### Abstract:

The restoring forces (kinetic variables) at a point and the deformations and strains (kinematic variables) are related to each other through the Constitutive Law. The Constitutive Law is most often an empirical relationship rather than one derived from first principles. It is an essential relation to solve equations of motion of the continuum system under consideration. In this thesis, we consider bio-filaments that are represented by rod model equations, which are useful in predicting the deformations under the action of different internal or external forces or moments. Many functions of micro-scale bio-filaments such as DNA molecules are dependent on the deformations of the filaments. Hence determination of the constitutive law is an important step. Though there are techniques available to determine the constitutive law from the force versus deformation data of macro-scale materials, the major challenge in estimation of constitutive law in case of micro filaments is due to the presence of non deterministic or stochastic thermal forces that arise as a result of bombardment from the neighboring solvent molecules. The exact magnitude of force that may have caused the deformation is uncertain, only statistical properties about the forces, arising out of every collision are known and has a Gaussian distribution, whose population mean is dependent on temperature, Boltzmann’s constant and Stoke’s drag coefficient.
Studying an ensemble of filaments instead of a single filament allows exploitation of the known statistical facts leading to the plausibility of two different approaches to estimate the constitutive law under the action of Langevin forces. These methods differ in how the mean is computed. Through either of the approaches there are three main aspects that we analyze. Firstly, what is the least number of filaments that we should analyze so as to obtain a reasonably good estimate? Secondly, is there any need for application of any external known force on the filament along with the exposure to the random forces? If so, what should be the range of magnitude of this force and how does accuracy vary with the magnitude of this external force? Lastly, what is likely the estimated error for different cases through both the approaches? The answer to the above questions will hopefully serve as a guide for experimentalists to work on the algorithms suggested in the thesis for the estimation of constitutive law of micro-scale biofilaments.