where the first two Gaussians terms of width σH1
centered at ZH1 and - ZH1 represent the hydrophilic
phospholipid polar headgroups. The third Gaussian term of width σC
at the centre of the bilayer shell accounts for the hydrophobic hydrocarbon chains
and ρr is the ratio of the electron density of the
hydrocarbon chains to that of the headgroups. Finally, the fourth Gaussian term
(with the width σH2 centered at ZH2 and the amplitude ratio A2)
can account for the asymmetry in the electron density profile which can be caused
either by the complicated rafted structure of the different types of lipids
and/or by the presence of the membrane protein on the inner or outer surface of the lipid vesicle.
The shape of the vesicle can also be approximated by an ellipsoid of revoluition
when instead of the form-factor of a thin spherical shell in SFF Formula (1)
one substitutes the formfactor of core-shell ellipsoid of revolution.
BILMIX allows one to model mixtures containing up to ten different
components (i.e. different types of particles)
Below there is the list of the fitting parameters for SFF approximation from ULV mixtures
using either the spherical or ellipsodial shape of the vesicle:
Parameters | Description |
Volume fraction |
0 < νk < 1 |
Internal Shell Radius |
Rin (if Rin=0, the sphere is solid) |
Internal Shell Contrast |
ρin (if ρin=ρout, the sphere is uniform) |
External Shell Radius |
Rout |
External Shell Contrast |
ρout |
Polydispersity |
dRout |
Volume concentration |
Proportion of volume taken by all particles (from 0 to 1), is needed to calculate structure factor |
Hard sphere radius |
Rhs: this should be fulfiled: Rout <= Rhs |
Stickiness parameter |
τ: 0.1<τ<100, τ=100 (Hard-sphere case), τ=0 no interactions |
Alternatively one can use an ellipsoidal shape of the vesicle
Parameters | Description |
Volume fraction |
0 < νk < 1 |
SemiAxis R for Ellipsoid (R,R,v*R) (inner value) |
R (if R=0 and t>0, the ellipsoid is solid) |
Ratio v of samiaxes for Ellipsoid, anisometry degree |
v (if v=1, the ellipsoid turns into sphere, v>1 prolate, v<1 oblate ) |
Thickness t of the Ellipsoid shell |
t (if t=0 and R>0, the ellipsoid is solid) |
Polydispersity of the Ellipsoid |
dRell |
Internal Shell Contrast |
ρin (if ρin=ρout, the ellipsoid is uniform) |
External Shell Contrast |
ρout |
Parameters | Description |
Volume fraction |
0 < νk < 1 |
Peak1 position of hydrophilic phospholipid polar headgroup(positive density) |
ZH1 |
Width of Peak1 of hydrophilic phospholipid polar headgroup(positive density) |
σH1 |
Peak2 position of hydrophilic phospholipid polar headgroup(positive density, inner surface of lipid vesicle) |
ZH2 (in) |
Width of Peak2 of hydrophilic phospholipid polar headgroup(positive density, inner surface of lipid vesicle) |
σH2 (in) |
Amplitude ratio of Peak2/Peak1 (positive density, left side) |
A2/A1 (in) |
Peak2 position of hydrophilic phospholipid polar headgroup(positive density, outer surface of lipid vesicle) |
ZH2 (out) |
Width of Peak2 of hydrophilic phospholipid polar headgroup(positive density, outer surface of lipid vesicle) |
σH2 (out) |
Amplitude ratio of Peak2/Peak1 (positive density, outer surface of lipid vesicle) |
A2/A1 (out) |
Width of Peak3 of the hydrophobic hydrocarbon chains (negative density) |
σC |
Amplitude ratio of Peak3/Peak1 (negative density) |
ρr |
Number of layers of lipid vesicles (1 - unilamellar lipid vesicles) |
L |
To start the program one has to have both bilmix.exe and bilmix.cmd
(see BILMIX Input Files ) files in one directory and
run with the following command: BILMIX.EXE < BILMIX.CMD
_/_/ _/_/_/_/_/ _/_/_/_/_/_/ _/_/_/_/_/_/ _/_/_/_/ _/_/_/_/_/
_/_/ _/_/ _/_/ _/_/ _/_/ _/_/ _/_/ _/_/
_/_/ _/_/ _/_/ _/_/ _/_/ _/_/ _/_/
_/_/ _/_/ _/_/ _/_/ _/_/ _/_/ _/_/_/_/_/
_/_/ _/_/ _/_/_/_/_/_/ _/_/ _/_/ _/_/
_/_/ _/_/ _/_/ _/_/ _/_/ _/_/ _/_/
_/_/_/_/_/ _/_/ _/_/ _/_/_/_/ _/_/_/_/_/ _/_/
DIALOGUE SERVICE PROGRAM FOR MINIMIZATION OF NONLINEAR MULTIVARIATE
FUNCTIONS.
Program BILMIX uses the optimization program suite OPTIS written
by V.V. Volkov (Institute of Crystallography, Moscow, Russia)
e-mail: vvo@ns.crys.ras.ru
Dialogue mode [ Interactive / PASSive ] ...... < Inter. >:
OPTIS: OPTIS: Welcome to user's problem
Enter the series title .................................: Enter comment line
.....................................: Enter Number of phases ................
< 0 >: Enter concentration of the system ...... < 0.0 >:
Choose type of data file format:.
1: OTOKO format
2: ASCII format
Enter the format code .................. < 2 >: Enter INTENSITY fil
name .............. < .dat >: 339703.100000000 10.4556000000
00
Angular units in the input file :
4*pi*sin(theta)/lambda [1/angstrom] (1)
4*pi*sin(theta)/lambda [1/nm ] (2)
2* sin(theta)/lambda [1/angstrom] (3)
2* sin(theta)/lambda [1/nm ] (4) < 1 >: Enter Fraction of t
e curve ............ < 0.0 >: 75 101
I(1), I(last) : 339703.100000000 10.7711200000000
s(1), s(last) : 3.229300000000000E-002 0.421600000000000
Err(1), Err(last): 10191.0900000000 0.323133500000000
15-dimensional user's problem
OPTIS: OPTIS: OPTIS: ///////////////////////////////////////////////////////
//////////////
This is the information which was read from the command file.
OPTIS --- Version 5.6 rev. 09 May 02 03:30 reports current information:
08-Oct-2009 time 10:42:37
Objective function -- : User's problem (15-dimensional): 0
Chosen algorithm ---- : B-F-G-S with simple bounds
number of performed function evaluations ----- : 1
number of performed iterations --------------- : 0
number of performed gradient estimations ----- : 0
maximum number of function evaluations ------- : 5250
maximum number of search iterations ---------- : 40
total number of problem variables ------------ : 15
number of problem variables to be varied ----- : 15
maximum allowed number of variables ---------- : 512
function value ------------------------------- : 5.678e-2
rel/abs function tolerance to be attained ---- : 0.0
tolerance multiplier switch ------------------ : 0.0 (OFF)
upper bound on hessian condition number ------ : 0.0 (OFF)
temperature and scedule factor --------------- : 0.0, 0.0
function computation noise (rel/abs) --------- : 0.0
relative machine precision ------------------- : 2.22e-16
message level -------------------------------- : 1
===> Press CR to continue: ---------------------------------------------------
------------------
fixative-vector
1 1 1 1 1 1 1 1 1 1
1 1 1 1 1
----------------------------------------------------------------------
argument values
0.500000 0.00000 0.00000 48.0000 1.00000
5.00000 55.0000 0.00000 0.500000 0.00000
0.00000 40.0000 1.00000 3.00000 300.000
----------------------------------------------------------------------
lower bounds on variables
0.00000 0.00000 0.00000 40.0000 1.00000
0.100000 48.0000 0.00000 0.00000 0.00000
0.00000 30.0000 1.00000 2.10000 300.000
----------------------------------------------------------------------
upper bounds on variables
1.00000 0.00000 0.00000 60.0000 1.00000
25.0000 85.0000 0.00000 1.00000 0.00000
0.00000 60.0000 1.00000 10.1000 300.000
//////////////////////////////////////////////////////////////////////
The information about minimization method
(it uses B-F-G-S with simple bounds) and the argument values
(its initial values, lower and upper boundaries).
CPU time used: 0 min 0.02 sec
The value of minimization tagret function at each iteration step.
This information is not shown any more to save computational time.
Continue? [ Y / N ]
E04JAF a|Itr.Rem.| No.of F | F | cond(H |
E04JAF a| 40| 0| 5.678e-2| 6.295|
E04JAF a| 39| 9| 3.509e-2| 8.380|
E04JAF a| 38| 18| 4.007e-3| 17.01|
E04JAF a| 37| 27| 3.483e-3| 2.358|
E04JAF a| 36| 36| 3.109e-3| 246.7|
E04JAF a| 35| 45| 3.099e-3| 170.6|
E04JAF a| 34| 56| 3.068e-3| 4.026|
E04JAF a| 33| 65| 3.046e-3| 2.668|
E04JAF a| 32| 78| 2.715e-3| 476.2|
E04JAF a| 31| 86| 2.651e-3| 447.6|
E04JAF a|Itr.Rem.| No.of F | F | cond(H) |
E04JAF a| 20| 185| 1.163e-3| 1680.|
E04JAF a| 19| 194| 1.162e-3| 1485.|
E04JAF a| 18| 202| 1.162e-3| 1383.|
E04JAF a| 17| 211| 1.162e-3| 1302.|
E04JAF a| 16| 227| 1.162e-3| 1444.|
E04JAF a| 15| 244| 1.162e-3| 1456.|
E04JAF b| 14| 260| 1.162e-3| 1731.|
E04JBM --- Local search attempt...
E04JAF c| 14| 268| 1.162e-3| 1731.|
OPTIS: OPTIS: 15-dimensional user's problem
Function no. 20, 301 value = 1.1621442432225761E-03
OPTIS: OPTIS:
Are you sure of the solution? [ Y / N ] < No >:
Remember to save your results before you leave "OPTIS"!
Exit now? [ Y / N ] .................... < No >:
Well, absolute accuracy requires infinite time...
----------------------------- END OF JOB -----------------------------
Final Function value. (In ideal fit it should approach to zero)
Produced function minimum is equal to 0.1162144E-02
at the point:
0.987769 0.00000 0.00000 54.1704 1.00000
7.11431 55.0000 0.00000 0.614604 0.00000
0.00000 45.2565 1.00000 4.02500 300.000
Optimized values for the arguments.
after 301 function evaluations,
0 gradient evaluations,
0 iterations
CPU time used: 0 min 4.61 sec
E4MAIN --- Normal termination
For running the program BILMIX it is necessary to prepare
the command file where one has to specify your model and
initial values of parameters for this model, i.e.
the number of "phases" (components), the type
for each component (SPHERE or ELLIPSOID, DIFFUSE),
dimension parameters (relative volume fraction, average
sphere radius, its polydispersity, type of distribution
function (Gauss or Schultz) etc., the upper and lower
boundary values for all fitting parameters.
Below there are two examples of command file for
the model containing different types of ULV particles (spherical and ellipsoidal),
one can design the model containing several components of the
same type (for example large and small unilamellar vesicles etc.),
but in all cases one has to follow
the thumb rule of describing the model parameters as it is done
in this example command files:
Below symbols !! are used for comments. For your convienence it is recommended
to keep the comments on each line, the program will automatically detect them.
Example Command File 1 (globular ULVs particles)
i !! Two initialising
!!!!!!!!!!!!!!!!!!!!!!!!! !! strings
pro us !! Command "Problem user"
Series titles !! Comment Line 1 (done by user )
Partial data titles !! Comment Line 2 (done by user )
EXPERIMENT !! MODE FOR FITTING DATA, "TEST" MODE FOR EXPERTS
2 !! Number of Phases (2 for this example)
0.2000 !! System Concentration
SPHERE !! Type of the first Phase (SFF approximation)
0.5000 0.0000 100.0000 !! Volume fraction of the component (vesicle/micelle)
0.0000 0.0000 0.0000 !! Inner (core) radius of the sphere
0.0000 0.0000 0.0000 !! Inner (core) contrast of the sphere
29.4598 20.7678 62.1518 !! Outer (core+shell) radius of the sphere
1.0000 1.0000 1.0000 !! Outer (shell) contrast of the sphere
2.6920 1.1730 17.3839 !! Polydisperstiry on the sphere radius
86.9196 86.9196 86.9196 !! Hard-sphere radius (for interactions only)
2 !! Schulz distribution 2 (Gauss distribution 1)
0.0000 0.0000 0.0000 !! stickiness parameter (for interactions only)
DIFFUSE !! Type of the second Phase (SFF approximation)
0.5 0.1 100.0 !! Volume fraction of the bilayer component
1.76 1.72 1.80 !! Peak1 position (positive density)
0.57 0.55 0.59 !! Width of peak1 position (positive density)
6.04 5.04 7.04 !! Peak2 position (positive density, inner surface of lipid vesicle)
1.13 0.83 1.13 !! Width of peak2 position (positive density, inner surface of lipid vesicle)
0.05 0.03 0.50 !! Amplitude ratio (Peak2/Peak1) (positive density, inner surface of lipid vesicle)
6.04 5.04 7.04 !! Peak2 position (positive density, outer surface of lipid vesicle)
1.13 0.83 1.13 !! Width of peak2 position (positive density, outer surface of lipid vesicle)
0.00 0.00 0.00 !! Amplitude ratio (Peak2/Peak1) (positive density, outer surface of lipid vesicle)
0.67 0.37 0.77 !! Width of peak3 position (negative density)
1.36 1.06 1.46 !! Amplitude ratio (Peak3/Peak1) (negative density)
1 !! Number of layers (1 - means only diffuse scattering from a single bilayer)
2 !! ASCII format file
test_bilmix.dat !! Experimental data file
1 !! Angular scale (1/2/3/4) as in GNOM
0.8 !! Exp. data portion to fit (from beginning)
meth sb !! Minimization method sb - "simple bounds"
loa maxit 1000 !! maximum number of iterations 40
run !! run minimisation process
y !! confirm running
y !! confirm running
mess 15 !! message for saving the output data
eva !! write data
mes 1 !! prepare for next set data
ex !! exit the program
y !! confirm exit
y !! confirm exit
Example Command File 2 (ellipsoidal ULVs particles)
i !! Two initialising
!!!!!!!!!!!!!!!!!!!!!!!!! !! strings
pro us !! Command "Problem user"
Series titles !! Comment Line 1 (done by user )
Partial data titles !! Comment Line 2 (done by user )
EXPERIMENT !! MODE FOR FITTING DATA, "TEST" MODE FOR EXPERTS
2 !! Number of Phases (2 for this example)
0.2000 !! System Concentration
ELLIPSOID !! Type of the fourth Phase ( sph, cyl, dmb, ell)
1.0 0.0 1.0 !! ellipsoid Fraction
60.0 50.0 70.0 !! SemiAxis for Ellipsoid (b,b,v*b) (inner value)
1.0 0.9 1.15 !! Ratio of SemiAxes for Ellipsoid
1.0 1.0 1.0 !! Thickness of the shell Ellipsoid
5.0 1.0 20.0 !! Polydispersity of the ellipsoid
1.0 1.0 1.0 !! Inner contrast of ellipsoid
1.0 1.0 1.0 !! Outer contrast of ellipsoid
1 !! Type distribution
DIFFUSE !! Type of the second Phase (SFF approximation)
0.5 0.1 100.0 !! Volume fraction of the bilayer component
1.76 1.72 1.80 !! Peak1 position (positive density)
0.57 0.55 0.59 !! Width of peak1 position (positive density)
6.04 5.04 7.04 !! Peak2 position (positive density, inner surface of lipid vesicle)
1.13 0.83 1.13 !! Width of peak2 position (positive density, inner surface of lipid vesicle)
0.05 0.03 0.50 !! Amplitude ratio (Peak2/Peak1) (positive density, inner surface of lipid vesicle)
6.04 5.04 7.04 !! Peak2 position (positive density, outer surface of lipid vesicle)
1.13 0.83 1.13 !! Width of peak2 position (positive density, outer surface of lipid vesicle)
0.00 0.00 0.00 !! Amplitude ratio (Peak2/Peak1) (positive density, outer surface of lipid vesicle)
0.67 0.37 0.77 !! Width of peak3 position (negative density)
1.36 1.06 1.46 !! Amplitude ratio (Peak3/Peak1) (negative density)
1 !! Number of layers (1 - means only diffuse scattering from a single bilayer)
2 !! ASCII format file
test_bilmix.dat !! Experimental data file
1 !! Angular scale (1/2/3/4) as in GNOM
0.8 !! Exp. data portion to fit (from beginning)
meth sb !! Minimization method sb - "simple bounds"
loa maxit 1000 !! maximum number of iterations 40
run !! run minimisation process
y !! confirm running
y !! confirm running
mess 15 !! message for saving the output data
eva !! write data
mes 1 !! prepare for next set data
ex !! exit the program
y !! confirm exit
y !! confirm exit
To start the program one has to have both bilmix.exe and bilmix.cmd
files in one directory and run with the following command:
BILMIX.EXE < BILMIX.CMD
The output files of the program BILMIX are the following:
3 files with extensions *.fit, *.den, *.vr, where
the names of files coincide with the file name of experimental data
as well as bilmix.log file
File extensions | Description |
*.fit |
In *.fit files there are three columns, the first column is
the S-vector axis, the second column - experimental data,
the third column - fit to the data.
|
*.den |
In *.den files the number of column is equal to two.
The first column is r-axis and the second
column contains the electron density of lipid bilayer profile of your model.
|
*.vr |
In *.vr files there is information about volume size distribution of ULV particles (globular or ellipsoidal).
The first column is r-axis, the other columns - the restored size distributions
of ULV particles.
|
bilmix.log file contains information about the file name of experimental data
and obtained fitting parameters for your model for each component.
After each run the program BILMIX adds information to this file, so you
will have the whole history of running the program in this file.
References:
[1] Pencer Jeremy, Krueger Susan, Adams Carl P., Katsaras John.
Method of separated form factors for polydisperse vesicles //
Journal of Applied Crystallography. 2006. Vol. 39, no. 3. Pp. 293-303.
[2] Kiselev M.A., Lesieur P., Kisselev A.M. et al.
Model of separated form factors for unilamellar vesicles //
Applied Physics A: Materials Science & Processing. 2002. Vol. 74.
Pp. s1654-s1656.
[3] Schulz G.V., Ueber die Beziehung zwischen Reaktiongeschwindigkeit und
Zusammensetzung des Reaktionproduktes Macropolymerisationsvorgaemgen //
Z. Phys. Chem. Abt. B (1935) 30, 379.
[4] Pabst Georg, Koschuch Richard, Pozo-Navas Beatriz et al.
Structural analysis of weakly ordered membrane stacks // Journal of Applied
Crystallography. 2003. Vol. 36, no. 6. Pp. 1378-1388.