## Origin of Life: Evolving Dynamical Systems

I am interested to answer the following question: What were the conditions on the earth around four billion years ago that led to a dynamical phase transition between the inanimate matter and the first forms of life? More generally, I am interested to know what are the general set of conditions required for a dynamical system to undergo a transition of this sort. To answer these questions, I am designing an ecosystem of coevolving computer meta-programs that modify each other. I use the mathematical techniques in recursion theory to analyze different dynamical regimes of such ecosystems.

## Homochirality and Noise Induced Bistability

In organic chemistry, molecules that are not superimposable on their mirror image are called chiral. Many of biological molecules such as sugars and amino acids are chiral, and their chirality plays an important role in structure and function of macromolecules that are built from them in living cells. It is remarkable that despite the diversity of such macromolecules in different living organisms, all of amino acids found in biological systems are $L$-chiral (left-handed) and all of the sugars are $D$-chiral (right-handed). The origin of the homochirality of sugars and amino acids in biology is not well understood.

In 1953, Charles Frank suggested that homochirality is a consequence of autocatalysis. To show this, he constructed a simple model in which $L$ and $D$ form of a chiral molecule are autocatalytically produced from an achiral molecule $A$, and they are consumed by reacting with each other. The latter reaction that has no biological justification is called chiral inhibition. By solving the rate equations for these reactions, Frank showed that there are three fixed points in the dynamics of this system: a fixed point at the racemic state ($50$% $D$ - $50$% $L$) which is unstable and two homochiral states that are stable. However, the existence of these fixed points is dependent on the chiral inhibition mechanism. By removing the chiral inhibition reaction, we lose the stability of the homochiral states. Moreover, the addition of even the slightest amount of non-autocatalytic production of $L$ and $D$ makes the racemic state the global attractor of the system.

In the absent of chiral inhibition, even with a small amount of non-autocatalytic production where the racemic state is the global attractor of the deterministic model, I have shown that homochirality can arise due to the stochastic nature of chemical reactions. In this model, the homochiral states are not fixed points, but when the autocatalysis is the dominant production mechanism, the functional form of the stochasticity of autocatalytic reactions stabilize the homochiral states.

Intuitively, autocatalysis in average, produces a proportionate amount of $L$ and $D$ molecules to the initial condition, leaving the chirality of the system unchanged at the deterministic limit. However, chemical reactions are stochastic, and every time an autocatalysis reaction happens, the chirality of the system changes by a small discrete amount. These zero mean random fluctuations that are caused by the stochasticity of the chemical reaction cause the chirality of the system to slowly drift away from the racemic state. However, the stochastic effect of the autocatalytic reactions is zero at the homochiral states (the autocatalysis produces more of the same chiral molecules that exist in the solution leaving the chirality of the system unchanged), while it is maximum at the racemic state. Therefore, the chirality of the system drifts away from the racemic state and gets trapped in one of the homochiral states.

Additionally, I have shown that when autocatalysis is dominant, coupling many well-mixed system of chemical reactions through diffusion results in synchronization of the homochiral states of all of the systems, indicating that the spatial extension of this model exhibits stable global homochirality

## Velocity Statistics of Dislocations in Plastic Flow

Plastic deformation of crystalline material under stress is known to be a smooth process in macroscopic scales. However, it has been shown that in smaller scales, the deformation of these material is not a smooth function of applied stress. In these systems, topological defects, such as dislocations, have a slowly evolving configurations with rapid collective rearrangements. These collective rearrangements are responsible for the intermittent character of the deformation (see figure below). Discrete dislocation dynamic simulations have shown a power-law scalings for the probability density function of velocity of the dislocations. The self similar characteristics of the velocity distribution is often attributed to the collective interaction of dislocations that leads to the intermittent dynamics, but the mathematical connection between these power laws and the intermittency of the deformation has not been established.

We have worked out the probability distribution of velocities of dislocations using methods from stochastic calculus by mapping the problem of motion of dislocations to Dyson's model of two dimensional electron gas confined to one dimension. Dyson's model was originally developed to calculate the statistical properties of energy levels of heavy nuclei. We have shown that the power-law distribution of velocities of dislocations is not a signature of collective dynamics. However, the deviation of the exponent of this power law from the exponent expected in the absence of collective dynamics can be used as a measure of collective interactions.