# Parametric disasters

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I recently got a new computer and I have been (slowly) going through my old files to try to maintain a little order around here. So, while I was doing the ol’ twenty-first century upkeep, I stumbled across a file called “nope_nope_nope.mov”. Here’s what was in that file:

There’s another one called “strand_fail.mov” that looks like this:

What each of these movies show is a failed initial attempt at extending the parametric bundle design methods used in Rosetta. A little while ago, Vikram Mulligan and I sat down to think about what would be needed to describe a β barrel. We started off noting that a strand could be thought of as a helix in which every residue flips 180º, and that we would need to describe a “squishing” parameter to describe non-circular barrel systems.

Vikram recently moved on to the Flatiron Institute in New York, but this was a fun little project that kind of went nowhere so I thought I would share it here. Also, he took really detailed notes and made a lot of pretty pictures and it would be a shame for the world to not have them.

## First, some background

We have had some success designing helical bundles from parametric equations first developed by Francis Crick. These equations enable us to calculate the coordinates of each helical residue’s α carbon using descriptors with clear physical meanings, which allows us to specify geometric properties or requirements of a helical bundle and quickly trace out a backbone based on those requirements.

The equations are:

\begin{align*} X(t) & = R_0 \cos(\omega_0 t + {\phi_0}^\prime) + R_1 \cos( \omega_0 t + {\phi_0}^\prime) \cos(\omega_1 t + \phi_1) \
&\qquad - R_1 \cos(\alpha) \sin(\omega_0 t + {\phi_0}^\prime) \sin(\omega_1 t + \phi_1) \\ \
Y(t) & = R_0 \sin(\omega_0 t + {\phi_0}^\prime) + R_1 \sin(\omega_0 t + {\phi_0}^\prime) \cos(\omega_1 t + \phi_1) \
&\qquad + R_1 \cos(\alpha) \cos(\omega_0 t + {\phi_0}^\prime) \sin(\omega_1 t + \phi_1) \\ \
Z(t) & = (\omega_0 R_0 / \tan(\alpha)) t - R_1 \sin(\alpha) \sin(\omega_1 t + \phi_1) + \Delta z \end{align*}

Grigoryan and DeGrado described the parameters as:

• Superhelical radius, $R_0$
• Superhelical frequency/twist, $\omega_0$
• Superhelical phase, $\phi_0$
• Helical radius, $R_1$
• Helical frequency/twist, $\omega_1$
• Helical phase, $\phi_1$
• Offset along the $z$ axis, $\Delta z$
• Pitch angle, $\alpha = \arcsin(R_0 \omega_0 / d)$, where $d$ is the distance between residues
• Superhelical phase (decoupled from the $z$ offset), ${\phi_0}^\prime = \phi_0 + \Delta z \tan(\alpha) / R_0$

Calculations are made a little simpler by holding $R_1$ and $d$ fixed at the values for ideal helices (1.51 and 2.26 Å), and by distributing helices evenly about the $z$ axis, which gives $\phi_0$ legal values of $\{0, 2\pi/n, 4\pi/n… 2(n-1) \pi/n\}$.

## How can we incorporate a “squish”?

One of the simplifying assumptions that is used in the equations above is that all parameters are constants. But when we want to model barrels with non-circular cross sections, that simplification leaves the room. In this case, the superhelical radius will depend on a new parameter, $\epsilon$, the eccentricity of the elipse. Here’s a photo of a white board where we derived this:

With the new $R_0$ term, we can stretch barrels in one dimension and evaluate metrics like hydrogen bonding along the way:

Here’s what it looks like normal to the barrel axis:

And then, finally, one can generate parameters and use some metric to perform monte carlo sampling of a parametrically-designed β barrel.

Thanks for all the fun times and good work, Vikram!

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