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…