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Chemoton model:. the shape . does matter. PACE – PA’s Coordination Workshop, Los Alamos 19-22 July 2005. Timoteo Carletti. Dipa rtimento di Statistica, Università Ca’ Foscari Venezia, ITALIA. [email protected] FP6. EU. summary. ► introduction.

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Chemoton model:

the shape

does matter

PACE – PA’s Coordination Workshop,

Los Alamos 19-22 July 2005

Timoteo Carletti

Dipartimento di Statistica, Università Ca’ Foscari

Venezia, ITALIA

[email protected]

FP6

EU


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summary

►introduction

►short description of the Chemoton original model

►a new model to overcome some drawbacks

►numerical analysis of the new model

►work in progress & perspectives

[email protected]


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1) membrane

2) metabolism

3) information

The membrane encloses the system and

separates it from environment.

It allows nutriment and waste material

to pass through.

The metabolic chemical system transforms

external energetically high materials

into internal materials needed to grow and

to duplicate templates

The double-stranded template (polymer)

is the information carrier. It can duplicate

itself if enough free monomers are available

the original model

► Gánti (1971) :

The Chemoton

once the membrane doubled its initial size the Chemoton

halves into two equal (smaller) units

► Csendes (1984) : first numerical simulation

[email protected]


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double-stranded template made of 2n monomers V0

pV2n

free monomers V0

template duplication: pV2n! 2pV2n (I)

template duplication

starts

……

if concentration

of V0 is larger than

a threshold V*

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chemical reactions:

duplication initiation

duplication propagation

final step

ki (direct) rate constant

ki0 (inverse) rate constant

pV2n concentration of double-stranded template

ki >> ki0

(pV2n¢ pVi) concentration of intermediate states

template duplication: pV2n! 2pV2n (II)

[email protected]


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chemical reactions:

Ai concentration of ith reagent

ki (direct) rate constant

ki>> ki0

ki0 (inverse) rate constant

metabolism, autocatalytic cycle : A1! 2A1

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chemical reactions:

T concentration of membrane molecules

T0 and T* concentration of

precursor of membrane molecules

ki (direct) rate constant

ki0 (inverse) rate constant

ki>>ki0

membrane growth

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cell surface growth

balance equation for free monomers

balance equation for R reagent

kinetic differential equations

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division

growth

growth

size

time

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

the original model : division (I)

►standard assumption: (Gánti, Csendes, Fernando & Di Paolo (2004))

when growing the Chemoton always keep a spherical shape

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at the division all concentrations increase (by a factor )

concentration generic ith reagent

(sphere hypothesis)

immediately after division

immediately before division

(doubling hypothesis)

(halving hypothesis)

the original model : division (II)

► remark: (Munteanu & Solé (2004))

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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

take care of the shape (I)

► we observe that the previous remark can be applied to include

the volume growth in the kinetic differential equations :

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when growing the Chemoton changes its shape passing from a

sphere to a sand-glass (eight shaped body), through a peanut.

shape

division

growth

growth

growth

growth

growth

time

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

take care of the shape (II)

►observations of real cells and their division process,

i.e. experiments, support the following working hypothesis:

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► the model depends on several parameters

(for instance )

polymer

membrane

model analysis

& it is high dimensional:5+2+ 4+2n

thus numerical simulations can help to understand its behaviour

… but …

What are we looking for?

Which are the “interesting” dynamics?

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A1(t)

S(t)

t

t

“The replication

Period”

TCi

let TCi be time interval between two

successive divisions at the ith generation

(ith replication time)

ith generation

regular behaviour

►”regular” behaviour: cell cycles repeat periodically

thus each generation starts with the same amount of internal materials

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S(t)

A1(t)

t

t

no replication period

can be defined

TCi

ith generation

non-regular behaviour

►”non-regular” behaviour:

replication times vary for each generation

each generation can start with different amount of internal materials

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we 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

regular behaviours vs parameters (I)

►determine how parameters affect the dynamics

[email protected]


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our new model

original model

TCi

TCi

V*

V*

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.

regular behaviours vs parameters (II)

[email protected]


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our new model

original model

TCi

TCi

N

N

long polymers need many free monomers V0 to duplicate themselves, thus many metabolic cycles A1! 2A1 have to be done, namely long replication period.

regular behaviours vs parameters (III)

[email protected]


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blue spot: more than one

or no replication period at all

V*

red spot: a unique

replication period

N

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

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blue spot: more than one

or no replication period at all

red spot: a unique

replication period

A1

V*

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

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work in progress

►use a more fine mathematical tool to study the stability

of a periodic orbit

in all modelsinformationis carried by thelengthof the polymer

►study family of Chemotons with different polymer lengths

& consider the previous picture (N,V*)

►introduce a divisions process where internal materials are not

equally shared in next generations & consider the previous

picture (A1,V0)

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perspectives

►use a stochastic integrator (Gillespie) & compare results

with our deterministic approach

►consider a “more realistic template” build with, at least, two

different monomers V0 and W0

, then it will be possible to

include mutations both in the length of the polymer and

in the copying fidelity

►introduce the space and consider competition for food

[email protected]


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Chemoton model:

the shape

does matter

PACE – PA’s Coordination Workshop,

Los Alamos 19-22 July 2005

Timoteo Carletti

Dipartimento di Statistica, Università Ca’ Foscari

Venezia, ITALIA

[email protected]

FP6

EU


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[email protected]


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