Conditions for emergent synchronization in
protocells
Roberto Serra2,1, Timoteo Carletti3,1, Irene Poli1, Marco Villani2 and
Alessandro Filisetti2
1 Dipartimento di Statistica, Universita` Ca’ Foscari, San Giobbe - Cannaregio 873,
30121 Venezia, Italy poli@unive.it
2 Dipartimento di Scienze Sociali, Cognitive e Quantitative, Universita` di Modena
e Reggio Emilia, via Allegri 9, 42100 Reggio Emilia, Italy rserra@unimore.it
, villani.marco@unimore.it , alessandro.filisetti@unimore.it
3 De´partement de mathe´matique, Universite´ Notre Dame de la Paix, rempart de
la Vierge 8, B 5000 Namur, Belgium timoteo.carletti@fundp.ac.be
Summary. A major aspect of the dynamics of a populations of protocells, is that,
in order to assure sustainable growth, the genetic material and container productions
must be synchronized if they are coupled. In a previous work [11] we have shown that
such synchronization is an emergent property in many relevant protocell models,
assuming both linear and nonlinear law for the replications rates. While in those
previous studies the replicators were assumed to compete for resources, without
any direct interaction among them, here we address a similar question in the case
where the replicators interact in a linear way (due to the fact that the protocell
divides into two when a certain critical size is reached, the overall model is definitely
nonlinear). Using the technique introduced in [11], we are able to give a quite general
analytical answer about the synchronization phenomenon even in the presence of
interactions. We also report on the results of numerical simulations to support the
theory, where applicable, and allow to investigate cases which are not amenable to
analytical calculations . A short comment on preliminary results concerning fully
nonlinear models is presented in the conclusions.
1 Introduction: the problem of synchronization
Several attempts are currently under way to obtain protocells capable of
growth and duplication, endowed with some limited form of genetics [8, 10, 13].
The interest for these systems is motivated either by the quest to understand
which are the minimal requirements for a life form to exist and evolve, or by
the search for indications about the way in which primitive life might have
developed on earth.
In order to study how protocells can develop, given that they do not yet
exist, it is necessary to consider simplified models able to capture universal

2 R. Serra et al.
behaviors, without carefully adding complicating details [4]. A protocell com-
prises at least one kind of “container”molecule (typically a lipid or amphiphile)
and one kind of replicator molecule–loosely speaking, “genetic material”. This
is typically a linear polymer which can be copied or a system of two or more
kinds of replicators which catalyze each other’s synthesis–like e.g. proteins
and nucleic acids. There are therefore two kinds of reactions which are cru-
cial for the working of the protocell, which will be called here “key”reactions:
those which synthesize the container molecules and those which synthesize
the replicators.
The two key reactions may take place at different rates. However, to
achieve sustained protocell growth and avoiding death by dilution, it is neces-
sary that the two are proceed at equal rate, i.e. that the genetic material has
doubled when the protocell splits into two–a condition referred to as synchro-
nization. Indeed, if replication were slower than duplication, the concentration
of genetic material would eventually vanish (we refer to the splitting of a pro-
tocell as duplication, and to the doubling of genetic polymers as replication).
In the opposite case its concentration would grow unbounded. Of course the
requirement of duplication of the genetic material at duplication time refers to
the average behavior, while each single event is affected by noise and random
fluctuations.
Note that synchronization has a further important property, namely that,
even in the case where the kinetic equations for the replicators have sub lin-
ear growth terms [9, 7], it leads to exponential growth of the population of
protocells (a straightforward consequence of constant doubling time 4) and
therefore to strictly Darwinian selection among protocells.
Most of the different protocell architectures which have been proposed
can be divided in two main families, according to the region of cell space
where these key reactions occur. Some proposals belong to a category which
can be termed Inner REeaction NEtworks models, since both types of key
reactions take place in the interior of a vesicle [8], while in other models (which
have been called Surface Reaction Models, SRM for short) the key reactions
take place on the surface of the cell membrane. In this model we address the
synchronization question for the latter, while the study of the IRENE models
has been performed elsewhere [2].
A well known model where the key reactions take place near the surface
is the so–called “Los Alamos bug”(briefly, Labug) [9, 10], where replicators
are PNAs which can be found in the membrane, either in its interior or on
its surface, but replication takes place when a single–stranded PNA on the
surface ligates (via Watson-Crick pairing) the corresponding nucleotides which
are supposed to be freely available. The PNA’s on the membrane surface
also catalyze the formation, from lipid precursors, of amphiphiles which are
4 We ignore here further terms which might limit the growth of the whole popula-
tion of protocells, like e.g. competition for limited resources or growth in a limited
volume.