Biomedical Engineering Reference
In-Depth Information
reaction rates has been observed. A recent theoretical study conducted in the labo-
ratory of A. Jasanoff suggested how several parameters, including particle concen-
tration, functional group density, and ratio of particle types, can be optimized to
achieve reaction rates on the order of seconds [59]. The group's simulations pre-
dicted that reaction rates could vary over three orders of magnitude within reason-
able activation kinetics, biomolecular on and off rates, particle concentrations, and
functionalization levels.
Shapiro
et al.
have proposed a two-step model for MRSw agglomeration, with
the fi rst step consisting of an activation of both species of nanoparticles due to the
presence of an analyte [59]. Such activation results from the analyte binding to or
analyte- induced modifi cation of the particle surface. The second step consisted of
agglomeration of the activated nanoparticles. These two steps are shown in Equa-
tion 1.4 :
+ →
∑
**
**
j
AB A B
+→
AB
j
(1.4)
where A and B denote nanoparticles of two different functionalities, A* and B*
represent activated A and B particles, respectively, and
AB
i
**
represent an aggre-
gate composed of
i
A particles and
j
B particles [59]. These authors assumed that
the rate of the fi rst step was much faster than that of the second step, thus causing
agglomeration to be the rate-limiting step. Interestingly, for sensors that are based
on nanoparticle dispersion, deactivation and dispersion are likely both to be fast
steps, which explains why for some sensors much faster rates are observed for
nanoparticle dispersion. Because T
2
measurements made by CPMG echos can be
less than seconds, the signal acquisition is rapid compared to the fi rst two steps,
and will not signifi cantly infl uence most observed reaction rates.
This two-step model was used to predict how changes in particle concentration,
the number of functional groups per particle, and also the ratio between particle
types, could infl uence the observed binding kinetics and particle size. Reaction
rates for different conditions were compared in terms of an observed time constant
(T
obs
). T
obs
is the time required for the reaction to reach 63% completion (one expo-
nential unit). A reaction following fi rst-order kinetics is 95% complete after 3
j
×
T
obs
,
and 99% complete after 5
×
T
obs
. Particle concentrations of 23 n
M
(iron content
10
g ml
- 1
), which are similar to those used for most MRSw studies, have a pre-
dicted T
obs
value of
μ
100 s. This suggests that these reactions should be complete
in less than 10 min, which is faster than many observed reaction times. This dis-
crepancy may arise from the number of functional groups for the experimental
results being much lower than those used for the simulations. Particle concentra-
tions as low as
<
10 p
M
have predicted T
obs
values longer than 1000 s, while concen-
trations as high as 0.1
∼
M
and 60 functional groups per particle have predicted T
obs
values of 2 s. Under all reaction conditions, the optimal ratio of particle types A and
B was predicted to be 1 : 1. According to these theoretical results, relatively fast reac-
tion times should be observed for particles at concentrations
μ
>
50 n
M
that have been
decorated with a high number of functional groups (
50). Many MRSw nanopar-
ticles have had much fewer functional groups on their surface, corresponding with
>
