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The twist energy parameter (ET) that governs the supercoiling free energy, and the linking difference (Δl) are measured for p30 δ DNA in solutions containing 0 to 40 w/v% ethylene glycol (EG). A plot of ET vs. −ln aw, where aw is the water activity, displays the full (reverse) sigmoidal profile of a discrete structural transition. A general theory for the effect of added osmolyte on a cooperative structural transition between two duplex states, 1□ 2, is formulated in terms of parameters applicable to individual base-pairs subunits. The resulting fraction of base-pairs in the 2-state ( ), is incorporated into expressions for the effective torsion and bending elastic constants, the effective twist energy parameter ( ), and the change in intrinsic twist (δl0). Fitting the expression for to the measured ET -values yields reasonably unambiguous estimates of ET1and ET2, the midpoint value (ln aw)1/2, and midpoint slope (ET/ln aw)1/2, but does not yield unambiguous estimates of the equilibrium constant ( K0), the difference in DNA-water preferential interaction coefficient (ΔΓ), or the inverse cooperativity parameter, J. Fitting a non-cooperative model (assumed J=1.0) to the data yields, K0 = 0.067, and ΔΓ = − 30.0 per base-pair (bp). Essentially equivalent fits are provided by models with a wide range of correlated J, ΔΓ, and K0 values. Other results favor ΔΓ in the range − 1.0 to 0, which then requires K0 ≥ 0.914, and a cooperativity parameter, 1/J ≥ 30.0 bp. The measured δl0 and circular dichroism (CD) at 272 nm are found to be compatible with curves predicted using the same -values that best-fit the ET -data. At least 7 to 10 % of the base-pairs are inferred to exist in the 2-state in 0.1 M NaCl in the complete absence of added osmolyte. Compared with the 1-state, the 2-state has a ~2.0- to 2.1-fold greater torsion elastic constant, a ~0.70-fold smaller bending elastic constant, a ~0.91-fold smaller ET -value, a ~0.2 % lower intrinsic twist, a somewhat lower CD near both 272 and 245 nm, and less water and/or more EG in its neighborhood. However, the relative change in preferential interaction coefficient associated with the transition is likely rather slight.
DNA secondary structures are typically classified into three main families, specifically the right-handed A and B families and the left-handed Z-family. Numerous conformational substates within the right-handed families have also been detected by various methods, although specific structural details are lacking in most cases.1 Average properties that reflect changes in secondary structure respond to practically every conceivable perturbation, including the presence of small neutral osmolytes.1–25 Such changes appear to arise from shifts of population (base-pairs) among different conformational substates. Conformation-sensitive properties include the elastic constants, α and κβ, for torsion and bending, respectively, the twist energy parameter, ET, that governs the supercoiling free energy, the intrinsic twist, l0 (turns), the intrinsic binding constant for ethidium, and the circular dichroism spectrum.1 All of these properties can be measured for samples as small as 20 μg of DNA.
At 37 °C, increasing concentrations of the neutral osmolytes, ethylene glycol (EG) and acetamide, caused initially sigmoidal decreases in the respective curves of ET vs. −ln aw, where aw is the water activity.17 These data, which previously extended only to ~20 w/v% osmolyte,17 could be satisfactorily modeled by a two-state transition of independent subunits (or domains) of unknown size. However, the terminal lower plateau of the transition could not be located from those data, and neither the value of (−ln aw)1/2 nor the slope, (ET/ (− ln aw))1/2, at the midpoint could be reliably assessed. Moreover, quantitative information about properties of the terminal state, which prevails in ~40 w/v% EG, could not be obtained.
A principal goal of the present study is to measure a complete curve of ET vs. −ln aw from 0 to 40 w/v% EG for a single preparation of p30δ DNA. Also obtained is a corresponding curve of δl0 vs. −ln aw, where δl0 is the change in intrinsic twist from the value prevailing in 0 w/v% EG. Circular dichroism (CD) spectra in 0, 10, 20, 30, and 40 w/v% EG were also recorded. Particular objectives are: (i) to develop a general model of a hydration-coupled two-state transition, which expresses the various thermodynamic quantities on a per base-pair basis and takes into account cooperativity of the transition; (ii) to fit and compare different versions of this model to the ET vs. −ln aw data to extract relevant model parameters; (iii) to identify the values of −ln aw and ET/ (−ln aw) at the midpoint of the transition; (iv) to estimate what fraction of the sequence exhibits the terminal state in the complete absence of added osmolyte; (v) to estimate the torsion and bending elastic constants, α and κβ, of the terminal state that predominates in ~40 w/v% EG from the measured values of α determined previously for 0 and 20 w/v% EG17,25 and the value of ET for 40 w/v% EG; (vi) to ascertain δl0 for that same terminal conformation; (vii) to show that a model, which gives a good fit of the ET vs. −ln aw data, also gives an acceptable fit of the somewhat noisier δl0 vs. −ln aw data with no further adjustment of parameters, when a suitable estimate is adopted for the difference in intrinsic twist between the two states involved; (viii) to show that the CD data are consistent with the same model; and (ix) to learn something about the cooperative domain size at the midpoint of the transition and the change, ΔΓ, in the preferential interaction coefficient per base-pair for water. Because our data lack sufficient precision, not all of these objectives are reached unambiguously. Nevertheless, significant new information emerges.
We consider a duplex DNA that contains N subunits, each of which consists of a base-pair (bp) plus its associated backbone atoms. Each subunit is assumed to exist in either of two conformational states, 1 or 2. The instantaneous conformational configuration of the entire DNA is specified by the N consecutive state labels, k = 1 or 2, of its N subunits, for example,
Such a specific configuration is denoted by the index I, and a DNA with the Ith configuration is denoted by DI. In purely aqueous buffer (0 w/v% neutral osmolyte), the standard state chemical potential of DI is , where denotes the concentration (molecules per unit volume) of water in that buffer. Adding a small neutral osmolyte (os) to concentration cos generally decreases both cw and the water activity, aw. It also typically alters the standard state molecular chemical potential of DI to . This standard state chemical potential is that part of the total chemical potential, μDI (cw,cos), that is independent of the concentration of DI. In its hypothetical 1.0 M standard state, each DI molecule experiences only the environment of the infinitely dilute solution, where . Under such conditions, the solution surrounding the DNA has only two components, w and os, and at constant T and P it has only a single independent chemical potential, which we take to be, . To first order, the variation of with ln aw is given by
where k is Boltzmann’s constant, T is absolute temperature, and
is a preferential interaction coefficient (PIC) of DI with water. In equation (1) it is assumed that ΓDI|w is approximately constant over the range of ln aw encountered in the present experiments, and that ln aw for the purely aqueous buffer is approximately zero. ΓDI|w can be expressed in molecular terms as, 26,27
where gDI|w (r) and gDI|os (r) are the DI − w and DI − os pair correlation functions, respectively, r is the distance from the (arbitrarily chosen) center of the macromolecule to the (arbitrarily chosen) centers of neighboring w or os molecules, and dr is a volume element.
We invoke the heuristic assumption that the integrand in equation (3) arises primarily from two additive contributions, namely (i) the different volumes that are totally excluded by DI to water and osmolyte centers, and (ii) osmolyte-water exchange (or displacement) reactions at osmolyte accessible sites outside those regions, but presumably still near the surface of DI. These assumptions yield,27
where is the difference between the volumes excluded by DI to osmolyte centers, , and water centers, , in units of the partial molecular volume of water, w. S is the osmolyte-water exchange contribution, and is given by
where the i-sum runs over a lattice of L sites of equal volume, os, that completely fill the space accessible to osmolyte centers, ν os/w, and Ki is the equilibrium constant for the exchange reaction, wherein a single osmolyte is removed from a site in the distant solution, and placed in the ith site near DI, while ν water molecules are transported in the reverse direction. Finally, aos is the osmolyte activity. X varies with the area and rugosity of the DNA surface and the osmolyte size, which define the region that is inaccessible to osmolyte centers due to hard-core repulsion, whereas S depends upon the net affinity of osmolyte for its accessible sites outside the region of total exclusion, and could in principle be either positive (net exclusion) or negative (net attraction). Some of the sites accessible to osmolyte centers permit surface contact between the osmolyte and DNA. An important point is that X and S in equation (4) arise from excess or deficit concentrations of water near the DNA surface relative to its bulk average concentration. Because both contributions, X and S, in equation (4) arise from a region close to the surface of DI, it is reasonable to assume that ΓDI|w is a sum of contributions originating from the surfaces of the individual subunits,
where Γkj|w is the contribution of the kth subunit, which is in state j (j=1 or 2) in the Ith conformational configuration, and is the PIC per bp.
It is also assumed that can be parsed into contributions from base-pairs in the 1-state and those in the 2-state, plus an additional free energy, FJ, associated with 12 or 21 junctions,
where is the contribution of the kth base-pair in state j, and nJ is the total number of 12 plus 21 junctions in the Ith conformational configuration. Incorporating equations (6) and (7) into (1) yields finally,
For simplicity, it is now assumed that the difference, , takes the same value, , regardless of whether the kth subunit is a GC or an AT base-pair. Likewise, the difference, Γk2|w − Γk1|w, is assumed to take the same value, ΔΓ, for both GC and AT base-pairs. These assumptions are approximations of unknown accuracy, and may need to be revised after data pertaining to the composition dependence of and ΔΓ become available.
The configuration, D1, wherein every base-pair is in state 1, is adopted as a reference state. The concentration of configuration DI relative to that of D1 is found to be:
The quantity, is the intrinsic equilibrium constant for the 1□ 2 transition of a single base-pair in the purely aqueous buffer with no osmolyte, ΔΓ is the difference in the PIC per base-pair between the 2 and 1 states, and J exp[−FJ/kT] is the inverse cooperativity parameter, which is assumed to be insensitive to aw.
By summing over all distinct configurations, I (n2), that exhibit exactly n2 subunits in state 2, we obtain the total concentration of DNAs with n2 subunits in state 2,
where nJ denotes the number of junctions in the I(n2)-th configuration. A further summation over n2 yields the total DNA concentration, , where
is the conformational grand partition function.
The average fraction of subunits in state 2 is given by
and the average number of 12 plus 21 junctions per DNA molecule is given by
The grand partition function, χ, in equation (12) is evaluated simply by summing over all possible 2N distinct states of the N subunits, which can be effected by using a transfer matrix, M. For a circular DNA,
are its eigenvalues. The last equality in equation (15) applies in the limit of large N.
The midpoint of the transition, , occurs when B=1.0, or equivalently when
At the midpoint the slope becomes . Using d ln B = ΔΓd ln aw, this becomes
Evidently, the midpoint slope in equation (20b) is equally sensitive to ΔΓ and 1/J. Experimental measurements of (ln aw)1/2 and provide only two numbers, which are insufficient to determine K0, ΔΓ, and 1/J. Estimation of all three parameters must rely on data in the wings of the transition. Unfortunately, our data lack sufficient precision to provide separate estimates of ΔΓ and 1/J. It can be shown from equations (10) and (19) that the curve of vs. ln aw is symmetric for all choices of K0, ΔΓ, and J, in the sense that , where Δ is any difference between and the midpoint value of 1/2 (unpublished analysis).
and the average size of domains of consecutive 1-states or 2-states at the midpoint is given in base-pairs by,
Clearly, the domain size is determined entirely by 1/J with no contribution from ΔΓ, as expected.
Equations (6)–(23) generalize linear lattice models of two-state structural transitions to incorporate the effects of a small cosolute (the osmolyte), which relative to water may be either net attracted to, or excluded from, the neighborhood of a subunit of the macromolecule in either of its two possible conformations. This theory could readily be extended to admit an effect of osmolyte on the junction free energy, FJ.
For a given topoisomer, the number of turns, l, of one strand around the other is an integral topological invariant called the linking number. The state of deformation of a supercoiled DNA is characterized by its linking difference, Δl = l − l0, where l0 is the generally non-integral intrinsic twist. The linking number is partitioned between twist (t) and writhe (w) according to, l = t + w,28,29 so
Both experiments17, 30–34 and simulations24,35,36 pertaining to long DNAs (N ≥ 2000 bp) in ~0.1 M monovalent salt indicate that the free energy change to vary the linking difference from Δm = m − l0 to Δl is given by
wherein ET is independent of Δl and Δm, especially over the thermally populated range of topoisomers. Equation (25) also accurately represents the present data for p30 δ in all concentrations of EG from 0 to 40 w/v%.
It was previously proposed that, for DNAs in ~0.1 M monovalent salt, ET could be approximated by,4
where αeff and are the effective elastic constants for torsion and bending of the springs that are imagined to exist between base-pairs, and Bw is a formerly unknown constant. Recent simulations of a 4910 bp model DNA under our standard topoisomerization conditions validated equation (26) within the simulation errors, and established that Bw 0.594.37
We imagine that the kth subunit is connected to the (k+1)th subunit by Hookean torsion and bending springs, whose minimum energy positions and torque constants depend upon the conformational state of the kth subunit. In regard to twisting springs, the minimum energy twists are, respectively, and , and the corresponding torque constants for twisting are α1 and α2. Similarly, in regard to bending springs, the minimum energy bends are, respectively, and , and the corresponding torque constants for bending are κβ1 and κβ2.
When the numbers of springs of types 1 and 2 are fixed at their unperturbed values, and , the effective elastic constants are found to be
where and are the unperturbed fractions of subunits (base-pairs) prevailing in the absence of any deformational strain. However, in reality the 1□ 2 equilibrium generally shifts in such a way as to reduce the deformational strain energy, so f2 is not exactly equal to its unperturbed value, , in equations (18), (27), and (28). The relevant theory to incorporate torsional strain into the present model of the conformational equilibrium is presented in the Appendix. The final result for f2 = n2/N is given by (A20) with F (n2) given by (A19) and λ given by (A17). The quantity, U (n2, Δt) n2, is the slope of the torsional strain energy for a given fixed net twist, Δt = t − l0, with respect to n2 (the number of subunits in state 2), and is given explicitly in terms of n2, α1, α2, , and Δt in equation (A21). In principle, equation (A20) can be solved by iteration to find f2 for any fixed choice of α1, α2, Δ0, and Δt. It is shown via equations (A22) and (A23) that in the present case of extremely small Δ0 and small Δt, corresponding to Δl = +2 turns, the value of n2 must lie very close to , so that the 1□ 2 equilibrium is practically unshifted by the net twist. The similarly small net bending strain in a topoisomer with Δl = +2 is also believed to exert no significant effect on the 1□ 2 equilibrium. Consequently, equations (27) and (28) should apply to very high accuracy in the present study.
where is the expected value of ET, when the entire DNA is in state j, j=1 or 2. Curves of vs. ln aw can be calculated by using equations (18) and (29) for arbitrary choices of ET1, ET2, K0, J, and ΔΓ. Fitting these equations to the measured values of ET vs. −ln aw in principle allows determination of all five of these parameters. However, the present data lack sufficient precision to determine both J and ΔΓ, which can be distinguished only in the wings of the transition, as noted earlier.
At the midpoint of the transition,
Fortunately, the best-fit values of ET1 and ET2 and also the best-fit values of (ln aw)1/2 and are rather insensitve to the large, but highly correlated, uncertainties in K0, ΔΓ, and 1/J, as shown below.
The p30δ plasmid was described previously.17 However, recent direct sequencing indicates unambiguously a length, N=4932 bp, rather than the 4752 bp inferred from restriction digests by the group that constructed the plasmid.38 Isolation and purification of the p30δ were carried out as described previously.17 The native p30δ was dialyzed into a storage buffer (50 mM KCl, 50mM Tris, 0.1 mM EDTA, pH 7.5) and kept at 4 °C. The A260/A280 was ~1.90. Analysis by gel electrophoresis indicated a supercoiled:nicked ratio greater than 3:1.
A portion of the p30δ sample was linearized by Eco RI. Completion of the circular → linear conversion was confirmed by gel electrophoresis in 0.8% agarose. The enzyme was extracted from the linearized DNA sample several times with buffered phenol, and the sample was then dialyzed into the CD buffer (100 mM NaCl, 10 mM Tris, 1mM EDTA, and 2.5 w/v% glycerol, plus various w/v% EG). The water activities of these buffers differ negligibly from those of the Topo I buffers with the same w/v% EG, which are described subsequently. CD spectra of these samples, corrected for any solvent or instrumental contribution, were measured at 37 °C, as described elsewhere.41 The absorbance of these CD samples at 258 nm was 0.204.
Supercoiled p30δ was relaxed at 37 °C for 6 hours by calf-thymus topoisomerase I (from Invitrogen) in the presence of 0 to 40 w/v% EG. This topoisomerase I remains active up to 43 w/v% EG, but is rendered inactive by 47 w/v% EG. The topoisomerase I was added in three separate aliquots at 0, 2.0, and 4.0 hours to ensure complete topisomerization of the sample. The total final concentration of topoisomerase I in the sample is 0.1 units/μL. The final concentrations of all other components of the Topo I buffer were: 50 mM KCl, 50mM Tris, 0.1 mM EDTA, 10 mM MgCl2, 30 μg/mL bovine serum albumin, 0.5 mM dithiothreitol, 2.5 w/v% glycerol, 25 μg/mL DNA, and various w/v% EG. Reactions were halted by extracting the enzyme once with a mixture of phenol, 0.1 w/v% 8-hydroxyquinoline, 48 v/v% chloroform, and 2 v/v% isoamyl alcohol. The phenol/8-hydroxyquinoline was preequilibrated with a buffer containing 0.2 mM NaCl and 0.1 mM Tris at pH 8.0. Every topoisomerization reaction was replicated at least four times. The ~60 μL volume of each replicate reaction yielded three 20 μL aliquots of reaction products, each of which was run in a separate lane in gel electrophoresis. The topoisomerization reactions performed for each particular condition thus yielded a total of at least 12 lanes for gel analysis. The large number of gel lanes was required to achieve acceptable statistical accuracy of the results.
Gel electrophoresis was performed in 0.8 w/v% agarose, as described previously.17 Sufficient (0.08 μg/mL) chloroquine was added to reduce the effective intrinsic twist, and thereby shift the linking differences of all visible topoisomers into the slightly positive range. At the conclusion of the run, the gel was soaked twice in distilled water (2L, 1 hr) to remove the chloroquine. The gel was then stained by soaking in 1 L of 0.5 μg/mL ethidium bromide for 45 minutes in the dark, and finally soaked twice in distilled water to reduce the background level of ethidium. Under the prevailing conditions, ethidium uptake and fluorescence are proportional to the local concentration of the topoisomers in each band.11
The gel was illuminated by 488 nm light, and a digital image of the ethidium fluorescence (λmax = 640 nm) was recorded by a Molecular Dynamics Fluoro Imager-SI Optical Scanner. Plots of fluorescence intensity vs. distance migrated (in pixels) are shown in Figure 1 for p30δ samples that were relaxed in 0 and 40 w/v% EG. The integrated fluorescence intensity of each band was determined with the aid of the ImageQuant 5.1 program from Molecular Dynamics. The baseline and limits of integration for each peak were set manually. The ratio of integrated intensities of any two bands in the same lane is taken to be equal to the ratio of their equilibrium concentrations under the conditions of their particular topoisomerization reaction.11 Reliable data were typically obtained for 6–8 topoisomers in each lane.
The linking number of the most populous topoisomer is the nearest integer to the intrinsic twist, l0, and is denoted by NINT(l0). The excess linking number, lex, of a given topoisomer with linking number, l, is defined by,
The bands in each gel lane are indexed according to their lex values (…−3, −2, −1,0,+1,+2,+3,…). The ratio, clex/c0, of the equilibrium concentration of a given topoisomer with lex to that of the most populous topoisomer is given by,
where Δl0 NINT (l0) −l0 is the linking difference of the most populous topoisomer. Equation (33) is fitted to the measured ratios of integrated intensities, Ilex/I0 = clex/c0, by an iterative nonlinear least-squares protocol with ET and Δl0 as adjustable parameters.17 Optimum fits for the two sets of data in Figure 1 are shown in Figure 2. Such good fits typify every data set analyzed. After fitting the clex/c0 data for each of the 10–15 satisfactory gel lanes for each reaction condition, the resulting ET and Δl0 values were averaged and reported along with the standard deviations of their means in Table 1.
Both l0 and Δl0 may vary with EG concentration (cEG (w/v%)), and are now written as l0 (cEG) and Δl0 (cEG) to indicate that. In the case of 0 w/v% EG, cEG = 0, and one has l0 (0) and Δl0 (0). The difference between experimental linking differences,
reflects a change in l0 (cEG) with increasing cEG. The change in intrinsic twist upon adding osmolyte is defined by17
The variation in NINT (l0 (cEG)) with EG concentration can be ascertained directly by running topoisomerization reaction products for the conditions cEG and 0 in the same gel, as shown in Figure 1. Although NINT (l0 (40)) − NINT (l0 (0))= 0 for the two lanes in Figure 1, only slight differences in integrated intensities could lead to values of −1 or −2. However, a shift of NINT (l0 (40)) − NINT (l0 (0)) to −1 would be accompanied by a change of −1.0 in the corresponding value of ΔΔl0, and a shift to −2 would be accompanied by a change of −2 in the corresponding value of ΔΔl0, so the value reckoned for δl0 would remain unaltered in any case. The mean values of the measured δl0 for each condition are listed in Table 1. Standard error propagation is used to calculated the standard deviations, σΔΔl0 and σδl0, of the mean ΔΔl0 and δl0 values from the standard deviations, σΔl0, of the mean Δl0 values for each condition. In calculating the mean Δl0 and standard deviation σΔl0 for all data sets at a given condition, the most populous topoisomer must be chosen to yield the same value of NINT (l0 (cEG)) for each of those data sets. The σδl0 values are all rather small, as indicated in Figure 4 by the error bars at ± σδl0, which in most cases are separated by less than the symbol diameter.
Thermodynamic activities (aw) of water in EG-containing solutions were estimated as follows. For EG concentrations below 15 w/v%, the aw were derived from vapor osmometry measurements of the topoisomerization (Topo I) buffers, as reported previously.17 The vapor pressure osmometry method apparently becomes less reliable for EG concentration ≥ 15 w/v%, so aw -values for those solutions were reckoned in the following way. First, ln aw was determined from published freezing point (Tf) data on EG-water solutions39 according to the following equation,
The constants, R=1.9872 cal/(mol-K), , and Δ fus (273)= 1435.6 cal/mol, were taken from the Handbook of Chemistry and Physics.40 The −ln aw values obtained from equation (36) apply strictly to EG-water solutions at 273 K, but are assumed to apply also at 310 K. The −ln aw vs. w/v% EG data were accurately fitted by the relation, f(x)=0.0035 + 0.0018 x + 0.0001 x2, where f −ln aw and x w/v%EG. This relation is expected to apply for EG-water solutions. However, the buffers used in topoisomerization experiments contained also 2.5 w/v% glycerol in addition to other buffer components. Consequently, the −ln aw values of those buffers exceed those of the corresponding EG-water mixtures. Thus, for Topo I buffers, the best-fit curve was shifted upward so as to practically coincide with the vapor pressure osmometry measurements at 0, 5, and 10 w/v% EG. The resulting curve, ftopo(x)=0.011 + 0.0018 x + 0.0001 x2, was used to estimate the −ln aw values for all Topo I buffers, which are presented in Table 1. Because the concentration of small-ions plus small neutral buffer components, including the 2.5 w/v% glycerol, are practically the same in the CD buffer as in the Topo I buffer, the same water activities are expected to apply to those solutions as well.
The best-fit values of ET are plotted vs. −ln aw in Figure 3, where the reverse sigmoidal shape is plainly apparent. It is desired to fit equation (29), with given by equation (18), to these data. There are five adjustable parameters, namely ET1, ET2, K0, ΔΓ, and J. From the shape of the data “curve”, one might presume that it could provide reasonable estimates of four quantities, namely the upper and lower plateau values, ET1 and ET2, the midpoint value, (−ln aw)1/2, and the slope at the midpoint, . This is indeed the case, as shown below.
Initially, it was assumed that J = 1.0, corresponding to a non-cooperative model, and a four-dimensional grid search was undertaken to find optimum values of ET1, ET2, ΔΓ, and K0. This yielded a fairly robust value, ET1 = 979, and a plausible range of ET2 -values from 880 to 894. Trial values of ET2 within this range were selected, the optimum values of K0 and ΔΓ were determined for each selection, and the corresponding value of was computed in each case. The minimum value, χ02 = 12.91 was found for ET1 = 979, ET2 =887, K0 = 0.067, and ΔΓ = −30.0. These best-fit parameters are collected in Table 2. The probabililty, F(χ2 > χ02) = 0.250, indicates an acceptable fit of the theory to the experimental data. This is confirmed by visual comparison of the best-fit curve with the data in Figure 3.
The standard deviation of ET2 was estimated as that deviation from the optimum value, 887, for which , where χ2 applies for the deviated value of ET2. This protocol yields σET2 5.0. It may be inferred that there is no statistically significant difference among ET2 -values in the range 884 to 890. The variances, and , and normalized covariance, , were estimated from elements of the curvature tensor of the χ2 surface (as a function of δΔΓ and δK0) at its minimum, as described elsewhere.41 The estimated relative error in K0 is, , and that in ΔΓ is, . However, these values cannot be simply interpreted, because the normalized covariance, τ −0.8, is large and negative, which implies that deviations of K0 from its most probable value are strongly anti-correlated with deviations in ΔΓ from its most probable value, as expected.41 Perhaps more relevant is the fact that the implied value of increases with decreasing ET2 from 0.083 (for ET2 = 892) to 0.090 (for ET2 = 887) to 0.097 (for ET2 = 882). Given the estimated uncertainty, σET2 5.0, in ET2, the associated relative uncertainty in is then ± 0.007/0.090 = ± 0.08.
Different models with the same values of ET1 and ET2, (−ln aw)1/2, and give almost equally good fits to our data, and are henceforth referred to as corresponding models. As can be seen from equations (20a,b) and (31), it is possible to vary J subject to the constraint of constant midpoint slope by introducing a compensating change in ΔΓ, and in turn to vary ΔΓ subject to the constraint of constant (ln aw)1/2 by introducing a compensating change in K0. Thus, there is a continuum of corrersponding models that all give nearly equivalent fits to our data. If the data in the wings of the transition were sufficiently precise, it would be possible to narrow considerably the allowed ranges of J, ΔΓ, and K0, but that is not the case here.
In all events, the negative midpoint slope, , indicates that ΔΓ must be negative, since J is inherently positive. Thus, there is evidently less water and/or more EG in the neighborhood of a subunit in state 2 than in the neighborhood of a subunit in state 1. In view of equation (4), either the exchange contribution (S) or the excluded volume contribution (X), or both, must be smaller in state 2 than in state 1. X is reduced primarily by decreasing the (hard-core) surface area of the DNA, and S is decreased either by increasing the equilibrium constants for osmolyte-solvent exchange at one or more sites, or by reducing the number of exchange sites, which is equivalent to a reduction in DNA surface area. Thus, DNA in state 2 has either a smaller hard-core surface area or a higher affinity of one or more exchange sites for EG, or both.
An absolute value, |ΔΓ| |Γ2|w − Γ1|w = 30.0 per bp, seems much too large for an osmolyte like EG, which is net excluded (relative to water) from the DNA surface.42 Studies of the effect of EG on the melting temperature indicated that ΔΓ −4.0 for that transition between very different structures with surfaces of rather different size and atomic composition.43 Because the present transition appears to involve only rather modest changes in structure, with doubtless much smaller changes in surface size and atomic composition, |ΔΓ| should be considerably less than 4.0. Studies of the effect of sucrose and other polyols on the B□ Z transition of poly(dG-dm5C) were combined with an independent estimate of J to infer that ΔΓsuc = −2.5, when the osmolyte is sucrose.44 The value of (ln aw)1/2 for EG exceeds that for sucrose by the ratio 8.5/3.9.44 In view of equation (20a), this suggests that for EG, ΔΓ = ΔΓsuc/(8.5/3.9) = −1.15. Because that B□ Z transition involves a much more extensive structural rearrangement than the present transition, it is most probable that ΔΓ for EG and the present transition lies in the range, −1.0 ≤ ΔΓ ≤ 0.0.
For illustrative purposes, we now assume that ΔΓ= −1.0 for the present transition, in which case the value, 1/J = 30.0, is required to match the midpoint slope of the best-fit non-cooperative curve of vs. −ln aw. A value, 1/J ≥ 1.0, implies that the transition is cooperative. In addition, −ln K0 must decrease by 30-fold in order to attain the same midpoint value, (ln aw)1/2 = −ln K0/ΔΓ= −0.09010, as that of the non-cooperative curve. This change yields, , for the cooperative model, which is denoted by the superscript c. The predicted curve of vs. −ln aw for the cooperative model with ΔΓ = −1.0, 1/J = 30.0, and is displayed in Figure 3, and the relevant parameters are listed in Table 2. The apparent fit to the data is practically as good as that for the original non-cooperative curve, despite the fact that it is merely a “prediction” that preserves the same values of ET1, ET2, (−ln aw )1/2, and , rather than a true fit.
If ΔΓ = −1.0, as assumed here, then 1/J = 30.0, and the average domain size at the midpoint is d = 31 base-pairs. There is no information to preclude the possibility that |ΔΓ| 1.0, in which case 1/J and d would exceed 30 base-pairs by many fold.
The fractions of base-pairs in the 2-state that are predicted by equation (18) for the best-fit non-cooperative model (J= 1.0, ΔΓ = −30.0, K0 = 0.067) and the corresponding cooperative model (ΔΓ = −1.0, J = 1/30.0, ) are compared in Figure 3. These curves are practically identical, except in the wings, and they also converge to identical limiting values sufficiently far in the wings.
We can estimate the fractions, , of base-pairs existing in the 2-state under standard 0.1 M NaCl conditions, where aw = 0.996 (i.e. in the complete absence of ethylene glycol and glycerol) in the following way. When this value of aw is substituted into equation (18), it yields, for the best-fit non-cooperative model, and for the corresponding cooperative model with ΔΓ = − 1.0. Thus, the best-fit non-cooperative model predicts that ~7% of the base-pairs exist in the 2-state under standard 0.1 M NaCl conditions, and this estimate rises with increasing assumed cooperativity of the transition to ~ 10% of the base-pairs when J = 1/30.0!
The experimental values of δl0 are plotted vs. −ln aw in Figure 4. The effect of increasing EG from 0 to 40 w/v% is to reduce the intrinsic twist by ~0.84 ± 0.1 turn out of ~472 total turns. Thus, the relative change in intrinsic twist (~0.18%) is very slight.
It was shown previously that, if l01 and l02 are the intrinsic twists of states 1 and 2, respectively, then17
where is given by equation (18), and , where is the value of , when −ln aw = 0.011, which corresponds to the Topo I buffer with 0 w/v% EG. With this value of Δa, equation (37) yields δl0= 0, when −ln aw = 0.011, as required.
Our next goal is to obtain a reasonable estimate of l02 − l01, and with that to show that the measured δl0 vs. −ln aw data are consistent with model parameters already determined by fitting equations (29) and (18) to the ET vs. −ln aw data. A theoretically predicted curve of δl0 vs. − ln aw is generated in the following way. First, is calculated for arbitrary values of −ln aw according to equation (18) for the non-cooperative model with its best-fit parameters (J=1.0, ΔΓ = −30.0, K0 =0.067). Then l02 − l01 is approximated simply by the measured value of δl0 at 40 w/v% EG, namely l02 − l01= − 0.836. Use of these results in equation (37) yields the solid curve in Figure 4. This “predicted” curve accounts fairly well for the observed trend of the data, despite the absence of adjustable parameters. The predicted |δl0| in 40 w/v% EG is ~ 10% smaller than the measured value, due to the fact that ~10 % of the total drop, |l02 − l01|, has already “occurred” by the first point at −ln aw = 0.011 (i.e. 0 w/v% EG), where δl0 is defined to vanish. Hence, the good agreement in the tail of the curve is due in part to the fact that the measured |δl0| at 40 w/v% EG, which was used to estimate |l02 − l01|, fortuitously exceeds the mean of the last three plateau |δl0|-values (at 35, 37.5, and 40 w/v% EG) by just the right amount. Any method of selecting δl0 for 35, 37.5, and 40 w/v% EG that yields a reasonable fit of the terminal plateau, together with values computed using the already determined best-fit parameters, will predict a δl0 vs. −ln aw curve in fairly good agreement with the observed trend of the data. When values are calculated for the corresponding cooperative model (ΔΓ = −1.0, J = 1/30.0, ) and inserted in equation (27), the dashed curve in Figure 4 is obtained. Again, the predicted curve follows the trend of the data rather well, although an increase in the assumed value of |l02 − l01| would improve the agreement with experiment. These results demonstrate that the δl0 vs. −ln aw data are consistent with the same model that fits the ET vs. −ln aw data.
Deviations of the δl0 data from any smooth curve are quite large compared to the statistical errors of the individual values for each concentration of EG. This suggests the presence of some unidentified and apparently random sample-to-sample error that greatly exceeds the statistical errors from fluctuations within the data set for any given concentration of EG. In this regard, the ET vs. −ln aw data appear at first glance to be different, with much less indication of excess variance beyond that due to statistical fluctuations within a given data set. However, in some cases the full data set for a given reaction condition consists of multiple data sets, each comprising results for ~12 gel lanes. In such cases, the variance of the mean ET-values of the multiple data sets significantly exceeds the variances of the ET-values for any single data set (~12 gel lanes). This implies a significant, but apparently random, reproducibility error from one 12-lane data set to another, even though the prevailing conditions are nominally identical. Conceivably, both the excess variance of δl0 and the reproducibility error in ET arise from some ultra-slowly relaxing metastable secondary structure(s) with different intrinsic twist(s) and ET -values that prevail in circular DNAs subsequent to relaxation of native superhelical stress. Our laboratory has previously reported many examples of such metastable secondary structure appearing subsequent to changes in superhelical stress caused by linearization, binding of intercalating dyes, or binding of single-strand binding protein from E. coli.2–8 The existence of metastable secondary structure in p30δ DNA immediately after relaxation of superhelical stress and acceleration of its equilibration by topiosomerase I41 will be described in a forthcoming paper.
Each raw CD spectrum was smoothed by standard Fourier filtering techniques, and both the raw and smoothed spectra were reported elsewhere.41 This filtering/smoothing yields good visual fits to the raw data at the maximum near 272 nm and the minimum near 245 nm, but not at wavelengths much longer than 272 nm or much shorter than 245 nm. The CD decreases with increasing w/v%EG at both 272 and 245 nm, as indicated in Table 3, and no isosbestic point was observed at any wavelength.
At all wavelengths ≥ 240 nm, the spectra for 10, 20, and 30 w/v% EG lie closer to each other than to the curves for 0 and 40 w/v% EG, which is contrary to expectation for a simple sigmoidal transition.
The failure of the present CD spectra to approach plateau values at low and high w/v% EG differs qualitatively from the behavior observed for previously studied structural transitions.44–50 One may ask whether and how it might be possible for such behavior to be compatible with a structural transition. One possibility is that the CD at 272 nm consists of an ordinary sigmoidal transition superposed on a linearly descending contribution. Unfortunately, the present CD data lack sufficient accuracy for quantitative analysis, except near 272 nm, where the separation between the spectra is the greatest. We consider the possibility that the CD values at 272 nm relative to that in 0 w/v% EG could be expressible as, , where y w/v% EG/100, −a is the slope of the linearly descending contribution, and b and c are the additional contributions per unit fraction of base-pairs in the 1- and 2-states that are superposed on the linear contribution. The S (y) CD272 (y)/CD272 (0) are tabulated in Table 3. The above expression can be simplified to give,
where d c−b is the amplitude of the sigmoidal transition. The are reckoned from equation (18) using the best-fit non-cooperative model (J = 1.0, ΔΓ =−30.0, K0 = 0.067) obtained for the ET vs. −ln aw data. The optimum parameters, a=1.7479, b=0.9639, d=0.2983 were obtained by a least-squares fit of equation (38) to the experimental S(y) values. The best-fit theoretical values, Sth(y), were computed from equation (38) using the optimum values of a, b and d, and are also presented in Table 2. In every case, the agreement with experiment is within ~1.5 % or less, which is comparable to the relative errors in the CD measurements. This indicates that equation (38) with -values reckoned from equation (18) for the best-fit non-cooperative model gives a satisfactory account of the CD data. In that sense, the CD272 data are consistent with the independently determined non-cooperative model.
The origin of the −ay term in equation (39) is unknown. In principle, it could arise from changes in coordinates to which the CD is sensitive, but the twisting and bending elastic constants and intrinsic twist are not. Perhaps such coordinates pertain to water molecules or ions in contact with the bases, whose displacement could significantly alter the excited and/or ground states of the bases, and thereby affect their CD spectra.
The present results differ quantitatively from those of an earlier study17 in two regards: (i) the present ET -values for 0 and 10 w/v% EG lie below the corresponding previous values by ~3–4 %; and (ii) the present ET -value for ~ 20 w/v% EG lies above the corresponding previous value by ~ 4 %. These deviations significantly exceed the estimated statistical errors, which are indicated by the error bars in Figure 3. Consequently, the midpoint slope, (ET/ln aw)1/2, in the present study is substantially lower than that obtained previously.
The origins of the differences in results between the two studies are unknown. It is noteworthy that the compositions of the present and previous Topo I buffers differ in certain regards, due primarily to a difference in composition of the supplier’s buffer in which the Topo I is shipped. Specifically, the present buffer contained more bovine serum albumin (30 vs. 19 μg/mL, less glycerol (2.5 vs. 4.7 w/v%), and less Topo I activity (0.1 vs. 0.65 units/μL).
Due to experimental difficulties, torsion elastic constants could not be measured directly in the Topo I buffer, or even in TE buffer (100 mM NaCl, 10 mM Tris-HCl, 1 mM EDTA, pH 8.0) with 40 w/v% EG, but could be measured in TE buffer with 0 and 20 w/v% EG. Moreover, these measurements were made at 40 °C, instead of 37 °C. The resulting αeff -values are listed in Table 4. All of the other αeff -values in Table 4 are estimated by correcting these two values. First, these αeff are multiplied by 1.034 to correct from 40 to 37 °C. This factor was obtained from the slope, αeff/T, defined by previous measurements of αeff vs. T in TE buffer with 0 w/v% EG.14 The Topo I buffer contains 9.4 mM MgCl2 in ~71.4 mM monovalent “salt”. It was found previously that adding 5.5 mM MgCl2 to 30 mM monovalent salt inceased αeff by 1.23-fold. The Topo I buffer has ~1.7-fold more Mg2+, but also 2.4-fold more monovalent cations, which act to reduce the concentration of the Mg2+ near the DNA. In the absence of further information, the αeff -values in TE buffer at 37 °C are simply multiplied by 1.23 to estimate the values in the corresponding Topo I buffer, which are also listed in Table 3. This procedure likely underestimates the actual αeff -values in the Topo I buffers. Implicit in such a correction are three assumptions: (i) Mg2+ affects the torsional rigidities of the 1- and 2- states similarly; (ii) Mg2+ does not alter the two-state equilibrium, apart from its (negligibly small) effect on the water activity; and (iii) the effect of Mg2+ is not significantly modulated by the presence of 20 w/v% EG.
Incorporating the αeff and values appropriate for Topo I buffer containing first 0 and then 20 w/v% EG into equation (27) yields two simultaneous equations that can be solved for α1 and α2. The prevailing fractions, , of base-pairs in the 2-state in the Topo I buffer at 37 °C can be estimated via equation (18) using the parameters for either the best-fit non-cooperative model or the corresponding cooperative model with ΔΓ = − 1.0, which are listed in Table 2. The resulting values of , α1, and α2 are listed under their respective categories in Table 4. In Topo I buffer, we find α2/α1= 1.97 for the best-fit non-cooperative model, and α2/α1= 2.11 for the corresponding cooperative model with ΔΓ = − 1.0.
It is also desired to estimate α1 and α2 in the TE buffer at 37 °C for purposes of comparison with previously measured torsion elastic constants. If the 1□ 2 equilibrium is not affected by the Mg2+ in the Topo I buffer, as assumed here, but only by the water activity, then the same values of the parameters (J, ΔΓ and K0) should also apply in the TE buffer at 37 °C. Hence, the -values at 37 °C in the TE buffer are reckoned from the known values of −ln aw via equation (18), and then α1, and α2 are reckoned as before. The resulting values of , α1, and α2 for the best-fit non-cooperative and corresponding cooperative models in TE buffer are also listed in Table 4. The estimated torsion elastic constant of the 2-state in TE buffer, namely α2 = (10.70–11.40) ×10−12 erg, lies in the range, (9.0–12.0)×10−12 erg, that has been observed for sufficiently small circular DNAs (N ≤ 250 bp) and for DNAs under tensile stress in similar buffers.24
The bending elastic constants in the Topo I buffer are estimated by rearranging equation (26) to yield (for j=1 or 2),
We consider the best-fit non-cooperative model first. Inserting ET1 =979, α1=6.08×10−12 erg, and Bw =0.594 yields κβ1 =5.49×10−12 erg, and inserting ET2 =887, α2 =11.99×10−12 erg, and Bw =0.594 yields κβ2 =3.86×10−12 erg. The ratio, κβ2/κβ1 = 0.70, is independent of the value of Bw, provided that Bw is truly constant. For the corresponding cooperative model with ΔΓ = −1.0, similar calculations are performed using α1=5.95×10−12 and α2 =12.54×10−12 erg to obtain κβ1 =5.56×10−12 erg, κβ2 =3.83×10−12 erg, and κβ2/κβ1 =0.69. The relative decrease in bending elastic constant is virtually the same for either model.
Whenever intrinsic bends contribute negligibly to the mean squared curvature, the effective persistence length can be related to the effective bending elastic constant according to, Peff = hκβ/kT, where h=3.4 ×10−8 cm is the rise per bp, which is assumed to be practically the same for the 2 -state as for the 1-state. Evidence that intrinsic bends in a native DNA contribute negligibly to its Peff has been presented,51 and that finding is assumed to apply here to both the 1- and 2-states. The estimated persistence lengths are then P1= 436 Å and P2 =307 Å for the non-cooperative model in the Topo I buffer at 37 °C.
Upon undergoing the transition, the change in intrinsic twist is extremely slight (~0.84 turn) and the changes in ET and the CD spectrum are relatively modest. These features would be consistent with a scenario, wherein only a small fraction of the total sequence undergoes the transition with increasing EG in the Topo I buffer. However, such a model cannot be reconciled with the observed substantial (2.0- to 2.1-fold) increase in torsion elastic constant from α1 to α2. An estimate of the absolute minimum fraction of the sequence undergoing the transition ( ) can be obtained by assuming that in ~ 40 w/v% EG a fraction ( ) has α1 =6.08×10−12 erg, and the fraction has α2 = ∞, so the effective torsion elastic constant is given by , which gives . At least 49 % of the sequence must undergo the transition! Moreover, if α2 takes any value between ∞ and 11.99×10−12 erg (Table III), then must take an appropriate value between 0.493 and 1.0. If α2 = 12.0×10−12 erg, corresponding to the largest observed values of the torsion elastic constant, which have been measured for both small circular DNAs with N ≤ 250 bp and large linear DNAs under tension,24 then the fraction, , of the entire sequence must undergo the transition. Thus the absolute minimum fraction undergoing the transition in any case is substantial, and with a reasonable estimated upper limit for α2, the corresponding lower limit fraction undergoing the transition is 1.00. In the absence of any information to the contrary, it is assumed in this work that the entire sequence undergoes the transition.
Structural transitions induced in dilute calf-thymus DNA at 27 °C by increasing concentrations of several salts were studied extensively by CD spectroscopy.47,50 CD spectra of the NaCl, KCl and less concentrated NH4Cl solutions could in each case be represented as linear combinations of two limiting spectra, which were qualitatively assigned as normal B (low to moderate salt) and normal C (high salt) spectra.47 Although these limiting B- and C-spectra varied somewhat from one salt to another, the C-spectra all exhibited modest negative values in the 260–278 nm ‘band’, instead of the large positive values exhibited by the B-form, but at 245 nm there was little or no significant difference between the B- and C-spectra. The CD spectra in LiCl and CsCl could not be simply represented in terms of two components, and it was suggested that a third A-type component was also involved.50 However, the CD spectrum inferred for the A-form is practically identical to that of the later discovered left-handed Z-helix, rather than that eventually associated with A-helix.49 Subsequent topological studies of PM2 DNA in 3.0 M CsCl, 6.2 M LiCl, and 5.4 M NH4Cl indicated that the decrease in CD at 275 nm was a universal function of the increase, Δ0, in the intrinsic succession angle above that in 0.05 M NaCl, regardless of which salt was used to induce the structural transition.48 Moreover, the net change in Δ0 to go from the B-spectrum in 0.05 M NaCl to the C-spectra in 3.0 M CsCl, 6.2 M LiCl, or 5.4 M NH4Cl was rather small, in the range +0.7 to +0.8 degree/bp, corresponding to a change of −0.20 to −0.23 bp/turn in the helix repeat. For this reason, the B- and C-conformations in solution are now commonly referred to as, respectively, the B(10.4) and B(10.2) conformations.
The disappearance of B-DNA was correlated with a decrease in the net preferential hydration, which occurs with increasing salt concentration and depends almost exclusively on the water activity, independent of the particular salt used.50 Sedimentation studies indicated that increasing salt concentration induced a more compact structure, due either to greater flexibility or intrinsic curvature or to weaker interduplex repulsions.50
One may inquire whether the present state 2 induced by 40 w/v% EG is related in any way to the B(10.2) and/or Z- or A-type structures induced by high concentrations of various salts. Although both EG and salts reduce aw, their effects on the CD spectrum differ significantly. Salts reduce the CD primarily at 260–278 nm, but not at 245 nm, whereas EG lowers the CD significantly in both regions, as indicated in Table 3. Also, for the same water activity, salts lower the CD at 272–275 nm considerably more than does EG. For example, aw is approximately 0.78 in both 40 w/v% EG and 5.70 molal (m) NaCl, yet the CD in 40 w/v% EG is reduced by ~0.59-fold, whereas that in 5.70 m NaCl is reduced by ~0.42-fold.47 Interestingly, the sigmoidal part of the change in CD272 associated with the 1 □ 2 transition is positive, although it is overwhelmed by the negatively sloped linear contribution, upon which it is superposed. The change in intrinsic twist induced by 40 w/v% EG, namely Δ0 = (0.84)(360)/4932=−0.061 degree/bp, is extremely small and negative, in contrast to the small (but much larger) positive changes, Δ0 =+0.07 to +0.08 degree/bp, induced by 3.0 M CsCl, 6.2 M LiCl, and 5.4 M NH4Cl.48 Finally, the torsion elastic constants of native and modified sequences from pBR322 declined by 0.60- to 0.73-fold between 0.1 M and 5.25 M NaCl.9 In contrast, addition of 40 w/v% EG to 0.1 M NaCl TE buffer causes a ~10.70/5.16 = 2.1-fold increase in torsion elastic constant. These findings indicate that the present 2-state is a distinct subconformation within the B-family with its own characteristic properties and is not obviously related to any previously reported structure induced by high concentrations of NaCl, KCl, LiCl, CsCl, or NH4Cl. The different states induced by high salt concentration on one hand and by 40 w/v% EG on the other both exhibit smaller apparent persistence lengths than that prevailing in 0.1 M NaCl with 0 w/v% EG, but the origins of such behavior are presently unknown, and could in principle differ between the two dehydration conditions.
The dielectric constant takes the values 72.7, 67.8, and 61.7 in, respectively, 0, 20 and 40 w/v% EG. Previous studies of the effects of ethylene glycol on the melting transition43 and of sucrose on the B □ Z transition44 suggested that the effect of such polyols to lower the dielectric constant did not contribute significantly, compared to simple dehydration, to stabilize the final states. This circumstance is assumed to prevail in the present work.
Increasing the EG concentration from 0 to 40 w/v% induces a sigmoidal transition of duplex DNA to an alternative duplex state. This transition is satisfactorily modeled by a two-state, 1 □ 2, transition. Compared to the 1-state, the 2-state exhibits a ~2.0- to 2.1-fold larger torsion elastic constant, a ~0.70-fold smaller bending elastic constant, a 0.2 % lower intrinsic twist, and a somewhat lower CD near both 272 and 245 nm. Even in the absence of EG, a small, but significant, fraction (7 to 10 %) of the base-pairs is predicted to exist in the 2-state. The change in preferential interaction coefficient, ΔΓ = Γ2|w − Γ1|w, is negative, which implies that the 2-state has less water and/or more EG in its neighborhood than does the 1-state. The present data lack sufficient precision to assess the cooperativity (1/J) of the transition, but when combined with other considerations, which argue that −1.0 ≤ ΔΓ ≤ 0, they suggest that 1/J ≥ 30.0 bp, which in turn suggests an average domain size, d ≥ 31 bp, at the midpoint of the transition.
Future studies of the effects of EG on DNAs with different sequences will likely require elaboration of the present model to incorporate sequence dependent variations of K0, ΔΓ, and J.
This work was supported in part by a grant R01 GM61685 from the National Institutes of Health.
We consider a chain of N subunits, each of which can undergo a 1□ 2 transition. Both the equilibrium twist, (rad), and the torsion elastic constant, αkj (erg), of the kth torsion spring kj between the kth and (k+1)th subunits depend upon the state, j (j=1 or 2), of the kth subunit. In the absence of twisting torques, the intrinsic twist of the chain (turns) is given by,
where and are the intrinsic twists (radians) of the springs in states 1 and 2, respectively, and is the number of subunits of the unperturbed chain that are in the 2-state. In the presence of a net torsional strain, the only stable configuration of the chain is the minimum energy configuration, wherein the net torque on every subunit vanishes, which requires that all 1-springs exhibit the same displacement, δ1, and that all 2-springs exhibit the same displacement, δ2, and furthermore that δ1 = (α2/α1) δ2. In this case, the net twist (turns) can be written as,
where n2 is the number of springs in state 2. In general, , because the imposed torsional strain shifts the 1□ 2 equilibrium, although this effect may be extremely slight in some cases. The relevant question is whether the net twisting strain suffices to significantly shift the 1□ 2 equilibrium. Combining equations (A1) and (A2) yields the net twisting strain (radians),
where is the difference in intrinsic twist per spring between the 2 and 1 states. Equation (A3) can be rearranged to yield
The total (minimum) strain energy arising from Δt is (with δ1 = (α2/α1) δ2),
In order to assess the effect of Δt on the equilibrium value of n2, we must incorporate U (n2, Δt) into the formalism for reckoning the mean value, 2. Specifically, we begin with the conformational partition function χ defined by equation (12) in the text. In the presence of net twist, Δt, this would have to be modified to,
where U(n2, Δt) has been explicitly incorporated. In fact, each kind of spring exhibits not only its minimum energy value for a given net twist, but also thermal fluctuations about that value.35 In a circular DNA with a fixed writhe and a given net twist, t, the fluctuations of individual springs are not entirely independent, but are constrained to sum to zero. The leading contribution of this constraint to the free energy per spring is ~ (kT/2N)ln N,4 which is negligibly small for N ≥ 1000 bp, and is ignored here. Hence, these fluctuations in the torsion springs may be regarded as effectively independent in sufficiently large circles. The contribution of thermal fluctuations in twist of each spring to the configuration integral and free energy of its associated subunit, relative to that of a 1-spring, is assumed to be already included in B, so only the minimum energy arising from the net twist, U(n2, Δt), must be incorporated into χ. Because U(n2, Δt) is not linear in n2, the matrix method cannot be employed directly. However, because U(n2, Δt) is independent of the arrangement of the n2 2-states among the subunits, the configuration sum,
is independent of, and unaffected by, U(n2, Δt), and must take the same value for a given n2, regardless of the value of Δt or U(n2, Δt).
For a sufficiently long chain, it is permissible to replace ln χ by ln(max term), where maxterm is the largest term in the sum.52 Thus,
where the subscript indicates the maximum value, which is assumed to occur, when n2 takes its mean value in the presence of Δt. From equation (13) we also have for the average value,
which remains valid, despite the incorporation of exp[−U(n2, Δt)/kT] into χ. The condition for the maximum term in equation (A8) is,
The solution of equation (A11) for n2 gives the mean value, 2.
The last term in equation (A11) can be evaluated by considering first the case, when U(n2, Δt) = 0 for all n2, so the second term can be omitted. Then after exponentiating both sides and multiplying by f2 = n2/N, we obtain f2 as the solution of
It is clear from equation (A7) that W(n2) depends only upon n2 and J, but not upon B or U(n2, Δt), hence F(n2) also depends only upon n2 and J.
When U(n2, Δt) = 0, the quantity F(n2) can be found in the following way. The characteristic equation for the transfer matrix, M, in equation (16) of the main text is
Implicit differentiation leads to
Solving (A16) for f2 yields,
where λ is given in terms of f2 and J by (A17). Thus, in the present large N limit, F(n2) is a function only of f2 = n2/N and J. Although F(n2) in equation (A19) was derived for the case U(n2, Δt) = 0, it must also apply for arbitrary U(n2, Δt), as noted above.
From equation (A5), we find,
When equation (A19) for F(n2), (A17) for λ, and (A21) are substituted into equation (A20), there results a transcendental equation, which can be solved numerically for n2 = 2. The in (A21) is the corresponding solution of equation (A12), when U(n2, Δt) = 0. Note that, when Δt = 0, both U(n2, Δt) and U(n2, Δt)/n2 vanish for , as expected.
The effect of the twisting strain present in a supercoiled DNA to shift n2 away from can be estimated in the following way. The linking difference, Δl = l − l0, is distributed between net twist, Δt = t − l0, and writhe, w (turns), according to Δl = Δt + w. Under standard conditions, Δt accounts for about 1/3 of Δl and w accounts for about 2/3 of Δl. The most prominent topoisomers in a thermally equilibrated population lie in the range Δl = −2 to +2. Under standard conditions (with Δl = +2), one would expect Δt □ 2/3 turn, whence 2πΔt = 4π/3 radians. The spring constants of the 1- and 2-states in Topo I buffer are estimated to be α1 = 6.08×10−12 erg and α2 = 11.99×10−12 erg, respectively (c.f. Table 3). It is also found 0 in the present study that (l20 − l10) −0.84 turns, so Δ0 = 2π(l20 − l10)/4932 = −0.0011 radians/bp. We assume that B=1.0, so that, when Δt = 0, the solution is , as is evident from equation (18) in the main text. We consider this choice (B=1.0), because many properties, such as the effective elastic constant, are maximally sensitive to perturbation of the chemical equilibrium at its midpoint. We now consider a small non-vanishing value of Δt, and imagine that equation (A20) is to be solved by iteration, beginning with the choice, on the right-hand side. With this choice, then, on the right-hand side one has, f2 = 1/2, λ =1+ J (from (A17)), and F(n2) = 1/2 (from (A19), so that (A20) now reads
where the exponential factor is to be evaluated at . Numerical evaluation of the exponent yields,
The second term in braces in equation (A21) yields a positive contribution that is about 4.0 times that of the first term for Δl = +2. The right-hand side of equation (A23) differs negligibly from the input value, f2 = 1/2, which indicates that the populations are not significantly affected by the net twisting strain, Δt, in this example. If the effective equilibrium constant, B, were increased from 1.0 to exp[+0.00022], then equation (A20) would be precisely satisfied by f2 = 1/2. Thus, the effect of the twisting strain in a topoisomer with Δl = +2, when the transition is near its midpoint, is equivalent to a 1.00022-fold decrease in the effective equilibrium constant, B, for the 1 □ 2 transition. The positive twisting strain evidently favors very slightly the 1-state, primarily because of its greater intrinsic twist, and to a lesser extent because of its smaller torsion elastic constant.
We have no information regarding the difference in curvature between states 2 and 1. Nevertheless, the net bending strain required to close the 4932 bp DNA into a circle and add 4/3 of a writhe turn is also extremely slight, and is believed not to significantly shift f2 away from .
Because neither the twisting nor bending strain significantly shifts the 1 □ 2 equilibrium, no significant error is encountered in the present study by assuming that .