epl draft
Residual entropy in a model for the unfolding of single polymer
chains
E. Van der Straeten
(a)(b) and J. Naudts(c)
1 Departement Fysica, Universiteit Antwerpen, Groenenborgerlaan 171, 2020 Antwerpen, Belgium
PACS 82.35.Lr – Physical properties of polymers
PACS 05.20.-y – Classical statistical mechanics
Abstract. - We study the unfolding of a single polymer chain due to an external force. We use
a simplified model which allows to perform all calculations in closed form without assuming a
Boltzmann-Gibbs form for the equilibrium distribution. Temperature is then defined by calcula-
ting the Legendre transform of the entropy under certain constraints. The application of the
model is limited to flexible polymers. It exhibits a gradual transition from compact globule to
rod. The boundary line between these two phases shows reentrant behavior. This behavior is
explained by the presence of residual entropy.
Introduction. – The unfolding of polymers has been
studied for many years. Recently, it became possible to
investigate the stability of a single polymer against un-
folding by applying a mechanical force to the end-point
of the molecule. Tools like optical tweezers, atomic force
microscopes and soft microneedles are used in this kind
of experiments. The unfolding transition of polymers is
a single- or a multi-step process, depending on the ex-
perimental conditions. For example, with a force-clamp
apparatus one is able to study the mechanical unfolding
at constant force. One applies a sudden force which is
then kept constant with feedback techniques. Numerical
simulations [1] show that a certain protein (called ubiqui-
tin) unfolds in a single step, while an other protein (called
integrin) unfolds in multiple steps. Ubiquitin is studied ex-
perimentally in [2]. The findings of [2] support the numer-
ical simulations rather well, because in 95% of the cases
a clear single-step unfolding process is observed. In [3, 4],
force-extension relations of single DNA molecules are ob-
tained in the fixed-stretch ensemble. One measures the
average applied force while keeping the extension of the
DNA molecule constant. Depending on the solvent condi-
tions, force plateaus or stick-release patterns are observed.
The unfolding transition from compact globule to rod
was already theoretically predicted 15 years ago [5], based
on heuristic arguments. Nowadays, the theoretical study
(a)Research Assistant of the Research Foundation - Flanders (FWO
- Vlaanderen)
Erik.VanderStraeten@ua.ac.be
Jan.Naudts@ua.ac.be
of this transition is dominated by numerical simulations.
The self-avoiding walk (SAW) in two dimensions is in-
tensively used to model the unfolding transition of poly-
mers [6–8]. The advantage of SAWs is that local interac-
tions like monomer-monomer attraction and the excluded
volume effect are taken into account. The disadvantage
is that one is limited to short walks due to the computa-
tional cost (up to chain length 55 [7]). In [6–8], a force-
temperature state diagram for flexible polymers in a poor
solvent is calculated in the fixed-force ensemble. The most
important property is that the force, at which the poly-
mer unfolds, goes through a maximum as a function of
the temperature. This so called reentrant behavior is ex-
plained by the presence of residual entropy [6].
Reentrant behavior has been observed at other occa-
sions as well. A realistic, analytical solvable model for the
unfolding transition is presented in [9]. The authors ob-
tain state diagrams for 4 different molecules. One of these
diagrams shows reentrant behavior. The authors of [9] do
not comment on this interesting feature. In [10], simpli-
fied, analytical solvable, lattice models are considered to
describe the unzipping of DNA. The obtained state dia-
gram shows reentrant behaviour. This is also caused by
the appearance of residual entropy in the model.
In [11], the present authors have proposed a simple
model to describe single polymers. This model has been
used in [12] to compare the outcome of two experiments,
which are performed in the fixed-force ensemble and the
fixed-extension ensemble. In the present letter, we focus
on the fixed-force ensemble, because the model is com-
http://arxiv.org/abs/0704.1233v3
E. Van der Straeten et al.
0 0.5 1 1.5
0 0.5 1 1.5
Fig. 1: Plot of the force-extension relation relation at constant
temperature, T = 0.1. The value of the parameter h is −1 for
the solid line and 0.01 for the dotted line. The value of a is
equal to 1 for both lines.
pletely solvable in closed form in this ensemble [12]. The
more general theory of [14] allows to simplify the calcula-
tions of [11, 12]. Of course, this simplified model is of a
qualitative nature. More sophisticated models are needed
to describe all the experimentally observed features of the
unfolding transition. The aim of the present letter is to
focus on one of these features, namely the consequences of
the appearance of residual entropy.
One dimensional random walk with memory. –
The model under study is a discrete-time random walk on
a one-dimensional lattice. It depends on two parameters
ǫ and µ, the probabilities to go straight on, when walk-
ing to the right, respectively to the left. This is not a
Markov chain since the walk remembers the direction it
comes from. The process of the increments is a Markov
chain with two states, → and ←. The stationary proba-
bility distributions of these two states are [11]
p(→) = 1− µ
2− ǫ− µ
and p(←) = 1− ǫ
2− ǫ− µ
. (1)
With these expressions, the average end postion 〈x〉 of the
walk and the average number of reversals of direction 〈K〉
of the walk can be written conveniently as [12, 14]
= a [p(→)− p(←)] ,
= 2(1− ǫ)p(→) = 2(1− µ)p(←), (2)
with a the lattice parameter and n the total number of
steps. Also, an expression for the entropy can be obtained
in closed form (neglecting boundary terms) [12, 14]
= p(→) [−ǫ ln ǫ− (1 − ǫ) ln(1− ǫ)]
+p(←) [−µ lnµ− (1− µ) ln(1− µ)] . (3)
0 0.5 1 1.5 2 2.5
Fig. 2: Plot of the force-entropy relation at constant tempera-
ture. The values of the temperature are from top till bottom
1; 0.5; 0.2; 0.1. The value of the parameters h and a are equal
to −1 and 1 respectively.
We use units in which kB = 1.
This one-dimensional random walk can be used as a sim-
ple model of a flexible polymer in the fixed-force ensemble.
The macroscopic observables are the position of the end
point and the number of reversals of direction (kinks) of
the walk. The position of the end point measures the effect
of an external force applied to the end point. The ground-
state of a flexible polymer in a poor solvent is a compact
globule. An obvious definition of the Hamiltonian is then
H = hK, with h a negative constant with dimensions of
energy and K the total number of kinks. A positive value
of the parameter h correponds with a polymer is a good
solvent. The strength of the solution determines the ab-
solute value of h. The contour length of the polymer is
equal to na.
Thermodynamics. – The Legendre transform of S
is the free energy G
G = inf
E − F 〈x〉 −
. (4)
The solution of the set of equations ∂G/∂ǫ = 0 and
∂G/∂µ = 0 gives relations for β and F as a function of
the model parameters
and β =
(1− ǫ)(1 − µ)
. (5)
The most general case when ǫ 6= µ corresponds with a
persistent random walk with drift. A persistent random
walk [13] without drift is obtained with the choice ǫ =
µ. This implies F = 0 but non-vanishing β. Also non-
persistent random walk with drift is a special case. This
corresponds with ǫ + µ = 1 and implies β = 0 but non-
vanishing F . Simple random walk is obtained with ǫ =
µ = 1/2. In this case both β and F equal zero.
Residual entropy in a model for the unfolding of single polymer chains
0.5 1 1.5 2
Fig. 3: (color online). Plot of the average number of kinks as
a function of the temperature and the force. The color code is
mentioned to the right. The black solid line, marks the gradual
transition from the compact phase to the stretched phases. The
black dotted line is an approximation for the solid line, valid
at low temperatures only. The value of the parameters h and
a are equal to −1 and 1 respectively.
The set of equations (5) can be inverted in closed form
and has a unique solution for every value of β and F . We
also calculated the eigenvalues of the matrix of the sec-
ond derivatives of the free energy. These eigenvalues are
always non-negative. We conclude that the present model
exhibits no phase transition. It is well known that no true
phase transition can occur, because of the finite size of
single molecules. This poses the problem of defining the
different phases of the model. In [6], the sudden change of
an appropriate average value is used to obtain the bound-
aries between the different phases in the state diagram.
We will use the same criterion to define the boundary line
between the different phases of the present model.
With expressions (2) and (5) one can calculate the force-
extension relation at constant temperature. It is shown in
figure 1 at low temperature and for two different values of
the parameter h. For polymers in a bad solvent (h < 0),
one observes a steep increase of the average end-to-end
distance at F ≈ 1. For polymers in a good solvent, this
steep increase occurs at vanishing force. This in qualita-
tive agreement with experimental observations [3, 4]. In
the present letter we will focus on polymers in a bad sol-
vent. At F ≈ 1 the shape of the polymer changes from
compact globule to rod. This becomes a real phase tran-
sition (a true force plateau) for T → 0 only. The smallest
eigenvalue of the matrix of the second derivatives of the
free energy vanishes at this moment. The steep increase
of the average end-to-end distance at F ≈ 1 disappears
for higher temperatures. Also the force-entropy relation
at constant temperature can be obtained. It is shown in
figure 2 for different values of the temperature. At low
temperatures, the entropy goes through a sharp maximum
with increasing force. This maximum disappears at higher
0.5 1 1.5 2
Fig. 4: (color online). Plot of the entropy as a function of the
temperature and the force. The color code is mentioned to
the right. The black solid line, marks the gradual transition
from the compact phase to the stretched phases. The black
dotted line is an approximation for the solid line, valid at low
temperatures only. The value of the parameters h and a are
equal to −1 and 1 respectively.
temperatures and the entropy becomes a monotonic de-
creasing function of the force.
In the present model, the sudden change of the aver-
age end-to-end distance can be used to define the gradual
transition from compact globule to rod. The boundary line
is obtained from the peak value of ∂〈x〉/∂F at constant
temperature. Figures 3 and 4 show the average number of
kinks and the entropy as a function of force and temper-
ature. The black solid line shows the boundary between
the compact phase and the stretched phases. The end
points of the boundary line are (Tb = 0, Fb = −h/a) and
(Tb = −2h/ ln 3, Fb = 0). At low temperatures the bound-
ary line is an increasing function of the temperature. At
intermediate temperatures the boundary line is a decreas-
ing function of the temperature. In the appendix an ap-
proximated expression for the boundary line at low tem-
peratures is calculated. One obtains F = 1+ 0.35T + . . ..
Figures 3 and 4 show the result of the latter expression
together with the exact boundary line. The two lines co-
incide up to a temperature of approximately 0.5.
In the appendix, we show that the positive slope of the
boundary line at low temperatures is due to the entropy.
This is unexpected, because the entropy is usually not im-
portant at low temperatures. However, the entropy of the
present model contains two contributions, the usual ther-
mal entropy and a configurational entropy, closely related
to the zero-temperature phase transition at F = 1. It’s
clear from figures 2 and 3, that there are three separate
phases at low temperatures. The globular phase (F < 1
and 〈K〉/n ≈ 1) and the stretched phase (F > 1 and
〈K〉 ≈ 0) have a small value of the residual entropy, be-
cause only a limited amount of configurations are allowed.
The intermediate phase at F ≈ 1 has 〈K〉/n ≈ 1/2. This
E. Van der Straeten et al.
means there are plenty of allowed configurations. The
residual entropy of this phase is very high. The reentrance
of the phase boundary is the consequence of a very sub-
tle asymmetry in the residual entropy, which prefers the
globular phase above the stretched phase. This asymme-
try can most clearly be seen in figure 2.
Discussion. – To summarize, we solve a simplified
model for the unfolding of a polymer in closed form in the
fixed-force ensemble. The force-extension relation shows
an approximate force plateau at low temperatures. The
boundary line between the globular and stretched phases
shows reentrant behavior. This reentrant behavior is ex-
plained by the presence of residual entropy in the model.
We want to stress that the present approach to intro-
duce the temperature deviates from the standard way of
introducing temperature in statistical mechanics. Usually,
one assumes that the equilibrium probability distribution
is of the Boltzmann-Gibbs form e−βH , with β the inverse
temperature. The present approach is different. We start
from a two-parameter model and calculate the average of
the macroscopic variables of interest as a function of these
two parameters, without assuming the Boltzmann-Gibbs
distribution. Then we define the temperature by calculat-
ing the Legendre transform (4) of the entropy. Usually,
the infimum is taken over all possible probability distribu-
tion. This results in the Boltzmann-Gibbs distribution.
We take the infimum only over the model parameters.
This adds an extra constraint on the equilibrium prob-
ability distribution. As a consequence, the resulting dis-
tribution is not necessarily of the Boltzmann-Gibbs form,
although we started from the Boltzmann-Gibbs definition
for the entropy [14]. Indeed, we already pointed out in [11]
that the joint probability distribution that after n steps
the walk is in x and changed its direction K times is not
of the Boltzmann-Gibbs form. The deviations from the
Boltzmann-Gibbs form are small and disappear for long
chains.
The results of the present paper are limited to the fixed-
force ensemble, although most experiments are performed
in the fixed-stretch ensemble. In [12] we show that it is
possible to extend the present model to the fixed-stretch
ensemble. However, in this ensemble almost all calcula-
tions have to be performed numerically. We also show
in [12] that the differences between the two ensembles van-
ish in the thermodynamic limit and are negligible for long
chains. So we expect only small, finite size corrections to
the state diagram after extending the present calculations
to the fixed-stretch ensemble. In [3, 4] force plateaus are
experimentally observed in stretching experiments in the
fixed-stretch ensemble. Following the previous reasoning,
this is in agreement with the present calculations up to
finite size corrections. To the best of our knowledge, the
temperature dependence of this plateau has not yet been
studied experimentally.
As mentioned in the introduction, SAWs are inten-
sively used to study the unfolding transition of polymers.
Reentrant behavior is also observed in these models [6–8].
Starting from a phenomenological expression for the free
energy near T = 0, one obtains the following boundary
line for flexible polymers [6, 15]
F = −α+ α√
+ ScT, (6)
with α a negative model parameter. The second term is
a surface correction term. The last term is a contribution
due to the residual entropy of the globular phase, with Sc
the entropy per monomer. The presence of the residual
entropy causes the reentrant behavior. The present model
does not contain the surface correction term, because lo-
cal interactions are not included. Formula (6) is similar to
expression (10) in the thermodynamic limit, because the
surface correction term disappears in this limit. A sim-
plified version of the SAW is the partially directed walk
(PDSAW). This means that steps with negative projection
along the x axis are forbidden. Analytical calculations in
the thermodynamic limit are possible for this simplified
model. The PDSAW is used in [16] as a model for a poly-
mer in the fixed-stretch ensemble. It exhibits a true phase
transition from a compact phase to a stretched phase. The
critical force as a function of the temperature can be ob-
tained in closed form in the thermodynamic limit and does
not show reentrant behavior. In [15], the PDSAW is stud-
ied numerically for finite chains. Reentrant behavior is
observed in contrast to the results of [16] in the thermo-
dynamic limit. It is argued in [15] that for the PDSAW,
the value of Sc is too small to cause reentrant behavior.
Reentrant behavior does show up in the numerical sim-
ulations because at small temperatures, there is a finite
entropy associated with the deformed globule. Together
with the surface term this gives rise to the observed reen-
trant behavior for finite walks. Our model predicts that
the reentrant behavior survives the thermodynamic limit,
in contrast to the results of the PDSAW. The reason for
this difference is that the restrictions of the PDSAW de-
crease the residual entropy.
Our model is basicly a two-state model. This kind of
model has been used before to model biopolymers, for ex-
ample to study the stress-induced transformation from B-
DNA to S-DNA. In an attempt to explain the obtained ex-
perimental data, the pure two-state model is used in [17].
The application of the two-state model is limited to the
region of the transformation from B-DNA to S-DNA. For
this reason, the model is combined in [18] with the well
known Worm Like Chain model. The combination of the
two models results in a reasonable fit to the experimen-
tal data. The present work uses a two-state model in an
other context, the unfolding transition of single polymers
instead of the transformation from B-DNA to S-DNA. To
the best of our knowledge, this is the first time that a
two-state non-Markovian random walk is used to study
the unfolding transition of single polymer chains. We ex-
pect that a three-dimensional version of the present model
can be used to study the transformation from B-DNA to
Residual entropy in a model for the unfolding of single polymer chains
S-DNA.
In conclusion, our model exhibits a gradual transition
from compact globule to rod in qualitative agreement with
experimental observations. The boundary line between
these two phases shows reentrant behavior in agreement
with numerical simulations. Our model predicts that this
reentrant behavior survives the thermodynamic limit, in
contrast to the results obtained for the partially directed
walk.
Appendix. – At low temperatures one can replace
the average number of kinks by 〈K〉 = n − 〈x〉/a. With
this approximation, the free energy becomes
G = inf
− F 〈x〉 −
, (7)
with the entropy approximately equal to
= − 1
2〈x〉 ln 2〈x〉+ (na− 〈x〉) ln (na− 〈x〉)
− (na+ 〈x〉) ln (na+ 〈x〉)
. (8)
The solution of the equation ∂G/∂〈x〉 = 0 gives the fol-
lowing force-extension relation
F = −h
4〈x〉2
n2a2 − 〈x〉2
. (9)
After inverting this relation, one can calculate the second
derivative of the average end-to-end distance with respect
to the force at constant temperature. This second deriva-
tive equals zero if the following equation holds
F = −h
T. (10)
This is a low temperature approximation for the boundary
line between the globular and stretched phases. The factor
ln 2/2a is clearly a contribution due to the entropy.
REFERENCES
[1] P. Szymczak, M. Cieplak, Stretching of proteins in a force-
clamp, J. Phys.:Condens. Matter 18, L21 (2006)
[2] M. Schlierf, H. Li, J. M. Fernandez, The unfolding kinet-
ics of ubiquitin captured with single-molecule force-clamp
techniques, PNAS 101, 7299 (2004)
[3] C. G. Baumann, V. A. Bloomfield, S. B. Smith, C. Bus-
tamante, M. D. Wang, S. M. Block, Stretching of Single
Collapsed DNA Molecules, Biophys. J. 78, 1965 (2000)
[4] Y. Murayama, Y. Sakamaki, M. Sano, Elastic Response
of Single DNA Molecules Exhibits a Reentrant Collapsing
Transition, Phys. Rev. Lett. 90, 018102 (2003)
[5] A. Halperin, E. B. Zhulina, On the Deformation Be-
haviour of Collapsed Polymers, Europhys. Lett. 15, 417
(1991).
[6] S. Kumar, D. Giri, Force-induced conformational transi-
tion in a system of interacting stiff polymers: Application
to unfolding, Phys. Rev. E. 72, 052901 (2005).
[7] S. Kumar, I. Jensen, J. L. Jacobsen, A. J. Guttmann,
Role of conformational entropy in force-induced biopoly-
mer unfolding, Phys. Rev. Lett. 98, 128101 (2007)
[8] D. Marenduzzo, A. Maritan, A. Rosa, F. Seno, Stretching
of a Polymer below the θ Point, Phys. Rev. Lett. 90,
088301 (2003)
[9] A. Imparato, A. Pelizzola, M. Zamparo, Protein mechan-
ical unfolding A model with binary variables, J. Chem.
Phys. 127, 145105 (2007)
[10] D. Marenduzzo, A. Trovato, A. Maritan, Phase diagram
of force-induced DNA unzipping in exactly solvable mod-
els, Phys. Rev. E 64, 031901 (2001)
[11] E. Van der Straeten, J. Naudts, A two-parameter random
walk with approximate exponential probability distribu-
tion, J. Phys. A: Math. Gen. 39, 7245 (2006).
[12] E. Van der Straeten, J. Naudts, A one-dimensional model
for theoretical analysis of single molecule experiments, J.
Phys. A: Math. Gen. 39, 5715 (2006).
[13] I. Claes, C. Van den Broeck, Random walk with persis-
tence, J. Stat. Phys. 49, 383 (1987)
[14] J. Naudts, E. Van der Straeten, Transition records of sta-
tionary Markov chains, Phys. Rev. E. 74, 040103 (2006).
[15] S. Kumar, D. Giri, Does changing the pulling direction
give better insight into biomolecules?, Phys. Rev. Lett.
98, 048101 (2007)
[16] A. Rosa, D. Marenduzzo, A. Maritan, F. Seno, Mechani-
cal unfolding of directed polymers in a poor solvent: Crit-
ical exponents, Phys. Rev. E 67, 041802 (2003)
[17] A. Ahsan, J. Rudnick, R. Bruinsma, Elasticity theory
of the B-DNA to S-DNA transition, Biophys. J. 74, 132
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[18] P. Cluzel, A. Lebrun, C. Heller, R. Lavery, J.-L. Viovy,
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ence 271, 792 (1996)
	Introduction. –
	One dimensional random walk with memory. –
	Thermodynamics. –
	Discussion. –
	Appendix. –

  We study the unfolding of a single polymer chain due to an external force. We
use a simplified model which allows to perform all calculations in closed form
without assuming a Boltzmann-Gibbs form for the equilibrium distribution.
Temperature is then defined by calculating the Legendre transform of the
entropy under certain constraints. The application of the model is limited to
flexible polymers. It exhibits a gradual transition from compact globule to
rod. The boundary line between these two phases shows reentrant behavior. This
behavior is explained by the presence of residual entropy.

Introduction. – The unfolding of polymers has been
studied for many years. Recently, it became possible to
investigate the stability of a single polymer against un-
folding by applying a mechanical force to the end-point
of the molecule. Tools like optical tweezers, atomic force
microscopes and soft microneedles are used in this kind
of experiments. The unfolding transition of polymers is
a single- or a multi-step process, depending on the ex-
perimental conditions. For example, with a force-clamp
apparatus one is able to study the mechanical unfolding
at constant force. One applies a sudden force which is
then kept constant with feedback techniques. Numerical
simulations [1] show that a certain protein (called ubiqui-
tin) unfolds in a single step, while an other protein (called
integrin) unfolds in multiple steps. Ubiquitin is studied ex-
perimentally in [2]. The findings of [2] support the numer-
ical simulations rather well, because in 95% of the cases
a clear single-step unfolding process is observed. In [3, 4],
force-extension relations of single DNA molecules are ob-
tained in the fixed-stretch ensemble. One measures the
average applied force while keeping the extension of the
DNA molecule constant. Depending on the solvent condi-
tions, force plateaus or stick-release patterns are observed.
The unfolding transition from compact globule to rod
was already theoretically predicted 15 years ago [5], based
on heuristic arguments. Nowadays, the theoretical study
(a)Research Assistant of the Research Foundation - Flanders (FWO
- Vlaanderen)
Erik.VanderStraeten@ua.ac.be
Jan.Naudts@ua.ac.be
of this transition is dominated by numerical simulations.
The self-avoiding walk (SAW) in two dimensions is in-
tensively used to model the unfolding transition of poly-
mers [6–8]. The advantage of SAWs is that local interac-
tions like monomer-monomer attraction and the excluded
volume effect are taken into account. The disadvantage
is that one is limited to short walks due to the computa-
tional cost (up to chain length 55 [7]). In [6–8], a force-
temperature state diagram for flexible polymers in a poor
solvent is calculated in the fixed-force ensemble. The most
important property is that the force, at which the poly-
mer unfolds, goes through a maximum as a function of
the temperature. This so called reentrant behavior is ex-
plained by the presence of residual entropy [6].
Reentrant behavior has been observed at other occa-
sions as well. A realistic, analytical solvable model for the
unfolding transition is presented in [9]. The authors ob-
tain state diagrams for 4 different molecules. One of these
diagrams shows reentrant behavior. The authors of [9] do
not comment on this interesting feature. In [10], simpli-
fied, analytical solvable, lattice models are considered to
describe the unzipping of DNA. The obtained state dia-
gram shows reentrant behaviour. This is also caused by
the appearance of residual entropy in the model.
In [11], the present authors have proposed a simple
model to describe single polymers. This model has been
used in [12] to compare the outcome of two experiments,
which are performed in the fixed-force ensemble and the
fixed-extension ensemble. In the present letter, we focus
on the fixed-force ensemble, because the model is com-
http://arxiv.org/abs/0704.1233v3
E. Van der Straeten et al.
0 0.5 1 1.5
0 0.5 1 1.5
Fig. 1: Plot of the force-extension relation relation at constant
temperature, T = 0.1. The value of the parameter h is −1 for
the solid line and 0.01 for the dotted line. The value of a is
equal to 1 for both lines.
pletely solvable in closed form in this ensemble [12]. The
more general theory of [14] allows to simplify the calcula-
tions of [11, 12]. Of course, this simplified model is of a
qualitative nature. More sophisticated models are needed
to describe all the experimentally observed features of the
unfolding transition. The aim of the present letter is to
focus on one of these features, namely the consequences of
the appearance of residual entropy.
One dimensional random walk with memory. –
The model under study is a discrete-time random walk on
a one-dimensional lattice. It depends on two parameters
ǫ and µ, the probabilities to go straight on, when walk-
ing to the right, respectively to the left. This is not a
Markov chain since the walk remembers the direction it
comes from. The process of the increments is a Markov
chain with two states, → and ←. The stationary proba-
bility distributions of these two states are [11]
p(→) = 1− µ
2− ǫ− µ
and p(←) = 1− ǫ
2− ǫ− µ
. (1)
With these expressions, the average end postion 〈x〉 of the
walk and the average number of reversals of direction 〈K〉
of the walk can be written conveniently as [12, 14]
= a [p(→)− p(←)] ,
= 2(1− ǫ)p(→) = 2(1− µ)p(←), (2)
with a the lattice parameter and n the total number of
steps. Also, an expression for the entropy can be obtained
in closed form (neglecting boundary terms) [12, 14]
= p(→) [−ǫ ln ǫ− (1 − ǫ) ln(1− ǫ)]
+p(←) [−µ lnµ− (1− µ) ln(1− µ)] . (3)
0 0.5 1 1.5 2 2.5
Fig. 2: Plot of the force-entropy relation at constant tempera-
ture. The values of the temperature are from top till bottom
1; 0.5; 0.2; 0.1. The value of the parameters h and a are equal
to −1 and 1 respectively.
We use units in which kB = 1.
This one-dimensional random walk can be used as a sim-
ple model of a flexible polymer in the fixed-force ensemble.
The macroscopic observables are the position of the end
point and the number of reversals of direction (kinks) of
the walk. The position of the end point measures the effect
of an external force applied to the end point. The ground-
state of a flexible polymer in a poor solvent is a compact
globule. An obvious definition of the Hamiltonian is then
H = hK, with h a negative constant with dimensions of
energy and K the total number of kinks. A positive value
of the parameter h correponds with a polymer is a good
solvent. The strength of the solution determines the ab-
solute value of h. The contour length of the polymer is
equal to na.
Thermodynamics. – The Legendre transform of S
is the free energy G
G = inf
E − F 〈x〉 −
. (4)
The solution of the set of equations ∂G/∂ǫ = 0 and
∂G/∂µ = 0 gives relations for β and F as a function of
the model parameters
and β =
(1− ǫ)(1 − µ)
. (5)
The most general case when ǫ 6= µ corresponds with a
persistent random walk with drift. A persistent random
walk [13] without drift is obtained with the choice ǫ =
µ. This implies F = 0 but non-vanishing β. Also non-
persistent random walk with drift is a special case. This
corresponds with ǫ + µ = 1 and implies β = 0 but non-
vanishing F . Simple random walk is obtained with ǫ =
µ = 1/2. In this case both β and F equal zero.
Residual entropy in a model for the unfolding of single polymer chains
0.5 1 1.5 2
Fig. 3: (color online). Plot of the average number of kinks as
a function of the temperature and the force. The color code is
mentioned to the right. The black solid line, marks the gradual
transition from the compact phase to the stretched phases. The
black dotted line is an approximation for the solid line, valid
at low temperatures only. The value of the parameters h and
a are equal to −1 and 1 respectively.
The set of equations (5) can be inverted in closed form
and has a unique solution for every value of β and F . We
also calculated the eigenvalues of the matrix of the sec-
ond derivatives of the free energy. These eigenvalues are
always non-negative. We conclude that the present model
exhibits no phase transition. It is well known that no true
phase transition can occur, because of the finite size of
single molecules. This poses the problem of defining the
different phases of the model. In [6], the sudden change of
an appropriate average value is used to obtain the bound-
aries between the different phases in the state diagram.
We will use the same criterion to define the boundary line
between the different phases of the present model.
With expressions (2) and (5) one can calculate the force-
extension relation at constant temperature. It is shown in
figure 1 at low temperature and for two different values of
the parameter h. For polymers in a bad solvent (h < 0),
one observes a steep increase of the average end-to-end
distance at F ≈ 1. For polymers in a good solvent, this
steep increase occurs at vanishing force. This in qualita-
tive agreement with experimental observations [3, 4]. In
the present letter we will focus on polymers in a bad sol-
vent. At F ≈ 1 the shape of the polymer changes from
compact globule to rod. This becomes a real phase tran-
sition (a true force plateau) for T → 0 only. The smallest
eigenvalue of the matrix of the second derivatives of the
free energy vanishes at this moment. The steep increase
of the average end-to-end distance at F ≈ 1 disappears
for higher temperatures. Also the force-entropy relation
at constant temperature can be obtained. It is shown in
figure 2 for different values of the temperature. At low
temperatures, the entropy goes through a sharp maximum
with increasing force. This maximum disappears at higher
0.5 1 1.5 2
Fig. 4: (color online). Plot of the entropy as a function of the
temperature and the force. The color code is mentioned to
the right. The black solid line, marks the gradual transition
from the compact phase to the stretched phases. The black
dotted line is an approximation for the solid line, valid at low
temperatures only. The value of the parameters h and a are
equal to −1 and 1 respectively.
temperatures and the entropy becomes a monotonic de-
creasing function of the force.
In the present model, the sudden change of the aver-
age end-to-end distance can be used to define the gradual
transition from compact globule to rod. The boundary line
is obtained from the peak value of ∂〈x〉/∂F at constant
temperature. Figures 3 and 4 show the average number of
kinks and the entropy as a function of force and temper-
ature. The black solid line shows the boundary between
the compact phase and the stretched phases. The end
points of the boundary line are (Tb = 0, Fb = −h/a) and
(Tb = −2h/ ln 3, Fb = 0). At low temperatures the bound-
ary line is an increasing function of the temperature. At
intermediate temperatures the boundary line is a decreas-
ing function of the temperature. In the appendix an ap-
proximated expression for the boundary line at low tem-
peratures is calculated. One obtains F = 1+ 0.35T + . . ..
Figures 3 and 4 show the result of the latter expression
together with the exact boundary line. The two lines co-
incide up to a temperature of approximately 0.5.
In the appendix, we show that the positive slope of the
boundary line at low temperatures is due to the entropy.
This is unexpected, because the entropy is usually not im-
portant at low temperatures. However, the entropy of the
present model contains two contributions, the usual ther-
mal entropy and a configurational entropy, closely related
to the zero-temperature phase transition at F = 1. It’s
clear from figures 2 and 3, that there are three separate
phases at low temperatures. The globular phase (F < 1
and 〈K〉/n ≈ 1) and the stretched phase (F > 1 and
〈K〉 ≈ 0) have a small value of the residual entropy, be-
cause only a limited amount of configurations are allowed.
The intermediate phase at F ≈ 1 has 〈K〉/n ≈ 1/2. This
E. Van der Straeten et al.
means there are plenty of allowed configurations. The
residual entropy of this phase is very high. The reentrance
of the phase boundary is the consequence of a very sub-
tle asymmetry in the residual entropy, which prefers the
globular phase above the stretched phase. This asymme-
try can most clearly be seen in figure 2.
Discussion. – To summarize, we solve a simplified
model for the unfolding of a polymer in closed form in the
fixed-force ensemble. The force-extension relation shows
an approximate force plateau at low temperatures. The
boundary line between the globular and stretched phases
shows reentrant behavior. This reentrant behavior is ex-
plained by the presence of residual entropy in the model.
We want to stress that the present approach to intro-
duce the temperature deviates from the standard way of
introducing temperature in statistical mechanics. Usually,
one assumes that the equilibrium probability distribution
is of the Boltzmann-Gibbs form e−βH , with β the inverse
temperature. The present approach is different. We start
from a two-parameter model and calculate the average of
the macroscopic variables of interest as a function of these
two parameters, without assuming the Boltzmann-Gibbs
distribution. Then we define the temperature by calculat-
ing the Legendre transform (4) of the entropy. Usually,
the infimum is taken over all possible probability distribu-
tion. This results in the Boltzmann-Gibbs distribution.
We take the infimum only over the model parameters.
This adds an extra constraint on the equilibrium prob-
ability distribution. As a consequence, the resulting dis-
tribution is not necessarily of the Boltzmann-Gibbs form,
although we started from the Boltzmann-Gibbs definition
for the entropy [14]. Indeed, we already pointed out in [11]
that the joint probability distribution that after n steps
the walk is in x and changed its direction K times is not
of the Boltzmann-Gibbs form. The deviations from the
Boltzmann-Gibbs form are small and disappear for long
chains.
The results of the present paper are limited to the fixed-
force ensemble, although most experiments are performed
in the fixed-stretch ensemble. In [12] we show that it is
possible to extend the present model to the fixed-stretch
ensemble. However, in this ensemble almost all calcula-
tions have to be performed numerically. We also show
in [12] that the differences between the two ensembles van-
ish in the thermodynamic limit and are negligible for long
chains. So we expect only small, finite size corrections to
the state diagram after extending the present calculations
to the fixed-stretch ensemble. In [3, 4] force plateaus are
experimentally observed in stretching experiments in the
fixed-stretch ensemble. Following the previous reasoning,
this is in agreement with the present calculations up to
finite size corrections. To the best of our knowledge, the
temperature dependence of this plateau has not yet been
studied experimentally.
As mentioned in the introduction, SAWs are inten-
sively used to study the unfolding transition of polymers.
Reentrant behavior is also observed in these models [6–8].
Starting from a phenomenological expression for the free
energy near T = 0, one obtains the following boundary
line for flexible polymers [6, 15]
F = −α+ α√
+ ScT, (6)
with α a negative model parameter. The second term is
a surface correction term. The last term is a contribution
due to the residual entropy of the globular phase, with Sc
the entropy per monomer. The presence of the residual
entropy causes the reentrant behavior. The present model
does not contain the surface correction term, because lo-
cal interactions are not included. Formula (6) is similar to
expression (10) in the thermodynamic limit, because the
surface correction term disappears in this limit. A sim-
plified version of the SAW is the partially directed walk
(PDSAW). This means that steps with negative projection
along the x axis are forbidden. Analytical calculations in
the thermodynamic limit are possible for this simplified
model. The PDSAW is used in [16] as a model for a poly-
mer in the fixed-stretch ensemble. It exhibits a true phase
transition from a compact phase to a stretched phase. The
critical force as a function of the temperature can be ob-
tained in closed form in the thermodynamic limit and does
not show reentrant behavior. In [15], the PDSAW is stud-
ied numerically for finite chains. Reentrant behavior is
observed in contrast to the results of [16] in the thermo-
dynamic limit. It is argued in [15] that for the PDSAW,
the value of Sc is too small to cause reentrant behavior.
Reentrant behavior does show up in the numerical sim-
ulations because at small temperatures, there is a finite
entropy associated with the deformed globule. Together
with the surface term this gives rise to the observed reen-
trant behavior for finite walks. Our model predicts that
the reentrant behavior survives the thermodynamic limit,
in contrast to the results of the PDSAW. The reason for
this difference is that the restrictions of the PDSAW de-
crease the residual entropy.
Our model is basicly a two-state model. This kind of
model has been used before to model biopolymers, for ex-
ample to study the stress-induced transformation from B-
DNA to S-DNA. In an attempt to explain the obtained ex-
perimental data, the pure two-state model is used in [17].
The application of the two-state model is limited to the
region of the transformation from B-DNA to S-DNA. For
this reason, the model is combined in [18] with the well
known Worm Like Chain model. The combination of the
two models results in a reasonable fit to the experimen-
tal data. The present work uses a two-state model in an
other context, the unfolding transition of single polymers
instead of the transformation from B-DNA to S-DNA. To
the best of our knowledge, this is the first time that a
two-state non-Markovian random walk is used to study
the unfolding transition of single polymer chains. We ex-
pect that a three-dimensional version of the present model
can be used to study the transformation from B-DNA to
Residual entropy in a model for the unfolding of single polymer chains
S-DNA.
In conclusion, our model exhibits a gradual transition
from compact globule to rod in qualitative agreement with
experimental observations. The boundary line between
these two phases shows reentrant behavior in agreement
with numerical simulations. Our model predicts that this
reentrant behavior survives the thermodynamic limit, in
contrast to the results obtained for the partially directed
walk.
Appendix. – At low temperatures one can replace
the average number of kinks by 〈K〉 = n − 〈x〉/a. With
this approximation, the free energy becomes
G = inf
− F 〈x〉 −
, (7)
with the entropy approximately equal to
= − 1
2〈x〉 ln 2〈x〉+ (na− 〈x〉) ln (na− 〈x〉)
− (na+ 〈x〉) ln (na+ 〈x〉)
. (8)
The solution of the equation ∂G/∂〈x〉 = 0 gives the fol-
lowing force-extension relation
F = −h
4〈x〉2
n2a2 − 〈x〉2
. (9)
After inverting this relation, one can calculate the second
derivative of the average end-to-end distance with respect
to the force at constant temperature. This second deriva-
tive equals zero if the following equation holds
F = −h
T. (10)
This is a low temperature approximation for the boundary
line between the globular and stretched phases. The factor
ln 2/2a is clearly a contribution due to the entropy.
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	Introduction. –
	One dimensional random walk with memory. –
	Thermodynamics. –
	Discussion. –
	Appendix. –
