Team:TUDelft/Modeling Conjugation

From 2009.igem.org

(Difference between revisions)
Line 1: Line 1:
{{Template:TUDelftiGEM2009_menu_Model}}
{{Template:TUDelftiGEM2009_menu_Model}}
-
=Conjugation Modeling=
+
='''Conjugation Modeling=
-
Using the paper "[http://www3.interscience.wiley.com/journal/118837812/abstract?CRETRY=1&SRETRY=0 A model for bacterial conjugal gene transfer on solid surfaces]", we established the assumptions and parameters that would be needed for the modeling of the conjugation.
+
Our conjugation modeling work is based on the paper [http://www3.interscience.wiley.com/journal/118837812/abstract?CRETRY=1&SRETRY=0 A model for bacterial conjugal gene transfer on solid surfaces]. Using a slightly modified model we simulated our [[Team:TUDelft/ConjugationProtocol | conjugation protocol]] for R751 cells conjugating on a 0.2 μm filter placed on agar for a period of one hour.
-
=='''Assumptions'''==
+
 
-
#Cells are distributed randomly on the agar when introduced to it
+
==''Assumptions''==
-
#Number of cells grows until one or more of the medium components are exhausted, unless the initial nutrient concentration is lager than the saturation constant for growth (i.e. the ability of the colony to expand)
+
# The cells are distributed randomly on the filter at the start.
-
#Donors, recipients andf transconjugants have identical colony growth rates, specific growth rates and cell yields on solidified LB
+
# The conjugation times used in our simulations are less than the time required to exhaust the nutrients in the medium.
-
#Conjugation occurs such that all recipients become transconjugants after a certain conjugation time
+
# All cells form a colony (are CFU).
-
#Plasmid loss is negligable
+
# All cell types (donors, recipients and transconjugants) have identical colony radial growth rates and specific growth rates.
-
#Cells only move on the surface through colony expansion
+
# Conjugation occurs instantly when a donor colony touches a recipient colony. Future versions of the model will integrate a conjugation delay time once two colonies meet.
-
=='''Parameters'''==
+
# Plasmid loss is not taken into account.
-
*Surface area of media (A)
+
# Cells can not move over the surface except through expansion of the colony.
-
*Initial colony radius (r_0)
+
 
-
*Specific growth rate (g_n)
+
==''Parameters''==
-
*Colony radial growth rate (g_r)
+
 
-
*Maximum numbers of cells sustained by system (N_max)
+
The parameters needed for this model are:
-
*Initial number of donors (N_d) and recipients (N_r)
+
 
-
All these parameters can be determined experimentally, or by literary research into previous conjugation experiments.
+
* Surface area of media (A)
-
=='''Spatial distribution of bacteria'''==
+
* Initial colony radius (r<sub>0</sub>)
 +
* Specific growth rate (g<sub>n</sub>)
 +
* Colony radial growth rate (g<sub>r</sub>)
 +
* Maximum numbers of cells sustained by system (N<sub>max</sub>)
 +
* Initial number of donors (N<sub>d</sub>) and recipients (N<sub>r</sub>)
 +
 
 +
Default values for these parameter can be found on the [[Team:TUDelft/Modeling_Parameters | parameters]] page.
 +
 
 +
==''Spatial distribution of bacteria''==
The bacteria are initially distributed at random on the medium, so a distribution function to find the closest cell r would be:<br>
The bacteria are initially distributed at random on the medium, so a distribution function to find the closest cell r would be:<br>
-
F(r)=1-exp(-λ*r^2), where λ is the intensity (the expected number of bacteria per unit area).
+
[[Image:TUDcolonydisteq.png]]
-
=='''Growth of colonies'''==
+
where &lambda; is the intensity (cells per unit area).
-
With g_r being the exponential increasing rate of the colony radius at a certain time period, we can express the radius of the colony at time t as rad(t)=r_0*exp(g_r*t), where r_0 is the initial radius.<br>
+
-
With g_n being the exponential increasing rate of the cell number within a colony at a certain time period, we can express the number of bacteria in a colony at time t as N(t)=exp(g_n*t).<br>
+
-
Finally, there is a pre-given maximum yield of cells N_max, and growth stops when the total number of cells reaches this number.
+
-
=='''Contact between colonies'''==
+
==''Growth of colonies''==
-
Two colonies with radius ''r'' will touch if their centres are less then ''2r'' apart (i.e. overlapping colonies). Therefore, we can express the probability that the colonies will not be interacting by the function<br>
+
While the original model presented in the paper used an exponentially increasing rate for the colony radius, this was changed to a linear rate of expansion for our model. The radius at a point in time is given by:<br>
-
p(Not touching)=p(radius1+radius2>2r)=p1=exp(-4*r^2*π*λ)<br>
+
[[Image:TUDcolonyradiuseq.png]]
-
When this is considered as a function of time, the function would be<br>
+
-
p1(t)=exp(-4*π*λ*(r_0*exp(g_r*t))^2)=exp(-4*π*λ*(r_0)^2*exp(2*g_r*t))<br
+
-
When we approximate the distribution of colony clumps, it depends on the parameter ''p1'', so the probability that a colony chosen at random from the plate is part of a clump containing ''n'' colonies is<br>
+
-
p_n=p1(1-p1)^(n-1)=exp(-4*r^2*π*λ)(1-exp(-4*r^2*π*λ))^(n-1)<br>
+
-
This function in terms of time would be expressed as<br>
+
-
p_n(t)=exp(-4*π*λ*(r_0)^2*exp(2*g_r*t))(1-exp(-4*π*λ*(r_0)^2*exp(2*g_r*t)))^(n-1)<br>
+
-
In this case, the intensity λ=(N_d+N_r)/A, where A is the area of the medium, and N_d and N_r are the number of donors and recipients at t=0, respectively.
+
-
=='''Conjugation modeling'''==
+
==''Contact between colonies''==
-
In order for conjugation to take place, the donor and recipient colonies must interact with each other and be in the same clump (depnds on parameter p_n). For a colony chosen at random to be a recipient at time t, it must initially be a recipient and within a clump of only recipient colonies. The probability that any bacterium is a recipient is N_r/(N_d+N_r), and the probability that a clump with n colonies has only recipients is (N_r/(N_d+N_r))^n. To find whether a colony is sampled and a recipient, its probability can be expressed as<br>
+
Two colonies with radius r touch if their centers are less then 2r apart. The model checks this through an iterative process were it examines the separation distance between recipients and donors/transconjugants. Once this distance is less than 2r the receiver colony is turned into a transconjugant.
-
p_r=p1*N_r/(N_d+p1*N_r)
+
 
 +
==''Conjugation modeling''==
 +
 
 +
 
 +
[[Image:Gr30nd10000anigood.gif]]
 +
 
 +
[[Image:Gr60nd10000anigood.gif]]
 +
 
 +
[[Image:TUDGr30ng099_1hr_ani_T_20000.gif]]
-
=='''Final cell number'''==
 
-
The assumptions state that cell numbers increase and conjugation can occur until either the end of the incubation period, t_max, or until all the nutrients have been fully utilised, whichever occurs first. This is modelled by continuing calculations until N_max cells are produced or t = t_max. This time, t_eff , is the minimum of t_max and {ln(N_max)-ln(N_d+N_r)}/gn. Numbers of donor (M_d), recipient (M_r), and transconjugant cells (M_t) at t_max are given
 
-
by:<br>
 
-
M_d = N_d*exp(g_n*t_eff)<br>
 
-
M_r = (N_d+N_r)*p1*N_r*exp(g_n*t_eff)/(N_d+p1*N_r)<br>
 
-
M_t = N_r*exp(g_n*t_eff)-M_r
 
-
<html>
 
-
<center>
 
-
<embed
 
-
  src="https://static.igem.org/mediawiki/2009/9/98/TUDSignalPropagation.swf"
 
-
  width="550"
 
-
  height="400"
 
-
  allowscriptaccess="always"
 
-
  allowfullscreen="true"
 
-
/>
 
-
</center>
 
-
</html>
 
[[Image:Transcon-time-varg.png]]
[[Image:Transcon-time-varg.png]]
[[Image:Transcon-time-ci.png]]
[[Image:Transcon-time-ci.png]]
 +
 +
==''Distribution of separation distances''==
[[Image:TUDelftDist-ci.png]]
[[Image:TUDelftDist-ci.png]]
Line 85: Line 76:
</gallery>
</gallery>
|}
|}
 +
 +
==''Conjugation Efficiecy''==
 +
 +
<html>
 +
<center>
 +
<embed
 +
  src="https://static.igem.org/mediawiki/2009/9/98/TUDSignalPropagation.swf"
 +
  width="550"
 +
  height="400"
 +
  allowscriptaccess="always"
 +
  allowfullscreen="true"
 +
/>
 +
</center>
 +
</html>
[[Image:TUD-coneff-colg.png]]
[[Image:TUD-coneff-colg.png]]
Line 90: Line 95:
[[Image:TUD-coneff-inte.png]]
[[Image:TUD-coneff-inte.png]]
-
[[Image:Gr30nd10000anigood.gif]]
 
-
[[Image:Gr60nd10000anigood.gif]]
 
-
[[Image:TUDGr30ng099_1hr_ani_T_20000.gif]]
+
==''Signal Propagation''==
[[Team:TUDelft/Modeling_Conjugation_Plots|More Conjugation Modeling Plots]]
[[Team:TUDelft/Modeling_Conjugation_Plots|More Conjugation Modeling Plots]]

Revision as of 18:20, 18 October 2009

Conjugation Modeling

Our conjugation modeling work is based on the paper [http://www3.interscience.wiley.com/journal/118837812/abstract?CRETRY=1&SRETRY=0 A model for bacterial conjugal gene transfer on solid surfaces]. Using a slightly modified model we simulated our conjugation protocol for R751 cells conjugating on a 0.2 μm filter placed on agar for a period of one hour.

Assumptions

  1. The cells are distributed randomly on the filter at the start.
  2. The conjugation times used in our simulations are less than the time required to exhaust the nutrients in the medium.
  3. All cells form a colony (are CFU).
  4. All cell types (donors, recipients and transconjugants) have identical colony radial growth rates and specific growth rates.
  5. Conjugation occurs instantly when a donor colony touches a recipient colony. Future versions of the model will integrate a conjugation delay time once two colonies meet.
  6. Plasmid loss is not taken into account.
  7. Cells can not move over the surface except through expansion of the colony.

Parameters

The parameters needed for this model are:

  • Surface area of media (A)
  • Initial colony radius (r0)
  • Specific growth rate (gn)
  • Colony radial growth rate (gr)
  • Maximum numbers of cells sustained by system (Nmax)
  • Initial number of donors (Nd) and recipients (Nr)

Default values for these parameter can be found on the parameters page.

Spatial distribution of bacteria

The bacteria are initially distributed at random on the medium, so a distribution function to find the closest cell r would be:
TUDcolonydisteq.png

where λ is the intensity (cells per unit area).

Growth of colonies

While the original model presented in the paper used an exponentially increasing rate for the colony radius, this was changed to a linear rate of expansion for our model. The radius at a point in time is given by:
TUDcolonyradiuseq.png

Contact between colonies

Two colonies with radius r touch if their centers are less then 2r apart. The model checks this through an iterative process were it examines the separation distance between recipients and donors/transconjugants. Once this distance is less than 2r the receiver colony is turned into a transconjugant.

Conjugation modeling

Gr30nd10000anigood.gif

Gr60nd10000anigood.gif

TUDGr30ng099 1hr ani T 20000.gif


Transcon-time-varg.png

Transcon-time-ci.png

Distribution of separation distances

TUDelftDist-ci.png

TUDhist1000.png
TUDhist5000.png
TUDhist10000.png
TUDhist20000.png
TUDhist40000.png

Conjugation Efficiecy

TUD-coneff-colg.png

TUD-coneff-inte.png


Signal Propagation

More Conjugation Modeling Plots

1donor gr30 ng099 set2.gif

1donor gr30 ng099 set3.gif


Retrieved from "http://2009.igem.org/Team:TUDelft/Modeling_Conjugation"