Scaling in Colloidal and Biological Networks

Folding-unfolding asymmetry and a RetroFold computational algorithm

We treat protein folding as molecular self-assembly, while unfolding is viewed as disassembly. Fracture is typically a much faster process than self-assembly. Self-assembly is often an exponentially decaying process, since energy relaxes due to dissipation, while fracture is a constant-rate process as the driving force is opposed by damping. Protein folding takes two orders of magnitude longer than unfolding. We suggest a mathematical transformation of variables, which makes it possible to view self-assembly as time-reversed disassembly, thus folding can be studied as reversed unfolding. We investigate the molecular dynamics modelling of folding and unfolding of the short Trp-cage protein. Folding time constitutes about 800 ns, while unfolding (denaturation) takes only about 5.0 ns and, therefore, fewer computational resources are needed for its simulation. This RetroFold approach can be used for the design of a novel computation algorithm, which, while approximate, is less time-consuming than traditional folding algorithms.

Topological bio-scaling analysis as a universal measure of protein folding

Scaling relationships for polymeric molecules establish power law dependencies between the number of molecular segments and linear dimensions, such as the radius of gyration. They also establish spatial topological properties of the chains, such as their dimensionality. In the spatial domain, power exponents α = 1 (linear stretched molecule), α = 0.5 (the ideal chain) and α = 0.333 (compact globule) are significant. During folding, the molecule undergoes the transition from the one-dimensional linear to the three-dimensional globular state within a very short time. However, intermediate states with fractional dimensions can be stabilized by modifying the solubility (e.g. by changing the solution temperature). Topological properties, such as dimension, correlate with the interaction energy, and thus by tuning the solubility one can control molecular interaction. We investigate these correlations using the example of a well-studied short model of Trp-cage protein. The radius of gyration is used to estimate the fractal dimension of the chain at different stages of folding. It is expected that the same principle is applicable to much larger molecules and that topological (dimensional) characteristics can provide insights into molecular folding and interactions.

Friction in Myocardial Anoxia Leads to Negative Excess Entropy Production, Self-Organization, and Dissipative Structures

Contraction of the heart is caused by actin filaments sliding along myosin filaments. This generates a frictional force inducing wear of the contractile apparatus. We postulated that this process could be exacerbated when the heart was submitted to severe anoxia. Anoxia induced dramatic abnormalities in the molecular properties of actin-myosin crossbridges. We applied the formalism of far-from-equilibrium thermodynamics to the left ventricular papillary muscles (LVPMs) of mammalian rat hearts which had been subjected to a prolonged anoxia (3 h). We showed that when subjected to prolonged anoxia, the heart operated far-from-equilibrium as evidenced by the non-linearity between thermodynamic force (F/T: Frictional force/Kelvin temperature) and thermodynamic flow (v0: myofilament sliding velocity). The rate of entropy production (EPR) was the product of (F/T) and v0. The excess entropy production (EEP) was equal to ∂δ2S∂t = ∂FTδvo; (S: entropy). The tribological system remained stable when EEP was positive and became unstable when EEP became negative, thus characterizing instability of the system and reflecting the occurrence of self-organization and possibly dissipative structures. After 3 h anoxia, re-oxygenation induced significant reversibility. About 20% of the myosin heads did not recover despite re-oxygenation. These results may be of importance in the context of heart transplantation where the delay between the time of sampling from the donor and the time of the graft installation in the recipient should be as short as possible.

Evolutionary Advantages of Stimulus-Driven EEG Phase Transitions in the Upper Cortical Layers

Spatio-temporal brain activity monitored by EEG recordings in humans and other mammals has identified beta/gamma oscillations (20-80 Hz), which are self-organized into spatio-temporal structures recurring at theta/alpha rates (4-12 Hz). These structures have statistically significant correlations with sensory stimuli and reinforcement contingencies perceived by the subject. The repeated collapse of self-organized structures at theta/alpha rates generates laterally propagating phase gradients (phase cones), ignited at some specific location of the cortical sheet. Phase cones have been interpreted as neural signatures of transient perceptual experiences according to the cinematic theory of brain dynamics. The rapid expansion of essentially isotropic phase cones is consistent with the propagation of perceptual broadcasts postulated by Global Workspace Theory (GWT). What is the evolutionary advantage of brains operating with repeatedly collapsing dynamics? This question is answered using thermodynamic concepts. According to neuropercolation theory, waking brains are described as non-equilibrium thermodynamic systems operating at the edge of criticality, undergoing repeated phase transitions. This work analyzes the role of long-range axonal connections and metabolic processes in the regulation of critical brain dynamics. Historically, the near 10 Hz domain has been associated with conscious sensory integration, cortical "ignitions" linked to conscious visual perception, and conscious experiences. We can therefore combine a very large body of experimental evidence and theory, including graph theory, neuropercolation, and GWT. This cortical operating style may optimize a tradeoff between rapid adaptation to novelty vs. stable and widespread self-organization, therefore resulting in significant Darwinian benefits.