1. M. D. Kvalheim and E. D. Sontag. Autoencoding dynamics: Topological limitations and capabilities, 2025. Note: Journal version submitted.[WWW] [PDF] Keyword(s): autoencoders, dynamical systems, encoding dynamics, differential geometry. Abstract:

   Given a "data manifold" $M\subset \mathbb{R}^n$ and "latent space" $\mathbb{R}^\ell$, an autoencoder is a pair of continuous maps consisting of an "encoder" $E\colon \mathbb{R}^n o \mathbb{R}^\ell$ and "decoder" $D\colon \mathbb{R}^\ell o \mathbb{R}^n$ such that the "round trip" map $D\circ E$ is as close as possible to the identity map $\mbox{id}_M$ on $M$. We present various topological limitations and capabilites inherent to the search for an autoencoder, and describe capabilities for autoencoding dynamical systems having $M$ as an invariant manifold.




2. Eduardo D. Sontag. Dynamics of binding three independent ligands to a single scaffold, 2025. [WWW] Keyword(s): bispecific antibodies, synthetic biology, immunology, dCAs9, CRISPR, CRN, chemical reaction networks, complex balanced, detail balanced. Abstract:

   This note considers a system in which three ligands can independently bind to a scaffold. Such systems arise in diverse applications, including immunotherapy and synthetic biology. It is shown that there are unique steady states in each conservation class, and these are asymptotically stable. The dependency of the steady-state amount of fully bound complex, as a function of total scaffold, is analyzed as well.




3. A.P. Tran, D.D. Jatkar M.A. Al-Radhawi, E. Ernst, and E.D. Sontag. Optimization of heuristic logic synthesis by iteratively reducing circuit substructures using a database of optimal implementations, 2025. Keyword(s): Heuristic logic minimizer, Boolean circuit reduction, optimal synthesis, logic optimization, synthetic biology. Abstract:

   Minimal synthesis of Boolean functions is an NP-hard problem, and heuristic approaches typically give suboptimal circuits. However, in the emergent field of synthetic biology, genetic logic designs that use even a single additional Boolean gate can render a circuit unimplementable in a cell. This has led to a renewed interest in the field of optimal multilevel Boolean synthesis. For small numbers (1-4) of inputs, an exhaustive search is possible, but this is impractical for large circuits. In this work, we demonstrate that even though it is challenging to build a database of optimal implementations for anything larger than 4-input Boolean functions, a database of 4-input optimal implementations can be used to greatly reduce the number of logical gates required in larger heuristic logic synthesis implementations. The proposed algorithm combines the heuristic results with an optimal implementation database and yields average improvements in fractional gate-count reduction relative to ABC of 5.16\% for 5-input circuits and 4.54\% for 6-input circuits on outputs provided by the logic synthesis tool ABC. In addition to the gains in the efficiency of the implemented circuits, this work also attests to the importance and practicality of the field of optimal synthesis, even if it cannot directly provide results for larger circuits. The focus of this work is on circuits made exclusively of 2-input NOR gates but the presented results are readily applicable to 2-input NAND circuits as well as (2-input) AND/NOT circuits. The framework proposed here is likely to be adaptable to other types of circuits. Moreover, a small computational pipeline is provided for integration with synthetic biology tools such as Cello. An implementation of the described algorithm, HLM (Hybrid Logic Minimizer), is available at https://github.com/sontaglab/HLM/.

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