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Natalia Komarova

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Darwinian evolution (of species) Time-scale: ... Somatic evolution. Cells reproduce and die inside an organ of one organism ... Cancer as somatic evolution ... – PowerPoint PPT presentation

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Title: Natalia Komarova


1
Review Cancer Modeling
  • Natalia Komarova
  • (University of California - Irvine)

2
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

3
Darwinian evolution (of species)
  • Time-scale hundreds of millions of years
  • Organisms reproduce and die in an environment
    with shared resources

4
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)

5
Somatic evolution
  • Cells reproduce and die inside an organ of one
    organism
  • Time-scale tens of years

6
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)

7
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

8
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

9
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

10
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

11
Progression to cancer
12
Progression to cancer
Constant population
13
Progression to cancer
Advantageous mutant
14
Progression to cancer
Clonal expansion
15
Progression to cancer
Saturation
16
Progression to cancer
Advantageous mutant
17
Progression to cancer
Wave of clonal expansion
18
Genetic pathways to colon cancer (Bert
Vogelstein)
Multi-stage carcinogenesis
19
Methodology modeling a colony of cells
  • Cells can divide, mutate and die

20
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)
21
Mutation-selection network
u8
(r3)
u8
(r2)
(r6)
u8
u5
(1)
(r4)
(r1)
(r6)
u2
u2
u5
u8
(r1)
(r5)
(r7)
22
Common patterns in cancer progression
  • Gain-of-function mutations
  • Loss-of-function mutations

23
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)
24
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
25
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

26
Gain-of-function mutations
Selection-mutation diagram
Number of is i
u
(1)
(r )
Number of is jN-i
Fitness 1
Fitness r 1
27
Evolutionary selection dynamics
Fitness 1
Fitness r 1
28
Evolutionary selection dynamics
Fitness 1
Fitness r 1
29
Evolutionary selection dynamics
Fitness 1
Fitness r 1
30
Evolutionary selection dynamics
Fitness 1
Fitness r 1
31
Evolutionary selection dynamics
Fitness 1
Fitness r 1
32
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
33
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
34
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
35
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
36
Loss-of-function mutations
37
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

38
Transition probabilities
39
A two-step process
40
A two-step process
41
A two step process
42
A two-step process
Scenario 1 gets fixated first, and then
a mutant of is created
Number of cells
time
43
Stochastic tunneling
44
Stochastic tunneling
Scenario 2 A mutant of is created before
reaches fixation
Number of cells
time
45
The coarse-grained description
Long-lived states x0 …all green x1 …all
blue x2 …at least one red
46
Stochastic tunneling
Neutral intermediate mutant

Disadvantageous intermediate mutant
Assume that and
47
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

48
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

49
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

50
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

51
Spatial dynamics
52
Spatial dynamics
53
Spatial dynamics
54
Spatial dynamics
55
Spatial dynamics
56
Spatial dynamics
57
Spatial dynamics
58
Spatial dynamics
59
Spatial dynamics
60
Gain-of-function probability to invade
  • In the spatial model, the probability to invade
    depends on the spatial location of the initial
    mutation

61
Probability of invasion
Neutral mutants, r 1
Advantageous mutants, r 1.2
Mass-action
Disadvantageous mutants, r 0.95
Spatial
62
Use the periodic boundary conditions
Mutant island
63
Probability to invade
  • For disadvantageous mutants
  • For neutral mutants
  • For advantageous mutants

64
Loss-of-function mutations
65
Transition probabilities
No double-mutants, j intermediate cells
At least one double-mutant
Mass-action
Space
66
Stochastic tunneling

67
Stochastic tunneling
Slower

68
Stochastic tunneling
Slower
Faster

69
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
70
Hierarchical model of cancer
71
Colon tissue architecture
72
Colon tissue architecture
Crypts of a colon
73
Colon tissue architecture
Crypts of a colon
74
Cancer of epithelial tissues
Gut
Cells in a crypt of a colon
75
Cancer of epithelial tissues
Cells in a crypt of a colon
Gut
Stem cells replenish the tissue asymmetric
divisions
76
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
77
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
78
Finite branching process
79
Cellular origins of cancer
Gut
If a stem cell tem cell acquires a mutation,
the whole crypt is transformed
80
Cellular origins of cancer
Gut
If a daughter cell acquires a mutation, it will
probably get washed out before a second mutation
can hit
81
Colon cancer initiation
82
Colon cancer initiation
83
Colon cancer initiation
84
Colon cancer initiation
85
Colon cancer initiation
86
Colon cancer initiation
87
First mutation in a daughter cell
88
First mutation in a daughter cell
89
First mutation in a daughter cell
90
First mutation in a daughter cell
91
First mutation in a daughter cell
92
First mutation in a daughter cell
93
First mutation in a daughter cell
94
First mutation in a daughter cell
95
First mutation in a daughter cell
96
First mutation in a daughter cell
97
First mutation in a daughter cell
98
First mutation in a daughter cell
99
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
100
Stochastic tunneling in a hierarchical model

101
Stochastic tunneling in a hierarchical model

The same
102
Stochastic tunneling in a hierarchical model

The same
Slower
103
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
104
Comparison of the models
Probability of mutant invasion for
gain-of-function mutations
r 1 neutral
105
Comparison of the models
The tunneling rate
(lowest rate)
106
The tunneling and two-step regimes
107
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
108
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
109
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
110
Summary
  • The details of population modeling are important,
    the simple mass-action can give wrong predictions

111
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

112
Summary
  • If the tissue is organized into compartments with
    stem cells and daughter cells, the risk of
    mutations is lower than in homogeneous populations

113
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

114
Summary
  • A more flexible homeostatic regulation mechanism
    with an increased cellular motility will serve as
    a protection against double-mutant generation

115
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…

116
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