6th St.Petersburg Workshop on Simulation (2009) 1-3
Synchronization phenomena in Non Linear1
Reaction Models of protocell12
Alessandro Filisetti2, Roberto Serra3, Marco Villani4, Timoteo3
Carletti5, Irene Poli64
Abstract5
The present work extends our previous studies which had considered syn-6
chronization in the classes of so{called \Surface Reaction Models" SRM [12,7
13, 3, 5, 2] and \Internal Reaction Models" IRM [3, 2] when linear kinetics8
were assumed for the relevant chemical reactions. Let us show here that sim-9
ilar results have been obtained also for the \Non linear Reaction Models",10
both SMRs and IRMs, hence the synchronization phenomenon seems to be11
very robust with respect to the chosen architecture once linear and non linear12
kinetics are considered. Also the case where the \genetic material"kinetic13
is chaotic has been taken into account and the result is that the coupling14
between genetic material and container kinetics seems to be able to rule the15
chaos leading to synchronization.16
1 Introduction17
Protocells are lipid vesicles (or, less frequently) micelles which are endowed with18
some rudimentary metabolism, contain \genetic" material, and which should be19
able to grow, reproduce and evolve. While viable protocells do not yet exist, their20
study is important in order to understand possible scenarios for the origin of life,21
as well as for creating new \protolife" forms which are able to adapt and evolve22
[7] . This endeavour has an obvious theoretical interest, but it might also lead to23
an entirely new \living technology", denitely dierent from conventional biotech-24
nology.25
1Support from the EU FETPACE project within the 6th Frame-work Programme
under contract FP6002035 (Programmable Articial Cell Evolution) and Fondazione
Venezia are gratefully acknowledged.
2European Centre for Living Technology, E-mail:
alessandro.filisetti@ecltech.org
3University of Modena and Reggio Emilia, E-mail: roberto.serra@unimore.it
4University of Modena and Reggio Emilia, E-mail: marco.villani@unimore.it
5Departement de mathematique, Facultes Universitaires Notre Dame de la Paix, E-
mail: timoteo.carletti@fundp.ac.be
6University Ca'Foscari of Venice, European Centre for Living Technology, E-mail:
irenpoli@unive.it

Theoretical models can be extremely useful to devise possible protocell architec-26
tures and to forecast their behaviour. What can be called the \genetic material" of27
a protocell is composed by a set of molecules which, collectively, are able to repli-28
cate themselves. At the same time, the whole protocell undergoes a growth process29
(its metabolism) followed by a break-up into two daughter cells. This break-up is30
a physical phenomenon which is frequently observed in lipid vesicles, and it has31
nothing to do with life, although it supercially resembles the division of a cell.32
In order for evolution to be possible, some genetic molecules should aect the rate33
of duplication of the whole container, and some mechanisms have been proposed34
whereby this can be achieved.35
In order to form an evolving protocells population it is necessary that the rhythms36
of the above mentioned two processes, i.e. metabolism and genetic replication, are37
synchronized and it has previously been shown that this may indeed happen when38
one takes into account successive generations of protocells [11, 12, 13, 10, 3] .39
2 A review of previous results40
Before going on with the discussion about the non linear reactions models it is41
interesting to consider which kind of behaviours one can expect to nd:42
1. Synchronization: in successive generations (as k ! 1 where k is the gen-43
eration number) the interval of time needed to duplicate the membrane44
molecules of the protocell between two consecutive divisions,
Tk, and the45
time required to duplicate the genetic material, again between two consecu-46
tive divisions,
T gk , approach the same value;47
2. as k !1 the concentration of the genetic material at the beginning of each48
division vanishes. In this case, given the above assumptions, the growth of49
the container ends and the whole process stops;50
3. as k !1 the concentration of the genetic material at the beginning of each51
division cycle, grows unbounded. This points to a limitation of the equations52
introduced before, that indeed lack a rate limiting term for the growth rate53
of X;54
4. the two intervals of time,
Tk and
T
g
k , oscillate in time with the same55
frequency . This condition is not equivalent to synchronization strictu sensu56
but it would nonetheless allow sustainable growth of the population of pro-57
tocells. Therefore this condition might be called supersynchronization. Note58
that in principle supersynchronization does not require equality of the two59
frequencies, but that their ratio be a rational number;60
5. the two intervals of times,
Tk and
T
g
k , change in time in a \chaotic way".61
3 Non linear interacting molecules62
While considering the linear models the analytical treatment was widely viable and63
numerical simulations were useful to support the analytical approach, in non lin-64
2

ear models simulations became fundamental to investigate the system behaviour.65
Through this approach several non linear kinetics for the genetic molecules have66
been taken into account. In some cases the non linearity is imposed starting from67
the linear models (e.g (1)) and introducing a limit in the growth rate of the genetic68
molecules (e.g. (2)).69
(
dC
dt = C

PiXi
dX
dt = C
MijXj ;
(1)
(
dC
dt = C

PiXi
dX
dt = C
 tanh(MijXjC ) ;
(2)
where C is the total quantity of the \container" material,  is a parameter that70
determines the thickness of the container (ranging between 2=3 for a micelle and71
1 for a very thin vesicle), X is the total quantity of the genetic memory molecule72
and
and  are positive parameters related to the rates of the chemical reactions.73
In the previous example the deviations from linearity were due to the squashing74
eect, but there were no real interactions among dierent molecules.75
Another model takes into account pairwise interactions between molecules (3),76
(
dC
dt = C

PiXi
dX
dt = C
XiXiXiXj iMijX
XiXi
i X
XiXj
j ;
(3)
This kinetic is very interesting to explore since it is frequently used in literature.77
The last part of the work concerns the coupling between the container and a chaotic78
set of equations used as molecules kinetic, like the well known Lorenz equations79
or the Williamowsky-Rossler system [1].80
4 Results81
Non linear reaction models of protocell, like the linear ones, show synchronization82
in many cases and this an indication of robustness with respect to the modication83
of the kinetic equations.84
Another important remark is that one of the models in which synchronization85
does not occur is the one ruled by a second order kinetic (3), a kind of interaction86
between molecules common in literature.87
The last important question addressed about the synchronization in non linear88
reaction models pertains to the ability of duplication to rule the chaos, even con-89
sidering a chaotic kinetic between molecules inside the protocell, the emergence of90
synchronization leads to a viable population of evolving protocells.91
References92
[1] B. D. Aguda and B. L. Clarke. Dynamic elements of chaos in the williamowski-93
rossler network. Phys, (89):12, December 1988.94
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[2] R. Serra, T. Carletti, A. Filisetti, and I. Poli. Synchronization phenomena in95
internal reaction models of protocell. WIVACE08 (Workshop Italiano sulla96
vita articiale), 2008.97
[3] T. Carletti, R. Serra, I. Poli, M. Villani, and A. Filisetti. Sucient conditions98
for emergent synchronization in protocell models. J Theor Biol, 254(4):741{99
751, 2008 Oct 21.100
[4] A. Filisetti. Dinamica di replicatori e sincronizzazione. M. Sc thesis, Dept.101
of Social, Cognitive and Quantitative Sciences, Modena and Reggio Emilia102
University, 2007.103
[5] A. Filisetti, R. Serra, T. Carletti, M. Villani, and I. Poli. Synchronization104
phenomena in protocell models. Biophysical Reviews and Letters (BRL),105
3(1/2):325{342, 2008.106
[6] T. Ganti. Chemoton theory, vol. i: Theory of
uyd machineries: Vol. ii:107
Theory of livin system. New York: Kluwer Academic/Plenum, 2003.108
[7] S. Rasmussen, L. Chen, D. Deamer, D. C.Krakauer, N. H. Packard, P. F.109
Stadler, and M. A. Bedau. Transitions from nonliving to living matter. Sci-110
ence, 303, 963-965, 2004.111
[8] S. Rasmussen, L. Chen, M. Nilsson, and S. Abe. Bridging nonliving and living112
matter. Articial Life, 9, 269-316, 2003.113
[9] S. Rasmussen, L. Chen, and B. M. Stadler. Proto-organism kinetics. Origins114
Life and Evolution of the biosphere, 34, 171-180, 2004.115
[10] R. Serra, T. Carletti, and I. Poli. Surface reaction models of protocells.116
BIOMAT2006, International Symposium on Mathematical and Computa-117
tional Biology, (World Scientic, ISBN 978-981-270-768-0), 2007.118
[11] R. Serra, T. Carletti, and I. Poli. Syncronization phenomena in surface-119
reaction models of protocells. Articial Life 13: 1-16, 2007.120
[12] R. Serra, T. Carletti, I. Poli, and A. Filisetti. The growth of popolation of121
protocells. In G. Minati and A. Pessa (eds): Towards a general theory of122
emergence. Singapore: World Scientic (in press), 2007.123
[13] R. Serra, T. Carletti, I. Poli, M. Villani, and A. Filisetti. Conditions for124
emergent synchronization in protocell. In J. Jost and D. Helbing (eds): Pro-125
ceedings of ECCS07: European Conference on Complex Systems. CD-Rom,126
paper n.68, 2007.127
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