Fractal Form of Proteins

Stapleton, H. J.; Allen, James P.; Flynn, C. P.; Stinson, Douglas G.; Kurtz, S. R.

doi:10.1103/physrevlett.45.1456

Cited by 195 publications

(77 citation statements)

References 12 publications

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“…Recently there has been an accumulation of evidence supporting the hypothesis that proteins are fractal-like objects (7)(8)(9)(10)(11)(12)(13)(14)(15)(16). The fractal nature of proteins manifests itself in the way proteins vibrate and in the manner in which they fill space.…”

Section: Introductionmentioning

confidence: 98%

“…Proteins are commonly envisioned vibrating under the influence of a complicated energy landscape (1) held responsible for their intricate dynamics (1)(2)(3)(4)(5)(6)(7)(8)(9). Despite its overall complexity the energy landscape is approximately harmonic near its minima, a fact that does not seem to coincide with the rich dynamics mentioned above.…”

Section: Introductionmentioning

confidence: 99%

“…Recent studies on proteins have already shown that gðωÞ obeys the above power law behavior (7,8,(11)(12)(13). However, in contrast to regular threedimensional structures, for proteins it is usually found that d s < 2 associating them to a realm of low dimensionality.…”

Section: Introductionmentioning

confidence: 99%

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Anomalies in the vibrational dynamics of proteins are a consequence of fractal-like structure

Reuveni

Granek

Klafter

2010

Proc. Natl. Acad. Sci. U.S.A.

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Proteins have been shown to exhibit strange/anomalous dynamics displaying non-Debye density of vibrational states, anomalous spread of vibrational energy, large conformational changes, nonexponential decay of correlations, and nonexponential unfolding times. The anomalous behavior may, in principle, stem from various factors affecting the energy landscape under which a protein vibrates. Investigating the origins of such unconventional dynamics, we focus on the structure-dynamics interplay and introduce a stochastic approach to the vibrational dynamics of proteins. We use diffusion, a method sensitive to the structural features of the protein fold and them alone, in order to probe protein structure. Conducting a large-scale study of diffusion on over 500 Protein Data Bank structures we find it to be anomalous, an indication of a fractal-like structure. Taking advantage of known and newly derived relations between vibrational dynamics and diffusion, we demonstrate the equivalence of our findings to the existence of structurally originated anomalies in the vibrational dynamics of proteins. We conclude that these anomalies are a direct result of the fractal-like structure of proteins. The duality between diffusion and vibrational dynamics allows us to make, on a single-molecule level, experimentally testable predictions. The time dependent vibrational mean square displacement of an amino acid is predicted to be subdiffusive. The thermal variance in the instantaneous distance between amino acids is shown to grow as a power law of the equilibrium distance. Mean first passage time analysis is offered as a practical tool that may aid in the identification of amino acid pairs involved in large conformational changes.protein dynamics | Gaussian network model | anomalous diffusion | intramolecular distance distributions | mean first passage time IntroductionProteins are commonly envisioned vibrating under the influence of a complicated energy landscape (1) held responsible for their intricate dynamics (1-9). Despite its overall complexity the energy landscape is approximately harmonic near its minima, a fact that does not seem to coincide with the rich dynamics mentioned above. Attacking the validity of the harmonic approximation is one way out of this puzzle. One might have suggested that the harmonic approximation is unable to capture the anomalous dynamics displayed by proteins since the latter is dominated by large excursions from the native state structure. The failure of the harmonic approximation then follows from the fact that it does not describe the energy of conformations which considerably deviate from the native state structure. In this paper we argue differently showing that the native state structure of proteins is fractal-like and hence anomalous in its basis. As a result one needs not go beyond the harmonic approximation in order to observe anomalous dynamics since anomalous structure leads to it even in the presence of the simplest interactions. Interestingly, when the structure is fractal-like, large con...

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Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Anomalies in the vibrational dynamics of proteins are a consequence of fractal-like structure

Reuveni

Granek

Klafter

2010

Proc. Natl. Acad. Sci. U.S.A.

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“…The phases of polymers and in particular proteins, have been presumed to display fractal geometry [10][11][12][13][14][15][16][17][18][19][20]. Traditionally, the following values of ν are assigned to these phases [1, 2]: Biologically active proteins are commonly in the space filling d H = 3 phase.…”

mentioning

confidence: 99%

“…The numerical value of ν = 1/d H serves as an order parameter of polymer phase structure: For a continuous self-nonintersecting chain in three space dimensions d H can in principle acquire any value between 1 and 3. Simple examples of fractal structures where d H is not an integer, include the piecewise linear Koch curve and attractors of chaotic equations such as the Lorenz and the Rössler equations [9].The phases of polymers and in particular proteins, have been presumed to display fractal geometry [10][11][12][13][14][15][16][17][18][19][20]. Traditionally, the following values of ν are assigned to these phases [1, 2]: Biologically active proteins are commonly in the space filling d H = 3 phase.…”

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confidence: 99%

Long range correlations and folding angle with applications to α-helical proteins

Krokhotin

Nicolis

Niemi

2014

The Journal of Chemical Physics

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The conformational complexity of linear polymers far exceeds that of point-like atoms and molecules. Polymers can bend, twist, even become knotted. Thus they may also display a much richer phase structure than point particles. But it is not very easy to characterize the phase of a polymer. Essentially, the only attribute is the radius of gyration. The way how it changes when the degree of polymerization becomes different, and how it evolves when the ambient temperature and solvent properties change, discloses the phase of the polymer. Moreover, in any finite length chain there are corrections to scaling, that complicate the detailed analysis of the phase structure. Here we introduce a quantity that we call the folding angle, a novel tool to identify and scrutinize the phases of polymers. We argue for a mean-field relationship between its values and those of the scaling exponent in the radius of gyration. But unlike in the case of the radius of gyration, the value of the folding angle can be evaluated from a single structure. As an example we estimate the value of the folding angle in the case of crystallographic α-helical protein structures in the Protein Data Bank (PDB). We also show how the value can be numerically computed using a theoretical model of α-helical chiral homopolymers.Despite substantial differences in their chemical composition, all linear polymers are presumed to share the same universal phase structure [1][2][3]. But the phase where a particular polymer resides depends on many factors including polymer concentration, the quality of solvent, ambient temperature and pressure. Three phases are commonly identified, each of them categorized by the manner how the polymer fills the space [1][2][3][4]: If the solvent is poor and the attractive interactions between monomers dominate, a single polymer chain is presumed to collapse into a space-filling conformation. In a good solvent environment or at sufficiently high temperatures, a single polymer chain tends to swell until its geometric structure bears similarity to a self-avoiding random walk (SARW). The collapsed phase and the SARW phase are separated by Θ-point where a polymer has the characteristics of an ordinary random walk (RW). Biologically active proteins are commonly presumed to reside in the space filling collapsed phase, under physiological conditions. We shall examine proteins as an important subset of polymers, for which a large amount of experimental data is available in PDB [5].The phase where a polymer resides can be determined from the value of the scaling exponent ν [1-3]. To define this quantity, we consider the asymptotic behavior of the radius of gyration when the number of monomers N is very large. With r i the coordinates of the skeletal atoms of the polymer, the radius of gyration becomes in this limit [1][2][3][6][7][8] The length scale R 0 is an effective Kuhn distance between the skeletal atoms in the large-N limit. It is in principle a computable quantity, that depends on all the atomic level details of the polymer and all...

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New dimension indices for the characterization of the solvent-accessible surface

2001

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ABSTRACT:A method for the computation of a dimension index D is implemented in program TOPO and applied to calculate the solvent-accessible surfaces of molecules. Our algorithm distinguishes external from internal atoms, and uses such a feature to give two fractal-like dimension indices, D and D . The D − D difference is a sensitive method to elucidate the occurrence of atoms that are hidden to solvents. For molecules with buried atoms this difference is great (e.g., faujasite). The procedure is compared with the GEPOL code, which provides high-quality results. TOPO systematic error can be easily corrected by simple addition of a small constant value (0.011). Correlation models between indices D and D , globularity, rugosity, dipole moment and other properties make clear the existence of a homogeneous molecular structure in each series. Additional applications are the extrapolation of D to infinite polymers, the variation of the D with generations of dendrimers and a revision of D for lysozyme.

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Fractal Form of Proteins

Cited by 195 publications

References 12 publications

Anomalies in the vibrational dynamics of proteins are a consequence of fractal-like structure

Anomalies in the vibrational dynamics of proteins are a consequence of fractal-like structure

Long range correlations and folding angle with applications to α-helical proteins

New dimension indices for the characterization of the solvent-accessible surface

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