natalia komarova (university of california - irvine) somatic evolution and cancer
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![Page 1: Natalia Komarova (University of California - Irvine) Somatic evolution and cancer](https://reader035.vdocuments.mx/reader035/viewer/2022062516/56649d575503460f94a36b49/html5/thumbnails/1.jpg)
Natalia Komarova
(University of California - Irvine)
Somatic evolution and cancer
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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)
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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 u3u4
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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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Stochastic dynamics on a selection-mutation network
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Number of is i
A birth-death process with 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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Two-hit process (Alfred Knudson 1971)
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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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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Two-hit process
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
10R21R 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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Stem cells, initiation of cancer and optimal tissue architecture
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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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What is known:• Normal cells undergo apoptosis at the top of the
crypt, the tissue is renewed and cell number is constant
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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
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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
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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?
?
?
?
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Colon cancer initiation
• Both copies of the APC gene must be mutated before a phenotypic change is observed (tumor suppressor gene)
APC+/+ APC+/- APC-/-
X XX
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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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What is the cellular origin of cancer?
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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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Cellular origins of cancer
• The prevailing theory is that the mutations leading to cancer initiation occur is stem cells
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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
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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
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Mathematical approach:
• Formulate a model which distinguishes between stem and differentiated cells
• Calculate the relative probability of various mutation patterns
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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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Stochastic tunneling in a heterogeneous population
1Nuu
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)
20R 1120 log uNuuR
) .( 1uNuRcf
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Stochastic tunneling in a heterogeneous population
1Nuu
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)
20R 1120 log uNuuR
) .( 1uNuRcf Lower rate
![Page 83: Natalia Komarova (University of California - Irvine) Somatic evolution and cancer](https://reader035.vdocuments.mx/reader035/viewer/2022062516/56649d575503460f94a36b49/html5/thumbnails/83.jpg)
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
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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.
Nuu
1log 11
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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!!!
![Page 86: Natalia Komarova (University of California - Irvine) Somatic evolution and cancer](https://reader035.vdocuments.mx/reader035/viewer/2022062516/56649d575503460f94a36b49/html5/thumbnails/86.jpg)
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?
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Optimal number of stem cells
m=1m=2
m=4m=8
Crypt size isn=16
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Probability to develop dysplasia
Time (individual’s age)
Pro
babi
lity
to d
evel
op d
yspl
asia
One stem cell
Many stem cells
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The optimal solution is time-dependent!
Time (individual’s age)
Pro
babi
lity
to d
evel
op d
yspl
asia
Optimum:one stemcell
Optimum:many stem cells
Many stem cells
One stem cell
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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
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Evolutionary compromiseP
roba
bili
ty to
dev
elop
dys
plas
ia
Time (individual’s age)
One stem cell
Many stem cells
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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!
Evolutionary compromiseP
roba
bili
ty to
dev
elop
dys
plas
ia
Time (individual’s age)
One stem cell
Many stem cells
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Cancer vs aging
• Cancer and aging are two sides of the same coin…..
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Drug therapy and generation of resistance
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Leukemia
• Most common blood cancer
• Four major types:
Acute Myeloid Leukemia (AML),
Chronic Lymphocytic Leukemia (CLL),
Chronic Myeloid Leukemia (CML),
Acute Lymphocytic Leukemia (ALL)
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Leukemia
• Most common blood cancer
• Four major types:
Acute Myeloid Leukemia (AML),
Chronic Lymphocytic Leukemia (CLL),
Chronic Myeloid Leukemia (CML),
Acute Lymphocytic Leukemia (ALL)
![Page 97: Natalia Komarova (University of California - Irvine) Somatic evolution and cancer](https://reader035.vdocuments.mx/reader035/viewer/2022062516/56649d575503460f94a36b49/html5/thumbnails/97.jpg)
CML• Chronic phase (2-5 years)
• Accelerated phase (6-18 months)
• Blast crisis (survival 3-6 months)
![Page 98: Natalia Komarova (University of California - Irvine) Somatic evolution and cancer](https://reader035.vdocuments.mx/reader035/viewer/2022062516/56649d575503460f94a36b49/html5/thumbnails/98.jpg)
Targeted cancer drugs
• Traditional drugs: very toxic agents that kill dividing cells
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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)
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Gleevec: a new generation drug
Bcr-Abl
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Gleevec: a new generation drug
Bcr-Abl Bcr-Abl
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Small molecule inhibitors
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Targeted cancer drugs
• Very effective
• Not toxic
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Targeted cancer drugs
• Very effective
• Not toxic
• Resistance poses a
problem
Bcr-Abl protein
Gleevec
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Targeted cancer drugs
• Very effective
• Not toxic
• Resistance poses a
problem
Bcr-Abl protein
Gleevec
Mutation
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Treatment without resistance
time
treatment
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Development of resistance
treatment
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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.
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Mutation network for developing resistance against n=3 drugs
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During a short time-interval, t, a cell of type Ai can:
• Reproduce faithfully with probability
Li(1-uj) t
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During a short time-interval, t, a cell of type Ai can:
• Reproduce faithfully with probability
Li(1-uj) t
• Produce one cell identical to itself, and a mutant cell of type Aj with probability Liuj t
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During a short time-interval, t, a cell of type Ai can:
• Reproduce faithfully with probability
Li(1-uj) t
• Produce one cell identical to itself, and a mutant cell of type Aj with probability Liuj t
• Die with probability Di t
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The method
]))((1)[()1()1(
])1)[(()1()1)(()(
ij1ji,j1,i
1-ji,j1,-iij
tjiDLttDjtDi
tiLuLjttuLittt
DyDLLuxyuLy
DxDLLxxt
)]([)1()( 22
Assume just one drug. ij(t) is the probability to have i susceptible and j resistantcells at time t.
x,y;tij(t)xjyi is the probability generating function.
))()(()1()1(
])1)[(()1()1)((
ij1ji,j1,i
1-ji,j1,-iij
jiDLtDjDi
iLuLjtuLit
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The method
]))((1)[()1()1(
])1)[(()1()1)(()(
ij1ji,j1,i
1-ji,j1,-iij
tjiDLttDjtDi
tiLuLjttuLittt
))()(()1()1(
])1)[(()1()1)((
ij1ji,j1,i
1-ji,j1,-iij
jiDLtDjDi
iLuLjtuLit
ij(t) is the probability to have i susceptible and j resistantcells at time t.
x,y;tij(t)xjyi is the probability generating function.
.)]([)1(
;)(2
2
DyDLLuxyuLy
DxDLLxx
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For multiple drugs:
niDxDLLiuxxiuLx
DxDLLxx
iiii
0 ,)]([)1(
;)(
12
02
00
i0, i1, …, im(t) is the probability to have is cells of type As at time t.
x0,x1,…,xm;ti0, i1, …, im(t) x0im …xm
i0
is the probability generating function.
0,1,…,1;tis the probability that at time t there are no cells of type Am
0,0,…,0;tis the probability that at time t the colony is extinct
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The method
.0)0(
,0 ,)]([)1(
;)(
12
02
00
i
iiii
x
niDxDLLiuxxiuLx
DxDLLxx
he probability that at time t the colony is extinct is (0,0,…,0;t) =xn
M(t),
where M is the initial # of cells and xn is the solution of
The probability of treatment failure is
)(lim1 txP Mntfail
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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?
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1. How important is pre-existence of mutants?
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Single drug therapy
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Single drug therapy
Pre-existance = Generation during treatment
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Single drug therapy
Pre-existance = Generation during treatment
Unrealistic!
![Page 122: Natalia Komarova (University of California - Irvine) Somatic evolution and cancer](https://reader035.vdocuments.mx/reader035/viewer/2022062516/56649d575503460f94a36b49/html5/thumbnails/122.jpg)
Single drug therapy
Pre-existance >> Generation during treatment
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Multiple drug therapies
Fully susceptible
Fully resistant
Partially susceptible
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Development of resistance
Fully susceptible
Partially susceptible
Fully resistant
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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
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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
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Single drug therapy
Low turnover cancer, D<<L
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Single drug therapy
High turnover cancer, D~L
More mutant colonies are produced, but theprobability of colony survival is proportionally smaller…
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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
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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
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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=0.01
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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=0.01
• The size of failure increases with # of drugs and decreases with mutation rate
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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
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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
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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
![Page 136: Natalia Komarova (University of California - Irvine) Somatic evolution and cancer](https://reader035.vdocuments.mx/reader035/viewer/2022062516/56649d575503460f94a36b49/html5/thumbnails/136.jpg)
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
![Page 137: Natalia Komarova (University of California - Irvine) Somatic evolution and cancer](https://reader035.vdocuments.mx/reader035/viewer/2022062516/56649d575503460f94a36b49/html5/thumbnails/137.jpg)
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
![Page 138: Natalia Komarova (University of California - Irvine) Somatic evolution and cancer](https://reader035.vdocuments.mx/reader035/viewer/2022062516/56649d575503460f94a36b49/html5/thumbnails/138.jpg)
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
![Page 139: Natalia Komarova (University of California - Irvine) Somatic evolution and cancer](https://reader035.vdocuments.mx/reader035/viewer/2022062516/56649d575503460f94a36b49/html5/thumbnails/139.jpg)
Log size of treatment failure
(a) 1 drug 2 drugs 3 drugs 4 drugs 5 drugs D/L=0.1 5.95 12.34 18.45 24.38 30.19 D/L=0.5 5.95 12.13 17.99 23.69 29.26 D/L=0.9 5.95 11.48 16.70 21.74 26.66 (b) 1 drug 2 drugs 3 drugs 4 drugs 5 drugs D/L=0.1 4.00 8.55 12.80 16.89 20.86 D/L=0.5 4.00 8.31 12.37 16.20 19.93 D/L=0.9 4.00 7.68 11.07 14.40 17.40
u=10-8
u=10-6
![Page 140: Natalia Komarova (University of California - Irvine) Somatic evolution and cancer](https://reader035.vdocuments.mx/reader035/viewer/2022062516/56649d575503460f94a36b49/html5/thumbnails/140.jpg)
Application for CML
• The model suggests that 3 drugs are needed to push the size of failure (1% failure) up to 1013 cells
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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…
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Multiple drug treatments
• For fast turnover cancers, adding more drugs will not prevent generation of resistance
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Size of failure for different turnover rates