Somatic evolution and cancer
Natalia Komarova
(University of California - Irvine)
Plan
•
•
Introduction: The concept of somatic evolution
Methodology: Stochastic processes on
selection-mutation networks
Two particular problems:
1. Stem cells, initiation of cancer and optimal
tissue architecture (with L.Wang and P.Cheng)
2. Drug therapy and generation of resistance:
neutral evolution inside a tumor (with
D.Wodarz)
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)
Stochastic dynamics on a
selection-mutation network
A birth-death process with
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 )
Two-hit process
(Alfred Knudson 1971)
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
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
…
Two-hit process
u
(1)
u1
(r)
(a)
Number of cells
Scenario 2:
A mutant of
is created before
reaches fixation
time
The coarse-grained description
R01
R12
R0 2
Long-lived states:
x0 …“all green”
x1 …“all blue”
x2 …“at least one red”
x0   R01 x0  R02 x0
x1  R01 x0  R12 x1
x 2  R01 x0  R12 x1
Stochastic tunneling
Nu
Nu1
Neutral intermediate mutant
R0 2
R02  Nu u1
| 1  r | u1
Nuu1r

1 r
| 1  r | u1
R02
Disadvantageous intermediate mutant
Assume that r  1
and a  1
Stem cells, initiation of cancer and
optimal tissue architecture
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
What is known:
• Normal cells undergo apoptosis at the top of the
crypt, the tissue is renewed and cell number is
constant
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
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
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?
Colon cancer initiation
• Both copies of the APC gene must
be mutated before a phenotypic
change is observed (tumor
suppressor gene)
X
APC+/+
APC+/-
XX
APC-/-
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
What is the cellular origin of
cancer?
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
Cellular origins of cancer
• The prevailing theory is that the mutations
leading to cancer initiation occur is stem
cells
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
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
Mathematical approach:
• Formulate a model which distinguishes
between stem and differentiated cells
• Calculate the relative probability of various
mutation patterns
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
Stochastic tunneling in a
heterogeneous population
u
Nu1
R0 2
R02  Nuu 1 log u1
(cf .
R  Nu u1 )
1) 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)
Stochastic tunneling in a
heterogeneous population
u
Nu1
R02  Nuu 1 log u1
R0 2
Lower rate
(cf .
R  Nu u1 )
1) 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)
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
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.
1
u 1 log u1 
N
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!!!
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 individual’s age?
Optimal number of stem cells
m=1
m=2
Crypt size is
n=16
m=4
m=8
Probability to develop dysplasia
Probability to develop dysplasia
One stem
cell
Many stem
cells
Time (individual’s age)
Probability to develop dysplasia
The optimal solution is timedependent!
Optimum:
many
stem
cells
One stem
cell
Many stem
Optimum:
cells
one stem
cell
Time (individual’s age)
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 timeindependent 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
Probability to develop dysplasia
Evolutionary compromise
Many
stem
cells
One stem
cell
Time (individual’s age)
Probability to develop dysplasia
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!
One stem
cell
Time (individual’s age)
Cancer vs aging
• Cancer and aging are two sides of the
same coin…..
Drug therapy and generation of
resistance
Leukemia
• Most common blood cancer
• Four major types:
Acute Myeloid Leukemia (AML),
Chronic Lymphocytic Leukemia (CLL),
Chronic Myeloid Leukemia (CML),
Acute Lymphocytic Leukemia (ALL)
Leukemia
• Most common blood cancer
• Four major types:
Acute Myeloid Leukemia (AML),
Chronic Lymphocytic Leukemia (CLL),
Chronic Myeloid Leukemia (CML),
Acute Lymphocytic Leukemia (ALL)
CML
• Chronic phase (2-5 years)
• Accelerated phase (6-18 months)
• Blast crisis (survival 3-6 months)
Targeted cancer drugs
• Traditional drugs: very toxic agents that kill
dividing cells
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)
Gleevec: a new generation drug
Bcr-Abl
Gleevec: a new generation drug
Bcr-Abl
Bcr-Abl
Small molecule inhibitors
Targeted cancer drugs
• Very effective
• Not toxic
Targeted cancer drugs
• Very effective
• Not toxic
• Resistance poses a
problem
Gleevec
Bcr-Abl protein
Targeted cancer drugs
• Very effective
• Not toxic
• Resistance poses a
problem
Mutation
Gleevec
Bcr-Abl protein
Treatment without resistance
treatment
time
Development of resistance
treatment
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.
Mutation network for developing
resistance against n=3 drugs
During a short time-interval, Dt, a
cell of type Ai can:
• Reproduce faithfully with probability
Li(1-Suj) Dt
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
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
The method
Assume just one drug. ij(t) is the probability to have
i susceptible and j resistantcells at time t.
ij (t  Dt )  i-1, j (t )(i  1) L(1  u)Dt  i, j-1 (t )[( j  1) L  iLu ]Dt
 i 1, j (i  1) DDt  i, j1 ( j  1) DDt  ij (t )[1  ( L  D)(i  j )Dt ]
ij   i-1, j (t )(i  1) L(1  u )   i, j-1 (t )[( j  1) L  iLu ]
  i 1, j (i  1) D   i, j1 ( j  1) D   ij (t )( L  D)(i  j )
x,y;tSij(t)xjyi is the probability generating function.
 


Lx 2  ( L  D) x  D 
L(1  u ) y 2  [ Lux  ( L  D)] y  D
t
x
y




The method
ij(t) is the probability to have i susceptible and j resistant
cells at time t.
ij (t  Dt )  i-1, j (t )(i  1) L(1  u)Dt  i, j-1 (t )[( j  1) L  iLu ]Dt
 i 1, j (i  1) DDt  i, j1 ( j  1) DDt  ij (t )[1  ( L  D)(i  j )Dt ]
ij   i-1, j (t )(i  1) L(1  u )   i, j-1 (t )[( j  1) L  iLu ]
  i 1, j (i  1) D   i, j1 ( j  1) D   ij (t )( L  D)(i  j )
x,y;tSij(t)xjyi is the probability generating function.
x  Lx 2  ( L  D) x  D;
y  L(1  u ) y 2  [ Lux  ( L  D)] y  D.
For multiple drugs:
i0, i1, …, im(t) is the probability to have is cells of type
As at time t.
x0,x1,…,xm;t  S i0, i1, …, im(t) x0im …xmi0
is the probability generating function.
0,1,…,1;t
is the probability that at time t there are no cells of type Am
0,0,…,0;t
is the probability that at time t the colony is extinct
2
x0  Lx0  ( L  D) x0  D;
2
xi  L(1  iu ) xi  [ Liuxi 1  ( L  D)] xi  D,
0in
The method
The probability that at time t the colony is
extinct is
(0,0,…,0;t) =xnM(t),
where M is the initial # of cells and xn is the solution of
2
x0  Lx0  ( L  D) x0  D;
2
xi  L(1  iu ) xi  [ Liuxi 1  ( L  D)] xi  D,
xi (0)  0.
The probability of treatment failure is
Pfail  1  lim t  xnM (t )
0  i  n,
The questions:
1. Does resistance mostly arise before or
after the start of treatment?
2. How does generation of resistance
depend on the properties of cancer
growth (high turnover D~L vs low
turnover D<<L)
3. How does the number of drugs influence
the success of treatment?
1. How important is pre-existence
of mutants?
Single drug therapy
Single drug therapy
Pre-existance =
Generation during treatment
Single drug therapy
Unrealistic!
Pre-existance =
Generation during treatment
Single drug therapy
Pre-existance >>
Generation during treatment
Multiple drug therapies
Fully susceptible
Partially susceptible
Fully resistant
Development of resistance
Fully susceptible
Partially susceptible
Fully resistant
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
2. How does generation of
resistance depend on the turnover
rate of cancer?
• Low turnover (growth rate>>death rate)
Fewer cell divisions needed to reach a
certain size
• High turnover (growth rate~death rate)
Many cell divisions needed to reach a
certain size
Single drug therapy
Low turnover cancer, D<<L
Single drug therapy
High turnover cancer, D~L
More mutant colonies
are produced, but the
probability of colony
survival is proportionally
smaller…
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
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
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 d=0.01
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 d=0.01
• The size of failure increases with # of
drugs and decreases with mutation rate
Minimum # of drugs for different
parameter values
1013 cells
u=10-8-10-9 is the basic point mutation rate, u=10-4 is associated with
genetic instabilities
Minimum # of drugs for different
parameter values
1013 cells
u=10-8-10-9 is the basic point mutation rate, u=10-4 is associated with
genetic instabilities
Minimum # of drugs for different
parameter values
1013 cells
u=10-8-10-9 is the basic point mutation rate, u=10-4 is associated with
genetic instabilities
Minimum # of drugs for different
parameter values
1013 cells
u=10-8-10-9 is the basic point mutation rate, u=10-4 is associated with
genetic instabilities
Minimum # of drugs for different
parameter values
1013 cells
u=10-8-10-9 is the basic point mutation rate, u=10-4 is associated with
genetic instabilities
CML leukemia
•
•
•
•
Gleevec
u=10-8-10-9
D/L between 0.1 and 0.5 (low turnover)
Size of advanced cancers is 1013 cells
Log size of treatment failure
u=10-8
(a)
D/L=0.1
D/L=0.5
D/L=0.9
1 drug
5.95
5.95
5.95
2 drugs
12.34
12.13
11.48
3 drugs
18.45
17.99
16.70
5 drugs
30.19
29.26
26.66
u=10-6
(b)
D/L=0.1
D/L=0.5
D/L=0.9
4 drugs
24.38
23.69
21.74
1 drug
4.00
4.00
4.00
2 drugs
8.55
8.31
7.68
3 drugs
12.80
12.37
11.07
4 drugs
16.89
16.20
14.40
5 drugs
20.86
19.93
17.40
Application for CML
• The model suggests that 3 drugs are
needed to push the size of failure (1%
failure) up to 1013 cells
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…
Multiple drug treatments
• For fast turnover cancers, adding more
drugs will not prevent generation of
resistance
Size of failure for different turnover
rates