Population and evolutionary dynamics of tumour growth on 1 , 2 Tom - - PowerPoint PPT Presentation

SLIDE 1

Introduction Motivation & model formulation Well-stirred systems Inhomogeneous systems Conclusions & summary

Population and evolutionary dynamics of tumour growth

Tom´ as Alarc´

• n1,2

1ICREA (Instituci´

• Catalana de Recerca i Estudis Avan¸

cats)

2Computational & Mathematical Biology Group

Centre de Recerca Matem` atica talarcon@crm.cat

• T. Alarc´
• n (ICREA & CRM, Barcelona, Spain)

Population dynamics & cancer ICMAT, Madrid, February 2016 1 / 37

SLIDE 2

Introduction Motivation & model formulation Well-stirred systems Inhomogeneous systems Conclusions & summary

Outline

Background Motivation & model formulation Well-stirred systems: From Invasion to Latency Systems with spatial inhomogeneities: The role of cell motility Conclusions & summary

• T. Alarc´
• n (ICREA & CRM, Barcelona, Spain)

Population dynamics & cancer ICMAT, Madrid, February 2016 2 / 37

SLIDE 3

Introduction Motivation & model formulation Well-stirred systems Inhomogeneous systems Conclusions & summary

Outline

Background Motivation & model formulation Well-stirred systems: From Invasion to Latency Systems with spatial inhomogeneities: The role of cell motility Conclusions & summary

• T. Alarc´
• n (ICREA & CRM, Barcelona, Spain)

Population dynamics & cancer ICMAT, Madrid, February 2016 3 / 37

SLIDE 4

Introduction Motivation & model formulation Well-stirred systems Inhomogeneous systems Conclusions & summary

Population-dynamical aspects of tumour growth2

1

Cancer is a disease of clonal evolution within the body1

1Nowell Nature (1976) 2Merlo et al. Nature Rev. Cancer (2006), Greaves & Maley Nature (2012), Gatenby et al. Nature Rev.

Cancer (2012)

• T. Alarc´
• n (ICREA & CRM, Barcelona, Spain)

Population dynamics & cancer ICMAT, Madrid, February 2016 4 / 37

SLIDE 5

Introduction Motivation & model formulation Well-stirred systems Inhomogeneous systems Conclusions & summary

Population-dynamical aspects of tumour growth2

1

Cancer is a disease of clonal evolution within the body1

2

Although this idea of cancer as an evolutionary problem is not new, it has received less attention than it perhaps deserves

1Nowell Nature (1976) 2Merlo et al. Nature Rev. Cancer (2006), Greaves & Maley Nature (2012), Gatenby et al. Nature Rev.

Cancer (2012)

• T. Alarc´
• n (ICREA & CRM, Barcelona, Spain)

Population dynamics & cancer ICMAT, Madrid, February 2016 4 / 37

SLIDE 6

Introduction Motivation & model formulation Well-stirred systems Inhomogeneous systems Conclusions & summary

Population-dynamical aspects of tumour growth2

1

Cancer is a disease of clonal evolution within the body1

2

Although this idea of cancer as an evolutionary problem is not new, it has received less attention than it perhaps deserves

3

The succession of somatic mutations to which cancer cells are subjected leads to clonal expansion and heterogeneity

1Nowell Nature (1976) 2Merlo et al. Nature Rev. Cancer (2006), Greaves & Maley Nature (2012), Gatenby et al. Nature Rev.

Cancer (2012)

• T. Alarc´
• n (ICREA & CRM, Barcelona, Spain)

Population dynamics & cancer ICMAT, Madrid, February 2016 4 / 37

SLIDE 7

Introduction Motivation & model formulation Well-stirred systems Inhomogeneous systems Conclusions & summary

Population-dynamical aspects of tumour growth2

1

Cancer is a disease of clonal evolution within the body1

2

Although this idea of cancer as an evolutionary problem is not new, it has received less attention than it perhaps deserves

3

The succession of somatic mutations to which cancer cells are subjected leads to clonal expansion and heterogeneity

4

Heterogeneity is a key aspect since it almost directly leads to drug resistance

1Nowell Nature (1976) 2Merlo et al. Nature Rev. Cancer (2006), Greaves & Maley Nature (2012), Gatenby et al. Nature Rev.

Cancer (2012)

• T. Alarc´
• n (ICREA & CRM, Barcelona, Spain)

Population dynamics & cancer ICMAT, Madrid, February 2016 4 / 37

SLIDE 8

Introduction Motivation & model formulation Well-stirred systems Inhomogeneous systems Conclusions & summary

Effects on targeted cance therapy3

Evolutionary dynamics of cancer poses a barrier to targeted therapy efficiency

3Gillies, Verduzco & Gatenby. Nature Rev. Cancer (2012)

• T. Alarc´
• n (ICREA & CRM, Barcelona, Spain)

Population dynamics & cancer ICMAT, Madrid, February 2016 5 / 37

SLIDE 9

Introduction Motivation & model formulation Well-stirred systems Inhomogeneous systems Conclusions & summary

Cancer as an evolutionary ecology problem

Competition between normal and cancer cells

• T. Alarc´
• n (ICREA & CRM, Barcelona, Spain)

Population dynamics & cancer ICMAT, Madrid, February 2016 6 / 37

SLIDE 10

Introduction Motivation & model formulation Well-stirred systems Inhomogeneous systems Conclusions & summary

Tumour dormancy4

Tumour dormancy in cancer refers to an extended period of growth restriction of undetected metastases

4Willis et al. Cancer Res. 70, 4310-4317 (2010)

• T. Alarc´
• n (ICREA & CRM, Barcelona, Spain)

Population dynamics & cancer ICMAT, Madrid, February 2016 7 / 37

SLIDE 11

Introduction Motivation & model formulation Well-stirred systems Inhomogeneous systems Conclusions & summary

Tumour dormancy4

Tumour dormancy in cancer refers to an extended period of growth restriction of undetected metastases Late relapse of breast cancer can occur as late as 25 years after resection of the primary tumour

4Willis et al. Cancer Res. 70, 4310-4317 (2010)

• T. Alarc´
• n (ICREA & CRM, Barcelona, Spain)

Population dynamics & cancer ICMAT, Madrid, February 2016 7 / 37

SLIDE 12

Introduction Motivation & model formulation Well-stirred systems Inhomogeneous systems Conclusions & summary

Tumour dormancy4

Tumour dormancy in cancer refers to an extended period of growth restriction of undetected metastases Late relapse of breast cancer can occur as late as 25 years after resection of the primary tumour Such long duration between resection and relapse is thought to be inexplicable from continual growth of secondary cancer

4Willis et al. Cancer Res. 70, 4310-4317 (2010)

• T. Alarc´
• n (ICREA & CRM, Barcelona, Spain)

Population dynamics & cancer ICMAT, Madrid, February 2016 7 / 37

SLIDE 13

Introduction Motivation & model formulation Well-stirred systems Inhomogeneous systems Conclusions & summary

Tumour dormancy4

Tumour dormancy in cancer refers to an extended period of growth restriction of undetected metastases Late relapse of breast cancer can occur as late as 25 years after resection of the primary tumour Such long duration between resection and relapse is thought to be inexplicable from continual growth of secondary cancer Three mechanisms for tumour dormancy have been hypothesised based on experimental models:

4Willis et al. Cancer Res. 70, 4310-4317 (2010)

• T. Alarc´
• n (ICREA & CRM, Barcelona, Spain)

Population dynamics & cancer ICMAT, Madrid, February 2016 7 / 37

SLIDE 14

Introduction Motivation & model formulation Well-stirred systems Inhomogeneous systems Conclusions & summary

Tumour dormancy4

Tumour dormancy in cancer refers to an extended period of growth restriction of undetected metastases Late relapse of breast cancer can occur as late as 25 years after resection of the primary tumour Such long duration between resection and relapse is thought to be inexplicable from continual growth of secondary cancer Three mechanisms for tumour dormancy have been hypothesised based on experimental models:

Solitary cells which persist in a quiescent state for months or even years post-resection

4Willis et al. Cancer Res. 70, 4310-4317 (2010)

• T. Alarc´
• n (ICREA & CRM, Barcelona, Spain)

Population dynamics & cancer ICMAT, Madrid, February 2016 7 / 37

SLIDE 15

Introduction Motivation & model formulation Well-stirred systems Inhomogeneous systems Conclusions & summary

Tumour dormancy4

Tumour dormancy in cancer refers to an extended period of growth restriction of undetected metastases Late relapse of breast cancer can occur as late as 25 years after resection of the primary tumour Such long duration between resection and relapse is thought to be inexplicable from continual growth of secondary cancer Three mechanisms for tumour dormancy have been hypothesised based on experimental models:

Solitary cells which persist in a quiescent state for months or even years post-resection Non-vascularised, non-angiogenic micro-metastases restricted to a size of 1 to 2 mm in diameter

4Willis et al. Cancer Res. 70, 4310-4317 (2010)

• T. Alarc´
• n (ICREA & CRM, Barcelona, Spain)

Population dynamics & cancer ICMAT, Madrid, February 2016 7 / 37

SLIDE 16

Introduction Motivation & model formulation Well-stirred systems Inhomogeneous systems Conclusions & summary

Tumour dormancy4

Tumour dormancy in cancer refers to an extended period of growth restriction of undetected metastases Late relapse of breast cancer can occur as late as 25 years after resection of the primary tumour Such long duration between resection and relapse is thought to be inexplicable from continual growth of secondary cancer Three mechanisms for tumour dormancy have been hypothesised based on experimental models:

Solitary cells which persist in a quiescent state for months or even years post-resection Non-vascularised, non-angiogenic micro-metastases restricted to a size of 1 to 2 mm in diameter Vascularised metastases that are held at an equilibrium size by the immune system

4Willis et al. Cancer Res. 70, 4310-4317 (2010)

• T. Alarc´
• n (ICREA & CRM, Barcelona, Spain)

Population dynamics & cancer ICMAT, Madrid, February 2016 7 / 37

SLIDE 17

Introduction Motivation & model formulation Well-stirred systems Inhomogeneous systems Conclusions & summary

Outline

Background Motivation & model formulation Well-stirred systems: From Invasion to Latency Systems with spatial inhomogeneities: The role of cell motility Conclusions & summary

• T. Alarc´
• n (ICREA & CRM, Barcelona, Spain)

Population dynamics & cancer ICMAT, Madrid, February 2016 8 / 37

SLIDE 18

Introduction Motivation & model formulation Well-stirred systems Inhomogeneous systems Conclusions & summary

Two strategies to use up available resources

Switch-like vs Bistable Response Curves

Switch-like Bistable

c0 c1 c2 Response Signal c0

1

c c2 Response Signal

• T. Alarc´
• n (ICREA & CRM, Barcelona, Spain)

Population dynamics & cancer ICMAT, Madrid, February 2016 9 / 37

SLIDE 19

Introduction Motivation & model formulation Well-stirred systems Inhomogeneous systems Conclusions & summary

Two cellular populations

Switch-like cells.

1

Birth probability    ∼ 1 when c > c2 ∼ 1/2 when c ≃ c0 ∼ 0 when c < c1 . Death probability    ∼ 0 when c > c2 ∼ 1/2 when c ≃ c0 ∼ 1 when c < c1

Bistable cells.

1

Two phenotypes: Proliferating and Quiescent

2

Proliferating cells duplicate or change phenotype to quiescent

3

Quiescent cells die or change phenotype to proliferating

4

The dynamics of the population is then controlled by the rates at which cells switch phenotype:

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.5 1

PAP(c)

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.5 1

c PPA(c)

• T. Alarc´
• n (ICREA & CRM, Barcelona, Spain)

Population dynamics & cancer ICMAT, Madrid, February 2016 10 / 37

SLIDE 20

Introduction Motivation & model formulation Well-stirred systems Inhomogeneous systems Conclusions & summary

Example: Model for the cell-cycle

Simple model for the G1/S transition

x1 =[Cdh1], x7 =[CycB/Cdk1], m = cell mass dx1 dt = (k31 + k32A)(1 − x1) J3 + 1 − x1 − k4mx7x1 J4 + x1 , dx7 dt = k1 − (k21 + k22x1)x7 m = 0.2 m = 1

0.2 0.4 0.6 0.8 1 10

−2

10

−1

10 x7 x1 0.2 0.4 0.6 0.8 1 10

−3

10

−2

10

−1

10 10

1

x7 x1

• T. Alarc´
• n (ICREA & CRM, Barcelona, Spain)

Population dynamics & cancer ICMAT, Madrid, February 2016 11 / 37

SLIDE 21

Introduction Motivation & model formulation Well-stirred systems Inhomogeneous systems Conclusions & summary

Population dynamics

The system we are considering is the competition between an incumbent population of switch-like cells and an invasor made out of bistable cells. These two populations compete by a shared resource, eg oxygen.

Oxygen concentration

∂c ∂τ = ∇2c − κc

NT

• k=1

χ(k), ∂c ∂x

• x=0

= h0, ∂c ∂x

• x=L

= −hL (1) with h0 − hL = Lh−1¯ κΩc0. The parameter Ω corresponds to the total cell population that a uniform concentration of oxygen c(x, t) = c0 is capable of sustaining. χ(k) is the population vector corresponding to the kth cellular type k = 1, . . . , NT. NT = 3: Switch-like cells, proliferating bistable cells, and quiescent bistable cells.

• T. Alarc´
• n (ICREA & CRM, Barcelona, Spain)

Population dynamics & cancer ICMAT, Madrid, February 2016 12 / 37

SLIDE 22

Introduction Motivation & model formulation Well-stirred systems Inhomogeneous systems Conclusions & summary

Population dynamics (cont.)

Master Equation for the cellular populations

∂P(χ(1), . . . , χ(NT ), t) ∂t =

NT

• k=1

N

• i=1
• E−

χ(k)

i

− 1

• Tχ(k)

i

+1|χ(k)

i P +

• E+

χ(k)

i

− 1

• Tχ(k)

i

−1|χ(k)

i P +

NT

• k=1

N

• i=1
• j=k
• E−

χ(k)

i

E+

χ(j)

i

− 1

• Tχ(k)

i

+1χ(j)

i

−1|χ(k)

i

χ(j)

i P+

• E+

χ(k)

i

E−

χ(j)

i

− 1

• Tχ(k)

i

−1χ(j)

i

+1|χ(k)

i

χ(j)

i P

• +

NT

• k=1

N

• i=1
• j∈i
• E−

χ(k)

i

E+

χ(k)

j

− 1

• Tχ(k)

i

+1χ(k)

j

−1|χ(k)

i

χ(k)

j P+

• E+

χ(k)

i

E−

χ(k)

j

− 1

• Tχ(k)

i

−1χ(k)

j

+1|χ(k)

i

χ(k)

j P

• (2)

where χ(k) is the population vector corresponding to the kth cellular type k = 1, . . . , NT and E±

χ(k)

i

f (χ(k)

i

) = f (χ(k)

i

± 1).

• T. Alarc´
• n (ICREA & CRM, Barcelona, Spain)

Population dynamics & cancer ICMAT, Madrid, February 2016 13 / 37

SLIDE 23

Introduction Motivation & model formulation Well-stirred systems Inhomogeneous systems Conclusions & summary

Population dynamics (cont.)

Dimensionless transition rates

Transition rate Event Tχ(1)

i

+1|χ(1)

i

= ǫ

eα0(c−1) eα0(c−1)+e−α0(c−1) χ(1) i

Switch-like cell division Tχ(1)

i

−1|χ(1)

i

= ǫ

e−α0(c−1) eα0(c−1)+e−α0(c−1) χ(1) i

Switch-like cell death Tχ(2)

i

+1|χ(2)

i

= ǫχ(2)

i

Bistable proliferating cell division Tχ(3)

i

+1χ(2)

i

−1|χ(3)

i

χ(2)

i

= ǫw0e−HPA(c)χ(2)

i

Proliferating-to-quiescent switch Tχ(2)

i

+1χ(3)

i

−1|χ(2)

i

χ(3)

i

= ǫw0e−HAP (c)χ(3)

i

Quiescent-to-proliferating switch Tχ(3)

i

−1|χ(3)

i

= ǫχ(3)

i

Bistable quiescent cell death Tχ(k)

i

−1χ(k)

j

+1|χ(k)

i

χ(k)

j

= ǫνkχ(k)

i

Cell migration ∀ k and ∀ j ∈< i >

Table : Dimensionless transition rates for our stochastic model of competition between switch-like and bistable populations.

• T. Alarc´
• n (ICREA & CRM, Barcelona, Spain)

Population dynamics & cancer ICMAT, Madrid, February 2016 14 / 37

SLIDE 24

Introduction Motivation & model formulation Well-stirred systems Inhomogeneous systems Conclusions & summary

Aims

Examine the effect of intracellular noise on the long-term behaviour of the competition between sitch-like and bistable populations Explore intracellular noise as a mechanism for latency (i.e. long-term survival of a non-invading mutant) Analyse the role of cell motility Study the therapeutic implications of our model

• T. Alarc´
• n (ICREA & CRM, Barcelona, Spain)

Population dynamics & cancer ICMAT, Madrid, February 2016 15 / 37

SLIDE 25

Introduction Motivation & model formulation Well-stirred systems Inhomogeneous systems Conclusions & summary

Outline

Background Motivation & model formulation Well-stirred systems: From Invasion to Latency Systems with spatial inhomogeneities: The role of cell motility Conclusions & summary

• T. Alarc´
• n (ICREA & CRM, Barcelona, Spain)

Population dynamics & cancer ICMAT, Madrid, February 2016 16 / 37

SLIDE 26

Introduction Motivation & model formulation Well-stirred systems Inhomogeneous systems Conclusions & summary

Low-noise behaviour: Invasion in well-stirred systems

Hybrid Gillespie simulation results

10 10

1

10

2

10

3

10

4

α 0,2 0,4 0,6 0,8 1 Fixation Probability

Figure : Simulation results for the fixation probability of the mutant (bistable) cell population as a function of the barrier-to-noise ratio α. The initial mutant-to-incumbent ratio y = 1/500. w0 = 102.

• T. Alarc´
• n (ICREA & CRM, Barcelona, Spain)

Population dynamics & cancer ICMAT, Madrid, February 2016 17 / 37

SLIDE 27

Introduction Motivation & model formulation Well-stirred systems Inhomogeneous systems Conclusions & summary

Supression of intracellular noise reduces population noise

Comparison between Gillespie simulations and the mean-field approximation

Ω = 100 Ω=1000

2 4 6 8 10 x 10

8

0.5 1 1.5 2 Time (dimensionless) c, np 2 4 6 8 10 x 10

8

0.5 1 1.5 2 Time (dimensionless) c, np

dc dτ = κΩ − κ (ns + np + nq) c (3) dns dτ = ǫ tanh (α0(c − 1)) ns (4) dnp dτ = ǫnp − ǫw0PPA(c)np + ǫw0PAP(c)nq (5) dnq dτ = −ǫnq + ǫw0PPA(c)np − ǫw0PAP(c)nq (6)

• T. Alarc´
• n (ICREA & CRM, Barcelona, Spain)

Population dynamics & cancer ICMAT, Madrid, February 2016 18 / 37

SLIDE 28

Introduction Motivation & model formulation Well-stirred systems Inhomogeneous systems Conclusions & summary

Linear analysis predicts abrupt transition between invasive a non-invasive regimes as a function of intracellular noise

Linear analysis

The ability of invasion by a small mutant population is provided by linearising of the dynamics of the mutant around the mutant-free steady state. In our case, this correponds to the steady state of Eqs. (3) and (4) with np = nq = 0, i.e. c = 1 and ns = Ω: dnp dτ = ǫnp − ǫw0e−α/2np + ǫw0e−α/2nq dnq dτ = −ǫnq + ǫw0e−α/2np − ǫw0e−α/2nq

10

−1

10 10

1

10

2

10

3

10

4

0.2 0.4 0.6 0.8 1 α λ

The blue line corresponds to w0 = 102, the green line to w0 = 103, and the red line to w0 = 104

• T. Alarc´
• n (ICREA & CRM, Barcelona, Spain)

Population dynamics & cancer ICMAT, Madrid, February 2016 19 / 37

SLIDE 29

Introduction Motivation & model formulation Well-stirred systems Inhomogeneous systems Conclusions & summary

High-noise behaviour: Emergence of latency

Hybrid Gillespie simulation results

10 10

1

10

2

10

3

10

4

α 0,2 0,4 0,6 0,8 1 Fixation Probability

Figure : Simulation results for the fixation probability of the mutant (bistable) cell population as a function of the barrier-to-noise ratio α. The initial mutant-to-incumbent ratio y = 1/500. w0 = 102.

• T. Alarc´
• n (ICREA & CRM, Barcelona, Spain)

Population dynamics & cancer ICMAT, Madrid, February 2016 20 / 37

SLIDE 30

Introduction Motivation & model formulation Well-stirred systems Inhomogeneous systems Conclusions & summary

The mutant population exhibits vanishing growth rate under elevated intracellular noise

Neutral invasion dynamics

2e+09 4e+09 6e+09 8e+09 1e+10 Time (rescaled) 0,2 0,4 0,6 0,8 1 c(t) 2e+09 4e+09 6e+09 8e+09 1e+10 Time (rescaled) 0,2 0,4 0,6 0,8 1 ns+np+nq, np+nq

We observe that c(t) = c0 Constant oxygen concentration implies that Switch-like cells+Proliferating cells+Quiescent cells ≃ Ω This allows us to reduce the system and study only the dynamics of the mutant population

• T. Alarc´
• n (ICREA & CRM, Barcelona, Spain)

Population dynamics & cancer ICMAT, Madrid, February 2016 21 / 37

SLIDE 31

Introduction Motivation & model formulation Well-stirred systems Inhomogeneous systems Conclusions & summary

Embedded branching process

In order to formulate an analytical theory, we consider the corresponding embedded branching process equivalent to our multi-type birth-death process.

Embedded branching process

1

The process of embedding consists of a coarse-graining of time variable carryied out in the following way.

2

After birth, each individual (of type j) lives for a length of time which is exponentially distributed with characteristic time τ (j)

c

=

• ǫ(1 + w0e−α/2)

−1 .

3

At the end of their life-span, each individual produces offspring according to the corresponding generating functions of the per-cell offspring probabilities. GP(x, y) = w0e−α/2 1 + w0e−α/2 y + 1 1 + w0e−α/2 x2 (7) GQ(x, y) = 1 1 + w0e−α/2 + w0e−α/2 1 + w0e−α/2 x (8)

4

For full specification of the age-dependent, embedded branching process, we need to give the age distribution which in this case is f (τ) = τ −1

c

e−τ/τc .

• T. Alarc´
• n (ICREA & CRM, Barcelona, Spain)

Population dynamics & cancer ICMAT, Madrid, February 2016 22 / 37

SLIDE 32

Introduction Motivation & model formulation Well-stirred systems Inhomogeneous systems Conclusions & summary

The embedded branching process is critical for high-noise intensity

Let M be the matrix whose entries, mi,j, correspond to the average offspring of type j produced by each individual of type j, simply by computing the derivatives of the generating functions: mPP = ∂xGP|x=y=1, mPQ = ∂yGP|x=y=1, mQP = ∂xGQ|x=y=1, and mQQ = ∂yGQ|x=y=1

Dominant eigenvalue of the offspring matrix

M =

• 2

1+ω ω 1+ω ω 1+ω

• whose dominant eigenvalue is:

λ1 = 1 +

• 1 + ω21/2

1 + ω where ω = w0e−α/2

10

−4

10

−3

10

−2

10

−1

10 10

1

10

2

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

ω λ1

• T. Alarc´
• n (ICREA & CRM, Barcelona, Spain)

Population dynamics & cancer ICMAT, Madrid, February 2016 23 / 37

SLIDE 33

Introduction Motivation & model formulation Well-stirred systems Inhomogeneous systems Conclusions & summary

Survival probability for critical branching processes (λ1 = 1)

Probability of eventual extincition (PE)

1

If λ1 > 1 ⇒ PE < 1

2

If λ1 ≤ 1 ⇒ PE = 1

Asymptotic behaviour of the survival probability (PS(t)) for critical branching processes

If λ1 = 1 then PS(t) ∼ σ−1t−1 as t → ∞ PS(t) ∼ t−1 implies that there is no characterstic survival time and that both short- and (aribitrarily) long-lived states are possible

• T. Alarc´
• n (ICREA & CRM, Barcelona, Spain)

Population dynamics & cancer ICMAT, Madrid, February 2016 24 / 37

SLIDE 34

Introduction Motivation & model formulation Well-stirred systems Inhomogeneous systems Conclusions & summary

The structure of the mutant population supresses population noise (σ)

σ is supressed as ω increases

By supressing population noise, PS(t), which is inversely proportional to σ, increases and, thus, so does the probability of latency, i.e. a state where growth is supressed by that can last for aribitraly long time

10

−4

10

−3

10

−2

10

−1

10 10

1

10

2

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

ω λσ

2

• T. Alarc´
• n (ICREA & CRM, Barcelona, Spain)

Population dynamics & cancer ICMAT, Madrid, February 2016 25 / 37

SLIDE 35

Introduction Motivation & model formulation Well-stirred systems Inhomogeneous systems Conclusions & summary

Summary of results for well-stirred systems

Analytical and numerical results show that:

1

Intracellular noise in the cells of the mutant (bistable) population controls their ability to invade the incumbent (switch-like) population

• T. Alarc´
• n (ICREA & CRM, Barcelona, Spain)

Population dynamics & cancer ICMAT, Madrid, February 2016 26 / 37

SLIDE 36

Introduction Motivation & model formulation Well-stirred systems Inhomogeneous systems Conclusions & summary

Summary of results for well-stirred systems

Analytical and numerical results show that:

1

Intracellular noise in the cells of the mutant (bistable) population controls their ability to invade the incumbent (switch-like) population

2

Bistable populations with low intracellular noise (i.e. transitions between phenotypes are determined by nutrient concentration) are very aggresive and invade the switch-like incumbent with high probability

• T. Alarc´
• n (ICREA & CRM, Barcelona, Spain)

Population dynamics & cancer ICMAT, Madrid, February 2016 26 / 37

SLIDE 37

Introduction Motivation & model formulation Well-stirred systems Inhomogeneous systems Conclusions & summary

Summary of results for well-stirred systems

Analytical and numerical results show that:

1

Intracellular noise in the cells of the mutant (bistable) population controls their ability to invade the incumbent (switch-like) population

2

Bistable populations with low intracellular noise (i.e. transitions between phenotypes are determined by nutrient concentration) are very aggresive and invade the switch-like incumbent with high probability

3

Bistable populations with high intracellular noise (i.e. transitions between phenotypes are erratic and less strongly dependent on nutrient concentration) are critical (in the sense of growth rate equal to zero) and thus the life expetancy of an invasion is arbitrarily long (latency)

• T. Alarc´
• n (ICREA & CRM, Barcelona, Spain)

Population dynamics & cancer ICMAT, Madrid, February 2016 26 / 37

SLIDE 38

Introduction Motivation & model formulation Well-stirred systems Inhomogeneous systems Conclusions & summary

Summary of results for well-stirred systems

Analytical and numerical results show that:

1

Intracellular noise in the cells of the mutant (bistable) population controls their ability to invade the incumbent (switch-like) population

2

Bistable populations with low intracellular noise (i.e. transitions between phenotypes are determined by nutrient concentration) are very aggresive and invade the switch-like incumbent with high probability

3

Bistable populations with high intracellular noise (i.e. transitions between phenotypes are erratic and less strongly dependent on nutrient concentration) are critical (in the sense of growth rate equal to zero) and thus the life expetancy of an invasion is arbitrarily long (latency)

4

The structure of the bistable population acts as a buffer for population noise, thus contributing to long-lived, growth supressed states to be more likely (latency)

• T. Alarc´
• n (ICREA & CRM, Barcelona, Spain)

Population dynamics & cancer ICMAT, Madrid, February 2016 26 / 37

SLIDE 39

Introduction Motivation & model formulation Well-stirred systems Inhomogeneous systems Conclusions & summary

Outline

Background Motivation & model formulation Well-stirred systems: From Invasion to Latency Systems with spatial inhomogeneities: The role of cell motility Conclusions & summary

• T. Alarc´
• n (ICREA & CRM, Barcelona, Spain)

Population dynamics & cancer ICMAT, Madrid, February 2016 27 / 37

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Introduction Motivation & model formulation Well-stirred systems Inhomogeneous systems Conclusions & summary

Low-intracellular noise: Mean-field inhomogeneous model

Mean-field inhomogeneous model

∂c ∂τ = ∂2c ∂x2 − κ(ns + np + nq)c, ∂c ∂x

• x=0

= h0, ∂c ∂x

• x=L

= −hL (9) ∂ns ∂τ = ǫν1 ∂2ns ∂x2 + ǫ tanh (α0(c − 1)) ns (10) ∂np ∂τ = ǫν2 ∂2np ∂x2 + ǫnp − ǫw0PPA(c)np + ǫw0PAP(c)nq (11) ∂nq ∂τ = ǫν3 ∂2nq ∂x2 − ǫnq + ǫw0PPA(c)np − ǫw0PAP(c)nq (12) with h0 − hL = Lh−1¯ κΩc0 no-flux boundary conditions for the equations for the cellular populations

Initial coniditions

c(x, t = 0) = 1, np(x, t = 0) = δ(x − L/2), nq(x, t = 0) = δ(x − L/2), and ns(x, t = 0) = Ω/L if x = L/2 and ns(x = L/2, t = 0) = Ω/L − 2

• T. Alarc´
• n (ICREA & CRM, Barcelona, Spain)

Population dynamics & cancer ICMAT, Madrid, February 2016 28 / 37

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Introduction Motivation & model formulation Well-stirred systems Inhomogeneous systems Conclusions & summary

Bistable mutant invades switch-like incumbent by migrating towards better-perfused regions

Numerical results

50 100 150 200 250 300 350 400 450 500 5 10 15 20 25 30 35 40 Time = 71607 x (cells diameters) c, ns, np, nq 50 100 150 200 250 300 350 400 450 500 5 10 15 20 25 30 35 Time = 99753 x (cells diameters) c, ns, np, nq 50 100 150 200 250 300 350 400 450 500 2 4 6 8 10 12 14 16 18 20 Time = 122872 x (cells diameters) c, ns, np, nq 50 100 150 200 250 300 350 400 450 500 5 10 15 20 25 Time = 163237 x (cell diameters) c, ns, np, nq

• T. Alarc´
• n (ICREA & CRM, Barcelona, Spain)

Population dynamics & cancer ICMAT, Madrid, February 2016 29 / 37

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Introduction Motivation & model formulation Well-stirred systems Inhomogeneous systems Conclusions & summary

Increased mutant motility increases its aggresiveness

Time to incumbent extinction as a function of mutant motility

¯ ν2 = ¯ ν3 = 10−1 ¯ ν2 = ¯ ν3 = 10−2

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 x 10

5

200 400 600 800 1000 1200 1400 Time (dimensionless)

Ns, Np, Nq

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 x 10

5

200 400 600 800 1000 1200 1400 1600 1800 2000 Time (dimensionless) Ns, Np, Nq

¯ ν2 = ¯ ν3 = 10−3 ¯ ν2 = ¯ ν3 = 10−4

1 2 3 4 5 6 7 x 10

5

200 400 600 800 1000 1200 1400 1600 1800 2000 Time (dimensionless) Ns, Np, Nq 2 4 6 8 10 12 x 10

5

200 400 600 800 1000 1200 1400 1600 1800 2000 Time (dimensionless) Ns, Np, Nq

• T. Alarc´
• n (ICREA & CRM, Barcelona, Spain)

Population dynamics & cancer ICMAT, Madrid, February 2016 30 / 37

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Introduction Motivation & model formulation Well-stirred systems Inhomogeneous systems Conclusions & summary

Non-motile mutants are non-invasive but generate long-term coexistence

Non-motile mutants: Stasis followed by transient bursts of mutant

1 2 3 4 5 6 7 x 10

5

100 200 300 400 500 600 700 800 900 1000 Time (dimensionless) Ns, Np, Nq

• T. Alarc´
• n (ICREA & CRM, Barcelona, Spain)

Population dynamics & cancer ICMAT, Madrid, February 2016 31 / 37

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Introduction Motivation & model formulation Well-stirred systems Inhomogeneous systems Conclusions & summary

Mere upregulation of motility is not enough for invasion

Competition between non-motile (incumbent) and motile (mutant) switch-like populations

¯ νm = 10−1 ¯ νm = 10−4

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 x 10

5

100 200 300 400 500 600 700 800 900 1000 Time (dimensionless) Ns, Nm 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 x 10

5

100 200 300 400 500 600 700 800 900 1000 Time (Dimensionless) Ns, Nm

• T. Alarc´
• n (ICREA & CRM, Barcelona, Spain)

Population dynamics & cancer ICMAT, Madrid, February 2016 32 / 37

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Introduction Motivation & model formulation Well-stirred systems Inhomogeneous systems Conclusions & summary

Summary of results for inhomogeneous systems

Numerical solution of our model equations shows that:

1

Spatial inhomogeneities have a non-trivial effect on the competition between our two cell types

• T. Alarc´
• n (ICREA & CRM, Barcelona, Spain)

Population dynamics & cancer ICMAT, Madrid, February 2016 33 / 37

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Introduction Motivation & model formulation Well-stirred systems Inhomogeneous systems Conclusions & summary

Summary of results for inhomogeneous systems

Numerical solution of our model equations shows that:

1

Spatial inhomogeneities have a non-trivial effect on the competition between our two cell types

2

Increased cell motility leads to more aggresive mutants

• T. Alarc´
• n (ICREA & CRM, Barcelona, Spain)

Population dynamics & cancer ICMAT, Madrid, February 2016 33 / 37

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Introduction Motivation & model formulation Well-stirred systems Inhomogeneous systems Conclusions & summary

Summary of results for inhomogeneous systems

Numerical solution of our model equations shows that:

1

Spatial inhomogeneities have a non-trivial effect on the competition between our two cell types

2

Increased cell motility leads to more aggresive mutants

3

Long-term coexistence between mutant and is possible provided mutant cells are not

• motile. This is in contrast to the well-stirred case, where no coexistence is possible

for low-noise mutant cells

• T. Alarc´
• n (ICREA & CRM, Barcelona, Spain)

Population dynamics & cancer ICMAT, Madrid, February 2016 33 / 37

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Introduction Motivation & model formulation Well-stirred systems Inhomogeneous systems Conclusions & summary

Outline

Background Motivation & model formulation Well-stirred systems: From Invasion to Latency Systems with spatial inhomogeneities: The role of cell motility Conclusions & summary

• T. Alarc´
• n (ICREA & CRM, Barcelona, Spain)

Population dynamics & cancer ICMAT, Madrid, February 2016 34 / 37

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Introduction Motivation & model formulation Well-stirred systems Inhomogeneous systems Conclusions & summary

Summary & Conclusions

Summary

Formulation, analytical and numerical results for a model of competition between two populations characterised by two different strategies (response curves) to turn nutrients into offspring Intracellular noise shown to be a major factor controlling the outcome of the competition Increased cell motility leads to more aggresive mutants although, on its own, it is insufficient to give the advantage to the mutant (previous transformation is required) Coexistence between the switch-like and low-noise, non-motile bistable cells is possible in spatially homogeneous systems. It consists of long periods of stasis followed by sudden, short-lived bursts of the mutant population High level of intracellular noise leads to critical growth dynamics: growth-supressed states but with aribitrarily long life span

Future work

1

Explore the therapeutic implications of our model

• T. Alarc´
• n (ICREA & CRM, Barcelona, Spain)

Population dynamics & cancer ICMAT, Madrid, February 2016 35 / 37

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Introduction Motivation & model formulation Well-stirred systems Inhomogeneous systems Conclusions & summary

Acknowledgements – Collaborators

People involved in this project

Pilar Guerrero (Dep. Mathematics, University College London, UK) Helen M Byrne (Computational Biology Group & Wolfson Centre for Mathematical Biology, University of Oxford, UK) Philip K Maini (Wolfson Centre for Mathematical Biology, University of Oxford, UK)

• T. Alarc´
• n (ICREA & CRM, Barcelona, Spain)

Population dynamics & cancer ICMAT, Madrid, February 2016 36 / 37

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Introduction Motivation & model formulation Well-stirred systems Inhomogeneous systems Conclusions & summary

Acknowledgements – Funding

Funding

Collaborative Mathematics Programme awarded by Obra Social La Caixa Foundation to the CRM Spanish Ministry of Science & Innovation (MICINN) under grant MTM2011-29342 Generalitat de Catalunya for funding under grant 2009SGR345 Mar´ ıa de Maeztu Unit of Excellence award to the Barcelona Graduate School of Mathematics

• T. Alarc´
• n (ICREA & CRM, Barcelona, Spain)

Population dynamics & cancer ICMAT, Madrid, February 2016 37 / 37