ICREA/Complex Systems Lab, Universitat Pompeu Fabra (UPF) Barcelona Biomedical Research Park (PRBB) Grup de Recerca en Informàtica Biomedica (GRIB) Main Researchers :  Ricard V. Solé  and  Josep Sardanyés Related Projects: PACE Introduction: The transition from non-living to living chemistry is one of the fundamental issues of prebiotic evolution. The formation of self-organized networks of molecular species might be one of the crucial steps towards the first living cell, where the collective emergent properties of such networks might give raise to the overall characteristics of living systems. Hypercycles are cyclically-coupled chains of information carriers organized in the form of nonlinear networks of self-replicating species. The cyclic coupling leads to the cooperative selection of all the hypercyclic units, which evolve as a single coherent unit.  Such an organization might be crucial in the evolution of earlier replicator molecules, which had to evolve under an informational constraint given by the error-threshold transition. Manfred Eigen and Peter Schuster are the parents of the hypercycle theory, they argued that primitive genomes must had been physically-unlinked in order to maintain below the critical string length imposed by the error threshold. The hypercycle allows coexistence among otherwise competing replicators, allowing the possibility to store a larger genetic message by integrating s set of cooperating replicators, which would  indiviually maintain below such critical length, ensuring a stable genetic message as a whole. We are working with the hypercycle dynamical system in order to explore both evolutionary and dynamical properties of these networks, which might be relevant in prebiotic evolutionary processes in the context of the origins of life as well as in the appearance of the first living cell. Our main analyses are focused on dynamical systems models, as well as on the development of spatial simulation models of hypercyclic macromolecules. We are also interested in the role of parasites in the stability, evolution and dynamics of hypercycles, as well as in the emergence of spatiotemporal phenomena associated to spatially-extended hypercycles within parasites. The dynamics of hypercycles is extremely interesting because can be generically extended to model catalytic viral growth phases as well as cooperative i.e. symbiotic, processes in the context of ecological dynamics.

NEWS 1/2/2006    First public release and publication of the Complex Systems Lab HYPERCYCLE website Publications: (pdf's available here) Sardanyés, J. and Solé, R. V. (2006) Bifurcations and phase transitions in spatially extended two-member hypercycles . J. theor. Biol. 243 (4) 468-482 Sardanyés, J. and Solé, R. V. (2006) Ghosts in the origins of life ?  Int. Journal of Bif. and Chaos 16 (9) 2761-2765. http://www.worldscinet.com/ijbc/16/1609/S02181274061609.html                                   Sardanyés, J. and Solé, R. V. (2007) Delayed transitions in nonlinear replicator networks: About ghosts and hypercycles. Chaos, Solitons & Fractals. 31 (2) 305-315 Sardanyés, J. and Solé, R. V. (2007) The role of cooperation and parasites in nonlinear replicator delayed extinctions. Chaos, Solitons & Fractals. 31 (2) 1279-1296 Sardanyés, J. and Solé, R. V.  Spatio-temporal dynamics in simple asymmetric hypercycles under weak parasitic coupling. Physica D 231(2)  116-129 2007-2008    Forthcoming papers: Sardanyés, J. Error threshold ghosts in a simple hypercycle with error prone self-replication. Chaos, Solitons & Fractals. In press. To appear in 2007 Sardanyés, J. Ghosts in high dimensional non-linear dynamical systems: the example of the Hypercycle. Chaos, Solitons & Fractals. In press. To appear in 2008

References of interest: - Eigen, M. and Schuster, P. (1979) The Hypercycle. A Pinciple of Natural Self-Organization. Springer-Verlag - Bernd-Olaf Küppers (1985) Molecular Theory of Evolution. Outline of a Physico-Chemical Theory of the Origin of Life - D. H. Lee, K. Severin, and M. Reza Ghadiri. Autocatalytic networks: the transition from molecular self-replication to molecular ecosystems. Curr. Opin. Chem. Biol., 1:491-496, 1997 - B.M.R Stadler and P. F. Stadler. Molecular Replicator Dynamics, Advances in Complex Systems 6: 47-77 2003 - Strogatz, S. H. and Westervelt, R. M. Predicted power laws for delayed switching of charge density waves. Physical Review B, 40(15):10501-10508, 1989 - D. H. Lee, K. Severin, Y. Yokobayashi, and M. Reza Ghadiri. Emergence of symbiosis in peptide self-replication through a hypercyclic network. Nature, 390:591-594, 1997  ABS - Boerlijst, M.C., Hogeweg, P. (1991) Spiral wave structure in pre-biotic evolution: Hypercycles stable against parasites. Physica D 48, pp. 17-28 - Cronhjort, M.B., Blomberg, C. (1997) Cluster compartimentalization may provide resistance to parasites for catalytic networks. Physica D 101, pp. 289-298 - Cronhjort, M.B., Blomberg, C. (1994) Hypercycles versus Parasites in a Two-Dimensional Partial Differential Equations Model. J. Theor. Biol. 169, pp. 31-49 - Cronhjort, M.B. (1995) Hypercycles versus parasites in the origin of life: model dependence in spatial hypercycle systems. Orig Life Evol Biosph. Jun;25(1-3):227-33 - Cronhjort, M. (1995) Models and Computer Simulations of Origins of Life and Evolution. In: Origins of Life. Ph.D.Thesis, 1-7. In: Hypercyles. Ph.D.Thesis, pp. 9-13 - Eigen, M., Biebricher, C. K., Gebinoga, M. (1991) The Hypercycle. Coupling of RNA and Protein Biosynthesis in the Infection Cycle of an RNA BActeriphage. Biochemistry, 30(46)    Hypercycle dynamical system: molecular ecosystems The cyclically-coupled array of seff-replicating species conceived by the hypercycle system (see Fig. 1) has been considered as a possible molecular network of prebiotic replicators in the context of the origins of life and has also been suggested as an important step in the transition from inanimate to living chemistry. The important selective and evolutionary properties of the hypercycle make such networks good candidates to explain key steps in prebiotic evolution. The hypercycle organization allows the cooperative selection of otherwise competing replicators, ensuring a larger genetic message able to overcome the error threshold transition as well as the transmission of a stable genetic message generation to generation. Schematic hypercycle reactions: The hypercycle supposes cross-catalytic interactions, involving the integration of the hypercyclic replicators into a single system that reproduces thorugh a second-order or higher form of nonlinear autocatalysis. The chemical kinetics for a general n-membered hypercycle can be represented with the next set of reactions: The first reaction represents the catalytically-assisted self-replication of molecule type i, which instructs the formation of a new i molecule at rate ki, by using some energy-rich building blocks s. The second reaction involves the spontaneous decay of molecules because of hydrolysis. Note that the hypercycle has cyclical structure. Mean field model: Hypercycle chemical kinetics can be studied by means of ordinary differential equations, according to: with the replication function: The right term in the firt equation represents some population limiting function. Here the quantity pai is an integer which gives the power to which the concentration variable xa is raised in the ith growth function of the hypercycle. The cyclical symmetry is introduced to the growth function, according to: Stochastic simulations: Spatial dynamics of hypercycles have been shown to be relevant for evolutionary processes as resistance to parasites. By means of cellular automata models, which can gather the effect of local interactions among replicators, we have shown the presence of  an absorbing first-order phase transition after a critical decay rate is overcome. Our probabilistic model includes replication, molecular decay and diffusion of molecules.                  Fig. 3. First-order phase transition for the surface-bonded two-member hypercycle extended on a surface. The phase transition is achieved by increasing decay rate. The insets show the first hypercycle replicator time evolution (left) below and (right) above the critical hydrolysis value.   click here to see an animation of the surface-bonded two-member hypercycle dynamics (the load of the simulation can be slow)    Nonlinear delayed transitions: saddle-node ghosts The analysis of the hypercycle mean field model have revealed a saddle-node bifurcation scenario associated to hypercycle extinction. Such bifurcation leaves a ghost in phase space, responsible of causing the so-called delayed transition. Delayed transitions have been reported in some driven dynamical systems, for instance in ferroelectrics, condensed matter physics or semiconductor lasers. Strogatz and Westervelt showed that such a phenomenon can be given by the ghost. The ghost is a saddle remnant that continue attracting trajectories towards a region in phase space where two fixed points have coalesced. This phenomena is typical from dynamical systems near to a saddle-node bifurcation. The delay time spent in the bottleneck region of the ghost generally follows a power law, in the case of the two-membered hypercycle, the time delay follows the square-root scaling law i.e. ß = -1/2.          Fig. 4. (left) Delayed extinction for the two-membered hypercycle, and (right) power law dependence of extinction time near threshold bifurcation threshold. The presence of such a phenomenon, presumably given by the inherent nonlinear nature of hypercycle networks, might be of relevance from an evolutionary point of view. Fig. 5. (a-C) Spiral trajectories towards the coexistence fixed point for the three-membered hypercycle. (d) The flow is sucked by the ghost once a saddle-node bifurcation has destroyed both coexistence and saddle points.    Hypercycle Evolution with Parasites Hypercycle network with a shortcut among I1 and I3, and with an attached parasite, P One of the major problems for hypercycles is given by the so-called molecular parasites, which can appear in a branching process. As previously mentioned, the hypercycle consists of catalytically-linked replicator species. Such an "altruistic" organization could be broken if a selfish replicator appeared. This parasite replicator receives catalytic help from the hypercycle but does not reciprocate catalysis. Hypercycles are sensitive to parasites because thuy destabilize the graph structure of the catalytic network. Nevertheless. possible solutions to the parasite problem have been reported, among them the presence of self-structuring in spatially extended hypercycles, as well as  compartmentalization of hypercycles. Mean field model: The hypercycle dynamical system with a parasite is characterized by a sharp selection depending on the relative values of replication rates for the hypercycle and for the parasite. Coexistence scenarios among replicators are of interest in future experimental research with such kind of systems. Fig. 6. Time dependent numerical solutions for the two-membered hypercycle with an attached parasite. Here coexistence among hypercycle members (solid line) and parasites (dotted line) is shown. Stochastic spatial dynamics: The role of space in hypercycle dynamics has been deeply explored during the last years. The presence of spatial patterns in hypercycle dynamics have revealed the importance of self-structuring as a way to avoid parasites. We are also exploring stochastic spatial dynamics of hypercycles under the presence of self-replicating parasites.  Our models include catalytic-assisted self-replication processes, molecular decay because of hydrolysis as well as diffusion processes. The interplay among these parameters might be useful in future experimental research. click here to see an animation of the surface-bonded dynamics of hypercycles with parasites    (the load of the simulation can be slow)