Abstract
From the systemic point of view protein aggregation is a compensatory mechanism allowing transition of a system (protein solution) from an initially stable equilibrium, which became unstable under a stress, to another stable equilibrium, which bifurcates from the initial one because of the stress. The simplest bifurcation of this type is Logistic bifurcation with a positive small parameter.
We realize this bifurcation as a model of protein aggregation through a large-dimensional Becker-Döring system with a one-dimensional Logistic attractor (BDL) containing two equilibria. BDL depends on the magnitude δ of stress as a small parameter. Kinetics on the attractor is transformed by the observable (which is a fewnomial, i.e., a high-degree polynomial with a number of terms that is small relative to the degree) into the observed kinetics of the experiment. This model explains Gompertzian growth, unimodality of size distribution of aggregates, and relations between Rate, Plateau and time elapsed from onset to inflection moments. The explanation is based on the existence of a nonequilibrium partition function. It exists under the assumption of formation of aggregation-competent monomer as a precursor of the aggregation.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Morris, A.M., Watzky, M.A., Finke, R.G.: Protein Aggregation Kinetics, Mechanism, and Curve Fitting: A Review of the Literature. Biochimica et Biophysica Acta 1794, 375–397 (2009)
Bernacki, J.P., Murphy, R.M.: Model Discrimination and Mechanistic Interpretation of Kinetic Data in Protein Aggregation Studies. Biophysical Journal 96, 2871–2887 (2009)
Frieden, C.: Protein aggregation processes: In search of the mechanism. Protein Sci. 16(11), 2334–2344 (2007)
Roberts, C.J.: Non-native protein aggregation: pathways, kinetics, and shelf-life prediction. In: Murphy, R.M., Tsai, A.M. (eds.) Misbehaving Proteins: Protein Misfolding, Aggregation, and Stability. Springer, New York (2006)
Becker, R., Döring, W.: Kinetische Behandlung der Keimbildung in ubersattigten Dampfern. Ann. Phys. (Leipzig) 24, 719–752 (1935)
Wegner, A., Engel, J.: Kinetics of the cooperative association of actin to actin filaments. Biophys. Chem. 3, 215–225 (1975)
Raibekas, A.A.: Estimation of protein aggregation propensity with a melting point apparatus. Analytical Biochemistry 380, 331–332 (2008)
Krishnan, S., Raibekas, A.A.: Multistep Aggregation Pathway of Human Interleukin-1 Receptor Antagonist: Kinetic, Structural, and Morphological. Biophysical Journal 96, 199–208 (2009)
Ball, J.M., Carr, J., Penrose, O.: The Becker-Döring Cluster Equations: Basic Properties and Asymptotic Behaviour of Solutions. Commun. Math. Phys. 104, 657–692 (1986)
Penrose, O.: The Becker-Döring equations for the kinetics of phase transitions, August 22 (2001)
Niethammer, B.: A Vanishing Excess Density Limit of the Becker-D’́oring Equations, http://sfb611.iam.uni-bonn.de/uploads/159-komplett.pdf
Wattis, J.A.D., Coveney, P.V.: Renormalisation-theoretic analysis of non-equilibrium phase transitions I:The Becker-D’́oring equations with power law rate coefficients, 1–18 (2001)
Arnold, V.I.: Geometrical Methods in the Theory of Ordinary Differential Equations. In: Grundlehren der mathematischen Wissenschaften, vol. 250. Springer, New York (1983)
Shoshitaishvili, A.N.: Bifurcations of topological type of a vector field near a singular point. Tr. Sem. Petrovskogo 1, 279–309 (1975); English translation in American Math. Soc. Translations 118(2) (1982)
Wattis, J.A., King, J.R.: Asymptotic solutions of Becker-Döring equations. J. Phys. A: Math. Gen. 31, 7169–7189 (1998)
Bishop, M.F., Ferrone, F.A.: Kinetics of Nucleation-controlled Polymerization, A Perturbation Treatment for Use with a Secondary Pathway. Biophysical Journal 46(5), 631–644 (1984)
Cooper, J.A., Loren Buhle Jr., E., Walker, S.B., Tsong, T.Y., Pollard, T.D.: Kinetic Evidence for a Monomer Activation Step in Actin Polymerization. Biochemistry 22, 2193–2202 (1983)
Powers, E.T., Powers, D.L.: The Kinetics of Nucleated Polymerizations at High Concentrations: Amyloid Fibril Formation Near and Above the Supercritical Concentration. Biophysical Journal 91, 122–132 (2006)
Kuret, J., Congdon, E.E., Li, G., Yin, H., Yu, X., Zhong, Q.: Evaluating Triggers and Enhancers of Tau Fibrillization. Microscopy Research and Technique 67, 141–155 (2005)
Winsor, C.P.: The Gompertz Curve as a Growth Curve. Proceedings of the National Academy of Sciences 18, 1 (1932)
Wang, W.: Protein aggregation and its inhibition in biopharmaceutics. International Journal of Pharmaceutics 289, 1–30 (2005)
Tanford, C.: Thermodynamics of Micelle Formation: Prediction of Micelle Size and Size Distribution. Proc. Nat. Acad. Sci. USA 71(5), 1811–1815 (1974)
Richards, F.J.: A flexible growth function for empirical use. J. Exp. Bot. 10, 290–300 (1959)
Wegner, A., Savko, P.: Fragmentation of Actin Filaments. Biochemistry 21, 1909–1913 (1982)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Shoshitaishvili, A., Raibekas, A. (2010). Bifurcations of Dynamical Systems, Logistic and Gompertz Growth Laws in Processes of Aggregation. In: Lévine, J., Müllhaupt, P. (eds) Advances in the Theory of Control, Signals and Systems with Physical Modeling. Lecture Notes in Control and Information Sciences, vol 407. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-16135-3_28
Download citation
DOI: https://doi.org/10.1007/978-3-642-16135-3_28
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-16134-6
Online ISBN: 978-3-642-16135-3
eBook Packages: EngineeringEngineering (R0)