Computational Simulation of Optical
Tracking of Cell Populations using
Quantum Dot Fluorophores
Martyn R. Brown*, Huw D. Summers†, Paul Rees*, Kerenza Njoh‡,
Sally C. Chappell‡, Paul J Smith‡ and Rachel J. Errington‡
*Multidisciplinary
†School
Nanotechnology Centre, Swansea University, Singleton
Park, Swansea, SA2 8PP, UK.
of Physics and Astronomy, Cardiff University, 5, The Parade, Cardiff,
CF24 3YB, UK.
‡School
of Medicine, Cardiff University, Heath Park, Cardiff, CF14 4XN, UK.
Talk Outline


Introduction
Cell Division and Cycle



Experimental Methods and Measurements





Stochastic Cell Splitting Model
Genetic Algorithm
Results


Imaging Quantum Dot Incorporation
Population Tracking with Quantum Dots
Flow Cytrometry
Theoretical Simulation


Population Studies
Endocytosis
Two-Four Parameter Optimisation
Summary
Introduction

The ability to track the evolution of large cell populations over time is crucial



Provides a means of monitoring the general health of a population of cells
Informing on the outcome of specific assays (e.g. pharmacodynamic assay)
Overall aim is to track the evolving generations within a growing cell culture and to
identify the influence of drug intervention on the cell cycle i.e. can the cell division
rate be slowed or even stopped

A key component to this has been the computational simulation of the QD dilution via
cell mitosis

Modeling of this kind provides detailed insights into the evolution of cell lineage

Provides insight at the individual cell level from whole population experiments i.e.
flow cytometry analysis as opposed to cell to cell tracking via time-lapse imaging

Traditional approaches used for determining cell proliferation require knowledge of
population size or the behavior of a cellular marker diluted on a cell-to-cell basis

The use of this type of modeling provides a new avenue for large population cellcycle analysis using flow cytometry
Cell Division and Cycle

Biology focus – interference / blocking of cell cycle by
drugs

Anti-cancer therapeutics

Currently done by time lapse microscopy – time
consuming

Exacerbated by the required statistical sampling of large
populations because of the heterogeneous response

E.g. if you treat a tumour with a drug many of the cell
lineages will die off but a few will be immune and it is
these that survive and proliferate
Quantum Dot Fluorophores

Recent developments in semiconductor physics have produced
a new class of fluorophore suitable for cytometry techniques

Nano-sized semiconductor crystals that emit specific
wavelengths of light when excited by an optical source

Surface of fluorophores can be functionalised to bond or dock
with specific sites within the biological cells

Provide long lived optical markers with the cell

Inorganic particles and so there is potential for them to be
biologically inert

Properties applicable
generations

to
tracking
Passive optical reporters within the cell
cell
dynamics
across
Endocytosis

Endocytosis - process whereby cells absorb material from the
outside by engulfing it with their cell membrane

QDot take-up in cell via endocytotic pathways


Surface functionalised with peptides
Specifically target the endosomal sites within cell

Concentration of nanoparticles within early / late endosomes

Process is repeatable, 5-6 runs and the same level of dot loading is
seen.

Confirmed that by 24 hours the QDs are located in the endosomes
and up until 72 hours the signal per dot is stable
from ‘Molecular Biology of the Cell’, Alberts et al
Imaging QD Incorporation


705 nm CdTe, QDs from Invitrogen (QTracker)
functionalised surface coating to ensure cell uptake
U2OS – Osteosarcoma cells
Images show QD uptake and evolution from
membrane localisation, 1-3 hours through to clear
compartmentalisation within the cell by 24hrs
1hr
3hr
5hr
24hr
Imaging Cell Mitosis

Typical approach to tracking cell cycle dynamics, involves huge
amounts of data collection and painstaking post-capture analysis

Movie shows an example of the current experimental approach

Images taken at 15 min intervals

Time lapse movie of QDs in growing cell population followed by time
consuming cell tracking done manually
Cell Population Tracking with Quantum Dots

The basic concept is illustrated below in plot (a)




(a)
QDs are loaded into an initial population of cells
As a cell undergoes mitosis the quantum dots are partitioned
into the two daughter cells
The optical intensity, I, is reduced due to the reduction of dot
density per cell, N (figure (b))
Optical signal can be directly related to the cell lineage.
N
(b)
1
1/2
1/4
I
Flow Cytometry – FACS Scan

Measurement of large data sets
(10,000 cells typically)

High measurement rates

> 103 cells/s

Cells channelled through an
interrogating laser beam

488 nm excitation of dots, fluorescence
monitored with 670 nm long pass filter


Scattered/emitted light by cells is
detected and used to analyse cell
structure and function
Forward and side scatter signals from
the cells used to gate a healthy
population

data sets represent only live cells
Experiment – Fluorescence Distributions
80

Figure displays three typical experimental
data sets, acquired from flow cytometry
measurements on a population of 104
cells
The data sets are presented in the form of
histograms derived by binning cells
according to their quantum dot
fluorescence intensity
24hr
60
Cell Count

48hr
40
72hr
20
0



Cells used are human osteosarcoma
(U-2 OS; ATCC HTB-96)
0
10
10
2
Fluorescence Intensity (a.u.)
These have a typical mean cell inter-mitotic time of ~22 hours and so
measurements at 24 hour intervals effectively sample sequential cell
generations
It is apparent that each successive generation has a lower fluorescence due
to the dilution of quantum dot number by cell mitosis
4
10
Theoretical Simulation and Optimisation

The computer simulation consists of two components



The aim of the CMM is to generate a theoretical
equivalent to the experimental fluorescence intensity
histograms


A cell mitosis model (CMM)
Genetic algorithm (GA)
The CMM is the function that the GA minimises, f(X)
Through the use of a GA the important ensemble
parameters are optimized


To obtain agreement with experimental data
Subsequently provide a more detailed picture of the quantum dot
partitioning during cell division
Cell Mitosis Model (CMM) – Two Parameters


Flowchart indicating main steps of the CMM
Two parameter version:



Mean partition ratio of parent to daughter
cells, μp, i.e. distribution of QDs
Associated standard deviation, σp
Firstly, the recorded data describing the
cellular fluorescence intensity from the
quantum dots within a population of 104
cells is taken as an input set for the
program

Measured 24 hours following QD loading
Input 24 hour
experimental data
Stochastically assign
local time to parent cell
Increment time
by 1 hour
Determine if parent
cell has split
Yes
Randomly determine how the
number of quantum dots is
distributed to the daughter cells
Reset mean lifetime
of daughters
Update cell
popu lation
Next parent cell
No
CMM – Two Parameters (2)







Each of the 104 input cells is stochastically allocated a time
within its cell-cycle
Randomly from a normal distribution centered on the mean
inter-mitotic time, μIMT with an associated standard
deviation, σIMT
This step mimics the fact that each of the 104 cells in the
experiment will be at different stages within the cell-cycle
For our model the cell-cycle is simply defined by an intermitotic time, i.e. a time relative to the cell’s birth at which
the cell will split into two daughter cells
Therefore, from birth the cell moves through its cycle
unchanged until it reaches its inter-mitotic time
The cell-cycle is far more complicated than this and
different compartments of the cycle can be included in the
model however, this is not required for this present
analysis
The variables μIMT and σIMT are the two other of the four
parameters to be optimized by the genetic algorithm
Input 24 hour
experimental data
Stochastically assign
local time to parent cell
Increment time
by 1 hour
Determine if parent
cell has split
Yes
Randomly determine how the
number of quantum dots is
distributed to the daughter cells
Reset mean lifetime
of daughters
Update cell
popu lation
Next parent cell
No
CMM – Two Parameters (3)




The next step of the algorithm determines if a
particular parent cell has split or not
Again this choice is stochastically determined
The previously assigned cycle time of a cell together
with the laboratory time is used to generate a
cumulative distribution specific to each individual cell
This choice is illustrated in the figure below where a
particular cell has been randomly given a cell-cycle
time of 12 hours
Input 24 hour
experimental data
Stochastically assign
local time to parent cell
Increment time
by 1 hour
Determine if parent
cell has split
Yes
Randomly determine how the
number of quantum dots is
distributed to the daughter cells
Reset mean lifetime
of daughters
Update cell
popu lation
Next parent cell
No
CMM – Two Parameters (4)




If for example μIMT is 23 hours and σIMT is 6 hours the resulting
cumulative distribution will be centered on 35 hours
A splitting event occurs if a random number, uniformly
distributed over the interval [0 1], lies below the cumulative
probability curve at the laboratory time
For example, the filled black circle indicates the probability of a
split occurring for this particular cell at a laboratory time of 27
hours, the graph indicates a 10% chance of this split occurring
This sampling occurs at every time interval (1 hour in our case)
Input 24 hour
experimental data
Stochastically assign
local time to parent cell
Increment time
by 1 hour
Determine if parent
cell has split
Yes
Randomly determine how the
number of quantum dots is
distributed to the daughter cells
Reset mean lifetime
of daughters
Update cell
popu lation
Next parent cell
No
CMM – Two Parameters (5)



If the parent cell has not split it is returned to
the populace
If a splitting event occurs the algorithm next
decides how the quantum dots are distributed
to its daughters
When splitting occurs we assume that the
number of quantum dots is always conserved


The total number of dots in each daughter cell is
equal to the number of dots in the parent cell
The number of dots allocated to each daughter
cell is chosen at random from a normal
distribution centered on a mean partition ratio,
μP, which has an associated standard deviation,
σP
Input 24 hour
experimental data
Stochastically assign
local time to parent cell
Increment time
by 1 hour
Determine if parent
cell has split
Yes
Randomly determine how the
number of quantum dots is
distributed to the daughter cells
Reset mean lifetime
of daughters
Update cell
popu lation
Next parent cell
No
CMM – Two Parameters (6)
Input 24 hour
experimental data



Once the daughter cells have been assigned their
respective quantum dot population, the algorithm
resets their cycle time equal to their parents plus the
value of μIMT
This action ensures that the probability of two newly
formed daughter cells splitting again in the immediate
future is small
The final stage of the algorithm simply stores both
daughter and the initial parent cells yet to split in the
first hour in the laboratory frame of reference
Stochastically assign
local time to parent cell
Increment time
by 1 hour
Determine if parent
cell has split
Yes
Randomly determine how the
number of quantum dots is
distributed to the daughter cells
Reset mean lifetime
of daughters
Update cell
popu lation
Next parent cell
No
CMM – Two Parameters (7)





The total population is now > 104
Laboratory and cycle time of the cells are incremented
by 1 hr
At the set ‘measurement’ time (typically a 24 hour
increment) a fluorescent histogram is calculated by
determining the number of dots in each cell from a
random sample population of 104
This histogram can then be compared directly with the
experimental data
Specifically, the Euclidean norm of the two histogram
curves is calculated and compared for particular
values of μp and σp
Input 24 hour
experimental data
Stochastically assign
local time to parent cell
Increment time
by 1 hour
Determine if parent
cell has split
Yes
Randomly determine how the
number of quantum dots is
distributed to the daughter cells
Reset mean lifetime
of daughters
Update cell
popu lation
Next parent cell
No
Genetic Algorithm (GA) (1)

Flowchart indicating main steps of the GA

Initial population of chromosomes randomly
generated to span the whole parameter space






10 chromosomes
8 genes per optimisation parameter
Each gene randomly given a 0 or 1
Fitness of the initial populace is evaluated by
running through the CMM
Fitness is determined by calculation of the Euclidean
norm of the experimental and simulated data over
the entire intensity range
Although, the simulated data does not produce a
fluorescent signal, but rather a number detailing the
number of quantum dots per cell, a meaningful
comparison between the experimental and
simulated data can be made on the supposition that
florescence intensity is proportional to cell dot
density
Generate random
population of
10 chromosomes
Evaluate
fitness
Mate chromosomes
according to fitness
Mutate chromosome
elements
Evaluate function, f(X)
No
Converged?
Yes
End
Genetic Algorithm (GA) (2)





The fitness of chromosome generation is analyzed to
see if a desired convergence criterion is met
If true the simulation is halted
The population is ranked in order of fitness and
chosen stochastically to generate the succeeding
generation
Generate random
population of
10 chromosomes
Evaluate
fitness
The simulation utilizes two methods to produce the
next chromosome generation, mating and elitism
Mate chromosomes
according to fitness
Chromosome mating, utilizes 65% gene crossover
rate between stochastically selected parents
Mutate chromosome
elements
Evaluate function, f(X)

The random choice of the parents is weighted in
favor of individual fitness

No
Converged?
Higher their fitness the more likely they will be chosen
to mate
Yes
End
Genetic Algorithm (GA) (3)



Elitism is included to ensure that the fittest individual
of one generation survives to the next without
modification
Also, in each new-generation there is a small
probability that a chromosome may undergo a
random mutation
This is set to occur to 5% of the total number of
genes available at each generation
Generate random
population of
10 chromosomes
Evaluate
fitness
Mate chromosomes
according to fitness
Mutate chromosome
elements

The new-generation of chromosomes is again
evaluated in the manner above until a suitable
convergence criterion is achieved
Evaluate function, f(X)
No

The magnitude of the optimized parameters varied
by less than 5% across the whole chromosome
population
Converged?
Yes
End
Results – Two Parameter Model

(a) Experimental and (b) simulated quantum
dot fluorescence intensity histograms taken at
24 hour intervals following take-up
(c) Computed (blue trace) and measured
(black trace) fluorescence histograms 72
hours after quantum dot uptake
60
24hr
(a)
48hr
72hr
40
Cell Count

20
0
60
(b)
24hr
48hr


Excellent fit between computed and measured
traces
The modeled fit has a peak probability of
partitioning ratio of 74:26 % with a 6 %
standard deviation

20
0 0
10
10
Asymmetry, verified using microscopic
techniques
Hypothesised to be due to the presence of
QDs within the cell
1
10
2
10
3
10
Fluorescence Intensity (a.u.)
50
The importance and relevance of the
asymmetric splitting is very unexpected and
the subject of much further work

72hr
(c)
40
Cell Count

40
30
20
10
0
100
101
102
103
104
Fluorescence Intensity (a.u.)
4
Results – Four Parameter Model (1)


In addition to the parent partition ratio
and its standard deviation we include
the mean inter-mitotic time, μIMT and its
deviation σIMT
Parameters
Sample
Space
μP, σP
[0 1]
μIMT
[0 48]
σIMT
[1 20]
Including these two supplementary parameters provides detailed
analysis of cell growth dynamics without the requirement of prior
knowledge of cell growth parameters other than the measurements
themselves

Initial population of 50 chromosomes

Each chromosomes with 32 genes split evenly between the 4 parameters
Results – Four Parameter Model (2)




Figure displays both the experimental (black
trace) and the simulated quantum dot
fluorescence intensity at 48 hours (blue
trace) using the 4 parameter cell-cycle
model in conjunction with the genetic
algorithm
The values of inter-mitotic time and its
associated standard deviation predicted by
the simulation are 22.5 and 4 hours
respectively
Using microscopic techniques the intermitosis time for the human osteosarcoma
cell line has been estimated at 21 hours with
a standard deviation of 4 hours
The values of the cell partitioning ratio and
its standard deviation are found to be 0.733
and 0.14
Again a strong asymmetry in the parent to
daughter portioning values is apparent
60
50
Cell Count

40
30
20
10
0
10
0
10
1
10
2
10
3
Fluorescence Intensity (a.u.)
10
4
Summary of Results




Outlined the use of a genetic algorithm coupled with a stochastic
cell-cycle model, which when compared with experimental flow
cytometry data enables tracking of quantum dot fluorophores within
large cell populations over multiple generations
The cell-cycle model complements the experimental investigations in
that it mimics the cell division behavior of individual cells within large
populations
By utilizing a genetic algorithm in conjunction with the cell-cycle
model we have been able to achieve excellent fits of the theoretically
predicted quantum dot distributions with that measured
experimentally
Using the genetic algorithm we obtain an inter-mitotic time of 22.5
hours with a standard deviation of 4 hours for the four parameter
version

We also obtain an asymmetric cell partition ratio of 73:27% with a
standard deviation of 14%

These results are in excellent agreement with single cell microscopic
studies
Importance of these Results



The ability of this computer model to fit to experimental flow
cytometry data provides a unique and novel analysis that
allows tracking of cell population growth and lineage whilst
maintaining information at the single cell level
It is also extremely powerful in that it provides the biologist
with a detailed analysis of cell growth dynamics without the
requirement of prior knowledge of the cell growth parameters
These results demonstrate that flow cytometry
measurements, of quantum dot intensity, in conjunction with
our model can give the single cell information required to
assess anti-cancer therapeutics