Quantitative Relationship between Analyte Concentration and

Quantitative Relationship between Analyte Concentration and Amplified Signal Intensity of a Molecular Wire Sensor. Jaeyoung Sung, and Robert J. Silbey...
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Anal. Chem. 2005, 77, 6169-6173

Quantitative Relationship between Analyte Concentration and Amplified Signal Intensity of a Molecular Wire Sensor Jaeyoung Sung† and Robert J. Silbey*,‡

Department of Chemistry, Chung-Ang University, Seoul 156-756, Korea, and Department of Chemistry, Massachusetts Institue of Technology, Cambridge Massachusetts 02139

The molecular wire approach has recently been proposed as a method to enhance the sensitivity of traditional chemosensors. In this paper, we present the exact quantitative relationship between analyte concentration and the signal for the ideal molecular wire sensor (MWS). The signal profile of a MWS differs from that of traditional chemosensors in that its Stern-Volmer curve has a positive curvature that increases with the number K of receptor units in each molecular wire composing the MWS. We find that the sensitivity of the MWS to the change of analyte concentration increases with K if K is less than a critical value K*; otherwise, it becomes a decreasing function of K. We also briefly comment on aspects of nonideal sensors. Quantitative detection of ultratrace analyte molecules is the ultimate goal of chemosensors. Of the previously reported methods of sensory signal amplification to achieve this goal,1-8 the molecular wire approach1 proposed by Swager and co-workers is particularly attractive because it offers a general method of sensory signal amplification applicable to a variety of analytereceptor systems. Applications of the molecular wire approach have been made recently in the detection of toxic ions,9 explosive compounds,10 nucleic acid,11 and so on.12-15 A molecular wire sensor (MWS) is composed of a number of molecular wires that are often conjugated semiconducting organic polymers. Each of the molecular wires is coupled to a number, κ, * To whom correspondence should be addressed. E-mail: [email protected]. † Chung-Ang University. ‡ Massachusetts Institue 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. (2) Blaedel, W. J.; Boguslaski, R. C. Anal. Chem. 1978, 50, 1026. (3) Shibata, T.; et al. J. Am. Chem. Soc. 1998, 120, 12157. (4) Ehret, A.; Spitler, M. T.; Stuhl, L. S. Comments Inorg. Chem. 2002, 23, 275. (5) Hartig, J. S.; Grune, I.; Najafi-Shoushtari, S. H.; Famulok, M. J. Am. Chem. Soc. 2004, 126, 722. (6) Brunner, J.; Mokhir, A.; Kremer, R. J. Am. Chem. Soc. 2003, 125, 12410. (7) Wu, Q.; Anslyn, E. V. J. Am. Chem. Soc. 2004, 126, 14682. (8) Ellington, A. D.; Levy, M. Proc. Natl. Acad. Sci. U.S.A. 2003, 100, 6416. (9) Kim, T.; Swager, T. M. Angew. Chem., Int. Ed. 2003, 42, 4803. (10) Kuroda, K.; Swager, T. M. Macromol. Symp. 2003, 201, 127. (11) Dore´, K.; et al. J. Am. Chem. Soc. 2004, 126, 4240. (12) Lee, D.; Swager, T. M. J. Am. Chem. Soc. 2003, 125, 6870. (13) Kim, Y.; Zhu, Z.; Swager, T. M. J. Am. Chem. Soc. 2004, 126, 452. (14) Zhou, X. H.; Yan, J. C.; Pei, J. Macromolecules 2004, 37, 7078. (15) Pohl, R.; et al. Chem. Commun. 2004, 1282. 10.1021/ac050659t CCC: $30.25 Published on Web 08/31/2005

© 2005 American Chemical Society

of receptor units as shown in Figure 1. The interactions between the molecular wire and the coupled receptor units are designed to develop a collective response from the entire molecular wire upon analyte binding to any one of the receptors on the molecular wire. For example, in a turnoff type MWS, each molecular wire in the MWS contributes a unit amount of signal, e.g., fluorescence or electric current, to the total signal of the MWS when none of receptors on the molecular wire is occupied, but stops signaling upon binding of analyte molecules to one or more of its κ receptor units. It is obvious that the response of such a polyreceptor molecular wire system is more sensitive to the presence of analyte molecules than that of an analogous monoreceptor system. Indeed, the signal amplification scheme described in the previous paragraph has been demonstrated for a real polyreceptor conjugated polymer system such as polypyrrole,12 poly(p-phenylenevinylene),13 polythiophene,16 polyaniline,17 and poly(phenyleneethynylene),18 to name a few,1 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. A variety of different types of receptors have been employed in MWS, which cannot be covered completely here. However, in many 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, thus reducing 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 ultimate signal amplification in a 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; therefore, chemical tactics have been developed to introduce steric repulsions between the polyreceptor polymer chains to minimize the (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) Zhou, Q.; Swager, T. M. J. Am. Chem. Soc. 1995, 117, 7017. (19) Zhou, Q.; Swager, T. M. J. Am. Chem. Soc. 1995, 117, 12593.

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Figure 1. Schematic illustration of a molecular wire sensor. A single molecular wire comprises κ number of receptor units, and L number of molecular wires compose a molecular wires 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.

communication between them.12,20,21Although the signal amplification concept of the molecular wire approach has been demonstrated in a variety of real systems and proved to be useful in ultrasensitive analyte detection, the quantitative relationship between signal intensity of a MWS and analyte concentration has not yet been established. Here, for the representative model of the MWS mentioned above, we derive an exact quantitative relationship between analyte concentration and the signal intensity of the MWS by considering the grand canonical ensemble22 of analyte molecules on a MWS in equilibrium with the analyte solution. The result shows that the dependence of the signal intensity of a MWS on analyte concentration has many features qualitatively different from those of a traditional chemosensor. The Stern-Volmer curve of a MWS has a positive curvature whose magnitude increases with the number κ of receptor units in each molecular wire. As expected, a MWS with more receptors in each molecular wire can detect the analyte molecule at lower concentration. In comparison, we find the sensitivity of the MWS to the change of analyte concentration has a nonmonotonic dependence on the number κ of receptors units in each molecular wire: it increases with κ if κ is less than a critical value; otherwise, it decreases with κ. We also find that the range of analyte concentrations that can be quantitatively measured by a MWS is strongly dependent on the structural features of the MWS, such as the (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) McQuarrie, D. A. Statistical Mechanics; Harper & Row: New York, 1976; Chapter 3.

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number of molecular wires composing the MWS and the number of receptors in each molecular wire. FORMULATION OF THE PROBLEM AND THE KEY RESULT The signal intensity, I of the MWS model described above is proportional to the number of molecular wires with unoccupied receptors only, among the L molecular wires in total. The mathematical expression for this is L

I(s) )

∑i δ j)1

0 sj 0

(1)

where i0 denotes the intensity of the sensory signal from each molecular wire on which none of receptor units is bound to analyte molecule and sj denotes the number of receptor units occupied by analyte molecules on the jth polymer chain of the MWS. s is the L-dimensional vector whose jth component is sj. In eq 1, δsj0 is the Kronecker delta function whose value is unity only if sj is 0, but zero otherwise. Note that sj is a stochastic variable ranging from zero to κ, 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 zero for high analyte concentration limit. The quantity of primary 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〉 ) ∑sI(s)P(s). Here P(s) denotes the probability that we find a configuration s[≡ {s1,s2, ...,sL}] over

molecular wire is the only mechanism that turns off the signal from the molecular wire, Stern-Volmer curves of the MWS with κ ) 1 (i.e., traditional chemosensors) are linear.23 The SternVolmer constant, K0SV, defined by K0SV ≡ lim[A]f0((I0/〈I 〉) - 1)/ [A], quantifies the sensitivity of sensory signal intensity for analyte concentration change in the low concentration limit, and as shown in Figure 2, the Stern-Volmer constant, K0SV, increases with κ,

K0SV ) κ/A1/2

Figure 2. Dependence of 〈I〉 on κ and [A]. (a) Plot of I0/〈I〉 vs [A]/A1/2 for various values of κ; (b) the same plotted in a log-log form.

the L molecular wire, and ∑s denotes the sum over all possible configurations. Given that analyte molecules in the sample are in equilibrium with those bound to receptors in the MWS, one can calculate P(s) as a function of analyte concentration [A] and the exact relation between 〈I〉 and [A] for the MWS described above by the application of equilibrium statistical thermodynamics (see Supporting Information). The result is

〈I〉 )

I0 (1 + [A]/A1/2)κ

(3)

Thus, compared to the conventional chemosensor corresponding to the case with κ ) 1, the MWS with κ receptor units in each single molecular wire is κ times more sensitive to the change of analyte concentration, when the analyte concentration is much lower than A1/2. The detection range of a MWS is not only dependent on the resolution of the signal detector of our sensor but also on the number κ of receptors on each molecular wire. If the minimum detectable change in the signal intensity is given by mi0, the minimum amount of detectable analyte concentration [A]min of the MWS is given by

[A]min m ) 1A1/2 L

(

-1/κ

)

-1

(4)

(2)

where I0 is the maximum signal intensity defined by i0L. In eq 2, A1/2 denotes the value of the analyte concentration at which the probability that a receptor unit on the MWS is occupied is half, and it is inversely proportional to the binding affinity of an analyte molecule to the receptor unit coupled to the molecular wire. Note that the signal profile given in eq 2 reduces to that of traditional chemosensors when the number κ of receptors in a single polymer chain is unity; i.e., when κ ) 1. As discussed in the following section, the detection capability of a MWS for analyte molecules always increases with κ, as expected; however, the sensitivity of the MWS to the change of analyte concentration, in comparison, is a increasing function of κ only if κ < [ln(1 + [A]/A1/2)]-1 ≡ κ* and is a decreasing function of κ for κ > κ*. Thus, the sensitivity of a MWS to the change of analyte concentration is greater than that of traditional chemosensor for 2 e κ e κ*. In the large κ limit, although the MWS may be an excellent detector of an analyte, it is insensitive to changes in analyte concentration. The range of analyte concentration that can be quantified by a MWS turns out to be dependent on the two key structural parameters of the MWS, the number of receptors in each molecular wire, κ, and the number of molecular wires, L, in the MWS (see eqs 4 and 6). Therefore, by adjusting these structural parameters of a MWS, one can control the range of useful quantitative detection of the MWS. DISCUSSION Equation 2 indicates that the Stern-Volmer curve or the dependence of I0/〈I〉 on [A] of a MWS with κ larger than 1 has a positive curvature whose magnitude increases with κ as shown in Figure 2. Provided that analyte binding to receptors on a

If the resolution of our signal detector is good enough to detect the signal change from single molecular wire, we should set m ) 1 in eq 4. When m/L is small enough, eq 4 becomes [A]min/A1/2 = m/(κL), which indicates that, for a given value of m, the MWS with larger values of κ and L is more sensitive to the presence of analyte molecule. However, when the analyte concentration is high enough, the sensitivity of the MWS to the change of analyte concentration decreases with κ. We will define the sensitivity S([A]) of a sensor as the inverse of the minimum analyte concentration change ∆[A]min from [A] that can be detected by the sensor. Mathematically, ∆[A]min is defined by -(∂I/∂[A])∆[A]min ) mi0. From eq 2, we obtain the expression for the sensitivity of the MWS to change of analyte concentration as

S([A]) )

(

)

[A] κ 1+ A1/2 A1/2

-(1+κ)

L m

(5)

The sensitivity S of the MWS given in eq 5 is a nonmonotonic function of κ, as shown in Figure 3. If we designate 1/ln(1 + [A]/A1/2) as κ*, the sensitivity increases with κ if κ < κ* but decreases with κ for κ > κ*. In the low analyte concentration limit, κ* becomes large and the sensitivity always increases with κ. However, as the value of [A]/A1/2 gets larger, κ* decreases so that the sensitivity becomes a decreasing function of κ as κ increases. (23) Stern-Volmer curves of a traditional sensor system can have positive curvature when the signal of the sensor can be turned off by more than one quenching mechanism. (a) Organic molecular photophysics; Birks, J. B., Ed.; 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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Equation 7 indicates that the relative fluctuation of signal intensity decreases with L but increases with κ. In practice, there can be other sources of signal fluctuations and the magnitude of the relative signal fluctuation will be greater than that given in eq 7.

Figure 3. Sensitivity, S, as a function of κ for various concentrations, [A].

Figure 4. Maximum detectable concentration, Amax, as a function of κ for different values of L.

On the other hand, the upper bound [A]max of the analyte concentration that can be quantitatively measured by the MWS is found to be

[A]max L ) A1/2 m

1/κ

()

-1

(6)

For analyte concentration larger than [A]max, the average signal intensity 〈I〉 of the MWS given in eq 2 becomes smaller than our detector resolution, mi0 so that it gets difficult to determine [A] quantitatively from a measurement of 〈I〉. In this case, the MWS may serve as a good detector for an analyte molecule but not as a good sensor for quantitative measurements. Note that [A]max/ A1/2 given in eq 6 increases with the number L of molecular wires composing the MWS, but decreases with the number κ of receptors in a molecular wire, as shown in Figure 4. In the large κ limit, eqs 4 and 6 indicate that [A]min ≈ [A]max = 0, which indicates that a MWS with infinite κ is an ideal detector for an analyte, but it does not give quantitative information about analyte concentration. This is an expected result since, in the large κ limit, even a minute amount of solute can switch off the signal from molecular wires, provided that the number of solute molecules is still much greater than that of molecular wires. In the Supporting Information, we present the derivation of the expression for the variance σ2I [≡ 〈(I(s) - 〈I〉)2〉] of the signal intensity I(s) resulting from the equilibrium fluctuation in the number of the analyte molecules bound to the receptors of the MWS:

σI 〈I〉 6172

)

1 xL

x(

1+

[A] A1/2

)

κ

-1

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(7)

NONIDEAL EFFECTS In the present work, we have considered an ideal MWS in which a signal from a molecular wire in MWS is switched off completely upon binding of an analyte molecule to any of receptor units. However, a large class of molecular wires would not have such an ideal turnoff behavior upon analyte binding; instead, the signal intensity of the molecular wire would reduce gradually as the number of analyte-bound receptor units increases. It is expected that both the minimum and maximum detectable analyte density [A]min/A1/2 and [A]max/A1/2 of the nonideal molecular wire would be larger than those of the ideal molecular wire, given in eqs 4 and 6. However, we expect that the sensitivity S([A]) of the nonideal MWS to the change of analyte concentration would have a nonmonotonic dependence on κ, qualitatively the same as that of the ideal MWS. For a simple model of the nonideal MWS, one can still obtain an exact relation between sensory signal intensity and the analyte density by straightforward generalization of the present work. The presentation of this model and its comparison to experimental results will be published in a separate article. It is possible that mechanisms exist that cause the signal of one molecular wire to be influenced by the state of the other molecular wires composing a MWS, in which case, the theory becomes more complex. However, because the signal intensity I(s) of the MWS is a function of the configuration vector s of the number of analyte-bound receptor units, one can calculate the moments 〈In〉 of the signal intensity numerically using the probability distribution P(s) of s given in eqs A5 and A9 in Supporting Information. One can think of even more complex situations in which strong interactions between polymer chains in a MWS make the number of analyte molecules bound in one molecular wire correlated to those bound in others, in which case P(s) is no longer given in a factorized form as in eqs A5 and A9, and the problem becomes more difficult to handle. Fortunately, ingenious chemical tactics have been developed to minimize the above-mentioned undesirable interactions between polymer chains in designing a MWS.12,20,21Although we here assume that the number of receptor units in each molecular wire is a constant κ, it can also be a stochastic variable with a probability distribution. The results presented in this work can be easily generalized to the latter case by performing an average over the distribution of κ. It should be mentioned that the key results, eqs 2 and 7, are valid only when we can neglect interactions between analyte molecules in solution compared to interactions between analyte molecules and solvent molecules. We considered a turnoff type MWS in the present work; however, by following a similar derivation, one can also obtain the corresponding results for a turn-on type MWS,9 which will be published separately. SUMMARY In this paper, we derived the exact relationship, eq 2, between the analyte concentration and the sensor signal intensity for a representative model of an ideal turnoff type MWS. The result shows that the Stern-Volmer curve of the MWS will have a

positive curvature whose magnitude increases with the number κ of receptors coupled to each molecular wire. For the case with κ ) 1, we recover the signal profile of traditional chemosensors along with the linear Stern-Volmer curve. Equations 4 and 5 indicate that, for the case where the analyte concentration is much lower than the half-coverage concentration, the MWS with a larger value of κ will be more sensitive to the presence of analyte molecules and to the change of analyte concentration than are traditional chemosensors. However, we also find that the sensitivity of the MWS to the change in analyte concentration is a nonmonotonic function of κ, given in eq 5: the sensitivity increases with κ for κ smaller than a critical value κ* but decreases with κ when κ is larger than κ*. The magnitude of κ* is found to be κ* ) 1/ln(1 + [A]/A1/2), dependent on the analyte concentration. We also found (eqs 4 and 6) the lower and upper limits of analyte concentration that can be quantitatively measured by the MWS.

In addition, we found that the relative magnitude of inherent signal fluctuation decreases with the number L of molecular wires composing MWS but decreases with κ. ACKNOWLEDGMENT This Research was supported by an NSF grant to MIT, CHE0306287. 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 April 18, 2005. Accepted July 29, 2005. AC050659T

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