Evolution of canalizing Boolean networks

Boolean networks with canalizing functions are used to model gene regulatory networks. In order to learn how such networks may behave under evolutionary forces, we simulate the evolution of a single Boolean network by means of an adaptive walk, which allows us to explore the fitness landscape. Mutations change the connections and the functions of the nodes. Our fitness criterion is the robustness of the dynamical attractors against small perturbations. We find that with this fitness criterion the global maximum is always reached and that there is a huge neutral space of 100% fitness. Furthermore, in spite of having such a high degree of robustness, the evolved networks still share many features with “chaotic” networks.     

Comparing the evolution of canalyzing and threshold networks

We study the influence of the type of update functions on the evolution of Boolean networks under selection for dynamical robustness. The chosen types of functions are canalyzing functions and threshold functions. Starting from a random initial network, we evolve the network by an adaptive walk. During the first time period, where the networks evolve to the plateau of 100 percent fitness, we find that both type of update functions give the same behavior, albeit for different network sizes and connectedness. However, on the long run, as the networks continue to evolve on the fitness plateau, the different types of update functions give rise to different network structure, due to their different mutational robustness. When both types of update functions occur together, none of them is preferred under long-term evolution.     

Emergent criticality through adaptive information processing in boolean networks

We study information processing in populations of boolean networks with evolving connectivity and systematically explore the interplay between the learning capability, robustness, the network topology, and the task complexity. We solve a long-standing open question and find computationally that, for large system sizes N, adaptive information processing drives the networks to a critical connectivity K(c)=2. For finite size networks, the connectivity approaches the critical value with a power law of the system size N. We show that network learning and generalization are optimized near criticality, given that the task complexity and the amount of information provided surpass threshold values. Both random and evolved networks exhibit maximal topological diversity near K(c). We hypothesize that this diversity supports efficient exploration and robustness of solutions. Also reflected in our observation is that the variance of the fitness values is maximal in critical network populations. Finally, we discuss implications of our results for determining the optimal topology of adaptive dynamical networks that solve computational tasks.     

Response of boolean networks to perturbations

We evaluate the probability that a Boolean network returns to an attractor after perturbing h nodes. We find that the return probability as function of h can display a variety of different behaviours, which yields insights into the state-space structure. In addition to performing computer simulations, we derive analytical results for several types of Boolean networks, in particular for Random Boolean Networks. We also apply our method to networks that have been evolved for robustness to small perturbations, and to a biological example.     

Robustness of Complex Networks under Attack and Repair

To study the robustness of complex networks under attack and repair, we introduce a repair model of complex networks. Based on the model, we introduce two new quantities, i.e. attack fraction fa and the maximum degree of the nodes that have never been attacked ~Ka, to study analytically the critical attack fraction and the relative size of the giant component of complex networks under attack and repair, using the method of generating function. We show analytically and numerically that the repair strategy significantly enhances the robustness of the scale-free network and the effect of robustness improvement is better for the scale-free networks with a smaller degree exponent. We discuss the application of our theory in relation to the
understanding of robustness of complex networks with reparability.     

Nonlinear mean field Fokker-Planck equations. Application to the chemotaxis of biological populations

We study a general class of nonlinear mean field Fokker-Planck equations in relation with an effective generalized thermodynamical (E.G.T.) formalism. We show that these equations describe several physical systems such as: chemotaxis of bacterial populations, Bose-Einstein condensation in the canonical ensemble, porous media, generalized Cahn-Hilliard equations, Kuramoto model, BMF model, Burgers equation, Smoluchowski-Poisson system for self-gravitating Brownian particles, Debye-Hückel theory of electrolytes, two-dimensional turbulence... In particular, we show that nonlinear mean field Fokker-Planck equations can provide generalized Keller-Segel models for the chemotaxis of biological populations. As an example, we introduce a new model of chemotaxis incorporating both effects of anomalous diffusion and exclusion principle (volume filling). Therefore, the notion of generalized thermodynamics can have applications for concrete physical systems. We also consider nonlinear mean field Fokker-Planck equations in phase space and show the passage from the generalized Kramers equation to the generalized Smoluchowski equation in a strong friction limit. Our formalism is simple and illustrated by several explicit examples corresponding to Boltzmann, Tsallis, Fermi-Dirac and Bose-Einstein entropies among others.     

Selective advantage of topological disorder in biological evolution

We examine a model of biological evolution of Eigen's quasispecies in a so-called holey fitness landscape, where the fitness of a site is either 0 (lethal site) or a uniform positive constant (viable site). The evolution dynamics is therefore determined by the topology of the genome space which is modelled by the random Bethe lattice. We use the effective medium and single-defect approximations to find the criteria under which the localized quasispecies cloud is created. We find that shorter genomes, which are more robust to random mutations than average, represent a selective advantage which we call “topological”. A way of assessing empirically the relative importance of reproductive success and topological advantage is suggested. Received 9 August 2002 / Received in final form 7 November 2002 Published online 14 February 2003 RID="a" ID="a"e-mail: slanina@fzu.cz     

A limited resource model of fault-tolerant capability against cascading failure of complex network

We propose a novel capacity model for complex networks against cascading failure. In this model, vertices with both higher loads and larger degrees should be paid more extra capacities, i.e. the allocation of extra capacity on vertex i will be proportional to k_i ^γ , where k_i is the degree of vertex i and γ > 0 is a free parameter. We have applied this model on Barabási-Albert network as well as two real transportation networks, and found that under the same amount of available resource, this model can achieve better network robustness than previous models.     

Adaptive walks and extreme value theory

We study biological evolution in a high-dimensional genotype space in the regime of rare mutations and strong selection. The population performs an uphill walk which terminates at local fitness maxima. Assigning fitness randomly to genotypes, we show that the mean walk length is logarithmic in the number of initially available beneficial mutations, with a prefactor determined by the tail of the fitness distribution. This result is derived analytically in a simplified setting where the mutational neighborhood is fixed during the adaptive process, and confirmed by numerical simulations.     

Effect of edge removal on topological and functional robustness of complex networks

We study the robustness of several network models subject to edge removal. The robustness is measured by the statistics of network breakdowns, where a breakdown is defined as the destroying of the total connectedness of a network, rather than the disappearance of the giant component. We introduce a simple traffic dynamics as the function of a network topology, and the total connectedness can be destroyed in the sense of either the topology or the function. The overall effect of the topological breakdown and the functional breakdown, as well as the relative importance of the topological robustness and the functional robustness, are studied under two edge removal strategies.     

Ageing of natural rubber under stress

We report a dynamical-mechanical study of stress relaxation at small deformation in a natural (polyisoprene) rubber well above its glass transition temperature . We find that an almost complete relaxation of stress takes place over very long relaxation periods, even though the elastic network is retained. The relaxation rate and the long-time equilibrium modulus are sensitive functions of temperature which do not follow time-temperature superposition. Many characteristic features of non-ergodic ageing response are apparent at both short and very long times. We interpret the observed behaviour in terms of the nature of rubber cross-links, capable of isomerisation under stress, and relate the results to recent models of slow glassy rheology. Received 22 November 1999 and Received in final form 18 January 2000     

Adaptation dynamics of the quasispecies model

We study the adaptation dynamics of an initially maladapted population evolving via the elementary processes of mutation and selection. The evolution occurs on rugged fitness landscapes which are defined on the multi-dimensional genotypic space and have many local peaks separated by low fitness valleys. We mainly focus on the Eigen’s model that describes the deterministic dynamics of an infinite number of self-replicating molecules. In the stationary state, for small mutation rates such a population forms a quasispecies which consists of the fittest genotype and its closely related mutants. The quasispecies dynamics on rugged fitness landscape follow a punctuated (or steplike) pattern in which a population jumps from a low fitness peak to a higher one, stays there for a considerable time before shifting the peak again and eventually reaches the global maximum of the fitness landscape. We calculate exactly several properties of this dynamical process within a simplified version of the quasispecies model.      

How non-uniform tolerance parameter strategy changes the response of scale-free networks to failures

In this paper, we introduce a non-uniform tolerance parameter (TP) strategy (the tolerance parameter is characterized by the proportion between the unused capacity and the capacity of a vertex) and study how the non-uniform TP strategy influences the response of scale-free (SF) networks to cascading failures. Different from constant TP in previous work of Motter and Lai (ML), the TP in the proposed strategy scales as a power-law function of vertex degree with an exponent b. The simulations show that under low construction costs D, when b>0 the tolerance of SF networks can be greatly improved, especially at moderate values of b; When b<0 the tolerance gets worse, compared with the case of constant TP in the ML model. While for high D the tolerance declines slightly with the b, namely b<0 is helpful to the tolerance, and b>0 is harmful. Because for smaller b the cascade of the network is mainly induced by failures of most high-degree vertices; while for larger b, the cascade attributes to damage of most low-degree vertices. Furthermore, we find that the non-uniform TP strategy can cause changes of the structure and the load-degree correlation in the network after the cascade. These results might give insights for the design of both network capacity to improve network robustness under limitation of small cost, and for the design of strategies to defend cascading failures of networks.     

Multispeckle diffusing wave spectroscopy of colloidal particles suspended in a random packing of glass spheres

We use a multispeckle diffusing wave spectroscopy (MSDWS) method to study the ensemble-averaged dynamics of the fluctuating speckle pattern when illuminating colloidal particles suspended in a static and opaque porous medium with a coherent light source. Experiments were performed with Brownian latex particles in a random packing of glass spheres. The mixing of the light scattered by the moving colloidal particles and the porous matrix gives rise to a plateau value of the intensity autocorrelation function in the long-waiting-time limit. From the plateau in the correlation function, we can determine the fraction of light scattered from moving particles and estimate the photon mean free path in the colloidal solution. The method opens up promising possibilities to probe the static fraction in semisolid materials.     

Direct imaging of dynamical heterogeneities near the colloid-gel transition

We observe the microscopic dynamics of a suspension of colloids with attractive interaction by confocal fluorescence microscopy to provide a deeper understanding of the relationship between local structure and dynamics near the gel transition. We study the distinct and self-parts of the van Hove density-density correlation function applied to our experimental data. Separable fast and slow populations emerge in the self-part, while the distinct part shows a pronounced signature of dynamic heterogeneities close to the gel transition, dominated by the fast particles. The slow population close to the gel transition shares features with an attraction-driven colloidal glass, including a plateau in the mean squared displacement that provides an estimate for the dynamical localization length.     

Computational study of impulsively generated standing slow acoustic waves in a solar coronal loop

We numerically investigated standing slow acoustic waves impulsively excited in a solar coronal loop by gas pressure and mass density perturbations in one-dimensional space. The corresponding computer model is described by the hydrodynamic equations that are solved numerically by means of the so-called flux limiters methods on uniformly structured mesh. We discuss the fundamental mode and the first harmonic mode which are generated in dependence on position of the initial perturbation in the numerical box. We show how the standing slow acoustic waves are generated in the corona, where they are trapped in space between two dense layers as in the resonator, and how their energy leaks from the corona to the dense layers. We found that this leakage increases with the decrease of the density jump at the transition region. We also studied the case when the perturbation is initiated at the transition region. We found that even in this case the standing wave is formed, but their energetics is influenced by the evaporation of the plasma from the transition region into the corona.     

Physics of evolution: Selection without fitness

Traditionally evolution is seen as a process where from a pool of possible variations of a population (e.g. biological species or industrial goods) a few variations get selected which survive and proliferate, whereas the others vanish. Survival probabilities and proliferation rates are typically associated with the ‘fitness’ of particular variations. In this paper we argue that the notion of fitness is an a posteriori concept, in the sense that one can assign higher fitness to species that survive but one can generally not derive or predict fitness per se. Proliferation rates can be measured, whereas fitness landscapes, i.e. the inter-dependence of proliferation rates, cannot. For this reason we think that in a physical theory of evolution such notions should be avoided. In this spirit, here we propose a random matrix model of evolution where selection mechanisms are encoded in interaction matrices of species, thereby extending the previous work of ours by a control parameter describing suppressors in the system. We are able to recover some key facts of evolution dynamics endogenously, such as punctuated equilibrium, i.e. the existence of intrinsic large extinction events, and, at the same time, periods of dramatic diversification, as known e.g. from the fossil record. Further, we comment on two fundamental technical problems of a ‘physics of evolution’, the non-closedness of its phase space and the problem of co-evolving boundary conditions, apparent in all systems subject to evolution.     

Efficiency based strategy spreading in the prisoner’s dilemma game

In contrast to well-mixed populations, discrete interaction patterns have been shown to support cooperation in the prisoner’s dilemma game, and a scale-free network topology may even lead to a dominance of cooperation over defection. The majority of studies assumes a strategy adoption scheme based on accumulated payoffs. The use of accumulated payoffs, however, is incompatible with the integral property of the underlying replicator dynamics to be invariant under a positive affine transformation of the payoff function. We show that using instead the payoff per interaction to determine the strategy spread, which has been suggested recently and recovers the required invariance, results in fundamentally different dynamical behavior under a synchronized strategy adoption considered here. Most notably, in such an efficiency based scenario the advantage of a scale-free network topology vanishes almost completely. We present a detailed explanation of the fundamentally altered dynamical behavior.