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 co-operate and perform their respective functions for the good of the whole organism

Cancer as somatic evolution Cells in a multicellular organism have evolved to co-operate 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 co-operate 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 co-operate 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 u2 u3 u4 (1) (r1) (r2) (r3) (r4)

Mutation-selection network (1) (r1) (r4) (r6) u2 u2 u5 u8 (r1) (r5) (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)

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 x x x (1) (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: Number of is i u (1) (r ) Number of is j=N-i 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 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? If r=1 then = 1/N If r<1 then < 1/N If r>1 then > 1/N If r then = 1 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? 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, we can show that Fitness = 1 Fitness = r >1

Loss-of-function mutations (1) (r) (a) What is the probability that by time t a mutant of has been created? Assume that and

1D Markov process j is the random variable, 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

A two-step process

A two-step process

A two step process … …

A two-step process (1) (r) (a) Scenario 1: gets fixated first, and then a mutant of is created; Number of cells time

Stochastic tunneling …

Stochastic tunneling (1) (r) (a) Scenario 2: A mutant of is created before reaches fixation Number of cells time

The coarse-grained description Long-lived states: x0 …“all green” x1 …“all blue” x2 …“at least one red”

Stochastic tunneling Assume that and Neutral intermediate mutant Disadvantageous intermediate mutant Assume that and

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 Neutral mutants, r = 1 Advantageous mutants, r = 1.2 Mass-action Disadvantageous mutants, r = 0.95 Spatial

Use the periodic boundary conditions Mutant island

Probability to invade For disadvantageous mutants For neutral mutants For advantageous mutants

Loss-of-function mutations (1) (r) (a) What is the probability that by time t a mutant of has been created? Assume that and

Transition probabilities No double-mutants, j intermediate cells At least one double-mutant Mass-action Space

Stochastic tunneling

Stochastic tunneling Slower

Stochastic tunneling Slower Faster

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 P

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 Cells in a crypt of a colon Gut Stem cells replenish the tissue; asymmetric divisions

Cancer of epithelial tissues Cells in a crypt of a colon Gut Proliferating cells divide symmetrically and differentiate Stem cells replenish the tissue; asymmetric divisions

Cancer of epithelial tissues Cells in a crypt of a colon Gut 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

Two-step process and tunneling time Number of cells First hit in the stem cell Second hit in a daughter cell Number of cells First hit in a daughter cell time

Stochastic tunneling in a hierarchical model

Stochastic tunneling in a hierarchical model The same

Stochastic tunneling in a hierarchical model The same Slower

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 P P

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 Large Interm. mutants Neutral (mass-action, spatial and hierarchical) All models give the same results Spatial model is the fastest Hierarchical model is the slowest Disadvantageous (mass-action and Spatial only) Spatial model is the fastest Mass-action model is faster Spatial model is slower

Production of double-mutants Population size Small Large Interm. mutants Neutral (mass-action, spatial and hierarchical) All models give the same results Spatial model is the fastest Hierarchical model is the slowest Disadvantageous (mass-action and Spatial only) Spatial model is the fastest Mass-action model is faster Spatial model is slower

The definition of “small” 1000 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

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-of-function and loss-of-function mutations There are many more questions in cancer research…