# SMC in Biological Problems

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## Molecular Simulation

**Protein folding problem** is the most well-known open problem in structural biology and biophysics, which one is required to predict the three-dimensional fold shape of a protein molecule based only on its sequence of amino acids. One simple model to imitate the real protein folding process in a 2-D or 3-D lattice-bead model. Compared to SAW model, they use a more complicated function for interactive energy between the “beads” on the lattice. In protein language, these beads correspond to amino acid residues, which are of 20 different types.

It is of interest to find out its most “favorable fold” in a 2-D lattice space. To define what is meant by a favorable fold, we define an energy function for each configuration of the bead sequence:

\[U_n(\mathbf x_n)=-\sum\limits_{\vert i-j\vert>1}c(x_i,x_j)\]where $c(x_i,x_j)=1$ if $x_i$ and $x_j$ are both black, and $c(x_i,x_j)=0$ otherwise.

With a target distribution,

\[\pi_n(\mathbf x_n)\propto exp\{-U_n(\mathbf x_n)/2\}\]Then apply the SIS, with resampling, rejection control, and one-step-look-ahead.

## References

Liu, Jun S. Monte Carlo strategies in scientific computing. Springer Science & Business Media, 2008.