# Helical Shifts, Turns and Repeats

I have been trying to get my head around something which is probably quite simple today - and was thinking that someone on this forum could probably give me an answer!

The question is how a shift along a helix translates into a rotation around the helix. In the Sachse et al. TMV paper a formula is used whereby (as i understand it) a shift from one turn to the next (approx 23 angstroms) translates into a rotation of 360 degrees. Surely this means that after a shift of 23 angstroms I should be looking at exactly the same thing? However I believe there to be a non integer number of subunits per turn in TMV, so that a shift of one turn cannot be exactly the same thing. I've seen references to the repeat distance in TMV being 3 turns - so shouldn't this mean that the angle should change 360 degrees over 3 turns?

On the over hand, the idea that the shift of one rise (~1.4 angstroms) corresponding to a rotation of ~22 degrees does make sense to me, but if this was true then the structure should be repeating after one turn?

I figure i must be missing something here that consolidates the two? Anyone care to tell me what? :)

Thanks,

Tim

## the idea that the shift of

the idea that the shift of one rise (~1.4 angstroms) corresponding to a rotation of ~22 degrees does make sense to me

Right, this is what I would start from.

but if this was true then the structure should be repeating after one turn?

If you chose to position helical subunit #1 at phi = 0.0, so that the next subunit (#2) is at phi = 22.03 and z += 1.4A, then "travel" along the helix a complete turn of the helix (i.e. phi += 360 degrees), you will "be" between subunits #17 and #18 (if I'm counting right), and at z 23A away from your starting point, subunit #1. "After one turn" of the helix therefore, the structure does not repeat.

is a shift of 0.7 angstroms along the helical axis equivalent to an 11 degree rotation?

Yes.

Does it only make sense to shift in whole helical rise units?

If for example you are "walking along" your filament and extracting segments to insert into a 3D reconstruction, then you would start at an arbitrary point along the filament (presumably near one end), call that subunit #1 and assign it phi=0.0, then move 1.4A along the filament, and call that subunit #2 and assign it phi=22.03, and so on... Therefore in that context it would only "make sense" to shift in whole helical rise units.

Hope this helps!

## Hey Alexis, I'm in the pub

In reply to the idea that the shift of by Alexis

Hey Alexis,

I'm in the pub right now so may need to re read this and change my mind tomorrow. However, above you say that following a 360 degree rotation, you end up between subunits - but you start on a subunit, so doesn't that mean something has gone wrong?

I'll re read this tomorrow!

Tim

## Ahhh... but I didn't say

In reply to Hey Alexis, I'm in the pub by timgrant

Ahhh... but I didn't say "rotate by 360", I said "travel" along the helix a complete turn of the helix. This means a rotation of 360 plus a shift.

Maybe this will make more sense after another pint?

## Yes, probably. Particularly

In reply to Ahhh... but I didn't say by Alexis

Yes, probably. Particularly at this point!

Let me ask it another way. If I start at an euler angle of 0, 90, 0 - and shift by exactly one turn. What is my new euler angle?

## If I start at an euler angle

In reply to Yes, probably. Particularly by timgrant

If I start at an euler angle of 0, 90, 0 - and shift by exactly one turn. What is my new euler angle?

If you project TMV at 0,90,0, then shift the projection by 23A along the helical axis, and you then wanted to, say, backproject this "new" projection into the 3D coherently, you'd have to use angles 0,90,gamma1 or 0,90,gamma2, and shift them by z1 or z2 respectively, where:

gamma1 = 16*22.03 = 352.48 degrees

gamma2 = 17*22.03 = 374.51 degrees

z1 = -[360-(16*22.03)]*(1.408/22.03) = (360-352.48)*0.0639 = -0.48 A

z2 = -[360-(17*22.03)]*(1.408/22.03) = (360-374.51)*0.0639 = 0.92 A

The above come with the caveat that I'm in a rush to catch a train, so I reserve the right to edit if I made mistakes.

## Hope you got your

In reply to If I start at an euler angle by Alexis

Hope you got your train...

Above you said "is a shift of 0.7 angstroms along the helical axis equivalent to an 11 degree rotation?

Yes."

Does this not mean that a shift == a rotation, without caveats? So if i start a 0,90,0 and shift by one turn, there is an angle of that image without a shift? If this is not the case then aren't you stating that you can only shift in "helical rise units"?

## When I said that "a shift of

In reply to Hope you got your by timgrant

When I said that "a shift of 0.7 angstroms along the helical axis is equivalent to an 11 degree rotation", I was referring the the helix itself, the continuous spiral Chuck talks about in his other post.

However, when you're doing 3D reconstruction, image alignment etc, you have to worry about where the helical subunits fall on that spiral, and in that context the type of calculations I was illustrating in my previous post, and the integer k that Chuck mentions in his post become necessary.

HTH