RNA Velocity: a mathematical model to predict cellular differentiations

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RNA Velocity: a mathematical model to predict

cellular differentiations

Loïc DEMEULENAERE

Comprehensible Seminars – Institute of Mathematics, ULiège

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Some notions of genetics and genomics RNA Velocity In practice Perspectives

Some notions of genetics and genomics

RNA Velocity RNA dynamics RNA Velocity

In practice

Estimation of γ

Short-term approximation and long-term prediction

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In practice

Estimation of γ

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Deoxyribonucleic acid (DNA)

([5])

• Basic units: nucleotides (sugar (=deoxyribose) + phosphate + nucleic base)

4 types: Adenine (A), Cytosine (C), Guanine (G), Thymine (T) • One DNA strand: concatenation of millions of nucleotides (on

average 140 × 106 nucleotides, 4.8 cm) ([3]) • Structure:

I Double-helix (2 strands, with links A-T or C-G)

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Deoxyribonucleic acid (DNA)

([5])

4 types: Adenine (A), Cytosine (C), Guanine (G), Thymine (T)

• One DNA strand: concatenation of millions of nucleotides (on average 140 × 106 nucleotides, 4.8 cm) ([3])

• Structure:

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Deoxyribonucleic acid (DNA)

([5])

average 140 × 106 _{nucleotides, 4.8 cm) ([3])}

• Structure:

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Deoxyribonucleic acid (DNA)

([5])

average 140 × 106 _{nucleotides, 4.8 cm) ([3])} • Structure:

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Genes and ribonucleic acid (RNA)

• Genes: fraction of DNA molecules leading to the synthesis of RNA or proteins

• Genes determine traits of an organism (colour of hair or eyes, height, etc.)

Caution!

• All the genes are present in every cell, but are not expressed in

every cell!

• Genetic activities →synthesis of (messenger) RNA

• Ribonucleic Acid (RNA):4 nucleotides (A,C,G, and Uracil (U)), with ribose as sugar (usually, one simple strand)

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Genes and ribonucleic acid (RNA)

Caution!

every cell!

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Genes and ribonucleic acid (RNA)

Caution!

every cell!

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Genes and ribonucleic acid (RNA)

Caution!

every cell!

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Genes and ribonucleic acid (RNA)

Caution!

every cell!

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Synthesis of RNA

1. Transcription: copy of a gene → production of precursor RNA

2. Splicing : maturation, excision of certain fragments of the precursor RNA → production of mature (i.e. functional) RNA

Coding genes and mRNA

• Among genes, there exist coding genes: they produce

messenger RNA (mRNA), which is used to build some proteins needed by the organism

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Synthesis of RNA

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Synthesis of RNA

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In practice

Estimation of γ

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In practice

Estimation of γ

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Purpose

• Usually, in genomics: clustering of cells ; study of differentiated cells...

• Other point of view: (continuous) differentiation of cells (from stem/precursor cells) −4 −2 0 2 4 −4 −2 0 2 4 Dim 1 Dim 2 ([1]) • Purpose: modelling this differentiation by study “RNA

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Purpose

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Purpose

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Purpose

• Other point of view: (continuous) differentiation of cells (from stem/precursor cells) −4 −2 0 2 4 −4 −2 0 2 4 Dim 1 Dim 2 ([1])

• Purpose: modelling this differentiation by study “RNA dynamics” (transcription + splicing)

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Purpose

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RNA dynamics

In one cell, for one gene ([1])...

DNA Transcription Rate α(t) *2Unspliced RNA Quantity u(t) Splicing Rate β(t) ∅ Spliced RNA Quantity s(t) Degradation Rate γ(t) nn        du dt(t) = α(t)−β(t)u(t) ds dt(t) = β(t)u(t) −γ(t)s(t)

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RNA dynamics

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RNA dynamics

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RNA dynamics

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RNA dynamics

DNA

Transcription Rate α(t)

*2Unspliced RNA_{Quantity u(t)}

Splicing Rate β(t) ∅ Spliced RNA Quantity s(t) Degradation Rate γ(t) nn        du dt(t) = α(t)−β(t)u(t) ds dt(t) = β(t)u(t) −γ(t)s(t)

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RNA dynamics

DNA

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RNA dynamics

DNA

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RNA dynamics

DNA

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Remarks

1. Parameters

• Assumption/simplification: all the parameters are constant and

α ≥ 0, β > 0, γ > 0

• Change of unit of time s.t. β = 1

2. Distribution

• u(t) and s(t) are continuous functions describing integer

numbers: expected values!

• Real integer values: (asymptotic) 2-dimensional Poisson

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Remarks

1. Parameters

α ≥ 0, β > 0, γ > 0

2. Distribution

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Remarks

1. Parameters

α ≥ 0, β > 0, γ > 0

2. Distribution

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Remarks

1. Parameters

α ≥ 0, β > 0, γ > 0

2. Distribution

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Solution of the first equation

If u0:= u(0), u(t) = α + (u0− α)e−t. u0< α 0 2 4 6 8 10 0.0 0.2 0.4 0.6 0.8 1.0 t u(t) α u0 > α 0 2 4 6 8 10 0.0 0.2 0.4 0.6 0.8 1.0 t u(t) α In all cases, lim t→∞u(t) = α.

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Solution of the second equation

If u0:= u(0) and s0 := s(0), s(t) =        α γ + u0− α γ − 1 e −t ₊ s0+α − u0 γ − 1 − α γ e−γt if γ 6= 1 α + [(u0− α)t + s0− α] e−t if γ = 1(= β). We always have lim t→∞s(t) = α γ and so lim t→∞ u(t) s(t) = γ.

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Solution of the second equation

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Solution of the second equation

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Solution of the second equation

u0= s0= 0, α = 0.25, γ = 0.75 0 2 4 6 8 10 0.00 0.05 0.10 0.15 0.20 0.25 0.30 t s(t) α/γ u0= 3, s0= 1, α = 0.25, γ = 0.75 0 2 4 6 8 10 0.5 1.0 1.5 t s(t) α/γ u0= 3, s0= 4, α = 30, γ = 5 0 2 4 6 8 10 2 3 4 5 6 t s(t) α/γ u0= 4, s0= 6, α = 1, γ = 1 0 2 4 6 8 10 0 1 2 3 4 5 6 t s(t) α/γ

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In practice

Estimation of γ

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RNA Velocity

In one cell, for p genes... ([1])

• sj(t)quantity of spliced RNA associated to the jth gene (at time t)

• sj has its own parameters αj ≥ 0, βj = 1, and γj > 0 Caution!

βj = 1 for all j : the rates of splicing are equal for all genes! • The cell is “characterized” (at time t) by

~

s(t) = (s1(t), · · · , sp(t))

in a p-dimensional space (space of spliced quantities) Definition

The RNA velocity of the cell is ~ v := d~s dt = ds1 dt , · · · , dsp dt .

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RNA Velocity

~

s(t) = (s1(t), · · · , sp(t))

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RNA Velocity

• sj has its own parameters αj ≥ 0, βj = 1, and γj > 0

Caution!

~

s(t) = (s1(t), · · · , sp(t))

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RNA Velocity

βj = 1 for all j : the rates of splicing are equal for all genes!

• The cell is “characterized” (at time t) by ~

s(t) = (s1(t), · · · , sp(t))

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RNA Velocity

~

s(t) = (s1(t), · · · , sp(t))

in a p-dimensional space (space of spliced quantities)

Definition

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RNA Velocity

~

s(t) = (s1(t), · · · , sp(t))

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Representations

Help for interpretation... Example (p = 2) α1 = 2, γ1 = 0.5; α2= 3, γ2 = 1 0 1 2 3 4 0.0 0.5 1.0 1.5 2.0 2.5 3.0 s1 s2 ● ● ● ● ● ● ● ● • Grey curve: trajectory of the cell in the space of spliced quantities (t 7→ (s1(t), s2(t))) • Vectors: RNA velocities • Red point: “steady” state

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Representations

For p > 3?

We need some dimensional reductions!

• Principle component analysis: quite natural, projection on P.C.;

• There exist some non-linear techniques (t-SNE ([4]), UMAP ([2])). Idea: arrows joining cells aligned along velocities...

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Representations

For p > 3?

• Principle component analysis

: quite natural, projection on P.C.;

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Representations

For p > 3?

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Representations

For p > 3?

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Representations

Example: Schwann cell precursors (coming from [1])

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Representations

Example: Schwann cell precursors (coming from [1])

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In practice

Estimation of γ

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In practice

Estimation of γ

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Estimation of γ

For one fixed gene and a given sample of cells (of size n)...

Assumption

Degradation coefficient γ depends on the gene, but not on the cell!

According to [1], 89 % of genes respect this property (filters for others).

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Estimation of γ

Assumption

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Estimation of γ

Assumption

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Phase portrait

• Reminder: in every cell, limt→∞(u(t)/s(t)) = γ so

u(t) ≈ γs(t)if t 0.

• Phase portrait: graphic s vs u Theoritical example (α = 3, γ = 0.75) 400cells, uniformly generated in time

0 1 2 3 4 0.0 0.5 1.0 1.5 2.0 2.5 3.0 s u • Straight line u = γs • Linear regression: γ ≈ 0.4803 • Extreme-quantile linear regression (1%): γ ≈ 0.73493

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Phase portrait

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Phase portrait

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Phase portrait

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Estimation of γ: caution!

300 from the 400 previous cells...

● ● ● ● ● ● ● ● ● ●● ● ●● ●●●●● ●●●● ●●●●● ●●● ●●● ●● ●●● ●● ●●●● ●●●●● ●●●●●●●●●●●● ●●●● ●●●● ●●●● ●●●● ●●●●●●●●●●● ●●●●●● ●●●●●●●●● ●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●● 0 1 2 3 4 0.0 0.5 1.0 1.5 2.0 2.5 3.0 s u γ ≈ 0.97433

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Estimation of γ: caution!

● ● ● ● ● ● ● ● ● ●● ● ●● ●●●●● ●●●● ●●●●● ●●● ●●● ●● ●●● ●● ●●●● ●●●●● ●●●●●●●●●●●● ●●●● ●●●● ●●●● ●●●● ●●●●●●●●●●● ●●●●●● ●●●●●●●●● ●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●● 0 1 2 3 4 0.0 0.5 1.0 1.5 2.0 2.5 3.0 s u ● ● ●● γ ≈ 0.97433

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Estimation of γ: caution!

● ● ● ● ● ● ● ● ● ●● ● ●● ●●●●● ●●●● ●●●●● ●●● ●●● ●● ●●● ●● ●●●● ●●●●● ●●●●●●●●●●●● ●●●● ●●●● ●●●● ●●●● ●●●●●●●●●●● ●●●●●● ●●●●●●●●● ●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●● 0 1 2 3 4 0.0 0.5 1.0 1.5 2.0 2.5 3.0 s u ● ● ●● γ ≈ 0.97433

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In practice

Estimation of γ

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Estimation of α and short-term approximation

• Estimation of α: Difficult, α depends on genes AND cells...

• Idea: short-term approximations (one cell, one gene):

I Model I: v := ds dt is ± constant: s(t) = vt + s0, with v := u0− γs0. I Model II u is ± constant: s(t) = u0 γ + s0−u0 γ e−γt.

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Estimation of α and short-term approximation

• Estimation of α: Difficult, α depends on genes AND cells... • Idea: short-term approximations (one cell, one gene)

: I Model I: v := ds dt is ± constant: s(t) = vt + s0, with v := u0− γs0. I Model II u is ± constant: s(t) = u0 γ + s0−u0 γ e−γt.

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Estimation of α and short-term approximation

• Estimation of α: Difficult, α depends on genes AND cells... • Idea: short-term approximations (one cell, one gene):

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Estimation of α and short-term approximation

• Estimation of α: Difficult, α depends on genes AND cells... • Idea: short-term approximations (one cell, one gene):

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Long-term prediction

Use of a Markov Model

Idea: flow of cells through manifold, following RNA velocities along close neighbours...

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Long-term prediction

Use of a Markov Model

Idea: flow of cells through manifold, following RNA velocities along close neighbours...

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In practice

Estimation of γ

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Research project

• Using RNA Velocity to detect genes significantly responsible for cellular differentiations

• 1st application: study of stem cells in retina cells and their differentiated states

• 2nd application: study of a genetic disease (“S-cone enhanced Syndrome”) and detection of genetic differences between healthy and ill patients

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Research project

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Research project

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References I

G. La Manno et al.

RNA velocity of single cells.

Nature, 560:494–516, 2018.

L. McInnes, J. Healy, and J. Melville.

UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction.

ArXiv e-prints, 2018.

T. Strachan, J. Goodship, and P. Chinnery. Genetics and genomics in medicine.

Garland Science, New York, 2015.

L. Van der Maaten and G. Hinton. Visualizing Data using t-SNE.

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References II

Wikipedia. DNA.