{\huge Experimental test of fluctuation theorems in dual trap setups and recovery of the full dissipation from partial work measurements.
M. Ribezzi-Crivellari
Departament de Fisica Fonamental
Universitat de Barcelona
Diagonal 647
08028 Barcelona
Spain
and
F. Ritort
Departament de Fisica Fonamental
Universitat de Barcelona
Diagonal 647
08028 Barcelona
Spain
Ciber-BBN de Bioingeneria
Biomateriales y Nanomedicina
Instituto de Salud Carlos III
Madrid
Spain
F. Ritort
Fluctuation relations (FRs) are among the few existing general results in non-equilibrium systems. Their verification requires the measurement of the total work (or entropy production) performed on a system. Nevertheless in many cases only a partial measurement of the work is possible. Here we consider FRs in dual-trap optical tweezers where two different forces (one per trap) are measured. With this setup we perform pulling experiments on single molecules by moving one trap relative to the other. We demonstrate that work should be measured using the force exerted by the trap that is moved. The force that is measured in the trap at rest fails to provide the full dissipation in the system leading to a (incorrect) work definition that does not satisfy the FR. The implications to single-molecule experiments and free energy measurements are discussed. In the case of symmetric setups a new work definition, based on differential force measurements, is introduced. This definition is best suited to measure free energies as it shows faster convergence of estimators. We discuss measurements using the (incorrect) work definition as an example of partial work measurement. We show how to infer the full work distribution from the partial one via the FR. The inference process does also yield quantitative information, e.g. the hydrodynamic drag on the dumbbell. Results are also obtained for asymmetric dual-trap setups. We suggest that this kind of inference could represent a new and general application of FRs to extract information about irreversible processes in small systems.
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\title{{\huge Free energy inference from partial work
    measurements in small systems}}

%\title{{\huge Differential work measurement in dual traps improves free energy estimates from single-molecule pulling experiments. }}

%\title{{\huge Experimental test of fluctuation theorems in dual trap setups and recovery of the full dissipation from partial work measurements.}}


\author{M. Ribezzi-Crivellari \\ 
\\
Departament de Fisica Fonamental,\\
Universitat de Barcelona,\\
Diagonal 647, 08028 Barcelona, Spain\\
\\
and\\
\\
F. Ritort \\
\\
Departament de Fisica Fonamental,\\
Universitat de Barcelona,\\
Diagonal 647, 08028 Barcelona, Spain\\
\\
Ciber-BBN de Bioingeneria, Biomateriales y Nanomedicina,\\
Instituto de Salud Carlos III,\\
Madrid, Spain}

\date{}
%\affil{Departament de Fisica Fonamental, Universitat de Barcelona, Av. Diagonal 647, 08028 Barcelona}
%\affil{Ciber-BBN de Bioingeneria, Biomateriales y Nanomedicina, Instituto de Salut Carlos III, 	Calle Sinesio Delgado, 4  28029 Madrid}

%\author{F. Ritort}
%\affiliation{Departament de Fisica Fonamental, Universitat de Barcelona}
%\affiliation{Ciber-BBN de Bioingeneria, Biomateriales y Nanomedicina}

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\begin{document}
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\begin{abstract}
Fluctuation relations (FRs) are among the few existing general results in
non-equilibrium systems. Their verification requires the measurement of
the total work (or entropy production) performed on a system. 
Nevertheless in many cases only a partial measurement of the
work is possible. Here we consider FRs in
dual-trap optical tweezers where two different forces (one per
trap) are measured. With this setup we perform pulling experiments on
single molecules by moving one trap relative to the other. We
demonstrate that work should be measured using the force exerted by the
trap that is moved. The force that is measured in the trap at rest fails to provide 
the full dissipation in the system leading to a (incorrect) work definition that does not 
satisfy the FR. The implications to single-molecule experiments
and free energy measurements are discussed. In the case of symmetric
setups a new work definition, based on differential force measurements,
is introduced. This definition is best suited to measure free energies
as it shows faster convergence of estimators. We
discuss measurements using the (incorrect) work definition as an example
of partial work measurement. We show how to infer the full work distribution 
from the partial one via the FR. The inference
process does also yield quantitative information, e.g. the hydrodynamic
drag on the dumbbell. Results are also obtained for asymmetric dual-trap
setups. We suggest that this kind of inference could represent a new and
general application of FRs to extract information about 
irreversible processes in small systems.
\end{abstract}
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\section*{Significance Statement}
Fluctuation Relations (FRs) provide general results about the full work
(or entropy production) distributions in non-equilibrium
systems. However, in many cases the full work is not measurable and only
partial work measurements are possible.  The latter do not fulfill a FR
and cannot be used to extract free energy differences from irreversible
work measurements. We propose a new application of FRs to infer the full
work distribution from partial work measurements. We prove this new type
of inference using dual-trap optical tweezers where two forces (one per
trap) are measured, allowing us to derive full and partial work
distributions. We derive a set of results of direct interest to single
molecule scientists and, more in general, to physicists and
biophysicists.

\section*{Introduction}
%
Fluctuation Relations (FRs) are mathematical equations connecting non
equilibrium work measurements to equilibium free energy differences. FRs, 
such as the Jarzynski Equality (JE) or the Crooks Fluctuation Relation (CFR)
have become a valuable tool in single-molecule biophysics where they are
used to measure folding free energies from irreversible pulling
experiments \cite{PNAS.hummer.2001,PNAS.hummer.2010}.  Such measurements
have been carried out with laser optical tweezers on different
nucleic acid structures such as hairpins 
\cite{liphardt2002equilibrium,nature.collin.2005,natphys.gupta.2011,natphys.alemany.2012},
G-quadruplexes \cite{dhakal2010coexistence,dhakal2013structural} and
proteins
\cite{cecconi2005direct,Nature.shank.2010,PNAS.gebhardt.2010,PNAS.yu.2012},
and with Atomic Force Microscopes on proteins \cite{prl.harris.2007} and
bi-molecular complexes \cite{PCCP.bizzarri.2010}.   
% The distribution of work values over repeated measurements can
% be used to infer the free energy difference between the folded and the
% unfolded states using the Jarzynsky equality (JE)
% \cite{jarzynski1997nonequilibrium} or the Crooks Fluctuation Relation
% (CFR) \cite{crooks1999entropy}. 
An important issue regarding FRs is the
correct definition of work, which rests on the correct identification of
configurational variables and control parameters. 
In the single-trap optical tweezers configuration this issue has been thoroughly discussed
\cite{douarche2005estimate,jchemphys.mossa.2009,alemany2011recent}.
% essential as wrong work measurements do not fulfill the CFR or the JE,
% leading to incorrect free energy estimates. In the single trap
% LOT configuration this issue has been thoroughly discussed
% \cite{douarche2005estimate,jchemphys.mossa.2009,alemany2011recent}. It
% was shown that when the optical trap is moved relative to the pipette the control parameter is
% given by the trap-pipette distance ($\lambda$) and the configurational
% variable by the molecular extension ($x$) (Figs. 1A and 1B).
\\
The situation about how to correctly measure work in small
systems becomes subtle when there are different forces applied to
the system. In this case theory gives the prescription to
correctly define the work ($W_{\Gamma}$) for a given trajectory ($\Gamma$): 
integrate the generalized force ($f_{\lambda}$)
(conjugated to the control parameter, $\lambda$) over $\lambda$ along $\Gamma$,
$W_{\Gamma}=\int_{\Gamma}f_{\lambda}d\lambda$. However, in some cases one
cannot measure the proper generalized force or has limited experimental access
to partial sources of the entropy production leading to what we call incorrect
or partial work measurements. A remarkable example of this situation are
dual-trap setups, mostly used as the high-resolution tool for single
molecule studies (Fig. \ref{Figure:opts}A). In this case a dumbbell formed by a
molecule tethered between two optically trapped beads is manipulated by
moving one trap relative to the other. In this setup two different forces (one per trap) can be measured
and at least two different work definitions are possible. In equilibrium
conditions, i.e. when the traps are not moved both forces are
equivalent: the forces acting on each bead have equal magnitude and
opposite sign. 
On the contrary, in pulling experiments, where
one trap is at rest (with respect to water) while the other is moved,
the two forces become inequivalent. This is so because the center of 
mass of the dumbbell drifts and the beads are affected by different
viscous drags (purple arrows in Fig. \ref{Figure:opts}B).
In such conditions theory prescribes that   
the full thermodynamic work must be defined on the force measured at the
moving trap whereas the force measured in the trap at rest (with respect to
water) leads to a {\it partial work measurement} which, as we show below, entails a systematic error in free energy
estimates. The difference between both works equals the dissipation by
the center of mass of the dumbbell, which is only correctly accounted for in the correct work definition. 
% This is not true anymore in a pulling experiment where
% one trap is at rest (with respect to water) while the other is moved.
% As the center of mass of the dumbbell drifts, the
%  viscous drag on each bead is different making the resultant forces exerted by
% the two traps nonequivalent (short purple arrows in Fig. 1C).  
%In such conditions theory prescribes that   
% the full thermodynamic work must be measured from the force measured at the
% moving trap whereas the force measured in the trap at rest (with respect to
% water) leads to partial work measurements and systematic error in free energy
% estimates. The difference between both works equals the dissipation by
% the center of mass of the dumbbell, only accounted for in the right work definition. 
\\ In this paper we combine theory and experiments in a dual-trap
  setup to demonstrate several results.  First, we show that if the
  wrong work definition is used, free energy estimates will be
  flawed. The error is especially severe in the case of unidirectional
  work estimates (e.g. JE) while it influences bi-directional estimates
  to a lesser extent. This fact is not purely academical: measuring the
  force in the trap at rest is experimentally easier in dual-trap setups
  and, in fact, many groups choose to do so
  \cite{natphys.gupta.2011,nature.van.2008,mmb.cisse.2011}.  For example
  if different lasers are used for trapping and detection, measuring the
  force in the moving trap poses the additional challenge of keeping the
  trapping and detection lasers aligned while moving them.  Second, we
  show how it is possible, by using the CFR, to {\it infer} the
  the full work distribution from partial work
  measurements.  We demonstrate this new type of inference in our
  dual-trap setup by showing how to reconstruct the correct work
  distribution (i.e the one we would have measured in the moving optical
  trap) from partial work measurements in the wrong optical trap
  (i.e. the one at rest with respect to water) by using the CFR.  In
  particular, for symmetric setups the correct work distribution can be
  directly inferred by simply shifting the partial work distribution.
  In asymmetric setups inference is still possible in the framework of a
  Gaussian approximation, but the knowledge of some equilibrium
  properties of the system is still required. This type of inference should be seen an example of a more general application of
  FRs which aims at extracting information about the total entropy
  production of a nonequilibrium system from partial entropy production
  measurements. This allows us to determine the average power dissipated by the center of mass
  of the dumbbell, from which we can extract the corresponding
  hydrodynamic coefficient thereby avoiding direct hydrodynamic
  measurements. Moreover, we argue this type of inference might find
  applicability to future
  biophysical experiments where the sources of entropy production are
  not directly measurable, e.g. ATP-dependent motor translocation where
  the hydrolysis reaction cycle cannot be followed one ATP at a
  time. Finally we show how, in symmetric setups, the work definition
  satisfying the CFR is not unique. In particular the {\it differential
    work}, based on differential force measurements
  \cite{PNAS.moffitt.2006}, still satisfies the CFR and leads to the
  least biased free energy estimates. The distinguishing feature of this
  work definition is that it completely filters out the dissipation due
  to the motion of the center of mass of the dumbbell.  Throughout the
  paper, and for simplicity and pedagogical reasons, most of the
  derivations, calculations and experiments are shown for symmetric
  dual-trap setups whereas the asymmetric case is discussed towards the
  end of the paper.




\section*{The model} In dual-trap setups a molecule is stretched by two optical traps, the control parameter $\lambda$ being the trap-to-trap distance. 
Let A,B denote the two optical traps. When one trap (say trap A) is
moved with respect to the bath while trap B is at rest, suitable
configurational variables are the positions of the beads, both measured
from the center of the trap at rest (trap B).  We shall denote these
variables by $y_A$ and $y_B$, Fig. \ref{Figure:opts}B. The total energy
of the system is composed of three terms:
%
\begin{equation}\label{pot}
U(y_A,y_B,\lambda)=U_m(y_A-y_B)+\frac{k_B}{2}y_B^2+\frac{k_A}{2}\left(\lambda-y_A\right)^2\,\,\,,
\end{equation} 
where the quadratic terms model the potential of the optical trap and $U_m$ describes the properties of the tether. 
%1
However one could measure the positions of the beads and the trap-to-trap distance in the moving frame of trap A ($x_A$, $x_B$ and $-\lambda$ in Fig. \ref{Figure:opts}B),
and the potential energy in Eq. \eqref{pot} would be written as:
\begin{equation}\label{p2}
U'(x_A,x_B,\lambda)=U_m(x_A-x_B)+\frac{k_A}{2}x_A^2+\frac{k_B}{2}\left(\lambda+x_B\right)^2\,\,\,,
\end{equation}
where $y_A-x_A=y_B-x_B=\lambda$ ($x_A$ and $x_B$ are negative).
%2
Central to our analysis will be the equation connecting the potential $U$ and the work performed on the system by changing the control parameter \cite{PNAS.hummer.2001}:
\begin{equation}\label{w}
 W=\int_{t_i}^{t_f} dt \partial_t U\,.
\end{equation}
From Eq. \eqref{pot} we get:
\begin{equation}\label{trtt}
W=\int_{t_i}^{t_f} f_A\dot\lambda dt,
\end{equation}
with $f_A=-k_A(y_A-\lambda)$.
% Now, let us consider a different work definition, based on the force measured in trap B:
% \begin{equation}\label{wsb}
% W'=-\int_{t_i}^{t_f} f_B\dot \lambda dt\qquad ,
% \end{equation}
% where the minus sign stems from the fact that $f_A$ and $f_B$ point into opposite directions.
% This work definition can be obtained using Eq. \eqref{w} in the moving (with respect to water) reference frame of trap A, i.e. using the position of trap A as origin. 
Inserting $U'$ instead of $U$ in \eqref{w} gives:
\begin{equation}\label{wsb}
W'=\int_{t_i}^{t_f} dt \partial_t U'=-\int_{t_i}^{t_f} f_B\dot \lambda dt,
\end{equation}
%
Despite of their similarity we will show that $W$ and $W'$ are remarkably different.
In fact, from the reference frame of trap A, the bath is seen to flow with velocity $-\dot
\lambda$ (Fig. \ref{Figure:opts}B). Because of this flow an experiment in which trap A
is moved is not Galilean equivalent to one in which trap B is moved.  In the presence of a flow the connection between potential
and work, Eq. \eqref{w}, is not valid anymore. In fact thermodynamic
work measurements must be based on the force measured in the trap being
moved. This fact has been discussed in \cite{prl.speck.2008}, yet the
implications to single-molecule experiments have never been pointed out.
Summarizing, if trap A is moved and B is at rest with respect to water,
using the work $W$ (Eq. \eqref{trtt}) in the JE leads to correct free
energy estimates whereas using $W'$ (Eq. \eqref{wsb}) in the JE leads to
a systematic error. Below we quantify such error in detail.
%
The difference between $W$ and $W'$ can be readily discussed in
symmetric setups ($k_A=k_B=k$) where calculations are much simpler.  To do this we switch to a new
coordinate system: $x_+=\frac{1}{2}(y_A+y_B),\,x_-=y_A-y_B$.  Here $x_+$
is the position of the geometric center of the dumbbell, while $x_-$ is
the differential coordinate \cite{PNAS.moffitt.2006}.  In this
coordinate system the potential Eq. \eqref{pot} reads
$U(x_-,x_+;\lambda)=U^-(x_-;\lambda)+U^+(x_+;\lambda)$:
%
\begin{equation}\label{potos}
\begin{split}
U^-(x_-,\lambda)&=U_m(x_-)+\frac{k}{4}\left(x_--\lambda\right)^2,\\U^+(x_+,\lambda)&=k\left(x_+-\frac{\lambda}{2}\right)^2.
\end{split}
\end{equation}
The potential energy term $U_+$ associated to $x_+$ is that of a moving trap, a problem that has been addressed both with experiments and theory \cite{mazonka1999exactly,wang2002experimental},  
while the potential energy term $U_-$ associated to $x_-$ corresponds to pulling experiments performed using a single trap and a fixed point.
The dumbbell is in contact with an isothermal bath where the equilibrium
state is described by the Boltzmann distribution and the corresponding
partition function (we assume a weak system-environment coupling,
 a situation satisfied in our experimental conditions, Section S2 in the SI).
Consequently, the two degrees of freedom are uncoupled and the total partition function for the system factorizes:
%
\begin{equation}\label{parti}
\begin{split}
Z(\lambda)&=Z^+(\lambda)Z^-(\lambda)\\ 
\qquad Z^\pm(\lambda)&=\int dx_\pm \exp\left(-\beta U^\pm\right)\,\,\, ,
\end{split}
\end{equation}
%
with $\beta=(K_BT)^{-1}$, $T$ being the temperature and $K_B$ being Boltzmann constant.
As a consequence free energy changes in the system can be decomposed into two contributions:
\begin{equation}\label{bl}
\Delta G=\Delta G^++\Delta G^-,\qquad \Delta G^{\pm}=-\beta^{-1}\log Z^\pm . 
\end{equation}
Work can also be decomposed into two contributions, each regarding one of the subsystems:
$ W^\pm=\int \partial_t U^\pm(x_\pm;\lambda)dt.$
Here $W^-$ contains the work done in stretching the molecule while $W^+$ is pure dissipation due to the movement of the center of mass of the dumbbell.
Note that:
\begin{equation}\label{wdecompo}
W=W^-+W^+\,\,, \qquad W'=W^--W^+\,\,,
\end{equation}
which shows that the difference between $W$ and $W'$ is entirely due to $W^+$.
The JE holds for $W$, the standard work definition, so that 
\begin{equation}\label{jazzy}
\Delta G=-\beta^{-1}\log\langle \exp{\left(-\beta W\right)}\rangle.
\end{equation}
Inserting Eqs. \eqref{bl},\eqref{wdecompo} in \eqref{jazzy} we get:
$\Delta G^++\Delta G^-=-\beta^{-1} \langle \exp{\left(-\beta
  \left(W^++W^-\right)\right)}\rangle$. In symmetric setups $W^+$ and
$W^-$ are independently distributed random variables (see Sections
  S1 and S3 in the SI) and we can conclude
that:
\begin{equation}\label{jzpm}
\Delta G^\pm=-\beta^{-1}\log\langle e^{-\beta W^\pm}\rangle\qquad.
\end{equation}
Using the JE on both $W$ and $W'$ we get two different free energy estimates: $\Delta G$ (Eq. \eqref{jazzy}) and
$\Delta G'=-\beta^{-1} \log\langle \exp\left(-\beta W'\right)\rangle$. The error $\mathcal{E}$ committed by using $W'$ instead of $W$  can be quantified as:
\begin{equation}\label{bias}
\mathcal{E}=\Delta G-\Delta G'=-\beta^{-1}\log\frac{\langle\exp(-\beta W)\rangle}{\langle\exp(-\beta W')\rangle}.
\end{equation}
From Eqs. \eqref{wdecompo},\eqref{jazzy} and again using the fact $W^+$ and $W^-$ are independently distributed random variables we get $\langle \exp(-\beta W)\rangle=\langle \exp(-\beta \left(W^++W^-\right))\rangle=\langle \exp(-\beta W^+)\rangle\langle \exp(-\beta W^-)\rangle$ and
similarly for $W'$. As a consequence $\frac{\langle\exp(-\beta W)\rangle}{\langle\exp(-\beta W')\rangle}=\frac{\langle\exp(-\beta W^+)\rangle}{\langle\exp(+\beta W^+)\rangle}$ and Eq. \eqref{bias} is reduced to 
\begin{equation}\label{biasp}
\mathcal{E}= -\beta^{-1}\log\frac{\langle\exp(-\beta W^+)\rangle}{\langle\exp(+\beta W^+)\rangle}.
\end{equation}
Since $x_+$ is subject to a quadratic potential (Eq. \eqref{potos}), we expect $W^+$ to be a Gaussian random variable. This is not true in general for $W^-$ given that $x_-$ feels the nonlinear term $U_m$.
For Gaussian Random Variables we have:
\begin{equation}\label{gigi}
\beta^{-1}\log\langle \exp(\pm\beta W^+)\rangle=\langle W^+\rangle\pm\frac{\beta}{2}\sigma^2_+, 
\end{equation}
where by $\sigma_+^2$ we denote the variance of $W^+$.  Moreover 
dragging a trapped bead in a fluid causes no free energy change, so that:
\begin{equation}\label{gromp}
\Delta G^+=-\beta^{-1} \log\langle \exp(-\beta W^+)\rangle=\langle W^+ \rangle-\frac{\beta}{2}\sigma_+^2=0 
\end{equation}
or $\langle W^+ \rangle=\frac{\beta}{2}\sigma_+^2$. 
Inserting Eqs. \eqref{gigi} and \eqref{gromp} in Eq. \eqref{biasp} we get:
\begin{equation}\label{ppo}
\mathcal{E}=\langle W^+\rangle+\frac{\beta}{2}\sigma_+^2=2\langle W^+\rangle.
\end{equation}
Equation \eqref{ppo} gives $\mathcal{E}=\beta\sigma_+^2>0$ showing that $\Delta G'$ is lower than $\Delta G$ (Eq. \eqref{bias}). Interestingly enough, using $W'$ instead of $W$ in the JE leads
to free energy estimates in apparent violation of the second law.
The error on free energy estimates obtained using $W'$ instead of $W$ is proportional to the mean work performed on the center of the dumbbell.
This mean work $\langle W^+\rangle$ is just the mean friction force times the total trap displacement $\Delta \lambda$:
\begin{equation}\label{defiw}
\langle W^+\rangle=\gamma_+\frac{\dot \lambda}{2} \Delta \lambda\qquad,
\end{equation}
where $\gamma_+$ is the friction coefficient of the drag force opposing
the movement of the geometric center of the dumbbell. The value of
$\gamma_+$ can be independently obtained from equilibrium measurements \cite{Prl.meiners.1999} (Sections S4,S5 in the SI).


\section*{Differential Work Measurements}

Equations \eqref{wdecompo} and \eqref{gromp} show that free energy estimates based on the standard work $W$ and the differential work $W^-$ are equivalent:
\begin{equation}\label{ddcc}
 \begin{split}
\Delta G&=-\beta^{-1}\log\langle \exp(-\beta W)\rangle=\\
&= -\beta^{-1}\log\langle \exp(-\beta W^-)\rangle-\beta^{-1}\log\langle \exp(-\beta W^+)\rangle=\\
&=-\beta^{-1}\log\langle \exp(-\beta W^-)\rangle=\Delta G^{-}.
\end{split}
\end{equation}
%
We stress that this is only true for symmetric setups where $W^+$ and
$W^-$ are independent random variables. The case of asymmetric setups
is discussed further below. Therefore $W^-$ can be used for free energy determination, as it has been done in \cite{PNAS.gebhardt.2010}, although without discussion.
Equation \eqref{ddcc} does only hold when the number of work measurements, $N$, tends to infinity. In all practical cases we deal with finite $N$ and the Jarzynski estimator is biased \cite{PNAS.gore.2003,prl.palassini.2011}. The bias is strongly 
linked to the typical dissipation $D_\text{typ}$ and a reliable estimate of free energy differences requires a number of work measurements which scales as $N\simeq \exp\left(D_\text{typ}\right)$ \cite{ritort2002two,PrE.jarzynski.2006},
so that even a small reduction in $D_\text{typ}$ entails a considerable improvement in the convergence of free energy estimators. Moreover the bias is superadditive. Let us consider for simplicity 
Gaussian Work Distributions.  In this case the bias, $B^\text{GWD}_N$, in the large $N$ limit is a function of the variance of the distribution $\sigma^2$ and of $N$ \cite{PNAS.gore.2003}:
\begin{equation}\label{fgf}
B^\text{GWD}_N=\frac{\exp{(\beta^2\sigma^2}-1)}{2\beta N}\qquad .
\end{equation}
$B^\text{GWD}_N$ is a convex function of $\sigma$, and is superadditive i.e. $B^\text{GWD}_N(\sigma^2+\phi^2)>B^\text{GWD}_N(\sigma^2)+B^\text{GWD}_N(\phi^2)$. 
This means that, should the work be the sum of two independent Gaussian contributions, the bias on the sum is greater than the sum of the biases. 
Although Eq. \eqref{fgf} was derived under strong assumptions, superadditivity does also hold for other theoretical expressions for the bias and has been checked in our experimental data (see below). 
Let us introduce the following Jarzynski estimators for finite $N$:
\begin{eqnarray}
\Delta G_N=-\beta^{-1}\log \frac{1}{N}\sum_{i=1}^Ne^{-\beta W_i},\label{e1}\\
\Delta G_N^\pm=-\beta^{-1}\log \frac{1}{N}\sum_{i=1}^Ne^{-\beta W_i^\pm}\label{e2}
\end{eqnarray}
and the corresponding bias functions:
\begin{eqnarray}
B_N&=&\Delta G_N-\Delta G \label{fffd}\\
B^{\pm}_N&=&\Delta G^\pm_N-\Delta G.\label{fffe}
\end{eqnarray}
Since $W=W^++W^-$, superadditivity guarantees:
\begin{equation}\label{hiera}
B_N\geq B^{-}_N+B^{+}_N\geq  B^{-}_N.
\end{equation}
Because of Eq. \eqref{hiera} differential work measurements always improve the convergence of free energy estimates in dual-trap setups. 
This is especially important in all those cases in which bidirectional methods (e.g. the CFR) cannot be used and one has 
to employ unidirectional methods. 



\section*{Pulling on ds-DNA} The theory discussed so far has been put to test in a series of pulling experiments performed in a recently developed dual-trap optical tweezers setup which directly measures force in each trap \cite{ribezzi2012force,ribezzi2013counter}.
The setup can move the two optical traps independently and measure their relative position with sub-nanometer accuracy, giving direct access to both $W$ and $W'$. 
In these experiments 3 kb ds-DNA tethers ($\simeq1$ $\mu$m in contour
length) were stretched between 1 and 3 pN (Fig. \ref{Figure:opts}C)
in a symmetric dual-trap
setup ($k_A=k_B=0.02$pN/nm) using 4 $\mu$m silica beads as force probes.  The experiments were performed moving one of the two traps (trap A) with respect to the lab frame 
and leaving trap B at rest. All experiments were performed in PBS buffer (pH 7.4, 1M NaCl).
We chose cyclical protocols ($\lambda_t:\, \lambda_0=\lambda_{T_{c}}$, where $T_{c}$ is the total duration of the cyclic protocol). The excursion of the control parameter, $\Delta \lambda=\lambda_{T_{c}/2}-\lambda_0$,  
was varied between 200, 400 and 600 nm, while the pulling speed was varied between 1.35$\pm$0.05, 4.3$\pm$0.1 and 7.2$\pm$1 $\mu$m/s.  Given the force-distance curves the total dissipation along cycles was measured:
\begin{equation}\label{dd}
D=\oint d\lambda f_A\,\,, \qquad D'=-\oint d\lambda f_B\,\,. 
\end{equation}
The CFR \cite{crooks1999entropy} is a symmetry relation between the work distribution associated to the forward ($P_F$) and time reversed ($P_R$) protocols:
\begin{equation}\label{crozza}
P_F(W)=P_R(-W)\exp(\beta\left(W-\Delta G\right)).
\end{equation}
In the case of cyclic protocols $P_F=P_R=P$ and $\Delta G=0$ so that the CFR takes the form:
\begin{equation}\label{cfr}
P(D)=\exp{\left(\beta D\right)}P(-D)\qquad.
\end{equation}
Such symmetry of the probability distribution for $D$ can be directly tested in cases where negative dissipation events are observed. In Fig. \ref{Figure:multi}A we show measured work histograms (solid points, left hand side of Eq. \eqref{cfr}) 
and reconstructed histograms (open points, right hand side of Eq. \eqref{cfr}). If Eq. \eqref{cfr} is fulfilled then the measured and reconstructed histograms match each other. A quantitative measure of the deviation from Eq. \eqref{cfr} can be obtained
from the ratio $P(D)/P(-D)$, as shown in Fig. \ref{Figure:multi}B. Experimental data shows that $D$ fulfills the FR whereas $D'$ does not. 
In Fig. \ref{Figure:multi}C we show that the pdfs of $D^+$ and $D^-$, with:
\begin{eqnarray}
D^-=\frac{D+D'}{2}\label{oro}\\
D^+=\frac{D-D'}{2}\label{sboro},
\end{eqnarray}
are experimentally found to satisfy a FR as in Eq. \eqref{cfr}. $D^-$ is
just the differential work, $W^-$, Eq. \eqref{wdecompo} evaluated on a
cyclic protocol, whereas $D^+$ is the dissipation due to the movement of
the center of mass of the dumbbell.  Summarizing, although in general
$W$ is the only observable we expect to fulfill a FR, in symmetric
setups two new FRs emerge, for $W^-$ and $W^+$.  In Fig. \ref{Figure:seifert}A,B we
compare the predictions of Eqs. \eqref{bias}, \eqref{defiw} with
experimental results for different pulling speeds, $\dot \lambda$, and
different displacements $\Delta \lambda$.  Equation \eqref{defiw} must
be used to correct free energy estimates obtained in all those dual-trap
setups which do not measure the force applied by the trap which is being
moved (as in \cite{natphys.gupta.2011,nature.van.2008,mmb.cisse.2011}).
The advantages of using $W^-$ in free energy estimates are shown in
Fig. \ref{Figure:seifert}C.  There we show the convergence of the
Jarzynski estimator with sample size for the cycles in
Fig. \ref{Figure:opts}C. Being evaluated over cycles, the expected free
energy change is zero.  The convergence of the estimator is faster for
$W^-$ than for $W$ in the three cases (Eq. \eqref{hiera}). The effect is
enhanced in our experiments by the high pulling speed (in the range 1-7
$\mu$m) and by the large bead radius (2 $\mu$m).  Let us note that due
to the finite lifetime of molecular tethers and unavoidable drift
effects, raising the pulling speed is a convenient strategy to improve
the quality of free energy estimates. Similar results have
been found also at low pulling speeds where, again, $W'$ does not
satisfy the CFR (Section S7 in the SI).

\section*{Experiments on DNA hairpins} 
Fluctuation theorems are used to extract folding free energies for nucleic acid secondary structures or proteins. 
We further tested the different work definitions by performing pulling experiments on a 20bp DNA hairpin (Fig. 4A) at a $0.96\pm0.02$ $\mu$m/s pulling speed in the same dual-trap
setup as in the previous dsDNA experiments. In this case the work performed during the unfolding and refolding of the molecule were 
considered separately, as it is customary for free energy determination. In Fig. 4B we present forward and reverse work histograms for $W,W'$ and $W^+$. Again $W$ 
and $W^-$ both fulfill the CFR  but $W$ shows higher dissipation than $W^-$, resulting in slower convergence of unidirectional free energy estimators (Fig. 4C). 
The difference between unidirectional free energy estimates
based on $W$ and $W^-$ is in this case $\simeq 1$ $K_BT$. 
As previously discussed for double-stranded DNA (Fig. 2A), $W'$ does not fulfill the CFR and,
as a consequence, unidirectional free energy estimates based on $W'$ are flawed. In our experimental conditions the error committed by using the wrong work definition 
is again positive and equal to $\mathcal{E}\simeq3$ $K_B T$. As previously discussed this leads to a negative average dissipated work, apparently violating the second law. 
It must be noted that the difference in free energy estimates based on $W^-$ and $W$ is a finite-size effect, whose magnitude decreases when an increasing number of work 
measurements is considered. On the contrary 
the error committed by using $W'$ does not vanish by increasing the number of work measurements. It can be noted from Fig 4B that, although $W'$ does not fulfill the 
CFR and gives wrong unidirectional estimates, its forward and reverse
distributions apparently cross at $W'=\Delta G$ within the experimental
error. Although this could be used for free energy determination
  the result should be taken with caution as we have no general proof
that this should happen in all cases.



\section*{Free energy inference from partial work measurements}

The JE and CFR are statements on the statistics of the total dissipation
in irreversible thermodynamic transformations.  The need to measure the
total dissipation limits the range of applicability of these and other FRs. 
For example testing FRs concerning the dynamics of
molecular motors would need the simultaneous measurement of both the
work performed by the motor and the number of hydrolyzed ATPs. Here we
demonstrate that, at least in some cases, a different approach is
possible. Let us start by considering the simple case of symmetric
setups as developed in the previous sections. In the experiments
discussed so far there are two sources of dissipation that we
were able to measure and characterize separately: the motion of the
dumbbell and the dissipation of the differential coordinate.We
already learnt that $W$ satisfies a FR while $W'$ does
not. Moreover we know that $W=(W_-+W_+)/2$ and $W'=(W_--W_+)/2$, and
that, being $W^+$ and $W^-$ uncorrelated random variables, $W$ and
$W'$ have the same variance. Imagine now to have only partial
information on the system.  For example, one could be able to measure
force only in the trap at rest, as many experimental setups do. With
this information, and in absence of any guiding principle, no statement
about the total entropy production is possible. We will find such
guiding principle if we assume the FR to hold for $W$.  Knowing
 that $W$ and $W'$ have the same variance, we just have to shift
the work distribution $P'(W')$ by $W=W'+\Delta$ to get a new
distribution that satisfies the CFR.  In practice this is done starting
from the set of $W'$ values and tuning the value of $\Delta$
(Fig. \ref{Figure5}A) until $P(W)=P'(W-\Delta)$ fulfills the CFR
(Fig. \ref{Figure5}B).  In the case of the hairpin, this same shifting
procedure is operated for both forward and reverse work
distributions. Again the value of $\Delta$ is tuned
(Fig. \ref{Figure5}C) until the CFR symmetry is recovered
(Fig. \ref{Figure5}D). The unique value of $\Delta$ that restores the
validity of the CFR equals the average work dissipated by the motion of
center of mass of the dumbbell, giving the hydrodynamic coefficient
$\gamma_+$ via Eq. \eqref{defiw}. Let us note that, once the work
distribution in the correct trap (i.e. the moving trap) has been
recovered, then we could also extract the correct free energy difference
(the value of $\Delta G$, Fig. \ref{Figure5}E) and infer the
distribution for the differential work $W_-$ by deconvolution.
\\
The extension of this analysis to the asymmetric case is more complex
but equally interesting (Fig. 6A). The decomposition of $W,W'$ in $W^+$ and $W^-$ (Eq. \eqref{wdecompo}) is
still possible although $W^+,W^-$ are not uncorrelated variables
anymore and neither $W^+$ nor $W^-$ satisfy a FR. In this
general case only $W$ satisfies a FR (but not $W'$, $W^+$,
$W^-$). Remarkably enough, in the framework of a Gaussian approximation, it is still possible to infer the correct work
distribution $P(W)$ out of partial work $W'$ measurements. The analysis
is presented in Section S6 of the SI. In this case it is enough to know the trap and molecular
stiffnesses $k_A,k_B,k_m$ i.e. some equilibrium properties of the system,
for a successful inference. To reconstruct $P(W)$ both the mean and
the variance of $P'(W')$ must be changed, which can be achieved by doing
a convolution between the $P'(W')$ and a normal distribution:
$P_{\Delta,\Sigma} =P'\star {\cal N}(\Delta,\Sigma)$ where
$\star$ denotes the convolution operator and ${\cal
  N}(\Delta,\Sigma)$ is a normal distribution with mean $\Delta$ and
standard deviation $\Sigma$. Starting from a distribution $P'(W')$ there
are infinitely many choices of $\Delta$ and $\Sigma$ which yield a
$P_{\Delta,\Sigma}(W)$ satisfying the CFR. Indeed, let us suppose that the
pair $\Delta^*,\Sigma^*$ is such that $P_{\Delta^*,\Sigma^*}(W)$
satisfies the CFR. Then it is easy to check that
$P_{\Delta^*+\phi,\sqrt{\Sigma^{*2}+2\phi K_BT}}$ will also satisfy the CFR
for any $\phi$ (Fig. 6B). In this situation the inference cannot rest on
the CFR alone. Explicit calculations in the Gaussian
case (section S6 in SI) show that variances ($\sigma^2$) and means
($\langle...\rangle$) of $P(W)$ and $P'(W')$ are related by an Asymmetry Factor (AF),
%
\begin{equation}
AF(k_A,k_m,k_B)=\frac{\sigma_W^2-\sigma_{W'}^2}{\langle W\rangle-\langle
  W'\rangle}=K_BT \frac{4k_m(k_A-k_B)}{k_A(k_B+2k_m)}
\label{eqAF}
\end{equation}
%
which only depends on {\it equilibrium properties} such as the stiffnesses of 
the different elements (section S6.3 in the SI). Knowing the AF allows us to select the unique pair
$\Delta,\Sigma$ such that $AF= \Sigma^2/\Delta$ with
$P_{\Delta,\Sigma}(W)$ satisfying the CFR. The inference procedure
can be described with a very simple formula (Section S6 in the SI). The key idea is to proceed as previously done in the case of
symmetric setups by just shifting the mean of $P'(W')$ by a parameter $\delta$
until the CFR is satisfied, i.e. $\Delta=\delta,\Sigma =0$. From the
values of $AF$ and $\delta$ we can reconstruct $P(W)$ by using the formulae,
%
\begin{equation}
\Delta=\frac{2\delta}{2-\beta AF}\,\,;\,\,\Sigma^2=\frac{2\delta AF}{2-\beta AF}
\label{eqdeltasigma}
\end{equation}
%
The inference procedure for an asymmetric setup is shown in Fig. 6C,6D
for cyclic ds-DNA pulling experiments.  For non-cyclic pulls ($\Delta
G\ne 0$) the procedure can be easily generalized in the line of what has
been shown for the case of hairpin in the symmetric setup (Fig. 5C).


\section*{Discussion}
%
% It is remarkable that the reconstruction of the correct work
% distribution $P(W)$ is at all possible without using any data measured
% in the correct trap. One could even replace the moving trap by a moving
% micropipette or any other solid object or substrate attached to the
% molecular system lacking any measurement capability and still infer the
% work distribution exerted by that \tcr{infinitely rigid substrate on the
% molecular system (this extremely asymmetric setup would still be
% described by Eq.(\ref{eqAF}), with
% $AF=4K_BTk_m/(k_B+2k_m)$).} Definitely, this looks counterintuitive. Yet
% it is no more counterintuitive than the consequences implied by the JE,
% or the possibility of extracting the reversible work in a thermodynamic
% transformation from irreversible work measurements. How can we
% rationalize this fact? In our dual-trap experiments all information
% contained in the spectrum of force fluctuations as measured in one trap
% appears correlated to the spectrum of force fluctuations present in the
% other optical trap (for the same, any anchoring site lacking measurement
% capability). In a nutshell, one can recover information about the
% entropy production in the double-trap system from only partial
% measurements of the dissipation (i.e. measurements taken in one of its
% % parts such as the wrong trap). 
% This result opens the exciting prospect
% of extending and applying these ideas to steady state systems, such as
% molecular motors, to extract useful information about their
% mechanochemical cycle.

FRs are among the few general exact results
in non-equilibrium statistical mechanics. Their validity has been
already extensively tested in different systems, ranging from single
molecules to single electron transistors, and in different conditions
(steady state dynamics, irreversible transformations between steady
states, transient nonequilibrium states). At the present stage, the main
widespread application of FR is free energy recovery from
non-equilibrium pulling experiments in the single molecule field. What
we are presenting here is a new application of FR for
inference. All FRs are statements about the statistics of the total
entropy production in a system plus the environment. If some part of
the entropy production is missed or inadequately considered FRs will
in general not hold. This is why, for irreversible transformations
between equilibrium states, we have a FR for the dissipated work
(which is the total entropy production) but not for the dissipated
heat (which is just the entropy production in the environment). The
main tenet is now that the violation of FRs in a given setting
provides useful information: it is an evidence that some
contribution to the total entropy production is being missed.
We have given rigorous examples in which the violation
of FRs can be used to characterize the missing entropy production.
Remarkably, in our model system, one could even replace the moving trap by a moving
micropipette, an object lacking any measurement capability, and still infer the
work distribution exerted by that object on the
molecular system (this extremely asymmetric setup would still be
described by Eq.(\ref{eqAF}), with $k_A\to\infty$ and $AF=4K_BTk_m/(k_B+2k_m)$).
These results open the exciting prospect of extending and applying these ideas 
to steady state systems, such as molecular motors, to extract useful information about their mechanochemical cycle.


\section*{Conclusions} 
%
In order to give it a clear and definite meaning to free energy inference we have discussed irreversible 
transformations between equilibrium states performed with dual-trap optical tweezers. In these
experiments a molecular tether is attached between two beads
which are manipulated with two optical traps. The irreversible
transformation is performed by increasing the trap-to-trap distance at
a finite speed. In this kind of transformations the dissipated work
equals the total entropy production leading to our first
result: in pulling experiments work $W$ must be defined on
the force measured in the trap which is moved with respect to the
thermal bath. The force measured in the trap at rest gives rise to a
work definition, $W'$, which does not satisfy the FR and is unsuitable to
extract free energy differences. We have called $W'$ a partial work measurement
because it misses part of the total dissipation. This result is of
direct interest to experimentalists: many optical tweezers setups are designed
so that they can only measure $W'$. We have thus imagined a situation in which 
$W'$ is measurable while $W$ is not and asked the question: can we infer the distribution of $W$ 
from that of $W'$? If the question is asked in full generality, without any system-specific
information, the answer is probably negative. Knowing only the extent of
violation of the FR will be of little use, in general some additional system specific information will 
be needed for a successful inference. Here we discussed free energy inference in the framework of a Gaussian
approximation, the extent to which such inference is generally possible should be the subject of future studies.
Let us summarize our main results:
%
\begin{itemize}
\item A symmetry of the system can be crucial for the inference. For
  left-right symmetric systems (as exemplified in our symmetric
  dual-trap setup) the $P(W)$ can be inferred from $P'(W')$ just by
  imposing that the former satisfies the CFR. When symmetry
  considerations cannot be used, the knowledge of some equilibrium
  properties of the system may suffice to successfully guide the
  inference (such as the stiffnesses for the asymmetric setup).

\item The inference process can be used both to recover the full
  dissipation spectrum plus additional information about the {\em
    hidden} entropy source. In our specific dual-trap example $W'$ does
  not account for the dissipation due to the movement of the
  center-of-mass of the dumbbell and the inference procedure can be seen
  as a method to measure the associated hydrodynamic drag.

\item We stress the benefits of using symmetric dumbbells
   in single molecule manipulation. In this case an alternative work definition, the
{\em differential work} $W^-$, fulfills the CFR and is thus suitable for free energy
measurements. Being $W^-$ less influenced by dissipation than $W$,  switching from $W$ to $W^-$ ensures faster
convergence of unidirectional free energy estimates.  For asymmetric
setups $W^-$ does not satisfy a FR anymore (only $W$ does) and cannot be used to extract free
energy differences.

\end{itemize}

A deep understanding of how to correctly define and measure
thermodynamic work in small systems (a long debated question in the past
20 years) is not just a fine detail for experimentalists and theorists
working in the single molecule field, but an essential question
pertaining to all areas of modern science interested in energy transfer
processes at the nanoscale. The new added feature of free energy inference
discovered in this paper paves the way to apply FRs to new problems and
contexts. This remains among the most interesting open problems in this exciting field.

\iffalse
Dual-trap setups are preferred in single-molecule applications due to
their higher resolution.  At equilibrium the forces exerted by the two
traps are statistically equivalent. In non-equilibrium pulling
experiments, due to the movement of the center of mass of the dumbbell,
the two forces become inequivalent. It has been shown that thermodynamic
work must be defined using the force measured in the trap which is being
moved with respect to water during the experiment. Work defined on the
force measured in the trap at rest does not satisfy the Crooks relation,
the Jarzinsky equality or even the second law.  This has important
consequences in free energy determination, as many dual-trap setups
cannot measure the force in the moving trap and therefore the
thermodynamic work. In all these cases unidirectional free energy
estimates are subject to an error that increases with pulling speed.
Such errors can be estimated and the free energy corrected when the
hydrodynamic parameters of the dumbbell are known.  One of the salient
features of equilibrium measurements in dual-trap setups is the
possibility of differential detection. In equilibrium experiments this
allows to filter out the fluctuations due to the movement of the center
of mass of the dumbbell. Differential detection proves to be useful in
irreversible pulling experiments: differential work $(W^-)$ measurements
fulfill fluctuation relations and are less influenced by
dissipation. This is achieved by filtering out the dissipation due to
the movement of the center of mass of the dumbbell.  The decrease in
dissipation obtained by switching from $W$ to $W^-$ ensures faster
convergence of unidirectional free energy estimates and $W^-$ should be
preferentially used to extract free energy differences in single
molecule pulling experiments.  Finally, a deep understanding of how to
correctly define and measure thermodynamic work in small systems (a long
debated question in the past 20 years) is not just a fine detail for
experimentalists and theorists working in the single molecule field, but
an essential issue in all areas of modern science interested in energy
transfer processes at the nanoscale.
\fi

%The magnitude of this effect grows linearly with the pulling speed and becomes considerable at high pulling speeds. 

% {\it Fluctuation theorems for composite variables - } Let $W_A$ and $W_B$ be two work observables fulfilling a fluctuation symmetry:
% \begin{equation}
% \frac{P_F(W_{A,B})}{P_R(W_{A,B})}=\exp(W_{A,B}-\Delta_{A,B}). 
% \end{equation}
% Where $P_F$($P_R$) is probability along the forward (reverse) protocol. The fluctuation symmetry can be expressed in terms of the generating function $G^A_{F,R}(\gamma)=\langle \exp(\gamma W_A)\rangle_{F,R}$ as:
% \begin{equation}
% G^A_{F}(\gamma)=G^A_{R}(-1-\gamma)\exp(-\Delta F_A).
% \end{equation}
% Using this expression and the composition property of generating functions it is now easy to prove a fluctuation relation for $W_A+W_B$ and to show that in general $W_A-W_B$ does not satisfy a FR:
% \begin{equation}
% \begin{split}
% G^{A+B}_F(\gamma)&=G^{A}_F(\gamma)G^{B}_F(\gamma)=\\
% &=G^{A}_R(-1-\gamma)G^{B}_R(-1-\gamma)\exp(-\Delta F_A-\Delta F_B)=\\
% &=G^{A}_R(-1-\gamma)\exp(-\Delta F_A-\Delta F_B)
% \end{split}
% \end{equation}
% 
% \begin{equation}
% \begin{split}
% G^{A-B}_F(\gamma)&=G^{A}_F(\gamma)G^{B}_F(-\gamma)=\\
% &=G^{A}_R(-1-\gamma)G^{B}_R(-1+\gamma)\exp(-\Delta F_A-\Delta F_B)=\\
% &=G^{A}_R(-1-\gamma)G^{B}_R(-(1-\gamma))\exp(-\Delta F_A-\Delta F_B)=\\
% &\neq G^{A-B}_R(-1-\gamma)\exp(-\Delta F_A-\Delta F_B)
% \end{split}
% \end{equation}

\section*{Methods}

{\bf Buffers and DNA substrates.} All experiments were
  performed in PBS Buffer 1M NaCl at 25$^\circ C$; 1 mg/ml BSA was added to
  passivate the surfaces and avoid nonspecific interactions. The dsDNA
  tether was obtained ligating a 1kb segment to a biotin-labeled
  oligo at one end and a dig-labeled oligo at the other end.  The DNA
  hairpin used in the experiments has short (20bp) molecular handles and
  was synthesized by hybridization and ligating three different
  oligos. One oligo is biotin-labeled and a second is
  dig-labeled. Details of the systhesis procedure are given in
  \cite{forns2011hand}.

{\bf Optical Tweezers Assay.} Measurements were
  performed with a highly stable miniaturized laser tweezers in the dual
  trap mode \cite{ribezzi2013counter}.  This instrument directly
  measures forces by linear momentum conservation. In all experiments we
  used silica beads with 4 $\mu$m diameter, which give a maximum
  trapping force around $20$ pN. Data is acquired at 1 kHz.

\section*{Acknowledgements}
We thank A. Alemany and M. Palassini for a critical reading of the manuscript. FR is supported by ICREA Academia 2008 grant. The research leading to these results 
has received funding from the European Union Seventh Framework Programme (FP7/2007-2013) under grant agreement $\text{n}^\text{o}$ 308850 INFERNOS.

%merlin.mbs apsrev4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked
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%\bibliographystyle{pnas}
%\bibliography{prl}

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\newblock (2009) Measurement of work in single-molecule pulling experiments.
\newblock \emph{The Journal of Chemical Physics} 130:234116--234125.

\bibitem{alemany2011recent}
Alemany A, Ribezzi-Crivellari M, Ritort F
\newblock (2011) \newblock {\it in} Nonequilibrium Statistical Physics of Small Systems:
  Fluctuation Relations and Beyond, \newblock eds Klages R, Just W, Jarzynski C (Wiley-VCH, Weinheim)
\newblock pp 155--179.

\bibitem{nature.van.2008}
van Mameren J, {et~al.}
\newblock (2008) Counting rad51 proteins disassembling from nucleoprotein
  filaments under tension.
\newblock \emph{Nature} 457:745--748.

\bibitem{mmb.cisse.2011}
Cisse I, Mangeol P, Bockelmann U
\newblock (2011) \newblock in Single Molecule Analysis, \newblock eds Peterman EJG, Wuite GJL (Humana Press, New York) pp 45-61.

\bibitem{PNAS.moffitt.2006}
Moffitt J, Chemla Y, Izhaky D, Bustamante C
\newblock (2006) Differential detection of dual traps improves the spatial
  resolution of optical tweezers.
\newblock \emph{Proceedings of the National Academy of Sciences}
  103:9006--9011.


\bibitem{prl.speck.2008}
Speck T, Mehl J, Seifert U
\newblock (2008) Role of external flow and frame invariance in stochastic
  thermodynamics.
\newblock \emph{Physical Review Letters} 100:178302--178305.

\bibitem{mazonka1999exactly}
Mazonka O, Jarzynski C
\newblock (1999) Exactly solvable model illustrating far-from-equilibrium
  predictions.
\newblock \emph{arXiv preprint cond-mat/9912121}.

\bibitem{wang2002experimental}
Wang G, Sevick EM, Mittag E, Searles DJ, Evans DJ
\newblock (2002) Experimental demonstration of violations of the second law of
  thermodynamics for small systems and short time scales.
\newblock \emph{Physical Review Letters} 89:50601--50604.

\bibitem{Prl.meiners.1999}
Meiners JC, Quake SR
\newblock (1999) Direct measurement of hydrodynamic cross correlations between
  two particles in an external potential.
\newblock \emph{Physical Review Letters} 82:2211--2214.

\bibitem{PNAS.gore.2003}
Gore J, Ritort F, Bustamante C
\newblock (2003) Bias and error in estimates of equilibrium free energy
  differences from nonequilibrium measurements.
\newblock \emph{Proceedings of the National Academy of Sciences}
  100:12564--12569.

\bibitem{prl.palassini.2011}
Palassini M, Ritort F
\newblock (2011) Improving free energy estimates from unidirectional work
  measurements: theory and experiment.
\newblock \emph{Physical Review Letters} 107:60601.

\bibitem{ritort2002two}
Ritort F, Bustamante C, Tinoco I
\newblock (2002) A two-state kinetic model for the unfolding of single
  molecules by mechanical force.
\newblock \emph{Proceedings of the National Academy of Sciences}
  99:13544--13548.

\bibitem{PrE.jarzynski.2006}
Jarzynski C
\newblock (2006) Rare events and the convergence of exponentially averaged work
  values.
\newblock \emph{Phys. Rev. E} 73:046105--046114.

\bibitem{ribezzi2012force}
Ribezzi-Crivellari M, Ritort F
\newblock (2012) Force spectroscopy with dual-trap optical tweezers: Molecular
  stiffness measurements and coupled fluctuations analysis.
\newblock \emph{Biophysical journal} 103:1919--1928.

\bibitem{ribezzi2013counter}
Ribezzi-Crivellari M, Huguet JM, Ritort F
\newblock (2013) Counter-propagating dual-trap optical tweezers based on linear
  momentum conservation.
\newblock \emph{Review of Scientific Instruments} 84:043104--043104.


\bibitem{forns2011hand}
Forns N, De Lorenzo S, Manosas M, Hayashi K, Huguet JM, Ritort F 
\newblock(2011) Improving signal/noise resolution in single-molecule experiments using molecular constructs with short handles. 
\newblock \emph{Biophysical journal} 100:1765--1774.

\end{thebibliography}


% \clearpage

\section*{Caption of Figure 1}


\begin{figure}[ht]
 \centering
 \includegraphics[width=.70\textwidth]{Fig1main.pdf}
\caption{\label{Figure:opts}}
\end{figure}


{\bf Pulling experiments with dual-trap optical tweezers.} A) Force-distance curves
in a pulling experiment on a 20 bp hairpin with our dual-trap setup. A molecular tether is attached
between two trapped beads. By increasing the distance, $\lambda$,
between the traps the tether is stretched or released until some
thermally activated reaction is triggered, e.g. the unfolding or folding
of a DNA hairpin, and detected as a force jump (black
arrows). The small force jump (0.2 pN) is due to the low-trap stiffness
 of our dual-trap setup ($\simeq$0.02 pN/nm). Inset: scheme of the
hairpin with color-coded sequence (A/T: yellow/green, G/C: red/blue). B)
Pulling experiments in a dual-trap setup where trap A is moved at speed
$\dot\lambda$ and trap B is at rest with respect to water. $\lambda$ is
the control parameter, $y_A$ and $y_B$ are the configurational
variables with respect to the moving trap A while $x_A$ and $x_B$ are the configurational
variables with respect to the trap at rest (trap B). C)
pulling curves (red stretching, black releasing) for a 3kb ds-DNA
tether in a dual-trap setup. Inset: the cyclic pulling protocol used in
the experiments.

% \clearpage 
% 
% \clearpage

\section*{Caption of Figure 2}


 \begin{figure}[h]
\centering
 \includegraphics[width=.70\textwidth]{Fig2main2-new.pdf}
 \caption{\label{Figure:multi}}
 \end{figure}
% 


{\bf Work Distributions.} Work statistics obtained on cyclic protocols
with a 200 nm excursion with different pulling speeds (columns). Four
different work observables are considered. In each case the solid points
are direct work measurements (l.h.s. of Eq. \eqref{cfr}).  In order to
improve statistics these distributions are calculated as the convolution
of the distribution of the work performed while stretching with that
performed while releasing. For that we took all forward
  and reverse work values $W_F,W_R$ and combined them $W=W_F+W_R$ in
  order to get a joint work distribution for the cycle. Open symbols
are the reconstructed histogram (r.h.s. of Eq. \eqref{cfr}). Different
columns refer to different pulling speeds,$\dot \lambda$ , as shown on
top.  A) Comparison of the measured and reconstructed distributions
according to the two definitions of Eq. \eqref{dd} ($D,D'$).  The
distribution for $D$ satisfies the CFR Eq.\eqref{cfr}, i.e. the measured
and reconstructed distributions superimpose. The distribution for $D'$
does not satisfy it. Horizontal error bars represent the systematic
error in work measurements, while vertical error bars denote statistical
errors.  B) The CFR, Eq. \eqref{cfr} is satisfied within the
experimental error for $D,D^+$ and $D^-$ but not for $D'$. C) Comparison
between the measured and reconstructed distributions for $D^-$ and $D^+$
(Eq.\eqref{oro} and Eq. \eqref{sboro}) both of which satisfy the
CFR. Horizontal error bars represent the systematic error in work
measurements, while vertical error bars denote statistical errors.



\section*{Caption for Figure 3}


% \clearpage 
% 
\begin{figure}[h]
  \centering
 \includegraphics[width=0.6\textwidth]{Fig3main-new.pdf}
 \caption{\label{Figure:seifert} }
 \end{figure}
% 
% \clearpage 

{\bf Bias in unidirectional free energy estimators.} A) Error
($\mathcal{E}$) on free energy estimates Eq. \eqref{bias} committed by
using JE for $W'$.  Circles, diamonds and squares refer to excursions of
200 nm, 400 nm and 600 nm. Three different pulling speeds were
considered in each case. Note that in this case work is evaluated over
closed cycles ($\Delta G=0$) and the error is defined as:
$\mathcal{E}=-\beta^{-1}\log\langle e^{-\beta W'}\rangle$.  B) $\langle
W^+\rangle$ displays a bilinear dependence on pulling speed
$\dot\lambda$ (Main Figure) and $\Delta \lambda$ (Inset) as expected
from Eq. \eqref{defiw}. Continuous lines are linear fits to the
experimental results which contains a single fitting parameter. The
shaded area in the inset corresponds to the region within one standard
deviation from the expected value of $\langle W^+\rangle$ based on
equilibrium measurements of $\gamma_+$ (see Section 4,5 in the SI).  C)
Experimental bias measurements from the cycles shown in
Fig. \ref{Figure:opts}C. The plots show the bias
(Eqs. \eqref{fffd},\eqref{fffe}) as a function of the number of work
measurements, $N$. The three plots correspond to different pulling
speeds (7.2 $\mu$m/s, 4.3 $\mu$m/s, 1.35 $\mu$m/s). Interestingly
$B^-_N\ll B_N$ which guarantees faster convergence of free energy
estimates.  Moreover $B_N$ is also larger than the sum $B^-+B^+$ (dashed
line) i.e. the bias is superadditive (cf. Eq. \eqref{hiera}). The error
bars represent the statistical error on free energy determination, not
including systematic calibration errors in force and distance.
Continuous lines show the theoretical predictions from
Ref. \cite{prl.palassini.2011} for Gaussian work distributions. Note
that these are not fits but predictions which only use the mean
dissipation as input parameter.
% 
% \clearpage
% 

% \clearpage 
\section*{Caption for Figure 4}

\begin{figure}[h]
  \centering
 \includegraphics[width=.7\textwidth]{Fig4main.pdf}
 \caption{\label{Figure:sm}}
 \end{figure}
% 
% 

{\bf Work measurements on a DNA hairpin.} A)Scheme of the experimental setup (beads and hairpin not to scale). The hairpin is presented with color-coded sequence (A/T: yellow/green, G/C: red/blue).  B) Work measurements
upon unfolding and refolding according to three different work definitions: $W$ (upper panel), $W'$ (central panel) and $W^-$ (lower panel). In this experiment pulling speed was $0.96\pm0.02$ $\mu$m/s. Open symbols show an estimate of the refolding work distributions
reconstructed from the unfolding distribution via the CFR (Eq. \eqref{crozza}).  $W$ and $W^-$ fulfill the fluctuation theorem while $W'$ does not. Horizontal error bars represent the systematic error on work measurements, while vertical error bars denote statistical errors. The contribution of trap, handles and single stranded DNA have been removed as detailed in \cite{natphys.alemany.2012}.
C) Unidirectional estimates for the free energy from the unfolding work distribution. The optimal estimator based on $W^-$ (red) converges to the correct value $\Delta G_0=51$ $K_B$T as measured from bi-directional estimates. The estimator based on $W$ (green) shows a larger bias 
and overestimates the free energy by $\simeq1\,\, K_BT$. The estimator based on $W'$ (blue) converges to a wrong free energy difference ($\Delta G-\mathcal{E}$) which is $\simeq 3$ $K_BT$ below the correct value, against the second law. Note that the error committed by using $W$
is due to finite-size effects and decreases when more unfolding curves are measured. In contrast the error committed by using $W'$ remains finite for all sample sizes. 
The error bars represent the statistical error on free energy determination and do not include the systematic error due to force and distance calibrations.



\section*{Caption for Figure 5}

% \clearpage 
\begin{figure}[h]
 \centering
\includegraphics[width=0.99\textwidth]{Fig5main-new.pdf}
\caption{\label{Figure5}}
 \end{figure}
% 
% \clearpage




{\bf Inference of $P(W)$ from partial
    work measurements in the symmetric case.} A) The distribution $P'(W')$  in the case of the dsDNA tether, for the work
  measured in the wrong trap, $W'$. The distribution does not fulfill the CFR. To recover the correct work distribution $P(W)$, 
  $W'$ is shifted by a constant amount $\Delta$. The shifted distribution is
  tested for the CFR by defining the function: $H(W')=\log
  (P'(W'-\Delta)/P'(-W'-\Delta))$.  The prediction by the CFR
  is $H(W')=\beta\left(W'-\Delta G\right)$ which can be tested
  by determining the $H$ function.  B) Evolution of  $H(W')$ as a function of $W'$ for different values of
  $\Delta$. The value $\Delta^*$ for which the slope of
  $H(W')$ is equal to one (work being measured in $K_BT$ units) determines the correct work
  distribution $P(W)$ ($\Delta\simeq 15$, inset). 
  C) In the case of bidirectional measurements both the forward and the reverse work
  distributions $P'_F(W'),P'_R(-W')$ are shifted by an amount $\Delta$
  in opposite directions. D) Evolution of the function
  $H=\log (P_F'(W'-\Delta)/P_R'(-W'-\Delta))$ as a function of
  $\Delta$. Again the CFR predicts $H$ should be linear in $W'$ 
  with slope one. E) Inference of the correct work distributions and
  $\Delta G$ measurement.  For each value of $\Delta$ a linear fit
  $A(\Delta)W+B(\Delta)$ to $H$ is
  performed. The value $\Delta^*$ for which $A(\Delta^*)=1$
  ($\Delta^*\simeq 7$ in the figure), is the shift needed in order to
  recover the full work distribution $P(W)$ from partial work measurements in
  the wrong trap. Moreover the CFR implies $B(\Delta^*)=-\Delta G$
  ($\Delta G\simeq 60K_BT$).


\section*{Caption for Figure 6}

%   \clearpage 
\begin{figure}[h]
 \centering
 \includegraphics[width=0.7\textwidth]{Fig6main.pdf}
 \caption{\label{Figure6}}
 \end{figure}
% \clearpage 

{\bf Inference of $P(W)$ from partial
    work measurements in the asymmetric case.} (A) Equilibrium force
distributions at $2$pN for the 3kb ds-DNA tether measured in an asymmetric dual-trap setup ($k_A = 0.012$
pN/nm,$k_B = 0.003$ pN/nm,$k_m = 0.0027$ pN/nm). (B) Convolution of
$P'(W')$ with different Gaussian distributions corresponding to
different pairs $\Delta,\Sigma$.We show $P'(W)$ (blue filled circles),
$P'(-W)\exp(\beta W)$ (blue empty circles), $P(W)$ (green filled circles).  
Among all different $P_{\Delta,\Sigma}$ only one matches the correct work distribution $P(W)$, i.e. only one reconstructed
distribution is physically correct (rightmost graph, with $ \Delta=7.5$
$K_B$T, $\Sigma^2 =8$ ($K_B$T$)^2$). In this
situation the inference cannot rest on the CFR alone,
and additional information is required to infer $P(W)$. (C) Asymmetry
factor (AF) as a function of $x = k_A/k_B$ for different values of $y =
k_m/k_B$. The blue (red) circles indicate the symmetric ($AF=0$) and
asymmetric ($AF\simeq 1$)
cases respectively. (D) The AF defined by $\Sigma^2=AF\times\Delta$ (red
line) and the CFR invariance
$\Delta=\Delta^*+\phi,\Sigma^2=\Sigma^{*2}+2\phi K_BT$ for any $\phi$
(blue line), do
select a narrow range of possible
pairs ($\Delta$,$ \Sigma$) at the intersection between the blue and red lines.
The intersection region is compatible with the parameters
($\Delta$ =8 $K_B$T, $ \Sigma^2$ =7.2 ($K_B$T)$^2$ ) describing the true
correct work distribution $P(W)$ (black point).


 \newpage
\clearpage
\includepdf[pages=-]{SI.pdf}

\end{document}


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