Scientific Research

paceI am doing a PhD at the Complex Systems Lab now placed at the Barcelona Biomedical Research Park (Universitat Pompeu Fabra - I.C.R.E.A. - G.R.I.B.) in the general subject of theoretical molecular biology and evolution under Ricard V. Solé 's supervision. We are studying the dynamics of nonlinear replicator networks by means of nonlinear dynamical systems theory, specifically focusing on the so-called molecular quasispecies and hypercycles, which represent, respectively, the molecular basis to study RNA viruses dynamics and prebiotic evolution in the framework of the origins of life problem. These systems can be described with nonlinear models which show a wide array of complex dynamics, caused by molecular cooperation, competition or by parasitism. I am also involved in the European PACE project within the 6th Framework Program under contract FP6-002035 (Programmable Artificial Cell Evolution).



          go   Hypercycles: nonlinear replicator networks
          go  Eigen's molecular quasispecies: evolution of RNA viruses and viroids
          goahead  Host-pathogen replicator dynamics: from molecular processes to ecological dynamics

atoms Introduction : about molecular biology, RNA viruses, origins of life and complex systems
In the last decades lots of discoveries have been done in the field of molecular biology. In the 1950's, Watson and Crick discovered the physical structure of the DNA. A double helix appeared from X-ray experiments and from that discovery molecular biology started a revolutionary evolution. On February 28, in 1953, Francis Crick walked into the Eagle pub in Cambridge, England, and, as James Watson later recalled, announced: "we had found the secret of life." Actually, they had. That morning, Watson and Crick had figured out the structure of deoxyribonucleic acid, DNA. And that structure — a "double helix" that can "unzip" to make copies of itself — confirmed suspicions that DNA carries life's hereditary information. Another important discovery directly related to the origins of life was made by Stanley Miller. This scientist and his teacher Harold C. Urey form Chicago University, demonstrated, empirically, that from inorganic compounds as methane ( CH4 ), ammonium ( NH3 ), hydrogen ( H2 ) and water (H2O),  simple organic molecules as alpha-aminoacids could be synthesized. The possible origin of life in our planet and in the Universe is one of the most exciting nowadays scientific subjects. Humanity has often asked how life emerged on the Earth 3800 milion years ago. Nowadays, we find milions of cells in a spoon full of sand extracted from a forest. These are procariotes, primitive organisms that still live at present. In the last centuries and specially in the XX century and at the beggining of the XXI century, lots of progresses have been done related to the origins of life. At the end of the 1920's, the britanic scientist John Haldane and russian scientist Aleksandr Oparin proposed the first scientific theory about the origin of life. According to these authors, complex molecules randomly and abiotically formed would build new complexes under ultraviolet light influences, until an organism persisted and replicated. As often happens in the scientific world, such a theory had no importance because no evidences could be shown. But that changed with the experiments of Miller and Urey. Miller-Urey These two scientists took the inorganic compounds previously mentioned, such compounds were supposed to be the most important ones in the Oparin's reductive primitive atmosphere. They put such compounds in a closed circuit submitted to electricity with a continuous electric current. After a week, about 10-15% of the total carbone had formed organic compounds. So, it was empirically demonstrated the possibility to get organic molecules from inorganic ones. Some of these new compounds were amino acids, the basic components of proteins found in all living forms. Other experiments similar to Miller's one have given more than 90 different amino acids, that is a very interesting result because organisms produce their proteins only from 20 amino acids. The famous primitive soup (aqueous medium with lots of hanging molecules) would be a good place because simplest molecules abiotically synthesized could interact one to each other initializing the path to the appearance of life.RNA  An inherent property of living systems is their self-replicating capacity. Molecules with self-replicating function are polinucleotides (DNA/RNA). As replication process is a physical process it is not free from mistakes (mutations). This is a very important feature that must be taken into account in the approaches to the origins of life (see next section). Another important discovery opened the doors to conduct origins of life research through a track related to the RNA world. It was shown, in 1982, that some RNA molecules had catalytic activity (rybozymes), under  favourable conditions, during replication. In other words, what was ascribed only to poteins a RNA molecule could also do. So the same molecule could be, at the same time, an information carrier and a replicator. RNA sequences have a limited calalytic power, but such discover was very important to understant origins of life. Another important hypothesis related to prebiotic organization and evolution was developed by Manfred Eigen and Peter Schuster (see references). They proposed a novel kind of matter self-organization, the hypercycle, that would involve a qualitative step related to the origins of life studies. These authors also proposed the quasispecies theory, also applied to replicator dynamics, which are nowadays giving lots of clues to understand RNA virus dynamics. The hypercycle is explained in next section.

related papers and books:
Haldane, J. B. S. "The Origin of Life," New Biology, 16, 12 (1954).
Haldane, J. B. S. Possible Worlds. New York: Hugh & Bros. (1928).
Oparin, A. I. The Origin of Life. New York: Dover (1952) (first published in 1938).
Oparin, A., and V. Fesenkov. Life in the Universe. New York: Twayne Publishers (1961).
Miller, S. L. "Production of Amino Acids Under Possible Primitive Earth Conditions," Science, 117, 528 (1953).
Miller, S. L., and Urey, H. C. "Organic Compound Synthesis on the Primitive Earth," Science, 130, 245 (1959).
SYSTEMS BIOLOGY: from simplicity to complexity
Complex systems theory or nonlinear science definitely gives the tools to study biological phenomena in the framework of the so-called systems biology. Such theory comes from an elegant branch of applied mathematics called dynamical systems theory. Steven H. Strogatz proposed a classification of general systems depending on their number of variables and their degree of nonlinearity. Such a classification is explained in the first chapter of his excellent book: Nonlinear Dynamics and Chaos, and gives a wide and very clear intuition about the expected dynamic al behaviors for several systems including physical, chemical or biological systems. For example, ecosystems, earthquakes, relativity, turbulence or life processes are placed in systems with n>>1 (here n represents de number of variables i.e. dimensionality of the system) or in systems with a continuum of variables and higher nonlinear degree (Strogatz, 2000). The dynamics of biological molecular interactions, self-organization and molecular evolution can also display complex dynamics and nonlinear mathematical techniques offer a very good chance to characterize and quantify such kind of behaviors. By using linear stability analysis and by exploring the topological properties of the phase space vector field for these systems we can characterize and understand both qualitatively and quantitatively the time-dependent behavior as the time variation of the concentration of several coupled chemical species. Chaotic dynamics is another kind of dynamical behavior of time-evolving systems. Apart from chaos, trajectories in phase space can be asymptotically captured by several kinds of attractors, as quasiperiodic, limit cycles or point attractors. Chaos becomes a fundamental science dealing to complexity in pure state. The discover of the so-called deterministic chaos actually caused a scientific revolution opening a new view of systems, nowadays this theory is used in lots of scientific fields as physics, non-equilibrium chemical kinetiks, ecology, physiology, economics, neural networks, immune systems, lasers and nonlinear optics. Such systems can actually show irregular, nonperiodic time evolution with the impossibility of, although in a deterministic world, predicting the long-term dynamics. Extremely complex systems like the atmosphere of Jupiter (turbulence problem), atmospheric convection (also turbulent flow) motions or population dynamics can be studied with simple models which might gahter the overall inherent properties of such systems in the form of nonlinear equations. Nonlinear dynamics systems theory includes concepts like strange attractor and fractal geometry, and all the qualitative and quantitative methods that such a theory claims can often be the only way to approach to such hard systems. Of course, theory should be considered as a complement of experimental research, allowing the development of general models able to explain what we see in nature. Theory allows scientists to gain a huge intuition in real systems dynamics, as well as the capability to make predictions.


Publications  of  the  COMPLEX  SYSTEMS  LAB  within  the  PACE Project

BiomodHypercycles: nonlinear replicator networks
Hypercycles are nonlinear catalytic networks composed by self-replicating units (macromolecules) which are able to catalyze the replication of one other unit, forming a closed loop (Eigen, 1971; Eigen and Schuster, 1979) (see Fig. 1).  Hypercycles were initially proposed by Manfred Eigen in the 1970's, and opened a novel framework to study the matter self-organization in the context of the origins of life in prebiotic evolution. One of the most important evolutionary properties of the hypercycle is that this system allows to overcome the Eigen's error catastrophe transition, which is present in earlier (error prone) replicators, which have a quasispecies distribution (see next section). Eigen (1971) argued that primitive genomes must have been segmented (consisting of physically unlinked genes) and that these single genes would had the tendency to compete with one other so, consequently, some mechanism ensuring their coexistence was needed. By establishing a cross-catalytic system, otherwise competing replicators could coexist, and by means of individual sequences, each one below the information error threshold, a larger genetic message could be stored. Hence the informational problem of earlier self-replicating molecules could have been solved. Up to now, only one hypercyclic system has been characterized in a real system (Eigen et al (1991), Biochemistry 30), although it is thought that such an organization could be found in other real molecular systems. Nevertheless, this elegant and novel theory has opened a genuine research field in prebiotic evolution, and has given a lot of scientific literature. There are different kinds of hypercycles: RNA hypercycles, DNA-protein hypercycles, first order, second order hypercycles, ...  but all of them share singular mathematical properties, translated to quite different dynamics, from cyclic to chaotic oscillations. Hypercycle kinetics is hyperbolic. Hyperbolic growth allows reaching infinite concentrations in finite time. Maynard Smith stated that hypercycles, as altruistic networks of replicators, might have a major problem, given by the so-called catalytic parasites (see Fig. 1(b)). The hypercycle parasite is a replicator attached to some unit/s of the hypercycle, receiving catalytic help but not recirpocating catalysis to any other unit of the network. If the parasitic molecular species gets more support than any other hypercycle replicator, the parasite will be selected and the hypercyclic chain broken (May 1991, Nature 353). The hypercycle has provided extremely intertesting scientific results. For instance, complex patterns in a reaction-diffusion model shown by Cronhjort and Blomberg in 1994. By using Boerlijst and Hogeweg equations (Boerlijst and Hogeweg, 1991) they showed the presence of spiral waves involved in the resistance against parasites, by considering a spatially-extended hypercycle of 5 members.

Fig. 1.  (a) general hypercycle formed by
n members. The set
of self-replicating macromolecules is coupled catalytically.
Circle arrows indicate self-replication and arrows between species indicate
the cross catalytic help in replication.  (b) a hypercycle with two members and a parasite

Boerlijst and Hogeweg equations (represented in a general form in Eq. 1) are partial differential equations (reaction-diffusion) describing spatial hypercycle dynamics. This model represents a macroscopic description and considers concentrations of hypercycle species instead of  taking individual molecules into account (Cronhjort and Blomberg, 1994). Such species are spatially distributed in a continuous, non well-mixed media. Without taking into account the linear growth, this dynamical system is given by:


Here xi is the concentration of a ith hypercycle molecule in a (r, s) lattice point (assuming a two-dimensional R x S spatial domain), k and g are the replication and decay rates, respectively, and D is a dimensionless unit defining the system's temporal scale.  Equations set (2) represents the hypercycle model of Fig. 1 (with n = i = 5), wich is extended in a given spatial domain (here on a surface with toroidal topology i.e. on a torus), and thus is described by partial differential equations (reaction-diffusion). Here i actually indicates the dimension of the system (in the mathematical sense), although spatially-extended systems have a continuum number of variables, and thus the system, rigorously speaking, is infinite-dimensional. Note that these are coupled equations so species 1 is catalysed by species 5 (here we close the loop).

                 RD2     (2)   
             random               random2  random3  spiral
            initial random pattern            spirals appers solving Eqs. 2 (when n = 5) numerically (here R = S = 50)
Fig. 2. Spatial patterns emergence from a 5 member hypercycle.

In Fig. 3 is possible to follow the time evolution of all the species of the hypercycle. After a first transient period, time series get coupled undergoing oscillatory dynamics which is governed by the limit cycle shown in Fig 3 (right). Here a trajectory flows oscillating until is confined into the limit cycle attractor. This attractor represents the asymptotic dynamic state of the system, which is asymptotically stable because it will be reached from any arbitrary initial condition in phase space.

oscil              limit cycle

                                                                Fig. 3. (left) Concentration time evolution for the 5 hypercycle members (in colours) , (right) periodic attractor (limit cycle) in phase space
Hypercycles have been very deeply studied since were proposed, and complex patterns like chaos pointed out. Schnabl, Stadler, Forst and Schuster published, in 1990,  an exceptional paper where they deeply studied a strange attractor in a system of Lotka-Volterra dynamics of dimension 4. So they studied a four-dimensional catalytic replication network. They characterized and described a very complex evolution in the chaotic regime finding bifurcations (see the meaning of bifurcation) driving to different chaotic regimes. Their greatest conclusion was that exists a gradual disappearance of complex dynamics with increasing mutation rates (Schnabl et al. 1990). Forst also characterized chaotic dynamics in a four-dimensional catalytic replicator network, his elegant paper titled Chaotic interactions of self-replicating RNA (published in Computers Chem. in 1996) also reviews chaotic dynamics in many other systems.

related papers:
Eigen, M. and Schuster, P. (1979) The Hypercycle. A Principle of Natural Self-Organization. Springer-Verlag
Boerlijst, M.C. and Hogeweg, P. (1991) Spiral wave structure in pre-biotic evolution: Hypercycles stable against parasites. Physica D 48, pp. 17-28
Complex Systems Lab papers:
Sardanyés, J. and Solé, R.V. (2006) Ghosts in the origins of life? International Journal of Bifurcation and Chaos 16(9), 1-5.
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 non-linear replicator delayed extinctions. Chaos, Solitons & Fractals 31(5), 1279-1296
Sardanyés, J. and Solé, R.V. (2006) Bifurcations and phase transitions in spatially extended two-member hypercycles. J. theor. Biol. In press.
Sardanyés, J. (2007) Error threshold ghosts in a simple hypercycle with error prone self-replication. Chaos, Solitons & Fractals. In press.
Sardanyés, J. (2007) Ghosts in high dimensional non-linear dynamical systems: The example of the Hypercycle.
Chaos, Solitons & Fractals. In press.

See The Complex Systems Lab HYPERCYCLES web site

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virusEigen's molecular quasispecies: evolution of RNA viruses and viroids

Replicator dynamics theory initially developed and formalized by Manfred Eigen and Peter Schuster is an excellent theoretical framework to approach RNA virus dynamics. Such viruses actually show a huge adaptability to changing environments because of their high mutability in replication. Viral genomes form the so-called molecular quasispecies, which are represented by a set of strands forming a cloud of mutants with extremely heterogeneous genotypes. The quasispecies structure actually provides to this set of strains with extremely large capacities to face changes in the environment, sometimes helping to avoid the immune system action. RNA virus actually mutate near the critical region of the so-called error catastrophe. The insight in the error catastrophe, as well as many other dynamical properties of RNA viruses, is important to understand their life cycle. Theoretical as well as empirical research may serve to search for medical or pharmaceutical strategies to face such viruses. Eigen's error catastrophe claims that information crashes in error-prone replicators  (information can not be mantained for Darwinian selection and becomes random) if we overcome a critical mutation rate (error threshold transition). Such catastrophe corresponds to a first-order phase transition, which separates the phase of stable avolution from the phase of random information. This phenomenon can be shown in a simple model considering viruses genomes as bit strings. catastrophe
The algorithm to simulate the error catastrophe is as follows:
we start from a propulation of n strings of size L. Each position of such strings can be occupied by a bit i.e. 0 or 1, simulating, for example, purines or pyrimidines. In the population there are strings that replicate with probability pm = 1 (these are called master strings in which all the positions are taken by the bit 1); the other strings (different to the master one) replicate with a lower probability (po with po<< pm).  Replicating processes involve mutation (µ), which means that the error probability for unit and for repeating cycle is µ for each string, thus mistakes are independent.

The model considers:

** Replication: Maximum fitness (pm) of strings with 1s in all sequence. So master string will always replicate. Other strings: strings with one or more 0s in the sequence will replicate with po << pm.
** Mutation: for each string position vi (i = 1, ..., L) for all strings, there is a probability to make mistakes during replication (mutation probability µ).
           vi   ---------> 1 - vi
 Errors are independent for each string position.

** Initial conditions: we start from a population where all strings are master strings.

In the simulation we choose a string of the population at random at each iteration and it copies itself onto another chosen randomly string. If we run this simulation under 104 iterations for different mutation rates µ (here µ acts as the order parameter) and compute the probability of finding as minimum one master string in population we can see that exists a critical mutation rate where no master strings exist (see Fig. 4). This sharp transition actually corresponds to a first-order phase transition.

Fig. 4.  Phase transition in the Swetina-Schuster algorithm showing
inheritance breack and extinction of best-adapted strings

The error catastrophe is crucially important in the context of the origins of life. In modern organisms, sophisticated proofreading and error-correction mechanisms are employed to keep the error rate down. Cells can call upon a suite of enzymes, evolved over billions of years, to finesse the copying process. No such enzymes would have been available to the first organisms. Their replication must have been extremely error-prone. According to Eigen's rule, this means that the genomes of the first organisms (or prebiotic replicons) must have been very short in length if they were to evade this error catastrophe. With too large replicators natural selection had nothing to do. But here we fall into the Eigen's paradox: if a genome is too short, it can not store enough information to build the copying machinery itself. Eigen believes that the simplest replication equipment requires much more information than could ever have been accommodated in a primitive nucleic-acid-like sequence. To reach the sort of length needed for the necessary copying enzymes, the genome risks falling foul of the very error catastrophe it is trying to combat. To link this informational restriction with the hypercycle we just only must imagine the hypercycle as an array of cyclically coupled replicators which can, individually, maintain below the critical size avoiding the error catastrophe: we obtain a connected system of unlinked replicators able to avoid the informational crisis but containing, as a network, large contents of information.

related papers:
Eigen, M. (2002) Error catastrophe and antiviral strategy (Commentary). PNAS, vol. 99, nº 21, 13374-13376

Complex Systems Lab papers:
Solé, R.V., Sardanyés, J., Díez, J. and Mas, A. (2006) Information catastrophe in RNA viruses through replication thresholds. Journal of theoretical Biology 240(3) 353-359.

See The Complex Systems Lab VIRUS DYNAMICS web site

Interesting related links:
Plant Virus Diversity and Evolution Group (Universitat de València - Group of Santiago F. Elena)
RNA viruses - Wikipedia
RNA plant and animal virus replication
Peter F. Stadler website
Centro de Biologia Molecular Severo Ochoa (Group of Esteban Domingo)

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host-pathogenHost-pathogen replicator dynamics: from molecular processes to ecological dynamics
Host-pathogen (or host-parasitoid, host-parasite or predator-prey) dynamics are ubiquitous in nature. We are exploring general mathematical and simulation models describing host-pathogen (e.g. predator-prey) dynamics of self-replicating macromolecules or entities which can also applied to characterize ecological dynamics (by considering for example that the strength of interactions amog predator-prey organisms is influenced by a pair of haplid di/multiallelic loci). Thus, these systems include some genetic traits involved in the coevolutionary outcomes. Our main goals are to investigate the role of self-replication i.e. intrinsic growth, properties as well as mutation processes in the asymptotic dynamics of such systems. We are specifially analyzing some models related to the so-called mathcing allele (MA) dynamics.

Gene-for-gene dynamics in coevolutionary processes:
Gene-for-gene (GFG) dynamics has been described in host-pathogen interactions in agriculture, natural plant populations, and has been extensively explored in mathematical models for coevolution. The GFG hypothesis, originally formulated by H. H. Flor, states that "for each gene determining resistance in the host there is a corresponding gene for avirulence in the parasite with which it specifically interacts". This kind of coevolution has been described in several agricultural plant-pathogen associations, and it is thought that GFG interactions among plants and pathogens as viruses, bacteria and fungi are likely to be found in nature.

related papers:
Thompson John N. and Burdon Jeremy, J. Gene-for-gene coevolution between plants and parasites. Nature 1992; 360:121-5

Complex Systems Lab papers:

Sardanyés, J. and Solé, R.V. (2007) Chaotic Stability in Spatially-Resolved Host-Parasite Replicators: The Red Queen on a Lattice. International Journal of Bifurcation and Chaos Vol. 17(2) 1-18.

Sardanyés, J. and Solé, R.V. (2007) Red Queen Strange Attractors in host-parasite replicator gene-for-gene coevolution. Chaos, Solitons & Fractals 32(5) p. 1666-1678

Sardanyés, J. and Solé, R.V. (2007) Matching allele dynamics and coevolution in a minimal predator-prey replicator model. Physics Letters A (2007) In press.

See SETH (Spatiotemporal Evolution Through Hypercubes) project

Interesting related links:

Program for evolutionary dynamics (Harvard University)

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Hypercycles are a novel class of nonlinear reaction networks with unique properties, amenable to a unified mathematical treatment

Eigen and Schuster, 1979 

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