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.