Supplementary Materials - University of...

59
Supplementary Materials 1 Integration of multiple STAT pathways (Fig 1A main text) This section provides more details regarding the interactions between multiple STAT pathways, schematically represented in Fig 1A in the main text. Several STATs, for example STAT3 and STAT5 (1, 2), lead to the production of IFN-γ and IL-10. The molecular mechanism of IFN-γ and IL-10 production via STAT3 and STAT5 is as follows. Extracellular IL-2, IL-6 and IL-21 bind to their complementary receptors. The receptors remain in complex with JAKs. Binding of the interleukins to their receptors induces the autophosphorylation of the receptors and the bound JAKs (3). The phosphorylated receptor-JAK complexes can be dephosphorylated by SHP-1 (4). Phosphorylation of STAT3 is performed by JAK in IL-2R:JAK and IL-6R:JAK complexes, while STAT5 is phosphorylated by IL-2R:JAK and IL-21R:JAK (5-7). STAT3 and STAT5 are dephosphorylated by SHP-1 and SHP-2 phosphatases respectively (8, 9). The phosphorylated STAT3 and STAT5 then can form either homo- or hetero- dimers (5, 6). The dimers translocate into the cell nucleus and promote transcription of genes responsible for IFN-γ and IL-10 production. Both IFN-γ and IL-10 are degraded by various factors of dissociation such as the cytokine-receptor binding, diffusion and cleavage by metalloproteases (10), which, in our model, we denote as Mp1 and Mp2 for IFN-γ and IL-10, respectively. Other STAT pathways, in addition to STAT3 and STAT5, can also lead to the induction of IFN-γ and IL-10 production. The production of IFN-γ is induced by the following interleukins: IL-12, IL-21, IL-2 and IL-35 (11-14). Interleukins IL-12 and IL-35 activate STAT4 through the JAK-STAT pathway (6) while IL-21 and IL-2 activate STAT5 (15, 16). The production of anti- inflammatory IL-10 is up-regulated by STAT1, STAT3 and STAT6 (2, 17, 18). In particular, STAT1 is activated by IL-6 and IL-35; STAT3 by IL-2 and IL-6; STAT6 by IL-3 and IL-4 (6, 7, 19, 20).

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Page 1: Supplementary Materials - University of Exeterempslocal.ex.ac.uk/people/staff/vnb262/publ/stats/stats-supp.pdf · P2 SHP-1 phosphatase RpJ2P2 Phosphorylated IL-2 Receptor:JAK complex

Supplementary Materials

1 Integration of multiple STAT pathways (Fig 1A main text)

This section provides more details regarding the interactions between multiple STAT pathways,

schematically represented in Fig 1A in the main text. Several STATs, for example STAT3 and

STAT5 (1, 2), lead to the production of IFN-γ and IL-10. The molecular mechanism of IFN-γ

and IL-10 production via STAT3 and STAT5 is as follows.

Extracellular IL-2, IL-6 and IL-21 bind to their complementary receptors. The receptors remain

in complex with JAKs. Binding of the interleukins to their receptors induces the

autophosphorylation of the receptors and the bound JAKs (3). The phosphorylated

receptor-JAK complexes can be dephosphorylated by SHP-1 (4). Phosphorylation of STAT3 is

performed by JAK in IL-2R:JAK and IL-6R:JAK complexes, while STAT5 is phosphorylated

by IL-2R:JAK and IL-21R:JAK (5-7). STAT3 and STAT5 are dephosphorylated by SHP-1 and

SHP-2 phosphatases respectively (8, 9). The phosphorylated STAT3 and STAT5 then can form

either homo- or hetero- dimers (5, 6). The dimers translocate into the cell nucleus and promote

transcription of genes responsible for IFN-γ and IL-10 production. Both IFN-γ and IL-10 are

degraded by various factors of dissociation such as the cytokine-receptor binding, diffusion and

cleavage by metalloproteases (10), which, in our model, we denote as Mp1 and Mp2 for IFN-γ

and IL-10, respectively.

Other STAT pathways, in addition to STAT3 and STAT5, can also lead to the induction of

IFN-γ and IL-10 production. The production of IFN-γ is induced by the following interleukins:

IL-12, IL-21, IL-2 and IL-35 (11-14). Interleukins IL-12 and IL-35 activate STAT4 through the

JAK-STAT pathway (6) while IL-21 and IL-2 activate STAT5 (15, 16). The production of anti-

inflammatory IL-10 is up-regulated by STAT1, STAT3 and STAT6 (2, 17, 18). In particular,

STAT1 is activated by IL-6 and IL-35; STAT3 by IL-2 and IL-6; STAT6 by IL-3 and IL-4 (6,

7, 19, 20).

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2 Mathematical model

In this section, mathematical details of the model for STAT-STAT interactions are provided.

We derived the equations employed in the STAT3-STAT5 (Fig 1B), STAT3-STAT4 (Fig 1C)

and the combined STAT3-STAT4-STAT5 (Fig 5A) circuits. Table S1 shows the short names

and abbreviations used in our model. Table S1. Abbreviations used in the STAT phosphorylation model.

Abbreviation Meaning

I2 IL-2

RJ2 IL-2 Receptor complex with JAK

I2RJ2 IL-2 Receptor:JAK complex with bound IL-2

RpJ2 Phosphorylated IL-2 Receptor:JAK complex

P2 SHP-1 phosphatase

RpJ2P2 Phosphorylated IL-2 Receptor:JAK complex with SHP-1

I6 IL-6

RJ6 IL-6 Receptor:JAK complex

I6RJ6 IL-6 Receptor:JAK complex with bound IL-6

RpJ6 Phosphorylated IL-6 Receptor:JAK complex

P6 SHP-2 phosphatase

RpJ6P6 Phosphorylated IL-6 Receptor:JAK complex with SHP-2

I12 IL-12

RJ12 IL-12 Receptor:JAK complex

I12RJ12 IL-12 Receptor:JAK complex with bound IL-12

RpJ12 Phosphorylated IL-12 Receptor:JAK complex

P12 JAK phosphatase

RpJ12P12 Phosphorylated IL-12 Receptor:JAK complex with P12

phosphatase

I35 IL-35

RJ35 IL-35 Receptor:JAK complex

I35RJ35 IL-35 Receptor:JAK complex with bound IL-35

RpJ35 Phosphorylated IL-35 Receptor:JAK complex

P35 JAK phosphatase

RpJ35P35 Phosphorylated IL-35 Receptor:JAK complex with P35

phosphatase

I21 IL-21

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RJ21 IL-21 Receptor:JAK complex

I21RJ21 IL-21 Receptor:JAK complex with bound IL-21

RpJ21 Phosphorylated IL-21 Receptor:JAK complex

P21 JAK phosphatase

RpJ21P21 Phosphorylated IL-21 Receptor:JAK complex with P21

phosphatase

S3 STAT3

RpJ2S3 Phosphorylated IL-2 Receptor:JAK complex with STAT3

RpJ6S3 Phosphorylated IL-6 Receptor:JAK complex with STAT3

S3p Phosphorylated STAT3

P3 SHP-1 phosphatase

P3S3p Phosphorylated STAT3 complex with SHP-1

S4 STAT4

S4p Phosphorylated STAT4

RpJ12S4 Phosphorylated IL-12 Receptor:JAK complex with STAT4

RpJ35S4 Phosphorylated IL-35 Receptor:JAK complex with STAT4

P4 PTP phosphatase

P4S4p Phosphorylated STAT4 complex with PTP phosphatase

S5 STAT5

S5p Phosphorylated STAT5

RpJ2S5 Phosphorylated IL-2 Receptor:JAK complex with STAT5

RpJ21S5 Phosphorylated IL-21 Receptor:JAK complex with STAT5

P5 SHP-2 phosphatase

P5S5p Phosphorylated STAT5 complex with SHP-2

phosphatase

S33 STAT3:STAT3 homodimer

S34 STAT3:STAT4 heterodimer

S44 STAT4:STAT4 homodimer

S35 STAT3:STAT5 heterodimer

S55 STAT5:STAT5 homodimer

Gg Gene responsible for IFN-γ production

S44Gg IFN-γ gene complex with STAT4:STAT4 homodimer

S55Gg IFN-γ gene complex with STAT5:STAT5 homodimer

S44S55Gg IFN-γ gene complex with STAT4:STAT4 and

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STAT5:STAT5

Ig IFN-γ

Mp1 Metalloprotease that cleaves IFN-γ

IgMp1 Metalloprotease complex with IFN-γ gene

Ign Non-active IFN-γ

G10 IL-10 gene

S33G10 IL-10 gene complex with STAT3:STAT3 homodimer

Sp1 SP1 transcription factor

Sp1a SP1 transcription factor in active form

S33Sp1aG10 IL-10 gene complex with STAT3:STAT3 and Sp1a

Sp1aG10 Complex of Sp1a with IL-10 gene

Mp2 Metalloprotease that cleaves IL-10

I10Mp2 Metalloprotease complex with IL-10 gene

I10n Non-active IL-10

2.1 Model for the STAT3-STAT5 circuit

The biochemical reactions involved in the STAT3-STAT5 circuit (Fig 1B) are given by:

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2

3

4 5

6

7 8

9

10 11

12

13 14

15

16

18

1

2

17

1

3

I2 RJ2 p

p P2 p P2 P2

I6 RJ6 p

p P6 p P6 P6

I2RJ2 R J2

R J2 R J2 RJ2

I6RJ6 R J6

R J6 R J6 RJ6

I21RJ21I21 RJ21 p

p P21 p P21 P21

R J21

R J21 R J21 RJ21

R J2 S3 R J 3p p p2S R

f

f

f

f

f

f

f

f

f

f

f

f

a

a

f

f

f

f

f

a

f

→←

→←

→←

→←

→←

→←

→←

+ →

+

+ +

+

+ +

+

+ +

4 5

6

7 8

9

10 11

12

13 14

15

17

18

20

21

2

16

1

2

9

J2 S3p

R J6 S3 R J6S3 R J6 S3p

P3 S3p P3S3p P3 S3

R J2 S5 R J2S5 R J2 S5p

R J21 S5 R J21S5 R J21 S5p

P5 S5p P5S5 P5 S5

S3 S3p S33

S

p p p

p p p

p p

3p S5p S35

S5 S5

p

p

p

p

a

a

a

a

a

a

a

a

a

a

a

a

a

a

a

a

a

a

a

→←

→←

→←

→←

→←

→←

→←

+

+ → +

+ → +

+ → +

+ → +

+ → +

+

+

+23

24

1

2

p

I2 Sp1

55

1

S

I2Sp

a

a

l

l

→←

→←+

(S1)

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3

4

3

4

1

2

2

5

3 4

1

5

6

7

8

9

8

9

6

7

I2Sp1 CD46 Sp1a

CD46 Sp1 CD46Sp1

CD46Sp1 I2 Sp1a

S55 Gg S55Gg

S55Gg Ig

Ig Mp1 IgMp1 Ign Mp1

S33 G10 S33G10

S33G10 Sp1a S33Sp1aG10

Sp1a G10 Sp1aG10

Sp1aG10 S33 S

l

l

l

l

l

l

k

k

k

k

k

k

k

k

k

k

k

k

l

k

→←

→←

→←

→←

→←

→←

→←

→←

→←

+

+

+

+

+ +

+

+

+

+

6

6

6

10 11

12

33Sp1aG10

S33G10 I10

Sp1aG10 I10

S33Sp1aG10 I10

I10 Mp2 I10Mp2 I10n Mp2k

k

l

l

l

k→←

+→+

The system of reactions (S1) can be divided into three major subsystems of interactions: i)

Cytokine-receptor interactions, ii) STAT phosphorylation and dimerization, iii) Cytokine

production.

2.1.1 Cytokine-receptor interactions In the most general case the reactions can be written as follows:

2

3

4 5

6

1

C RJ

Rp

CRJ RpJ,

J P RpJP RJ P,

q

q

q

q

q

q

→←

→←

+

+ +

(S2)

where C is cytokine, RJ is Receptor:JAK complex, P is phosphatase and small p denotes

phosphorylated state.

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The ODEs for the system (S2):

[ ] [ ][ ] ( )[ ]

[ ] [ ] [ ][ ] [ ]

[ ] [ ][ ] ( )[ ]

2 3

2 4 6

4 5 6

1 ,

,

.

C q q

R

d CRJ q RJ CRJdtd RpJ q CRJ q P q RpJPdtd RpJP q RpJ P

pJ

q q RpJPdt

+

+

= −

= − +

= −

(S3)

Corresponding conservation equations:

[ ] [ ] [ ] [ ][ ] [ ]

,

,T

T

R RJ RCRJ RpJ

P P

pJP

RpJP

+ +

= +

= + (S4)

where TR and TP are the total amounts of receptors and phosphatase, respectively. Here we

neglected STAT-receptor interactions since STAT proteins do not have a significant impact on

receptor dephosphorylation.

Equations (S4) can be written as follows:

,,T

T

PR

pα β γ ω

ω

+ +

=

+

+

= (S5)

where [ ] [ ] [ ] [ ] [ ] [ ] , , , , , .RJ CRJ RpJ RpJP p P c Cα β γ ω= = = = = =

The ODEs (S3) can be rewritten in the following way:

( )

( )

2 3

2 4 6

6

1

4 5

,

,

.

d qdtd q q p qdtd

c q

q

q

q qpdt

β α β

γ β γ ω

ω γ ω

= −

= − +

+

+= −

(S6)

We need to find steady-state solutions of Equations (S6):

( )

( )

2 3

2 4 6

4 5 6

10 ,0 ,0 ,

,.

T

T

qq q p qc q q

q qq

P pR

p

α β

β γ ω

γ

α β γ ω

ω

ω

= −

= − +

= −

+ +

=

+

=

+

+

+

(S7)

We found the concentrations of the complexes:

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( )

( )

5 6 2

4

2 3 2

5 5

2 2

2 3 1

1 3 2

,

,

,

TT

T T

T

PP

qq q

q q Q

Q Q Q

Q

P Pq qq q Q Pq c Qc

γ γω

γγ

γ γβ ω

γ γ

β γα

γ

=

= = =

= =

=+ ++

+ +

++

(S8)

where 2 31

1qq qQ +

= and 5 62

4qq qQ +

= are the Michaelis constants for phosphorylation and

dephosphorylation, respectively, and 23

5

Q qq

= .

We can write the following equation using the conservation Equation for the receptor (S5):

1

2 3 3

1 1 .TT

QRQ Q QP

γγ

+ +⎞⎛

= + ⎟⎜+ ⎝ ⎠ (S9)

As a result, we obtained a quadratic equation: 20 ,γ γ δχ+= − (S10)

where 12

3 3

1 1TTQQ RQc Q

Pχ⎞⎛

= +− + ⎜⎝

+ ⎟⎠, 2TR Qδ = .

The solution of Equation (S10) is:

2 42 2

χ δγ

χ+

+= − (S11)

Equation (S11) can be rewritten as follows:

21 4 ,2 c c

ρ ργ λ δ λ

⎞⎛ ⎞⎛ ⎟⎜ + + − −⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠= (S12)

where 23

1 1T TQ R PQ

λ =⎞⎛

+ ⎟⎜⎝

+⎠

− and 1

3

TP QQ

ρ = .

Using Equation (S12) we can now write for [ ]2RpJ ,[ ]6RpJ and [ ]21RpJ in non-dimensional

form respectively:

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[ ][ ]

[ ]

[ ][ ]

[ ]

[ ][ ]

1

1 1

2

1

1 1

3

2 2

2

3

2 2

5

2

2 2

4

4 4

36

3

1 12

22

1 1 42

,2

1 16

62

1 1 4

2 2

2 2 2

6 6

6 6 6

21

6,

2

1 12

21

211

2

t t

t t t

t t

t t t

t t

M r p

M r p r M

M r p

M r

Mn i n

w

Mn i n

Mn i n

w

Mn i n

Mn i

M

wM r p

n

p r

⎞⎛+ + ⎟⎜

⎝ ⎠= − +

⎞⎛ ⎞⎛+ + +⎟⎜ ⎟⎜⎜ ⎟⎝ ⎠⎝ ⎠+

⎞⎛+ + ⎟⎜

⎝ ⎠= − +

⎞⎛ ⎞⎛+ + +⎟⎜

− +

− +

− +

− + ⎟⎜⎜ ⎟⎝ ⎠⎝ ⎠+

⎞⎛+ + ⎟⎜

⎝=

−⎠

+

− +

[ ]5

6 6

2

3 3

1 1 421

,

21 21 2

2

1t t tM

n i nM r p r M

⎞⎛ ⎞⎛+ + +⎟⎜ ⎟⎜⎜ ⎟⎝

− +⎠⎝ ⎠+

(S13)

where

[ ] [ ] [ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ]

[ ] [ ]

2 3 5 6 22 1

1 4 5

8 9 11 12 83 4 2

7 10 11

12 62 2, , ,3 3 3 3 3

216 6 , ,3 3 3 3

213

2 2 2 , 2 , 2 , , , 6

3 36

6 , 6 , 6 , , , 213 3

2121 , 21

T T

T T T T T T T

T T

T T T

t t

t t

t

T T T

T

T

RpJi w r p M M n i

f S f S fRpJ

w r p M M n if S f

I IR

S f

P f f f f fS S S S S

IR P f f f f

RpJ

fS S S S

RS

w r

= = = = = =

= = = = = =

= =

+ += =

+ +=

14 15 17 18 145 6 3

13 16 17

, 21 ,21 .3 3

, ,3 3

T

T T T Ttp M M n

f SP f f f f f

S S f S f+

= = =+

=

2.1.2 STAT phosphorylation and dimerization The ODEs describing biochemical reactions in the STAT subsystem:

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[ ] [ ][ ] ( )[ ]

[ ] [ ][ ] ( )[ ]

[ ] [ ] [ ] [ ][ ] [ ]

[ ] [ ] [ ][ ] [ ]

[ ] [ ][ ] ( )[ ]

[ ] [ ][ ] ( )[ ]

1 2 3

2

1

4 5 6

5 7 9

220 21 22

7 8 9

10 11 1

9

2

2 3 2 3 2 3 ,

6 3 6 3 6 3 ,

3 2 3 6 3 3 3 3 3

2 3 2 33 3 5 35 ,

3 3 3 3 3 3 ,

2 5 2 5 2 5 ,

p p

p p p

d RpJ S a RpJ S a a RpJ Sdtd RpJ S a RpJ S a a RpJ Sdtd S a RpJ S a RpJ S a P S p a P Sdt

a S a S a S S a Sd P S a P S a P Sdtd RpJ S a RpJ S a a RpJ Sdt

p p a p

= − +

= − +

= + − + −

− + − +

= − +

= − +

[ ] [ ][ ] ( )[ ]

[ ] [ ] [ ] [ ][ ] [ ]

[ ][ ] [ ] [ ] [ ]

[ ] [ ][ ] ( )[ ]

[ ] [ ] [ ]

[ ] [ ][ ] [ ]

13 14 15

11 14 18

21 22 23 24

17 18

20

21

16

2

1

22

6

219

21 5 21 5 21 5 ,

5 2 5 21 5 5 5 5 5

3 5 35 2 5 2 55 ,

5 5 5 5 5 5 ,

33 3 33 ,

35 5 35 ,

5

3

d RpJ S a RpJ S a a RpJ Sdtd S a RpJ S a RpJ S a P S a P Sdt

a S S a S a S p a Sd P S a P S a a P Sdtd S a S p a Sdtd S a S p a Sdtd Sdt

p p p

p p

p p p

S p

= − +

= + − + −

− + − +

= − +

= −

= −

[ ] [ ] [ ]223 245 5 55 .a S p a S= −

(S14)

Conservation equations are given by:

[ ] [ ] [ ] [ ] [ ] [ ][ ] [ ] [ ] [ ] [ ] [ ][ ][ ]

3 3 3 2 33 35 2 3 6 3 [ 3 3 ],

5 5 5 2 55 35 2 5 21 5 [ 5 5 ],

3 3 [ 3 3 ],

5 5 [ 5 5 ].

T

T

T

T

S S S p S S RpJ S RpJ S P S

S S S p S S RpJ S RpJ S P S

P P P S

P P P S

p

p

p

p

= + + + + + +

= + + + + + +

= +

= +

(S15)

Then we can normalize Equations (S15) to 3TS :

[ ] [ ] [ ] [ ] [ ] [ ] [ ][ ] [ ] [ ] [ ] [ ] [ ] [ ][ ] [ ][ ] [ ]

1 3 3 2 33 35 2 3 6 3 3 3

5 5 5 2 55 35 2 5 21 5 5 5

3 3

,

,

3 3

5 5 5 ,

,

5

t

t

t

s s s s w s w s p s

s s s s s w s w s p s

p p p

p p

p

s

p p p

p

s

p

p

= + + + + + +

= + + + + + +

= +

= +

(S16)

where

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[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]3 3 2 3 6 3 3 3 53 , 3 , 2 3 , 6 33 3 3 3

, 3 3 ,3

,3

5 T

T T T T T Tt

p pp p

SS S RpJ S RpJ

S SS P S Ss s w s w s p s s

S S S= = = = = =

[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]5 5 2 5 21 5 5 5 5 , 5 , 2 5 , 21 5 ,

3 35

35 ,

3 3T T T T T

S S RpJ S Rpp J S P Ss s w s w

pp p

S S S S Ss p s= = = = =

[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]33 35 55 3 533 , 35 , 5

3 3 3 3, , 5 ,

35 3

T T T T TS S S SS S S P P

s s s p pS

= = = = = 33 5 .

33 , 5T T

Tt

Tt

P Pp pS S

= =

The ODEs (S14) can be written in non-dimensional form as follows:

[ ] [ ][ ] [ ]

[ ] [ ][ ] ( )[ ]

[ ] [ ] [ ] [ ][ ] [ ]

[ ] [ ] [ ][ ] [ ]

[ ] [ ][ ] ( )[ ]

[ ] [ ][ ] ( )[ ]

[ ] [ ][ ] ( )

1

3 4 5

2 4 6 8

18 19 20 21

6 7 8

9 10 11

12 1 1

2

3 4

2 3 2 3 2 3 ,

6 3 6 3 6 3 ,

3 2 3 6 3 3 3 3 3

2 3 2 33 3 5 35 ,

3 3 3 3 3 3 ,

2 5 2 5 2 5 ,

21 5 21 5 21

d w s m w s w sdd w s m w s m m w sdd s p m w s m w s m p s m p sd

m s p m s m s p s p m sd p s p m p s p m m p s pdd w s m w s m m w sdd w s m

p p

w s m m wd

τ

τ

τ

τ

τ

τ

= −

= − +

= + − + −

− + − +

= − +

= − +

= − + [ ]

[ ] [ ] [ ] [ ][ ] [ ]

[ ][ ] [ ] [ ] [ ]

[ ] [ ][ ] ( )[ ]

[ ] [ ] [ ]

[ ] [ ][ ] [ ]

[ ] [ ] [ ]

10 13 15 17

16 17

2

220 21 22 23

15

18

20 21

222 23

19

5 ,

5 2 5 21 5 5 5 5 5

3 5 35 2 2 55 ,

5 5 5 5 5 5 ,

33 3 33 ,

35 3 5 35 ,

55 5 5

5

5 ,

s

d s m w s m w s m p s m p sd

m s s m s m m sd p s p m p s m m p sdd s m s m sdd s m s p s

p p p

p p s p

p

p m sdd s m

p

m

p

sd

ps

τ

τ

τ

τ

τ

= + − + −

− − +

= − +

= −

= −

+

=

(S17)

where

( ) 12 3 1

2 3 2 3 2 3 2 3 2 3

2 3 2 3 2

2 4 5 62 3 4 5

7 8 9 10 11 126 7 8 9 10 11

13

3 2 3 2 3 2 3

2 3

14

2

1512 13 14

3 2

, 3 , , 3 , , ,

3 , , , 3 , , ,

3 , ,

T T

T T

T

t a a m S m m S m ma a a a a a a a a a

m S m m m S m ma a a a a a a a a a a a

m

a a a a a

a a a a a a

a S m ma a a

a aa a

τ = + = = = = =+ + + + +

= = = = = =+ + + + + +

= = =+ + +

16 17 1815 16 17

19 20 21 22 23 2418 19 20 21

3

22 23

2 3 2 3 2 3

2 3 2 3 2 3 2 3 2 3 2 3

, 3 , , ,

3 , , 3 , , 3 , .

T

T T T

m S m ma a a a a a a

m S m m S m m S ma a a a a a a a

a a a

a a a a a aa a a a

= = =+ + +

= = = = = =+ + + + + +

We need to find steady-state solutions of Equations (S17):

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[ ][ ] [ ][ ][ ] ( )[ ]

[ ] [ ] [ ] [ ][ ] [ ]

[ ] [ ] [ ][ ] [ ][ ][ ] ( )[ ][ ][ ] ( )[ ][ ][ ] ( )[ ]

[ ] [ ] [ ] [ ]

1

3 4 5

2 4 6 8

18 19 20 21

6 7 8

9 10 11

12 13 14

10 1

2

13 5

0 2 3 2 3 ,

0 6 3 6 3 ,

3 2 3 6 3 3 3 3 3

2 3 2 33 3 5 35 ,

0 3 3 3 3 ,

0 2 5 2 5 ,

0 21 5 21 5 ,

5 2 5 21 5 5

m w s w s

m

p

w s m m w sd s p m w s m w s m p s m p sd

m s p m s m s p s p m s

m p s p m m p s p

m w s m m w s

m w s m m w sd s m w s m w s m p

p

dp

τ

τ

= −

= − +

= + − + −

− + − +

= − +

= − +

= − +

= + − [ ] [ ]

[ ][ ] [ ] [ ] [ ][ ][ ] ( )[ ][ ] [ ][ ][ ] [ ][ ] [ ]

220 21 22 23

15

18

20 21

1

2

7

16 17

219

22 23

5 5 5

3 5 35 2 2 55 ,

0 5 5 5 5 ,

0 3 33 ,

0 3 5 35 ,

0 5 55 .

5

s m p s

m s s m s m m s

m p s m m p s

m s m s

m s p s p m s

m s m

p p

p p s p

p

p

s

p

p

+ −

− − +

= − +

= −

= −

= −

+

(S18)

Equations (S18) can be simplified as follows:

[ ][ ] [ ][ ][ ] ( )[ ]

[ ] [ ] [ ] [ ]

[ ][ ] ( )[ ][ ][ ] ( )[ ][ ][ ] ( )[ ]

[ ] [ ] [ ] [ ]

[ ][ ] ( )[ ][ ]

1

3 4 5

2 4 7

6 7 8

9 10 11

12 13 14

16

16 17

2

10 13

15

8 191

0 2 3 2 3 ,

0 6 3 6 3 ,

3 2 3 6 3 3 3 ,

0 3 3 3 3 ,

0 2 5 2 5 ,

0 21 5 21 5 ,

5 2 5 21 5 5 5 ,

0 5 5 5 5 ,

0 3 3

m w s w s

m w s m m w sd s p m w s m w s m p sdm p s p m m p s p

m w s m m w s

m w s m m w sd s m w s m w s m p s

p

p p

p p

p

dm p s m m p s

m s m s

τ

τ

= −

= − +

= + −

= − +

= − +

= − +

= + −

= − +

= − [ ][ ][ ] [ ][ ] [ ]

20 21

222 23

3 ,

0 3 5 35 ,

0 5 55 .p

m s p s p m s

m s m s

= −

= −

(S19)

Then using conservation Equations (S16) and System (S19) we obtained:

[ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ]

2 4 7

10 1 63 1

3 2 3 6 3 3 3 ,

5 2 5 21 5 5 5 ,

pd s p m w s m w s m p sdd s m w s m w s sp pm pd

τ

τ

= + −

= + −

(S20)

where

Page 13: Supplementary Materials - University of Exeterempslocal.ex.ac.uk/people/staff/vnb262/publ/stats/stats-supp.pdf · P2 SHP-1 phosphatase RpJ2P2 Phosphorylated IL-2 Receptor:JAK complex

[ ] [ ]

[ ]

[ ] [ ]

[ ]

[ ] [ ]

[ ] [ ][ ]

[ ] [ ]

[ ] [ ][ ]

[ ] [ ][ ]

[ ] [ ][ ]

[ ] [ ][ ]

[ ] [ ] [ ] [ ] [ ]

[ ]

7 8

6

16 17

15

18

20

21

22

23

3

4 5

9

10 11

12

13 14

2

9

3

1

2

1

1

33 3 3 ,

3

55 5 5 ,

5

33 3 ,

35 3 5 ,

55 5 ,

2 3 2 3 ,

6 3 6

2 5 2

21 5 21

3 2 33 35 3 33

1 2

3 ,

5 ,

5 ,

1

t

t

s pp s p m m s p

ms p

p s p m m s pm

ms s pmms s s pmms s pm

w s m wmw s w

m mmw s w

m mmw s w

m ms s s p s

s

p

p

p

s

s

s

s

p pmm w

=+

+

=+

+

=

=

=

=

=+

=+

=+

− − −=

+ +

[ ]

[ ] [ ] [ ] [ ] [ ]

[ ] [ ]

4 5

9 12

10 11 13 14

,6

5 5 2 55 35 5 55 .

1 2 21

t

wm m

s s s s p ss m mw w

m m m m

p p+

− − −

+

−=

+ ++

(S21)

We can rewrite Equations (S21):

Page 14: Supplementary Materials - University of Exeterempslocal.ex.ac.uk/people/staff/vnb262/publ/stats/stats-supp.pdf · P2 SHP-1 phosphatase RpJ2P2 Phosphorylated IL-2 Receptor:JAK complex

[ ] [ ][ ]

[ ] [ ][ ]

[ ] [ ]

[ ] [ ][ ]

[ ] [ ]

[ ] [ ][ ]

[ ] [ ][ ]

[ ] [ ][ ]

[ ] [ ][ ]

[ ] [ ] [ ] [ ] [ ][ ] [ ]

[ ] [ ] [ ] [ ] [ ][ ] [ ]

9

12

2

13

142

15

7

8

10

11

7 8

10

33 3 3 ,

3

55 5 5 ,

5

333 ,

3 535 ,

555 ,

2 32 3 ,

66 3 ,

22 5 ,

2121 5 ,

3 2 33 35 3 33 ,

2 61

5 5 2 55 35 5 55

2 211

3

5

5

1

t

t

t

s pp s p

M s p

s pp s p

M s p

s ps

Ms s p

sM

s ps

Mw

w sM

ww s

Mw

w sMw

w sM

s s s p ss

w wM M

s s s

p

p

p

s

s

s

s

p p

p s p ss

w wM

p

=+

=+

=

=

=

=

=

=

=

− − −=

+ +

+ +

− −=

11

,

M

(S22)

where we denoted the Michaelis constants

2 3 5 6 8 9 11 12 14 15 17 187 8 9 10 11 12

1 4 7 10 13 16

20 22 2413 14 15

19 21 23

3 3 3 3 3, , , , , ,

, , .

3

3 3 3

T T T T T T

T T T

a a a a aM M M M M Ma S a S a S a S a S a Sa a aM M Ma S a

a a a a a a a

S a S

= = = = = =

= = =

+ + + + + +

We look for steady-state solutions of System (S20):

[ ] [ ] [ ][ ] [ ] [ ]

2 4 7

10 13 16

0 2 3 6 3 3 3 ,

0 2 5 21 5 5 5 .

m w s m w s m p s

m

p

pw s m w s m p s

= + −

= + − (S23)

We can rewrite Equations (S23) as follows:

[ ] [ ] [ ][ ] [ ] [ ]4 5

6 7

0 2 3 6 3 3 3 ,

0 2 5 21 5 5 5 ,

n w s n w s p s

n w s n w s ps

p

p

= + −

= + − (S24)

Page 15: Supplementary Materials - University of Exeterempslocal.ex.ac.uk/people/staff/vnb262/publ/stats/stats-supp.pdf · P2 SHP-1 phosphatase RpJ2P2 Phosphorylated IL-2 Receptor:JAK complex

where 2 5 11 144 5 6 7

8 8 17 17

, , ,a a a an n n na a a a

= = = = .

System (S24) can be rewritten after substituting solutions from Equation (S22):

[ ] [ ] [ ][ ]

[ ][ ]

[ ] [ ][ ] [ ]

[ ] [ ] [ ][ ]

[ ][ ]

[ ] [ ][ ] [ ]

2

13 14

7 8 8 7

9 8 7

2

15 14

10 11 11 10

12 11 10

4 5

6 7

3 3 50 3 2

3 2 63 1 1,

3 2 6

5 3 50 5 2

5 2 215 1 5 ,

5 2 21

t

t t

s p s s ps p

M M

s p M M M w M wpM s p n M w n M w

s p s s ps p

M M

s p M M M w M wp sM s p n M w n M w

p

p

= + + +

⎞⎛ + ++ + −⎟⎜

+ +⎝ ⎠

= + + +

⎞⎛ + ++ + −⎟⎜

+ +⎝ ⎠

(S25)

We solved System (S25) for and numerically.

2.1.3 Cytokine production In general case, transcription factor T can activate gene G by forming a complex with the gene

TG:

(S26)

The ODEs for Equation (S26):

(S27)

Conservation equation that follows from Equations (S27):

(S28)

where is the total concentration of the gene.

Equation (S28) can be written as follows:

(S29)

where .

[ ]3s p [ ]5s p

2

1

G TGTh

h

→←+

[ ] [ ][ ] [ ]

[ ] [ ][ ] [ ]

1

1

2

2

,

.

d G h G h TGdtd

T

TTG h G h TGdt

= − +

= −

[ ] [ ],TG GG T= +

TG

,TG λ θ= +

[ ] [ ] ,G TGλ θ= =

Page 16: Supplementary Materials - University of Exeterempslocal.ex.ac.uk/people/staff/vnb262/publ/stats/stats-supp.pdf · P2 SHP-1 phosphatase RpJ2P2 Phosphorylated IL-2 Receptor:JAK complex

The ODEs (S27) can be rewritten in the following way:

(S30)

where .

We need to find steady-state solutions of Equation (S30):

(S31)

We can find from Equations (S31):

(S32)

where is the Michaelis constant.

In the most general case the reactions of the activation of a gene G by two transcription factors

T1 and T2 are:

(S33)

The ODEs for System (S33):

(S34)

1

21

2 ,

,

d h hdtd h hdt

T

T

λ λ θ

θ λ θ

= − +

= −

[ ]T T=

210 ,.T

TGh hλ θ

λ θ

=

=

+

θ

,TTG

Qh Tθ =

+

2

1

hQhh

=

2

3

4

1

3

4

1

2

T1 G

T1G T2 T1T2G

T

T1G

T22 G

T2G

G

T1 T1T2G

b

b

b

b

b

b

b

b

→←

→←

→←

→←

+

+

+

+

[ ] [ ][ ] [ ] [ ][ ] [ ]

[ ] [ ][ ] [ ] [ ][ ] [ ]

[ ] [ ][ ] [ ] [ ][ ] [ ]

[ ] [ ][ ] [ ] [ ][ ] [ ]

2 4

2 3 4

1 3

3 4 1 2

1

3 4 1 2

1 2 ,

1 1 2 1 2 ,

2 2 1 1 2 ,

1 2 2 1 2 1 1

1 2

1 1

2 2

1 2 2 .

d G b G b T G b G b T Gdtd T G b G b T G b T T T Gdtd T G

T T

T T G b

T T G b

T G

b G b T G b T T T Gdtd T T G b T T T G b T T T Gb T G bdt

= − + − +

= − − +

= − − +

= − + −

Page 17: Supplementary Materials - University of Exeterempslocal.ex.ac.uk/people/staff/vnb262/publ/stats/stats-supp.pdf · P2 SHP-1 phosphatase RpJ2P2 Phosphorylated IL-2 Receptor:JAK complex

Conservation equation that follows from Equations (S34):

(S35)

where is the total amount of the gene.

Equation (S35) can be written as follows:

(S36)

where

The ODEs (S34) can be rewritten in the following way:

(S37)

where .

We found steady-state solutions of Equations (S37):

(S38)

Next, we found concentrations of the complexes:

[ ] [ ] [ ] [ ]1 2 ,1 2T T G TG TGG T G= + + +

TG

,TG ψ ξ ϕ ν+ + +=

[ ] [ ] [ ] [ ], 1 , 2 , 1 2 .G T G T G T T Gψ ξ ϕ ν= = = =

2 4

2 3 4

3 4 1 2

3 4

1 3

1 2

1

,

2 ,

1 2

,

1 ,

2 1

1

2

d b b b bdtd b b b Tdtd b b b Tdt

T T

T b

T b

b bd b T b Tdt

ψ ψ ξ ψ ϕ

ξ ψ ξ ν

ϕ ψ

ξ

ϕ

ξ

ϕ ν

ν ν νϕ

= − + − +

= − − +

= − − +

= − + −

[ ] [ ]1 1 , 2 2T T T T= =

2 3 4

3 4 1

1

2

3 4 1 2

0 2 ,0 1

12 ,

0 2 1 ,.T

b b b Tb b b Tb T b

Tb

bG

T

bT

b

ψ ξ ν

ψ ϕ ν

ν ν

ν

ξ

ϕ

ξ ϕ

ψ ξ ϕ

= − − +

= − − +

=

+

+=

− +

+

Page 18: Supplementary Materials - University of Exeterempslocal.ex.ac.uk/people/staff/vnb262/publ/stats/stats-supp.pdf · P2 SHP-1 phosphatase RpJ2P2 Phosphorylated IL-2 Receptor:JAK complex

(S39)

where are the Michaelis constants.

If a protein is activated by the first and the second transcription factors at the same time, then

its concentration is proportional to the concentration only:

(S40)

which is a probability of the two transcription factors to be bound to the same gene.

If a protein is activated by the first or the second transcription factors, then it is proportional to

the sum of concentrations :

(S41)

which is a probability of either of the two transcription factors to be bound to the same gene.

( )( )

( )( )

( )( )

4

23

1 22 4

1 3

2

11

1 22 4

1 3

1 22 4

1 3

11

1 21 2

22

1 21 2

1 2 1 2 ,1 2

1

,

2

,

T T

T T

T T

b TQb TbG G

Qb T Qb Tb bT Tb b

b TQbTbG G

Qb T Qb Tb bT Tb b

T T T TG GQb T Qb Tb bT T

b b

ξ

ϕ

ν

= =⎞ + +⎞⎛⎛

+ + ⎟⎟⎜⎜⎝ ⎝⎠ ⎠

= =⎞ + +⎞⎛⎛

+ + ⎟⎟⎜⎜⎝ ⎝⎠ ⎠

= =⎞ + +⎞⎛⎛

+ + ⎟⎟⎜⎜⎝ ⎝⎠ ⎠

1 31

42

2 ,bQb Qb bb b

= =

ν

1 2

1 2 ,1 2T

T TGQb T Qb T

ν = ⋅+ +

ξ ϕ ν+ +

( )( )

( )( )( ) ( )

( )( )

( )( )

2 1

1 2

2 1

1 2

2 1

1 2

1 2 1 2

1 2

1 2 1 21 2

1 2 1 2 1 2 1 2 ,1 2

1 2 2 1 1 2,

1 2

1 2 1 2 ,1 2 1 2

1

,

21

T

T

T

T

T

Qb T QbT T TGQb T Qb TQb T QbT T T T T T TG

Qb T Qb T

T Qb T T Qb T T TG

Qb T Qb T

T T T TGQb T Qb T Qb T Qb T

T TGQb T Qb

ξ ϕ ν

ξ ϕ ν

ξ ϕ ν

ξ ϕ ν

ξ ϕ ν

+ ++ + =

+ +

+ + + −+ + =

+ +

+ + + −+ + =

+ +

⎞⎛+ + = + − ⎟⎜ ⎟+ + + +⎝ ⎠

+ + = ++ + 1 2

1 2 ,2 1 2

T TT Qb T Qb T

⎞⎛− ⎟⎜ + +⎝ ⎠

Page 19: Supplementary Materials - University of Exeterempslocal.ex.ac.uk/people/staff/vnb262/publ/stats/stats-supp.pdf · P2 SHP-1 phosphatase RpJ2P2 Phosphorylated IL-2 Receptor:JAK complex

CD46 The full mechanism of how CD46 enhances IL-10 production is still not clear. We assumed

here that the mechanism of reactions is similar to the one described in Equation (S33). Thus it

can be written for the concentration of SP1 in non-dimensional form according to Equation

(S40):

(S42)

where , , , , and .

We assumed here that the gene interaction with the transcription factor and its subsequent

expression lead to the mRNA translation and certain cytokine secretion. The produced cytokine

then can be degraded by a metalloprotease. In this case the biochemical reactions can be written

as follows:

(S43)

where TG is the transcription factor complex with gene, C is the active cytokine, Mp is the

metalloprotease, CMp is cytokine-metalloprotease complex and Cn is a non-active cytokine.

The ODEs for the reactions in System (S43):

(S44)

Conservation equation that follows from Equations (S44):

(S45)

We found steady-state solutions of System (S44):

[ ] [ ][ ]

[ ][ ]16 17

2 461 1 ,

2 46ti cd

sp a spM i M cd

= ⋅+ +

[ ] [ ]113T

SP asp a

S=

113T

tT

SPspS

= [ ] [ ]46463T

CDcd

S= 2

161 3T

lMl S

= 417

3 3T

lMl S

=

1 2

3

TG C,

C Mp CMp Cn Mp,u

u

KP

u→← →+ +

[ ] [ ] [ ][ ] [ ]

[ ] [ ][ ] ( )[ ]

[ ] [ ][ ] ( )[ ]

1 3

1 2 3

1 2 3

,

,

.

d C KP TG u C Mp u CMpdtd CMp u C Mp u u CMpdtd Mp u C Mp u u CMpdt

= − +

= − +

= − + +

[ ] [ ]TMp Mp CMp= +

Page 20: Supplementary Materials - University of Exeterempslocal.ex.ac.uk/people/staff/vnb262/publ/stats/stats-supp.pdf · P2 SHP-1 phosphatase RpJ2P2 Phosphorylated IL-2 Receptor:JAK complex

(S46)

Thus, we obtain a system of equations:

(S47)

where

We can find from this system of Equations (S47):

, (S48)

where is the Michaelis constant.

We next substituted from equation (S48) to the first equation in System (S47):

(S49)

Since C should be positive and the maximum value for is 1, , which implies that

the rate of the cytokine production should be less than its maximum rate of the degradation by

metalloprotease.

Equation (S49) can be written as follows:

(S50)

[ ] [ ][ ] [ ][ ][ ] ( )[ ]

1 3

1 2 3

0 ,

0 .

KP TG u C Mp u CMp

u C Mp u u CMp

= − +

= − +

( )2

1 2 3

0 ,0 ,

,T

KP uu C u u

Mp

ϑ σ

ζ σ

ζ σ

= −

= − +

= +

[ ] [ ] [ ] [ ],, , .C C CG MT Mp pζ σϑ = = ==

σ

TCMp

Qu Cσ =

+

2 3

1

u uQuu+

=

σ

2

2

2

0 ,

,

.1

T

T

T

CKP u MpQu C

KP QuCu Mp KPQuC u Mp

KP

ϑ

ϑϑ

ϑ

= −+

=−

=−

TMpϑ 2KP u<

,1T

QuC MpQpϑ

=−

Page 21: Supplementary Materials - University of Exeterempslocal.ex.ac.uk/people/staff/vnb262/publ/stats/stats-supp.pdf · P2 SHP-1 phosphatase RpJ2P2 Phosphorylated IL-2 Receptor:JAK complex

where , .

When the cytokine production is up-regulated by one transcription factor only, from

Equation (S32) and thus it can be written:

(S51)

If the cytokine production is up-regulated by two transcription factors at the same time,

as shown in Equation (S40), it can be written:

(S52)

If the cytokine production is up-regulated by either of the two transcription factors,

as shown in Equation (S41), and thus it can be written:

(S53)

IFN-γ and IL-10 production Since IFN-γ is activated by STAT55 only (Fig 1B) we can write using Equation (S51):

(S54)

where .

IL-10 gene can be activated by either STAT33 or CD46. Thus, it can be written according to

Equation (S53):

2

KPQpu

= 1Qp <

ϑ θ=

.1T

TTG

Qh

QuC Mp

QpT+

=−

ϑ ν=

1 2

21

.11

2

T

T

QuC

T TGQb T Qb T

Mp

Qp

=

+ +

ϑ ξ ϕ ν= + +

1 2 1 2

1 2 1 21 2

.

2

1

1

T

TT T T TG

Qb T Qb T Qb

QuC Mp

QT b T

pQ

=

⎞⎛+ − ⎟⎜ + +⎝ ⎠

+ +

[ ]

[ ][ ]

18

819

555

1

5

,1

t

t

Mig mp

ns

gsM

g+

=−

[ ] [ ] 4 5 2 518 19 8 8

3 1 4

1, 1 , , , , , 13 3 3 3 3

T Tt t

T T T T T

Ig Mp Gg k k k lig mp gg M M n nS S S k S k S k

+= = = = = = <

Page 22: Supplementary Materials - University of Exeterempslocal.ex.ac.uk/people/staff/vnb262/publ/stats/stats-supp.pdf · P2 SHP-1 phosphatase RpJ2P2 Phosphorylated IL-2 Receptor:JAK complex

(S55)

where .

2.2 Model for the STAT3-STAT4 circuit

The biochemical reactions involved in the STAT3-STAT4 circuit (Fig 1C) are as follows:

[ ]

[ ][ ]

[ ][ ]

[ ][ ]

[ ][ ]21

20

22 21 229

1

33 1 33 110

33 1

,2

3

0

1

1

3

t

ts sp a s sp a

M s M sp a M s M

Mi mp

n gsp a

⎞⎛+ − ⎟⎜

+ + + + ⎠

=

11 12 7 9 620 21 22 9 9

10 6 8 11

2 102 , 10 , , , , , 13 3 3 3 3

t tt t

T T T T T

Mp G k k k k lmp g M M M n nS S k S k S k S k

+= = = = = = <

2

3

4 5

6

7 8

9

10 11

12

13 14

15

16 1

18

1

7

19

21

I2 RJ2 p

p P2 p P2 P2

I6 RJ6 p

p P6 p P6 P6

I2RJ2 R J2

R J2 R J2 RJ2

I6RJ6 R J6

R J6 R J6 RJ6

I12RJ12 R J12

R J1

I12 RJ12 p

p P12 R J12 RJ12

I

2 p P1

35RJ

2 P12

I35 RJ35 35

f

f

f

f

f

f

f

f

f

f

f

f

f

f

f

f

f

f

f

f

→←

→←

→←

→←

→←

→←

→←

+

+ +

+

+

+

+

+ +

+

→20

R 5pJ3f

Page 23: Supplementary Materials - University of Exeterempslocal.ex.ac.uk/people/staff/vnb262/publ/stats/stats-supp.pdf · P2 SHP-1 phosphatase RpJ2P2 Phosphorylated IL-2 Receptor:JAK complex

(S56)

22 23

24

4 5

6

7 8

9

10 11

12

13 14

1

1

3

5

2

R J35 R J35 RJ35

R J2 S3 R J2S3 R J2 S3p

R J6 S3 R J6S3 R J6 S3p

P3 S3p P3S3p P3 S3

R J12 S4 R J12S4 R J12 S4p

R J35 S4

p P35 p P35 P35

p

R J35S4 R J35 S4p

P4 S4p

p p

p p p

p p p

p p p

f

a

a

a

a

a

a

f

a

a

a

a

a

a

a

f

a

a

→←

→←

→←

→←

→←

→←

+ +→

+ → +

+ → +

+ → +

+ → +

+ → +

+17

18

20

21

22

23

24

1

2

3

4

3

4

1

2

2

16

19

1

5

3

5

p

p

p p

I2 Sp1 I2Sp1

I2Sp1 CD46 Sp1a

CD46 Sp1 CD46Sp1

CD46Sp1 I2 Sp1a

S44 Gg S44Gg

S

P4

44Gg I

S4 P4 S4

S3 S3p S33

S3p S4p S34

S4 S4 S44

g

Ig Mp1 IgMp1

a

a

l

l

l

l

l

l

l

l

k

k

k

k

a

a

a

a

a

a

k

a

l

→←

→←

→←

→←

→←

→←

→←

→←

→←

→←

+

+

+

+

+

+

+

+

+

+

→4

6

7

Ign Mp1

S33 G10 S33G10k

k

→←

+

+

Page 24: Supplementary Materials - University of Exeterempslocal.ex.ac.uk/people/staff/vnb262/publ/stats/stats-supp.pdf · P2 SHP-1 phosphatase RpJ2P2 Phosphorylated IL-2 Receptor:JAK complex

2.2.1 Cytokine-receptor interactions According to Equations (1.12) we can write for , , and in

non-dimensional form respectively:

8

9

8

9

6

7

6

6

6

10 11

12

S33G10 Sp1a S33Sp1aG10

Sp1a G10 Sp1aG10

Sp1aG10 S33 S33Sp1aG10

S33G10 I10

Sp1aG10 I10

S33Sp1aG10 I10

I10 Mp2 I10Mp2 I10n Mp2

k

k

k

k

k

k

k

k

l

l

l

k

→←

→←

→←

→←

+

+

+

+ +→

[ ]2RpJ [ ]6RpJ [ ]21RpJ [ ]35RpJ

[ ][ ]

[ ]

[ ][ ]

[ ]

1

1 1

2

1

1 1

3

2

2

2 2

2

42

3

24 4

2

1 12

22

1 1 42

,21 1

66

2

1 1 46

2 2

2 2 2

6 6

6 6 6

,2

t t

t t t

t t

t t t

Mn i n

w

Mn i n

Mn i n

w

M r p

M r p r M

M r p

M ri

r MMpn n

⎞⎛+ + ⎟⎜

⎝ ⎠= − +

⎞⎛ ⎞⎛+ + +⎟⎜ ⎟⎜⎜ ⎟⎝ ⎠⎝ ⎠+

⎞⎛+ + ⎟⎜

⎝ ⎠= − +

⎞⎛ ⎞⎛+ + +⎟⎜ ⎟⎜⎜ ⎟⎝ ⎠

− +

− +

+

+⎝+

Page 25: Supplementary Materials - University of Exeterempslocal.ex.ac.uk/people/staff/vnb262/publ/stats/stats-supp.pdf · P2 SHP-1 phosphatase RpJ2P2 Phosphorylated IL-2 Receptor:JAK complex

(S57)

where

2.2.2 STAT phosphorylation and subsequent dimerization ODEs for the STAT phosphorylation and dimerization module are given by:

[ ][ ]

[ ]

[ ][ ]

[ ]

5

3 3

2

5

3 3

7

4 4

2

7

6

84

6 6

8

48

12 12

12 12 1

1 112

122

1 1 412

,2

1 135

352

1 1 43

2

35 35

35 5 35

5

,2

3

t t

t t t

t t

t t t

Mn i n

w

Mn i n

Mn i n

w

M r p

M r p r M

M r p

M r p r MMn i n

− +

− +

− +

− +

⎞⎛+ + ⎟⎜

⎝ ⎠= − +

⎞⎛ ⎞⎛+ + +⎟⎜ ⎟⎜⎜ ⎟⎝ ⎠⎝ ⎠+

⎞⎛+ + ⎟⎜

⎝ ⎠= − +

⎞⎛ ⎞⎛+ + +⎟⎜ ⎟⎜⎜ ⎟⎝ ⎠⎝ ⎠+

[ ] [ ] [ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ]

[ ] [ ]

2 3 5 6 22 1

1 4 5

8 9 11 12 83 4 2

7 10 11

12 62 2, , ,3 3 3 3 3

126 6 , ,3 3 3 3

123

2 2 2 , 2 , 2 , , , 6

3 36

6 , 6 , 6 , , , 123 3

1212 , 12

T T

t T T T T T T

T T

T T T

t t

t t

t

T T T

T

T

RpJi w r p M M n i

f S f S fRpJ

w r p M M n if S f

I IR

S f

P f f f f fS S S S S

IR P f f f f

RpJ

fS S S S

RS

w r

= = = = = =

= = = = = =

= =

+ += =

+ +=

[ ] [ ] [ ] [ ]

14 15 17 18 145 6 3

13 16 17

20 21 23 24 207 8 4

19 22 23

12 ,3 3

35

, 12 , , ,3 3

3535 35 , 35 , 35 , , 35 35, .

3 3 33,

33

T

T T T T

T T

T T T T T

t

t tT

p M M nf S f S f

Rp

P f f f f fS S

I R P f f fJi w r p M M n

f S f S ff f

S S S S

= = =+ +

+

=

= == = = = =+

Page 26: Supplementary Materials - University of Exeterempslocal.ex.ac.uk/people/staff/vnb262/publ/stats/stats-supp.pdf · P2 SHP-1 phosphatase RpJ2P2 Phosphorylated IL-2 Receptor:JAK complex

(S58)

Conservation equations:

(S59)

Conservation Equations (S59) in non-dimensional form:

(S60)

where

[ ] [ ][ ] ( )[ ]

[ ] [ ][ ] ( )[ ]

[ ] [ ] [ ] [ ][ ] [ ]

[ ] [ ] [ ][ ] [ ]

[ ] [ ][ ] ( )[ ]

[ ] [ ][ ] ( )

1 2 3

2

1

4 5 6

5 7 9

220 21 22

7 8 9

10 11 1

9

2

2 3 2 3 2 3 ,

6 3 6 3 6 3 ,

3 2 3 6 3 3 3 3 3

2 3 2 33 3 5 35 ,

3 3 3 3 3 3 ,

12 4 12 4 12

p p

p p p

d RpJ S a RpJ S a a RpJ Sdtd RpJ S a RpJ S a a RpJ Sdtd S a RpJ S a RpJ S a P S p a P Sdt

a S a S a S S a Sd P S a P S a P Sdtd Rp

p p a

J S a RpJ S a a Rp

p

Jdt

= − +

= − +

= + − + −

− + − +

= − +

= − + [ ]

[ ] [ ][ ] ( )[ ]

[ ] [ ] [ ] [ ][ ] [ ]

[ ][ ] [ ] [ ] [ ]

[ ] [ ][ ] ( )[ ]

[ ] [ ] [ ]

[ ] [ ][ ] [ ]

13 14 15

11 14 18

21 22 23 24

17 18

20

21 22

16

2

16

219

4 ,

35 4 35 4 35 4 ,

4 12 4 35 4 4 4 4 4

3 4 34 2 4 2 44 ,

4 4 4 4 4 4 ,

33 3 33 ,

34 34 ,3 4

S

d RpJ S a RpJ S a a RpJ Sdtd S a RpJ S a RpJ S a P S a P Sdt

a S S a S a S p a Sd P S a P S a a P Sdtd S a S p a

p p p

p

Sdtd S a S p a Sdt

p

p

d

p p

S p

= − +

= + − + −

− + − +

= − +

= −

= −

[ ] [ ] [ ]2 242

344 4 44 .S a S p a Sdt

= −

[ ] [ ] [ ] [ ] [ ] [ ][ ] [ ] [ ] [ ] [ ] [ ][ ][ ]

3 3 3 2 33 34 2 3 6 3 [ 3 3 ],

4 4 4 2 44 34 12 4 35 4 [ 4 4 ],

3 3 [ 3 3 ],

4 4 [ 4 4 ].

T

T

T

T

S S S p S S RpJ S RpJ S P S

S S S p S S RpJ S RpJ S P S

P P P S

P

p

P

p

p

pP S

= + + + + + +

= + + + + + +

= +

= +

[ ] [ ] [ ] [ ] [ ] [ ] [ ][ ] [ ] [ ] [ ] [ ] [ ] [ ][ ] [ ][ ] [ ]

1 3 3 2 33 34 2 3 6 3 3 3

4 4 4 2 44 34 12 4 35 4 4 4

3 3 3 3

4 4

,

,

,

,4 4

t

t

t

s s s s w s w s p s

s s s s s w s w s p s

p p p s

p p p s

p p

p p

p

p

= + + + + + +

= + + + + + +

= +

= +

Page 27: Supplementary Materials - University of Exeterempslocal.ex.ac.uk/people/staff/vnb262/publ/stats/stats-supp.pdf · P2 SHP-1 phosphatase RpJ2P2 Phosphorylated IL-2 Receptor:JAK complex

ODEs (S58) in non-dimensional form can be written as follows:

(S61)

where

[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]3 3 2 3 6 3 3 3 43 , 3 , 2 3 , 6 33 3 3 3

, 3 3 ,3

,3

4 T

T T T T T Tt

p pp p

SS S RpJ S RpJ

S SS P S Ss s w s w s p s s

S S S= = = = = =

[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]3 34 4 12 4 35 4 4 4

4 , 4 , 12 4 , 35 4 , 3

43

3

4 ,T T T T T

pS S RpJ S RpJ S P Ss s w s w s p

pp p

S S S S Ss= = = = =

[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]33 34 44 3 4 3 433 , 34 , 44 ,3 3 3

3 ,3 3 3 3

4 , 3 , 4 .T T

T T T T T Tt t

T

S S S PS S S

P P Ps s s p p p pS S S S

= = = = = = =

[ ] [ ][ ] [ ]

[ ] [ ][ ] ( )[ ]

[ ] [ ] [ ] [ ][ ] [ ]

[ ] [ ] [ ][ ] [ ]

[ ] [ ][ ] ( )[ ]

[ ] [ ][ ] ( )[ ]

[ ] [ ][ ] ( )[ ]

1

3 4 5

2 4 6 8

18 19 20 21

6 7 8

9 10 11

12 13 14

2

2 3 2 3 2 3

6 3 6 3 6 3

3 2 3 6 3 3 3 3 3

2 3 2 33 3 4 34

3 3 3 3 3 3

12 4 12 4 12 4

35 4 35 4 35 4

d w s m w s w sdd w s m w s m m w sdd s p m w s m w s m p s m p sd

m s p m s m s p s p m sd p s p m p s p m m p s pdd w s m w s m m w sdd w s m w s m m w s

p

d

p

τ

τ

τ

τ

τ

τ

= −

= − +

= + − + −

− + − +

= − +

= − +

= − +

[ ] [ ] [ ] [ ][ ] [ ]

[ ][ ] [ ] [ ] [ ]

[ ] [ ][ ] ( )[ ]

[ ] [ ] [ ]

[ ] [ ][ ] [ ]

[ ] [ ] [ ]

10 13 15

220 21 22 23

15

18

20 21

17

16 17

219

222 23

4 12 4 35 4 4 4 4 4

3 4 34 2 2 44

4 4 4 4 4 4

33 3 33

34 3 4 34

44 4 4

4

4

d s m w s m w s m p s m p sd

m s s m s m m sd p s p m

p p p

p p s p

p s m m p sdd s m s m sdd s m s p s p

p p

p

p

m sdd s m s m sd

τ

τ

τ

τ

τ

= + − + −

− − +

= −

=

= −

+

+

=

( ) 12 3 1

2 3 2 3 2 3 2 3 2 3

2 3 2 3 2

2 4 5 62 3 4 5

7 8 9 10 11 126 7 8 9 10 11

13

3 2 3 2 3 2 3

2 3

14

2

1512 13 14

3 2

, 3 , , 3 , , ,

3 , , , 3 , , ,

3 , ,

T T

T T

T

t a a m S m m S m ma a a a a a a a a a

m S m m m S m ma a a a a a a a a a a a

m

a a a a a

a a a a a a

a S m ma a a

a aa a

τ = + = = = = =+ + + + +

= = = = = =+ + + + + +

= = =+ + +

16 17 1815 16 17

19 20 21 22 23 2418 19 20 21

3

22 23

2 3 2 3 2 3

2 3 2 3 2 3 2 3 2 3 2 3

, 3 , , ,

3 , , 3 , , 3 , .

T

T T T

m S m ma a a a a a a

m S m m S m m S ma a a a a a a a

a a a

a a a a a aa a a a

= = =+ + +

= = = = = =+ + + + + +

Page 28: Supplementary Materials - University of Exeterempslocal.ex.ac.uk/people/staff/vnb262/publ/stats/stats-supp.pdf · P2 SHP-1 phosphatase RpJ2P2 Phosphorylated IL-2 Receptor:JAK complex

We need to find steady-state solution of System (S61):

(S62)

We can simplify System (S62):

(S63)

Then using conservation Equations (S60) and System (S63) we obtained:

(S64)

[ ][ ] [ ][ ][ ] ( )[ ]

[ ] [ ] [ ] [ ][ ] [ ]

[ ] [ ] [ ][ ] [ ][ ][ ] ( )[ ][ ][ ] ( )[ ][ ][ ] ( )[ ]

[ ] [ ] [ ] [ ][ ]

1

3 4 5

2 4 6 8

18 19 20 21

6 7 8

9 10 11

12 13 14

10 13 15

2

0 2 3 2 3

0 6 3 6 3

3 2 3 6 3 3 3 3 3

2 3 2 33 3 4 34

0 3 3 3 3

0 12 4 12 4

0 35 4 35 4

4 12 4 35 4 4 4

m w s w s

m w s m m w sd s p m w s m w s m p s m p sd

m s p m s m s p s p m s

m p s p m m p s p

m w s m m w s

m w s m m w sd s m w s

p

m w s mp ppd

p

s

τ

τ

= −

= − +

= + − + −

− + − +

= − +

= − +

= − +

= + − [ ]

[ ][ ] [ ] [ ] [ ][ ][ ] ( )[ ][ ] [ ][ ][ ] [ ][ ] [ ]

220 21 22 23

15

18

20 21

2

17

16 17

2

22 23

19

4 4

3 4 34 2 2 44

0 4 4 4 4

0 3 33

0 3 4 34

0 4 44

4

m p s

m s s m s m m s

m p s m m p s

m s m s

m s p s p m s

m s m s

p

p p s p

p p

p

p

+ −

− − +

= − +

= −

= −

= −

+

[ ][ ] [ ][ ][ ] ( )[ ]

[ ] [ ] [ ] [ ]

[ ][ ] ( )[ ][ ][ ] ( )[ ][ ][ ] ( )[ ]

[ ] [ ] [ ] [ ]

[ ][ ] ( )[ ][ ] [ ]

1

3 4 5

2 4 7

6 7 8

9 10 11

12 13 1

16

16 1

4

10 13

15

1

7

298 1

2

0 2 3 2 3

0 6 3 6 3

3 2 3 6 3 3 3

0 3 3 3 3

0 12 4 12 4

0 35 4 35 4

4 12 4 35 4 4 4

0 4 4 4 4

0 3 33

0

m w s w s

m w s m m w sd s p m w s m w s m p sdm p s p m m p s p

m w s m m w s

m w s m m w sd s m w s m w s m p sdm p s m m p s

m s

p

s

p p

m

m

p p

p

τ

τ

= −

= − +

= + −

= − +

= − +

= − +

= + −

= − +

= −

= [ ][ ] [ ][ ] [ ]0 21

222 23

3 4 34

0 4 44p

s p s p m s

m s m s

= −

[ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ]

2 4 7

10 13 16

3 2 3 6 3 3 3 ,

4 12 4 35 4 4 4 ,

d s p m w s m w s m p sdd s m

p

p w s m w s m p sd

p

τ

τ

= + −

= + −

Page 29: Supplementary Materials - University of Exeterempslocal.ex.ac.uk/people/staff/vnb262/publ/stats/stats-supp.pdf · P2 SHP-1 phosphatase RpJ2P2 Phosphorylated IL-2 Receptor:JAK complex

where

Or we can rewrite it as follows:

[ ] [ ]

[ ]

[ ] [ ]

[ ]

[ ] [ ]

[ ] [ ][ ]

[ ] [ ]

[ ] [ ][ ]

[ ] [ ][ ]

[ ] [ ][ ]

[ ] [ ][ ]

[ ] [ ] [ ] [ ] [ ]

[ ] [ ]

7 8

6

16 17

15

18

20

21

22

23

3

4 5

9

10 11

12

13 14

3

4 5

2

19

2

1

1

33 3 3

3

44 4 4

4

33 3

34 3 4

44 4

2 3 2 3

6 3 6

12 4 12

35 4 35

3 2 3

3

4

4

3 34 3 33

1 2 6

1

t

t

s pp s p m m s p

ms p

p s p m m s pm

ms s pmms s s pmms s pm

w s m w

p

p

p

s

s

s

s

p

mw s wm m

mw s wm mmw s w

m ms s s p s

s mm w wm m

p

=+

+

=+

+

=

=

=

=

=+

=+

=+

− − −=

+ ++

[ ] [ ] [ ] [ ] [ ]

[ ] [ ]9 12

10 11 13 14

4 4 2 44 34 4 44

1 12 35

t p ps s s s p ss m mw w

m m m m

− − −=

+ ++ +

Page 30: Supplementary Materials - University of Exeterempslocal.ex.ac.uk/people/staff/vnb262/publ/stats/stats-supp.pdf · P2 SHP-1 phosphatase RpJ2P2 Phosphorylated IL-2 Receptor:JAK complex

(S65)

where we denote the Michaelis constants

When considering steady-state solutions of System (S64) we can write:

(S66)

Or we can rewrite Equations (S66) as follows:

(S67)

[ ] [ ][ ]

[ ] [ ][ ]

[ ] [ ]

[ ] [ ][ ]

[ ] [ ]

[ ] [ ][ ]

[ ] [ ][ ]

[ ] [ ][ ]

[ ] [ ][ ]

[ ] [ ] [ ] [ ] [ ][ ] [ ]

[ ] [ ] [ ] [ ] [ ][ ] [ ]

11

14

2

15

162

17

9

10

12

13

9 10

12 13

33 3 3

3

44 4 4

4

333

3 434

444

2 32 3

66 3

1212 4

3535 4

3 2 33 34 3 33

2 61

4 4 2 44 34 4 4

3

412

4

1

351

4

t

t

t

s pp s p

M s p

s pp s p

M s p

s ps

Ms s p

sM

s ps

Mw

w sM

ww s

Mw

w s

p

p

p

s

s

s

s

p p

p

Mw

w sM

s s s p ss

w wM M

s s s s p ss

w wM

p

M

=+

=+

=

=

=

=

=

=

=

− − −=

+ +

+ +

− − −=

2 3 5 6 8 9 11 12 14 159 10 11 12 13

1 4 7 10 13

17 18 20 22 2414 15 16 17

16 19 21 23

, , , , ,

,

3 3 3 3 3

3 3 3, , .

3

T T T T T

T T T T

a a a a a aa a a aM M M M Ma S a S a S a S a Sa a a aM M M Ma S a S a S a S

a

= = = = =

= =

+ +

+=

+

=

+ +

[ ] [ ] [ ][ ] [ ] [ ]

2 4 7

10 13 16

0 2 3 6 3 3 3 ,

0 12 4 35 4 4 4 .

m w s m w s m p s

m

p

pw s m w s m p s

= + −

= + −

[ ] [ ] [ ][ ] [ ] [ ]5 6

7 8

0 2 3 6 3 3 3 ,

0 12 4 35 4 4 4 ,

n w s n w s p s

n w s

p

n w s p s p

= + −

= + −

Page 31: Supplementary Materials - University of Exeterempslocal.ex.ac.uk/people/staff/vnb262/publ/stats/stats-supp.pdf · P2 SHP-1 phosphatase RpJ2P2 Phosphorylated IL-2 Receptor:JAK complex

where .

We can rewrite System (S67) substituting Equations (S65):

(S68)

We find and in System (S68) numerically.

CD46 It can be written for SP1 in non-dimensional form according to Equation (S40):

(S69)

where , , , , and .

2.2.3 IFN-γ and IL-10 production Since IFN-γ is activated in this module by STAT44 only (Fig 1C) we can write using Equation

(S51):

(S70)

where .

According to Equation (S53) we can write:

2 5 11 145 6 7 8

8 8 17 17

, , ,a a a an n n na a a a

= = = =

[ ] [ ] [ ][ ]

[ ][ ]

[ ] [ ][ ] [ ]

[ ] [ ] [ ][ ]

[ ][ ]

[ ] [ ][ ] [ ]

2

15 16

9 10 10 9

11 10 9

2

17 16

12 13 13 12

14 13

6

8 1

5

27

3 3 40 3 2

3 2 63 1 1,

3 2 6

4 3 40 4 2

4 12 354 1 4 ,

4 12 35

t

t t

s p s s ps p

M M

s p M M M w M wpM s p n M w n M w

s p s s ps p

M M

s p M M M w M wp s

M s p n M w n M w

p

p

= + + +

⎞⎛ + ++ + −⎟⎜

+ +⎝ ⎠

= + + +

⎞⎛ + ++ + −⎟⎜

+ +⎝ ⎠

[ ]3s p [ ]4s p

[ ] [ ][ ]

[ ][ ]18 19

2 461 1 ,

2 46ti cd

sp a spM i M cd

=+ +

[ ] [ ]113T

SP asp a

S=

113T

tT

SPspS

= [ ] [ ]46463T

CDcd

S= 2

181 3T

lMl S

= 419

3 3t

lMl S

=

[ ]

[ ][ ]

20

921

444

1

4

,1

t

t

Mig mp

ns

gsM

g+

=−

[ ] [ ] 4 5 2 520 21 9 9

3 1 4

1, 1 , , , , , 13 3 3 3 3

T Tt t

T T T T T

Ig Mp Gg k k k lig mp gg M M n nS S S k S k S k

+= = = = = = <

Page 32: Supplementary Materials - University of Exeterempslocal.ex.ac.uk/people/staff/vnb262/publ/stats/stats-supp.pdf · P2 SHP-1 phosphatase RpJ2P2 Phosphorylated IL-2 Receptor:JAK complex

(S71)

where

2.3 Combined STAT3-STAT4-STAT5 model

The reactions involved in STAT3-STAT4-STAT5 circuit (Fig 5A):

[ ]

[ ][ ]

[ ][ ]

[ ][ ]

[ ][ ]

22

23 24 23 2410

33 1 33 110

33

10 2

1 3

1

3

,

1t

t

s sp a s sp aM s M

Mi mp

nsp a M s M sp a

g⎞⎛

+ − ⎟⎜+ + +⎝ ⎠

+

=

11 12 7 9 622 23 24 10 10

10 6 8 11

2 102 , 10 , , , , , 1.3 3 3 3 3T T

t tT T T T T

Mp G k k k k lmp g M M M n nS S k S k S k S k

+= = = = = = <

2

3

4 5

6

7 8

9

10 11

12

13 14

15

1

I2RJ2 R J2

R J2 R J2 RJ2

I6RJ6 R J

I2 RJ2 p

p P2 p P2 P2

I6 RJ6 p

p

6

R J6 R J6 RJ6

I12

P6 p

RJ12 R

P6 P6

I12 RJ1 J1p 22

f

f

f

f

f

f

f

f

f

f

f

f

f

f

f

→←

→←

→←

→←

→←

+

+ +

+

+ →

+

+

Page 33: Supplementary Materials - University of Exeterempslocal.ex.ac.uk/people/staff/vnb262/publ/stats/stats-supp.pdf · P2 SHP-1 phosphatase RpJ2P2 Phosphorylated IL-2 Receptor:JAK complex

(S72)

16 17

18

19 20

21

22 23

24

25 26

27

28

3

29

2

30

1

R J12 R J12 RJ12

I35RJ35 R J35

R J35 R J35 RJ35

I21RJ21 R J21

R J

p P12 p P12 P12

I35 RJ35 p

p P35 p P35 P35

I21 RJ21 p

p P21 p P21 P221 R J21 RJ21

R J2 S3 R J2S3

1

p p RpJ

f

f

f

f

f

f

f

f

f

f

a

a

f

f

f

a

f

f

→←

→←

→←

→←

→←

→←

+ +

+

+ +

+

+ →

+

+→

4 5

6

7 8

9

10 11

12

13 14

15

17

1

1

0

2

6

8

19 2

1

2 S3p

R J6 S3 R J6S3 R J6 S3p

P3 S3p P3S3p P3 S3

R J12 S4 R J12S4 R J12 S4p

R J35 S4 R J35S4 R J35 S4p

P4 S4p P4S4 P4 S4

R J2 S5 R J2S5 R J2 S5p

p p p

p p p

p p p

p

p

R J21 S

p p

p

a

a

a

a

a

a

a

a

a

a

a

a

a

a

a

a

a

a

→←

→←

→←

→←

→←

→←

+

+ → +

+ → +

+ → +

+ → +

+ → +

+ → +

+22 23

24

25 26

27

28

29

30

31

5 R J21S5 R J21 S5p

P5 S5p P5S5 P5 S5

S3 S3

p p

p S33

S3p S4p S

p

4

p

3

a

a

a

a

a

a

a

a

a

a

→←

→←

→←

→←

→ +

+ → +

+

+

Page 34: Supplementary Materials - University of Exeterempslocal.ex.ac.uk/people/staff/vnb262/publ/stats/stats-supp.pdf · P2 SHP-1 phosphatase RpJ2P2 Phosphorylated IL-2 Receptor:JAK complex

32

33

34

35

36

37

1

2

3

4

3

4

1

2

1

1

1

5 6

7

p p

p p

S44 Gg S44Gg

S44Gg S55 S44S55Gg

S55 Gg S55Gg

S55Gg S44 S44S55Gg

S44Gg Ig

S55Gg Ig

S44S55Gg Ig

Ig Mp1

S4 S4 S4

IgMp1 Ign Mp1

I2 Sp1

4

S3p S5p S35

S5 S5 S55

a

a

a

k

k

k

k

k

k

k

a

a

k

k

k

a

l

l

l

k

→←

→←

→←

→←

→←

→←

→←

→←

+

+

+

+

+ +

+

+

+

+

2

3

4

5

4

5

2

3

8

9

10

11

10

11

8

9

6

6

I2Sp1

I2Sp1 CD46 Sp1a

CD46 Sp1 CD46Sp1

CD46Sp1 I2 Sp1a

S33 G10 S33G10

S33G10 Sp1a S33Sp1aG10

Sp1a G10 Sp1aG10

Sp1aG10 S33 S33Sp1aG10

S33G10 I10

Sp1aG10 I10

S33Sp

l

l

l

l

l

l

l

l

k

k

k

k

k

k

k

k

l

l

→←

→←

→←

→←

→←

→←

→←

→←

+

+

+

+

+

+

+

→6

12 13

14

1aG10 I10

I10 Mp2 I10Mp2 I10n Mp2k

k

l

k→← →+ +

Page 35: Supplementary Materials - University of Exeterempslocal.ex.ac.uk/people/staff/vnb262/publ/stats/stats-supp.pdf · P2 SHP-1 phosphatase RpJ2P2 Phosphorylated IL-2 Receptor:JAK complex

2.3.1 Cytokine-receptor interactions According to Equation (S12) we can write for , , , and

in non-dimensional form respectively:

(S73)

[ ]2RpJ [ ]6RpJ [ ]21RpJ [ ]35RpJ

[ ]21RpJ

[ ][ ]

[ ]

[ ][ ]

[ ]

[ ][ ]

1

1 1

2

1

1 1

3

2 2

2

3

2 2

5

2

2 2

4

4 4

36

3

1 12

22

1 1 42

,21 1

66

2

1 1 4

2 2

2 2 2

6 6

6 6 6

12

6,

21 1

11

212

2

2

t t

t t t

t t

t t t

t t

M r p

M r p r M

M r p

M r

Mn i n

w

Mn i n

Mn i n

w

Mn i n

Mn i

M

wM r p

n

p r

⎞⎛+ + ⎟⎜

⎝ ⎠= − +

⎞⎛ ⎞⎛+ + +⎟⎜ ⎟⎜⎜ ⎟⎝ ⎠⎝ ⎠+

⎞⎛+ + ⎟⎜

⎝ ⎠= − +

⎞⎛ ⎞⎛+ + +⎟⎜

− +

− +

− +

− + ⎟⎜⎜ ⎟⎝ ⎠⎝ ⎠+

⎞⎛+ + ⎟⎜

⎝=

−⎠

+

− +

[ ]

[ ][ ]

[ ]

[ ][ ]

[ ]

2

5

3 3

7

4 4

2

7

4 4

6 6

8

8 8

10

10

9

5 5

9

5

12 12 12

35 35

35 35 35

21

1 1 412

,2

1 135

352

1 1 435

,2

1 121

212

2

21

21 211

t t t

t t

t t t

t t

t t

Mn i n

Mn i n

w

Mn i n

Mn i n

w

Mn i

M r p r M

M r p

M r p r M

M r p

M r p

⎞⎛ ⎞⎛+ + +⎟⎜ ⎟⎜⎜ ⎟⎝ ⎠⎝ ⎠+

⎞⎛+ + ⎟⎜

⎝ ⎠= − +

⎞⎛ ⎞⎛+ + +⎟⎜ ⎟⎜⎜ ⎟⎝ ⎠

− +

− +

− +

− +

⎝ ⎠+

⎞⎛+ + ⎟⎜

⎝ ⎠= +

+

+ 10

2

5

1 14

,2

21 tr Mn

⎞⎛ ⎞⎛+ + +⎟⎜ ⎟⎜⎜ ⎟⎝ ⎠⎝ ⎠

Page 36: Supplementary Materials - University of Exeterempslocal.ex.ac.uk/people/staff/vnb262/publ/stats/stats-supp.pdf · P2 SHP-1 phosphatase RpJ2P2 Phosphorylated IL-2 Receptor:JAK complex

where

2.3.2 STAT phosphorylation and dimerization ODEs for the STAT module are given by:

[ ] [ ] [ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ]

[ ] [ ]

2 3 5 6 22 1

1 4 5

8 9 11 12 83 4 2

7 10 11

12 62 2, , ,3 3 3 3 3

126 6 , ,3 3 3 3

123

2 2 2 , 2 , 2 , , , 6

3 36

6 , 6 , 6 , , , 123 3

1212 , 12

T T

T T T T T T T

T T

T T T

t t

t t

t

T T T

T

T

RpJi w r p M M n i

f S f S fRpJ

w r p M M n if S f

I IR

S f

P f f f f fS S S S S

IR P f f f f

RpJ

fS S S S

RS

w r

= = = = = =

= = = = = =

= =

+ += =

+ +=

[ ] [ ]

[ ] [ ]

[ ] [ ] [ ] [ ]

14 15 17 18 145 6 3

13 16 17

20 21 23 24 207 8 4

19 22 23

, 12 , , , 353 3

3535 , 35 , 35 , , ,

3 321

3512 , ,3 3 3

35 35 ,3 3 3

21 221 21 , 21 , 1,3 3 3

T

T T T T T

T T

T

t

t t

t

T T T T

T

T T T

p M M n if S f S f

RpJw r p M M n

f S

IP f f f f fS S S

R P f f f f fS S S f S f

RpJI RS S S

i w r p

= = = =

= = = = =

+ +=

+ +=

= == 26 27 29 30 269 10 5

25 28 29

21 , ,21 .3

,3 3

T

T Tt

T

M M nf S f

P f f fS ff f

S= = = =

+ +

[ ] [ ][ ] ( )[ ]

[ ] [ ][ ] ( )[ ]

[ ] [ ] [ ] [ ][ ] [ ]

[ ] [ ] [ ][ ] [ ][ ][ ] [ ]

[ ] [ ][ ] ( )[ ]

1 2 3

4 5 6

5 7 9

228 29 30 31

34 35

7 8 9

2

2 3 2 3 2 3 ,

6 3 6 3 6 3 ,

3 2 3 6 3 3 3 3 3

2 3 2 33 3 4 34

3 5 35 ,

3 3 3 3 3 3 ,

d RpJ S a RpJ S a a RpJ Sdtd RpJ S a RpJ S a a RpJ Sdtd S a RpJ S a RpJ S a P S p a P Sdt

a S a S a S S a S

a S S a S

p p

p p p

p p

p pd P S a P S a P Sdt

a p

= − +

= − +

= + − + −

− + − + −

− +

= − +

Page 37: Supplementary Materials - University of Exeterempslocal.ex.ac.uk/people/staff/vnb262/publ/stats/stats-supp.pdf · P2 SHP-1 phosphatase RpJ2P2 Phosphorylated IL-2 Receptor:JAK complex

(S74)

Conservation equations:

(S75)

Conservation Equations (S75) in non-dimensional form:

[ ] [ ][ ] ( )[ ]

[ ] [ ][ ] ( )[ ]

[ ] [ ] [ ] [ ][ ] [ ]

[ ][ ] [ ] [ ] [ ]

[ ] [ ][ ] ( )[ ]

[ ]

10 11 12

13 14 15

11 14 18

30 31 32

16

2

16

1

1

9

33

17 8

12 4 12 4 12 4 ,

35 4 35 4 35 4 ,

4 12 4 35 4 4 4 4 4

3 4 34 2 4 2 44 ,

4 4 4 4 4 4 ,

2 5

d RpJ S a RpJ S a a RpJ Sdtd RpJ S a RpJ S a a RpJ Sdtd S a RpJ S a RpJ S a P S a P Sdt

a S S a S a S p a Sd P S a P

p

S a a P Sdt

p p

p p

p

d

p p

RpJ S adt

= − +

= − +

= + − + −

− + − +

= − +

= [ ][ ] ( )[ ]

[ ] [ ][ ] ( )[ ]

[ ] [ ] [ ] [ ][ ] [ ]

[ ][ ] [ ] [ ] [ ]

[ ] [ ][ ] ( )[ ]

[ ] [ ] [ ]

[ ]

20

20

21

22 23 24

23 25 27

234 35 36 37

25 26 27

28 292

2 5 2 5 ,

21 5 21 5 21 5 ,

5 2 5 21 5 5 5 5 5

3 5 35 2 5 2 55 ,

5 5 5 5 5 5 ,

33 3 33 ,

34

RpJ S a a RpJ S

d RpJ S a RpJ S a a RpJ Sdtd S a RpJ S a RpJ S a P S p a P Sp p

pdt

a S S a S a S a Sd P S a P S a P Sdtd S a S p a Sdtd

p p

p p

S adt

a p

− +

= − +

= + − + −

− + − +

= − +

= −

= [ ][ ] [ ]

[ ] [ ] [ ]

[ ] [ ][ ] [ ]

[ ] [ ] [ ]

30 31

32 33

34 35

26

2

3 37

4 34 ,

44 4 44 ,

3

3

35 5 35 ,

55 5 55 .

S p a S

d S a S p a Sdtd S a S p a Sdtd S a S p a Sdt

S p

S p

= −

= −

= −

[ ] [ ] [ ] [ ] [ ] [ ] [ ][ ] [ ] [ ] [ ] [ ] [ ][ ] [ ] [ ] [ ] [ ] [ ][ ][ ][ ]

3 3 3 2 33 34 35 2 3 6 3 [ 3 3 ],

4 4 4 2 44 34 12 4 35 4 [ 4 4 ],

5 5 5 2 55 35 2 5 21 5 [ 5 5 ],

3 3 [ 3 3 ],

4 4 [ 4 4 ],

5 5 [ 5 5 ].

T

T

T

T

T

T

S S S p S S S RpJ S RpJ S P S

S S S p S S RpJ S RpJ S P S

S S S p S S RpJ

p

p

p

p

S RpJ S P S

P P P S

P P P S

P P P S

p

p

= + + + + + + +

= + + + + + +

= + + + + + +

= +

= +

= +

Page 38: Supplementary Materials - University of Exeterempslocal.ex.ac.uk/people/staff/vnb262/publ/stats/stats-supp.pdf · P2 SHP-1 phosphatase RpJ2P2 Phosphorylated IL-2 Receptor:JAK complex

(S76)

where

ODEs (S74) in non-dimensional form are given by:

[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ][ ] [ ] [ ] [ ] [ ] [ ] [ ][ ] [ ] [ ] [ ] [ ] [ ] [ ][ ] [ ][ ] [ ][ ] [ ]

1 3 3 2 33 34 35 2 3 6 3 3 3

4 4 4 2 44 34 12 4 35 4 4 4

5 5 5 2 55 35 2 5 21 5 5 5

3 3 3 3

4 4 4 4

5 5 5 5

,

,

,

,

,

,

t

t

t

t

t

s s s s s w s w s p s

s s s s s w s w s p s

s s s s s w s w s p s

p p p s

p p

p p p s

p

p p

p p

s pp

p

p

p

= + + + + + + +

= + + + + + +

= + + + + + +

= +

= +

= +

[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]3 3 2 3 6 3 3 3 43 , 3 , 2 3 , 6 33 3 3 3

, 3 3 ,3

,3

4 t

T T T T T Tt

p pp p

SS S RpJ S RpJ

S SS P S Ss s w s w s p s s

S S S= = = = = =

[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]3 34 4 12 4 35 4 4 4

4 , 4 , 12 4 , 35 4 , 3

43

3

4 ,T T T T T

pS S RpJ S RpJ S P Ss s w s w s p

pp p

S S S S Ss= = = = =

[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]5 5 2 5 21 5 5 555 , 5 , 5 , 2 5 , 21 5 ,3 3

5 5 ,3 3 3 3

T

T T T T T Tt

S S RpJp pp p

S S S S SS RpJ S P SSs s s w s w s p s

S= = = = = =

[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]33 34 44 35 55 3 433 , 34 , 44 , 35 , 55 , 3 , 4 ,

3 3 3 3 3 3 3T T T T T T TSS S S S S P P

s sS S S S S

s pS

s s p= = = = = = =

[ ] [ ]5 3 4 55 , 3 , 43 3 3

.3, 5T T T

T Tt t t

T T

PS S S

P P Pp p pS

p = = = =

Page 39: Supplementary Materials - University of Exeterempslocal.ex.ac.uk/people/staff/vnb262/publ/stats/stats-supp.pdf · P2 SHP-1 phosphatase RpJ2P2 Phosphorylated IL-2 Receptor:JAK complex

(S77)

where

[ ] [ ][ ] [ ]

[ ] [ ][ ] ( )[ ]

[ ] [ ] [ ] [ ][ ] [ ]

[ ] [ ] [ ][ ] [ ][ ][ ] [ ]

[ ] [ ][ ] ( )[ ]

[ ] [ ][ ] ( )[ ]

[ ]

1

3 4 5

2 4 6 8

27 28 292

30

33 34

6 7 8

9 10 11

12

2 3 2 3 2 3

6 3 6 3 6 3

3 2 3 6 3 3 3 3 3

2 3 2 33 3 4 34

3 5 35

3 3 3 3 3 3

12 4 12 4 12 4

35 4

d w s m w s w sdd w s m w s m m w sdd s p m w s m w s m p s m p sd

m s p m s m s p s p m s

m s p s p m sd p s p m p s p m m p s pdd w s m w s m m w sdd w sd

p

m

p

τ

τ

τ

τ

τ

τ

= −

= − +

= + − + −

− + − + −

− +

= − +

= − +

= [ ][ ] ( )[ ]

[ ] [ ] [ ] [ ][ ]

[ ] [ ][ ][ ] [ ] [ ]

[ ] [ ][ ] ( )[ ]

[ ] [ ][ ] ( )[ ]

[ ] [ ][ ] ( )[ ]

[ ] [ ]

17

16 1

13 14

10 13 15

29

230 31 32

15 7

1918 20

21 22 23

19 2

35 4 35 4

4 12 4 35 4 4 4

4 4 3 4

34 2 2 44

4 4 4 4 4 4

2 5 2 5 2 5

21 5 21 5 21 5

5 2 5

4

w s m m w s

d s m w s m w s m p sd

m p s m s s

m s m m sd p s p m p s m m p sdd w s m w s m m w sdd w s m w s m m

p p

p p p

s p

w sdd s p m w s md

p p

τ

τ

τ

τ

τ

− +

= + − +

+ −

+ − +

= − +

=

− +

+

− +

=

= + [ ] [ ][ ] [ ]

[ ][ ] [ ] [ ] [ ]

[ ] [ ][ ] ( )[ ]

[ ] [ ] [ ]

[ ] [ ][ ] [ ]

[ ] [ ] [ ]

[ ] [ ][ ] [ ]

[ ] [ ] [ ]

2 24 26

33 34 35 36

24 25 26

27 28

29 30

231 32

33 34

235 36

2

2

21 5 5 5 5 5

3 5 35 2 5 2 55

5 5 5 5 5 5

33 3 33

34 3 4 34

44 4 44

35 3 5 35

55 5 55

w s m p s m p s

m s p s p m s m s p m sd p s p m p s p m m p s pdd s m s m sdd s m s p s p m sdd s m s m sdd s m s p s p m sdd s m s m

p

p

sd

p

p

p

τ

τ

τ

τ

τ

τ

− + −

− + − +

= − +

= −

= −

= −

= −

= −

Page 40: Supplementary Materials - University of Exeterempslocal.ex.ac.uk/people/staff/vnb262/publ/stats/stats-supp.pdf · P2 SHP-1 phosphatase RpJ2P2 Phosphorylated IL-2 Receptor:JAK complex

We need to find a steady-state solution of System (S77):

(S78)

We can simplify System (S78):

( ) 12 3 1

2 3 2 3 2

2 4 5 6 72 3 4 5 6

8 9 10 11

3 2 3 2 3 2 3

2 3 2 3 2 3 2 3 2 3 2 3 2 3

2

12 13 147 8 9 10 11 12 13

1514

, 3 , , 3 , , , 3 ,

, , 3 , , , 3 , ,

t t t

t t

t a a m S m m S m m m Sa a a a a a a a a a a a

m m m S m m m S ma a a a a a a a a a a a a a

m

a a a a a a

a a a a a a

aa a

a

τ = + = = = = = =+ + + + + +

= = = = = = =+ + + + + + +

=+

16 17 18 19 20 2115 16 17 18 1

3 2 3 2 3 2 3 2 3 2 39 20

22 23 24 25 26 2

2 3

2 3 2 3 2 3 2 3

721 22 23 24 25 26

2827 2

2 3 2 3

2 3

, 3 , , , 3 , , ,

3 , , , 3 , , ,

3 ,

t t

t t

t

m S m m m S m ma a a a a a a a a a a a

m S m m m S m ma

a a a a a a

a a a a a aa a a a a a a a a a a

m S ma aa

= = = = = =+ + + + + +

= = = = = =+ + + + + +

=+ 2 3 2 3 2 3 2 3 2 3

2 3 2

29 30 31 32 338 29 30 31 32

34 35

3 2 3

36 3733 34 35 3

2 36

, 3 , , 3 , ,

3 , , 3 , .

t t

t t

m S m m S ma a a a a a a a a a

m S m m S ma a a a a

a a

a a

a a a

a a a aa

= = = = =+ + + + +

= = = =+ + + +

[ ][ ] [ ][ ][ ] ( )[ ]

[ ] [ ] [ ] [ ][ ] [ ] [ ]

[ ] [ ][ ] [ ][ ][ ] [ ]

[ ][ ] ( )[ ][ ][ ] ( )[ ][ ][ ] ( )[ ]

[ ] [ ]

1

3 4 5

2 4 6 8 27

28 29 30

33 34

6 7 8

9 10 11

12 13 14

10

2

0 2 3 2 3

0 6 3 6 3

3 2 3 6 3 3 3 3 3 2 3

2 33 3 4 34

3 5 35

0 3 3 3 3

0 12 4 12 4

0 35 4 35 4

4 12 4

m w s w s

m w s m m w sd s p m w s m w s m p s m p s m s pd

m s m s p s p m s

m s p s p m s

m p s p m m p s p

m w s m m w s

m w s m m w sd s m w s

p p

pd

τ

τ

= −

= − +

= + − + − +

+ − + −

− +

= − +

= − +

= − +

= [ ] [ ][ ] [ ] [ ][ ]

[ ] [ ] [ ][ ][ ] ( )[ ][ ][ ] ( )[ ][ ][ ] ( )[ ]

[ ] [ ] [ ] [ ][ ] [ ]

[ ][ ] [ ] [ ]

13 15 29

230 31 32

15

18 20

21

1

22 23

19 22 24 26

33 34 3

7

16 17

1

5

9

2

35 4 4 4 4 4 3 4

34 2 2 44

0 4 4 4 4

0 2 5 2 5

0 21 5 21 5

5 2 5 21 5 5 5 5 5

3 5 35 2 5

4

2

m w s m p s m p s m s s

m s m m s

m p s m m p s

m w s m m w s

m w s m m w sd s p m w s m w s m p s m p sd

m s p

p p p

s p m s m s p m

p

s p

p p

p pτ

+ − + −

+ − +

= − +

= − +

= − +

= + − + −

− + − +

+

[ ][ ][ ] ( )[ ][ ] [ ][ ][ ] [ ][ ] [ ][ ][ ] [ ][ ] [ ]

36

24 25 26

27 28

29 30

231 32

33 34

235

2

36

55

0 5 5 5 5

0 3 33

0 3 4 34

0 4 44

0 3 5 35

0 5 55

s

m p s p m m p s p

m s m s

m s p s p m s

m s m s

m s p s p m s

m s m s

p

p

p

= − +

= −

= −

= −

= −

= −

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(S79)

Then using conservation Equations (S76) and System (S79) we obtained:

(S80)

where

[ ][ ] [ ][ ][ ] ( )[ ]

[ ] [ ] [ ] [ ]

[ ][ ] ( )[ ][ ][ ] ( )[ ][ ][ ] ( )[ ]

[ ] [ ] [ ] [ ]

[ ][ ] ( )[ ][ ][ ] ( )

1

3 4 5

2 4 7

6 7 8

9 10 11

12 13 1

16

16 1

4

10 13

15

1

7

218 9 0

0 2 3 2 3

0 6 3 6 3

3 2 3 6 3 3 3

0 3 3 3 3

0 12 4 12 4

0 35 4 35 4

4 12 4 35 4 4 4

0 4 4 4 4

0 2 5 2

m w s w s

m w s m m w sd s p m w s m w s m p sdm p s p m m p s p

m w s m m w s

m w s m m w sd s m w s m w s m p sdm p s m m

p

p p

p pp s

m w s m m w s

τ

τ

= −

= − +

= + −

= − +

= − +

= − +

= + −

= − +

= − + [ ][ ][ ] ( )[ ]

[ ] [ ] [ ] [ ]

[ ][ ] ( )[ ][ ] [ ][ ][ ] [ ][ ] [ ][ ][ ] [ ][ ] [ ]

21 22 23

19 22 25

24 25 26

27 28

29 30

231 32

33 34

235 36

2

5

0 21 5 21 5

5 2 5 21 5 5 5

0 5 5 5 5

0 3 33

0 3 4 34

0 4 44

0 3 5 35

0 5 55

m w s m m w sd s p m w s m w s m p sdm p s p m m p s p

m

p

ps m s

m s p s p m s

m s m s

m s p s p m s

m s m s

p

p

τ

= − +

= + −

= − +

= −

= −

= −

= −

= −

[ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ]

2 4 7

10 13

19 22 25

16

3 2 3 6 3 3 3

4 12 4 35 4 4 4

5 2 5 21 5 5 5

d s p m w s m w s m p sdd s m w s m w s m p sdd s p m w s m w s m p sd

p

p p

p

τ

τ

τ

= + −

= + −

= + −

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Or we can rewrite it in the following way:

[ ] [ ]

[ ]

[ ] [ ]

[ ]

[ ] [ ]

[ ]

[ ] [ ]

[ ] [ ][ ]

[ ] [ ]

[ ] [ ][ ]

[ ] [ ]

[ ] [ ][ ]

[ ] [ ][ ]

[ ] [ ][ ]

7 8

6

16 17

15

25 26

24

27

28

29

30

31

32

33

34

35

36

3

4 5

2

2

9

10 11

2

1

33 3 3

3

44 4 4

4

55 5 5

5

33 3

34 3 4

44 4

35 3 5

55 5

2 3 2 3

6 3 6

12

3

4 412

35

t

t

t

s pp s p m m s p

ms p

p s p m m s pms p

p s p m m s pm

ms s pmms s s pmms s pmms s s p

p

p

p

p

p

s

s

mms s pm

w s m wmw s w

m mmw ss w

m m

w s

=+

+

=+

+

=+

+

=

=

=

=

=

=

=+

=+

[ ] [ ][ ]

[ ] [ ][ ]

[ ] [ ][ ]

[ ] [ ] [ ] [ ] [ ] [ ]

[ ] [ ]

[ ] [ ] [ ] [ ] [ ]

[ ] [ ]

[ ] [ ] [ ] [ ] [ ]

[ ]

12

13 14

18

19 20

21

22 23

3

4 5

9 12

10 11 13 14

18 21

19 20 22

1

4 35

2 5 2

21 5 21

3 2 33 34 35 3 33

1 2 6

4 4 2 44 34 4 44

1 12 35

5 5 2 5

4

5

5 35 5 55

1

1 2

5

t

t

m wm mmw s w

m mmw s w

m ms s s s p s

s mm w wm m

s s s s p ss m mw w

m m m ms s s s p s

s m mw

s

s

s

p

m m

p

p

m

p p

p

=+

=+

=+

− − − −=

+ ++

− − −=

+ ++ +

− − −

+

++

−=

+[ ]

23

21wm

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(S81)

where we denoted the Michaelis constants

[ ] [ ][ ]

[ ] [ ][ ]

[ ] [ ][ ]

[ ] [ ]

[ ] [ ][ ]

[ ] [ ]

[ ] [ ][ ]

[ ] [ ]

[ ] [ ][ ]

[ ] [ ][ ]

[ ] [ ][ ]

[ ] [ ][ ]

[ ] [ ][ ]

[ ] [ ][ ]

[ ]

13

16

19

2

20

212

22

232

24

11

12

14

15

17

18

33 3 3

3

44 4 4

4

55 5 5

5

333

3 434

444

3 535

555

2 32 3

66

3

4

4

3

1212 4

3535 4

22 5

2121 5

3

5

5

1

t

t

t

s pp s p

M s p

s pp s p

M s p

s pp s p

M s p

p

p

p

p

p

s

s

s

s

s ps

Ms s p

sM

s ps

Ms s p

sM

s ps

Mw

w sMw

w sMw

w sM

ww s

Mw

w sM

s

s

s

ww s

Ms

=+

=+

=+

=

=

=

=

=

=

=

=

=

=

=−

=

[ ] [ ] [ ] [ ] [ ][ ] [ ]

[ ] [ ] [ ] [ ] [ ][ ] [ ]

[ ] [ ] [ ] [ ] [ ][ ] [ ]

11 12

14 15

17 18

3 2 33 34 35 3 32 6

1

4 4 2 44 34 4 44

12 351

5 5 2 55 35 5 55

2 211

t

t

s s s p sw wM M

s s s s p s

p p

p ps

w wM M

s s s s p ss

w wM

p p

M

− − − −

+ +

− − −=

+ +

+

− − −=

+

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We looked for steady-state solutions of System (S80):

(S82)

We can rewrite Equations (S82) as follows:

(S83)

where .

We can rewrite System (S83) substituting Equations (S81):

(S84)

We found , and in System (S84) numerically.

2 3 5 6 8 9 11 12 14 15 17 1811 12 13 14 15 16

1 4 7 10 13 16

20 21 23 24 26 27 29 31 3317 18 19 20 21 22

19 22 25 28 30 32

23

, , , , , ,

, , ,

3 3 3 3 3 3

3,

3 3,

3 3 3,

T T T T T T

T T T T T T

a a a a a a a

a a

a a a a aM M M M M Ma S a S a S a S a S a Sa a a a a aM M M M M Ma S

aa S a S a S a S a S

aM

= = = = =+ +

=

= =

+ + + +

+= =

+=

+=

= 35 3724

34 363, .

3T T

aMa S a S

=

[ ] [ ] [ ][ ] [ ] [ ][ ] [ ] [ ]

2 4 7

10 13

1

1

22

6

9 25

0 2 3 6 3 3 3 ,

0 12 4 35 4 4 4 ,

0 2 5 21 5 5 5 .

m w s m w s m p s

m w s m w s m p s

m w s m w s

p

p

pm p s

= + −

= + −

= + −

[ ] [ ] [ ][ ] [ ] [ ][ ] [ ] [ ]

6 7

8 9

10 11

0 2 3 6 3 3 3 ,

0 12 4 35 4 4 4 ,

0 2 5 21 5 5 5 .

n w s n w s p s

n w s n w s p s

n w s n

p

s pw s

p

p

= + −

= + −

= + −

2 5 11 14 20 236 7 8 9 10 11

8 8 17 17 26 26

, , , , ,a a a a a an n n n n na a a a a a

= = = = = =

[ ] [ ] [ ][ ] [ ][ ]

[ ][ ]

[ ] [ ][ ] [ ]

[ ] [ ] [ ][ ]

[ ][ ]

[ ] [ ][ ] [ ]

[ ] [ ] [ ]

2

20 21 23

11 12 12 11

13 12 11

2

6 7

8 9

22 21

14 15 15 14

16 15 14

2

24

3 3 4 3 50 3 2

3 2 63 1 1,

3 2 6

4 3 40 4 2

4 12 354 1 4 ,

4 12 35

5 30 5 2

t

t t

s p s s p s s ps p

M M M

s p M M M w M wpM s p n M w n M w

s p s s ps p

M M

s p M M M w M wp s

M s p n M w n M

p p

p

w

p sp

Mps

s

= + + + +

⎞⎛ + ++ + −⎟⎜

+ +⎝ ⎠

= + + +

⎞⎛ + ++ + −⎟⎜

+ +⎝ ⎠

= + +[ ]

[ ][ ]

[ ] [ ][ ] [ ]

23

17 18 18 17

19 110 18 71 1

5

5 2 215 1 5 ,

5 2 21t t

s pM

s p M M M w M wp sM s p n M w n M w

⎧⎪⎪⎪⎪⎪

+

⎞⎛ + ++ + −⎟⎜

⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ + +⎝ ⎠

[ ]3s p [ ]4s p [ ]5s p

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CD46 It can be written for SP1 in non-dimensional form according to Equation (S40):

(S85)

where , , , , and .

2.3.3 IFN-γ and IL-10 production Since IFN-γ gene is activated by either STAT44 or STAT55, it can be written according to

Equation (S53):

(S86)

where

and

.

The production of IL-10 can be activated by either STAT33 or SP1 through CD46. Thus,

according to Equation (S53), it can be written:

(S87)

where

where .

[ ] [ ][ ]

[ ][ ]25 26

2 461 1 ,

2 46ti cd

sp a spM i M cd

=+ +

[ ] [ ]113T

SP asp a

S=

113T

tT

SPspS

= [ ] [ ]46463T

CDcd

S= 3

252 3T

lMl S

= 526

4 3T

lMl S

=

[ ]

[ ][ ]

[ ][ ]

[ ][ ]

[ ][ ]

27

1228 29 28 29

55 5

,1 144 444

555 544 54t

t

s s s sgg

s s

Mig mp

nM M s sM M

+ −+ +

=−

⎞⎛⎟⎜

⎝ ⎠+ +

[ ] [ ] 6 7 2 4 127 28 29 12

5 1 3 6

1, 1 , , , , , ,3 3 3 3 3 3

T Tt t

T T T T T T

Ig Mp Gg k k k k lig mp gg M M KM nS S S k S k S k S k

+= = = = = = =

12 1n <

[ ]

[ ][ ]

[ ][ ]

[ ][ ]

[ ][ ]

30

31 32 31 3213

33 1 33 110

33

10 2

1 3

1

3

,

1t

t

s sp a s sp aM s M

Mi mp

nsp a M s M sp a

g⎞⎛

+ − ⎟⎜+ + +⎝ ⎠

+

=

13 14 9 11 630 31 32 13

12 8 10 13

2 102 , 10 , , , , ,3 3 3 3 3T T

t tT T T T T

Mp G k k k k lmp g M M M nS S k S k S k S k

+= = = = = =

13 1n <

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3 Model parameters

3.1 Parameter fitting

For the parameter fitting in the STAT3-STAT5 model, we used the Genetic Algorithm (GA)

tool integrated to MATLAB. As the criterion for fitting we chose the squared error, which can

be described as ( )21

N

i ii

SM E M=

= −∑ , where iE is experimental data for cytokine concentration

(IFN-γ or IL-10) corresponding to the i-th value of IL-2 concentration, M is the model

predictions for cytokine concentration corresponding to the same IL-2 concentration, i is the

number of experimental data point, 4N = is the total number of experimental data points. The

GA tool allows minimizing the squared error using the integrated algorithms for the optima

search.

We selected a "nominal" set of parameters "by hand", which qualitatively demonstrates the

switching between IFN-γ and IL-10. This set of parameters is represented as "Nom" in Table

S2. The sets of the optimized parameters and corresponding squared errors are also shown in

Table S2. We performed 15 optimization tests setting the allowable ranges for the parameters

ten-fold either side of the nominal values of parameters and chose the best fitting (set "O3" in

Table S2) with the smallest squared error 77.34 10SM −= ⋅ . Fig S1 shows the distribution of

five parameter sets with the closest minimum squared errors SM , namely sets "O2", "O3",

"O9", "O10" and "O12".

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Table S2. Nominal, optimized parameters and squared error SM.

Fig S1. The distribution of the optimized parameters. The parameter sets with the closest minimum squared errors , namely "O2", "O3", "O9", "O10" and "O12". SM

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Fig S2. Model predictions for the swapped parameters. A. STAT monomers. B. STAT homodimers. C. Produced cytokines.

Table S3. The effects of the parametric changes on the concentration of produced IFN-γ and IL-10 shown in Fig 3.

Fig Parameters

Perturbation

thin line

(optimized)

3A 9n 4163 10−⋅ 4191 10−⋅ 4220 10−⋅

20M 4119 10−⋅ 4149 10−⋅ 4179 10−⋅

3B

5ts 4198 10−⋅ 4247 10−⋅ 4297 10−⋅

18M 7.93 9.913 11.896

8n 481 10−⋅ 4101 10−⋅ 4121 10−⋅

14M 0.068 0.1 0.145

9M 69.185 47.714 32.445

3C 2tr

42.7 10−⋅ 427 10−⋅ 4268 10−⋅

1n 11.842 118.42 1184.2

3D 7M 23.63 10−⋅ 43.63 10−⋅ 63.63 10−⋅

3E 10M 470 447 10−⋅ 947 10−⋅

3F, G 6Q 41 10−⋅ 414 10−⋅ 4209 10−⋅

Table S4. Parameters in the STAT3-STAT4 subsystem and their correspondence to the

parameters in the STAT3-STAT5 subsystem.

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3.2 Parameter sensitivity analysis

In the main text, we suggested that IFN-γ to IL-10 switching (Fig 2) is due to the competition

between STAT3 and STAT5 proteins. These model predictions are not obvious from the model

structure due to the following. The structure of our model presented in Fig 1 is symmetrical in

relation to the varied IL-2, i.e. IL-2 activates both STAT3 and STAT5. The model predictions

for the phosphorylated states of STAT3, STAT5 and their homodimers could be swapped in

Fig 2 if the parameters for STAT3 and STAT5 are swapped (Fig S2). However, in this case

there would be no cytokine switching (Fig S2C). Thus the model predictions depend on the

assumed structure of our model as well as on the chosen set of parameters. In this section, we

performed parameter sensitivity analysis (SA) (21, 22) to identify the parameter conditions for

the conclusions to hold.

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First, we tested if the IFN-γ to IL-10 and STAT5 to STAT3 switchings are still present when

we vary the parameters compared to their optimal values (set "O3" in Table S3). Quantitatively,

we defined the switching of either IFN-γ to IL-10 or STAT5 to STAT3 as follows. We assumed

that the switching occurs when the 2-fold changes take place, which is typically considered as

significant change in Biology (23, 24). Thus, the changes should include at least 2-fold increase

in IL-10 and STAT3 as well as decrease of the peak of IFN-γ and STAT5 concentrations within

the range of the tested IL-2 concentrations.

We varied the parameters up to 10-fold either side of their values in the optimized set. We used

the Latin Hypercube Sampling (LHS), which is considered as one of the most effective

strategies for sampling the parameters (22, 25). We performed the LHS sampling and checked

the assumed condition for switching for 1000 samples of the optimized parameters. Fig S3

illustrates the probability of the cases, in percent, where the switching of both cytokine and

STAT (C+S+), cytokine but not STAT (C+S-), not cytokine but STAT (C-S+), neither cytokine

nor STAT (C-S-) occurs for 1-10 fold change of the optimized parameters. The obvious result

that follows from Fig S3 is that when the parameters are not perturbed, which corresponds to

the 1-fold change, the probability of the presence of both cytokine and STAT switching is

100%. However, with an increase of the fold change up to ten, the probability for switching of

both cytokine and STAT decreases down to 2.1% (data shown in Table S5). The probability of

either of the cases (cytokine or STAT switching) is almost equal with an increase of the fold

change as it can be seen from Fig S3. Thus, we can conclude that the model with the optimized

parameters demonstrates both IFN-γ to IL-10 and STAT5 to STAT3 switching with higher

probability (more than 50%) when the parameters are perturbed within 2-fold, with modest

probability (between 10% and 50%) when the parameters are perturbed within 3-6-fold and

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with low probability (less than 10%) when the parameters are perturbed within 7-fold and

higher.

Next, we identified the most sensitive parameters that have the greatest impact on the steady-

state concentrations of IFN-γ, IL-10, STAT5 and STAT3 for the three concentrations of IL-2:

10-10, 10-6 and 10-1, which correspond to the three assumed T cell phenotypes shown in Fig 1A:

Th1, Th1/Tr1 and Tr1 respectively. We performed the SA using the eFast method (26), because

it was reported as one of the most reliable methods of parameter sensitivity analysis (25). As a

tool for the eFast sensitivity analysis, we used the SBToolbox software (27). We performed the

SA over one order of magnitude of perturbation for 10000 simulations.

Fig S4-Fig S6 illustrate the results of the sensitivity analysis for IFN-γ, IL-10, STAT5 and

STAT3 by the SBToolbox for the non-dimensional IL-2 concentrations [ ] 102 10i −= (Fig S4),

[ ] 62 10i −= (Fig S5) and [ ] 12 10i −= (Fig S6). The bars indicate the sensitivity indices for each

of the parameters of our model. Here we classified the parameters as sensitive if the

corresponding sensitivity index is more than 0.5 .

It can be seen from Fig S4A, Fig S5A and Fig S6A that IFN-γ production is the most sensitive

to the following parameters: 18M , tgg , 8n , 19M , 1tmp and 5ts , which demonstrated high

sensitivity indices (more than 0.5 ) for all the three IL-2 concentrations. Parameters 1n , 2 ,tr 6n

and 2M are sensitive for [ ] 102 10i −= (Fig S4A) and [ ] 62 10i −= (Fig S5A). Parameter 14M is

sensitive only for [ ] 62 10i −= (Fig S5A) and [ ] 12 10i −= (Fig S6A). Another group of IFN-γ-

sensitive parameters that includes 5ts and 6n is involved in the STAT5 pathway activation,

which leads to the production of IFN-γ as shown in Fig 1B and Equations (6). There is also the

third group of IFN-γ-sensitive parameters consisting of 1n , 2tr and 2M that are involved in

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the upstream activation of IL-2 receptor described by Equation (1). Finally, IFN-γ is sensitive

to the Michaelis constant of heterodimerization 14M .

The parameter sensitivity analysis revealed that the concentration of IL-10 is the most sensitive

to 10tg and 9n for [ ] 102 10i −= (Fig S4B), [ ] 62 10i −= (Fig S5B) and [ ] 12 10i −= (Fig S6B).

Parameter 9M is the most sensitive for the phosphorylated STAT3 for all the three IL-2

concentrations as it is shown in Fig S4C, Fig S5C and Fig S6C. Two parameters, namely, 5n

and 6Q , showed high sensitivity only for [ ] 102 10i −= (Fig S4C) and [ ] 62 10i −= (Fig S5C).

The concentration of phosphorylated STAT5 is the most sensitive to the total amount of

STAT5, modeled by parameter 5ts , for all the three tested IL-2 concentrations as it is shown in

Fig S4D, Fig S5D and Fig S6D. Parameter 6n demonstrated high sensitivity indices for

STAT5p for [ ] 102 10i −= (Fig S4D) and [ ] 62 10i −= (Fig S5D). This parameter is involved in

the activation of STAT5 pathway as shown in Fig 1B and Equations (6). The concentration of

STAT5p is sensitive to the Michaelis constant of heterodimerization 14M for [ ] 62 10i −= (Fig

S5D) and [ ] 12 10i −= (Fig S6D). Three parameters, namely 1n , 2tr and 2M , are sensitive for

[ ] 102 10i −= (Fig S4D) and [ ] 62 10i −= (Fig S5D) involved in the activation of the IL-2 receptor

and described by Equation (1).

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Fig S3. The probability of cases when the switching occurs. The percentage of cases when the switching of both cytokine and STAT (C+S+), cytokine but not STAT (C+S-), not cytokine but STAT (C-S+), neither cytokine nor STAT (C-S-) occurs for 1-10 fold change of the optimized parameters.

Table S5. The percentage of cases when the switching of both cytokine and STAT (C+S+), cytokine but not STAT (C+S-), not cytokine but STAT (C-S+), neither cytokine nor STAT (C-S-) occurs for 1-10 fold change of the optimized parameters.

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Fig S4. Parameter sensitivity analysis performed by eFAST for low concentrations of IL-2. Sensitivity indicators for the developed model for IFN-γ (A), IL-10 (B), STAT3 (C) and STAT5 (D) for non-dimensional IL-2 concentration [ ] 102 10i −= .

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Fig S5. Parameter sensitivity analysis performed by eFAST for medium concentrations of IL-2. Sensitivity indicators for the developed model for IFN-γ (A), IL-10 (B), STAT3 (C) and STAT5 (D) for non-dimensional IL-2 concentration [ ] 62 10i −= .

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Fig S6. Parameter sensitivity analysis performed by eFAST for high concentrations of IL-2. Sensitivity indicators for the developed model for IFN-γ (A), IL-10 (B), STAT3 (C) and STAT5 (D) for non-dimensional IL-2 concentration [ ] 12 10i −= .

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4 FACS analysis

Fig S1. FACS analysis for STAT3p- and STAT5p-positive cells. Purified human CD4+ T cells were activated with a-CD3/CD46 antibodies in the presence of 50U/ml

rhIL-2 for 36 hours and percentage of STAT3p- and STAT5p-positive cells assessed by FACS analysis.

Cells were analyzed either as bulk population (bulk CD4+) or after subsorting populations into the three

IFN-γ +/- IL-10-secreting populations generated by CD3/CD46-activation. STAT3p and STAT5p values

were normalized against respective non-phosphorylated STAT protein levels. NA, non-activated, n=3;

shown are mean values ± SD of three independently performed experiments using a different donor

each time.

5 References

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