CATH (https://www.cathdb.info) identifies domains in protein structures from wwPDB and classifies these into evolutionary superfamilies, thereby providing structural and functional annotations. There are two levels: CATH-B, a daily snapshot of the latest domain structures and superfamily assignments, and CATH+, with additional derived data, such as predicted sequence domains, and functionally coherent sequence subsets (Functional Families or FunFams). The latest CATH+ release, version 4.3, significantly increases coverage of structural and sequence data, with an addition of 65,351 fully-classified domains structures (+15%), providing 500 238 structural domains, and 151 million predicted sequence domains (+59%) assigned to 5481 superfamilies. The FunFam generation pipeline has been re-engineered to cope with the increased influx of data. Three times more sequences are captured in FunFams, with a concomitant increase in functional purity, information content and structural coverage. FunFam expansion increases the structural annotations provided for experimental GO terms (+59%). We also present CATH-FunVar web-pages displaying variations in protein sequences and their proximity to known or predicted functional sites. We present two case studies (1) putative cancer drivers and (2) SARS-CoV-2 proteins. Finally, we have improved links to and from CATH including SCOP, InterPro, Aquaria and 2DProt.

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We study the runtime in probabilistic programs with unbounded recursion. As underlying formal model for such programs we use probabilistic pushdown automata (pPDA) which exactly correspond to recursive Markov chains. We show that every pPDA can be transformed into a stateless pPDA (called "pBPA") whose runtime and further properties are closely related to those of the original pPDA. This result substantially simplifies the analysis of runtime and other pPDA properties. We prove that for every pPDA the probability of performing a long run decreases exponentially in the length of the run, if and only if the expected runtime in the pPDA is finite. If the expectation is infinite, then the probability decreases "polynomially". We show that these bounds are asymptotically tight. Our tail bounds on the runtime are generic, i.e., applicable to any probabilistic program with unbounded recursion. An intuitive interpretation is that in pPDA the runtime is exponentially unlikely to deviate from its expected value.A pPDA consists of a finite set of control states, a finite stack alphabet, and a finite set of rules of the form pX x ֒→ qα, where p, q are control states, X is a stack symbol, α is a finite sequence of stack symbols (possibly empty), and x ∈ (0, 1] is the (rational) probability of the rule. We require that for each pX, the sum of the probabilities of all rules of the form pX x ֒→ qα is equal to 1. Each pPDA ∆ induces an infinite-state Markov chain M ∆ , where the states are configurations of the form pα (p is the current control state and α is the current stack content), and pXβ x → qαβ is a transition of M ∆ iff pX x ֒→ qα is a rule of ∆. We also stipulate that pε 1 → pε for every control state p, where ε denotes the empty stack. For example, consider the pPDA∆ with two control states p, q, two stack symbols X, Y , and the rules pX ֒

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