Minimization - Example Calculation

Substrate Binding Study of DHFR-Trimethoprim

Roberts et al. (1986) reported a comprehensive energetic and structural analysis of the dihydrofolate reductase (DHFR)-trimethoprim system. The objective of the study was to understand, in molecular terms, why trimethoprim binds 100,000 times more strongly to bacterial DHFR than to vertebrate DHFR. This selectivity is what makes trimethoprim useful as an antibiotic. This study provides an excellent example of the strategy involved in constructing appropriate molecular structures, minimizing the structures, and extracting and interpreting both structural and energetic information from the minimized systems.

Strategic approach to studying an enzyme-ligand complex

The central objective of the DHFR-trimethoprim study was to probe the energetic basis for trimethoprim selectivity between bacterial and vertebrate DHFR, as well as account for the cooperativity (enhanced trimethoprim binding) observed in the bacterial system in the presence of NADPH. The approach was to subject four different complexes to rigorous energy minimization beginning from crystal structures and then to thoroughly analyze the energetic components of the minimized structures in order to discover the structural basis for the observed selectivity. The four systems minimized were:

  1. E. coli DHFR + trimethoprim.
  2. E. coli DHFR + trimethoprim + NADPH.
  3. Chicken liver DHFR + trimethoprim.
  4. Chicken liver DHFR + trimethoprim + NADPH.
Crystal structures were available only for the E. coli binary and the chicken liver ternary complexes (1 and 4). The E. coli ternary complex (2) was constructed by using the NADPH coordinates from the known structure of L. casei DHFR with bound methotrexate and NADPH. The chicken liver binary system (3) was built by replacing the NADPH of system 4 with water molecules.

Binding of ligands to enzymes depends strongly on solvent and ionic interactions. Therefore, one of the first steps is choosing the ionization state of side chains and the treatment of solvent. In this study, the side chains of Glu, Asp, Arg, and Lys were charged, as were the N- and C-terminal amine and carboxylate groups. In addition, a His at position 45 is known to be protonated when NADPH is bound, so this amino acid was treated as a special case. Waters that appeared to play a structural role in the protein structure and any other crystallographic waters within 7 Å of trimethoprim and NADPH were retained. Additional waters were added to fill the volume within 7 Å of trimethoprim and NADPH and within 3 Å of all charged residues. All hydrogens were included explicitly, and a dielectric constant of 1 was used throughout.

A common and important modeling objective is to relax poorly defined regions of a protein structure without disrupting well defined regions. Above, we explained how template forcing and tethering can be used for this. In this study, the protein was tethered in stages to relax the least well refined parts of the model first, starting with the solvent. The backbone of the protein was restrained next to allow the side chains to make minor adjustments. Eventually, the entire system was relaxed unrestrained, until the minimization converged (at an average absolute derivative of 0.0002 kcal mol-1 Å-1).

Structural analysis

Following minimization of structures from the crystal coordinates, it is necessary to confirm that the model is sufficiently accurate to justify detailed energy analysis. Unrealistic deviations from the experimental structure may indicate that the model is improperly defined or that the energy expression is inaccurate, and compromises subsequent structural or energetic analysis. A common measure of the fit of two protein structures is the rms deviation of corresponding atoms in the two superimposed structures. Minimized structures are superimposed onto the corresponding initial crystal structure using a least-squares fit of the heavy atoms of their secondary structures, i.e., the helices and strands. By using only atoms in the relatively rigid secondary structures, you avoid introducing artifacts in the superposition that arise from bias due to poorly defined regions or loops at the surface, which is where the model is expected to be less rigorous because bulk solvent, counterions, and crystal contacts are ignored. Given this preliminary superposition, the rms deviation can be calculated for the entire protein or for local regions such as the active site, to quantify the relative movement throughout the protein complex.

Table 4-1 summarizes the results of several rms deviation analyses for each of the DHFR complexes. Figure 4-9 graphically represents the rms deviations for the E. coli DHFR + trimethoprim system. As expected, internal regions of the protein, such as the active site, were found to fit the experimental structure more accurately than areas on the surface of the protein. The well defined areas of the protein, such as the active site and the C carbons of the helices and strands, show rms deviations substantially lower than does the overall protein. The pyrimidine ring of trimethoprim, deep in the active site cleft, deviates from the experimentally observed structure less than does the trimethoxyphenyl moiety, which extends out towards the surface of the protein. Similarly, in the chicken liver ternary complex, the nicotinamide ring, which extends into the active site, has a much smaller rms deviation (0.34 Å) than the NADPH molecule as a whole, which generally lies on the surface of the protein. Thus, the crucial area for this study, the active site, shows good agreement with the experimental structure and was judged adequate for further detailed structural and energetic analysis.

Structural and energetic analysis

Roberts et al. (1986) analyzed the energetics of the minimized structures by breaking down the total energy into various components, including intramolecular strain (bond lengths and angles, etc.) and intermolecular energies (van der Waals and electrostatic interactions) between the various molecules. The ability to consider the energetic contributions due to various kinds of interactions is a crucial part of any energetic modeling strategy. For the purpose of illustration, this discussion focuses only on the analysis of how NADPH enhances the binding of trimethoprim and, in particular, does so to different degrees in the vertebrate and bacterial systems.

The selectivity of bacterial versus vertebrate DHFR for trimethoprim binding depends partially on cooperativity with respect to NADPH. Specifically, Baccanari et al. (1982) showed that the E. coli DHFR-trimethoprim dissociation constant is 40-fold less in the presence of NADPH, whereas for rodent lymphoma DHFR the corresponding decrease is only a factor of 2.8. By examining the two DHFR systems both with and without NADPH, the energetic basis underlying any cooperativity was investigated. Initially, two general mechanisms for cooperativity appeared possible: an indirect mechanism, in which the NADPH changes the protein so that it has higher affinity for trimethoprim; or a direct mechanism, in which favorable interactions between NADPH and trimethoprim directly stabilize the ternary complex. If the indirect mechanism is responsible, the interaction energy between the protein and trimethoprim would be expected to improve in the presence of NADPH. The calculated intermolecular energy between trimethoprim and the protein in the binary complex was -214.4 kcal mol-1 compared with -188.7 kcal mol-1 in the presence of NADPH. The major difference in binding energy arose because His45 becomes protonated (Poe et al. 1979) upon binding of NADPH, as indicated by proton magnetic resonance studies.

The interaction of this protonated histidine with the positively charged trimethoprim accounts for 20.9 kcal mol-1 of the 25.7 kcal mol-1 difference in the interaction energy. (Note the importance of correctly identifying the charge state of His45.)

Thus, the energy results indicate that NADPH binding does not predispose the enzyme to interact more favorably with trimethoprim, and protein-ligand interactions cannot explain the observed cooperativity in E. coli DHFR.

Comparison of the interactions of NADPH with trimethoprim in the E. coli ternary complex and the chicken liver ternary complex provides a clue as to the source of the enhanced cooperativity of the bacterial enzyme with NADPH. The energy of this interaction in the minimized structures is 6.6 kcal mol-1 more favorable in the E. coli complex. This is not sufficient to overcome the 25.7 kcal mol-1 increase in energy experienced upon binding of NADPH and indicates that quantitative energy comparisons may be unreliable in this model. Nevertheless, the qualitative trend is still worth investigating more closely. By partitioning these interaction energies into contributions from the various functional groups, the specific interactions responsible for this difference can be determined (Table 4-2). The favorable interactions arise predominantly from the spatial relationship between the trimethoxyphenyl ring of trimethoprim and NADPH. The trimethoxyphenyl ring occupies a different cleft in the chicken liver enzyme, so that the ring is closer to the nicotinamide of NADPH and its associated ribose in the E. coli system than in the chicken liver ternary complex. Table 4-2 shows that the van der Waals interactions between these groups of the NADPH and the trimethoprim phenyl ring are 2.2 kcal mol-1 more favorable in the E. coli complex. Coulombic interactions between the entire NADPH molecule and the trimethoxyphenyl ring are 3.1 kcal mol-1 more favorable in the E. coli ternary complex. This suggests that it is the difference in trimethoprim-NADPH interactions that underlies the cooperativity in the E. coli complex. Specifically, the cooperativity appears to be due to better van der Waals interactions between NADPH and the trimethoxyphenyl ring of trimethoprim in the E. coli complex than in the vertebrate DHFR.

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