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Centro Ricerche Ambientali Montecatini Tumors as complex systems Roberto Serra Centro Ricerche Ambientali Montecatini rserra@cramont.it n introduction n in vitro tests n models of foci formation n differential equation models n comparison with experimental data n CA models of foci formation n indication for further tests

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Centro Ricerche Ambientali Montecatini determining whether a substance is carcinogen n epidemiological studies on humans ndifficulties in the formation of different groups; “small” effects may pass unnoticed; very important as it concerns humans directly n molecular biology studies on humans or animals navailable only in a small number of cases; very informative n laboratory tests on animals nhigh doses, high costs; relationship between animal and human; ethical issues n in vitro tests ndifferent doses; low cost; amenable to detailed studies at a molecular level; extrapolation to humans n the contribution of molecular biology will further improve the use of in-vitro tests

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Centro Ricerche Ambientali Montecatini a typical in-vitro test n a definite cell line is used (e.g. Balb/c 3T3 from mouse) n a given number of cells are initially plated on a Petri dish ne.g. 30.000 n the cells are allowed to adapt to the new environment for some days n the cells are then exposed to the suspect carcinogen ne.g. for 2 days n the suspect carcinogen is washed away and the cells are cultured for some weeks (changing the culture medium every 2-3 days) nafter about two weeks the normal cells have reached confluenced, i.e. have built a monolayer which covers the bottom of the plate

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Centro Ricerche Ambientali Montecatini comparing carcinogens n at the end of the experiments, cells are stained n transformed cells - unlike normal cells - do not feel contact inhibition; they give rise to transformation foci (dark spots) n foci are counted nconsiderable care must be taken to identify proper foci n the number of foci is compared with those obtained using well known carcinogens and using innocuous substances

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Centro Ricerche Ambientali Montecatini a population dynamics model:growth of normal cells n M(t) = number of cells at time t n classical growth equations include Verhulst dM(t)/dt = aM(t) - bM(t) 2 M(0) = M 0 Gomperz dM/dt = q(t)M(t) dq(t)/dt = -a M(0) = M 0, q(0)=q 0 n the growth curves are similar, Gomperz fits better in vivo data

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Centro Ricerche Ambientali Montecatini cell growth in culture plates n in cell cultures the limitations to cell growth become apparent only after a certain density dM/dt = [Gs(M)- ]M G, are constants n s(M) can be a piecewise linear function; s(M) = n 1 (0

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Centro Ricerche Ambientali Montecatini carcinogenesis is a multi- step process

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Centro Ricerche Ambientali Montecatini the role of the carcinogen n carcinogenesis is a multi-step process n the cell cultures which are in use have already undergone some of the genetic changes leading to cancer n what does the carcinogen do? nit is believed that it does not directly provide the “final push” nif the carcinogen were directly responsible of the final genetic change then, as long as the number of transformation foci F is low, we would expect a linear growth of F with the number of initial cells - not observed n the carcinogen is likely to induce some (inheritable) change in a fraction of the seeded cells n some of these “activated” cells undergo a further change during the culture period, leading them to malignancy

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Centro Ricerche Ambientali Montecatini a minimal model n carcinogenesis requires two steps n some normal (B-type) cells become activated (A-type) during the exposure period n A-type cells are phenotipically indistinguishable from normal cells nbut cell repair mechanisms may lead them to death with higher probability n A-type cells may undergo a further change, leading them to fully transformed cells (T-type) B -> A -> T n each newly formed T cell gives rise to a focus nF = number of transformations A->T (apart from coalescence of nearby foci)

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Centro Ricerche Ambientali Montecatini population dynamics model equations n let B = number of B cells per plate, A = number of A cells per plate, M=A+B, F = number of foci per plate dB/dt = [Gs(A+B)- ]B dA/dt = [Gs(A+B)- -p]A n dF/dt = p’(dA/dt) + =p’Gs(A+B)A n comments: p represents the sum of nthe disappearance of A due to transformation, i.e. p’Gs(A+B)A nthe extra death term for activated cells wrt to normal cells; nexperimentally, F grows slowly with M 0 ; if A cells had the same dynamics as B cells, the dependence of F upon M 0 would be flat, if A were more resistant it would be decresing: therefore p>0 n the study starts at t=0, when the carcinogen has been washed away and cells have received their nutrients n details of the culture method are not taken into account

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Centro Ricerche Ambientali Montecatini behaviour of the population dynamics model n analytical results for (A,B) the state (0,0) is a fixed point, but an unstable one (in the intersting case Gs(M 0 ) > +p) nno fixed point with A#0, B#0 can exist nif B 0 =0, the final state is of the form (A ,, 0); if A 0 =0, the final state is of the form (0,B ) nif A 0 #0 and B 0 #0 (the interesting case) then the final state is (0,B ) and n y(t) = A(t)/B(t) = y 0 e -pt n experiments never reach the asymptotic state n approximate dynamics dM/dt = [Gs(M)- ]M - pA [Gs(M)- ]M n this equation can be easily integrated nlinear until M(t*)=s 2 nVerhulst after t*

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Centro Ricerche Ambientali Montecatini

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n let, as usual, y=A/B; then A=zM, where z=y/(1+y); n if y is small, z y and n A yM = y 0 e -pt M(t) n the total number of foci formed is n F(t) = F 0 + 0 t p’Gs(M(t))A(t)dt n the integral can be broken into two pieces, from 0 to t* and from t* to t : F(t) = F 0 + F(t*,0) + F(t,t*)

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Centro Ricerche Ambientali Montecatini power law dependenncy of F(T*,0) upon M 0

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F can be written as above, with (s) independent of M 0 and decreasing exponentially as e -ps ; therefore vanishes for t>>t* (i.e. t-t* >>1/p)

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Centro Ricerche Ambientali Montecatini therefore also F(t,t*) depends upon M 0 with the same power law as F(t*,0) F = cost*M 0 p/(G- ) n this result is in agreement with a previous, crude model by Fernandez et al we may suppose that the death rate in cell cultures is much smaller than the maximum possible growth rate, i.e. G>> nFernandez et al estimated that the “repair rate” is of the order of about 30% per generation; this provides an order of magnitude guess for p/G n there is a reasonable agreement between model and data

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n but warning: direct observations show that cells grow in approximately circular clusters around the initial seed ndue to contact inhibition, the cells in the interior of the cluster do not reproduce (unless they have already undergone transformation!) ntherefore the effective exponent for the growth of cells is smaller than one and changes in time dB/dt = [Gs(A+B)- ]B dA/dt = [Gs(A+B)- -p]A numerical simulations show that dF/dM 0 is positive if is close to 1, but that it becomes negative if takes values which are slightly smaller than one

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Centro Ricerche Ambientali Montecatini effective exponent: analytical study n let us consider what happens to y=A/B at the beginning of the experiment nmost foci are formed before confluence n a necessary condition for dF/dM 0 >0 is that dy/dt <0 in the initial phase ncompare two experiments, say one with 1000 initial cells, the other with 5000, with the same y 0 =A 0 /B 0 nat time T the cells of the first experiment become 5000; nif dy/dt > 0 in [0,T], then y(T) > y 0, so more foci will be formed in the first experiment (from 5000 to confluence) nmoreover some foci have been formed in [0,T] ntherefore F in the second experiment is smaller than in the first, i.e. dF/dM 0 <0 n let us then study the initial value dy/dt

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Centro Ricerche Ambientali Montecatini if M~~
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Centro Ricerche Ambientali Montecatini estimating the critical estimating the critical n close to t=0, dy/dt (G- )(1-y 1- ) < p n which implies that < th = 1 + [ln(p/(G- ))]/lny 0 with <

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Centro Ricerche Ambientali Montecatini modelling cluster growth with ODE n a possible approach is that of writnig down a set of equations which explicitly contains the clusters n let us suppose that in the beginning there are A 0 clusters of A cells, and B 0 clusters of B cells; each of these clusters is composed by a single cell n the cell growth processes involve only cells at the surface, so, without taking into account the inhibition by neighbouring clusters, the equations would be of the type n dA/dt proportional to A 0 f(A/A 0 ) n dB/dt proportional to B 0 f(B/B 0 )

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n the exact shape of the dependency of F upon M 0 is complicated to interpret n a robust feature, which persists over a wide range of “reasonable” parameter values, is that F is very weakly affected by M 0, i.e. dF/dM 0 0 nfor example, in a typical simulation, varying M 0 by two orders (from 500 to 64000) led to values for the number of new A’s between 5000 and 10000 n try a different modelling approach which directly takes into account the local interactions between different cell types

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Centro Ricerche Ambientali Montecatini cellular automata n the space is divided into a discrete set of cells (or lattice sites) n the time evolves in discrete steps, equally spaced n one or more state variables, belonging to a finite set (e.g. {0,1}), are associated to each cell n a topology is defined n the value of the state veriable of a given cell at time t+1 depends only upon the values, at time t, of the variables of the cells belonging to its neighbourhood

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Centro Ricerche Ambientali Montecatini how a CA evolves n discrete time steps t, t+1, t+2 … n let x i (t) be the state of the i-th cell at time t n let N i be the (fixed) neighbourhood of the i-th cell n x i (t+1) = F({x k (t)|k N i }) n the evolution law is the same for every cell n(although parameters may vary in generalized CA) n local evolution: the future state depends only upon interactions with the neighbours n let X(t) = [x 1 (t), x 2 (t), …] be the state of the whole automaton n X(t+1) = F(X(t)): the CA rule defines a dynamical system ni.e. a trajectory in state space

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Centro Ricerche Ambientali Montecatini CA story n introduced by von Neumann nin his quest for the logical features of self-reproducing systems n for a long time they remained a mathematical curiosity nthe “Game of Life” n in the 80’s they were applied to nthe simulation of physical systems (Toffoli, FHP) nthe study of complex systems (Wolfram) nartificial life (Langton) nparallel computation n their applications are growing nsimulation of the remediation of contaminated sites ncoffee percolation nimmune system simulation ntraffic simulation nimage processing

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Centro Ricerche Ambientali Montecatini the CA model of cell cultures n the model describes the growth of normal and activated cells and the birth of new foci nit does not describe the growth of foci n monolayer growth of normal cells => two-dimensional CA nsquare topology, 9-membered neighbourhood n the state space is the cartesian product [biological state] X [reproductive state] n the set of biological states of each CA site is {E, B, A, T} nempty, or occupied by A-type, B-type, or T-type cells n if the state is B or A, a boolean variable determines whether the biological cell will attempt reproduction

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Centro Ricerche Ambientali Montecatini the CA model (2) n a cell which attempts reproduction will succeed only if its offspring can occupy a free CA site n a two-step procedure nfirst, for each CA cell which is either in A or B state, and which attempts reproduction, and which has at least an empty neighbour, a tentative location of the offspring is determined nsecond, for each empty cell where at least two offsprings tend to be placed, a stochastic choice of the parent is performed nthe procedure is iterated to allow reproduction, if further free cells are available, of those cells which have lost the previous competition for parentship n each time a new A cell is generated, it has a fixed probability of becoming a T cell nit is believed that genetic changes are more likely to take place during cell reproduction

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Centro Ricerche Ambientali Montecatini model variants n some observations indicate limited mobility of newborn mouse fibroblasts n in order to simulate this phenomenon, it has been assumed that newborn cells walk away from the parents, if there is room, for two lattice spaces nin this way crowding effects are delayed n a further variant requires two generations to produce a fully transformed cell nDNA damage is likely to take place at reproduction time nfirst a single DNA strand is damaged in one of the two daughter cells; this cell has both a correct and a damaged strand: none of the two offsprings of such a cell is “fully damaged” nmore space is needed for transformation, as only one out of four offsprings is transformed

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Centro Ricerche Ambientali Montecatini parameters n automaton nsquare grid 400*400 n10 run for each set of parameter values (initial conditions at random) nsimulations stop after 80 generations n initial conditions: the simulations start after the carcinogen has been washed away nM A+B; M 0 between 160 and 32000 ny A/M; y 0 between 0.025 and 1 n model nprobability to attempt reproduction at each time step for B cells =1, for A ranging between 0.7 and 0.1 nprobability that B type cells die at each time step = 0 nprobability that A type cells die at each time step between 0 and 0.3 nprobability of transition B to A, without carcinogen = 0 nprobability of repair, from A back to B = 0 nprobability of transformation from “newly generated A” to T = 10 -3

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Centro Ricerche Ambientali Montecatini andamento nel tempo delle popolazioni cellulari

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Centro Ricerche Ambientali Montecatini dependency upon A 0

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Centro Ricerche Ambientali Montecatini T fin vs. M 0 (single-cell seeds)

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Centro Ricerche Ambientali Montecatini using the number of new A’s instead of T fin

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Centro Ricerche Ambientali Montecatini cells: C3H10T1/2; carcinogen: MCA; slope 0.4

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Centro Ricerche Ambientali Montecatini cells: CH310T1/2 carcinogen: BP; slope 0.33

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Centro Ricerche Ambientali Montecatini cells: C3H10T1/2 carcinogen: MCA; slope 0.4

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Centro Ricerche Ambientali Montecatini initial conditions n experimental data concerning cells treated with carcinogens, albeit noisy, show an increase of F with growing M 0 n the CA model provides an approximately flat diagram, similar to that of VEE models n the initial conditions in our model were based upon random placement of B-type or A-type seeds on the CA sites n but if the carcinogen acts by converting the offspring of a B-type into an A-type cell, then each initial A-type is close to at least one B-type (its parent cell) n therefore further experiments were performed using “coupled nuclei”, where the initial seeds are composed either by two B cells or by a B and an A cell

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Centro Ricerche Ambientali Montecatini log T fin vs. log M 0 (coupled nuclei); initial slope 0.4

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Centro Ricerche Ambientali Montecatini total number of new A cells instead of T fin

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Centro Ricerche Ambientali Montecatini further tests n coupled nuclei account for the gross features of the dependency of F upon M 0, in the case of basic tests with chemical carcinogens n further tests would be possible if we could find experimental cases related to the case of single seeds n INIT cells have been identified by Mordan et al, and seeded together with C3H10T1/2 n re-seeding: confluent cells are detached and re-seeded nisolated A-type cells should be the new initial condition nhowever, tranformed cells may be seeded as well n also Kennedy, Little and co-workers performed extensive re-seeding experiments with cells which had been activated by X-rays

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Centro Ricerche Ambientali Montecatini INIT cells seeded with C3H10T1/2

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Centro Ricerche Ambientali Montecatini re-seeding with a high number of foci

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Centro Ricerche Ambientali Montecatini conclusions n CA models allow to describe, in a natural way, the processes related to cell replication n building the model sharpens the analysis of the phenomena involved in in-vitro tests n the model displays robust behaviours which can be associated to experimental observations n the model suggests further experiments as well as re- interpretation of old ones n in-vitro models can be tested more accurately than in vivo models n extrapolation to in-vivo cases is not obvious - but some features are likely to be the same

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Centro Ricerche Ambientali Montecatini Acknowledgments n model development with Marco Villani (CRA, Ravenna) and Annamaria Colacci (INRC-IST, Bologna) n contributions from several colleagues at CRA and IST n very useful discussions with Sandro Grilli (University of Bologna) and David Lane (University of Modena and Santa Fe Institute)

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