Today we will create our own math model

1. Flagellar length control in Chlamydomonus

2. Lotka-Volterra Model

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 flagellar length is controlled in Chlamydomonus?

How are the assembly and disassembly ratescontrolled to achieve this balance at the correct length?

Balance-point model:a) shortening rate is constantb) elongation rate decreases with length

Shortening rate is constant (measurement before mitosis,when IFT transport stops):

Elongation rate is ~ 1/L

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; = (2L/V) – time for one IFT trip; thenM / = (0.5MV) / L – length increase per second

Adding two terms,

Rate of change of length: DL

A

dt

dL

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.)

Code will be saved in the Mathmodel.

This window will pop up. We will write our code in this window.

• 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 Lengthcontrol.

Steps:

MathDescription {

1. Constant Declaration (end with ;)

2. VolumeVariable declartion

3. Function declaration (end with ;)

4. CompartmentSubDomain Compartment { ODE declaration } }

Math Description

Sequentially: ConstantVolumeVariable Function ODEs

Constant Declaration:

Format is : Constant Parameter Name Value ;

L in the volume of consideration which is varying with time:

Afetr constant declaration write VolumeVariable L

Constant A 7.0;Constant D 1.0;Constant L_init 1.0;

VolumeVariable:

Function Declaration:

Write: Function J_length (( A/L) - D) ;

Format is: Function Functionname ( Function expression) ;

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;

}}

DL

A

dt

dL

This part means

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.

When you click run to the simulation,the software willask 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 FileOpenMathModel (and click the model of your interest)

Remember

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 equations describe the dynamics of the biological

systems, where two species interact, one is predator and one its prey.

Lotka-Volterra Model(Alfred J. Lotka in 1925 and Vito Volterra in 1926.)

Consider,

R= number of prey (e.g Rabbits)

W= number of predator (e.g Wolves)

dt

dWdt

dRGrowth of Rabbit‘s population against time

Growth of of Wolf‘s population against time

dt

dWdt

dR= rabbit‘s growth – rabbits killed by Wolves

= Wolf‘s growth – wolf‘s death

Mathematically:

WRbRadt

dR... Equation for Rabbit

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.

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.

1st term

2nd term

Equation for Wolf WcWRddt

dW...

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

WcWRddt

dW...

WRbRadt

dR...

We will solve these equtions using Vcell, and analyze Results.Let us start:

FileNewMathModelNon-spatial

This window will pop up. We will write our code in this window.

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 RVolumeVariable W

Functions

1.J_wolfgrowth

2.J_predation

WcWRd ...

WRbRa ...

In mathmodel write—

Function J_wolfgrowth ((R * d * W) - (c * W));Function J_predation ((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;

}}

WcWRd ...

WRbRa ...

The window will look like

Click Apply changes and then go to simulation Text.

For R(0)=10.0, W(0)=5.0 , a=b=c=d=1

Result:

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

Experiment when one flagellum is amputated

1 212

VT L L M

L - Rate of growth of 1st flag.

1 222

VT L L M

L - Rate of growth of 2nd flag.

1 2

1 1

1dL LT

A Ddt L L

2 1

2 2

1dL LT

A Ddt L L

Solve the system of differential equation using VCell

1 2

1 1

302

dL L

dt L L 2 1

2 2

302

dL L

dt L L

1, 30, 1A T D

Initial condition: 1 2(0) 10, (0) 1L L

‘available length’

‘length pool’

After you solve the equations, think about the results. Do they agree with experimental observations at the top of this slide? Why?

Exercise:

Results:

A=D=1.0, T=30, L1_init= 10, L2_init= 1.

L1_init= 15, L2_init=1, A = D= 1.0

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. 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.

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.

rate of growth of x

rate of growth of y

1

1

dxx x ay

dt

dyy y ax

dt

Exercise

/dx dt rx

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.