Mathematical modeling of
nano-particle toxicology in
living cell systems
1
,Contents
1 Introduction
1.2 Cell cycle
1.3 Flow cytometry
1.4 Cell generation tracking and typical results
1.41 Computer program for histogram
1.42 Transfer function
2 Theoretical background & Computer codes
2.1 Fourier transforms, FFT and convolution.
2.12 Discrete Fourier transforms (DFT)
2.13 Fast Fourier transform (FFT) Algorithm
2.14 Convolution theorem
2.2 Information theory
2.22 Entropy
2.23 Applying Information theory and entropy to QD cell population
2.24 Computer algorithm to calculate entropy and information
2.25 Future study involving Information theory and entropy
2.3 Statistical mechanics
2.31 Microstates and Macrostate
2.312 Stirling’s theorem
2.32 Statistical mechanics relation to information theory
2.321 Computer to program to solve for entropy,
3 Study of cell division and proliferation
3.1 Result for transfer function
3.2 Entropy results
3.3 Results for information transmitted
3.4 Results for statistical mechanics entropy
3.41 lnW results
4 Theoretical and analytical models
4.1 Theoretical model for lnW
4.2 Analytical and information theory used on different cell models
4.21 First cell model (symmetrical spitting)
4.22 2nd cell model (asymmetrical spitting)
5 Conclusion
6 REFERENCES
7 APPENDIX
7.1 FFT program and its inverse (downloaded)
7.2 Histogram program
7.3 lnW program
7.4 Program to calculate Shannon’s entropy and information
7.5 Program to justify Stirling’s formula
7.6 Program to produce a Gaussian function
2
,Project aim:
Quantum dots in cells are used as indicators to track cell lineage.
Fluorescence is emitted by the dots upon excitation and is collected using
flow cytometry experiments. The fluorescence data obtained gives
indication of the quantity of cells within each cell and the amount by
which the cells have been diluted due to cell division. Using these data
sets, statistical mechanics and information theory will be used to develop
mathematical models describing this dilution of dots. Such a
mathematical model will help in tracking cells and understanding cell
evolution.
1. Introduction
Cells divide by the process of mitosis. Tracking the division of many cells
over time will enable one to trace the health of cells, and can help in
understanding the effects of drugs on cell production. In order to produce
realistic models of cell evolution, a large number of cells must be studied.
Flow cytometry is a technique which provides a means to analyse many
cells.1
(a) (b) N
1
1/2
1/4 I
FIG 1: Schematic diagrams show the partitioning of quantum dots due to
cell division.
The graphs show a decrease in fluorescence intensity as a result of cell
mitosis.1
Quantum dot (QD) fluorophores can be used as optical markers in a cell
and thus can aid in tracking cell populations using flow cytometry.
Quantum dots are nano-sized semiconductor crystals that emit light when
excited by an optical source; their unique properties such as narrow
emission spectra and photostability make them ideal for cell tracking 2.
Initially quantum dots are placed in a cell population, as the cells divide
by mitosis the quantum dots will split unevenly into the two daughter
cells. This means the optical intensity emitted by each cell will be
reduced due to the decrease in quantum dot density in each cell as a
result of division. Hence, the optical intensity signal can be linked to the
3
, cell lineage. The graphs in fig 1displays the decrease in fluorescence
intensity as a result of cell mitosis.1
1.2 Cell cycle
In order to develop a mathematical technique modelling cell dilution it is
important to firstly understand how cells divide and the process by which
QDs are absorbed by cells.
A human cell on average divides once every 24 hour. The cell cycle
consists of two main phases: interphase and the mitotic phase. During the
cell division process many regulatory steps are used to prevent cells
dividing uncontrollably, while detecting and repairing genetic damage.
Tumour formation can occur when a gene mutates causing cells to divide
rapidly. 3
Fig 2: Schematic diagram of the cell cycle. 5
Mitosis
Mitosis consists of predominantly two procedures, nuclear division occurs
during this process in which the cells chromosomes are divided between
the two daughter cells. The cells cytoplasm is also divided. 3,4
Interphase
During this period the cell prepares itself for cell division and gathers
nutrients for growth, mitosis and duplicating DNA. Interphase composes
of three phases;
G1 phase - After mitosis is complete the cell enters the next stage of the
cycle. During this phase many enzymes are synthesised which will be
required for the next phase (S phase), majority of these enzymes will be
used for DNA replication.
S phase - DNA replication occurs in this phase. All of the chromosomes
are replicated, leaving the cell with double the number of chromosomes.
4
nano-particle toxicology in
living cell systems
1
,Contents
1 Introduction
1.2 Cell cycle
1.3 Flow cytometry
1.4 Cell generation tracking and typical results
1.41 Computer program for histogram
1.42 Transfer function
2 Theoretical background & Computer codes
2.1 Fourier transforms, FFT and convolution.
2.12 Discrete Fourier transforms (DFT)
2.13 Fast Fourier transform (FFT) Algorithm
2.14 Convolution theorem
2.2 Information theory
2.22 Entropy
2.23 Applying Information theory and entropy to QD cell population
2.24 Computer algorithm to calculate entropy and information
2.25 Future study involving Information theory and entropy
2.3 Statistical mechanics
2.31 Microstates and Macrostate
2.312 Stirling’s theorem
2.32 Statistical mechanics relation to information theory
2.321 Computer to program to solve for entropy,
3 Study of cell division and proliferation
3.1 Result for transfer function
3.2 Entropy results
3.3 Results for information transmitted
3.4 Results for statistical mechanics entropy
3.41 lnW results
4 Theoretical and analytical models
4.1 Theoretical model for lnW
4.2 Analytical and information theory used on different cell models
4.21 First cell model (symmetrical spitting)
4.22 2nd cell model (asymmetrical spitting)
5 Conclusion
6 REFERENCES
7 APPENDIX
7.1 FFT program and its inverse (downloaded)
7.2 Histogram program
7.3 lnW program
7.4 Program to calculate Shannon’s entropy and information
7.5 Program to justify Stirling’s formula
7.6 Program to produce a Gaussian function
2
,Project aim:
Quantum dots in cells are used as indicators to track cell lineage.
Fluorescence is emitted by the dots upon excitation and is collected using
flow cytometry experiments. The fluorescence data obtained gives
indication of the quantity of cells within each cell and the amount by
which the cells have been diluted due to cell division. Using these data
sets, statistical mechanics and information theory will be used to develop
mathematical models describing this dilution of dots. Such a
mathematical model will help in tracking cells and understanding cell
evolution.
1. Introduction
Cells divide by the process of mitosis. Tracking the division of many cells
over time will enable one to trace the health of cells, and can help in
understanding the effects of drugs on cell production. In order to produce
realistic models of cell evolution, a large number of cells must be studied.
Flow cytometry is a technique which provides a means to analyse many
cells.1
(a) (b) N
1
1/2
1/4 I
FIG 1: Schematic diagrams show the partitioning of quantum dots due to
cell division.
The graphs show a decrease in fluorescence intensity as a result of cell
mitosis.1
Quantum dot (QD) fluorophores can be used as optical markers in a cell
and thus can aid in tracking cell populations using flow cytometry.
Quantum dots are nano-sized semiconductor crystals that emit light when
excited by an optical source; their unique properties such as narrow
emission spectra and photostability make them ideal for cell tracking 2.
Initially quantum dots are placed in a cell population, as the cells divide
by mitosis the quantum dots will split unevenly into the two daughter
cells. This means the optical intensity emitted by each cell will be
reduced due to the decrease in quantum dot density in each cell as a
result of division. Hence, the optical intensity signal can be linked to the
3
, cell lineage. The graphs in fig 1displays the decrease in fluorescence
intensity as a result of cell mitosis.1
1.2 Cell cycle
In order to develop a mathematical technique modelling cell dilution it is
important to firstly understand how cells divide and the process by which
QDs are absorbed by cells.
A human cell on average divides once every 24 hour. The cell cycle
consists of two main phases: interphase and the mitotic phase. During the
cell division process many regulatory steps are used to prevent cells
dividing uncontrollably, while detecting and repairing genetic damage.
Tumour formation can occur when a gene mutates causing cells to divide
rapidly. 3
Fig 2: Schematic diagram of the cell cycle. 5
Mitosis
Mitosis consists of predominantly two procedures, nuclear division occurs
during this process in which the cells chromosomes are divided between
the two daughter cells. The cells cytoplasm is also divided. 3,4
Interphase
During this period the cell prepares itself for cell division and gathers
nutrients for growth, mitosis and duplicating DNA. Interphase composes
of three phases;
G1 phase - After mitosis is complete the cell enters the next stage of the
cycle. During this phase many enzymes are synthesised which will be
required for the next phase (S phase), majority of these enzymes will be
used for DNA replication.
S phase - DNA replication occurs in this phase. All of the chromosomes
are replicated, leaving the cell with double the number of chromosomes.
4
