media:Modelling_Growth_Curve.pdf]]. For more information on our modelling strategy, please click on [[Tutorial for Growth Curve]].
{{Imperial/Box1|Key Phases of Bacterial Growth|
===Lag Phase===
During the lag phase, the rate of growth is slow due to two main reasons, ''B. subtilis'' is absorbing nutrients in the medium and the replication machinery is being switched on. The higher the concentration of nutrients in the medium, the faster the rate of bateria growth.  


== The Model ==
As a result, the volume of the bacteria increases, followed by an increase in the number of bacteria.
[[Image:Nutrient ft.TIF]]
[[Image:Growth_model_dia.TIF]]
 
The model illustrates the main growth phases ''B. subtilis'' undergoes.  These are identified as the '''lag phase''', the '''exponential phase''' and the '''stationary phase'''.  The death phase is a constitutive event and it is possible that it exerts an influence on the three phases discussed below.  However, to simplify a complicated model, it is less relevant in this case and therefore, is not included in our model.
 
====== LAG PHASE ======
 
During the lag phase, the rate of growth is slow due to two main reasons. ''B. subtilis'' is absorbing nutrients in the medium and the replication machinery is switched on. The higher the concentration of nutrients in the medium, the faster the rate of bateria growth.
 
As a result, the volume of the bacteria increases, followed by an increases in the number of bacteria.
 
====== EXPONENTIAL PHASE ======


=== Exponential Phase ===
Both colony number and cell volume increase exponentially during this phase. Our model assumes concentration of the nutrients inside the bacteria is constant.
Both colony number and cell volume increase exponentially during this phase. Our model assumes concentration of the nutrients inside the bacteria is constant.


====== STATIONARY PHASE ======
=== Stationary Phase ===
The growth of the colony ceases in number and in volume due to a finite concentration of nutrients, hence it does not have a gradient.  Other causes may be death and cell division.


The growth of the colony ceases in number and in volume due to a finite concentration of nutrients, hence its does not have a gradient.  Other causes may be death and cell division.
According to the model, the maximum growth of the bacteria is determined by the concentration of nutrients available initially.


According to the model, the maximum growth of the bacteria is determined by the value of C<sub>max</sub> (from M-file regarding ''A plot of nutrient concentration vs time'').
To further enhance the accuracy of the model, the following information will be extracted from experimental data:
 
To further enhance the accuracy of the model, the following information will be extracted from experimental data


*Time span of lag phase, stationary phase and exponential phase
*Time span of lag phase, stationary phase and exponential phase
*The growth rate
*The growth rate}}
*The error of each experiments
 
== Results ==
 
The variables that affect bacterial growth rate versus time were adjusted to yield a curve similar to that given by experimental data, illustrated as the red curve on the graph.
 
[[Image:Model_fitting.tif]]


The values that result in the above models are:


GROWTH CONSTANT: k = 2.03
{{Imperial/Box1|The Model|
The model illustrates the main growth phases the ''B. subtilis'' undergoes.  These are identified as the '''lag phase''', the '''exponential phase''' and the '''stationary phase'''.  The death phase is a constitutive event and it is possible that it exerts an influence on the three phases discussed below.  However, to simplify a complicated model, it is less relevant in this case and therefore is not included in this model.


MAXIMUM CONCENTRATION: Cmax = 5
The M-file used to generate the model below is located in the Appendices section of the Dry Lab hub.


HILL COEFFICIENT: n = 3.05
<center>[[Image:Nutrient ft.TIF|500px]][[Image:Label_model.TIF‎|500px]]</center>}}


Ka = 5


With regards to the concentration of nutrients at certain time points:
{{Imperial/Box1|Results|
The model for the growth curve was fitted to the experimental results as shown below.  The experimental results is depicted by the red curve, while our model is shown by the green curve. The resource curve was also plotted as a function of time and is shown below. Based on our experimental results from the Wet Lab, a log graph was plotted to determine the growth rate.  The growth rate was then determined from the gradient of the log graph.  This value was included when simulating the growth model using MATLAB.
<html><table border="0" cellpadding="5" cellspacing="0" align="center"><tr><td width=33%></html>[[Image:Experimental_Result.JPG|thumb|center]]<html>
</td><td></html>[[Image:Fitted_Curve.JPG‎|thumb|center]]<html>
</td><td></html>[[Image:Resource_Curve.JPG‎|thumb|center]]<html>
</td></tr><tr><td><center>Experimental Results</center>
</td><td><center>Fitted Curve</center>
</td><td><center>Resource Curve</center>
</td></tr></table></html>


t0 = 0.5 s
The following constants used to generate the model were found to yield the best fit to experimental results.


t1 = 2 s


t2 = 2.03 s
GROWTH CONSTANT (A): 1.3494


t3 = 7 s
INITIAL NUTRIENT CONCENTRATION (R<sub>0</sub>): 2


==Discussion==
HILL COEFFICIENT (n): 1.25


[[IGEM:IMPERIAL/2008/New/Dry_lab | Dry lab hub]]
INITIAL OD: 0.4


Growth Curve
CONSTANT (<math>\alpha</math>): 0.64516


[[IGEM:IMPERIAL/2008/New/Genetic_Circuit | Genetic Circuit]]
}}


[[IGEM:IMPERIAL/2008/New/Motility | Motility]]


[[IGEM:IMPERIAL/2008/New/Appendices | Appendices]]


{{Imperial/EndPage|Dry_Lab|Genetic_Circuit}}
{{Imperial/EndPage|Dry_Lab|Genetic_Circuit}}

Latest revision as of 05:50, 25 October 2008

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Growth Curve

Part of our dry lab team concentrated on modelling the growth curve of B. subtilis. This is important to characterise the chassis; particularly, the growth of subtilis is a vital parameter for planning experiments for future projects. Characterisation increases the predictability of the growth of B. subtilis by determining, for example, its growth rate and the duration of its distinctive growth phases. In order to model the growth of B. subtilis, the process was broken down into three main steps, where a separate submodel is produced in MATLAB for each step. Each submodel is an ODE model, which can be simulated using MATLAB. The variables in each submodel can be adjusted according to the boundary conditions (from experimental results).

In the final step, a combination of Submodels 1 and 2 are superposed with Submodel 3, resulting in a more complex model which enhances the accuracy of illustrating bacterial growth. For more details about the submodels, please click on the following links:


Tutorial for Growth Curve (More details about each submodel)

Media:Modelling_Growth_Curve.pdf (The tutorial questions)

The graph on the right shows the log graph used to determine the growth rate.

Log-Graph used to determine the growth rate


Key Phases of Bacterial Growth

Lag Phase

During the lag phase, the rate of growth is slow due to two main reasons, B. subtilis is absorbing nutrients in the medium and the replication machinery is being switched on. The higher the concentration of nutrients in the medium, the faster the rate of bateria growth.

As a result, the volume of the bacteria increases, followed by an increase in the number of bacteria.

Exponential Phase

Both colony number and cell volume increase exponentially during this phase. Our model assumes concentration of the nutrients inside the bacteria is constant.

Stationary Phase

The growth of the colony ceases in number and in volume due to a finite concentration of nutrients, hence it does not have a gradient. Other causes may be death and cell division.

According to the model, the maximum growth of the bacteria is determined by the concentration of nutrients available initially.

To further enhance the accuracy of the model, the following information will be extracted from experimental data:

  • Time span of lag phase, stationary phase and exponential phase
  • The growth rate


The Model

The model illustrates the main growth phases the B. subtilis undergoes. These are identified as the lag phase, the exponential phase and the stationary phase. The death phase is a constitutive event and it is possible that it exerts an influence on the three phases discussed below. However, to simplify a complicated model, it is less relevant in this case and therefore is not included in this model.

The M-file used to generate the model below is located in the Appendices section of the Dry Lab hub.



Results

The model for the growth curve was fitted to the experimental results as shown below. The experimental results is depicted by the red curve, while our model is shown by the green curve. The resource curve was also plotted as a function of time and is shown below. Based on our experimental results from the Wet Lab, a log graph was plotted to determine the growth rate. The growth rate was then determined from the gradient of the log graph. This value was included when simulating the growth model using MATLAB.

<html><table border="0" cellpadding="5" cellspacing="0" align="center"><tr><td width=33%></html>
<html> </td><td></html>
<html> </td><td></html>
<html>

</td></tr><tr><td><center>Experimental Results</center> </td><td><center>Fitted Curve</center> </td><td><center>Resource Curve</center> </td></tr></table></html>

The following constants used to generate the model were found to yield the best fit to experimental results.


GROWTH CONSTANT (A): 1.3494

INITIAL NUTRIENT CONCENTRATION (R0): 2

HILL COEFFICIENT (n): 1.25

INITIAL OD: 0.4

CONSTANT ([math]\displaystyle{ \alpha }[/math]): 0.64516





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