We show that aggregation plays a major role in seeded growth of protein crystals. The seeded batch approach provides the opportunity to set the starting conditions for protein crystallization by adding a defined amount of wellcharacterized seed particles. The experimental observations for tetragonal hen egg-white lysozyme (LSZ) confirm the concept of the oriented aggregation of larger building blocks to form a protein crystal. It was shown that the aggregation of the seed particles/bioconjugates is advantageous for the product quality in terms of larger and more defined LSZ crystals and in terms of accelerated reaction kinetics. We present a population balance (PB) model for the seeded batch crystallization of LSZ considering the aggregation of growth units to form protein crystals. For the modeling of crystal growth, evolving particle size distributions (PSDs) of agglomerating LSZ molecules were measured by dynamic light scattering (DLS). Moreover, the aggregation of seed particles in LSZ solutions under crystallization conditions was investigated by DLS. In line with our expectations, the number of seeds was found to be important as it strongly affects the collision frequency in the aggregation term of our PB model. Finally, the applied model gives trends of the supersaturation depletion curves and orders of magnitude of the measured CSDs in particle size correctly, ranging from only a few nanometers up to micrometer-sized particles/crystals. Thus, by the combination of PB modeling and experimentally determined crystallization parameters, insights into the crystal formation mechanism were obtained. To the best of our knowledge, this is the first attempt to model growth within a crystal population by an aggregation mechanism induced by seeding with foreign particles.
Motivated by research on gender identity norms and the distribution of the woman's share in a couple's total labor income, we consider functional additive regression models for probability density functions as responses with scalar covariates. To preserve nonnegativity and integration to one under summation and scalar multiplication, we formulate the model for densities in a Bayes Hilbert space with respect to an arbitrary finite measure. This enables us to not only consider continuous densities, but also, e.g., discrete or mixed densities. Mixed densities occur in our application, as the woman's income share is a continuous variable having discrete point masses at zero and one for single-earner couples. We discuss interpretation of effect functions in our model in terms of odds-ratios. Estimation is based on a gradient boosting algorithm that allows for a potentially large number of flexible covariate effects. We show how to handle the challenge of estimation for mixed densities within our framework using an orthogonal decomposition. Applying this approach to data from the German Socio-Economic Panel Study (SOEP) shows a more symmetric distribution in East German than in West German couples after reunification, differences between couples with and without minor children, as well as trends over time.
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