POLAND-SCHERAGA model and renewal theory Maha Khatib Supervised by - PowerPoint PPT Presentation

The Poland-Scheraga model The generalized Poland-Scheraga model POLAND-SCHERAGA model and renewal theory Maha Khatib Supervised by Giambattista Giacomin LPMA - University of Paris 7-Denis Diderot 22 Avril 2016 Colloque Jeunes Probabilistes et Statisticiens Maha Khatib POLAND-SCHERAGA model and renewal theory

The Poland-Scheraga model The generalized Poland-Scheraga model Plan The Poland-Scheraga model 1 Definition The homogeneous pinning model The generalized Poland-Scheraga model 2 Biophysics version Renewal process viewpoint Large deviation The localization/delocalization transition Transitions in the localized regime Maha Khatib POLAND-SCHERAGA model and renewal theory

The Poland-Scheraga model Definition The generalized Poland-Scheraga model The homogeneous pinning model A DNA molecule is composed of an alternating sequence of bound and denaturated states. 5 10 6 4 1 8 12 11 2 7 7 3 9 3 1 2 7 6 8 11 9 12 4 10 5 Maha Khatib POLAND-SCHERAGA model and renewal theory

The Poland-Scheraga model Definition The generalized Poland-Scheraga model The homogeneous pinning model A DNA molecule is composed of an alternating sequence of bound and denaturated states. 5 10 6 4 1 8 12 11 2 7 7 3 9 3 1 2 7 6 8 11 9 12 4 10 5 Maha Khatib POLAND-SCHERAGA model and renewal theory

The Poland-Scheraga model Definition The generalized Poland-Scheraga model The homogeneous pinning model A DNA molecule is composed of an alternating sequence of bound and denaturated states. 5 10 6 4 1 8 12 11 2 7 7 3 9 3 1 2 7 6 8 11 9 12 4 10 5 The statistical weight: bound sequence of length k: exp( − kE b / T ). loop of length k: As k / k c . Maha Khatib POLAND-SCHERAGA model and renewal theory

The Poland-Scheraga model Definition The generalized Poland-Scheraga model The homogeneous pinning model Link with the pinning model ? 2 3 1 4 10 11 5 6 7 8 9 12 Maha Khatib POLAND-SCHERAGA model and renewal theory

The Poland-Scheraga model Definition The generalized Poland-Scheraga model The homogeneous pinning model A discrete renewal issued from the origin is a random walk τ = { τ n } n =0 , 1 ,... where τ 0 = 0 and, for n ∈ N , τ n is a sum of n independent identically distributed random variables taking values in N 2 . Maha Khatib POLAND-SCHERAGA model and renewal theory

The Poland-Scheraga model Definition The generalized Poland-Scheraga model The homogeneous pinning model A discrete renewal issued from the origin is a random walk τ = { τ n } n =0 , 1 ,... where τ 0 = 0 and, for n ∈ N , τ n is a sum of n independent identically distributed random variables taking values in N 2 . Let P ( τ 1 = n ) = K ( n ) := L ( n ) n 1+ α , where L ( · ) is a slowly varying function and α > 0. Maha Khatib POLAND-SCHERAGA model and renewal theory

The Poland-Scheraga model Definition The generalized Poland-Scheraga model The homogeneous pinning model A discrete renewal issued from the origin is a random walk τ = { τ n } n =0 , 1 ,... where τ 0 = 0 and, for n ∈ N , τ n is a sum of n independent identically distributed random variables taking values in N 2 . Let P ( τ 1 = n ) = K ( n ) := L ( n ) n 1+ α , where L ( · ) is a slowly varying function and α > 0. Without loss of generality, we suppose that � K ( n ) = 1 , n ≥ 1 Maha Khatib POLAND-SCHERAGA model and renewal theory

The Poland-Scheraga model Definition The generalized Poland-Scheraga model The homogeneous pinning model The polymer measure P c N , h is defined as � � d P c N � 1 N , h := exp h 1 n ∈ τ 1 N ∈ τ , Z c d P N , h n =1 Maha Khatib POLAND-SCHERAGA model and renewal theory

The Poland-Scheraga model Definition The generalized Poland-Scheraga model The homogeneous pinning model The polymer measure P c N , h is defined as � � d P c N � 1 N , h := exp h 1 n ∈ τ 1 N ∈ τ , Z c d P N , h n =1 The partition function N n � � � Z c N , h = exp( h ) K ( l i ) = exp ( NF ( h )) P ( N ∈ � τ h ) , l ∈ N n : n =1 i =1 | l | = N Maha Khatib POLAND-SCHERAGA model and renewal theory

The Poland-Scheraga model Definition The generalized Poland-Scheraga model The homogeneous pinning model The polymer measure P c N , h is defined as � � d P c N � 1 N , h := exp h 1 n ∈ τ 1 N ∈ τ , Z c d P N , h n =1 The partition function N n � � � Z c N , h = exp( h ) K ( l i ) = exp ( NF ( h )) P ( N ∈ � τ h ) , l ∈ N n : n =1 i =1 | l | = N with P ( � τ h = n ) = K ( n ) exp( − F ( h ) n + h ) , Maha Khatib POLAND-SCHERAGA model and renewal theory

The Poland-Scheraga model Definition The generalized Poland-Scheraga model The homogeneous pinning model The polymer measure P c N , h is defined as � � d P c N � 1 N , h := exp h 1 n ∈ τ 1 N ∈ τ , Z c d P N , h n =1 The partition function N n � � � Z c N , h = exp( h ) K ( l i ) = exp ( NF ( h )) P ( N ∈ � τ h ) , l ∈ N n : n =1 i =1 | l | = N with P ( � τ h = n ) = K ( n ) exp( − F ( h ) n + h ) , F ( · ) is the free energy: the unique solution of � K ( n ) exp ( − F ( h ) n + h ) = 1 . n ≥ 1 Maha Khatib POLAND-SCHERAGA model and renewal theory

The Poland-Scheraga model Definition The generalized Poland-Scheraga model The homogeneous pinning model Renewal theorem: N →∞ P ( N ∈ � lim τ h ) = 1 / E [ � τ h , 1 ] . Maha Khatib POLAND-SCHERAGA model and renewal theory

The Poland-Scheraga model Definition The generalized Poland-Scheraga model The homogeneous pinning model Renewal theorem: N →∞ P ( N ∈ � lim τ h ) = 1 / E [ � τ h , 1 ] . τ 0 , 1 ] = � If h = 0 and E [ � n ≥ 1 nK ( n ) = ∞ (implied by α ∈ (0 , 1)) then α sin(Π α ) 1 τ h ) N →∞ P ( N ∈ � ∼ N 1 − α . Π Maha Khatib POLAND-SCHERAGA model and renewal theory

The Poland-Scheraga model Definition The generalized Poland-Scheraga model The homogeneous pinning model Renewal theorem: N →∞ P ( N ∈ � lim τ h ) = 1 / E [ � τ h , 1 ] . τ 0 , 1 ] = � If h = 0 and E [ � n ≥ 1 nK ( n ) = ∞ (implied by α ∈ (0 , 1)) then α sin(Π α ) 1 τ h ) N →∞ P ( N ∈ � ∼ N 1 − α . Π The free energy 1 N log Z c F ( h ) = lim N , h . N →∞ Maha Khatib POLAND-SCHERAGA model and renewal theory

The Poland-Scheraga model Definition The generalized Poland-Scheraga model The homogeneous pinning model We define the critical point h c := sup { h : F ( h ) = 0 } , (1) at which a localization/ delocalization transition takes place in the system. Maha Khatib POLAND-SCHERAGA model and renewal theory

The Poland-Scheraga model Definition The generalized Poland-Scheraga model The homogeneous pinning model We define the critical point h c := sup { h : F ( h ) = 0 } , (1) at which a localization/ delocalization transition takes place in the system. Theorem For every choice of α ≥ 0 and L ( · ) , there exists a slowly varying function ˆ L ( · ) such that F ( h ) = h 1 / min(1 ,α ) ˆ L (1 / h ) . Maha Khatib POLAND-SCHERAGA model and renewal theory

Biophysics version Renewal process viewpoint The Poland-Scheraga model Large deviation The generalized Poland-Scheraga model The localization/delocalization transition Transitions in the localized regime The generalized Poland-Scheraga model: The possibility of formation of non-symmetrical loops in the two strands (i.e., the contribution to a loop, in terms of number of bases, from the two strands is not necessarily the same). Maha Khatib POLAND-SCHERAGA model and renewal theory

Biophysics version Renewal process viewpoint The Poland-Scheraga model Large deviation The generalized Poland-Scheraga model The localization/delocalization transition Transitions in the localized regime The generalized Poland-Scheraga model: The possibility of formation of non-symmetrical loops in the two strands (i.e., the contribution to a loop, in terms of number of bases, from the two strands is not necessarily the same). The two strands may be of different lengths. Maha Khatib POLAND-SCHERAGA model and renewal theory

Biophysics version Renewal process viewpoint The Poland-Scheraga model Large deviation The generalized Poland-Scheraga model The localization/delocalization transition Transitions in the localized regime The generalized Poland-Scheraga model: The possibility of formation of non-symmetrical loops in the two strands (i.e., the contribution to a loop, in terms of number of bases, from the two strands is not necessarily the same). The two strands may be of different lengths. Each base pair is energetically favored and carries an energy 1 − E b < 0; A base which is not in pair is in a loop: we associate to a loop 2 of length ℓ an entropy factor B ( ℓ ) := s ℓ ℓ − c , (2) where s ≥ 1 and c > 2. Maha Khatib POLAND-SCHERAGA model and renewal theory

Biophysics version Renewal process viewpoint The Poland-Scheraga model Large deviation The generalized Poland-Scheraga model The localization/delocalization transition Transitions in the localized regime Maha Khatib POLAND-SCHERAGA model and renewal theory