of structures. An RNAγ-structure is a partial chord diagram S, such that the genera of the irreducible components of π(S) is bounded byγ, i.e.
g(S0)≤γ ∀S0 ∈ {S0 ∈π(S)|S0 is an irreducible component inπ(S)}. Reidys et al. [77] have devised a folding algorithm for RNAγ-structures withγ≤ 1, based on the decomposition into irreducible shadows. This allowed an efficient implementation of topology-dependent energy penalties for pseudoknots. The resulting software achieved 10-20% increases in the prediction accuracy of base pairs.
Up to now the discussion has been on RNA structures with one backbone component. But the concepts of shadows, irreducibility and henceγ-structures apply equally well to the structures with more than one backbone components.
In particular the analysis of structures over two backbone is relevant for the modelling of RNA-RNA interaction structures. A decomposition grammar for theγ-structures over two backbones withγ= 0 has been developed by Andersen et al. [7].
by a chord or backbone underside, thend2 marked points followed by a chord or backbone underside, and so on all the way around the boundary component.
Letej denote the sequence (0, . . . ,0,1,0, . . .), where 1 appears at thej’th entry and all other entries are 0. We say a diagram is of a certain type if it has the specified parameters and spectra; for example, a diagram of type{g, k, l;b,m} has genusg,kchords, lmarked points, with the backbone spectrumband the boundary length and point spectrum m(see figure 17 for an example).
Figure 17: This chord diagram has 2 backbones, 4 chords, 5 marked points and 4 boundary components. By the Euler’s formula, its genus is 0. It has the backbone spectrum (e5+e8), the boundary point spectrum (3e0+e5), the boundary length spectrum (e1+ 2e2+e5), and the boundary length and point spectrum (m(0)= 1, m(0,0)= 2, m(1,2,0,2,0)= 1).
The following relations follow immediately from the definitions.
b=X
i≥0
bi, n=X
i≥0
li =X
i≥1
ni = X
K≥1
X
dK
mdK, 2k+l=X
i≥0
ibi, l=X
i≥0
ili = X
K≥1
X
dK
|dK|mdK, 2k+b=X
i≥1
ini= X
K≥1
X
dK
KmdK, nK =X
dK
mdK, li= X
K≥1
X
|dK|=i
mdK,
where|dK|=PK j=1dj.
Let Ng,k,l(b,l,n,m) be the number of distinct connected partial chord diagrams of type {g, k, l,b,l,n,m}. Note the parameters are not indepen- dent, as evident from the above relations. We set Ng,k,l(b,l,n,m) = 0, if a partial chord diagram of the given type is not possible. We will also con- sider the number of distinct partial chord diagrams, where we sum one or more of the parameters, such as Ng,k,l(b,l,n) = P
mNg,k,l(b,l,n,m) and Ng,k,l(b,m) = P
l
P
nNg,k,l(b,l,n,m). For a sequence indexed by integer such asb,l, andn, and a variablet= (t0, t1, . . .), we denote
tb=Y
i≥0
tbii =tb00tb11· · ·.
For the variablem, indexed by ordered setsdK = (d1, . . . , dK), we consider the variable indexed by the same sets u= (udK) and denote
um= Y
K≥1
Y
dK
umdKdK.
With these notations, the orientable, multi-backbone, boundary point spectrum generating functionF(x, y;s;t) is given by
F(x, y;s;t) =X
b≥1
Fb(x, y;s;t) (31)
=X
b≥1
1 b!
X∞ k=b−1
X
l
X
Pbi=b
Ng,k,l(b,l)x2g−2yksltb. (32)
F satisfies the following differential equations. We also present its proof to illustrate the cut and join method.
Theorem 3.4. (Alexeev et al. [3])Consider the linear differential operators L0= 1
2 X∞ i=0
Xi j=0
(i+ 2)sjsi−j ∂
∂si+2,
L2= 1 2
X∞ i=2
si−2 Xi−1 j=1
j(i−j)∂s∂2
j∂si−j
and the quadratic differential operators
QF = 1 2
X∞ i=2
si−2 i−1
X
j=1
j(i−j)∂ F
∂sj
∂ F
∂si−j. Then the following differential equations hold;
∂ F1
∂y = (L0+x2L2)F1,
∂ F
∂y = (L0+x2L2+x2Q)F. (33)
These equations, together with the initial condition given by x−2P
i≥1siti for y= 0, determines the generating functions F1 andF uniquely.
Proof. We note first that equation (33) is equivalent to the following recursion
relation forNg,k,l(b,l):
kNg,k,l(b,l) = 1
2 X∞ i=0
Xi j=0
(i+ 2)(li+2+ 1)Ng,k−1,l+2(b,l−ej−ei−j+ei+2) + 1
2 X∞ i=0
Xi+1 j=1
j(i+ 2−j)(lj+ 1 +δj,i+2−j−δi,j)(li+2−j+ 1−δj,2)× Ng−1,k−1,l+2(b,l+ej+ei+2−j−ei) +
1 2
X∞ i=0
Xi+1 j=1
X
g1+g2=g
X
k1+k2=k−1
X
l(1)+l(2)=l−ei
X
b(1)+b(2)=b
j(i+ 2−j)(l(1)j + 1)(l(2)i+2−j+ 1) b!
b(1)!b(2)! ×
Ng1,k1,l1+j(b(1),l(1)+ej)Ng2,k2,l2+i+2−j(b(2),l(2)+ei+2−j), (34) where
b(r)= X∞ i=1
b(r)i , lr= X∞ i=1
il(r)i , X∞ i=0
li(r)=kr−2gr−b(r)+ 2,
forr= 1,2. Checking this is a straightforward computation looking at the coef- ficients of x2−2gyk−1sltb in both sides of equation (33). The recursion relation (34) is then proved by the cut and join method, similarly to theorem 4.1. The idea is to express the number of diagrams of type{g, k, l,b,l} with one marked chord in two different ways, one by simply marking a chord on a diagram of type {g, k, l,b,l}, and the other by adding a marked chord to a diagram, such that the resulting diagram is of type{g, k, l,b,l}. The former is straightforward; given a diagram of type {g, k, l,b,l}, there are kchords to choose from for marking, so this gives the l.h.s. in equation (34). We do the latter by first removing a chord from a diagram of type {g, k, l,b,l}(“cut”), then adding an appropriate marked chord (“join”). See theorem 4.1 for more details in proof.
The recursion relations for other spectra, such as the boundary length spec- trum (for chord diagrams) [3], and the boundary length and point spectrum (for partial chord diagrams) [15] have been established using the same method.
Recursion relation for non-oriented chord diagrams
Non-oriented (partial) chord diagrams can be considered by assigning an extra binary data to each chord, showing whether the chord is twisted or not. IfC is such a (partial) chord diagram, on the associated surfaceF =F(C), the twisted chord then become twisted bands. Clearly this construction produces 2k dif- ferent orientable and non-orientable partial chord diagrams from one orientable partial chord diagram with k chords. We have the following definition for the Euler characteristic in the non-oriented case. TheEuler characteristic χof the non-oriented surfaceF is give by
χ(F) = 2−h,
wherehis the number of cross-caps and we have Euler’s relation 2−h=b−k+n.
We also need to introduce a small change for the boundary point and length spectrum m. Since we are no longer able to traverse a boundary component using the induced orientation in the non-orientable case, we also need to consider the order-reversing, as well as cyclic, permutation. So we have
dK = (d1, d2, . . . , dK) = (dK, . . . , d2, d1).
With these definitions, it is possible to derive the recursion relations for the number of orientable and non-orientable (partial) chord diagrams, filtered by different spectra. The specific expressions and derivations can be found in [3, 15].