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# Generalized Born using Molecular Volume (GBMV) Solvation Energy and Forces Module and Surface Area¶

```Questions and comments regarding GBMV should be directed to

Michael S. Lee or Michael Feig c/o
Charles L. Brooks, III (brooks@scripps.edu)```

## Background¶

The GBMV module is a Generalized Born method for mimicking the Poisson-Boltzmann (PB) electrostatic solvation energy. The PB method for obtaining solvation energies is considered a benchmark for implicit solvation calculations. However, the PB method is slow and the derivatives, i.e. forces, are ill-defined unless one changes the definition of the molecular volume.

The Generalized Born equation, as prescribed by Still, et. al. allows one to compute solvation energies very similar to the PB equations. As it is an analytical expression, forces are available as well:

```                                                 q q
N   N                   i j
G   =  -C  (1-1/eps){1/2 sum sum ------------------------------------ }
pol     el              i=1 j=1 [r^2 + alpha *alpha exp(-D  )]^(0.5)
ij        i      j      ij

D   = r^2 / (K_s * alpha * alpha )
ij    ij               i       j```

where K_s = 4 for Still’s original equation, or 8 for modified equation.

The only problem is that one needs to calculate the alpha’s, a.k.a. Born radii for each atom, accurately. There are various methods available, such as the GBORN, ACE, and GBSW modules in CHARMM.

The GBMV method obtains the Born radii very accurately, i.e, w/ greater than 0.99 correlation. It is available as three approaches:

1. grid-based (Most accurate)
2. analytical method I (Least accurate, fastest)
3. analytical method II (preferred for dynamics)

The analytical method has derivatives and thus can be used in molecular dynamics simulations. The grid-based method has no derivatives, however, it is the most accurate and still can be used in energy ranking and Monte-Carlo methods.

### When should you use GBMV?¶

Because the analytical and grid-based methods are quite accurate, the parameters change very little when optimized for a particular force-field. Hence, forcefields besides those of CHARMM can be used with GBMV without refitting of parameters. The GBMV method II approximates the molecular surface directly. The agreement with respect to electrostatic solvation energies from standard Poisson theory is very good (<1% relative error). The higher accuracy comes at a price, however, and GBMV is slower than other GB methods in CHARMM.

Papers related to GBMV method:

1. M. S. Lee, F. R. Salsbury, Jr., and C. L. Brooks III. J. Chem. Phys.(2002),116, 10606
2. M. S. Lee, M. Feig, F. R. Salsbury, Jr., and C. L. Brooks III. J. Comp. Chem. (2003), 24, 1348
3. M. Feig, A. Onufriev, M. S. Lee, W. Im, D. A. Case, C. L. Brooks III, J. Comp. Chem. (2004), 25, 265-284
4. S. Tanizaki, M. Feig, J. Chem. Phys. (2005), 122, 124706
5. J. Chocholousova, M. Feig, J. Comp. Chem. (2006) 27, 719-729

### Surface Area¶

A solvent accessible surface area (SASA) calculation is implemented within the GB module. There is essentially no additional cost compared to the GB calculation itself and about 5 times faster than an exact, analytical calculation (e.g. from the ASP module). It is accurate to within 1% of the exact surface and is much more accurate than the SASA module within CHARMM.

## Syntax of Generalized Born Molecular Volume (GBMV) Solvation commands¶

### method I: faster but less accurate¶

```GBMV { P1 <real> P2 <real> LAMBda1 <real> DN <real> SHIFT <real>
WATR <real> BETA <real> EPSILON <real>
SA <real> SB <real> SCUT <real>}
[WEIGHT]
[GBVDW]
CORR <int>
{ ESHIFT <real> SHIFT <real> TT <real>   (CORR = 0) }
{ SHIFT <real> SLOPE <real>              (CORR = 1) }
}```

### HDGB method: heterogeneous dielectric / membrane model¶

```GBMV { { GBMV II options }
CORR <int> (CORR = 3, 4, 5)
A1 <real> A3 <real> A4 <real> A5 <real>
UNEPS <int>
ZS <real> ZM <real> ZT <real> ST0 <real>
HDGBRC
}```

### grid-based method¶

```GBMV GRID { [GEOM] [ARITH] [CONV] [WEIGHT]
EPSILON <int> DN <real> WATR <real>
P6 <real>
KAPPA <real>
WTYP <int> NPHI <int>
CORR <int>
{ SHIFT <real> SLOPE <real>     (CORR = 1) }
{ ESHIFT <real> SHIFT <real>    (CORR = 0) }         }
}```

### free-up memory and/or start over¶

`GBMV CLEAr`

## Function¶

### Parameters of the Generalized Born using Molecular Volume Model common to all methods¶

 WTYP Angular integration grid type: 0 - Dodecahedron 1 - Spherical polar 2 - Lebedev (DEFAULT) 3 - Alternating octahedron/cube NPHI Used when WTYP equals 1 or 2. When WTYP=1, it corresponds to number of phi angles. When WTYP=2, it corresponds to size of Lebedev grid, which can only have values of 6,26 (Default), and 38 at the present time. CUTA Extent of radial integration points in Angstroms. (Default 20) GCUT radial spacing of integration grid 1 - default spacing: ```0.1 0.2 0.3 0.4 0.5 0.75 1.0 1.25 1.5 1.75 2.0 2.5 3.0 3.5 4.0 5.0 6.0 7.0 8.0 10.0 12.0 16.0 20.0``` 2 - finer spacing for small radii ```0.1 0.2 0.3 0.4 0.5 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.7 4.1 5.1 6.1 7.1 8.1 10.1 12.1 16.1 20.1``` 3 - custom grid, specify with RADG RADG custom grid spacing, first argument is number of intervals following arguments are interval limits CORR Coloumb field correction method: 0 - for R^5 method. use: SHIFT/ESHIFT/TT ```alpha(i) = - 1/( r4 - TT * r5 + ESHIFT ) + SHIFT ``` 1 - for R^7 method (default) use: SHIFT/SLOPE ```alpha(i) = SLOPE/( (1-1/sqrt(2)) * r4 + r7) + SHIFT ``` 2 - for R^5/R^7 use: A1/A2/A3/SHIFT/SLOPE/ESHIFT ```alpha(i) = SLOPE/( A1 * r4 + A2 * r5 + A3 * r7 + ESHIFT) + SHIFT ``` this mode is intended for the calculation of Born radii in different dielectric environments 3 - for R^5/R^7 use: A1/A3/A4/A5/SLOPE ```alpha(i) = SLOPE/( A1 * r4 + A3(i) * r7) + A4 + A5/(eps(i) + 1) A3(i) = A3 * 3 * eps(i) / (3 * eps(i) + 2 * EPS)``` this mode is intended for the implicit membrane model(see below) 4 - Same as CORR = 3 except that the local dielectric constant is modulated spherically. 5 - Same as CORR = 3 except that the local dielectric constant is modulated cylindrically. where r4 is volume integral over 1/r^4, r5 is square root of integral over 1/r^5 and r7 is integral over 1/r^7 to the power of 1/4. TT Multiplicative factor for correction term (CORR = 0 only). SHIFt The shifting factor of Alpha(i). MUST be set! ESHIft Energy shifting factor of the self-polarization energies: 1/Alpha(i). CORR=0 or 2 only. (Default 0.0) SLOPE Multiplicative factor of the Alpha(i). CORR=1 or 2 only. (Default 1) A1,A2, A3 Multiplicative factors in calculation of Alpha(i). WATR The radius of the water probe. Usually this is set to 1.4 Angstroms. If this were changed, other parameters would have to be modified. EPSILON This is the value of the dielectric constant for the solvent medium. The default value is 80. KAPPA Debye-Huckel ionic term: Units of inverse length (Angs). Default is 0 (no salt). GEOM Select geometric cross-term in Still equation (default). ARITH Select arithmetic cross-term in Still equation. P6 Exponent in exponential of Still equation. Default is 4, for historical reasons. Value of 8 is RECOMMENDED for GEOM, 6.5 for ARITH. WEIGHT Use WMAIN array for radii. (Default uses vdW radii array) CLEAr Clear all arrays and logical flags used in Generalized Born calculation. Use command by itself.

### Parameters specific to GBMV I and II¶

 FIXA Update alphas only if coordinates have changed more than expected for finite differences. Useful for static pka calculations. With FIXA keyword, finite-difference wouldn’t work correctly, hence it must be specified. Not on by default. ALFRQ Update frequency of Born radii. Use with great caution! One of LIMP,IMP, or EMP options must be selected. (Default 1) Values other 1 not generally recommended. LIMP Use ALFRQ*(dE/dalpha)(dalpha/dx) part of GB force every ALFRQ steps. For ALFRQ <= 5. EMP Decay constant of the impulse force. Default is 1.5, which is meant for ALFRQ of 5. Generally, EMP ~= ALFRQ/4. For ALFRQ <= 10. (Recommended option) IMP Use (dE/dalpha)(dalpha/dx) part of GB force every ALFRQ steps. Any ALFRQ can be used. Only meant for equilibrium calculations. DN The cell width of the lookup grid. Larger values make program slower. Smaller values use up more memory. Default of 1.0 A is best compromise between speed and memory. BETA Smoothing factor for tailing off of volume. Values of around -100 are fine for GBMV I. Values between -8 to -50 are reasonable for GBMV II. (Default -20) Smaller values of beta lead to more stable dynamics, but compromise the agreement with Poisson theory. In GBMV II the choice of BETA also affects P3. Good pairs of values for GBMV II are: ```BETA = -20, P3 = 0.70 BETA = -12, P3 = 0.65 * recommended as best compromise BETA = -10, P3 = 0.57 BETA = -8, P3 = 0.35``` LAMBda The threshold value for the atomic volumes. In GBMV I, smaller values produces shorter Born radii and wide variance w/respect to accurate PB radii. Large values produce larger radii but smaller variance. In GBMV II, value should be kept at 0.5. BUFR Distance that any atom is allowed to move before lookup table is rebuilt. Larger values lead to less lookup table update but larger memory usage. Use 0.0 for static structure. Values between 0.2 and 1.0 Angstrom. (Default 0.5) MEM Percentage extra memory beyond hypothetical calculation of table size. (Default 10) TOL Accuracy of the switching function used to determine accuracy of the first derivatives, i.e. forces. (Default 1e-8) SA Surface area coefficient (KCAL/(MOL*A**2)). (Default 0.0) SASA Energy term shows up under EXTERN/ASP. SB Surface area constant (KCAL/MOL) (no effect on forces) (Default 0.0) SON The startpoint for the switching function of each hard sphere. (Default 1.2) Units in Angstroms SOFF The endpoint for the switching function of each hard sphere. (Default 1.5) P1 The multiplicative factor for the exponent of the quartic exponential atomic function: `Gamma(i) = P1 * log(lambda)/(Rad(i)^4)` Parameters specific to GBMV II: P1,P2 Variables which affect the shape of the VSA atomic function in the region of R to R+2. ```F(x) = A^2 / (A + x^2 - R^2)^2 where A = P1 * R + P2 (Defaults: P1 = 1.25/P2 = 0.45)``` P3 Scaling factor of VSA function. Default = 0.7 This factor depends on the value chosen for BETA (see description above) P4 Scaling coefficient for correction term to Still’s equation. (set to 0.0 for now) P5 Exponent to the Still correction term. (use default for now) HSX1/HSX2 Start and stop of hard-sphere tail with R(vdW) as origin. (Defaults: -0.125/0.25). ONX/OFFX Start and stop of VSA tail. Increasing values up to 2.8 A makes better accuracy, however slows calculation. Compromise of 1.9/2.1 is default. FAST Turns on fast GBMV routine. SGBFRQ Update frequency of internal lookup list in fast GBMV mode (Default 1). Values between 1 and 10 are recommended. SXD Delta used in fast GBMV mode lookup buffer. (Default 0). Recommended values between 0.1 and 0.5. Requires ‘FAST 1’ GBVDW If present, the VDW dispersion term is turned on.

### Parameters specific to HDGB (CORR = 3, 4, 5)¶

 UNEPS Unit number of an input file holding the delectric profile values (Use -1 for the default profile). The format of this input file is restricted. Comments are not allowed in a file. The dielectric profile must be sampled in equal intervals. The first line needs the number of sampling points n and the sampling interval h (Angstrom). Two columns of the z coordinates (Angstrom) and dielectric constants. The example is given in test/data/hdgb_eps.dat. A4,A5 Parameters in calculation of Alpha(i). A4 and A5 correspond to the parameter D and E of Equation (15) in the reference (3) respectively. ZS,ZM,ZT Parameters for a switching function for the nonpolar energy. ST0 ZS, ZM, ZT, and ST0 corresponds to Za, Zb, Zc, and C of Equation (11) in the reference (4). HDGBRC If this flag is specified, the radius will be corrected upon insertion to an implicit membrane. (Only availabe for CORR = 3)

### Parameters specific to Grid-based GBMV¶

 ML Number of surface points to carve out re-entrant surface CONV Smear grid with cross-shaped blur function to improve accuracy

## Usage Examples and Compatibility¶

The examples below illustrate some of the uses of the generalized Born Molecular Volume (GBMV) module. See c29test/gbmvtest.inp for more examples.

Warning

THERE ARE TWO REQUIREMENTS TO RUN GBMV

1. Coordinates MUST be defined for all atoms before invoking the GBMV keyword. Otherwise, “infinite” grid is established which uses too much memory.
2. CUTOFF Parameters MUST be defined. For non-infinite cutoffs, “switch” in nonbonded parameters is NECESSARY.

### Example 1¶

```!To perform a single-point energy calculation w/infinite cutoffs using
!GBMV I algorithm (any forcefield):

GBMV BETA -100 EPSILON 80 DN 1.0 WATR 1.4 TT 2.92 -
SHIFT -0.5 ESHIFT 0.0 LAMBDA1 0.1 P1 0.44 -
BUFR 0.5 Mem 20 CUTA 20 WTYP 0 -

ENERGY ctonnb 979 ctofnb 989 cutnb 999```

### Example 2¶

```!To perform a single-point energy calculation w/infinite cutoffs using
!the GBMV II algorithm (any forcefield):

GBMV BETA -20 EPSILON 80 DN 1.0 watr 1.4 GEOM -
TOL 1e-8 BUFR 0.5 Mem 10 CUTA 20 HSX1 -0.125 HSX2 0.25 -
ALFRQ 1 EMP 1.5 P4 0.0 P6 8.0 P3 0.70 ONX 1.9 OFFX 2.1 -
WTYP 2 NPHI 38 SHIFT -0.102 SLOPE 0.9085 CORR 1

ENERGY ctonnb 979 ctofnb 989 cutnb 999

GBMV CLEAR ! Clear GB arrays```

### Example 3¶

```!Recommended setup for molecular dynamics simulations with
!the GBMV II algorithm:

UPDATE atom CDIE eps 1 cutnb 21 ctofnb 18 ctonnb 16 switch vswitch

GBMV EPSILON 80 BUFR 0.2 MEM 20 CUTA 20 ALFRQ 1 -
GEOM BETA -12 P1 0.45 P2 1.25 P3 0.65 P6 8.0 -
CORR 1 SHIFT -0.1 SLOPE 0.9 WTYP 1 NPHI 5 -
FAST 1 SGBFRQ 4 SXD 0.3

!You should use Langevin dynamics and a 1.5 fs time step (with SHAKE)
!is recommended for optimal stability. Many applications will also
!tolerate 2 fs time step

SHAKE BONH TOL 1E-08 PARAM

SCALAR FBETA SET 10 SELECT .not. TYPE H* END

DYNAMICS LEAP LANG START TIMESTEP 0.0015 NSTEP 1000 -
FIRSTT 298 FINALT 298 BYCB -
INBFREQ -1 IASORS 1 IASVEL 1 NPRINT 100 IPRFRQ 100 NSAVC 100 -
ECHECK 20 TBATH 298 RBUF 0 ILBFREQ 50 -
IUNVEL -1 IUNREA 11 IUNWRI 12 IUNCRD 13 KUNIT -1```

### Example 4¶

```!Grid-based GBMV:

GBMV GRID EPSILON 80 DN 0.2 watr 1.4 GEOM P6 8.0 -
WTYP 0 NPHI 10 SHIFT -0.007998 SLOPE 0.9026 CORR 1 CONV

ENERGY ctonnb 979 ctofnb 989 cutnb 999```

### Example 5¶

```! HDGB DPPC membrane
! If you want the default DPPC profile used in the reference (4),
! comment out the open file statement and set UNEPS to -1.

! The input file will be closed automatically, so you don't need
! the explicit close statement.
open unit 1 name eps.dat read form

GBMV A1 0.3255 A3 1.085 A4 -0.14 A5 -0.15 -
UNEPS 1 -
ZS 0.5 ZM 9.2 ZT 25 ST0 0.32

ENERGY ctonnb 979 ctofnb 989 cutnb 999```

## Issues¶

### Known Compatible with¶

• PARALLEL
• CONS FIX
• INTE
• PHMD
• VIBRAN (finite difference second derivatives)
• MMFF (WEIGHT keyword must be used)

### Known Incompatible with (so far)¶

• VIBRAN (no analytic second derivatives)
• BLOCK (hence not compat. w/ PERT/PIMPLEM/PERTURB/REPLICA)
• IMAGE/CRYSTAL
• EWALD
• multiple dielectric
• QUANTUM* (single energy with original charges is ok)
• FLUCQ
• GAMESS
• GENBORN
• GRID
• PRESSURE
• SBOUND

• GENBORN
• GBMV
• GBMVFAST
• HDGB