Statistical mechanics of protein folding with separable energy functions

We have initiated an entirely new approach to statistical mechanical models of strongly interacting systems where the configurational parameters and the potential energy function are both constructed so that the canonical partition function can be evaluated analytically. For a simplified model of proteins consisting of a single, fairly short polypeptide chain without cross-links, we can adjust the energy parameters to favor the experimentally determined native state of seven proteins having diverse types of folds. Then 497 test proteins are predicted to have stable native folds, even though they are also structurally diverse, and 480 of them have no significant sequence similarity to any of the training proteins.     

The combinatorial distance geometry method for the calculation of molecular conformation. I. A new approach to an old problem

A new approach to the long-standing local minimum problem of molecular energy minimization is proposed. The approach relies upon a field of computer mathematics known as combinatorial optimization, together with methods of conformational analysis derived from distance geometry. The advantages over the usual numerical techniques of optimization are, first, that the algorithms derived are globally convergent, and second, that the mathematical problems involved are well-posed and suitable for study within the modern theory of computational complexity. In this paper we introduce the approach, and describe a computer program based on it.     

Topology of globular proteins. II.

This paper continues the inquiry into the likelihood of occurrence of protein structures which are “knotted”. The probability of knotted con formations is calculated by a simple Monte Carlo method for the proteins lysozyme and ribonuclease and for an elementary bicyclic polymer. Ribonuclease shows no measurable inclination towards tangling, but some 10%. of the lysozyme conformations are knotted. The implications of this result for the role of disulfide bridge formation in protein renaturation are discussed.     

Size-independent comparison of protein three-dimensional structures

Protein structures are routinely compared by their root-mean-square deviation (RMSD) in atomic coordinates after optimal rigid body superposition. What is not so clear is the significance of different RMSD values, particularly above the customary arbitrary cutoff for obvious similarity of 2–3 Å. Our earlier work argued for an intrinsic cutoff for protein similarity that varied with the number of residues in the polypeptide chains being compared. Here we introduce a new measure, ρ, of structural similarity based on RMSD that is independent of the sizes of the molecules involved, or of any other special properties of molecules. When ρ is less than 0.4–0.5, protein structures are visually recognized to be obviously similar, but the mathematically pleasing intrinsic cutoff of ρ>1.0 corresponds to overall similarity in folding motif at a level not usually recognized until smoothing of the polypeptide chain path makes it striking. When the structures are scaled to unit radius of gyration and equal principle moments of inertia, the comparisons are even more universal, since they are no longer obscured by differences in overall size and ellipticity. With increasing chain length, the distribution of ρ for pairs of random structures is skewed to higher values, but the value for the best 1% of the comparisons rises only slowly with the number of residues. This level is close to an intrinsic cutoff between similar and dissimilar comparisons, namely the maximal scaled ρ possible for the two structures to be more similar to each other than one is to the other's mirror image. The intrinsic cutoff is independent of the number of residues or points being compared. For proteins having fewer than 100 residues, the 1% ρ falls below the intrinsic cutoff, so that for very small proteins, geometrically significant similarity can often occur by chance. We believe these ideas will be helpful in judging success in NMR structure determination and protein folding modeling. © 1995 Wiley-Liss, Inc.     

Topology of globular proteins

This paper inquires whether it is reasonable to expect the native structure of proteins to be “knotted”. To this end, some topological properties of polypeptides containing disulfide bridges are discussed using notions from mathematical knot theory and graph theory. The probability of occurrence of knots in random cyclic polymers is calculated as a function of chain length by elementary Monte Carlo methods. The implications of this for protein renaturation and for determining the tertiary structure of proteins are discussed.     

The theory and practice of distance geometry

The mathematics of distance geometry constitutes the basis of a group of algorithms for revealing the structural consequences of diverse forms of information about a macromolecule's conformation. These algorithms are of proven utility in the analysis of experimental conformational data. This paper presents the basic theorems of distance geometry in Euclidean space and gives formal proofs of the correctness and, where possible, of the complexity of these algorithms. The implications of distance geometry for the energy minimization of macromolecules are also discussed.     

Dimensional oscillation. A fast variation of energy embedding gives good results with the AMBER potential energy function.

The structure of the AMBER potential energy surface of the cyclic tetrapeptide cyclotetrasarcosyl is analyzed as a function of the dimensionality of coordinate space. It is found that the number of local energy minima decreases as the dimensionality of the space increases until some limit at which point equipotential subspaces appear. The applicability of energy embedding methods to finding global energy minima in this type of energy-conformation space is explored. Dimensional oscillation, a computationally fast variant of energy embedding is introduced and found to sample conformation space widely and to do a good job of finding global and near-global energy minima.     

Cell proliferation in the bone marrow, thymus and spleen of mice studied by continuous, in vivo bromodeoxycytidine labelling and flow cytometric analysis

Abstract We have applied the technique of labelling dividing cells with bromodeoxyuridine (BrdUrd) in combination with in vivo continuous labelling, propidium iodide (PI) staining for DNA content, and flow cytometric analysis, for the determination of cell proliferation in bone marrow, thymus and spleen of mice. The percentage of BrdUrd labelled cells increased as a function of exposure time in a tissue specific manner for each of the three tissues. Thymus and bone marrow had cell populations which exhibited different kinetics for the accumulation of label: (1) those that cycled and became labelled within 2–3 days (88% in 2 days for bone marrow, 84% in 3 days for thymus); (2) those that cycled during the remainder of the 6 day infusion period (11% of bone marrow and 13% of thymus cells); and (3) those that did not cycle during the 6 day period studied (<2% of bone marrow and 3% of thymus cells). In contrast, the spleen exhibited a slower, constant accumulation of labelled cells. After six days of infusion a large proportion of spleen cells (50%) had not become labelled. These results suggest that a larger proportion of spleen cells are long lived than indicated by other methods. We also have found that the period of labelling with BrdUrd extended several days beyond the period of infusion. This method will be very useful in studying perturbations of cell populations induced in mice exposed to toxic agents.