natalia komarova (university of california - irvine) review: cancer modeling
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Natalia Komarova
(University of California - Irvine)
Review: Cancer Modeling
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.
(1) (r1) (r2) (r3) (r4)
u1 u2 u3 u4
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
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
(1) (r)
u
geneper division cellper 10 9u
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
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
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
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
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?
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
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?
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
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
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
Transition probabilities
jP
jP
jP
Ej
jj
jj
1
1
A two-step process1uu
A two-step process1uu
A two step process
…
…
1uu
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
Stochastic tunneling
…
1uu
Stochastic tunneling
time
Num
ber
of c
ells
Scenario 2:A mutant of is created before reaches fixation
1uu
(1) (r) (a)
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”
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
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
Disadvantageousmutants, r = 0.95
Advantageousmutants, r = 1.2
Neutralmutants, r = 1
510
Mass-action
Spatial
Use the periodic boundary conditions
Mutant island
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|
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
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
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
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
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
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
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
Cells in a crypt of a colon
Gut
Cancer of epithelial tissues
Stem cells replenish thetissue; asymmetric divisions
Cells in a crypt of a colonGut
Cancer of epithelial tissues
Stem cells replenish thetissue; asymmetric divisions
Gut
Proliferating cells dividesymmetrically and differentiate
Cells in a crypt of a colon
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
Finite branching process
Cellular origins of cancer
If a stem cell tem cell acquires a mutation, the whole crypt is transformed
Gut
Cellular origins of cancer
If a daughter cell acquiresa mutation, it will probablyget washed out beforea second mutation can hit
Gut
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
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
Stochastic tunneling in a hierarchical model
1Nuu
20R
1120 log uNuuR
) .( 1uNuRcf
Stochastic tunneling in a hierarchical model
1Nuu
20R
1120 log uNuuR
) .( 1uNuRcf
The same
Stochastic tunneling in a hierarchical model
1Nuu
20R
1120 log uNuuR
) .( 1uNuRcf
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
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-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
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
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
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…
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