Some mathematical aspects of RNA velocity

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Some mathematical aspects of RNA velocity

Loïc DEMEULENAERE

Université de Liège – GIGA-Genomics

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Some recalls A mathematical model RNA velocity Estimation of parameters and prediction Appendix

Purpose: summary of the mathematical aspects of

the paper RNA velocity of single cells ([1])

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Some recalls

A mathematical model RNA velocity

Estimation of parameters and prediction Appendix

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Some recalls

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The notion of derivatives

Let f be a function and t0 a point of its domain.

0 2 4 6 8 10 0.0 0.5 1.0 1.5 t f(t) ● ● t0 df dt(t0) := limt→0 f (t₀+ t) − f (t₀) t

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The notion of derivatives

0 2 4 6 8 10 0.0 0.5 1.0 1.5 t f(t) ● ● t0 df dt(t0) := limt→0 f (t₀+ t) − f (t₀) t

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The notion of derivatives

0 2 4 6 8 10 0.0 0.5 1.0 1.5 t f(t) ● ● t0 t0+t t df dt(t0) := limt→0 f (t₀+ t) − f (t₀) t

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The notion of derivatives

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The notion of derivatives

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The notion of derivatives

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The notion of derivatives

0 2 4 6 8 10 0.0 0.5 1.0 1.5 t f(t) ● ● t0 df dt(t0) :=t→lim0 f (t₀+ t) − f (t₀) t

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The notion of derivatives

0 2 4 6 8 10 0.0 0.5 1.0 1.5 t f(t) ● ● t0 df dt(t0) :=t→lim0 f (t₀+ t) − f (t₀) t

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Variations with respect to time

df

dt(t0) := t→lim0

average change during t units of time

z }| {

f (t₀+ t) − f (t₀)

t

| {z }

Instantaneous rate of varation with respect to time

Examples (Physics)

f (t) = x (t): motion (1D) of a particle; the velocity of the

particle (at time t) is

v (t) := dx

dt(t)

In 3D: likewise! If ~x(t) := (x(t), y(t), z(t)) is the position of a

particle in the space, then its velocity is ~ v (t) := d~x dt(t) := dx dt(t), dy dt(t), dz dt(t) .

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Variations with respect to time

df

dt(t0) := t→lim0

z }| {

f (t₀+ t) − f (t₀)

t

| {z }

Examples (Physics)

v (t) := dx

dt(t)

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Variations with respect to time

df

dt(t0) := t→lim0

z }| {

f (t₀+ t) − f (t₀)

t

| {z }

Examples (Physics)

v (t) := dx

dt(t)

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Variations with respect to time

df

dt(t0) := t→lim0

z }| {

f (t₀+ t) − f (t₀)

t

| {z }

Examples (Physics)

f (t) = x (t): motion (1D) of a particle;

the velocity of the particle (at time t) is

v (t) := dx

dt(t)

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Variations with respect to time

df

dt(t0) := t→lim0

z }| {

f (t₀+ t) − f (t₀)

t

| {z }

Examples (Physics)

v (t) := dx

dt(t)

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Variations with respect to time

df

dt(t0) := t→lim0

z }| {

f (t₀+ t) − f (t₀)

t

| {z }

Examples (Physics)

v (t) := dx

dt(t)

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Some recalls

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RNA dynamics: the idea

In one cell, for one gene...

DNA Transcription Rate α(t) ..Unspliced 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: the idea

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RNA dynamics: the idea

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RNA dynamics: the idea

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RNA dynamics: the idea

DNA

Transcription Rate α(t)

..Unspliced 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: the idea

DNA

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RNA dynamics: the idea

DNA

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RNA dynamics: the idea

DNA

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RNA dynamics

In this context

u(t) and s(t) are the expected values of the numbers of

molecules of unspliced and spliced RNA (at time t)

Real numbers of molecules (at time t) have a bivariate Poisson

distribution with parameters (expected values) u(t) and s(t). Our equations        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

In this context

distribution with parameters (expected values) u(t) and s(t).

Our equations        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

In this context

distribution with parameters (expected values) u(t) and s(t). Our equations        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

Assumptions

The rates α, β, γ are constant: α ≥ 0, β > 0, γ > 0.

β =1 (all units expressed in terms of β, i.e. everything divided

by β). Final equations        du dt(t) = α − u(t) ds dt(t) = u(t) − γs(t)

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RNA dynamics

Assumptions

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RNA dynamics

Assumptions

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RNA dynamics

Assumptions

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Solution of the rst equation

du dt(t) = α − u(t) Solution 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

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Solution of the rst equation

du dt(t) = α − u(t) Solution 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

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Solution of the rst equation

du dt(t) = α − u(t) Solution 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

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Solution of the rst equation

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Solution of the rst equation

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Solution of the second equation

ds dt(t) = u(t) − γs(t) Solution 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.

Many graphical possibilities... But we always have

lim t→∞s(t) = α γ, i.e. s(t) ≈ α γ if t 0.

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Solution of the second equation

ds dt(t) = u(t) − γs(t) Solution 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.

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Solution of the second equation

ds dt(t) = u(t) − γs(t) Solution 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.

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Solution of the second equation

ds dt(t) = u(t) − γs(t) Solution 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).

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Solution of the second equation

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Solution of the second equation

Many graphical possibilities...

But we always have

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Solution of the second equation

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Solution of the second equation: graphical examples

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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RNA dynamics: summary

There exist solutions u, s, depending on u0, s0 (initial conditions)

and on α, γ (parameters), with lim

t→∞u(t) = α and t→∞lim s(t) =

α γ. In particular, lim t→∞ u(t) s(t) = γ. Steady state

When t 0, the system reaches a steady state, with

u(t) ≈ α, s(t) ≈ α

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RNA dynamics: summary

u(t) ≈ α, s(t) ≈ α

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RNA dynamics: summary

u(t) ≈ α, s(t) ≈ α

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Phase portrait

Graphic "spliced vs. unspliced"

u₀=0, s₀=0, α = 3, γ = 0.75 0 1 2 3 4 0.0 0.5 1.0 1.5 2.0 2.5 3.0 s u u ≥ γs =⇒ u₀=3, s0=4, α = 0, γ = 0.75 0 1 2 3 4 0.0 0.5 1.0 1.5 2.0 2.5 3.0 s u u ≤ γs

The system reaches thesteady state, i.e. the straight lineu = γs.

0 1 2 3 4 0.0 0.5 1.0 1.5 2.0 2.5 3.0 s u

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Phase portrait

Graphic "spliced vs. unspliced"

u₀=0, s₀=0, α = 3, γ = 0.75 0 1 2 3 4 0.0 0.5 1.0 1.5 2.0 2.5 3.0 s u u ≥ γs =⇒ u₀=3, s0=4, α = 0, γ = 0.75 0 1 2 3 4 0.0 0.5 1.0 1.5 2.0 2.5 3.0 s u u ≤ γs

The system reaches thesteady state, i.e. the straight lineu = γs.

0 1 2 3 4 0.0 0.5 1.0 1.5 2.0 2.5 3.0 s u

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Some recalls

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RNA velocity

Context

Here, we consider one cell, with p genes.

Let sj(t)be the (expected value of the) quantity of spliced

RNA associated to the jth gene (at time t).

Each sj(t) veries the previous equations, with its own

parameters αj ≥0, βj =1, and γj >0.

Warning! Implicit assumption!

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

Denition

The RNA velocity of the cell (at time t) is

d~s dt(t) := ds₁ dt (t), ..., dsp dt (t) .

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RNA velocity

Context

Each sj(t) veries the previous equations, with its own

Denition

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RNA velocity

Context

Each sj(t)veries the previous equations, with its own

Denition

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RNA velocity

Context

Denition

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RNA velocity

Context

Denition

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Representation: an unreal world...

A cell with 2 genes...

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 ((s1(t), s₂(t))). • Arrows: RNA velocity • Red point: steady state Physical velocity in RNA's space!

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Representation: an unreal world...

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 ((s1(t), s₂(t))). • Arrows: RNA velocity • Red point: steady state Physical velocity in RNA's space!

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Representation: an unreal world...

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 ((s1(t), s₂(t))). Arrows: RNA velocity • Red point: steady state Physical velocity in RNA's space!

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Representation: an unreal world...

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 ((s1(t), s₂(t))). Arrows: RNA velocity • Red point: steady state Physical velocity in RNA's space!

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Representation: an unreal world...

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 ((s1(t), s₂(t))). Arrows: RNA velocity • Red point: steady state Physical velocity in RNA's space!

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Representation: an unreal world...

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 ((s1(t), s₂(t))). Arrows: RNA velocity • Red point: steady state Physical velocity in RNA's space!

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Representation: an unreal world...

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 ((s1(t), s₂(t))). Arrows: RNA velocity • Red point: steady state Physical velocity in RNA's space!

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Representation: an unreal world...

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 ((s1(t), s₂(t))). Arrows: RNA velocity • Red point: steady state Physical velocity in RNA's space!

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Representation: an unreal world...

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 ((s1(t), s₂(t))). Arrows: RNA velocity • Red point: steady state Physical velocity in RNA's space!

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Representation: an unreal world...

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 ((s1(t), s₂(t))). Arrows: RNA velocity Red point: steady state

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Representation: an unreal world...

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 ((s1(t), s₂(t))). Arrows: RNA velocity Red point: steady state

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Representation of RNA velocity

And if p > 3?

Principle component analysis: quite natural, projection on

P.C.;

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Representation of RNA velocity

And if p > 3?

P.C.;

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Representation of RNA velocity

And if p > 3?

P.C.;

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Representation of RNA velocity

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

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Representation of RNA velocity

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

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Some recalls

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Estimation of γ

We study one gene (i.e. its parameters) through a sample of several cells.

Assumptions

The sample of cells is suciently large to cover all the RNA

cycle (from beginning of production to steady state).

The rate of degradation γ of the gene is the same in all cells.

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Estimation of γ

Assumptions

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Estimation of γ

Assumptions

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Theoritical estimation of γ

● ● ●● ● ●● ●● ● ●● ●● ●● ●● ●●● ●●● ●●●● ●●●●●●● ●●●●● ●●●●● ●● ●●● ●●●●●● ●●●● ●●●●●●●●●●●●●●●● ●● ●●●●●●●●●●● ●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● 0 1 2 3 4 0.0 0.5 1.0 1.5 2.0 2.5 3.0 s u α =3, γ = 0.75 Steady state: u = γs 400 cells, uniformly generated in time. Process

Selection of the extreme cells (here smallest and greatest 1%'s) Linear regression on the extreme cells

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Theoritical estimation of γ

● ● ●● ● ●● ●● ● ●● ●● ●● ●● ●●● ●●● ●●●● ●●●●●●● ●●●●● ●●●●● ●● ●●● ●●●●●● ●●●● ●●●●●●●●●●●●●●●● ●● ●●●●●●●●●●● ●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● 0 1 2 3 4 0.0 0.5 1.0 1.5 2.0 2.5 3.0 s u ● ● ●● ● ●●● α =3, γ = 0.75 Steady state: u = γs 400 cells, uniformly generated in time. Process

1. Selection of theextreme cells(here smallest and greatest 1%'s)

Linear regression on the extreme cells Here, estimation of γ (slope): 0.73493

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Theoritical estimation of γ

2. Linear regression on theextreme cells

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Theoritical estimation of γ

2. Linear regression on theextreme cells

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Diculties of estimation for γ...

Assumption 1 not respected...

● ● ●● ● ●● ●● ●● ●● ●●● ●●● ●● ●● ●●● ●●●●●●●●●●● ●●●● ●●● ●●●●● ●●●●●●●●● ●●●●●●●●●●●●●●●●●●●● ●●●● ●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●● 0 1 2 3 4 0.0 0.5 1.0 1.5 2.0 2.5 3.0 s u α =3, γ = 0.75 Steady state: u = γs 150 cells, uniformly generated in time. Estimation of γ: 0.97433... Corrections?

Estimation on very correlated genes, clusters of very correlated cells...

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Diculties of estimation for γ...

● ● ●● ● ●● ●● ●● ●● ●●● ●●● ●● ●● ●●● ●●●●●●●●●●● ●●●● ●●●●● ●●● ●●●●●●●●● ●●●●●●●●●●●●●●●●●●●● ●●●●●●●● ●●●●●●●●●●●● ●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●● 0 1 2 3 4 0.0 0.5 1.0 1.5 2.0 2.5 3.0 s u ● ● ●● α =3, γ = 0.75 Steady state: u = γs 150 cells, uniformly generated in time. Estimation of γ: 0.97433... Corrections?

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Diculties of estimation for γ...

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Diculties of estimation for γ...

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Multiple splicing

In [1], ± 89 % of studied genes showed a unique degradation

rate γ...

but 11 % showed several degradation rates!

Example from [1]:

Ntrk2

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Multiple splicing

In [1], ± 89 % of studied genes showed a unique degradation

rate γ... but 11 % showed several degradation rates!

Example from [1]:

Ntrk2

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Estimation of α

According to [1], it is very dicult to estimate α... Two approximations are considered:

Model I: v := ds

dt is assumed to be constant; then,

s(t) = vt + s₀,

with v := u0− γs0.

Model II: u is assumed to be constant; then,

s(t) = u0 γ + s₀−u0 γ e−γt.

These two models are correct in the short term; they have to be used step by step to predict the future (Markov process).

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Estimation of α

Model I: v := ds

s(t) = vt + s₀,

s(t) = u0 γ + s₀−u0 γ e−γt.

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Estimation of α

Model I: v := ds

s(t) = vt + s₀,

s(t) = u0 γ + s₀−u0 γ e−γt.

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Estimation of α

Model I: v := ds

s(t) = vt + s₀,

s(t) = u0 γ + s₀−u0 γ e−γt.

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Some recalls

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And if the parameters are non-constant?

Much more complex...

Example

Assume thatα(t) =1 − cos(t), β = 1, and γ > 0 is constant.

0 5 10 15 20 25 0.0 0.5 1.0 1.5 2.0 t 1 − cos(t) RNA equations        du dt(t) = [1 − cos(t)]− u(t) ds dt(t) = u(t) − γs(t)

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And if the parameters are non-constant?

Much more complex...

Example

Assume thatα(t) =1 − cos(t), β = 1, and γ > 0 is constant.

0 5 10 15 20 25 0.0 0.5 1.0 1.5 2.0 t 1 − cos(t) RNA equations        du dt(t) = [1 − cos(t)]− u(t) ds dt(t) = u(t) − γs(t)

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Solutions

Unspliced RNA u(t) =1 − 1 2(cos(t) + sin(t)) + u₀−1 2 e−t. u₀=0 0 5 10 15 20 25 0.0 0.5 1.0 1.5 t u(t) u₀ =3 0 5 10 15 20 25 0.5 1.0 1.5 2.0 2.5 3.0 t u(t)

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Solutions

Spliced RNA If γ 6= 1, s(t) = 1 γ − 1 2(1 + γ2₎((γ −1) cos(t) + (γ + 1) sin(t)) +u0−1/2 γ −1 e −t + s₀− 1 γ + 1/2 − u0 γ −1 + γ −1 2(1 + γ2₎ e−γt and, if γ = 1, s(t) =1 − 1 2sin(t) + u₀−1 2 t + s0 − 1 e−t.

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Solutions

u₀= s₀=0, γ = 0.2 0 5 10 15 20 25 0 1 2 3 4 5 t s(t) u₀=7, s0=5, γ = 0.2 0 5 10 15 20 25 5 6 7 8 t s(t) u₀=1, s0=2, γ = 2.5 0 5 10 15 20 25 0.5 1.0 1.5 2.0 t s(t) u₀=20, s₀=5, γ = 1 0 5 10 15 20 25 2 4 6 8 t s(t)

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Phase portraits

u₀= s₀=0, γ = 0.2 0 1 2 3 4 5 0.0 0.5 1.0 1.5 s(t) u(t) u₀=7, s0=5, γ = 0.2 5 6 7 8 1 2 3 4 5 6 7 s(t) u(t) u₀=1, s0=2, γ = 2.5 0.5 1.0 1.5 2.0 0.4 0.6 0.8 1.0 1.2 1.4 1.6 s(t) u(t) u₀=20, s₀=5, γ = 1 2 4 6 8 0 5 10 15 20 s(t) u(t)

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References I

G. La Manno et al.

RNA velocity of single cells.

Some mathematical aspects of RNA velocity