Theoretical
models can be extremely useful to devise possible protocell architectures and
to forecast their behaviour. In this project we have addressed an important
issue in protocell research. 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 undergoes 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 superficially
resembles the division of a cell. In order for evolution to be possible, some
genetic molecules should affect the rate of duplication of the whole container,
and some mechanisms have been proposed whereby this can be achieved.
But then a new
problem arises: the genetic material duplicates at a certain rate, while the
lipid container grows, in general, at another rate. When the container splits
into two, it may be that the genetic material has not yet doubled: in this case
its density would be lower in the daughter protocells. Through generations,
this density might eventually vanish. On the other hand, if the genetic
material were faster than the container, it would accumulate in successive
generations.
So, in order for
a viable population of evolving protocells to form, it is necessary that the
rhythms of the two processes are synchronized. In some models (like the
Chemoton) this is imposed a priori in the kinetic equations, but it is unlikely
that such a set of exactly coupled reactions springs up spontaneously in a
single step. It is therefore interesting to consider the possibility that such
synchronization be an emergent phenomenon, without imposing it a priori.
Note also that
the possibility to use evolving
populations of protocells, subject to some form of variation and
selection, for information processing tasks requires such a sustainable
population growth; therefore synchronization (in the sense given above) is a
preprequisite for the use of protocells in ICT.
Synchronization
has been studied here using abstract models which can be grouped in two
classes : surface reaction models (briefly, SRMs) and internal reaction
models (IRMs). The difference is that, in the former case, the reactions which
lead to the formation of the new genetic material and those which lead to the
formation of the new membrane molecules take place close to the protocell outer
surface, while in IRMs they both take place in the interior of the vesicle. The
modelling level is fairly abstract, so the results should hold for different
detailed protocell architectures. SRMs are inspired by the the so-called
"Los Alamos bug", a model of protocells where the genetic material is
composed by strands of PNA which should be found in the vesicle membrane. Internal
reaction models are related to other detailed models of protocell architecture,
like for example the one proposed by Luisi.
Initially we have
concentrated our studies on SRMs. By introducing suitable hypotheses, we have
described the system by a set of coupled ordinary differental equations. We
made the assumption that the replication molecules which collectively carry the
"genetic memory" of the system (briefly, GMMs; sometimes they will be
referred to also as "replicators") can be treated by the methods of
chemical kinetics (while stochastic generalizations might be introduced with
standard techniques if needed). Note that there is considerable arguing in the
literature concerning the chemical nature of these molecules: the hypotheses
which are most often discussed are that they are i) nucleic acids, able to
drive their own duplication by Watson-Crick base pairing or ii) sets of
polypeptides which catlayze each
other's formation or iii) lipids, as in the Gard model. An interesting feature
of our methods is that the techniques are general, and can be applied to theese
different cases, while the different specific hypotheses show up in the form of
the kinetic equations.
The are two kinds
of equations in the system: the first one describes the growth of the container
which, for simplicity, is supposed to be composed by one single type of lipids
(again, generalizing this assumption would be straightforward) and has the general
form
[1]
Where C is
the total quantity (moles) of the lipid container of a protocell, S is its
surface and 
are the quantities
(moles) of the various GMMs in that protocell. Note that Eq.1 provides the
necessary coupling between the growth rate of the protocell and the GMMs.
It is
assumed that, once the quantity C ha reached a threshold value q, the protocell splits into two.
Therefore Eq.1 describes the continuous growth phase of each generation. The
equations for the GMMs in the same continuous growth phase are then
[2]
Various
specific models give rise to different forms for the function f. We will below
refer to the "linear case" when f is linear, being however understood
that the overall model is fully nonlinear due to the doubling and halving
hypothesis. The continuous growth phase ends when the cells reaches the
critical size q; at that time
the quantities of GMMs can be denoted by 
, the subscript meaning "final". At that point it
splits into two, and each daughter cell starts a new growth phase, with initial
conditions where C equals 
, and the quantities of GMMs are equal to 
.
We have
introduced a nice mathematical technique to deal with this system, which works
well when the function f is such that one or more quantities are conserved in
the continuous growth phase. This allows one to derive a discrete map which
relates the initial values of the GMMs at generation k+1 to those at generation
k
[3]
From the
fact that the doubling time Td is determined by the initial values
of the GMMs (and by the initial quantity of the container, which is however
always equal to q/2) it follows
that synchronization, which means that Td tends to become equal in
successive generations, is equivalent to the statement that the initial quantities
of GMMs tend to become equal in successive generations. This is the basis for
the analytical treatment of the system, which can be developed in several
interesting cases.
Whenever
such an analysis is impossible, one resorts to computer simulations: for this
purpose we devloped a simulation software particularly suited to our system
(where one must be careful since disconinuities occur at C=q).
The first case
which has been considered is that of a single type of GMM. If Eq. 1 is linear
in X, and the growth eqs 2 are also linear with respect to X, synchronization
is always achieved (provided of course that the kinetic constant is positive). Interestingly,
the same holds also if one considers a nonlinear growth for X which has the
form of a power law, with an exponent smaller than 2. This encompasses also the
case of parabolic growth, which is believed to provide a good description of the kinetics of
molecules which replicate by passing from a double strand to a single one, and
then attaching precursors to the single stranded form.
The property of
synchronization turns out to be robust with respect to different variations of
the basic model, like the introduction of realistic vesicle geometry, and that
of a term which slows the growth of C and X. It also holds for more general
forms of the function which describes the growth of the protocell or that of
the replicators.
The same
formalism allows one to consider tha case of a spherical micelle and that of
a vesicle, but it has also been
proven that, by properly rescaling time, one can always modify the vesicle
equations in such a way that they take the same form as those of a micelle,
which are simpler to treat. Since we are interested in the determining the
asymptotic properties, and not the duration of the transients, this simplifies
the analysis in a remakable way.
A further
generalization is worth discussing. In writing Eq 2 we have assumed that [X]=X/
VL, where square brackets denote concentrations and VL is
the volume of the lipid phase, proportional to C. This is correct as long as X
itself does not appreciably contribute to the volume of the lipid phase. But if
the quantity of X becomes large, and X itself is a lipophilic compound which
contributes to the container, this
formula should be substituted by [X]=X/(VL+VX),
where VX is the contribution to the volume of the lipid phase of the
GMMs . By rescaling time, it has been analytically shown and numerically
confirmed that also in this case synchronization is achieved. Therefore our
model can handle also the case where the protocell architecture is similar to
that of the GARD model.
If there are
several replicators in the same cell, but they do not interact directly, one
again finds synchronization if the kinetic constants are positive. In the
linear case, only the fastest replicator survives in the final population of
protocells, while if the kinetcis is parabolic all the GMMs survive, their
relative proportion being a function of the ratio between their kinetic
constants. This is consistent with similar behaviours observed in population
dynamics.
Let us then turn
the interaction on, and let us first consider the linear replicator case. This
can be studied analytically, and a complete discussion of the different cases has
been given. The most relevant results are that the behaviour of the system is
ruled, in the long time limit, by the eigenvalue of the kinetic matrix M of the
GMMs with largest real part (ELRP). If Re(ELRP)>0, and if the corresponding
eigenvector (
) is nonnegative, synchronization is achieved. The asymptotic
value of 
is a multiple of and the division
time is related to the ELRP. In order to physically interpret these results it
is then necessary that the ELRP be real and positive, and that the components
of be real and nonnegative.
A sufficient
condition to guarantee that ELRP is real and positive, and that , for every
j=1…N, 
, is that the matrix elements Mij are never
negative. This is an important case, where all the replicator molecules which
do interact directly contribute to the synthesis of the others (mutual
calatalysis).
It is however
possible to imagine also cases where the network of reactions includes some
negative terms (if they were all ≤0 the system would of course die out) and to
analyze these cases.
In the
linear case we also found and studied an interesting phenomenon which we termed
supersynchronization, where the initial quantities of replicators in successive
generations (and therefore also the doubling times) regularly oscillate in
time. This regime does not correspond to synchronization strictu sensu, but it also allows a sustainable
growth of the protocell population.
The most striking
result of the analysis of linear replicators is that they behave in a way
similar to that of a CSTR: here vesicle splitting limits the asymptotic values,
while in CSTR it is the outflow which provides such limitation. But the ratio
of various replicators is the same in the two cases. The couplin to the
container growth rate does not influence this ratio, nor the asymptotic
division time, although it affects the actual value of the asymptotic
quantities.
We also
performed an extensive analysis of various cases with direct nonlinear
interactions among replicators. Since few results can be obtained analytically,
the study has relied mainly on numerical experiments performed with our
simulator. It has been shown that in most cases one finds either
synchronization or extinction (which is of course always possible depending
upon the kinetic coefficients), so the former seems a widespread property.
However, this is not always the case: for example, with quadratic interactions
(where dX/dt and dY/dt are proportional to the product XY) synchronization is
not achieved. This is particularly important in view of the widespread use of
quadratic interaction in various models. However, synchronization is again
found if, besides the quadratic term, also a linear term is present in the
kinetic equations.
It is also
interesting to observe that if the interactions are quasi-linear (i.e. they are
proportional to a sigmoid function of a linear combination of their inputs)
then the behaviour is similar to that of the corresponding linear system;
supersynchronization is sometimes observed also in this case.
Impressed
by the wide diffusion of synchronization, we considered some particular forms
of the equations for the interaction among replicators, which are known to lead to chaotic behaviours. So, if
replicators were to interact according to these rules in the bulk, chaos should
be observed. But in a protocell the situation is different. Due to the coupling
with the container growth and splitting, we observed that chaos is often
suppressed. Only for very small coupling coefficients can chaotic behaviour
still be observed. This opens the way to interesting speculations about a possible
role of replication in taming chaos, theferore allowing also kinds of
interactions which would lead to uncontrollable behaviours in the bulk. The
phenomena have been so far observed in
models related to those of Lorenz and Roessler, adapted to the case
where the variables can take only non negative values (as it is necessary since
they describe quantities of various species of chemicals).
The models
considered initially were of the "surface reaction" type, like the
Los Alamos bug, where all the key reactions involved in the growth of the container
and of the replicators take place near the vesicle membrane. However, several
proposed protocell architectures assume that these reactions take place in the
aqueous interior of a vesicle. We therefore introduced another class of
abstract models, which we call "internal reaction" models, to deal
with this kind of hypothesis, and we investigated synchronization in these
cases.If we assume that precursors (of both replicators and lipids) can diffuse
rapidly through the membrane, then the equations turn out to be similar to
those of the surface reaction models (indeed, depending upon the specific
kinetic assumptions, they may even be exactly the same). The system with
internal reaction can then be (and it has been) analyzed with the same
techniques as the previous one. It turns out that sysnchronization is frequent
also in this case (being however excluded in the case of quadratic
replication).
We have
also developed a model of internal reactions where the diffusion rate of
precursors through the membrane is slow, and we have shown that synchronization
can be achieved also in this case.
In the end
we have come to a fairly clear picture of synchronization as a widespread
property of protocells, when the kinetic is fast enough (otherwise the growth
of the number of protocells halts).
Supersynchronization can also be observed in some cases of linear
interaction. Synchronization may be impossible in some models, like the
quadratic one; but also in this case a small change in the model equations
(i.e. adding a linear term) can restore synchronization. Chaotic behaviour
seems to be possible only when the coupling of replicators and container is
very small: however this kind of systems require further analysis.
We have also
addressed some relevant questions concerning the well know “Chemoton” model
introduced by Ganti in the 70's as a possible model to describe a protocell.
Despite its long history very few simulations of this model are available in
literature, we thus developed technical tools to analyse the behaviour of the
Chemoton, with a twofold aim : to study the possible dynamics of a single
protocell and the evolution of
population of Chemotons, subjected to genetic mutations and to the pressure of
the environment.
Our first
contribution has been to consider that the relevant chemical reactions involved
in the Chemoton model occur in a varying volume, e.g. the protocell is growing,
and moreover this volume variation has not been imposed a priori as previously done
in literature, but it is intrinsically determined by the protocell membrane
growth. We then studied the dependence of the division time, the most important phenotype
property of the model, on the involved parameters. We show that it can exhibit
a complex behaviour as the original model did, where bifurcation diagrams and
period doubling cascades are present.
Once we obtained a full understanding of a
single “generic” orbit, we addressed the question concerning the stability of the regular behaviours, thus to
characterise a large set of orbits in a neighbourhood of a given periodic
solution. We realise that ultimate fate of the protocell, once the initial
conditions are specified, is completely predicted by a Stability Function, which allows in particular to
define basins of regular or irregular (chaotic) dynamics. Roughly speaking
given the initial value of reagents concentrations the Stability Function tells
us if the resulting solution is regular or not.
Using this Stability Function we
have been able to introduce in the model a non-perfect halving mechanism at the division and thus to study
the evolution of a large population over long periods of time. Our main result
is that the model reproduces, as an emergent property the formation of stable non–homogeneous (in size) families starting from a homogeneous one.
We also introduced a mutation mechanism, acting on the polymer: during the
template duplication extra monomers are added or removed according to a
pre-assigned probability. Once again the time evolution of a single protocell
is monitored as functions of some polymers related parameters, and this
knowledge translates into a Division Function which determines the future behaviour of the
protocell. Using the Division Function we have been able to obtain the first
simulation concerning the evolution of a large family of protocells over long
times, subjected to genetic mutations and to the pressure of the environment. Our main results are that the
model can evolve toward stable families with modified genome or into new species: speciation is an emergent property of our model.
In this
project we have also developed the idea of a process which might help to
concentrate the required chemicals in the interior of the vesicle (this is a
major problem in making a protocell work). We have shown theoretically that, if
a given reaction takes place on the surface (e.g. because it is catalyzed by
molecules on the surface), and if the reactants can freely pass the membrane,
but some products can not, then one may concentrate these products in the
interior. If the system is open (like for example a continuously stirred tank
reactor) the concentration difference can persist in the steady state.
The work has been described in the following papers:
Papers in international journals
1. Carletti, T., Serra, R., Poli, I., Villani, M. & Filisetti, A. (2008): Sufficient conditions for emergent synchronization in protocell models. Journal of Theoretical Biology (accepted)
2. Serra, R., Carletti, T. & Poli, I. (2007): Synchronization phenomena in surface reaction models of protocells. Artificial Life 13, 1-16
3. Carletti, T. & Fanelli, D. (2007): From chemical reactions to evolution: emergence of species. Europhysics Letters 77, p. 18005.
Conference proceedings
4. Serra, R., Carletti, T., Poli, I., & Filisetti, A. (2008): Synchronization phenomena in internal reaction models of protocells. In R. Serra, I. Poli & M. Villani (eds): Proceedings of Wivace 2008 (accepted)
5. Serra, R. & Villani, (2008): A CA model of spontaneous formation of concentration gradients. In H. Umeo (ed): ACRI 2008. Berlin: Springer Lecture Notes in Computer Science (in press)
6. Serra, R., Carletti, T., Poli, I., & Filisetti, A. (2008): The growth of populations of protocells. In G. Minati & A. Pessa (eds): Towards a general theory of emergence. Singapore: World Scientific (in press)
7. Filisetti, A., Serra, R., Carletti, T., Poli, I. & Villani, M. (2008): Synchronization phenomena in protocell models. In R. Mondaini (ed): BIOMAT2007, International Symposium on Mathematical and Computational Biology, 373-389. Singapore, World Scientific
8. Serra, R., Carletti, T., Poli, I., Villani, M. & Filisetti, A. (2007): Conditions for emergent synchronization in protocells. In J. Jost & D. Helbing (eds): Proceedings of ECCS07: European Conference on Complex Systems. CD-Rom, paper #68
9. Serra, R., Carletti, T. & Poli, I. (2007): Surface reaction models of protocells. In R. Mondaini (ed): BIOMAT2006, International Symposium on Mathematical and Computational Biology. Singapore: World Scientific
10. Carletti, T. & Fanelli, D. (2007): Evolution of a population of protocells: Emergence of species. In R. Mondaini (ed): BIOMAT2006, International Symposium on Mathematical and Computational Biology. Singapore: World Scientific.
11. Serra, R., Carletti, T. & Poli, I. (2006): Emergent synchronization in protocell models. In A. Acerbi, S. Giansante e D. Marocco (eds.): Proceedings of Wiva 3 (http://laral.istc.cnr.it/wiva3/atti/wiva3/home.htm; ISSN 1970 50 77)
Other papers concerning the dynamics with nonlinear interactions, the suppression of chaos in protocells and the concentration effect are in preparation