Egyptian Candidates Ahmed Faramawy T.A in ASU, Cairo, Egypt Hadeer ELHabashy TA in AUC, Cairo, Egypt Mostafa Abo Elsoud National Research Center Supervised By Marina lyashko & SvetLana Accenova

 The SOS response in Escerichia coli bacteria is a set of inducible physiological reactions that help a cell to servive after the treatment with various DNA-damaging agents, such as ultraviolet and ionizing radiation and some chemicals.  More than 40 genes are induced in response to DNA damage as part of the SOS regulon in Escherichia coli.  SOS repair may result in SOS mutagenesis due to the inhibition of the proofreading activity of the epsilon subunit of DNA Pol III.

SOS gene box Umuc, UmuD, DinI Lex A RecA Pyrimidine photodimers

.. umuC umuD LexA ssDNA+RecA+ATP RecA* dinI UmuC DinI UmuD’ UmuD’2 UmuD2 UmuDD’ UmuD2CUmuD’2CUmuDD’C Pol V UmuD

Mathematical Model of SOS-induced mutagenesis in bacteria Escherichia coli under ultraviolet irradiation By: Hadeer El Habashy

Contents 1.Why?, Why?, Why? And why? 2. Developing Mathematical model

Mathematical Model WHY? of SOS-induced mutagenesis WHY? in bacteria Escherichia coli under WHY? ultraviolet irradiation & WHY?

Object for study Escherichia coli bacteria – colibacillus cells – play an important role among the traditional biological objects used for studying the fundamental mechanisms of induced mutagenesis. Using these cells as an object of study allows us to study the structural and functional organization of the genetic apparatus and the biochemical processes controlling the mutation process in details.

T-T and T-C dimers: bases become cross-linked, T-T more prominent, caused by UV light (UV- C(<280 nm) and UV-B ( nm Excision Repair SOS Repair

The biological mechanism of SOS Reponce in E.Coli

Developing the Mathematical model 1.Developing a system of Molecular Equations 2.Developing a system of Differential Equations 3.Developing a system of Normalized Differential Equations 4.Finding the constants

1. Developing a system of molecular equations

2.Developing the non-normalized differential equations The regulatory protein intracellular concentration the regulatory accumulation protein rate. the regulatory protein degradation rate.

Equation for RecA protein

Normalization process WHY? We non-dimensionalize the model equations : 1. To facilitate analysis and solution correctly 2. To reduce the parameters in the problem (Aksenov 1999 ) How? By dividing the parameters by constants that have the same dimensions

3. Developing a system of normalized differential equations

Developing a system of Normalized Differential Equation for each protein of the SOS response LexA RecA UmuD

The normalized( dimensionless) questions for each protein of the SOS response UmuC UmuD’

UmuDD’ UmuDD’C DinI

Finding the constants

References Aksenov, S.V., Dynamics of the inducing signal for the SOS regulatory system in Escherichia coli after ultraviolet irradiation. Belov, O.V., Time dependence of the inducing signal of the E. coli SOS system under ultraviolet irradiation. Part. Nucl. Lett. 4, 519–523.

MATHEMATICA What it can do for you ? Ahmed Faramawy Ahmed Faramawy (T.A in ASU, Cairo, Egypt ) 25

Background Created by Stephen Wolfram and his team Wolfram Research. Version 1.0 was released in Latest version is Mathematica 8.0 – released last year. Stephen Wolfram: creator of Mathematica 26

Q: What is Mathematica? A: An interactive program with a vast range of uses: -Numerical calculations to required precision -Symbolic calculations/ simplification of algebraic expressions -Matrices and linear algebra -Graphics and data visualisation -Calculus -Equation solving (numeric and symbolic) -Optimization -Statistics -Polynomial algebra -Discrete mathematics -Number theory -Logic and Boolean algebra -Computational systems e.g. cellular automata 27

Structure Composed of two parts: Kernel: -interprets code, returns results, stores definitions (be careful) Front end: - provides an interface for inputting Mathematica code and viewing output (including graphics and sound) called a notebook - contains a library of over one thousand functions - has tools such as a debugger and automatic syntax colouring 28

More on notebooks Notebooks are made up of cells. There are different cell types e.g. “Title”, “Input”, “Output” with associated properties To evaluate a cell, highlight it and then press shift-enter To stop evaluation of code, in the tool bar click on Kernel, then Quit Kernel 29

Language rules ; is used at the end of the line from which no output is required Built-in functions begin with a capital letter [ ] are used to enclose function arguments { } are used to enclose list elements ( ) are used to indicate grouping of terms expr /. x y means “replace x by y in expr ” expr /. rules means “apply rules to transform each subpart of expr ” (also see Replace ) = assigns a value to a variable == expresses equality := defines a function x_ denotes an arbitrary expression named x 30

Language rules (2) Any part of the code can be commented out by enclosing it in (* *). Variable names can be almost anything, BUT - must not begin with a number or contain whitespace, as this means multiply (see later) - must not be protected e.g. the name of an internal function BE CAREFUL - variable definitions remain until you reassign them or Clear them or quit the kernel (or end the session). 31

Mathematica as a calculator Contains mathematical and physical constants e.g. i (Imag), e (Exp) and  (Pi) Addition + Subtraction - Multiplication * or blank space Division / Exponentiation ^ SimplifyFullSimplify. Can do symbolic calculations and simplification of complicated algebraic expressions – see Simplify and FullSimplify. 32

Calculus See D to Differentiate. Can do both definite and indefinite integrals – see Integrate For a numeric approximation to an integral use NIntegrate. 33

Equation solving Use Solve to solve an equation with an exact solution, including a symbolic solution. Use NSolve or FindRoot to obtain a numerical approximation to the solution. Use DSolve or NDSolve for differential equations. To use solutions need to use expr /. x y. 34

Creating your own functions Plot3D equation “as example” Plot3D equation “as example” 35

36 NDSolve equation “as example”

Graphics Mathematica allows the representation of data in many different formats: -1D list plots, parametric plots -3D scatter plots -3D data reconstruction -Contour plots -Matrix plots -Pie charts, bar charts, histograms, statistical plots, vector fields (need to use special packages) Numerous options are available to change the appearance of the graph. Use Show to display combined graphics objects 37

Taking it further Mathematica has an excellent help menu (shift- F1) Can get help within a notebook by typing? Function Name(e.g : NDSolve ) Website: a/index.html To use Mathematica for parallel programming, look up Grid Mathematica. 38

The Basic Of Mathematical Modeling The development of mathematical models of the genetic regulation and repair process in bacterial cells is caused by the necessity to study the structure and functioning of the genetic apparatus and biochemical mechanisms controlling the mutation process. 39

Experimental data Sequence of Reactions Reaction’s code Run Output Results Steps For Building Up The Model 40

All reactions were simulated using Mathematica software, using two approaches: 1.Stochastic approach 2.Deterministic approach Outputs we obtained, characterized DNA repair steps as well as enzyme’s concentration changes. 41

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Result s Lex A protein 2D plotting for Lex A 3D plotting for Lex A Blue1 J /m 2 Pink5 J /m 2 yellow20 J /m 2 Green100 J /m 2 43

Rec A protein 3D plotting for Rec A & Rec A * Rec A * protein 44

Blue1 J /m 2 Pink5 J /m 2 yellow20 J /m 2 Green100 J /m 2 min 45 UmuD’ 2 C protein (pol V) 3D plotting for UmuD’ 2 C 2D plotting for UmuD’ 2 C

DinI protein 2D plotting for DinI min 3D plotting for DinI Blue1 J /m 2 Pink5 J /m 2 yellow20 J /m 2 Green100 J /m 2 46

Conclus ion  Using mathematical approaches 1.The model adequately describes the basic processes of the SOS response, 2.we consider how this model could be applied for the estimation of the mutagenic effect of UV irradiation and radiation, 3.A model of describing the dynamics of DinI- protein is developed, 47

4. The role of the DinI-proteins in the basic life processes of cells during the formation of mutations is studied, 5. Graphs were obtained, characterizing the concentration dynamic of DinI-proteins over time and depending on the dose of UV irradiation 48

Acknowledgments o Dr. Oleg Belov, LRB, JINR o Marina lyashko, LRB, JINR o SvetLana Aksenova, LRB, JINR

Thank You For Your Attention “ спасибо” “ спасибо” 50 Дубна