Transcript Lab3

Today we will create our own math model
1. Flagellar length control in Chlamydomonus
2. Lotka-Volterra Model
How flagellar length is controlled in Chlamydomonus?
1.Flagellum is dynamic and undergoes continuous
turnover even after the flagellum has fully formed.
2. Site of the turnover is the distal end of the flagellum.
3. Intraflagellar transport (IFT) – driven by molecular
motors - is required for flagellum assembly.
Reference: http://www.molbiolcell.org/cgi/reprint/16/1/270
How are the assembly and disassembly rates
controlled to achieve this balance at the correct length?
Balance-point model:
a) shortening rate is constant
b) elongation rate decreases with length
Shortening rate is constant (measurement before
mitosis,when IFT transport stops):
Elongation rate is ~ 1/L
aM - length increase per one trip of all IFTs.
M - number of IFTs moving within a flagellum;
aM – length increase per one trip;
v - speed of IFT particle movement;
t = (2L/V) – time for one IFT trip; then
aM /t = (0.5aMV) / L – length increase per second
Adding two terms,
Rate of change of length:
dL A
 D
dt L
Where, A and D are Assembly and Disassembly constant.
We will solve this problem creating our own Mathematical description in Vcell.
Create our own Math Model:
Steps:
File>New>MathModel>Non-Spatial
and enter
(Non-Spatial for solving ODEs.)
This window will pop up. We will write our code in this window.
Code will be saved in the Mathmodel.
• We will write our code in VCML
Editor.
• We can view Equations.
• Run simulations as before.
First we will create our own Math model to solve the problem of Flagellar Length
control.
Math Description
Steps:
MathDescription {
1. Constant Declaration (end with ;)
2. VolumeVariable declartion
3. Function declaration (end with ;)
4. CompartmentSubDomain Compartment {
ODE declaration }
}
Sequentially:
ConstantVolumeVariable Function ODEs
Constant Declaration:
Format is : Constant Parameter Name Value ;
Constant A 7.0;
Constant D 1.0;
Constant L_init 1.0;
VolumeVariable:
L in the volume of consideration which is varying with time:
Afetr constant declaration write
VolumeVariable L
Function Declaration:
Format is: Function Functionname ( Function expression) ;
Write: Function J_length (( A/L) - D) ;
Declaration of ODE:
In VCML editor we will set our ODE inside these „{}“ brackets
Format:
CompartmentSubDomain Compartment {
OdeEquation L {
Rate
J_reaction;
Initial
L_init;
}
}
This part means
dL A
 D
dt L
Thats all !!!
VCML editor will look like this.
Click Apply Changes, If don‘t get error message, then run simulation.
Click Simulations, it will look like this
Now we know all the steps to run simulations.
Remember
When you click run to the simulation,the software will
ask you to save the model with a name.
Your own math description will be saved as a separate
document in Mathmodel.
You can reuse and update your model whenever you want,
by FileOpenMathModel (and click the model of your interest)
Play with your model:
- Check how length L changes with time.
- Check how rate of change of length varies with L
- Check assembly and disassembly rate with length.
Length-Time graph. tend= 10 sec, check for tend= 60. what do you see?
Length Vs Reaction rate
Elongation (assembly) rate Vs L (when D=0)
Disassembly rate Vs Length.
Lotka-Volterra Model(Alfred J. Lotka in 1925
and Vito Volterra in 1926.)
Lotka-Volterra equations describe the dynamics of the biological
systems, where two species interact, one is predator and
one its prey.
Consider,
R= number of prey (e.g Rabbits)
W= number of predator (e.g Wolves)
dR
dt
dW
dt
Growth of Rabbit‘s population against time
Growth of of Wolf‘s population against time
dR
= rabbit‘s growth – rabbits killed by Wolves
dt
dW
= Wolf‘s growth – wolf‘s death
dt
Mathematically:
Equation for Rabbit
dR
 a.R  b.R.W
dt
1st term
The prey are assumed to reproduce exponentially unless subject to predation;
this exponential growth is represented in the equation above
by the term a.R. Here a is a constant at which Rabbits grows.
2nd term
The rate of predation upon the prey is assumed to be proportional to the rate at which
the predators and the prey meet; this is represented above by b.R.W.
If either R or W is zero then there can be no predation. Here b is constant at which predation
occurs.
Equation for Wolf
dW
 d .R.W  c.W
dt
1st term
d.R.W is growth of wolf population. Note the similarity to the predation rate;
however, a different constant d is used as the rate at which the predator
population grows is not necessarily equal to the rate at which it consumes the prey.
2nd term
c.W represents natural death of wolves. It shows the exponential decay. C is the rate
constant at which wolves die.
So equtions are
dR
 a.R  b.R.W
dt
dW
 d .R.W  c.W
dt
We will solve these equtions using Vcell, and analyze
Results.
Let us start:
FileNewMathModelNon-spatial
This window will pop up. We will write our code in this window.
Constants
Constant
Constant
Constant
Constant
Constant
Constant
VolumeVariable
R_init
W_init
d
c
b
a
10.0;
5.0;
1.0;
2.0;
1.0;
15.0;
(R and W are the parameter which are varying)
VolumeVariable R
VolumeVariable W
Functions
1.J_wolfgrowth
d.R.W  c.W
2.J_predation
a.R  b.R.W
In mathmodel write—
Function J_wolfgrowth
Function J_predation
((R * d * W) - (c * W));
((a * R) - (R * b * W));
ODEs
We have 2 differential equations for rabbit and Wolf.
CompartmentSubDomain Compartment {
OdeEquation R {
Rate
J_predation;
Initial
R_init;
}
OdeEquation W {
Rate
J_wolfgrowth;
Initial
W_init;
}
}
a.R  b.R.W
d.R.W  c.W
The window will look like
Click Apply changes and then go to simulation Text.
Result:
For R(0)=10.0, W(0)=5.0 , a=b=c=d=1
Results:
R(0)=10.0, W(0)=5.0 , a=10.0, c=5.0
R Vs W Plot for a=10.0, c=5.0
R(0)=20.0, W(0)=5.0, a=10.0, b=2.0, c=7.0,d=1.5
Exercise:
Experiment when one flagellum is amputated
‘length pool’
a T  L1  L2  M
‘available length’
V
2 L1
- Rate of growth of 1st flag.
V
- Rate of growth of 2nd flag.
a T  L1  L2  M
2 L2
T
T
dL1
L2 
dL2
L1 
 A 1    D
 A 1    D
dt
L1 
dt
L2 
 L1
 L2
A  1, T  30, D  1
dL1 30 L2
dL2 30 L1

 2

 2
dt
L1 L1
dt
L2 L2
Initial condition: L1 (0)  10, L2 (0)  1
Solve the system of differential equation using VCell
After you solve the equations, think about the results. Do they agree with experimental
observations at the top of this slide? Why?
Results:
A=D=1.0, T=30, L1_init= 10, L2_init= 1.
L1_init= 15, L2_init=1, A = D= 1.0
Exercise
Consider similar, and equally famous model for when two species (say, rabbits
and sheep) compete for the same resource (say, grass): Equations of this model
have a very simple form:
dx / dt  rx
where r is growth/death rate. Let’s say there is no y (y = 0). Then,
r = 1 – x: if x <1, x grows; if x >1, x dies (x ‘eats’ its own resources). Now, if y
also ‘eats’ x’s resources, the growth rate becomes:
r = 1 – x – ay ; hence the first equation. The second equation follows the same
logic.
dx
 x  1  x  ay 
dt
rate of growth of x
dy
 y  1  y  ax 
dt
rate of growth of y
Exercise: solve this system of equations with VCell with initial conditions
x(0) = 0.5, y(0) = 0.7 first at a = 2, then at a = 0.5.
Describe the results in words. Think how to explain these results in words.
Results
X_init=0.5, Y_init=0.7, a=2.0,
co-existance impossible. The species with less initial
concentration decline to zero.
For a= 0.5, Rabbit and sheep co-exist happily.