## Final Activity Report Summary - IIF-NATIRAM (Dynamical Aspects of Cytoskeletal Motility and Associated Studies in Critical Phenomena)

We elaborated a theoretical study on the dynamical aspects of chosen biological systems using techniques from statistical physics, non-linear dynamics and numerical simulation. The considered biological systems were active ones, i.e. stochastic non-equilibrium systems driven by thermal and cytoskeletal noise.

We mostly focussed on three such systems, namely molecular motor driven systems, membrane fluctuations in immunological systems and bio-polymeric dynamics. Using techniques that were developed in these studies and in specific cases in order to test the veracity of developed techniques we extended our studies in complex systems to the study of confined quantum dots and developmental economics.

Regarding molecular motors, which were groups of proteins converting chemical energy to mechanical energy, our attention was twofold. Firstly, we aimed to understand the biomechanical nature of this energy conversion process and, secondly to study the dynamics of hair-like appendages called flagella or cilia which were responsible for the fertilisation process in mammals as well as in their hearing mechanism. Using relevant stochastic ratchet models, we showed that the underlying mechanism of energy conversion could be explained through stochastic resonance, a technical name given to a process meaning, in simple terms, a bulk energy transfer in between two modes at a certain specific value of the noise strength. Regarding the flagellar issue, we exactly solved the non-linear problem with most agreeable results when compared to experimental observations.

The immunological problem that we studied during the course of the fellowship concerned the dynamics of the TCR-APC bond formation that was generally referred to as immunological synapse in biological parlance. Using a linearised stochastic model of membrane fluctuations, we predicted that the sizes of TCR-APC bonds, the chief architect in the immunological defence mechanism of vertebrates, were of the order of 0.5 microns while the time of their contact could be up to a few seconds. As an extension of the same theory that exploited the statistical mechanics theory of persistence, we also calculated the average time for which a stock price could be probabilistically expected to lie in between an upper and a lower limit. Such results could prove crucial to the values projected during stock market calculations.

Our interest in bio-polymeric problems led us to study model systems involving semi-flexible polymers subjected to varying boundary conditions. We showed that the presence of hysteresis in semi-flexible polymers could be the explanation for the chaotic trajectories observed in F-actin polymers. Work was also ongoing, by the time of the project completion, to calculate the strength of interaction of such polymers against a semi-infinite wall, an allusion to a different boundary condition that was sometimes seen in experimental setups.

To summarise, we set out to solve certain specific problems concerning molecular biology using techniques from theoretical physics and eventually ended up achieving much more. During the process we extended the applications of our methods to problems in varying fields, like developmental economics, where we tackled the key problem of pinpointing the ideal income distribution function for a developing economy that would satisfy all three economics' axioms. Related works, involving quantitative predictions of the poverty functions were ongoing and served once again as a testimony of the strength of the statistical methods that we developed in context.

As to the most important scientific achievement made during my fellowship term, I would not refer to any individual project accomplished but rather to the fact that we were able to develop and successfully apply our techniques to a range of diverse fields, spanning biological physics, statistical physics and, ultimately, economics and stock market predictions with a remarkable success when compared with real data. In other words, our greatest accomplishment was the successful study of complex systems as a whole.

We mostly focussed on three such systems, namely molecular motor driven systems, membrane fluctuations in immunological systems and bio-polymeric dynamics. Using techniques that were developed in these studies and in specific cases in order to test the veracity of developed techniques we extended our studies in complex systems to the study of confined quantum dots and developmental economics.

Regarding molecular motors, which were groups of proteins converting chemical energy to mechanical energy, our attention was twofold. Firstly, we aimed to understand the biomechanical nature of this energy conversion process and, secondly to study the dynamics of hair-like appendages called flagella or cilia which were responsible for the fertilisation process in mammals as well as in their hearing mechanism. Using relevant stochastic ratchet models, we showed that the underlying mechanism of energy conversion could be explained through stochastic resonance, a technical name given to a process meaning, in simple terms, a bulk energy transfer in between two modes at a certain specific value of the noise strength. Regarding the flagellar issue, we exactly solved the non-linear problem with most agreeable results when compared to experimental observations.

The immunological problem that we studied during the course of the fellowship concerned the dynamics of the TCR-APC bond formation that was generally referred to as immunological synapse in biological parlance. Using a linearised stochastic model of membrane fluctuations, we predicted that the sizes of TCR-APC bonds, the chief architect in the immunological defence mechanism of vertebrates, were of the order of 0.5 microns while the time of their contact could be up to a few seconds. As an extension of the same theory that exploited the statistical mechanics theory of persistence, we also calculated the average time for which a stock price could be probabilistically expected to lie in between an upper and a lower limit. Such results could prove crucial to the values projected during stock market calculations.

Our interest in bio-polymeric problems led us to study model systems involving semi-flexible polymers subjected to varying boundary conditions. We showed that the presence of hysteresis in semi-flexible polymers could be the explanation for the chaotic trajectories observed in F-actin polymers. Work was also ongoing, by the time of the project completion, to calculate the strength of interaction of such polymers against a semi-infinite wall, an allusion to a different boundary condition that was sometimes seen in experimental setups.

To summarise, we set out to solve certain specific problems concerning molecular biology using techniques from theoretical physics and eventually ended up achieving much more. During the process we extended the applications of our methods to problems in varying fields, like developmental economics, where we tackled the key problem of pinpointing the ideal income distribution function for a developing economy that would satisfy all three economics' axioms. Related works, involving quantitative predictions of the poverty functions were ongoing and served once again as a testimony of the strength of the statistical methods that we developed in context.

As to the most important scientific achievement made during my fellowship term, I would not refer to any individual project accomplished but rather to the fact that we were able to develop and successfully apply our techniques to a range of diverse fields, spanning biological physics, statistical physics and, ultimately, economics and stock market predictions with a remarkable success when compared with real data. In other words, our greatest accomplishment was the successful study of complex systems as a whole.