- where n_tdenotes the total number of target molecules, n_b(i) and n_b(i+1) are the numbers of bound target molecules at t=iΔt and t=(i+1)Δt, respectively, and where p_b(i) and p_r(i) denote the probabilities of a target molecule binding to and releasing from a capturing probe during the i^thtime interval, respectively. Hence,

\begin{matrix} \frac{n_{b} (i + 1) - n_{b} (i)}{Δ t} = [n_{t} - n_{b} (i)] p_{b} (i) - n_{b} (i) p_{r} (i) . & (1) \end{matrix}

It is reasonable to assume that the probability of the target release does not change between time intervals, i.e., p_r(i)=p_r, for all i. On the other hand, the probability of forming a target-probe pair depends on the availability of the probes on the surface of the array. If we denote the number of probes in a spot by n_p, then we can model this probability as

\begin{matrix} p_{b} (i) = (1 - \frac{n_{b} (i)}{n_{p}}) - p_{b} = \frac{n_{p} - n_{b} (i)}{n_{p}} p_{b}, & (2) \end{matrix}

- where p_bdenotes the probability of forming a target-probe pair assuming an unlimited abundance of probes.

By combining (1) and (2) and letting Δt→0, we arrive to

\begin{matrix} \frac{d n_{b}}{d t} = (n_{t} - n_{b}) \frac{n_{p} - n_{b}}{n_{p}} p_{p} - n_{b} p_{r} = n_{t} p_{b} - [(1 + \frac{n_{t}}{n_{p}}) p_{b} + p_{r}] n_{b} + \frac{p_{b}}{n_{p}} {n_{b}}^{2} & (3) \end{matrix}

Note that in (3), only nb=nb(t), while all other quantities are constant parameters, albeit unknown. Before

proceeding any further, we will find it useful to denote

\begin{matrix} α = (1 + \frac{n_{t}}{n_{p}}) p_{b} + p_{r}, β = n_{t} p_{b}, γ = \frac{P^{b}}{n_{p}} . & (4) \end{matrix}

Clearly, from (4),

p_{b} = \frac{β}{n_{t}}, n_{p} = \frac{p_{b}}{γ}, p_{r} = α - (1 + \frac{n_{t}}{n_{p}}) p_{b} .

Using (4), we can write (3) as

\begin{matrix} \frac{{dn}_{b}}{d t} = β - α n_{b} + γ n_{b}^{2} = γ (n_{b} - λ_{1}) (n_{b} - λ_{2}), & (5) \end{matrix}

where λ₁and λ₂are introduced for convenience and denote the roots of

β−αn_b+γn_b²=0.

Note that γ=β/(λ₁λ₂). The solution to (5) is found as

n_{b} (t) = λ_{1} + \frac{λ_{1} (λ_{1} - λ_{2})}{λ_{2} e^{β (\frac{1}{λ_{1}} - \frac{1}{λ_{2}}) t} - λ_{1}} .

We should point out that (3) describes the change in the amount of target molecules, nb, captured by the probes in a single probe spot of the microarray. Similar equations, possibly with different values of the parameters np, nt, pb, and pr, hold for other spots and other targets.

Estimating Parameters of the Model

The following is an outline of a procedure for estimation of the parameters. Ultimately, by observing the hybridization process, nt, np, pb, and pr may be obtained. However, we do not always have direct access to nb(t) in (6), but rather to yb(t)=knb(t), where k denotes a transduction coefficient. In particular, we observe

\begin{matrix} y_{b} (t) = λ_{1}^{*} + \frac{λ_{1}^{*} (λ_{1}^{*} - λ_{2}^{*})}{λ_{2}^{*} e^{β (\frac{1}{λ_{1}^{*}} - \frac{1}{λ_{2}^{*}}) t} - λ_{1}^{*}}, & (7) \end{matrix}

where λ₁*=kλ₁, λ₂*=kλ₂, and β*=kβ.

For convenience, we also introduce

\begin{matrix} γ^{*} = \frac{β^{*}}{λ_{1}^{*} λ_{2}^{*}} = \frac{γ}{k}, α^{*} = γ^{*} (λ_{1}^{*} + λ_{2}^{*}) = α . & (8) \end{matrix}

From (5), it follows that

\begin{matrix} β^{*} = \frac{d y_{b}}{d t} ❘_{t = 0} . & (9) \end{matrix}

Assume, without a loss of generality, that λ₁* is the smaller and λ₂* the larger of the two, i.e., λ₁*=min(λ₁, λ₂), and λ₂*=max(λ₁, λ₂). From (7), we find the steady-state of yb(t),

λ₁*=limy_b(t),t→∞. (10)

So, from (9) and (10) we can determine β* and λ₁*, two out of the three parameters in (7). To find the remaining one, λ₂*, one needs to fit the curve (7) to the experimental data.

Having determined β*, λ₁*, and λ₂*, we use (8) to obtain α* and γ*. Then, we should use (4) to obtain pb, pr, np, and nt from α*, β*, and γ*. However, (4) gives us only 3 equations while there are 4 unknowns that need to be determined. Therefore, we need at least 2 different experiments to find all of the desired parameters. Assume that the arrays and the conditions in the two experiments are the same except for the target amounts applied. Denote the target amounts by nt₁and nt₂; on the other hand, pb and pr remain the same in the two experiments. Let the first experiment yield α₁*, β₁*, and γ₁*, and the second one yield α₂*, β₂*, and γ₂* (we note that γ₁*=γ₂*). Then it can be shown that

\begin{matrix} p_{b} = \frac{β_{1}^{*} γ_{1}^{*} - β_{2}^{*} γ_{2}^{*}}{α_{1}^{*} - α_{2}^{*}}, & (11) \\ and \\ p_{r} = α_{1}^{*} - p_{b} - \frac{β_{1}^{*} γ_{1}^{*}}{p_{b}} . & (12) \end{matrix}

Moreover,

\begin{matrix} n_{p} = \frac{p_{b}}{k γ_{1}^{*}}, & (13) \\ and \\ n_{t_{1}} = \frac{β_{1}^{*} γ_{1}^{*}}{p_{b}^{}} n_{p}, n_{t_{2}} = \frac{β_{2}^{*} γ_{2}^{*}}{p_{b}^{}} n_{p} . & (14) \end{matrix}

We note that quantities (13)-(14) are known within the transduction coefficient k, where k=yb(0)/np. To find k and thus unambiguously quantify np, nt1, and nt2, we need to perform a calibration experiment (i.e., an experiment with a known amount of targets nt).

Experiments are designed to test the validity of the proposed model and demonstrate the parameter estimation procedure. To this end, two DNA microarray experiments are performed. The custom 8-by-9 arrays contain 25mer probes printed in 3 different probe densities. The targets are Ambion mRNA Spikes, applied to the arrays with different concentrations. The concentrations used in the two experiments are 80 ng/50 μl and 16 ng/50 μl. The signal measured in the first experiment, where 80 ng of the target is applied to the array, is shown inFIG. 17. The smooth line shown in the same figure represents the fit obtained according to (7). In the second experiment, 16 ng of the target is applied to the array. The measured signal, and the corresponding fit obtained according to (7), are both shown inFIG. 18.

Applying (11)-(14), we obtain

p_b=1.9×10⁻³, p_r=2.99×10⁻⁵.

Furthermore, we find that

n_t1/n_t2=β₁*/β₂*=3.75. (15)

Note that the above ratio is relatively close to its true value, 80/16=5. Finally, assuming that one of the experiments is used for calibration, we find that the value of the transduction coefficient is k=4.1×10⁻⁴, and that the number of probe molecules in the observed probe spots is np=1.6×10⁻¹¹.

Example 3

This example shows how the methods and systems of the present disclosure can be used for measurement of gene expression. Real-time microarray technology can measure, for example, expression level differences for different cell types or tissues, distinct developmental stages, cancerous versus normal cells or tissues, treated versus untreated cells or tissues, mutant versus wild-type cells, tissues or organisms.

The gene expression profiles of inflorescences of theArabidopsis thalianafloral homeotic mutants apetala1, apetala2, apetala3, pistillata, and agamous can be compared with that of wild-type plants. By combining the data sets from the individual mutant/wild type comparisons, it is possible to identify a large number of genes that are, within flowers, predicted to be specifically or at least predominantly expressed in one type of floral organ. For each sample, floral buds from approximately 50 plants are collected, and RNA is isolated from 100 mg of tissue with the RNeasy RNA Isolation Kit (Qiagen). To prepare labeled target material from those samples, an in vitro transcription amplification method followed by aminoallyl-UTP-mediated labeling is used. In brief, first and second strand cDNA is synthesized from 3 μg of total RNA using a polyA-primer with a T7 promoter sequence. Then in vitro transcription is performed using the Megascript T7 kit (Ambion), in the presence of aminoallyl-UTP and of a reduced amount of UTP, to incorporate the modified aaUTP into the aRNA during the transcription process. Finally, dye molecules (Cy3 Mono-Reactive Dye, Cy5 Mono-Reactive Dye, Amersham) are coupled to the amplified RNA and the dye-labeled RNA is fragmented before hybridization. These and similar protocols are well established in the microarray field and are well known to those skilled in the art, and commercial kits are available for cDNA synthesis, in vitro transcription amplification, and amino-allyl labeling, such as the Amino Allyl MessageAmp™ II aRNA Amplification Kit, from Ambion. Aminoallyl UTP contains a reactive primary amino group on the C5 position of uracil that can be chemically coupled to N-hydroxysuccinimidyl ester-derivatized reactive dyes (NHS ester dyes), in a simple efficient reaction.

Amine-reactive quenchers are commercially available, forexample QSY® 9 carboxylic acid, succinimidyl ester, from Invitrogen. The real-time microarray technology is used with minimal modifications to the sample preparation and labeling procedures: namely, with the simple substitution of the QSY-9 ester for the Cy-dye ester. In this case, total RNA samples are prepared as described above. The purified total RNA is then amplified using the Amino Allyl MessageAmp™ II aRNA Amplification Kit, from Ambion, and the resulting aRNA is labeled with QSY9. This labeled RNA population is then used in hybridization withArabidopsisreal-time microarrays. Such arrays consist of an arranged collection of probes that correspond to all (or a subset of) the genes in theArabidopsisgenome. The oligonucleotide probes are labeled with a fluorescent moiety (such as Cy3) and printed by contact deposition onto CodeLink slides (GE Healthcare), which are processed and blocked after printing following manufacturer's instructions.

Example 4

This example relates to using an algorithm to measure cross-hybridization. A variety of different techniques to recover the signal, including, but not limited to, total least squares, ESPRIT, and Prony's method, (see Dowling et. al. IEEE Trans. on Antennas and Propag., vol. 42, no. 5, 1994 and van der Veen et al. Proc. of the IEEE, 81(9):1277-1308, 1993).

In this example we study the performance of one such algorithm in simulation and illustrate the results inFIG. 19. In particular, we consider the so-called total least squares (TLS) algorithm in the situation where two target analytes bind to the same probe spot—one due to hybridization, and the other due to cross-hybridization. Parameters of the system (probabilities of hybridization, cross-hybridization, release, etc.) are chosen so as to mimic realistic experimental scenarios. The probability of hybridization is assumed to be 5 times greater than the probability of cross-hybridization (i.e., p_h/p_c=5). The number of hybridizing target is $n_h=10⁹$, while the number of cross-hybridizing molecules is varied. InFIG. 19, we plot the relative mean-square error of estimating n_h(averaged over many realizations of noise) as a function of the ratio n_h/n_c. The simulation results indicate potentially successful suppression of cross-hybridization over 3 orders of magnitude of n_h/n_c.

Example 5

This example illustrates how an optical filter layer in the form of a band-pass filter on the surface of the CMOS chip enhances optical performance of the integrated biosensor array. A challenge of designing any fluorescent-based detector is the proper optical excitation and detection of fluorescent labels.FIG. 33 illustrates the absorption and emission spectra of Cy3 molecule which is an example fluorophore of a system. The absorbed photon density for Cy3, denoted by, exposed to the incident photon flux, obeys the Beer-Lambert law. For a thin layer of diluted absorbing media with Cy3 as in microarray applications we have

A=F_X[1−e^−a⁰^(λ)N]≈F_Xa₀(λ)N (16)

where a₀(λ) and N are the extinction coefficient in wavelength λ and surface concentration of Cy3 respectively. Considering Q_Y, the fluorescence quantum yield of Cy3, we can calculate I_E, the total emitted photons per surface area by

I_E=Q_YA≈Q_YF_Xa₀(λ)N (17)

As evident inFIG. 33, the photon emission is, to first order, proportional to N which is parameter of interest in microarrays. The major impediment for measuring I_Eand therefore N, is the presence of F_Xduring detection. Although F_Xhas a different wavelength from I_E, it is very typical for F_Xto be to 4-5 orders of magnitude larger than I_Ein microarray applications. To block F_Xin our system, a multi-layer dielectric Fabry-Perot optical band-pass filter has been fabricated on the surface of a CMOS chip. This emission filter can block F_Xby 100 dB in the Cy3 excitation band of 545-550 nm, while only loosing 25% of the signal in the pass-band.

Example 6

This example describes the construction of a fully integrated biosensor array of the present disclosure. To design the CMOS photo-detector a Nwell/Psub photodiode array is used in the 0.35 μm CMOS process. Each diode is 50 μm×50 μm and the array pitch is 250 μm. This dimension is compatible with commercial microarray specifications and also minimizes the optical cross-talk between photodiodes while providing sufficient space to integrate in-pixel a photocurrent detector and an analog to digital converter (ADC).

Most of the CMOS image sensors use direction integration, where the photocurrent is directly integrated over a reverse bias photodiode capacitor. In the design, a capacitive transimpedance amplifier (CTIA) in the pixel is used as a photocurrent integrator. Comparing with a CTIA, a direct integrator suffers from the junction capacitance variation as the reverse bias voltage changes depending on the output signal level. A CTIA does not have any such problems since the photodiode bias is regulated by an operational transimpedance amplifier (OTA) used in the CTIA and it is set to VR1 as shown inFIG. 34. Photocurrent input in the system is integrated using the feedback capacitor.

In order to take full advantage of integration capability of CMOS, an ADC is also included in the pixel level. Pixel level processing relaxes the speed requirement of ADC while removing all analog signal bus lines in the array and significantly reduces the cross-talk issues. To suppress the low frequency noise of the comparator, a chopper stabilized preamplifier has been implemented with overall voltage gain of approximately 60 dB gain.

A measurement of the concentration of Cy3 fluorophores on the surface of the chip using our integrated microarray system can be performed to determine the performance compatibility of this system with fluorescent-based microarrays.

FIG. 35 shows the die photo and the I/O pin allocation of the integrated 7 by 8 pixel biosensor array. Auxiliary circuits beside the biosensor array include row and column decoder and column switches which are necessary for the functionality of the array.

While preferred embodiments of the present invention have been shown and described herein, it will be obvious to those skilled in the art that such embodiments are provided by way of example only. Numerous variations, changes, and substitutions will now occur to those skilled in the art without departing from the invention. It should be understood that various alternatives to the embodiments of the invention described herein may be employed in practicing the invention. It is intended that the following claims define the scope of the invention and that methods and structures within the scope of these claims and their equivalents be covered thereby.