Reconfigurable ionchannel based biosensor: Input excitation design and analyte classification
ABSTRACT This paper considers modeling and signal processing of a biosensor incorporating gramicidin A (gA) ion channels. The gA ion channel based biosensor provides improved sensitivity in rapid detection of biological analytes and is easily adaptable to detect a wide range of analytes. In this paper, the electrical dynamics of the biosensor are modeled by an equivalent second order linear system. The chemical dynamics of the biosensor response to analyte concentration are modeled by a twotime scale nonlinear system of differential equations. An optimal input excitation is designed for the biosensor to minimize the covariance of the channel conductance estimate. By using the theory of singular perturbation, we show that the channel conductance varies according to one of three possible modes depending on the concentration of the analyte present. A multihypothesis testing algorithm is developed to classify the analyte concentration in the system as null, medium or high. Finally experimental data collected from the biosensor in response to various analyte concentrations are used to verify the modeling of the biosensor as well as the performance of the multihypothesis testing algorithm.

Conference Paper: Chemical kinetics and mass transport in an ion channel based biosensor
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ABSTRACT: This paper deals with the construction and analysis of distributed dynamical models for a novel biosensor that exploits the molecular switching mechanism of biological ion channels. The rate of change of the concentration of the chemical species in the biosensor are given by a system of nonlinear ordinary differential equations in the reactionrate limited region of operation. When the transport rate of analyte to the biosensor surface is comparable to the intrinsic reaction rates, the dynamics of the biosensor are explained accurately using a two dimensional advection diffusion parabolic partial differential equation. When the rate of transport of analyte to the biosensor surface is much slower than the intrinsic reaction rates the biosensor is said to be operating under mass transport limited conditions. Under these conditions a system of coupled ordinary differential equations and the mass transport coefficient model the dynamics of the biosensor accurately. The equivalent mathematical models are shown to produce accurate data under the required operating conditions by comparison with experimental data.Decision and Control (CDC), 2010 49th IEEE Conference on; 01/2011
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Reconfigurable IonChannel based Biosensor: Input Excitation Design and
Analyte Classification
Sahar M.Monfared,Vikram Krishnamurthy Fellow, IEEE,Bruce Cornell
Abstract—This paper considers modeling and signal process
ing of a biosensor incorporating gramicidin A (gA) ion chan
nels. The gA ion channel based biosensor provides improved
sensitivity in rapid detection of biological analytes and is easily
adaptable to detect a wide range of analytes. In this paper,
the electrical dynamics of the biosensor are modeled by an
equivalent second order linear system. The chemical dynamics
of the biosensor response to analyte concentration are modeled
by a twotime scale nonlinear system of differential equations.
An optimal input excitation is designed for the biosensor to
minimize the covariance of the channel conductance estimate.
By using the theory of singular perturbation, we show that the
channel conductance varies according to one of three possible
modes depending on the concentration of the analyte present.
A multihypothesis testing algorithm is developed to classify
the analyte concentration in the system as null, medium or
high. Finally experimental data collected from the biosensor in
response to various analyte concentrations are used to verify
the modeling of the biosensor as well as the performance of the
multihypothesis testing algorithm.
I. INTRODUCTION
Biological ion channels are waterfilled subnanosized
pores formed by protein molecules in the membranes of all
living cells [8], [1]. Ion channels in cell membranes play a
crucial role in living organisms. They selectively regulate
the flow of ions into and out of a cell and regulate the
cell’s biochemical activities. This paper deals with model
ing and signal processing of a biosensor that exploits the
selective conductivity of ion channels. Such ion channel
based biosensors can detect target molecular species of
interest across a wide range of applications. These include
medical diagnostics, environmental monitoring and general
biohazard detection.
The ionchannel based biosensors we focus on are built
using gramicidin A. Gramicidin A was one of the first
antibiotics isolated in the 1940s [7, pp.130] and has a
low molecular weight. In [3], a novel biosensor, which
incorporates gramicidin A ion channels into an artificial cell
membrane was developed by our coauthor, see [13], [4], [2].
This paper describes how such ion channel biosensors can be
modeled as a stochastic dynamical system, how their input
can be dynamically adapted to minimize the detection error
covariance, and finally how maximum likelihood classifiers
can be used to detect analyte presence and concentration.
The main results of this paper are as follows:
1) Dynamical Response of Biosensor: In Section II the
electrical dynamics of the ion channel biosensor are
modeled by an equivalent second order linear system,
see [13],[11]. We then formulate the dynamics of the
biosensor response to analyte concentration as a two
time scale nonlinear dynamical system. The presence
of analyte decreases the concentration of the dimers
formed in the biosensor, thus decreasing the channel
admittance. In order to study the evolution of the
channel admittance in response to the introduction of
analyte, the chemical kinetics of the biosensor are mod
eled as a singularly perturbed system, see also [9],[5].
The channel concentration evolves according to three
regimes depending on the concentration of the analyte
in the system.
2) Optimal Input Design: An input controller is designed
to optimize the input excitation to the biosensor, by
minimizing the covariance of the biosensor impedance
estimate, see [10]. The optimal input excitation is
found to be independent of channel conductance. As
a result we can decouple the biosensor input design
problem from the analyte concentration classification.
3) Analyte Concentration Classification: The final con
tribution of the paper is to devise a multihypothesis
Kalman filtering algorithm to classify the concentration
of the analyte present in the system as null, medium or
high. The derived equations, describing the evolution
of the channel conductance as a function of analyte
concentration, is used in this stage.
The remainder of this paper is organized as follows. Section
II describes the construction of the biosensor as well as the
electrical and chemical modeling of the biosensor. Section
III focuses on optimal input excitation design as well as
the maximum likelihood classifier algorithm. Section IV
includes an experimental study of the classification algorithm
described in Section III on the actual ion channel biosensor,
using Streptavidin as analyte and Biotin as binding site.
II. MODELING THE DYNAMICS OF THE ION CHANNEL
BIOSENSOR
The construction of the ion channel biosensor developed
by [3] involves sophisticated concepts in biochemistry. How
ever, for our purposes its operation can be simply described
as follows. First an artificial ‘tethered’ lipid monolayer is
constructed containing tethered gramicidin channels. ‘Teth
ered’ means that the inner layer of the membrane is fixed to
a gold substrate (using a disulphide bond) and is no longer
mobile. Then a second outer mobile monolayer comprising
of lipids and gramicidin channels is introduced. These com
ponents “selfassemble” in water to form a lipid bilayer that
mimics a cell membrane. The lipid bilayer is 4nm thick and
is tethered 4nm away from the gold electrode. A voltage of
Joint 48th IEEE Conference on Decision and Control and
28th Chinese Control Conference
Shanghai, P.R. China, December 1618, 2009
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typically 100 to 300 mV is applied across the membrane. The
gramicidin channels (each with conducting pore of diameter
0.4nm and length 2.8nm) act as subnanosized pipes that
move randomly along the outer monolayer of the bilayer.
The channels in the inner layer are tethered and hence cannot
move. As the mobile gramicidin channels in the outer layer
diffuse, occasionally a channel in the outer layer will align
exactly with a channel at the inner level of the membrane,
thereby forming a single longer pipe known as a dimer. When
a dimerforms, ions travel along it, thereby resulting in a small
current. At any given time instant, several such pairs of pipes
can align (forming dimer) or disassociate (breaking dimers),
since the outer layer diffuses randomly. Therefore the current
recorded at the output of the biosensor is a random process.
In the construction of the biosensor, specific antibodies
(binding sites/receptor molecules) that recognize specific
analyte molecules are attached to the mobile outer layer
channels, see Fig. 1(b). Electrolyte that may or may not
contain the analyte is introduced. If no analyte molecules
are present, the biosensor operates as above. On the other
hand, if analyte is present in the electrolyte, then the arrival
of analyte crosslinks antibodies attached to the mobile outer
layer channels, to those attached to membrane spanning lipid
tethers, Fig. 2(b). Due to the low density of tethered channels
within the inner membrane, this disrupts their ability to
align with their immobilized inner layer channel partners.
Gramicidin dimer conduction is thus prevented and the
ionic admittance of the membrane decreases. The resulting
decrease in current signals the presence of analyte. The mode
of increase of resistance (decrease of admittance) with time
provides a means to estimate the concentration of the analyte.
A. Dynamical Response of Biosensor
The ion channel biosensor can also be viewed as a switch
that acts as a biological transistor. Fig.1(a) illustrates the
equivalent circuit of the switch before introduction of analyte
and Fig. 2(a) shows the equivalent circuit diagram after the
detection of analyte,. The broken RGsignals the disruption
of dimer formation and the decrease in channel conductance.
The resistor RG = 1/G, where G denotes the channel
conductance, models the membrane resistance and increases
with the presence of analyte. CM denotes the membrane
capacitance and is typically 100nF. CIdenotes the interfacial
capacitance of the gold substrate. Note that one face of the
capacitor CI is charged due to ions, the other face is due
to electrons that form the output current of the biosensor.
Thus CIprovides the interface between the biological sensor
and the electrical instrumentation. REdenotes the electrolyte
resistance – this is typically known to be around 200 Ω. The
300mV bias voltage controls the value of the capacitor CI.
Typically CI can change from 1 µF at 300mV to 0.01µF
at no bias. The biosensor is usually deployed with a 300mV
bias. The admittance transfer function of the equivalent
circuit parameterized by G is
H(s;G) =
I
Vout
=
s2+ saG
s2RE+ s(b2+ b1G) + b3G
(1)
(a) (b)
Fig. 1.
symbolic representation of the structure, when dimer is formed. No analyte
is present.
Ion channel switch biosensor equivalent electrical circuit and the
(a)(b)
Fig. 2.
symbolic representation of the structure. Analyte is present and the dimer
formation is prevented.
Ion channel switch biosensor equivalent electrical circuit and the
The constants in H(s) above are
b3=
a =
1
CM, b1 =
RE
CM,
1
CMCI, b2=
1
CM+
1
CI.
The system transfer function is bandlimited using a third
order Butterworth antialiasing filter with cutoff frequency
of fcutoff = 1500Hz and discretized using a bilinear
transformation with sampling frequency of Fs = 3000Hz.
The discretization results in the fifthorder discrete time
ARMA process;
?
i=1
?
i=1
y(k)=−
5
?
q1iy(k − i) +
5
?
i=0
q2iu(k − i)
?
+
G−
5
?
q4iy(k − i) +
5
?
i=0
q3iu(k − i)
?
(2)
where u is the input to the system and q1i,q2i,q3i,q4i are
constants linear in G, with values depending on the elements
of the circuit.
B. Dynamics of Biosensor Response to Analyte
From an abstract point of view, we can describe the biosen
sor response to an analyte by a two time scale stochastic
dynamical system: Let k = 1,2... denote the fast time scale.
Typically, with a sampling frequency of 3 kHz, k evolves on
the millisecond time scale. Let n = 1,2,... denote the slow
time scale. We will refer to n as batch number. So for fixed
large positive integer T, the nth batch on the slow time
scale corresponds to times k ∈ [(n−1)T +1,...,nT] in the
fast time scale. Typically, T is chosen in the order of a few
seconds.
Let A denote the unknown concentration of analyte. From
detailed experimental analysis of the biosensor and reaction
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rate dynamics analysis presented in Section IIC, it is known
that the conductance G of the biosensor evolves on the
slow time scale according to one of 3 different concentration
modes; M
Gn+1= fM(A∗)(Gn) + wn,
Here κ1,κ2,κ3,κ4are constants, with κ2 < 1. wnis a zero
mean white Gaussian noise process with small variance and
models our uncertainty in the evolution of G. The function
fM(A∗)models the decay of the membrane conductance,
Gnaccording to one of 3 distinct modes depending on the
analyte concentration A∗. For no analyte present (M = 1),
the admittance remains constant. For medium (A ≥ 10−9M)
and high (A ≥ 10−8M) concentrations (M = 2 , M = 3),
the decrease in admittance is exponential with different decay
rates.
Let uk denote the applied input excitation voltage to the
biosensor at time k. For practical purposes, in the ion channel
biosensor, we restrict ukto a pseudo random binary sequence
(PRBS) that is symmetric with uk ∈ {−300mV,300mV}.
For simplicity of hardware implementation, we further re
strict the PRBS sequence uk to a twostate Markov chain
with symmetric transition probability matrix
?
The measured output current of the biosensor on each T
length interval in the fast time scale is given by applying
an antialiasing filter and then sampling the continuous time
system (1) as mentioned in Section IIA.
(3)
M =
1
2
3
if A∗= 0 (no analyte) : fM(A∗)(Gn) = Gn
if A∗is medium : fM(A∗)(Gn) = κ1Gn+ κ2
if A∗is high : fM(A∗)(Gn) = κ3Gn+ κ4
P =
p1 − p
p1 − p
?
,0 ≤ p ≤ 1.
(4)
C. Chemical Dynamics of Biosensor
To study the evolution of the channel conductance as a
function of time we need to analyze the evolution of the
dimer concentration in the biosensor. Since the dynamics of
the biosensor including the dimer concentration are governed
by many chemical species and their reactions, we analyze the
dynamics of these reactions.
The reactions involved in the biosensor mainly stem
from the binding of the primary species and the secondary
species or complexes. The primary species are Analyte, a,
of concentration A, binding site b of concentration B, free
moving monomeric ion channel c of concentration C, and
tethered monomeric ion channel s of concentration S. While
the complexes are w, x, y and z with concentrations, W, X,
Y and Z and are formed according to the following equations
a+b ⇋f1
s ⇋f5
b ⇋f4
r1w,a + c ⇋f2
a + d ⇋f6
r2x,w + c ⇋f3
x + c ⇋f7
r3y
c+
r5d,
r6z
r7z
(5)
x+
r4y
where, fiand ri for i = {1,2,3,4,5,6,7} are respectively
the forward and reverse reaction rate constants with units of
(M−1s−1( for 3D reactions and (cm2s−1) for 2D reactions.
The rate of these chemical reactions can be computer as
below
R1
R3
R5
R7
=
f1AB − r1W,
f3WC − r3Y,
f5CS − r5D,
f7XS − r7Z
R2= f2AC − r2X
R4= f4XB − r4Y,
R6= f6AD − r6Z
=
=
=
(6)
In general the analyte concentration, A, changes throughout
the biosensor chamber as a result of mass transport due to
diffusion. However, at reasonably high analyte concentra
tions, (A ≥ 1µM), since the time scale for A to reach a
steady state distribution throughout the chamber is shown
in [12] to be typically substantially less than the time scale
associated with the chemical reactions, it is reasonable to
assume a constant analyte concentration through out the
chamber. Ignoring the mass transport in the biosensor we
denote the constant analyte concentration as, A∗.
Defining u = {B, C, D, S, W, X, Y, Z}Tand
r(u(t)) = {R1,R2,R3,R4,R5,R6,R7}T,where T denotes
transpose and Ri are defined in (6), the nonlinear system
of differential equations describing the evolution of the
chemical species can be written as
d
dt(u) = Mr(u(t))
(7)
where M is a matrix defined by
Although the system of differential equations, denoted
in (6),(7), is solvable numerically, we require an analytical
solution to (7) which describes the evolution of the channel
conductance as a function of time.
Eigenvalues of the linearized version of (7) for typical pa
rameter values of the biosensor satisfy λ7,8 ≫ λ1,2,3,4,5,6.
Therefore the species Y and Z decay at a rate much faster
than the other species. Accordingly define the fast species
β = {Y,Z}, and slow species α = {B,C,D,S,W,X}.
Let g(α,β) denote the vector field of the fast variables and
f(α,β) the vector field of slow variables. Equations (6),(7)
can be expressed as a twotime scale system
M=
−1
0
0
0
1
0
0
0
00−1
0
0
0
0
−1
1
0
00
0
0
0
0
−1
0
0
0
1
0
0
−1
0
0
−1
0
1
0
−1
1
−1
0
0
0
0
−1
0
0
0
0
1
−1
0
−1
0
1
dα
dt= f(α,β)ǫdβ
dt= g(α,β)
(8)
Here ǫ =
of the governing differential equations.
The following theorem uses basic singular perturbation
theory, specifically Tikhonov’s theorem, [9, Sec.11.1] to
simplify the above twotime scale nonlinear system.
1
λ7= 10−2is chosen as the smallest time constant
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Theorem 2.1:
depicted by the two time scale system (8). Then as ǫ → 0, the
trajectories of (8) converge to the trajectories of the following
system:
dα
dt= f(¯ α,h(¯ α)),
Here ¯ α denotes the solution to ˙ α = f(α,h(α)) where h(α)
denotes the solution of the algebraic equation g(α,β) =
0. More specifically, trajectories of (8) starting in an O(ǫ)
neighborhood of β = h(α) satisfy
Consider the chemical species dynamics
¯β = h(¯ α)
(9)
α(t) − ¯ α(t)=O(ǫ),z(t) − ¯ z = O(ǫ)
(10)
for all t ∈ [0,T] where T denotes a finite time horizon.
The proof follows from verifying the conditions of Theo
rem 11.1 in [9, Sec.11.1]. We can now use the above theorem
to show that the biosensor responds in three distinct modes
depending on the analyte concentration. (Recall these modes
were described in SectionII). However, first notice the
following conservation equations;
C + D + X + Y + Z
B + W + Y
=
=
C(0)
B(0)
D + S + X=S(0)
(11)
which when used along with the empirical fact that
C(0)≫S(0)≥B(0), leads to the approximate
relation S ≈ S(0). This relation will be used to further
simplify the differential equation describing evolution of
dimer concentration with time.
Corollary 2.1:
S ≈ S(0), a simplified equation describing the evolution
of the dimer concentration with time is obtained. Using
Euler’s method with stepsize h we arrive at a discrete time
expression;
By Theorem 2.1 and the approximation
Dn+1= (1 − (r5+ f6A∗)h)Dn+
?
hS(0)
?f5C(r6+ r7) + r6f7X
r6+ r7
??
(12)
Therefore the dimer concentration of the biosensor, D,
evolves according to one of the following three modes,
depending on the concentration of the analyte, A∗;
Dn+1= fM(A∗)(Dn) + wn,
where the constants κifor i ∈ {1,2,3,4} are calculated from
the following equations using medium A∗for κ1and κ2and
high A∗for κ3and κ4;
(13)
M =
1
2
3
if A∗= 0 (no analyte) : fM(A∗)(Dn) = Dn
if A∗is medium : fM(A∗)(Dn) = κ1Dn+ κ2
if A∗is high : fM(A∗)(Dn) = κ3Dn+ κ4
κ1= κ3= (1 − (r5+ f6A∗)h),
?
κ2= κ4= hS(0)
?f5C(r6+ r7) + r6f7X
r6+ r7
??
(14)
Using
is directly proportional to dimer
derive equations explaining the evolution of the channel
conductance on the slow time scale.
(13) and noting that the channel conductance
concentration, we can
III. JOINT INPUT EXCITATION DESIGN AND ANALYTE
CONCENTRATION CLASSIFICATION
The binding of analyte and receptor molecules in the
biosensor results in a slow decrease in the admittance G.
By optimizing the applied input voltage u to the biosensor,
we can infer the presence and concentration of analyte more
rapidly. The goal of this section is to describe how to choose
an optimal pseudorandom binary sequence of applied volt
ages u (parameterized by p in (4)) to the biosensor to mini
mize the detection time for the presence and concentration of
analytes. In other words we have a combined optimal input
design and multiple hypothesis testing problem. Moreover,
by exploiting the two time scale nature of the biosensor
dynamics and the characteristics of its response, the problem
can be decoupled into optimizing u and then doing a multi
hypothesis test.
A. Optimal Input Excitation Design
First consider the problem of optimizing the PRBS input
u. Our approach is based on optimal input design for
open loop experiments, and is well known in the systems
identification literature [10, Sec.13.3]. The open loop design
is approximately justified due to the twotime scale nature
of the biosensor dynamics described above, and because we
adjust the parameter p batchwise on the slow time scale
n. Assuming that a consistent estimator is available for
G, then the asymptotic covariance of the estimate when a
prediction error method is applied for estimation is (see [10,
Eq.(13.26)])
?π
Here κ and Me denotes terms independent of u. Also
Su(ejω;p) denotes the spectral density of the input PRBS
sequence u with transition probabilities given above. The co
variance E{u0uk} of the Markov chain is straightforwardly
evaluated as u2(2p−1)k, where u denotes the magnitude
of binary symmetric signal uk. In discrete time, the spectral
density of the input PRBS (Markov chain) u is
¯
M = κ
−π
H(ejω;G)2Su(ejω;p)dω + Me.
Su(z;p) =
2u2
1 − (2p − 1)z−1
(15)
The optimal choice of input that minimizes the asymptotic
covariance of the estimate of G is
?π
The above optimization problem is easily solved numerically.
Fig.3 shows how the optimal probability p∗(G) of the PRBS
input u varies with admittance G. The really nice property of
p∗(G) is that it varies very little with G. Therefore, p∗(G)
varies very little with M – since G evolves according to
M. As a result we can decouple the optimal input design
p∗(G) = argmaxp
−π
H(ejω;G)2Su(ejω;p)dω
(16)
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and the multihypothesis test. This result was also confirmed
by computing the least squares estimate of the channel
conductance when the statistical properties of the input to the
system, u, were changed according to varying probabilities.
The results are denoted as p∗
LS(G) and are superimposed on
Fig.3. It can be seen that the two results are similar.
Fig. 3.
power spectrum calculations as well as least square simulations is plotted
and it can be concluded that the two methods results in similar optimal
probabilities
Optimal probability for each RGvalue found from both analytical
B. Maximum Likelihood Classifier
Equation (12) in Section IIC describes D(n,A∗).
Since channel conductance is directly proportional to the
dimer concentration it can be concluded that the channel
conductance also changes according to one of three possible
modes, denoted previously in (3), depending on the analyte
concentration.
We use Multiple Modal Adaptive Estimation (MMAE)
algorithm to classify the concentration of the analyte in the
system as medium, high or null. Details of this algorithm
can be found in [6].
In the MMAE algorithm three Kalman filters are used with
each filter; KFi where i ∈ {1,2,3}, tuned to one possible
model of the system, denoted by
G(n + 1)=
F(n)G(n) + ω(n)
Y (n)=
H(n)G(n) + V(n)
(17)
where {ω(n)} and {V(n)} are mutually independent zero
mean white Gaussian sequences with
E{ω(n)ω(n)T} = Q,
E{V(n)V(n)T} = R
Note that the observation equation in (17) is arrived at by
manipulating the 5th order ARMA model process denoted
in (2) in Section IIA.
In equation(17) the observation vector, Y (n) and the
observation matrix H(n) are large in size due to the fact that
for each increment of the slow time scale, n, the fast time
scale is incremented by hFs. Therefore the simulation of
the MMAE algorithm is hindered by numerical problems.
To circumvent these numerical problems we average the
observation vector and the observation matrix. Comparing
the Kalman filter estimation for the original model, (17), and
the averaged system depicted in (18), shows that averaging
does not affect the optimality of the Kalman filter estimates
in the MMAE algorithm. The averaged system is denoted
below
G(n + 1)
Hl(n)TYl(n)
Hl(n)THl(n)
?
=
F(n)G(n) + ω(n)
G(n) +Hl(n)TVl(n)
Hl(n)THl(n)
?
???
Yaverage(n)
=
???
Vaverage(n)
(18)
where, Yaverage(n) and Vaverage(n) are the averaged obser
vation matrix and noise vector, and the averaged noise vector
has the following new noise variance
σ2
new=
σ2
hn(1)2+ hn(2)2+ ... + hn(hFs)2
(19)
Using the new averaged system denoted in
assuming1
3as the prior probability of each mode, the MMAE
algorithm produces the posterior probabilities of each mode.
Therefore allowing ℘nto denote the active model at time n
and P to denote probability, the algorithm calculates Pl(n) =
P{℘n = ℘lY (n)} . At each time instant the mode with
maximum posterior probability, P(n) is chosen as the true
model.
The averaged system (18) with noise variance (19) is
used to test the validity of the the MMAE algorithm with
empirical data in the next section.
(19) and
IV. EXPERIMENTAL RESULTS
Using the averaged discrete time state space representation
of the system depicted in(18) in Section IIIB and the
empirical data obtained from an experiment run conducted
of the biosensor, we test the MMAE algorithm in classifying
the concentration of the analyte.
A. Experimental Setup
The experimental data used are the measurements of
channel conduction when Streptavidin is used as the analyte.
Streptavidin is a tetrameric protein which binds very tightly
to the small molecule, Biotin. Therefore Biotin is used
as the biding site. The experiment was run for various
concentrations of Streptavidin, including medium, high and
null levels.
B. Maximum likelihood analyte classifier
Fig.4 shows the response of the classification algorithm to
experimental data. As shown the algorithm performs well in
detecting the active mode of the system when real data are
used.
In each of the subplots of Fig.4, the channel conductance
is evolving according to one of the three possible regimes
depicted in (3) in Section IIB, depending on the analyte
concentration. It can be seen that as expected the probability
of each of the models except for the true model approaches
0 while the probability of the true model approaches 1.
For example in the first subplot the analyte concentration
is high and since the first model corresponds to high ana
lyte concentration state equation, lim
n→∞(Pactive mode(n) =
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