In many molecular dynamics simulations, water is explicitly modeled along with the protein. In the absence of explicit water, proteins modeled with the energy function in Equation 2.4 unfold over the course of a molecular dynamics trajectory. By including water along with the protein, one can maintain proteins’ folded states. So in addition to the thousands of atoms in a single protein, a biophysicist must include tens of thousands of water molecules (Berne, 1977). Getting the right water model has been an important topic in molecular dynamics (Jorgenson et al., 1983; Hermans et al., 1988; Silverstein et al., 1998). Modeling water explicitly costs much more than modeling a protein by itself and considerable effort has been directed at simulating only as much water as is necessary (Brooks and Karplus, 1983; Brungler et al., 1985; Brooks and Karplus, 1989). In protein design, the expense of modeling water explicitly would be prohibitive. Designers do not run molecular dynamics simulations during the course of their design calculations. To avoid the costs associated with modeling water explicitly, attempts have been made to express water’s effects implicitly.
The amount of buried hydrophobic surface area has proven the most accurate metric to gage the stability conferred by the hydrophobic effect (Dill, 1990; Dahiyat and Mayo, 1996; Kortemme et al., 1998). To measure surface burial requires a representation of the protein’s solvent-accessible surface (SAS) (Lee and Richards, 1971). Surface burial cannot be accurately modeled by a pairwise decomposable function. Once one
Figure 2.10: Pairwise Surface Burial. (a) The portion of residue i that is not buried from solvent by the backbone (b) Residuej buries some ofi’s surface from water (blue dots). (c) Residuekburies some ofi’s surface from water. (d) Together, residuesjandk
bury some ofi’s surface doubly (red dots). In Street and Mayo’s pairwise decomposable energy function, this region is doubly counted, and over-counting in general is treated by scaling.
atom buries a certain portion of the surface of another atom, that surface portion cannot become more buried by the approach of a third atom. Because the solvent- accessible-surface area (SASA) is not decomposable into a sum of pairs, exact SASA- based solvation models have not been previously incorporated into the optimization step of protein design software.
Street and Mayo described an approximate SASA-based solvation model that is pairwise decomposable on the level of rotamer pairs (Street and Mayo, 1998). In their technique, they placed rotamer pairs onto a backbone from which all other side chains had been removed. They then measured the surface area that each rotamer buried of the other (Figure 2.10). Because the surface of one rotamer can be buried multiple times by other rotamers, this technique over-counts the degree of surface burial. They correct the over-counting of surface area burial by a scaling constant. Surfaces that are fully covered but only by a single other rotamer will be counted as somewhat exposed. Zhang, Zeng and Wingreen improved upon Street and Mayo’s approximate SASA- based solvation model. They placed rotamer pairs onto a backbone from which all other side chains had been replaced by large spheres that approximated the side chains’ presence, and measured the surface area each rotamer buried of the other (Zhang et al., 2004). Because these extra spheres buried certain portions of each rotamer, there was less double-counting of surface burial. Instead their approximation under-counts burial in some areas and over-counts the burial in others. The average error in the approximation is neither positive nor negative; it is not biased towards under-counting or over-counting surface area. This is not to say that the approximation usually gets the right answer; their approximation regularly miscalculates SASA by 10 ˚A2.
Lazaridis and Karplus proposed an implicit solvation model that is decomposable into the sum of atom pair interactions (Lazaridis and Karplus, 1999). They conceived of a field, Fi, that surrounds atom i where the field describes for a point in space the solvation energy contributed by that point if it were occupied by water. An atomj that approaches i excludes water from occupying a region of space surrounding i (Figure 2.11). They define the change in the free energy of solvation of atom i, ∆Gslv
i , caused by placing a single atom j in i’s solvation field as the difference between a reference solvation energy, ∆GREF
i and a volume integral over this field: ∆Gslvi = ∆G REF i − Z Vj Fi(r)dr. (2.4)
They define the change in free energy of solvation induced by all atomsj that surround
i as ∆Gslv i = ∆G REF i − X j Z Vj Fi(r)dr (2.5)
Having decided upon the form of a volume integral, they needed a function F to describe the field. They observed that the first solvation sphere surrounding an atom accounted for∼84% of the solvation energy. They also noticed that the error function, which is given by erf(y) = √2 π Z y 0 e−x2dx.
evaluates to 84% for erf(1). Thus to describe the field, thesolvation free energy density, for a point at a distance ofr from the center of atomi with radiusRi, with correlation distanceλi, and with a field intensity of αi they chose the following function:
Fi(r) = αie
−(r−λRi)2
4πr2 (2.6)
Both αi and λi depend on what the type of atom i is (aromatic carbon, carbonyl oxygen, etc). I should note here, though it I discuss it again in Chapter 8, that water is usually approximated as a sphere with a radius of 1.4 ˚A– the radius of an oxygen atom. Therefore, for most atoms, they set λi = Ri+ 2.8 ˚A so that the correlation distance represented the first solvation sphere surrounding atomi. If the entire volume surrounding atom i out to 2.8 ˚A past its van der Waals surface were occupied by other atoms, then the volume integral in Equation 2.5 would reflect an 84% loss in solvation free energy for the atom. Without getting into the details, the constants
Figure 2.11: Pairwise Decomposable Solvation Model. A two-dimensional analogy for Lazaridis’ and Karplus’ solvation free energy density. The density is shown in gray scale surrounding the atom i in the center. Atoms j and k prevent water molecules from interacting withiin the regionsj and koccupy, causing a loss of solvation energy. This loss is given by the integral of the density inside the volumes (areas) excluded by the two atoms. Atoms j and k effectively punch holes in the solvation field; one could imagine using a hole punch on this piece of paper to carve out some additional gray area. The volume of each atom is set so that volumes of overlapping (bonded) atoms are not double-counted.
αi were chosen to penalize the burial of hydrophilic groups, and reward the burial of hydrophobic groups.
Lazaridis and Karplus tested their implicit solvation model with molecular dynam- ics. As mentioned before, in molecular dynamics simulations that do not include explicit or implicit solvent, the protein molecules tend to unfold over time. The force field de- scribed in Equation 2.4 is unable to keep a protein folded in vacuo. When Lazaridis and Karplus included their implicit solvation model, the proteins they simulated did not unfold.