Solution of the Quasispecies Model for an Arbitrary Gene Network

In this paper, we study the equilibrium behavior of Eigen's quasispecies equations for an arbitrary gene network. We consider a genome consisting of $N$ genes, so that each gene sequence $\sigma$ may be written as $\sigma = \sigma_1 \sigma_2 ... \sigma_N$. We assume a single fitness peak (SFP) model for each gene, so that gene $i$ has some ``master'' sequence $\sigma_{i, 0}$ for which it is functioning. The fitness landscape is then determined by which genes in the genome are functioning, and which are not. The equilibrium behavior of this model may be solved in the limit of infinite sequence length. The central result is that, instead of a single error catastrophe, the model exhibits a series of localization to delocalization transitions, which we term an ``error cascade.'' As the mutation rate is increased, the selective advantage for maintaining functional copies of certain genes in the network disappears, and the population distribution delocalizes over the corresponding sequence spaces. The network goes through a series of such transitions, as more and more genes become inactivated, until eventually delocalization occurs over the entire genome space, resulting in a final error catastrophe. This model provides a criterion for determining the conditions under which certain genes in a genome will lose functionality due to genetic drift. It also provides insight into the response of gene networks to mutagens. In particular, it suggests an approach for determining the relative importance of various genes to the fitness of an organism, in a more accurate manner than the standard ``deletion set'' method. The results in this paper also have implications for mutational robustness and what C.O. Wilke termed ``survival of the flattest.''Comment: 29 pages, 5 figures, to be submitted to Physical Review

Neutral Evolution of Mutational Robustness

We introduce and analyze a general model of a population evolving over a network of selectively neutral genotypes. We show that the population's limit distribution on the neutral network is solely determined by the network topology and given by the principal eigenvector of the network's adjacency matrix. Moreover, the average number of neutral mutant neighbors per individual is given by the matrix spectral radius. This quantifies the extent to which populations evolve mutational robustness: the insensitivity of the phenotype to mutations. Since the average neutrality is independent of evolutionary parameters---such as, mutation rate, population size, and selective advantage---one can infer global statistics of neutral network topology using simple population data available from {\it in vitro} or {\it in vivo} evolution. Populations evolving on neutral networks of RNA secondary structures show excellent agreement with our theoretical predictions.Comment: 7 pages, 3 figure

Coupled Replicator Equations for the Dynamics of Learning in Multiagent Systems

Starting with a group of reinforcement-learning agents we derive coupled replicator equations that describe the dynamics of collective learning in multiagent systems. We show that, although agents model their environment in a self-interested way without sharing knowledge, a game dynamics emerges naturally through environment-mediated interactions. An application to rock-scissors-paper game interactions shows that the collective learning dynamics exhibits a diversity of competitive and cooperative behaviors. These include quasiperiodicity, stable limit cycles, intermittency, and deterministic chaos--behaviors that should be expected in heterogeneous multiagent systems described by the general replicator equations we derive.Comment: 4 pages, 3 figures, http://www.santafe.edu/projects/CompMech/papers/credlmas.html; updated references, corrected typos, changed conten

A recent appreciation of the singular dynamics at the edge of chaos

We study the dynamics of iterates at the transition to chaos in the logistic map and find that it is constituted by an infinite family of Mori's $q$-phase transitions. Starting from Feigenbaum's $\sigma$ function for the diameters ratio, we determine the atypical weak sensitivity to initial conditions $\xi _{t}$ associated to each $q$-phase transition and find that it obeys the form suggested by the Tsallis statistics. The specific values of the variable $q$ at which the $q$-phase transitions take place are identified with the specific values for the Tsallis entropic index $q$ in the corresponding $\xi_{t}$. We describe too the bifurcation gap induced by external noise and show that its properties exhibit the characteristic elements of glassy dynamics close to vitrification in supercooled liquids, e.g. two-step relaxation, aging and a relationship between relaxation time and entropy.Comment: Proceedings of: Verhulst 200 on Chaos, Brussels 16-18 September 2004, Springer Verlag, in pres

Information-theoretic approach to the study of control systems

We propose an information-theoretic framework for analyzing control systems based on the close relationship of controllers to communication channels. A communication channel takes an input state and transforms it into an output state. A controller, similarly, takes the initial state of a system to be controlled and transforms it into a target state. In this sense, a controller can be thought of as an actuation channel that acts on inputs to produce desired outputs. In this transformation process, two different control strategies can be adopted: (i) the controller applies an actuation dynamics that is independent of the state of the system to be controlled (open-loop control); or (ii) the controller enacts an actuation dynamics that is based on some information about the state of the controlled system (closed-loop control). Using this communication channel model of control, we provide necessary and sufficient conditions for a system to be perfectly controllable and perfectly observable in terms of information and entropy. In addition, we derive a quantitative trade-off between the amount of information gathered by a closed-loop controller and its relative performance advantage over an open-loop controller in stabilizing a system. This work supplements earlier results [H. Touchette, S. Lloyd, Phys. Rev. Lett. 84, 1156 (2000)] by providing new derivations of the advantage afforded by closed-loop control and by proposing an information-based optimality criterion for control systems. New applications of this approach pertaining to proportional controllers, and the control of chaotic maps are also presented.Comment: 18 pages, 7 eps figure

Intermittency at critical transitions and aging dynamics at edge of chaos

We recall that, at both the intermittency transitions and at the Feigenbaum attractor in unimodal maps of non-linearity of order $\zeta >1$, the dynamics rigorously obeys the Tsallis statistics. We account for the $q$-indices and the generalized Lyapunov coefficients $\lambda_{q}$ that characterize the universality classes of the pitchfork and tangent bifurcations. We identify the Mori singularities in the Lyapunov spectrum at the edge of chaos with the appearance of a special value for the entropic index $q$. The physical area of the Tsallis statistics is further probed by considering the dynamics near criticality and glass formation in thermal systems. In both cases a close connection is made with states in unimodal maps with vanishing Lyapunov coefficients.Comment: Proceedings of: STATPHYS 2004 - 22nd IUPAP International Conference on Statistical Physics, National Science Seminar Complex, Indian Institute of Science, Bangalore, 4-9 July 2004. Pramana, in pres

Chance and Necessity in Evolution: Lessons from RNA

The relationship between sequences and secondary structures or shapes in RNA exhibits robust statistical properties summarized by three notions: (1) the notion of a typical shape (that among all sequences of fixed length certain shapes are realized much more frequently than others), (2) the notion of shape space covering (that all typical shapes are realized in a small neighborhood of any random sequence), and (3) the notion of a neutral network (that sequences folding into the same typical shape form networks that percolate through sequence space). Neutral networks loosen the requirements on the mutation rate for selection to remain effective. The original (genotypic) error threshold has to be reformulated in terms of a phenotypic error threshold. With regard to adaptation, neutrality has two seemingly contradictory effects: It acts as a buffer against mutations ensuring that a phenotype is preserved. Yet it is deeply enabling, because it permits evolutionary change to occur by allowing the sequence context to vary silently until a single point mutation can become phenotypically consequential. Neutrality also influences predictability of adaptive trajectories in seemingly contradictory ways. On the one hand it increases the uncertainty of their genotypic trace. At the same time neutrality structures the access from one shape to another, thereby inducing a topology among RNA shapes which permits a distinction between continuous and discontinuous shape transformations. To the extent that adaptive trajectories must undergo such transformations, their phenotypic trace becomes more predictable.Comment: 37 pages, 14 figures; 1998 CNLS conference; high quality figures at http://www.santafe.edu/~walte

Feigenbaum graphs: a complex network perspective of chaos

The recently formulated theory of horizontal visibility graphs transforms time series into graphs and allows the possibility of studying dynamical systems through the characterization of their associated networks. This method leads to a natural graph-theoretical description of nonlinear systems with qualities in the spirit of symbolic dynamics. We support our claim via the case study of the period-doubling and band-splitting attractor cascades that characterize unimodal maps. We provide a universal analytical description of this classic scenario in terms of the horizontal visibility graphs associated with the dynamics within the attractors, that we call Feigenbaum graphs, independent of map nonlinearity or other particulars. We derive exact results for their degree distribution and related quantities, recast them in the context of the renormalization group and find that its fixed points coincide with those of network entropy optimization. Furthermore, we show that the network entropy mimics the Lyapunov exponent of the map independently of its sign, hinting at a Pesin-like relation equally valid out of chaos.Comment: Published in PLoS ONE (Sep 2011

Complex dynamics of defective interfering baculoviruses during serial passage in insect cells

Defective interfering (DI) viruses are thought to cause oscillations in virus levels, known as the 'Von Magnus effect'. Interference by DI viruses has been proposed to underlie these dynamics, although experimental tests of this idea have not been forthcoming. For the baculoviruses, insect viruses commonly used for the expression of heterologous proteins in insect cells, the molecular mechanisms underlying DI generation have been investigated. However, the dynamics of baculovirus populations harboring DIs have not been studied in detail. In order to address this issue, we used quantitative real-time PCR to determine the levels of helper and DI viruses during 50 serial passages of Autographa californica multiple nucleopolyhedrovirus (AcMNPV) in Sf21 cells. Unexpectedly, the helper and DI viruses changed levels largely in phase, and oscillations were highly irregular, suggesting the presence of chaos. We therefore developed a simple mathematical model of baculovirus-DI dynamics. This theoretical model reproduced patterns qualitatively similar to the experimental data. Although we cannot exclude that experimental variation (noise) plays an important role in generating the observed patterns, the presence of chaos in the model dynamics was confirmed with the computation of the maximal Lyapunov exponent, and a Ruelle-Takens-Newhouse route to chaos was identified at decreasing production of DI viruses, using mutation as a control parameter. Our results contribute to a better understanding of the dynamics of DI baculoviruses, and suggest that changes in virus levels over passages may exhibit chaos.The authors thank Javier Carrera, Just Vlak and Lia Hemerik for helpful discussion. MPZ was supported by a Rubicon Grant from the Netherlands Organization for Scientific Research (NWO, www.nwo.nl) and a 'Juan de la Cierva' postdoctoral contract (JCI-2011-10379) from the Spanish 'Secretaria de Estado de Investigacion, Desarrollo e Innovacion'. 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