On The Asymptotic Distribution of Nucleation Times of - PowerPoint PPT Presentation

On The Asymptotic Distribution of Nucleation Times of Polymerization Processes SUN Wen Joint work with Philippe Robert Les probabilit´ es de demain, Paris, May 2018

Overview Nucleation phenomenon Observations from biological experiments Mathematical literature Model A Markovian model for nucleation Basic Assumptions The math problem Results Main Results Sketch of proofs Future work and references

Polymerization & Nucleation synthesis − − − − − − − − ⇀ small particles decomposition Big (stable) clusters . ↽ − − − − − − − − ◮ Physics: Aerosols... ◮ Chemistry: Polymers/monomers ◮ Biology: Protein/Peptide v.s. amino acid monomers Figure: Flyvbjerg, Jobs, and Leibler’s model (96’ PNAS) for the self-assembly of microtubules, retrieved from Morris et al. (09’ Biochimica et Biophysica Acta)

Experiments: large variability in nucleation 240 1 . 1 220 1 Observations: 200 0 . 9 180 0 . 8 Quantity of Polymers (normalised) 0 . 7 ◮ sharp curve; 160 0 . 6 140 ◮ huge variance 0 . 5 120 0 . 4 in time. 100 0 . 3 80 0 . 2 60 0 . 1 0 40 − 0 . 1 20 0 10 20 30 40 50 60 70 80 90 100 Time (hours) Figure: Experiments for the evolution of polymerized mass. From data published in Xue et al.(08’ PNAS).

Goals of our study ◮ Explain sharp phase transition in nucleation; ◮ Explain high variance of the transition moment.

Literature: coagulation and fragmentation models ◮ Particles are identified by their sizes. ◮ Reactions: for m = � n i =1 m i , � ( m 1 )+( m 2 )+ . . . +( m n ) → ( m ) ( coagulation ) , ( m ) → ( m 1 )+( m 2 )+ . . . +( m n ) ( fragmentation ) , ◮ Binary reaction: Smoluchowski Model K ( i , j ) − − − ⇀ ( i ) + ( j ) F ( i , j ) ( i + j ) . ↽ − − − where ( F ( i , j )), ( K ( i , j )) are reaction rates.

Literature: coagulation and fragmentation models ◮ Deterministic studies: Oosawa et al. (75), Ball et al. (86’), Penrose (89’,08’), Jabin et al. (03’), Niethammer (04’) . . . ◮ Stochastic studies: Jeon (98’), Durrett et al. (99’), Norris (99’), Ranjbar et al. (10’), Bertoin (06’,17’), Calvez et al. (12’), Sun (18’) . . . ◮ Survey: Aldous (99’), Hingant & Yvinec (16’) Can not explain the high variance observed in the experiments! (CLT is not enough!)

Model with the nucleus ◮ Reaction:  κ k + (1)+( k ) − → ( k + 1) ,   κ k , a  ( k ) − → ( a 1 )+( a 2 )+ · · · +( a p ) , ∀ p ≥ 2 , a 1 + · · · + a p = k , −  ◮ Critical Nucleus size: n c ◮ Polymers larger than the nucleus are more stable than the smaller polymers: ∀ s < n c < ℓ , κ s ≫ κ ℓ − − . κ s κ ℓ + + a κ k , a where κ k − = � − .

Assumptions & Markovian description ◮ Only monomers at t = 0 with total mass N ; ◮ Scaling assumption: for two positive sequences ( λ k ), ( µ k ) and µ > 0, � N µ k , if k < n c κ k κ k + = λ k , − = µ, if k ≥ n c ◮ U N k ( t ) := number of polymers of size k at time t ; ◮ Markov process ( U N k ( t ) , k ∈ N ) with generator + ∞ u 1 � Ω N ( f )( u ) = λ k u k N [ f ( u + e k +1 − e k − e 1 ) − f ( u )] k =1 + ∞ � � � � + N µ k ✶ { k < n c } + µ ✶ { k ≥ n c } u k [ f ( u + y − e k ) − f ( u )] ν k ( d y ) S k k =2 where ( ν k ) are fragmentation measures and ( S k ) are the set of all possible fragmentations.

Mathematical Interpretation ◮ Lag time: for any fraction δ ∈ (0 , 1), L N � kU N δ := inf { t ≥ 0 : k ( t ) ≥ δ N } . k ≥ n c ◮ Observations in terms of lag time: ◮ sharp phase transition: 240 1 . 1 220 1 for any δ 1 , δ 2 ∈ (0 , 1), 200 0 . 9 0 . 8 180 Quantity of Polymers (normalised) 0 . 7 160 0 . 6 140 L N δ 1 ∼ L N 0 . 5 120 δ 2 0 . 4 100 0 . 3 80 0 . 2 60 0 . 1 0 40 − 0 . 1 20 ◮ high variance: 0 10 20 30 40 50 60 70 80 90 100 Time (hours) Figure: Xue et al.(08’ PNAS). �� � � � E ( L N Var ( L N O δ ) ∼ O δ )

Main Results (for n c ≥ 3 ) ◮ The moment of the first nucleus: T N := inf { t ≥ 0 : U N n c ( t ) = 1 } . With high probability, for any δ ∈ (0 , δ 0 ), T N + log ( N ) � � L N δ ∼ O . ◮ For the convergence in probability � � T N lim = E ρ , N n c − 3 N →∞ where E ρ is an exponential random variable with parameter ρ only depends on ( λ k , µ k , k ≤ n c − 1).

Main Results (for n c ≥ 3 ) ◮ The moment of the first nucleus: T N := inf { t ≥ 0 : U N n c ( t ) = 1 } . With high probability, for any δ ∈ (0 , δ 0 ), E ρ N n c − 3 + log ( N ) � � L N Therefore, δ ∼ O . ◮ For the convergence in probability � � T N lim = E ρ , N n c − 3 N →∞ where E ρ is an exponential random variable with parameter ρ only depends on ( λ k , µ k , k ≤ n c − 1).

Main Results (for n c ≥ 3 ) ◮ The moment of the first nucleus: T N := inf { t ≥ 0 : U N n c ( t ) = 1 } . With high probability, for any δ ∈ (0 , δ 0 ), L N δ If n c > 3 , N n c − 3 ∼ O ( E ρ ) ← − Not depends on δ & Large var! ◮ For the convergence in probability � T N � lim = E ρ , N n c − 3 N →∞ where E ρ is an exponential random variable with parameter ρ only depends on ( λ k , µ k , k ≤ n c − 1).

Sketch of proofs (Step I) Before T N , U N k ( t ) ≡ 0, for all k > n c + 1. A simple example, Becker-D¨ oring reactions: λ k − − − − ⇀ (1) + ( k ) ( k + 1) . ↽ − − − − N µ k +1 fast process U N 1 λ k U N k U N 1 / N ∼ O (1) U N k U N U N U N U N slow process · · · nc − 1 n c 2 3 N µ k +1 U N k +1 ∼ O ( N ) U N k +1

Sketch of proofs (Step I) ◮ Distribution of T N only depends on the fast-slow system ( U N 1 ( t ) , . . . , U N n c ( t )). ◮ Study the dynamic on the very large time interval [0 , N n c − 3 t ] by using marked Poisson point processes; ◮ Main Difficulties: ◮ very large fluctuations (Time scale N n c − 3 v.s. Space scale N ). ◮ multi-dimensional stochastic averaging system: hard to identify the limit of occupation measures ◮ Techniques: coupling, flow balance equations... ◮ Proofs work for general fragmentation measures under reasonable conditions.

Sketch of proofs (Step II) After time T N , by coupling, number of stable polymers ( U N n c ( t ) , U N n c +1 ( t ) , . . . ) could be lower bounded by a supercritial branching process. ◮ The lag time of the branching process is less than K log N with probability p 0 > 0. ◮ Therefore, stochastically G p 0 T N ≤ L N � ( T N δ ≤ + K log N ) i i =1 where G p 0 is a geometric random variable.

Future work ◮ Experiments: fragmentation rates are more likely sublinear for smaller polymers, i.e. , for k small, κ k − = O ( N α ) , for an 0 < α < 1 . See the biology review Morris et al. (09’). ◮ In the general case κ − /κ + ∼ φ ( N ), the nucleation time should be O ( φ ( N ) n c − 2 / N ). ◮ Nucleation in a multi-type polymers environment.

References ene, Xue, Robert & Doumic, 16’) Insights into the variability of ◮ (Eug` nucleated amyloid polymerization by a minimalistic model of stochastic protein assembly. (Journal of Chemical Physics) ene & Robert, 16’) Asymptotics of stochastic protein ◮ (Doumic, Eug` assembly models. (SIAM Journal on Applied Mathematics) ◮ (Robert & Sun, 17’) On the Asymptotic Distribution of Nucleation Times of Polymerization Processes. (arXiv:1712.08347)

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