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The Born ion

One of the canonical examples for polar solvation is the Born ion: a nonpolarizable sphere with a single charge at its center surrounded by an aqueous medium. In the absence of external ions, the polar solvation energy for this system is given by

where q is the ion charge, a is the ion radius, and the two ε variables denote the internal and external (solution) dielectric constants. This model assumes zero ionic strength.

We can setup a PQR file for the Born ion for use with APBS with the contents:
   REMARK  This is an ion with a 3 A radius and a +1 e charge
   ATOM      1   I  ION     1 0.000   0.000   0.000  1.00 3.00
We're interested in performing two APBS calculations for the charging free energies in homogeneous and heterogeneous dielectric coefficients. We'll assume the internal dielectric coefficient is 1 (e.g., a vacuum) and the external dielectric coefficient is 78.54 (e.g., water). for these settings, the polar Born ion solvation energy expression has the form

where z is the ion charge in electrons and R is the ion size in Å.

This solvation energy calculation can be setup in APBS with the following input file:
       mol pqr born.pqr
   elec name solv # Electrostatics calculation on the solvated state
       mg-manual # Specify the mode for APBS to run
       dime 97 97 97 # The grid dimensions
       nlev 4 # Multigrid level parameter
       grid 0.33 0.33 0.33 # Grid spacing
       gcent mol 1 # Center the grid on molecule 1
       mol 1 # Perform the calculation on molecule 1
       lpbe # Solve the linearized Poisson-Boltzmann equation
       bcfl mdh # Use all multipole moments when calculating the potential
       pdie 1.0 # Solute dielectric
       sdie 78.54 # Solvent dielectric
       chgm spl2 # Spline-based discretization of the delta functions
       srfm mol # Molecular surface definition
       srad 1.4 # Solvent probe radius (for molecular surface)
       swin 0.3 # Solvent surface spline window (not used here)
       sdens 10.0 # Sphere density for accessibility object
       temp 298.15 # Temperature
       calcenergy total # Calculate energies
       calcforce no # Do not calculate forces
   elec name ref # Calculate potential for reference (vacuum) state
       dime 97 97 97
       nlev 4
       grid 0.33 0.33 0.33
       gcent mol 1
       mol 1
       bcfl mdh
       pdie 1.0
       sdie 1.0
       chgm spl2
       srfm mol
       srad 1.4
       swin 0.3
       sdens 10.0
       temp 298.15
       calcenergy total
       calcforce no
   # Calculate solvation energy
   print energy solv - ref end
Running this example with a recent version of APBS should give an answer of -229.59 kJ/mol which is in good agreement with the -230.62 kJ/mol predicted by the analytic formula above.

Note that the Born example above can be easily generalized to other polar solvation energy calculations. For example, ions could be added to the solv ELEC, dielectric constants could be modified, surface definitions could be changed (in both ELEC sections!), or more complicated molecules could be examined. Many of the examples included with APBS (e.g., solv and ionize) also demonstrate solvation energy calculations.

Note that, as molecules get larger, it is important to examine the sensitivity of the calculated polar solvation energies with respect to grid spacings and dimensions.