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J Phys Chem B. Author manuscript; available in PMC 2010 March 30.

Published in final edited form as:

Published online 2008 September 30. doi: 10.1021/jp802139a

PMCID: PMC2847613

NIHMSID: NIHMS178775

Department of Chemistry, University of Washington, Box 351700, Seattle, WA 98195-1700

Email: ude.notgnihsaw.mehc@rruhcs, Telephone: 206 543 6681, FAX: 206 585 8665

The publisher's final edited version of this article is available at J Phys Chem B

The twist energy parameter (*E _{T}*) that governs the supercoiling free energy, and the linking difference (Δ

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 *κ _{β}*, for torsion and bending, respectively, the twist energy parameter,

At 37 °C, increasing concentrations of the neutral osmolytes, ethylene glycol (EG) and acetamide, caused initially sigmoidal decreases in the respective curves of *E _{T}* vs. −ln

A principal goal of the present study is to measure a complete curve of *E _{T}* vs. −ln

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,

…1111222222211111111112222211111112222…

Such a specific configuration is denoted by the index I, and a DNA with the Ith configuration is denoted by *D _{I}*. In purely aqueous buffer (0 w/v% neutral osmolyte), the standard state chemical potential of

$${\mu}_{{D}_{I}}^{0}\left({c}_{w},{c}_{os}\right)={\mu}_{{D}_{I}}^{0}\left({c}_{w}^{\square}\right)-{\mathrm{\Gamma}}_{{D}_{I}\mid w}\xb7\mathit{kT}ln{a}_{w}$$

(1)

where k is Boltzmann’s constant, T is absolute temperature, and

$${\mathrm{\Gamma}}_{{D}_{I}\mid w}\equiv -{\left(\partial {\mu}_{{D}_{I}}^{0}/\partial {\mu}_{w}\right)}_{T,P,{c}_{{D}_{I}}^{\infty}}=-(1/kT){\left(\partial {\mu}_{{D}_{I}}^{0}/\partial ln{a}_{w}\right)}_{T,P,{c}_{{D}_{I}}^{\infty}}$$

(2)

is a *preferential interaction coefficien*t (PIC) of *D _{I}* with water. In equation (1) it is

$${\mathrm{\Gamma}}_{{D}_{I}\mid w}={c}_{w}\left({\int}_{0}^{\infty}d\mathbf{r}\left({g}_{{D}_{I}\mid w}(r)-{g}_{{D}_{I}\mid os}(r)\right)\right)$$

(3)

where *g*_{DI|w} (*r*) and *g*_{DI|os} (*r*) are the *D _{I}* −

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 *D _{I}* 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

$${\mathrm{\Gamma}}_{{D}_{I}\mid w}=X+S$$

(4)

where
$X=\left({V}_{os}^{ex}-{V}_{w}^{ex}\right)/{\overline{V}}_{w}$ is the difference between the volumes excluded by *D _{I}* to osmolyte centers,
${V}_{os}^{ex}$, and water centers,
${V}_{w}^{ex}$, in units of the partial molecular volume of water,

$$S=\sum _{i=1}^{L}\left(\nu -({c}_{w}/{c}_{os}){K}_{i}{a}_{os}{({a}_{w})}^{-\nu}/\left(1+{K}_{i}{a}_{os}{({a}_{w})}^{-\nu}\right)\right)$$

(5)

where the i-sum runs over a lattice of L sites of equal volume, * _{os},* that completely fill the space accessible to osmolyte centers,

$${\mathrm{\Gamma}}_{{D}_{I}\mid w}=\sum _{k=1}^{N}{\mathrm{\Gamma}}_{kj\mid w}$$

(6)

where Γ_{kj}_{|}* _{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
${\mu}_{{D}_{I}}^{0}\left({c}_{w}^{\square}\right)$ can be parsed into contributions from base-pairs in the 1-state and those in the 2-state, plus an additional free energy, *F _{J},* associated with 12 or 21 junctions,

$${\mu}_{{D}_{I}}^{0}\left({c}_{w}^{\square}\right)=\sum _{k=1}^{N}{\mu}_{kj}^{0}\left({c}_{w}^{\square}\right)+{n}_{J}{F}_{J}$$

(7)

where
${\mu}_{kj}^{0}\left({c}_{w}^{\square}\right)$ is the contribution of the *k*th base-pair in state *j*, and *n _{J}* is the total number of 12 plus 21 junctions in the Ith conformational configuration. Incorporating equations (6) and (7) into (1) yields finally,

$${\mu}_{{D}_{I}}^{0}({c}_{w},{c}_{os})=\sum _{k=1}^{N}\left({\mu}_{kj}^{0}\left({c}_{w}^{\phantom{\rule{0.16667em}{0ex}}}\right)-{\mathrm{\Gamma}}_{kj\mid w}\xb7\mathit{kT}ln({a}_{w})\right)+{n}_{J}{F}_{J}$$

(8)

For simplicity, it is now assumed that the difference,
${\mu}_{k2}^{0}\left({c}_{w}^{\square}\right)-{\mu}_{k1}^{0}\left({c}_{w}^{\square}\right)$, takes the same value,
$\mathrm{\Delta}{\mu}^{0}\left({c}_{w}^{\square}\right)$, regardless of whether the kth subunit is a GC or an AT base-pair. Likewise, the difference, Γ_{k}_{2|}* _{w}* − Γ

The configuration, *D*_{1}, wherein every base-pair is in state 1, is adopted as a reference state. The concentration of configuration *D _{I}* relative to that of

$$\begin{array}{l}{c}_{{D}_{I}}/{c}_{{D}_{I}}=exp\left[-\left({\mu}_{{D}_{I}}^{0}({c}_{w},o{c}_{s})-{\mu}_{{D}_{I}}^{0}({c}_{w},{c}_{os})\right)\right]\\ ={(B)}^{{n}_{2}}{J}^{{n}_{J}}\end{array}$$

(9)

wherein

$$B={K}_{0}{({a}_{w})}^{\mathrm{\Delta}\mathrm{\Gamma}}$$

(10)

The quantity,
${K}_{0}=exp\left[-\mathrm{\Delta}{\mu}^{0}\left({c}_{w}^{0}\right)\right]$ 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[−*F _{J}/kT*] is the inverse cooperativity parameter, which is assumed to be insensitive to

By summing over all distinct configurations, *I* (*n*_{2}), that exhibit exactly *n*_{2} subunits in state 2, we obtain the total concentration of DNAs with *n*_{2} subunits in state 2,

$${c}_{{n}_{2}}=\sum _{I({n}_{2})}{c}_{{D}_{I}}={c}_{{D}_{I}}\xb7{B}^{{n}_{2}}\xb7\sum _{I({n}_{2})}{J}^{{n}_{J}}$$

(11)

where *n _{J}* denotes the number of junctions in the I(n

$$\chi \equiv \sum _{{n}_{2}\ge 0}{B}^{{n}_{2}}\sum _{I({n}_{2})}{J}^{{n}_{J}}$$

(12)

is the conformational grand partition function.

The average fraction of subunits in state 2 is given by

$${f}_{2}^{0}\equiv (1/{Nc}_{\mathit{tot}})\sum _{{n}_{2}\ge 0}{n}_{2}{c}_{{n}_{2}}=(1/N)B\partial ln\chi /\partial B$$

(13)

and the average number of 12 plus 21 junctions per DNA molecule is given by

$$\langle {n}_{J}\rangle =J\partial ln\chi /\partial J$$

(14)

The grand partition function, *χ*, in equation (12) is evaluated simply by summing over all possible 2* ^{N}* distinct states of the N subunits, which can be effected by using a transfer matrix,

$$\chi =Tr\left({\mathbf{M}}^{N}\right)={{\lambda}_{+}}^{N}+{{\lambda}_{-}}^{N}\cong {{\lambda}_{+}}^{N}$$

(15)

where

$$\mathbf{M}=\left(\begin{array}{cc}1& JB\\ J& B\end{array}\right)$$

(16)

and

$${\lambda}_{\pm}=(1/2)\left(B+1\pm {\left({(B-1)}^{2}+4{BJ}^{2}\right)}^{1/2}\right)$$

(17)

are its eigenvalues. The last equality in equation (15) applies in the limit of large N.

Use of equations (17) and (15) in (13) yields the fraction of subunits in the 2-state,

$$\begin{array}{l}{f}_{2}^{0}=(1/2)\left(1+\frac{B-1}{{\left[{(B-1)}^{2}+4{BJ}^{2}\right]}^{1/2}}\right)\\ =(1/2)\left(1+\frac{{K}_{0}{({a}_{w})}^{\mathrm{\Delta}\mathrm{\Gamma}}-1}{{\left[{\left({K}_{0}{({a}_{w})}^{\mathrm{\Delta}\mathrm{\Gamma}}-1\right)}^{2}+4{K}_{0}{({a}_{w})}^{\mathrm{\Delta}\mathrm{\Gamma}}{J}^{2}\right]}^{1/2}}\right)\end{array}$$

(18)

When J=1, equation (18) simplifies to the non-cooperative expression employed previously.^{17} From equation (18) one obtains the slope,

$$\partial {f}_{2}^{0}/\partial lnB=B(B+1){J}^{2}/{\left[{(B-1)}^{2}+4{BJ}^{2}\right]}^{3/2}$$

(19)

The midpoint of the transition, ${f}_{2}^{0}=1/2$, occurs when B=1.0, or equivalently when

$${\left(ln{a}_{w}\right)}_{1/2}=-(ln{K}_{0})/\mathrm{\Delta}\mathrm{\Gamma}$$

(20a)

At the midpoint the slope becomes
${\left(\partial {f}_{2}^{0}/\partial lnB\right)}_{1/2}=1/4J$. Using *d* ln *B =* ΔΓ*d* ln *a _{w}*, this becomes

$${\left(\partial {f}_{2}^{0}/\partial ln{a}_{w}\right)}_{1/2}=\mathrm{\Delta}\mathrm{\Gamma}/4J$$

(20b)

Evidently, the midpoint slope in equation (20b) is equally sensitive to ΔΓ and 1/*J*. Experimental measurements of (ln *a _{w}*)

Use of equations (17) and (15) in (14) gives the average number of 12 plus 21 junctions,

$$\langle {n}_{J}\rangle =N2{BJ}^{2}/\left({\lambda}_{+}{\left[{(B-1)}^{2}+4{BJ}^{2}\right]}^{1/2}\right)$$

(21)

At the midpoint, when B = 1.0, equations (17) and (21) give,

$${\langle {n}_{J}\rangle}_{1/2}=NJ/(1+J)$$

(22)

and the average size of domains of consecutive 1-states or 2-states at the midpoint is given in base-pairs by,

$$d=N/\langle {n}_{J}\rangle =1+1/J$$

(23)

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, *F _{J}*.

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, Δ*l = l* − *l*_{0}, where *l*_{0} 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

$$\mathrm{\Delta}l=t-{l}_{0}+w$$

(24)

Both experiments^{17}^{, }^{30}^{–}^{34} and simulations^{24}^{,}^{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 Δ*m = m* − *l*_{0} to Δ*l* is given by

$$\mathrm{\Delta}{G}_{sc}=kT({E}_{T}/N)\left(\mathrm{\Delta}{l}^{2}-\mathrm{\Delta}{m}^{2}\right)$$

(25)

wherein *E _{T}* is independent of Δ

It was previously proposed that, for DNAs in ~0.1 M monovalent salt, *E _{T}* could be approximated by,

$${E}_{t}^{\mathit{eff}}=\left({(2\pi )}^{2}/2kT\right){\alpha}^{\mathit{eff}}{B}_{w}{\kappa}_{\beta}^{\mathit{eff}}/\left({\alpha}^{\mathit{eff}}+{B}_{w}{\kappa}_{\beta}^{\mathit{eff}}\right)$$

(26)

where *α ^{eff}* and
${\kappa}_{\beta}^{\mathit{eff}}$ are the effective elastic constants for torsion and bending of the springs that are imagined to exist between base-pairs, and

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,
${\phi}_{1}^{0}$ and
${\phi}_{2}^{0}$, and the corresponding torque constants for twisting are *α*_{1} and *α*_{2}. Similarly, in regard to bending springs, the minimum energy bends are, respectively,
${\beta}_{1}^{0}$ and
${\beta}_{2}^{0}$, and the corresponding torque constants for bending are *κ*_{β1} and *κ _{β}*

When the numbers of springs of types 1 and 2 are fixed at their unperturbed values, ${n}_{1}^{0}$ and ${n}_{2}^{0}$, the effective elastic constants are found to be

$$1/{\alpha}^{\mathit{eff}}=\left(1-{f}_{2}^{0}\right)/{\alpha}_{1}+{f}_{2}^{0}/{\alpha}_{2}$$

(27)

and

$$1/{\kappa}_{\beta}^{\mathit{eff}}=\left(1-{f}_{2}^{0}\right)/{\kappa}_{{\beta}_{1}}+{f}_{2}^{0}/{\kappa}_{{\beta}_{2}}$$

(28)

where
$1-{f}_{2}^{0}$ and
${f}_{2}^{0}={n}_{2}/N$ 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 *f*_{2} is not exactly equal to its unperturbed value,
${f}_{2}^{0}$, 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 *f*_{2} = *n*_{2}/*N* is given by (A20) with *F* (*n*_{2}) given by (A19) and *λ* given by (A17). The quantity, *U* (*n*_{2}, Δ*t*) *n*_{2}, is the slope of the torsional strain energy for a given fixed net twist, Δ*t = t* − *l*_{0}, with respect to *n*_{2} (the number of subunits in state 2), and is given explicitly in terms of *n*_{2}, *α*_{1}, *α*_{2},
$\mathrm{\Delta}{\phi}^{0}\equiv {\phi}_{2}^{0}-{\phi}_{1}^{0}$, and Δ*t* in equation (A21). In principle, equation (A20) can be solved by iteration to find *f*_{2} for any fixed choice of *α*_{1}, *α*_{2}, Δ^{0}, and Δ*t*. It is shown via equations (A22) and (A23) that in the present case of extremely small Δ^{0} and small Δ*t,* corresponding to Δ*l* = +2 turns, the value of *n*_{2} must lie very close to
${n}_{2}^{0}$, so that the 1□ 2 equilibrium is practically unshifted by the net twist. The similarly small net bending strain in a topoisomer with Δ*l* = +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.

Substituting equations (27) and (28) into the inverse of equation (26) and rearranging somewhat yields,

$$1/{E}_{T}^{\mathit{eff}}=\left(1-{f}_{2}^{0}\right)/{E}_{{T}_{1}}+{f}_{2}^{0}/{E}_{{T}_{2}}$$

(29)

where
${E}_{{T}_{j}}=\left({(2\pi )}^{2}/2kT\right)\left({\alpha}_{j}{B}_{w}{\kappa}_{j}/\left({\alpha}_{j}+{B}_{w}{\kappa}_{j}\right)\right)$ is the expected value of *E _{T}*, when the entire DNA is in state j, j=1 or 2. Curves of
${E}_{T}^{\mathit{eff}}$ vs. ln

At the midpoint of the transition,

$${\left(1/{E}_{T}^{\mathit{eff}}\right)}_{1/2}=(1/2)\left(1/{E}_{{T}_{1}}+1/{E}_{T2}\right)$$

(30)

and

$$\begin{array}{l}{\left(\partial {E}_{T}^{\mathit{eff}}/\partial ln{a}_{w}\right)}_{1/2}={\left(\partial {E}_{t}^{\mathit{eff}}/\partial {f}_{2}^{0}\right)}_{1/2}\xb7{\left(\partial {f}_{2}^{0}/\partial ln{a}_{w}\right)}_{1/2}\\ ={\left(0.5/{E}_{{T}_{1}}+0.5/{E}_{{T}_{2}}\right)}^{-2}\xb7\left(1/{E}_{{T}_{1}}-1/{E}_{{T}_{2}}\right)\xb7\mathrm{\Delta}\mathrm{\Gamma}/4J\end{array}$$

(31)

Fortunately, the best-fit values of *E*_{T1} and *E*_{T2} and also the best-fit values of (ln *a _{w}*)

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 MgCl_{2}, 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 (*λ*_{max} = 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, *l*_{0}, and is denoted by NINT(*l*_{0}). The excess linking number, *l _{ex}*, of a given topoisomer with linking number, l, is defined by,

$${l}_{ex}=l-\text{NINT}({l}_{0})$$

(32)

The bands in each gel lane are indexed according to their *l _{ex}* values (…−3, −2, −1,0,+1,+2,+3,…). The ratio,

$${c}_{{l}_{ex}}/{c}_{0}=exp\left[-({E}_{T}/N)\left({({l}_{ex}+\mathrm{\Delta}{l}_{0})}^{2}-{(\mathrm{\Delta}{l}_{0})}^{2}\right)\right]$$

(33)

where Δ*l*_{0} NINT (*l*_{0}) −*l*_{0} is the linking difference of the most populous topoisomer. Equation (33) is fitted to the measured ratios of integrated intensities, *I _{lex}*/

Both *l*_{0} and Δ*l*_{0} may vary with EG concentration (*c _{EG}* (w/v%)), and are now written as

$$\mathrm{\Delta}\mathrm{\Delta}{l}_{0}\equiv \mathrm{\Delta}{l}_{0}({c}_{EG})-\mathrm{\Delta}{l}_{0}(0)=\text{NINT}({l}_{0}({c}_{EG}))-{l}_{0}({c}_{EG})-\left(\text{NINT}({l}_{0}(0))-{l}_{0}(0)\right)$$

(34)

reflects a change in *l*_{0} (*c _{EG}*) with increasing

$$\delta {l}_{0}\equiv {l}_{0}({c}_{EG})-{l}_{0}(0)=\text{NINT}({l}_{0}({c}_{EG}))-\text{NINT}({l}_{0}(0))-\mathrm{\Delta}\mathrm{\Delta}{l}_{0}$$

(35)

The variation in NINT (*l*_{0} (*c _{EG}*)) with EG concentration can be ascertained directly by running topoisomerization reaction products for the conditions

Thermodynamic activities (*a _{w}*) of water in EG-containing solutions were estimated as follows. For EG concentrations below 15 w/v%, the

$$-\mathit{RT}ln{a}_{w}=-\underset{273}{\overset{{T}_{f}}{\int}}dT\mathrm{\Delta}{\overline{H}}_{\mathit{fus}}(273)/T-\underset{273}{\overset{{T}_{f}}{\int}}dT\left({C}_{P}^{\mathit{liq}}-{C}_{P}^{\mathit{ice}}\right)+\underset{273}{\overset{{T}_{f}}{\int}}dT\left({C}_{P}^{\mathit{liq}}-{C}_{P}^{\mathit{ice}}\right)273/T$$

(36)

The constants, R=1.9872 cal/(mol-K),
${C}_{P}^{\mathit{liq}}=18.18\phantom{\rule{0.16667em}{0ex}}\text{cal}/(\text{mol}-\text{K}),{C}_{P}^{\mathit{ice}}=8.64\phantom{\rule{0.16667em}{0ex}}\text{cal}/(\text{mol}-\text{K})$, and Δ* _{fus}* (273)= 1435.6 cal/mol, were taken from the Handbook of Chemistry and Physics.

The best-fit values of *E _{T}* are plotted vs. −ln

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 *E*_{T1}, *E*_{T2}, ΔΓ, and *K*_{0}. This yielded a fairly robust value, *E*_{T1} = 979, and a plausible range of *E*_{T2} -values from 880 to 894. Trial values of *E*_{T2} within this range were selected, the optimum values of *K*_{0} and ΔΓ were determined for each selection, and the corresponding value of
${\chi}^{2}\equiv {\sum}_{j=1}^{10}{\left({E}_{{T}_{j}}^{exp}-{E}_{{T}_{j}}^{th}\right)}^{2}/{\sigma}_{j}^{2}$ was computed in each case. The minimum value, *χ*_{0}^{2} = 12.91 was found for *E*_{T1} = 979, *E*_{T2} =887, *K*_{0} = 0.067, and ΔΓ = −30.0. These best-fit parameters are collected in Table 2. The probabililty, *F*(*χ*^{2} > *χ*_{0}^{2}) = 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.

Comparison of best-fit parameters of the non-cooperative model with the implied parameters of a corresponding cooperative model.

The standard deviation of *E*_{T2} was estimated as that deviation from the optimum value, 887, for which
$exp\left[-{\chi}^{2}/2\right]/exp\left[-{{\chi}_{0}}^{2}/2\right]=exp[-1/2]=0.607$, where *χ*^{2} applies for the deviated value of *E*_{T2}. This protocol yields *σ*_{ET2} 5.0. It may be inferred that there is no *statistically significant* difference among *E*_{T2} -values in the range 884 to 890. The variances,
$\langle {(\delta \mathrm{\Delta}\mathrm{\Gamma})}^{2}\rangle $ and
$\langle {(\delta {K}_{0})}^{2}\rangle $, and normalized covariance,
$\tau \equiv \langle (\delta \mathrm{\Delta}\mathrm{\Gamma})(\delta {K}_{0})\rangle /\left({\langle {(\delta {K}_{0})}^{2}\rangle}^{1/2}{\langle {(\delta \mathrm{\Delta}\mathrm{\Gamma})}^{2}\rangle}^{1/2}\right)$, were estimated from elements of the curvature tensor of the *χ*^{2} surface (as a function of *δ*ΔΓ and *δK*_{0}) at its minimum, as described elsewhere.^{41} The estimated relative error in *K*_{0} is,
${\langle {(\delta {K}_{0})}^{2}\rangle}^{1/2}/{K}_{0}\cong 0.23$, and that in ΔΓ is,
${\langle {(\delta \mathrm{\Delta}\mathrm{\Gamma})}^{2}\rangle}^{1/2}/\mathrm{\Delta}\mathrm{\Gamma}\cong 0.08$. However, these values cannot be simply interpreted, because the normalized covariance, *τ* −0.8, is large and negative, which implies that deviations of *K*_{0} 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
${\left(-ln{a}_{w}\right)}_{1/2}$ increases with decreasing *E*_{T2} from 0.083 (for *E*_{T2} = 892) to 0.090 (for *E*_{T2} = 887) to 0.097 (for *E*_{T2} = 882). Given the estimated uncertainty, *σ*_{ET2} 5.0, in *E*_{T2}*,* the associated relative uncertainty in
${\left(-ln{a}_{w}\right)}_{1/2}$ is then ± 0.007/0.090 = ± 0.08.

Different models with the same values of *E*_{T1} and *E*_{T2}, (−ln *a _{w}*)

In all events, the *negative* midpoint slope,
${\left(\partial {f}_{2}^{0}/\partial ln{a}_{w}\right)}_{1/2}=\mathrm{\Delta}\mathrm{\Gamma}/4J=-30.0$, 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| |Γ_{2|}* _{w}* − Γ

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
${E}_{T}^{\mathit{eff}}$ vs. −ln *a _{w}*. A value, 1/

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, *K*_{0} = 0.067) and the corresponding cooperative model (ΔΓ = −1.0, *J* = 1/30.0,
${K}_{0}^{c}=0.914$) 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,
${f}_{2}^{0}$, of base-pairs existing in the 2-state under standard 0.1 M NaCl conditions, where *a _{w}* = 0.996 (i.e. in the complete absence of ethylene glycol and glycerol) in the following way. When this value of

The experimental values of *δl*_{0} are plotted vs. −ln *a _{w}* 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 *l*_{01} and *l*_{02} are the intrinsic twists of states 1 and 2, respectively, then^{17}

$$\delta {l}_{0}=-{\mathrm{\Delta}}_{a}+({l}_{02}-{l}_{01}){f}_{2}^{0}$$

(37)

where
${f}_{2}^{0}$ is given by equation (18), and
${\mathrm{\Delta}}_{a}\equiv ({l}_{02}-{l}_{01}){f}_{20}^{0}$, where
${f}_{20}^{0}$ is the value of
${f}_{2}^{0}$, when −ln *a _{w}* = 0.011, which corresponds to the Topo I buffer with 0 w/v% EG. With this value of Δ

Our next goal is to obtain a reasonable estimate of *l*_{02} − *l*_{01}, and with that to show that the measured *δl*_{0} vs. −ln *a _{w}* data are consistent with model parameters already determined by fitting equations (29) and (18) to the

Deviations of the *δl*_{0} 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 *E _{T}* vs. −ln

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,
$S(y)=-ay+{bf}_{1}^{0}+{cf}_{2}^{0}$, 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*) *CD*_{272} (*y*)/*CD*_{272} (0) are tabulated in Table 3. The above expression can be simplified to give,

$$S(y)=-ay+b+d{f}_{2}^{0}(y)$$

(38)

where *d* *c*−*b* is the amplitude of the sigmoidal transition. The
${f}_{2}^{0}(y)$ are reckoned from equation (18) using the best-fit non-cooperative model (J = 1.0, ΔΓ =−30.0, *K*_{0} = 0.067) obtained for the *E _{T}* vs. −ln

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 study^{17} in two regards: (i) the present *E _{T}* -values for 0 and 10 w/v% EG lie

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 *α ^{eff}* -values are listed in Table 4. All of the other

Incorporating the *α ^{eff}* and
${f}_{2}^{0}$ 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

It is also desired to estimate *α*_{1} and *α*_{2} 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 Mg^{2+} in the Topo I buffer, as assumed here, but only by the water activity, then the same values of the parameters (J, ΔΓ and *K*_{0}) should also apply *in the TE buffer* at 37 °C. Hence, the
${f}_{2}^{0}$-values at 37 °C *in the TE buffer* are reckoned from the known values of −ln *a _{w}* via equation (18), and then

The bending elastic constants in the Topo I buffer are estimated by rearranging equation (26) to yield (for j=1 or 2),

$${\kappa}_{{\beta}_{j}}=(1/{B}_{w})(2kT){E}_{{T}_{j}}{\alpha}_{j}/\left({(2\pi )}^{2}{\alpha}_{j}-2{\mathit{kTE}}_{{T}_{j}}\right)$$

(40)

We consider the best-fit non-cooperative model first. Inserting *E*_{T1} =979, *α*_{1}=6.08×10^{−12} erg, and *B _{w}* =0.594 yields

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, *P ^{eff}* =

Upon undergoing the transition, the change in intrinsic twist is extremely slight (~0.84 turn) and the changes in *E _{T}* 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)

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 NH_{4}Cl 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 NH_{4}Cl indicated that the decrease in CD at 275 nm was a universal function of the *increase*, Δ^{0}, 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 Δ^{0} 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 NH_{4}Cl 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 *a _{w}*, 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,

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 transition^{43} and of sucrose on the B □ Z transition^{44} 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 = Γ_{2|}* _{w}* − Γ

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 *K*_{0}, 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,
${\phi}_{kj}^{0}$ (rad), and the torsion elastic constant, *α _{kj}* (erg), of the kth torsion spring

$${l}_{0}=\left[\left(N-{n}_{2}^{0}\right){\phi}_{1}^{0}+{n}_{2}^{0}{\phi}_{2}^{0}\right]/2\pi $$

(A1)

where
${\phi}_{1}^{0}$ and
${\phi}_{2}^{0}$ are the intrinsic twists (radians) of the springs in states 1 and 2, respectively, and
${n}_{2}^{0}$ 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, *δ*_{1}, and that all 2-springs exhibit the same displacement, *δ*_{2}, and furthermore that *δ*_{1} = (*α*_{2}/*α*_{1}) *δ*_{2}. In this case, the net twist (turns) can be written as,

$$t=\left[(N-{n}_{2})\left({\phi}_{1}^{0}+\delta {\phi}_{1}\right)+{n}_{2}\left({\phi}_{2}^{0}+\delta {\phi}_{2}\right)\right]/2\pi $$

(A2)

where *n*_{2} is the number of springs in state 2. In general,
${n}_{2}\ne {n}_{2}^{0}$, 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),

$$2\pi (\mathrm{\Delta}t)=2\pi (t-{l}_{0})=\left({n}_{2}-{n}_{2}^{0}\right)\mathrm{\Delta}{\phi}^{0}+((N-{n}_{2}){\alpha}_{2}/{\alpha}_{1}+{n}_{2})\delta {\phi}_{2}$$

(A3)

where $\mathrm{\Delta}{\phi}^{0}\equiv {\phi}_{2}^{0}-{\phi}_{1}^{0}$ is the difference in intrinsic twist per spring between the 2 and 1 states. Equation (A3) can be rearranged to yield

$$\delta {\phi}_{2}=\left(2\pi \mathrm{\Delta}t-\left({n}_{2}-{n}_{2}^{0}\right)\mathrm{\Delta}{\phi}^{0}\right)/((N-{n}_{2}){\alpha}_{2}/{\alpha}_{1}+{n}_{2})$$

(A4)

The total (minimum) strain energy arising from Δ*t* is (with *δ*_{1} = (*α*_{2}/*α*_{1}) *δ*_{2}),

$$\begin{array}{l}U({n}_{2},\mathrm{\Delta}t)={n}_{2}({\alpha}_{2}/2)\delta {{\phi}_{2}}^{2}+(N-{n}_{2})({\alpha}_{2}/{\alpha}_{1})({\alpha}_{2}/2)\delta {{\phi}_{2}}^{2}\\ =(1/2)\left\{\frac{1}{(N-{n}_{2})(1/{\alpha}_{1})+{n}_{2}/\alpha 2}\right\}{\left(2\pi \mathrm{\Delta}t-\left({n}_{2}-{n}_{2}^{0}\right)\left(\mathrm{\Delta}{\phi}^{0}\right)\right)}^{2}\end{array}$$

(A5)

The second line in equation (A5) follows from the first by using equation (A4) for *δ*_{2}.

In order to assess the effect of Δ*t* on the equilibrium value of *n*_{2}, we must incorporate *U* (*n*_{2}, Δ*t*) 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, Δ*t*, this would have to be modified to,

$$\chi \equiv \sum _{{n}_{2}\ge 0}{B}^{{n}_{2}}{e}^{-U({n}_{2},\mathrm{\Delta}t)/kT}\sum _{I({n}_{2})}{J}^{{n}_{J}}$$

(A6)

where *U*(*n*_{2}, Δ*t*) 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 sprin*g is ~ (*kT*/2*N*)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*(*n*_{2}, Δ*t*), must be incorporated into *χ*. Because *U*(*n*_{2}, Δ*t*) is not linear in *n*_{2}, the matrix method cannot be employed directly. However, because *U*(*n*_{2}, Δ*t*) is independent of the arrangement of the *n*_{2} 2-states among the subunits, the configuration sum,

$$W({n}_{2})\equiv \sum _{I({n}_{2})}{J}^{{n}_{J}}$$

(A7)

is independent of, and unaffected by, *U*(*n*_{2}, Δ*t*), and must take the same value for a given *n*_{2}, regardless of the value of Δ*t* or *U*(*n*_{2}, Δ*t*).

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,

$$ln\chi =ln{\left[{B}^{{n}_{2}}{e}^{-U({n}_{2},\mathrm{\Delta}t)/kT}W({n}_{2})\right]}_{max}$$

(A8)

where the subscript indicates the maximum value, which is assumed to occur, when *n*_{2} takes its mean value in the presence of Δ*t*. From equation (13) we also have for the average value,

$${\overline{n}}_{2}=\left(1/{c}^{\mathit{tot}}\right)\sum _{{n}_{2}\ge 0}{n}_{2}{c}_{{n}_{2}}=B(\partial ln\chi /\partial B)$$

(A9)

which remains valid, despite the incorporation of exp[−*U*(*n*_{2}, Δ*t*)/*kT*] into *χ*. The condition for the maximum term in equation (A8) is,

$$(\partial /\partial {n}_{2})({n}_{2}lnB-U({n}_{2},\mathrm{\Delta}t)/kT+lnW({n}_{2}))=0$$

(A10)

which becomes

$$lnB-(\partial U({n}_{2},\mathrm{\Delta}t)/\partial {n}_{2})/kT+\partial lnW({n}_{2})/\partial {n}_{2}=0$$

(A11)

The solution of equation (A11) for *n*_{2} gives the mean value, _{2}.

The last term in equation (A11) can be evaluated by considering first the case, when *U*(*n*_{2}, Δ*t*) = 0 for all *n*_{2}, so the second term can be omitted. Then after exponentiating both sides and multiplying by *f*_{2} = *n*_{2}/*N*, we obtain *f*_{2} as the solution of

$${f}_{2}=BF({n}_{2})$$

(A12)

where

$$F({n}_{2})\equiv {f}_{2}exp\left[\partial lnW({n}_{2})/\partial {n}_{2}\right]$$

(A13)

It is clear from equation (A7) that *W*(*n*_{2}) depends only upon *n*_{2} and J, but not upon *B* or *U*(*n*_{2}, Δ*t*), hence *F*(*n*_{2}) also depends only upon *n*_{2} and J.

When *U*(*n*_{2}, Δ*t*) = 0, the quantity *F*(*n*_{2}) can be found in the following way. The characteristic equation for the transfer matrix, **M**, in equation (16) of the main text is

$${\lambda}^{2}-\lambda (B+1)+B\left(1-{J}^{2}\right)=0$$

(A14)

Implicit differentiation leads to

$$\partial \lambda /\partial B=\left(\lambda +{J}^{2}-1\right)/(2\lambda -(B+1))$$

(A15)

By using the relation, *f*_{2} = (*B*/*λ*)*λ*/*B*, which follows from equations (13) and (15) in the main text, *B* can be expressed in terms of *λ* and *f*_{2} as,

$$B=\lambda {f}_{2}(2\lambda -1)/\left(\lambda (1+{f}_{2})+{J}^{2}-1\right)$$

(A16)

By substituting (A16) into (A14), it is possible to eliminate *B*, and express *λ* in terms of *f*_{2} and *J* as,

$$\lambda =\left\{\left(1-{J}^{2}\right)(1-2{f}_{2})+1+{\left[{\left(\left(1-{J}^{2}\right)(1-2{f}_{2})+1\right)}^{2}-4\left(1-{J}^{2}\right){(1-{f}_{2})}^{2}\right]}^{1/2}\right\}/(2(1-{f}_{2}))$$

(A17)

Solving (A16) for *f*_{2} yields,

$${f}_{2}=B\left(\lambda (1+{f}_{2})+{J}^{2}-1\right)/(\lambda (2\lambda -1))$$

(A18)

Comparison of (A18) with (A12) implies that

$$F({n}_{2})=\left(\lambda (1+{f}_{2})+{J}^{2}-1\right)/(\lambda (2\lambda -1))$$

(A19)

where *λ* is given in terms of *f*_{2} and *J* by (A17). Thus, in the present large N limit, *F*(*n*_{2}) is a function only of *f*_{2} = *n*_{2}/*N* and *J*. Although *F*(*n*_{2}) in equation (A19) was derived for the case *U*(*n*_{2}, Δ*t*) = 0, it must also apply for arbitrary *U*(*n*_{2}, Δ*t*), as noted above.

When both sides of (A11) are exponentiated and multiplied by *f*_{2}, and use is made of (A13), we obtain finally,

$${f}_{2}=B\xb7exp\left[-(\partial U({n}_{2},\mathrm{\Delta}t)/\partial {n}_{2})/kT\right]\xb7F({n}_{2})$$

(A20)

From equation (A5), we find,

$$\begin{array}{l}(\partial U({n}_{2},\mathrm{\Delta}t)/\partial {n}_{2})=\\ (1/2)\left\{\frac{-(1/{\alpha}_{2}-1/{\alpha}_{1}){\left(2\pi \mathrm{\Delta}t-\left({n}_{2}-{n}_{2}^{0}\right)\mathrm{\Delta}{\phi}^{0}\right)}^{2}}{{\left[(N-{n}_{2})/{\alpha}_{1}+{n}_{2}/{\alpha}_{2}\right]}^{2}}-\frac{2\mathrm{\Delta}{\phi}^{0}\left(\left(2\pi \mathrm{\Delta}t-\left({n}_{2}-{n}_{2}^{0}\right)\mathrm{\Delta}{\phi}^{0}\right)\right)}{\left[(N-{n}_{2})/{\alpha}_{1}+{n}_{2}/{\alpha}_{2}\right]}\right\}\end{array}$$

(A21)

When equation (A19) for *F*(*n*_{2}), (A17) for *λ*, and (A21) are substituted into equation (A20), there results a transcendental equation, which can be solved numerically for *n*_{2} = _{2}. The
${n}_{2}^{0}$ in (A21) is the corresponding solution of equation (A12), when *U*(*n*_{2}, Δ*t*) = 0. Note that, when Δ*t* = 0, both *U*(*n*_{2}, Δ*t*) and *U*(*n*_{2}, Δ*t*)/*n*_{2} vanish for
${n}_{2}={n}_{2}^{0}$, as expected.

The effect of the twisting strain present in a supercoiled DNA to shift *n*_{2} away from
${n}_{2}^{0}$ can be estimated in the following way. The linking difference, Δ*l* = *l* − *l*_{0}, is distributed between net twist, Δ*t* = *t* − *l*_{0}, and writhe, *w* (turns), according to Δ*l* = Δ*t* + *w*. Under standard conditions, Δ*t* accounts for about 1/3 of Δ*l* and *w* accounts for about 2/3 of Δ*l*. The most prominent topoisomers in a thermally equilibrated population lie in the range Δ*l* = −2 to +2. Under standard conditions (with Δ*l* = +2), one would expect Δ*t* □ 2/3 turn, whence 2*π*Δ*t* = 4*π*/3 radians. The spring constants of the 1- and 2-states in Topo I buffer are estimated to be *α*_{1} = 6.08×10^{−12} erg and *α*_{2} = 11.99×10^{−12} erg, respectively (c.f. Table 3). It is also found 0 in the present study that (*l*_{20} − *l*_{10}) −0.84 turns, so Δ^{0} = 2*π*(*l*_{20} − *l*_{10})/4932 = −0.0011 radians/bp. We assume that *B*=1.0, so that, when Δ*t* = 0, the solution is
${n}_{2}={n}_{2}^{0}=N/2$, 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 Δ*t*, and imagine that equation (A20) is to be solved by iteration, beginning with the choice,
${n}_{2}={n}_{2}^{0}=N/2$ on the right-hand side. With this choice, then, on the right-hand side one has, *f*_{2} = 1/2, *λ* =1+ *J* (from (A17)), and *F*(*n*_{2}) = 1/2 (from (A19), so that (A20) now reads

$${f}_{2}=(1/2)exp\left[-(\partial u({n}_{2},\mathrm{\Delta}t)/\partial {n}_{2})/kT\right]$$

(A22)

where the exponential factor is to be evaluated at ${n}_{2}={n}_{2}^{0}=N/2$. Numerical evaluation of the exponent yields,

$${f}_{2}=(1/2)exp[-0.00022]$$

(A23)

The second term in braces in equation (A21) yields a positive contribution that is about 4.0 times that of the first term for Δ*l* = +2. The right-hand side of equation (A23) differs negligibly from the input value, *f*_{2} = 1/2, which indicates that the populations are not significantly affected by the net twisting strain, Δ*t*, 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 *f*_{2} = 1/2. Thus, the effect of the twisting strain in a topoisomer with Δ*l* = +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 *f*_{2} away from
${f}_{2}^{0}$.

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 ${f}_{2}={f}_{2}^{0}$.

1. Schurr JM, Delrow JJ, Fujimoto BS, Benight AS. Biopolymers. 1997;44:283. [PubMed]

2. Shibata JH, Wilcoxon J, Schurr JM, V, Knauf V. Biochemistry. 1984;23:1188. [PubMed]

3. Langowski J, Benight AS, Fujimoto BS, Schurr JM, Schomburg U. Biochemistry. 1985;24:4022. [PubMed]

4. Wu PG, Song L, Clendenning JB, Fujimoto BS, Benight AS, Schurr JM. Biochemistry. 1988;27:8128. [PubMed]

5. Wu P, Schurr JM. Biopolymers. 1989;28:1695. [PubMed]

6. Song L, Fujimoto BS, Wu JM, Thomas JC, Shibata JH, Schurr JM. J Mol Biol. 1990a;214:307. [PubMed]

7. Wu PG, Fujimoto BS, Song L, Schurr JM. Biophys Chem. 1991;41:217. [PubMed]

8. Schurr JM, Fujimoto BS, Wu P-G, Song L. Topics in fluorescence spectroscopy. In: Lakowicz JR, editor. Biochemical applications. Vol. 3. Plenum Press; New York: 1992. p. 137.

9. Kim US, Fujimoto BS, Furlong CE, Sundstrom JA, Humbert R, Teller DC, Schurr JM. Biopolymers. 1993;33:1725. [PubMed]

10. Clendenning JB, Schurr JM. Biopolymers. 1994;34:849. [PubMed]

11. Clendenning JB, Naimushin AN, Fujimoto BS, Stewart DW, Schurr JM. Biophys Chem. 1994;52:191. [PubMed]

12. Naimushin AN, Clendenning JB, Kim US, Song L, Fujimoto BS, Stewart DW, Schurr JM. Biophys Chem. 1994;52:219. [PubMed]

13. Heath PJ, Clendenning JB, Fujimoto BS, Schurr JM. J Mol Biol. 1996;260:718. [PubMed]

14. Delrow JJ, Heath PJ, Schurr JM. Biophys J. 1997;73:2688. [PubMed]

15. Delrow JJ, Heath PJ, Fujimoto BS, Schurr JM. Biopolymers. 1998;45:503. [PubMed]

16. Naimushin AN, Fujimoto BS, Schurr JM. Biophys J. 2000;78:1498. [PubMed]

17. Rangel DP, Sucato CA, Spink CH, Fujimoto BS, Schurr JM. Biopolymers. 2004;75:291. [PubMed]

18. Suh D, Sheardy RD, Chaires JB. Biochemistry. 1991;30:8722. [PubMed]

19. Riccelli PV, Vallone PM, Kashin I, Faldasz BD, Lane MJ, Benight AS. Biochemistry. 1999;38:11197. [PubMed]

20. Owczarzy R, Vallone PM, Goldstein RF, Benight AS. Biopolymers. 1999;52:29. [PubMed]

21. Vallone PM, Benight AS. Biochemistry. 2000;39:7835. [PubMed]

22. Qu X, Ren J, Riccelli PV, Benight AS, Chaires JB. Biochemistry. 2003;42:11960. [PubMed]

23. Mandell K, Vallone PM, Owczarzy R, Riccelli PV, Benight AS. Biopolymers. 2005;82:199. [PubMed]

24. Fujimoto BS, Brewood GP, Schurr JM. Biophys J. 2006;91:4166. [PubMed]

25. Rangel DP, Brewood GP, Fujimoto BS, Schurr JM. Biopolymers. 2007;85:222. [PubMed]

26. Chitra R, Smith PE. J Phys Chem B. 2001;105:11513.

27. Schurr JM, Rangel DP, Aragon SR. Biophys J. 2005;89:2258. [PubMed]

28. White JH. Am J Math. 1969;91:693.

29. Fuller FB. Proc Nat Acad Sci USA. 1971;68:815. [PubMed]

30. Depew RE, Wang JC. Proc Natl Acad Sci USA. 1975;72:4275. [PubMed]

31. Pulleyblank DE, Shure M, Tang J, Vinograd J, Vosberg HP. Proc Natl Acad Sci USA. 1975;72:4280. [PubMed]

32. Horowitz D, Wang JC. J Mol Biol. 1984;173:75. [PubMed]

33. Shore D, Baldwin RL. J Mol Biol. 1983;170:983. [PubMed]

34. Clendenning JB, Naimushin AN, Fujimoto BS, Stewart DW, Schurr JM. Biophys Chem. 1994;52:191. [PubMed]

35. Gebe JA, Allison SA, Clendenning JB, Schurr JM. Biophys J. 1995;68:619. [PubMed]

36. Sucato CA, Rangel DP, Aspleaf D, Fujimoto BS, Schurr JM. Biophys J. 2004;86:3079. [PubMed]

37. Rangel DP, Fujimoto BS, Schurr JM. Biophys Chem. 2007 submitted.

38. di Mauro E, Caserta M, Negri R, Carnevalli F. J Biol Chem. 1985;260:152. [PubMed]

39. Flick E. Industrial Solvents Handbook. William Andrew Publishing; Noyes: 1998.

40. Weast RC, editor. CRC Handbook of Chemistry and Physics. Boca Raton, Florida: CRC Press, Inc; 1980.

41. Brewood GP. PhD Thesis. University of Washington; 2006.

42. Stanley C, Ran DC. Biophys J. 2006;91:912. [PubMed]

43. Spink CH, Chaires JB. Biochemistry. 1999;38:496. [PubMed]

44. Preisler RS, Chen HH, Colombo MF, Choe Y, Short BJ, Jr, Rau DC. Biochemistry. 1995;34:14400. [PubMed]

45. Tunis-Schneider MJB, Maestre MF. J Mol Biol. 1970;52:521. [PubMed]

46. Nelson RG, Johnson WC., Jr Biochem Biophys Res Commun. 1970;41:211. [PubMed]

47. Hanlon S, Brudno S, Wu TT, Wolf B. Biochemistry. 1975;14:1648. [PubMed]

48. Baase WC, Johnson WC., Jr Nucleic Acids Res. 1979;6:797. [PMC free article] [PubMed]

49. Tinoco I, Jr, Sauer K, Wang JC, Puglisi JD. Physical Chemistry, Principles and Applications in the Life Sciences. Prentice-Hall; Upper Saddle River, N. J: 2002. p. 575.

50. Wolf B, Hanlon S. Biochemistry. 1975;14:1661. [PubMed]

51. Vologodskaia M, Vologodski AV. J Mol Biol. 2002;317:205. [PubMed]

52. Hill TL. An Introduction to Staistical Thermodynamics. Addison-Wesley Publishing Co., Inc; Reading, Massachusetts: 1962.

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