Timoteo Carletti

1. 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 t.carletti@sns.it FP6 EU

2. 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 t.carletti@sns.it

3. 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 t.carletti@sns.it

4. 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* t.carletti@sns.it

5. 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) t.carletti@sns.it

6. chemical reactions: Ai concentration of ith reagent ki (direct) rate constant ki>> ki0 ki0 (inverse) rate constant metabolism, autocatalytic cycle : A1! 2A1 t.carletti@sns.it

7. 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 t.carletti@sns.it

8. cell surface growth balance equation for free monomers balance equation for R reagent kinetic differential equations t.carletti@sns.it

9. 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 t.carletti@sns.it

10. 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)) t.carletti@sns.it

11. 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 : t.carletti@sns.it

12. 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: t.carletti@sns.it

13. ► 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? t.carletti@sns.it

14. 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 t.carletti@sns.it

15. 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 t.carletti@sns.it

16. 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 t.carletti@sns.it

17. 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) t.carletti@sns.it

18. 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) t.carletti@sns.it

19. 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 t.carletti@sns.it

20. 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 t.carletti@sns.it

21. 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) t.carletti@sns.it

22. 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 t.carletti@sns.it

23. 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 t.carletti@sns.it FP6 EU

24. t.carletti@sns.it