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

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.

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

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

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 to tracking cell dynamics across generations

Passive optical reporters within the cell

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 24hr5hr3hr

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)

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.

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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

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)

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Fluorescence Intensity (a.u.)

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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

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

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

Yes

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Input 24 hourexperimental data

Stochastically assignlocal time to parent cell

Determine if parentcell has split

Reset mean lifetimeof daughters

Update cellpopulation

Next parent cell

Increment timeby 1 hour

Randomly determine how the number of quantum dots is

distributed to the daughter cells

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 inter-mitotic 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

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Next parent cell




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 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)

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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

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Next parent cell





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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Next parent cell





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

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Next parent cell




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 randompopulation of

10 chromosomes

Evaluatefitness

Mate chromosomesaccording to fitness

Mutate chromosomeelements

Evaluate function, f(X)

Converged?

End

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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


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Evaluatefitness




Converged?

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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


10 chromosomes

Evaluatefitness




Converged?

End

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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 74:26 % 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

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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

ParametersSample 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 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 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

Documents

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