- Nonlinear elasticity, proteinquakes, and the energy landscapes of functional transitions in proteins

Nonlinear elasticity, proteinquakes, and the energy landscapes of functional transitions in proteins

O. Miyashita {dagger} {ddagger}, J. N. Onuchic {dagger} {ddagger}, and P. G. Wolynes {dagger} {ddagger} § 

{dagger}Center for Theoretical Biological Physics and Departments of {ddagger}Physics and §Chemistry and Biochemistry, University of California at San Diego, 9500 Gilman Drive, La Jolla, CA 92093

Large-scale motions of biomolecules involve linear elastic deformations along low-frequency normal modes, but for function nonlinearity is essential. In addition, unlike macroscopic machines, biological machines can locally break and then reassemble during function. We present a model for global structural transformations, such as allostery, that involve large-scale motion and possible partial unfolding, illustrating the method with the conformational transition of adenylate kinase. Structural deformation between open and closed states occurs via low-frequency modes on separate reactant and product surfaces, switching from one state to the other when energetically favorable. The switching model is the most straightforward anharmonic interpolation, which allows the barrier for a process to be estimated from a linear normal mode calculation, which by itself cannot be used for activated events. Local unfolding, or cracking, occurs in regions where the elastic stress becomes too high during the transition. Cracking leads to a counterintuitive catalytic effect of added denaturant on allosteric enzyme function. It also leads to unusual relationships between equilibrium constant and rate like those seen recently in single-molecule experiments of motor proteins.

Source: PNAS vol. 100, no. 22,  2003


The regulation of biological machinery through allostery is a dominant theme in our modern molecular understanding of life. Allostery requires a biomolecule to have at least a pair or, more likely, a multiplicity of conformational states of nearly equal free energy. How can we describe movement between such states? When the pair of states exhibits large-scale structural differences, it is tempting to connect the states by routes using the low-frequency collective elastic vibrations around each structure, the normal modes with the smallest restoring forces. Even in its simplest form, the notion of normal modes is remarkably successful for visualizing and predicting the character of the motions. The motions are in reality overdamped, but their structure often parallels the low-frequency normal modes (1). Yet clearly, a linear normal mode description cannot be complete because the very existence of the two low-lying conformations requires us to acknowledge considerable anharmonicity. The normal mode picture strictly describes the excitations about a single minimum. The limited adequacy of a normal mode description becomes even more apparent when we try to embed our picture of the motion between two dominant conformational states in the complete energy landscape of a biomolecule, which is replete with a myriad of local minima, ranging from the more subtle conformational substrates apparent in kinetic experiments (2) to the still more disordered states that are partially unfolded. Our goal in this article is to describe how allosteric conformational switches function by using a theoretical framework that unites an energy landscape description with the elastic model based on normal modes. To do so we need to go beyond the usual approaches that describe only the geometry of allosteric structural changes to estimate the energetics, i.e., the free energy barriers between different minima. In the simplest interpolation between initial and final structures this barrier depends on the relative stabilities of the two forms and the elastic, geometric properties of the structures, discernible by crystallography. We also argue that allosteric changes need not always go by way of a single transition-state structure but rather sometimes pass through a transition-state ensemble of structures that are partially unfolded.

Our approach will be to see, first, how far the idea of a single distortion path can be pushed by following the normal modes from each of the dominant conformational states of an allosteric protein as determined crystallographically. The strict harmonic picture holds only very close to each minimum. On the other hand, modest adjustments of the small amplitude motions allow the protein to follow adiabatically the large-scale movements quite well for substantial distances (1). This range might be called the regime of nonlinear elasticity. As the amplitude of the displacement grows, however, the elastic limit is reached (3). At this point, one possibility is that a glissile motion much like dislocation flow in a defective solid will occur. This requires special structural elements preexisting in the native structure. Looking for such pivot points is, of course, very worthwhile. When such a pivot exists, the resulting dislocational motion can be energetically modeled by simply switching from the elastic model that starts from one dominant conformation to that from the other structure (4). Another possibility is that the biomolecule will "crack" under the stress of the restoring forces of deforming along these low-frequency modes, just as titin unfolds when externally pulled in single-molecule experiments (5, 6). In this case, the molecule will locally unfold and partially reassemble before following a low-frequency elastic motion downhill to the product. The activated transition then does not occur through a single path but a multiplicity of detailed routes, involving partially unfolded states of the stressed regions. Such cracking would be disastrous for macroscopic machines, but unlike macroscopic machinery, a biological machine can break during its ordinary function and still complete its task because biomolecules can unfold and later refold properly as needed. The possibility of such a mechanism [a "proteinquake" (7)] has recently been invoked in describing myosin conformational change by Terada et al. (8), where they elegantly reconcile several single-molecule experiments that cannot be explained by the rigid lever arm model. Here we explore how the structural details of such cracking can be inferred from models like those already used to describe the free energy profile for the folding of proteins (911). Cracking need not occur in all conformational transitions, but when it does occur it leads to a predictable relationship between the thermodynamics of folding and the kinetics of allosteric conformational changes that can be tested in the laboratory. The linkage between partial protein unfolding and functional cooperativity in thermodynamics is well established in the work of Hilser, Freire, and collaborators (12, 13). The focus of the present article is on kinetics and mechanism where unfolding is less expected, because both initial and final structures may be completely folded in the critical regions.

We illustrate the ideas by using adenylate kinase. Kinases are an important family of proteins, constituting {approx}1.7% of human genes (14). Their critical role in signal pathways is made possible by an allosteric conformational change, which occurs when they are phosphorylated or bind ATP or other proteins. We chose adenylate kinase because it has rather well resolved crystal structures for both dominant conformational states, here called "open" (15) and "closed" (16). One feature, common to other allosteric proteins, is that even in the x-ray structures some parts of the protein are "disordered," perhaps partially unfolded. That observation alone makes clear the overlap of the energy landscapes for folding and functional motion.

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