Computational Simulation of Optical Tracking of Cell Populations using Quantum Dot Fluorophores

1
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 Nanotechnology Centre, Swansea
  University, Singleton Park, Swansea, SA2 8PP, UK.
• School of Physics and Astronomy, Cardiff
  University, 5, The Parade, Cardiff, CF24 3YB, UK.
• School of Medicine, Cardiff University, Heath
  Park, Cardiff, CF14 4XN, UK.

2
Talk Outline

• Introduction
• Cell Division and Cycle
• Population Studies
• Endocytosis
• Experimental Methods and Measurements
• Imaging Quantum Dot Incorporation
• Population Tracking with Quantum Dots
• Flow Cytrometry
• Theoretical Simulation
• Stochastic Cell Splitting Model
• Genetic Algorithm
• Results
• Two-Four Parameter Optimisation
• Summary

3
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 cell-cycle analysis
  using flow cytometry

4
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

5
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 to tracking cell dynamics
  across generations
• Passive optical reporters within the cell

6
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

7
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

8
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

9
Cell Population Tracking with Quantum Dots

• The basic concept is illustrated below in plot
  (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)
(a)
1
1/2
I
1/4
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Flow Cytometry FACS Scan

• Measurement of large data sets (10,000 cells
  typically)
• High measurement rates
• gt 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

11
Experiment Fluorescence Distributions

• 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
• Cells used are human osteosarcoma (U-2 OS
  ATCC HTB-96)

• 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

12
Theoretical Simulation and Optimisation

• The computer simulation consists of two
  components
• A cell mitosis model (CMM)
• Genetic algorithm (GA)
• The aim of the CMM is to generate a theoretical
  equivalent to the experimental fluorescence
  intensity histograms
• 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

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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, sp
• 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

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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, sIMT
• 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 inter-mitotic time, i.e. a time relative to
  the cells 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 sIMT are the two other of
  the four parameters to be optimized by the
  genetic algorithm

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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

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CMM Two Parameters (4)

• If for example µIMT is 23 hours and sIMT 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)

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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,
  sP

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CMM Two Parameters (6)

• 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

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CMM Two Parameters (7)

• The total population is now gt 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 sp

20
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

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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
• The simulation utilizes two methods to produce
  the next chromosome generation, mating and
  elitism
• Chromosome mating, utilizes 65 gene crossover
  rate between stochastically selected parents
• The random choice of the parents is weighted in
  favor of individual fitness
• Higher their fitness the more likely they will be
  chosen to mate

22
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
• The new-generation of chromosomes is again
  evaluated in the manner above until a suitable
  convergence criterion is achieved
• The magnitude of the optimized parameters varied
  by less than 5 across the whole chromosome
  population

23
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
• Excellent fit between computed and measured
  traces
• The modeled fit has a peak probability of
  partitioning ratio of 7426 with a 6 standard
  deviation
• The importance and relevance of the asymmetric
  splitting is very unexpected and the subject of
  much further work
• Asymmetry, verified using microscopic techniques
• Hypothesised to be due to the presence of QDs
  within the cell

(c)
24
Results Four Parameter Model (1)
Parameters Sample Space
µP, sP 0 1
µIMT 0 48
sIMT 1 20

• In addition to the parent partition ratio and its
  standard deviation we include the mean
  inter-mitotic time, µIMT and its deviation sIMT

• 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

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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 inter-mitosis
  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

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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 7327 with a standard deviation of 14
• These results are in excellent agreement with
  single cell microscopic studies

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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