L. J. Prins. P. Scrimin et al.
drimers. However, before interpreting these correlations it is
important to first analyze which kind of trends in kcat and
KM are expressed intrinsically by this kind of multivalent
systems.
local pH, etc.) will additionally affect the “overall” Michae-
lis–Menten parameters (either positively or negatively). To
correctly interpret the data of the NP-based systems pre-
sented in this study we need first to establish the intrinsic
changes of the “overall” Michaelis–Menten parameters, KM
and kcat, as a function of the composition of the mixed mon-
olayers. It is evident that these changes will also depend on
the structural order of the monolayer, that is, whether the
two thiols 1 (catalytic unit) and 2 (inert unit) are randomly
distributed or present in clusters or homodomains. These
boundary conditions will both be addressed. To evaluate the
correlation between KM, kcat, and the mole fraction x1 on the
surface, we took a truncated icosahedron as a simple model
for a spherical nanoparticle.[55] A truncated icosahedron is
an Archimedean solid composed of 12 pentagonal and 20
hexagonal faces. It was assumed that each face can accom-
modate one unit of 1 or 2. This implies that at full surface
coverage 32 units are bound. In this model each unit has
five or six neighboring units, which corresponds reasonably
to the packing of SAMs of thiols on Au surfaces. The only
catalytically relevant species is the dimeric site 1–1 formed
between two neighboring units 1. In all simulations the 1–1
site has constant KM and kcat values. Contributions from the
single catalytic unit 1 and the dimeric site 1–2 to catalysis
were not included.
A statistical distribution of units of 1 on the surface was
simulated by the stepwise random insertion of 1 on the
facets of the model completely covered with 2 (x1 =0). This
was repeated until all facets were occupied with 1 (x1 =1).
After each addition, the number of catalytic dimeric sites 1–
1 was counted and also the number of dimeric sites 1–1 that
could be occupied simultaneously. It is evident that the first
number is higher than the second, because a single catalytic
unit 1 can potentially form a catalytic site with each of its 1
neighbors, but effectively can participate in the binding of
one substrate molecule only. This is exemplified in Fig-
ure 6a, which depicts the number of potential catalytic sites
and the effective catalytic sites for x1 =0.5. Assuming that
for a given amount of 1 and 2 SAMs are formed according
to a binomial distribution and normalizing for the amount
of 1 present (meaning that the concentration of 1 is a con-
stant) gives the profiles for the number of initial binding
sites and for the maximum number of substrate molecules
bound as a function of the mole fraction x1 as depicted in
Theoretical analysis: Enzyme-like behavior of a catalyst im-
plies the occurrence of an initial binding event between the
catalyst and the substrate, characterized by a dissociation
constant KM, followed by the conversion of substrate into
product, characterized by a first-order rate constant kcat. The
consequence is a saturation profile when the initial rate of
reaction (ninit) is plotted as a function of substrate concentra-
tion, [S]. This graph is defined by a second-order regime at
low substrate concentration ([S]!KM for which ninit =(kcat
/
KM)[E][S]; [E]=“enzyme” or catalyst concentration) and a
first-order regime at high substrate concentrations ([S]@KM
for which ninit =kcat[E]). The Michaelis–Menten equation is
designed to describe the behavior of a single substrate–
enzyme interaction. The interpretation of the Michaelis–
Menten parameters for multivalent enzyme-like catalysts is
not straightforward, because in such systems multiple sub-
strate–enzyme interactions take place simultaneously. The
experimentally obtained values are therefore average values
of all single binding and catalytic events. To have a meaning-
ful discussion of these “averaged” values, one has to know
the intrinsic effect originating from the multivalency of the
system. Previously we have performed such an analysis on a
series of homofunctionalised dendrimers of increasing valen-
cy.[42] Rather interestingly, we observed that the clustering of
catalytic units on a multivalent scaffold gives spontaneously
rise to a positive dendritic effect; that is, an increased cata-
lytic efficiency (kcat/KM) as a function of the dendrimer va-
lency on the condition that catalysis requires the simultane-
ous action of two catalytic units. A theoretical analysis
showed that this intrinsic dendritic effect originates from the
fact that the concentration of catalytic sites (composed of
two units) increases more than linearly as a function of the
valency of the system. In other words, the same number of
catalytic units can create more catalytic sites in case the va-
lency is higher. Consequently, the “apparent” concentration
of catalytic sites is much higher than the nominal concentra-
tion of catalytic units. This implies that fitting of the satura-
tion profile using the classical Michaelis–Menten equation
(taking the concentration of catalytic units as reference)
yields an “overall” value for KM that decreases as a function
of the valency (suggesting stronger binding). This decrease
in the “overall” KM value is an intrinsic property of multiva-
lent systems and has nothing to do with the actual binding
affinity between the catalytic site and the substrate (which
remains constant). Additionally, we showed that the catalyt-
ic efficiency of the multivalent system at saturation (where
&
Figure 7 ( ). Next, homodomain formation of 1 was simulat-
ed by inserting new 1 units on the facets of the model on po-
sitions neighboring other 1 units (Figure 6b). Also here, it
was assumed that SAMs are formed according to a binomial
distribution of 1 and 2 on different nanoparticles. This as-
sumption implies that clustering occurs after monolayer for-
mation. The resulting profiles are also depicted in Figure 7
(&). Clustering during monolayer formation effectively
means the tendency of the thiols to self-sort preferentially
on different nanoparticles, which in an extreme case means
that only homomeric 1 and 2 nanoparticles are present. Nor-
malized on the concentration of 1 would give constant KM
and kcat values independent of x1. This situation is added to
v
init =kcat[E]) is determined by the actual number of catalytic
units that are participating in the formation of catalytic sites.
The “overall” kcat value is lowered by the presence of isolat-
ed catalytic units unable to form a dimeric catalytic site.
Furthermore, an alteration of the catalytic site as a function
of valency (distance between the catalytic units, polarity,
4884
ꢁ 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
Chem. Eur. J. 2011, 17, 4879 – 4889