Tumors as complex systems

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

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

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

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

5. M(t) = number of cells at time t classical growth equations include Verhulst dM(t)/dt = aM(t) - bM(t)2 M(0) = M0 Gomperz dM/dt = q(t)M(t) dq(t)/dt = -a M(0) = M0, q(0)=q0 the growth curves are similar, Gomperz fits better in vivo data a population dynamics model:growth of normal cells Centro Ricerche Ambientali Montecatini

6. in cell cultures the limitations to cell growth become apparent only after a certain density dM/dt = [Gs(M)-w]M G, w are constants s(M) can be a piecewise linear function; s(M) = 1 (0<Ms2) 1-b(M-s2) (s2<Ms3) 0 (s3<M) cell growth in culture plates s M Centro Ricerche Ambientali Montecatini

7. carcinogenesis is a multi-step process Centro Ricerche Ambientali Montecatini

8. the role of the carcinogen • carcinogenesis is a multi-step process • the cell cultures which are in use have already undergone some of the genetic changes leading to cancer • what does the carcinogen do? • it is believed that it does not directly provide the “final push” • if 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 • the carcinogen is likely to induce some (inheritable) change in a fraction of the seeded cells • some of these “activated” cells undergo a further change during the culture period, leading them to malignancy Centro Ricerche Ambientali Montecatini

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

10. population dynamics model equations • 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)-w]B • dA/dt = [Gs(A+B)-w-p]A • dF/dt = p’(dA/dt)+ =p’Gs(A+B)A • comments: p represents the sum of • the disappearance of A due to transformation, i.e. p’Gs(A+B)A • the extra death term for activated cells wrt to normal cells; • experimentally, F grows slowly with M0; if A cells had the same dynamics as B cells, the dependence of F upon M0 would be flat, if A were more resistant it would be decresing: therefore p>0 • the study starts at t=0, when the carcinogen has been washed away and cells have received their nutrients • details of the culture method are not taken into account Centro Ricerche Ambientali Montecatini

11. behaviour of the population dynamics model • analytical results for (A,B) • the state (0,0) is a fixed point, but an unstable one (in the intersting case Gs(M0) > w+p) • no fixed point with A#0, B#0 can exist • if B0=0, the final state is of the form (A,, 0); if A0=0, the final state is of the form (0,B ) • if A0#0 and B0#0 (the interesting case) then the final state is (0,B ) and • y(t) = A(t)/B(t) = y0e-pt • experiments never reach the asymptotic state • approximate dynamics • dM/dt = [Gs(M)-w]M - pA  [Gs(M)-w]M • this equation can be easily integrated • linear until M(t*)=s2 • Verhulst after t* Centro Ricerche Ambientali Montecatini

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13. let, as usual, y=A/B; then A=zM, where z=y/(1+y); • if y is small, z  y and • A  yM = y0e-ptM(t) • the total number of foci formed is • F(t) = F0 + 0t p’Gs(M(t))A(t)dt • the integral can be broken into two pieces, from 0 to t* and from t* to t : • F(t) = F0 + DF(t*,0) + DF(t,t*) Centro Ricerche Ambientali Montecatini

14. power law dependenncy of DF(T*,0) upon M0 Centro Ricerche Ambientali Montecatini

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17. DF can be written as above, with f(s) independent of M0 and decreasing exponentially as e-ps; therefore vanishes for t>>t* (i.e. t-t* >>1/p) Centro Ricerche Ambientali Montecatini

18. therefore also DF(t,t*) depends upon M0 with the same power law as DF(t*,0) • F = cost*M0p/(G-w) • 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>>w • Fernandez 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 • there is a reasonable agreement between model and data Centro Ricerche Ambientali Montecatini

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20. but warning: direct observations show that cells grow in approximately circular clusters around the initial seed • due to contact inhibition, the cells in the interior of the cluster do not reproduce (unless they have already undergone transformation!) • therefore the effective exponent for the growth of cells is smaller than one and changes in time • dB/dt = [Gs(A+B)-w]Bn • dA/dt = [Gs(A+B)-w-p]An • numerical simulations show that dF/dM0 is positive if n is close to 1, but that it becomes negative if n takes values which are slightly smaller than one Centro Ricerche Ambientali Montecatini

21. effective exponent: analytical study • let us consider what happens to y=A/B at the beginning of the experiment • most foci are formed before confluence • a necessary condition for dF/dM0>0 is that dy/dt <0 in the initial phase • compare two experiments, say one with 1000 initial cells, the other with 5000, with the same y0=A0/B0 • at time T the cells of the first experiment become 5000; • if dy/dt > 0 in [0,T], then y(T) > y0, so more foci will be formed in the first experiment (from 5000 to confluence) • moreover some foci have been formed in [0,T] • therefore F in the second experiment is smaller than in the first, i.e. dF/dM0<0 • let us then study the initial value dy/dt Centro Ricerche Ambientali Montecatini

22. if M<s2 and therefore s(M)=1: dy/dt may be positive if n<1 and y<1 (remember that G>p) if s2<M<s3: also in this case dy/dt may be positive! Centro Ricerche Ambientali Montecatini

23. estimating the critical n • close to t=0, • dy/dt < 0 <=> (G-w)(1-y1-n) < p • which implies that • n < nth = 1 + [ln(p/(G-w))]/lny0 • with w<<G, p/G = 0.3 and y0=0.1, nth = 0.87 • in agreement with simulations • slightly changing the exponent changes the shape of the function • so the agreement between the population dynamics model and the data seems fortuitous Centro Ricerche Ambientali Montecatini

24. modelling cluster growth with ODE • a possible approach is that of writnig down a set of equations which explicitly contains the clusters • let us suppose that in the beginning there are A0 clusters of A cells, and B0 clusters of B cells; each of these clusters is composed by a single cell • 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 • dA/dt proportional to A0f(A/A0) • dB/dt proportional to B0f(B/B0) Centro Ricerche Ambientali Montecatini

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26. the exact shape of the dependency of F upon M0 is complicated to interpret • a robust feature, which persists over a wide range of “reasonable” parameter values, is that F is very weakly affected by M0, i.e. dF/dM0 0 • for example, in a typical simulation, varying M0 by two orders (from 500 to 64000) led to values for the number of new A’s between 5000 and 10000 • try a different modelling approach which directly takes into account the local interactions between different cell types Centro Ricerche Ambientali Montecatini

27. the space is divided into a discrete set of cells (or lattice sites) the time evolves in discrete steps, equally spaced one or more state variables, belonging to a finite set (e.g. {0,1}), are associated to each cell a topology is defined 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 cellular automata Centro Ricerche Ambientali Montecatini

28. how a CA evolves • discrete time steps t, t+1, t+2 … • let xi(t) be the state of the i-th cell at time t • let Ni be the (fixed) neighbourhood of the i-th cell • xi(t+1) = F({xk(t)|k  Ni}) • the evolution law is the same for every cell • (although parameters may vary in generalized CA) • local evolution: the future state depends only upon interactions with the neighbours • let X(t) = [x1(t), x2(t), …] be the state of the whole automaton • X(t+1) = F(X(t)): the CA rule defines a dynamical system • i.e. a trajectory in state space Centro Ricerche Ambientali Montecatini

29. CA story • introduced by von Neumann • in his quest for the logical features of self-reproducing systems • for a long time they remained a mathematical curiosity • the “Game of Life” • in the 80’s they were applied to • the simulation of physical systems (Toffoli, FHP) • the study of complex systems (Wolfram) • artificial life (Langton) • parallel computation • their applications are growing • simulation of the remediation of contaminated sites • coffee percolation • immune system simulation • traffic simulation • image processing Centro Ricerche Ambientali Montecatini

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

31. the CA model (2) • a cell which attempts reproduction will succeed only if its offspring can occupy a free CA site • a two-step procedure • first, 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 • second, for each empty cell where at least two offsprings tend to be placed, a stochastic choice of the parent is performed • the procedure is iterated to allow reproduction, if further free cells are available, of those cells which have lost the previous competition for parentship • each time a new A cell is generated, it has a fixed probability of becoming a T cell • it is believed that genetic changes are more likely to take place during cell reproduction Centro Ricerche Ambientali Montecatini

32. model variants • some observations indicate limited mobility of newborn mouse fibroblasts • 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 • in this way crowding effects are delayed • a further variant requires two generations to produce a fully transformed cell • DNA damage is likely to take place at reproduction time • first a single DNA strand is damaged in one of the two daughter cells; this cell has both a correct and a damaged strand: • one of the two offsprings of such a cell is “fully damaged” • more space is needed for transformation, as only one out of four offsprings is transformed Centro Ricerche Ambientali Montecatini

33. automaton square grid 400*400 10 run for each set of parameter values (initial conditions at random) simulations stop after 80 generations initial conditions: the simulations start after the carcinogen has been washed away M  A+B; M0 between 160 and 32000 y  A/M; y0 between 0.025 and 1 model probability to attempt reproduction at each time step for B cells =1, for A ranging between 0.7 and 0.1 probability that B type cells die at each time step = 0 probability that A type cells die at each time step between 0 and 0.3 probability of transition B to A, without carcinogen = 0 probability of repair, from A back to B = 0 probability of transformation from “newly generated A” to T = 10-3 parameters Centro Ricerche Ambientali Montecatini

34. andamento nel tempo delle popolazioni cellulari Centro Ricerche Ambientali Montecatini

35. dependency upon A0 Centro Ricerche Ambientali Montecatini

36. Tfin vs. M0 (single-cell seeds) Centro Ricerche Ambientali Montecatini

37. using the number of new A’s instead of Tfin Centro Ricerche Ambientali Montecatini

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

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

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

41. initial conditions • experimental data concerning cells treated with carcinogens, albeit noisy, show an increase of F with growing M0 • the CA model provides an approximately flat diagram, similar to that of VEE models • the initial conditions in our model were based upon random placement of B-type or A-type seeds on the CA sites • 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) • 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 Centro Ricerche Ambientali Montecatini

42. log Tfin vs. log M0 (coupled nuclei); initial slope 0.4 Centro Ricerche Ambientali Montecatini

43. total number of new A cells instead of Tfin Centro Ricerche Ambientali Montecatini

44. further tests • coupled nuclei account for the gross features of the dependency of F upon M0, in the case of basic tests with chemical carcinogens • further tests would be possible if we could find experimental cases related to the case of single seeds • INIT cells have been identified by Mordan et al, and seeded together with C3H10T1/2 • re-seeding: confluent cells are detached and re-seeded • isolated A-type cells should be the new initial condition • however, tranformed cells may be seeded as well • also Kennedy, Little and co-workers performed extensive re-seeding experiments with cells which had been activated by X-rays Centro Ricerche Ambientali Montecatini

45. INIT cells seeded with C3H10T1/2 Centro Ricerche Ambientali Montecatini

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

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

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