This grant funds a collaboration between Sara Walker, a Professor in the School of Earth and Space Exploration and Deputy Director of the Beyond Center for Fundamental Concepts in Science at Arizona State University, and Leroy Cronin, the Regius Professor of Chemistry at the University of Glasgow, to advance the development of Assembly Theory (AT) by formalizing the mathematical structure of the theory and extending its applicability. Assembly Theory is a framework developed to quantify the complexity of molecules and objects by assessing the minimal number of steps required to assemble them from fundamental building blocks. The theory assigns an assembly index to objects, which serves as a measure of their structural complexity. It is one promising framework for quantifying, understanding, and predicting the emergence and evolution of varied types of complex objects. Walker and Cronin will work towards a general AT framework by determining a suite of mathematical relationships that hold across any assembly space. This will involves applying AT to other substrates, in this case, minerals and genomes / proteomes. The team will develop an assembly theory for minerals and they propose to trace out the evolutionary history of minerals on Earth by combining mineral AT with existing phylogenetic methods that reveal evolutionary connections between objects. The team will also apply AT to large molecules made from nucleic acid building blocks (e.g. DNA / RNA) and from amino acid building blocks (proteins). The plan is to combine AT with phylogenetic techniques to gain insights into the evolutionary history of the modern-day transcription and translation systems used by all known life. Walker and Cronin will also attempt to establish connections between assembly theory and thermodynamics. By developing a bridged framework—Assembly Thermodynamics—they expect to quantify how limits on free energy constrain when a selection phase transition takes place, and to make predictions about such transitions that can be tested in laboratory experiments. If successful, this project will uncover mathematical relationships that apply to all versions of AT (irrespective of the type of object undergoing complexification), allow AT to describe complex minerals and polymers (DNA, RNA, proteins), and make connections between AT and thermodynamics.