Helix Coil Transition

1. The helix coil transition is not a real phase transition but is a cooperative transition between a molecule being in a mostly helical state, and a mostly coiled state. This is discussed in Nelson Chapter 9. It is often described by the Zimm Bragg Links to an external site. model which is equivalent to the one dimensional Ising model. Using the attached file, we can simulate this model. In the ising model, the energy depends on the spin coupling $J$ $J$ , and magnetic field $h$ $h$

$E = \sum_{i = 1}^{n} [- J s_{i} s_{i + 1} - h s_{i}]$ $E = \sum_{i=1}^n [-J s_i s_{i+1} - h s_i]$

where $s_{i} = \pm 1$ $s_i = \pm 1$ . Because the partition function involves $J / (k_{B} T)$ $J/(k_B T)$ , we can rescale our units so that $T \to k_{B} T / J$ $T \rightarrow k_B T/J$ , $h \to h / J$ $h \rightarrow h/J$ and eliminate $J$ $J$ .

Use hw7/helix_coil.py to find the magnetization as a function of $T$ $T$ and $h$ $h$ (the mac version will only be useful for old systems). Compare this with the theoretical result for the Ising model given by Nelson "Your Turn 9H" Links to an external site. and eqn (9.25) or by looking here. Links to an external site.(Note that their expression for the magnetization can be further simplified algebraically to yield a simpler formula.), or here on their page 129 Links to an external site..

To find the relationship between the parameters $J$ $J$ and $h$ $h$ , you can use the discussion in Nelson. He has a parameter that is the propensity to form an alpha helix, that he calls $α$ $\alpha$ . This is related to $h$ $h$ in the above formula as $h = k_{B} T α$ $h = k_B T \alpha$ . The propensity to form a helix depends on temperature. Around the temperature of the transition, he writes

$k_{B} T α = \frac{Δ E_{b o n d}}{2 T_{m}} (T - T_{m})$ $k_B T \alpha = \frac{\Delta E_{bond}}{2 T_m}(T-T_m)$
So that $α$ $\alpha$ passes through zero at $T = T_{m}$ $T=T_m$ . So there are two independent parameters you need to consider $T_{m}$ $T_m$ and $A \equiv Δ E_{b o n d} / 2 T_{m}$ $A \equiv \Delta E_{bond}/2 T_m$ . $T_{m}$ $T_m$ is the absolute position on the temperature axis of where the transition takes place, and $A$ $A$ is the scale relative to $T_{m}$ $T_m$ .

The other parameter $γ$ $\gamma$ that Nelson uses, is related to $J$ $J$ in the Ising model by $J = k_{B} T γ$ $J = k_B T \gamma$ . This defines how cooperative the transition is.

Looking at the above theoretical result for the magnetization of the Ising model mentioned above Links to an external site., try to understand the optical rotation $θ$ $\theta$ , (i.e. helix fraction) as as the temperature is changed for a fixes $γ$ $\gamma$ . In other words, suppose the cooperativity is small, $γ ≪ 1$ $\gamma \ll 1$ , you can examine what happens as you change the temperature. Changing the temperature around $T_{m}$ $T_m$ is the same as changing $α$ $\alpha$ . So as $α$ $\alpha$ goes from negative to positive values, plot $θ$ $\theta$ . Now do the same for a larger value of $γ$ $\gamma$ . What qualitatively can you say about the shape of the curves? How does this compare with the simulation?

[Note: if you examine the code, you will see that there is a factor of two difference in the temperature (or equivalently the Hamiltonian) used there and the definition used in the analytical formulas. So it is important to take this into account when doing a comparison.]

2. Biological questions:

Give examples of two proteins that are primarily composed of alpha helices. What are their functions?
What is a "coiled coil"?
In an alpha helix, are some amino acids more commonly seen than others? Give examples of some that are frequently found, and others that are unlikely to be in such a secondary structure. For some of those unlikely to be in alpha helices, give reasons why they are disfavored.