Enhancing the performance of DNA surface-hybridization biosensors through target depletion
Stefanos K. Nomidis, Michal Szymonik, Tom Venken, Enrico Carlon, Jef Hooyberghs
EEnhancing the performance of DNAsurface-hybridization biosensors throughtarget depletion
Stefanos K. Nomidis, † , ‡ Michal Szymonik, ‡ Tom Venken, ¶ , § Enrico Carlon, † andJef Hooyberghs ∗ , ‡ , (cid:107) † Laboratory for Soft Matter and Biophysics, KU Leuven, Celestijnenlaan 200D, 3001Leuven, Belgium ‡ Flemish Institute for Technological Research (VITO), Boeretang 200, B-2400 Mol,Belgium ¶ Center for Cancer Biology, VIB, 3000 Leuven, Belgium § Laboratory of Translational Genetics, Department of Human Genetics, KU Leuven, 3000Leuven, Belgium (cid:107)
Theoretical Physics, Hasselt University, Campus Diepenbeek, B-3590 Diepenbeek, Belgium
E-mail: [email protected]
Abstract
DNA surface-hybridization biosensors utilize the selective hybridization of targetsequences in solution to surface-immobilized probes. In this process, the target is usu-ally assumed to be in excess, so that its concentration does not significantly vary whilehybridizing to the surface-bound probes. If the target is initially at low concentrationsand/or if the number of probes is very large and have high affinity for the target, theDNA in solution may get depleted. In this paper we analyze the equilibrium and kinetics a r X i v : . [ q - b i o . B M ] S e p f hybridization of DNA biosensors in the case of strong target depletion, by extendingthe Langmuir adsorption model. We focus, in particular, on the detection of a smallamount of a single-nucleotide “mutant” sequence (concentration c ) in a solution, whichdiffers by one or more nucleotides from an abundant “wild-type” sequence (concentra-tion c (cid:29) c ). We show that depletion can give rise to a strongly-enhanced sensitivityof the biosensors. Using representative values of rate constants and hybridization freeenergies, we find that in the depletion regime one could detect relative concentrations c /c that are up to three orders of magnitude smaller than in the conventional ap-proach. The kinetics is surprisingly rich, and exhibits a non-monotonic adsorption withno counterpart in the no-depletion case. Finally, we show that, alongside enhanceddetection sensitivity, this approach offers the possibility of sample enrichment, by sub-stantially increasing the relative amount of the mutant over the wild-type sequence. Introduction
DNA hybridization, the binding of two single-stranded DNA molecules to form a double-stranded helix, is a physico-chemical process of broad interest to disciplines ranging fromfundamental to applied sciences and engineering. It is also central to many applicationswhere detection or enrichment of specific target DNA molecules is required. E.g. in clinicaldiagnostics, which is typically targeted and not hypothesis-free, the detection of known DNAvariants is of high importance. These variants, e.g. DNA from a tumor, can sometimes differin only a single nucleotide from the wild-type DNA of the healthy cells. For non-invasive testsfrom peripheral blood, devices must be specific enough to detect mutated DNA molecules ina background of wild-type DNA down to frequencies of 0.1% or less. This challenge drivesnew detection principles and enrichment strategies among which hybridization-based. Inthese applications, single-stranded DNA probes are designed to bind to the target moleculesduring a hybridization process. Often the probe molecules are immobilized on a surface fordetection purposes or for further processing. Using the sequence-specific properties of the2rocess, specificity and sensitivity of the binding are two important characteristics that canbe aimed for. This is often challenging due to the presence of cross-hybridization, whichoccurs when DNA molecules resembling the sequence of the target molecules hybridize tothe probes and blur the detection or poison the enrichment.Hybridization of targets to surface-immobilized probes can be physically described bythe Langmuir adsorption model, used extensively to predict the equilibrium state of typicalsystems.
In a standard Langmuir approach, the target concentration is assumed to beconstant, which is the case when it is large enough not to be depleted due to hybridizationwith the probe molecules. In experimental applications this assumption may be violated,and corrections need to be applied to incorporate the reduced target concentration into themodel.This paper builds upon three previous works that considered such a target-depletioneffect on surface hybridization. Michel et al. and Ono et al. independently calculated theequilibrium intensity for the case of one target (Ono et al. also for two) hybridizing witha single probe.
Then, Burden and Binder performed a more systematic analysis, bydistinguishing between local and global depletion, depending on whether depletion by a probeaffects only itself or all other probes too, respectively. The hybridization model by Michelet al. and Ono et al. fall under the former category. Our work assumes that hybridization isoperating in a non diffusion limited regime and that depletion is global. For global depletionBurden and Binder presented a numerical scheme to calculate the equilibrium solution, byassuming the probe concentration to be identical among different probes.Our work extends this result, by analytically deriving the equilibrium solution for anarbitrary number of targets and probes, under a realistic assumption (no probe saturation),and allowing for varying probe concentration. This allowed us to design an experimentalsetup that exploits target depletion, so as to enhance the performance of DNA biosensors.In particular, we focus on typical situations interesting for diagnostic purposes, where thesample to be analyzed contains a large amount of “wild type” sequence at concentration3 and a much smaller amount of “mutant” sequence, differing by a single nucleotide. The latter is at a concentration c (cid:28) c . We discuss a minimal-design strategy (Fig. 1)and show how the depletion of the wild type sequence may lead to an increased sensitivity(as confirmed by a practical demonstration), where the detection of the mutant becomespossible even for very small ratios c /c . We also show that this method can be utilized inorder to achieve sample enrichment, by increasing the ratio of the captured mutant overthe wild type target. Finally, we calculated both analytically and numerically the kineticsof the process and analyzed the rich resulting behavior. Materials and methods
In what follows, we will first review the standard Langmuir adsorption model, and thenpresent a simple extension, which accounts for the depletion of the target sequence. Finally,we discuss how this problem can be analytically approached by introducing some usefulapproximations, without much loss of generality.
Langmuir adsorption model
The Langmuir adsorption model treats hybridization as a two-state process. Among theseveral simplifications, such as the homogeneity of the surface and the lack of interactionsamong adsorbates, the model assumes that the concentration of the target sequences insolution is so large, that it practically remains unchanged throughout the process. Let usconsider the simple case of one target type in solution, brought into contact with a singleprobe type. Denoting by θ the fraction of hybridized probes, i.e. the number of hybridizedprobes divided by the total number of probes, the kinetics of the process is described byd θ d t = k + (1 − θ ) c − k − θ, (1)4here k + and k − are the association and dissociation constants, respectively, and c thetarget concentration. The first term on the right-hand side of Eq. (1) is the hybridizationrate, which is partially controlled by the fraction − θ of available probes, whereas the secondterm is the denaturation rate. The solution of Eq. (1) with initial condition θ (0) = 0 is θ ( t ) = (cid:101) θ (1 − e − t/τ ) , (2)where τ ≡ ( k + c + k − ) − is the relaxation time and (cid:101) θ = cK cK (3)the value of θ at equilibrium, where we also introduced the equilibrium constant, K ≡ k + /k − ,of the reaction. The Langmuir isotherm (3) has been successfully employed in the past for thedescription and quantification of DNA hybridization on a surface at chemical equilibrium. This relation becomes linear in the target concentration, (cid:101) θ ≈ cK , when the probes are farfrom chemical saturation, i.e. cK (cid:28) [or θ (cid:28) in Eq. (1)]. Target depletion
In the case of target depletion the hybridization kinetics is described byd θ d t = k + (1 − θ )( c − aθ ) − k − θ = ak + ( θ − θ + )( θ − θ − ) , (4)where a is the probe concentration, and θ ± the two fixed points, given by θ ± = 12 aK (cid:20) aK + cK ± (cid:113) (1 + aK + cK ) − acK (cid:21) . (5)Note that, the hybridization rate is now additionally controlled by the amount of the re-maining target in solution, i.e. c − aθ . Since θ ≤ , it follows that target depletion may besafely neglected as long as c (cid:29) a , i.e. the initial target concentration is greater than the5robe concentration. Equation (4) can be solved through separation of variables. Using theinitial condition θ (0) = 0 , one obtains θ = θ + θ − (1 − e − t/τ ) θ + − θ − e − t/τ , (6)where the characteristic time now is τ ≡ [ ak + ( θ + − θ − )] − . At long times t (cid:29) τ , thesolution (6) converges to θ − , which is a stable fixed point of Eq. (4), whereas θ + is unstable.The approach to the stable fixed point is monotonic in t , as expected for a single first-orderordinary differential equation (ODE). Moreover, in the limit a → , one finds θ − = (cid:101) θ [givenby Eq. (3)] and θ + → ∞ . Finally, note that this equilibrium solution [smallest root inEq. (5)] is identical to Eq. (6) of Ref. 13 and Eq. (7) of Ref. 14, apart from some constantfactors.Equation (4) may be generalized, so as to describe the hybridization of n t different targetswith n p different probes. The fraction θ ij of the i -th probe hybridized with the j -th targetsatisfies the differential equationd θ ij d t = k + ij (cid:18) − n t (cid:88) m =1 θ im (cid:19)(cid:18) c j − n p (cid:88) n =1 a n θ nj (cid:19) − k − ij θ ij . (7)Here k + ij and k − ij are the association and dissociation constants, respectively, whereas c j and a n are the total concentrations of the j -th target and the n -th probe, respectively.Equations (7) constitute a set of coupled nonlinear equations, which, in general, cannot besolved analytically. In order to proceed, we will assume that the probes remain far fromchemical saturation i.e. (cid:80) n t m =1 θ im (cid:28) , which leads to the following set of linear equationsd θ ij d t ≈ k + ij (cid:18) c j − n p (cid:88) n =1 a n θ nj (cid:19) − k − ij θ ij . (8)The equations for θ ij no longer couple the different targets in solution (second index j in θ ij ). As the spots are not saturated, each target sequence has always probe sequences at its6isposal for hybridization, hence θ ij and θ ij (cid:48) evolve independently from each other for j (cid:54) = j (cid:48) .The equilibrium hybridization fraction is given by (details are given in Appendix) (cid:101) θ ij = c j K ij (cid:80) n p n =1 a n K nj , (9)where we have defined K ij ≡ k + ij /k − ij , in analogy with the case of a single probe/target pair.For the numerical solution of Eq. (7), we used the Python implementation of the LSODAODE solver, using time steps. The latter were chosen to be evenly spaced on a logarithmicscale (further supported by the exponential nature of the solution), allowing for the accuratesampling of both the short- and long-time behavior, while keeping the number of time stepsat a minimum. The kinetics can be solved analytically in the case target depletion occursdue to a single probe, which is an interesting case for application purposes. Results
Here, we discuss the consequences of target depletion in conventional hybridization experi-ments. In particular, we show that depletion can significantly improve the performance ofhybridization biosensors. For this purpose, we consider the setup shown in Fig. 1, which issimple enough to capture the basics of the process, yet, at the same time, relevant for diag-nostic applications: detecting small amounts of mutant DNA in a sample with a majority ofwild type DNA.The sample in solution contains two types of target DNA, a wild-type and a mutantsequence, the latter having a point mutation with respect to the former. The two sequenceshave initial concentrations c and c , respectively. Particularly interesting for diagnosticpurposes is the detection of mutants at very low abundance, e.g. c /c (cid:28) . To this end, weemploy a collection of three types of probes, a wild-type ( wt ), a mutant ( mut ) and a reference( ref ) probe, immobilized on a surface at concentrations a , a and a , respectively (see Fig. 1).While wt and mut are the perfect complements of their target counterparts, ref contains7igure 1: A minimal experimental setup for the study of target depletion. The samplesolution (top) contains two targets, a wild-type (blue) and a mutant sequence (red), withconcentrations c and c , respectively. We assume the former to be in abundance, and thelatter to be present in small traces, i.e. c (cid:29) c . The two targets come into contact withthree probes spotted on a surface (middle), with concentrations a , a and a . The first twoprobes (blue and red) are the perfect complements of the two targets, while the third probe(green) is used as a reference for the detection of the mutant target. In order to achievetarget depletion, we propose the use of a large concentration, a , of wild-type probes. Finally,at the bottom all possible duplexes are shown, together with the notation we employ.contains one and two mismatches relative to the wild-type and mutant targets, respectively.Further, the hybridization affinity of ref to the wild-type target is designed to be equal tothat of mut . When a sample contains only wild-type target the equilibrated hybridizationsignal θ from ref equals the θ from mut , hence the name reference probe. When the samplealso contains a trace of mutant target DNA, θ but not θ will be significantly affected.More quantitively, the signal measured from each probe is the sum of the contributionsfrom the wild type and the mutant, i.e. θ i = θ i + θ i . Probes mut and ref both have a singlemismatch with respect to the wild-type target. By design we assume that their hybridizationaffinity to the wild type is similar, hence θ ≈ θ , which can be achieved with a properchoice of the reference probe. In case a mutant target is present in solution ( c > ), one has θ (cid:29) θ , due to its much higher affinity for the second probe (perfect complement) than8he third probe (two mismatches). Following Ref. 21, we define the detection signal as S ≡ log θ θ = log θ + θ θ + θ ≈ log (cid:18) θ θ (cid:19) , (10)which will be zero when c = 0 ( θ = 0 ) and positive otherwise. Clearly, for diagnostic pur-poses we wish to have a large value of S for small ratios c /c . Note that cross-hybridizationcan cause θ to be much larger than θ , especially when c (cid:29) c , hence obscuring thedetection of the mutant target. In order to address this issue, we propose the use of a largeconcentration a of wt , which will deplete the corresponding target, hence leading to a cleanersignal from mut . Though perhaps evident, this approach will also deplete the mutant target,and a profound quantitative analysis is needed to investigate this issue. In what follows, wewill quantify this effect by considering both the equilibrium and kinetics of the hybridizationprocess. Equilibrium properties
We will first focus on the equilibrium aspects of target depletion. For a system with threeprobes, the hybridized probe fraction at equilibrium is given by [see Eq. (9)] (cid:101) θ j = c j K j a K j + a K j + a K j . (11)The detection signal is then given by S = log (cid:32) (cid:101) θ (cid:101) θ (cid:33) ≈ log (cid:18) c c K PM K a K PM + a K + a K a K + a K PM + a K (cid:19) . (12)For simplicity, we have assumed that K = K ≡ K PM , K = K = K ≡ K and K ≡ K , associated with the perfect-match, single-mismatch and two-mismatchhybridizations, respectively. It is important to stress that the above relations are introducedfor convenience and do not affect the main conclusions of this work. In absence of depletion9 a i = 0 ), the detection signal becomes S = log (cid:18) c c K PM K (cid:19) = log (cid:18) c c e ∆∆ G /RT (cid:19) , (13)where we have used the thermodynamic relation K = e − ∆ G/RT , with ∆ G the hybridizationfree energy, R the gas constant and T the temperature (note that by convention ∆ G < ). Wehave also introduced ∆∆ G ≡ ∆ G − ∆ G PM , the free-energy difference between theperfect-match and one-mismatch hybridizations which depends on the mismatch identityand on flanking nucleotides, according to the nearest-neighbor model of DNA hybridiza-tion. Equation (13) has been experimentally verified in the past (see e.g. Fig. 3 of Ref.21), and shows that there are two factors controlling the detection limit of the device. Oneis the relative abundance, i.e. it is easier to detect mutants at high relative concentrations( c /c ). The other factor is the relative affinity ∆∆ G > , i.e. a large free energy penaltyfor mismatched hybridization leads to suppression of cross hybridization, and hence facilitatesthe detection of the mutant. Since typical values of ∆∆ G lie in the range − kcal/mol, and using the fact that the signal is detectable only when S ≥ S min = 0 . , it follows thatthe minimum relative concentration, c /c , that can be measured with this method lies inthe range 0.17% to 15%, in agreement with previous reports. In this calculation we used T = 65 ◦ C as a typical system temperature.
Next, we consider the other limit of strong depletion. We will assume the target depletionto occur only due to the wild-type probe, which can be tuned by choosing a large-enoughprobe concentration so that the condition a K (cid:29) a K PM is met. Moreover, by fullyexploiting the effect of target depletion, so that a K (cid:29) , we obtain the followingelegant expression S ≈ log (cid:34) c c (cid:18) K PM K (cid:19) (cid:35) = log (cid:18) c c e G /RT (cid:19) . (14)By comparing Eqs. (13) and (14), we see that depleting the abundant wild-type target indeed10eads to higher S (additional factor of two in the exponent). Performing the same analysisas above, we find that the minimum relative concentration, c /c , that is experimentallydetectable is in the range 0.00044% to 3.3%. This corresponds to an enhancement of thedetection sensitivity by one to three orders of magnitude, owing to target depletion.In order to experimentally realize the aforementioned detection enhancement, two con-ditions need to be met, as mentioned above. First, the relative concentration a of thewild-type probes has to be much larger than those of the mutant and reference probes, sothat a a n (cid:29) K PM K = e ∆∆ G /RT , (15)with n = 2 , . Using typical values for hybridization free energies of single base pair mis-matches (see above) we estimate (cid:46) exp(∆∆ G /RT ) (cid:46) . Thus, the larger the free-energy penalty, ∆∆ G , of a mismatch is, the larger the ratio a /a n ( n = 2 , ) needed.Moreover, the absolute value of a needs to be large enough, so as to maximize the contri-bution of depletion. The precise condition is a (cid:29) K = e ∆ G /RT . (16)The precise value of ∆ G /RT depends strongly on the DNA sequence, and can be esti-mated based on the nearest-neighbor model of DNA. As a practical evaluation of these predictions, we also performed a DNA microarray exper-iment. The microarray contained two drastically-different groups of sequences, allowing usto study DNA hybridization both in absence and presence of target depletion (see Appendixfor details). The solution contained both a wild-type and a mutant target, with relativeproportion equal to c /c = 5% . In absence of depletion, the observed detection signal, asdefined in Eq. (10), was found to be S = 0 . ± . . On the other hand, target depletion wasfound to yield the value S = 4 . ± . , corresponding to a five-fold enhancement of the de-tection signal (See details in Appendix). Interestingly, by taking ∆∆ G = 2 . kcal/mol,11hich is a reasonable value for the cost of a mismatch, Eqs. (13) and (14) yield S ≈ . and S ≈ . , respectively. These values are remarkably close to the experimental ones, giventhe presence of a single free parameter.Performing again the same analysis as above this leads to an estimated detection limitof the relative concentration c /c of 1.8% for the non-depletion case and 0.052% for thedepletion case. Hence the sensitivity is increased by a factor of 35 through target depletion. Hybridization Kinetics
To investigate the kinetics of the system, we have numerically solved the coupled ODE (7).The wild-type concentration was fixed at the experimentally-realistic value of c = 50 pM,while to obtain strong depletion we have set a = 800 pM and a = a = 4 pM. We consideredon-rates identical for all sequences, which is supported to a good extent by experimentalobservations. The value was set to k + = 10 s − M − . The off-rates were then fixed bythe equilibrium condition K ≡ k + /k − = e − ∆ G/RT . For the perfect-match, one- and two-mismatch hybridizations we used ∆ G PM = − kcal/mol, ∆ G = − . kcal/mol and ∆ G = − kcal/mol, respectively, based on nearest-neighbors data for a 15-mer at T = 65 ◦ C. Figure 2 shows the hybridization kinetics for θ i in the case of no-depletion (a and b)and of strong depletion (c and d). The dots are the analytical solution, while the solid linesare obtained from the numerical integration of Eq. (7). In (a) and (c) the solution containswild-type target at concentration c = 50 pM and no mutant ( c = 0 ). The signals measuredfrom mut and ref perfectly overlap, since we have assumed equal hybridization free energy ∆ G for the two sequences. In (b) and (d) the solution additionally contains c = 0 . pMmutant (corresponding to a ratio c /c = 0 . , i.e. of the wild-type concentration).Figure 2 indicates that the presence of the mutant is more easily detectable in the case ofstrong depletion as the gap between mut and ref is much more pronounced. The kinetics isalso remarkably different: in absence of depletion the signal increases monotonically in time,12 Figure 2: Hybridization evolution of a wild-type and a mutant target with an array of threeprobes, a wild-type (blue line), a mutant (red line) and a reference one (green line), fortwo values of the relative target concentration r ≡ c /c . Panels (a) and (b) correspondto the case where no depletion of the wild-type target takes place, whereas (c) and (d)to the strong depletion case, where the wild-type probe is in excess concentration. In thelatter case, besides the numerical solution of Eqs. (7) (solid lines), we also plot the analyticalsolution (17) (points). The dashed vertical lines correspond to the three characteristic times, t , t and t , discussed in the main text. The inset in (d) shows a magnification of the wt signal, revealing a very small overshoot.whereas in the strong depletion case we observe a nonmonotonic behavior and even a dip inthe signal from mut .The long time behavior shown in Fig. 2 corresponds to the equilibrium solution given byEq. (9). In order to understand the observed rich kinetics, one can use a simplified solvablecase in which we assume that the depletion occurs due to the wt probe alone, i.e. a , a ≈ and a ≡ a (cid:54) = 0 . Under this approximation, the solution θ i = θ i + θ i is found to be [see13q. (31) of Appendix] θ = c K PM aK PM (cid:2) − e − ( ak + + k PM ) t (cid:3) + c K aK (cid:2) − e − ( ak + + k ) t (cid:3) ,θ = c K aK PM (cid:26) − e − k t + K PM K ak + ak + + k PM − k (cid:2) e − k t − e − ( ak + + k PM ) t (cid:3)(cid:27) + c K PM aK (cid:26) − e − k PM t + K K PM ak + ak + + k − k PM (cid:2) e − k PM t − e − ( ak + + k ) t (cid:3)(cid:27) ,θ = c K aK PM (cid:26) − e − k t + K PM K ak + ak + + k PM − k (cid:2) e − k t − e − ( ak + + k PM ) t (cid:3)(cid:27) + c K aK (cid:26) − e − k t + K K ak + ak + + k − k (cid:2) e − k t − e − ( ak + + k ) t (cid:3)(cid:27) , (17)where we used k PM , k and k to denote the off-rates, while K PM = k + /k PM , K = k + /k and K = k + /k . Equations (17) are shown in Fig. 2 as dotted lines and arein excellent agreement with numerics. One can further simplify them using the assumption aK PM > aK (cid:29) , which corresponds to the limit of strong depletion. This condition issatisfied for the parameters used in Fig. 2. Under this assumption, Eqs. (17) reduce to θ ≈ c + c a (cid:0) − e − ak + t (cid:1) ,θ ≈ c K aK PM (cid:20) − e − k t + K PM K (cid:0) e − k t − e − ak + t (cid:1)(cid:21) + c K PM aK (cid:20) − e − k PM t + K K PM (cid:0) e − k PM t − e − ak + t (cid:1)(cid:21) ,θ ≈ c K aK PM (cid:20) − e − k t + K PM K (cid:0) e − k t − e − ak + t (cid:1)(cid:21) . (18)In the last expression we have neglected the contribution θ of the mutant target to theprobe ref , as the corresponding hybridization involves two mismatches and c (cid:28) c . We,thus, identify three characteristic times, t ≡ /ak + , t ≡ /k and t ≡ /k PM , whichare ordered as t < t < t and are shown as dashed vertical lines in panels (c) and (d)of Fig. 1. We note that, while θ is clearly a monotonic function of time, there are several14ime-dependent factors with opposite signs in θ and θ , giving rise to nonmonotonic timeevolution.To analyze this time dependence in more detail, we first consider the regime in which t (cid:46) t . In this time interval we approximate exp( − ak + t ) ≈ − ak + t and exp( − k t ) ≈ exp( − k PM t ) ≈ , so as to get θ ≈ θ ≈ θ ≈ ( c + c ) k + t, (19)which indicates that at short time scales the kinetics is characterized by an identical bindingrate to wt , mut and ref . This is because we have assumed equal attachment rate k + for allprobes and targets, which is a reasonable approximation. This, however, does not influencethe main features of the kinetics at the subsequent time scales. In the next interval t (cid:28) t (cid:46) t , we approximate exp( − ak + t ) ≈ and exp( − k PM t ) ≈ . In this case the wt probe signalreaches a stationary value θ ≈ ( c + c ) /a , which can also be obtained from the equilibriumsolution (9), while θ ≈ c a (cid:20) K K PM + (cid:18) − K K PM (cid:19) e − k t (cid:21) + c a ,θ ≈ c a (cid:20) K K PM + (cid:18) − K K PM (cid:19) e − k t (cid:21) , (20)which, as K PM > K , are decreasing functions of time. In this regime the wild-type targetstarts dissociating at the same rate k from mut and ref probes. This leads to a very weakincrease in the hybridization of the wt probe, which is not detectable in the scale of Fig. 2(see inset of panel d), and also not present in the approximated solution (18). This weakincrease is, however, present in the full solution (17). Finally, at even longer times, i.e. for t (cid:28) t ∼ t , one has exp( − ak + t ) ≈ exp( − k t ) ≈ . The ref probe reaches a steady state15 ≈ c K / ( aK PM ) , while the mut probe increases monotonically as θ ≈ c a K K PM + c a (cid:20) K PM K − (cid:18) K PM K − (cid:19) e − k PM t (cid:21) . (21)This increase takes place only if c > , while in absence of mutant target ( c = 0 ) thisthird timescale is absent, and mut reaches a steady state value from above as ref . In thislast regime the wild-type target has completely equilibrated, and the mutant target getsredistributed from the wt probe to the mut probe. This gives rise to a monotonic increase inthe hybridization of the latter, until the complete equilibration of the system. The turnovertime at which θ is minimal can be calculated from Eqs. (18) and is given by t min = log( c /c ) k − k PM . (22)Next, we show how target depletion can be used for sample enrichment, i.e. increasingthe relative amount of mutant DNA over wild-type DNA. From an application point of view,this is an important issue, and can lead to an increased performance for mutant detectionby other techniques, such as sequencing. Hereto, we focus on the hybridized material onthe mut probe (probe number 2) and study two important quantities (see Fig. 3): the ratioof mutant over wild-type target attached to mut (a,c) and the absolute amount of mutanttarget (b,d). The former quantity determines whether we can achieve enrichment, the latteris needed as a measure of capture efficiency. In Fig. 3 (a) and (b) these quantities are plottedas functions of depletion (i.e. wt probe concentration, a ), for a sample with starting targetmutant ratio of c /c = 0 . . These plots quantitatively show how depletion leads to atrade-off between yield and sample enrichment. As an example, indicated with the dashedvertical line, is a regime where a yield of about 4% gives a mutant enrichment of a factor ( θ /θ ) / ( c /c ) ≈ . Finally, the kinetics shown in Fig. 3 (c) and (d) indicates that thequantities evolve monotonically in time, hence equilibrium conditions can be used to achievethe best results. 16 E n r i c h m e n t [ θ / θ ] (a) (c)0 5 10 wt concentration (nM)0 . . . A tt a c h e d m u t [ a θ / c ] (b) 10 − Time (hours) c /c = 0 . Figure 3: Upper panels: Mutant/wild-type target ratio, θ /θ , attached to the mut probe.Higher values of the ratio enable the further enrichment of the sample, by increasing therelative population of the mutant with respect to the wild-type target. Lower panels: Frac-tion, a θ /c , of the mutant target that has hybridized with the mut probe. For applicationpurposes, the concentration, a θ , of the captured mutant target should be comparable tothe initial one, c , in solution. These quantities are plotted both (a,b) as a function of the wt probe concentration at equilibrium and (c-d) as a function of time at a fixed wt probeconcentration ( a = 2 . nM, corresponding to the dashed vertical lines). The small devia-tion between analytics and numerics arises from the approximation a , a ≈ included inthe former. Conclusion
In this paper we have analyzed the equilibrium and kinetics of hybridization in DNA biosen-sors under the effect of strong target depletion. This is a condition which has been consideredonly in limited prior studies since the typical assumption behind hybridization models inDNA biosensors, as the Langmuir adsorption model, is that the target sequences in solutionare in excess. Target concentration is then considered to be constant throughout the dura-tion of the experiment. To fulfill this condition one needs a sufficient amount of hybridizingmaterial to start with. Although target depletion is typically avoided, our analysis shows17hat one can turn it, in some applications, into an advantageous condition, leading to anincrease of the performance of the biosensor.We focused on the problem of detection of small amount of mutant sequence (with con-centration c ) diluted in a highly-abundant wild type (with concentration c ), and specificallyaddressed the case of a single nucleotide difference between the two. An example where thisis an important diagnostic problem is in liquid biopsy, where one examines a mixture of“healthy” molecules in majority, with a small subpopulation of molecules carrying a specificpathogenic property.The minimal design employed in this study involves three probe sequences, which wereferred to as wild-type ( wt ), mutant ( mut ) and reference ( ref ) probe. The presence of themutant in solution is inferred by the ratio of hybridization signals measured from ref and mut . We have presented a quantitative analysis of the hybridization kinetics and shown thatin presence of wild-type depletion one can decrease the detection limit up to three ordersof magnitudes in the ratio c /c [as revealed by a comparison between Eqs. (13) and (14)].Note that the only sequence-dependent parameter controlling the detection limit is the freeenergy penalty associated to a single mismatch. With the same design we showed that, nextto detection, also target enrichment can be enhanced.Finally, our analysis of the kinetics revealed a rich behavior, with interplay between theinitial strong binding of target, followed by unbinding and redistribution of the sequencesbetween the different probes. This resulted in three different time scales and a mut signalthat exhibits a nonmonotonic behavior: an increase followed by a decrease and then by afinal increase towards equilibrium. We expect that this distinct feature, which we have foundto take place only when c > , should be observable in experiments which have access tothe kinetics of hybridization. cknowledgement We acknowledge financial support from the Research Funds Flanders (FWO Vlaanderen)Grant No. VITO-FWO 11.59.71.7N.
Appendix
DNA microarray experiment
To confirm the detection enhancement predicted by Eqs. (13) and (14), we performed amicroarray experiment. We designed two sets of wild-type, mutant and reference probes,shown in Table 1. The first probe set was based on previously-published data, from whichwe selected the optimal double-mismatch ref probe for the wt and mut pair, i.e. one forwhich | θ − θ | is minimised. (Note that the selected ref probe which best fulfilled thiscriteron actually contains two mismatches with respect to the wild-type target and three tothe mutant target, in contrast to the one- and two-mismatch case described in the main text.The number of mismatches is not critical here, but rather the relative ∆∆ G values.)The second probe set was designed to maintain the sequences and positions of the variabletriplets from the first set, as well as a similar overall ∆ G PM , while the remainder of thesequence was kept distinct to avoid cross-hybridisation.The six probes were laid out on the array, which contained 10 spots of the wt probe and . × spots of the wt dep probe, corresponding to the no depletion and strong depletionregimes. The array was incubated with a mixture of wt and mut targets for each probe set(Table 2), with c /c = 0 . . Targets for the two probe sets were labelled with differentfluorophores, allowing them to be measured independently on the same array. The resultsare shown in Figure 4. 19 t mut ref C y i n t e n s i t y log( I /I )0 . ± . No depletion wt dep mut dep ref dep C y i n t e n s i t y log( I /I )4 . ± . Strong depletion
Figure 4: Microarray fluorescence intensity data showing the effect of depletion on the relativemutant hybridisation signal.Table 1: Microarray probe sequences used in the experiment.Probe Sequence (5’ - 3’) Array replicatesDepletion wt dep GTTGGAGCTGGTGGCGTAGGCAA mut dep
GTTGGAGCTGCTGGCGTAGGCAA ref dep GTTGGGGCTGGTGGCGAAGGCAA wt CGCCGAGTCGGTCATGTACTGGC mut CGCCGAGTCGCTCATGTACTGGC ref CGCCGGGTCGGTCATGAACTGGC Materials and methods
A custom 8x15K Agilent microarray slide was used (Agilent Technologies, US). Microarrayprobes all included a ( dA ) linker sequence on the 3’ end. Target oligonucleotides (IDT,Germany) included a ( dA ) linker and a Cy3 or Cy5 fluorescent dye on the 3’ end. Targetoligonucleotides were mixed to final concentrations shown in Table 2 in × Agilent GExTable 2: Target sequences used in the microarray experiment.Target Sequence (5’ - 3’) ConcentrationDepletion T wt, dep TTGCCTACGCCACCAGCTCCAAC +Cy3 95 pM T mut, dep TTGCCTACGCCAGCAGCTCCAAC +Cy3 5 pMNo depletion T wt, GCCAGTACATGACCGACTCGGCG +Cy5 95 pM T mut, GCCAGTACATGAGCGACTCGGCG +Cy5 5 pM20ybridisation buffer HI-RPM with × Agilent GE blocking agent. The slide was incubatedwith 40 µ L of this target mixture in an Agilent hybridisation oven for 17 hours at 65 ◦ C withrotor setting 10, followed by washing according to manufacturer instructions. An AgilentG2565BA scanner was used to image the microarray, with a 5-µ m resolution and 100% gain.Image analysis was carried out using the Agilent Feature Extraction software, version 10.7.The signal was background-corrected by subtraction of the global minimum signal. Equilibrium isotherm and the Sherman-Morrison formula
In order to compute the fraction of hybridized probes at equilibrium, it is convenient tointroduce a vector θ , defined as θ ≡ { θ , θ , . . . , θ n p , θ , θ , . . . , θ n p , . . . } . With thisdefinition, one can cast Eq. (8) in matrix formd θ d t = − M θ + b , (23)where M is a block diagonal matrix. The j-th block, M j , corresponds to the contributionfrom a single target j and mixes the elements of the subvector θ j ≡ { θ j , θ j , . . . , θ n p j } . Itsentries are M jnm = k + nj a m + k − nj δ nm , (24)where δ nm indicates the Kronecker δ function. In the j -th block, the constant vector is givenby b jn = k + nj c j . The equilibrium value (cid:101) θ is obtained by inverting the matrix M as (cid:101) θ = M − b , (25)which can be performed independently for each block. In order to calculate M − , we noticethat the j-th block of M is the sum of a diagonal matrix and the outer product of twovectors, i.e. M j = D + uv T , with D diagonal and uv T ≡ u ⊗ v . This allows us to use the21herman-Morrison formula, which reads (cid:0) D + uv T (cid:1) − = D − − D − uv T D − v T D − u , (26)(note that, while uv T is an n p × n p matrix, v T D − u is a scalar). In the present case D = diag { k − j , k − j , k − j , . . . } , while the two vectors are u = { k +1 j , k +2 j , k +3 j , . . . } and v = { a , a , a , . . . } . Using the above definitions, together with b j = c j u , we find the follow-ing equilibrium solution of the j -th block: (cid:101) θ j = (cid:0) M j (cid:1) − b j = c j (cid:0) D + uv T (cid:1) − u = c j D − u v T D − u . (27)A simple calculation gives v T D − u = n p (cid:88) n =1 a n K nj and D − u = { K j , K j , K j , . . . } , (28)where K ij ≡ k + ij /k − ij is the equilibrium constant. Combining Eqs. (27) and (28), and recallingthat (cid:101) θ ji = (cid:101) θ ij , we finally obtain Eq. (9). Hybridization kinetics under depletion by a single sequence
Equation (23) can be analytically solved when depletion occurs due to a single probe. Inthis case we can writed θ ij d t = k + ij (cid:32) c j − n p (cid:88) n =1 a n θ nj (cid:33) − k − ij θ ij ≈ k + ij ( c j − a θ j ) − k − ij θ ij , (29)where we have assumed that a n (cid:28) c j for n > . This condition can be experimentallyrealized through a proper design of the probes and choice of target concentrations. Under22his assumption, one has a set of independent equations for θ j that can be easily solved θ j = c j K j a K j (cid:104) − e − ( a k +1 j + k − j ) t (cid:105) , (30)which is monotonically growing in time and approaches the stationary value (cid:101) θ j = c j K j / (1+ a K j ) . One can plug Eq. (30) in (29) to solve for the remaining θ ij with i > . The resultis θ ij = c j K ij a K j (cid:34) − e − t/τ ij + k − ij k − j a k +1 j a k +1 j + k − j − k − ij (cid:0) e − t/τ ij − e − t/τ j (cid:1)(cid:35) , (31)where τ ij ≡ ( a k + ij δ i + k − ij ) − is a characteristic time. Note that by setting i = 1 in Eq. (31),one recovers Eq. (30), as the third term within the square brackets vanishes. Thus, Eq. (31)can be used as a general solution of the problem for all i and j . References (1) Naef, F.; Lim, D. A.; Patil, N.; Magnasco, M. DNA hybridization to mismatched tem-plates: a chip study.
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