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EMBL Hamburg Biological
Small Angle Scattering
BioSAXS

BILMIX manual

bilmix

Written by P.V. Konarev, and D.I. Svergun.
Post all your questions about BILMIX to the ATSAS Forum.

© ATSAS Team, 2003-2020

Table of Contents

Manual

Program BILMIX makes allows one to restore the electron density of a lipid bilayer and simultaneously generate the corresponding size distribution of the unilamellar lipid vesicles. The presence of the proteins on the inner or outer surface of the unilamellar lipid vesicle can be modelled by the introduction of asymmetry in the electron density profile.

If you use results from BILMIX in your own publication, please cite:

P.V. Konarev, M.V. Petoukhov, L.A.Dadinova, N.V. Fedorova, P.E. Volynsky, D.I. Svergun, O.V.Batishchev, E.V.Shtykova (2020). BILMIX: a new approach to restore the size polydispersity and electron density profiles of lipid bilayers from liposomes using small-angle X-ray scattering data Journal of Applied Crystallography Vol. 53 (in press)

Introduction

The SAXS data collected from globular lipid vesicles can be well approximated by a product of the formfactor of a thin spherical shell FTS (defining the vesicle size) and the formfactor of a flat lipid bilayer FFB (containing information about the electron density across the bilayer). This so-called separated form factor (SFF) approximation is valid when the vesicle size is much larger than the bilayer thickness [1-2].

The intensity from a dilute polydisperse mixture of the unilamellar vesicles (ULVs) can be represented as follows:

where s=4πsin(θ) / λ, 2θ is the scattering angle, λ is the X-ray wavelength, N is the number of ULV particles with different bilayer structures, νk and Ik(s) are the volume fractions and the partial scattering intensities from these ULVs, respectively. Using the above SFF approximation and taking into account vesicle size polydispersity, each partial intensity can be expressed as

where DV(r) is the volume size distribution of vesicles, FTS(s,r) is the form-factor of a thin spherical shell with radius r, FFB(s,r) is the form-factor of the flat lipid bilayer. The volume distribution of ULVs DV(r) can be parametrized by a monomodal Gauss or Schulz distribution with a mean radius R and width σ [3].

The form-factor FFB(s,r) is the Fourier transform of the electron density profile of the bilayer, which can be approximated by four Gaussian functions similar to [4]:

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)

Running BILMIX

Configuration

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:

SPHERE, FTS(s) term from SFF approximation

ParametersDescription
Volume fraction 0 < νk < 1
Internal Shell Radius Rin (if Rin=0, the sphere is solid)
Internal Shell Contrast ρin (if ρinout, 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

ELLIPSOID, FTS(s) term from SFF approximation

ParametersDescription
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 ρinout, the ellipsoid is uniform)
External Shell Contrast ρout

DIFFUSE, FFB(s) term from SFF approximation

ParametersDescription
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

Running Output

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

BILMIX Input Files

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

BILMIX Output Files

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 extensionsDescription
*.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.


  Last modified: February 12, 2020

© BioSAXS group 2020