Reconstruction of cell lineage trees from single-cell DNA sequencing data, has the potential to become a fundamental tool in study of development of disease, in particular cancer. For cells without copy number alterations that has not been exposed to specific marking techniques, that is normal cells, lineage tracing is naturally based on somatic point mutations. Current single cell sequencing techniques applicable to such cells require an amplification step, which introduces errors, and still often suffer from so-called allelic dropout. We present a detailed model of current technologies for the purpose of estimating the distance between cells without copy number changes, based on single-cell DNA sequencing data. The model is well suited for full Bayesian analysis by introducing prior probabilities for key parameters as well as maximum a posteriori estimation using expectation maximization algorithm. Our model outputs distance between two cells, simultaneously taking all the other cells into account. In particular, the model contains variables associated with pairs of loci, of which one is homozygous and the other heterozygous, and has the capacity to perform Bayesian probabilistic read phasing. By applying a fast distance based method, such as FNJ, to the estimated distance, a cell lineage tree can be obtained. In contrast to MCMC based methods, FNJ can easily handle data sets with tens of thousands of taxa. The high accuracy of the so obtained method, called SCuPhr, is shown in studies of several synthetic data set.

We propose HAMSI (Hessian Approximated Multiple Subsets Iteration), which is a provably convergent, second order incremental algorithm for solving large-scale partially separable optimization problems. The algorithm is based on a local quadratic approximation, and hence, allows incorporating curvature information to speed-up the convergence. HAMSI is inherently parallel and it scales nicely with the number of processors. Combined with techniques for effectively utilizing modern parallel computer architectures, we illustrate that the proposed method converges more rapidly than a parallel stochastic gradient descent when both methods are used to solve large-scale matrix factorization problems. This performance gain comes only at the expense of using memory that scales linearly with the total size of the optimization variables. We conclude that HAMSI may be considered as a viable alternative in many large scale problems, where first order methods based on variants of stochastic gradient descent are applicable.

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Phylogenetics is a classical methodology in computational biology that today has become highly relevant for medical investigation of single-cell data, e.g., in the context of development of cancer. The exponential size of the tree space is unfortunately a formidable obstacle for current Bayesian phylogenetic inference using Markov chain Monte Carlo based methods since these rely on local operations. And although more recent variational inference (VI) based methods offer speed improvements, they rely on expensive auto-differentiation operations for learning the variational parameters. We propose VaiPhy, a remarkably fast VI based algorithm for approximate posterior inference in an augmented tree space. VaiPhy produces marginal log-likelihood estimates on par with the state-of-the-art methods on real data, and is considerably faster since it does not require auto-differentiation. Instead, VaiPhy combines coordinate ascent update equations with two novel sampling schemes: (i) SLANTIS, a proposal distribution for tree topologies in the augmented tree space, and (ii) the JC sampler, the, to the best of our knowledge, first ever scheme for sampling branch lengths directly from the popular Jukes-Cantor model. We compare VaiPhy in terms of density estimation and runtime. Additionally, we evaluate the reproducibility of the baselines. We provide our code on GitHub: https://github.com/ Lagergren-Lab/VaiPhy.