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

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

Sep 21, 2022 263 likes •664 views

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

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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

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The Poland-Scheraga model The generalized Poland-Scheraga model

Plan

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The Poland-Scheraga model Definition The homogeneous pinning model

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The generalized Poland-Scheraga model Biophysics version Renewal process viewpoint Large deviation The localization/delocalization transition Transitions in the localized regime

Maha Khatib POLAND-SCHERAGA model and renewal theory

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The Poland-Scheraga model The generalized Poland-Scheraga model Definition The homogeneous pinning model

A DNA molecule is composed of an alternating sequence of bound and denaturated states.

1 1 2 3 4 5 6 7 8 9 10 11 12 2 3 5 4 6 7 7 8 9 10 11 12

Maha Khatib POLAND-SCHERAGA model and renewal theory

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The Poland-Scheraga model The generalized Poland-Scheraga model Definition The homogeneous pinning model

A DNA molecule is composed of an alternating sequence of bound and denaturated states.

1 1 2 3 4 5 6 7 8 9 10 11 12 2 3 5 4 6 7 7 8 9 10 11 12

Maha Khatib POLAND-SCHERAGA model and renewal theory

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The Poland-Scheraga model The generalized Poland-Scheraga model Definition The homogeneous pinning model

A DNA molecule is composed of an alternating sequence of bound and denaturated states.

1 1 2 3 4 5 6 7 8 9 10 11 12 2 3 5 4 6 7 7 8 9 10 11 12

The statistical weight:

bound sequence of length k: exp(−kEb/T). loop of length k: Ask/kc.

Maha Khatib POLAND-SCHERAGA model and renewal theory

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The Poland-Scheraga model The generalized Poland-Scheraga model Definition The homogeneous pinning model

Link with the pinning model ?

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Maha Khatib POLAND-SCHERAGA model and renewal theory

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The Poland-Scheraga model The generalized Poland-Scheraga model Definition The homogeneous pinning model

A discrete renewal issued from the origin is a random walk τ = {τn}n=0,1,... where τ0 = 0 and, for n ∈ N, τn is a sum of n independent identically distributed random variables taking values in N2.

Maha Khatib POLAND-SCHERAGA model and renewal theory

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The Poland-Scheraga model The generalized Poland-Scheraga model Definition The homogeneous pinning model

A discrete renewal issued from the origin is a random walk τ = {τn}n=0,1,... where τ0 = 0 and, for n ∈ N, τn is a sum of n independent identically distributed random variables taking values in N2. Let P(τ1 = n) = K(n) := L(n) n1+α , where L(·) is a slowly varying function and α > 0.

Maha Khatib POLAND-SCHERAGA model and renewal theory

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The Poland-Scheraga model The generalized Poland-Scheraga model Definition The homogeneous pinning model

A discrete renewal issued from the origin is a random walk τ = {τn}n=0,1,... where τ0 = 0 and, for n ∈ N, τn is a sum of n independent identically distributed random variables taking values in N2. Let P(τ1 = n) = K(n) := L(n) n1+α , where L(·) is a slowly varying function and α > 0. Without loss of generality, we suppose that

  • n≥1

K(n) = 1 ,

Maha Khatib POLAND-SCHERAGA model and renewal theory

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The Poland-Scheraga model The generalized Poland-Scheraga model Definition The homogeneous pinning model

The polymer measure Pc

N,h is defined as

dPc

N,h

dP := 1 Z c

N,h

exp

  • h

N

  • n=1

1n∈τ

  • 1N∈τ ,

Maha Khatib POLAND-SCHERAGA model and renewal theory

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The Poland-Scheraga model The generalized Poland-Scheraga model Definition The homogeneous pinning model

The polymer measure Pc

N,h is defined as

dPc

N,h

dP := 1 Z c

N,h

exp

  • h

N

  • n=1

1n∈τ

  • 1N∈τ ,

The partition function Z c

N,h = N

  • n=1
  • l∈Nn:

|l|=N n

  • i=1

exp(h)K (li) = exp (NF(h)) P (N ∈ τh) ,

Maha Khatib POLAND-SCHERAGA model and renewal theory

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The Poland-Scheraga model The generalized Poland-Scheraga model Definition The homogeneous pinning model

The polymer measure Pc

N,h is defined as

dPc

N,h

dP := 1 Z c

N,h

exp

  • h

N

  • n=1

1n∈τ

  • 1N∈τ ,

The partition function Z c

N,h = N

  • n=1
  • l∈Nn:

|l|=N n

  • i=1

exp(h)K (li) = exp (NF(h)) P (N ∈ τh) , with P( τh = n) = K(n) exp(−F(h)n + h) ,

Maha Khatib POLAND-SCHERAGA model and renewal theory

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The Poland-Scheraga model The generalized Poland-Scheraga model Definition The homogeneous pinning model

The polymer measure Pc

N,h is defined as

dPc

N,h

dP := 1 Z c

N,h

exp

  • h

N

  • n=1

1n∈τ

  • 1N∈τ ,

The partition function Z c

N,h = N

  • n=1
  • l∈Nn:

|l|=N n

  • i=1

exp(h)K (li) = exp (NF(h)) P (N ∈ τh) , with P( τh = n) = K(n) exp(−F(h)n + h) , F(·) is the free energy: the unique solution of

  • n≥1

K(n) exp (−F(h)n + h) = 1 .

Maha Khatib POLAND-SCHERAGA model and renewal theory

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The Poland-Scheraga model The generalized Poland-Scheraga model Definition The homogeneous pinning model

Renewal theorem: lim

N→∞ P (N ∈

τh) = 1/E[ τh,1] .

Maha Khatib POLAND-SCHERAGA model and renewal theory

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The Poland-Scheraga model The generalized Poland-Scheraga model Definition The homogeneous pinning model

Renewal theorem: lim

N→∞ P (N ∈

τh) = 1/E[ τh,1] . If h = 0 and E[ τ0,1] =

n≥1 nK(n) = ∞ (implied by

α ∈ (0, 1)) then P (N ∈ τh) N→∞ ∼ α sin(Πα) Π 1 N1−α .

Maha Khatib POLAND-SCHERAGA model and renewal theory

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The Poland-Scheraga model The generalized Poland-Scheraga model Definition The homogeneous pinning model

Renewal theorem: lim

N→∞ P (N ∈

τh) = 1/E[ τh,1] . If h = 0 and E[ τ0,1] =

n≥1 nK(n) = ∞ (implied by

α ∈ (0, 1)) then P (N ∈ τh) N→∞ ∼ α sin(Πα) Π 1 N1−α . The free energy F(h) = lim

N→∞

1 N log Z c

N,h .

Maha Khatib POLAND-SCHERAGA model and renewal theory

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The Poland-Scheraga model The generalized Poland-Scheraga model Definition The homogeneous pinning model

We define the critical point hc := 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

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The Poland-Scheraga model The generalized Poland-Scheraga model Definition The homogeneous pinning model

We define the critical point hc := 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) = h1/min(1,α)ˆ L(1/h) .

Maha Khatib POLAND-SCHERAGA model and renewal theory

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The Poland-Scheraga model The generalized Poland-Scheraga model Biophysics version Renewal process viewpoint Large deviation 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

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The Poland-Scheraga model The generalized Poland-Scheraga model Biophysics version Renewal process viewpoint Large deviation 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

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The Poland-Scheraga model The generalized Poland-Scheraga model Biophysics version Renewal process viewpoint Large deviation 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.

1

Each base pair is energetically favored and carries an energy −Eb < 0;

2

A base which is not in pair is in a loop: we associate to a loop

  • f length ℓ an entropy factor

B(ℓ) := sℓℓ−c , (2) where s ≥ 1 and c > 2.

Maha Khatib POLAND-SCHERAGA model and renewal theory

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

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The Poland-Scheraga model The generalized Poland-Scheraga model Biophysics version Renewal process viewpoint Large deviation The localization/delocalization transition Transitions in the localized regime

Bivariate renewal theory: A discrete two-dimensional renewal issued from the origin is a random walk τ = {τn}n=0,1,... = (τ (1), τ (2)) = {(τ (1)

n , τ (2) n )}n=0,1,... where

τ0 = (0, 0) and, for n ∈ N := {1, 2, . . .}.

Maha Khatib POLAND-SCHERAGA model and renewal theory

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The Poland-Scheraga model The generalized Poland-Scheraga model Biophysics version Renewal process viewpoint Large deviation The localization/delocalization transition Transitions in the localized regime

Bivariate renewal theory: A discrete two-dimensional renewal issued from the origin is a random walk τ = {τn}n=0,1,... = (τ (1), τ (2)) = {(τ (1)

n , τ (2) n )}n=0,1,... where

τ0 = (0, 0) and, for n ∈ N := {1, 2, . . .}. We set K(n, m) := P(τ1 = (n, m)) = L(n + m) (n + m)1+α , with α ≥ 1.

Maha Khatib POLAND-SCHERAGA model and renewal theory

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The Poland-Scheraga model The generalized Poland-Scheraga model Biophysics version Renewal process viewpoint Large deviation The localization/delocalization transition Transitions in the localized regime

Bivariate renewal theory: A discrete two-dimensional renewal issued from the origin is a random walk τ = {τn}n=0,1,... = (τ (1), τ (2)) = {(τ (1)

n , τ (2) n )}n=0,1,... where

τ0 = (0, 0) and, for n ∈ N := {1, 2, . . .}. We set K(n, m) := P(τ1 = (n, m)) = L(n + m) (n + m)1+α , with α ≥ 1. The polymer measure Pc

N,M,h is defined as

dPc

N,M,h

dP := 1 Z c

N,M,h

exp

  • h

N

  • n=1

M

  • m=1

1(n,m)∈τ

  • 1(N,M)∈τ .

Maha Khatib POLAND-SCHERAGA model and renewal theory

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

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The Poland-Scheraga model The generalized Poland-Scheraga model Biophysics version Renewal process viewpoint Large deviation The localization/delocalization transition Transitions in the localized regime

The partition function Z c

N,M,h = exp ((N + M)g) N∧M

  • n=1
  • l∈Nn:

|l|=N

  • t∈Nn:

|t|=M n

  • i=1
  • Kh (li, ti)

= exp ((N + M)g) P ((N, M) ∈ τh) ,

Maha Khatib POLAND-SCHERAGA model and renewal theory

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The Poland-Scheraga model The generalized Poland-Scheraga model Biophysics version Renewal process viewpoint Large deviation The localization/delocalization transition Transitions in the localized regime

The partition function Z c

N,M,h = exp ((N + M)g) N∧M

  • n=1
  • l∈Nn:

|l|=N

  • t∈Nn:

|t|=M n

  • i=1
  • Kh (li, ti)

= exp ((N + M)g) P ((N, M) ∈ τh) , where Kh(n, m) = exp(h − (n + m)g)K(n + m) and g = g(h) is the only solution to

  • n,m

K(n + m) exp(h − (n + m)g) = 1 , (3) when such a solution exists (that is, when h ≥ 0), and g = 0

  • therwise.

Maha Khatib POLAND-SCHERAGA model and renewal theory

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The Poland-Scheraga model The generalized Poland-Scheraga model Biophysics version Renewal process viewpoint Large deviation The localization/delocalization transition Transitions in the localized regime

Theorem (Borovkov, Mogulskil 1996) For every θ = (θ1, θ2) ∈ R2 lim

t→∞

1 t log P (⌈tθ⌉ ∈ τ) = −D (θ) , where D(θ) = sup

λ∈A

λ, θ = sup

λ∈∂A

λ, θ , and A is the closed convex set {λ ∈ R2 : E[exp(λ, τ)] ≤ 1} and ∂A is the boundary of A.

Maha Khatib POLAND-SCHERAGA model and renewal theory

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The Poland-Scheraga model The generalized Poland-Scheraga model Biophysics version Renewal process viewpoint Large deviation The localization/delocalization transition Transitions in the localized regime

Theorem For every γ ≥ 1 , fγ(h) := lim

N,M→∞:

M N →γ

1 N log Z c

N,M,h

with fγ(h) =

  • if h ≤ 0 ,

(1 + γ)g(h) − Dh(1, γ) if h > 0 , The critical point is hc := inf{h : fγ(h) > 0} = max{h : fγ(h) = 0} = 0 ,

Maha Khatib POLAND-SCHERAGA model and renewal theory

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The Poland-Scheraga model The generalized Poland-Scheraga model Biophysics version Renewal process viewpoint Large deviation The localization/delocalization transition Transitions in the localized regime

We have Dh(1, γ) = max

λ∈Bh

(λ1 + γλ2) , where Bh =

  • λ :
  • n,m

K(n + m)eh−g(h)(n+m)eλ1n+λ2m = 1

  • .

Maha Khatib POLAND-SCHERAGA model and renewal theory

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The Poland-Scheraga model The generalized Poland-Scheraga model Biophysics version Renewal process viewpoint Large deviation The localization/delocalization transition Transitions in the localized regime

We have Dh(1, γ) = max

λ∈Bh

(λ1 + γλ2) , where Bh =

  • λ :
  • n,m

K(n + m)eh−g(h)(n+m)eλ1n+λ2m = 1

  • .

Let λ1(h) := sup

  • λ1 < 0 :
  • n,m

K(n + m)eh−(g(h)−λ1)n ≤ 1

  • ,

then λ1 ≤ λ1 ≤ 0, 0 ≤ λ2 ≤ g. (4)

Maha Khatib POLAND-SCHERAGA model and renewal theory

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The Poland-Scheraga model The generalized Poland-Scheraga model Biophysics version Renewal process viewpoint Large deviation The localization/delocalization transition Transitions in the localized regime

We have Dh(1, γ) = max

λ∈Bh

(λ1 + γλ2) , where Bh =

  • λ :
  • n,m

K(n + m)eh−g(h)(n+m)eλ1n+λ2m = 1

  • .

Let λ1(h) := sup

  • λ1 < 0 :
  • n,m

K(n + m)eh−(g(h)−λ1)n ≤ 1

  • ,

then λ1 ≤ λ1 ≤ 0, 0 ≤ λ2 ≤ g. (4) For γ = 1, we have Dh(1, 1) = 0. Then f1(h) = 2g(h) .

Maha Khatib POLAND-SCHERAGA model and renewal theory

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The Poland-Scheraga model The generalized Poland-Scheraga model Biophysics version Renewal process viewpoint Large deviation The localization/delocalization transition Transitions in the localized regime

Theorem For every choice of α > 1 and L(·), there exists a slowly varying function ˆ L(·) such that f1(h) = h1/min(1,α−1)ˆ L(1/h) . Moreover there exists cα,γ such that fγ(h)

hց0

∼ cα,γf1(h) .

Maha Khatib POLAND-SCHERAGA model and renewal theory

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The Poland-Scheraga model The generalized Poland-Scheraga model Biophysics version Renewal process viewpoint Large deviation The localization/delocalization transition Transitions in the localized regime

We set, always for h > 0 γc(h) :=

  • n,m m K(n + m) exp
  • −n
  • g(h) − λ1(h)
  • n,m n K(n + m) exp
  • −n
  • g(h) − λ1(h)

.

Maha Khatib POLAND-SCHERAGA model and renewal theory

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The Poland-Scheraga model The generalized Poland-Scheraga model Biophysics version Renewal process viewpoint Large deviation The localization/delocalization transition Transitions in the localized regime

We set, always for h > 0 γc(h) :=

  • n,m m K(n + m) exp
  • −n
  • g(h) − λ1(h)
  • n,m n K(n + m) exp
  • −n
  • g(h) − λ1(h)

. In the Cram´ er regime (the maximum is achieved in the interior: when γc(h) > γ) fγ(h) =

  • g(h) − ˆ

λ1(h)

  • + γ
  • g(h) − ˆ

λ2(h)

  • .

Maha Khatib POLAND-SCHERAGA model and renewal theory

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The Poland-Scheraga model The generalized Poland-Scheraga model Biophysics version Renewal process viewpoint Large deviation The localization/delocalization transition Transitions in the localized regime

We set, always for h > 0 γc(h) :=

  • n,m m K(n + m) exp
  • −n
  • g(h) − λ1(h)
  • n,m n K(n + m) exp
  • −n
  • g(h) − λ1(h)

. In the Cram´ er regime (the maximum is achieved in the interior: when γc(h) > γ) fγ(h) =

  • g(h) − ˆ

λ1(h)

  • + γ
  • g(h) − ˆ

λ2(h)

  • .

Out of the Cram´ er regime (the maximum is achieved at the boundary: when γc(h) ≤ γ) fγ(h) = g(h) − λ1(h) .

Maha Khatib POLAND-SCHERAGA model and renewal theory

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The Poland-Scheraga model The generalized Poland-Scheraga model Biophysics version Renewal process viewpoint Large deviation The localization/delocalization transition Transitions in the localized regime

Theorem Fix γ ≥ 1. fγ(·) is analytic on {h : h > 0 such that γc(h) − γ = 0} and fγ(·) is not analytic for the values h > 0 at which γc(h) − γ changes sign. However, f′

γ(·) is continuous on the

positive semi-axis.

Maha Khatib POLAND-SCHERAGA model and renewal theory

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The Poland-Scheraga model The generalized Poland-Scheraga model Biophysics version Renewal process viewpoint Large deviation The localization/delocalization transition Transitions in the localized regime

Thank you for your attention :)

Maha Khatib POLAND-SCHERAGA model and renewal theory