timoteo carletti [email protected] pace – pa’s coordination workshop, los alamos 19-22 july 2005...
TRANSCRIPT
Timoteo Carletti
PACE – PA’s Coordination Workshop, Los Alamos 19-22 July 2005
Dipartimento di Statistica, Università Ca’ FoscariVenezia, ITALIA
FP6 EU
PACE
summary
►introduction
►short description of the Chemoton original model
►work in progress & perspectives
►numerical analysis of the new model
►a new model to overcome some drawbacks
1) membrane
The membrane encloses the system andseparates it from environment. It allows nutriment and waste material to pass through.
PACE
the original model
► Gánti (1971) :
► Csendes (1984) : first numerical simulation
once the membrane doubled its initial size the Chemotonhalves into two equal (smaller) units
The metabolic chemical system transformsexternal energetically high materials into internal materials needed to grow and to duplicate templates
2) metabolism3) information
The double-stranded template (polymer)is the information carrier. It can duplicate
itself if enough free monomers are availableThe Chemoton
PACE
template duplication: pV2n! 2pV2n (I)
free monomers V0
double-stranded template made of 2n monomers V0
pV2n
template duplicationstarts
if concentrationof V0 is larger than
a threshold V*
……
PACE
template duplication: pV2n! 2pV2n (II)
chemical reactions:duplication initiation
duplication propagation
final step
pV2n concentration of double-stranded template
ki (direct) rate constant
ki0 (inverse) rate constant
(pV2n¢ pVi) concentration of intermediate states ki >> ki0
PACE
metabolism, autocatalytic cycle : A1! 2A1
chemical reactions:
Ai concentration of ith reagent
ki (direct) rate constant
ki0 (inverse) rate constant ki>> ki
0
PACE
membrane growth
chemical reactions:
T concentration of membrane molecules
ki (direct) rate constant
ki0 (inverse) rate constant
T0 and T* concentration of precursor of membrane molecules
ki>>ki0
PACE
kinetic differential equations
cell surface growth
balance equation for free monomers
balance equation for R reagent
time
size
growth growthdivision
PACE
the original model : division (I)
►standard assumption: (Gánti, Csendes, Fernando & Di Paolo (2004))
when growing the Chemoton always keep a spherical shape
when the surface size doubled its initial value (cell cycle),suddenly the Chemoton divides into two equal smaller spheres,
preserving total number of T molecules and halving all the contained materials
PACE
the original model : division (II)
► remark: (Munteanu & Solé (2004))
at the division all concentrations increase (by a factor )
(sphere hypothesis)
concentration generic ith reagent
immediately before division
(doubling hypothesis)
immediately after division
(halving hypothesis)
PACE
take care of the shape (I)
► we observe that the previous remark can be applied to includethe volume growth in the kinetic differential equations :
the kinetic differential equation for the generic concentration ci
has to be modified by the addition of the term
►the shape, hence the volume, changes concentrations, thus the dynamics is affected by the chosen shape
PACE
take care of the shape (II)
►observations of real cells and their division process,i.e. experiments, support the following working hypothesis:
when growing the Chemoton changes its shape passing from a sphere to a sand-glass (eight shaped body), through a peanut.
growth growth growth growth growth division
time
sha
pe
once the surface size doubled its initial value (cell cycle),the eight shaped Chemoton naturally divides into two equal smal spheres,
preserving total number of T molecules and halving all the contained materials
PACE
model analysis
& it is high dimensional: 5+2+ 4+2n
► the model depends on several parameters (for instance )
membrane
polymer
thus numerical simulations can help to understand its behaviour
What are we looking for? Which are the “interesting” dynamics?
… but …
PACE
regular behaviour
t
S(t)
t
A1(t)
►”regular” behaviour: cell cycles repeat periodically
thus each generation starts with the same amount of internal materials
let TCi be time interval between two
successive divisions at the ith generation(ith replication time)
TCi
ith generation
“The replicationPeriod”
PACE
non-regular behaviour
►”non-regular” behaviour: replication times vary for each generation
each generation can start with different amount of internal materials
TCi
ith generation
t
S(t)A1(t)
t
no replication period can be defined
PACE
regular behaviours vs parameters (I)
►determine how parameters affect the dynamicswe fix two parameters between and we study the dependence
of the replication time on the third free parameter
TCi
TCi
zoom
high concentrations of X induce a faster dynamics, thus shorter replication period,and instabilities can be found for small concentrations
PACE
regular behaviours vs parameters (II)
TCi
V*
our new model
TCi
V*
original model
high values of V* implies that polymerization (and thus all the growth process) can start only after many metabolic cycles A1! 2A1 (to produce enough V0), Namely long replication period.
At lower values, polymerization can (almost) always be done, thus the (eventually) bottleneck in the growth process must be found elsewhere & the replication period is
independent of V*. Intermediate values can give rise to instabilities.
PACE
regular behaviours vs parameters (III)
N
TCi
original model
TCi
N
our new model
long polymers need many free monomers V0 to duplicate themselves, thus many metabolic cycles A1! 2A1 have to be done, namely long replication period.
V*
N
PACE
a global picture
►to better understand the interplay of N and V* in determining regular behaviours, we fix X & for several (N,V*) we look for a unique replication period
blue spot: more than one or no replication period at all
red spot: a unique replication period
PACE
stability of regular behaviours
►once we determine a unique replication period, some natural questions arise: is this dynamics stable? Are there other
regular behaviours close to this one?
we fix N=25, V*=50 & X=100 and we consider the role of A1 and V0
blue spot: more than one or no replication period at all
red spot: a unique replication period
A1
V*
PACE
work in progress
►use a more fine mathematical tool to study the stability of a periodic orbit
in all models information is carried by the length of the polymer
►introduce a divisions process where internal materials are not equally shared in next generations & consider the previous picture (A1,V0)
►study family of Chemotons with different polymer lengths & consider the previous picture (N,V*)
PACE
perspectives
►use a stochastic integrator (Gillespie) & compare results with our deterministic approach
, then it will be possible to include mutations both in the length of the polymer and in the copying fidelity
►consider a “more realistic template” build with, at least, two different monomers V0 and W0
►introduce the space and consider competition for food
Timoteo Carletti
PACE – PA’s Coordination Workshop, Los Alamos 19-22 July 2005
Dipartimento di Statistica, Università Ca’ FoscariVenezia, ITALIA
FP6 EU
