Logistic map analysis of biomolecular network evolution

Research about the Characteristics of Chaotic Systems Based on Multi-Scale Entropy

The logistic chaotic system, as a classical complex phenomenon of nonlinear dynamic systems, has received extensive attention in the field of secure communication. It is generally believed that the characteristics of chaos are suitable for the needs of encryption systems. In this paper, a multi-scale entropy theory analysis and statistical analysis are carried out on the chaotic sequences produced by different parameters and different initial values of logistic systems. According to the simulation results, the complexity of the chaotic system represented by the logistic system is mainly decided by parameter μ. Not all characteristic parameters of the chaotic system depend on the initial values. It is possible to make a reasonable estimation and prediction of the chaotic system from a macroscopic level. A variance estimation method for the parameter μ is proposed and applied to a logistic system and to another chaotic system, which is equally effective.

Using rotation number to detect sticky orbits in Hamiltonian systems

In Hamiltonian systems, depending on the control parameter, orbits can stay for very long times around islands, the so-called stickiness effect caused by a temporary trapping mechanism. Different methods have been used to identify sticky orbits, such as recurrence analysis, recurrence time statistics, and finite-time Lyapunov exponent. However, these methods require a large number of map iterations and to know the island positions in the phase space. Here, we show how to use the small divergence of bursts in the rotation number calculation as a tool to identify stickiness without knowing the island positions. This new procedure is applied to the standard map, a map that has been used to describe the dynamic behavior of several nonlinear systems. Moreover, our procedure uses a small number of map iterations and is proper to identify the presence of stickiness phenomenon for different values of the control parameter.

Neutral aggregation in finite-length genotype space

The advent of modern genome sequencing techniques allows for a more stringent test of the neutrality hypothesis of Darwinian evolution, where all individuals have the same fitness. Using the individual-based model of Wright and Fisher, we compute the amplitude of neutral aggregation in the genome space, i.e., the probability of finding two individuals at genetic (Hamming) distance k as a function of the genome size L, population size N, and mutation probability per base ν. In well-mixed populations, we show that for Nν<1/L, neutral aggregation is the dominant force and most individuals are found at short genetic distances from each other. For Nν>1, on the contrary, individuals are randomly dispersed in genome space. The results are extended to a geographically dispersed population, where the controlling parameter is shown to be a combination of mutation and migration probability. The theory we develop can be used to test the neutrality hypothesis in various ecological and evolutionary systems.

Influence of homology and node age on the growth of protein-protein interaction networks

Proteins participating in a protein-protein interaction network can be grouped into homology classes following their common ancestry. Proteins added to the network correspond to genes added to the classes, so the dynamics of the two objects are intrinsically linked. Here we first introduce a statistical model describing the joint growth of the network and the partitioning of nodes into classes, which is studied through a combined mean-field and simulation approach. We then employ this unified framework to address the specific issue of the age dependence of protein interactions through the definition of three different node wiring or divergence schemes. A comparison with empirical data indicates that an age-dependent divergence move is necessary in order to reproduce the basic topological observables together with the age correlation between interacting nodes visible in empirical data. We also discuss the possibility of nontrivial joint partition and topology observables.