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In this article, we develop a computational approach for estimating the most likely trajectories describing rare events that correspond to the emergence of non-dominant genotypes. This work is based on the large deviations approach for discrete Markov chains describing the genetic evolution of large bacterial populations. We demonstrate that a gradient descent algorithm developed in this article results in the fast and accurate computation of most likely trajectories for a large number of bacterial genotypes. We supplement our analysis with extensive numerical simulations demonstrating the computational advantage of the designed gradient descent algorithm over other, more simplified, approaches. 

Let $$\mathcal{D}$$ be a dataset of smooth {3D}-surfaces, partitioned into disjoint classes $$\mli{CL}_j$$, $$j= 1, \ldots, k$$. We show how \emph{optimized diffeomorphic registration} applied to large numbers of pairs $$S,S' \in \mathcal{D}$$ can provide descriptive feature vectors to implement automatic classification on $$\mathcal{D}$$, and generate classifiers invariant by rigid motions in $$\mathbb{R}^3$$. To enhance accuracy of automatic classification, we enrich the smallest classes $$\mli{CL}_j$$ by diffeomorphic interpolation of smooth surfaces between pairs $$S,S' \in \mli{CL}_j$$. We also implement small random perturbations of surfaces $$S\in \mli{CL}_j$$ by random flows of smooth diffeomorphisms $$F_t:\mathbb{R}^3 \to \mathbb{R}^3$$. Finally, we test our automatic classification methods on a cardiology data base of discretized mitral valve surfaces. 

Our work targets automated analysis to quantify the growth dynamics of a population of bacilliform bacteria. We propose an innovative approach to frame-sequence tracking of deformable-cell motion by the automated minimization of a new, specific cost functional. This minimization is implemented by dedicated Boltzmann machines (stochastic recurrent neural networks). Automated detection of cell divisions is handled similarly by successive minimizations of two cost functions, alternating the identification of children pairs and parent identification. We validate the proposed automatic cell tracking algorithm using (i) recordings of simulated cell colonies that closely mimic the growth dynamics of E. coli in microfluidic traps and (ii) real data. On a batch of 1100 simulated image frames, cell registration accuracies per frame ranged from 94.5% to 100%, with a high average. Our initial tests using experimental image sequences (i.e., real data) of E. coli colonies also yield convincing results, with a registration accuracy ranging from 90% to 100%.