Page 1

Reconfigurable Ion-Channel 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 two-time 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 multi-hypothesis 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

multi-hypothesis testing algorithm.

I. INTRODUCTION

Biological ion channels are water-filled sub-nano-sized

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

bio-hazard detection.

The ion-channel 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 multi-hypothesis

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 “self-assemble” 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 16-18, 2009

FrB11.2

978-1-4244-3872-3/09/$25.00 ©2009 IEEE7698

Page 2

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 sub-nano-sized 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 dis-associate (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 cross-links 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 band-limited using a third

order Butterworth anti-aliasing filter with cutoff frequency

of fcutoff = 1500Hz and discretized using a bilinear

transformation with sampling frequency of Fs = 3000Hz.

The discretization results in the fifth-order 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 milli-second 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 n-th 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

FrB11.2

7699

Page 3

rate dynamics analysis presented in Section II-C, 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 two-state 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 anti-aliasing filter and then sampling the continuous time

system (1) as mentioned in Section II-A.

(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 two-time 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 two-time scale nonlinear system.

1

λ7= 10−2is chosen as the smallest time constant

FrB11.2

7700

Page 4

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 pseudo-random 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 two-time scale nature

of the biosensor dynamics described above, and because we

adjust the parameter p batch-wise 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 |u|2(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) =

2|u|2

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)

FrB11.2

7701

Page 5

and the multi-hypothesis 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 II-C 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 5-th order ARMA model process denoted

in (2) in Section II-A.

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 = ℘l|Y (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 III-B 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 II-B, 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) =

FrB11.2

7702