Understanding NK cell development using mathematical modeling

Supervisors: Prof. Ramit MehrSimon Michal

Students: Nissim Pinhas Hila Rothschild

Final project 2012

Background• Natural killer (NK) cells are lymphocytes of the innate

immune system, whose cellular actors also include granulocytes, macrophages, dendritic cells and mast cells.

• Some principle functions of NK cells include:– participate in the early control of viral infection.– participate in regulation of immune responses. – participate in tumor immunosurveillance.

• The developing NK cell population can be subdivided into 4 stages according to the expression of specific markers (CD11b and CD27).

NK education stages

• According to Vivier’s work, describing 4 stages of NK cells development, it was found that we can distinct between cells in different stages based on molecular expression.

• These cells also differ by their gene expression at the different stages.

CD11blow CD27low

CD11blow CD27high

CD11bhigh CD27high

CD11bhigh CD27low

Nb Mb Pb Cb

Model Scheme

Goals• Validate the 4-stage model suggested by Vivier and elucidate

the rates of proliferation, death and transition between the stages in the bone-marrow.

• Investigate the option in which NK cells may skip maturation steps, and transfer directly from DN (CD11blowCD27low) to CD11bhighCD27low.

CD11blow CD27low

CD11blow CD27high

CD11bhigh CD27high

CD11bhigh CD27low

Nb Mb Pb Cb

Methodology

• DT - Diphtheria toxin – was injected to the mice in order to eliminate the NK population cell in order to monitor development.

• BrdU staining data– BrdU is a thymidine analog that incorporates into

the cell's DNA when the cell is dividing, providing visual evidence of cell division.

Experiment timeline

DT

BrdUBrdU BrdUBrdU BrdU

6 hours

6 hours

6 hours

6 hours

6 hours

Total NK cell numbers were measured at days: 3, 6, 7, 8, 10, 13, 17, 24, 31.

Model’s mathematical equations

• Bone marrow population – total cell numbers:

dNb/dt = S1 + (γNb *(1-(Nb/KNb))– δNMb – δNCb – δNbp – µNb)*Nb

dMb/dt = S2 +δNMbNb + (γMb *(1-(Mb/KMb))– δMPb – δMbp – µMb)*Mb

dPb/dt = δMPbMb + (γPb*(1-(Pb/KPb))– δPCb – δPbp – µPb)*Pb

dCb/dt = δPCbPb + δNCbNb + (γCb *(1-(Cb/KCb)- δCbp – µCb)*Cb

Modeling labeled cells

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Nb Mb Pb Cb

Unlabeled

Labeled

S

Modeling labeled cells

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Nb Mb Pb Cb

Unlabeled

Labeled

γ

Modeling labeled cells

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Nb Mb Pb Cb

Unlabeled

Labeled γ

Modeling labeled cells

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Nb Mb Pb Cb

Unlabeled

Labeled

δ

δ

μ

Simulation• Out simulation was constructed in Matlab

Initialization – set parameter ranges

Create all combinations of parameter values

Are there any combinations

left?

Solve differential equations. calculate AIC score

(Based on MLE).

Yes

Select best parameters according to the AIC score (based on the labeling data)

Test that total number of cells in each combination is reasonable. Remove invalid combinations

Plot the best result

End

No

Fitting the model

• Fitting methods– RMS (root mean square).

– MLE (maximum likelihood estimation).

– AIC (Akaike information criterion).Enables to receive a confident interval in order to distinguish a moresuitable model.

1st and 2nd population1st - Total cell numbers 1st – Labeled cells fraction

2nd - Total cell numbers 2nd – Labeled cells fraction

3rd population3rd - Total cell numbers 3rd – Labeled cells fraction

Findings

SM

b

Hypotheses

1. Cells may express CD27 prior to expressing molecules that identify the cells as NK cells (e.g. NK1.1)

Stem cell

DN

MbCD27+ (NK1.1+)

CD27↑

NK 1.1 ↑

CD27+ (NK1.1-)

CD27↑

NK 1.1 ↑

Hypotheses 2. Perhaps the DN population (1st population) does not

evolve and could be served as a different position rather then be involved in the NK maturation process.

Stem cell

DNMb

Pb

Cb

Hypotheses

3. Perhaps the DN population skips the second stage and differentiate directly into the 4th population (Cb).

Stem cell

DNMb

Pb

Cb

Further work

• Finish the search for the 3rd and 4th populations.• Conclude the results and compare them to

Vivier’s model.• Continue with investigating NK cell dynamics in

additional organs.• Check correctness of our results in compare

with Vivier’s development model through research.

Project

Research

Acknowledgements

• We would like to thank:– Prof. Mehr Ramit– Simon Michal and the lab.

For the caring and the guidance along the way.