Protein Dynamics, Entrophy & Function 

The formation of protein complexes involves a complicated manifold of intermolecular interactions that often includes dozens of amino acids and thousands of square Ångstroms of contact area. The energetic origins of high affinity interactions are potentially quite diverse and complex. This complexity is reflected in the general difficulty of computing the energetics of high affinity interactions between proteins based on molecular structure alone. Indeed, with a few notable exceptions, structurebased design of pharmaceuticals has been largely impeded by this barrier. Our central interest is the role of protein conformational entropy in modulating the free energy of the association of a protein with a ligand. Simple decomposition of the free energy emphasizes the fact that the entropy of binding is comprised of contributions from the protein, the ligand and the solvent: 

Historically, the contributions of solvent entropy to the energetics of proteinligands interactions have taken center stage and are framed in terms of the socalled hydrophobic effect. Hydrophobic solvation by water has been and continues to be the subject of extensive analysis. In principle, the entropic contributions of a structured protein to the binding of a ligand (DSprotein) includes both changes in its internal conformational entropy (DSconf) as well as changes in rotational and translational entropy (DSRT). The equation above emphasizes that the measurement of the entropy of binding does not resolve contributions from internal protein conformational entropy. Indeed it is only recently that experimental evidence has been obtained that suggests the conformational entropy of structured proteins is significant and, more importantly, sufficiently responsive to influence the thermodynamics of protein association. A few years ago we found an intriguing correlation between the change in dynamics of the signal transduction protein calmodulin upon binding of the domains of proteins that it regulates in a calciumdependent manner (figure on right). This data was interpreted using a harmonic oscillator model to general a parametric relationship between motion over conformational states and the conformational entropy that it inherently implies. This has a number of obvious limitations and potential defects (subsequently avoided, see below) but did provide a powerful insight into the largely ignored potential for conformational entropy of proteins to be significantly involved in the entropy underlying the free energy of protein function such as ligand binding. More recently we have used the calmodulin system to calibrate the "entropy meter" (right figure). This required determining the dynamics of the bound target domains. Using a few simple and seemingly reasonable assumptions, we arrived at a very simple linear relationship: The total binding entropy was measured by ITC, the solvent entropy was calculated from the structures of free CaM and the complexes and using a completely solvated model for the unstructured free domains. The changes in order parameters are residueweighted i.e. they are assumed to reflect the entire protein. The rotational and translation entropy and "other" entropy (e.g. from electrostriction arising from breaking ion pairs) was assumed to be constant across the complexes. See Marlow et al. (2010) Nature Chemical Biology for more details. The obtained calibration was remarkable (below figure). The ability to "rigorously" calibrate the dynamical proxy allows us to skip over the objections to using an "oscillator inventory", which is modeldependent, to estimate conformational entropy. The obtained calibration indicates that conformational entropy does indeed contribute significantly to the overall binding entropy of the calmodulin complexes. Even more interesting is the observation that it is the conformational entropy of calmodulin that is the most variable component and in effect tunes the entropy of binding. This unprecedented view strongly suggests a role for evolutionary variation of entropy in the development of high affinity binding partners, which is a current focus of our work in this area. Overall, our general goals are to:
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Useful papers: Marlow et al. (2010) Nature Chem. Biol.

Correlation of apparent contribution of conformational entropy of calmodulin to the total free energy of binding of target domains by calmodulin. The apparent conformational entropy of binding is estimated from changes in fast internal dynamics reported by methylbearing amino acids and employed deuterium relaxation methods. See Frederick et al. (2007) Nature for more details. Calibration of the dynamical proxy for protein conformational entropy. Simple considerations lead to the prediction of a quantitative linear relationship between the total binding entropy and the entropy of solvent to the conformational entropy by NMR relaxation parameters derived from methyl bearing amino acids (see Equation above). The error bars represent the average standard deviation of the difference of the average parameters between free CaM and the complex with each target. Each parameter was derived from T1 and T1ρ values obtained at two magnetic field strengths. The average dynamics of wildtype and E84K CaM were based on 52 resolved methyl sites. The average dynamics of the six complexes shown were based on 73 to 88 resolved methyl sites. The lower CaM:CaMKKα(p) datum is a clear outlier (Jackknife distance 8.8, all others < 2.3). The upper CaM:CaMKKα(p) point results from a simple correction to the solvent entropy arising from a postulated hydrophobic cluster in the free state of this target domain. Excluding the CaM:CaMKKα(p) points results in a linear regression statistic R of 0.95. This regression line is shown. The slope (m = 0.037 ± 0.007 kJ K1 mol res1) allows for empirical calibration of the conversion of changes in sidechain dynamics to a quantitative estimate of changes in conformational entropy. The ordinate intercept is 0.26 ± 0.18 kJ K1 mol res1. See Marlow et al (2010) Nature Chemical Biology for further details. Decomposition of the entropy of binding of target domains to calciumsaturated calmodulin. Solid diamonds are the solvent entropies calculated from the changes in accessible surface area and include the correction resulting from the postulated hydrophobic cluster of the free CaMKα(p) target domain. The uncorrected value for CaMKα(p) is shown as an open diamond. No structure is available for the CaM:PDE(p) complex so the corresponding solvent entropy cannot be calculated. Solid circles and triangles are the contributions to the binding entropy by the conformational entropy of CaM and the target domains, respectively. Solid squares are the contributions to the binding entropy not reflected in the measured dynamics, which is obtained from linear regression (see Fig. 3). Though not required, there are interesting linear correlations between the total binding entropy and its components. There is a significant (R = 0.94) but weakly dependent (slope = 0.25 ± 0.04) negative linear correlation between solvent entropy and total binding entropy. In contrast, there is a significant (R = 0.91) and relatively strong positive dependence (slope = +1.0 ± 0.2) observed between the change of conformational entropy of CaM and the total binding entropy. The apparent negative correlation between target binding entropy and the total binding entropy is not statistically significant (R = 0.045). See Marlow et al (2010) Nature Chemical Biology for further details. 