Mathematical Models of Cancer. People. Oncogenes. Tumor suppressor genes

Transcription

1 Mathematical Models of Cancer Franziska Michor Harvard University People Martin Nowak (Princeton) Yoh Iwasa (Kysh) Natalia Komarova (Princeton) Steven Frank (Irvine) Christoph Lengaer (Johns Hopkins) ert Vogelstein (Johns Hopkins) Oncogenes Tmor sppressor genes one copy of the gene gets mtated in a specific way the cell has an increased reprodctive rate both copies of the gene mst be inactivated the cell has an increased reprodctive rate Normal cells are diploid Most cancer cells are aneploid

2 Genetic instability genes Mitosis in the frican blood lily Haemanths katherinae prodce proteins that maintain the genomic material of a cell. If sch genes are mtated, then the rate of accmlating frther mtations can increase. Chromosomal instability () Increased rate of losing (arms of) chromosomes. Loss of heterozygosity, LOH Colon cancer Loss of the maternal or paternal copy of a gene. normal adenoma cancer metastasis PC RS.. p53 Colon cancer Colon cancer normal adenoma cancer metastasis normal adenoma cancer metastasis PC RS.. p53 PC RS.. p53 somewhere: Does occr before PC?

3 The biggest qestion Is genetic instability necessary for the somatic evoltion of most cancers? Is genetic instability an early event, a driving force of cancer progression? David von Hansemann reports abnormal mitoses in cancer cells. 89 Mtation and Cancer 94 Theodor overi proposes that chromosomal abnormalities are fndamental to cancers. Ernest Tyzzer applies the term somatic mtation to cancer events Hermann J. Mller proposes that cancer reslts when a single cell receives mltiple mtations. lfred Kndson discovers two hits in retinoblastoma. Mtation and Cancer Discovery of tmor sppressor genes. Kndson s two hit hypothesis Discovery of oncogenes. Eric Fearon and ert Vogelstein find specific seqential mtations in carcinoma. Tmor sppressor genes are inactivated by hits tmor sppressor gene These cells have normal reprodctive rates. LOH p p nd mtation 3

4 These cells have increased reprodctive rates. p Colon cancer arises in a crypt crypt consists of -4 cells. poptosis on top of crypt 36 hors small nmber of stem cells replenishes the whole crypt. 7 The colon contains crypts. Colon cancer arises in a crypt The effective poplation size of a single crypt is small. N stochastic description is necessary. pproximation of homogeneos crypts t any one, all cells in a crypt have either, or inactivated copies of PC. This is a good approximation, if << / N. 7 N Rate of evoltion N ( + p)...mtation rate p...rate of LOH N...compartment size N ρ( + p ) ρ...probability of fixation If X X t << : X ( t) X ( t) t ( t) N( + p hits... t X X ) t X& = X X& = X N( + p) X X& = N( + p ) X Kolmogorov forward eqation 4

5 Chromosomal instability, mtations + p + p p case very LOH r c Somatic fitness r c + p + p a ar r < a > X Homogenos compartments ) X X ( p N + X X X N ( + p) hits Y Y Y Y t Y t c c Y t c Y Y Y X N( + p Y t c ) t X Y X X N ( + p) hits Y Y N is er if > ( + p ) Tnneling N( + p) ρ( rp N ρ (r) N ρ( ar) r N pρ( 5

6 Tnneling Tnneling fixation fixation state of compartment state of compartment next mtation state of compartment Tnneling next mtation and fixation Netral X N ( + p ) ρ( N c pρ ( X N( + p) ρ( c pρ( N X Y Y Y N pρ( N pρ( Komarova et al., 3 Costly X X N ( + p ) ρ( N( + p) ρ( N ρ(r) N ρ(r) r rp N pρ ( ar ) Nc ρ( ar) / X Y Y Y N p ( r Nc pρ( ar) Komarova et al., 3 Timing Whether a mtation occrs before the inactivation of the first depends on parameters. Natral choices of parameters sggest that occrs before. 6

7 Two tmor sppressor genes Two tmor sppressor genes +/ / +/ 4 hits +/ / Tnnel +/ 3 hits +/ / +/ +/ / +/ 3 hits 3 hits wins it depends on parameters Many genes The effect of tisse architectre on cancer initiation With the addition of each new, it is more likely that occrs before the inactivation of the first. Somatic selection advantageos: r > Small compartments protect against tmor sppressor genes disadvantageos: small compartments accelerate this step r < bt favor. 7

8 Tisses evolved to minimize the risk of developing cancer t the optimm compartment size, the contribtion of cells to the total risk can be sbstantial. Smmary- In many cases of tmorigenesis, mtations might precede the inactivation of the first gene is compatible with Kndson s two hit hypothesis Smmary- Stochastic elimination of cancer cells Somatic selection works for and against cancer Small compartments protect against mtations in genes and oncogenes bt favor There is an optimm compartment size diff. cells stem cells Stochastic elimination of cancer cells diff. cells diff. cells stem cells stem cells 8

9 or diff. cells stem cells or or p x diff. cells stem cells p y p x 9

10 Optimm abndance of stem cells depends on the somatic fitness of the mtated cells and the mtation rates in stem cells and differentiated cells. Tisse architectre with stem cells evolved to wash ot cells with advantageos mtations. It has no conseqence for netral mtations. The rate of netral evoltion is independent of the poplation size (Kimra 968). Stack design Stack design Restriction Smmary- Tisse design can redce the rate of somatic evoltion that leads to cancer Stem cells can prevent the accmlation of mtations in differentiated cells (wash ot) Stack design garanteeing wash ot can frther redce the risk of neoplastic growth

11 Smmary- dvantageos mtations are best contained in small compartments that allow wash ot Netral mtations are naffected by tisse design People Martin Nowak (Princeton) Yoh Iwasa (Kysh) Natalia Komarova (Princeton) Steven Frank (Irvine) Christoph Lengaer (Johns Hopkins) ert Vogelstein (Johns Hopkins) Stochastic elimination of cancer cells Stochastic elimination of cancer cells x& y& x& y& = r ( ) x = rx = dx = dx + s( v) y + svy dx + rα x dx Ψ x Φ y + sα y Φ y Ψ x Ψ = ( rx + rα x dx ) / x x = x + x Φ = ( dx + sy + sα y ) / y y = y + y z = x x and w y / y / = z& = r( z) + r( α )( z) + ξ ( t) w& = sv( w) + dx / y( z w) + s( α ) w( w) + ξ ( t) = rz( z) / x = sw( w) / y ξ x ( t), ξ ( t)... white noise y x y Stochastic elimination of cancer cells wash ot condition : α ri xi = α α α discarding rate : c = r x α optimal stack design : r x = c α c ri xi = α α n n n i i r x for i =,,..., n for i =,,..., n