Bifurcations of Dynamical Systems, Logistic and Gompertz Growth Laws in Processes of Aggregation

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.