Thermodynamic relations for DNA phase transitions

0.2 MB·

by  Poulomi Sadhukhan

Thermodynamic relations for DNA phase transitions
Poulomi Sadhukhan∗
Institut fu¨r Theoretische Physik, Universita¨t G¨ottingen,
Friedrich-Hund-Platz 1, 37077 G¨ottingen, Germany
Somendra M. Bhattacharjee†
Institute of Physics, Bhubaneswar 751 005, India
4
1
The force induced unzipping transition of a double stranded DNA is considered from a purely
0
thermodynamicpoint of view. This analysis providesuswith a set of relations that can beused to
2
testmicroscopictheoriesandexperiments. Thethermodynamicapproachisbasedonthehypothesis

r of impenetrability of the force in the zipped state. The melting and the unzipping transitions are
p
considered in the same framework and compared with the existing statistical model results. The
A
analysis is then extendedto a possible continuousunzippingtransition.

2
2 PACSnumbers: 87.15.Zg,05.70.Fh,05.70.Jk,87.14.gk
Keywords: DNAunzipping,meltingandforceinducedtransitions,thermodynamicrelations

]
t
f
o 1. INTRODUCTION The co-operativity in melting comes from the entropy
s (S) of the DNA through the correlations introduced by
t. To read the genetic information encoded in the base thestrandsaslongpolymers[23,24]. Theunzippingtran-
a
sitionisduetothecompetitionbetweenthepairingofthe
m sequence, hidden in the helical structure of a DNA, it
strandsandthe stretchingofthe unboundstrands[2,10–
is necessary to break the hydrogen bonds of the base
- 12]. The work done in stretching the free polymers pro-
d pairs[1]. The mechanism for doing so is the unzipping
n by a force [2, 3] of a double stranded DNA (dsDNA), videsthe costofunpairingthe strands. This costatzero
o or a thermal melting [4]. In the melting transition, the temperature is only the pairing energy, but, because of
c entropy, the critical unzipping force vanishes as one ap-
hydrogen bonds of base pairing are broken by thermal
[ proaches the melting temperature. The thermodynamic
energy,whileintheunzippingtransition,itisbyapulling
2 force at one end of the DNA. In both cases, the strands conjugatepairforthetransitionisg,theunzippingforce,
v remain intact. and x, the separation of the two strands at the point
1 of application of force (see Fig. 1). It transpires that
While there is a long history of experimental stud-
5 gross quantities like the entropy, the specific heat, and
ies of the melting transition[4], the investigations of the
4
the response function for force, are the relevant thermo-
5 unzipping transition or responses to external forces are
dynamic quantities to study, especially as the transition
. of more recent origin[5]. Pioneering calorimetric studies
1 point is approached. The advantage in the thermody-
were done over a large range of temperature (T) from
0 namic approach is that the results obtained are valid
2K to 400K under different solution conditions[6]. So
4
under quite general conditions without getting into the
1 far as force is concerned, isotherms of DNA, like the re-
microscopic details of DNA.
: sponseunderaforcehavebeenobtainedinmanydifferent
v
types of single molecule experiments[7–9]. Nevertheless,
i g
X calorimetry in presence of a force is still not available. f
r It is known from various theoretical models that, for
a x
both melting and unzipping, the nature of the transi-
f
tion depends on the aspects of the DNA captured in a R
model[2, 7, 10–20]. Any natural DNA, because of its
(a) Unzipping force (b) Stretching force
large length, is expected to show the characteristic fea-
tures of the transitions; however, the situation is not so
clearontheexperimentalfront. In-vitroexperimentsare FIG. 1: Two types of external forces on a DNA. (a) Un-
generally restricted to short chains. Consequently, very zipping force where theendsof thetwo strands are pulled in
opposite directions. (b) Stretching force where the two ends
little is known about the role of sequence variation, e.g.,
are pulled in thesame direction.
as seen across species, vis-a-vis the true transition be-
haviour expected of long chains. As a matter of fact, we
evenlacka clearexperimentalansweraboutthe orderof Besides the conjugate pairs (T,S), and (g,x), there
the melting transition [21, 22]. could be other types of external forces, e.g., isotropic
hydrostatic or osmotic pressure affecting the volume of
the polymer, and a stretching force (f) that distorts and
elongates the chain. Although there are evidences of
∗Electronicaddress: [email protected] hydrostatic or osmotic pressure affecting protein-DNA
†Electronicaddress: [email protected] interaction[56–61],thereisonlyaveryweakeffectonthe
2
melting ofDNA. In contrast,a stretchingforcemay lead relations for DNA unzipping are listed in Appendix B.
to an“overstretching”transitionwhere the lengthofthe The specific heatrelationforacontinuoustransitioncan
DNAincreasesbyafactorof1.7[8,17,25,26]. Whether be found in Appendix C.
itisanequilibrium(meaningthermodynamic)transition
is still debated.
2. THERMODYNAMIC DESCRIPTION
Theunzippingtransitionwasfirstestablishedinacon-
tinuummodelinRef. [2,3]. Itwasalsoprovedbystudy-
ing the dynamics of pulling in Ref. [27] and by several Our main concern is in the unzipping transition and
exactlysolvablemodels[10,11,13]. Variousaspectsofthe therefore we restrict ourselves to the g and x pair. In
unzippingtransition,andinthatcontextthecorrespond- absenceofanyotherinformation,wemayallowboththe
ing melting transition, have been studied. These include unzipping and the melting transition to be either first
the effects of randomnessininteraction,or force [28–31], order or continuous. Both cases are discussed here.
semiflexibility[32], and finite length [33, 34]. Many de- What makes the problem different from others is the
tailsofthetransitionhavealsobeenstudied,likevarious fact that the unzipping force does not affect the bound
distributions[35], temperature dependence[36], different state for small forces. In fact only other system that
types of noise[37, 38], role of ensembles[39]. The depen- showssimilarthermodynamicrelationsisasuperconduc-
denceofmeltingonthenatureofthespacehasalsobeen tor with the Meissner phase not allowing the external
studied via the choice of different fractal lattices[40–44], magnetic field to penetrate[50]. In that analogy,a paral-
showingthepossiblevariationsinthe meltingtransition. lel scenario for DNA would be the case where the force
ThemappingoftheDNAmeltingproblemtoaquantum penetrates for an intermediate range of force, leading to
problem revealedthe connection between the bubble en- a continuous transition[49].
tropy of DNA and the quantum transition [45], and to One may consider two mutually exclusive situations,
the Efimov-physics[46]. Biologicalapplications have also either g orx is fixed. Thesecorrespondtothe twopossi-
beenconsidered,especiallythemotionoftheinterfaceor bleensemblesinthestatisticalmechanicalapproach. The
the Y-fork[47, 48]. fixed-force case described by the Helmholtz free energy
F(T,x)andthefixed-distanceensemble,describedbythe
Our purpose in this paper is to consider the melting
Gibbs free energy G(T,g). These are in addition to the
and the unzipping transitions from a purely thermody-
usual canonical (fixed-T) and micro-canonical (fixed-S)
namic point of view, without any consideration of any
ensembles. The free energies are given by
microscopic models. This way we derive the relevant
thermodynamic relations applicable to these transitions. F(T,x) = U T S, (1a)
−
Obviously such predictions are independent of the mi-
G(T,g) = U T S g x=F g x, (1b)
croscopic details. Our basic hypothesis is that the bound − − −
double-stranded state of DNA is a thermodynamic phase where U is the internal energy. Henceforth, we use
that does not allow penetration of an unzipping force. In F,G,U,S to mean the corresponding quantities per
otherwords,thelinearresponsefunctionforaweakforce monomer or base pair. The differential form for G is
in the zipped state is strictly zero. (The words “zipped”
dG= S dT xdg. (2)
and “bound” are to be used as synonymous.) We start − −
withthedefinitionsandstandardrelationsintermsofthe It is possible to extend the thermodynamic formulation
DNA variablesinSec. 2. The case ofa firstorderunzip- to include other external forces. Some details may be
ping transition is discussed in Sec. 3. Here we consider found in Appendix A.
the case of no penetration of force in the bound state. By integrating Eq. (2) at constant temperature, one
Inother wordsthe bound state remainsthe sametill the gets the Gibbs free energy at a force g as
critical unzipping force is reached. The thermodynamic
g
predictionsarethencompared,inSec. 4,withtheknown G(T,g)=G(T,0) xdg. (3)
exact solutions in certain class of models. Although all −Z0
theoreticalstudiesbasedonsimplecoarse-grainedmodels
This form is valid for equilibrium with x = x(g) as the
predict a first-order unzipping transition, there is a pro-
equilibrium isotherm of a DNA and is used extensively
posal that local penetration of forces may lead to a con-
in this paper. The formula for work done in Eq. (3) is
tinuous transition[49]. Thermodynamics does not rule
different from the mechanical definition of work ( gdx).
out any continuous unzipping transition, but, instead,
A justification is as follows. In a nonequilibrium situa-
allowsadifferentphasewithpartialpenetrationofforce. R
tion,tochangetheforcefromzerotog,theworkdoneon
Athermodynamicanalysisofsuchacaseofacontinuous g
the DNA is xdg for a trajectory. For example, an in-
transition is discussed in Sec. 5. In this case we assume 0
stantaneouschangeinforcewouldrequireaworkw=xg
that for a range of force g g g , there is a change R
c1 ≤ ≤ c2 if the distance remains fixed at x. Then the histogram
in the DNA bound state by the external force. A few
transformation in statistical mechanics gives us the free
details can be found in the Appendices. The additions
energy difference as [51]
of other forces like hydrostatic pressure and a stretching
forcearediscussedinAppendixA.TherelevantMaxwell G(T,g) G(T,0)= k T ln exp( βw) , (4)
B
− − h − i
3
where β = (k T)−1, and the angular bracket indicates HereG (T,0)isthe freeenergyofthe unzippedphasein
B u
averaging over all possible trajectories starting with the zeroforceifithadexisted. OnewayofobtainingG (T,0)
u
equilibrium distribution at zero force. In this particular isbyextrapolationofthehighforcefree-energy,assuming
case of instantaneous increase, the averaging is over all that the extrapolation is thermodynamically admissible,
values of x with the equilibrium probability distribution or from the free energy of a single stranded DNA.
P (x)attheinitialforce. Foraninfinitesimalincrement It is known that for the first order unzipping transi-
T,g
dg, from g to g + dg, we may expand exp( βxdg) tion (Fig. 2a), the force does not penetrate the bound
− ≈
1 β(x dg). Therefore, for small dg, the equivalent of state for g <g (T). As mentioned in the introduction,
c
−
Eq. (4) is we take this as the starting hypothesis in the thermody-
namic analysis. Therefore, effectively, x =0, and
z
∆G(T,g) = −kBT ln(cid:18)1−βZ dx(xdg)PT,g(x)(cid:19) Gz(T,g)=Gz(T,0), (g ≤gc). (8)
= x(g) dg, (5) This equation is valid at g = g because of coexistence
c
of phases. At this point the force-dependent unzipped
where the average value of x is denoted by x(g). On
phase has the same G as the zipped phase. In the linear
successive integration, one recovers the thermodynamic
response regime, x = χ g where χ , the extensibility,
u T T
formula of Eq. (3) (with no angular bracket). Inciden- may be taken to be a constant. (See Appendix B for
tally, the mechanical work done in stretching or unzip-
definitions.) Eq. (7) then simplifies to
ping has been used in other contexts too, as, e.g., to
obtain and use the hysteresis around the transition for 1
G (T,0)=G (T,0) χ g2, (9)
thermodynamicfreeenergies[52]andassociateddynamic z u − 2 T c
transitions[53–55].
where the last term is the work W(g ). A more useful
The notations we are using are as follows. The zero c
form is obtained by combining Eqs. (3), (8), and (9),
forcethermalmeltingtemperatureisdenotedbyT . The
c as
unzipping transitionby aforce g attemperature T takes
placeatatemperaturedependentforceg =g (T)sothat 1
c G (T,g)=G (T,g)+ χ (g2 g2), (10)
gc(Tc)=0. z u 2 T − c
in principle, valid for all g. This shows that for g < g ,
c
(a) (b)
x x the zipped phase is more stable than the unzipped one
and vice versa.
A. Entropy
g g g g g
c c1 c2 The entropy difference, from Eq. (9), (see Appendix
B) comes out to be
FIG. 2: Possible isotherms (constant T). (a) A first-order
∂g (T)
unzipping transition at g = gc. There is a jump in x. (b) S (T,g ) S (T,g ) = χg (T) c (11)
z c u c c
Two continuous transitions at g = gc1 and g = gc2. The − ∂T
force does not affect the DNA for g < gc1 but penetrates ∂gc(T)
and modifies the bound state continuously from g > gc1 to = x(gc) ∂T , (12)
g < gc2. The unbound or stretched denatured phase occurs
for g>gc2. where the second form, a more general one, follows by
notingthat∂W(g)/∂g =x. Fornotationalsimplicity,we
omitthesubscriptsofχ. Theentropydifferenceisrelated
to the latent heat L = T(S S ) at the transition.
z u
−
3. FIRST ORDER UNZIPPING TRANSITION Except for g = 0, energy is required to unzip a DNA.
In real situations this energy is supplied by nonthermal
sources like ATP etc.
For the unzipping transition, G(T,g) is continuous
The continuity of the Gibbs free energy at the unzip-
across the phase boundary. This implies
ping transition point in a fixed force (dG = dG along
z u
thephaseboundary),givestheClausius-Clapeyronequa-
G (T,g )=G (T,g ), (6)
z c u c
tion as
where subscripts z and u indicate the zipped and the ∂g S S
c u z
unzipped phases. Eq. (3) therefore allows us to write = − , (13)
∂T x x
u z
−
gc whereallthequantitiesontherighthandsideareonthe
G (T,0) G (T,0)= (x x )dg. (7)
z − u Z0 z− u phase boundary. The impenetrability condition, xz = 0
4
with the linear response relation x = χg , yields the Depending on the value of κ, a few cases can be consid-
u c
entropy relation of Eq. (11). ered as g ∂g /∂T T T 2κ−1.
c c c
∼| − |
The sign of the right hand side in Eq. (11), i.e., the
slope of the phase boundary, is not fixed a priori. This 1. If κ >1/2, then ∂gc(T) remains finite. At the zero
∂T
is important for identification of the state which is more force melting point, there is no change in entropy
ordered. In a temperature driven transition,the entropy or no latent heat. In this situation, the melting
increases as one crosses a phase transition line from the transition is continuous.
low to the high temperature side. If the zipped phase is
2. If κ = 1/2, there is a latent heat and the melting
more ordered then S <S requiring ∂g /∂T <0.
z u c transition is first order.
3. Since infinite latent heat is not possible, there is a
B. Specific heat strict lower bound: κ 1/2.
≥
The shape of the phase boundary, as determined by the
The specific heat relation for g = 0 follows from Eq.
exponent κ, is linked to the order of the melting transi-
(11) as
tion.
∂g (T) 2
C (T ,0) C (T ,0)=T χ c , (14) g
z c u c
− (cid:18) ∂T (cid:19)g=0 A C B
g
where χ is the extensibility of the unzipped chain at the m
meltingpointT =T . Eq. (14)givesthediscontinuityin
c
the specific heatexpected atthe melting point, provided
∂g /∂T is finite. If the entropy change is finite, there is
c
a latent heat which contributes a δ-function peak at the
transitionpoint. Eq. (14) is a specialcase of the general T
T
formula valid for all g, viz., m
Cz(T,gc) Cu(T,gc) FIG.4: AmaximumatC=(Tm,gm)intheT-gphasebound-
−
∂g 2 ∂2g ∂χ ∂g ary. ApathACBseescertain specialfeaturesatC.Azipped
= T "χ(cid:18)∂Tc(cid:19) +x(gc)∂T2c + ∂Tgc∂Tc#, (15) dpihsacsoentaitnuTit=y iTnmenctarnopbye.unzippedbyaforce gm without any
with an extra latent heat contribution. The derivatives
Away from melting, in general, the right hand side of
appearing in Eq. (15) may conspire to make the RHS
Eq. (11) is not zero, unless ∂g /∂T = 0. The force in-
zero. The specific heat curve will then have only a delta c
ducedunzipping transitionisnecessarilyfirstorder. The
function at the transition point superposed on a contin-
extremum of the phase boundary (Point C at (T ,g )
uous specific heat curve. m m
in Fig. 4) is a specialcase. In absence of any nonanalyt-
icity in the phase boundary, both phases will have same
entropy but with a discontinuity in the specific heat as
C. Phase boundary
per Eq. (15). Since both g and T are intensive vari-
ables, every point in the T-g plane represents a unique
The shape of the unzipping phase boundary near the phase of the DNA, except on the transition line. Along
zeroforce melting point canbe describedasymptotically a path ACB, there is no real change in phase and no la-
by (Fig. 3) tentheatis expected. There ishoweverthe possibilityof
occurrence of the zipped phase at C. One may therefore
g (T) T T κ, for g (T) 0, T T . (16)
c c c c measure either C or C . If T is kept constant at T ,
∼| − | → → u z m
specific heat will show a discontinuity as we cross C in
thephasediagramvertically. Thislookslikeacontinuous
(a) κ>1 (b) κ=1 (c) κ=1/2 transition.
g g g When ∂gc/∂T >0, the unzipped phase becomes more
ordered than the zipped phase. This counter-intuitive
behaviouris anexampleofare-entrantphasetransition.
It occurs because the unzipping force acts as stretching
T T T T T T forces on the two unbound chains, orienting them at low
c c c temperatures in the direction of the force reducing the
entropy,whiletheflexibilityofthezippedphase,because
FIG.3: Theunzippingphaseboundarynearthemeltingpoint of the impenetrability of the force, contributes to the
for different valuesof κ, theshape exponent. entropy.
5
4. COMPARISON WITH EXACT RESULTS The unzipping phase boundary is given by
g (T)=T cosh−1(p(β) 1), p(β)=(2z )−1, (18)
c z
There are several models for which exact solutions for −
the unzipping transition are known. We compare the obtained by equating the two free energies at the unzip-
thermodynamics results with a few such cases. ping transition, i.e., from Gz(T,gc) = Gu(T,gc). Close
to T where z 1/4, and g 0. The shape is
c u c
→ →
2e−1/Tc
A. Continuum Gaussian models g (T) (T T), (19)
c c
≈ √1 e−1/TcT −
c
−
TheunzippingtransitionwasfirstprovedinRef. [2,3] i.e., κ=1 (see Fig. 3).
for Gaussian polymers interacting with same monomer The extensibility comes from the derivative of G as
u
indexasinDNA.Thetransitionlinein(d+1)-dimensions
1
is given by gc(T) ∼ |T − Tc|1/(d−2), i.e., κ = d−12 for χ= 2T sech2(g/2T). (20)
2 d 4. The zero force melting is continuous but the
≤ ≤
unzippingtransitionisfirstorder. Ford>4,themelting Linearresponseisexpectedinthesmallforcelimit,when
transitionisfirstorderandthereis aκ=1/2behaviour, x=g/(2T). At the transition
as also found in the lattice modes of Ref. [11] discussed
g
c
below. xc =tanh . (21)
2T
The model showed that the bound, zipped state does (cid:16) (cid:17)
not allow the force to penetrate and after the unzipping The free energy near an unzipping point (T,gc) can be
transition the strands are stretched by the pulling force. written as
∂gc/∂T <0. zu(T,g)
G (T,g) = G (T,g) T ln (22)
A necklace model analysis shows that κ is determined z u − z (T)
z
by the size exponent of the polymer provided there is
z (T,g)
no other length scale, i.e., κ = ν, where ν is the size = G (T,g) T ln u , (23)
u
− z (T,g )
exponent[12, 15]. For a first order melting point, since u c
all other length scales remain finite the relevant length by the continuity of the free energy at the transition
scale is the size of the polymer. For Gaussian polymers point. Close to the melting point T = T , g is small.
c c
ν = 1/2, giving κ = 1/2, as we saw above for d > 4. In this region, for a small g, an expansion gives
For continuous melting. the thermal correlation length
1
is going to play an important role, giving a different κ. G (T,g)=G (T,g)+ χ(T )(g2 g2), (24)
z u 2 c − c
consistent with Eq. (9) based on thermodynamic work.
B. Lattice models with bubbles: continuous Forspecific heat,the discontinuityatT =T forg =0
c
melting is just the specific heat of the bound state because the
unboundstateatg =0haszerospecificheat. Adifferen-
The unzipping transition problem can be solved ex- tiationofEq. (17a)showstheagreementwiththeRHSof
actlyforaclassoflatticemodelsinvolvingdirectedpoly- Eq. (14). Ref. [33] shows the behaviour of specific heat
mers in d+ 1 dimensions[10, 11]. For the model with foraforcethatshowsreentrance. Foranonzeroforce,the
bubbles, there is a continuous melting transition in di- latent heat from Eq. (11) can be verified directly. The
mensions d<4 as for the continuum case. phase diagram shows reentrance and an extrema as in
Fig. 4, recovering the features discussed in the previous
section. Fig. 5 shows the specific heat as we go through
1. d=1 the peak C in the verticaldirection keeping T =Tm and
horizontally by keeping g = g . The entropy is contin-
m
uous. The specific heat shows a discontinuity along the
For the 1+1 dimensionalmodel if the two strandsare
vertical direction. Along the horizontal direction of the
not allowed to cross, the free energies are[10]
phasediagram,theentropyiscontinuousbutthespecific
G (T,g) = k T lnz (T), (17a) heat has one single point for the zipped phase. There is
z B z
no identifiable critical region. The results are fully con-
z (T) = 1 e−β 1+e−β, (17b)
z sistent with our discussions in Sec. II.
− −
Gu(T,g) = pkBT lnzu(T,g), (17c)
z (T,g) = [2+2cosh(βg)]−1, (17d)
u
C. Y-model: first order melting
where β = (k T)−1. We choose k = 1. Here G is in-
B B z
dependentofg becauseoftheimpenetrationoftheforce. A model of DNA that does not allow any bubble is
Themeltingtransition(g =0)atT =[ln(4/3)]−1iscon- also exactly solvable[10, 11]. The thermal melting corre-
c
tinuous with a finite discontinuity of the specific heat. spondstoanallornonetypebehaviour,allbasepairsare
6
(a) (b) der a force,it is possible to get a double strandedbound
1 state at high temperatures from an unzipped state.
0.8 C S Another special situation is the 2 + 1 dimensional
g) u g) 0.8 z modelwithoutcrossingofchains. Thephaseboundaryis
,m0.6 ,m S
T T u gc(T) exp[ a/(T Tc)] with ∂gc/∂T 0 as T Tc.
∼ − − → →
C( 0.4 C S( 0.6 It is possible to unzip for T close to but below Tc by
z an arbitrarily small force. Even though this case does
0.2 0.4 not correspond to the power law form of Eq. (16), the
0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3
g g g g thermodynamic relations can still be verified including
m m
the reentrancebehaviourand the behaviournear the ex-
(c) (d)
trema of the phase boundary.
2
0.8
) ) 1.5
m0.6 m E. Summary of model comparison
g g S
T, 0.4 C T, 1 S u
C( 0.2 C u S( 0.5 z As mentioned exact results are available for a large
z class of models in various dimensions. In all these cases,
00 0.5 1 1.5 2 2.5 3 00 0.5 1 1.5 2 2.5 3 wefindthatthegeneralthermodynamicpredictionsbased
T T T T on impenetrability of the force below the unzipping tran-
m m
sition are consistent with the extant results. This lends
credence to a generalthermodynamic analysisof various
FIG. 5: Specific heat for the d = 1 exactly solvable model
thermodynamic functions near the transition and phase
withnocrossing. ThephasediagramissimilartoFig. 4with
boundary.
Tm = 0.904642475... and gm = 1.358806498... in the units
chosen. C(Tm,g) [in (a)] and S(Tm,g) [in (b)] vs g. In (a)
thereisadiscontinuityaswegothroughthepeakofthephase
5. CONTINUOUS UNZIPPING TRANSITION
boundary vertically by changing g at T =Tm. In (b), we see
that theentropy is continuousat thetransition point. There
isnolatentheat. In(c)C(T,gm)vsT andin(d)S(T,gm)vs Thermodynamics, by itself, does not exclude the pos-
T areshownfor afixedg=gm,i.e., alongthehorizontal line sibility ofa continuoustransitionunder forceofthe type
ofFig. 4. Thezippedphaseoccursonlyatonepointwithout showninFig. 2(b). Inthissectionweobtainthethermo-
anysignatureelsewhere. Thiscontributionisshownbyastar.
dynamic relations for such a continuous transition. We
Theentropyremainscontinuousandanalyticthroughoutbut
hereassumethat the forceaffects the bound stateovera
thespecificheathasoneextradiscontinuouspointatT =Tm.
range g g g , but leaves the bound state as it is
c1 c2
≤ ≤
for smaller forces. In other words, a DNA in its bound
stateisresilienttosmallforcesbutallowsittopenetrate
eitherformedorbroken. Inthe boundstate,the number
and alter its nature over a range of forces.
of configurations is 2N for N bonds, while it is 2N for
Instead of a jump discontinuity in the isotherm, we
each strand in the unzipped state. The free energies are
allow a continuous transition at a force g where the
of the form of Eqs. (17a), (18), except c1
DNA goes from the bound to a phase different from the
unzipped phase. The intermediate phase is further as-
p(β)=exp( β). (25)
− sumed to undergo a transition to the unzipped phase at
The all-or-none melting transition is first order with a gc2. In case, gc2 , there is only one phase in the
→ ∞
latent heat at k T =1/ln2. high force regime. However, since single stranded DNA
B c
Near the melting at p(β) = 2, the phase boundary under a force is a stable thermodynamic system, we ex-
behaves as gc(T) 2√Tc T, matching with κ = 1/2 pect gc2 to remain finite. There is therefore a range of
behaviour of Fig. ≈3 for a fi−rst order transition. Eq. (24) forces gc1 < g < gc2 for which the DNA is affected in
is valid with appropriate T and g . Other relations like a nontrivial way by the force. Such a scenario was con-
c c
specificheat,entropyandlatentheatcanbedirectlyver- sidered in Ref. [49]. The intermediate state is called a
ified. mixed state.
Near the two transition points, we take the isotherm
to behave as
D. Other special cases
a (g g )β, for g g +,
x 1 − c1 → c1 (26)
≈ χg a(gc2 g), for g gc2 ,
Ref. [11] considers several exactly solvable models. (cid:26) − − → −
Reentranceisobservedinallsituationsconsideredexcept with β,a,a > 0 (β is not to be confused with 1/k T).
1 B
for the case of two strands with crossing in 1+1 dimen- Theunzippedphaseforg >g istaken,forsimplicity,to
c2
sions. In this case ∂g /∂T > 0 for all T with T . be in the linear response regime, x = χg. The exponent
c c
→ ∞
Therefore,as in Eq. (11), by increasing temperature un- β, by universality, is same along the transition line.
7
With the help of the formula for work at a constant forcetopenetrate. We proceedwiththe generalformsof
temperature from g to g , the Gibbs free energy, Eq. the isotherms. This phenomenon of partial penetration
c2
(7), can be written as of force looks very much like type II superconductors.
Even though the variables and the microscopic origins
1
G (T,g) = G (T,g )+ χ(g2 g2) are different in the two cases, there is a striking simi-
m m c2 2 c2− larity between the relations obtained here for DNA and
1a(g g)2, (27) thermodynamicrelationsforsuperconductors. Lastly,we
c2
−2 − restrictedourselvesto the unzipping force here but simi-
laranalysiscanbedonefortheotherforceslikestretching
where the subscript m indicates the mixed or the inter-
forceandpressure. Itwouldbeinterestingtoobservethe
mediate state. By continuity, G (T,g ) = G (T,g ),
m c2 u c2
effect of these forces on such transitions.
and therefore,
1
G (g,T)=G (g,T) a(g g)2, (28)
m u − 2 c2− Appendix A: Other external forces
for g close to but smaller than g . This form not only
c2 As discussed in the Introduction, there could be other
shows that the mixed state has a lower free energy than
forces like pressure and the stretching force. To include
the unzipped state, but also gives the specific heat be-
these external forces, the Gibbs free energy of Eq. (1a)
haviour at g , as (see Eq. (C8))
c2 needs two additional terms
∂gc2 2 G(T,g,P,f)=U T S g x + P V f R, (A1)
C (T,g ) C (T,g )= Ta . (29)
m c2 u c2 − − − ·
− − ∂T
(cid:18) (cid:19) where V refers to the volume and R = R + R to
1 2
The specific heat relation derived in the appendix is the end to end distances of the chains. In this nota-
also applicable for the z to m transition. In this case the tion x= R R with the subscript denoting the two
1 2
zipped phase has zero extensibility and therefore chains. T|he c−onjug|ate variable is volume (V), but it is
thevolumeofthepolymerwiththesurroundingdistorted
∂g 2
C (T,g ) C (T,g )= T c1 a β(g g )β−1, solvent layers. For simplicity we ignored the terms in-
m c1 − z c1 − ∂T 1 − c1 volvingf andP,andconsideredonlytheunzippingforce
(cid:18) (cid:19)
(30) g. The extended form of the free energy shows that it is
indicating the possibility of a diverging specific heat if possibletohavecross-effectslikethef andP dependence
β <1. of x.
Different studies looked at the effect of hydrostatic
pressure, e.g., the stability of hairpins, B-DNA, the
6. CONCLUSION stalling of transcription elongation complexes[56–61].
The melting temperature T seems to depend on the hy-
c
In this paper the thermodynamic description of the drostatic pressure, P, only at a very high P. It is pos-
DNA unzipping phase transition is discussed. Without sible to measure the adiabatic compressibility (constant
considering any microscopic details, we show that the entropy)ofaDNAmoleculebymeasuringthevelocityof
thermodynamicrelationsinthefixedforceensemblehave ultrasonic waves. A different way of exerting a pressure
all the important features of the phase transition. Here is to use osmolytes like polyethylene glycol (PEG) and
we concentrate only on the force induced unzipping by other molecules that cannot penetrate the DNA. There
pullingthetwostrandsapart. Alinearresponsehasbeen are reportsof hydrostatic pressurereversingthe effect of
used for evaluating the work by force, but the analysis osmotic pressure in protein-DNA interaction.
canbecarriedoutkeepingthefullform. Althoughafirst There can be a stretching force (f) that distorts the
order phase transition is observedin various models, the shape and tries to elongate the chain. What one finds is
possibility of a continuous transition did not get much atransformationofadsDNAtoan“overstretched”state
attention. Thermodynamics does not discard this pos- with its length increasing by a factor of 1.7 [8, 17, 26].
sibility, and hence we extend our study to the case of Whether it is an thermodynamic (meaning equilibrium)
continuous transition. The only information we use as transition is still debated. The conjugate variable is the
an input to our analysis is that the zipped phase does end-to-end distance (R), if the end points are tied to-
not allow the force to penetrate below a certain crit- gether. The conjugate variable becomes the length of
ical force in the first order phase transition. The be- thepolymerfortheoverstretchingtransition. Thisisthe
haviourofthe changeinentropyandthe specific heatgo usualforceconsideredforapolymer[23,24],butcoupling
in accordance with the observedfeatures in some known to unzipping seems to lead to the new feature of over-
models. Various cases of the phase boundary line near stretching. As a perturbation to an entropy-dominated
the melting point are also analyzed. For the continuous polymer configuration, the response to the stretching
transition there is an additional region in the phase dia- force need not be linear, if the chain does not behave as
gram, showing a possible mixed phase, which allows the Gaussian,buttheoverstretchingtransitionisbeyondthis
8
regimewherethefiniteextensionsofthebondsneedtobe entropies of the two phases A and B are the same, i.e.,
takenintoaccount. Thereareevidencesofoverstretching
beingcoupledwithunzippingmakingcrosstermsimpor- G (g∗,T) = G (g∗,T), (C1)
A B
tant. Theresponsefunctionsneededforsuchcrosseffects and
would be χ = ∂x/∂f ,χ = ∂x/∂P with other appro-
i i P S (g∗,T) = S (g∗,T). (C2)
priate variables kept constant. A B
Along the phase boundary, at a neighbouring point,
G (g∗ +dg∗,T +dT) = G (g∗ +dg∗,T +dT). An ex-
Appendix B: Maxwell relations A B
pansion gives
The differential relations of the free energy are of the ∂G ∂G ∂G ∂G
expected type A dT + A dg∗ = B dT + A dg∗, (C3)
∂T ∂g∗ ∂T ∂g∗
which tells us that at the transitionpoint, the conjugate
dU = T dS+g dx, (B1)
variable x is continuous, because of the continuity of the
dF = S dT +g dx, (B2) entropy (S = ∂G/∂T).
− −
The constant force specific heat is given by
In the canonical fixed-g case, the conjugate parame-
ters and the response functions are the first and sec- ∂S
A
ond derivatives respectively of the appropriate thermo- CA =T . (C4)
∂T
dynamic potential as (cid:12)g
(cid:12)
(cid:12)
The derivative of the entropy can(cid:12)be expressed in terms
∂G 1 ∂S ∂2G
S = , C = = , (B3a) of the derivative along the transition line as
− ∂T T g ∂T − ∂T2
(cid:12)g (cid:12)g (cid:12)g
∂G(cid:12)(cid:12) ∂x (cid:12)(cid:12) ∂2G (cid:12)(cid:12) dSA ∂SA ∂SA ∂g∗
x= (cid:12) χ = (cid:12)= (cid:12), (B3b) = + , (C5)
− ∂g T ∂g − ∂g2 dT ∂T ∂g ∂T
(cid:12)T (cid:12)T (cid:12)T (cid:12)g (cid:12)T
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
where Cg is the c(cid:12)onstant force h(cid:12)eat capacity(cid:12) and χT is and a similar relation for(cid:12)phase B.(cid:12)Since at each point
the extensibilityatconstanttemperature,thelocalslope on the transition line entropy is continuous, dSA = dSB.
dT dT
ofa g-xisotherm. As usual, the positivity ofCg andχT, Eq. (C4) can now be used to express the constant force
needed for stability, are related to the convexity condi- specific heat difference as
tions satisfied by G.
The differential forms yield the Maxwell relations for ∂g∗ ∂S ∂S
A B
C C =T . (C6)
DNA as A− B ∂T ∂g − ∂g
(cid:20) (cid:12)T (cid:12)T (cid:21)
(cid:12) (cid:12)
∂x ∂S ∂x ∂S (cid:12) (cid:12)
= , and = , (B3c) A further simplification can be(cid:12) achieved(cid:12)by using one of
∂T ∂g ∂T ∂g
(cid:12)g (cid:12)T (cid:12)S (cid:12)x the Maxwell relations, Eq. (B3d),
(cid:12) (cid:12) (cid:12) (cid:12)
∂S (cid:12) ∂g(cid:12) ∂(cid:12)T ∂(cid:12)x
(cid:12)= (cid:12) , and (cid:12) = (cid:12) ,(B3d) ∂S ∂S ∂x ∂g∗
∂x(cid:12)T − ∂T(cid:12)x − ∂g(cid:12)S ∂S(cid:12)g A = A = χA, (C7)
(cid:12) (cid:12) (cid:12) (cid:12) ∂g ∂x ∂g ∂T
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12)T (cid:12)T (cid:12)T
ofwhicht(cid:12)hefirsttwo(cid:12)relatethetherma(cid:12)lexpansi(cid:12)onofthe (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12)
open fork to the heat flow for change in force. where χA is the(cid:12)extensibil(cid:12)ity of p(cid:12)hase A. With a similar
relation for phase B, we obtain
Appendix C: Specific heat near a line of continuous ∂g∗ 2
C C = T (χ χ ). (C8)
transition A B A B
− − ∂T −
(cid:18) (cid:19)
Let us consider a phase boundary g = g∗(T), where Ifthe extensibilityofphaseAislessthanthatofB,then
at any point (T,g∗(T)), the Gibbs free energies and the the specific heat of phase A is higher than that of B.
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