To understand and engineer biological and artificial nucleic acid systems, algorithms are employed for prediction of secondary structures at thermodynamic equilibrium. Dynamic programming algorithms are used to compute the most favoured, or Minimum Free Energy (MFE), structure, and the Partition Function (PF), a tool for assigning a probability to any structure. However, in some situations, such as when there are large numbers of strands, or pseudoknoted systems, NP-hardness results show that such algorithms are unlikely, but only for MFE. Curiously, algorithmic hardness results were not shown for PF, leaving two open questions on the complexity of PF for multiple strands and single strands with pseudoknots. The challenge is that while the MFE problem cares only about one, or a few structures, PF is a summation over the entire secondary structure space, giving theorists the vibe that computing PF should not only be as hard as MFE, but should be even harder. We answer both questions. First, we show that computing PF is #P-hard for systems with an unbounded number of strands, answering a question of Condon Hajiaghayi, and Thachuk [DNA27]. Second, for even a single strand, but allowing pseudoknots, we find that PF is #P-hard. Our proof relies on a novel magnification trick that leads to a tightly-woven set of reductions between five key thermodynamic problems: MFE, PF, their decision versions, and #SSEL that counts structures of a given energy. Our reductions show these five problems are fundamentally related for any energy model amenable to magnification. That general classification clarifies the mathematical landscape of nucleic acid energy models and yields several open questions.

翻译：为理解和设计生物及人工核酸系统，算法被用于预测热力学平衡下的二级结构。动态规划算法用于计算最有利的（即最小自由能，MFE）结构以及配分函数（PF）——一种为任意结构分配概率的工具。然而，在某些情况下（例如存在大量链或假结系统时），NP难性结果表明此类算法难以实现，但这仅针对MFE问题。值得注意的是，PF的算法复杂性结果尚未得到证明，这留下了关于多链及含假结单链系统PF复杂性的两个开放性问题。其挑战在于：MFE问题仅关注一个或少数结构，而PF需对整个二级结构空间进行求和，这使得理论研究者普遍认为PF计算不仅应至少与MFE问题同等困难，甚至可能更为复杂。本文对这两个问题作出解答：首先，我们证明对于具有无界数量链的系统，PF计算是#P难的，这回答了Condon、Hajiaghayi和Thachuk[DNA27]提出的问题；其次，对于允许假结的单链系统，我们发现PF计算同样是#P难的。我们的证明依赖于一种新颖的放大技巧，通过该技巧在五个关键热力学问题之间建立了紧密的归约关系：MFE、PF、它们的判定版本，以及计算给定能量结构数量的#SSEL问题。这些归约表明，对于任何适用于放大技巧的能量模型，这五个问题具有本质关联。这一普适性分类阐明了核酸能量模型的数学图景，并衍生出若干有待探索的新问题。