Hi Reggie,

  I measure the volume inside the capsid of bacteriophage P22 from Electron Microscopy Databank entry 1220 as about 77,000 cubic nanometers.  Here’s how I calculated it in Chimera.  The idea is to fit a surface that is something between an icosahedron and a sphere to the inside of the capsid then measure the volume within that surface.  The Chimera Icosahedron Surface tool (menu Tools / Higher-Order Structure) can make this surface.  But the problem is that the EMD 1220 density map is not in a standard icosahedral orientation.  The closest orientation is 2-fold symmetry axis along x, 3-fold symmetry axis along z (called 2n3r in Chimera).  That orientation looks to be about 5 degrees out of alignment with the virus.  So I set the radius to 320 (about the size of the capsid), sphere factor to 0.5 to approximate the spherical shape of the virus, and subdivision factor to 100.  Then I created a density map that is ones inside this surface and zeros outside with the command “mask ones #1”  (the icosahedron surface is model #1, as shown in the Model Panel dialog, Favorites menu).  Then I fit this icosahedron map to EMD 1220 to get the correct alignment using the Fit in Map tool (menu Tools / Volume Data / Fit in Map, fit #2 in #0).  I judge the fit is good by seeing that the virus capsid proteins stick out uniformly from the surface of the fit mask map.  Then I align the original icosahedron surface to the fit mask using “matrixcopy #1 #2”.  Then I adjust the icosahedron surface radius and sphere factor (285 and 0.54) to visually match the inner boundary of the EMD 1220 capsid.  To see that I clip the EMD 1220 map in half using Per-Model Clipping (menu Tools / Depiction).  Then I measure the enclosed volume of that aligned icosahedron surface using the Measure Volume and Area tool (menu Tools / Volume Data), and I get 77 million cubic Angstroms, or 77,000 cubic nanometers.  See attached picture with aligned icosahedron surface shown transparent and brown.

Tom

  
On Oct 13, 2014, at 10:40 AM, Reginald McNulty  wrote:

Hi Tom,
Can you tell me the tricks of how to calculate the volume of just the interior for EMD-1220? 
Reggie

On Oct 9, 2014, at 5:58 PM, Tom Goddard  wrote:

Hi Reggie,

  What are you trying to calculate?  Chimera can calculate the volume inside your density map surface if that is what you want.  If it is just a capsid map then it won’t include the hollow interior.  There are tricks to include the interior.  If you really care about icosahedron volumes Chimera gives the exact value by summing polyhedral volumes. The Chimera icosahedron radius is the distance from the center to a vertex.  The measure volume command reports the enclosed volume of the icosahedron with its flat faces as 58.7e6 for radius 285.  As you note a sphere of radius 285 has volume 95e6.

Tom


On Oct 9, 2014, at 5:34 PM, Reginald McNulty  wrote:

Dear Tom,

Eager to hear your thoughts on what’s below. 

I’ve done the math by hand for how to calculate the volume of an icosahedron. One of them (exsphere) gives the exact answer shown in Chimera with 'measure volume'. I’ll give a brief definition below and then show all calculations. The volume (V) of an icosahedron depends on the length of the edge (distance between five folds). My understanding is that there are different ways of drawing the radius (R) to calculate the length of an edge (a). Assuming an exsphere radius of 285 angstroms, I can calculate a volume of 58709669.80741614 cubic angstroms (or 59 million cubic angstroms) which is exactly the answer shown in chimera. However, I suspect the midsphere radius of 285 angstroms, yielding a volume of 95379640.39879936 cubic angstroms  (or 95 million cubic angstroms) is a closer approximation, as this answer is very close to that obtained assuming V=4/3*(pi)*r^3 (97 million cubic angstroms).

Definitions:


Assuming a radius of 285 angstroms, here are the calculations for volume 3 different ways:

Method A
R(m)- Midsphere radius= 285 Å

R(m)=a / 4 * ( 1 + √5 )
---------------------------------------
multiply both sides by 4
---------------------------------------
285(4)= a(1+√5)
          = a(3.23606797749979)
----------------------------------------
a=285(4)/3.23606797749979
----------------------------------------
a=352.27937358744003
----------------------------------------
Now that we have the icosahedral edge length (a), we can calculate the volume of the virus with:

V= 5 / 12 * a³ * ( 3 + √5 )

= 5/12 * 43718136.95711854 *(3 + √5)
= 5/12 *43718136.95711854 5.23606797749979
=95379640.39879936
---------------------------------
or 95 million cubic Å
---------------------------------



Method B
R(e)- Exsphere radius- 285

R(e) = a / 4 * √(10 + 2 * √5) 
----------------------------------
writing in python was easier for me here:

285=a/4 * math.sqrt(10+2*math.sqrt(5))
285=a/4 * 3.804226065180614
----------------------------------------------

solve for a:
a=(285/ 3.804226065180614)*4
a=299.6667339079062

--------------------------------------------
Now that we have the icosahedral edge length (a), we can calculate the volume of the virus with:

V= 5 / 12 * a³ * ( 3 + √5 )
--------------------------------------------------
writing in python is a little easier here:

V= ((5/12.0)*(299.6667339079062**3)*(3+math.sqrt(5)))

= 58709669.80741614
--------------------------------
or 58 million cubic Å
---------------------------------



Method C
R(i)- Insphere radius (tangent to face of icosahedran)

R(i) = a / 12 * √3 * ( 3 + √5 )

-----------------------------

285 = a/12 *((math.sqrt(3))*(3+math.sqrt(5)))  

285= a/12 * 9.069135768914048

a= (285/9.069135768914048)*12
  = 377.1031868022763
----------------------------------------

Use edge of 377.1031868022763 to calculate volume

V= 5 / 12 * a³ * ( 3 + √5 )
-------------------------------------------

V= ((5/12.0)*(377.1031868022763**3)*(3+math.sqrt(5)))
V= 116996977.425189
---------------------------------------------

or 116 million cubic Å

------------------------------------------

Reggie McNulty

On Oct 9, 2014, at 4:49 PM, Tom Goddard <goddard@sonic.net> wrote:

 When you use the “measure volume” command on an icosahedral cage it is measuring the volume enclosed in the polygonal solid defined by the cage, not the volume of a sphere.  I don’t know what you mean by “exsphere”, “midsphere”, “calculate by hand”.

Tom


On Oct 9, 2014, at 1:01 PM, Reginald McNulty  wrote:

I manually fixed the cage and shrunk it to encompass the internal DNA using a radius of 285. I used the measure volume command to measure the volume inside the cage. Can Chimera show sphere it is using to calculate calculate the volume? A cage with radius 285 angstroms gives a volume of 58 million cubic angstroms. By hand, I can calculate volume of also 58 million cubic angstroms using an exsphere radius of 285; edge is 300 angstroms. However, a midsphere radius of 285 gives me an edge of 352 angstroms and volume of 95 million cubic angstroms.
Reggie

On Oct 8, 2014, at 2:24 PM, Tom Goddard wrote:

Yes you could probably fix the alignment of a cage by hand in Chimera so it looked decent.  If you want to do some analysis using the symmetry that is harder.

Tom

On Oct 8, 2014, at 2:14 PM, Reginald McNulty wrote:

Never mind. I think it’s close enough with 2fold on x, 3 fold on z, with extra rotation.
On Oct 8, 2014, at 2:07 PM, REGINALD MCNULTY wrote:

Thanks. It works.
I’m trying to do the same thing now for a virus with a tail. It was refined with C1 symmetry. The icosahedral symmetry in the capsid is evident. But I can’t get the icosahedal surface to match the symmetry exactly. Can you take a look at this map: emd_1220.map?
Reggie
On Oct 8, 2014, at 11:34 AM, Tom Goddard  wrote:

Hi Reggie,

  I used the following Chimera commands to create an icosahedral cage for virus capsid 2xyz and density map for full capsid, image attached.

Tom

open 2xyz
rainbow chain
sym #0 surf true

# Used Tools / Higher-Order Structure / Icosahedron Surface to figure out orientation is n25r (no sym on x, 2-fold on y, 5-fold on z, with extra rotation).

shape icos radius 400 orient n25r lattice 2,1 color blue linewidth 5
molmap #0 10 sym i,n25r


<2xyz.jpg>

On Oct 8, 2014, at 10:40 AM, Reginald McNulty wrote:

Dear Tom,

I’ve imported a virus capsid shell pdb 2xyz to chimera. I want to measure the symmetry and make a cage based on that symmetry. Molmap seems to only produce a map of the asymmetric unit, not the entire capsid. Any thoughts?
I’m currently following directions that are here: http://www.cgl.ucsf.edu/chimera/videodoc/IcosWedge/index.html

All the best,
-Reggie

--
Reginald McNulty, Ph.D.
Postdoctoral Research Associate
The Scripps Research Institute
Johnson Lab
Department of Integrative Structural and Computational Biology
10550, N. Torrey Pines Road, MB-31
La Jolla, California 92037













All the best,
-Reggie

--
Reginald McNulty, Ph.D.
Postdoctoral Research Associate
The Scripps Research Institute
Johnson Lab
Department of Integrative Structural and Computational Biology 
10550, N. Torrey Pines Road, MB-31 
La Jolla, California 92037