Transcript r<1
Review: Cancer Modeling
Natalia Komarova
(University of California - Irvine)
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 cooperate and perform their respective functions for the
good of the whole organism
Cancer as somatic evolution
• Cells in a multicellular organism have evolved to cooperate 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 cooperate 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 cooperate 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.
u1
(1)
u2
(r1)
u4
u3
(r2)
(r3)
(r4)
Mutation-selection network
(1)
u8
(r2)
u8
(r3)
u1 u
1
u1
u3
u8
(r4)
u3
u4
(r1)
u2
(r1)
u5
u2
u5
(r5)
u8
(r6)
(r6)
(r7)
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
u
(1)
(r)
u 109 per cell division per gene
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+/+
TSP+/-
TSP-/-
u
u
xx
x
(1)
u 107 per cell division per gene copy
(r<1)
(R>1)
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
Gain-of-function mutations
Selection-mutation diagram:
u
(1)
Fitness = 1
Fitness = r >1
(r )
Number of
is i
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
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
Evolutionary selection
dynamics
Start from only one cell of the second type.
What is the chance that it will take over?
1/ r 1
(r )
N
1/ r 1
Fitness = 1
Fitness = r >1
If
If
If
If
r=1 then = 1/N
r<1 then < 1/N
r>1 then > 1/N
then = 1
r
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
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,
u 1/ N
we can show that
Fitness = 1
Fitness = r >1
T
1
Nu (r )
Loss-of-function mutations
u1
u
(1)
(r)
(a)
What is the probability that by time t a mutant of
has been created?
Assume that r 1
and a 1
1D Markov process
• j is the random variable,
j {0,1,..., N , E}
• 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
Transition probabilities
Pj j 1 j
Pj j 1 j
Pj E j
A two-step process
u
u1
A two-step process
u
u1
A two step process
u
u1
…
…
A two-step process
u
(1)
u1
(r)
(a)
Number of cells
Scenario 1:
gets fixated first, and then a mutant of
is created;
time
Stochastic tunneling
u
u1
…
Stochastic tunneling
u
(1)
u1
(r)
(a)
Number of cells
Scenario 2:
A mutant of
is created before
reaches fixation
time
The coarse-grained description
R01
R12
R0 2
Long-lived states:
x0 …“all green”
x1 …“all blue”
x2 …“at least one red”
x0 R01 x0 R02 x0
x1 R01 x0 R12 x1
x 2 R01 x0 R12 x1
Stochastic tunneling
Nu
Nu1
Neutral intermediate mutant
R0 2
R02 Nu u1
| 1 r | u1
Nuu1r
1 r
| 1 r | u1
R02
Disadvantageous intermediate mutant
Assume that r 1
and a 1
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
Advantageous
mutants, r = 1.2
10 5
Neutral
mutants, r = 1
Mass-action
Disadvantageous
mutants, r = 0.95
Spatial
Use the periodic boundary
conditions
Mutant island
Probability to invade
• For disadvantageous
mutants
2r
space
1 r
r 1, | 1 r | 1 / N
• For neutral mutants
| 1 r | 1 / N
• For advantageous
mutants
r 1, | 1 r | 1 / N
1
space
N
2r
space
3r 1
Loss-of-function mutations
u1
u
(1)
(r)
(a)
What is the probability that by time t a mutant of
has been created?
Assume that r 1
and a 1
Transition probabilities
j {0,1,..., N , E}
No double-mutants,
j intermediate cells
Mass-action
At least one double-mutant
Space
Pj j 1 j
Pj j 1
Pj j 1 j
Pj j 1
Pj E j
Pj E j
Stochastic tunneling
Nu space
Nu1
R0 2
R02 uN (9u1 )
1/ 3
R02
(2 / 3)
; (mass act. Nu u1 )
(1 / 3)
(r 1) 2 r 2
Nuu1r
3rNuu1
; (mass act.
)
2
(r 1)
1 r
Stochastic tunneling
Slower
Nu space
Nu1
R0 2
R02 uN (9u1 )
1/ 3
R02
(2 / 3)
; (mass act. Nu u1 )
(1 / 3)
(r 1) 2 r 2
Nuu1r
3rNuu1
; (mass act.
)
2
(r 1)
1 r
Stochastic tunneling
Slower
Nu space
Nu1
Faster
R0 2
R02 uN (9u1 )
1/ 3
R02
(2 / 3)
; (mass act. Nu u1 )
(1 / 3)
(r 1) 2 r 2
Nuu1r
3rNuu1
; (mass act.
)
2
(r 1)
1 r
The mass-action model is
unrealistic
• All cells are assumed to interact with each
other, regardless of their spatial location
P• 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
Gut
Cells in a crypt of a colon
Cancer of epithelial tissues
Gut
Cells in a crypt of a colon
Stem cells replenish the
tissue; asymmetric divisions
Cancer of epithelial tissues
Gut
Cells in a crypt of a colon
Proliferating cells divide
symmetrically and
differentiate
Stem cells replenish the
tissue; asymmetric divisions
Cancer of epithelial tissues
Gut
Cells in a crypt of a colon
Differentiated cells get
shed off into the lumen
Proliferating cells divide
symmetrically and
differentiate
Stem cells replenish the
tissue; asymmetric divisions
Finite branching process
Cellular origins of cancer
Gut
If a stem cell tem cell
acquires a mutation,
the whole crypt is
transformed
Cellular origins of cancer
Gut
If a daughter cell acquires
a mutation, it will probably
get washed out before
a second mutation can hit
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
Number of cells
Two-step process and tunneling
First hit in the stem cell
Number of cells
time
Second hit in a
daughter cell
First hit in a daughter cell
time
Stochastic tunneling in a
hierarchical model
u
Nu1
R0 2
R02 Nuu 1 log u1
(cf .
R Nu u1 )
Stochastic tunneling in a
hierarchical model
The same
u
Nu1
R0 2
R02 Nuu 1 log u1
(cf .
R Nu u1 )
Stochastic tunneling in a
hierarchical model
The same
u
Nu1
R0 2
Slower
R02 Nuu 1 log u1
(cf .
R Nu u1 )
The mass-action model is
unrealistic
• All cells are assumed to interact with each
other, regardless of their spatial location
P• Spatial model of cancer
• All cells of the same type are identical
P• 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-mutants
Population size
Small
Interm. mutants
Large
Neutral
(mass-action,
spatial and
hierarchical)
Disadvantageous
(mass-action and
Spatial only)
All models give
the same results
Spatial model is the fastest
Hierarchical model is the
slowest
Mass-action model is
faster
Spatial model is
slower
Spatial model is the
fastest
Production of double-mutants
Population size
Small
Interm. mutants
Large
Neutral
(mass-action,
spatial and
hierarchical)
Disadvantageous
(mass-action and
Spatial only)
All models give
the same results
Spatial model is the fastest
Hierarchical model is the
slowest
Mass-action model is
faster
Spatial model is
slower
Spatial model is the
fastest
The definition of “small”
1000
N
r=1
Spatial model is the fastest
r=0.99
500
r=0.95
r=0.8
1
2
3
4
5
6
7
8
9
log 10 (u1 )
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-offunction and loss-of-function mutations
• There are many more questions in cancer
research…