natalia komarova (university of california - irvine) somatic evolution and cancer

Natalia Komarova

(University of California - Irvine)

Somatic evolution and cancer

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 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




• The offspring of such a cell may spread

• This is a beginning of cancer

Progression to cancer


Constant population


Advantageous mutant


Clonal expansion


Saturation


Advantageous mutant


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 u3u4

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

Stochastic dynamics on a selection-mutation network

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

Evolutionary selection dynamics

Fitness = 1

Fitness = r >1


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?


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


Fitness = 1

Fitness = r >1

Start from zero cell of the second type.What is the expected time until the second type takes over?


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

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

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

Two-hit process

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

10R21R 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

Stem cells, initiation of cancer and optimal tissue architecture

Colon tissue architecture

Colon tissue architecture

Crypts of a colon

Cancer of epithelial tissues

Cells in a crypt of a colon

Gut


Stem cells replenish thetissue; asymmetric divisions

Cells in a crypt of a colonGut



Gut

Proliferating cells dividesymmetrically and differentiate




Gut

Proliferating cells dividesymmetrically and differentiate

Differentiated cells get shed off into the lumen


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



• One of the earliest events in colon cancer is inactivation of the APC gene



• 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)

APC+/+ APC+/- APC-/-

X XX

Cellular origins of cancer

If a stem cell tem cell acquires a mutation, the whole crypt is transformed

Gut


If a daughter cell acquiresa mutation, it will probablyget washed out beforea second mutation can hit

Gut

What is the cellular origin of cancer?

Colon cancer initiation

First mutation in a daughter cell


• 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



• 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

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

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


• If the tissue is organized into compartments with stem cells and daughter cells, the risk of mutations is lower than in homogeneous populations


• 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


• 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=1m=2

m=4m=8

Crypt size isn=16

Probability to develop dysplasia

Time (individual’s age)

Pro

babi

lity

to d

evel

op d

yspl

asia

One stem cell

Many stem cells

The optimal solution is time-dependent!


Pro

babi

lity

to d

evel

op d

yspl

asia

Optimum:one stemcell

Optimum:many stem cells

Many stem cells

One stem cell

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

Evolutionary compromiseP

roba

bili

ty to

dev

elop

dys

plas

ia


One stem cell

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!

Evolutionary compromiseP

roba

bili

ty to

dev

elop

dys

plas

ia


One stem cell

Many stem cells

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)

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


• Very effective

• Not toxic


• Very effective

• Not toxic

• Resistance poses a

problem

Bcr-Abl protein

Gleevec


• Very effective

• Not toxic

• Resistance poses a

problem

Bcr-Abl protein

Gleevec

Mutation

Treatment without resistance

time

treatment

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, t, a cell of type Ai can:

• Reproduce faithfully with probability

Li(1-uj) t



Li(1-uj) t

• Produce one cell identical to itself, and a mutant cell of type Aj with probability Liuj t



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

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

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

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

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

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

Pre-existance = Generation during treatment

Unrealistic!

Single drug therapy

Pre-existance >> Generation during treatment

Multiple drug therapies

Fully susceptible

Fully resistant

Partially susceptible

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 theprobability of colony survival is proportionally smaller…


rate of cancer?

• Single drug therapies: the production of mutants is independent of 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=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=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

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

(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

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

natalia komarova (university of california - irvine) somatic evolution and cancer

Documents

organism slide

u2u2 slide

fittest slide

cancer saturation slide

beginning of cancer

wodarz slide

cancer clonal expansion

somatic evolution cells