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Clinical Chemistry 50: 2254-2262, 2004. First published September 23, 2004; 10.1373/clinchem.2004.037226
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Right arrow Molecular Diagnostics and Genetics
(Clinical Chemistry. 2004;50:2254-2262.)
© 2004 American Association for Clinical Chemistry, Inc.

Molecular Diagnostics and Genetics

Hybridization Isotherms of DNA Microarrays and the Quantification of Mutation Studies

Avraham Halperin1,a, Arnaud Buhot1 and Ekaterina B. Zhulina2

1 Unité Mixte de Recherche (UMR) 5819 [Université Joseph Fourier (UJF), Centre National de la Recherche Scientifique (CNRS), Commissariat à l’Energie Atomique (CEA)], DRFMC/SI3M, CEA Grenoble, Grenoble, France.
2 Institute of Macromolecular Compounds of the Russian Academy of Sciences, Bolshoy, Russia.

aAddress correspondence to this author at: UMR 5819 (UJF, CNRS, CEA), DRFMC/SI3M, CEA Grenoble, 17 rue des Martyrs, 38054, Grenoble cedex 9, France. Fax 33-4-3878-5691; e-mail ahalperin{at}cea.fr.


   Abstract
Top
 Abstract
Introduction
 The Adsorption Isotherm
 The Two-Spot Approach
 The Two-Temperature Approach
 Discussion
 References
 
Background: Diagnostic DNA arrays for detection of point mutations as markers for cancer usually function in the presence of a large excess of wild-type DNA. This excess can give rise to false positives as a result of competitive hybridization of the wild-type target at the mutation spot. Analysis of the DNA array data is typically qualitative, aimed at establishing the presence or absence of a particular point mutation. Our theoretical approach yields methods for quantifying the analysis to obtain the ratio of concentrations of mutated and wild-type DNA.

Method: The theory is formulated in terms of the hybridization isotherms relating the hybridization fraction at the spot to the composition of the sample solutions at thermodynamic equilibrium. It focuses on samples containing an excess of single-stranded DNA and on DNA arrays with a low surface density of probes. The hybridization equilibrium constants can be obtained by the nearest-neighbor method.

Results: Two approaches allow acquisition of quantitative results from the DNA array data. In one, the signal of the mutation spot is compared with that of the wild-type spot. The implementation requires knowledge of the saturation intensity of the two spots. The second approach requires comparison of the intensity of the mutation spot at two different temperatures. In this case, knowledge of the saturation signal is not always necessary.

Conclusions: DNA arrays can be used to obtain quantitative results on the concentration ratio of mutated DNA to wild-type DNA in studies of somatic point mutations.


   Introduction
Top
 Abstract
 Introduction
The Adsorption Isotherm
 The Two-Spot Approach
 The Two-Temperature Approach
 Discussion
 References
 
Cancers are attributed to accumulation of somatic mutations (1). In turn, the mutated DNA can provide diagnostically useful molecular markers (2). Among them are point mutations, which involve a change in a single base pair in the DNA, such as those occurring in the p53 and K-ras genes (3)(4)(5). The detection of such point mutations is useful in screening for cancers (6) as well as for typing of cancers to optimize treatment protocols (7). DNA microarrays (8)(9), also called "DNA chips", are among the analytical techniques with demonstrated potential to this end (6)(10). The detection of somatic point mutations is hampered by the presence of an excess of wild-type DNA. This favors hybridization of the wild-type single-stranded DNA (ssDNA),1 with mismatched sequences causing false positives. A similar problem occurs for analysis of single-nucleotide polymorphisms in pooled samples (2). Here we present a theoretical analysis, based on equilibrium thermodynamics, of the errors introduced by such mishybridization and suggest methods for quantifying point mutations detected by DNA chips. In particular, these methods allow calculation of the ratio of concentrations of mutated and wild-type DNA in the sample. This ratio is of diagnostic interest and provides a systematic method for the minimization of false positives. The numerical calculations illustrating this approach are based on recent DNA chip studies of point mutation in the K-ras gene (6)(10).

DNA chips consist of a support surface carrying "spots" (8)(9). Each spot comprises numerous oligonucleotides of identical and known sequence, "probes", that are terminally anchored to the surface. The spots are placed in a checkered pattern such that each sequence is allocated a unique site. Each probe hybridizes preferentially with a ssDNA containing a complementary sequence, referred to as "target". In a typical experiment, the DNA microarray is immersed in a solution containing a mixture of labeled ssDNA fragments of unknown sequence. The presence of a given sequence is signaled by hybridization on the corresponding spot as monitored by correlating the strength of the label signal with the position of the spot.

False positives can occur because each probe can also hybridize with a mismatched sequence. When the DNA microarray is designed to investigate gene expression, it is possible to optimize the probe design to minimize this effect (11)(12); however, implementation of this strategy is more difficult for studies of point mutations. When studying somatic point mutations, the situation is further complicated by an excess of wild-type, nonmutated, DNA. Two observations substantiate this point. The first observation is that solid tumors are heterogeneous, containing a mixture both cancerous and healthy, stromal cells. The cancer cells are a minority, and the fraction of mutated DNA obtained from homogenized tumor biopsy can be as small as 15% (2). The second observation is that the fraction of mutated DNA is much smaller for noninvasive testing for early-stage cancers using body fluids such as urine (5), serum (4), or stool (3)(6). Because the hybridization is controlled by the law of mass action, the excess of wild-type sequence will typically contribute to the hybridization on the spots allocated to the point mutations. Accordingly, the ratio of intensities of different spots may not reflect the ratio of concentrations of DNA species in the sample. This mishybridization contribution will increase with the ratio of wild-type DNA to mutated DNA, thus diminishing the efficacy of the early-stage screening. A similar situation is encountered in the analysis of pooled samples for single-nucleotide polymorphisms.

This problem can be resolved by determining the contribution of the wild-type DNA to the signal of the mutation spots. As we shall discuss, this is possible when three conditions are fulfilled: (a) The hybridization is allowed to reach equilibrium. (b) The equilibrium hybridization isotherm, relating the hybridization fraction on the spot to the sample composition, is of the Langmuir form, i.e., x/(1 – x) = Kc, where x is the fraction of surface sites that bind a reactant of concentration c, and K is the equilibrium constant of the reaction (13). This regime is expected when the surface density of probes is sufficiently low (14). (c) The sample contains an excess of ssDNA as is the case when using asymmetric PCR amplification (15)(16) or after digestion by lambda exonuclease (17). Under these conditions, the fraction of correctly hybridized probes on a spot is obtainable, thus also yielding the concentrations of mutated and wild-type DNA in the sample. Two approaches are possible: (a) comparing the signals of two spots with probes that match, respectively, the wild-type and mutated sequences; and (b) comparing the signals of the spot corresponding to the mutation of interest at two different temperatures, T1 and T2. The first approach requires measurement of the saturation signal of the two spots, whereas for the second approach, this step may be eliminated. Importantly, the experimental set up reported by Fotin et al. (18) satisfies the three conditions listed above and allows implementation of the two-temperature approach.

Our analysis is based on the fact that the hybridization isotherms of DNA chips allow for two types of competitive hybridization (14). Competitive surface hybridization occurs when two different targets can hybridize with the same probe, whereas competitive bulk hybridization takes place when the target can hybridize with a complementary sequence in the bulk. The second process is dominant when the samples are produced by symmetric PCR amplification. When asymmetric PCR (15)(16) is used, the sample contains an excess of ssDNA and does not experience the effect of competitive bulk hybridization. This situation can also be obtained by digesting the product of symmetric PCR with lambda exonuclease (17). Under these conditions, the excess of ssDNA dominates the hybridization with the probes, and competitive surface hybridization becomes the major source of error. In the general case, the hybridization isotherm reflects the electrostatic penalty incurred because each hybridization event increases the charge of the probe layer (14)(19). We will focus on systems in which these interactions are screened and the hybridization isotherm assumes the Langmuir form (13), which facilitates quantitative analysis of the data. The use of PCR amplification renders the absolute concentrations of DNA species meaningless. Our analysis is thus concerned with the ratio cm/cw of the excess concentrations of the mutated (cm) and wild-type (cw) forms of ssDNA.

Our discussion focuses on the situation after amplification by asymmetric PCR (Fig. 1 ) and concerns two spots: a mutation spot carrying m' probes and a wild-type spot carrying w' probes. The sample solution contains a wild-type ssDNA fragment w and a mutated fragment m as well as their complementary fragments, w' and m'. The number of w and m fragments is, however, larger. We assume that all free w' and m' fragments hybridize with the free complementary w and m ssDNA fragments. Our analysis focuses on the excess of unhybridized m and w fragments, whose concentrations are denoted by cm and cw, respectively. We consider the small spot limit, where hybridization at the spot has a negligible effect on bulk concentrations. In this limit, the double-stranded DNA (dsDNA) ww' and mm' pairs in the bulk do not affect the hybridization equilibrium at the spot. In other words, the hybridization isotherm relates cm and cw to the measured hybridization signal of a spot. The targets are actually much larger than the probes. In the K-ras studies we cite, the target is composed of 117 (10) to 157(6) monomers (nucleotides), whereas the probes contain 13–14 monomers. As we shall discuss, this "size asymmetry" mostly affects the boundary of the Langmuir regime. Its effect on the hybridization isotherms of spots of high probe density is beyond the scope of this report. In the following section, we present the necessary background on the hybridization isotherm when competitive surface hybridization is dominant. The following two sections discuss the two-spot and two-temperature approaches, respectively, for the determination of cm/cw. Experimental considerations, the relation to existing experimental studies, and open questions are presented in the Discussion.



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  		Figure 1. Competitive surface hybridization.

The m' probes at the mutation spot can hybridize with both the complementary m targets and the mismatched wild-type w targets. The fraction of mishybridized, m'w, probes is high when the w targets are present in excess. Double-stranded targets are not shown.


   The Adsorption Isotherm
Top
 Abstract
 Introduction
 The Adsorption Isotherm
The Two-Spot Approach
 The Two-Temperature Approach
 Discussion
 References
 
The hybridization isotherm relates the equilibrium fraction of hybridized probes on the spot to the composition of the sample (14). A simple derivation of this isotherm is possible on equating of the rates of hybridization and denaturation at the surface of the spot (13). In the following discussion, we consider the isotherm obtained when competitive surface hybridization is important. In particular, both the wild-type sequence, w, and the mutated one, m, can hybridize at the mutated m' spot. Consider a spot carrying a total of nT m' probes in a sample containing unhybridized m and w targets of concentrations cm and cw mol/L, respectively. The total hybridization reflects both hybridization of the perfectly matched m targets and of the mismatched w ones. The fractions of m'm and m'w hybridized probes are denoted, respectively, by x and y. The rate of denaturation of mismatched m'w probes is kdmynT, whereas the rate of hybridization of m' probes with w targets is khmcw(1 – x – y)nT, where kdm and khm are the corresponding rate constants. When electrostatic interactions within the layer are strongly screened, the rate constants do not depend on the probe density and the fraction of hybridized probes. At equilibrium, the two rates are equal, i.e., kdmynT = khmcw(1 – x – y)nT, leading to:

(1)
where khm/kdm = Km'w is the equilibrium constant (20). Similarly, the two rates for the perfectly matched m'm probes are kdpxnT and khpcm(1 – x – y)nT. Their equality at equilibrium, kdpxnT = khpcm(1 – x – y)nT, leads to:

(2)
Here, Km'w and Km'm, respectively, are the equilibrium constants of hybridization between m' probes and w and m targets; {Delta}Gm'w0 and {Delta}Gm'm0are the corresponding standard Gibbs free energies; T is the absolute temperature; and R is the gas constant. This kinetic derivation recovers the results of a rigorous thermodynamic analysis (14) with one caveat: The molar concentrations should be replaced by dimensionless activities, ai = {gamma}ci, where {gamma} is the activity coefficient (20). Because {gamma} approaches 1 as ci approaches 0, we will retain the expressions used above, noting that the ci are dimensionless, having the numerical value of the molar concentrations of the ith species. The two isotherms (Eqs. 1Up and 2Up ) are not helpful because PCR amplification of a given sample does not allow for different labels of the m and w targets. Because both are labeled identically, the measurement yields the total hybridization fraction {theta} = x + y rather than x and y separately. Accordingly, the observed isotherm for the m' spot is:

(3)
as obtained by summing Eqs. 1Up and 2Up . The fraction of mishybridized probes on the m' spot is:

(4)
Here and in the following, the subscript i = m', w' of {Omega}i, Pi, and so forth identifies the spot. Pm' = 1/2 when cw = cmKm'm/Km'w and Pm' > 1/2 for cw > cmKm'm/Km'w.In the case of interest, when cw >> cm, Pm' of the m' spot is large, whereas Pw' of the w' spot:

(5)
is always small. The values of Pm' (Fig. 2 ) for a typical situation (Table 1 ) confirm that competitive surface hybridization is important only when the competing species is present in large excess. As noted earlier, this is the case in studies of somatic point mutations when the wild-type ssDNA is a majority component. The wild-type ssDNA then contributes to the hybridization on all mutation spots. In contrast, the concentrations of the different mutated ssDNA species are much smaller. As a result, their contributions to the hybridization on the m' spot and on other mutation spots are negligible. This observation justifies limiting the analysis to the competition between two species, m and w.



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  		Figure 2. Fraction of mishybridized probes on the mutation spot as a function of the concentration of the wild-type target.

Pm' vs cw curves calculated at T = 47 °C for cm = 10–9 mol/L (curve 1), cm = 10–10 mol/L (curve 2), cm = 10–11 mol/L (curve 3), and cm = 0 (curve 4).


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  		Table 1. Thermodynamic parameters used in the numerical calculations.1

When competitive hybridization is negligible, Eq. 3Up reduces to {Omega}m' = Km'mcm, and the corresponding isotherm for the wild type is {Omega}w' = Kw'wcw. In such cases, cm/cw can be determined from:

(6)
This situation can be realized in gene expression studies with proper probe design and in the absence of overwhelming excess of one target species. Note that {theta} of the mutation spot can be tuned to be low, {theta} << 1, by adjusting the hybridization temperature. In such regimes, one may invoke a "weak spot" approximation:

(7)
As we shall discuss, when applicable, this simplifies the analysis of the two-temperature experiments. Finally, in most cases, the signal of the spot i at equilibrium, Ii, is proportional to {theta}inT and:

(8)
where Iimax is the saturated signal of the i spot as obtained on equilibration with a concentrated solution of the perfectly matched target. It is important to note that Iimax can depend on temperature.

The melting temperatures of dsDNA are used as design criteria for probes (6)(12). It is thus useful to note that competitive surface hybridization affects the effective melting temperature, TM, which is defined by the condition {theta} = 1/2 or {Omega}i = 1. In general, TM depends on both cm and cw; i.e., TM = TM(cm, cw). Note that there are two targets that can hybridize with the same probe. This situation, involving three ssDNA species and two different dsDNA species, is different from the one invoked in the definition of the melting temperature of a dsDNA. In this last case, two complementary ssDNA species hybridize to form one type of dsDNA, and the total number of species is three rather than five. Furthermore, we are considering the small spot limit, where hybridization with the probes has negligible effect on the bulk composition. In the absence of competitive surface hybridization, TM reduces to its familiar forms: for cw = 0, this leads to Tm'm(cm) = {Delta}Hm'm0/({Delta}Sm'm0 + Rln cm), whereas for cm = 0, it leads to Tm'w(cw) = {Delta}Hm'w0/({Delta}Sm'w0 + R ln cw). It is not possible to obtain an explicit analytical expression for TM(cm, cw), but in the regime of interest, cm/cw << 1, the melting temperature is well approximated by the first two terms of the Taylor expansion in cm around cm = 0, i.e.,

thus specifying the melting temperature at the mutation spot:

(9)
Importantly, in this regime, TM increases steeply with cm (Fig. 3 ), and its initial value, for cm = 0, is Tm'w rather than Tm'm. In marked distinction, for cm/cw = 1, the effect of the competitive surface hybridization is negligible, and TM {approx} Tm'm(cm) with a weak logarithmic dependence on cm.



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  		Figure 3. Dependence of the melting temperature of the mutation spot on the concentration of the mutation target.

TM(cw, cm) in °C is plotted vs cm/cw for various cm + cw = c: (curve 1), c = 10–6 mol/L; (curve 2), c = 10–7 mol/L; (curve 3), c = 10–8 mol/L. • correspond to the melting temperatures [Tm'w(cw)].

When the probe density is relatively high and the salt concentration is sufficiently low, it is necessary to allow for the electrostatic penalty incurred on hybridization because of the extra charge deposited at the probe layer. In this regime, the hybridization isotherms assume a different form exemplified by:

(10)
For the case of probes and targets of equal and short length, {Gamma}m' = constant x {sigma}m' where {sigma}m' is the charge density of the unhybridized m' spot and the constant is set by the ionic strength and the length of the probe (14). In the following, we consider the opposite limit, when the electrostatics interactions are screened and the chains do not interact. The hybridization is then described by a simple Langmuir isotherm, i.e., the case of {Gamma}i {approx} 0. Importantly, in studies of point mutations, the targets are typically longer than the probes. Usually, the hybridization site is situated roughly in the middle of the target. Accordingly, a hybridized probe incorporates two unhybridized target sections of similar length. The span of such segments is R0 {approx} (nal)1/2, where n is the number of bases in the section, a is the base size, and l is the persistence length. Typical values are a {approx} 0.6 nm and l {approx} 0.75 nm (21). The Langmuir regime occurs when the area per probe, {Sigma}, exceeds R02, thus ensuring that the hybridized probes do not interact. For the target–probe pairs considered, this condition is satisfied when {Sigma} {approx} 65 nm2 or less than 1.5 x 104 grafted probes per µm2. Operation in this range allows us to benefit from the absence of the nonlinear behavior introduced by the exp[–{Gamma}m'(1 + {theta}m')]term.

The hybridization isotherm is applicable only at equilibrium. In turn, this implies two conditions: stationary signal and path independence, i.e., independence of the preparation method and sample history. The reported times required to attain a stationary signal vary between minutes (18) and 14 h (22)(23). Importantly, the equilibration time may depend on the probe density, hybridization conditions, the equilibrium {theta}i, and other factors. Path independence requires that the measured signal at equilibrium will not change with the thermal history (heating and cooling cycles). As stated earlier, these conditions are satisfied by the system of Fotin et al. (18). Under conditions of thermodynamic equilibrium, the quantitative methods of analysis we describe below are applicable. However, these methods require knowledge of {theta}i and {Omega}i. In turn, to obtain {theta}i and {Omega}i, it is necessary to ascertain the saturation value of the signal of the i spot, Iimax, or equivalently, the nT of the i spot.

Numerical implementation of the analysis we describe requires knowledge of the equilibrium constants Km'm and Km'w as well as Kw'm and Kw'w. These can be best obtained experimentally from the hybridization isotherm of a spot in contact with single-component samples. However, this approach is time-consuming when several equilibrium constants are required. Fortunately, in the Langmuir regime, it is reasonable to approximate the equilibrium constants by their bulk values. The nearest-neighbor model, with the unified set of parameters compiled by Santa Lucia’s group (24)(25)(26), allows calculation of {Delta}H0 and {Delta}S0 for the hybridization of perfectly matched oligonucleotide pairs as well as for pairs containing a single mismatch. In the numerical calculations, we use probe–target pairs incorporating the 12th and 13th K-ras codons. The {Delta}H0 and {Delta}S0 values are obtained by use of HYTHERTM software implementing the nearest-neighbor approach (27). The identity of the probes and targets considered in the numerical calculations as well as the corresponding standard enthalpies and entropies are listed in Table 1Up , which also lists the free energies of formation and the equilibrium constants of hybridization at 37 °C. The hybridization temperatures considered were chosen in view of the conditions in the cited experiments. Fotin et al. (18) studied hybridization at the temperature range of –20 to 60 °C. In the K-ras studies, the hybridization temperatures were 37 (6) and 50 °C (10). The cm values in the numerical examples vary around 3 x 10–10 mol/L, whereas the concentration of the control target cw is 102- to 103 -fold larger (6).


   The Two-Spot Approach
Top
 Abstract
 Introduction
 The Adsorption Isotherm
 The Two-Spot Approach
The Two-Temperature Approach
 Discussion
 References
 
When two spots carrying, respectively, wild-type (w') and mutation (m') probes are placed in contact with a solution of w and m targets, the corresponding isotherms are:

(11)
and

(12)
where {Omega}i = {theta}i/(1 – {theta}i) of spot i = w', m'; and {theta}i is the corresponding fraction of hybridized probes, irrespective of their identity (mismatched or perfectly matched). These two equations immediately determine cw and cm:

(13)
and

(14)
Accordingly:

(15)
where, in the regime considered, {alpha} = {Omega}w'/{Omega}m' >> 1. The range of {alpha} values varies between {alpha}max = Kw'w/Km'w >> 1, corresponding to cm = 0 and {alpha}min = Kw'm/Km'm << 1when cw = 0. In the realistic limit of {alpha}Km'w << Kw'w, this expression reduces to cm/cw {approx} Kw'w/{alpha}Km'm, that is:

(16)
where Im' and Iw' are the measured intensities of the m' and w' spots, whereas Im'max and Iw'max are the corresponding saturation values. Thus, the implementation of this approach requires knowledge of nT, or equivalently Imax, for the two spots. In the general case it is necessary to know four equilibrium constants, whereas in the simplest case, of {alpha}Km'w << Kw'w, knowledge of Km'm and Kw'w is sufficient.

The two-spot approach is feasible when the values of {theta}m' and {theta}w' for different sets of cm and cw are distinguishable. In practical terms, this imposes two requirements: (a) it is necessary to avoid the saturation regimes of the melting curve (Fig. 4 ) and the hybridization isotherm (Fig. 5 ); and (b), the {theta}i values corresponding to the sample composition must be large enough in comparison with the experimental errors. One may optimize the performance by tuning the hybridization temperature T: Increasing T lowers the extent of hybridization, thus preventing saturation at the price of weaker {theta}i and a higher noise-to-signal ratio.



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  		Figure 4. Melting curves of the mutation spot.

Curves of {theta}m' vs temperature are plotted for cm + cw = c = 10–7 mol/L and cm/cw = 0 (curve 1), cm/cw = 10–4 (curve 2), cm/cw = 10–3 (curve 3), and cm/cw = 10–2 (curve 4). For a given cw {approx} c, the melting temperature and the saturation temperature increase with cm/cw.



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  		Figure 5. Hybridization isotherm of the mutation spot.

{theta}m' vs cm curves are plotted at T = 37 °C (A) and T = 47 °C (B) for cw = 10–7 mol/L (curve 1), cw = 10–8 mol/L (curve 2), cw = 10–9 mol/L (curve 3), and cw = 0 (curve 4). The saturation regime can be avoided by increasing the hybridization temperature.

The operational conditions are thus determined by two inputs: the typical cw of the amplicon and the desired range of cm/cw. Once these two variables are specified, it is possible to calculate the relevant melting curves and hybridization isotherms to choose the hybridization temperature. As noted before (Eq. 15Up ), cm/cw is extracted from {alpha} = {Omega}w'/{Omega}m'. The temperature dependence of {alpha} is relatively weak (Fig. 6 ). The accessible range of {alpha}, however, depends on the detection limit of {theta}m', which sets a lower bound of {theta}m', {theta}m' (min). Because {Omega}w' is ≤1, the maximum range of {alpha} is roughly 1/{theta}m'(min).



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  		Figure 6. Ratios of concentrations obtainable from a two-spot experiment.

Curves for cm/cw vs {Omega}w'/{Omega}m' are plotted at T = 37 °C (curve 1) and T = 47 °C (curve 2) in the range 0.005% ≤ cm/cw ≤ 1%. (Inset), range 1% ≤ cm/cw ≤ 10%.


   The Two-Temperature Approach
Top
 Abstract
 Introduction
 The Adsorption Isotherm
 The Two-Spot Approach
 The Two-Temperature Approach
Discussion
 References
 
An alternative approach involves equilibration of the DNA chip with the biological sample at two different temperatures, T1 and T2. Focusing on the m' spot, we have:

(17)
and

(18)
Solving Eqs. 17Up and 18Up leads to:

(19)

(20)
and

(21)
To simplify this equation, it is convenient to express

as

where T2 = T1 + {Delta}T. On defining {lambda} = {Omega}m'(T2)/{Omega}m'(T1), we obtain cm/cw in the form:

(22)
Eq. 22Up can be simplified further when {Delta}T/T1 << 1, thus allowing approximation of

In the weak spot limit,

and the saturation signals cancel when Im'max(T) is independent of T. When Im'max(T) does vary with T, it is possible to eliminate this contribution by use of an appropriate calibration method (18). Hence, measurement of the saturation values can be eliminated in the weak spot regime. This is of interest when the measurement technique allows study of the {theta}i << 1 range.

As we discussed, optimization of the two-spot approach is achieved by tuning a single hybridization temperature. In the two-temperature approach, it is attained by choosing T1 and {Delta}T = T2 – T1. The general requirements are the same: avoiding the saturation regimes on one hand and the high noise-to-signal ratios on the other. Two additional observations merit comments: (a) increasing {Delta}T magnifies the range of {lambda} corresponding to the cm/cw range of interest (Fig. 7 ); and (b) it also allows one to avoid the divergence regions in the cm/cw vs {lambda} curves (Fig. 8 ) where ultraprecise values of {lambda} are required.



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  		Figure 7. Effect of concentration ratio on the temperature dependence of {Omega}m'(T1)/{Omega}m'(T2).

{lambda} = {Omega}m'(T1)/{Omega}m'(T2) is plotted vs {Delta}T for the interval 10 ≤ {Delta}T ≤ 15 °C for T1 = 37 °C. The concentration ratios are cm/cw = 10–2 (curve 1), cm/cw = 10–3 (curve 2), cm/cw = 10–4 (curve 3), and cm/cw = 0 (curve 4).



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  		Figure 8. Ratio of concentrations obtainable from a two-temperature experiment.

Plots of cm/cw vs {Omega}m'(T1)/{Omega}m'(T2) for T1 = 37 °C and T2 = 47 °C (curve 1), T2 = 49 °C (curve 2), and T2 = 51 °C (curve 3).


   Discussion
Top
 Abstract
 Introduction
 The Adsorption Isotherm
 The Two-Spot Approach
 The Two-Temperature Approach
 Discussion
References
 
Studies of somatic point mutations inevitably concern samples with a large excess of wild-type DNA. This excess is propagated by the PCR amplification unless peptide nucleic acid (PNA) clamps (28) are used. The use of DNA chips to characterize the composition of such samples is hampered by competitive surface hybridization attributable to the pairing of the wild-type target with the mutation probe. This mishybridization contributes significantly to the intensity of the mutation spot signal and thus gives rise to false positives. Our theoretical analysis suggests a systematic approach to the minimization of such errors. At best one can extract cm/cw, thus obtaining additional clinical data. At least, this method provides a rational approach for devising criteria for identification of false positives.

DNA chips designed to detect point mutation in solid tumors were developed by Lopez-Crapez et al. (10). The sample preparation in this case involves asymmetric PCR or symmetric PCR followed by digestion with lambda exonuclease. DNA chips of higher sensitivity for the detection of K-ras mutations in stool were reported recently by Prix et al. (6). The amplification method uses PNA-mediated PCR clamping (28). The PNA binds to the wild-type codons of interest, thus inhibiting their amplification. The PNA binding to the mutated DNA is weaker and does not inhibit its amplification. The overall result is a selective amplification of the mutated DNA. The methods we propose can be used to improve the performance of the approach of Lopez-Crapez et al. (10). In this case, the cm/cw values may provide supplementary information of diagnostic value. In marked contrast, our methods are of little use within the approach of Prix et al. (6), which relies on preferential amplification of the mutated DNA thus avoiding problems attributable to excess of wild-type DNA.

Our analysis suggested two methods to obtain cm/cw. The advantage of the two-spot approach is that it does not involve a change in T. This approach, however, requires knowledge of the saturation value of the two spots. Obtaining the saturation values can be time-consuming, but for chips designed for multiple use, this step need be performed only once provided the saturation value does not change with the hybridization regeneration cycles. The two-temperature approach is attractive when a label-free detection scheme such as surface plasmon resonance is used. In this case, the two measurements can be carried out with no washing steps. Importantly, Fotin et al. (18) demonstrated the feasibility of studying the temperature dependence of hybridization isotherms even for targets labeled with fluorescent tags.

Thermodynamic equilibrium is a necessary condition for implementation of the quantification methods proposed in this report. The equilibration times of DNA chips and their variation with the hybridization conditions are not yet fully understood. The reported equilibration times for hybridization on DNA chips vary widely. In comparing the results, it is important to note differences in hybridization conditions, length of probes, type of chip, and detection techniques. Two studies are of special interest. Peterson and coworkers (22)(23) reported equilibration times of up to 14 h. In marked contrast, the results of Fotin et al. (18) suggest equilibration within minutes. In the present context, the work of Fotin et al. (18) is of special interest because it reports equilibrium hybridization isotherms at different temperatures. Importantly, the isotherms indeed satisfy the conditions of equilibrium: stationary state and lack of hysteresis. Furthermore, the system exhibits a Langmuir-type hybridization isotherm, and the thermodynamic parameters extracted from the temperature dependence of the hybridization isotherms are in good agreement with the reported bulk values. Note, however, that the hybridization constants describing other types of DNA chips do not always exhibit such agreement (29)(30).

The proposed methods require DNA chips that obey Langmuir-type hybridization isotherms. In turn, this requires spots with a relatively low probe density. Clearly, there are advantages to chips carrying spots with a high probe density. One is that high density enables smaller spots, thus allowing for greater number of different spots. Importantly, in DNA microarrays, high probe density gives rise to an electrostatic modification of the isotherms. The resulting nonlinearities are undesirable for obtaining quantitative results.


   Acknowledgments
 
We benefited from insightful discussions with T. Livache. The work of E.B.Z. was funded by the Centre National de la Recherche Scientifique with additional support from the Russian Fund Fundamental Research (RFBR 02-0333127).


   Footnotes
 
1 Nonstandard abbreviations: ssDNA, single-stranded DNA; dsDNA, double-stranded DNA; and PNA, peptide nucleic acid.


   References
Top
 Abstract
 Introduction
 The Adsorption Isotherm
 The Two-Spot Approach
 The Two-Temperature Approach
 Discussion
 References
 

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A. Halperin, A. Buhot, and E. B. Zhulina
Brush Effects on DNA Chips: Thermodynamics, Kinetics, and Design Guidelines
Biophys. J., August 1, 2005; 89(2): 796 - 811.
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