Electrostatics of Channels—pK’s and potential of mean force for permeation

1. Electrostatics of Channels—pK’s and potential of mean force for permeation Sameer Varma, NCSA/UIUC/Beckman Institute Computational Biology/Nanoscience Group

2. Inputs to the calculation of pK’s of titratable residues in proteins • Inherent pK’s of residues in solution • Shape of the protein and position of titratable residues

3. H Amino Acid R1 R2 N CA C N CA C H H O O H H Peptide linkage Basic principle of pKa calculation (Titration curve calculation) • Some amino acid side chains are titratable. • They have hydrogen atoms that stay bonded to the side chain with a certain probability chemically defined as pKa • This probability depends on the following :- • Local electric field at the hydrogen atom – E • Presence of other such titratable groups in the vicinity • Concentration of hydrogen ions in the solvent – pH

4. First stage of the calculation Use the Poisson-Boltzmann equation to calculate the Born Energy for moving the titratable residue to the surface of the protein. From the Born energy calculate the new pK and the ionization state of each residue over a range of pH’s

5. Gs(AH,A) AsH Hs As + GS,P(AH) GS,P(A) GP(AH,A) ApH Ap + Hs pKa calculation protocol • Step 1 : Change in local E field at titratable site • Presence of protein charges and dielectric boundaries in proximity alter local electric fields seen by titratable residues. • Following the thermodynamic cycle of ionizing a residue in presence and absence of the the protein system: • Thus, change in local E field  change in free energy G (with no change in entropy) • This change in free energy can be related to shift in pKa as: • Intrinsic pKa shift

6. Here, • GIONIZATION = GP(AH,A) - Gs(AH,A) • Assuming a single site protonation model, it may be shown that the above ionization energy differences can be written as • Where, • 1Kand 1K are the potentials at the location of charge K created by a unit positive charge at site 1, the site of the titratable atom. • qK' is the charge of the site after ionization and qKis that before ionization. • n and N are the atoms in the amino acid and the protein.

7. Important observation regarding ionization energy difference equations: • Difference in free energy is independent of all the charges in the amino acid or the protein except the charge at the titrating site! • Implementation • The desired potentials can be easily obtained by performing 2 Finite Difference Poisson Boltzmann (FDPB) calculations for each titratable residues: • With a single unit charge on the ionization site of the group in isolation to obtain the values of 1K • With a single unit charge on the same site for the group in the neutral protein to get the values of 1K • Since these FDPB calculations are mutually independent, these calculations have been implemented to run in parallel on multiple processors.

8. H+ solvent R1 R2 R3 R4 protein interior • Step 2 : Presence of other titratable sites • Titratable residues in proximity contend for an acquisition/release of hydrogen ions giving rise to inter-dependent titration behaviors. • The activity that each residue exhibits in this behavior depends on: • a. availability of hydrogen ions  pH • b.Intrinsic pKa’s of the titratable residues. • c. proximity of these residues  interaction energy between these residuesW(calculated by solving the PB equation)

9. A statistical distribution that incorporates this effect can be based on a probability that is proportional to the Boltzmann-weighted sum over all protonation states at each pH (Karplus and Bashford, 1990). • The fraction of molecules having site iprotonated, or a point in the titration curve of site i for a given pH is then: • Where, • x is an N element protonation state vector whose elements are zero or one according to whether site  is protonated or unprotonated, • qo is the charge of the unprotonated state of site  • {x} indicates summation over all 2N possible protonation states and • W are the interactions energy between titrating sites  and .

10. ImplementationFor a protein with N titratable sites, this equation has 2N terms in the summation. To generate a titration curve,  needs to be solved for every titratable residue in the protein for a range of pH values.The solution of  for every titratable site yields qo for that site which needs to be plugged back into the equation and iterated for all  to achieve self- consistency.The algorithm for solving this equation is based on a “clustering method” where small clusters of titratable sites, based on an interaction energy cut-off, are independently solved before solving the complete system.Interaction energies W1Kare obtained by performing a single FDPB calculation with a single unit charge at site 1 for the group in the neutral protein. (NOTE: W1K = WK1. This calculation is similar to the previously mentioned FDPB calculation required for determining free energy differences. So these energies can be obtained from the results of the previous FDPB calculations.)

11. Result of implementation of the theory: Titration curves for selected amino acids in the ompF porin channel at 150mM KCl. (These titration curves are used to calculate the charges on titratable residues for the PNP calculations.)

12. Second use of electrostatics is to compute potential of mean force for ion motion, to put into either Brownian dynamics or PNP solution of flux equation.A problem emerges in very narrow channels, as shown in the next slide.

13. The problem, as shown above, is that in very narrow regions, such as the selectivity filter of the potassium channel shown above, spurious features arise due to unphysical proximity of ionic charge to dielectric boundary.

14. Analysis of problem: The use of the crystal structure for the protein does not take into account the deformation of the protein as the ion moves through the selectivity filter. Possible solution (work in progress): Use molecular dynamics to calculate how the structure deforms with ion passage, use the appropriately deformed structure to do the Poisson-Boltzmann calculation for each ion position.