Upcoming SlideShare
×

# More Stochastic Simulation Examples

1,626

Published on

Published in: Education
0 Likes
Statistics
Notes
• Full Name
Comment goes here.

Are you sure you want to Yes No
• Be the first to comment

• Be the first to like this

Views
Total Views
1,626
On Slideshare
0
From Embeds
0
Number of Embeds
1
Actions
Shares
0
20
0
Likes
0
Embeds 0
No embeds

No notes for slide

### More Stochastic Simulation Examples

1. 1. Computer Science Large Practical: More Stochastic Simulation Examples Stephen Gilmore School of Informatics Friday 2nd November, 2012Stephen Gilmore (School of Informatics) Stochastic simulation examples Friday 2nd November, 2012 1 / 26
2. 2. A reaction network: the cascade Often one chemical species transforms into another, which transforms into a third, which transforms into a fourth, and so on. Events such as these are the basis of signalling processes which occur within living organisms. A series of reactions such as A becoming B, B becoming C , and so forth is called a cascade. The reactions in the cascade may occur at diﬀerent rates. This will aﬀect the dynamics of the process.Stephen Gilmore (School of Informatics) Stochastic simulation examples Friday 2nd November, 2012 2 / 26
3. 3. 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 = 0.0625Stephen Gilmore (School of Informatics) Stochastic simulation examples Friday 2nd November, 2012 3 / 26
4. 4. A simulation script, cascade.txt (2/3) # 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 = 0Stephen Gilmore (School of Informatics) Stochastic simulation examples Friday 2nd November, 2012 4 / 26
5. 5. A simulation script, cascade.txt (3/3) # 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 -> EStephen Gilmore (School of Informatics) Stochastic simulation examples Friday 2nd November, 2012 5 / 26
6. 6. 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, then later C increases, and later still D increases. No molecules of E were produced during the ﬁrst second of this simulation.Stephen Gilmore (School of Informatics) Stochastic simulation examples Friday 2nd November, 2012 6 / 26
7. 7. 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"Stephen Gilmore (School of Informatics) Stochastic simulation examples Friday 2nd November, 2012 7 / 26
8. 8. Visualising the results of a cascade simulation 1000 800 600 molecule count A B C D E 400 200 0 0 20 40 60 80 100 timeStephen Gilmore (School of Informatics) Stochastic simulation examples Friday 2nd November, 2012 8 / 26
9. 9. Adding a reaction: allowing E to decay Now we make a slight change to the model, adding a reaction which decays E. We need a new reaction constant for this new reaction. We have assigned reaction e the slowest rate. Our intuition should be that this does not make much diﬀerence to the proﬁle of chemical species A, B, C and D in the output, but it should aﬀect the proﬁle of species E .Stephen Gilmore (School of Informatics) Stochastic simulation examples Friday 2nd November, 2012 9 / 26
10. 10. A simulation script, cascade-decay.txt (1/3) # The simulation stop time (t) is 100 seconds 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 = 0.03125Stephen Gilmore (School of Informatics) Stochastic simulation examples Friday 2nd November, 2012 10 / 26
11. 11. A simulation script, cascade-decay.txt (2/3)This part is exactly the same as cascade.txt # 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 = 0Stephen Gilmore (School of Informatics) Stochastic simulation examples Friday 2nd November, 2012 11 / 26
12. 12. A simulation script, cascade-decay.txt (3/3) # The five 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 # decays without producing another output. a : A -> B b : B -> C c : C -> D d : D -> E e : E ->Stephen Gilmore (School of Informatics) Stochastic simulation examples Friday 2nd November, 2012 12 / 26
13. 13. Visualising the results of a cascade-decay simulation 1000 800 600 molecule count A B C D E 400 200 0 0 20 40 60 80 100 timeStephen Gilmore (School of Informatics) Stochastic simulation examples Friday 2nd November, 2012 13 / 26
14. 14. About the cascade-decay simulation Our intuition was correct. The proﬁles of A, B, C , and D are very similar to previously. Because this is a stochastic simulation which involves pseudo-random number generation the results will not be exactly the same but they will be very similar. We can see that reactions are still occurring right up to the stop-time of this simulation (t = 100 seconds). That is perfectly OK in the results. We simulate up to the stop-time and no further.Stephen Gilmore (School of Informatics) Stochastic simulation examples Friday 2nd November, 2012 14 / 26
15. 15. 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 the rate of reaction e. Our intuition should be that this again does not make much diﬀerence to the proﬁle of chemical species A, B, C and D in the output, but it should aﬀect the proﬁle of species E .Stephen Gilmore (School of Informatics) Stochastic simulation examples Friday 2nd November, 2012 15 / 26
16. 16. A simulation script, cascade-decay-fast.txt (1/3) # The simulation stop time (t) is 100 seconds 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.Stephen Gilmore (School of Informatics) Stochastic simulation examples Friday 2nd November, 2012 16 / 26
17. 17. A simulation script, cascade-decay-fast.txt (2/3)This part is exactly the same as cascade-decay.txt # 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 = 0Stephen Gilmore (School of Informatics) Stochastic simulation examples Friday 2nd November, 2012 17 / 26
18. 18. 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 # 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 # decays without producing another output. a : A -> B b : B -> C c : C -> D d : D -> E e : E ->Stephen Gilmore (School of Informatics) Stochastic simulation examples Friday 2nd November, 2012 18 / 26
19. 19. Visualising the results of a cascade-decay-fast simulation 1000 800 600 molecule count A B C D E 400 200 0 0 20 40 60 80 100 timeStephen Gilmore (School of Informatics) Stochastic simulation examples Friday 2nd November, 2012 19 / 26
20. 20. About the cascade-decay-fast simulation Our intuition was correct again. The proﬁles of A, B, C , and D are very similar to previously. We can see that very little E builds up in the system (because it decays away much faster than it is produced). The proﬁle for E hovers around zero throughout the simulation run.Stephen Gilmore (School of Informatics) Stochastic simulation examples Friday 2nd November, 2012 20 / 26
21. 21. A dimerisation example We saw earlier that dimerisation is a special case for the Gillespie simulation algorithm. Let’s consider an example which uses dimerisation and also includes a decay reaction.Stephen Gilmore (School of Informatics) Stochastic simulation examples Friday 2nd November, 2012 21 / 26
22. 22. A simulation script, dimer-decay.txt (1/3) # The simulation stop time (t) is 20 seconds 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 = 0.04Stephen Gilmore (School of Informatics) Stochastic simulation examples Friday 2nd November, 2012 22 / 26
23. 23. A simulation script, dimer-decay.txt (2/3) # 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 = 0Stephen Gilmore (School of Informatics) Stochastic simulation examples Friday 2nd November, 2012 23 / 26
24. 24. A simulation script, dimer-decay.txt (3/3) # 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 -> ZStephen Gilmore (School of Informatics) Stochastic simulation examples Friday 2nd November, 2012 24 / 26
25. 25. Visualising the results of a dimer-decay simulation 10000 8000 6000 molecule count X Y Z 4000 2000 0 0 5 10 15 20 timeStephen Gilmore (School of Informatics) Stochastic simulation examples Friday 2nd November, 2012 25 / 26
26. 26. Summary We have seen some examples of simulation scripts involving cascades and 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.Stephen Gilmore (School of Informatics) Stochastic simulation examples Friday 2nd November, 2012 26 / 26
1. #### A particular slide catching your eye?

Clipping is a handy way to collect important slides you want to go back to later.