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Next, we present some ideas that emerged when we used our model to analyze the DNA overstretching curves of various groups:

(i) Regardless of temperatureT, lengthLor ionic concentrationI, for experimental curves where the change in extension is ¯L ≈ 1.7L the cooperativity of the system is given approximately by

l ∈ [22−25]bp. Although T, I and L are different among the data sets presented in Fig. 6.2, the three sets are quantitatively reproduced by our model using l ≈ 25bp. Similarly, Fig. 6.3(a) shows that using l≈22bp accurately reproduces Zhang et al. [32] experiments atI = 150mM and

T = 22C. Furthermore , usingl ≈22bp in Fig.E.1, we show that Bianco et al. [35] and Bongini et al. [34] experiments atI= 150mM andT = 25C are consistent with experiments in [32, 33].

Experiment T[◦_{C] I[mM] L[kbp] Description L¯}

Zhang et al. [32] Fig. 3b 24 150 ∼7.3 n-h (S-DNA) ∼1.7

Zhang et al. [32] Fig. 3b 24 1.0 ∼7.3 h (M-DNA) ∼1.5

Zhang et al. [32] Fig. 3c 12 3.5 ∼7.3 n-h (S-DNA) ∼1.7

King et al. [33] Fig 1A 22 50 ∼48.5 n-h (M-DNA) ∼1.7

King et al. [33] Fig S4 22 150 ∼25 n-h (S-DNA) ∼1.7

Bongini et al. [34] 10-25 150 ∼48.5 n-h and h ∼1.7

Zhang et al. [30] 10-20 500 ∼48.5 n-h (S-DNA) ∼1.7

Table 6.1: Summary of recent DNA overstretching experiments used throughout this study. Lis the length in the B-DNA state and ¯Lis the length of the DNA filament after the transition measured inLunits. The abbreviations n-h and h stand for non-hysteretic and hysteretic respectively.

(a)

(b)

(c)

(d)

Figure 6.3: DNA overstretching atT = 24C for different ionic concentrations. Red Markers correspond to Zhang et al. [32] experiments and solid lines are theoretical predictions from our model. As the ionic strength decreases going from (a) trough (d),A2 decreases andlincreases. We useA4 ∼500pN andFc as measured in experiments: Fc= [68.3,63.5,58.8,50.5]pN going from panel (a) through (d).

(ii) The experimental data in [32] showed a gradual change inhzias a function of ionic concen- tration. This is shown in Fig. 6.3 where we fit data from figure 3b in [32]. In Zhang et al. [32], at the low ionic concentrations, there is an asymmetric pattern in the force-extension curves when the force increases or decreases. This hysteretic behavior observed during overstretching is due to the slow convergence to equilibrium [159, 169]. The asymmetric hysteresis at I = 1mM and

I= 5mM in [32], where the system is out of equilibrium mainly during unloading, is consistent with previous stretching experiments that depict marked hysteresis during the decreasing force regime [34, 35, 164, 170, 180, 181]. During the loading phase in some of these experiments there are no hysteretic effects, while in others, if hysteresis is present, its effect is significantly less pronounced than during unloading. Therefore, for the lowI=1 and 5mM we fit only to the pulling data in [32]. As shown in Fig. 6.3,l decreases gradually with increasingI, ranging froml ≈60bp at I = 1mM down tol ≈22[bp] atI= 150mM. The smaller cooperativity values are consistent with ¯L→1.7L. Given that experiments [32, 33, 171] confirm that the S-form is preferred at high salt concentrations

and that the S form is mainly responsible for the 1.7 times change in extension [34], we conclude based on our fittings that forl <30bp the predominant state in the overstretched form is S-DNA.

(iii) As the transition becomes less cooperative, the increase inlof each subsystem has the same theoretical effect as drastically decreasing the temperature T, as evidenced from the definition of the partition function given in Eq. (6.9). A change of 4 times in the value ofl from 25bp to 100bp is equivalent to a decrease of the absolute temperature from room temperature to T = 75K, which effectively yields less global statistical fluctuations. Thus, one can expect the sigmoidal curve to sharpen and become closer to an abrupt first order phase transition. In figure 6.2(a) we have plotted the curve for l = 100bp next to the 25bp solution, so that the difference in width of the curves is apparent.

(iv) Although moderate changes in temperatureT ∈[10−25]C affect the critical forceFc value [32, 34], the change of temperature in this range does not seem to affect the extension of the molecule up to T =TM at fixedI (see Fig 3(c) in [32]). But onceT ≥TM, there is a sudden change in the extension profile of the overstretching curves [32]. Bongini et al. [34] and Zhang et al. [30] data support the idea that at a fixed I for a range of temperaturesT ∈∼[10−25], the extension of the overstretched form remains approximately the same. This would imply that given a fixedI, there is a single transition class to the S-form (or at least closer to pure S) for T < TM and a melting transition for T > TM. Hence, we think of l as independent of T for each transition class, and making use of the phenomenological model forC(F, T) as described in section 6.1.3, we predict the behavior of the overstretching transition as a function ofT. A sample of the results is shown in Fig. E.3, where we show that introducing the temperature effects throughC(F, T) captures what is seen in experiment.

(v) At higher ionic concentrations (I= 500mM [30]), we found that althoughl= 22bp is a good average fit to the experimental data, the curve is not symmetric about the midpoint of the transition and the data is better fit byl≈15bp near the overstretched state (See fig 6.4). Similar behavior is found in King et al. [33] overstretching curve atT = 22C andI= 1M. But, this asymmetric aspect of the overstretching transition is much more evident in Fig. 6.5 where we present the comparison of Zhang et al. [30] variance measurements with our theoretical predictions. Since the n segments making up the entire chain are independent of one another, the system is analogous to a random walk ofn steps. In this analogy the average step size of the walker ishuliand the variance of each step equals the variance of one segment of lengthl :

σl2=l2 u2 − hui2, (6.23) where u2

=d(lnZm)/dbis the second moment of the partition function andbis defined in expres- sion Eq. (6.10). Then the variance for the entire chain (n-steps) is [146] :

σv2=nσl2=L

u2

− hui2l. (6.24)

For a given force, since L is fixed, the variance grows linearly with the cooperativity length. In Fig. 6.5 gray circles correspond to Zhang et al. [30] experimental measurements of the variance at I = 500mM. Lines correspond to our theoretical predictions for different values of l using Eq. Eq. (6.24). The red solid line (forl= 15bp) agrees strongly with the experimental data for ∆F >0 (right side of the graph) , while on the left side of the graph the blue solid line (forl= 30bp) provides a better fit. Black solid line (l = 22bp) in Fig. 6.5 is shown as an average fit for both sides of the graph.

(vi) An alternative method to quantify the cooperativity of the DNA overstretching transition is to use the Zimm-Bragg parameterσF [7]:

σF = exp(−2βEs), (6.25)

l[bp] Fig. σF·10−3 δF[pN] δz[nm] 55 Fig. 6.3(d) 0.3 1.8 0.16 31 Fig. 6.3(c) 1.0 2.8 0.19 25 Fig. 6.2(a) 1.6 3.1 0.20 22 Fig. 6.3(a) 2.0 3.4 0.21 15 Fig. 6.4(b) 4.3 4.5 0.23

Table 6.2: Calculation of the Zimm-Bragg cooperativityσFas a function oflusing relation Eq. (6.26). As a reference value, Rouzina and co-workers [7, 8] measured σ≈10−3 _{in DNA overstretching experiments at room temperature} andI= 150mM.

qualitative terms, large values of a cooperative unit l are analogous to small values of σF [173], but a quantitative relation can be obtained by the following procedure. The parameter σF reflects the widthδF of the overstretching transition in terms of the force [7, 182]. δF can be determined by the midpoint slope of the plot PSvs.F [7, 182], where Ps is the fraction of the filament in the overstretched state. Then the force transition width is [7]:

δF = ∂f ∂Ps _F=Fc = 4σF1/2 kBT δz , (6.26)

where δz is the change in extension per basepair during the transition. Instead of using δz as approximated before in the zero-temperature calculations, the model presented in section 6.2 with statistical fluctuations allows to provide a more exact estimate forδz:

δz=hufi − huii, (6.27)

where the subscriptsiandf stand for initial and final point of the transition respectively. Using the definition of Ps given by Eq. (6.39) we can directly compute δF by the left equality in expression Eq. (6.26). Next, making use of expression Eq. (6.21) we can evaluate huii and and hufiat Fi = Fc−δF/2 and Ff = Fc +δF/2 respectively. By doing so, expression Eq. (6.26) directly links our methods to the Zimm-Bragg cooperativity model, and we can calculate the parameterσF as a function ofl. In table 6.2 we presentδFandσF for several sets of experimental data used throughout this chapter, where we see thatσF is of the order of 10−3in agreement with the reported values in [7, 8].

Since the model based on subsystems of cooperative lengthlaccurately describes the quasi-static overstretching experiments, in section 6.4 we extend our methods to study the kinetics of a system with sharp interfaces, meaning the phase transition takes place in a spatially homogeneous way.