1

Mathematical Modeling in Cancer Cell Biology

Takashi Suzuki

Division of Mathematical Science Department of Systems Innovation

Graduate School of Science Osaka University

Benign Tumor and Malignant TumorBenign Tumor and Malignant TumorBenign Tumor and Malignant TumorBenign Tumor and Malignant Tumor

■ Adenoma ： glandular system (benign)

■ Adeno-carcinoma ：invasion to surrounding tissues (malignant)

■ Stroma：tissue surrounding tumor, ex-cellular matrix, blood cell, glandular cell

2

Hematogenous Metastasis Hematogenous Metastasis Hematogenous Metastasis Hematogenous Metastasis

Figure 20-17 Molecular Biology of the Cell (© Garl and Science 2008)

cell deformationECM degradation

in-traverse

extra-traverse

basement membrane

epithelial cell

stroma

blood vessel

Cell MovementCell MovementCell MovementCell Movement

HT1080Human Adenosarcoma Cell

Physiological

• morphogenesis

• wound healing

• cellular immunity

Pathological

• inflammation

• arteriosclerosis

• cancer invasion,

metastasis

3

Moving Cell Moving Cell Moving Cell Moving Cell

moving direction

filopodia and lamellipodia on cell tip

Moving ProcessMoving ProcessMoving ProcessMoving Process

actin reorganization inside cell close to the membrane ~ a driving force of extrusion

4

Figure 16-7: Molecular Biology of the Cell (© Garla nd Science 2008)

Actin (cell bone) reorganization Actin (cell bone) reorganization Actin (cell bone) reorganization Actin (cell bone) reorganization

C. Sarmiento et al. J Cell Biol 180: 1219-1232 (2008)T. Oikawa et al. Nature Cell Biol 6: 420-426 (2004)M. Martinez-Quiles et al. Nature Cell Biol 3: 484-491 (2001)H. Miki et al. Nature 408: 732-735 (2000)H. Yamaguchi et al. PNAS 97: 12631-12636 (2000)

IRSp53

WIP

PIP3

WASP family proteinsactin control signal network

5

→identify factors

top down modeling

Insight from experiments

→integrated formulae →simulation check →understand the evens as a system

Lead the study beyond reductionism

Complicated networkCutting individual pathways may cause opposite effects

6

Model cascade related to cell dynamics

7

1. Quantized blowup mechanism 2. recursive hierarchy

3. field-particle duality

T. SuzukiMean Field Theories andDual Variation Atlantis Press Amsterdam-Paris, 2008

T. SuzukiFree Energy and Self-Interacting ParticlesBirkhauser, Boston, 2005

Non-linear/non-equilibrium～principles established by analysis

4. Nonlinear spectral mechanics

General model of cell dynamics (1)

Smoluchowski-ODE system

8

Function ODEPDE

General model of cell dynamics (2)

signal transmission, sensitivitychemical reaction material transport

Chaplain-Anderson 98

1) ODE-PDE system 2) diffusion, haptotaxis 3) chemical reaction 4) production, consumption

Invasion ～～～～ tissue level model

No peak formation in tissue level

9

basementmembrane

Blood bessel

EC (n)Endotherial cell

Avascular tumor

TAF (c)Tumor angiogenesis factor

receptor

haptotaxis

chemotaxis

ECM (f) Extra-cell matrix

hybrid simulation

Angiogenesis top down model

Individual (particles) - continuous (field) hybrid simulation

10

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m(x

,t)

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"solution-m.txt"

angiogenesismalignant

Invasion to Extra Cell Matrix

chemical substances, foods, X-rays, ultra-violet, radial rays, virus

gene mutation

Normal

clonalgrowth

Invasion to Blood Bessel

Metastasis to Organs

Cancer Events and Biological Hierarchy

organs

tissues

cells

organelles

proteins

DNA

Primarycarcinoma

Top down

Bottom up

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Individual (particles) - continuous (field) hybrid simulation

→ individual based simulation

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Summary

1. Cancer control is nowadays theme. Its growth is a combination of time series events occurring in biological hierarchy. Malignant tumor is characterized by metastasis which begins with invasion. There is formation of peaks called invadepodia in early stage of invasion

2. In the top down modeling one picks up several factors confirmed experimentally to combine a system of partial differential equations. Then one understands leading principles of the event using hybrid simulation

3. A chemtaxis system has evoked a huge mathematical studies whereby significant analytic methods and mathematical principles are formulated

Physical law

Mathematical model

Molecular interaction

precisioncoarsening

First principle

data

In vivo/in vitro

Experiment

Hybrid simulation

Top down modeling

Bottom up modeling

Cell organelle

protein

DNA

organ

tissue

Mean field

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Abstract

Several positive feedback loops are observed in early state of invasion. We focus on ECM degradation – actin reorganization and show how a bottom up model to search key paths in molecular level

1. protein signal network (14) 2. basement membrane degradation (2) 3. bottom up modeling (4) 4. key path search (2)

Malignant Tumor

１） infinite reproduction

２） motility

３) invasion to normal tissue

metastasis to other organs

1. Protein signal network

1/22

14

上皮細胞上皮細胞上皮細胞上皮細胞

がん細胞がん細胞がん細胞がん細胞

基底膜基底膜基底膜基底膜 基底膜基底膜基底膜基底膜

内皮細胞内皮細胞内皮細胞内皮細胞

血管血管血管血管

転移したがん転移したがん転移したがん転移したがん

原発巣から

の離脱

組織間移動

再増殖

Metastasis

2/22

XX

XX

XX

XX

Invasion beginning of metastasis

invadepodia

basement membrane

3/22

15

integrincadherin

ECM

A. Weaver (Vanderbilt) 06

Invasion - cell level

peaks②ECM degradation③adhesion regulation

①actin reorganization

F-actin

Invadopodia

4/22

Invadopodia

� cell membrane structure observed in vitro invasive cancer on ECM

� full of actin filaments active to ECM degradation

� observed in cell adenose acted by v-Src, 1989, Wen-Tien Chen

� Actin, adhesion, signal transmission, membrane transport, ECM degradation

� Important in cancer cell invasion, metastasis

5/22

16

Protein signals

6/22

MT1-MMP（（（（membrane type-1 matrix metalloproteinase) is a proteinase casting

tumor invasion apparatus

１．１．１．１．Manifestation in cancer cells Manifestation in cancer cells Manifestation in cancer cells Manifestation in cancer cells

２．２．２．２．Malignant cancer front Malignant cancer front Malignant cancer front Malignant cancer front

３．３．３．３．Collagen degradation for cancer cell Collagen degradation for cancer cell Collagen degradation for cancer cell Collagen degradation for cancer cell breeding and movement breeding and movement breeding and movement breeding and movement

４．４．４．４．MT1MT1MT1MT1----MMP induced cancer cells begin MMP induced cancer cells begin MMP induced cancer cells begin MMP induced cancer cells begin metastasis metastasis metastasis metastasis

Drill at the front of cancer cell

7/22

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MT1-MMP research

1. Manifestation from cancer cell, activate MMP2 (secretion type basement membrane protainase) H. Sato, T. Takino, Y. Okada, J. Cao, A. Shinagawa, E. Yamamoto, M. Seiki, Nature1994

2. MT1-MMP polymer + TIMP2-proMMP2 produces activated MMP2 A.Y. Strongin et. al. 1995, Imai-Ohuchi-Aoki-Fujii-Sato-Seiki-Okada, Cancer Res. 1996

3. [MT1-MMP] polymers Y. Itoh, A. Tamura, N. Ito, Y. Maru, H. Sato, T. Aoki, M Seiki, EMBO J. 2001K. Lehti et. al. 2002

4. collagen I, II, III, laminin 1, 5 degradation Ohuchi-Imai-Fujii-Seiki-Okada 1997Koshikawa-Gianneli-Cirulli-Miyazaki-Quaranta, Cell Biol. 2000

5. collagen IV（basement membrance）degradation by activated MMP2 Okada, et.al. 1990

6. EGF receptor activation by DIII filament produced by laminin gamma 2 chain cut by MT1-MMP N. Koshikawa et. al. to appear

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MT1-MMP (f)

EGFR

Actin (n)

ECM (c)

ECM* degradation

ECM

transport

Membrane

signal

F-actin G-actin

9/22

1, 6

4

2, 3

5

Cell

1. Schenk S, Hintermann E, Bilban M, et al. JCell Biol 2003; 161: 197-209.2. Koshikawa N, Minegishi T, Sharabi A,Quaranta V, Seiki M. J Biol Chem 2005; 280:88-933. Koshikawa N, Giannelli G, Cirulli V,Miyazaki K, Quaranta V. J Cell Biol 2000;148: 615-244. Sakurai-Yageta, M. et.al. J. Cell Biol.2008;181: 985-985. den Hartigh JC, van Bergen enHenegouwen PM, Verkleij AJ, Boonstra J. JCell Biol 1992; 119: 349-556. Koshikawa, N. et al. to appear

adhesion

ECM degradation

actin reorganization

Positive feedback loop

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Cell

Sub-cell

molecular switch

Allen-Cahn model

degradation activityPositive feedback

-1 +1

fluctuation

10/22

slow dynamics

c

ECM degradation

f*10

f**

c*

→activation switch →fluctuation

→localization

→invadopodia?

Activation loop

MT1-MMP EGFR

Actin (n)

ECM (c)

ECM* (c*)

degradation (3)

transport (2)

-1 1

f*

polymerization (4)

Extinction (1)

f

c=1

c=0

reinforce

n* f*

nucleus

MMP(f)

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(4)

(3)

(1)(2)

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ＥＧＦＲＥＧＦＲＥＧＦＲＥＧＦＲ ⇒⇒⇒⇒ MMP production

MT1-MMP EGFR

actin (n)

ECM (c)

ECM* (c*)

degradation

Cell Extra cell

c

f

transport

n

membrane

f*

polymerization

adhesion

extinction

f

n*

f*

n* c*f*

nucleus

MMP(f)

12/22

activatedMMP2

production

c*production

c*-signal

EGFR signal from ECM fragments

another signal

actin reorganization - ECM degradation positive feedback loop

MT1-MMP EGFR

Actin (n)

ECM (c)

ECM* (c*)

degradation (3)

inside cell

switching

ccell outside

c

f

ECM degradation

Interface

outside cell

transport (2)

n

membrane

membrane-1 1

f*

polymerization (4)

f*

extinction (1)c=1

n*

f**

c*

n* c*f*

nucleous

MMP(f)

c=0n

c

activatedMMP2

production (5)

separation (6)

(4)(6)

(3)

(3)

(1)(2)(5)

Double swicth

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"solution.txt" u 1:2:3 0.2 0.1

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"solution.txt" u 1:2:4 0.9 0.5 0.2 0.1

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"solution.txt" u 1:2:6 0.5 0.2 0.1

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"solution.txt" u 1:2:5 0.5 0.2 0.1

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"solution.txt" u 1:2:4 0.9 0.5 0.2 0.1

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"solution.txt" u 1:2:5 0.02

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"solution.txt" u 1:2:3 0.2 0.1

0.02

MMP concentration to membrane

2D simulation

14/22

n: actin c: ECMf: MMP

c*: ECM fragment

c*→f … localization (peak)c*→n→f … actin extension

Zn

Zn

Pro

ProMMP-2

MT1-MMP

N C

TIMP-2

“ Receptor ”

Zn

TIMP-2-free MT1-MMP

Zn

Active MMP-2

“ Activator ”

MT1-MMP activates ProMMP-2 at the moving front

15/22

2. Basement Membrane Degradation – Molecular Level

Collagen IV（basement membrane）

Collagen I, II, IIIlaminin 1, 5

Cell membrane

inside cell

outside cell

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ZnZn

MT1-MMP

Fibroblasts

Tumor Cells

Zn

proMMP-2

Zn

Zn

Degradation of Collagen I and IVPromotion of tumor growth and invasion

How MMP-2 expression is induced by tumor cells？

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(active)

MT1-MMP_MT1-MMP_TIMP2_MMP2

17/22

MT1-MMP_MT1-MMP

MT1-MMP, TIMP2, pro-MMP2 → activated MP2

secrete type basement membrane degradation protainase

TIMP2 initial[M]

MT1-MMP-MT1-MMP-TIMP2-MMP2 final[M]

Experimental insights → mathematical model →MMP2 activation research

Key path search → control factor identification

control pathway simulation → new clinical application

3. bottom up modeling

threshold

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a b c

ab bc cc

abc bcc

abcc bccb

abccb

abccba

hierarchy

Rules

・a→b・b→a,c・c→b, c・one direction ・two connectors in each protein・up to two polymers of ｃ

(a:MMP2 b:TIMP2 c:MT1-MMP)

a b

b a

c

c bc

articulatio

detachment

one peak

18/22

Mass reaction ⇒ ODE system

化学反応での物質濃度の時化学反応での物質濃度の時化学反応での物質濃度の時化学反応での物質濃度の時間変化を記述間変化を記述間変化を記述間変化を記述

初期値を与えると各時刻の物質濃度は

微分方程式に従って決定される

ある程度時間が経過すると

物質濃度の時間変化のない平衡状態に到達する

化学反応： A+B → C

[A],[B],[C] : concentration [M]

Initial concentration

reaction rate

19/22

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・ make the curve flat by mathematical operation (pathway cut)

20/22

activated MMP2 production

robust network

・total mass conservation of a,b,c elements

・bind-detachment dynamics of a(MMP2), b(TIMP2), c(MT1-MMP) based 12 proteins

・equilibrium realized as three-dimensional algebraic manifold embedded in 9-dimensional Euclidean space

・one peak of b-abcc curve on the manifold

⇒

key path search control pathway simulation

Path cut （１）

Initial b concentration [M]

Final abcc concentration [M]

Bind path ①①①① ②②②② ③③③③ ④④④④ cut

MMP1

4

bMMP2

MMP14･TIMP2

MMP14･TIMP2･MMP2

ab MMP14･MMP14

MMP14･MMP14･TIMP2

abcc MMP14･TIMP2･MMP14･TIMP2

MMP14･TIMP2･MMP14･TIMP2･MMP2

MMP14･TIMP2･MMP2･MMP14･TIMP2･MMP2

①①①①

②②②②

②②②②

③③③③③③③③

④④④④

④④④④

①①①①T2M2 +M14 → M14T2M2②②②②T2M2 +M14M14 → M14M14T2M2③③③③T2M2 +M14M14T2 → M14T2M14T2M2④④④④T2M2 +M14M14T2M2 → M14T2M2M14T2M2

original

peak extinction

21/22

4. Key path search

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Path cut (2)

æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ

æ

ææ æ æ æ æ æ æ æ

10-9 10-8 10-7 10-6 10-5

0

5.´10-8

1.´10-7

1.5 ´10-7

2.´10-7

Initial b concentration [M]

Final abcc concentration [M]

bind path ①①①① ②②②② ④④④④ cut

Narrow peak

MMP1

4

bMMP2

MMP14･TIMP2

MMP14･TIMP2･MMP2

ab MMP14･MMP14

MMP14･MMP14･TIMP2

abcc MMP14･TIMP2･MMP14･TIMP2

MMP14･TIMP2･MMP14･TIMP2･MMP2

MMP14･TIMP2･MMP2･MMP14･TIMP2･MMP2

①①①①

①①①①

②②②②

②②②②

③③③③③③③③

④④④④

④④④④

①①①①T2M2 +M14 → M14T2M2②②②②T2M2 +M14M14 → M14M14T2M2③③③③T2M2 +M14M14T2 → M14T2M14T2M2④④④④T2M2 +M14M14T2M2 → M14T2M2M14T2M2

original

22/22

• ①①①① ②②②② ③③③③ ④④④④ bind path cut → b-abcc peak extinction

• ①①①① ②②②② ④④④④ bind path cut → narrow peak

• any other pathway cut other than ①①①① ②②②② ③③③③ ④④④④ is non- effevtive

①①①① ②②②② ③③③③ ④④④④ … ab bind path

mathematically captured control protein

support medical insight

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Summary

1．MT1-MMP appears at invasion front, which is not only the origin of ECM degradation but also and activates the secrete basement membrane protainase MMP2

2. ECM fragments cut by MT1-MMP attach EGFR

3．This attachment induces at least two positive feedback loops inside cell, actin polymerization→MMP transport, direct MMP up-regulate

Conclusion

1．The above feedback fluctuation realizes invadopodia formation by a top down mathematical model

2. The roles of two feedback are clarified; actin polymerization-MMP transport …actin extension, direct MMP up-regulate … actin localization

3．MT1-MMP, TIMP2, MMP2 pathway mathematical analysis picks up key paths of MMP2 activation, which opens mathematical methods to create new clinics

References

1. T. Suzuki, Parallel optimization applied to MEG, JCAM 183 (2005) 177-190

2. T. Saito, M. Rouzimaimaiti, N. Koshikawa, M. Seiki, K. Ichikawa, T. Suzuki, A sub-cell mathematical model for initial stage of cancer invasion, preprint

3. K. Ichikawa, M. Rouzimaimaiti, T. Suzuki, Reaction diffusion equation with non-local term arises as a mean field limit of the master equation, to appear