A robust spectral method for finding lumpings and meta stable states of non-reversible Markov chains

A spectral method for identifying lumping in large Markov chains is presented. Identification of meta stable states is treated as a special case. The method is based on spectral analysis of a self-adjoint matrix that is a function of the original transition matrix. It is demonstrated that the technique is more robust than existing methods when applied to noisy non-reversible Markov chains.Comment: 10 pages, 7 figure

A method for inferring hierarchical dynamics in stochastic processes

Complex systems may often be characterized by their hierarchical dynamics. In this paper do we present a method and an operational algorithm that automatically infer this property in a broad range of systems; discrete stochastic processes. The main idea is to systematically explore the set of projections from the state space of a process to smaller state spaces, and to determine which of the projections that impose Markovian dynamics on the coarser level. These projections, which we call Markov projections, then constitute the hierarchical dynamics of the system. The algorithm operates on time series or other statistics, so a priori knowledge of the intrinsic workings of a system is not required in order to determine its hierarchical dynamics. We illustrate the method by applying it to two simple processes; a finite state automaton and an iterated map.Comment: 16 pages, 12 figure

A Design Path for Hierarchical Self-Assembly of Patchy Colloids

Patchy colloids are promising candidates for building blocks in directed self-assembly. To be successful the surface patterns need to both be simple enough to be synthesized, while feature-rich enough to cause the colloids to self-assemble into desired structures. Achieving this is a challenge for traditional synthesis methods. Recently it has been suggested that the surface pattern themselves can be made to self-assemble. In this paper we show that a wide range of functional structures can be made to self-assemble using this approach. More generally we present a design path for hierarchical targeted self-assembly of patchy colloids. At the level of the surface structure, we use a predictive method utilizing universality of patterns of stripes and spots, coupled with stoichiometric constraints, to cause highly specific and functional patterns to self-assemble on spherical surfaces. We use a minimalistic model of an alkanethiol on gold as a model system and demonstrate that, even with limited control over the interaction between surface constituents, we can obtain patterns that causes the colloids themselves to self-assemble into various complex geometric structures. We demonstrate how variations of the same design path cause in-silico self-assembly of strings, membranes, cubic and spherical aggregates, as well as various crystalline patterns.Comment: 8 pages, 5 figure

Predicting self-assembled patterns on spheres with multi-component coatings

Interactions between the components in many-body systems can give rise to spontaneous formation of complex structures. Usually very little is known about the connection between the interactions and the resulting structure. Here we present a theory for self-assembling pattern formation in multi-component systems, formulated as an analytic technique that predicts morphologies directly from the interactions in an effective model. As a demonstration we apply the method to a model of alkanethiols on spherical gold particles, successfully predicting its morphologies and transitions as a function of the interaction parameters. This system is interesting because it has been suggested to provide an effective route to produce patchy colloids.Comment: 5 pages, 4 figure

Quasi-Species and Aggregate Dynamics

At an early stage in pre-biotic evolution, groups of replicating molecules must coordinate their reproduction to form aggregated units of selection. Mechanisms that enable this to occur are currently not well understood. In this paper we introduce a deterministic model of primitive replicating aggregates, proto-organisms, that host populations of replicating information carrying molecules. Some of the molecules promote the reproduction of the proto-organism at the cost of their individual replication rate. A situation resembling that of group selection arises. We derive and analytically solve a partial differential equation that describes the system. We find that the relative prevalence of fast and slow replicators is determined by the relative strength of selection at the aggregate level to the selection strength at the molecular level. The analysis is concluded by a preliminary treatment of finite population size effects.Comment: 6 page