The understanding, and even the description of protein folding is impeded by the complexity of the process. Much of this complexity can be described and understood by taking a statistical approach to the energetics of protein conformation, that is, to the energy landscape. The statistical energy landscape approach explains when and why unique behaviors, such as specific folding pathways, occur in some proteins and more generally explains the distinction between folding processes common to all sequences and those peculiar to individual sequences. This approach also gives new, quantitative insights into the interpretation of experiments and simulations of protein folding thermodynamics and kinetics. Specifically, the picture provides simple explanations for folding as a two-state first-order phase transition, for the origin of metastable collapsed unfolded states and for the curved Arrhenius plots observed in both laboratory experiments and discrete lattice simulations. The relation of these quantitative ideas to folding pathways, to uniexponential vs. multiexponential behavior in protein folding experiments and to the effect of mutations on folding is also discussed. The success of energy landscape ideas in protein structure prediction is also described. The use of the energy landscape approach for analyzing data is illustrated with a quantitative analysis of some recent simulations, and a qualitative analysis of experiments on the folding of three proteins. The work unifies several previously proposed ideas concerning the mechanism protein folding and delimits the regions of validity of these ideas under different thermodynamic conditions. © 1995 Wiley-Liss, Inc.

Funnels, pathways, and the energy landscape of protein folding: A synthesis

The dynamics of a folded globular protein (bovine pancreatic trypsin inhibitor) have been studied by solving the equations of motion for the atoms with an empirical potential energy function. The results provide the magnitude, correlations and decay of fluctuations about the average structure. These suggest that the protein interior is fluid-like in that the local atom motions have a diffusional character.

Dynamics of folded proteins

Noonan syndrome (MIM 163950) is an autosomal dominant disorder characterized by dysmorphic facial features, proportionate short stature and heart disease (most commonly pulmonic stenosis and hypertrophic cardiomyopathy). Webbed neck, chest deformity, cryptorchidism, mental retardation and bleeding diatheses also are frequently associated with this disease. This syndrome is relatively common, with an estimated incidence of 1 in 1,000-2,500 live births. It has been mapped to a 5-cM region (NS1) [corrected] on chromosome 12q24.1, and genetic heterogeneity has also been documented. Here we show that missense mutations in PTPN11 (MIM 176876)-a gene encoding the nonreceptor protein tyrosine phosphatase SHP-2, which contains two Src homology 2 (SH2) domains-cause Noonan syndrome and account for more than 50% of the cases that we examined. All PTPN11 missense mutations cluster in interacting portions of the amino N-SH2 domain and the phosphotyrosine phosphatase domains, which are involved in switching the protein between its inactive and active conformations. An energetics-based structural analysis of two N-SH2 mutants indicates that in these mutants there may be a significant shift of the equilibrium favoring the active conformation. This implies that they are gain-of-function changes and that the pathogenesis of Noonan syndrome arises from excessive SHP-2 activity.

Mutations in PTPN11, encoding the protein tyrosine phosphatase SHP-2, cause Noonan syndrome.

An efficient methodology, further referred to as ICM, for versatile modeling operations and global energy optimization on arbitrarily fixed multimolecular systems is described. It is aimed at protein structure prediction, homology modeling, molecular docking, nuclear magnetic resonance (NMR) structure determination, and protein design. The method uses and further develops a previously introduced approach to model biomolecular structures in which bond lengths, bond angles, and torsion angles are considered as independent variables, any subset of them being fixed. Here we simplify and generalize the basic description of the system, introduce the variable dihedral phase angle, and allow arbitrary connections of the molecules and conventional definition of the torsion angles. Algorithms for calculation of energy derivatives with respect to internal variables in the topological tree of the system and for rapid evaluation of accessible surface are presented. Multidimensional variable restraints are proposed to represent the statistical information about the torsion angle distributions in proteins. To incorporate complex energy terms as solvation energy and electrostatics into a structure prediction procedure, a “double-energy” Monte Carlo minimization procedure in which these terms are omitted during the minimization stage of the random step and included for the comparison with the previous conformation in a Markov chain is proposed and justified. The ICM method is applied successfully to a molecular docking problem. The procedure finds the correct parallel arrangement of two rigid helixes from a leucine zipper domain as the lowest-energy conformation (0.5 A root mean square, rms, deviation from the native structure) starting from completely random configuration. Structures with antiparallel helixes or helixes staggered by one helix turn had energies higher by about 7 or 9 kcal/mol, respectively. Soft docking was also attempted. A docking procedure allowing side-chain flexibility also converged to the parallel configuration starting from the helixes optimized individually. To justdy an internal coordinate approach to the structure prediction as opposed to a Cartesian one, energy hypersurfaces around the native structure of the squash seeds trypsin inhibitor were studied. Torsion angle minimization from the optimal conformation randomly distorted up to the rms deviation of 2.2 A or angular rms deviation of l0° restored the native conformation in most cases. In contrast, Cartesian coordinate minimization did not reach the minimum from deviations as small as 0.3 A or 2°. We conclude that the most promising detailed approach to the protein-folding problem would consist of some coarse global sampling strategy combined with the local energy minimization in the torsion coordinate space. © 1994 by John Wiley & Sons, Inc.

ICM—a new method for protein modeling and design: applications to docking and structure prediction from the distorted native conformation

The deformability of double helical DNA is critical for its packaging in the cell, recognition by other molecules, and transient opening during biochemically important processes. Here, a complete set of sequence-dependent empirical energy functions suitable for describing such behavior is extracted from the fluctuations and correlations of structural parameters in DNA–protein crystal complexes. These elastic functions provide useful stereochemical measures of the local base step movements operative in sequence-specific recognition and protein-induced deformations. In particular, the pyrimidine-purine dimers stand out as the most variable steps in the DNA–protein complexes, apparently acting as flexible “hinges” fitting the duplex to the protein surface. In addition to the angular parameters widely used to describe DNA deformations (i.e., the bend and twist angles), the translational parameters describing the displacements of base pairs along and across the helical axis are analyzed. The observed correlations of base pair bending and shearing motions are important for nonplanar folding of DNA in nucleosomes and other nucleoprotein complexes. The knowledge-based energies also offer realistic three-dimensional models for the study of long DNA polymers at the global level, incorporating structural features beyond the scope of conventional elastic rod treatments and adding a new dimension to literal analyses of genomic sequences.

https://www.pnas.org/content/pnas/95/19/11163.full.pdf

DNA sequence-dependent deformability deduced from protein–DNA crystal complexes

A powerful Monte Carlo method is described to simulate thermal conformational fluctuations in native proteins by using an empirical conformational energy function in which bond lengths and bond angles are kept fixed and only dihedral angles are independent variables. In this method, collective variables corresponding to eigenvectors of the second-derivative matrix of the energy function at its minimum point are scaled according to corresponding eigenvalues in such a way that the energy function in terms of the scaled collective variables is isotropic at the minimum point. Simulation is carried out with an isotropic step size in the space of these scaled collective variables. This simulation method is applied to a small protein, bovine pancreatic trypsin inhibitor (BPTI), and its model harmonic system defined by a quadratic energy function with the same second-derivative matrix as that of BPTI at its minimum point. Efficiency of the simulation method with an isotropic step size in the space of the scaled collective variables is found to be about 500–50 times greater than the conventional method with with an isotropic step in the space of the usual nonscaled variables. One step of this new method generates conformational changes that occur in the real-time range of 0.05 ps. In a record of 5 × 105 step simulation, the BPTI molecule is observed to migrate beyond a single minimum-energy region.

Efficient Monte Carlo method for simulation of fluctuating conformations of native proteins.

Fluctuations in backbone dihedral angles in the α-helical conformation of homopolypeptides are studied based on an assumption that the conformational energy function of a polypeptide consisting of n amino-acid residues can be approximated by a 2n-dimensional parabola around the minimum point in the range of fluctuations. A formula is derived that relates 〈ΔθiΔθj〉, the mean value of the product of deviations of dihedral angles ϕi and ψi (collectively designated by θi) from their energy minimum values, with a matrix inverse to the second derivative matrix F,n of the conformational energy function at the minimum point. A method of calculating the inverse matrix Fn−1 explicitly is given. The method is applied to calculating 〈ΔθiΔθj〉 for the α-helices of poly(L-alanine) and polyglycine. The autocorrelations 〈(Δϕi)2〉 and 〈(Δψi)2〉 at 300°K are found to be about 66 deg2 and 49 deg2, respectively, for poly(L-alanine), and 84 deg2 and 116 deg2, respectively, for polyglycine. The length of correlations of fluctuations along the chain is found for both polypeptides to be about eight residues long.

Fluctuations of an α‐helix

By using the static correlations of fluctuations in the dihedral angles of the α-helices of polyglycine and poly(L-alanine) calculated previously, geometrical fluctuations of a section (consisting of up to 18 peptide units) of the α-helices of infinite length are calculated. These fluctuations are found to differ in some respects (e.g., the dependence of amplitudes on the length of section) from those of a circular rod made of homogeneous continuous material. However, the moduli of the mechanical strengths (tensile Young's modulus, bending Young's modulus, and the shear modulus) of a circular rod are calculated, whose geometrical fluctuations are approximately equal to the fluctuations of a section consisting of 18 peptide units. They are of the order of 1011 dyn/cm2. The tensile rigidity, flexural rigidity, and torsional rigidity are calculated to be 1.20 × 10−3 dyn, 2.46 × 10−19 dyn·cm2 and 1.79 × 10−19 dyn·cm2 for polyglycine, and 1.96 × 10−3 dyn, 4.05 × 10−19 dyn·cm2 and 3.28 × 10−19 dyn·cm2 for poly(L-alanine), respectively.

Fluctuations and mechanical strength of α-helices of polyglycine and poly(L-alanine)

A method of calculating time correlation functions from records of computer simulated equilibrium conformational fluctuations in a globular protein is discussed. Use of the calculated time correlation function for discussions of dynamics of folding and unfolding transition in the two-dimensional lattice model of proteins. The time correlation functions can be approximated in general by a sum of two simple exponential terms. The relaxation time of the slower mode does not depend on the nature of the physical quantity with respect to which the time correlation function is calculated. This time characterizes the overall folding and unfolding transition. The relaxation time of the faster mode depends on the nature of the physical quantity and characterizes conformational fluctuations within each of the native and denatured states. The mechanism of a previously observed phenomenon of the acceleration of the folding and unfolding transition by short-range interactions is discussed.

Dynamics of folding and unfolding transition in a globular protein studied by time correlation functions from computer simulation.

A theoretical analysis is given of the triple-helix–random-coil transition in a mixed solution of poly(Pro-Pro-Gly)n with two different but defined degrees of polymerization n and n′. Because of the highly cooperative nature of this helix–coil transition, each polypeptide chain tends to form a triple helix with other polypeptide chains with the same degree of polymerization (size recognition). Occurrence of triple helices consisting of polypeptide chains with different degrees of polymerization (error in recognition) is studied in detail as a function of the cooperativity, and n and n′. Implication of this analysis for molecular recognition in general is discussed.

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Papers

Efficient Monte Carlo method for simulation of fluctuating conformations of native proteins.

Fluctuations of an α‐helix

Fluctuations and mechanical strength of α-helices of polyglycine and poly(L-alanine)

Dynamics of folding and unfolding transition in a globular protein studied by time correlation functions from computer simulation.

The formation of the triple helix in a mixed solution of (Pro‐Pro‐Gly)n and (Pro‐Pro‐Gly)n′. A model of molecular size recognition