The protein matrix model

Figure 30: A fatgraph model forα-helix.

Figure 31: A fatgraph model forβ-sheet.

For each boundary component in the fatgraph, we have a (possibly empty) sequence of unpaired hydrogens and oxygens, represented by the external ma- trices Λ1 and Λ2. To record these sequences, we introduce a combinatorial parameter, which we call the boundary point type spectrum, `ij = {`ij}. It is a sequence of numbers `ij, indexed by two sequences i = (i1, . . . , iK) and j= (j1, . . . , jK). For each boundary component,irecords the numbers of con- secutive unpaired oxygens, andjrecords the numbers of consecutive hydrogens.

So for example, the diagram in figure 30 has the boundary point type spectrum (`(1),(2)= 1, `(2),(0)= 1, `(0),(1) = 1), while the diagram in figure 31 has the spec- trum (`(0),(0)= 1, `(1,1),(1,1)= 1, `(1),(1) = 1). The total number 2l of unpaired hydrogens and oxygens is given by

l = X

K≥1

X

(i1,...,iK)

X

(j1,...,jK)

XK L=1

iL`(i1,...,iK)(j1,...,jK)

= X

K≥1

X

(i1,...,iK)

X

(j1,...,jK)

XK L=1

jL`(i1,...,iK)(j1,...,jK).

We also require two backbone spectra, a = {ai} and b = {bi}, for the numbers of backbone segments of each type with i peptide units. For the di- agram in figure 30, we have {ai} = e5,{bi} = 0, and for figure 31 we have {ai}= 0,{bi}=e5.

Leta=P

i≥1be the total number of peptide units in theα-helix backbones, andb=P

i≥1 be the total number in theβ-sheet backbones.

Definition 4. LetN^g,k,l(a,b,`ij) denote the number of protein fatgraphs with genus g, k propagators, 2l marked points, a backbone spectrum for the un- twisted backbones (i.e.α-helix),bbackbone spectrum for the twisted backbones (i.e.β-sheet), and`ij boundary point spectrum. The generating function of the

number N^g,k,l(a,b,`ij) is defined by F(x, y;α,β,rij) =X

b≥0

Fa,b(x, y;α,β,rij), Fa,b(x, y;α,β,rij)

= 1 a!b!

X

g≥0

X

k≥a+b−1

X

`ij

X

Pai=a

X

Pbi=b

N^g,k,l(a,b,`ij)x^2g−2y^k

×Y

i≥0

α^a_iⁱβ_i^biY

ij

r_ij^`ij.

Using Wick’s theorem with the Wick conrtaction hAabBcdi= 1

Vol(H^N)² Z

HN

dAdB AabBcde^−TrAB=δadδbc, (42) we can express the generating function as a hermitian matrix integral.

Theorem 4.5. Let ZN(y;α,β,rij) denote the exponential of the generating function:

ZN(y;α,β,rij) = exp

F(1/N, y;α,β,rij) .

ZN(y;α,β,rij) is given as the partition function of the hermitian 2-matrix model with external fields Λ1 andΛ2:

ZN(y;α,β,rij)

= 1

Vol²_N Z

H^⊗N²

dAdB exp

"

−NTr

AB−X

i≥0

αiyⁱ(A+y^−1/2Λ1)ⁱ(B+y^−1/2Λ2)ⁱ

−X

i≥0

βiyⁱ((A+y^−1/2Λ1)(B+y^−1/2Λ2))ⁱ

#

= 1

Vol²_N Z

H^⊗2N

dAdBe^{−NTrVy,α,β(A,B;Λ1,Λ2)}, (43)

whereVolN =N^N(N+1)/2Vol(H^N), andrij’s are defined by the single trace for a product of Λ1’s andΛ2’s as

r(i1,...,iK),(j1,...,jK)= 1

NTr(Λⁱ₁¹Λ^j₂¹Λⁱ₁²Λ^j₂²· · ·Λⁱ₁^KΛ^j₂^K).

Proof. The construction is done similarly to section 3.3, by assigning the ap- propriate elements to the diagram elements as described above.

This generating function obeys the heat equation.

Theorem 4.6. The generating functionZN(y;α,β,rij)satisfies the heat equa- tion:

∂

∂yZN(y;α,β,rij)

= 1

2N Tr ∂²

∂Λ1∂Λ2

+ Tr ∂²

∂Λ2∂Λ1

!

ZN(y;α,β,rij), (44)

whereTr_∂Λ^∂₁²_∂Λ2 denotes

Tr ∂²

∂Λ1∂Λ2

= XN a,b=1

∂²

∂Λ1ab∂Λ2ba

.

Proof. The heat equation for the partition functionZN(y;α,β,rij) is obtained by the shift invariance of the matrix integral measuredAanddB.

∂

∂Λ1ba

ZN(y;α,β,rij)

= 1

Vol²_N Z

H^⊗_N²

dAdB NX

i≥1

y^i−1/2

×

αi i−1

X

k=0

(A+y^−1/2Λ1)^k(B+y^−1/2Λ2)ⁱ(A+y^−1/2Λ1)^i−k−1

ab

+iβi (B+y^−1/2Λ2)((A+y^−1/2Λ1)(B+y^−1/2Λ2))ⁱ⁻¹

ab

×e^{−NTrVy,α,β(A,B;Λ1,Λ2)}

= 1

Vol²_N Z

H^⊗_N²

dXdY NX

i≥1

y^i−1/2 αi

Xi−1 k=0

(X^kYⁱX^i−k−1) +iβiY(XY)ⁱ⁻¹

!

ab

×e^{−NTrWy,α,β(X,Y;Λ1,Λ2)},

whereX =A+y^−1/2Λ1, Y =B+y^−1/2Λ2, and Wy,α,β(X, Y; Λ1,Λ2)

= (X−y^−1/2Λ1)(Y −y^−1/2Λ2)−X

i≥0

αiyⁱXⁱYⁱ−X

i≥0

βiyⁱ(XY)ⁱ.

We then compute the derivativePN

a,b=1∂/∂Λ2ab to find Tr ∂²

∂Λ1∂Λ2

ZN(y;α,β,rij)

= 1

Vol²_N Z

H^⊗2N

dXdY N²X

i≥1

yⁱ⁻¹

×Tr 

αi i−1

X

k=0

X^kYⁱX^i−k−1+iβiY(XY)ⁱ⁻¹



(X−y^−1/2Λ1)

!

×e^{−NTrWy,α,β(X,Y;Λ1,Λ2)}.

Exchanging the role of (X,Λ1) and (Y,Λ2), we find Tr ∂²

∂Λ2∂Λ1

ZN(y;α,β,rij)

= 1

Vol²_N Z

H^⊗N²

dXdY N²X

i≥1

yⁱ⁻¹

×Tr 

αi i−1

X

k=0

Y^kXⁱY^i−k−1+iβi(XY)ⁱ⁻¹X



(Y −y^−1/2Λ2)

!

×e^{−NTrWy,α,β(X,Y;Λ1,Λ2)}. (45)

On the other hand, the derivative with respect toyis

∂

∂yZN(y;α,β,rij)

= 1

Vol²_N Z

H^⊗N²

dAdB N 2

X

i≥1

yⁱ⁻¹

×Tr

"

αi i−1

X

k=0

(A+y^−1/2Λ1)^kA(A+y^−1/2Λ1)^i−k−1(B+y^−1/2Λ2)ⁱ

+(B+y^−1/2Λ2)^kB(B+y^−1/2Λ2)^i−k−1(A+y^−1/2Λ1)ⁱ +iβi

A(B+y^−1/2Λ2) + (A+y^−1/2Λ1)B

(A+y^−1/2Λ1)(B+y^−1/2Λ2) i−1#

×e^{−NTrVy,α,β(A,B;Λ1,Λ2)}

= 1

2N Tr ∂²

∂Λ1∂Λ2

+ Tr ∂²

∂Λ2∂Λ1

!

ZN(y;α,β,rij).

The initial condition for this heat equation is found by settingy= 0 in (43).

ZN(y= 0;α,β,rij) = e^{NPi≥0Tr(αiΛi1Λi2+βi(Λ1Λ2)i)}.

The above heat equation can be expressed as a cut-and-join equation.

Theorem 4.7. Let L0 and L2 denote the following differential operators with

respect to parametersrij;

L0= X

K≥1

X

{iL,jL}^KL=1

XK L=1

XK M=1

iL

X

k=1 jM

X

`=1

r(iL−k−1,iL+1,...,iM),(jL,...,j_M−1,`)r(k,iM+1,...,i<sub>L−1</sub>),(jM−`−1,jM+1,...,j_L−1)

× ∂

∂r(i1,...,iK),(j1,...,jK)

,

L2= X

K,V≥1

X

{iL,jL}^KL=1

X

{sQ,tQ}^VQ=1

XK L=1

XV Q=1

iL

X

k=1 tQ

X

u=1

r(iL−k−1,iL+1,...,i_L−1,k,sQ+1,...,s_Q−1,sQ),(jL,jL+1,...,j_L−1,tQ−u−1,tQ+1,...,t_Q−1,u)

× ∂²

∂r(i1,...,iK),(j1,...,jK)∂r(s1,...,sV),(t1,...,tV)

,

where the labelsL, M are defined modulo K, and the labelQ is defined modulo V.

The heat equation (44) is rewritten as the cut-and-join equation:

∂ZN(y;α,β,rij)

∂y =

L0+ 1

N²L2

ZN(y;α,β,rij).

Proof. By the chain rule applied to the right hand side of the heat equation (44), we find

Tr ∂²

∂Λ1∂Λ2

ZN(y;α,β,rij)

= X

K≥1

X

{iL,jL}^K_L=1

Tr∂²r(i1,...,iK),(j1,...,jK)

∂Λ1∂Λ2

∂ZN(y;α,β,rij)

∂r(i1,...,iK),(j1,...,jK)

+ X

K,V≥1

X

{iL,jL}^KL=1

X

{sQ,tQ}^VQ=1

Tr∂r(i1,...,iK),(j1,...,jK)

∂Λ1

∂r(s1,...,sV),(t1,...,tV)

∂Λ2

× ∂²ZN(y;α,β,rij)

∂r(i1,...,iK),(j1,...,jK)∂r(s1,...,sV),(t1,...,tV)

.(46)

The coefficients in (46) are;

Tr∂²r(i1,...,iK),(j1,...,jK)

∂Λ1∂Λ2

= 1 N

XK L,M=1

iL

X

k=1 jM

X

`=1

Tr(Λⁱ₁^L−k−1Λ^j₂^L· · ·Λⁱ₁^MΛ^`₂)

×Tr(Λ^j₂^M−`−1Λⁱ₁^M+1Λ^j₂^M+1· · ·Λⁱ₁^L−1Λ^j₂^L−1Λ^k₁)

=N XK L,M=1

iL

X

k=1 jM

X

`=1

r(iL−k−1,iL+1,...,iM),(jL,...,jM−1,`)

×r(k,iM+1,...,iL−1),(jM−`−1,jM+1,...,jL−1), Tr∂r(i1,...,iK),(j1,...,jK)

∂Λ1

∂r(s1,...,sV),(t1,...,tV)

∂Λ2

= 1 N²

XK L=1

XV Q=1

iL

X

k=1 tQ

X

u=1

Tr(Λⁱ₁^L−k−1Λ^j₂^LΛⁱ₁^L+1Λ^j₂^L+1· · ·Λⁱ₁^L−1Λ^j₂^L−1Λ^k₁

·Λ^t₂^Q−u−1Λ^s₁^Q+1Λ^t₂^Q+1· · ·Λ^s₁^Q−1Λ^t₂^Q−1Λ^s₁^QΛ^u₂)

= 1 N

XK L=1

XV Q=1

iL

X

k=1 tQ

X

u=1

r(iL−k−1,iL+1,...,iL−1,k,sQ+1,...,sQ−1,sQ),(jL,jL+1,...,jL−1,tQ−u−1,tQ+1,...,tQ−1,u). Thus we find the operators L0 and L2. Summing with the results of Tr_Λ^∂2

2Λ1

accounts for the factor _2N¹ in (44).

Merging backbones

We will now slightly relax the initial requirement that a backbone can only contain one type of connection between the peptide units by introducing another matrix M. To the endpoints of backbones, we attach the matrix M, with propagators connecting these endpoints to create a loop structure (figure 32).

Figure 32: Connectingα-helix andβ-sheet backbones.

Definition 5. The partition function of the protein matrix model for merged backbones is defined as the following hermitian 3-matrix model:

ZN(y;α⁽¹⁾,α⁽²⁾,β⁽¹⁾,β⁽²⁾,rij)

= 1

Vol(H^N)³ Z

H^⊗N³

dAdBdM exp

"

−NTr AB+1 2M²

+X

i≥0

yⁱ (

α⁽¹⁾_i M(A+y^−1/2Λ1)ⁱ(B+y^−1/2Λ2)ⁱM +α⁽²⁾_i M(B+y^−1/2Λ2)ⁱ(A+y^−1/2Λ1)ⁱM +β_i⁽¹⁾M (A+y^−1/2Λ1)(B+y^−1/2Λ2)i

M +β_i⁽²⁾M (B+y^−1/2Λ2)(A+y^−1/2Λ1)i

M )!#

= 1

Vol(H^N)³ Z

H^⊗_N³

dAdBdM e^{−NTrVα,β(A,B,M;Λ1,Λ2)}.

The propagatorhAabBcdiof this matrix model represents the hydrogen bond- ings and the propagator hMabMcdi represents the the loop that connects the α-helices andβ-sheets in the backbones.

The heat equation for the model is as follows.

Theorem 4.8. The partition functionZN(y;α⁽¹⁾,α⁽²⁾,β⁽¹⁾,β⁽²⁾,rij)obeys the heat equation,

∂ZN(y;α⁽¹⁾,α⁽²⁾,β⁽¹⁾,β⁽²⁾,rij)

∂y

= 1

2NTr ∂²

∂Λ1∂Λ2

+ ∂²

∂Λ2∂Λ1

!

ZN(y;α⁽¹⁾,α⁽²⁾,β⁽¹⁾,β⁽²⁾,rij).

Proof. First, we consider the derivative∂/∂Λ1of the partition functionZN(y;α⁽¹⁾,α⁽²⁾,β⁽¹⁾,β⁽²⁾,rij).

∂

∂Λ1baZN(y;{α⁽¹⁾_i },{α⁽²⁾_i },β⁽¹⁾,β⁽²⁾,rij)

= 1

Vol³_N Z

H^⊗3N

dAdBdM NX

i≥1

y^i−1/2

×

α⁽¹⁾_i

i−1

X

k=0

(A+y^−1/2Λ1)^k(B+y^−1/2Λ2)ⁱM²(A+y^−1/2Λ1)^i−k−1

ab

+α⁽²⁾_i

i−1

X

k=0

(A+y^−1/2Λ1)^kM²(B+y^−1/2Λ2)ⁱ(A+y^−1/2Λ1)^i−k−1

ab

+β_i⁽¹⁾ Xi−1 k=0

(B+y^−1/2Λ2)((A+y^−1/2Λ1)(B+y^−1/2Λ2))^kM²

×((A+y^−1/2Λ1)(B+y^−1/2Λ2))^i−k−1

ab

+β_i⁽²⁾

i−1

X

k=0

((B+y^−1/2Λ2)(A+y^−1/2Λ1))^kM²

×((B+y^−1/2Λ2)(A+y^−1/2Λ1))^i−k−1(B+y^−1/2Λ2)

ab

×e^{−NTrVy,α,β(A,B,M;Λ1,Λ2)}

= 1

Vol³_N Z

H^⊗N³

dXdY dM NX

i≥1

y^i−1/2

+α⁽¹⁾_i

i−1

X

k=0

(X^kYⁱM²X^i−k−1) +α⁽²⁾_i

i−1

X

k=0

(X^kM²YⁱX^i−k−1)

+β_i⁽¹⁾ Xi−1 k=0

Y(XY)^kM²(XY)^i−k−1+β_i⁽²⁾ Xi−1 k=0

(Y X)^kM²(Y X)^i−k−1Y

!

ab

×e^{−NTrWy,α,β(X,Y,M;Λ1,Λ2)},

whereX =A+y^−1/2Λ1, Y =B+y^−1/2Λ2, and Wy,α,β(X, Y, M; Λ1,Λ2)

= (X−y^−1/2Λ1)(Y −y^−1/2Λ2) +1 2M²

−X

i≥0

yⁱ(α⁽¹⁾_i M XⁱYⁱM +α⁽²⁾_i M YⁱXⁱM)

−X

i≥0

yⁱ(β_i⁽¹⁾M(XY)ⁱM+β_i⁽²⁾M(Y X)ⁱM).

We now compute the derivativePN

a,b=1∂/∂Λ2abto find Tr ∂²

∂Λ1∂Λ2ZN(y;α⁽¹⁾,α⁽²⁾,{β⁽¹⁾_i },{β⁽²⁾_i },rij)

= 1

Vol³_N Z

H^⊗N³

dXdY dM N²X

i≥1

yⁱ⁻¹

×Tr α⁽¹⁾_i

Xi−1 k=0

X^kYⁱM²X^i−k−1+α_i⁽²⁾ Xi−1 k=0

X^kM²YⁱX^i−k−1

+β_i⁽¹⁾

i−1

X

k=0

Y(XY)^kM²(XY)^i−k−1+β_i⁽²⁾

i−1

X

k=0

(Y X)^kM²(Y X)^i−k−1Y

×(X−y^−1/2Λ1)

!

e^{−NTrWy,α,β(X,Y,M;Λ1,Λ2)}

= 1

Vol³_N Z

H^⊗N³

dXdY dM N²X

i≥1

yⁱ⁻¹e^{−NTrWy,α,β(X,Y,M;Λ1,Λ2)}

×Tr α⁽¹⁾_i

i−1

X

k=1

M X^i−k−1(X−y^−1/2Λ1)X^kYⁱM

+α⁽²⁾_i Xi−1 k=1

M YⁱX^i−k−1(X−y^−1/2Λ1)X^kM

+β_i⁽¹⁾

i−1

X

k=1

M(XY)^i−k−1(X−y^−1/2Λ1)Y(XY)^kM

+β_i⁽²⁾ Xi−1 k=1

M(Y X)^i−k−1Y(X−y^−1/2Λ1)(Y X)^kM

!

. (47)

Exchanging the role of (X,Λ1) and (Y,Λ2), we find Tr ∂²

∂Λ2∂Λ1ZN(y;α⁽¹⁾,α⁽²⁾,{β_i⁽¹⁾},{β_i⁽²⁾},rij)

= 1

Vol³_N Z

H^⊗3N

dXdY dM N²X

i≥1

yⁱ⁻¹e^{−NTrWy,α,β(X,Y,M;Λ1,Λ2)}

×Tr α⁽¹⁾_i

i−1

X

k=1

M Xⁱy^i−k−1(Y −y^−1/2Λ2)Y^kM

+α⁽²⁾_i

i−1

X

k=1

M Y^i−k−1(Y −y^−1/2Λ2)Y^kXⁱM

+β_i⁽¹⁾ Xi−1 k=1

M(XY)^i−k−1X(Y −y^−1/2Λ2)(XY)^kM

+β_i⁽²⁾

i−1

X

k=1

M(Y X)^i−k−1(Y −y^−1/2Λ2)X(Y X)^kM

! . (48)

Finally, we compute the derivative with respect toy to find

∂

∂yZN(y;{αi},{βi},rij)

= 1

Vol³_N Z

H^⊗N³

dAdBdM N 2

X

i≥1

yⁱ⁻¹

×Tr

"

α⁽¹⁾_i Xi−1 k=0

M(A+y^−1/2Λ1)^kA(A+y^−1/2Λ1)^i−k−1(B+y^−1/2Λ2)ⁱM

+M(A+y^−1/2Λ1)ⁱ(B+y^−1/2Λ2)ⁱB(B+y^−1/2Λ2)^i−k−1M +α⁽²⁾_i

i−1

X

k=0

M(B+y^−1/2Λ2)^kB(B+y^−1/2Λ2)^i−k−1(A+y^−1/2Λ1)ⁱM

+M(B+y^−1/2Λ2)ⁱ(A+y^−1/2Λ1)^kA(A+y^−1/2Λ1)^i−k−1 +β_i⁽¹⁾

Xi−1 k=0

(A+y^−1/2Λ1)(B+y^−1/2Λ2) k

×

A(B+y^−1/2Λ2) + (A+y^−1/2Λ1)B

(A+y^−1/2Λ1)(B+y^−1/2Λ2) i−k−1

+β_i⁽²⁾ Xi−1 k=0

(B+y^−1/2Λ2)(A+y^−1/2Λ1) k

×

B(A+y^−1/2Λ1) + (B+y^−1/2Λ2)A

(B+y^−1/2Λ2)(A+y^−1/2Λ1)

i−k−1#

×e^{−NTrVy,α,β(A,B;Λ1,Λ2)}

= 1

2N Tr ∂²

∂Λ1∂Λ2 + Tr ∂²

∂Λ2∂Λ1

!

ZN(y;{αi},{βi},rij). (49)

For the initial condition of the heat equation, we find ZN(y= 0;α⁽¹⁾,α⁽²⁾,β⁽¹⁾,β⁽²⁾,rij)

= 1

Vol(H^N) Z

HN

dM e^−N2TrM

IN−2P

i≥0(α⁽¹⁾_i Λⁱ₁Λⁱ₂+α⁽²⁾_i Λⁱ₂Λⁱ₁+β⁽¹⁾_i (Λ1Λ2)ⁱ+β_i⁽²⁾(Λ2Λ1)ⁱ) M

= det



IN−2X

i≥0

(α⁽¹⁾_i Λⁱ₁Λⁱ₂+α⁽²⁾_i Λⁱ₂Λⁱ₁+β_i⁽¹⁾(Λ1Λ2)ⁱ+β⁽²⁾_i (Λ2Λ1)ⁱ)





−1/2

= exp



 X∞ n=1

1 nTr



X

i≥0

(α⁽¹⁾_i Λⁱ₁Λⁱ₂+α⁽²⁾_i Λⁱ₂Λⁱ₁+β_i⁽¹⁾(Λ1Λ2)ⁱ+β_i⁽²⁾(Λ2Λ1)ⁱ)





n

,

where the Plemelj’s formula is used det(IN +X) = exp



X^∞

n=1

(−1)ⁿ⁻¹ n TrXⁿ



.

Introducing N- and C-termini

We extend the protein matrix model further by introducing yet another exter- nal matrix Λ, which labels N- and C-termini of the backbones (figure 33 and figure 34). The boundary cycles containingpbackbone ends are labelled by the set of numbers (i⁽¹⁾₁ , . . . , i⁽¹⁾_K1 : · · ·: i^(p)₁ , . . . , i^(p)_Kp) that count the number of un- paired carboxyl oxygens (Λ1) and (j₁⁽¹⁾, . . . , j_K⁽¹⁾₁ :· · ·:j₁^(p), . . . , j_K^(p)_p) that count the number of unpaired amino hydrogens (Λ2) keeping their ordering on the boundary cycle.

Figure 33: Adding C- and N-ends of backbone

Figure 34: Two backbones with C- and N-ends

Let nij,p denote the extended boundary point type spectrum that counts the number nij;p of boundary components containing a sequence of i Λ1’s a sequence ofj Λ2’s, andpbackbone end points.

Definition 6. The partition function of the protein matrix model with back-

bone endpoints is defined as the following hermitian 3-matrix model;

ZN(y, η;α⁽¹⁾,α⁽²⁾,β⁽¹⁾,β⁽²⁾,ri,j;p)

= 1

Vol(H^N)³ Z

H^⊗3N

dAdBdM exp

"

−NTr AB+1 2M²

−X

i≥0

yⁱη(M +η^−1/2Λ) (

+α⁽¹⁾_i (A+y^−1/2Λ1)ⁱ(B+y^−1/2Λ2)ⁱ +α⁽²⁾_i (B+y^−1/2Λ2)ⁱ(A+y^−1/2Λ1)ⁱ +β_i⁽¹⁾ (A+y^−1/2Λ1)(B+y^−1/2Λ2)i

+β_i⁽²⁾ (B+y^−1/2Λ2)(A+y^−1/2Λ1)i

)

(M +η^−1/2Λ)

!#

= 1

Vol(H^N)³ Z

H^⊗3N

dAdBdM e^{−NTrVy,η,α,β(A,B,M;Λ1,Λ2,Λ)}. (50) The parameterri,j;,p is given by

ri,j;,p

=r_(i⁽¹⁾

1 ,...,i⁽¹⁾_K

1:i⁽²⁾₁ ,...,i⁽²⁾_K

2:...:i^(p)₁ ,...,i^(p)_Kp),(j₁⁽¹⁾,...,j⁽¹⁾_K

1:j₁⁽²⁾,...,j_K⁽²⁾

2:...:j^(p)₁ ,...,j_Kp^(p))

= 1 NTr

Λⁱ

(1) 1

1 Λ^j

(1) 1

2 · · ·Λⁱ

(1) K1

1 Λ^j

(1) K1

2 ΛΛⁱ

(2) 1

1 Λ^j

(2) 1

2 · · ·Λⁱ

(2) K2

1 Λ^j

(2) K2

2 Λ· · ·Λⁱ

(p) 1

1 Λ^j

(p) 1

2 · · ·Λⁱ

(p) Kp

1 Λ^j

(p) Kp

2 Λ

. The heat equations are as follows.

Theorem 4.9. The partition functionZN(y, η;α⁽¹⁾,α⁽²⁾,β⁽¹⁾,β⁽²⁾,rij;p)obeys heat equations:



∂

∂y − 1 2N

∂²

∂Λ1∂Λ2

+ ∂²

∂Λ2∂Λ1

!

ZN(y, η;α⁽¹⁾,α⁽²⁾,β⁽¹⁾,β⁽²⁾,rij;p) = 0, (51)

∂

∂η − 1 2N

∂²

∂Λ²

!

ZN(y, η;α⁽¹⁾,α⁽²⁾,β⁽¹⁾,β⁽²⁾,rij;p) = 0. (52) Proof. The first equation is proven in the same way as the previous model (i.e.

Λ = 0). Here we focus on the proof of the second equation (52).

Consider the derivative with respect to Λ Tr ∂²

∂Λ²ZN(y, η;α⁽¹⁾,α⁽²⁾,β⁽¹⁾,β⁽²⁾,rij;p)

= 1

Vol³_N Z

H^⊗N³

dXdY dT N²X

i≥0

yⁱe^{−NTrWy,η,α,β(X,Y,T;Λ1,Λ2,Λ)}

× (

α⁽¹⁾_i XⁱYⁱ+α⁽²⁾_i YⁱXⁱ+β_i⁽¹⁾(XY)ⁱ+β⁽²⁾_i (Y X)ⁱ )

×

(T−η^−1/2Λ)T+T(T−η^−1/2Λ)

!#) ,

whereT =M +η^−1/2Λ and

Wy,η,α,β(X, Y, T; Λ1,Λ2,Λ)

= (X−y^−1/2Λ1)(Y −y^−1/2Λ2) +1

2(T−η^−1/2Λ)²

−X

i≥0

yⁱ(α⁽¹⁾_i T XⁱYⁱT +α⁽²⁾_i T YⁱXⁱT)

−X

i≥0

yⁱ(β_i⁽¹⁾T(XY)ⁱT+β_i⁽²⁾T(Y X)ⁱT).

The derivative with respect toη is given by

∂

∂ηZN(y, η;α⁽¹⁾,α⁽²⁾,β⁽¹⁾,β⁽²⁾,rij;p)

= 1

Vol³_N Z

H^⊗3N

dAdBdM N 2

X

i≥0

yⁱe^{−NTrVy,ζ,α,β(A,B;Λ1,Λ2,Λ)}

×Tr

"(

α⁽¹⁾_i (A+y^−1/2Λ1)ⁱ(B+y^−1/2Λ2)ⁱ +α⁽²⁾_i (B+y^−1/2Λ2)ⁱ(A+y^−1/2Λ1)ⁱ +β_i⁽¹⁾ (A+y^−1/2Λ1)(B+y^−1/2Λ2)ⁱ +β_i⁽²⁾ (B+y^−1/2Λ2)(A+y^−1/2Λ1)i

)

×(M(M +η^−1/2Λ) + (M+η^−1/2Λ)M)

# . Comparing these two results, we obtain the heat equation (51).

For the initial condition withy= 0 andη= 0, we find

ZN(y= 0, η= 0;{α⁽¹⁾_i },{α⁽²⁾_i },{β_i⁽¹⁾},{β⁽²⁾_i },{r_{i},{j};{K},p})

= exp



X

i≥0

TrΛ(α_i⁽¹⁾(Λⁱ₁Λⁱ₂) +α⁽²⁾_i (Λⁱ₂Λⁱ₁) +β_i⁽¹⁾(Λ1Λ2)ⁱ+β_i⁽²⁾(Λ2Λ1)ⁱ)Λ



.

The initial condition that keepsη can also be considered as follows:

ZN(y= 0, η;{α_i⁽¹⁾},{α⁽²⁾_i },{β_i⁽¹⁾},{β⁽²⁾_i },{r_{i},{j};{K},p})

= 1

Vol(H^N) Z

HN

dM exp

"

−NTr (M²

2 −(M +η^−1/2Λ)

α_i⁽¹⁾(Λⁱ₁Λⁱ₂) +α⁽²⁾_i (Λⁱ₂Λⁱ₁)

+β_i⁽¹⁾(Λ1Λ2)ⁱ+β_i⁽²⁾(Λ2Λ1)ⁱ

(M +η^−1/2Λ) )#

. Finally, we express the heat equations as the cut-and-join equations. The indexing of r makes the notation cumbersome, but a systematic computation gives the following result.

Theorem 4.10. Let L0 and L2 denote the derivatives following differential operators;

L0=X

p≥1

X

{K}

X

{i},{j}

Xp q=1

Xp r=1

Kq

X

L=1 Kr

X

M=1 i^(q)_L −1

X

`=0 j^(r)_M−1

X

m=0

r_(i^(r)

1 , . . . , i(r) M, `, i(q)

L+1, . . . , i(q) Kq:i(q+1)

1 , . . .:· · ·:. . . , i(r−1) Kr−1) ,(j(r)

1 , . . . , j(r) M−1, j(r)

M −m−1, j(q) L , . . . , j(q)

Kq:j(q+1)

1 , . . .:· · ·:. . . , j(r−1) Kr−1)

×r_(i^(q)

1 , . . . , i(q) L−1, i(q)

L −`−1, i(r)

M+1, . . . , i(r) Kr:i(r+1)

1 , . . .:· · ·:. . . , i(q−1) Kq−1) ,(j(q)

1 , . . . , j(q) L−1, m, j(r)

M+1, . . . , j(r) Kr:j(r+1)

1 , . . . ,:· · ·:. . . , j(q−1) Kq−1)

× ∂

∂r_(i(1)

1 ,...:···:...,i^(p)

Kp),(j⁽¹⁾₁ ,...:···:...,j^(p)

Kp)

,

L2= X

p,u≥1

X

{K},{V}

X

{i,j}

X

{s,t}

Xp q=1

Xu w=1

Kq

X

L=1 Vw

X

R=1 i^(q)_L −1

X

`=0 t^(w)_R −1

X

b=0

r_(s^(w)

1 , . . . , s(w) R , `, i(q)

L+1, . . . , i(q) Kq:i(q+1)

1 , . . .:· · ·:. . . , i(q−1) Kq−1: i(q)

1 , . . . , i(q) L−1, i(q)

L −`−1, s(w)

R+1, . . . , s(w) Vw:s(w+1)

1 , . . .:· · ·:. . . , s(w−1) Vw−1 :) ,(t(w)

1 , . . . , t(w) R−1, t(w)

R −b−1, j(q) L , . . . , j(q)

Kq:j(q+1)

1 , . . .:· · ·:. . . , j(q−1) Kq−1: j(q)

1 , . . . , j(q) L−1, b, t(w)

R+1, . . . , t(w) Vw:t(w+1)

1 , . . .:· · ·:. . . , t(w−1) Vw−1:)

× ∂²

∂r_(i⁽¹⁾

1 ,...:···:...,i^(p)_Kp),(j⁽¹⁾₁ ,...:···:...,j_Kp^(p))∂r_(s⁽¹⁾

1 ,...:···:...,s^(u)_Vu),(t⁽¹⁾₁ ,...:···:...,t^(u)_Vu)

.

Let M0 andM2 denote the following differential operators;

M0=1 2

X

p≥1

X

{K}

X

{i},{j}

Xp q=1

Xp

r=1

r_(i^(r−1)

1 , . . . , i(r−1) Kr−1, i(q)

1 , . . . , i(q) Kq :i(q+1)

1 , . . . ,:· · ·:, . . . , i(r−2) Kr−2:) ,(j(r−1)

1 , . . . , j(r−1) Kr−1, j(q)

1 , . . . , j(q) Kq:j(q+1)

1 , . . . ,:· · ·:, . . . , j(r−2) Kr−2:)

×r_(i^(q−1)

1 , . . . , i(q−1) Kq−1, i(r)

1 , . . . , i(r) Kr:i(r+1)

1 , . . . ,:· · ·:, . . . , i(q−2) Kq−2:) ,(j(q−1)

1 , . . . , j(q−1) Kq−1, j(r)

1 , . . . , j(r) Kr:j(r+1)

1 , . . . ,:· · ·:, . . . , j(q−2) Kq−2:)

× ∂

∂r_(i⁽¹⁾

1 ,...:···:...,i^(p)_Kp),(j₁⁽¹⁾,...:···:...,j_Kp^(p))

,

M2=1 2

X

p,u≥1

X

{K},{V}

X

{i},{j}

X

{s},{t}

Xp q=1

Xu

w=1

r_(s^(w−1)

1 , . . . , s(w−1) Vw−1, i(q)

1 , . . . , i(q) Kq :i(q+1)

1 , . . .:· · ·:, . . . , i(q−2) Kq−2: i(q−1)

1 , . . . , i(q−1) Kq−1, s(w)

1 , . . . , s(w) Vw :s(w+1)

1 , . . .:· · ·:. . . , s(w−2) Vw−2 :) ,(t(w−1)

1 , . . . , t(w−1) Vw−1, j(q)

1 , . . . , j(q) Kq :j(q+1)

1 , . . .:· · ·:, . . . , j(q−2) Kq−2: j(q−1)

1 , . . . , j(q−1) Kq−1, t(w)

1 , . . . , t(w) Vw :t(w+1)

1 , . . .:· · ·:. . . , t(w−2) Vw−2 :)

× ∂²

∂r_(i⁽¹⁾

1 ,...:···:...,i^(p)_Kp),(j₁⁽¹⁾,...:···:...,j_Kp^(p))∂r_(s⁽¹⁾

1 ,...:···:...,s^(u)_Vu),(t⁽¹⁾₁ ,...:···:...,t^(u)_Vu)

.

The heat equations (51) and (52) can be rewritten as the cut-and-join equa-

tions:

∂ZN(y, η;α⁽¹⁾,α⁽²⁾,β⁽¹⁾,β⁽²⁾,rij;p)

∂y

=

L0+ 1 N²L2

ZN(y, η;α⁽¹⁾,α⁽²⁾,β⁽¹⁾,β⁽²⁾,rij;p), (53)

∂ZN(y, η;α⁽¹⁾,α⁽²⁾,β⁽¹⁾,β⁽²⁾,rij;p)

∂η

=

M0+ 1 N²M2

ZN(y, η;α⁽¹⁾,α⁽²⁾,β⁽¹⁾,β⁽²⁾,rij;p). (54) Proof. First we will derive L0 and L2 operators from the chain rule. The L0

operator comes from the following derivative:

Tr ∂²rij;p

∂Λ1∂Λ2

= 1 N

Xp q=1

Xp r=1

Kq

X

L=1 Kr

X

M=1 i^(q)_L −1

X

`=0 j_M^(r)−1

X

m=0

Tr

Λ^`₁Λ^j

(q) L

2 Λⁱ

(q) L+1

1 Λ^j

(q) L+1

2 · · ·Λⁱ

(q) Kq

1 Λ^j

(q) Kq

2 Λ· · ·ΛΛⁱ

(r) 1

1 Λ^j

(r) 1

2 · · ·Λⁱ

(r) M

1 Λ^j

(r) M−m−1 2

×Tr

Λ^m₂Λⁱ

(r) M+1

1 Λ^j

(r) M+1

2 · · ·Λⁱ

(r) Kr

1 Λ^j

(r) Kr

2 Λ· · ·ΛΛⁱ

(q) 1

1 Λ^j

(q) 1

2 · · ·Λⁱ

(q) L −`−1 1

=N Xp q=1

Xp r=1

Kq

X

L=1 Kr

X

M=1 i^(q)_L −1

X

`=0 j^(r)_M−1

X

m=0

r_(i^(r)

1 , . . . , i(r) M, `, i(q)

L+1, . . . , i(q) Kq:i(q+1)

1 , . . .:· · ·:. . . , i(r−1) Kr−1) ,(j(r)

1 , . . . , j(r) M−1, j(r)

M −m−1, j(q) L , . . . , j(q)

Kq:j(q+1)

1 , . . .:· · ·:. . . , j(r−1) Kr−1)

×r_(i^(q)

1 , . . . , i(q) L−1, i(q)

L −`−1, i(r)

M+1, . . . , i(r) Kr:i(r+1)

1 , . . .:· · ·:. . . , i(q−1) Kq−1) ,(j(q)

1 , . . . , j(q) L−1, m, j(r)

M+1, . . . , j(r) Kr:j(r+1)

1 , . . . ,:· · ·:. . . , j(q−1) Kq−1)

.

TheL2operator is from Tr∂rij;p

∂Λ1

∂rst;u

∂Λ2

= 1 N²

Xp q=1

Xu w=1

Kq

X

L=1 Vw

X

R=1 i^(q)_L −1

X

`=0 t^(w)_R −1

X

b=0

Tr

Λ^`₁Λ^j

(q) L

2 Λⁱ

(q) L+1

1 Λ^j

(q) L+1

2 · · ·Λⁱ

(q) Kq

1 Λ^j

(q) Kq

2 Λ· · ·ΛΛⁱ

(q) 1

1 Λ^j

(q) 1

2 · · ·Λⁱ

(q) L −`−1 1

·Λ^b₂Λ^s

(w) R+1

1 Λ^t

(w) R+1

2 · · ·Λ^s

(w) Vw

1 Λ^t

(w) Vw

2 Λ· · ·ΛΛ^s

(w) 1

1 Λ^t

(w) 1

2 · · ·Λ^s

(w) R

1 Λ^t

(w) R −b−1 2

= 1 N

Xp q=1

Xu w=1

Kq

X

L=1 Vw

X

R=1 i^(q)_L −1

X

`=0 t^(w)_R −1

X

b=0

r_(s^(w)

1 , . . . , s(w) R , `, i(q)

L+1, . . . , i(q) Kq:i(q+1)

1 , . . .:· · ·:. . . , i(q−1) Kq−1: i(q)

1 , . . . , i(q) L−1, i(q)

L −`−1, s(w)

R+1, . . . , s(w) Vw :s(w+1)

1 , . . .:· · ·:. . . , s(w−1) Vw−1 :) ,(t(w)

1 , . . . , t(w) R−1, t(w)

R −b−1, j(q) L , . . . , j(q)

Kq :j(q+1)

1 , . . .:· · ·:. . . , j(q−1) Kq−1: j(q)

1 , . . . , j(q) L−1, b, t(w)

R+1, . . . , t(w) Vw:t(w+1)

1 , . . . ,:· · ·:. . . , t(w−1) Vw−1 :)

.

For the second heat equation, theM0 operator is from Tr∂²rij;p

∂Λ²

= 1 N

Xp q=1

Xp r=1

Tr

Λⁱ

(q) 1

1 Λ^j

(q) 1

2 · · ·Λⁱ

(q) Kq

1 Λ^j

(q) Kq

2 Λ· · ·ΛΛⁱ

(r−1) 1

1 Λ^j

(r−1) 1

2 · · ·Λⁱ

(r−1) Kr−1

1 Λ^j

(r−1) Kr−1

2

×Tr

Λⁱ

(r) 1

1 Λ^j

(r) 1

2 · · ·Λⁱ

(r)

1KrΛ^j

(r)

2KrΛ· · ·ΛΛⁱ₁^(q−1)1 Λ^j₂^(q−1)1 · · ·Λⁱ

(q−1) Kq−1

1 Λ^j

(q−1) Kq−1

2

=N Xp q=1

Xp r=1

r_(i^(r−1)

1 , . . . , i(r−1) Kr−1, i(q)

1 , . . . , i(q) Kq :i(q+1)

1 , . . . ,:· · ·:, . . . , i(r−2) Kr−2:) ,(j(r−1)

1 , . . . , j(r−1) Kr−1, j(q)

1 , . . . , j(q) Kq :j(q+1)

1 , . . . ,:· · ·:, . . . , j(r−2) Kr−2:)

×r_(i^(q−1)

1 , . . . , i(q−1) Kq−1, i(r)

1 , . . . , i(r) Kr:i(r+1)

1 , . . . ,:· · ·:, . . . , i(q−2) Kq−2:) ,(j(q−1)

1 , . . . , j(q−1) Kq−1, j(r)

1 , . . . , j(r) Kr:j(r+1)

1 , . . . ,:· · ·:, . . . , j(q−2) Kq−2:)

.

Finally, theM2operator is from Tr∂rij;p

∂Λ

∂rst;u

∂Λ

= 1 N²

Xp q=1

Xu w=1

Tr

Λⁱ

(q) 1

1 Λ^j

(q) 1

2 · · ·Λⁱ

(q) Kq

1 Λ^j

(q) Kq

2 Λ· · ·ΛΛⁱ

(q−1) 1

1 Λ^j

(q−1) 1

2 · · ·Λⁱ

(q−1) Kq−1

1 Λ^j

(q−1) Kq−1

2

×Λ^s

(w) 1

1 Λ^t

(w) 1

2 · · ·Λ^s

(w)

1VwΛ^t

(w)

2VwΛ· · ·ΛΛ^s

(w−1) 1

1 Λ^t

(w−1) 1

2 · · ·Λ^s

(w−1) Vw−1

1 Λ^t

(w−1) Vw−1

2

= 1 N

Xp q=1

Xu w=1

r_(s^(w−1)

1 , . . . , s(w−1) Vw−1, i(q)

1 , . . . , i(q) Kq :i(q+1)

1 , . . .:· · ·:, . . . , i(q−2) Kq−2: i(q−1)

1 , . . . , i(q−1) Kq−1, s(w)

1 , . . . , s(w) Vw :s(w+1)

1 , . . .:· · ·:. . . , s(w−2) Vw−2 :) ,(t(w−1)

1 , . . . , t(w−1) Vw−1, j(q)

1 , . . . , j(q) Kq :j(q+1)

1 , . . .:· · ·:, . . . , j(q−2) Kq−2: j(q−1)

1 , . . . , j(q−1) Kq−1, t(w)

1 , . . . , t(w) Vw :t(w+1)

1 , . . .:· · ·:. . . , t(w−2) Vw−2 :)

.

We note the first cut-and-join equation (53) expresses the cut/join manipu- lation of the hydrogen bonds, while the second cut-and-join equation (54) is for loops (or turns) in the backbones.

5 Topology of protein β-sheets

5.1 β-sheet topology

The α-helix and the β-sheet are two common protein secondary structures.

While theα-helix is essentially a local structure with the participating residues all lying together along the backbone, theβ-sheet involves interactions between residues which are far apart in the backbone (section 4.1). It is also more heterogeneous as a structure, consisting of both parallel and anti-parallel con- figurations of the participatingβ-strands. Furthermore,β-sheet has an intrinsic structural flexibility compared to α-helix, complicating the structural analyses [30]. A better understanding of their structures and foldings is therefore crucial, if we are to understand the folding mechanism of entire proteins.

The configurations of β-strands in a protein, often called β-sheet topolo- gies, have been studied since the 1970’s [79]. Early studies ([79, 78, 87]) have identified some general rules (such as the preference for the right-handedness in parallel β-sheets) from investigation of individual proteins. As the amount of available data increased, studies have used computer programmes to survey the database and found frequent patterns in the β-strand configurations [99, 80]. The information can be used to filter and rank a series of candidate struc- tures by computing probabilities for different patterns [80]. Another approach is to assign pseudoenergy to each pair ofβ-strand residues and solve theβ-sheet topology prediction problem as an optimisation problem [29]. At least one study [44] has compared the two methods, and found that the latter’s performance to be better. One may also combine the two methods by, for example, forbidding certain β-strand configurations that are not found in the database [89], or by incorporating the two in Bayesian modelling [18]. Other studies used integer programming techniques to predict β-sheet topologies [81, 34].

In order to study β-sheet topology of proteins, we introduce a new model inspired by the protein fatgraph model described in section 4, which we call pro- tein metastructure. This model greatly simplifies the study ofβ-sheet topologies by amalgamating consecutive residues belonging to the same secondary struc- ture, but still retains the information needed to understand the configuration of β-strands. We give a detailed definition in section 5.2. Furthermore, each metas- tructure corresponds to a fatgraph, and this transition to fatgraphs allows us to compute topological invariants such as the number of boundary components and the genus associated to each protein. The details of this correspondence are described in section 2.1. Compared to the model described in section 4, our construction is much simpler, and only takes into account the hydrogen bonds that are part of β-sheets. In the following sections, we will analyse the topol- ogy of fatgraphs associated to proteins and suggest potential applications in the study ofβ-sheet topology.

No documento q g m - centre f o r quan t u m g e o m etry o f m o dulis p a c es (páginas 69-87)