Martin Nilsson Jacobi

Determining interaction rules in animal swarms

In this paper, we introduce a method for determining local interaction rules in animal swarms. The method is based on the assumption that the behavior of individuals in a swarm can be treated as a set of mechanistic rules.The principal idea behind the technique is to vary parameters that define a set of hypothetical interactions, as for example, a rule for aligning with neighbors. The parameter values are optimized so that the deviation between the observed movements in an animal swarm and the movements predicted by the assumed rule set is minimal. We demonstrate the method by reconstructing the interaction rules from the trajectories produced by a computer simulation. Copyright 2010, Oxford University Press.

Optimal networks of nature reserves can be found through eigenvalue perturbation theory of the connectivity matrix

Conservation and management of natural resources and biodiversity need improved criteria to select functional networks of protected areas. The connectivity within networks due to dispersal is rarely considered, partly because it is unclear how connectivity information can be included in the selection of protected areas. We present a novel and general method that applies eigenvalue perturbation theory (EPT) to select optimum networks of protected areas based on connectivity. At low population densities, characteristic of threatened populations, this procedure selects networks that maximize the growth rate of the overall network. This method offers an improved link between connectivity and metapopulation dynamics. Our framework is applied to connectivities estimated for marine larvae and demonstrates that, for open populations, the best strategy is to protect areas acting as both strong donors and recipients of recruits. It should be possible to implement an EPT framework for connectivity analysis into existing holistic tools for design of protected areas.

Emergence of protocellular growth laws

Template-directed replication is known to obey a parabolic growth law due to product inhibition (Sievers & Von Kiedrowski 1994 Nature 369, 221; Lee et al. 1996 Nature 382, 525; Varga & Szathmáry 1997 Bull. Math. Biol. 59, 1145). We investigate a template-directed replication with a coupled template catalysed lipid aggregate production as a model of a minimal protocell and show analytically that the autocatalytic template–container feedback ensures balanced exponential replication kinetics; both the genes and the container grow exponentially with the same exponent. The parabolic gene replication does not limit the protocellular growth, and a detailed stoichiometric control of the individual protocell components is not necessary to ensure a balanced gene–container growth as conjectured by various authors (Gánti 2004 Chemoton theory). Our analysis also suggests that the exponential growth of most modern biological systems emerges from the inherent spatial quality of the container replication process as we show analytically how the internal gene and metabolic kinetics determine the cell populations generation time and not the growth law (Burdett & Kirkwood 1983 J. Theor. Biol. 103, 11–20; Novak et al. 1998 Biophys. Chem. 72, 185–200; Tyson et al. 2003 Curr. Opin. Cell Biol. 15, 221–231). Previous extensive replication reaction kinetic studies have mainly focused on template replication and have not included a coupling to metabolic container dynamics (Stadler et al. 2000 Bull. Math. Biol. 62, 1061–1086; Stadler & Stadler 2003 Adv. Comp. Syst. 6, 47). The reported results extend these investigations. Finally, the coordinated exponential gene–container growth law stemming from catalysis is an encouraging circumstance for the many experimental groups currently engaged in assembling self-replicating minimal artificial cells (Szostak 2001 et al. Nature 409, 387–390; Pohorille & Deamer 2002 Trends Biotech. 20 123–128; Rasmussen et al. 2004 Science 303, 963–965; Szathmáry 2005 Nature 433, 469–470; Luisi et al. 2006 Naturwissenschaften 93, 1–13).1

A Spectral Method For Aggregating Variables In Linear Dynamical Systems With Application To Cellular Automata Renormalization

We present a method for identifying coarse-grained dynamics through aggregation of variables or states in linear dynamical systems. The condition for aggregation is expressed as a permutation symmetry of a set of dual eigenvectors of the matrix that defines the dynamics. The applicability of the condition is illustrated in examples from three different generic classes of reducible Markov chains: systems consisting of independent subsystems, dynamics with symmetries, and nearly decoupled Markov chains. Furthermore we show how the method can be used to coarse-grain cellular automata.

A METHOD FOR INFERRING HIERARCHICAL DYNAMICS IN STOCHASTIC PROCESSES

Complex systems may often be characterized by their hierarchical dynamics. In this paper 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 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.

Effective thermostat induced by coarse graining of simple point charge water

We investigate how the transport properties of a united atom fluid with a dissipative particle dynamics thermostat depend on the functional form and magnitude of both the conservative and the stochastic interactions. We demonstrate how the thermostat strongly affects the hydrodynamics, especially diffusion, viscosity, and local escape times. As model system we use simple point charge (SPC) water, from which projected trajectories are used to determine the effective interactions in the united atom model. The simulation results support our argument that the thermostat should be viewed as an integral part of the coarse-grained dynamics rather than a tool for approaching thermal equilibrium. As our main result we show that the united atom model with the adjusted effective interactions approximately reproduces the diffusion constant and the viscosity of the underlying detailed SPC water model.

Designing Isotropic Interactions for Self-Assembly of Complex Lattices

We present a direct method for solving the inverse problem of designing isotropic potentials that cause self-assembly into target lattices. Each potential is constructed by matching its energy spectrum to the reciprocal representation of the lattice to guarantee that the desired structure is a ground state. We use the method to self-assemble complex lattices not previously achieved with isotropic potentials, such as a snub square tiling and the kagome lattice. The latter is especially interesting because it provides the crucial geometric frustration in several proposed spin liquids.

A method for finding aggregated representations of linear dynamical systems

A central problem in the study of complex systems is to identify hierarchical and intertwined dynamics. A hierarchical level is defined as an aggregation of the systems variables such that the aggregation induces its own closed dynamics. In this paper, we present an algorithm that finds aggregations of linear dynamical systems, e.g. including Markov chains and diffusion processes on weighted and directed networks. The algorithm utilizes that a valid aggregation with n states correspond to a set of n eigenvectors of the dynamics matrix such that these respect the same permutation symmetry with n orbits. We exemplify the applicability of the algorithm by employing it to identify coarse grained representations of cellular automata.

Using force covariance to derive effective stochastic interactions in dissipative particle dynamics

There exist methods for determining effective conservative interactions in coarse-grained particle-based mesoscopic simulations. The resulting models can be used to capture thermal equilibrium behavior, but the model system we study does not correctly represent transport properties. We suggest the use of force covariance to determine the full functional form of dissipative and stochastic interactions. We show that a combination of the RDF and a force covariance function can be used to determine all interactions in dissipative particle dynamics (DPD). Furthermore, we use the method to test whether the effective interactions in DPD can be adjusted to produce a force covariance consistent with the projection of a microscopic Lennard-Jones simulation. The results indicate that the DPD ansatz may not be consistent with the underlying microscopic dynamics. We discuss how this result relates to theoretical studies reported in the literature.

A method for estimating the interactions in dissipative particle dynamics from particle trajectories

We introduce a method for determining the functional form of the stochastic and dissipative interactions in a dissipative particle dynamics (DPD) model from projected phase space trajectories. The DPD model is viewed as a coarse graining of a detailed dynamics that displays a clear timescale separation. Based on the Mori-Zwanzig projection operator method we derive a consistency equation for the stochastic interaction in DPD. The consistency equation can be solved by an iterative bootstrapping procedure. Combined with standard techniques for estimating the conservative interaction, our method makes it possible to reconstruct all the forces in a coarse-grained DPD model. We demonstrate how the method works by recreating the interactions in a DPD model from its phase space trajectory. Furthermore, we discuss how our method can be used in realistic systems with finite timescale separation.