Recurrence quantification analysis can reveal subtle aspects of dynamics not easily appreciated by other methods such as the Fourier transform. Laminarity and maxline are two RQA variables that can demonstrate the presence of unstable singularities which are often found in biological dynamics. Examples are presented and their implications are discussed relative to deterministic dynamics and stochastic processes.

PURPOSE Historically, F-waves have been classified by various linear descriptors like persistence, latency, duration, amplitude, chrono-dispersion and number of repeater waves. But because physiological signals are notoriously nonlinear in nature, the objective of this study was to apply modern nonlinear methodology to F-waves sequences to assess the presence of underlying deterministic structures. Subtle changes in these sensitive markers could give early warnings for neurological problems. METHODS F-waves were elicited in the left abductor pollicis breivs muscle by supra-maximally stimulating the median nerve percutaneously at the wrist. Approximately 200 stimuli were applied (0.5 Hz) to three subjects for at least four trials each. F-wave latencies were measured and assembled into sequences in proper order. Recurrence quantification analysis (RQA) was applied to these F-wave sequences from different dimensional perspectives. Controls were constructed by randomly shuffling the ordered sequences. RQA has a theoretical mathematical foundation and practical performance record on numerous other physiological systems. RESULTS Recurrence analysis showed that sequential F-waves form recurrent patterns with parallel trajectories with deterministic and laminated structures. These features could be destroyed by randomizing the sequential orders of F-waves, upholding the hypothesis that sequences of F-waves are deterministically formed from underlying physiological rules. CONCLUSIONS F-wave time series are fully amenable to recurrence analysis which provides a higher-dimensional perspective on the physiological dynamic. The recurrent patterns are complex, but not random, meaning that physiological rules dominate the sequence of F-waves. Disease processes within the central or peripheral nervous system may alter F-wave patterns. If so, RQA potentially may be a diagnostic tool to help discern subtleties between altered deterministic rules operating in disease.

The presence of partially folded intermediates along the folding funnel of proteins has been suggested to be a signature of potentially aggregating systems. Many studies have concluded that metastable, highly flexible intermediates are the basic elements of the aggregation process. In a previous paper, we demonstrated how the choice between aggregation and folding behavior was influenced by hydrophobicity distribution patterning along the sequence, as quantified by recurrence quantification analysis (RQA) of the Myiazawa-Jernigan coded primary structures. In the present paper, we tried to unify the "partially folded intermediate" and "hydrophobicity/charge" models of protein aggregation verifying the ability of an empirical relation, developed for rationalizing the effect of different mutations on aggregation propensity of acyl-phosphatase and based on the combination of hydrophobicity RQA and charge descriptors, to discriminate in a statistically significant way two different protein populations: (a) proteins that fold by a process passing by partially folded intermediates and (b) proteins that do not present partially folded intermediates.

A statistical model describing the propensity for protein aggregation is presented. Only amino-acid hydrophobicity values and calculated net charge are used for the model. The combined effects of hydrophobic patterns as computed by the signal analysis technique, recurrence quantification, plus calculated net charge were included in a function emphasizing the effect of singular hydrophobic patches which were found to be statistically significant for predicting aggregation propensity as quantified by fluorescence studies obtained from the literature. These results suggest preliminary evidence for a mesoscopic principle for protein folding/aggregation.

The problem of protein folding vs. aggregation was investigated in acylphosphatase and the amyloid protein Abeta(1-40) by means of nonlinear signal analysis of their chain hydrophobicity. Numerical descriptors of recurrence patterns provided the basis for statistical evaluation of folding/aggregation distinctive features. Static and dynamic approaches were used to elucidate conditions coincident with folding vs. aggregation using comparisons with known protein secondary structure classifications, site-directed mutagenesis studies of acylphosphatase, and molecular dynamics simulations of amyloid protein, Abeta(1-40). The results suggest that a feature derived from principal component space characterized by the smoothness of singular, deterministic hydrophobicity patches plays a significant role in the conditions governing protein aggregation.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Complex Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Recurrence Quantification Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Mathematical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Biological Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Bioinformatic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Other Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13

AbstractChemInform is a weekly Abstracting Service, delivering concise information at a glance that was extracted from about 100 leading journals. To access a ChemInform Abstract of an article which was published elsewhere, please select a “Full Text” option. The original article is trackable via the “References” option.

The complexity, nonlinearity and nonstationarity of the cardiovascular system typically defy comprehensive and deterministic mathematical modeling, except from a statistical perspective. Living systems are governed by numerous, continuously changing, interacting variables in the presence of noise. Cardiovascular signals can be shown to be discontinuous alternations between deterministic trajectories and stochastic pauses (terminal dynamics). One promising approach for assessing such nondeterministic complexity is recurrence quantification analysis (RQA). As reviewed in this paper, strategies implementing quantification of recurrences have been successful in diagnosing changes in nonstationary cardiac signals not easily detected by traditional methods. It is concluded that recurrence quantification analysis is a powerful discriminatory tool which, when properly applied to cardiac signals, can provide objectivity regarding the degree of determinism characterizing the system, state changes, as well as degrees of complexity and/or randomness.

Subtle nonlinear behaviors of fluid-coupled mechanical oscillators at low and medium viscosities were better detected by cross recurrence analysis than spectral analysis. Cross recurrence with its high sensitivity to nonlinear dynamics may have applicability to weakly coupled oscillators prevalent in biology and physiology.

The generation of a global “complexity” score for numerical series was derived from a principal components analysis of a group of nonlinear measures of experimental as well simulated series. The concept of complexity was demonstrated to be independent from other descriptors of ordered series such as the amount of variance, the departure from normality and the relative nonstationarity; and to be mainly related to the number of independent elements (or operations) needed to synthesize the series. The possibility of having a univocal ranking of complexity for diverse series opens the way to a wider application of dynamical systems concepts in empirical sciences.

The potential energy time series obtained from molecular dynamics simulations of the B1 domain of protein G and plastocyanin both in vacuo and in water were analyzed by means of recurrence quantification analysis. This methodology is robust for nonlinear, nonstationary processes, and demonstrated the existence of a flat recurrence spectrum occurring beyond a previously described scaling region of protein dynamics, as well as the existence of clustered modes of very long period (approximately 500 ps) elicited by the solvent. The number of these modes was approximately related to the number of structural domains of the studied proteins. Thus the methodology may be useful to distinguish processes intrinsic to protein folding dynamics from those which develop from hydration.

Secondary structures of proteins were studied by recurrence quantification analysis (RQA). High‐resolution, 3‐dimensional coordinates of alpha‐carbon atoms comprising a set of 68 proteins were downloaded from the Protein Data Bank. By fine‐tuning four recurrence parameters (radius, line, residue, separation), it was possible to establish excellent agreement between percent contribution of alpha‐helix and beta‐sheet structures determined independently by RQA and that of the DSSP algorithm (Define Secondary Structure of Proteins). These results indicate that there is an equivalency between these two techniques, which are based upon totally different pattern recognition strategies. RQA enhances qualitative contact maps by quantifying the arrangements of recurrent points of alpha carbons close in 3‐dimensional space. For example, the radius was systematically increased, moving the analysis beyond local alpha‐carbon neighborhoods in order to capture super‐secondary and tertiary structures. However, differences between proteins could only be detected within distances up to about 6–11 Å, but not higher. This result underscores the complexity of alpha‐carbon spacing when super‐secondary structures appear at larger distances. Finally, RQA‐defined secondary structures were found to be robust against random displacement of alpha carbons upwards of 1 Å. This finding has potential import for the dynamic functions of proteins in motion. Proteins 2001;44:292–303. © 2001 Wiley‐Liss, Inc.

We use recurrence quantification, which does not require assumptions about stationarity, length or noise, and apply it to epileptic EEG activity. We found that both preictal and EEG segments free of seizure activity exhibit significant transients; but preictal transients consistently occur in a context of unstable EEG activity.

It has been suggested that the number and strength of local contacts are the major factors governing conformation accessibility of model two ground-state polypeptide chains. This phenomenology has been posed as a possible factor influencing prion folding. To test this conjecture, recurrence quantification analysis was applied to two model 36mers, and the Syrian hamster prion protein. A unique divergence of the radius function for the recurrence quantification variable %DET of hydrophobicity patterns was observed for both 36mers, and in a critical region of the hamster prion protein. This divergence suggests a partition between strong short- and long-range hydrophobicity patterns, and may be an important factor in prion phenomenology, along with other global thermodynamic factors.

Deterministic, chaotic, nonlinear dynamics have enjoyed considerable popularity in the analysis of physiological systems. Many models, however, fail to incorporate some basic features of the involved physiology. The authors' research using experimental data from ECGs analyzed by recurrence quantification analysis (RQA) suggest that some of these dynamics can be better modeled by singularities of differential equations, alternating with oscillations which result in multi-choice responses to excitations. ECGs of 34 healthy volunteers were recorded for 512 consecutive beats. Recurrence plots suggested discontinuities at the T-P interval, and the durations of the T-P and P-T intervals were manually calculated. Although the distributions of these two intervals were found to be similar (KS two sample test, p=NS), RQA demonstrated that the T-P Interval approached the embedding limit of pseudo random numbers [pseudo random=4; T-P=4; (95% CI=3.64.3)]. This suggests the primary stochastic process of ECGs is located In the T-P interval and represents a singularity of the dynamics alternating with the determinism of the P-T interval. The dynamics are thus discontinuous and poorly represented by FFT and other, nonlinear transform techniques

A simple data analysis strategy based on the scoring of recurrences, allowed us to distinguish relatively short runs of (i) a chaotic system (logistic map), (ii) transcendental numbers (π), (iii) pseudo-random numbers, and (iv) physical random numbers (atmospheric noise). Beside the theoretical implications of distinguishing transcendental and pseudo-random numbers from physical noise, these results suggest a test

Annals of the New York Academy of SciencesVolume 879, Issue 1 p. 258-266 Recurrence Quantification Analysis in Molecular Dynamics CESARE MANETTI, Corresponding Author CESARE MANETTI Department of Chemistry, University of Rome "La Sapienza," Piazzale Aldo Moro, 5-00185 Rome, Italy Address for correspondence: [email protected] (e-mail).Search for more papers by this authorMARC-ANTOINE CERUSO, MARC-ANTOINE CERUSO Department of Chemistry, University of Rome "La Sapienza," Piazzale Aldo Moro, 5-00185 Rome, ItalySearch for more papers by this authorALESSANDRO GIULIANI, ALESSANDRO GIULIANI Istituto Superiore di Sanitá, TCE Lab, Rome 00161, Italy Department of Chemistry, University of Rome "La Sapienza," Piazzale Aldo Moro, 5-00185 Rome, ItalySearch for more papers by this authorCHARLES L. WEBBER JR., CHARLES L. WEBBER JR. Department of Physiology, Loyola Univ. Medical Center, 2160 South 1st Avenue, Maywood, Illinois 60153, USA Department of Chemistry, University of Rome "La Sapienza," Piazzale Aldo Moro, 5-00185 Rome, ItalySearch for more papers by this authorJOSEPH P. ZBILUT, JOSEPH P. ZBILUT Department of Molecular Biophysics and Physiology, Rush University, 1653 W. Congress, Chicago, Illinois 60612, USA Department of Chemistry, University of Rome "La Sapienza," Piazzale Aldo Moro, 5-00185 Rome, ItalySearch for more papers by this author CESARE MANETTI, Corresponding Author CESARE MANETTI Department of Chemistry, University of Rome "La Sapienza," Piazzale Aldo Moro, 5-00185 Rome, Italy Address for correspondence: [email protected] (e-mail).Search for more papers by this authorMARC-ANTOINE CERUSO, MARC-ANTOINE CERUSO Department of Chemistry, University of Rome "La Sapienza," Piazzale Aldo Moro, 5-00185 Rome, ItalySearch for more papers by this authorALESSANDRO GIULIANI, ALESSANDRO GIULIANI Istituto Superiore di Sanitá, TCE Lab, Rome 00161, Italy Department of Chemistry, University of Rome "La Sapienza," Piazzale Aldo Moro, 5-00185 Rome, ItalySearch for more papers by this authorCHARLES L. WEBBER JR., CHARLES L. WEBBER JR. Department of Physiology, Loyola Univ. Medical Center, 2160 South 1st Avenue, Maywood, Illinois 60153, USA Department of Chemistry, University of Rome "La Sapienza," Piazzale Aldo Moro, 5-00185 Rome, ItalySearch for more papers by this authorJOSEPH P. ZBILUT, JOSEPH P. ZBILUT Department of Molecular Biophysics and Physiology, Rush University, 1653 W. Congress, Chicago, Illinois 60612, USA Department of Chemistry, University of Rome "La Sapienza," Piazzale Aldo Moro, 5-00185 Rome, ItalySearch for more papers by this author First published: 06 February 2006 https://doi.org/10.1111/j.1749-6632.1999.tb10429.xCitations: 9Read the full textAboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL REFERENCES 1 Garcia, A.E. 1992. Large-amplitude nonlinear motions in proteins. Phys. Rev. Lett. 68: 2696– 2699. 2 Amadei, A., A.B.M Linssen & H.J.C. Berendsen. 1993. Essential dynamics of proteins. Proteins 17: 412– 425. 3 Karpen, M.E., D.J. Tobias & C.L. Brooks, III. 1993. Statistical clustering techniques for the analysis of long molecular dynamics trajectories: analysis of 2.2-ns trajectories of YPGDV. Biochemistry 32: 412– 420. 4 Giuliani, A. & C. Manetti. 1996. Hidden peculiarities in the potential energy time series of a tripeptide highlighted by a recurrence plot analysis: a molecular dynamics simulation. Phys. Rev. E 53: 6336– 6340. 5 Mestivier, D., N.P. Chau, X. Chanudet, P. Baudeceau & P. Larroque. 1997. Relationship between diabetic autonomic dysfunction and heart rate variability assessed by recurrence plot. Am. J. Physiol. 272: H1094– 1099. 6 Faure, P. & H. Korn. 1997. A nonrandom dynamic component in the synaptic noise of a central neuron. Proc. Natl. Acad. Sci. USA 94: 6506– 6511. 7 Eckmann, J.-P., S.O. Kamphorst & D. Ruelle. 1987. Recurrence plots of dynamical systems. Europhys. Lett. 4: 973– 977. 8 Webber, C.L.Jr., & J.P. Zbilut. 1994. Dynamical assessment of physiological systems and states using recurrence plot strategies. J. Appl. Physiol. 76: 965– 973. 9 Webber, C.L., M.A. Schmidt & J.M. Walsh. 1995. Influence of isometric loading on biceps EMG dynamics as assessed by linear and nonlinear tools. J. Appl. Physiol. 78: 814– 822. 10 Nienhaus, G.U., J.D. Müller, B.H. McMahon & H. Frauenfelder. 1997. Exploring the conformational energy landscape of proteins. Physica D 107: 297– 311. 11 Verlet, L. 1967. Computer "experiments" on classical fluids. I. Thermodynamical properties of Lennard-Jones molecules. Phys. Rev. 159: 98– 103. 12 van der Spoel, D., H.J.C. Berendsen, A.R. van Buuren, E. Apol, P.J. Meulenhoff, A.L.T.M. Sijbers & R. van Drunen. 1995. Gromacs User Manual. Nijenborgh 4, 9747 AG Groningen, The Netherlands. Internet:http://rugmd0.chem.rug.nlgmx. 13 vanBuuren, A.R., S.-J. Marrink & H.J.C. Berendsen. 1993. A molecular dynamics study of the decane/water interface. J. Phys. Chem. 97: 9206– 9212. 14 van Gunsteren, W.F. & H.J.C. Berendsen. 1987. Gromos Manual. BIOMOS, Biomolecular Software, Laboratory of Physical Chemistry, University of Groningen, The Netherlands. 15 van Gunsteren, W.F., S.R. Billeter, A.A. Eising, P.H. Hünenberger, P. Krüger, A.E. Mark, W.R.P. Scott & I.G. Tironi. 1996. Biomolecular Simulation: The GROMOS96 Manual and User's Guide. Biomos b.v. Zürich, Groningen. 16 Ryckaert, J.-P., G. Ciccotti & H.J.C. Berendsen. 1977. Numerical integration of the cartesian equations of motion of a system with constraints: molecular dynamics of n-alkanes. J. Comp. Phys. 23: 327– 341. 17 Gallagher, P.T., P.B. Alexander & G.L. Gilliand. 1994. Two crystal structures of the B1 immunoglobulin-binding domain of streptococcal protein G and comparison with NMR. Biochemistry 33: 4721– 4729. 18 Berendsen, H.J.C., J.P.M. Postma, W.F. van Gunsteren & J. Hermans. 1981. In Intermolecular Forces. B. Pullman, Ed.: 331–342. D. Reidel Publishing Company. Dordrecht, The Netherlands. 19 Berendsen, H.J.C., J.P.M. Postma, W.F. van Gunsteren, A. Di Nola & J.R. Haak. 1984. Molecular dynamics with coupling to an external bath. J. Chem. Phys. 81: 3684– 3690. 20 Broomhead, D.S. & G.P. King. 1986. Extracting qualitative dynamics from experimental data. Physica D 20: 217– 236. 21 Takens, F. 1980. In Dynamical Systems and Turbulence. D.A. Rand & L.-S. Young, Eds.: 366–381. Springer-Verlag. New York, Heidelberg, Berlin. 22 Zak, M., J.P. Zbilut & R.E. Meyers. 1997. In From Instability to Intelligence: Lecture Notes in Physics, M49. Springer. Heidelberg, Germany. 23 Zbilut, J.P., M. Zak & R.E. Meyers. 1996. A terminal dynamics of heartbeat. Biol. Cybern. 75: 277– 280. 24 Zbilut, J.P., A. Giuliani A. & C.L. Webber, Jr. 1998. Recurrence quantification analysis and principal components in the detection of short complex signals. Phys. Lett. A 237: 131– 135. 25 Shannon, C.E. 1948. A mathematical theory of communication. Bell. Syst. Tech. J. 27: 379– 423. 26 Lligona-Trulla, L., A. Giuliani, J.P. Zbilut & C.L. Webber, Jr. 1996. Recurrence quantification analysis of the logistic equation with transients. Phys. Lett. A 223: 255– 260. 27 Rahman, A. 1964. Correlation in the motion of atoms in liquid argon. Phys. Rev. 136: A405– 411. 28 Haile, J.M. 1992. In Molecular Dynamics Simulation: Elementary Methods: 53. John Wiley. New York. 29 Lebowitz, J.L., J.K. Percus & L. Verlet. 1967. Ensemble dependence of fluctuations with application to machine computations. Phys. Rev. 153: 250– 254. 30 Birkhoff, G.D. 1931. Proof of the ergodic theorem. Proc. Natl. Acad. Sci. USA 17: 656– 659. 31 V. Neumann, J. 1932. Physical Applications of the Ergodic Hypothesis. Proc. Natl. Acad. Sci. USA 18: 263– 266. 32 Ford, J. 1973. The transition from analytic dynamics to statistical mechanics, Adv. Chem. Phys. 24: 155– 185. 33 Honeycutt, J.D. & D. Thirumalai. 1990. Metastability of the folded states of globular proteins. Proc. Nat. Acad. Sci. USA. 87: 3526– 3529. 34 Eckmann, J.-P., S.O. Kamphorst, D. Ruelle & S. Ciliberto. 1986. Liapunov exponents from time series. Phys. Rev. A 34: 4971– 4979. 35 Ansari, A., J. Berendzen, S.F. Bowne, H. Frauenfelder, I.E.T. Iben, T.B. Sauke, E. Shyamsunder & R.D. Young. 1985. Protein states and proteinquakes. Proc. Natl. Acad. Sci. USA 82: 5000– 5004. 36 Clarage, J.B., T. Romo, B.K. Andrews, B.M. Pettitt & G.N. Phillips, Jr. 1995. A sampling problem in molecular dynamics simulations of macromolecules. Proc. Natl. Acad. Sci. USA 92: 3288– 3292. 37 Steinbach, P.J. & B.R. Brooks. 1996. Hydrated myoglobin's anharmonic fluctuations are not primarily due to dihedral transitions. Proc. Natl. Acad. Sci. USA 93: 55– 59. 38 Straub, J.E. & D. Thirumalai. 1993. Theoretical probes of conformational fluctuations in S-peptide and RNase A/3′-UMP enzyme product complex. Proteins 15: 360– 373. 39 Hodel, A., T. Simonson, R.O. Fox & A.T. Brunger. 1993. Conformational substates and uncertainty in macromolecular free energy calculations. J. Phys. Chem. 97: 3409– 3417. Citing Literature Volume879, Issue1TEMPOS IN SCIENCE AND NATURE: STRUCTURES, RELATIONS, AND COMPLEXITYJune 1999Pages 258-266 ReferencesRelatedInformation

A molecular dynamics simulation of a Lennard-Jones fluid and a trajectory of the B1 immunoglobulin G-binding domain of streptococcal protein G (B1-IgG) simulated in water are analyzed by recurrence quantification, which is noteworthy for its independence from stationarity constraints, as well as its ability to detect transients and both linear and nonlinear state changes. The results demonstrate the sensitivity of the technique for the discrimination of phase sensitive dynamics. The physical interpretation of the recurrence measures is also discussed.

Protein structure-function relationships have been increasingly scrutinized by a variety of correlational and information theoretic measures. In an effort to extend this methodology, a technique originally developed in non-linear science, recurrence quantification analysis, was combined with traditional principal components analysis to study a large number (56) of TEM-1 beta-lactamase mutants. The hydrophobicity profiles corresponding to the primary structure of 13 naturally occurring mutations partially impairing function, together with 43 artificial non-tolerated mutations were subjected to discriminant analysis, derived from the results of recurrence quantification analysis, coupled to a principal exponents extraction. Eleven (85%) of the naturally occurring mutants and 36 (84%) of the artificial mutants were correctly classified (p < 0.0001). We conclude that this technique may be useful in protein engineering and, in general, in structure-function studies of biopolymers.