Today we will create our own math model

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Today we will create our own math model

• Flagellar length control in Chlamydomonus
• Lotka-Volterra Model

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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/reprin
t/16/1/270
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• How are the assembly and disassembly rates
• controlled to achieve this balance at the correct
  length?
• Balance-point model
• shortening rate is constant
• 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
• M 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

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Adding two terms, Rate of change of length
Where, A and D are Assembly and Disassembly
constant.
We will solve this problem creating our own
Mathematical description in Vcell.
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Create our own Math Model
Steps FilegtNewgtMathModelgtNon-Spatial and
enter (Non-Spatial ?for solving ODEs.)
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This window will pop up. We will write our code
in this window.
Code will be saved in the Mathmodel.
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• 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.
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Math Description

• Steps
• MathDescription
• Constant Declaration (end with )
• 2. VolumeVariable declartion
• 3. Function declaration (end with )
• 4. CompartmentSubDomain Compartment
• ODE declaration

Sequentially Constant?VolumeVariable
?Function ?ODEs
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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
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Function Declaration
Format is Function Functionname ( Function
expression)
Write Function J_length (( A/L) - D)
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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
Thats all !!!
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VCML editor will look like this.
Click Apply Changes, If dont get error message,
then run simulation.
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Click Simulations, it will look like this
Now we know all the steps to run simulations.
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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)
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• 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.

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Length-Time graph. tend 10 sec, check for tend
60. what do you see?
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Length Vs Reaction rate
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Elongation (assembly) rate Vs L (when D0)
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Disassembly rate Vs Length.
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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.
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Consider, R number of prey (e.g Rabbits) W
number of predator (e.g Wolves)
Growth of Rabbits population against time
Growth of of Wolfs population against time
rabbits growth rabbits killed by Wolves
Wolfs growth wolfs death
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Mathematically
Equation for Rabbit?
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.
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Equation for Wolf?
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.
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So equtions are
We will solve these equtions using Vcell, and
analyze Results. Let us start
File?New?MathModel?Non-spatial
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This window will pop up. We will write our code
in this window.
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Constants
Constant R_init 10.0 Constant W_init
5.0 Constant d 1.0 Constant c 2.0 Constant
b 1.0 Constant a 15.0
VolumeVariable
(R and W are the parameter which are varying)
VolumeVariable R VolumeVariable W
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Functions
1.J_wolfgrowth? 2.J_predation?
In mathmodel write Function J_wolfgrowth ((R
d W) - (c W)) Function J_predation ((a
R) - (R b W))
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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
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The window will look like
Click Apply changes and then go to simulation
Text.
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Result
For R(0)10.0, W(0)5.0 , abcd1
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Results
R(0)10.0, W(0)5.0 , a10.0, c5.0
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R Vs W Plot for a10.0, c5.0
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R(0)20.0, W(0)5.0, a10.0, b2.0, c7.0,d1.5
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Exercise
Experiment when one flagellum is amputated
length pool
- Rate of growth of 1st flag.
available length
- Rate of growth of 2nd flag.
Initial condition
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?
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Results
AD1.0, T30, L1_init 10, L2_init 1.
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L1_init 15, L2_init1, A D 1.0
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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 where r is
growth/death rate. Lets say there is no y (y
0). Then, r 1 x if x lt1, x grows if x gt1,
x dies (x eats its own resources). Now, if y
also eats xs resources, the growth rate
becomes r 1 x ay hence the first
equation. The second equation follows the same
logic. 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.
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X_init0.5, Y_init0.7, a2.0, co-existance
impossible. The species with less initial
concentration decline to zero.
Results
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For a 0.5, Rabbit and sheep co-exist happily.