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 Approach Stochastic 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 Algorithm

General Form of Algorithm Input c ʋ ( ʋ =1,…,M) initi. Of X i (i=1,…,N) Set t=0 & n=0 Generate random numbers r 1 and r 2 Calculate a 1 = h v c ʋ ( ʋ =1,…,M) a 0 =  a ʋ Update t = t +  Update X = [X 1, X 2, …X C ] Update n= n + 1 Generate random numbers r 1 and r 2 Take Entire Simulation Stop If t > t stop OR no more Reactants Remain (h v =0)

7 Step 1: Given the system state, determine the rate of each reaction, a ʋ. Reaction 1: S 1 + S 2  S 3, with rate constant c 1 –X 1, X 2 are the numbers of the reactant molecules –Define the stoichiometry: h 1 = X 1 X 2 ; this will give dependence on amounts of molecules. –Then a 1 = h 1 c 1 = k 1 X 1 X 2 = rate for this reaction. Reaction 2: S 1 + S 1  S 2, –h 2 = X 1 (X 1 -1)/2 Finally, define: a 0 =  a ʋ ( ʋ = 1 to M) –This is the combined rate of all possible reactions

8 Step 2 When does the next reaction occur … Pick r 1, 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.)

9 Step 2 …and which reaction is it? Determine which reaction occurs at time  : Pick r 2, another uniform random number from 0 to 1 Find, such that: Think about dividing a 0 into M pieces of length a ʋ

10 Step 3 Update the System State Update t = t +  Update X = [X 1, X 2, …X C ] according to the reaction stoichiometry Update reaction step counter. If t > t stop or if no more reactions remain ( all (h v =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

DNA LigaseComplex 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 N N t t 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.