Timoteo Carletti PACE – PA’s Coordination Workshop, Los Alamos July 2005 Dipartimento di Statistica, Università Ca’ Foscari Venezia, ITALIA FP6 EU

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 and separates it from environment. It allows nutriment and waste material to pass through. the original model ► Gánti (1971) : ► Csendes (1984) : first numerical simulation once the membrane doubled its initial size the Chemoton halves into two equal (smaller) units The metabolic chemical system transforms external energetically high materials into internal materials needed to grow and to duplicate templates 2) metabolism 3) information The double-stranded template (polymer) is the information carrier. It can duplicate itself if enough free monomers are available The Chemoton

template duplication: pV 2n ! 2 pV 2n (I) free monomers V 0 double-stranded template made of 2n monomers V 0 pV 2n template duplication starts if concentration of V 0 is larger than a threshold V* … …

template duplication: pV 2n ! 2 pV 2n (II) chemical reactions: duplication initiation duplication propagation final step pV 2n concentration of double-stranded template k i (direct) rate constant k i 0 (inverse) rate constant (pV 2n ¢ pV i ) concentration of intermediate states k i >> k i 0

metabolism, autocatalytic cycle : A 1 ! 2A 1 chemical reactions: A i concentration of i th reagent k i (direct) rate constant k i 0 (inverse) rate constant k i >> k i 0

membrane growth chemical reactions: T concentration of membrane molecules k i (direct) rate constant k i 0 (inverse) rate constant T 0 and T * concentration of precursor of membrane molecules k i >>k i 0

kinetic differential equations cell surface growth balance equation for free monomers balance equation for R reagent

time size growth division 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

the original model : division (II) ► remark: (Munteanu & Solé (2004)) at the division all concentrations increase (by a factor ) (sphere hypothesis) concentration generic i th reagent immediately before division (doubling hypothesis) immediately after division (halving hypothesis)

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 : the kinetic differential equation for the generic concentration c i 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 (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 division time shape 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

model analysis & it is high dimensional: n ► 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 …

regular behaviour t S(t) t A 1 (t) ►”regular” behaviour: cell cycles repeat periodically thus each generation starts with the same amount of internal materials let  T C i be time interval between two successive divisions at the i th generation (i th replication time) TCiTCi i th generation “The replication Period”

non-regular behaviour ►”non-regular” behaviour: replication times vary for each generation each generation can start with different amount of internal materials TCiTCi i th generation t S(t) A 1 (t) t no replication period can be defined

regular behaviours vs parameters (I) ►determine how parameters affect the dynamics we fix two parameters betweenand we study the dependence of the replication time on the third free parameter TCiTCi 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 (II) TCiTCi V*V* our new model TCiTCi V*V* original model high values of V * implies that polymerization (and thus all the growth process) can start only after many metabolic cycles A 1 ! 2A 1 (to produce enough V 0 ), 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 (III) N TCiTCi original model TCiTCi N our new model long polymers need many free monomers V 0 to duplicate themselves, thus many metabolic cycles A 1 ! 2A 1 have to be done, namely long replication period.

V*V* 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 blue spot: more than one or no replication period at all red spot: a unique replication period

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 A 1 and V 0 blue spot: more than one or no replication period at all red spot: a unique replication period A1A1 V*V*

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 (A 1,V 0 ) ►study family of Chemotons with different polymer lengths & consider the previous picture (N,V * )

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 V 0 and W 0 ►introduce the space and consider competition for food

Timoteo Carletti PACE – PA’s Coordination Workshop, Los Alamos July 2005 Dipartimento di Statistica, Università Ca’ Foscari Venezia, ITALIA FP6 EU