The abbreviated name,‘mfold web server’,describes a number of closely related software applications available on the World Wide Web (WWW) for the prediction of the secondary structure of single stranded nucleic acids. The objective of this web server is to provide easy access to RNA and DNA folding and hybridization software to the scientific community at large. By making use of universally available web GUIs (Graphical User Interfaces),the server circumvents the problem of portability of this software. Detailed output,in the form of structure plots with or without reliability information,single strand frequency plots and ‘energy dot plots’, are available for the folding of single sequences. A variety of ‘bulk’ servers give less information,but in a shorter time and for up to hundreds of sequences at once. The portal for the mfold web server is http://www.bioinfo.rpi.edu/applications/ mfold. This URL will be referred to as ‘MFOLDROOT’.

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Mfold web server for nucleic acid folding and hybridization prediction

Background
Secondary structure forms an important intermediate level of description of nucleic acids that encapsulates the dominating part of the folding energy, is often well conserved in evolution, and is routinely used as a basis to explain experimental findings. Based on carefully measured thermodynamic parameters, exact dynamic programming algorithms can be used to compute ground states, base pairing probabilities, as well as thermodynamic properties.

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ViennaRNA Package 2.0

Computer codes for computation and comparison of RNA secondary structures, the Vienna RNA package, are presented, that are based on dynamic programming algorithms and aim at predictions of structures with minimum free energies as well as at computations of the equilibrium partition functions and base pairing probabilities. An efficient heuristic for the inverse folding problem of RNA is introduced. In addition we present compact and efficient programs for the comparison of RNA secondary structures based on tree editing and alignment. All computer codes are written in ANSI C. They include implementations of modified algorithms on parallel computers with distributed memory. Performance analysis carried out on an Intel Hypercube shows that parallel computing becomes gradually more and more efficient the longer the sequences are.

Fast folding and comparison of RNA secondary structures

An algorithm and a computer program have been prepared for determining RNA secondary structures within any prescribed increment of the computed global minimum free energy. The mathematical problem of determining how well defined a minimum energy folding is can now be solved. All predicted base pairs that can participate in suboptimal structures may be displayed and analyzed graphically. Representative suboptimal foldings are generated by selecting these base pairs one at a time and computing the best foldings that contain them. A distance criterion that ensures that no two structures are "too close" is used to avoid multiple generation of similar structures. Thermodynamic parameters, including free-energy increments for single-base stacking at the ends of helices and for terminal mismatched pairs in interior and hairpin loops, are incorporated into the underlying folding model of the above algorithm.

https://science.sciencemag.org/content/sci/244/4900/48.full.pdf

On finding all suboptimal foldings of an RNA molecule

A novel application of dynamic programming to the folding problem for RNA enables one to calculate the full equilibrium partition function for secondary structure and the probabilities of various substructures. In particular, both the partition function and the probabilities of all base pairs are computed by a recursive scheme of polynomial order N3 in the sequence length N. The temperature dependence of the partition function gives information about melting behavior for the secondary structure. The pair binding probabilities, the computation of which depends on the partition function, are visually summarized in a “box matrix” display and this provides a useful tool for examining the full ensemble of probable alternative equilibrium structures. The calculation of this ensemble representation allows a proper application and assessment of the predictive power of the secondary structure method, and yields important information on alternatives and intermediates in addition to local information about base pair opening and slippage.

The results are illustrated for representative tRNA, 5S RNA, and self-replicating and self-splicing RNA molecules, and allow a direct comparison with enzymatic structure probes. The effect of changes in the thermodynamic parameters on the equilibrium ensemble provides a further sensitivity check to the predictions.

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The equilibrium partition function and base pair binding probabilities for RNA secondary structure

Given a simple graph $G = ( V,E )$ and a positive number d, an $L_d ( 2,1 )$-labelling of G is a function $f:V ( G ) \to [ 0,\infty )$ such that whenever $x,y \in V$ are adjacent, $| f ( x ) - f ( ...

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Labelling graphs with a condition at distance 2

A simplified (two-base) version of the problem of planar folding of long chains (e.g., RNA and DNA biomolecules) is formulated as a matching problem. The chain is prescribed as a loop or circular sequence of letters A and B, n units long. A matching here means a set of A-B base pairings or matches obeying a planarity condition: no two matches may cross each other if drawn on the interior of the loop. Also, no two adjacent letters may be matched. We present a dynamic programming algorithm requiring $O( {n^3 } )$ steps and $O( {n^2 } )$ storage which computes the size of the maximum for the given A-B base sequence and which also allows reconstructing a particular folded form of the original string which realizes the maximum matching size. The algorithm can be adapted to deal with sequences with larger alphabets and with weighted matchings.An algorithm is also presented for a modified problem closer to the biochemical problem of interest: We demand that every match must be adjacent to another match, forcing ...

Algorithms for Loop Matchings

The interval number $i( G )$ of a simple graph G is the smallest number t such that to each vertex in G there can be assigned a collection of at most t finite closed intervals on the real line so that there is an edge between vertices v and w in G if and only if some interval for v intersects some interval for w. The well known interval graphs are precisely those graphs G with $i ( G )\leqq 1$. We prove here that for any graph G with maximum degree $d, i ( G )\leqq \lceil \frac{1}{2} ( d + 1 ) \rceil $. This bound is attained by every regular graph of degree d with no triangles, so is best possible. The degree bound is applied to show that $i ( G )\leqq \lceil \frac{1}{3}n \rceil $ for graphs on n vertices and $i ( G )\leqq \lfloor \sqrt{e} \rfloor $ for graphs with e edges.

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Jerrold R. Griggs

Papers

Labelling graphs with a condition at distance 2

Algorithms for Loop Matchings

Extremal Values of the Interval Number of a Graph

Interval graphs and maps of DNA

The number of maximal independent sets in a connected graph