Research Article: RNA Thermodynamic Structural Entropy

Date Published: November 10, 2015

Publisher: Public Library of Science

Author(s): Juan Antonio Garcia-Martin, Peter Clote, Emanuele Paci.


Conformational entropy for atomic-level, three dimensional biomolecules is known experimentally to play an important role in protein-ligand discrimination, yet reliable computation of entropy remains a difficult problem. Here we describe the first two accurate and efficient algorithms to compute the conformational entropy for RNA secondary structures, with respect to the Turner energy model, where free energy parameters are determined from UV absorption experiments. An algorithm to compute the derivational entropy for RNA secondary structures had previously been introduced, using stochastic context free grammars (SCFGs). However, the numerical value of derivational entropy depends heavily on the chosen context free grammar and on the training set used to estimate rule probabilities. Using data from the Rfam database, we determine that both of our thermodynamic methods, which agree in numerical value, are substantially faster than the SCFG method. Thermodynamic structural entropy is much smaller than derivational entropy, and the correlation between length-normalized thermodynamic entropy and derivational entropy is moderately weak to poor. In applications, we plot the structural entropy as a function of temperature for known thermoswitches, such as the repression of heat shock gene expression (ROSE) element, we determine that the correlation between hammerhead ribozyme cleavage activity and total free energy is improved by including an additional free energy term arising from conformational entropy, and we plot the structural entropy of windows of the HIV-1 genome. Our software RNAentropy can compute structural entropy for any user-specified temperature, and supports both the Turner’99 and Turner’04 energy parameters. It follows that RNAentropy is state-of-the-art software to compute RNA secondary structure conformational entropy. Source code is available at; a full web server is available at, including source code and ancillary programs.

Partial Text

Conformational (or configurational) entropy is defined by
where kB denotes the Boltzmann constant, and the sum is taken over all structures. As shown experimentally to be the case for calmodulin [1], conformational entropy plays an important role for the discrimination observed in protein-ligand binding. Since conformational entropy is well-known to be difficult to measure, this recent experimental advance involves using NMR relaxation as a proxy for entropy, a technique reviewed in [2].

In this section, we describe the two novel algorithms to compute RNA thermodynamic structural entropy using the Turner energy model [5]. Section “Statistical mechanics” describes the relation between entropy and expected energy, and provides two variants of a simple sampling method to approximate the value of structural entropy. The approximation does not yield accurate entropy values, so two accurate methods are described: (1) formal temperature derivative (FTD) method, (2) dynamic programming (DP) method. An overview of both algorithms is provided in this section. Full details of each algorithm are then provided in Sections “Entropy by statistical physics” and “Entropy by dynamic programming”.

In this section, we describe a detailed comparison of our thermodynamic entropy algorithms FTD and DP, both implemented in the publicly available program RNAentropy, with the algorithm of Manzourolajdad et al. [24] which computes the derivational entropy for trained RNA stochastic context free grammars. Subsequently, we show that by accounting for structural entropy, there is an improvement in the correlation between hammerhead ribozyme cleavage activity and total free energy, extending a result of Shao et al. [32].

In this paper, we have introduced two cubic time algorithms, both implemented in the publicly available program RNAentropy, to compute the RNA thermodynamic structural entropy, H = −∑sp(s) ln p(s), where p(s) = exp(−E(s)/RT)/Z is the Boltzmann probability of secondary structure s, and the sum is taken over all structures of a given RNA sequence a=a1,…,an. This answers a question raised by M. Zuker (personal communication, 2009). Taking a benchmarking set that consists of the first RNA from each of the 2450 families from database Rfam 11.0 [40], we determined the correlation of thermodynamic structural entropy with a variety of other measures used in the computational design and experimental validation of synthetic RNA [14, 31].