The Hydrophobic-Polar protein lattice model

1The main assignment¶

The Hydrophobic-Polar (HP) model is a toy model that has been used to show that hydrophobic interactions greatly reduces the number of low-free-energy compact configurations, and therefore greatly contribute to selecting the native conformation. The mandatory part of this assignment is to reproduce (some of) the results reported in Chan & Dill (1989) and Lau & Dill (1989). The most important data to reproduce are the numbers reported in Table I of Chan & Dill (1989) and the distributions of the number of sequences that have $s$ native states (the panels with the $g(s)$ label on the x axis of figures 8, 12, and 13 of Lau & Dill (1989)).

For this assignment you can use the code introduced in the lectures, or write your own code (perhaps derived from mine). Remember: in either case you will have to document your code.

2Possible extensions¶

• Modify the code so that it also computes the compactness ρ. You will then be able to reproduce more interesting figures (figure 2 of Chan & Dill (1989) and more panels from the figures of Lau & Dill (1989)).
• Extend the code so that it supports also the 3D HP model.
• Extend the code so that it can simulate other versions of the HP model (e.g. the one used in Li et al. (1996), where $\epsilon = -2.3$, $\epsilon_{HP} = -1$ and $\epsilon_{PP} = 0$).
• When $N$ is too large, enumerating all the conformations become impossible. Extend the code so that, if $N$ is too big, the code confines the generation of the conformations on the lattice with the algorithm described in Lau & Dill (1989) (pag. 3988, equations (1) and (2)).
• Likewise, for large $N$ it also becomes impossible to take averages over all the sequences. Modify the code so that it averages the interesting properties over $M$ randomly generated sequences instead of all the full sequence space.

3Additional details¶

The compactness is defined for a given chain length as the ratio of the number of topological contacts for a conformation relative to the maximum number of contacts attainable:

See the lecture notes for a way of computing the average compactness of all conformations, or of the conformations with lowest-energy.

Note that figure 8 of Lau & Dill (1989) was generated by considering all conformations, while figures 12, 13, 14 and 15 were generated by analysing only the most compact conformations. Moreover, figures 13, 14 and 15 used a subset of random sequences rather than the full sequence space.