The kinetic properties in biological systems are important for understanding biomolecular functions and are of great importance for drug discovery. However, how to precisely calculate the kinetics of rare events occurring in biomolecular systems with computational methods remains a big challenge. Recently, a variant of metadynamics, namely, frequency-adaptive metadynamics (FAMetaD), has been developed to calculate the unbiased transition time from biased molecular dynamics simulations with high efficiency. In this chapter, we provided a detailed tutorial about how to perform and analyze FAMetaD simulations by taking a simple pentapeptide as the model system which however has a complex free energy landscape with multiple conformational states. With the help of this tutorial, we hope that non-experienced users can master the skill of performing and analyzing FAMetaD simulations and apply to their own biological problems.

In this paper we provide a unifying description of different types of semantics of modal logic found in the literature via the framework of topological categories. In the style of categorical logic, we establish an exact correspondence between various syntactic extensions of modal logic on one hand, including modal dependence, group agent structures, and logical dynamic, and semantic structures in topological categories on the other hand. This framework provides us a uniform treatment of interpreting these syntactic extensions in all different types of semantics of modal logic, and it deepens our conceptual understanding of the abstract structure of modal logic.

In this paper, we provide an alternative description of the duality result for distributive lattices and coherent locales using ultraposet. In particular, we show that there are fully faithful embeddings from the opposite of the category of distributive lattices into the category of ultraposets with ultrafunctors, and from the category of coherent locales into the category of ultraposets with left ultrafunctors. We also define the notion of zero-dimensional ultraposets, which characterises the essential image of these embeddings.

This paper investigates the conditions under which diagonal sentences can be taken to constitute paradigmatic cases of self-reference. We put forward well-motivated constraints on the diagonal operator and the coding apparatus which separate paradigmatic self-referential sentences, for instance obtained via Gödel’s diagonalization method, from accidental diagonal sentences. In particular, we show that these constraints successfully exclude refutable Henkin sentences, as constructed by Kreisel.