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* Institute for Advanced Study, Princeton, New Jersey 08540 and Computational Biology Center, IBM Research, Yorktown Heights, New York 10598; Department of Molecular Biology, Princeton University, Princeton, NJ 08540; and Department of Biomedical Engineering, Boston University, Boston, Massachusetts 02215
Correspondence: Address reprint requests to Gyan V. Bhanot, IBM Research, T. J. Watson Research Center, 347 Dodds Lane, Princeton, NJ 08540. Tel.: 609-497-0241; E-mail: gyan{at}us.ibm.com, gyanbhanot{at}hotmail.com.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONDISCUSSION AND CONCLUSIONSREFERENCES |
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12–15 h for experimental target concentrations (10-13 - 10-14)M, but the time is greater for lower target concentrations in the range (10-15–10-16)M. The result points to an asymmetry in the accuracy with which up- and downregulated genes are identified. |
INTRODUCTION |
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TOPABSTRACTINTRODUCTIONDISCUSSION AND CONCLUSIONSREFERENCES |
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), genotyping and resequencing, and drug discovery and disease stratification (Eisen et al., 1998). The potential impact of micro-arrays on basic and applied biology is so important that an entire industry has been spawned, using any of dozens of variants of two generic methods to fabricate arrays—either direct deposition of probes (Schena et al., 1998; DeRisi et al., 1996; Duggan et al., 1999) or covalent attachment by in situ synthesis (Hughes et al., 2001; LeProust et al., 2000; Lipshutz et al., 1999; Singh-Gasson et al., 1999). The former method allows a wide range of substances such as presynthesized oligomers, proteins, cloned DNA, etc., to be used as probes. The latter is generally restricted to oligonucleotides but offers higher specificity. The central theme of this article is the physical chemical limits of specificity; i.e., conditions that allow the best specificity we consider mainly, though not exclusively, arrays of 20–30 nucleotides long probes, manufactured by in situ synthesis. These conditions minimize false hybridizations resulting from the slow equilibration that is characteristic of long probes, and avoid competition between surface-bound and solubilized probes.
Typically an array of tens to hundreds of thousands of different pixels, each consisting of a homogeneous set of 1–10 million oligonucleotide probes, is used to determine the expression levels of genes of known sequence. The molecules to be assayed, e.g., cDNA, are hybridized, during a 12–15 h incubation, with probes chosen to be their reverse complements The most common detection method relies on fluorescence. Usually molecules from the target and reference cells are labeled with red and green dyes respectively; pixels are then scanned at the two distinct wavelengths to determine expression changes. Genes that are up- or downregulated in response to drugs, hormones, or other environmental influences are thus quickly identified.
Although micro-array assays are high throughput in the sense that in excess of 10,000 genes at a time are probed, the number of false-positives is high, even for arrays prepared by in situ synthesis. Increased specificity is typically achieved by sacrificing sensitivity: only genes with a pronounced change in expression level, typically in the fifth percentile, are scored as having changed. The screened set, or a select group of the screened set, is then investigated further using traditional methods such as Northern blotting.
Increased throughput is generally achieved by increased array density. However, as the above remarks imply, a substantial increase in throughput can be achieved by a well validated, high-specificity system. To increase specificity by rational design procedures, it is helpful to have a clear understanding of the physical limitations of the assay. This includes understanding the conditions that will provide the best specificity, the robustness to deviations from optimal conditions, the relation of optimal conditions to those prevalent in the most common experimental procedures, and strategies for optimization.
This article is divided into two broad components: equilibrium and kinetic. In the first section, we outline the thermodynamics of hybridization. Specificity and sensitivity are maximum when equilibrium has been achieved, but even under this ideal condition the method used to select probes affects the formation of crosshybrids, and thus it affects specificity. Probe selection is a large optimization problem. We discuss this below, and present a new probe selection method. Further below, we use this method to select probes for the full set of yeast genes and compare the specificities obtained at equilibrium where both specificity and sensitivity are maximum. This has particular implications for long probes inasmuch as length substantially reduces the rate at which equilibrium is approached, and consequently increases false-positives if equilibrium is not achieved.
Thermodynamics of hybridization
Melting profiles
As temperature is increased, an initially fully intact hybrid will gradually destabilize, and at high enough temperature, the strands will separate. Approximately 90% of the transition occurs over a temperature range of 10–15 degrees for 25-mers, with the range narrowing as length increases. The so-called melting curve, determined under equilibrium conditions, is cooperative and has an inflection point which is referred to as the melting temperature, Tm.
The melting temperature is defined as the temperature at which half the total number of strands are free (i.e., not hybridized). In general the population of hybridized strands will have a distribution of intact basepairs, and the arrangement of a given number of pairs will also be distributed. The common practice of neglecting partially hybridized states reduces a very complex multistage model to a two state model, eliminates the physical basis for cooperativity, and broadens the melting profile. For short chains, however, it has little affect on the midpoint of the transition, introducing an error that is within the error caused by experimental uncertainty in the stacking free energy.
For this two-state model in which partially hybridized states are neglected, a sequence-dependent expression for the melting temperature is easily obtained. Define ß as the equilibrium constant for bimolecular nucleation (formation of the first bond) in units of inverse concentration, and let K be the (dimensionless) equilibrium constant for the formation of the remainder of the helix. For a helix with n bases, there will be n-1 stacking interactions. We write the sum of the standard Gibbs free energies for the n-1 stacks as 
H-TS, so that the corresponding intramolecular equilibrium constant is 
, where H and S are the sums of the standard enthalpies and entropies for base stacking, in accordance with the base sequence.
The free energy of the nucleation event also, to some extent, depends on the basepairs that nucleate dimerization. If A be the free strand concentration and B the concentration of hybrids, and we assume the molecules are either fully hybridized or completely separated, then,
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In addition, at the melting temperature Tm we have by definition 2B = A. Substituting these relations in the equation for B, and utilizing the definition of K, we have that,
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The presence of a surface
The formation of a DNA hybrid consists of a bimolecular nucleation event followed by formation of a double helix. The main effect of the surface is to freeze the rigid body translational energy and entropy of half the free strands, and to restrict the approach between opposing strands to a half space. The result is to multiply all equilibrium constants by the same constant factor, which is entirely independent of oligonucleotide sequence. This will shift the temperatures at which helices destabilize by some sequence-independent factor. To first order, therefore, the presence of the surface does not affect conclusions about specificity. As we will show via simulation, the effect of the surface on kinetics is crucial, and has a pronounced influence on specificity if equilibrium is not reached.
Probe selection
To be specific, we consider arrayed probes to be 25 nucleotides (nt) long that hybridize to complementary targets from genes in the cells of interest. We will consider one target region per gene, although that restriction is easily relaxed.
For a gene N long and a target L long, there are N - L + 1 potential targets, each potential target being the exact reverse complement of a probe that recognizes it. To understand how choice of target affects accuracy, consider the extreme case in which targets are selected at random. Hybrids would then cover a wide range of melting temperatures. The experimental temperature (at which hybridization is carried out) must be chosen low enough to assure stability of all hybrids. But with that requirement met, the wide range of melting temperatures would result in many hybrids having melting temperatures well above the experimental temperature, and more importantly sequences that are similar to the targets (differing by one base, say), would also be stable at the experimental temperature, as would complexes between target regions and certain incorrect probes. The problem is exacerbated when the expression levels of the spurious targets are higher than the correct targets, and made even worse when equilibrium is not achieved.
At the chosen experimental temperature, therefore, we not only want to choose the target regions so that the reverse complements (i.e., the probes) have a small enough binding free energy to assure stable hybridization, but we also want the free energy of potentially incorrect hybrids to be high enough to assure that they do not form at the experimentally chosen temperature. It is evident from the previous paragraph that we can screen out large numbers of false hybrids and choose targets by minimizing the dispersion in correct probe–target melting temperatures, subject to the constraint that the free energies of correct hybrids render them stable at the experimental temperature, whereas the free energy of incorrect hybrids renders them unstable. The two constraints place upper and lower bounds on the free energies (Li and Stormo, 2001)
Screening the pool of potential probes
A genome with M bases and N genes will provide a pool of M - N(L - 1)segments of length L, from which N probes are to be selected, one per gene. For the yeast genome we use a number of screening procedures to focus on high-quality probe sets. Initial pruning is achieved by a suitable choice of melting temperature.
It is important to take care in setting the hybridization temperature. Choosing an experimental temperature low enough to assure stability of correct hybrids is important for good sensitivity, whereas choosing a temperature high enough to eliminate spurious hybrids is required for specificity. To find a suitable experimental temperature, we first obtain the distribution of melting temperatures of the entire potential pool of M - N(L - 1) correct hybrids (Fig. 1). For the yeast system (http://genome-www.stanford.edu/Saccharomyces), N = 6280 and M = 12,057,500, so that the required calculations can be easily done on a PC.
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This constraint has two effects. First, it speeds probe selection by reducing the search space. Second, it guides the selection of the experimental temperature, which we take to be a few degrees below the lowest acceptable value of Tm. Using this restricted pool of probes, and with melting temperature set, we generate probe sets to simulate hybridization experiments, to compare specificities, and especially to understand the implications of failing to achieve equilibrium.
Probe sets
We used both randomly generated and optimized probe sets. The algorithm to generate an optimized probe set (OPS) is divided into two stages. In the first stage, probes are screened on the basis of binding free energies and other properties that depend only on their sequence, and not the global characteristics of the set. This stage is similar to existing methods. In the second stage, the remaining probe candidates are screened further, using a target function designed to minimize cross-reactivity and maximize specificity between the probe and its complement.
Typical Stage I strategies to reduce crosshybridization are based on tuple frequency (personal communication, Olympus, http://www.olympus.co.jp) and the BLAST sequence search algorithm (C. Sugnet, E. Rice, and T. Clark, 1999, personal communication, http://arrayit.com/Services/ArrayDesign/arraydesign. html, http://www.basic.nwu.edu/biotools/Primer3.html). We use these strategies to make a first cut at the number of probe candidates. Specifically, in our Stage I screening, we eliminate sequences based on the following considerations.
Self-complementarity
It is particularly important to eliminate self-complementarity when insufficient time is allowed to reach equilibrium (as is almost always the case). At equilibrium, and depending on concentration, the ratio of bimolecular to intramolecular complexes might be high, but intermolecular reactions will always slow the kinetics of binding, thereby affecting sensitivity and possibly specificity.
Base composition
We exclude probes that are particularly AT- or GC-rich, in accordance with empirically based guidelines developed by Affymetrix (Lockhart et al., 1996).
Stability
Probes are selected so that the hybridization free energy is lower than some threshold. If experiments achieve thermodynamic equilibrium, this threshold determines the sensitivity to expression level. If G* is the maximum allowed standard free energy of the duplex, relative to the singly nucleated dimer, the lowest detectable expression level will vary as e-G*/RT, where T is the temperature at which hybridization is performed.
3' bias
Dye is generally incorporated during reverse transcription when cDNA is synthesized. Reverse transcription rarely results in complete transcripts of the message; i.e., a substantial amount of 3' bias is typical. For sequences N long, we eliminate from consideration as probe candidates, the N/3 bases closest to the 5' end of the chain. In the second stage, we select probes to minimize crosshybridization. False-positives due to crosshybridization are often minimized empirically by adding pixels with probes that differ in a single basepair from the perfect complement. Although this procedure is helpful, there are problems with it of both a fundamental and pragmatic nature. The latter include cost and (for cells from most human tissue) insufficient quantities of pure mRNA.
The most direct way to choose the best probe set is to compute every crosshybrid free energy, and pick the probes that crosshybridize the least. It is, however, unnecessary to follow such a costly procedure. In particular we need only evaluate free energies of crosshybrids whose stability exceeds some reasonable threshold.
We generate a list of binding energies for all target–probe hybrids that have not been eliminated by restricting the melting-temperature range.
We let 
be the free energy of probe i for target k, with Gkk the binding energy of the correct target to the probe k. We discriminate against probes that are more likely to crosshybridize by using the reciprocal of the correct binding fraction at equilibrium, with all genes referred to a common expression level. Thus, we define the quantity C(k) as,
 | (3) |
, before melting-temperature pruning. This makes exhaustive computation of crosshybrid free energies prohibitive. We have mitigated this problem by the algorithm outlined below, which uses a combination of lookup tables, and a very fast dead-end elimination procedure to obtain free energies. Our binding strand search consists of a fast heuristic step that narrows the search space, followed by a slower evaluation (dynamic programming) on high-ranking candidates. We break the query sequence into overlapping k-mers, where k is the minimum number of basepairs necessary to form stable duplexes (typically 6–12). Candidates are quickly located through k-mer indexes of the entire gene set and a synonym table, both of which have to be prepared once before any search is performed. An extension step is then performed to get the entire duplex.
Step 1: Index the entire gene set; create a list of the occurrence sites for each of the 4k unique k-mers. Step 2: For each of the 4k unique k-mers, find a list of k-mers, called synonyms, that have no more than M mismatches with the given k-mer. We calculate the synonym score, i.e., the base-stacking free energy for the k-duplex. We compute the base stacking energy with a nearest-neighbors model (Fotin and Mirzabekov, 1998), using SantaLucia's parameters (SantaLucia, 1998; Seneviratne et al., 1999). We only need to do this once for a given k. Step 3: Decompose the query sequence into overlapping k-mers. A query sequence of length L has L - K + 1 overlapping k-mers. Find all synonyms for each k-mer in the synonym table prepared in Step 2. For each synonym, every entry in the index table represents a potential site that binds the query sequence with high affinity. Step 4: Use dynamic programming to extended a potential binding strand at both ends, coupled with calculation of binding free energy, following Eq. (3).
We allow mismatches during this step, but stop when long mismatched segments (e.g., three mismatches out of four consecutive basepairs) are encountered, due to unavailability of parameters to predict free energies of such long loops and bulges. The hits are restored in a hash table, using the site of the hit as the key. Whenever two alignments are obtained for the same site, we keep the one with more favorable binding free energy.
Our heuristic search algorithm focuses on short matched segments, which actually forms the "core" of the final duplex, inasmuch as the binding energy is more sensitive to the number of contiguous matching pairs than to the total number of matching pairs. This seamlessly combines a BLAST-like DNA sequence search with a calculation of binding free energies in such a way that the scores are no longer sequence similarities and E-values, but the G values that are used to model the hybridization process. The specificity of the OPS compared to the random probe set (RPS) is evident from Figs. 2 and 3, where we show a histogram of the number of probes binding to a given target.
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Kinetic simulations
Our in silico gene array consists of a two-dimensional square of size 2 cm x 2 cm which is divided into 80 x 80 pixels making up the x-y plane of the experiment. Each pixel has 107 probes tethered to it, which are assumed to be identical and equivalent with respect to their binding properties to presented targets.
For concreteness, we will assume a liquid film of thickness 1/4 mm divided into five equal layers. The bottom layer is where the hybridization between targets and probes takes place. In the remaining layers above the bottom layer, the targets merely diffuse. Thus, our modeling space is made up of a regular grid of boxes of size 
x,y, x,y, z in the x, y, z directions where xy = 2 cm/80 = 0.025 cm and z = 0.025 cm/5 = 0.005 cm.
We first experimented on the RPS to set limits on experimental parameters. The probes were placed on the two-dimensional grid at random locations. This is just a convenient placement of probes—and not necessarily optimal. The experiment consists of following the targets in time, allowing them to diffuse and hybridize. We track the number of targets of each type that are bound to each probe as a function of time.
To avoid issues with target–target interactions, we chose to model a middle range of target concentration of (10-15 - 10-13)M. The most favorable target–target binding energy is -35 kJ, whereas the most favorable binding energy between a probe and its appropriate target is -85 kJ. Hence, the ratio of affinities at T = 315K between probe–target and target–target interactions is greater than 2 x 108. Moreover, the target concentration in our modeling is less than or equal to the probe concentration. From this one can conclude that the total target–target binding rate is negligible compared to the probe–target binding rate and we can neglect target–target binding.
We can then consider each target as if it were diffusing on its own. If NT(x, y, z, t) is the number of target molecules of a specific type in a unit box centered at (x, y, z) at time t, then, the continuum diffusion equation for NT can be written as:
 | (4) |
Inasmuch as our targets are of size less than 100 nt we use a diffusion coefficient D = 10-6cm2/s (Chan et al., 1995). As we will show from our simulation, the precise value of D is not important. We rewrite the continuum diffusion equation as a difference equation. After some simple rearrangements, one gets:
 | (5) |
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The diffusion equation can be interpreted as a molecular dynamics evolution for which we define an updating procedure at each time step as follows: for each box centered at (x, y, z), the average number of target molecules that are entering the box from the +x direction is 
and the number that are leaving the box to enter the box in the +x direction is 
We generate Poisson variables A+ and A- with mean 
and 
bounded from above by NT(x + xy, y, z, t) and NT(x, y, z, t) respectively (inasmuch as we cannot have more particles diffuse out than are present in the box). Thus the net flow into the box at (x, y, z) from the +x direction is = A+ - A-. The quantity is calculated for each direction for each box and the value of NT is updated using these when they have all been calculated. For the z direction, the dimensionless coefficient Dz is used in place of Dxy.
This procedure works if the dimensionless constants Dxy and Dz are small enough so that they can be legitimately interpreted as probabilities. In our modeling, we chose Dz to be 0.2. This determines t = 5s as our time step and Dxy = 0.008. Note that the probability to diffuse in the x-y plane is smaller by a factor of 25 compared to the probability to diffuse in the z direction. Thus, we are led to the approximation that once a target has diffused out of the bottom layer in the z direction, we may assume that it mixes perfectly with the layers above. This allows us to simulate a single layer in the z direction. After each time step, we recalculate the number of targets in the layers above the bottom layer and assume that they are distributed evenly in the layers above.
The diffusion step is followed by a binding step. Once again, this happens only in the bottom layer. The equation governing the binding step will now be considered. If [TP] is the number of target–probe complexes per mol, [T] is the number of free targets per mol and [P] is the number of free probes per mol, then,
 | (7) |
It is easy to show that this equation can be transformed into an equation for the particle number. Thus if NT, NP, and NTP are the number of particles of target, probe and target–probe pairs in the interaction region, then,
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 | (9) |
NA = Avogadro's number, V = volume of liquid in which targets are placed = 10-4 liters, Rr = rr, and I and J label the targets and probes, respectively.
The binding and unbinding process is now modeled by interpreting the above equations as follows. Each xy pixel corresponds to a single probe. For each probe, the average number of new target–probe pairs formed in a time step t is 
where NT(I) is the number of available targets of type I in the box and NP(J) is the number of unbound probes in the box. We assume that there are no steric effects and all unbound probes have an equal chance to bind to the available probes. The actual new target–probe pairs formed in time t is computed by generating a Poisson variable with mean NT(I)P(J). Similarly, one can compute the number of target–probe pairs unbinding by generating a Poisson variable with mean RrNT(I)P(J)t.
Repeating this step over all the targets for all the boxes in the bottom layer of the array completes the binding–unbinding. A probe length of 25 nt gives an average melting temperature of 334 K at 0.3 M salt. The experimental temperature for our simulation, following the empirical rules described previously, is therefore set to 315 K.
Effect of diffusion
We first modeled the kinetics of hybridization for t varying from 0 to 150,000 s for the RPS. Fig. 4 shows the fraction of targets correctly bound (bound to the probe for which they have the lowest binding energy) as a function of time. We have separated the targets into two sets—those that bind to more than 15 probes and those that bind to less than 16 probes. These numbers were chosen to divide the target set into two sets which bind an equal number of probes in total. It is evident from Fig. 4 that even at t = 150,000 s, a significant fraction of the targets are bound to the wrong probes. This error is exacerbated in the targets that bind individually to a larger number of probes. In a normal gene array experiment, the hybridization is allowed to proceed for 12–15 h (40,000–50,000 s). Thus, for the RPS, our experiment shows that the "usual" procedure would yield a significant error.
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Effect of target concentration
Another parameter that might affect the hybridization rate is the concentration of targets. We raised/lowered the target concentration by a factor of 10 and reran the simulation. The comparison of these with the original middle range concentration is shown in Fig. 6. It is clear from this that target concentration does play an important role in hybridization error. The higher the target concentration, the better the hybridization.
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The role of probe specificity
Crosshybridization is a serious problem in determining the expression levels (concentrations) of targets from the hybridization levels. Intuitively, it is clear that one will get more crosshybridization if a given probe binds to many targets. Fig. 7 shows the average fraction of properly bound probes (those bound to the targets that bind to them with lowest binding energy) for different target concentrations for the RPS as a function of target concentration. We have separated the probes into two sets of approximately the same size—those that bind to fewer than 21 targets and those that bind to more than 20. Fig. 7 shows clearly that at all target concentrations, the probes that have the largest number of targets binding to them show the biggest errors in hybridization. For the high concentration experiment, the fraction of correctly bound probes for the two sets merged into a single curve after 50,000 s. However, at the other two concentrations, the two sets do not come together even after 150,000 s (42 h). This indicates the need to optimize probe–target specificity, as was done in selecting the OPS. We would expect, and we will show that this is indeed true, that these types of errors will be much reduced with the OPS.
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DISCUSSION AND CONCLUSIONS |
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TOPABSTRACTINTRODUCTIONDISCUSSION AND CONCLUSIONSREFERENCES |
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Using the RPS and OPS, we studied the dynamics of the hybridization process by computer simulation. We first analyzed the RPS to study the effects of crosshybridization, diffusion, and target concentration. We found that for the RPS, the fact that many probes bind to a given target, results in unacceptably high levels of crosshybridization (Fig. 4) which prevent reaching an equilibrium distribution. Next (Fig. 5), by comparing finite diffusion with instantaneous diffusion, we showed that the diffusion coefficient is large enough not to pose any essential problem. This is partly due to the fact that our targets are small, For larger targets, we expect diffusion to play a bigger role which may require other protocols, such as stirring or thermal annealing during hybridization, to reach an equilibrium distribution.
Finally (Figs. 6 and 7), by studying three different target concentration levels, we showed that after 50,000 s, which is the typical hybridization time in a gene-array experiment, there is a significant effect of target concentration. Targets that have a high concentration level are closer to equilibrium (have a greater fraction bound to correct target) than targets that have a low concentration level. This suggests a serious systematic bias in real experimental situations. In any experiment run for a finite time, downregulated gene levels would be measured lower than their actual downregulation and upregulated gene levels would be measured higher than their actual upregulation levels.
For the OPS, where the crosshybridization is very low, the problems from target concentration effects are less severe (Fig. 8), although they are not completely eliminated. To study these effects in more detail, we conducted a series of computer experiments (Figs. 9–11) on the OPS. We up-/downregulated targets by different amounts and attempted to identify both the targets that were up-/downregulated as well as their level of regulation. This was done by looking at their hybridization level compared to the baseline (unregulated) targets after 50,000 s of simulation.
These experiments confirmed our observation that downregulation is undermeasured and upregulation is overmeasured. Additionally, we found that it is very difficult to measure small variations in target regulation. In other words, for any set of experimental parameters (temperature, target size, probe set choice, hybridization time), there is some value of regulation below which it is impossible to measure regulation because the error rate is too large. This error has two components. One is the error in determining the number of up- or downregulated genes. The other is the error in identifying precisely which gene was up- or downregulated. The first error is significantly smaller than the second (Fig. 11). Thus, one can measure the number of genes which are up-/downregulated by a certain amount much more accurately than one can identify which genes they are.
Gene arrays are being used for an extraordinary range of applications that affect humans directly. These include cancer identification and staging, identifying individuals at risk for genetic disorders, drug regimens specific to the genetic signature of patients, etc. They have also become almost ubiquitous tools in pharmaceutical companies and research labs. It is therefore important to be able to determine the accuracy of the results of such gene array experiments. We view our computer experiment as a first step in this direction. It provides a relatively inexpensive and accurate method for studying the kinetics of gene array experiments to optimize parameter values and experimental protocols for more accurate predictions. Our methods can clearly be generalized to other genomes and experimental situations, such as more complex gene arrays, more sophisticated data collection methods, other parameter values, annealing, washing and stirring protocols, and so on.
Gene array experiments are ever more widely used. It is necessary that their results be validated by some process which has a high degree of credibility. We believe that computer simulations, if they were sufficiently detailed, could play this role. It is our hope that such computer modeling will become an integral part of the validation process of gene array results. We have devised a simulation tool, which may be used to plan experiments and validate their results.
Submitted on July 12, 2002; accepted for publication September 27, 2002.
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J. Bishop, C. Wilson, A. M. Chagovetz, and S. Blair Competitive Displacement of DNA during Surface Hybridization Biophys. J., January 1, 2007; 92(1): L10 - L12. [Abstract] [Full Text] [PDF]
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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. [Abstract] [Full Text] [PDF]
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H. Binder and S. Preibisch Specific and Nonspecific Hybridization of Oligonucleotide Probes on Microarrays Biophys. J., July 1, 2005; 89(1): 337 - 352. [Abstract] [Full Text] [PDF]
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D. Alcor, V. Croquette, L. Jullien, and A. Lemarchand Molecular sorting by stochastic resonance PNAS, June 1, 2004; 101(22): 8276 - 8280. [Abstract] [Full Text] [PDF]
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A. Halperin, A. Buhot, and E.B. Zhulina Sensitivity, Specificity, and the Hybridization Isotherms of DNA Chips Biophys. J., February 1, 2004; 86(2): 718 - 730. [Abstract] [Full Text] [PDF]
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