Johannes W. R. Martini

Epistasis and covariance: how gene interaction translates into genomic relationship

Key messageModels based on additive marker effects and on epistatic interactions can be translated into genomic relationship models. This equivalence allows to perform predictions based on complex gene interaction models and reduces computational effort significantly.AbstractIn the theory of genome-assisted prediction, the equivalence of a linear model based on independent and identically normally distributed marker effects and a model based on multivariate Gaussian distributed breeding values with genomic relationship as covariance matrix is well known. In this work, we demonstrate equivalences of marker effect models incorporating epistatic interactions and corresponding mixed models based on relationship matrices and show how to exploit these equivalences computationally for genome-assisted prediction. In particular, we show how models with epistatic interactions of higher order (e.g., three-factor interactions) translate into linear models with certain covariance matrices and demonstrate how to construct epistatic relationship matrices for the linear mixed model, if we restrict the model to interactions defined a priori. We illustrate the practical relevance of our results with a publicly available data set on grain yield of wheat lines growing in four different environments. For this purpose, we select important interactions in one environment and use this knowledge on the network of interactions to increase predictive ability of grain yield under other environmental conditions. Our results provide a guide for building relationship matrices based on knowledge on the structure of trait-related gene networks.

A mathematical view on the decoupled sites representation

The decoupled sites representation (DSR) is a theoretical instrument which allows to regard complex pH titration curves of biomolecules with several interacting proton binding sites as composition of isolated, non-interacting sites, each with a standard Henderson–Hasselbalch titration curve. In this work, we present the mathematical framework in which the DSR is embedded and give mathematical proofs for several statements in the periphery of the DSR. These proofs also identify exceptions. To apply the DSR to any molecule, it is necessary to extend the set of binding energies from

On the interaction of two different types of ligands binding to the same molecule part I: basics and the transfer of the decoupled sites representation to systems with n and one binding sites

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Contrasting biodiversity–ecosystem functioning relationships in phylogenetic and functional diversity

to a stripe within

A derivation of the Grand Canonical Partition Function for systems with a finite number of binding sites using a Markov chain model for the dynamics of single molecules

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The effect of the H−1 scaling factors τ and ω on the structure of H in the single-step procedure

. An important observation in this context is that even positive interaction energies (repulsion) between the binding sites will not guarantee real binding energies in the decoupled system, at least if the molecule has more than four proton binding sites. Moreover, we show that for a given overall titration curve it is not only possible to find a corresponding system with an interaction energy of zero but with any arbitrary fix interaction energy. This result also effects practical work as it shows that for any given titration curve, there is an infinite number of corresponding hypothetical molecules. Furthermore, this implies that—using a common definition of cooperative binding on the level of interaction energies—a meaningful measure of cooperativity between the binding sites cannot be defined solely on the basis of the overall titration. Consequently, all measures of cooperativity based on the overall binding curve do not measure the type of cooperativity commonly defined on the basis of interaction energies. Understanding the DSR mathematically provides the basis of transferring the DSR to biomolecules with different types of interacting ligands, such as protons and electrons, which play an important role within electron transport chains like in photosynthesis.