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Generic properties of combinatory maps: Neutral networks of RNA secondary structures

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Abstract

Random graph theory is used to model and analyse the relationship between sequences and secondary structures of RNA molecules, which are understood as mappings from sequence space into shape space. These maps are non-invertible since there are always many orders of magnitude more sequences than structures. Sequences folding into identical structures formneutral networks. A neutral network is embedded in the set of sequences that arecompatible with the given structure. Networks are modeled as graphs and constructed by random choice of vertices from the space of compatible sequences. The theory characterizes neutral networks by the mean fraction of neutral neighbors (λ). The networks are connected and percolate sequence space if the fraction of neutral nearest neighbors exceeds a threshold value (λ>λ*). Below threshold (λ<λ*), the networks are partitioned into a largest “giant” component and several smaller components. Structure are classified as “common” or “rare” according to the sizes of their pre-images, i.e. according to the fractions of sequences folding into them. The neutral networks of any pair of two different common structures almost touch each other, and, as expressed by the conjecture ofshape space covering sequences folding into almost all common structures, can be found in a small ball of an arbitrary location in sequence space. The results from random graph theory are compared to data obtained by folding large samples of RNA sequences. Differences are explained in terms of specific features of RNA molecular structures.

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Abbreviations

v[G]:

Vertex set of graphG

e[G]:

Edge set of graphG

ω(X):

Cardinality ofX as a set

δ v :

Vertex degree in a corresponding graphG

Q n α :

Generalized hypercube

\(\hat X\) :

X is a random variable

E[\(\hat X\)]:

Expectation value of the random variable\(\hat X\)

V[\(\hat X\)]:

Variance of\(\hat X\)

E[\(\hat X\)] r :

rth factorial moment of\(\hat X\)

μ n, λ,μ n :

Measure\(\mu _n (\Gamma _n )\mathop = \limits^{def} \lambda ^{\omega (v[\Gamma ])} (1 - \lambda )^{\omega (v[H]) - \omega (v[\Gamma ])} \)

Ω n :

Probability space ({Γ n },μ n, λ

\(\hat X_{n,k} \) :

Number of vertices in a random graph Γ n having degreek

\(\hat I_n (\Gamma _n )\) :

=ω({v∈v[Γ n ]|∂{v}∩v[Γ n ]=∅}), i.e., the number of isolated vertices in a random graph Γ n

\(\hat Z_n (\Gamma _n )\) :
figure 1

i.e. the number of vertices inQ n α that are at least of distance 2 w.r.t. a random graph Γ n

\(M_{n,k}^{\upsilon ,\upsilon '} (\Gamma _n )\) :

Set of paths {π(υ1)|π(υ1)∈Π(Γ n )}

\(\hat Y_{n,k}^{\upsilon ,\upsilon '} \) :

\( = \omega (M_{n,k}^{\upsilon ,\upsilon '} (\Gamma _n ))\) and 0 otherwise

\(\hat \Lambda _{n,k} \) :

Random variable that is 1 if all pairs υ,υ′∈v[Γ n ] withd(v, v′)<k occur in a path ofd(υ,υ′)<k and 0 otherwise

∂ G V :

Set of adjacent vertices w.r.t. a vertex setV⊂v[G] in a graphG

\(\bar V\) :

v[V]∪∂V, i.e. the closure ofV

ℬ r (v):
figure 2

the “ball” with radiusr and centerv

n :

Chain length

n u ,n p :

Number of unpaired and paired based of a certain secondary structure

γ n :

(α−1)n, i.e. the vertex degrees ofQ n α

s :

RNA secondary structure inn vertices

Π(s :

\(\mathop = \limits^{def} \left\{ {\left[ {i,k} \right]|a_{i,k} = 1,k \ne i - 1,i + 1} \right\}\) i.e. the set of contacts of the secondary structures

L n :

Shape space, in particular, the space of RNA secondary structures inn vertices

C[s]:

Graph of compatible sequences with respect tos

C[s]:

v[C[s]], the set of compatible sequences

S n :

Permutation group ofn letters

D m :

Dihedral group of order2m

Φ Γ x :

G 1 ×G 2{y∈v[G 2]⋎(x, y)∈v[⩾s]}], the fiber inx

Φ Γ y :

G 1 ×G 2[{x∈v[G 1]⋎(x, y)∈v[⩽s]}], the fiber iny

Γ A n [s]:

Random induced subgraph of

figure 3

according to model A

Γ B n [s]:

Random induced subgraph of

figure 4

according to model B

dist(Γ1, Γ2):

Minimum Hamming distance between the graph Γ1 and Γ2 considered as subgraph ofQ n α

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Authors and Affiliations

  1. Santa Fe Institute, 87501, Santa Fe, NM, U.S.A.

    Christian Reidys, Peter F. Stadler & Peter Schuster

  2. Los Alamos National Laboratory, 87545, Los Alamos, NM, U.S.A.

    Christian Reidys, Peter F. Stadler & Peter Schuster

  3. Institut für Theoretische Chemie der Universität Wien, A-1090, Wien, Austria

    Peter Schuster

  4. Institut für Molekulare Biotechnologie, D-07708, Jena, Germany

    Peter Schuster

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  1. Christian Reidys
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  2. Peter F. Stadler
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  3. Peter Schuster
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Deicated to professor Manfred Eigen

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Reidys, C., Stadler, P.F. & Schuster, P. Generic properties of combinatory maps: Neutral networks of RNA secondary structures. Bltn Mathcal Biology 59, 339–397 (1997). https://doi.org/10.1007/BF02462007

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