September 9, 2017
by Danilo Roccatano
Scientists have found inspiration for their ideas by observing the nature. The symmetries and regularity in the forms and patterns in Nature have been very likely the stimulus towards the foundation of the Euclidian geometry. In the 14th century, several mathematicians start to find a relation between the geometry, mathematics and natural patterns. More recently, in the 20th century, the French-American mathematician Benoit Mandelbrot (1924–2010) has introduced the concept of fractal geometry to describe forms and phenomena in science and nature that cannot otherwise be classified. It was soon clear that fractal pattern occurs everywhere in Nature. From majestic galaxies to infinitesimal molecular worlds, fractal geometry characterizes the structural assembly of stars and molecules as a consequence of invariance of scale present in these structures
All these studies evidenced that simple but powerful mathematical properties and geometric forms are recurrently used by Nature in the shape of organisms, and other natural objects and environments. You can find the golden ratio, the Fibonacci number, spirals shaped forms, and fractals everywhere even in your home garden.
In this presentation, I made an overview of the different pattern observed in Nature from a mathematical and geometric perspective. In the first, I focused the attention on the mathematical properties of the golden ratio and Fibonacci sequence exploring its relation with the plane and spatial geometry and with the number theory. In the second part, an overview of the concept of fractals with some examples of applications are provided.
The PDF version can be downloaded here: Pattern_in_Nature_8_7_2017
October 20, 2016
by Danilo Roccatano
In my recent public talk at the Gravity Fields Festival 2016, I have shown several models of molecular machines, I have added some of them to this blog with the details on their construction.
The models have been generated using a small script in awk language and represented using the program VMD (http://www.ks.uiuc.edu/Research/vmd/).
The Photosynthetic Apparatus of Rhodospirillum photometricum
The photosynthetic apparatus of purple photosynthetic bacteria is particularly simple and it is located in a specialized membrane system that develops in the bacterial cytoplasm. They are composed of four integral membrane protein complexes: a peripheral LH2 antennae complexes that serve to collect light and transfer the absorbed energy to the second complex, the core-complex, which is constituted of an antenna complex (LH1) associated with the photochemical reaction center (RC). The LH1 serves to funnel the light energy to the RC where a charge separation takes place catalyzing the oxidation of a water soluble carrier. The different crystallographic structures of these components are available in the Protein Data Bank. Based on the supramolecular assembly of these proteins observed using Atomic Force Topographic (Scheuring et al. 2007), it was possible to generate a molecular model of the photosynthetic apparatus of Rsp. photometricum. The model comprises one RC-LH1 core complex surrounded by several LH2 complexes. The model is a chimera of crystallographic structures from different organism. In particular, for the LH2 was used the nonameric Rps. acidophila structure [PDB id: 1KZU (McDermott et al., 1995)].
The LH1-RC complex is based on the recent crystal structure from Thermochromatium tedium [PDB id: 4V8K (Niwa et al. Nature, 2014)]. LH2 and the core complex were first centered and aligned and then translated and rotated to their relative positions using as reference the AFM topography reported by in Scheuring et al. 2007. A DPPC lipid bilayer was also generated using the tools in the VMD program and added to the model by removing the lipid molecules overlapping with the proteins.
Two views of the model are reported below. The complete fly-by animation is here.
The Yeast V-ATPase
Vacuolar-ATPase (V-ATPase) is an ancient enzyme with remarkably diverse functions in eukaryotic organisms. It is used to acidify different organelles and as a proton pump across the plasma membranes of numerous cell types. For its functions, V-ATPases use the energy produced by ATP hydrolysis. It is generally seen as the polar opposite of ATP Synthase that uses the energy from a proton gradient to produce ATP.
The structure of the model of the Yeast V-ATPase was generated by Zhao et al. using electron microscopy add homology modeling (PDB Id: 3J9U, Zhao et al. Nature, 2015). I have just generated a DPPC lipid bilayer using the tools in the VMD program and just added to the model by removing the lipid molecules overlapping with the proteins.
A view of the model is reported below. The complete fly-by animation is here.