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.
Lex A
RecA
Pyrimidine photodimers
SOS gene
boxDinI
Umuc,
UmuD,
ssDNA+RecA+ATP
RecA*
DinI
LexA
.
UmuD’
UmuD
umuC
umuD
dinI
.
UmuD2
UmuD’2
UmuC
UmuD2C
UmuD’2C
Pol V
UmuDD’C
UmuDD’
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.
Excision Repair
SOS Repair
T-T and T-C dimers: bases become cross-linked,
T-T more prominent, caused by UV light (UVC(<280 nm) and UV-B (280-320 nm
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., 1999. Dynamics of the inducing signal for the
SOS regulatory system in Escherichia coli after ultraviolet
irradiation.
• Belov, O.V., 2007. 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
(T.A in ASU, Cairo, Egypt )
25
Background
• Created by Stephen
Wolfram and his team
Wolfram Research.
• Version 1.0 was
released in 1988.
• 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 p (Pi)
• Addition
+
Subtraction
Multiplication
* or blank space
Division
/
Exponentiation ^
• 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 Evaluate X10 NA x1 a1 t,Dz .sol1 , t,0,150 , Dz,0.5,100 , PlotLabel Style "LexA", 16 ,ColorFunction "Aquamarine",
AxesLabel Style "мин.",14,Black ,Style "Дж м2",14,Black ,Style "N",14,Black ,LabelStyle Directive Black
Ticks 20,40,80,100 , 0,20,40,60,80,100 , 400,800,1300
35
NDSolve equation “as example”
sol1 NDSolve D x1 t, Dz , t
D x2 t, Dz , t
1 k5^h1 1 k5 x1 t, Dz ^h1 x1 t, Dz 1 k6 x3 t, Dz ,
1 k7^h2 1 k7 x1 t, Dz ^h2 x2 t, Dz 1 k8 OD t, Dz k1 x3 t, Dz ,
D x3 t, Dz , t k8 OD t, Dz x2 t, Dz k1 x3 t, Dz 1 x4 t, Dz k9 x3 t, Dz k1 x1 t, Dz x3 t, Dz ,
D x4 t, Dz , t k10 1 k11^h4 1 k11 x1 t, Dz ^h4 x4 t, Dz k12 x3 t, Dz x4 t, Dz k14 k13 x4 t, Dz ^2,
D x5 t, Dz , t k15 1 k16^h5 1 k16 x1 t, Dz ^h5 k17 x7 t, Dz x5 t, Dz k18 x8 t, Dz x5 t, Dz k19 x9 t, Dz x5 t, Dz k20 x5 t, Dz ,
D x7 t, Dz , t k24 x4 t, Dz ^2 k25 x7 t, Dz 0 k34 x5 t, Dz x7 t, Dz ,
D x6 t, Dz , t k12 x4 t, Dz x3 t, Dz k22 x6 t, Dz ^2 k21 x8 t, Dz x4 t, Dz k23 x6 t, Dz ,
D x8 t, Dz , t k22 x6 t, Dz ^2 k21 x8 t, Dz x4 t, Dz k26 x8 t, Dz 0 k35 x5 t, Dz x8 t, Dz ,
D x9 t, Dz , t k27 x4 t, Dz x6 t, Dz k21 x8 t, Dz x4 t, Dz k28 x9 t, Dz 0 k36 x5 t, Dz x9 t, Dz , D x10 t, Dz , t k29 x7 t, Dz x5 t, Dz k30 x10 t, Dz ,
D x11 t, Dz , t k18 x8 t, Dz x5 t, Dz k31 x11 t, Dz x4 t, Dz k32 x11 t, Dz , D x12 t, Dz , t k19 x9 t, Dz x5 t, Dz k31 x11 t, Dz x4 t, Dz k33 x12 t, Dz ,
D x13 t, Dz , t k37 1 k39^h6 1 k39 x1 t, Dz ^h6 x13 t, Dz k38 x3 t, Dz x13 t, Dz k37, x1 0, Dz 1, x2 0, Dz 1, x3 0, Dz 0,
x4 0, Dz 1, x5 0, Dz 1, x7 0, Dz 1, x6 0, Dz 0, x8 0, Dz 0, x9 0, Dz 0, x10 0, Dz 1, x11 0, Dz 0, x12 0, Dz 0, x13 0, Dz 1 ,
x1, x2, x3, x4, x5, x7, x6, x8, x9, x10, x11, x12, x13 , t, 0, 20 , Dz, 0.5, 100 , MaxStepSize 0.8
36
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 (shiftF1)
• Can get help within a notebook by typing?
Function Name(e.g : NDSolve )
• Website:
http://www.wolfram.com/products/mathematic
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
Steps For Building Up The Model
Experimental data
Reaction’s code
Sequence of
Reactions
Output
Run
Results
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
42
Lex A protein
Result
s
lex A
N 10
1400
4
1200
1000
800
600
Blue
1 J /m2
Pink
5 J /m2
yellow
20 J /m2
Green
100 J /m2
400
200
0
50
100
2D plotting for Lex A
150
200
time min
3D plotting for Lex A
43
Rec A protein
Rec A* protein
3D plotting for Rec A & Rec A*
44
UmuD’2C protein (pol V)
UmuD2'c
N
500
min
400
300
200
Blue
1 J /m2
Pink
5 J /m2
yellow
20 J /m2
Green
100 J /m2
3D plotting for UmuD’2C
100
0
50
100
150
2D plotting for UmuD’2C
200
time min
45
DinI protein
DinI
N
800
600
3D plotting for DinI
400
200
0
50
Blue
1 J /m2
Pink
5 J /m2
yellow
20 J /m2
Green
100 J /m2
100
150
2D plotting for DinI
200
time min
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”
Дубна
50