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S U P P L E M E N TA RY I N F O R M AT I O N

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Figure S1. Clustering of K-ras and Raf proteins on the plasma membrane. a Plasma membrane sheet prepared from BHK cells co-expressing mRFP-KrasG12V and mGFP-Raf1 labelled with anti-GFP 5nm gold and imaged by electron microscopy. No significant gold labelling of the plasma membrane was evident in cells expressing mGFP-Raf1 alone.b Plasma membrane sheet from BHK cells co-expressing mRFP-KrasG12V and mGFP-Raf1 co-labelled with anti-GFP-6nm gold and anti-RFP-2nm gold and imaged by electron microscopy. (Bars=200nm) c Plasma membrane sheets prepared from BHK cells transiently expressing GFP-Raf-tK or GFP-KrasG12V were immunogold labeled and imaged as in a. The resulting gold point patterns were analyzed using Ripley’s K-function and bootstrap tests used to evaluate differences between the weighted mean K-functions shown 20, 22. There was no significant difference between the clustering behavior of Raf-tK and K-rasG12V (p=0.347). Mean number of gold particles per plasma membrane sheet evaluated was 1772 m-2 (n=8) and 1845 m-2 (n=12) for K-rasG12V and Raf-tK respectively. Similar results (not shown) were obtained by comparing Raf-tK (1.0) and Raf-tK (0.03). We conclude that the K-ras C-terminal anchor attached to Raf operates as in full length K-ras. Therefore it is reasonable to apply the previously described spatial organization and stoichiometry of K-ras nanoclusters to those generated and populated by Raf-tK

c

r(nm)

L(r)-r

K-RasG12VRaf-tK

a b

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Figure S2. Comparison of signal outputs at equilibrium and 2min.All simulation results presented in the main paper are ERKpp concentrations when the system reaches a steady state. However, given that the experimental results in Fig.5b were measured at 2min we explored the effect of terminating the simulations at the same 2min time point. a The EGF dose response simulation shown in Fig.5a at steady state (blue) is compared with that obtained when the simulation is run for exactly 2min (red). The results are not significantly different, essentially because when nanoswitches are operating, as is the case here, steady state is reached very rapidly (generally in <1min). Similar results were also obtained at 2min for other experimental conditions including those shown in Fig.5c and Fig.S3 b Similar comments do not apply to a rerun of the EGF dose response simulation shown in Fig.6a where nanoclusters operate as analog circuits. Here there is divergence of the curves, predominantly because the maximal output at 2min is much weaker than at equilibrium. The results still support our conclusion that signal output of the MAPK pathway is nonlinear if the ERKpp output from each nanocluster is analog, but highlight an additional potential problem, namely slow responsiveness. In contrast if the ERKpp output from each nanocluster is switch-like the signaling output of the MAPK pathway is both graded and rapidly responsive.

a

EGF input EGF input

b







Figure S3. Influence of the number and total area of nanoclusters on the output from the MAPK module We realized the graded signal output of the MAP kinase pathway using Raf activities and numbers of nanoclusters that are directly proportional to EGF concentration. Here, we explore how the system responds to changing the output, numbers, or size of Ras nanoclusters. a The system activates decreasing numbers of nanoclusters, but with maximal Raf activity in each nanocluster. This is equivalent to evaluating the maximum ERKpp response that could be generated by EGF as the Ras clustered fraction ( ) is progressively decreased. b Both Raf activity and the number of nanoclusters are decreased together. The simulations in aand b generate essentially identical outputs showing that it is the number of nanoclusters, rather than Raf kinase activity, that is the dominant factor in the generation of a graded signal output. c The radius of clusters are decreased until they are reduced to an area equal to the cross sectional area of their constituent Ras proteins (~0.12% of the plasma membrane as calculated in xxv). To realize this simulation we used the maximal number of nanoclusters and maximal Raf activity in the nanoclusters but recalculated the effective recruitment rates for Raf and KSR/MEK/ERK (=a1 and a2) to reflect the reduced probability of a successful collision with a nanocluster as the domains decrease in size (see xxiii for details). There is an approximately linear decline in ERKpp generation as the fractional nanocluster area on the cell surface decreases. Strikingly, when nanoclusters disappear altogether and Ras functions as individual molecules, signal output drops to 3% of the maximal ppERK. Thus without functional nanoclusters the cell cannot be activated irrespective of the strength of input. d compares the results in a (black) and c (red: the x co-ordinates of the graph in c have been

multiplied by a factor of 0.0104 ). The results are very close, highlighting the broadly similar concept being tested but the different approaches being used. Both mechanisms clearly show that as nanoclusters diminish, so too does the maximal possible signal output.

a b

c d






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Figure S4. Experimental validation of the effect of reducing Ras nanoclustering a. The algorithm for simulating nanoclustered point patterns described by Plowman et al (Ref 6)was used to obtain an estimate for the clustered fraction ( ) of K-ras in BHK cells treated for 5min with 1 M latrunculin. The results are presented as a 99% confidence interval for the value of Lmax for a range of values of (0.125 - 0.5). Lmax is a summary statistic for clustering (defined in Ref 6) and is equal to the peak value of an L(r)-r curve of the type shown in Fig.S1c. Ref 6

shows that 1 M latrunculin reduces Lmax for K-ras clustering from 2.9 to 1.83, with no change in the radius of the nanoclusters. The results of the clustering simulation show that this equates to a reduction in from 0.42 to 0.20. b. Serum starved BHK cells were treated with 1 M latrunculin for 5min, or left untreated and stimulated with EGF (0-15ng/ml) for 2min. Total ERKpp response was measured as in Fig.5. Experimental results are shown as data points of mean values ± s.e.m. The experimental results are compared with the simulated ERKpp response to EGF (solid lines) as shown in Fig.5c for values of equivalent to those estimated for the control and latrunculin treated cells.

control

1 M = 0.42

= 0.20

Lmax

a

clustered fraction

EGF input

= 0.42

= 0.20

b

1 M Latrunccontrol

ERKpp







Figure S5. Function of ERK phosphatase on the plasma membrane In our model we assumed that the activities of phosphatases are very low on the plasma membrane compared to that in the cytosol. In order to explore the consequence of this assumption, we developed a second mathematical model that incorporates ERK phosphatase (ERKPase) into the plasma membrane subsystem. MEK phosphatase is still excluded from the model because of the weak activity that we have measured in BHK cells (=10% of that in the cytosol) and the small molecular number on the plasma membrane. The graphs compare signal outputs of the two systems with and without ERKPase on the plasma membrane. a Comparison of the Raf-tK system with switch like signal outputs (compare with Fig.4b) The results overlap. b Comparison of the modified Raf-tK system with graded signal outputs (compare with Fig.4d) c Comparison of the Ras-NC system with nanoswitches operating (compare with Fig.5a) d Comparison of the Ras-NC system with analog NC output (compare with Fig.6a). In all these different conditions, the signal outputs of these two systems are very close to each other. Numerical results therefore confirm that the function of phosphatases is not significant on the plasma membrane. The major reason for this negligible effect of ERKPase in nanoclusters is its low activity. Compared with the disassociation rates (d21 = 0.6 and d22 = 0.5) dephosphorylation rates (k21 = 0.01 and k22 = 0.01) are relatively small. In addition, the limited lifetime of nanoclusters helps minimize the function of ERKPase on the plasma membrane.

a

ERKPase on PM No ERKPase on PM

b

Raf input

c

EGF input EGF input

d




Raf input





Table S1. Initial conditions, molecular and nanocluster numbers in the system

Number Concentration

Ras on plasma membrane 71014.1 !

K-Ras on plasma membrane 61087.3 !

Raf in cytosol 5107.3 !

Maximum number of K-Ras nanoclusters 46440

Maximum number of Raf-tK nanoclusters 22230

MEK in cytosol 7102.2 ! 8.73 µM

ERK in cytosol 7101.2 ! 8.33 µM

MEKPase in cytosol 4100.4 ! 0.016 µM

ERKPase in cytosol 71000.1 ! 3.97 µM

Note: All other initial values are set to zero.





Table S2. Reaction rates in the plasma membrane subsystem

Base rate Ras nanocluster Raf-tK nanocluster Comment

1a 2.5 8.75 17.5

1d 0.2 0.7 1.4

2a 2.5 8.75 17.5

2d 0.01833 0.064 0.128

1k 48 168 336

2k 48 168 336

0k 2000 7000 14000

1a 262.5 525 Rate in the modified system

!

a2 262.5 525 Rate in the modified system

1k 0.8 1.6 Rate in the modified system

2k 0.8 1.6 Rate in the modified system

Note:

[1] Nanocluster reaction rates are the product of the base reaction rate and the averaged Raf

number per nanocluster, i.e. 3.5 and 7 for Ras and Raf-tK nanoclusters respectively.

[2] The lower part of this table only gives reaction rates of the modified system that are different

from those in top part. The modified system yields an approximately linear nanocluster output.





Table S3. Reaction rates in the subsystem of the cytosol.

Constant Rate Comments 3a 0.9450 1/6 of the value of the corresponding rate in Ref 8

4a 0.0165 The same comment for

3a

5a 0.0436

6a 0.9582 The same comment for

3a

7a 0.0436

!

a8 0.3304 The same comment for

3a

!

a9 0.2396 =

6a /4

10a 0.0826 =

!

a8/4

3d 0.8

4d 0.5

5d 0.033

6d 0.6

7d 0.033

8d 0.5

9d 0.6 The same value as

6d

10d 0.5 The same value as

8d

3k 0.0097 The same comment for

3a


3a

!

k5(0) 16


3a

)0(

7k 5.7


3a

9k 0.05 The same value as

6k

10k 0.05 The same value as

8k

1µ 0.08 Numerical estimation in this work




1µ 0.007 Rate in the modified system




Note: [1] Except where indicated reaction rates have the same values as in Ref 8. [2] The unit of binding reactions rates

ia is 11

sec!!

Mµ , that of1

µ and 2

µ is 12sec

!!Mµ , that

of3

µ and 4

µ is 11sec

!!Mµ , while that of all other rates is 1

sec! .

[3] All parameters in the cytosolic subsystem are the same for the Ras and the Raf-tK systems. [4] The lower part of this table only gives reaction rates of the modified system that are different from those in top part. The modified system yields an approximately linear nanocluster output.





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