Supplementary Material for:
Analytical Reduction of Combinatorial Complexity Arising from Multiple Protein
Modification Sites
Marc R. Birtwistle1
1
Department of Pharmacology and Systems Therapeutics, Icahn School of Medicine at Mount
Sinai, One Gustave L. Levy Place, New York, New York, 10029
++1-(212)-241-2051; marc.birtwistle@mssm.edu
Analysis of Assumptions for the EGFR System
Assumption 1. Adaptor binding is slow relative to the phosphorylation/dephosphorylation cycles,
such that these cycles reach quasi-equilibrium.
To evaluate Assumption 1 we must look at the rates of three processes: phosphorylation,
dephosphorylation, and binding.
There is a reasonably rich literature on catalytic rates for ErbB kinases. In an in vitro
setting, kcat’s are in the range from 0.01 to 0.2 s-1 (1-3). It has been shown that including the
juxtamembrane domain and membranes can increase the catalytic efficiency 70-fold (4), so these
kcat’s may be an underestimate for the in vivo situation. Effective first order rate constant
measurements for ErbB1 kinase activity from cell-based assays (which based on our derivation
below should be equivalent to kcat) are estimated to be between 0.05 and 0.1 s-1 (5). The rapid
temporal sampling needed to estimate these rates (seconds), coupled with the difficulty of
obtaining such rapid sampling from lysate-based assays, may well have led to underestimation.
Nevertheless, based on this data, a reasonable, order of magnitude estimate for the first order
kinase rate constants is ~0.1 s-1.
For ErbB phosphotyrosine phosphatases, estimated first order phosphatase rate constant for
ErbB1 receptors are between 0.1 and 0.17 s-1 in a cell-based setting (5). Again, the rapid
temporal sampling needed to estimate these cell-based quantities from lysate-based assays may
have led to underestimation of these rates. Thus, a reasonable, low-side order of magnitude
estimate for the first order phosphatase rate constant is also ~0.1 s-1.
For binding rates to phosphotyrosines of a variety of Src homology 2 (SH2) domains,
dissociation constants (Kd) are in the range of 0.1 to 10 M (6, 7), and their off rate constants are
estimated at ~0.1 s-1 from in vitro surface plasmon resonance experiments (8, 9). Given these
parameters, on rate constants are on the order of ~0.001 nM-1s-1 (koff/Kd = 0.1 s-1/100 nM—
considering the high range of affinity, a worst case scenario in terms of our assumption). We
must note that in purified, EGF-stimulated plasma membrane fractions on glass cover slips, Grb2
has been observed to have an off rate constant as high as 8 s-1 (10). However, live cell data
contradict this to some extent; the effective off rate constant for Grb2 was estimated as 0.5 s-1
when one considers potential effects of fast rebinding events (11), which is more in line with
previous estimates cited above. In live cell experiments using light-induced binding of a
cytoplasmic protein to a plasma membrane protein—a situation analogous to recruitment of
binding proteins to phosphorylated EGFR—complex formation reaches steady state over a time
course of ~5-10 seconds (12-14). Simulations of this complex formation using off rate constants
of 0.1 s-1 are in line with such observed live-cell dynamics, whereas those using off rate
constants of 10 s-1 are much faster, inconsistent with observed live-cell dynamics
(Supplementary Fig. S1). Thus, although the binding rate may be faster than we assumed above
for the case of Grb2, assuming an off rate constant ~ 0.1 s-1 is more consistent with a broad range
of data. The total binding rate depends on binder concentrations. In mouse fibroblasts, absolute
expression levels of common binding proteins (Shc, PLC, PI-3K, SOS) are around 1-10 nM,
with Grb2 being higher at ~ 400 nM (assuming a cell volume of 2000 m3) (15). The Kd of Grb2
for various phosphotyrosines is somewhat high (~1 M) (6), giving it lower on rate constant
which counteracts the effect of higher concentration. Multiplying the on rate constants by the
given 10 nM concentration (again conservative for our assumption) gives total binding rate
estimates ~0.01 s-1.
Based on these numbers, kinase and phosphatase reaction rate time scales are of roughly equal
magnitude, with the upper range of reasonable binding rate estimates at least an order of
magnitude lower. Thus, the assumption that kinase and phosphatase rates are faster than binding
rates is supported by data. Most importantly, these numbers are within the range where reduced
model simulations match well to simulations with models considering full combinatorial
complexity (see Fig. 2 above).
Assumption 2. The phosphorylation sites are located in close physical proximity to one another
relative to the size of the adaptors, such that adaptor binding to one site hinders binding events
on other sites.
The main argument for this assumption is protein size. The EGFR C-terminal tail does not have a
crystal structure and is thought to be largely unstructured. Crystal structures of the EGFR kinase
domain reveal some fixed structure to at least amino acid 1022 (PDB entries 2ITZ, 2ITX, 2EB2),
and the last phosphorylation site we consider is residue 1197 (1173 if the 24 amino acid
localization sequence signal is taken into account). Thus, the unstructured region consists of
perhaps ~150 amino acids. Previous work was done to estimate the lengths of such unstructured
protein regions by coupling the unstructured region between two FRET-compatible fluorescent
proteins (16). These data are well fit by a power law regression model, which predicts a ~100150 amino acid stretch of unstructured protein corresponds to ~6-7 nm (Supplementary Figure
S2). Full length Grb2 (~ 4 nm in length based on crystal structure) is only slightly smaller than
this, and Grb2 is one of the smallest of the common EGFR binding proteins (~25 kDa). Other
common proteins which bind to the receptor are much larger than Grb2 and include Shc (~62
kDa), the p85 subunit of PI-3K (~85 kDa), PLC (~148 kDa), and Stat5 (~92 kDa). It is
important to also consider that Grb2 and these other adaptors are, in most situations, necessarily
bound to other large proteins such as SOS (~170 kDa), the p110 catalytic subunit of PI-3K (~110
kDa), c-Cbl (~120 kDa) and Gab1 (~110 kDa), to mediate their biological effects. This
substantially increases the size of their binding footprint. Moreover, the available binding space
is constrained by the presence of two EGFR cytoplasmic kinase domains as well as the plasma
membrane (see Fig. 3A).
When one considers the large size of these binding protein complexes, along with the
approximate size of the EGFR C-terminal tail and the constraints of the kinase domains and
plasma membrane, the assumption that “phosphorylation sites are located in close physical
proximity to one another relative to the size of the adaptors, such that adaptor binding to one
phosphorylation site hinders binding events on other phosphorylated sites” seems reasonable.
However, we must emphasize that there is not conclusive experimental evidence in favor of this
assumption; it is quite possible that the C-terminal tail can exhibit stretching longer than the
average lengths suggested by Fig. S2, and it has been shown that isolated SH2 domains can bind
to phosphotyrosines in relatively close proximity (17). One alternative assumption would be to
consider that several binding proteins can all directly bind simultaneously to the same EGFR Cterminal tail on all phosphosites. Based on the sizes discussed above, however, that scenario
seems more unlikely than our assumed competitive situation. The other alternative would be to
consider complex combinatorics of direct binding, e.g. Shc can bind to pY1173 when Grb2 is
bound to pY1068 but not when PLC1 is bound. Besides the fact that no primary experimental
data exist to support such considerations and also would be very difficult to gather, the situation
would be extremely difficult and complex to model computationally. Thus, although the
competitive binding assumption is probably not entirely correct, it is reasonable when compared
either to the other extreme of non-competitive binding, or to a variety of complex steric
constraints for simultaneous binding for which no primary data exist.
Assumption 3. Binding of adaptor j occurs with rate constants konj and koffj to one or more
phosphorylated sites. The assumption simplifies the initial derivation and can be relaxed to allow
different on rate constants.
As discussed above, affinities and kinetics of binders to phosphorylated receptor have been
studied (6, 7), yielding well-defined values for phosphorylation site/adaptor pairs.
Assumption 4. Phosphorylation and dephosphorylation of site i occur as effective first-order
processes with rate constants kfi and kri when adaptor is not bound.
First we analyze phosphorylation. For EGFR (and several RTK systems in general) the kinase is
localized with the substrate in the same molecule. One kinase domain of a receptor dimer serves
as an allosteric activator of the other, leading to a “trans” mechanism of phosphorylation (Fig.
3A) (18). Due to the tethering of the substrate to the kinase complex in a small volume in the
same molecular complex (dotted lines in Fig. 3A), binding of unphosphorylated substrate to the
kinase domain should be on the time scale of intramolecular protein structural rearrangements (~
ns to s), extremely fast relative to kcat (~ 0.1 s-1) (1-3). Thus we expect substrate binding to
reach a quasi-equilibrium, giving
Kd 
koff
kon

K [ ES ]
[ S ][ E ]
 [E]  d
[ ES ]
[S ]
(S1)
where S denotes unphosphorylated substrate on the cytoplasmic tail of the “activator” receptor, E
denotes free active site on the “receiver” receptor, and ES denotes the enzyme-substrate complex
(Fig. 3A). Given moiety conservation of enzyme,
[ E ]T  [ E ]  [ ES ] 
K d [ ES ]
[ E ]T [S ]
.
 [ ES ]  [ ES ] 
[S ]
K d  [S ]
(S2)
Substituting into the total rate of phosphorylation vk = kcat[ES] gives
vk 
k cat [ E ]T [ S ]
.
K d  [S ]
(S3)
Now, the enzyme and substrate are in the same dimer, and each cytoplasmic tail on the activator
EGFR potentially contains multiple unphosphorylated substrate sites. We denote the number of
these unphosphorylated sites per receptor ns. Because the reaction is confined to the small
volume around the receptor (denoted by the dotted box labeled VR on Fig. 3A), if there are ND
total active receptor dimers in the cell, then the effective concentrations are given by
[ E ]T 
n N
ND
1

; [ S ]  s D  n s [ E ]T
( N DV R ) V R
( N DV R )
(S4)
Thus, we have
vk 
k cat [ E ]T ns [ E ]T
K d  ns [ E ]T
(S5)
The scale of Kd relative to ns[E] T can determine the effective phosphorylation rate. A single
kinase domain occupies approximately 60 nm3 (~5-by-4-by-3 nm rectangle) (18). We assume VR
is approximately 4 times this volume, giving 240 nm3 and [E] T on the order of ~10 mM.
Although ns is clearly time dependent as phosphorylation reactions proceed, a lower bound for
the term ns[E] T would be when ns=1, giving a low conservative estimate for ns[E] T as 10 mM.
Measured Michaelis constants for tyrosine kinases are on the order of 0.01-1 mM (1-3), and
Michaelis constants ((koff+kcat)/kon) are always greater than Kd (koff/kon). Thus, ns[E] T >> Kd is a
reasonable approximation. That gives
vk  k cat [ E ]T .
(S6)
So it is clear that the effective first-order rate constant for kinases, given these arguments, is
equal to the catalytic constant, giving kf=kcat.
Now we analyze dephosphorylation. It has been shown that a basic Michaelis-Menten
mechanism is a reasonable approximation so long as [D] 0/(Kdm+[pS] 0) << 1, where the subscript
0 denotes initial value, D denotes phosphatase, pS denotes phosphorylated substrate, and Kdm is
the Michaelis constant for the phosphatase (19). EGFR is the substrate here and its concentration
is typically ~100,000 molecules/cell (20), on the order of 10-100 nM. An upper bound for the
number of tyrosine phosphorylated sites on the receptor is ~10, giving [pS]0 as approximately
0.1-1 M. It is rare for any protein to have a cellular concentration greater than 1 M (15),
giving an approximate upper bound for phosphatase concentration. Typical Michaelis constants
for protein tyrosine phosphatases are ~ 1 mM (21, 22). Thus the above condition for MichaelisMenten validity is satisfied ([D] 0/(Kdm+[pS] 0)~0.001<<1), and we have the phosphatase rate as
vd 
k dcat [ D][ pS ]
K dm  [ pS ]
(S7)
where kdcat is the catalytic rate constant. Moreover, Kdm >> [pS], and therefore Eq. S7 can be
reduced to
vd 
k dcat [ D][ pS ]
.
K dm
(S8)
Thus, the dephosphorylation rate can be reasonably approximated as first-order with respect to
phosphorylated substrate, giving kr=kdcat[D]/Kdm.
Extending the Model to Include Binding Cooperativity Between Grb2 and c-Cbl
Adding c-Cbl and Y1045 to the model were straightforward and done as already described for
other single adaptors and sites. We assumed c-Cbl binds phoshphorylated Y1045 with 2.6 M
affinity to be on par with that of Grb2 for its site. We assumed that c-Cbl and Grb2-c-Cbl
complexes were present at the level of 0.44 nM as estimated similar to the other protein levels.
The key model alteration here was creation of a pseudosite that is the combination of Y1045 and
Y1068 to which the Grb2-c-Cbl complex binds with high affinity (0.1 M). We accomplished
this by re-deriving Eq. 21 and leaving out the first term for singly phosphorylated sites, starting
only with the doubly phosphorylated sites by searching for those terms having the site indices for
both considered sites. The implementation is in the MATLAB code file
‘CalculateBeta3TwoSite’.
Figure S1. Binding Kinetics for Different Off Rate Constants. We modeled a simple
bimolecular association reaction between a receptor and its binding partner using mass action
kinetics, with a total receptor concentration of 100 nM, a binder concentration of 10 nM, a
dissociation constant (Kd) of 100 nM, and an off rate constant of either 0.1 s-1 (solid) or 10 s-1
(dash-dot). The reaction kinetics proceed to steady-state over ~10 sec for koff=0.1 s-1, which is
more consistent with live-cell experimental measurements of light-induced membrane
recruitment kinetics (12-14).
Figure S2. Regression Between Number of Amino Acids and Physical Length of
Unstructured Protein Regions. Data were taken from (16), and fit to a power law model as
indicated.
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