Applicability of Birth–Death Markov Modeling for Single-Molecule Counting Using Single-Walled Carbon Nanotube Fluorescent Sensor Arrays

Abstract

A n important development in nanotechnology is the emergence of sensor transducers capable of single-molecule resolution at room temperature. SWNTs alone account for at least three sensor types capable of this important property. One sensor type is an electrically contacted field effect transistor (FET) with a single electrochemically induced catalytic defect, inducing a defection in the channel current in response to reactions at the defect site. 1À3 A second sensor type uses the interior of the SWNT as a nanopore for Coulter detection of single cations. 4 A third sensor type consists of a near-infrared fluorescent semiconducting SWNT where adsorption and desorption of a fluorescence quencher cause discretized and stochastic fluctuations of the intensity from the single nanotube. We have developed several near-infrared fluorescent SWNT sensors selective for glucose, 5À8 DNA, 9À12 ATP, 13 H 2 O 2 , 14,15 and recently NO. 16,17 For H 2 O 2 and NO, we first introduced the idea of a selective sensor interface able to count single analyte molecules, following the pioneering experiments of Cognet and Weismann demonstrating stochastic fluorescence quenching of SWNT excitons. 18 A central challenge in the theory of these single-molecule sensors is how to relate intensity fluctuations to the local analyte concentration of interest and/or its flux to the sensor. We have used recently, without rigorous proof, a birthÀdeath Markov model to accomplish this. This Letter develops a mathematical test for the agreement of molecular adsorption dynamics for this Markov process.When molecules bind to the surface of a SWNT, the fluorescence is partially quenched, resulting in a step decrease in intensity. Similarly, the intensity increases when molecules unbind from the surface. By counting the transient change in the number of step changes in intensity, the number of adsorption and desorption events, the total number of adsorbed molecules, and the incident flux can be estimated.Exciton quenching by adsorbed analyte molecules limits the number of observable adsorption and desorption states. For a typical SWNT sensor, the number of states will be about 10, based on a mean nanotube length of 1 μm and an estimated exciton diffusion length of 100 nm. 17 More detailed physical models of exciton dynamics propose that the diffusion length is limited by pre-existing or static defects, 19 but the analysis of adsorption and desorption kinetics in this work applies to these physical models as well. Due to this small number of observable states, adsorption and desorption events will be stochastic, which results in a deviation in the observed number of bound analyte molecules from the number predicted by the continuum approximation or average sensor. These stochastic deviations can cause the apparent reaction rates to be different for various SWNTs ABSTRACT: In recent work, we have shown that d(AT) 15 DNA-wrapped single-walled carbon nanotubes (SWNTs) are able to detect the adsorption and desorption of single molecules of nitric oxide (NO) from ...

Estimation of local concentration from measurements of stochastic adsorption dynamics using carbon nanotube-based sensors

“…(12) is proportional to the local concentration of target molecules, C(t), surrounding the sensor [16]. This assumption means that accurate estimation of k' A (t) translates directly into accurate estimation of the local concentration (with known adsorption parameter k A ), which is the desired system information.…”

“…In developing a kinetic model for the SWNT sensor [16], free target molecules, A, in a surrounding liquid are assumed to adsorb onto available sites of the nanotube segment, θ, to form bound molecules, Aθ, through the reversible adsorption reaction:…”

“…The adsorption rate is generally considered to be a pseudo-first-order in the concentration of target molecules proximate to the sensor and the number of unoccupied sites, and the desorption rate is assumed to be a pseudo-first-order in the number of occupied sites [16], so the associated rates are expressed as…”

“…The birth-and-death model, which was studied by [14], provides a path to an analytic solution. The adsorption and desorption process can be considered as a birth-and-death process system [16]. Birth corresponds to a desorption event and death to an adsorption event.…”

“…The analytical form of the solution makes possible an efficient estimation of parameters appearing in the CME. Based on this idea, maximum likelihood estimation (MLE) of the adsorption rate constant for the SWNT sensor system has been proposed [15][16][17].…”