k 2 k −2 configuration 1 configuration 2 k 1 k −1 3 The model is thus characterized by k$_{1}$ , k$_{2}$ and γ . To gain some more insight we consider the limit of fast equilibration for Eq. (2). First we obtain the equilibrium value for θ$_{1}$ by setting the right hand sides of Eqs. (2) and (3) to zero in the limit ce − Δ G/RT ≪ 1. We then solve Eq. (3) replacing for θ$_{1}$ its equilibrium value θ (eq) 1 = ce − Δ G $^{$^{′}$/RT}$. Setting the initial condition θ$_{2}$ (0) = 0 we get FIG. 4: The three state model for hybridization in DNA mi- croarrays is specified by the four rate constants. θ$_{2}$ ( t ) = ce − Δ G/RT ( 1 − e − t/τ ) , (6) τ − 1 = k$_{−}$$_{2}$ = k$_{2}$e (Δ G − Δ G $^{′}$) /RT = k$_{2}$e (1 − γ )Δ $^{G/RT}$. (7) The relaxation time, τ , depends on Δ G : weakly bounded sequences (small | Δ G | ) equilibrate faster than strongly bounded ones (large | Δ G | ). For fast equilibrating se- quences ( τ ≪ t ) one recovers Eq. (1) from Eq. (6); for sequences with long equilibration times τ ≫ t we expand Eq. (6) to lowest order in t/τ . With this approximation we find that for a given time t k$_{−}$$_{1}$ , k$_{2}$ and k$_{−}$$_{2}$ the four rates involved (see Fig. 4). For simplicity we have assumed that at most a single target molecule can bind to a given probe. θ$_{1}$ is the average occupation fraction over several configurations in which partial binding can occur at different positions of the probe sequence. The rate constants, using a two state model descrip- tion, have been measured in several microarray experi- ments. In Ref. [16] the hybridization of a common tar- get sequence to a perfect match probe and to a probe containing one mismatch were considered. The following rates were measured (at 45 ◦ C): k (PM) 1 = 19 · 10 $^{4}$M − $^{1}$s − $^{1}$, k (MM) 1 = 21 · 10 $^{4}$M − $^{1}$s − $^{1}$, k (PM) − 1 = 12 · 10 − $^{4}$s − 1 and k (MM) − 1 = 29 · 10 − $^{4}$s − $^{1}$. While there is more than a fac- tor two of difference in the detachment rates, the at- tachment rates differ only by 10%. These results are in agreement with observation for kinetic behavior in bulk solution [18]. The probes in our experiment differ by at most two nucleotides out of 30. We will then take k$_{1}$ as sequence independent. Consider now the reaction rate k$_{2}$ . In the partially hybridized state the target strand binds over a stretch of nucleotides with one probe se- quence (primary contact) but it can also bind to a second neighboring probe (secondary contact). The rate limit- ing step is the unbinding from secondary contacts and the strand contraction so that the target probe can over- come steric hindrance and wind up into a fully formed helix all along its length. k$_{2}$ will depend on target length and probe length and density. We will assume k$_{2}$ to be the same for the probes of Table I. The reverse rates are then fixed by the thermodynamics relations where Δ G ∗ is a crossover free energy that depends on time and is obtained by setting τ = t in Eq. (7). After hybridization the slides undergo washing steps according to the standard Agilent protocol, which are expected to remove weakly bound target molecules from the slide and have been also included in thermodynamics models of arrays [19]. In the present setup there is only one target sequence in solution and washing is likely to affect the partial hybridized state. In this case one can assume that the measured intensity is given by I ≈ Aθ$_{2}$ (in the model the typical free energies of the partially hybridized states are such that θ$_{1}$ ≪ θ$_{2}$ ). Equation (8) reproduces the two slopes in the log I vs. ΔΔ G plots as seen in the experiments (Fig. 1). It shows that the non-equilibrium regime is characterized by a slope equal to γ/RT . The solid lines in Fig. 2(b,c,d) are plots of the inten- sity I = Aθ$_{2}$ obtained from the solution of Eqs. (2) and (3). The parameters used are given in the caption of Fig. 2. For the choice of parameters given, the fast equi- libration limit (Eq.(6)) approximates very well the full solution of Eqs. (2) and (3). There is a reasonable agree- ment between the experiments and the kinetic model. We note though that the crossover between the two regimes is somewhat sharper in experiments. In addition, the experimental I vs. ΔΔ G data show a slight sigmoidal trend which is not present in the kinetic model. Within the two state model kinetics, and using the assumption k$_{1}$ as sequence independent, one arrives to a solution sim- ilar to Eq. (8) although with γ = 0. Therefore the two state model cannot account for a second finite slope as observed in the experiments. Note that the fit of the kinetic model to the data requires also an estimate of where Δ G ′ and Δ G are the free energy differences be- tween configurations 1 and 2, and the unbound state, respectively. Next we link Δ G ′ to Δ G . Weak total bind- ing (small | Δ G | ) caused by the presence of multiple mis- matches should also correspond to weak partial binding (small | Δ G $^{′}$| ). As a simple approximation we will assume that the two free energies are monotonically linked as k$_{−}$$_{1}$ = k$_{1}$e Δ G $^{$^{′}$/RT}$, k$_{−}$$_{2}$ = k$_{2}$e (Δ G − Δ G $^{′}$) /RT (4) Δ G ′ ≈ γ Δ G ( γ < 1) . (5) θ$_{2}$ ( t ) = { ce − Δ G/RT | Δ G | ≪ | Δ G $^{∗}$| ctk$_{2}$e − γ Δ G/RT | Δ G | ≫ | Δ G $^{∗}$| (8)