Transcript Introduction to Mathematica

Simulation of Biochemical Reactions
for Modeling of Cell DNA Repair
Systems
Dr. Moustafa Mohamed Salama
Laboratory of Radiation Biology, JINR
Supervisor : Dr. Oleg Belov
Simulation of Biochemical
Reactions
Stochastic approach
Master Equation
Deterministic Approach
Exact Stochastic Simulation
Reaction-Based Solving Methods:
• We are used to writing differential equations from
chemical reactions.
Is converted to
• For example:
X+Y  Z (rate a)
ZY
(rate b)
dX/dt = -aXY;
dY/dt = -aXY +bZ;
dZ/dt = aXY-bZ;
• But in stochastic systems the actual “events” or
“reactions” is stochastic.
• And, when a reaction occurs, it affects many
“chemicals” at once.
3
Stochastic?
• “Random or Probabilistic“
• Stochastic simulation:
uses a random number generator to
produce one or more possible time
courses.
Monte Carlo Simulations:
Stochastic Simulation Algorithm
General Form of Algorithm
Entire Simulation
Input cʋ (ʋ=1,…,M)
initi . Of Xi (i=1,…,N)
Set
t=0 & n=0
Generate random numbers r1 and r2
Calculate
a1= hvcʋ (ʋ=1,…,M)
a0 = aʋ
Generate random numbers r1 and r2
t 
Take
 1


 1
•
•
•
1
a0
 1 

ln 
 r 
 1 
a i  r2 a 0 

a
i
 1
Update t = t + t
Update X = [X1, X2, …XC]
Update n= n + 1
OR
Stop
If t > tstop
no more Reactants Remain (hv =0)
Step 1: Given the system state, determine
the rate of each reaction, aʋ .
• Reaction 1: S1 + S2  S3, with rate constant c1
– X1, X2 are the numbers of the reactant molecules
– Define the stoichiometry: h1 = X1X2 ; this will give
dependence on amounts of molecules.
– Then a1= h1c1= k1 X1X2 = rate for this reaction.
• Reaction 2: S1 + S1  S2,
– h2 = X1(X1-1)/2
• Finally, define: a0 = aʋ (ʋ = 1 to M)
– This is the combined rate of all possible reactions
7
Step 2 When does the next reaction
occur …
r1
8
1
0.8
0.6
0.4
0.2
• This is time of the next event.
• (Note that the time step
doesn’t have to be
predetermined, and is exact.)
t
• Let
1
t 
ln  
a 0  r1 
1
16
14
12
10
8
6
4
2
0
0
• Pick r1, a uniform random
number from 0 to 1
Step 2 …and which reaction is it?
• Determine which reaction occurs at time t:
• Pick r2, another uniform random number from 0 to 1
 1

• Find  , such that:
a
 1
i
 r2 a 0 
a
i
 1
• Think about dividing a0 into M pieces of length aʋ
9
Step 3 Update the System State
Step 3 is to determine how each of C chemicals are affected
• Update t = t + t
• Update X = [X1, X2, …XC] according to the
reaction stoichiometry
• Update reaction step counter.
• If t > tstop or if no more reactions remain ( all
(hv =0)), terminate the calculations ;
otherwise, return to step1.
10
Why consider Mathematica?
• Powerful system for symbolic
mathematical but also handles numerical
mathematics, graphics, data visualization,
simulation.
• Larger community of users comparing with
others.
• Containing the toolkits of Stochastic
Simulation Algorithm (SSA)
Example in Mathematica
Example in Mathematica
Example in Mathematica
Mathematical modeling of repair of DNA Single
strand breaks in Escherichia coli bacterial cells
By: Mohamed Abd Elmoez
Type I Repair
Complex between un legated
DNA and Ligase
DNA Ligase
Repaired DNA
Mathematical modeling of recombination repair mechanism for
Double strand DNA breaks in Escherichia coli bacterial cells
by : Alla Mohamed
RecBCD complex concentration change
N
N
t
t
Conclusion and Future work
• We learned here how to make a Mathematical
modeling for the chemical reactions.
• Know more features about Tools in
Mathematica software toolkits of Stochastic
Simulation Algorithm.
• We discussed developing a new algorithm for
Stochastic approach for range in rate of
reactions.
Acknowledgment
•I ‘d like to thank JINR especially Summer
school members.
•I also wish to thank Dr. Belov for Fruitful
discussions on Mathematical modeling in
radiation biology.