arXiv subjects
Publications and source records attributed to Wenxue Xu.
The Gaussian surface area measure and the Gaussian cone measure for $C$-pseudo-cones are introduced and their corresponding Gaussian Minkowski problem and Gaussian log-Minkowski problem are proposed, respectively. The existence and uniqueness of solutions to these problems for $C$-pseudo-cones are established.
In this paper, we study the non-degenerated $C$-pseudo-cones which can be uniquely decomposed into the sum of a $C$-asymptotic set and a $C$-starting point. Combining this with the novel work in \cite{Schneider-A_weighted_Minkowski_theorem}, we introduce the asymptotic weighted co-volume functional $T_Θ(E)$ of the non-degenerated $C$-pseudo-cone $E$, which is also a generalized function with the singular point $o$ (the origin). Using our convolution formula for $T_Θ(E)$, we establish a decay estimate for $T_Θ(E)$ at infinity and present some interesting results. As applications of this asymptotic theory, we prove a weighted Brunn-Minkowski type inequality and study the solutions to the weighted Minkowski problem for pseudo-cones. Moreover, we pose an open problem regarding $T_Θ(E)$, which we call the asymptotic Brunn-Minkowski inequality for $C$-pseudo-cones.
Designing protein mutants of both high stability and activity is a critical yet challenging task in protein engineering. Here, we introduce PRIME, a deep learning model, which can suggest protein mutants of improved stability and activity without any prior experimental mutagenesis data of the specified protein. Leveraging temperature-aware language modeling, PRIME demonstrated superior predictive power compared to current state-of-the-art models on the public mutagenesis dataset over 283 protein assays. Furthermore, we validated PRIME's predictions on five proteins, examining the top 30-45 single-site mutations' impact on various protein properties, including thermal stability, antigen-antibody binding affinity, and the ability to polymerize non-natural nucleic acid or resilience to extreme alkaline conditions. Remarkably, over 30% of the AI-recommended mutants exhibited superior performance compared to their pre-mutation counterparts across all proteins and desired properties. Moreover, we have developed an efficient, and successful method based on PRIME to rapidly obtain multi-site mutants with enhanced activity and stability. Hence, PRIME demonstrates the general applicability in protein engineering.