This thesis work is based on single molecule experiments, performed by hopping measurements via Optical Tweezers at the University of Padua.
Optical Tweezers are an amazing tool which uses light to manipulate biomolecule such as DNA hairpins or proteins; using two counterpropagating laser beams it is possible to trap these molecules and to perform hopping experiments, i.e. to observe the biosensor passing throug its equilibrium configurations.
Hidden Markov Model
For a DNA hairpin, two are the main states of equilibrium, which are the native state (folded configuration) and the unfolded one. In order to find the hidden configurations, an hidden Markov model (HMM) is performed.
The 3 problems in HMM Δ are:
- evaluation problem: how to calculate the probability P(O|Δ) of the observation sequence, indicating how much the HMM Δ parameters affects the sequence O;
- uncovering problem: how to find the sequence of states X ={x_1, x_2, ....., x_T} so that it is more likely to produce the observation sequence O;
- learning problem: how to adjust parameters of Δ such as initial state distribution
$\Pi$ , transition probability matrix$A$ and observation probability matrix$B$ .
To solve these problems, 3 algorithm have been introduced: Forward-backward algorithm, Viterbi algorithm and Expectation Maximization (EM) algortihm (also known as Baum-Welch algorithm).
The picture illustrates the 2-hidden states in an hopping plot.