more stochastic simulation examples

Computer Science Large Practical:More Stochastic Simulation Examples

Stephen Gilmore

School of Informatics

Friday 2nd November, 2012

Stephen Gilmore (School of Informatics) Stochastic simulation examples Friday 2nd November, 2012 1 / 26

A reaction network: the cascade

Often one chemical species transforms into another, which transformsinto a third, which transforms into a fourth, and so on.

Events such as these are the basis of signalling processes which occurwithin living organisms.

A series of reactions such as A becoming B, B becoming C , and soforth is called a cascade.

The reactions in the cascade may occur at different rates. This willaffect the dynamics of the process.


A simulation script, cascade.txt (1/3)

# The simulation stop time (t) is 100 seconds

t = 100

# The kinetic real-number rate constants of the four

# reactions: a, b, c, d

a = 0.5

b = 0.25

c = 0.125

d = 0.0625



# The initial integer molecule counts of the five species,

# A, B, C, D, and E. Only A is present initially.

# (A, B, C, D, E) = (1000, 0, 0, 0, 0)

A = 1000

B = 0

C = 0

D = 0

E = 0



# The four reactions. The reaction ‘a’ transforms

# A into B. The reaction ’b’ transforms B into C, and

# so on through the cascade. The cascade stops

# with E.

# A has a special role because it is only consumed,

# never produced. E has a special role because it

# is only produced, never consumed.

a : A -> B

b : B -> C

c : C -> D

d : D -> E


A simulation of the first second of the cascade example

The columns are time, and the molecule counts of A, B, C, D, E.

0.0, 1000, 0, 0, 0, 0

0.1, 949, 51, 0, 0, 0

0.2, 888, 112, 0, 0, 0

0.3, 843, 154, 3, 0, 0

0.4, 791, 203, 6, 0, 0

0.5, 756, 232, 12, 0, 0

0.6, 707, 273, 20, 0, 0

0.7, 674, 302, 22, 2, 0

0.8, 644, 322, 32, 2, 0

0.9, 615, 339, 44, 2, 0

From this we can see (as expected) that A decreases and B increases, thenlater C increases, and later still D increases. No molecules of E wereproduced during the first second of this simulation.


Visualising the results using GNUplotStore as “cascade.gnu”, plot using “gnuplot cascade.gnu” if results are in “cascade.csv”

set terminal postscript color

set output "cascade.ps"

set key right center

set xlabel "time"

set ylabel "molecule count"

set datafile separator ","

plot \"cascade.csv" using 1:2 with linespoints title "A", \"cascade.csv" using 1:3 with linespoints title "B", \"cascade.csv" using 1:4 with linespoints title "C", \"cascade.csv" using 1:5 with linespoints title "D", \"cascade.csv" using 1:6 with linespoints title "E"


Visualising the results of a cascade simulation

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ABCDE


Adding a reaction: allowing E to decay

Now we make a slight change to the model, adding a reaction whichdecays E.

We need a new reaction constant for this new reaction. We haveassigned reaction e the slowest rate.

Our intuition should be that this does not make much difference tothe profile of chemical species A, B, C and D in the output, but itshould affect the profile of species E .


A simulation script, cascade-decay.txt (1/3)


t = 100

# The kinetic real-number rate constants of the five

# reactions: a, b, c, d, e

a = 0.5

b = 0.25

c = 0.125

d = 0.0625

e = 0.03125


A simulation script, cascade-decay.txt (2/3)This part is exactly the same as cascade.txt



# (A, B, C, D, E) = (1000, 0, 0, 0, 0)

A = 1000

B = 0

C = 0

D = 0

E = 0


A simulation script, cascade-decay.txt (3/3)

# The five reactions. The reaction ‘a’ transforms



# with E.



# decays without producing another output.

a : A -> B

b : B -> C

c : C -> D

d : D -> E

e : E ->


Visualising the results of a cascade-decay simulation

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About the cascade-decay simulation

Our intuition was correct. The profiles of A, B, C , and D are verysimilar to previously.

Because this is a stochastic simulation which involves pseudo-randomnumber generation the results will not be exactly the same but theywill be very similar.

We can see that reactions are still occurring right up to the stop-timeof this simulation (t = 100 seconds).

That is perfectly OK in the results. We simulate up to the stop-timeand no further.


Changing a rate in the model

We set the new reaction, e, to be the slowest reaction in the model,but what if we had chosen it to be the fastest reaction instead?

We can find out how this would affect the results by changing therate of reaction e.

Our intuition should be that this again does not make muchdifference to the profile of chemical species A, B, C and D in theoutput, but it should affect the profile of species E .


A simulation script, cascade-decay-fast.txt (1/3)


t = 100

# The kinetic real-number rate constants of the five

# reactions: a, b, c, d, e

a = 0.5

b = 0.25

c = 0.125

d = 0.0625

e = 1.0

# The fastest reaction is e, the decay reaction for E.

# The slowest reaction here is d.


A simulation script, cascade-decay-fast.txt (2/3)This part is exactly the same as cascade-decay.txt



# (A, B, C, D, E) = (1000, 0, 0, 0, 0)

A = 1000

B = 0

C = 0

D = 0

E = 0


A simulation script, cascade-decay-fast.txt (3/3)This part is exactly the same as cascade-decay.txt

# The five reactions. The reaction ‘a’ transforms



# with E.



# decays without producing another output.

a : A -> B

b : B -> C

c : C -> D

d : D -> E

e : E ->


Visualising the results of a cascade-decay-fast simulation

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ABCDE


About the cascade-decay-fast simulation

Our intuition was correct again. The profiles of A, B, C , and D arevery similar to previously.

We can see that very little E builds up in the system (because itdecays away much faster than it is produced).

The profile for E hovers around zero throughout the simulation run.


A dimerisation example

We saw earlier that dimerisation is a special case for the Gillespiesimulation algorithm.

Let’s consider an example which uses dimerisation and also includes adecay reaction.


A simulation script, dimer-decay.txt (1/3)


t = 20

# The kinetic real-number rate constants of the four

# reactions: d, x, y, z

d = 1.0

x = 0.002

y = 0.5

z = 0.04



# The initial integer molecule counts of the three

# species, X, Y, and Z. Only X is present initially.

# (X, Y, Z) = (10000, 0, 0)

X = 10000

Y = 0

Z = 0



# The four reactions:

# (d), X can decay to nothing;

# (x), two molecules of X can bind to form Y;

# (y), Y can unbind to give two molecules of X; and

# (z), a molecule of Y can produce a molecule of Z.

d : X ->

x : X + X -> Y

y : Y -> X + X

z : Y -> Z


Visualising the results of a dimer-decay simulation

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Summary

We have seen some examples of simulation scripts involving cascadesand dimerisation.

Try creating some of your own. For example:

A cascade which involves more species.A cascade where every species can decay, not just the last one.A dimerisation example without a decay reaction.


more stochastic simulation examples

Education

b c c

c d d

b c d e

d e e

reaction b transforms

simulation script

b increases

reaction e