RNAlib-2.4.14
Partition Function and Equilibrium Probabilitiy Algorithm(s)

Table of Contents

Equilibrium Ensemble Statistics

In contrast to methods that compute the property of a single structure in the ensemble, e.g. Minimum Free Energy Algorithm(s) , the partition function algorithms always consider the entire equilibrium ensemble. For that purpose, the McCaskill algorithm [17] and its variants can be used to efficiently compute

  • the partition function, and from that
  • various equilibrium probabilities, for instance base pair probabilities, probabilities of individual structure motifs, and many more.

The principal idea behind this approach is that in equilibrium, statistical mechanics and polymer theory tells us that the frequency or probability $p(s)$ of a particular state $s$ depends on its energy $E(s)$ and follows a Boltzmann distribution, i.e.

\[ p(s) \propto e^{-\beta E(s)} \text{ with } \beta = \frac{1}{kT} \]

where $k \approx 1.987 \cdot 10^{-3} \frac{kcal}{mol~K}$ is the Boltzmann constant, and $T$ the thermodynamic temperature. From that relation, the actual probability of state $s$ can then be obtained using a proper scaling factor, the canonical partition function

\[ Z = \sum_{s \in \Omega} e^{-\beta E(s)} \]

where $\Omega$ is the finite set of all states. Finally, the equilibrium probability of state $s$ can be computed as

\[ p(s) = \frac{e^{-\beta E(s)}}{Z} \]

Instead of enumerating all states exhaustively to compute $Z$ one can apply the Secondary Structure Folding Recurrences again for an efficient computation in cubic time. An outside variant of the same recursions is then used to compute probabilities for base pairs, stretches of consecutive unpaired nucleotides, or structural motifs.

See also
Further details of the Partition function and Base Pair Probability algorithm can be obtained from McCaskill 1990 [17]

Partition Function and Equilibrium Probability API

We implement a wide variety of variants of the partition function algorithm according to McCaskill 1990 [17]. See the corresponding submodules for specific implementation details.

See also
Partition Function and Equilibrium Properties, consensus_pf_fold, Partition Function for Two Hybridized Sequences, Partition Function for two Hybridized Sequences as a Stepwise Process, local_pf_fold, Computing Partition Functions of a Distance Based Partitioning

Go to the next Chapter "Suboptimals and (other) Representative Structures"