simulation of biochemical reactions for modeling of cell dna repair systems
DESCRIPTION
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. Deterministic Approach. Master Equation. - PowerPoint PPT PresentationTRANSCRIPT
Simulation of Biochemical Reactions for Modeling of Cell DNA Repair
Systems
Dr. Moustafa Mohamed Salama
Laboratory of Radiation Biology, JINR
Supervisor : Dr. Oleg Belov
Deterministic ApproachStochastic approach
Master Equation
Simulation of Biochemical Reactions
Exact Stochastic Simulation
3
Reaction-Based Solving Methods:• We are used to writing differential equations
from chemical reactions. • For example: Is converted to
dX/dt = -aXY;dY/dt = -aXY +bZ;dZ/dt = aXY-bZ;
X+Y Z (rate a)Z Y (rate b)
• But in stochastic systems the actual “events” or “reactions” is stochastic.
• And, when a reaction occurs, it affects many “chemicals” at once.
Stochastic?
• “Random or Probabilistic“
• Stochastic simulation:
uses a random number generator to produce one or more possible time
courses.
Monte Carlo Simulations: Stochastic Simulation AlgorithmMonte Carlo Simulations: Stochastic Simulation Algorithm
General Form of AlgorithmInput cʋ (ʋ=1,…,M) initi . Of Xi (i=1,…,N)Set t=0 & n=0Generate random numbers r1 and r2
Calculate a1= hvcʋ (ʋ=1,…,M)a0 = aʋ
• Update t = t + • Update X = [X1, X2, …XC]• Update n= n + 1
Generate random numbers r1 and r2
Take
Entire Simulation
10
1ln
1
ra
102
1
1ii aara
Stop If t > tstop
OR no more Reactants Remain (hv =0)
7
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
8
Step 2 When does the next reaction occur …
• Pick r1, a uniform random number from 0 to 1
• Let
• This is time of the next event.• (Note that the time step
doesn’t have to be predetermined, and is exact.)
10
1ln
1
ra
02468
10121416
0
0.2
0.4
0.6
0.8 1
r1
9
Step 2 …and which reaction is it?
• Determine which reaction occurs at time :
• Pick r2, another uniform random number from 0 to 1
• Find , such that:
• Think about dividing a0 into M pieces of length aʋ
102
1
1ii aara
10
Step 3 Update the System State
• Update 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.
Step 3 is to determine how each of C chemicals are affected
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
DNA Ligase
Complex between un legated DNA and Ligase
Repaired DNA
Type I Repair
Mathematical modeling of repair of DNA Single strand breaks in Escherichia coli bacterial cells
By: Mohamed Abd Elmoez
Mathematical modeling of recombination repair mechanism for Double strand DNA breaks in Escherichia coli bacterial cells
by : Alla Mohamed
RecBCD complex concentration change
NN
tt
NN
tt
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.