Practical Model for Imperfect Conductometric Molecular Wire Sensors

Dec 17, 2008 - It turns out that, whereas the Stern−Volmer curve of the ideal MWS always has a positive curvature, the Stern−Volmer curve of the i...
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Anal. Chem. 2009, 81, 578–583

Practical Model for Imperfect Conductometric Molecular Wire Sensors Je Hyun Bae,† Yu Rim Lim,† Won Jung,† Robert J. Silbey,‡ and Jaeyoung Sung*,† Department of Chemistry, Chung-Ang University, Seoul 156-756 Korea, and Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 We present a theoretical model for description of real polyreceptor molecular wire sensors (MWS), whose conductance signal may dramatically reduce upon analyte binding to one of the receptors coupled to the molecular wire but may not vanish as completely as assumed in the ideal MWS model. For the present nonideal MWS model, we establish the exact relationship between analyte concentration and the sensory signal intensity. It turns out that, whereas the Stern-Volmer curve of the ideal MWS always has a positive curvature, the Stern-Volmer curve of the imperfect MWS can have a negative curvature, consistent with experimental data. We find that the MWS still performs better than the corresponding ideal monoreceptor sensor, unless the nonideality of the imperfect MWS is egregiously large. We establish the conditions for the imperfect polyreceptor MWS to have a sensitivity and detection limit superior to the traditional monoreceptor sensor. The molecular wire approach1 is one of the attractive approaches toward development of ultrasensitive sensors for quantification of ultratrace analyte.1–7 It provides a general signal amplification scheme to a variety of analyte-receptor systems. The molecular wire approach has been applied to the detection of toxic ions,8 explosive compounds,9 biomolecules,10 and so on.11-14 * To whom correspondence should be addressed. E-mail: [email protected]. † Chung-Ang University. ‡ Massachusetts Institute of Technology. (1) (a) Swager, T. M. Acc. Chem. Res. 1998, 31, 201. (b) McQuade, D. T.; Pullen, A. E.; Swager, T. M. Chem. Rev. 2000, 100, 2537. (c) Thomas, S. W., III; Joly, G. D.; Swager, T. M. Chem. Rev. 2007, 107, 1339. (2) Blaedel, W. J.; Boguslaski, R. C. Anal. Chem. 1978, 50, 1026. (3) Shibata, T.; Yamamoto, J.; Matsumoto, N.; Yonekubo, S.; Osanai, S.; Soai, K. J. Am. Chem. Soc. 1998, 120, 12157. (4) Hartig, J. S.; Grune, I.; Najafi-Shoushtari, S. H.; Famulok, M. J. Am. Chem. Soc. 2004, 126, 722. (5) Brunner, J.; Mokhir, A.; Kremer, R. J. Am. Chem. Soc. 2003, 125, 12410. (6) Wu, Q.; Anslyn, E. V. J. Am. Chem. Soc. 2004, 126, 14682. (7) Ellington, A. D.; Levy, M. Proc. Natl. Acad. Sci. U.S.A. 2003, 100, 6416. (8) Kim, T.; Swager, T. M. Angew. Chem., Int. Ed. 2003, 42, 4803. (9) (a) Kuroda, K.; Swager, T. M. Macromol. Symp. 2003, 201, 127. (b) Narayanan, A.; Varnavski, O. P.; Swager, T. M.; Goodson, T., III. J. Phys. Chem. C 2008, 112, 881. (10) (a) Dore, K.; Dubus, S.; Ho, H.-A.; Levesque, I.; Brunette, M.; Corbeil, G.; Boissinot, M.; Boivin, G.; Bergeron, M. G.; Boudreau, D.; Leclerc, M. J. Am. Chem. Soc. 2004, 126, 4240. (b) Kim, Y.; Swager, T. M. Macromolecules 2006, 39, 5177. (c) Zheng, J.; Swager, T. M. Macromolecules 2006, 39, 6781. (11) Lee, D.; Swager, T. M. J. Am. Chem. Soc. 2003, 125, 6870.

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Figure 1. Schematic illustration of an ideal molecular wire sensor: A single molecular wire is composed of κ receptor units, and L molecular wires comprise a molecular wire sensor. Each molecular wire gives off a signal with intensity i0 only if none of the receptors on the molecular wire is occupied by an analyte molecule (ref 15).

The signal amplification achieved in these molecular wire sensors (MWS) is directly related to their structural characteristics such as the number, L, of molecular wires composing the MWS and the number, κ, of receptor units coupled to each molecular wire. The structure of the MWS is schematically depicted in Figure 1. The molecular wires composing the MWS are often conjugated semiconducting organic polymers. The essence of the signal amplification of MWS lies in the special interactions between the molecular wire and the receptor units coupled to the molecular wire, which are designed so as to develop a collective response from the entire molecular wire upon analyte binding to any one of the receptors coupled to the molecular wire. In the ideal MWS model, only if none of κ receptors coupled to the molecular wire is occupied, the molecular wire emits a unit intensity of signal such as fluorescence or electric current; otherwise, the signal from the molecular wire vanishes completely. It is obvious that the response of such ideal polyreceptor MWS is more sensitive to the presence of analyte molecules than the (12) Kim, Y.; Zhu, Z.; Swager, T. M. J. Am. Chem. Soc. 2004, 126, 452. (13) Zhou, X. H.; Yan, J. C.; Pei, J. Macromolecules 2004, 37, 7078. (14) Pohl, R.; Aldakov, D.; Kuba´t, P.; Jursı´kova´, K.; Marquez, M.; Anzenbacher, P., Jr. Chem. Commun. 2004, 1282. 10.1021/ac801715x CCC: $40.75  2009 American Chemical Society Published on Web 12/17/2008

corresponding monoreceptor sensor in which signal change for analyte binding to one receptor is independent of that for analyte binding to another. For the ideal MWS model, a quantitative relationship between the analyte concentration and the amplified signal intensity of the MWS was established.15 The signal amplification scheme described in the previous paragraph has been demonstrated for real polyreceptor conjugated polymer systems such as polypyrrole,11 poly(p-phenylenevinylene),12 polythiophene,16 polyaniline,17 and poly(phenyleneethynylene),18 to name a few, whose electric conductivity or fluorescence quantum yield is dramatically diminished upon binding of an analyte molecule to any of the receptor units in the polymer. In conductometric MWS, a structural reorganization of a macrocyclic receptor unit such as crown ether or calyx[4]arene upon analyte binding produces enough strain to deform the planar structure of a π-conjugated polymer backbone significantly, which reduces the electronic conductivity along the polymer chain.1 In a fluorescence-based MWS, in comparison, analyte binding to a receptor unit provides an efficient fluorescence quenching pathway on the host polymer chain, which prevents electronic excitation created by an external illumination from migrating along the polymer chain to a light-emitting or reporter unit appended near the terminus of the polymer.19 To achieve signal amplification in the MWS with these mechanisms, it is crucial to suppress electron or excitation transfer between polymer chains through, for example, π-π stacking or excimer formation, so chemical tactics have been developed to introduce steric repulsions between the polyreceptor polymer chains to minimize the communication between them.11,20,21 However, even with the above-mentioned ingenious chemical strategy, it would be difficult to realize the ideal MWS in which the signal from the molecular wire vanishes completely upon the very first binding of an analyte molecule to any one of the many receptor units coupled to the molecular wire. In practice, the first binding of an analyte molecule may make the signal from a molecular wire diminish significantly, but it may not make the signal vanish completely. For a conductometric MWS, it would be difficult to realize the molecular wire whose electric resistance becomes infinite upon binding of a single analyte molecule to the molecular wire. For the fluorescence-based MWS, the nonideal behavior emerges whenever the single occupied receptor is not able to extinguish the optical excitation of the molecular wire completely. As we will show in this paper, the signal profile of the MWS with the latter nonideality has unique features missing in that of the ideal MWS, which can be identified in previously reported experimental data for real MWS.22-26 Whereas the Stern-Volmer (SV) curve of the ideal MWS always has a positive (15) Sung, J.; Silbey, R. J. Anal. Chem. 2005, 77, 6169. (16) Yu, H.; Xu, B.; Swager, T. M. J. Am. Chem. Soc. 2004, 125, 1142. (17) MacDiarmid, A. G.; Chiang, J.-C.; Richter, A. F.; Epstein, A. J. Synth. Met. 1987, 18, 285. (18) (a) Zhou, Q.; Swager, T. M. J. Am. Chem. Soc. 1995, 117, 7017. (b) Liao, J. H.; Swager, T. M. Langmuir 2007, 23, 112. (19) Zhou, Q.; Swager, T. M. J. Am. Chem. Soc. 1995, 117, 12593. (20) Yang, J.-S.; Swager, T. M. J. Am. Chem. Soc. 1998, 120, 5321. (21) Yang, J.-S.; Swager, T. M. J. Am. Chem. Soc. 1998, 120, 11864. (22) Jiang, D.-L.; Choi, C.-K.; Honda, K.; Li, W.-S.; Yuzawa, T.; Aida, T. J. Am. Chem. Soc. 2004, 126, 12084. (23) Cabarcos, E. L.; Carter, S. A. Macromolecules 2005, 38, 4409. (24) Xue, C.; Jog, S. P.; Murthy, P.; Liu, H. Biomacromolecules 2006, 7, 2470. (25) Tang, Y.; He, F.; Yu, M.; Wang, S.; Li, Y.; Zhu, D. Chem. Mater. 2006, 18, 3605.

curvature, that of the imperfect MWS has a negative curvature in the high analyte density regime; in the low analyte density regime, its curvature is mostly positive unless the nonideality of the MWS is too large. We find that the imperfect MWS still performs better than the corresponding ideal monoreceptor sensor, unless the nonideality of the imperfect MWS is egregiously large. We note that, for turn-off-type optical MWS, the fluorescence signal attenuation of a molecular wire occurs not only by quenching of the optical excitation by the analyte-occupied receptor but also by the first-order nonradiative decay.27 In the present work, we do not take into account the latter mechanism working in the fluorescence-based MWS; instead, we focus on the conductometric MWS, free of complication due the spontaneous decay of the signal intensity.28 However, we note that a slight modification of the present work is enough for description of simple models of the fluorescence-based MWS in which nonequilibrium exciton dynamics along molecular wires does not have to be taken into account. This paper is organized as follows: In the Theoretical Model and Key Result section, we present a detailed description of our theoretical model for the imperfect MWS, and then, for the model, we present the exact quantitative relationship between analyte concentration and sensory signal intensity. The result is obtained by considering the grand canonical ensemble of analyte molecules on the MWS in equilibrium with the analyte molecules in solution.29 In the Discussion section, we discuss the shape of SV curve,30 the detection limit, the sensitivity to the change of analyte concentration of the MWS on the basis of the result presented in the Theoretical Model and Key Result section. We also discuss the conditions for the imperfect MWS to have a detection limit and a sensitivity superior to the corresponding ideal monoreceptor sensor. In the Conclusion section, we summarize the present work. Finally, in the Supporting Information, we detail the derivation of the key results presented in the Theoretical Model and Key Result section. THEORETICAL MODEL AND KEY RESULT In our imperfect MWS, the conductance i(s) of a molecular wire is related to the number, s, of the receptor units occupied by analyte molecules in the molecular wire, by i(s) ) i0(1 - γ)s. Here, i0 denotes the conductance of each molecular wire on which none of the receptor units is occupied by analyte molecule, and the signal intensity diminishes per each binding of an analyte molecule to one of receptors in the molecular wire so that i(s + 1)/i(s) is given by a constant 1 - γ (0 < γ < 1). Throughout the present work γ is to be designated by the signal attenuation factor, which is a measure of the ideality (26) Joly, G. D.; Geiger, L.; Kooi, S. E.; Swager, T. M. Macromolecules 2006, 39, 7175. (27) Swager, T. M. In Redox Systems under Nano-Space Control; Hirao, T., Ed.; Springer: Berlin-Heiderberg-New York, 2006; Chapter XVIII, p 295. (28) Swager, T. M., MIT, Cambridge, Massachusetts. Personal communication, 2008. (29) McQuarrie, D. A. Statistical Mechanics; Harper & Row: New YorkEvanston-San Francisco-London, 1976. (30) Throughout the present work, the Stern-Volmer curve denotes the dependence of I(0)/I([A]) on the analyte density [A] where I([A]) denotes the average strength of a general sensory signal as a function of analyte density, [A]. We note that the conventional usage of the Stern-Volmer curve has been made specifically for characterization of such photophysical processes as fluorescence quenching and phosphorescence quenching.

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Figure 2. Schematic illustration of the nonideal molecular wire whose signal intensity decreases with a constant ratio, 1 - γ, per each binding of an analyte molecule to a molecular wire. All the other symbols have the same meaning as those used in Figure 1.

of a MWS. The MWS with γ being one is the ideal MWS in which the signal completely vanishes upon the first binding of an analyte molecule. For the other extreme where γ is zero, the signal from MWS would not respond at all to analyte binding events. The model described in the previous paragraph is schematically depicted in Figure 2. As the total signal intensity, I, of a MWS is given by the sum of signal intensities from molecular wires composing the MWS, we have L

I(S) )

∑ i (1 - γ)

sj

(1)

0

j)1

for the MWS composed of L identical molecular wires. Here, sj denotes the number of receptor units occupied by analyte molecules on the jth molecular wire of our MWS. S is the L-dimensional vector whose jth component is sj. Note that sj is a stochastic variable ranging from zero to κ, the number of receptor units coupled to each molecular wire, so the sensory signal intensity, I, given in eq 1 is also a stochastic variable ranging from i0L for the low analyte concentration limit to i0L(1 - γ)κ for high analyte concentration limit. The quantity of interest is the average value 〈I〉 of the signal intensity as a function of analyte concentration [A] in the sample, which is defined as 〈I〉 ) ∑S I(S)P(S). Here P(S) denotes the equilibrium probability that we find a configuration S [≡ {s1, s2,..., sL}] over the L molecular wires, and ∑S denotes the sum over all possible configurations. The expression for P(S) as a function of analyte density in bulk media can be obtained readily by considering a grand canonical ensemble of analyte molecules on MWS. We postpone the derivation of the result to the Supporting Information; here, we only present the final expression for the average signal intensity, 〈I〉, of the imperfect MWS:

〈I 〉

(

) I0

1 + (1 - γ)[A]/A1/2 1 + [A]/A1/2

)

κ

(2)

In eq 2 I0 and [A] denote the maximum signal intensity defined by i0L and the analyte density in bulk media, respectively. A1/2 denotes the value of the analyte concentration at which the probability that a receptor unit on the MWS is occupied 580

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becomes half, so it is inversely proportional to the binding affinity of an analyte molecule to the receptor unit. Equation 2 is the key result that expresses the quantitative relationship between analyte concentration and the signal intensity of our model MWS. For the ideal MWS with γ being equal to one, eq 2 correctly reduces to the known result:15 〈I〉id ) I0(1 + [A]/A1/2)-κ

(3)

When the value of κ is further set equal to one, eq 2 reduces to the well-known result for the signal intensity of a traditional monoreceptor sensor.31 〈I〉mono ) I0(1 + [A]/A1/2)-1

(4)

DISCUSSION From eq 2, we can predict the shape of the SV curve, the dependence of I0/〈I〉 on [A], for our imperfect MWS. In the low analyte concentration limit ([A]/A1/2 , 1), the relationship simplifies as follows:32 I0 〈I 〉

) 1 + κγ

( ) [( ) ]

[A] [A] 1 + κγ[γ(κ + 1) - 2] A1/2 2 A1/2

2

+O

[A] A1/2

3

(5) In the low analyte density regime, the curvature of the SV curve (1) 2 is determined by the coefficient K SV A1/2 of ([A]/A1/2)2 in eq 5, (1) 2 given by K SV A1/2 ≡ κγ[γ(κ + 1)/2 - 1]. Note that, for the ideal (1) MWS with γ being equal to one, the curvature, K SV of the SV curve becomes κ(κ - 1)/2, which is always positive for any number κ of receptors. In comparison, the SV curve of the imperfect MWS with γ less than one does not always have a positive curvature. Equation 5 tells us that, as long as the signal attenuation factor, γ, is greater than 2/(κ + 1), the SV curve of the imperfect MWS has a positive curvature in the low analyte concentration regime; in comparison, when the signal attenuation (1) factor, γ, is smaller than 2/(κ + 1), the curvature, K SV of the SV curve of the imperfect MWS has a negative value in the low (31) Janata, J. Principles of Chemical Sensors; Plenum Press: New York, 1989. (32) In eqs 5, 6, and 11, O[xn] represents a term whose magnitude is order of xn.

Figure 3. Nonlinear part in the Stern-Volmer relation for various values of η in the low analyte concentration regime. η is defined by 0 η ≡ γ - 2/(κ + 1). Equations 2 and 7 are used to plot the lines. KSV in the y-axis title denotes the Stern-Volmer constant.

analyte concentration regime. This is shown in Figure 3 where we have depicted the nonlinear part of the SV curve of the imperfect MWS. For a monoreceptor sensor, κ is equal to one, so eq 5 reduces to I0 〈I 〉

)1+γ

( ) [( ) ]

[A] [A] 1 - γ(1 - γ) A1/2 2 A1/2

2

+O

[A] A1/2

3

κγ A1/2

If the resolution of our signal detector is good enough, i.e., if m/L is small enough, eq 8 becomes, [A]min/A1/2 = (m/κL)/{1 - (1 - γ)[1 + (m/κL)]}. The latter result shows that, for a given value of m, the MWS with greater values of κ, L, and γ is more sensitive to the presence of analyte molecules. This is shown in Figure 4. In comparison, the detection limit of the ideal monoreceptor sensor is given by

(6)

Equation 6 indicates that the SV curve of the traditional monoreceptor sensor will have a negative curvature in the low analyte density limit, whenever signal from a sensor unit does not vanish completely upon analyte binding, i.e., unless γ is one. 0 The SV constant, defined by KSV ≡ lim[A]f0(I0/〈I〉 - 1)/[A], is the initial slope of the SV curve that quantifies the sensitivity of sensory signal intensity to a change in analyte concentration in the low-concentration limit. From eq 5, one can easily obtain 0 the expression for KSV for the imperfect MWS as follows: 0 ΚSV )

Figure 4. Relationship between [A]min/A1/2 (detection limit) and the number κ of receptors coupled to each molecular wire, for various values of γ. The detection limit of the imperfect MWS is lower than that of the traditional monoreceptor sensor in most cases.

(7)

0 Note that KSV increases with κγ for the imperfect MWS. The SV constant for the ideal monoreceptor sensor is given by eq 7 with κ ) 1 and γ ) 1. Therefore, in spite of the presence of the 0 nonideality, KSV of the MWS is still greater than that of the traditional monoreceptor sensor as long as the value of κγ is greater than one. The detection limit of a MWS is dependent on the resolution or the minimum detectable signal intensity of the signal detector of our sensor, the number κ of receptors coupled to each molecular wire, and the signal attenuation factor, γ. Given that the resolution of the signal detector is given by mi0, one can obtain the expression for the minimum amount of the detectable analyte concentration [A]min of the MWS from eq 2 as follows:

mono [A]min m ) A1/2 L-m

(9)

From the comparison between eqs 8 and 9, one can see that the imperfect MWS is more sensitive to the presence of analyte molecules than the ideal monoreceptor sensor, as long as the value of signal attenuation factor, γ, is greater than γ* defined by γ* )

[ (

L m 1- 1m L

) ] 1/κ

(10)

When the resolution of the detector is good enough or when m/L , 1, γ* given in eq 10 can be approximated to γ* ≈ κ-1. In the large κ limit, the value of γ* given in eq 10 becomes zero so that the condition, γ > γ* is always satisfied and, in spite of the nonideality, the imperfect MWS always has a greater sensitivity to the presence of analyte molecules than the ideal monoreceptor sensor. As shown in Figure 5, when the number L of molecular wires composing the MWS is large, the value of γ* is weakly dependent on L/m, but it rapidly decreases with κ. Let us now discuss the SV curve of the MWS in the high analyte concentration regime where [A] . A1/2. From eq 2, we obtain the approximate expression for I0/〈I〉 in the high-density regime as follows: I0 1 ) 〈I 〉 1-γ

(

){ κ

1-

( )

( ) [( ) ]}

κγ A1/2 κγ(κγ - γ - 2) A1/2 + 1 - γ [A] [A] 2(1 - γ)2 O

[A] A1/2

2

+

3

(11)

-1/κ

[A]min ) A1/2

(1 - mL ) - 1 m 1 - (1 - γ)(1 - ) L

-1/κ

(8)

Equation 11 indicates that the SV curve converges to (1 - γ)-κ (≡ I0/I∞) in the high analyte density limit. This is expected because the minimum signal intensity imin from a molecular Analytical Chemistry, Vol. 81, No. 2, January 15, 2009

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Figure 5. Relationship between κ and γ* for various values of L/m. The imperfect polyreceptor MWS with signal attenuation factor γ greater than γ* is more sensitive to the presence of analyte molecules than the ideal monoreceptor counterpart. [A]min (MWS) and [A]min (TMS) represent the detection limit of the imperfect MWS and that of the transitional monoreceptor sensor without the nonideality, respectively.

wire is (1 - γ)κi0, and the minimum signal intensity 〈I〉min from our MWS in the high analyte density limit is simply iminL. Since I0/I, an increasing function of [A], converges to the constant value (1 - γ)-κ in the high analyte density limit, the SV curve of our MWS should have a negative curvature in the high analyte density regime. The latter feature in the SV curves of the nonideal MWS cannot be found in the SV curves of the ideal MWS with γ ) 1.15 As mentioned before, the SV curve of the MWS with γ(κ + 1) greater than 2 has a positive curvature when [A] , A1/2 according to eq 5, but it has a negative curvature when [A] . A1/2, according to eq 11. Therefore, the SV curve of the MWS with γ(κ + 1) greater than 2 should have an inflection point at some density [A]# at which the curvature of the SV curve changes its sign. In comparison, the SV curve of the MWS with γ(κ + 1) less than 2 always has a negative curvature so that it does not have any inflection point. This is shown in Figure 6, where we show the SV curves and the curvatures of the SV curves given by ∂2(I0/I)/∂x2 ) κγ/(1 + γ)4(I0/I)1+(2/κ){[γ(1 + κ) - 2] - 2(1 - γ)x} with x being [A]/A1/2. From the latter equation, we obtain the expression for the inflection point as follows:

[A]#/A1/2 )

γ(1 + κ) - 2 2(1 - γ)

Whereas the SV curves of the traditional monoreceptor sensors are mostly linear,33 those of the MWS are quite nonlinear as shown in Figures 3 and 6. Nevertheless, ln(I0/〈I〉) of the present model of imperfect MWS is linearly proportional to ln{(1 + [A]/A1/2)/[1 + (1 - γ)[A]/A1/2]} with proportionality constant κ, as shown in Figure 7. Let us now discuss the sensitivity of the MWS to the change of analyte density. We define the sensitivity S([A]) to the change of analyte density as the inverse of the minimum detectable analyte density change ∆[A]min from [A]. Mathematically, ∆[A]min is (33) (a) Birks, J. B., Ed. Organic Molecular Photophysics; Wiley: New York, 1975; pp 409-613. (b) Nemzek, T. L.; Ware, W. R. J. Chem. Phys. 1975, 62, 477. (c) Sung, J.; Shin, K. J.; Lee, S. Chem. Phys. 1992, 167, 17.

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Figure 6. (a) Stern-Volmer (SV) curves of the MWS for various values of κ. The value of signal attenuation factor γ is set equal to 0.5. At high analyte density regime the SV curves have negative curvatures for every κ value. (b) The curvatures of SV curves shown in (a). If γ(κ + 1) is greater than 2, the curvature of the SV curve of the MWS changes its sign from positive to negative with increasing [A]; otherwise, the curvature is always negative for any value of [A].

Figure 7. ln(I0/〈I〉) as a function of ln[(1 + [A]/A1/2)/(1 + (1 - γ)[A]/ A1/2)] for various values of κ. The data depicted here are the same as those in Figure 6. ln(I0/〈I〉) is linearly proportional to ln[(1 + [A]/ A1/2)/(1 + (1 - γ)[A]/A1/2)] with the slope being κ.

defined by -(∂I/∂[A])∆[A]min ) mi0 where mi0 denotes the minimum detectable signal intensity change of the signal detector. With the latter definition and eq 2, we obtain the expression for the sensitivity S of the MWS to change of the analyte concentration as

S)

I0/〈I 〉 κγL A1/2m (1 + [A]/A1/2)[1 + (1 - γ)[A]/A1/2]

(12)

with 〈I〉 given in eq 2. The sensitivity S of the imperfect MWS given in eq 12 is a nonmonotonic function of κ as shown in Figure 8, and it becomes the maximum when κ is equal to κ* (≡ 1/ln{(1 + [A]/A1/2)/[1 + (1 - γ)[A]/A1/2]}). In the low analyte density limit κ* becomes infinite so that sensitivity, S, to the change of analyte concentration increases with κ monotonously. However, as the value of [A]/A1/2 increases, κ* gets smaller as shown in Figure 8, so the MWS with a greater number of receptors does not always have a greater sensitivity to the change of analyte density. It should be mentioned that the nonmonotonic dependence of S(κ) on κ does not result from the

different from those of turn-off-type MWS.8,35 Quantitative models for the turn-on-type MWS are currently under our investigation.

Figure 8. Sensitivity S as a function of κ for various values of analyte density, [A]. The sensitivity is a nonmonotonic function of κ with the maximum at critical value κ* defined in the text. The sensitivity S of the MWS decreases with analyte density. The values of γ and L/m used are given by 0.9 and 100, respectively.

imperfect quenching effects of the nonideal MWS; it is still there in the corresponding plot of the ideal MWS as long as the MWS is a turn-off type. Note that, in the turn-off-type MWS, one analyte binding to the molecular wire tends to block the effects of other analyte bindings on the signal of the MWS. In the present work, we consider a simple model of the imperfect MWS in which the signal intensity from a molecular wire reduces by a constant signal attenuation factor γ upon each analyte binding to receptors in the molecular wire. In practice, nonideal behaviors of the MWS can be more complex due to additional sources of nonideality. Examples for the latter include the binding-site-dependent signal attenuation, the electron transfer between molecular wires, the heterogeneity in the structure and binding affinities of molecular wires, and so on. However, one can generalize the present model to encompass more complex types of nonideal MWS.34 Finally, it should be mentioned that there are multiple examples of turn-on-type MWS whose sensitivity curves and signal responses to analyte binding are qualitatively (34) See also the NONIDEAL EFFECTS section in ref 15.

CONCLUSION We propose a theoretical model for the real polyreceptor MWS, in which the conductance of a molecular wire may reduce significantly upon analyte binding to one of the receptors coupled to the molecular wire but may not vanish as completely as assumed in the ideal MWS model, previously proposed. For the imperfect MWS model, we obtain the exact relationship between analyte concentration and the sensory signal intensity, which is given in eq 2. We find that the SV curve of the imperfect MWS can have a negative curvature, whereas that of the ideal MWS always has a positive curvature. The feature in signal profiles of the imperfect MWS model can also be identified in previously reported experimental data. In spite of the nonideality, the polyreceptor MWS can have a lower detection limit and a greater sensitivity to the change of analyte density than the ideal monoreceptor counterpart has. We establish the conditions that should be satisfied for the nonideal MWS to have performance superior to the ideal monoreceptor sensor. ACKNOWLEDGMENT The authors gratefully acknowledge Professor T. M. Swager for helpful comments. This work was supported by a Korea Research Foundation Grant funded by the Korean Government (MOEHRD) (KRF-2005-070-C00065) and by the Korea Science and Engineering Foundation (KOSEF) funded by the Korea government (MOST) (Grant No. R01-2005-000-10558-0). SUPPORTING INFORMATION AVAILABLE Additional information as noted in text. This material is available free of charge via the Internet at http://pubs.acs.org. Received for review August 15, 2008. Accepted November 3, 2008. AC801715X (35) Crawford, K. B.; Goldfinger, M. B.; Swager, T. M. J. Am. Chem. Soc. 1998, 120, 5187.

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