Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Review: Cancer Modeling Natalia Komarova (University of California - Irvine) Plan • Introduction: The concept of somatic evolution • Loss-of-function and gain-of-function mutations • Mass-action modeling • Spatial modeling • Hierarchical modeling • Consequences from the point of view of tissue architecture and homeostatic control Darwinian evolution (of species) • Time-scale: hundreds of millions of years • Organisms reproduce and die in an environment with shared resources Darwinian evolution (of species) • Time-scale: hundreds of millions of years •Organisms reproduce and die in an environment with shared resources • Inheritable germline mutations (variability) • Selection (survival of the fittest) Somatic evolution • Cells reproduce and die inside an organ of one organism • Time-scale: tens of years Somatic evolution • Cells reproduce and die inside an organ of one organism • Time-scale: tens of years • Inheritable mutations in cells’ genomes (variability) • Selection (survival of the fittest) Cancer as somatic evolution • Cells in a multicellular organism have evolved to cooperate and perform their respective functions for the good of the whole organism Cancer as somatic evolution • Cells in a multicellular organism have evolved to cooperate and perform their respective functions for the good of the whole organism • A mutant cell that “refuses” to co-operate may have a selective advantage Cancer as somatic evolution • Cells in a multicellular organism have evolved to cooperate and perform their respective functions for the good of the whole organism • A mutant cell that “refuses” to co-operate may have a selective advantage • The offspring of such a cell may spread Cancer as somatic evolution • Cells in a multicellular organism have evolved to cooperate and perform their respective functions for the good of the whole organism • A mutant cell that “refuses” to co-operate may have a selective advantage • The offspring of such a cell may spread • This is a beginning of cancer Progression to cancer Progression to cancer Constant population Progression to cancer Advantageous mutant Progression to cancer Clonal expansion Progression to cancer Saturation Progression to cancer Advantageous mutant Progression to cancer Wave of clonal expansion Genetic pathways to colon cancer (Bert Vogelstein) “Multi-stage carcinogenesis” Methodology: modeling a colony of cells • Cells can divide, mutate and die Methodology: modeling a colony of cells • Cells can divide, mutate and die • Mutations happen according to a “mutation-selection diagram”, e.g. u1 (1) u2 (r1) u4 u3 (r2) (r3) (r4) Mutation-selection network (1) u8 (r2) u8 (r3) u1 u 1 u1 u3 u8 (r4) u3 u4 (r1) u2 (r1) u5 u2 u5 (r5) u8 (r6) (r6) (r7) Common patterns in cancer progression • Gain-of-function mutations • Loss-of-function mutations Gain-of-function mutations • Oncogenes • K-Ras (colon cancer), Bcr-Abl (CML leukemia) • Increased fitness of the resulting type Wild type Oncogene u (1) (r) u 109 per cell division per gene Loss-of-function mutations • Tumor suppressor genes • APC (colon cancer), Rb (retinoblastoma), p53 (many cancers) • Neutral or disadvantageous intermediate mutants • Increased fitness of the resulting type Wild type TSP+/+ TSP+/- TSP-/- u u xx x (1) u 107 per cell division per gene copy (r<1) (R>1) Stochastic dynamics on a selection-mutation network • Given a selection-mutation diagram • Assume a constant population with a cellular turn-over • Define a stochastic birth-death process with mutations • Calculate the probability and timing of mutant generation Gain-of-function mutations Selection-mutation diagram: u (1) Fitness = 1 Fitness = r >1 (r ) Number of is i Number of is j=N-i Evolutionary selection dynamics Fitness = 1 Fitness = r >1 Evolutionary selection dynamics Fitness = 1 Fitness = r >1 Evolutionary selection dynamics Fitness = 1 Fitness = r >1 Evolutionary selection dynamics Fitness = 1 Fitness = r >1 Evolutionary selection dynamics Fitness = 1 Fitness = r >1 Evolutionary selection dynamics Start from only one cell of the second type; Suppress further mutations. What is the chance that it will take over? Fitness = 1 Fitness = r >1 Evolutionary selection dynamics Start from only one cell of the second type. What is the chance that it will take over? 1/ r 1 (r ) N 1/ r 1 Fitness = 1 Fitness = r >1 If If If If r=1 then = 1/N r<1 then < 1/N r>1 then > 1/N then = 1 r Evolutionary selection dynamics Start from zero cell of the second type. What is the expected time until the second type takes over? Fitness = 1 Fitness = r >1 Evolutionary selection dynamics Start from zero cell of the second type. What is the expected time until the second type takes over? In the case of rare mutations, u 1/ N we can show that Fitness = 1 Fitness = r >1 T 1 Nu (r ) Loss-of-function mutations u1 u (1) (r) (a) What is the probability that by time t a mutant of has been created? Assume that r 1 and a 1 1D Markov process • j is the random variable, j {0,1,..., N , E} • If j = 1,2,…,N then there are j intermediate mutants, and no double-mutants • If j=E, then there is at least one double-mutant • j=E is an absorbing state Transition probabilities Pj j 1 j Pj j 1 j Pj E j A two-step process u u1 A two-step process u u1 A two step process u u1 … … A two-step process u (1) u1 (r) (a) Number of cells Scenario 1: gets fixated first, and then a mutant of is created; time Stochastic tunneling u u1 … Stochastic tunneling u (1) u1 (r) (a) Number of cells Scenario 2: A mutant of is created before reaches fixation time The coarse-grained description R01 R12 R02 Long-lived states: x0 …“all green” x1 …“all blue” x2 …“at least one red” x0 R01 x0 R02 x0 x1 R01 x0 R12 x1 x2 R01 x0 R12 x1 Stochastic tunneling Nu Nu1 Neutral intermediate mutant R02 R02 Nu u1 | 1 r | u1 Nuu1r 1 r | 1 r | u1 R02 Disadvantageous intermediate mutant Assume that r 1 and a 1 The mass-action model is unrealistic • All cells are assumed to interact with each other, regardless of their spatial location • All cells of the same type are identical The mass-action model is unrealistic • All cells are assumed to interact with each other, regardless of their spatial location • Spatial model of cancer • All cells of the same type are identical The mass-action model is unrealistic • All cells are assumed to interact with each other, regardless of their spatial location • Spatial model of cancer • All cells of the same type are identical • Hierarchical model of cancer Spatial model of cancer • Cells are situated in the nodes of a regular, one-dimensional grid • A cell is chosen randomly for death • It can be replaced by offspring of its two nearest neighbors Spatial dynamics Spatial dynamics Spatial dynamics Spatial dynamics Spatial dynamics Spatial dynamics Spatial dynamics Spatial dynamics Spatial dynamics Gain-of-function: probability to invade • In the spatial model, the probability to invade depends on the spatial location of the initial mutation Probability of invasion Advantageous mutants, r = 1.2 10 5 Neutral mutants, r = 1 Mass-action Disadvantageous mutants, r = 0.95 Spatial Use the periodic boundary conditions Mutant island Probability to invade • For disadvantageous mutants 2r space 1 r r 1, | 1 r | 1 / N • For neutral mutants | 1 r | 1 / N • For advantageous mutants r 1, | 1 r | 1 / N 1 space N 2r space 3r 1 Loss-of-function mutations u1 u (1) (r) (a) What is the probability that by time t a mutant of has been created? Assume that r 1 and a 1 Transition probabilities j {0,1,..., N , E} No double-mutants, j intermediate cells Mass-action At least one double-mutant Space Pj j 1 j Pj j 1 Pj j 1 j Pj j 1 Pj E j Pj E j Stochastic tunneling Nu space Nu1 R02 R02 uN (9u1 ) 1/ 3 R02 (2 / 3) ; (mass act. Nu u1 ) (1 / 3) (r 1) 2 r 2 Nuu1r 3rNuu1 ; (mass act. ) 2 (r 1) 1 r Stochastic tunneling Slower Nu space Nu1 R02 R02 uN (9u1 ) 1/ 3 R02 (2 / 3) ; (mass act. Nu u1 ) (1 / 3) (r 1) 2 r 2 Nuu1r 3rNuu1 ; (mass act. ) 2 (r 1) 1 r Stochastic tunneling Slower Nu space Nu1 Faster R02 R02 uN (9u1 ) 1/ 3 R02 (2 / 3) ; (mass act. Nu u1 ) (1 / 3) (r 1) 2 r 2 Nuu1r 3rNuu1 ; (mass act. ) 2 (r 1) 1 r The mass-action model is unrealistic • All cells are assumed to interact with each other, regardless of their spatial location P• Spatial model of cancer • All cells of the same type are identical • Hierarchical model of cancer Hierarchical model of cancer Colon tissue architecture Colon tissue architecture Crypts of a colon Colon tissue architecture Crypts of a colon Cancer of epithelial tissues Gut Cells in a crypt of a colon Cancer of epithelial tissues Gut Cells in a crypt of a colon Stem cells replenish the tissue; asymmetric divisions Cancer of epithelial tissues Gut Cells in a crypt of a colon Proliferating cells divide symmetrically and differentiate Stem cells replenish the tissue; asymmetric divisions Cancer of epithelial tissues Gut Cells in a crypt of a colon Differentiated cells get shed off into the lumen Proliferating cells divide symmetrically and differentiate Stem cells replenish the tissue; asymmetric divisions Finite branching process Cellular origins of cancer Gut If a stem cell tem cell acquires a mutation, the whole crypt is transformed Cellular origins of cancer Gut If a daughter cell acquires a mutation, it will probably get washed out before a second mutation can hit Colon cancer initiation Colon cancer initiation Colon cancer initiation Colon cancer initiation Colon cancer initiation Colon cancer initiation First mutation in a daughter cell First mutation in a daughter cell First mutation in a daughter cell First mutation in a daughter cell First mutation in a daughter cell First mutation in a daughter cell First mutation in a daughter cell First mutation in a daughter cell First mutation in a daughter cell First mutation in a daughter cell First mutation in a daughter cell First mutation in a daughter cell Number of cells Two-step process and tunneling First hit in the stem cell Number of cells time Second hit in a daughter cell First hit in a daughter cell time Stochastic tunneling in a hierarchical model u Nu1 R02 R02 Nuu 1 log u1 (cf . R Nu u1 ) Stochastic tunneling in a hierarchical model The same u Nu1 R02 R02 Nuu 1 log u1 (cf . R Nu u1 ) Stochastic tunneling in a hierarchical model The same u Nu1 R02 Slower R02 Nuu 1 log u1 (cf . R Nu u1 ) The mass-action model is unrealistic • All cells are assumed to interact with each other, regardless of their spatial location P• Spatial model of cancer • All cells of the same type are identical P• Hierarchical model of cancer Comparison of the models Probability of mutant invasion for gain-of-function mutations r = 1 neutral Comparison of the models The tunneling rate (lowest rate) The tunneling and two-step regimes Production of double-mutants Population size Small Interm. mutants Large Neutral (mass-action, spatial and hierarchical) Disadvantageous (mass-action and Spatial only) All models give the same results Spatial model is the fastest Hierarchical model is the slowest Mass-action model is faster Spatial model is slower Spatial model is the fastest Production of double-mutants Population size Small Interm. mutants Large Neutral (mass-action, spatial and hierarchical) Disadvantageous (mass-action and Spatial only) All models give the same results Spatial model is the fastest Hierarchical model is the slowest Mass-action model is faster Spatial model is slower Spatial model is the fastest The definition of “small” 1000 N r=1 Spatial model is the fastest r=0.99 500 r=0.95 r=0.8 1 2 3 4 5 6 7 8 9 log 10 (u1 ) Summary • The details of population modeling are important, the simple mass-action can give wrong predictions Summary • The details of population modeling are important, the simple mass-action can give wrong predictions • Different types of homeostatic control have different consequence in the context of cancerous transformation Summary • If the tissue is organized into compartments with stem cells and daughter cells, the risk of mutations is lower than in homogeneous populations Summary • If the tissue is organized into compartments with stem cells and daughter cells, the risk of mutations is lower than in homogeneous populations • For population sizes greater than 102 cells, spatial “nearest neighbor” model yields the lowest degree of protection against cancer Summary • A more flexible homeostatic regulation mechanism with an increased cellular motility will serve as a protection against double-mutant generation Conclusions • Main concept: cancer is a highly structured evolutionary process • Main tool: stochastic processes on selection-mutation networks • We studied the dynamics of gain-offunction and loss-of-function mutations • There are many more questions in cancer research…