natalia komarova (university of california - irvine) review: cancer modeling
TRANSCRIPT
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Natalia Komarova
(University of California - Irvine)
Review: Cancer Modeling
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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.
(1) (r1) (r2) (r3) (r4)
u1 u2 u3 u4
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Mutation-selection network
1u1u
4u
1u
(1) (r1) 3uu2
u5
(r2)(r3)
(r4)
(r5)
(r6)
u8
(r7)u8(r1)
u5
u8
u8
(r6)3u
u2
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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
(1) (r)
u
geneper division cellper 10 9u
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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
Wild type TSP+/-
(1) (r<1)
uTSP-/-TSP+/+
(R>1)
copy geneper division cellper 10 7u
ux x x
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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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Number of is i
Gain-of-function mutations
Fitness = 1
Fitness = r >1
u
Selection-mutation diagram:
(1) (r ) Number of is j=N-i
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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
Fitness = 1
Fitness = r >1
Start from only one cell of the second type; Suppress further mutations.What is the chance that it will take over?
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Evolutionary selection dynamics
Fitness = 1
Fitness = r >1
Start from only one cell of the second type.What is the chance that it will take over?
1/1
1/1)(
Nr
rr
If r=1 then = 1/NIf r<1 then < 1/NIf r>1 then > 1/NIf r then = 1
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Evolutionary selection dynamics
Fitness = 1
Fitness = r >1
Start from zero cell of the second type.What is the expected time until the second type takes over?
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Evolutionary selection dynamics
Fitness = 1
Fitness = r >1
Start from zero cell of the second type.What is the expected time until the second type takes over?
)(1 rNuT
In the case of rare mutations,
Nu /1we can show that
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Loss-of-function mutations
1uu
(1) (r) (a)
1r
What is the probability that by time t a mutant of has been created?
Assume that and 1a
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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 j=E, then there is at least one double-mutant
• j=E is an absorbing state
},,...,1,0{ ENj
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Transition probabilities
jP
jP
jP
Ej
jj
jj
1
1
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A two-step process1uu
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A two-step process1uu
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A two step process
…
…
1uu
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A two-step process1uu
(1) (r) (a)
Scenario 1: gets fixated first, and then a mutant of is created;
time
Num
ber
of c
ells
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Stochastic tunneling
…
1uu
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Stochastic tunneling
time
Num
ber
of c
ells
Scenario 2:A mutant of is created before reaches fixation
1uu
(1) (r) (a)
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The coarse-grained description
1210102
1210101
0200100
xRxRx
xRxRx
xRxRx
20R
10R 21R Long-lived states:x0 …“all green”x1 …“all blue”x2 …“at least one red”
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Stochastic tunneling
1NuNu
Assume that and 1r 1a
120 uNuR
r
rNuuR
1
120
1|1| ur
1|1| ur
20RNeutral intermediate mutant
Disadvantageous intermediate mutant
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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
Disadvantageousmutants, r = 0.95
Advantageousmutants, r = 1.2
Neutralmutants, r = 1
510
Mass-action
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
r
rspace 1
2
13
2
r
rspace
Nspace
1
Nrr /1|1| ,1
Nrr /1|1| ,1
Nr /1|1|
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Loss-of-function mutations
1uu
(1) (r) (a)
1r
What is the probability that by time t a mutant of has been created?
Assume that and 1a
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Transition probabilities
jP
jP
jP
Ej
jj
jj
1
1
jP
P
P
Ej
jj
jj
1
1
Mass-action Space
},,...,1,0{ ENj
At least one double-mutantNo double-mutants,j intermediate cells
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Stochastic tunneling
1NuspaceNu
) act. (mass ;)3/1(
)3/2()9( 1
3/1120 uNuuuNR
)1
act. (mass ;)1(
)1(3 1
2
22
120 r
rNuu
r
rrrNuuR
20R
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Stochastic tunneling
1NuspaceNu
) act. (mass ;)3/1(
)3/2()9( 1
3/1120 uNuuuNR
)1
act. (mass ;)1(
)1(3 1
2
22
120 r
rNuu
r
rrrNuuR
20R
Slower
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Stochastic tunneling
1NuspaceNu
) act. (mass ;)3/1(
)3/2()9( 1
3/1120 uNuuuNR
)1
act. (mass ;)1(
)1(3 1
2
22
120 r
rNuu
r
rrrNuuR
20R
Faster
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
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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
Cells in a crypt of a colon
Gut
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Cancer of epithelial tissues
Stem cells replenish thetissue; asymmetric divisions
Cells in a crypt of a colonGut
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Cancer of epithelial tissues
Stem cells replenish thetissue; asymmetric divisions
Gut
Proliferating cells dividesymmetrically and differentiate
Cells in a crypt of a colon
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Cancer of epithelial tissues
Stem cells replenish thetissue; asymmetric divisions
Gut
Proliferating cells dividesymmetrically and differentiate
Differentiated cells get shed off into the lumen
Cells in a crypt of a colon
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Finite branching process
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Cellular origins of cancer
If a stem cell tem cell acquires a mutation, the whole crypt is transformed
Gut
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Cellular origins of cancer
If a daughter cell acquiresa mutation, it will probablyget washed out beforea second mutation can hit
Gut
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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
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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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Two-step process and tunneling
time
Num
ber
of c
ells
time
Num
ber
of c
ells
First hit in the stem cell
First hit in a daughter cell
Second hit in adaughter cell
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Stochastic tunneling in a hierarchical model
1Nuu
20R
1120 log uNuuR
) .( 1uNuRcf
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Stochastic tunneling in a hierarchical model
1Nuu
20R
1120 log uNuuR
) .( 1uNuRcf
The same
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Stochastic tunneling in a hierarchical model
1Nuu
20R
1120 log uNuuR
) .( 1uNuRcf
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
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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-mutantsPopulation size
Interm. mutantsSmall Large
Neutral
(mass-action,spatial andhierarchical)
Disadvantageous
(mass-action andSpatial only)
All models givethe same results
Spatial model is the fastest
Hierarchical model is theslowest
Mass-action model isfaster
Spatial model is slower
Spatial model is thefastest
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Production of double-mutantsPopulation size
Interm. mutantsSmall Large
Neutral
(mass-action,spatial andhierarchical)
Disadvantageous
(mass-action andSpatial only)
All models givethe same results
Spatial model is the fastest
Hierarchical model is theslowest
Mass-action model isfaster
Spatial model is slower
Spatial model is thefastest
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The definition of “small”
500
1000
1 2 3 4 5 6 7 8 9 )(log 110 u
r=1
r=0.99
r=0.95
r=0.8
Spatial model is the fastest
N
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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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