Natalia Komarova

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Review Cancer Modeling

• Natalia Komarova
• (University of California - Irvine)

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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

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Darwinian evolution (of species)

• Time-scale hundreds of millions of years
• Organisms reproduce and die in an environment
  with shared resources

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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)

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Somatic evolution

• Cells reproduce and die inside an organ of one
  organism
• Time-scale tens of years

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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)

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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

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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

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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

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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

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Progression to cancer
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Progression to cancer
Constant population
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Progression to cancer
Advantageous mutant
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Progression to cancer
Clonal expansion
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Progression to cancer
Saturation
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Progression to cancer
Advantageous mutant
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Progression to cancer
Wave of clonal expansion
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Genetic pathways to colon cancer (Bert
Vogelstein)
Multi-stage carcinogenesis
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Methodology modeling a colony of cells

• Cells can divide, mutate and die

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Methodology modeling a colony of cells

• Cells can divide, mutate and die
• Mutations happen according to a
  mutation-selection diagram, e.g.

u1
u4
u2
u3
(r3)
(r4)
(r2)
(1)
(r1)
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Mutation-selection network
u8
(r3)
u8
(r2)
(r6)
u8
u5
(1)
(r4)
(r1)
(r6)
u2
u2
u5
u8
(r1)
(r5)
(r7)
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Common patterns in cancer progression

• Gain-of-function mutations
• Loss-of-function mutations

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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)
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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

TSP/
TSP/-
Wild type
TSP-/-
u
u
x
x
x
(1)
(R1)
(r
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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

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Gain-of-function mutations
Selection-mutation diagram
Number of is i
u
(1)
(r )
Number of is jN-i
Fitness 1
Fitness r 1
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Evolutionary selection dynamics
Fitness 1
Fitness r 1
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Evolutionary selection dynamics
Fitness 1
Fitness r 1
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Evolutionary selection dynamics
Fitness 1
Fitness r 1
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Evolutionary selection dynamics
Fitness 1
Fitness r 1
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Evolutionary selection dynamics
Fitness 1
Fitness r 1
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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
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Evolutionary selection dynamics
Start from only one cell of the second type. What
is the chance that it will take over?
If r1 then 1/N If r1/N If r1 then 1/N If r
then 1
Fitness 1
Fitness r 1
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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
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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
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Loss-of-function mutations
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1D Markov process

• j is the random variable,
• If j 1,2,,N then there are j intermediate
  mutants, and no double-mutants
• If jE, then there is at least one double-mutant
• jE is an absorbing state

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Transition probabilities
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A two-step process
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A two-step process
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A two step process
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A two-step process
Scenario 1 gets fixated first, and then
a mutant of is created
Number of cells
time
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Stochastic tunneling
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Stochastic tunneling
Scenario 2 A mutant of is created before
reaches fixation
Number of cells
time
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The coarse-grained description
Long-lived states x0 all green x1 all
blue x2 at least one red
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Stochastic tunneling
Neutral intermediate mutant

Disadvantageous intermediate mutant
Assume that and
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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

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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

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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

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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

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Spatial dynamics
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Spatial dynamics
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Spatial dynamics
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Spatial dynamics
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Spatial dynamics
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Spatial dynamics
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Spatial dynamics
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Spatial dynamics
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Spatial dynamics
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Gain-of-function probability to invade

• In the spatial model, the probability to invade
  depends on the spatial location of the initial
  mutation

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Probability of invasion
Neutral mutants, r 1
Advantageous mutants, r 1.2
Mass-action
Disadvantageous mutants, r 0.95
Spatial
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Use the periodic boundary conditions
Mutant island
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Probability to invade

• For disadvantageous mutants
• For neutral mutants
• For advantageous mutants

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Loss-of-function mutations
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Transition probabilities
No double-mutants, j intermediate cells
At least one double-mutant
Mass-action
Space
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Stochastic tunneling

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Stochastic tunneling
Slower

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Stochastic tunneling
Slower
Faster

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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
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Hierarchical model of cancer
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Colon tissue architecture
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Colon tissue architecture
Crypts of a colon
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Colon tissue architecture
Crypts of a colon
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Cancer of epithelial tissues
Gut
Cells in a crypt of a colon
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Cancer of epithelial tissues
Cells in a crypt of a colon
Gut
Stem cells replenish the tissue asymmetric
divisions
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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
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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
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Finite branching process
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Cellular origins of cancer
Gut
If a stem cell tem cell acquires a mutation,
the whole crypt is transformed
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Cellular origins of cancer
Gut
If a daughter cell acquires a mutation, it will
probably get washed out before a second mutation
can hit
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Colon cancer initiation
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Colon cancer initiation
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Colon cancer initiation
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Colon cancer initiation
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Colon cancer initiation
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Colon cancer initiation
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First mutation in a daughter cell
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First mutation in a daughter cell
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First mutation in a daughter cell
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First mutation in a daughter cell
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First mutation in a daughter cell
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First mutation in a daughter cell
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First mutation in a daughter cell
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First mutation in a daughter cell
95
First mutation in a daughter cell
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First mutation in a daughter cell
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First mutation in a daughter cell
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First mutation in a daughter cell
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Two-step process and tunneling
First hit in the stem cell
Second hit in a daughter cell
Number of cells
First hit in a daughter cell
time
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Stochastic tunneling in a hierarchical model

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Stochastic tunneling in a hierarchical model

The same
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Stochastic tunneling in a hierarchical model

The same
Slower
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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
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Comparison of the models
Probability of mutant invasion for
gain-of-function mutations
r 1 neutral
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Comparison of the models
The tunneling rate
(lowest rate)
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The tunneling and two-step regimes
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Production of double-mutants
Population size
Small
Large
Interm. mutants
Neutral (mass-action, spatial and hierarchical)
Spatial model is the fastest Hierarchical model
is the slowest
All models give the same results
Disadvantageous (mass-action and Spatial only)
Spatial model is the fastest
Mass-action model is faster Spatial model is
slower
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Production of double-mutants
Population size
Small
Large
Interm. mutants
Neutral (mass-action, spatial and hierarchical)
Spatial model is the fastest Hierarchical model
is the slowest
All models give the same results
Disadvantageous (mass-action and Spatial only)
Spatial model is the fastest
Mass-action model is faster Spatial model is
slower
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The definition of small
N
1000
r1
Spatial model is the fastest
r0.99
500
r0.95
r0.8
1 2 3 4 5
6 7 8 9
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Summary

• The details of population modeling are important,
  the simple mass-action can give wrong predictions

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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

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Summary

• If the tissue is organized into compartments with
  stem cells and daughter cells, the risk of
  mutations is lower than in homogeneous populations

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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

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Summary

• A more flexible homeostatic regulation mechanism
  with an increased cellular motility will serve as
  a protection against double-mutant generation

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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

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