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Synchronization phenomena in Internal Reaction Models of
protocells
Roberto Serra
Dipartimento di Scienze Sociali, Cognitive e Quantitative
Universita di Modena e Reggio Emilia
via Allegri 9, 42100 Reggio Emilia, Italy
E-mail: rserra@unimore.it
Timoteo Carletti
Departement de Mathematique, Facultes Universitaires Notre Dame de la Paix
8 rempart de la Vierge, B5000 Namur, Belgium
E-mail: timoteo.carletti@fundp.ac.be
Alessandro Filisetti
European Center for Living Technology
Ca'Minich, Calle del Clero, S.Marco 2940 - 30124 Venice, Italy
E-mail: alessandro.lisetti@ecltech.org
Irene Poli
Dipartimento di statistica, Universita Ca' Foscari
San Giobbe - Cannaregio 873, 30121 Venezia, Italy
E-mail: irenpoli@unive.it
1. Introduction
Protocells are lipid vesicles (or, less frequently) micelles which are endowed
with some rudimentary metabolism, contain \genetic"material, and which
should be able to grow, reproduce and evolve. While viable protocells do not
yet exist, their study is important in order to understand possible scenarios
for the origin of life, as well as for creating new \protolife"forms which are
able to adapt and evolve1 . This endeavor has an obvious theoretical inter-
est, but it might also lead to an entirely new \living technology", denitely
dierent from conventional biotechnology.
Theoretical models can be extremely useful to devise possible protocell ar-

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chitectures and to forecast their behavior. What can be called the \genetic
material"of a protocell is composed by a set of molecules which, collectively,
are able to replicate themselves. At the same time, the whole protocell un-
dergoes a growth process (its metabolism) followed by a breakup into two
daughter cells. This breakup is a physical phenomenon which is frequently
observed in lipid vesicles, and it has nothing to do with life, although it
supercially resembles the division of a cell. In order for evolution to be
possible, some genetic molecules should aect the rate of duplication of the
whole container, and some mechanisms have been proposed whereby this
can be achieved.
In order to form an evolving protocells population it is necessary that the
rhythms of the above mentioned two processes, i.e. methabolism and ge-
netic replication, are synchronized and it has previously been shown that
this may indeed happen when one takes into account successive generations
of protocells2{6 .
The present paper presents and extends our previous studies which had
considered synchronization in the class of so{called \Internal Reaction Mod-
els", IRM for short6 when linear kinetics were assumed for the relevant
chemical reactions. Let us stress here that similar results have been ob-
tained also for the \Surface Reaction Models", SRM for short2{6 , hence
the synchronization phenomenom seems to be very robust with the respect
to the chosen architecture once linear kinetics are considered.
The IRMs are roughly inspired by the so{called RNA{cell7,8 whereas the
modelization of SRMs arises from the so-called \Los Alamos bug"9,10 . The
paper is organized as follows. In Sec. 2 we report a review of our previous
results, in Sec. 3 we describe the main features of IRMs and discuss the
behaviors of this class of models. Finally, in Sec. 4 some critical comments
and indications for further work are reported.
2. A review of previous results
As already explained in our previous works2{6 starting from a set of simpli-
ed hypotheses and considering a protocell endowed with only one kind of
genetic memory molecule, where the relevant reactions occurs on the exter-
nal protocell surface one can describe the container and genetic molecule
behavior by Eqs. (1).
(
dC
dt = C
X
dX
dt = CX ;
(1)

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where C is the total quantity of the \container"material,  is a parameter
that determines the thickness of the container (ranging between 2=3 for a
micelle and 1 for a very thin vesicle), X is the total quantity of the genetic
memory molecule and
and  are positive parameters related to the rates
of the chemical reactions.
Before going on with the discussion about the internal reactions models
it is interesting to consider which kind of behaviors one can expect to nd:
(1) Synchronization: in successive generations (as k ! 1 where k is the
generation number) the interval of time needed to duplicate the mem-
brane molecules of the protocell between two consecutive divisions,

Tk, and the time required to duplicate the genetic material, again
between two consecutive divisions,
T gk , approach the same value;
(2) as k !1 the concentration of the genetic material at the beginning of
each division vanishes. In this case, given the above assumptions, the
growth of the container ends and the whole process stops;
(3) as k ! 1 the concentration of the genetic material at the beginning
of each division cycle, grows unbounded. This points to a limitation of
the equations introduced before, that indeed lack a rate limiting term
for the growth rate of X;
(4) the two intervals of time,
Tk and
T gk , oscillate in time (we will
provide some examples in the following) with the same frequency . This
condition is not equivalent to synchronization strictu sensu but it would
nonetheless allow sustainable growth of the population of protocells.
Therefore this condition might be called supersynchronization. Note
that in principle supersynchronization does not require equality of the
two frequencies, but that their ratio be a rational number;
(5) the two intervals of times,
Tk and
T gk , change in time in a \chaotic
way".
3. Internal Reaction Models
Let us now consider the synchronization problem in a model where the
relevant chemical reactions are supposed to run inside the protocell vesicle,
such models have been named internal reaction models (IRMs)6 .
Assuming once again that the genetic memory molecules induce the
container growth via the production of lipids from precursors, one can de-
scribe the amount of container C in time by some non{linear function of

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the concentration [X] = X=V :
dC
dt = X
V 1
; (2)
where
> 0 is a parameter which determines the strength of the in
uence
of X on C and where V is the whole protocell volume, V , namely we are
here assuming that the volume occupied by the amount of C is really small
with respect to the volume determined by X
Let us now consider some possible replication rates for the genetic
molecules. The simplest one is the Linear replicator kinetics; in this case
the amount of the X molecules is proportional to the number of existing
ones (given that precursors are not limiting), so:
dX
dt = X : (3)
A straightforwardly generalization can be obtained assuming some power
law with exponent , of the concentration [X], hence we get:
dX
dt = [X]
V = XV 1 : (4)
The behavior of a protocell during the continuous growth phase is thus
described by:
(
dC
dt = X
V 1
dX
dt = [X]V = XV 1 :
(5)
In order to complete the treatment it is necessary to express V as a function
of C, and this depends upon geometry. One can assume a generic functional
relation of the form V = g(C), for some positive and monotone increasing
function g.
To provide an explicit example we will now compute such function g
under the assumption of spherical vesicle with very thin membrane.
Remark 3.1 (Spherical very thin vesicle). Let us suppose that the
vesicle is spherical, with internal radius ri and with a membrane of constant
width  (a reasonable assumption if it is a bilayer of amphiphilic molecules).
Then starting from V = Vi + VC we get:
Vi = 43r3i and VC = 43(ti + )3 43r3i ) VC = 4ri2 + 4r2i  + 433 :(6)
We can thus express ri as a function of C and then using the formula for the
sphere volume we can express V as a function of C, through its dependence

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on ri. One can easily obtain:
ri =
2 +
q
134 + VC4
2 ; (7)
we can nally assume  << 1 (thin membrane), to get
VC = 4r2i  = S (8)
where S is the surface area, S = 4r2i , and nally assuming VC << 1,
V  Vi = 43r
3
i = 43
 S
4
 3
2
= 43
 VC
4
 3
2 = 43
 C
4
 3
2
: (9)
Thus the required function is V = aC3=2.
By incorporating the constants into the kinetic constants and renaming
them, the model (5) can thus be described by
(
dC
dt = X
C3(1
)=2
dX
dt = XC3(1)=2 :
(10)
The model described by Eq. 5, or by 10, can be studied via an analytical
technique presented in2 and6 . Here we propose an alternative approach that
enables us to obtain the same results and also some explanations.
The division event can be seen as a map that to the amount of the
X{molecule at the beginning of the k{th generation arising at time Tk, as-
sociates the same quantity, say Xk+1 at the beginning of the next protocell
cycle:
F : (Xk; Tk) 7! F (Xk; Tk) = (Xk+1; Tk+1) : (11)
Then synchronization is equivalent to determine a xed point for this map,
if moreover we are interested in the possibility to reach this xed point
following the dynamics, this xed point must be a stable one.
The map F can be obtained by integrating Eqs. (5). To simplify the
successive computations we rst introduce an analytical trick, consisting
in a non{linear reparametrization of time. In fact from the rst relation of
Eq. (5) we can conclude that C is a monotone increasing function of time,
i.e. its derivative is strictly positive, hence we can introduce a new time
variable,  , dened by:
d
dt =  [g(C)]
1
X
dt ; (12)

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where V = g(C) denotes the generic dependence of the volume on the
container C. Using this new variable the system (5) can be rewritten as:
(
dC
d = 1
dX
d = X
g
 :
(13)
In this way the behavior of C is trivial and the division event is just k+1 =
k + =2. This simplies the map F that becomes a function of Xk only.
Moreover during the continuous growth we have C() = =2 + ( k) and
thus the second relation of (13) rewrites:
dX
d =

X

[g (=2 + ( k))]
 ; (14)
This equation can be solved explicitly thus providing the map F :
F (Xk) =
"
1
2
+1X
+1
k + (
 + 1)


Z =2
0
g(=2 + s) ds
#1=(
+1)
if
 + 1 6= 0 :
(15)
The case
 + 1 = 0 can be solved as well but one can show that in
this case synchronization is possible only for special values of the involved
parameters and thus it is not generic. For this reason we will not develop
further this case.
The function F admits a positive xed point if and only if
+ 1 > 0
which is given by:
F (X) = X ) X =

 + 1
2
+1 1

()
1=(
+1)
; (16)
where we denoted by () the constant integral in the right hand side
of (15).
To determine the stability character of this xed point we have to com-
pute the rst derivate of F and evaluate it at X, that is:
dF
dX (X) =
X1=(
+1) =2
+1
() +X1=(
+1) =2
+1
; (17)
and we can easily check that under the assumption
+1 > 0 and X > 0
this derivative is always smaller than 1, ensuring thus the stability of the
xed point.
This result can thus be restated by saying that synchronization is pos-
sible only if the genetic replication rate is small enough with respect to the
container growth:  <
+ 1.

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Remark 3.2. The very widely used model of quadratic growth for genetic
memory molecules doesn't allow for synchronization if the container growth
is linear, in fact here the previous relation doesn't hold, i.e. 2 =  6
+1 =
2. But observe that if the container growth were slightly faster, say
> 1,
then synchronization will be obtained.
(
dC
dt = X
dX
dt = X
2
V :
(18)
0 50 100 150 2000
5
10
15
20
25
30
Generations elapsed from T0
CellDivis
ionTime

Numerical CDT
0 100 200 300 400 5000
24
68
1012
14x 106
time
Amounto
fX i

X
Fig. 1. An example of a system ruled by Eqs. (18) where synchronization is not achieved.
On the left panel cell division time in function of generations elapsed from T(0) is shown
while on the right panel the total amount of replicators in function of time for each
generation is shown.
Remark 3.3 (Various kinetic equations). In the case where more than
one kind of genetic memory molecules is present in the same protocell, one
can consider of course more general situations as already done for the SRM
case.2,4 Under the assumption of very thin spherical vesicle we have con-
sidered the following two cases of second order interaction between dierent
genetic memory molecules:
(1) without cross-catalysis: no synchronization is observed for the stud-
ied set of parameters:
(
C
dt = 1X1 +


+ nXn
dXi
dt = C
3
2
PN
k=1MikXiXk
(19)
(2) with cross-catalysis :

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(
C
dt = 1X1 +


+ nXn
dXi
dt = C
3
2
PN
k=1MijkXjXk :
(20)
The behaviour is not completely understood. Varying the kinetic coe-
cients sometimes we observe synchronization but more often extinction.
Similar results hold true for the SRM case as well, but on dierent time
scales and somehaow SRM are more robust infact synchronization in SRMs
does not necessarily implies synchronization in IRMs.
3.1. Finite diusion rate of precursors through the
membrane
In this last section we will take into account the fact that the crossing
of the membrane from precursors may be slow. We suppose like in the
previous sections that the key reactions (i.e. synthesis of new C and new X)
take place in the interior of the cell, and that diusion in the water phase
(internal and external) is innitely fast. It is assumed that X molecules
do not permeate the membrane, but that precursors of C and X can. The
external concentration of these precursors is buered to xed values EC and
EX , while the internal concentrations can vary, their values being [PC ] =
PC=V and [PX ] = PX=V , where V is the inner volume, thus once again
we assume that the membrane volume to be negligeable. Note that, for
convenience, the xed external concentrations are indicated without square
brackets, while PC and PX denote internal quantities.
Precursors can cross the membrane at a nite rate; if D denotes diusion
coecient per unit membrane area, then the inward
ow of precursors of
C (quantites/time) is DCS(EC [PC ]), and a similar rule holds for X.
X catalyzes the formation of molecules of C, therefore we assume that
the rate of growth of C is proportional to the number of collisions of X
molecules with C precursors in the interior of the vesicle. It is therefore a
second order reaction. Reasoning as it was done in the case of Sec. 3 one
gets
(
dC
dt = 0hCV 1XPC
dX
dt = 0hXV 1i XPX :
(21)
Note that it might happen that more molecules of precursors are used
to synthesize one molecule of product (the number of precursor molecules
per product molecule can be called hX and hC).

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(
dPX
dt = SDX

EX PXV


0hXV 1XPX
dPC
dt = SDC

EC PCV


0hCV 1XPC ;
(22)
Equations (21) and (22) provide a complete description of the dynamics.
Note that by dening  = 0hX and  = 0hC one can eliminate the
stoichiometric coecients from these equations.
As done before, in order to complete the study it is necessary to express
V and the surface S as a functions of C, and this depends upon geometry.
Under the assumption of spherical very thin vesicle we obtain
V = aC 32 and S = bC ; (23)
for some positive constants a and b. The second relation of Eq. (23), in-
serted in Eq. (22), complete the model. The behavior of this model has
been studied with numerical methods, Fig. 2, and it has been numerically
veried that this model shows synchronization in the range of considered
parameters.
0 20 40 60 80 1000.2
0.250.3
0.350.4
0.450.5
0.550.6
0.65
Generations elapsed from T0
CellDivis
ionTime


0 5 10 15 2080
90100
110120
130140
150160
time
Amounto
fX i


X
Fig. 2. An example of a system ruled by Eqs. (21 and (22) where synchronization is
achieved. On the left panel cell division time in function of generations elapsed from T(0)
is shown while on the right panel the total amount of replicators in function of time for
each generation is shown.
4. Conclusion
In this paper we have addressed some relevant questions about synchro-
nization in a class of abstract models of protocell called Internal Reaction

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Models (IRMs) where key reactions occur within the vesicle, this complete
our previous works where all reactions occurred on the surface of the pro-
tocell (SRMs).
Comparing the two classes of models we observe that the behavior is very
similar in the two cases so synchronization is an emergent property also of
IRMs.
We also demonstrated that synchronization is an emergent property inde-
pendently from the geometry of the container if the genetic replication rate
is small enough respect to the container growth, Sec. 3.1.
Most of the analyses have been carried on under the simplifying assump-
tion that diusion through the membrane is fast with respect to the kinetic
constants of the other processes. Since this may be unrealistic in some real
cases, we have also considered a case with nite diusion rate, showing the
way in which such a case can be modelled and demonstrating, under the
particular kinetic model considered, that synchronization is also achieved.
It is worth remarking that, although the properties which have been shown
in this paper provide a clear picture of synchronization, further studies are
needed in order to consider more general cases.
Acknowledgments
Support from the EU FET{PACE project within the 6th Framework Pro-
gram under contract FP6{002035 (Programmable Articial Cell Evolution)
and from Fondazione Venezia are gratefully acknowledged. We had stim-
ulating and useful discussions with Nobuto Takeuchi during the summer
school \Blueprint for an articial cell" in beautiful San Servolo, Venice.
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