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Mathematics of genomic profiling of astrocytes

Dávid Džamba24.2.2015

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• Faculty of Mathematics and Physics of

the Charles University in Prague

• Physical institute UK, Ke Karlovu 5,

Praha

• Specialization: Biophysics and chemical

physics

Where do I come from

Measurement of membrane potential by means of TPP electrode

BACHELOR THESIS

Membrane potential measurements in Saccharomyces cerevisiae mutant strains deficient in

various membrane transporters

DIPLOMA THESIS

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• 2nd Faculty of Medicine, Charles University

• Institute of Experimental Medicine, EU Centre

of Excellence, The Czech Academy of Sciences

• Laboratory of Cellular Neurophysiology

PhD studies

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Laboratory of Cellular Neurophysiology

• Study of glial cells, especially astrocytes• Pathophysiology of cerebral ischemia, Alzheimer’s disease and

ageing• Gene expression profiling in collaboration with Institute of

Biotechnology, AS CR

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• GFAP/EGFP mice = mice with “green” astrocytes

• Collection of astrocytes - FACS

Collection of astrocytes

30 µm

single-cells bulk samples (103-104 cells)state of the art

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• DNA RNA protein• PCR - The polymerase

chain reaction

Gene expression

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Cq value

Cq value = number of cycle when threshold is reached

threshold

13 18 26,5Low Cq value

= high gene expression

gene1gene2gene3gene4gene5gene6gene7gene8gene9gene10gene11gene12gene13gene14gene15gene16gene17gene18gene19

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1. Bulk samples: average Cq value from all cells

2. Single-cells:

PCR results

% of cells expressing given gene advantage of single-cells

threshold

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Validate PCR results obtained from single-cells by comparison with commonly used bulk samples

containing thousands of cells

Mission

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• Need for collection of both bulk samples and single cells for comparison

• Together 84 genes tested in 12 mice cca 1000 data points

• For each mouse and each gene we have:• Bulk sample: Cq value • Single cells: % of cells expressing given gene

Data

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Theoretical dependence# of transcripts 1 2 4 8 16 32 64 128 256 512 1024

# of cells containing at least one transcript 1 2 4 8 15 28 50 74 90 98 100

• Given that we have 100 cells

Cq value 21 20 19 18 17 16 15 14 13 12 11% of gene expressing cells 1 2 4 8 15 28 50 74 90 98 100

𝒚=𝟏

𝟏+𝐞𝐱𝐩 ( (𝑬𝑪𝟓𝟎−𝒙 )∗𝒔𝒍𝒐𝒑𝒆)

Sigmoid formula:

Fit: EC50 = 15 slope = -1

Precondition: transcripts are divided between the cells randomly!

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Theoretical dependence

Fit: EC50 = 14,5 slope = -1,5

Fit: EC50 = 14,5 slope = -1

Fit: EC50 = 15 slope = -1

Fit: EC50 = 15 slope = -1

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Data

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Data - curve fitting

Fit: EC50 = 14,11 slope = -0,658

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Least square curve fitting• Least square method - most widespread • Zero x-axis uncertainty precondition• Total least square method (error-in-variables method or

orthogonal regression method) – should be used when both x and y axis data have some uncertainty

• For linear regression – Deming regression

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Problem

Non-linear total least square curve fitting

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Least square curve fitting

Fit: EC50 = 14,11 slope = -0,658

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Total least square curve fitting

Fit: EC50 = 14,92 slope = -1,15

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Problem solution

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Thank you for your attention.

W. Edwards Deming(1900-1993)

“Without data you’re just another person with an

opinion”

“Learning is not compulsory… Neither is survival.”