Parametric disasters

3 minute read

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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:

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

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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:

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

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

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

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