Concepts of generic rigidity are being applied to protein structure. Many exact mechanical properties of a network of distance constraints can be calculated exactly. In the simplest case, a protein structure is modeled as a set of quenched distance constraints that define a mechanical framework. Without using molecular dynamics or Monte Carlo simulation techniques, the protein can be substructured into rigid and flexible regions, where long-time motion of the protein can be addressed in real-time. The algorithm that performs this calculation is called FIRST (Floppy Inclusion and Rigid Substructure Topography). FIRST is now available as a Web based tool at Michigan State University.

Since arriving at CSUN, the concept of a constraint has been generalized to define network rigidity at finite temperatures. A very interesting statistical mechanical formalism has been developed where the competition between enthalpic and entropic contributions from a specified mechanical framework is balanced via network rigidity. A protein is modeled as a Gibbs ensemble of accessible mechanical frameworks, allowing microscopic interactions to be modeled as thermally fluctuating constraints. For example, hydrogen bonds are allowed to break and reform. Probability measures have been defined over the ensemble of mechanical frameworks that quantitatively characterize the conformational flexibility of a protein. The interest of the group is to compare model predictions with experimental measures of protein flexibility, such as NMR protection factors.

The importance of studying protein conformational flexibility lies in its role in governing function. There are numerous known mechanisms involving protein-mediated interactions where conformational flexibility at the mechanistic level underlies a biochemical process. Quantitative profiles (or flexprints) have been constructed to characterize flexibility. Flexprints may prove insightful in understanding protein function. In the process of developing the network rigidity model, protein function is compared to flexibility predictions using a bioinformatic approach. Parameters of the network rigidity model are being optimized over a variety of protein sets. Subsets of homologous proteins are considered to address questions about precision of theoretical predictions and sensitivity in model parameters. More interestingly, preliminary results show that differences in function between homologous proteins, all with the same fold, correlate reasonably well with differences found in flexibility profiles. Using a diverse set of proteins (mesophile diversity, including extremophiles of various kinds) allows the accuracy of predictions to be addressed. These issues are being actively pursued.

(Thesis project by Dang Hong Huynh): The conformational flexibility of four bacterial periplasmic binding proteins was analyzed using network rigidity within a mean field approximation. Four proteins were studied and compared consisting of glutamine-binding protein (GBP), histidine-binding protein (HBP), lysine-arginine-ornithine-binding protein (LAOBP) and phosphate-binding protein (PBP). The calculations started with known three-dimensional structure determined by X-ray crystallography, and obtained from the Protein Data Bank. The liganded conformation was studied for all four proteins, and the unliganded conformation was available only for GBP and LAOBP. All four proteins have similar sequence, structure and function. It has been established that all four proteins exhibit similar overall flexibility characteristics. However, specific differences were detected, especially in the measure for the degree of rigid and flexible cooperativity between pairs of residues. Variations in the measure for rigid/flexible cooperativeness among the four proteins and between open/closed conformations of the same protein, compare favorably with known differences in function, which seem to depend on their respective specificity for ligand(s).

The concept of network rigidity at finite temperature has proved to be powerful in understanding protein stability. In general, it is not correct to add individual free energy contributions from various interactions to obtain the total free energy. The total energy may well be additive, but the entropic component to free energy is non-additive. The non-additive property of entropic contributions is a direct result from not knowing which degrees of freedom in the system are independent or redundant. Network rigidity is a non-local interaction (both in sequence and spatially) that answers the question about which degrees of freedom are independent. Entropic contributions from independent degrees of freedom for a given mechanical framework are additive. A partition function is constructed as a sum over the Gibbs ensemble of all accessible mechanical frameworks. Monte Carlo simulations are used to perform these calculations. The interest of the group is to compare model predictions with calorimetric measurements on proteins and hydrogen-bond stability measurements. Protein folding temperatures (hot and cold) are used to determine adjustable model parameters.

It has been demonstrated that application of network rigidity at finite temperature can account for the high degree of cooperativity exhibited by proteins. Through the long-range nature of network rigidity, mutations or other local perturbations (such as docking of a small molecule) can have drastic effect (but often showing little effect) on the sub-ensemble of most stable conformations explored by the protein. Cooperativity exhibited by conformational changes triggered by a specific process, or by changes in the thermodynamic environment leading to protein folding/unfolding events --- are being interpreted as manifestations of topological re-arrangement of constraints. Self-organized structures derive from characteristic patterns of optimally placed constraints associated with the most probable microstates.

(Thesis project by Alicia Heckathorne): Network rigidity at finite temperature is used to model polypeptide chains in solution that undergo a helix to coil transition. Exact results are obtained by a transfer matrix method using a minimalist model. The cooperative interaction between hydrogen bonds is explicitly modeled using network rigidity. Network rigidity model parameters are compared to Lifson-Roig model parameters. The concept of a nucleation and propagation parameter is eliminated in the network rigidity model. The calculated partition function describing the thermodynamic states requires no specific reference to a nucleation process. Instead, the nucleation process is a consequence of the properties of network rigidity. The conceptual advantage of the network rigidity approach is that the network rigidity model parameters are expected to be transferable, unlike the nucleation and propagation parameters of previous helix-coil theories.

Protein unfolding with increase in temperature (or hot denaturation) is easy to obtain in the network rigidity model, because thermal energy becomes available to break constraints. However, solvent effects are built directly into the network rigidity model, where the affect of hydration is being explicitly modeled. Cold denaturation is the result of re-arrangement of optimally placed constraints as the temperature is lowered. Under mixed solvent conditions, the network rigidity model applied to the helix-coil transition describes a polypeptide chain in an alpha-helical state that is subject to hot and cold thermal denaturing. Although cold denaturation has not been a focus of the group, the affect simply falls out of the calculations. Therefore, the interest of the group is to compare model predictions with calorimetric measurements on cold (and hot) denaturation.

The interest of the group has recently expanded to address the correspondence of conformational flexibility to that of stability. Network rigidity at finite temperatures allows one to estimate (using Monte Carlo sampling starting from known 3D native structures) the stability of the protein. The model is far from being at an ab initio level. The model parameters need to be further optimized and cross-correlated with experimental stability measurements. Although this is a long term goal, the beginnings of this process is underway, with conformational flexibility predictions being the main focus.

(Thesis project by Vahan Minassian): The flexibility profiles serve multiple purposes. For the biochemist, they may potentially serve as a high-throughput bioinformatic finger-printing system. For the biophysicist, they serve as a roadmap in a high dimensional conformational space where the lowest energy deformations can take place. The flexibility profile becomes a time-dependent reaction-coordinate that effectively reduces the dynamical manifold down to a small subset of essential degrees of freedom, which govern biologically important correlated motions. Although the lowest energy paths (actually free energy) are specified through these flexibility profiles, nevertheless in general the protein will dynamically climb over barriers to make conformational changes. These issues are being addressed using toy models.