Studying the history of tumor evolution from single-cell sequencing data by exploring the space of binary matrices

Single-cell sequencing data has great potential in reconstructing the evolutionary history of tumors. Rapid advances in single-cell sequencing technology in the past decade were followed by the design of various computational methods for inferring trees of tumor evolution. Some of the earliest of these methods were based on the direct search in the space of trees. However, it can be shown that instead of this tree search strategy we can perform a search in the space of binary matrices and obtain the most likely tree directly from the most likely among the candidate binary matrices. The search in the space of binary matrices can be expressed as an instance of integer linear or constraint satisfaction programming and solved by some of the available solvers, which typically provide a guarantee of optimality of the reported solution. In this review, we first describe one convenient tree representation of tumor evolutionary history and present tree scoring model that is most commonly used in the available methods. We then provide proof showing that the most likely tree of tumor evolution can be obtained directly from the most likely matrix from the space of candidate binary matrices. Next, we provide integer linear programming formulation to search for such matrix and summarize the existing methods based on this formulation or its extensions. Lastly, we present one use-case which illustrates how binary matrices can be used as a basis for developing a fast deep learning method for inferring some topological properties of the most likely tree of tumor evolution.