INTRODUCTION




	The first part of this document is to introduce a principal math of calculating
the lattice parametters of thin 3-d crystals, which is based on the Ca-ATPase 3-d crystal 
with C2 symmetry. The experimental data was collected by taking tilt series (5 - 7 
films/crystal) from cryo-em, the increanments of each tilt was about 2-5 degree. All of 
the data has been processed by mrc program to refine the 2d parameters for each individual
film. To calculate the 3-d lattice parameters, two more steps have to be done:

	1) using program cstardir/cstarfit to obtain c* direction, real tilt axis and 
	tilt angle on first film, and relative tilt axes and tilt angles of each films in 
	the tilt series to untilted coordinator system.   
	2) using program linefit to calaulate the 3-d lattice parametters which includes 
	a*, b* and c* (vectors).

The second part will describe more detail about the programs, and how to use them.

				1. PRINCPLE

1). C* direction searching.  The goal of this program is to obtain the tilt axis 
of EM for each tilt series and C* vector direction, afterward, calculates the 
effective tilt axes and tilt angle of the films in tilt series, which are relative 
to the A* and C* in reciprocal space. The main point is that h,k of the reflections 
recorded by tilt series would be kept the same for the pile of films during 
the crystal was seriesly tilted. That means

	Rn(h,k,ln)-R1(h,k,l1)=(ln-l1)*Cn*, n=2,2,... 5 or 7

the Rn, R0 and C* are vectors in 3-d space. It is not nessecery for ln-l1 
being integer. The Cn*(n=2,...5 or 7) were averaged to obtain one C* vector
relative to the film coordinator system. The C* would be affected by tilt
axis' choosing. All the reflections of electron beam suppose to fall on the 
lines which are along the C* direction, and cross (h,k,l0) on the first film 
(this is arbitrary, you can choose other film). Based on this theory, the 
theoretic reflection position recorded by films could be solved the equations
of line (along c* direction) and planes (recorded by films) in 3-d space. Next step 
is to calculate the residue of observed reflections which are determined by a*, b* on 
each films in tiltseries and tilt angles   and theoretic positions, and to 
do this,  a big range of tilt axes was searched to find a tilt axis,  which is 
corresponding to the minimum residue, and the tilt axis was supposed to be the real one
relative to the first film's coordinator system. 


	The new method is to use non-linear least square fit to calculate tilt-axis and
c* direction, the program is called cstarfit, and it includes two maijor subroutines.
The first one is called residue, which calculates the standard residue of observation 
and theoretic c* parameter, the princepl is similar to the old method, but just use a*,
b* (not all the reflections), and tilt angles in a tilt series  as input data. The second 
subroutine called lmdiff to do nonlinear least square  fitting to minimize the residue. 
The program would outputs the final unit vector along c* and the tilt axis of EM 
goniemeter for the tilt series. 
 
	Once the tilt axis of EM and C* have been obtained, next step is to calculate the 
effective tilt axes and angles for each film. The princeple is to use one tilt add another
tilt, if you know the angle between two tilt axes, tilt angles, you can calculate the 
effective tilt axis and tilt angle. The first tilt axis is the cross line of first film 
and untilted crystal's a-b plne, it can be can calculated by fomula:

	alpha=arctan(cy*/cx*)+90.0 (degree)

and the the tilt angle:

	 delta=arccos(cz*/|c*|). 

The second tiltaxis is the EM goniemeter's tilt axis, and the tilt angle is read from 
goniemeter. To get the effective tilt axis and tilt angle, we can rotate the y-axis of the 
untilted system to the first tilt axis, tilt the crystal/film delta degree, then rotate 
the new y-axis called y'-axis) to second tilt axis, and tilt the crystal/film again. 
Finaly, rotate the coordinator system back.

2). Refinment of C*, A* and B*. This step is going to find a set of lattice 
parameters (A*,B* and C*) to fit the observation. Based on last step, the 
reflections which have the same (k,h) should fall on a line along C* vector,
first of all is based on last step to search the peak center positions of the 
reflections, which are manually choosen falling into separated peaks, and the 
peak shape is assumed as a sinc function. The thickness of crystal can be choosen 
as different unit cells, the centers of the peaks are searched within a range, 
and the intensities of the peaks are released for best fitting. After having 
obtained the peaks' positions, the C* length could be calculated by using the peaks
which have the same (h,k) but different l. Then the A* can be theoretically calculated 
by following formula: 

		A*=[R1(h1,k,l1)-R2(h2,k,l2)-(l1-l2)C*]/(h1-h2)  
		
		(in case of h1>h2,l1>l2) 

in which A*, R1 and R2 are vectors in 3d space, and k is kept the same for R1 and R2. 
Because l1 and l2 are unknown so far, we use m replace l1-l2, m=(-5,-4,...,4,5) to 
obtain a batch of possible A*'. A batch of possible B*' could be obtained by the same 
way, just exchanging h's and k's situation in above equation. Next thing is to find 
the real one set of A*, B* and C* which could fit all the observations. The formula

		A*=A*'+delta*C*

in which A*'=[R1(h1,k,l1)-R2(h2,k,l2)-mC*]/(h1-h2), m=-5,-4,...4,5, delta=-0.5,....,0.5,
is being used for searching the A*, the same way for B*, to find best fitting of A*, B*
and C* for observation data, that gives the minimum residue between lattice positions 
(R=hA*+kB*+lC*) and the sync function peaks' positions .    


			2. INSTRUCTION FOR PROGRAM

1.) Program cstardir is for searching the tilt axis of EM with certain camera length 
and lenses current, which gives the minimum residue for the experimental data of tilt 
series, and calculates the effective tilt axes to a* in the reciprocal space coordinator
system, in which the a-b* plane is  x*-y* plane and c* is parallel to z*, and the tilt
angles for the films in tiit series respectively.

The input data includes 
	(a) Tilt angle, a*, b* on each film which are obtained
	from mrc program pickauto 
	(b) Reflection list files named as *.spt.
	(c) Tilt angles in tilt series
	The program will creat some files to store the input data they are 
	calst - reflection data, sttheta - tilt angle, lattmeter - a* and b*
	translat is a* and b* parameter after rotated the y axis to em tilt axis.

	The control parameter
	(a) c* cutoff to control minimum z* used for calculating c* direction, but you
	can type 0.0.
	(b) Intensity cutoff was used to control reflections used for calculating, now
	just type 0.
	(c) Tilt axis, type in a estimated angle from tiltaxis to x-axis on film, for
	example, cm200-feg at 200 kv camera length at 2500 mm, tiltaxis was around 45
	degree.
	(d) job is an integer to control read input data
	(e) The number of tilt axes to be searched.
	(f) Step size for searching tiltaxis, you can choose a large value for first 
	time, and small value for second time.
 	

	The output files are:
	(a) Unitcell, a*, b* on the first film,  unit vector along c* relatted to 
	first film, and tilt-axis, last one is the minimum residue of tiltaxes 
	searched by the program. This data will be read by linefit.
	(b) Gunitcell lists all the a*, b*, unit vectors, tiltaxes and corresponding
	residues.
    	(c) Rltltanax list the real tilt axex and tilt angles for each films in tilt 
	series, the tilt axes are the angles of the projection of a* onto untilted plane
	and tilt axes, positive direction is from a* to b*. These parameters will be used 
	merging.

To run:

	(a) enter number of films in tilt series
	(b) control value for zstar, 0.0 - 60.0
	(c) enter a reasonable value of the angle between tilt axis and x-axis.
	(d) if this is the first time to run the program, and have to  enter tilt 
	angles, a*, b* and reflection data, enter 1 (job=1). if you have type in tilt 
	angle, and just need to type in a*, b* and reflections' data, enter 2 as job=2,
	if you have typed in a*, b*, and need to enter tilt angles and reflections' data
	type 3 in. if you have run this tilt series one time already, enter 4, go to 
	next step.
	(e) control value for intensity, type 0 in.
	(f) control number for tiltaxis searching.
	(g) control value for searching stepsize.
 
2.) Program cstarfit's goal is the same as above program, but it is more powerful, this
program calls a subroutine lmdif to do non-linear least-square fitting to find out c*
direction and tilt axis.

Input data includes: 

	(a) All of a* and b* measured from each films of a tilt series, which
	now is read in from a data file called 'lattmeter', and has been transfered to
	C2 symmetry, the length unit is in pixel.
	(b) tilt angle also is read in from a data file called sttheta, using degree as
	unit.
	(c) Initial tilt axis, read in from key board in degree, the angle from x-axis
	to tiltaxis clockwise as positive.
	(d) x* and y* components of unit vector along c*, usually type in from keyboard
	0.0 and 0.0.

Output data 
	
	(a) Tilt axis output on screen.
	(b) Unit vector along c* output on screen.
	(c) tltaxang is the file including (a) and (b)

To run:

	(a) Type the number of films in a tilt series.
	(b) Enter the data file's name of tilt angle
	(c) Enter the file's name of a* and b*
	(d) type in tilt axis (in degree), the x, y components of the unit vector along
	c* direction.

To obtain c*, tilt axis and final tilt angles and tilt axes for each films,  we should run
both cstardir and cstarfit:

	First run cstarfit to get c* and tilt axis of EM goniemeter, then run cstardir to 
get tilt angles and tilt axes for each film.

3.) Program linefit1 is a program to calculate the unit cell in reciprocal space, for 3d 
crystals. 

Subroutines included:

	(a) searchexp calculates the positions of all reflections in a tilt series, 
	along c* lines, i.e. (h,k,z*)
	(b) Leastsq using linear least-square method fits sync-function to obtain
	 peak's positions along c*.
	(c) abcstarcal roughly calculates the a*, b* and c* based on the theory 
	described above
	(d) searchucl is for searching unitcell parameters (a*, b* and c*) around 
	the values got from subroutine abcstarcal to obtain a minimum residue.
	(e) rvctout will output the observed peaks' coordinators in 3-d reciprocal
	space.

Input data:
	
	(a) a*, b* measured from first film, and unit vector along c* are included
	in data file unitcell.
	(b) reflections list was stored in calst.
	(c) if you run the tilt series first time, you have to selecte the peaks by 
	yourself, and at second time you can read in the peak positions from file 
	peaknum,
	(d) reflections list used for calculating lattice parameter, if you run this 
	tilt series first time, you should selecte the reflections youself, at second
	time, read from file hkrflct.

Control data:

	(a) intensity cutoff is the amplitude limitation for the peaks
	(b) character n is used to control if you don't want to use reflections for 
	fitting or you have selected the reflections.
	(c) character yes or no are used for controling first or second time processing
	the same tilt series.
	(d) c* increament is to set the searching stepsize along c*.

Output data:

	(a) bestabc includes a*, b* and length of c*.
	(b) ucellvct lists the x, y, and z components of a* in 3-d space,  c*
	and beta* which is the angle of a* and c* against residues.
	(c) residual table for peak's fitting, the file name can be choosen by user.
	(d) residr is the residual table for reflections' fitting.
	(e) acstarnew is the file having a*, c* and beta* obtained from reflection
	fitting.
	(f) angstrom lists the positions along c* direction against intesities  for all
	the reflections in a tilt series along c* direction

Creat a file for peaks choosen from cstarseries, the file name is peaknum with format
2I5,I6,2I5, and the title lists the spt file.

To run:
	
	(a) type in the number of films in one tilt series.
	(b) enter thickness of the crystals, which is number of unit cell along c.
	(c) type in the amplitude cutoff.
	(d) type in the file name for output residual table
	(e) type in the title for the table
	(f) if you have selected the reflections for fitting, type in n or following the
	instruction appearing on screen.
	(g) if you run this tilt series first time, type in no, otherwise type yes.
	(h) enter the increament of c* for peaks' fitting, first time running, the 
	increament should
	be bigger than
	1/400 (enter a number less than 400), second time, can enter a number bigger than 
	400.
	(i) enter a* increament, 20 - 60 for first running, 40 - 100 for second time.
	(j) enter c* search range for reflections' fitting.
	(k) enter a* search range for reflections' fitting.
	(l) enter (h,k,l) to calculate the angle of c* and vector (h,k,l), if you want to 
	stop, enter 100 0 0.



*********************************************************
Programs used:

cstarfit
cstardir
linefit
tseriefit

*********************************************************


To conver residual file to postscript format:

1. generate a residual file by linefit program

2. run residimg to convert ASCII file to MRC file, you can routate the image
by shifting x and y

3. use pltresid to convert MRC file to countor map
 
To plot tilt series *.spt file:

1. run tseriesfit, which generates a file called tseriesfit.plt to plot tilt 
series, this program will read a data file of reflections intensities, 
positions along c* and hk0 position, in current case, the file name is called 
positions generated by linefit.  

2. convert tiltseries.plt to postscript format by running mrcplt2ps program.