I 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 socalled 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 FP6002035 (Programmable
Artificial
Cell Evolution).


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 Xray 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 ( CH
_{4} ),
ammonium ( NH
_{3} ),
hydrogen ( H
_{2} )
and water (H
_{2}O),
simple organic molecules as alphaaminoacids 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.
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 1015% 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.
An inherent property of living systems
is their selfreplicating capacity. Molecules with selfreplicating
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 selforganization, 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 socalled 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,
selforganization 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 timedependent
behavior as the time variation of the concentration of several coupled
chemical species.
Chaotic dynamics is another kind of dynamical behavior of timeevolving
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 socalled 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, nonequilibrium 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 longterm 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.
Hypercycles:
nonlinear replicator networks
Hypercycles are nonlinear
catalytic networks composed by selfreplicating 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 selforganization 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
crosscatalytic 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 selfreplicating 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, DNAprotein 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 socalled 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
reactiondiffusion 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 spatiallyextended
hypercycle of 5 members.
Fig. 1. (a)
general hypercycle formed by n members.
The set
of selfreplicating macromolecules is coupled catalytically.
Circle arrows indicate selfreplication 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
(reactiondiffusion) 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 wellmixed
media. Without taking into account the linear growth, this dynamical
system is
given by:
Here x_{i}
is the concentration of a ith
hypercycle molecule in a (r, s) lattice point (assuming a
twodimensional 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 (reactiondiffusion). Here i
actually indicates the dimension of the system (in the mathematical
sense), although spatiallyextended systems have a continuum number of
variables, and thus the system, rigorously speaking, is
infinitedimensional. Note that these are coupled equations
so species 1 is catalysed by
species 5 (here we close the loop).
(2)
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.
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
LotkaVolterra dynamics of dimension 4. So they studied a
fourdimensional 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 fourdimensional catalytic
replicator network, his elegant paper titled Chaotic interactions of
selfreplicating 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 SelfOrganization. SpringerVerlag
Boerlijst, M.C. and
Hogeweg, P. (1991) Spiral wave structure in
prebiotic evolution: Hypercycles stable against parasites. Physica D
48, pp. 1728
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),
15.
Sardanyés, J.
and
Solé, R.V. (2007) Delayed
transitions in nonlinear replicator networks: About ghosts and
hypercycles. Chaos, Solitons &
Fractals 31(2), 305315
Sardanyés, J.
and
Solé, R.V. (2007) The role of cooperation and parasites in
nonlinear replicator delayed
extinctions.
Chaos, Solitons &
Fractals 31(5), 12791296
Sardanyés, J. and
Solé, R.V. (2006) Bifurcations and phase transitions in
spatially extended twomember hypercycles. J. theor. Biol. In press.
Sardanyés, J.
(2007) Error threshold ghosts in a simple hypercycle with error prone
selfreplication. Chaos, Solitons &
Fractals. In press.
Sardanyés, J. (2007) Ghosts in high dimensional nonlinear
dynamical systems: The example of the Hypercycle. Chaos, Solitons &
Fractals. In press. 
See
The Complex Systems Lab HYPERCYCLES web site
return up
Eigen'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 socalled
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
socalled 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
errorprone 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 firstorder 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.
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 p
_{m} = 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 (p
_{o} with p
_{o}<<
p
_{m}).
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 (p_{m}) 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 p_{o}
<<
p_{m}.
**
Mutation: for each string
position v_{i} (i = 1,
..., L) for all
strings, there is a probability to make mistakes during replication
(mutation probability µ).
µ
v_{i}
> 1  v_{i}
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
10^{4} 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
firstorder phase transition.
Fig. 4. Phase transition
in the SwetinaSchuster algorithm showing
inheritance breack and
extinction of bestadapted strings
The
error
catastrophe is crucially important in the context of the origins of
life. In modern organisms,
sophisticated proofreading and errorcorrection 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 errorprone. 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 nucleicacidlike 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, 1337413376
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) 353359.

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)
return up
Hostpathogen
replicator dynamics: from molecular processes to ecological dynamics
Hostpathogen
(or hostparasitoid, hostparasite or predatorprey) dynamics are
ubiquitous in nature. We are
exploring general mathematical and simulation models describing
hostpathogen (e.g. predatorprey) dynamics of
selfreplicating macromolecules or entities which can also applied to
characterize ecological dynamics (by considering for example that the
strength of interactions amog predatorprey 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 selfreplication 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 socalled mathcing allele (MA) dynamics.
Geneforgene dynamics in
coevolutionary processes:
Geneforgene
(GFG) dynamics has been described in hostpathogen 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
plantpathogen 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. Geneforgene coevolution
between plants and parasites. Nature 1992; 360:1215
Complex
Systems Lab papers:
Sardanyés,
J. and
Solé, R.V. (2007) Chaotic
Stability in SpatiallyResolved HostParasite Replicators: The Red
Queen on a Lattice. International
Journal of Bifurcation and Chaos Vol. 17(2) 118.
Sardanyés,
J. and
Solé, R.V. (2007) Red Queen
Strange Attractors in hostparasite replicator geneforgene
coevolution. Chaos, Solitons &
Fractals 32(5) p. 16661678.
Sardanyés,
J. and
Solé, R.V. (2007) Matching allele
dynamics and coevolution in a minimal predatorprey replicator model. Physics Letters A (2007) In press. 
See
SETH (Spatiotemporal
Evolution Through
Hypercubes) project
Interesting related links:
Program
for evolutionary dynamics (Harvard University)
return up
Hypercycles are a novel class of
nonlinear reaction networks with unique properties, amenable to a
unified mathematical treatment
Eigen and Schuster, 1979