Bayesian inference of a non-local proliferation model.

From a systems biology perspective, the majority of cancer models, although interesting and providing a qualitative explanation of some problems, have a major disadvantage in that they usually miss a genuine connection with experimental data. Having this in mind, in this paper, we aim at contributing to the improvement of many cancer models which contain a proliferation term. To this end, we propose a new non-local model of cell proliferation. We select data that are suitable to perform Bayesian inference for unknown parameters and we provide a discussion on the range of applicability of the model. Furthermore, we provide proof of the stability of posterior distributions in total variation norm which exploits the theory of spaces of measures equipped with the weighted flat norm. In a companion paper, we provide detailed proof of the well-posedness of the problem and we investigate the convergence of the escalator boxcar train (EBT) algorithm applied to solve the equation.

A Geometric Definition of Short to Medium Range Hydrogen-Mediated Interactions in Proteins.

We present a method to rapidly identify hydrogen-mediated interactions in proteins (e.g., hydrogen bonds, hydrogen bonds, water-mediated hydrogen bonds, salt bridges, and aromatic π-hydrogen interactions) through heavy atom geometry alone, that is, without needing to explicitly determine hydrogen atom positions using either experimental or theoretical methods. By including specific real (or virtual) partner atoms as defined by the atom type of both the donor and acceptor heavy atoms, a set of unique angles can be rapidly calculated. By comparing the distance between the donor and the acceptor and these unique angles to the statistical preferences observed in the Protein Data Bank (PDB), we were able to identify a set of conserved geometries (15 for donor atoms and 7 for acceptor atoms) for hydrogen-mediated interactions in proteins. This set of identified interactions includes every polar atom type present in the Protein Data Bank except OE1 (glutamate/glutamine sidechain) and a clear geometric preference for the methionine sulfur atom (SD) to act as a hydrogen bond acceptor. This method could be readily applied to protein design efforts.

Self-analysis of repeat proteins reveals evolutionarily conserved patterns.

BACKGROUND: Protein repeats can confound sequence analyses because the repetitiveness of their amino acid sequences lead to difficulties in identifying whether similar repeats are due to convergent or divergent evolution. We noted that the patterns derived from traditional "dot plot" protein sequence self-similarity analysis tended to be conserved in sets of related repeat proteins and this conservation could be quantitated using a Jaccard metric. RESULTS: Comparison of these dot plots obviated the issues due to sequence similarity for analysis of repeat proteins. A high Jaccard similarity score was suggestive of a conserved relationship between closely related repeat proteins. The dot plot patterns decayed quickly in the absence of selective pressure with an expected loss of 50% of Jaccard similarity due to a loss of 8.2% sequence identity. To perform method testing, we assembled a standard set of 79 repeat proteins representing all the subgroups in RepeatsDB. Comparison of known repeat and non-repeat proteins from the PDB suggested that the information content in dot plots could be used to identify repeat proteins from pure sequence with no requirement for structural information. Analysis of the UniRef90 database suggested that 16.9% of all known proteins could be classified as repeat proteins. These 13.3 million putative repeat protein chains were clustered and a significant amount (82.9%) of clusters containing between 5 and 200 members were of a single functional type. CONCLUSIONS: Dot plot analysis of repeat proteins attempts to obviate issues that arise due to the sequence degeneracy of repeat proteins. These results show that this kind of analysis can efficiently be applied to analyze repeat proteins on a large scale.

Sensitivity equations for measure-valued solutions to transport equations.

We consider the following transport equation in the space of bounded, nonnegative Radon measures $\mathcal{M}^+(\mathbb{R})$:$$ ∂_t\mu_t + ∂_x(v(x) \mu_t) = 0.$$We study the sensitivity of the solution $\mu_t$ with respect to a perturbation in the vector field, $v(x)$. In particular, we replace the vector field $v$ with a perturbation of the form $v^h = v_0(x) + h v_1(x)$ and let $\mu^h_t$ be the solution of $$ ∂_t\mu^h_t + ∂_x(v^h(x)\mu^h_t) = 0.$$We derive a partial differential equation that is satisfied by the derivative of $\mu^h_t$ with respect to $h$, $∂artial_h(\mu_t^h)$. We show that this equation has a unique very weak solution on the space $Z$, being the closure of $\mathcal{M}(\mathbb{R})$ endowed with the dual norm $(C^{1,\alpha}(\mathbb{R}))^*$. We also extend the result to the nonlinear case where the vector field depends on $\mu_t$, i.e., $v=v[\mu_t](x)$.