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Ion-Channel Biosensors: Construction andDynamic Modeling

Vikram Krishnamurthy Fellow, IEEE, Sahar M.Monfared, Bruce Cornell

Abstract—This paper deals with the construction anddynamic modeling of a novel biosensor that exploitsthe selective conductivity of biological ion channels. Thebiosensor comprises gramicidin A channels embedded ina synthetic tethered lipid bilayer. It provides highly sen-sitive and rapid detection of a wide variety of analytes.In this paper we outline the principle of operation ofthis ion channel biosensor including the fabrication ofbiochip arrays and reproducibility/stability issues. Then theelectrical dynamics and chemical dynamics of the biosensorare modeled. The electrical dynamics are modeled bya second order linear system. The chemical dynamicsof the biosensor response to analyte concentration inthe reaction-rate-limited regime are modeled by a two-time scale nonlinear system of differential equations. Alsothe analyte concentration in the mass-transport-influencedregime is modeled by a partial differential equation subjectto a mix of Neumann and Dirichlet boundary conditions.By using the theory of singular perturbation, we developmodels that predict the performance of the biosensor intransient and steady-state regimes. Finally, we outline theuse of statistical signal processing algorithms that exploitthe biosensor dynamics to classify analyte concentration.

Glossary• ICS–Ion Channel Switch• hCG–Human chorionic gonadotropin (glycoprotein hor-

mone is produced during pregnancy and can be used forearly detection).

• IgG–Immunoglobin G, the most common human im-munoglobin

• βhCG–β subunit of hCG gonadotropin• PVP–Polyvinylpyrrolidone• ELISA–Enzyme Linked Immunosorbent Assay (a bio-

chemical technique used mainly in immunology to detectthe presence of an antibody or an antigen in a sample)

• BLM–Bilayer lipid membrane• X≡aminocaproyl group• 4X≡four aminocaproyl groups• 4XB≡four aminocaproyl-linked biotin• MSL–Membrane spanning lipid• MSL4XB–Membrane spanning lipid with four

aminocaproyl-linked biotin

Vikram Krishnamurthy and Sahar M.Monfared are with the Depart-ment of Electrical and Computer Engineering, University of BritishColumbia, Vancouver, V6T 1Z4, Canada. email: vikramk@ece.ubc.caand smonfare@ece.ubc.ca. Bruce Cornell is with Surgical DiagnosticsLtd., Unit 2/12 Frederick St, St Leonards NSW 2065 Australia. email:BruceC@surgicaldiagnostics.com

• SAM–Self assembled monolayer• DI–Deionized water• gA–Gramicidin A ion channel• gAglyB–Gramicidin with a glycine group linker to Biotin• gA5XB–Gramicidin with five aminocaproyl group linker• Fab–Fragment antigen binding portion of antibody

I. INTRODUCTION

Biological ion channels are water-filled sub-nano-sized pores formed by protein molecules in the mem-branes of all living cells [17], [5]. Ion channels play acrucial role in living organisms by selectively regulatingthe flow of ions into and out of a cell thereby controllingthe cell’s electrical and biochemical activities. This paperdeals with modeling the dynamics of a novel biosensorthat exploits the selective conductivity of ion channels.Such ion channel based biosensors can detect targetmolecular species of interest across a wide range ofapplications. These include medical diagnostics, environ-mental monitoring and general bio-hazard detection.

In [7], a novel biosensor which incorporates gram-icidin A ion channels into a tethered synthetic cellmembrane was developed by our coauthor and publishedin Nature. Over the last few years, [48], [8], [6], [41],many novel functionalities have been added to successivegenerations of this biosensor. Throughout this paper, werefer to this ion channel biosensor as the Ion ChannelSwitch (ICS) biosensor.

CHECK Biological sensory systems function by reg-istering changes in the electrical conductivity of special-ized cell membranes [2]. A well documented exampleis the acetylcholine triggered cation channel involvedin cross synaptic nerve conduction [36]. An importantfeature of this mechanism is its inherent amplificationwith a single molecule potentially triggering the passageof up to a million ions per second across an otherwiseimpermeable membrane. In [24] a light sensitive bac-terial proton pump, bacteriorhodopsin, was incorporatedinto liposomes, which was stabilized with polymerisablephosphatidylcholines. These were observed to pumpprotons upon being photocycled. CHECK

The ICS biosensor provides an interesting example ofengineering at the nano-scale. It is significant that the

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functionality of the device depends on approximately100 lipids, and a single ion channel modulating theflow of billions of ions in a typical sensing event ofapproximately 5 minutes. Since the gramicidin channels(each with conducting pore of diameter 0.4 nm andlength 2.8 nm) move randomly in the outer lipid leaflet ofthe membrane (1.4 nm thick), we can view the biosensoras a fully functioning nano-machine with moving parts.Indeed, each individual gramicidin channel diffuses ran-domly in the order of 1 micron per second. Furthermore,the 4 nm thick lipid bilayer is tethered 4 nm away fromthe gold surface by hydrophylic spaces thereby allowingions to diffuse between the membrane and gold. Thispermits a flux in excess of 106 ions per second to traverseeach channel.

A. Main ResultsThis paper describes the construction and operation

of the ICS biosensor along with signal processing ofits measured signals. The ICS biosensor has severalinteresting properties (from a fabrication and modelingpoint of view) that make it a worthwhile case study.

1. Construction and Operation of the Biosensor: TheICS biosensor incorporates a self assembled monolayerproviding enhanced stability (see literature review belowfor a comparison with other schemes). The tetheredbilayer permits two-dimensional diffusion of gramicidinchannels which provides a remarkable gating mecha-nism. Since gramicidin has an inherent chemistry thatpermits attachment of a wide range of receptors, thebiosensor can be used to detect many different analytes.The ICS sensing mechanism does not require washing(unlike ELISA (see Glossary)), provides large amplifica-tion (millions of ions for every channel dimerization) andsensitivity since a single channel can diffuse and identifyanalyte molecules bound to capture sites, and providesan objective electrical readout which is intrinsically dig-ital. The digital output permits the use of sophisticatedstatistical signal processing algorithms to estimate thetype and concentration of analyte.

In Section II, the construction of the biosensor is dis-cussed in detail. The interface between the biochemicalpart of the biosensor and the electrical measurementsystem is provided via a gold surface to which thelipid bilayer containing the ion channels is tethered anda silver coated return electrode which is immersed inthe electrolyte above the biomimetic surface. We alsodiscuss how the biosensor can be readily adapted tominiaturization and multiplexing (multiple electrodes).

2. Dynamical Response of Biosensor: In modelingthe performance of the ICS biosensor, it is necessary

to consider the chemical and mass transfer dynamicsto predict performance. Because the sensor can detectanalyte concentrations smaller than 1 pico-molar, masstransfer of analyte over the electrodes becomes thedominant design criterion. This in turn requires carefulmodeling of the chemical dynamics (typically modeledby a two-time scale system of ordinary differential equa-tions) together with the mass transport dynamics (partialdifferential equations of fluid flow). In Section III theelectrical dynamics of the ion channel biosensor aremodeled by an equivalent second order linear system,see [48],[41]. We then formulate the biosensor responseto analyte concentration as a two-time scale nonlineardynamical system. The presence of analyte decreasesthe concentration of the dimers formed in the biosensor,thus decreasing the channel admittance. In order to studythe evolution of the channel admittance in response tothe introduction of analyte, the chemical kinetics ofthe biosensor are modeled as a singularly perturbedsystem [20], [9]. The analyte concentration in the mass-transport-influenced region of operation of the biosensoris modeled by a partial differential equation subject toa mix of Neumann and Dirichlet boundary conditions,[27], [10], [15]. Analysis of the chemical kinetics ofthe biosensor as a Singularly Perturbed system leads tothe result that the channel conduction evolves accordingto three regimes depending on the concentration of theanalyte in the system.

3. Analyte Concentration Classification: The eventualgoal of detailed modeling of the biosensor is to designsophisticated statistical signal processing algorithms thatcan exploit these model dynamics to rapidly classifyanalytes and estimate their concentrations. Using thederived equations describing the evolution of channelconductance as a function of analyte concentration, weillustrate how multi-hypothesis classifiers (via Kalmanand Hidden Markov Model filters) can be used to classifythe concentration of the analyte present in the system.

B. Related WorkThe literature on biosensors is vast. We refer the

reader to [32] for an interesting overview of the inter-face between ion channels and microelectronics. Severalcompanies/research groups have developed biosensorsbased on synthetic lipid monolayers and bilayers. Forexample OhmX Corporation is currently developinga reagentless biosensor system using self assembledmonolayers (SAMs) tethered to a gold surface for theelectronic detection of biomarkers in clinical samples[13]. Also Bayley’s group at Oxford has made substantialcontributions in ion channel biosensors [33],[19]. We

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focus on the ICS biosensor developed by AMBRI in thispaper as one example of an ion channel based biosensor

Below we provide a highly focused review of relatedwork in biosensors involving ion channels and tetheredlipid membranes. The first attempt at developing apractical membrane-based biosensor device was reportedin [24]. The poor stability of the receptor-membranecomplex limited the range of applications of the device.An active cytochrome C has also been incorporated intoa tethered membrane in [30] and is one of the firstexamples of a functionally active biomimetic surface.

The stabilization [23] of the bilayer lipid membrane(BLM) has been a central theme in the developmentof ion channel biosensors. Many strategies have beendeveloped, but these primarily focus on physisorbing orchemically attaching a layer of hydrocarbon to a silicon[16], hydrogel [26], polymer [29] or metal surface [43].Subsequently a second layer of mobile lipids is fusedonto the tethered monolayer to form a tethered bilayerlipid membrane. The papers [38] and [34] review earlierworks on BLM stabilization. In [28] peptide nanotubeshave been fabricated within supported SAMs. The ICSbiosensor employs an alkane disulphide bond to stabilizethe bilayer lipid membrane at the electrode surface.

Attempts to engineer receptor-based gated ion chan-nels have also been reported, see [41]. Mechanisms rangefrom anti-channel antibodies that disrupt ion transport[4], to molecular plugs that block the channel entrance[25]. Ompf porin channels from E coli, were incorpo-rated into a tethered BLM and their conduction alteredusing the channel blocker colicin [44]. The detection ofsingle gramicidin channel currents in a tethered mem-brane is described in [13]. All mechanisms proposedso far have had a very limited range of applicationand require re-engineering for each new analyte. TheICS, whilst using ion channel transduction provides amechanism that may be adapted to many different classesof target.

Miniaturization and patterning are two major oppor-tunities for tethered membrane technologies [12], [42].The functionalities which may be brought to tethered bi-layers are becoming extensive. Topographical templatesfor chemiselective ligation of antigenic peptides to selfassembled monolayers have been fabricated in [40].

to fix: ICS biosensor advances: covalent linkage ofFab to ion channel, use of flow cells, miniaturization ofelectrode dimension from 1 mm to 20 microns.

The remainder of this paper is organized as follows.Section II explains the details of the fabrication andoperation of the biosensor in single and multi-arrayformats. Section III describes the electrical modeling

of the biosensor. Section IV deals with modeling thechemical dynamics and analyte mass transport in thebiosensor. Finally, Section V presents an overview ofstatistical signal processing algorithms for detecting thepresence of analyte given noisy measurements from thebiosensor. It is hoped that these three aspects, namelyconstruction, modeling, and signal processing, providethe reader with an interesting case-study that can besuitably modified to other types of biosensors.

II. CONSTRUCTION AND OPERATION OF ICSBIOSENSOR

In this section we discuss the fabrication and operationof the ICS biosensor. This discussion forms the basis formodeling the biosensor dynamics in subsequent sections.This section is organized as follows. First the gatingmechanism caused by random movement of gramicidinchannels is explained. Then we discuss how the sensoroperation depends on the analyte size. Fabrication ofarrays of such sensors is then outlined along with multi-analyte detection. Finally we discuss scalability issuesof the biosensor in terms of reproducibility of sensorperformance and stability/storage.

A. Principle of Operation

The low molecular weight bacterial ion channel gram-icidin ([18], [47]) has been used, (see [7], [49], seealso [6] for a review), as the basis of a biosensorplatform with a range of applications for the detection oflow molecular weight drugs, large proteins and micro-organisms. The ICS biosensor employs a lipid monolayertethered via a disulphide group to a gold surface. Themembrane is separated from the gold surface by anethylene glycol spacer which provides a reservoir forions permeating through the membrane. The transductionmechanism depends on the properties of gramicidin Awithin a BLM. gramicidin monomers diffuse withinthe individual monolayers of the BLM. The flow ofions through gramicidin only occurs when two non-conducting monomers align to form a conducting dimer.The gramicidins within the tethered inner leaflet of thelipid bilayer are also tethered. Also attached to the goldsurface as part of the inner leaflet are membrane span-ning lipids. The arrival of analyte cross-links antibodiesattached to the mobile outer layer channels, to thoseattached to membrane spanning lipid tethers. Due tothe low density of tethered channels within the innermembrane leaflet, this anchors them distant, on average,from their immobilized inner layer channel partners.Gramicidin dimer conduction is thus prevented and the

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ionic admittance of the membrane decreases. Applyinga small alternating potential between the gold substrateand a reference electrode in the test solution generatesa charge at the gold surface which causes electrons toflow in an external circuit.

The membrane stability is primarily enhanced bytethering the inner membrane leaflet to the gold surface.However additional stability is achieved by substituting amajor fraction of the tethered lipids with archaebacteriallipids. These are lipids modeled on constituents foundin bacteria capable of surviving extremes of temperatureand hostile chemical environments. Characteristics ofthese lipids are that the hydrocarbon chains span theentire membrane, [22] and that all ester linkages arereplaced with ethers [37]. BLM films have previouslybeen formed from archaebacterial lipids [14] and resultedin membranes that are stable to temperatures in excessof 90C. A stable membrane incorporating ion channelscan be self-assembled [31] on a clean, smooth goldsurface using a combination of sulphur-gold chemistryand physisorption. Most studies of the ICS biosensorhave used antibody Fab (see Glossary) fragments as thereceptor; however, the approach has also been demon-strated to operate using oligonucleotide probes, heavymetal chelates and cell surface receptors.

B. Large and Small Analyte detection

Large analytes include proteins, hormones, polypep-tides, microorganisms, oligonucleotides, DNA segmentsand polymers. In the same manner that an ELISA sand-wich assay may be developed based on a complementaryantibody pair, the ICS biosensor may be adapted to thedetection of any antigenic target for which a suitableantibody pair is available. The bacterial ion channelgramicidin A is assembled into a tethered lipid mem-brane and coupled to an antibody targeting a compoundof diagnostic interest. The binding of the target moleculecauses the conformation of the gramicidin channels toswitch from predominantly conducting dimer to predom-inantly non-conducting monomers as shown in Fig. 1.For target analytes with low molecular weights such astherapeutic drugs where the target is too small to use atwo site sandwich assay, a competitive adaptation of theICS is available. This is shown in Fig. 2.

C. Biochip arrays

Sensor arrays have been fabricated using silicon ni-tride, silicon carbide and glass substrates. Using thisformat a multi-analyte detection capability is demon-strated. Multi-analyte detection is an advantage as it

(a) (b) (c)

Fig. 1. Large analyte transduction mechanism. The binding of analyte(green) to the antibody fragments (Fab) (red) causes the conformationof gramicidin A to shift from conductive dimers to non-conductivemonomers. This causes a loss of conduction of ions across themembrane. The scale can be visualized by the fact that the tetheredlipid bilayer is 4nm thick.

(a) (b)

Fig. 2. Small analyte transduction mechanism. In the absence ofanalyte, the mobile channels cross link to anti-target Fabs anchoredat the tether sites. Dimer formation is prevented and the conductanceof the membrane falls. The introduction of analyte competes off thehapten (target analogue) and increases the membrane conductance

permits on board calibration to correct for systematicvariations which can occur across an electrode array andto correct for electrode to electrode variation betweendifferent sensors. A novel element in the design ofthese arrays is the use of a titanium oxide ring at theperimeter of the electrode opening. The titanium ringis designed to provide a mechanical seal for the outerleaflet preventing it from diffusing beyond the area ofthe tethered inner leaflet lipids. In addition, the titaniumseal retains water during the patterning of antibodies,and during the dry down process for storage. Examplesof electrode arrays are shown in Fig. 3, recorded by bothoptical and scanning electron microscopy.

In Fig. 3(a) a 4 x 4, 16 element array of 150µmdiameter electrodes is shown and Fig. 3(b) a test array offour electrodes ranging from 150µm to 20µm diameter.A consequence of reducing the electrode diameter from150µm to 20µm is a reduction of the membrane capac-itance and an increase in the membrane resistance. Bothmeasures scale with membrane area, the capacitancelinearly and the impedance inversely.

Although the impedance of these electrodes is de-pendent on area, the time constant of the response isindependent of area. A dependence is expected, however,when the spacing of the ion channels or antibodies arecomparable to the electrode dimensions. In the present

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(a) (b) (c)

Fig. 3. (a) Optical microscopy image of a 16 element sensor arraywith 150µm diameter electrodes. The apparently square geometry ofthe sensor elements arises from the gold being patterned as rectanglesand the silicon nitride openings being round. The thin (100 nm) andtransparent silicon nitride allows the gold to be viewed through thenitride layer. Also visible is the light grey 2µm wide Titanium oxidering. (b) Optical microscopy image of a test array with four electrodeelements of 150µm, 100µm, 50µm and 20µm diameter. (c) Scanningelectron microscopy image of the edge of one of the 150µm electrodeopenings showing the Titanium oxide ring and the gold to siliconnitride interface. The etch angle at the gold to silicon nitride edgeis 65.

Fig. 4. Change of admittance versus time over an array of 16electrodes. The abscissa (time) axis spans 8 minand the admittanceaxis spans a FSD of 70ns. The schematic of the array in the centerof the trace shows the electrodes from which each trace was recorded.The CVs for the amplitude of the response across the 16 electrodes is2.8% and to the τ of an exponential fit to the decay is 2.1%.

case these spacings are approximately 0.1µm -1µm, farsmaller than the smallest 20µm diameter electrode re-ported here. One consequence of using an electrode arrayto measure the target species concentration rather than asingle electrode of comparable area is the improvementin the quality of estimating the response rate. This isshown in Fig. 4. An exponential fit to the admittancedecay curve across the 16 electrodes yields coefficientsof variation (standard-deviation/mean) of well below10% for a strong response. This indicates that the siliconchip fabrication procedures can provide a highly repro-ducible electrode geometry and structure. The distributedsensing array possesses a statistical improvement of 16independent measures rather than one.

D. Multi-analyte detection

Sensor arrays also have the ability to measure multipletarget concentrations from a single sample addition to

Fig. 6. From left to right, the first set of bars correspond toelectrode with no receptors, second set correspond to electrodes withhGG receptors, third and fourth set contain influenza A receptors.Chip challenged with single sample possessing 150mIU/ml hCG and100ng/mL influenza A virus.

the sensor. A key problem when fabricating an arraycapable of multi-analyte detection is the site specificdecoration of the chip with different antibodies. Theapproach used here is shown in Fig. 5. A fluid handlingspotter (sciFlexarrayer leased from Scienion AG Berlin)was loaded with the appropriate antibody solution anddirected to a chip surface that had been partially driedfrom glycerol, trehalose, PVP or their combinations.

The lower size limit of the electrodes was deter-mined by the resolution achievable by the spotter-surfacecharacteristics and not the constraints of the electrode-membrane characteristics. The limiting dimension setby the spotter-surface combination was 80µm diameterwhereas the limit set by the electrode-membrane com-bination was 10µm. Fig. 5(a & b) shows an exampleof decorating a chip array with four arbitrary antibodyreceptors. That is, one quadrant, contains four elec-trodes, with the pregnancy hormone (human chorionicgonadotrophin – hCG), two quadrants have antibodyreceptors for influenza A and a further quadrant usesthe reference receptor to a target not in the test sample.Figure 6 shows the response to four samples containingeither 150mIU/ml hCG, 100ng/L influenza A virus orneither. The reference electrode cluster yielded a nullresult to all three challenges; the βhCG cluster yieldeda positive response (reduction in admittance read asa negative slope) to the 150mIU/ml hCG sample butzero to the influenza A challenge, whereas the influenzaA clusters yielded a positive result to challenge withinfluenza A but zero to βhCG. These data show theability of an electrode array to detect multiple targetspecies in one sample addition. In this case the samplevolumes used were 100µL but these can be reduced to10− 20µL using a coplanar return electrode.

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(a) (b) (c)

Fig. 5. (a) Site specific coating was achieved by directing a metered volume of biotinylated antibody Fab onto an electrode element (230µmdiameter) in the array. The electrode surface was pretreated with 2-5% trehalose solution and dried prior to spotting the surface. Prior to spottingthe membranes had been assembled to a common structure across all electrodes, including a streptavidin linker to the ion channels and membranespanning lipids. (b) The sciFlexarrayer (Scienion AG Berlin) provided a stream of 20pL volume drop. Typically 15 drops were applied to eachelectrode resulting in 12nL per quadrant of four electrodes. Each quadrant received a different biotinylated antibody fragment. (c) The chipused here was a cluster of four 2x2 electrode arrays - each 230µm diameter on either a glass or silicon substrate.

E. Scale-up issues

Reproducibility of results is essential for calibrating asensor. Also stability and storage are critical issues forbiosensors. We discuss these issues below.

1) Reproducibility: Variation in the sensor perfor-mance occurs between test elements within a singlechip and between chip to chip. Custom fabricated chipswere supplied by Micralyne, Edmonton, Canada. Theline resolution of standard silicon foundry techniquessubstantially exceeds the requirements of the BioChipsused here. Silicon was chosen as a platform for scalingup sensor production since it was viewed as a maturetechnology that operated within a highly controlled,clean environment. However, an unexpected problemarose in the adaptation of standard photolithographicapproaches to the fabrication of patterned surfaces foruse in Au-alkanethiol based sensors. The etch proce-dures employed to expose the patterned gold seriouslycontaminates the gold surface in a manner that we havebeen unable to totally reverse. Cleaning solutions suchas Piranha followed by deionised (DI) water and ethanolcould achieve a functioning device but the longer termstability was seriously compromised. High energy ArgonIon milling of the surface was necessary immediatelyprior to coating the thiol or disulphide species regardlessof the wet cleaning process employed. The optimalapproach is to vary the lithographic process steps topermit a deposit gold last sequence, which althoughrequiring a final masking step, achieves the best results.

Monitoring sensor performance over many batchesprovides an estimate of the batch to batch variation inperformance. Figure 7 shows a one month record ofresponse to 60mIU hCG, run without calibrators or anyinternal corrections for an array of sixteen measures

One Month Response to 60mIU/L hCGn=16 each day

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on a plastic slide. Each day represents an average ofsixteen measures from one array and over the monthfrom sixteen separate arrays. A batch is complicatedto define as a new batch is initiated when any of theelectrodes, chemistries or biochemistries is altered. Thesample challenge was a standard solution of 60mIU/LhCG. The average daily CV over 16 measures was 12%and the CV of the daily means over the 16 arraysmeasured during the month was 4.7%. These measureswere performed in PBS at 30C on sensors made that day.

A major step towards achieving a quantitative test isto provide an on board calibrator to correct for dailyvariation. This is particularly important when measure-ments are made from serum or blood where matrix vari-ations cause substantial changes to the standard curve.In the absence of a calibrator the variation essentiallyeliminates the possibility of all but semi-quantitativemeasurements. Figure 8(a & b) show an example ofthe potential improvement in reproducibility that can

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be achieved by employing a calibrator. The approachdepends on a multi-analyte capability. Figure 8(a) showsa series of individual chip arrays. The horizontal scaledescribes particular chip arrays. The vertical scale is ameasure of the response rate to a challenge of analyte.In Fig. 8(a) the response rate has been normalised tothe gating amplitude of each measure. This is equivalentto reporting on the inverse of the time constant of anexponential fit to the response. In Fig. 8(b) the same datais processed differently. Calibration data is now used toscale the test data resulting in a substantial improvementin reproducibility. The provision of a calibrator channelis a significant benefit brought by simultaneous Multi-Analyte Detection.

2) Stability and Storage: A key requirement for adiagnostic technology is an ability to be stored forextended periods with minimal degradation in perfor-mance. A serendipitous observation has permitted in-sights into some of the mechanisms that cause agedependent changes in the ion channel switch sensor.Figure 9(a) shows a schematic of a sensor containing thebiotinylated gramicidin analogue, gaglyB, being gatedby the multivalent protein, Streptavidin (MW 60kDa).It was found that if the linker attaching the biotins to theion channel was reduced in length to a glycine group,then the streptavidin on rate to the biotin was ineffectivein achieving gating on the 5-10 minute timescale. How-ever, if as shown in Fig. 9(b) the membrane spanninglipid, MSL4XB is present, possessing a far longer linkerbetween the biotin and the tethered lipid, a strong rapidgating occurs. The experimental evidence for this isshown in Fig. 10(a and b). This suggests that streptavidinbinds to the biotin attached by the longer linker on

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the MSL4XB but not to the biotin attached by the farshorter linker on the gaglyB. However, once bound toMSL4XB, the streptavidin is presented in a way topermit gating of the gaglyB. It is proposed that gatingof the gaglyB only occurs from the gaglyB diffusingto a site where streptavidin has already bound to theMSL4XB. This provides a method of determining theaverage diffusion distance of the ion channel in themembrane and a means of probing whether this distancealters on storage. By titrating the gating reaction rate tothe concentration of the streptavidin challenge, the onrate or f1 of streptavidin to the MSL4XB is determined.This may be contrasted with the far slower on rate, f1

of the streptavidin to the gaglyB. Comparing these f1

values with those obtained from directly cross linkingion channels possessing longer 5XB linkers it is possibleto determine the diffusion distance. The various combi-nations of f1 are given in Table I. In Table I it is evidentthat when using the longer 5XB linkers the dependenceon the MSL4XB is almost eliminated.

Taking the ratio of the second and third entries ofTable I provides a measure of the reaction rate f1

of the mobile channel, gA5XB density to the density

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Case Configuration Slope= f1 (M−1s−1)1 gA5XB + MSL4XB 7.40× 106

2 gAglyB + MSL4XB 1.19× 106

3 gA5XB No MSL4XB 6.90× 106

4 gAglyB No MSL4XB 0.43× 106

TABLE IEXPERIMENTALLY DETERMINED f1 VALUES FOR THE VARIOUSCOMBINATIONS OF ION CHANNEL AND MSL4XB DESCRIBED

ABOVE.

of the tethered membrane spanning, MSL4XB sites.The ratio is 0.17 meaning that the density of mobilechannels is greater than the density of the tetheredmembrane spanning lipid MSL4XB capture sites. Thisis necessary if Streptavidin is used as a linker specieslinking the channel or the tethered lipid to the biotiny-lated antibodies. Higher levels of MSL4XB results intoo great a cross linking of the gA5XB to MSL4XBby the Streptavidin. This problem is eliminated whenusing a covalent linkage to the antibody fragment andlarger densities of MSL4XB are possible resulting insignificantly higher sensitivities. However, in the presentstudy, using the density of channels added in the mobilelayer as 3 × 108molecules/cm2 we may determine theMSL4XB density as 6.7 × 109molecules/cm2. Vary-ing the MSL4XB density relative to this estimate nowpermits a measure of the diffusion distance of the ionchannels. This is seen in Fig. 11 in which the MSL4XBdensity is plotted relative to the value used above. Thisvalue is described as the Standard (Stnd) value. It canbe seen that when the sensors are fresh, the gatingamplitude rises from a very low value at zero MSL4XBto a maximal value at approximately the standard densityof MSL4XB. This indicates a diffusion distance of thechannels of approximately (6.7 × 108) − 0.5 = 0.5µm.This further suggests a complex structure to the mem-brane surface restricting the diffusion distance. If thediffusion were unrestricted a diffusion distance ×10 thisvalue would be expected based on reported values of theself diffusion coefficient of gramicidin in lipid bilayers.The alternative explanation that the diffusion coefficientis slowed is not supported by the kinetics of the gatingresponse. When the sensors are dried and stored, thediffusion distance is reduced by 40% suggesting thedrying and storage have further restricted the averagediffusion distance (Fig. 11).

To maintain a gating response for long periods ofstorage requires increasing the density of MSL4XB.Figure 12 shows storage times at 20C for 28 days.After 3 months at 20C, the titration response deviates

Fig. 11. Plot of gating magnitude versus MSL4XB density relative tothe standard density of 6.7×108 MSL4XB/cm2. Drying and storagefor two weeks at 20C causes a need for higher densities of MSL4XBto achieve the same gating magnitude. Increasing capture site densityimproves both storage and sensitivity of the sensor.

Fig. 12. Plot of titration gating rate versus hCG concentration for asensor stored at 20C dry, for up to one month.

outside acceptable limits. Storage at 4C significantlyextends the acceptable titration response and can survivebeyond six months. Factors causing failure appear tobe associated with the diffusion of the mobile species.In all cases the loss of responsiveness is at high targetconcentrations, leaving the low concentration region ofthe standard curve substantially unaltered. This furthersuggests a diffusion mechanism as an important cause ofperformance loss and more work is required to under-stand and control these effects. Whether or not the slopeof the titration curve is lost, the membrane integrity doesnot fail. When stored dry, the membrane metrics indicatethat the membranes remain sealed or conducting closeto their initial value over extended periods of time.

9

Element ValueMembrane Capacitance C1 0.5µF/cm2

Interfacial Capacitance C2 3.5µF/cm2

Ion Channel Resistance C1 60kΩ–600kΩElectrolyte Resistance R2 200Ω

TABLE IITYPICAL VALUES FOR THE COMPONENTS OF THE EQUIVALENT

ELECTRICAL SYSTEM OF THE BIOSENSOR, DEPICTED IN FIG. 13.THE VALUES ARE GIVEN IN TERMS OF PER AREA UNITS AND THE

AREA OF THE ELECTRODE IS 0.03cm2 .

III. MODELING THE DYNAMICS OF THE IONCHANNEL BIOSENSOR

With the above description of the operation and con-struction of the ICS biosensor, this section and the nextsection deal with constructing mathematical models forthe biosensor’s electrical response and chemical dynam-ics. Obtaining an accurate mathematical model of thechemical kinetics and electrical response of the biosensoris important in the design, operation and analysis ofthe biosensor. For example, an accurate model of thebiosensor dynamics allows us to determine the flowrate for various analyte concentrations to optimize thesensor response, see Section IV-A. Also detailed model-ing/analysis of the chemical kinetics of the biosensor canbe exploited in designing sophisticated statistical signalprocessing algorithms for rapidly classifying analytes,see Section V.

A. Electrical Dynamics of the Biosensor

The gramicidin A Ion channel biosensor can be mod-eled as a biological transistor. Fig.13(a) illustrates theequivalent circuit of the switch before and after thedetection of analyte. Fig.13(b) details the componentsof the equivalent circuit. The resistor R = 1/G modelsthe biosensor resistance and increases with the presenceof analyte. C1 denotes the capacitance of the membranewhile C2 denotes the interfacial capacitance of the goldsubstrate. Note that one face of the capacitor C2 ischarged due to ions, the other face is due to electronsthat form the output current of the biosensor. Thus C2

provides the interface between the biological sensor andthe electrical instrumentation. R2 denotes the resistanceof the electrolyte and its value varies depending onthe type of electrolyte and the dimensions of the flowchamber. Finally, R3 denotes the input resistance of theamplifier at the next stage (which is ideally very large).Value of the equivalent electrical system components arefunctions of electrode area and flow chamber dimen-sions. Typical values are listed in Table II.

Digital Ion Channel Switch Device For Biothreat Detection

DARPA DSO BAA05-016: Engineered Bio-Molecular Nano-Devices/Systems

Principal Investigator: Becker N. T.

Biosensor Enterprises 19

CONFIDENTIAL

Gated Open (gA) Gated Closed (gA)

Figure 2. The ICS technology acts like a biological transistor and uses ~ 100x less

antibody than conventional immunoassay. The equivalent electrical circuit is shown in

Figure 2 for the ion channel switch closed and open.

Figure 3. Comparison of the ICS ! structure with that of an ELISA architecture

shows the common elements of a capture antibody species and a reporter antibody species. In the ELISA the readout is optical. In ICS it is a direct electrical readout.

1(a) The Tethered Lipid Membrane:

(a)

(b)

Fig. 13. The ICS biosensor comprises of an ion channel switchthat acts like a biological transistor. Fig.13(a) denotes the switched-on state when the ion channels are conducting. Fig.13(b) denotesthe switched-off state when the channels are not conducting. Theequivalent electrical circuit shown in Fig.13(b) results in a second ordersystem.

The admittance transfer function of the equivalent cir-cuit parameterized by G is (where s represents Laplacetransform variable)

H(s;G) =I(s)

Vout(s)=

s2 + s a G

s2R2 + s(b2 + b1G) + b3G(1)

The constants in H(s) above are a = 1C1

, b1 = R2C1

,b3 = 1

C1C2, b2 = 1

C1+ 1

C2.

Before proceeding we remark that the biosensorwe consider below comprises of electrodes with area0.03cm2. Each such electrode contains several millionsof ion channels. Therefore the measured current is theaverage effect of the formation and dis-association ofthousands of dimers and is approximately continuous-valued. Specifically each channel dimer posses a re-sistance to sodium ion flow of 1011Ω. In the presentdata, electrodes of 2mm diameters result in an area of0.03cm2, which for 108 channels per cm2 results in3 × 106 channels per electrode. It is worth noting thatthe density of the mobile and tethered channels is suchthat approximately 50% of the channels are dimerized.Taking the parallel resistance of all the ion channels intoconsideration the ion channel resistance is approximately60kΩ. In contrast, for micro-size electrodes of area10−10m2 (or smaller), the current would be a finite state

10

process since only a few dimers would form and dis-associate; such micro-sized electrodes are studied in ourrecent work [21].

B. Dynamics of Biosensor Response to Analyte

From an abstract point of view, we can describe thebiosensor response to an analyte by a two time scalestochastic dynamical system: Let k = 1, 2 . . . denotethe fast time scale. Typically, with a sampling frequencyof 3kHz, k evolves on the milli-second time scale. Letn = 1, 2, . . . denote the slow time scale. We will referto n as batch number. So for fixed large positive integerT , the n-th batch on the slow time scale corresponds totimes 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 concentration of analyte. From de-tailed experimental analysis of the biosensor and reactionrate dynamics analysis presented in Section IV, it isknown that the conductance G of the biosensor evolveson the slow time scale according to one of 3 differentconcentration modes; M:

Gn+1 = fM(Gn, A) + wn, (2)

M =

1 A = 0 : fM(Gn, A) = Gn

2 Amedium : fM(Gn, A) = κ1Gn + κ2

3 Ahigh : fM(Gn, A) = κ3Gn + κ4

Here κ1, κ2, κ3, κ4 are constants, with |κ2| < 1. wn isa zero mean white Gaussian noise process with smallvariance and models our uncertainty in the evolution ofG. The function fM models the fact that the admittanceGn of the membrane decreases according to one of 3distinct modes depending on the concentration A of theanalyte. For no analyte present (M = 1), the admittanceremains constant. As described earlier, when analyte ispresent the admittance G in (1) decreases due to analytemolecules binding to the antibodies in the biosensor. Formedium and high concentration (M = 2 , M = 3),the decrease in admittance is exponential with differ-ent decay rates. High analyte concentration refers to(A ≥ 10−8M) and medium analyte concentration refersto (A ≥ 10−9M). Change in channel conduction as afunction of analyte concentration is discussed in moredetail in section IV-B.

Fig.14 shows examples of the biosensor response toStreptavidin with different concentrations; see [6], [48]for details. Using Streptavidin as the protein antigen andbiotin as a low molecular weight receptor provides arapid and clear demonstration of the different kineticregimes of the sensor function. The Streptavidin-biotinbinding pair is one of the strongest and best characterized

Fig. 14. Biosensor response to Streptavidin; see [6], [48] for details

interactions available today and is used as a modelsystem in Fig.14 to permit full characterization of thesensor.

In the next section we will derive models according towhich the channel conductance evolves for various ana-lyte concentrations, by analyzing in detail the chemicaldynamics of the biosensor.

IV. CHEMICAL DYNAMICS OF BIOSENSOR

As described in the previous section, the resistance ofthe biosensor channel is determined by the concentrationof the gramicidin A dimers formed in the biosensor. Asthe concentration of the dimers decreases the channeladmittance decreases. Therefore to study the evolution ofthe channel conductance as a function of time we need toanalyze the evolution of the dimer concentration in thebiosensor. Since the dynamics of the biosensor includingthe dimer concentration are governed by many chemicalspecies and their reactions, we analyze the dynamics ofthese reactions.

The reactions involved in the biosensor mainly stemfrom the binding of the primary species and the sec-ondary species or complexes. The primary species areAnalyte, a, of concentration A, binding site b of con-centration B, free moving monomeric ion channel c ofconcentration C, and tethered monomeric ion channel sof concentration S. While the complexes are w, x, y andz with concentrations, W , X , Y and Z and are formedaccording to the following equations;

a + b f1r1

w a + c f2r2

x w + c f3r3

y

x + b f4r4

y c + s f5r5

d a + d f6r6

z

x + c f7r7

z (3)

where, fi denote the forward reaction rate constantswith units of (M−1s−1) in case of 3D reactions, suchas binding of analyte with capture sites, and (cm2s−1)in case of 2D reactions, such as dimerization of the

11

Initial Concentration of Species ValueMobile gramicidin A monomers C(0) 108molecules/cm2

Tethered gramicidn A monomersS(0) 109molecules/cm2

Tethered Capture Site B(0) 109/cm2–1011/cm2

Analyte A∗ µM–fM

TABLE IIITYPICAL CONCENTRATIONS OF PRIMARY SPECIES IN THE ICS

BIOSENSOR.

Reaction Rate Streptavidin-Biotin hCG-IgGf1 = f2 = f6 = f7 7× 106 (M−1s−1) 5× 105 (M−1s−1)

f3 = f4 10−10 (cm2s−1) 5× 10−11 (cm2s−1)f5 10−11 (cm2s−1) 10−11 (cm2s−1)

r1 = r2 = r6 = r7 10−6 (s−1) 10−4 (s−1)r3 = r4 10−6 (s−1) 10−4 (s−1)

r5 10−2 (s−1) 10−2 (s−1)

TABLE IVTYPICAL VALUES OF THE REACTION RATES FOR

ANTIGEN-ANTIBODY PAIR HGG-IGG AND STREPTAVIDIN-BIOTINPAIR. REFER TO (3) TO UNDERSTAND THE MEANING OF EACH

REACTION RATE.

ion channel, and ri for i = 1, 2, 3, 4, 5, 6, 7 denotebackward reaction rate constants with units of (s−1). Thetotal reaction rates can be computed as below,

R1 = f1AB − r1W R2 = f2AC − r2X

R3 = f3WC − r3Y R4 = f4XB − r4Y

R5 = f5CS − r5D R6 = f6AD − r6Z

R7 = f7XS − r7Z (4)

The initial concentrations of the primary species, A∗,B(0), C(0), and S(0), are given in Table III. Note thatin Table III, S(0) denotes the initial concentration of thetethered gramicidin A monomers, and also that S ≈ S(0)through out the experiment.

As mentioned before the ICS biosensor converts abiological binding event to a decrease in channel con-ductance, in case of large analyte molecules. So assuggested by the dynamics of the sensor, the forwardand backward reaction rates, denoted above as fi and ri,depend on the type of analyte and capture sites used inthe biosensor. An example of antigen-antibody detectionis the detection of hCG, using IgG as the capture site(hCG and IgG are defined in Glorssary). The biosensorcan be used to detect specific types of proteins such asStreptavidin as well. In case of streptavidin, Biotin isused as the capture site. Typical values of fi and ri forthese two cases are shown in Table IV.

The analyte concentration A is generally a functionof both space and time. In a flow cell Diffusion andadvection carry analyte molecules to the biomimetic

surface where binding occurs according to the chemicalkinetics defined in (3). Three operating regimes for thesystem can be distinguished: (i) reaction-rate-limited ki-netics,(ii) mass-transport-limited kinetics, or (iii) kineticsinfluenced by both mass-transport and reaction-rate [46].

In reaction-rate-limited kinetics regime, the depletionof analyte molecules due to reaction at the biomimeticsurface is compensated for rapidly. Therefore for analyteconcentrations in the order of µM it is reasonable toassume, A(x1, x2, x3, t) = A∗, where A∗ denotes aconstant analyte concentratoin and x1, x2 and x3 re-spectively denote the x, y and z coordinate axes. In thisregion depletion of analyte molecules is compensated forby either large flow rates with low analyte concentrationor average flow rates with high analyte concentration.

In (ii) and (iii) where chemical kinetics are influ-enced by both mass-transport and reaction-rates, thelocal concentration of analyte varies due to the chemicalreactions between the analyte and the binding sites, thusA(x1, x2, x3, t). At very low analyte concentrations veryhigh flow rates are required to get the highest responserate of the system. Dependence of response rate on flowrate and analyte concentration is discussed in more detailin section IV-A. The modeling of the chemical dynamicsof the system are explained in the following sections.

A. Comprise between Capture Density and Analyte FlowRate

A high capture density is essential for high sensitivity.With high capure density, target molecules collide morefrequently with receptors and are thus captured morequickly. The greater the ratio of capture sites to diffusingion channels, the faster the increase in resistance.

To illustrate the requirements for modeling both thereaction-rate limited kinetics and mass-transport limitedkinetics, we first illustrate these effects via numericalsimulation.

Fig. 15 shows quantitative predictions of the changein the resistance per unit time for various capture sitedensities and sample flow rates, for high analyte concen-trations (100pM) and low analyte concentrations (10fM).As apparent in this figure at high concentrations theresponse is insensitive to flow rate, whereas at lowconcentrations of analyte and high capture densitiesa high flow rate is required to achieve a measurableresponse.

The reaction-rate-limited chemical kinetics are mod-eled in the next section. In the reaction-rate-limitedregime of operation, as explained above mass transporteffects are negligible.

12

Fig. 15. Simulations of the response to 100pM and 10fM Streptavidinfor various capture site to ion channel densities and flow rates. At100pM and low capture densities (nearest trace) increasing flow rate(left to right) has little effect. At 10fM and high capture densities (reartrace), increasing flow rate has a big effect. High flow limits are shownas dashed lines at the left.

B. Reaction-Rate-Limited Detection Kinetics

In this section the chemical kinetics in the reaction-rate-limited regime are modeled as a two time scalesystem. Linearizing the nonlinear system of differentialequations describing the chemical dynamics of the sys-tem, the system is rewritten in terms of fast and slow dy-namics. Then using Theorem 4.1, an equation describingthe evolution of dimer concentration with time is derived.Therefore it is found that the dimer concentration evolvesaccording to three modes depending on the analyte con-centration. Keeping in mind that the channel conductanceis directly proportional to the dimer concentration in thesystem, we arrive at (2), in section III-B, describing theevolution of the channel conductance as a function oftime and analyte concentration.

As explained in the previous section, in the reaction-rate-limited regime, (A∗ > 1µM), analyte concentrationis constant. We will denote the constant analyte concen-tration as A∗.

Defining u = B, C, D, S, W, X, Y, ZT ,and r(u(t)) = R1, R2, R3, R4, R5, R6, R7T ,where Tdenotes transpose and Ri are defined in (4), we canwrite the nonlinear differential equation describing theevolution of the chemical species as;

d

dt(u) = Mr(u(t)) (5)

where M is a matrix defined by;

M =

−1 0 0 −1 0 0 00 −1 −1 0 −1 0 00 0 0 0 1 −1 00 0 0 0 −1 0 −11 0 −1 0 0 0 00 1 0 −1 0 0 −10 0 1 1 0 0 00 0 0 0 0 1 1

Clearly the system of differential equations given by(5,4), is solveable numerically. However, in order tomake further design and statistical signal analysis ofthe system possible, analytical equations describing theevolution of the system state are required. In this sectionwe will try to analyze the system of differential equationsdescribing the chemical kinetics of the biosensor, inorder to arrive at these analytical equations.

Analyzing the linearized version of (5) and estimatingthe eigenvalues with the typical parameter values for thebiosensor given in table IV and table III, it is foundthat |λ7,8| |λ1,2,3,4,5,6|. Therefore the eigenvaluesassociated with the species Y and Z are larger inabsolute value and it can be concluded that these speciesdecay at a rate much faster than the rest and thus it isreasonable to place these species in the “fast” group andthe rest in the “slow” group.

Define, β = Y, Z, and α = B,C,D, S,W,X andlet g(α, β) denote the vector field of the fast variablesand f(α, β) the vector field of slow variables. Equations(4),(5) can be expressed as a two-time scale system;

α = f(α, β) (6)εβ = g(α, β) (7)

Here ε > 0 and as indicated in [20] is chosen to bethe smallest time constant of the governing differentialequations, therefore ε ≈ 1

|λ7| = 10−2.The following theorem uses basic singular pertur-

bation theory, specifically Tikhonov’s theorem, [20,Sec.11.1], as well as the approximate relation, S ≈ S(0),to simplify the above two-time scale nonlinear system.The resulting simplified system yields the evolution ofthe admittance of the biosensor versus analyte concen-tration according to the two modes described in SectionIII-B, namely linear, and exponential decay.

Theorem 4.1: Consider the chemical species dy-namics depicted by the two time scale system (6),(7).Then as ε → 0, the fast time-scale variables arrive atthe quasi-steady state and the dimer concentration, ¯D(t),

13

converges to the trajectory of the following system:

d

dt(D(t)) = −D(r5 + f6A

∗)+(

f5C +r6f7Xr6 + r7

)S(0)

(8)

Specifically the trajectories that start in an O(ε) neigh-borhood of β = h(α), where h(α) denotes the solutionof the algebraic equation g(α, β) = 0 satisfy;

| ¯D(t)−D(t)| = O(ε), |z(t)− z| = O(ε) (9)

for all t ∈ [0, T ] where T denotes a finite time horizon.Eq. 8 can be solved using Euler’s method with step

size h to arrive at;

Dn+1 = (1− (r5 + f6A∗)h) Dn+

h

(S(0)

(f5C(r6 + r7) + r6f7X

r6 + r7

))(10)

The proof of Theorem 2.1 is in the appendix.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(Dn, A∗) + wn, (11)

M =

1 A∗ = 0 : fM(Dn, A∗) = Dn

2 A∗medium : fM(Dn, A∗) = κ1Dn + κ2

3 A∗high : fM(Dn, A∗) = κ3Dn + κ4

where wn is as defined in (2) and the constants κi for i ∈1, 2, 3, 4 are calculated from the following equationsusing medium A∗ for κ1 and high A∗ for κ3;

κ1,3 = (1− (r5 + f6A∗)h) ,

κ2,4 = h

(S(0)

(f5C(r6 + r7) + r6f7X

r6 + r7

))(12)

Note that the dimer concentration evolves accordingto three modes defined in (11), depending on the analyteconcentration A∗ which is constant in the reaction-rate-limited regime of operation. Also recall that the biosen-sor is in the reaction-rate-limited regime of operationwhen A∗ > 1µM.

Using (11) and noting that the channel conductanceis directly proportional to dimer concentration, we canderive equations explaining the evolution of the channelconductance on the slow time scale.

C. Mass Transport Influenced Detection Kinetics

As explained in Section IV-A, when the ratio ofanalyte concentration to binding sites is small (e.g.10−11 moles per 1010 binding sites per cm2) then mass

Analyte

x3

x1

x2 Biomimetic Surface

Fig. 16. A schematic of the flow chamber. The reactive surface islocated at x3 = 0. The solution containing the analyte molecules entersat x1 = 0 and flows along the x-axis

transport effects in the ICS biosensor become dominant.In this section the partial differential equation governingthe mass transport of analyte molecules is derived. Inthe mass transport influenced region of operation, theanalyte concentration is no longer a constant A∗ but isa function of time and space, A(x1, x2, x3, t).

Analyte is transported to the reacting surface of theICS biosensor, by diffusion and flow, where it reactswith the immobilized receptors. The flow chamber hasa rectangular cross section. A schematic of the flowchamber is shown in Fig. 16, where the height of thechamber, along the z-axis, is h = 0.1mm, the length ofthe chamber along the x-axis is L = 6mm and the widthof the chamber along the y-axis is W = 2mm.

In [3] it is shown that for aspect ratio h/W = 0.1and in general small, the variations along the width ofa flow chamber are not noticleable. In the case of thisbiosensor with flow chamber geometry as denoted inFig. 16, the aspect ratio is 0.05, hence we can ignorethe variations along the y-axis and the analyte concen-tration is A(x1, x3, t). In the flow chamber the analyteconcentration A(x1, x3, t) is governed by the followingreaction-diffusion partial differential equation [10]

∂A

∂t= γ(

∂2A

∂x21

+∂2A

∂x23

) + υ∂A

∂x1(13)

Here γ is the diffusivity constant of the analyte (e.g.,γ ≈ 10−6cm2s−1 for streptavidin or hCG). There arefour boundary conditions that need to be considered:(i) The chamber boundary at x3 = h is reflective and themass flux must equal zero. This results in the boundarycondition:

∂A

∂x3|(x1,x3=h,t) = 0 (14)

(ii) At the biomimetic surface x3 = 0, the mass fluxmust equal the time rate of change of the concentration

14

Fig. 17. Finite element solution of flow snapped at five time intervals.

of the species that combine with a. This results in theboundary condition

∂A

∂x3|(x1,x3=0,t) =

∂X

∂t+

∂W

∂t(15)

where, ∂X∂t and ∂W

∂t are nonlinear functions definedin (5).(iii) At the entry to the flow cell x1 = 0, the analyteconcentration is equal to the injection concentration,A and at x1 = L, where the analyte exits, the massflux is zero. That yields the third and fourth boundaryconditions

A(x1 = 0, x3, t) = A,∂A

∂x1|(x1=L,x3,t) = 0 (16)

The analyte concentration A(x1, x3, t) is therefore foundby solving (13) subject to boundary conditions (14-16).

To simulate mass transport effects, we solved (13)subject to boundary conditions (14-16) via the finiteelement method on the rectangular flow cell shown inFig.16. Fig.17 shows a typical solution obtained fromthe finite element analysis. A bolus of sample flowingleft to right, shown at five time intervals. The bottomof the flow cell has a high capture density of receptors.Whereas at the top surface the sample is retarded bythe usual flow hydrodynamics at a wall. At the bottomsurface sample is depleted by binding to the capture sitesdelaying the passage of the sample.

Fig.18(a) shows the increase in resistance of thechannel (decrease in channel conductance) as the proteinstreptavidin is captured on the membrane for a sampleflow of 100µL/min. It is significant that when the flowis stopped (shown by arrows in the figure), the resistancechange ceases. This arises from flow dependent analytedepletion at the electrode surface. Fig. 18(b) shows thesuperposition of the resistance increase resulting from achallenge of 10fM streptavidin at various flow rates from0µL/min to 100µL/min. Also shown is the predicted

Fig. 18. (a) Response of ICS sensor for the concentrations shownand 100ul/min flow rate. Note that when the flow stops the responsestops at all concentrations. This is due to analyte depletion at the sensorsurface. (b) Dependency of response of sensor to flow rate (black dots).The lines (purple) through the data points are predicted by the model.The straight line at 150 is the high flow limit. The red diamondsare PBS controls. (c) Predicted and experimental titration curves forthe ICS sensor response to Streptavidin in the range 1fM -100fM at150ul/min. The triangles are experimental data and the diamonds arefrom the model.

high flow resistance change which is three times themagnitude of change experienced at even 100µL/min.This difference shows the utility of sample flow whenoperating at high capture densities and low analyteconcentrations. Fig. 18(c) shows a superposition of thepredictive and experimental titration curve in the rangeof 0fM to 100fM streptavidin.

V. STATISTICAL SIGNAL PROCESSING WITHBIOSENSOR

With the construction and modeling of the ICS biosen-sor described above, our goal in this section is to describehow the measurements from the biosensor can be usedto detect the presence and concentration of analytes.Based on the size of the sensing electrodes, there aretwo possible cases:

15

a) Medium-sized electrodes: The current genera-tion ICS biosensor has electrode sizes of 1 mm radius.Thus the sensor response is dependent on the net currentchanges from millions of individual gramicidin chan-nels. The measured current is in the order of microamperes. The measurement noise is insignificant (apartfrom a slow baseline drift). So for the ICS biosensor, theconcentration of the analyte can be determined straight-forwardly from the three dynamic modes described in(2). In future generation ICS biosensors, the electrodeswill be miniaturized. Electrode sizes of 10 µm radiuscomprise of a few hundred gramicidin channels. Bycentral limit theorem arguments, the net current can beapproximated by a Gaussian probability distribution. Inthis case, the measurement noise can not be ignored.Because of the linear Gaussian dynamics of the under-lying current, a Kalman filter based algorithm [1] canbe derived to classify the concentration of the analyte inthe system as medium, high or null.

b) Micro-electrodes: Electrode sizes of radius1 µm comprise of only a few ion channels. In thiscase, the current pulses from individual channels can beresolved and the biosensor records a finite state “digital”output. The arrival of individual analyte molecules canthen be detected at individual electrodes. This allowsfor exploitation of the analyte flow equations in SectionIV-C for measurements at multiple electrodes therebyresulting in enhanced sensitivity. The biosensor outputcan be modeled as a noisy finite state Markov chain,i.e., a Hidden Markov Model [11]. The noise levels aresubstantial in this case and careful modeling of the noisedistribution is required.

Construction of ICS biosensors with such small elec-trodes is the subject of our current research. Here weprovide a simple proof-of-concept of the signal pro-cessing capabilities by using a simpler biosensor setup,see [21] for experimental details. We use a covalentlylinked dimer of gramicidin ion channels (called bis-gramicidin A) incorporated into an untethered bilayermembrane excised from a giant lipid vesicle seen inFigure 19. The bilayer is supported over the 1 µmdiameter opening of a micropipette as shown Figure 20.For illustrative purposes we used a low molecular weightanalyte methylbenzthonium chloride (MBC) shown inFigure 21 that blocks conduction of the gramicidinchannel.

The construction of the HMM detector for MBCwas done as follows. First maximum likelihood modelparameters of the transition probability, state levels of theMarkov chain and parameters of noise distribution areestimated from the data. This is done in the presence or

Fig. 19. Fluorescence image (left) of biosensor’s horizontal opticalsection shows the bis-gramicidin A channels labelled using FITC andidentified by the green color. Phase-contrast image (right) of the samehorizontal slice shows the overall shape of the biosensor

pippette electrode

reference electrode

Bias potential

ADC

Computer

1um contact area between glass

micropipette and liposome

2

Fig. 20. Photo of glass micropipette and liposome, with block diagramof the electrical detection scheme. The solution in the recording pipettewas 0.5 M KCl with the liposomes suspended in a 0.5 M NaCl solution.The conductance of bis-gramicidin channels under these conditions isapproximatately 20 pS.

absence of the channel blocker MBC. The detection ofthe channel blocker is now reduced to a classificationproblem. Modeling the noise was a challenging task.This arises from thermal noise, the anti-aliasing effectfrom sampling and an open channel noise with its powerproportional to the inverse of frequency. Figure 22 showsthe power spectral density of a typical sequence ofbiosensor recordings and clearly shows that the powerdecreases at a rate of -10 dB/dec at low frequenciesindicating the presence of 1/f noise. This 1/f noise isdiscussed in other studies of bis-gramicidin A ion chan-nels, see [39]. To model this correlated noise process,we used an auto-regressive (AR) Gaussian process thatcomprises white Gaussian noise process Wk filtered byan all-pole filter, see [21] for the Ljung-Box test used asa portmanteau lack of fit test for model adequacy.

16

Fig. 21. Chemical Structure of Methylbenzthonium Chloride (MBC)

10000 1000 100 10 1!40

!30

!20

!10

0

10

20

30

40

50

Frequency (Hz)

Pow

er S

pect

rum

of M

easu

red

Cur

rent

(dB)

Fig. 22. Power spectral density of biosensor response clearly showsthe 1/f open channel noise and the anti-aliasing effect in the samples.

Figure 23(a) shows the output of the biosensor beforeand after the analyte MBC is added. A Hidden MarkovModel (HMM) maximum likelihood classifier is thenused to detect the presence of the target molecules, seeFigure 23(b). While it is difficult to identify the transi-tion in the recordings in 23(a), the detection algorithmdetected a model change shortly after MBC was addedas shown in Figure Figure 23(b). The detection tracein Figure 23(b) indicates a switch in the most likelymodel from θ1 (no MBC present) to θ2 (MBC present)at k = 59.865 seconds, approximately 0.2 seconds afterMBC is added. We refer the reader to [21] for a completedescription.

Remark: The current pulses arising from these chan-nels are a consequence of a different mechanism tothat of monomeric gramicidin diffusing in the inner andouter bilayer membrane of the ICS biosensor. Currentfluctuations have been reported by [?] for bis-gramicidin.These were thought to arise from conformational inter-conversion in the bis-gramicidin A secondary structure.The mechanism generating the current fluctuations isunimportant for our description here.

VI. CONCLUSIONS AND EXTENSIONS

This paper began with a description of constructionand operation of an ion channel switch (ICS) biosensor.The electrical dynamics of the biosensor were mod-eled by a second order linear system. Furthermore, thechemical kinetics of the biosensor were modeled as atwo-timescale nonlinear dynamical system in case of

0 2 4 6 8 10 1240

50

60

70

80

90

100

110

120

Time (s)

Curr

ent (p

A)

MBC

no MBC

0 2 4 6 8 10 12Time (s)

!

!"

!1

!2

^

Fig. 23. (a) Current pulse recordings on an expanded scale before(blue) and after (red) channel blocker MBC. (b) Maximum likelihooddetection generated using Hidden Markov Model estimation algorithm.While it is difficult to identify the transition by eyeballing the record-ings in Figure 23(a), the detection algorithm detected a model changeshortly after MBC was added.

reaction-rate limited dynamics. Using Singular Perturba-tion theory, an analytical equation expressing the evolu-tion of the dimer concentration with time as a functionof analyte concentration was derived. A partial differ-ential equation discribing the evolution of the analyteconcentration in time and space in case of mass transportinfluenced chemical kinetics was derived. Finally, forthe case of micro-sized electrodes, we described howstatistical signal processing algorithms can be derivedfor detecting the presence of analyte and classifying theanalyte concentration.

There are several extensions that we are currentlyconsidering. When employing antibodies or other welldefined receptors much of the sensitivity advantages ofstochastic sensing, occurs when using stochastic detec-tion in conjunction with the spatial analysis across anelectrode arrays. Exploiting the arrival statistics of atarget molecule on the array, in addition to the stochasticion current properties within each element permits arapid estimate of the spontaneous concentration. Whenusing a large number of channels the benefit of measure-ment redundancy in an array is the classical

√N , where

17

N is the number of independently read electrodes in thearray. It is worthwhile examining if the dynamics of thefluid flow (given by a partial differential equation) of theanalyte can be exploited to devise a more efficient de-tector for the analyte and estimator for its concentration.

APPENDIX

A. Singular Perturbation Modeling of the Biosensor

The presence of “fast” and “slow” species in thenonlinear set of equations describing the evolution ofthe chemical species in the biosensor system, depicted in(5), allows us to rewrite the system dynamics in a multi-time scale setting as done in Section IV-B and denotedin (6,7).

Setting ε = 0 in (7), we can solve for the quasi steadystate as follows;

g(α, β) = 0 ⇒ β = h(α) =

(f4XB+f3WC

r3+r4f7XS+f6A∗D

r6+r7

)Replacing β in (6) by h(α), we arrive at the reducedmodel;

α = f(α, h(α)) (17)

Let α denote the solution to (17), and define y =β − h(α). Then letting dy/dτ = εdy/dt and noting thatsetting ε = 0 freezes the α variables at t0 which has nowbeen shifted to the origin we arrive at the boundary-layer model which as explained in detail in [20] is ashifted version of the quasi-steady-state solution of thefast dynamics h(α);

d

dτ(y) = g(α(t0), y + h(α(t0)))

=(

0 −(r3 + r4)−(r6 + r7) 0

)(y1

y2

)(18)

If the conditions of Theorem 11.1 in [20, Sec.11.1] aresatisfied, we can conclude the proof of Theorem 4.1 inSection IV. Next we will show that these conditions arein fact satisfied for the biosensor model.• In (6),(7) the functions f, g, their first par-

tial derivatives with respect to (α, β) and ∂g/∂tare continuous; the function h(α), and the Jaco-bian [∂g(α, β)/∂(β)] have continuous first partialderivatives with respect to their arguments.

• The component functions of the reduced model in(17) and their partial derivatives with respect to thecomponents of the functions are each continuous onR, therefore the component functions are each lo-cally Lipschitz in α on R. Thus the vector function

f(α, h(α)) is Lipschitz continuous in α on R, andhas a unique solution α on R.

• The origin is an exponentially stable equilibriumpoint of the boundary-layer model. To analyze thestability of the origin for the boundary-layer modelwe look at the eigenvalues of the matrix defined in(18). The eigenvalues are found to be λ1 = −0.011and λ2 = −0.002. Therefore the origin is an expo-nentially stable equilibrium point of the boundary-layer model.

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