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 0 1 2 time 3 4 5 exponential lifetimes (source : http://ljk.imag.fr/membres/Bernard.Ycart/) Adrien Mazoyer (LJK) Mutation models JPS 2016 2 / 17

Motivations N mut N f  2 1.36e9       3 1.05e9    Parameters of interest:    0 4.28e8    → π : Probability of mutation    0 6.24e8    → α : Mean number of mutations   5 7.36e8 → ρ : “Fitness”    6 4.90e8    . . .    110 1.36e9       1 9.56e8        0 6.82e8  Adrien Mazoyer (LJK) Mutation models JPS 2016 3 / 17

Mutation model  Number of divisions tends to ∞     ⇒ Asymptotic model Mutation probability tends to 0   Observation at time which tends to ∞   Adrien Mazoyer (LJK) Mutation models JPS 2016 4 / 17

Mutation model  Number of divisions tends to ∞     ⇒ Asymptotic model Mutation probability tends to 0   Observation at time which tends to ∞   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 ∞     ⇒ Asymptotic model Mutation probability tends to 0   Observation at time which tends to ∞   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 ∞     ⇒ Asymptotic model Mutation probability tends to 0   Observation at time which tends to ∞   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 ∞     ⇒ Asymptotic model Mutation probability tends to 0   Observation at time which tends to ∞   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 ∞     ⇒ Asymptotic model Mutation probability tends to 0   Observation at time which tends to ∞   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 ∞     ⇒ Asymptotic model Mutation probability tends to 0   Observation at time which tends to ∞   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. Adrien Mazoyer (LJK) Mutation models JPS 2016 6 / 17

Stastical model of LD  Number of divisions tends to ∞     ⇒ Asymptotic model Mutation probability tends to 0   Observation at time which tends to ∞   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). Adrien Mazoyer (LJK) Mutation models JPS 2016 7 / 17

Results Asymptotic assumptions Let t n et π n two sequences and α > 0 such that : n →∞ π n n e νt n = α n →∞ π n = 0 , lim lim n →∞ t n = + ∞ , lim Adrien Mazoyer (LJK) Mutation models JPS 2016 8 / 17

Results Asymptotic assumptions Let t n et π n two sequences and α > 0 such that : n →∞ π n n e νt n = α n →∞ π n = 0 , lim lim n →∞ t n = + ∞ , lim Initial result As n → ∞ , the final number of mutants at time t n , starting with n normal cells, converges to the distribution with probability generating function g α,ρ ( z ) = exp( α ( h ρ ( z ) − 1)) where h ρ ( 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 . Adrien Mazoyer (LJK) Mutation models JPS 2016 9 / 17

Estimation Estimation methods for α and ρ Maximum Likelihood estimators (but heavy tail distribution...) ; Compoud Poisson ⇒ Generating function estimators ; p 0 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 ; p 0 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 ; p 0 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. Adrien Mazoyer (LJK) Mutation models JPS 2016 10 / 17

Bias sources: fluctuation of final count N Instead of being constant, N is a random variable. Link between α and π � e − zN � If L [ z ] = E 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 � e − zN � If L [ z ] = E 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 � e − zN � If L [ z ] = E 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 . 0 0 1 1 2 2 time time 3 3 4 4 5 5 log−normal lifetimes exponential lifetimes Adrien Mazoyer (LJK) Mutation models JPS 2016 12 / 17