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Natalia Komarova

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Title: Natalia Komarova


1
Somatic evolution and cancer
  • Natalia Komarova
  • (University of California - Irvine)

2
Plan
  • Introduction The concept of somatic evolution
  • Methodology Stochastic processes on
    selection-mutation networks
  • Two particular problems
  • Stem cells, initiation of cancer and optimal
    tissue architecture (with L.Wang and P.Cheng)
  • Drug therapy and generation of resistance
    neutral evolution inside a tumor (with D.Wodarz)

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
Stochastic dynamics on a selection-mutation
network
23
A birth-death process with mutations
Selection-mutation diagram
Number of is i
u
(1)
(r )
Number of is jN-i
Fitness 1
Fitness r gt1
24
Evolutionary selection dynamics
Fitness 1
Fitness r gt1
25
Evolutionary selection dynamics
Fitness 1
Fitness r gt1
26
Evolutionary selection dynamics
Fitness 1
Fitness r gt1
27
Evolutionary selection dynamics
Fitness 1
Fitness r gt1
28
Evolutionary selection dynamics
Fitness 1
Fitness r gt1
29
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 gt1
30
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 rlt1 then lt
1/N If rgt1 then gt 1/N If r
then 1
Fitness 1
Fitness r gt1
31
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 gt1
32
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 gt1
33
Two-hit process (Alfred Knudson 1971)
34
A two-step process
35
A two-step process
36
A two step process
37
A two-step process
Scenario 1 gets fixated first, and then
a mutant of is created
Number of cells
time
38
Stochastic tunneling
39
Two-hit process
Scenario 2 A mutant of is created before
reaches fixation
Number of cells
time
40
The coarse-grained description
Long-lived states x0 all green x1 all
blue x2 at least one red
41
Stochastic tunneling
Neutral intermediate mutant

Disadvantageous intermediate mutant
Assume that and
42
Stem cells, initiation of cancer and optimal
tissue architecture
43
Colon tissue architecture
44
Colon tissue architecture
Crypts of a colon
45
Colon tissue architecture
Crypts of a colon
46
Cancer of epithelial tissues
Gut
Cells in a crypt of a colon
47
Cancer of epithelial tissues
Cells in a crypt of a colon
Gut
Stem cells replenish the tissue asymmetric
divisions
48
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
49
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
50
Finite branching process
51
What is known
  • Normal cells undergo apoptosis at the top of the
    crypt, the tissue is renewed and cell number is
    constant

52
What is known
  • Normal cells undergo apoptosis at the top of the
    crypt, the tissue is renewed and cell number is
    constant
  • One of the earliest events in colon cancer is
    inactivation of the APC gene

53
What is known
  • Normal cells undergo apoptosis at the top of the
    crypt, the tissue is renewed and cell number is
    constant
  • One of the earliest events in colon cancer is
    inactivation of the APC gene
  • APC-/- cells do not undergo apoptosis at the top
    of the crypt

54
What is NOT known
  • What is the cellular origin of cancer?
  • Which cells harbor the first dangerous mutaton?
  • Are the stem cells the ones in danger?
  • Which compartment must be targeted by drugs?

?
?
?
55
Colon cancer initiation
  • Both copies of the APC gene must be mutated
    before a phenotypic change is observed (tumor
    suppressor gene)

X
X
X
APC-/-
APC/-
APC/
56
Cellular origins of cancer
Gut
If a stem cell tem cell acquires a mutation,
the whole crypt is transformed
57
Cellular origins of cancer
Gut
If a daughter cell acquires a mutation, it will
probably get washed out before a second mutation
can hit
58
What is the cellular origin of cancer?
59
Colon cancer initiation
60
Colon cancer initiation
61
Colon cancer initiation
62
Colon cancer initiation
63
Colon cancer initiation
64
Colon cancer initiation
65
First mutation in a daughter cell
66
First mutation in a daughter cell
67
First mutation in a daughter cell
68
First mutation in a daughter cell
69
First mutation in a daughter cell
70
First mutation in a daughter cell
71
Cellular origins of cancer
  • The prevailing theory is that the mutations
    leading to cancer initiation occur is stem cells

72
Cellular origins of cancer
  • The prevailing theory is that the mutations
    leading to cancer initiation occur is stem cells
  • Therefore, all prevention and treatment
    strategies must target the stem cells

73
Cellular origins of cancer
  • The prevailing theory is that the mutations
    leading to cancer initiation occur is stem cells
  • Therefore, all prevention and treatment
    strategies must target the stem cells
  • Differentiated cells (most cells!) do not count

74
Mathematical approach
  • Formulate a model which distinguishes between
    stem and differentiated cells
  • Calculate the relative probability of various
    mutation patterns

75
First mutation in a daughter cell
76
First mutation in a daughter cell
77
First mutation in a daughter cell
78
First mutation in a daughter cell
79
First mutation in a daughter cell
80
First mutation in a daughter cell
81
Stochastic tunneling in a heterogeneous population
  • At least one mutation happens in a stem cell (cf.
    the two-step process)
  • 2) Both mutations happen in a daughter cell no
    fixation of an intermediate mutant (cf tunneling)

82
Stochastic tunneling in a heterogeneous population

Lower rate
  • At least one mutation happens in a stem cell (cf.
    the two-step process)
  • 2) Both mutations happen in a daughter cell no
    fixation of an intermediate mutant (cf tunneling)

83
Cellular origins of cancer
  • If the tissue is organized into compartments with
    stem cells and daughter cells, the risk of
    mutations is lower than in homogeneous populations

84
Cellular origins of cancer
  • If the tissue is organized into compartments with
    stem cells and daughter cells, the risk of
    mutations is lower than in a homogeneous
    population
  • Cellular origin of cancer is not necessarily the
    stem cell. Under some circumstances, daughter
    cells are the ones at risk.

85
Cellular origins of cancer
  • If the tissue is organized into compartments with
    stem cells and daughter cells, the risk of
    mutations is lower than in a homogeneous
    populations
  • Cellular origin of cancer is not necessarily the
    stem cell. Under some circumstances, daughter
    cells are the ones at risk.
  • Stem cells are not the entire story!!!

86
Optimal tissue architecture
  • How does tissue architecture help protect against
    cancer?
  • What are parameters of the architecture that
    minimize the risk of cancer?
  • How does protection against cancer change with
    the individuals age?

87
Optimal number of stem cells
m1
m2
Crypt size is n16
m4
m8
88
Probability to develop dysplasia
One stem cell
Probability to develop dysplasia
Many stem cells
Time (individuals age)
89
The optimal solution is time-dependent!
Optimum many stem cells
One stem cell
Probability to develop dysplasia
Many stem cells
Optimum one stem cell
Time (individuals age)
90
Optimization problem
  • The optimum number of stem cells is high in young
    age, and low in old age
  • Assume that tissue architecture cannot change
    with time must choose a time-independent
    solution
  • Selection mostly acts upon reproductive ages, so
    the preferred evolutionary strategy is to keep
    the risk of cancer low while the organism is
    young

91
Evolutionary compromise
Many stem cells
Probability to develop dysplasia
One stem cell
Time (individuals age)
92
Evolutionary compromise
Many stem cells
While keeping the risk of cancer low at the
young age, the preferred evolutionary strategy
works against the
older age, actually
increasing
the
likelihood of cancer!
Probability to develop dysplasia
One stem cell
Time (individuals age)
93
Cancer vs aging
  • Cancer and aging are two sides of the same coin..

94
Drug therapy and generation of resistance
95
Leukemia
  • Most common blood cancer
  • Four major types
  • Acute Myeloid Leukemia (AML),
  • Chronic Lymphocytic Leukemia (CLL),
  • Chronic Myeloid Leukemia (CML),
  • Acute Lymphocytic Leukemia (ALL)

96
Leukemia
  • Most common blood cancer
  • Four major types
  • Acute Myeloid Leukemia (AML),
  • Chronic Lymphocytic Leukemia (CLL),
  • Chronic Myeloid Leukemia (CML),
  • Acute Lymphocytic Leukemia (ALL)

97
CML
  • Chronic phase (2-5 years)
  • Accelerated phase (6-18 months)
  • Blast crisis (survival 3-6 months)

98
Targeted cancer drugs
  • Traditional drugs very toxic agents that kill
    dividing cells

99
Targeted cancer drugs
  • Traditional drugs very toxic agents that kill
    dividing cells
  • New drugs small molecule inhibitors
  • Target the pathways which make cancerous cells
    cancerous (Gleevec)

100
Gleevec a new generation drug
Bcr-Abl
101
Gleevec a new generation drug
Bcr-Abl
Bcr-Abl
102
Small molecule inhibitors
103
Targeted cancer drugs
  • Very effective
  • Not toxic

104
Targeted cancer drugs
  • Very effective
  • Not toxic
  • Resistance poses a
  • problem

Gleevec
Bcr-Abl protein
105
Targeted cancer drugs
  • Very effective
  • Not toxic
  • Resistance poses a
  • problem

Mutation
Gleevec
Bcr-Abl protein
106
Treatment without resistance
treatment
time
107
Development of resistance
treatment
108
How can one prevent resistance?
  • In HIV treat with multiple drugs
  • It takes one mutation to develop resistance of
    one drug. It takes n mutations to develop
    resistance to n drugs.
  • Goal describe the generation of resistance
    before and after therapy.

109
Mutation network for developing resistance
against n3 drugs
110
During a short time-interval, Dt, a cell of type
Ai can
  • Reproduce faithfully with probability
  • Li(1-Suj) Dt

111
During a short time-interval, Dt, a cell of type
Ai can
  • Reproduce faithfully with probability
  • Li(1-Suj) Dt
  • Produce one cell identical to itself, and a
    mutant cell of type Aj with probability Liuj Dt

112
During a short time-interval, Dt, a cell of type
Ai can
  • Reproduce faithfully with probability
  • Li(1-Suj) Dt
  • Produce one cell identical to itself, and a
    mutant cell of type Aj with probability Liuj Dt
  • Die with probability Di Dt

113
The method
Assume just one drug. xij(t) is the probability
to have i susceptible and j resistantcells at
time t.
F(x,yt)Sxij(t)xjyi is the probability
generating function.
114
The method
xij(t) is the probability to have i susceptible
and j resistant cells at time t.
F(x,yt)Sxij(t)xjyi is the probability
generating function.
115
For multiple drugs
xi0, i1, , im(t) is the probability to have is
cells of type As at time t.
F(x0,x1,,xmt) S xi0, i1, , im(t) x0im
xmi0 is the probability generating function.
F(0,1,,1t) is the probability that at time t
there are no cells of type Am
F(0,0,,0t) is the probability that at time t
the colony is extinct
116
The method
The probability that at time t the colony is
extinct is F(0,0,,0t)
xnM(t), where M is the initial of cells and
xn is the solution of
The probability of treatment failure is
117
The questions
  • Does resistance mostly arise before or after the
    start of treatment?
  • How does generation of resistance depend on the
    properties of cancer growth (high turnover DL vs
    low turnover DltltL)
  • How does the number of drugs influence the
    success of treatment?

118
1. How important is pre-existence of mutants?
119
Single drug therapy
120
Single drug therapy
Generation during treatment
Pre-existance
121
Single drug therapy
Unrealistic!
Generation during treatment
Pre-existance
122
Single drug therapy
Pre-existance gtgt Generation during
treatment
123
Multiple drug therapies
Fully susceptible
Partially susceptible
Fully resistant
124
Development of resistance
Fully susceptible
Partially susceptible
Fully resistant
125
1. How important is pre-existence of resistant
mutants?
  • For both single- and multiple-drug therapies,
  • resistant mutants are likely to be produced
    before start of treatment, and not in the
    course of treatment

126
2. How does generation of resistance depend on
the turnover rate of cancer?
  • Low turnover (growth rategtgtdeath rate)
  • Fewer cell divisions needed to reach a certain
    size
  • High turnover (growth ratedeath rate)
  • Many cell divisions needed to reach a certain
    size

127
Single drug therapy
Low turnover cancer, DltltL
128
Single drug therapy
High turnover cancer, DL
More mutant colonies are produced, but
the probability of colony survival is
proportionally smaller
129
2. How does generation of resistance depend on
the turnover rate of cancer?
  • Single drug therapies the production of mutants
    is independent of the turnover

130
2. How does generation of resistance depend on
the turnover rate of cancer?
  • Single drug therapies the production of mutants
    is independent of the turnover
  • Multiple drug therapies the production of
    mutants is much larger for cancers with a high
    turnover

131
3. The size of failure
  • Suppose we start treatment at size N
  • Calculate the probability of treatment failure
  • Find the size at which the probability of failure
    is d0.01

132
3. The size of failure
  • Suppose we start treatment at size N
  • Calculate the probability of treatment failure
  • Find the size at which the probability of failure
    is d0.01
  • The size of failure increases with of drugs and
    decreases with mutation rate

133
Minimum of drugs for different parameter values
1013 cells
u10-8-10-9 is the basic point mutation rate,
u10-4 is associated with genetic instabilities
134
Minimum of drugs for different parameter values
1013 cells
u10-8-10-9 is the basic point mutation rate,
u10-4 is associated with genetic instabilities
135
Minimum of drugs for different parameter values
1013 cells
u10-8-10-9 is the basic point mutation rate,
u10-4 is associated with genetic instabilities
136
Minimum of drugs for different parameter values
1013 cells
u10-8-10-9 is the basic point mutation rate,
u10-4 is associated with genetic instabilities
137
Minimum of drugs for different parameter values
1013 cells
u10-8-10-9 is the basic point mutation rate,
u10-4 is associated with genetic instabilities
138
CML leukemia
  • Gleevec
  • u10-8-10-9
  • D/L between 0.1 and 0.5 (low turnover)
  • Size of advanced cancers is 1013 cells

139
Log size of treatment failure
u10-8
u10-6
140
Application for CML
  • The model suggests that 3 drugs are needed to
    push the size of failure (1 failure) up to 1013
    cells

141
Conclusions
  • Main concept cancer is a highly structured
    evolutionary process
  • Main tool stochastic processes on
    selection-mutation networks
  • We addressed questions of cellular origins of
    cancer and generation of drug resistance
  • There are many more questions in cancer research

142
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143
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144
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145
Multiple drug treatments
  • For fast turnover cancers, adding more drugs will
    not prevent generation of resistance

146
Size of failure for different turnover rates
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