Mutation models: probabilistic study and parameter estimation - - PowerPoint PPT Presentation

Mutation models: probabilistic study and parameter estimation

Adrien Mazoyer, supervised by Bernard Ycart

Laboratoire Jean Kuntzmann, UGA GRENOBLE

JPS 2016

Adrien Mazoyer (LJK) Mutation models JPS 2016 1 / 17

Example

mutation rate = 0.05, fitness = 1, death = 0, cells = 143, mutants = 30 exponential lifetimes time 5 4 3 2 1

(source : http://ljk.imag.fr/membres/Bernard.Ycart/)

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Motivations

Nmut Nf 2 1.36e9 3 1.05e9 4.28e8 6.24e8 5 7.36e8 6 4.90e8 110 1.36e9 1 9.56e8 6.82e8                                                Parameters of interest: → π : Probability of mutation → α : Mean number of mutations → ρ : “Fitness” . . .

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Mutation model

Number of divisions tends to ∞ Mutation probability tends to 0 Observation at time which tends to ∞          ⇒ Asymptotic model

Adrien Mazoyer (LJK) Mutation models JPS 2016 4 / 17

Mutation model

Number of divisions tends to ∞ Mutation probability tends to 0 Observation at time which tends to ∞          ⇒ Asymptotic model

3 ingredients

Occurrences of random mutations during a growth process.

Adrien Mazoyer (LJK) Mutation models JPS 2016 4 / 17

Mutation model

Number of divisions tends to ∞ Mutation probability tends to 0 Observation at time which tends to ∞          ⇒ Asymptotic model

3 ingredients

Occurrences of random mutations during a growth process. ⇒ Poisson distribution (law of small numbers)

Adrien Mazoyer (LJK) Mutation models JPS 2016 4 / 17

Mutation model

Number of divisions tends to ∞ Mutation probability tends to 0 Observation at time which tends to ∞          ⇒ Asymptotic model

3 ingredients

Occurrences of random mutations during a growth process. ⇒ Poisson distribution (law of small numbers) Growth of a clone starting from a mutant cell for a random time.

Adrien Mazoyer (LJK) Mutation models JPS 2016 4 / 17

Mutation model

Number of divisions tends to ∞ Mutation probability tends to 0 Observation at time which tends to ∞          ⇒ Asymptotic model

3 ingredients

Occurrences of random mutations during a growth process. ⇒ Poisson distribution (law of small numbers) Growth of a clone starting from a mutant cell for a random time. ⇒ Sequence of independent exponential times for each clone.

Adrien Mazoyer (LJK) Mutation models JPS 2016 4 / 17

Mutation model

Number of divisions tends to ∞ Mutation probability tends to 0 Observation at time which tends to ∞          ⇒ Asymptotic model

3 ingredients

Occurrences of random mutations during a growth process. ⇒ Poisson distribution (law of small numbers) Growth of a clone starting from a mutant cell for a random time. ⇒ Sequence of independent exponential times for each clone. Number of cells in a mutant clone that develops for a finite time.

Adrien Mazoyer (LJK) Mutation models JPS 2016 4 / 17

Mutation model

Number of divisions tends to ∞ Mutation probability tends to 0 Observation at time which tends to ∞          ⇒ Asymptotic model

3 ingredients

Occurrences of random mutations during a growth process. ⇒ Poisson distribution (law of small numbers) Growth of a clone starting from a mutant cell for a random time. ⇒ Sequence of independent exponential times for each clone. Number of cells in a mutant clone that develops for a finite time. ⇒ Depends on model assumptions.

Adrien Mazoyer (LJK) Mutation models JPS 2016 4 / 17

The Luria-Delbr¨ uck model (LD)

Assumptions

At time 0 a homogeneous culture of n normal cells. The generation time of any normal cell is a random variable with Malthusian parameter ν. A splitting normal cell is replaced by :

One normal and one mutant cell with probability π Two normal cells with probability 1 − π.

The generation time of any mutant cell is exponentially distributed with parameter µ. A splitting mutant cell is replaced by two mutant cells. All random variables and events (division times and mutations) are mutually independent.

Adrien Mazoyer (LJK) Mutation models JPS 2016 5 / 17

The Luria-Delbr¨ uck model (LD)

Assumptions

At time 0 a homogeneous culture of n normal cells. The generation time of any normal cell is a random variable with Malthusian parameter ν. A splitting normal cell is replaced by :

One normal and one mutant cell with probability π Two normal cells with probability 1 − π.

The generation time of any mutant cell is exponentially distributed with parameter µ. A splitting mutant cell is replaced by two mutant cells. All random variables and events (division times and mutations) are mutually independent.

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Stastical model of LD

Number of divisions tends to ∞ Mutation probability tends to 0 Observation at time which tends to ∞          ⇒ Asymptotic model

3 ingredients

Occurrences of random mutations during a growth process. ⇒ Poisson distribution (law of small numbers) Growth of a clone starting from a mutant cell for a random time. ⇒ Sequence of independent exponential times for each clone. Number of cells in a mutant clone that develops for a finite time. ⇒ Sequence of independent geometric numbers (Yule process).

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Results

Asymptotic assumptions

Let tn et πn two sequences and α > 0 such that : lim

n→∞ πn = 0, lim n→∞ tn = +∞, lim n→∞ πnneνtn = α

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Results

Asymptotic assumptions

Let tn et πn two sequences and α > 0 such that : lim

n→∞ πn = 0, lim n→∞ tn = +∞, lim n→∞ πnneνtn = α

Initial result

As n → ∞, the final number of mutants at time tn, starting with n normal cells, converges to the distribution with probability generating function gα,ρ(z) = exp(α(hρ(z) − 1)) where hρ(z) is the probability generating function of the Yule distribution with parameter ρ = ν/µ.

Adrien Mazoyer (LJK) Mutation models JPS 2016 8 / 17

Features of LD

An explicit asymptotic distribution

Compound Poisson of an exponential mixture of geometric distributions; Two parameters:

α: the mean number of mutations; ρ: “fitness” parameter.

Heavy tail distribution with tail exponent ρ.

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Estimation

Estimation methods for α and ρ

Maximum Likelihood estimators (but heavy tail distribution...) ; Compoud Poisson ⇒ Generating function estimators ; p0 estimators (relies on P[0 mutants] = e−α).

Adrien Mazoyer (LJK) Mutation models JPS 2016 10 / 17

Estimation

Estimation methods for α and ρ

Maximum Likelihood estimators (but heavy tail distribution...) ; Compoud Poisson ⇒ Generating function estimators ; p0 estimators (relies on P[0 mutants] = e−α). Deduce ˆ π dividing ˆ α by the final count of cells.

Adrien Mazoyer (LJK) Mutation models JPS 2016 10 / 17

Estimation

Estimation methods for α and ρ

Maximum Likelihood estimators (but heavy tail distribution...) ; Compoud Poisson ⇒ Generating function estimators ; p0 estimators (relies on P[0 mutants] = e−α). Deduce ˆ π dividing ˆ α by the final count of cells.

Bias sources

Ignoring cells death; Fluctuations of the final count of cells; Exponentially distributed lifetime; Time homogeneity.

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Bias sources: fluctuation of final count N

Instead of being constant, N is a random variable.

Link between α and π

If L[z] = E

• e−zN

g(z) = L [π(h(z) − 1)] ;

Adrien Mazoyer (LJK) Mutation models JPS 2016 11 / 17

Bias sources: fluctuation of final count N

Instead of being constant, N is a random variable.

Link between α and π

If L[z] = E

• e−zN

g(z) = L [π(h(z) − 1)] ; Instead of π = α/N: π = L−1 [exp (α(h(z) − 1))] (h(z) − 1) ;

Adrien Mazoyer (LJK) Mutation models JPS 2016 11 / 17

Bias sources: fluctuation of final count N

Instead of being constant, N is a random variable.

Link between α and π

If L[z] = E

• e−zN

g(z) = L [π(h(z) − 1)] ; Instead of π = α/N: π = L−1 [exp (α(h(z) − 1))] (h(z) − 1) ; In practice: only empirical mean and variance of are known; Reduce the bias using approximation of L.

Adrien Mazoyer (LJK) Mutation models JPS 2016 11 / 17

Bias sources: lifetime distribution

Lifetime is not exponentially distributed with rate µ.

log−normal lifetimes time 5 4 3 2 1 exponential lifetimes time 5 4 3 2 1

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Bias sources: lifetime distribution

Lifetime is not exponentially distributed with rate µ.

log−normal lifetimes time 5 4 3 2 1 exponential lifetimes time 5 4 3 2 1

Features of the new distribution

µ is the Malthusian parameter of new lifetime distribution. Explicit distribution only for exponential and constant division times.

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Bias sources: time homogeneity

The lifetime distribution depends on the birth time of the cell.

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Bias sources: time homogeneity

The lifetime distribution depends on the birth time of the cell.

New assumptions

The generation time of a normal (resp. mutant) cell born at time s ≥ 0 has cumulative distribution function Fν(s, t) = 1 − e−ν(s,t)

• resp. Fµ(s, t) = 1 − e−µ(s,t)

where ν (resp. µ), is positive, differentiable and increasing, such that lim

t→∞ ν(s, t) = +∞

and ∀ t ∈ [0 ; s] , ν(s, t) = 0 .

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Results

Main Result (M.A.)

As n → ∞, the final number of mutants at time tn, starting with n normal cells, converges to the distribution with probability generating function g(z) = exp {α(I(z, +∞) − 1)} , where I(z, t) depends on ν and µ.

Adrien Mazoyer (LJK) Mutation models JPS 2016 14 / 17

Results

Main Result (M.A.)

As n → ∞, the final number of mutants at time tn, starting with n normal cells, converges to the distribution with probability generating function g(z) = exp {α(I(z, +∞) − 1)} , where I(z, t) depends on ν and µ.

Particular case : µ(s, t) = ν(s,t)

ρ

As n → ∞, the final number of mutants at time tn, starting with n normal cells, converges to the distribution with probability generating function gα,ρ(z) = exp(α(hρ(z) − 1)) where hρ(z) is the probability generating function of the Yule distribution with parameter ρ.

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Current and future works

Theoretical work

Results for non-trivial cases. Extend to multitype branching processes. Including normal cell deaths.

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Current and future works

Theoretical work

Results for non-trivial cases. Extend to multitype branching processes. Including normal cell deaths.

Practical work

Implementation of the R package flan including Simulation ; Estimation ; Statistical tests for the different possible approaches.

Adrien Mazoyer (LJK) Mutation models JPS 2016 15 / 17

R´ ef´ erences

• W. Angerer, An explicit representation of the Luria-Delbr¨

uck distribution, J. Math. Biol., 42 (2001), pp. 145–174.

• K. Athreya and P. Ney, Branching processes, Springer Berlin

Heidelberg, 1972.

• R. Bellman and T. Harris, On age-dependent binary branching

processes, Ann. Math., 55 (1952), pp. 280–295.

• A. Hamon and B. Ycart, Statistics for the Luria-Delbr¨

uck distribution, Elect. J. Statist., 6 (2012), pp. 1251–1272.

• S. Luria and M. Delbr¨

uck, Mutations of bacteria from virus sensitivity to virus resistance, Genetics, 28 (1943), pp. 491–511.

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Thank you for your attention !

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