Fluctuation-dissipation relations & fluctuation theorems 18.S995 - L11 1.7thermal Fluctuation-dissipation relation pumping allows particles to climb up-hill (see slides for an illustration). Until now, we focused primarily on over-damped Brownian motion processes, as sufficient to describe low-Reynolds number object. When inertia is not negligible, the above concepts can be easily extended by adding friction and noise to the Hamiltonian equation of motions. Considering a Hamiltonian H(x1 , . . . , xN , p1 , . . . , pN ), the corresponding system of SDEs Until now, we focused primarily on over-damped Brownian motion processes, as su reads 1.7 Fluctuation-dissipation relation to describe low-Reynolds number object. When inertia is not negligible, the above co @H dxi = dt friction and noise to the Hamiltonian (1.132a) can be easily extended by adding equation of m @pi Considering a Hamiltonian@H H(x1 , . . . , xNp , p1 , . . . , pN ), the corresponding system o dpi = dt pi dt + 2D dBi (t). (1.132b) @xi reads where (B1 (t), . . . , BN (t)) are standard Brownian @H motions, is the Stokes friction coefficient dt The last two terms in Eq. (1.132b) ( i = and D the di↵usion constant in dx momentum space. @pi provide an e↵ective description of the momentum transfer with ap surrounding heat bath. @H If the Hamiltonian has the standard form dpi = dt pi dt + 2D dBi (t). (1 @xi X p2 i H= + U (x1 , . . . , xN ), (1.133) 2m where (B1 (t), . . . , BN (t)) are standard Brownian motions, is the Stokes friction coe i and D the di↵usion constant in momentum space. The last two terms in Eq. (1 corresponding to momentum coordinates pi = mẋi , then the overdamped SDE is formally provide an e↵ective of the momentum transfer with a surrounding hea recovered by assuming dpi 'description 0 in Eq. (1.132b) and dividing by m , yielding If the Hamiltonian has the standardsform 1 @U X 2D dxi = dt + p2i2 2 dBi (t). (1.134) m @x m + U (x1 , . . . , xN ), H i= i 2m We see that the spatial di↵usion constant D and the momentum di↵usion constant D are related by corresponding to momentum coordinates pi = mẋi , then the overdamped SDE is fo recovered by assuming dpi ' 0 in Eq. D (1.132b) and dividing by m , yielding D = 2 2. (1.135) m s 1.7 Fluctuation-dissipation relation 1.7 Fluctuation-dissipation relation thermal pumping allows particles to climb up-hill (see slides for an illustration). Until we focused focusedprimarily primarily over-damped Brownian motion processes, as sufficient Until now, now, we onon over-damped Brownian motion processes, as sufficient totodescribe low-Reynoldsnumber number object. When inertia not negligible, the above concepts describe low-Reynolds object. When inertia is notisnegligible, the above concepts canbe be easily easily extended friction andand noise to the equation of motions. can extendedbybyadding adding friction noise to Hamiltonian the Hamiltonian equation of motions. Considering aa Hamiltonian systemsystem of SDEs 1 , .1., .. ,.x Considering HamiltonianH(x H(x . N, x, pN1,, p. .1., ,.p.N. ,),pNthe ), corresponding the corresponding of SDEs reads Until now, we focused primarily on over-damped Brownian motion processes, as su reads 1.7 Fluctuation-dissipation relation @H to describe low-Reynolds number object. When inertia is not (1.132a) negligible, the above co @H dxi = dt dxi = dt friction and noise to the Hamiltonian (1.132a) i can be easily extended by@padding equation of m @pi p @H Considering a Hamiltonian H(x1p,i dt . . +. , x2D , dB p1 ,i (t). . . . , pN ), the corresponding system o dpi = dt (1.132b) @H Np dpi = @xi dt pi dt + 2D dBi (t). (1.132b) @xi reads where (B1 (t), . . . , BN (t)) are standard Brownian motions, is the Stokes friction coefficient where . . . , BNconstant (t)) are standard Brownian is the Stokes friction coefficient @H motions, 1 (t), and D(B the di↵usion in momentum space. The last two terms in Eq. (1.132b) dx = dt The imomentum and D the di↵usiondescription constant of inthe momentum space. two terms heat in Eq. (1.132b) provide an e↵ective transfer withlast a surrounding bath. @pi If the Hamiltonian the standard provide an e↵ectivehas description of form the momentum transfer with ap surrounding heat bath. @H If the Hamiltonian has the standard Xdppi2form = dt pi dt + 2D dBi (t). i H= + U (x , . . . , x ), (1.133) 1 @xi N 2 X 2m pi i H= 2m + U (x1 , . . . , xN ), ( (1 (1.133) where (Bto . . . , BNcoordinates (t)) are standard Brownian motions, the Stokes friction coe 1 (t), i corresponding momentum pi = mẋi , then the overdamped SDE isisformally recovered assuming dpi ' 0 constant in Eq. (1.132b) and dividing byspace. m , yielding and Dby the di↵usion in momentum The last two terms in Eq. (1 corresponding to momentum coordinates pi = mẋi , then the overdamped SDE is formally s the momentum transfer with a surrounding hea provide an e↵ective description of recovered by assuming dpi ' 0 in1 Eq. and dividing by m , yielding @U (1.132b)2D dxi has = the standard dt + dBi (t). (1.134) If the Hamiltonian form s 2 2 m @xi m 1 @U X 2D 2 dx = dt + dBi (t).di↵usion constant D are(1.134) p i 2 2 i We see that the spatial di↵usion constant D and the momentum m @x m + U (x1 , . . . , xN ), H i= related by i 2m We see that the spatial di↵usion constantDD and the momentum di↵usion constant D are D = 2 2. (1.135) related by corresponding to momentum coordinates pi = mẋi , then the overdamped SDE is fo m recovered by assuming dpi ' 0 in Eq. (1.132b) and dividing by m ,. ,yielding Dthe phase The Fokker-Planck equation (FPE) governing space PDF f (t, x , . . . , x , p , . . pN ) 1 N 1 D = 2 2. (1.135) m s of the stochastic process (1.132) reads 1.7thermal Fluctuation-dissipation relation pumping allows X particles to climb up-hill (see slides for an illustration). p2 i H= + U (x1 , . . . , xN ), (1.133) 2m Until now, we focused primarily on i over-damped Brownian motion processes, as sufficient to describe low-Reynolds number object. When inertia is not negligible, the above concepts corresponding to momentum coordinates pi = mẋi , then the overdamped SDE is formally can be easily extended by adding friction and noise to the Hamiltonian equation of motions. recovered by assuming dpi ' 0 in Eq. (1.132b) and dividing by m , yielding Considering a Hamiltonian H(x1 , . . . , xN , p1 , . . . , pN ), the corresponding system of SDEs Until now, we focused primarily onsover-damped Brownian motion processes, as su reads 1 @U 2D dx = dt + dBi (t).inertia is not negligible, (1.134)the above co to describe low-Reynolds number object. When i 2 2 @H m @xdti m dx = (1.132a) i can be easily extended by adding friction and noise to the Hamiltonian equation of m @pi We Considering see that the spatial di↵usion constant momentum constant D are system o a Hamiltonian H(xD . . . ,the xNp , p1 , . . . , pNdi↵usion ), the corresponding @H 1 , and dpi = dt pi dt + 2D dBi (t). (1.132b) related by @xi reads D motions, is the Stokes friction coefficient where (B1 (t), . . . , BN (t)) are standardDBrownian = 2@H . (1.135) 2 dt m dx = ( i and D the di↵usion constant in momentum space. The last two terms in Eq. (1.132b) 1.7 Fluctuation-dissipation relation @pi provide an e↵ective description of thegoverning momentum transfer withPDF ap surrounding heat bath. The Fokker-Planck equation (FPE) the phase space f (t, x , . . . , x , p 1 N 1 , . . . , pN ) @H If the has the standard of the Hamiltonian stochastic process (1.132) reads dpi form = dt pi dt + 2D dBi (t). (1 X p2 ◆ @xi ✓ ✓ ◆ X @HH@f= @H i@f+ U (xX @f , xN ), (1.133) 1, . . .@ @ f + = p f + D (1.136) i motions, 2m where (B1t (t), . . . , B are Brownian is the Stokes friction coe Ni (t)) i @xstandard @p @x @p @p @p i i i i i i i and D the di↵usion constant in momentum space. The last two terms in Eq. (1 corresponding to momentum coordinates pi = mẋi , then the overdamped SDE is formally provide an e↵ective of29the momentum transfer with a surrounding hea recovered by assuming dpi 'description 0 in Eq. (1.132b) and dividing by m , yielding If the Hamiltonian has the standardsform 1 @U X 2D dxi = dt + p2i2 2 dBi (t). (1.134) m @x m + U (x1 , . . . , xN ), H i= i 2m We see that the spatial di↵usion constant D and the momentum di↵usion constant D are related by corresponding to momentum coordinates pi = mẋi , then the overdamped SDE is fo recovered by assuming dpi ' 0 in Eq. D (1.132b) and dividing by m , yielding D = 2 2. (1.135) m s 1.7thermal Fluctuation-dissipation relation pumping allows X particles to climb up-hill (see slides for an illustration). p2 i H= + U (x1 , . . . , xN ), (1.133) 2m Until now, we focused primarily on i over-damped Brownian motion processes, as sufficient to describe low-Reynolds number object. When inertia is not negligible, the above concepts corresponding to momentum coordinates pi = mẋi , then the overdamped SDE is formally can be easily extended by adding friction and noise to the Hamiltonian equation of motions. recovered by assuming dpi ' 0 in Eq. (1.132b) and dividing by m , yielding Considering a Hamiltonian H(x1 , . . . , xN , p1 , . . . , pN ), the corresponding system of SDEs Until now, we focused primarily onsover-damped Brownian motion processes, as su reads 1 @U 2D dx = dt + dBi (t).inertia is not negligible, (1.134)the above co to describe low-Reynolds number object. When i 2 2 @H m @xdti m dx = (1.132a) i can be easily extended by adding friction and noise to the Hamiltonian equation of m @pi We Considering see that the spatial di↵usion constant momentum constant D are system o a Hamiltonian H(xD . . . ,the xNp , p1 , . . . , pNdi↵usion ), the corresponding @H 1 , and dpi = dt pi dt + 2D dBi (t). (1.132b) related by @xi reads D motions, is the Stokes friction coefficient where (B1 (t), . . . , BN (t)) are standardDBrownian = 2@H . (1.135) 2 dt m dx = ( i and D the di↵usion constant in momentum space. The last two terms in Eq. (1.132b) 1.7 Fluctuation-dissipation relation @pi provide an e↵ective description of thegoverning momentum transfer withPDF ap surrounding heat bath. The Fokker-Planck equation (FPE) the phase space f (t, x , . . . , x , p 1 N 1 , . . . , pN ) @H If the has the standard of the Hamiltonian stochastic process (1.132) reads dpi form = dt pi dt + 2D dBi (t). (1 X p2 ◆ @xi ✓ ✓ ◆ X @HH@f= @H i@f+ U (xX @f , xN ), (1.133) 1, . . .@ @ f + = p f + D (1.136) i motions, 2m where (B1t (t), . . . , B are Brownian is the Stokes friction coe Ni (t)) i @xstandard @p @x @p @p @p i i i i i i i and D the di↵usion constant in momentum space. The last two terms in Eq. (1 corresponding to momentum coordinates pi = mẋi , then the overdamped SDE is formally provide an e↵ective of29the momentum transfer with a surrounding hea recovered by assuming dpi 'description 0 in Eq. (1.132b) and dividing by m , yielding the vanishes Hamiltonian the standard form TheIflhs. if f is has a function of the s Hamiltonian H. The rhs. vanishes for the 1 @U X 2D particular ansatz dxi = dt + p2i2 2 dBi (t). (1.134) m @x m ◆ i= ✓ H + U (x1 , . . . , xN ), 1 H (1.137) f = exp i 2m . Z D and kBthe T momentum di↵usion constant D are We see that the spatial di↵usion constant related by corresponding to momentum coordinates ẋisee , then SDE is fo i = m where T is the temperature of the surrounding heat pbath. To this, the noteoverdamped that recovered by assuming dpi ' 0 in✓Eq. and dividing by m , yielding D (1.132b) ◆ . (1.135) @f 1 @HD = m2 2H 1 pi = exp = s f (1.138) related by thermal pumping allows particles to climb up-hill (see slides for an illustration). D D = 2 2. m (1.135) 1.7 Fluctuation-dissipation relation The Fokker-Planck equation (FPE) governing the phase space PDF f (t, x , . . . , x 1 N , p1 , . . . , p N ) of the stochastic process (1.132) reads Until now, we focused primarily ◆ on over-damped Brownian motion processes, as su ✓ ✓ ◆ X @H @f number X @When inertia @H @f object. @f is not negligible, the above co to describe low-Reynolds @t f + = pi f + D (1.136) @pi @x @xi @pi friction @p i adding i i can be easily extended by and noise to@pthe Hamiltonian equation of m i i Considering a Hamiltonian H(x , . . . , xN , p1 , . . . , pN ), the corresponding The lhs. vanishes if f is a function of the129 Hamiltonian H. The rhs. vanishes for the reads ansatz particular ✓ ◆ @H 1 H (1.137) f= dxZi exp = k T dt. B @p system o ( i @Hbath. To see this,p where T is the temperature of the surrounding heat note that dpi =✓ dt pi dt + 2D dBi (t). ◆ @f 1 @H H@xi 1 pi = exp = f (1.138) @pi kB T @pi kB T kB T m (1 where (B1 (t), . . . , BN (t)) are standard Brownian motions, is the Stokes friction coe soand that D the the components of theconstant dissipative in momentum current,space. The last two terms in Eq. (1 di↵usion momentum ✓ ◆ ✓ ◆ ✓ ◆ @fdescription of D D provide an e↵ective thepi momentum transfer with a surrounding hea Ji = pi f + D = pi f f = pi f (1.139) kB T form m mkB T i If the Hamiltonian@phas the standard vanishes if X p2 i H= + U (x1 , . . . , xN ), kB T 2m D = mkB T , D= . (1.140) i m Equation (1.140) is the relation, p connecting the di↵usion constant corresponding to fluctuation-dissipation momentum coordinates i = mẋi , then the overdamped SDE (strength of the fluctuations) and the friction coefficient (dissipation) through the temperrecovered by assuming dpi ' 0 in Eq. (1.132b) and dividing by m , yielding ature of the bath. s is fo related by thermal pumping allows particles to climb up-hill (see slides for an illustration). D D = 2 2. m (1.135) 1.7 Fluctuation-dissipation relation The Fokker-Planck equation (FPE) governing the phase space PDF f (t, x , . . . , x 1 N , p1 , . . . , p N ) of the stochastic process (1.132) reads Until now, we focused primarily ◆ on over-damped Brownian motion processes, as su ✓ ✓ ◆ X @H @f number X @When inertia @H @f object. @f is not negligible, the above co to describe low-Reynolds @t f + = pi f + D (1.136) @pi @x @xi @pi friction @p i adding i i can be easily extended by and noise to@pthe Hamiltonian equation of m i i Considering a Hamiltonian H(x , . . . , x , p , . . . , p ), the corresponding system o N 1 The lhs. vanishes if f is a function of the129 Hamiltonian H. The N rhs. vanishes for the reads ansatz particular The lhs. vanishes if f is a function of the H. The rhs. vanishes for the ✓ Hamiltonian ◆ @H 1 H particular ansatz (1.137) f= dxZi exp = k T dt. ( B ✓ @p ◆ i 1 H , (1.137) f = exp p @H where T is the temperature of the surrounding bath. To see this, note that Z kheat T dpi =✓ B ◆ dt pi dt + 2D dBi (t). (1 @xi bath. 1To psee @f 1the@H Hheat i where T is the temperature of surrounding = exp = f this, note that (1.138) @pi kB T @pi kB T m ✓ kB T ◆ where (B1 (t), . .@f. , BN (t)) are1 standard 1 @H H Brownian 1 pimotions, is the Stokes friction coe = exp = f (1.138) soand that D the the components of the dissipative momentum current, @pi kconstant kB T kB Tspace. m B T @pi Z di↵usion in momentum The last two terms in Eq. (1 ✓ ◆ ✓ ◆ ✓ ◆ D provide an e↵ective description of Dthepi momentum transfer with a surrounding hea so thatJ the components of@fthe dissipative momentum current, = p f + D = p f f = p f (1.139) i i i i @p k T m mk T ✓ ◆ ✓ ◆ ✓ ◆ i B B If the Hamiltonian @f has the standard form D pi D J = pi f + D = pi f f = pi f (1.139) vanishes ifi 2 @pi kB T m mk T X B p vanishes if D = mkB T i H= , 2m D= + U (x1 , . . . , xN ), kB T . (1.140) m kB T D = mkB T , D= . (1.140) Equation (1.140) is the fluctuation-dissipation relation, connecting the di↵usion constant m corresponding to momentum coordinates pi = mẋi , then the overdamped SDE (strength of the fluctuations) and the friction coefficient (dissipation) through the temperrecovered by assuming dpi ' 0 in Eq. (1.132b) andthe dividing m , yielding Equation relation, connecting di↵usion by constant ature of the(1.140) bath. is the fluctuation-dissipation (strength of the fluctuations) and the friction coefficient (dissipation) through the tempers i is fo related by thermal pumping allows particles to climb up-hill (see slides for an illustration). D D = 2 2. m (1.135) 1.7 Fluctuation-dissipation relation The Fokker-Planck equation (FPE) governing the phase space PDF f (t, x , . . . , x 1 N , p1 , . . . , p N ) of the stochastic process (1.132) reads Until now, we focused primarily ◆ on over-damped Brownian motion processes, as su ✓ ✓ ◆ X @H @f number X @When inertia @H @f object. @f is not negligible, the above co to describe low-Reynolds @t f + = pi f + D (1.136) @pi @x @xi @pi friction @p i adding i i can be easily extended by and noise to@pthe Hamiltonian equation of m i i Considering a Hamiltonian H(x , . . . , x , p , . . . , p ), the corresponding system o N 1 The lhs. vanishes if f is a function of the129 Hamiltonian H. The N rhs. vanishes for the reads ansatz particular The lhs. vanishes if f is a function of the H. The rhs. vanishes for the ✓ Hamiltonian ◆ @H 1 H particular ansatz (1.137) f= dxZi exp = k T dt. ( B ✓ @p ◆ i 1 H , (1.137) f = exp p @H where T is the temperature of the surrounding bath. To see this, note that Z kheat T dpi =✓ B ◆ dt pi dt + 2D dBi (t). (1 @xi bath. 1To psee @f 1the@H Hheat i where T is the temperature of surrounding = exp = f this, note that (1.138) @pi kB T @pi kB T m ✓ kB T ◆ where (B1 (t), . .@f. , BN (t)) are1 standard 1 @H H Brownian 1 pimotions, is the Stokes friction coe = exp = f (1.138) soand that D the the components of the dissipative momentum current, @pi kconstant kB T kB Tspace. m B T @pi Z di↵usion in momentum The last two terms in Eq. (1 ✓ ◆ ✓ ◆ ✓ ◆ D provide an e↵ective description of Dthepi momentum transfer with a surrounding hea so thatJ the components of@fthe dissipative momentum current, = p f + D = p f f = p f (1.139) i i i i @p k T m mk T ✓ ◆ ✓ ◆ ✓ ◆ i B B If the Hamiltonian @f has the standard form D pi D J = pi f + D = pi f f = pi f (1.139) vanishes ifi 2 @pi kB T m mk T X B p vanishes if D = mkB T i H= , 2m D= + U (x1 , . . . , xN ), kB T . (1.140) m kB T D = mkB T , D= . (1.140) Equation (1.140) is the fluctuation-dissipation relation, connecting the di↵usion constant m corresponding to momentum coordinates pi = mẋi , then the overdamped SDE (strength of the fluctuations) and the friction coefficient (dissipation) through the temperrecovered by assuming dpi ' 0 in Eq. (1.132b) andthe dividing m , yielding Equation relation, connecting di↵usion by constant ature of the(1.140) bath. is the fluctuation-dissipation (strength of the fluctuations) and the friction coefficient (dissipation) through the tempers i is fo 1.8 Fluctuation theorems 20 Microbiological systems often perform ‘thermodynamic’ oper number of degrees of freedom. To characterize biological motor in terms of thermodynamic quantities (work, entropy, etc.), an thermodynamic concepts to non-equilibrium processes is has bee decade. Theoretical work in this direction was triggered by the perimental techniques [BSLS00] that make it possible to probe characteristics of individual DNA molecules with the help of opt for a simple schematic). These e↵orts led, amongst others, to t of exact fluctuation theorems (FTs) for non-equilibrium system which we will discuss below. The total Hamiltonian comprising the system of interest, e.g. a by coordinates x(t)), its environment y and mutual interactions http://hansmalab.physics.ucsb.edu/forcespec.html H(x, y; (t)) = H(x; (t)) + Henv (y) + Hint (x wherehttp://www.microscopy-analysis.com (t) denotes one or more external control parameters (e.g Single-molecule force measuring experiments by using AFM (a) and laser tweezers (b). Hummer G , and Szabo A PNAS 2001;98:3658-3661 ©2001 by National Academy of Sciences winding. We constructed a simple ‘toy’ model in which the radius of the helix was allowed to vary as the molecule was stretched, and asked whether this model could capture the twist–stretch coupling and other mechanical properties of DNA. The model consists of an elastic rod with a stiff helical ‘wire’ (analogous to the sugar-phosphate backbone) affixed to the outside surface (Fig. 3a). The inner rod is constructed from a material with a Poisson’s ratio n ¼ 0.5, so that it conserves volume under stress23. As this system is stretched, the inner rod decreases in diameter. When the model is stretched without constraining twist, changes in helicity arise from the tendency of the stiff outer ‘wire’ to resist changes in contour length (Fig. 3b). The inner rod alone has stretch modulus Br ¼ p2,4 R4r Y r =4 and torsional rigidity Sr ¼ pR2r Y r , bending rigidity 1 2 LETTERS torsion and also generates a non-zero twist–stretch coupling, g. With the three free parameters fitted to R r ¼ 0.924 nm (close to the crystallographic radius of DNA), S h ¼ 965 pN, and Y r ¼ 0.393 GPa (similar to estimated Y values for DNA and within the range of Vol 442|17 August 2006|doi:10.1038/nature04974 measured Y values for bulk polymeric materials14,24,26), we obtain the correct experimentally measured values for the mechanical parameters of DNA: B eff ¼ 225 pN nm 2 , C eff ¼ 460 pN nm 2 , g ¼ 290 pN nm and S eff ¼ 1,081 pN. Although construction of this toy DNA model was motivated by the discovery of negative twist–stretch coupling, it also provides a possible explanation for the anomalously large torsional rigidity of DNA. For an isotropic rod, the torsional rigidity C must be smaller than the bending rigidity B (see equation for C above) unless the rod 2 1–4 r DNA overwinds when stretched Jeff Gore †, Zev Bryant †, Marcelo Nöllmann , Mai U. Le2, Nicholas R. Cozzarelli ‡ & Carlos Bustamante DNA is often modelled as an isotropic rod1–4, but its chiral structure suggests the possible importance of anisotropic mechanical properties, including coupling between twisting and stretching degrees of freedom. Simple physical intuition predicts that DNA should unwind under tension, as it is pulled towards a denatured structure4–8. We used rotor bead tracking to directly measure twist–stretch coupling in single DNA molecules. Here we show that for small distortions, contrary to intuition, DNA overwinds under tension, reaching a maximum twist at a tension of ,30 pN. As tension is increased above this critical value, the DNA begins to unwind. The observed twist–stretch coupling predicts that DNA should also lengthen when overwound under constant tension, an effect that we quantitatively confirm. We present a simple model that explains these unusual mechanical properties, and also suggests a possible origin for the anomalously large torsional rigidity of DNA. Our results have implications for the action of DNA-binding proteins that must stretch and twist DNA to compensate for variability in the lengths of their binding sites9–11. The requisite coupled DNA distortions are favoured by the intrinsic mechanical properties of the double helix reported here. Many cellular proteins bend or wrap DNA upon binding, loop DNA to make contact with two non-adjacent binding sites, or twist DNA during translocation along the double helix12. The energetics of these distortions are governed by the mechanical properties of DNA, which have been investigated using a variety of bulk1 and singlemolecule techniques2,3. For small deformations, DNA in physiological buffer is modelled as an isotropic rod with bending rigidity B ¼ 230 ^ 20 pN nm2, twist rigidity C ¼ 460 ^ 20 pN nm2, and stretch modulus S ¼ 1,100 ^ 200 pN (refs 3, 13–16; and Z.B, J.G., N.R.C. and C.B, manuscript in preparation). A fourth mechanical parameter is allowed in the linear theory of a deformable rod5,6,17: the twist–stretch coupling g, which specifies how the twist of the helix changes when the molecule is stretched. At forces sufficient to stretched. suppress bending fluctuations, theconstruct energy of afor Figure 1 | DNA overwinds when a, The molecular 5,6,17 : sites and stretched and twisted DNA molecule may be written as rotor bead tracking experiments contains three distinct attachment DNA:protein complexes showed a weak positive correlation between twist and rise, implying a negative g value9. All-atom simulations have likewise suggested that twist and rise may be positively correlated for small distortions10,19. Finally, in contrast to the overstretching transition8,13,14 (in which DNA unwinds as it extends), the B–A transition involves a slight unwinding coupled to compression of the DNA helix20. Conclusive determination of the sign and magnitude of g requires direct measurement in isolated DNA molecules. To measure the twist–stretch coupling of DNA, we used the rotor bead tracking technique13,15,21, in which a submicrometre ‘rotor’ bead is attached to the middle of a stretched DNA molecule, immediately below a free swivel consisting of an engineered single strand nick (Fig. 1a, b). Tension is applied to the molecule using magnetic tweezers2,4. Changes in the rotor bead angle reflect changes in the twist of the lower DNA segment. Under fixed tension, the rotor bead fluctuated around a mean angle as a result of thermal noise (Fig. 1c). As the molecule was stretched by increasing the magnetic force, the mean angle of the rotor bead increased. A ,1% stretching of DNA led to an increase in the twist of ,0.1% (Fig. 1d). Analysis of this data yielded a twist– stretch coupling constant g ¼ 290 ^ 20 pN nm (N ¼ 4 molecules), opposite in sign to most previous estimates5–7 but consistent with twist–rise correlations in crystal structures9 and molecular simulations10,19. Our measurement of g was robust to alterations in the length and sequence of the torsionally constrained DNA segment (Fig. 1d). For small deformations, we have shown that DNA overwinds when stretched. However, in the limit of high forces, DNA must eventually unwind as the backbone is pulled straight. To test for sign reversal of g at elevated tensions, we monitored the twist of DNA molecules while gradually increasing the magnetic force. We found that the twist of DNA increases until the tension reaches a critical value F c < 30 pN, beyond which the DNA begins to unwind (Fig. 1e). The negative twist–stretch coupling g observed at lower tensions implies that DNA should overwound (Fig. 2a, b). Tolinearly with applied overwinding of thelengthen DNA. when d, The overwinding scales predict the expected magnitude of this effect, we write the total tension and with the length of the torque-bearing DNA segment. Plotted perimental techniques ature of the bath. [BSLS00] that make it possible to probe the folding and twisting characteristics of individual DNA molecules with the help of optical tweezers (see Fig. 1.1 for a simple schematic). These e↵orts led, amongst others, to the discovery of a number of1.8 exact fluctuation theorems (FTs) for non-equilibrium systems, the simplest version of Fluctuation theorems which we will discuss below. 20 Microbiological systems often perform ‘thermodynamic’ withdescribed a mesoscopic The total Hamiltonian comprising the system of interest, e.g. aoperations DNA molecule by coordinates x(t)), of itsfreedom. environment and mutual biological interactions reads protein energetics, etc. number of degrees To ycharacterize motors, in terms of thermodynamic quantities (work, entropy, etc.), an extension of traditional H(x, y; (t)) = H(x; (t)) + Henv (y) + Hint (x, y) thermodynamic concepts to non-equilibrium processes is has been developed (1.141) over the last decade.(t)Theoretical in this direction was triggered(e.g., by the where denotes onework or more external control parameters thedevelopment force exerted of by new a experimental [BSLS00] that make possible to probe the folding andthe twisting tweezer in a techniques molecule pulling experiments, see itFig. 1.1). The function (t) defines characteristics of individual DNA molecules with the help of optical tweezers (see Fig. 1.1 20 discussion in this section closely follows that in the Christopher Jarzynski’s review article [Jar11]. forThe a simple schematic). These e↵orts led, amongst others, to the discovery of a number of exact fluctuation theorems (FTs) for non-equilibrium systems, the simplest version of Figure 1.1: Schematic representation of a 29 molecule pulling experiment (from Ref. [Jar11 which we will discuss below. The molecule modeled by acomprising collectiontheofsystem masses connected springs. tweezer ac Theis total Hamiltonian of interest, e.g. aby DNA moleculeThe described on one end of the molecule, via anyattached gold bead (blue), by coordinates x(t)), its e.g., environment and mutual interactions reads whereas the other en is attached to surface. H(x, y; (t)) = H(x; (t)) + Henv (y) + Hint (x, y) (1.141) denotes one or more external control parameters (e.g.,we thewill force exerted that by a there protocol where of the (t) control parameter variation and, for simplicity, assume tweezer in a molecule pulling experiments, see Fig. 1.1). The function (t) defines the only a single control parameter from now on. For example, for the toy model in Fig. 1 The(z discussion section we have x20= , pthis andfollows that in the Christopher Jarzynski’s review article [Jar11]. 1 , z2 , z3in 1, p 2 , p3 )closely Figure 1.1: Schematic representation of a molecule pulling experiment (from Ref. [Jar11]). The molecule is modeled by a X collection of connected by springs. The tweezer acts 3 2 masses 29 2 X p on one end of the molecule, gold zbead (blue), zwhereas the other(1.14 end H(x; (t)) =e.g., viai an+ attached u(zk+1 k ) + u( 3) 2m k=0 is attached to surface. i=1 where u is of interaction and z0 (t) ⌘and, 0 theforposition of the The work protocol the controlpotential parameter variation simplicity, we wall. will assume that performe there is ature of the bath. characteristics of individual DNA molecules with the help of optical tweezers (see Fig. 1.1 for a simple schematic). These e↵orts led, amongst others, to the discovery of a number of exact fluctuation theorems theorems (FTs) for non-equilibrium systems, the simplest version of 1.8 Fluctuation which we will discuss below. 20 The total Hamiltonian comprising the system of interest, e.g. a DNA molecule described Microbiological systems often perform ‘thermodynamic’ operations with a mesoscopic by coordinates x(t)),ofitsfreedom. environment y and mutual interactions readsprotein energetics, etc. number of degrees To characterize biological motors, in terms of thermodynamic quantities (work, entropy, etc.), an extension of traditional H(x, y; to (t)) = H(x; (t)) +processes Henv (y) +isHhas y) developed over (1.141) int (x, thermodynamic concepts non-equilibrium been the last decade.(t)Theoretical work in this direction was parameters triggered by the the development of new where denotes one or more external control (e.g., force exerted by aexperimental [BSLS00] that makesee it Fig. possible probe the folding and twisting tweezer in atechniques molecule pulling experiments, 1.1).toThe function (t) defines the characteristics of individual DNA molecules with the help of optical tweezers (see Fig. 1.1 20 discussion in this section closelye↵orts followsled, that amongst in the Christopher review article for The a simple schematic). These others,Jarzynski’s to the discovery of a[Jar11]. number of exact fluctuation theorems (FTs) for non-equilibrium systems, the simplest version of 29 which we will discuss below. The total Hamiltonian comprising the system of interest, e.g. a DNA molecule described by coordinates x(t)), its environment y and mutual interactions reads H(x, y; (t)) = H(x; (t)) pulling + Henv (y) + H int (x, Figure 1.1: Schematic representation of a molecule experiment (from Ref.y) [Jar11]). The molecule is modeled by a collection of masses connected by springs. The tweezer acts on one end of the molecule, e.g., via an attached gold bead (blue), whereas(e.g., the other (t) denotes one or more external control parameters theendforce is attached to surface. where tweezer in a molecule pulling experiments, see Fig. 1.1). The function 20 (1.141) exerted by a (t) defines the protocol of the control parameter variation and, for simplicity, we will assume that there is The discussion in this section closely follows that in the Christopher Jarzynski’s review article [Jar11]. only a single control parameter from now on. For example, for the toy model in Fig. 1.1 we have x = (z1 , z2 , z3 , p1 , p2 , p3 ) and 29 3 2 X X p2i H(x; (t)) = + u(zk+1 2m i=1 k=0 zk ) + u( z3 ) (1.142) where u is interaction potential and z0 (t) ⌘ 0 the position of the wall. The work performed during an infinitesimal parameter variation d is defined by @H ature of the bath. attached to surface. characteristics of individual DNA molecules with the help of optical tweezers (see Fig. 1.1 ed to surface. for a simple schematic). These e↵orts led, amongst others, to the discovery of a number of exact fluctuation theorems theorems (FTs) for non-equilibrium systems, the simplest version of 1.8 Fluctuation otocol of which the control parameter we will discuss below. variation and, for simplicity, we will assume that ofy the control parameter variation and, for simplicity, we will assume tha 20 control The total Hamiltonian comprising the system of interest, e.g. a DNA molecule described a single parameter from now on. For example, for the toy model in Microbiological systems often perform ‘thermodynamic’ operations with a mesoscopic by coordinates x(t)),ofitsfreedom. environment y and mutual interactions readsprotein ngle parameter from on. For example, for the toy model in number of ,degrees Tonow characterize biological motors, energetics, etc. havecontrol x= (z , z z , p , p , p ) and 1 2 3 1 2 3 in terms of thermodynamic quantities (work, entropy, etc.), an extension of traditional x = (z1 , thermodynamic z2 , z3 , p1 , p2concepts , p3 )y;and H(x, (t)) = H(x; (t)) +processes Henv (y) +isHhas y) developed over (1.141) int (x, to non-equilibrium been the last 3 2 2direction X X decade. Theoretical work in this was parameters triggered by the the development of new p where (t) denotes one or more external control (e.g., force exerted by aexi H(x; (t))pulling +2makeseeitu(z zprobe +the u( folding z3defines ) twisting 3= perimental [BSLS00] thatX possible toThe and k+1 k ) function 2experiments, tweezer in atechniques moleculeX Fig. 1.1). (t) the p 2m i characteristics of individuali=1 DNA molecules with the help of optical tweezers (see Fig. 1.1 k=0 H(x; (t)) = + u(z z ) + u( z ) 20 k+1 k 3 The discussion in this section closely follows that in the Christopher Jarzynski’s review article [Jar11]. for a simple schematic). These e↵orts led, amongst others, to the discovery of a number 2m i=1 k=0 of exact fluctuation theorems (FTs) for ⌘ non-equilibrium systems, the simplest versionwork of ere u is interaction potential and z0 (t) 0 the position of the wall. The per 29 which we will discuss below. ring an infinitesimal parameter variation d ofisinterest, defined by The total Hamiltonian comprising the system e.g. described s interaction potential and z0 (t) ⌘ 0 the position ofa DNA the molecule wall. The work p by coordinates x(t)), its environment y and mutual interactions reads n infinitesimal parameter variation d @H is defined by W := d (x; ). H(x, y; (t)) = H(x; (t)) pulling + Henv (y) + H int (x, Figure 1.1: Schematic representation of a molecule experiment (from Ref.y) [Jar11]). The molecule is modeled by a collection of masses connected by springs. The tweezer acts on one end of the molecule, e.g., via an attached gold bead (blue), whereas(e.g., the other (t) denotes one or more external control parameters theendforce is attached to surface. @ (1.141) @H where exerted by a Winitial := d condition (x;Fig. (0) ). in a molecule pulling experiments, see 1.1).=The0 function (t) defines the ⌧ , giventweezer protocol (t) with and final (⌧ ) = @ ra protocol of the control parameter variation and, for simplicity, we will assume that there is 20 rk performed the system The on discussion in this sectionisclosely follows that in the Christopher Jarzynski’s review article [Jar11]. only a single control parameter from now on. For example, for the toy model in Fig. 1.1 have with x = (z , z ,initial zZ, p , p , p ) and en protocol we(t) condition (0) = 0 and final (⌧ ) = Z ⌧ 29 @H X p X ˙ (t) formed on the system is (t)) W W = = dt (t)) H(x; = + u(z z ) + u( z (x(t); ) (1.142) 2m @ 0 1 2 3 1 2 3 3 Z i=1 Z 2 i th ⌧, 2 k+1 k 3 k=0 where u is interaction potential and z0⌧(t) ⌘ 0 the position of the wall. The work performed during an infinitesimal parameter variation d is defined by @H ˙ ere the integralWis=computed the (t) trajectory x(t) (t)) realized by the system W =along dt (x(t); @H ing an infinitesimal parameter@ variation d is defined by For a given(t)protocol with initial (0) = 0 and final (⌧ ) = ⌧ , iven protocol with initial(t) condition (0) = condition 0 and final (⌧ ) = ⌧ , the total @H work performed onis the system is erformed on1.8 the system Fluctuation theorems W := d (x; ). @⌧ Z Z ⌧ Z Z 20 @Hperform ‘thermodynamic’ Microbiological systems often operations with a mesoscop ˙ W = W =W = dt (t) W = (x(t); dt (t))˙ (t) @H (x(t); (1.144) (t)) @To number of process degrees of0 freedom. characterize biological motors, protein(⌧energetics, et a givenRepeat protocol (t) with initial condition (0) = and final ) = ⌧ , th 0 and measure @ 0 in terms of thermodynamic quantities (work, entropy, etc.), an extension of traditiona kheperformed on the system is integralthermodynamic is computed along the trajectory x(t) realized by the system. concepts to non-equilibrium processes is has been That developed over the las whererealization the integral is computed along the trajectory x(t) realized by the syste a given W depends not only on the protocol but also on the initial Z Z ⌧ decade. Theoretical work in this direction was triggered by the development of new ex @H and realization the initial state ydepends ofthat the make environment, if we assume that is,the forsystem a given W not only on the protocol but also on t 0 of ˙ 0 perimental techniques [BSLS00] it possible to probe the folding and twistin W = W = dt (t) (x(t); (t)) y)state > 0 during the process. If we repeat this process many times forenvironment, the same @y 0theof xcharacteristics and the initial state theof if we 0molecules 0 of the system of individual DNA with help optical tweezers (see Fig.ass 1. l, we will observe di↵erent values of These work {W 1 , W2 , . . . , } that will be governed by schematic). e↵orts to the discovery of a numbe Hint (x,for y) a>simple 0 during the process. If led, we amongst repeat others, this process many times for n probability density ⇢(W ). FTs arealong exact equalities and inequalities for certain ere the integral is computed the trajectory x(t) realized by the system. of exact fluctuation theorems (FTs) for non-equilibrium systems, the simplest version o protocol, we will observe di↵erent values of work {W , W , . . . , } that will be go 1 2 tion valueswhich we will discuss below. fora acertain given probability realization density WZ depends theequalities protocol and but inequalities also on thef ⇢(W ).not FTsonly are on exact total Hamiltonian comprising the system of interest, e.g. a DNA molecule describe te expectation x0 of theThe system and the initial state y of the environment, if we assum 0 values hG(W )i := dW ⇢(W ) G(W ), (1.145) by coordinates x(t)), its environment y and mutual interactions reads t (x, y) > 0 during the process. If we Zrepeat this process many times for th H(x, y; (t)) = H(x; (t)) + {W Henv1(y) +2 ,H. int (x, y) (1.141 tocol, we will observe di↵erent values of work , W . . , } that will be gover hG(W )i := dW ⇢(W ) G(W ), 30 ertain probability density ⇢(W ). FTs are exact equalities and inequalities for where (t) denotes one or more external control parameters (e.g., the force exerted by ectation tweezer values in a molecule pulling experiments, see Fig. 1.1). The function (t) defines th 30 certain G(W) Z for 20 FTs = exact (in)equalities The discussion in this section closely follows that in the Christopher Jarzynski’s review article [Jar11 hG(W )i := dW ⇢(W ) G(W ), Figure 1.1: Schematic representation of a molecule pulling experiment (from Ref. [Jar11]). The molecule is modeled by a collection of masses connected by springs. The tweezer acts on one end of the molecule, e.g., via an attached gold bead (blue), whereas the other end is attached to surface. protocol of the control parameter variation and, for simplicity, we will assume that there is only a single control parameter from now on. For example, for the toy model in Fig. 1.1 we have x = (z1 , z2 , z3 , p1 , p2 , p3 ) and H(x; (t)) = 3 2 X X p2i + u(zk+1 2m k=0 i=1 zk ) + u( z3 ) (1.142) where u is interaction potential and z0 (t) ⌘ 0 the position of the wall. The work performed during an infinitesimal parameter variation d is defined by W := d @H (x; ). @ For a given protocol (t) with initial condition (0) = 0 and final (⌧ ) = work performed on the system is Z Z ⌧ @H W = W = dt ˙ (t) (x(t); (t)) @ 0 (1.143) ⌧, the total (1.144) where the integral is computed along the trajectory x(t) realized by the system. That is, for a given realization W depends not only on the protocol but also on the initial state x0 of the system and the initial state y 0 of the environment, if we assume that Hint (x, y) > 0 during the process. If we repeat this process many times for the same protocol, we will observe di↵erent values of work {W1 , W2 , . . . , } that will be governed by a certain probability density ⇢(W ). FTs are exact equalities and inequalities for certain expectation values Z hG(W )i := dW ⇢(W ) G(W ), (1.145) 30 29 30 t fluctuation theorems (FTs) for non-equilibrium systems, the simplest version we will discuss below. Reminder: Canonical freee.g.energy total Hamiltonian comprising the system of interest, a DNA molecule describ dinates x(t)), its environment y and mutual interactions reads H(x, y; (t)) = H(x; (t)) + Henv (y) + Hint (x, y) (1.14 0 (t) denotes one or more external control parameters (e.g., (weak the force exerted by coupling) in a molecule experiments, Fig. 1.1). function (t). (t) defines t that, underpulling very general conditions, holdsee regardless of exactThe time dependence To simplify the subsequent discussion, let us assume that we are able to decouple the discussionsystem in this section closely follows thattin Christopher article 21 from the environment at time = the 0, and assume thatJarzynski’s at time t = review 0 the PDF of [Jar1 the system state is given by a canonical distribution 291 H(x0 ; 0 ) f (x0 ; 0 , T ) = exp , (1.146a) Z( 0 , T ) kB T where T is the initial equilibrium temperature of system and environment at t = 0, and Z H(x0 ; 0 ) Z( 0 , T ) = dx0 exp (1.146b) kB T the classical partition function. In this case, the initial free energy of the system is given by F0 = kB T ln Z( 0 , T ). Moreover, since the dynamics for t > 0 is completely Hamiltonian, we have (1.147) f (x0 ; 0, T ) = exp , (1.146a) Z( , T ) k T 0 where T is the initial equilibrium temperature ofBsystem and environment at t = 0, and and environment at t = 0, and Z of system where T is the initial equilibrium temperature H(x0 ; 0 ) Z Z( 0 , T ) = dx0 exp H(x0 ; 0 ) kB T Z( 0 , T ) = dx0 exp (1.146b) kB T (1.146b the classical partition function. In this case, the initial free energy of the system is give the classical partition function. In this case, the initial free energy of the system is given by by Figure 1.1: Schematic representation of a molecule pulling experiment (from Ref. [Jar11]). The molecule is modeled by a collection of masses connected by springs. The tweezer acts on one end of the molecule, e.g., via an attached gold bead (blue), whereas the other end 0 B B 0 0 0 is attached to surface. F = F k=T lnkZ(T ln , TZ( ). , T ). (1.147) (1.147 protocol of the control parameter variation and, for simplicity, we will assume that there is Moreover,since since for >completely is completely Moreover, the for t from > now 0 tis Hamiltonian, we have we have onlythe adynamics singledynamics control parameter on. For0 example, for the toy model in Fig.Hamiltonian, 1.1 we have x = (z , z , z , p , p , p ) and ✓ ✓ ◆ ◆ X X X X dH dHH(x; (t)) = @H @H @H p @H + @H u(z z ) + u( @H z) (1.142) = = ż + 2m ṗi + i ṗii + ż@ti + dt dt @p @z i @t i where u is interaction potential and z (t) ⌘ @p 0 the iposition of the@z wall.i The work performed ✓ ◆ ✓ ◆ i during an infinitesimalX parameter variation d is defined by @H @H✓ @H◆ @H ✓ @H◆˙ X @H @H @H(1.143)+@H @H ˙ = + W := d @H (x; ). @ @zi @zi +@pi @ =i @pi + 1 2 3 1 2 3 3 2 2 i i=1 k+1 k 3 k=0 0 @p @z For a given protocol (t) with initial condition i (0) = 0 and final i (⌧ ) = work performed on the system is i Z Z ⌧ @H W = W = dt ˙ (t) (x(t); (t)) @ 0 = @H ˙ @H ˙ @ = @z , thei total @pi @ ⌧ (1.148) (1.144) (1.148 @ and, therefore, and, therefore, where the integral is computed along the trajectory x(t) realized by the system. That is, for a given realization W depends not only on the protocol but also on the initial state x0 of the system and the initial state y 0 of the environment, if we assume that Hint (x, y) > ⌧0 during the process. If we ⌧repeat this process many times for the same protocol, we will observe di↵erent values of work {W1 , W2 , . . . , } that will be governed by ⌧ a certain probability density ⇢(W ). FTs are exact equalities and ⌧inequalities for certain 0 ⌧ expectation0values ⌧ 0 Z ⌧(1.145)⌧ hG(W )i := dW ⇢(W ) G(W ), W = Z W = Z @H dt ˙ = dHZ = H(x ; Z @ @H dt ˙ = ) H(x ; dH = H(x ; ) 0) H(x0 ; (1.149) 0) where x(⌧ ) = x⌧ . Now consider the@expectation value of the function G(W ) = e , 0 0 30 which can be expressed as where x(⌧ ) = x⌧ . Now Z consider the expectation value of the function G(W ) = e ⌦ W/(k T ) ↵ B expressed as which ecan be = dx0 f (x0 ; 0 , T ) e W/(kB T ) Z Z ⌦ W/(k T ) ↵ [H(x⌧ ; ⌧ )W/(k H(xB0 T ; )0 )]/(kB T ) B = dx f (x ; , T ) e 0 0 0 e = dx0 f (x0 ; 0 , T ) e W/(kB T ) (1.149 W/(kB T i the integral is computed along That @Hthe ˙ trajectory x(t) realized by the system.(1.148) = @H ˙ @ not only on the protocol but also on the initial a given realization W depends = (1.148) of the system and the initial state y of the environment, if we assume that @ and, therefore, 0 0 Z ⌧ Z ⌧ y) > 0 during the process. If@H we repeat this process many times for the same herefore, W = dt ˙ dH{W = H(x H(x ) (1.149)by 0 ; 0will ol, we will observe di↵erent values of= work that be governed 1 , W⌧2; , .⌧ .) . , } @ 0 0 Z ⌧ ⇢(W Zare ⌧ exact equalities and inequalities for certain in probability density ). FTs @H W/(kB T ) ˙ representation of a molecule pulling experiment (from Ref. [Ja where W x(⌧= ) = x⌧ . dt Now consider value , (1.149) = the expectation dH = H(x H(x0 ; G(W ⌧ ; of⌧ )the function 0 ) ) = eFigure 1.1: Schematic ation values The molecule is modeled by a collection of masses connected by springs. The tweeze @ 0 0 which can be expressed as on one end of the molecule, e.g., via an attached gold bead (blue), whereas the oth Z is attached to surface. Z ⌦ W/(k T ) ↵ W/(kB T ) B T )), of the function G(W )(1.145) hG(W := ⇢(W )W/(k G(W x(⌧ ) = x⌧ . e NowB consider the value e control , variation and, for simplicity, we will assume that th = )i dx (xdW protocol= of the parameter 0 f expectation 0; 0, T ) e only a single control parameter from now on. For example, for the toy model in F Z can be expressed as we have x = (z , z , z , p , p , p ) and [H(x ; ) H(x ; )]/(k T ) dx0 f (x0 ; 0 , T ) e Z = X p X 30 H(x; (t)) = + u(z z ) + u( z) ( ⌦ W/(k T ) ↵ BT ) Z 2m W/(k B 1 0; 0, T ) e H(x0 ; 0 ) e = dx f (x T) =0 dx0 exp e [H(x ; ) H(x ; )]/(kwhere u is interaction potential and z (t) ⌘ 0 the position of the wall. The work perf Z( 0 , T ) kB T Z during an infinitesimal parameter variation d is defined by Z 1 [H(x ; ⌧ )T )H(x0 ; 0 )]/(kB T ) H(x ; ⌧)/(k @H = 0 f (x0 ; 0 ,dx (1.150) W := d (x; ). ( = dx T ) 0 ee @ Z( 0 , T ) For a given protocol (t) with initial condition (0) = and final (⌧ ) = , the Z 21 performed Similar results hold for remains coupled work to the bath on the system is 1 more complex dynamical models H(xwhere ) system 0 ; 0the [H(x⌧ ; ⌧ ) H(x0 ; 0 )]/(kB T ) Z Z = process; see discussiondx exp e throughout the in0Ref. [Jar11] and references therein. @H W = W = dt ˙ (t) (x(t); (t)) ( Z( 0 , T ) kB T @ Z where the integral is computed along the trajectory x(t) realized by the system. 1 H(x31 is, for a given realization W depends not only on the protocol but also on the ⌧ ; ⌧ )/(kB T ) = dx0 e (1.150) state x of the system and the initial state y of the environment, if we assum Z( 0 , T ) H (x, y) > 0 during the process. If we repeat this process many times for the 1 ⌧ ⌧ 0 0 2 3 1 2 3 B 3 2 2 i k+1 i=1 ⌧ ⌧ 0 0 k 3 k=0 B 0 ⌧ ⌧ B 0 ⌧ ⌧ 0 0 0 int protocol, we will observe di↵erent values of work {W1 , W2 , . . . , } that will be govern a certain probability density ⇢(W ). FTs are exact equalities and inequalities for c expectation values Z hG(W )i := dW ⇢(W ) G(W ), ( milar results hold for more complex dynamical models where the system remains coupled to the bath hout the process; see discussion in Ref. [Jar11] and references therein. 31 30 ✓ realized ◆ by the✓system. ◆ That i the integral is computed along trajectory x(t) @Hthe X ˙ @H @H @H @H (1.148)@H ˙ = @H a given realization W depends not only on the protocol + but also on the initial + (1.148) ˙@ = = @pi environment, @zi @zifi we @p @ i assume that and,system therefore,and the@ initial state iy 0 of the 0 of the Z ⌧ Z@H y) > 0 during the process. If@H we repeat this process many times for the same ⌧ herefore, ˙ W = dt ˙ =of = work dH{W = H(x H(x ) (1.149)by (1.148) 0 ; 0will ol, we will observe di↵erent values that be governed 1 , W⌧2; , .⌧ .) . , } @ 0 0 Z ⌧ ⇢(W Zare ⌧ @exact equalities and inequalities for certain in probability density ). FTs @H W/(k ) ˙ BT 1.1: Schematic representation of a molecule pulling experiment (from Ref. [Ja where W x(⌧= ) = x⌧ . dt Now consider the expectation value of⌧ )the function G(W ) = eFigure , (1.149) = dH = H(x ; H(x ; ) ⌧ 0 0 ationand, values therefore, The molecule is modeled by a collection of masses connected by springs. The tweeze @ 0 0 which can be expressed as on one end of the molecule, e.g., via an attached gold bead (blue), whereas the oth Z is attached to surface. Z Z Z ⌧ ⌧ ⌦ W/(k T ) ↵ W/(kB T ) @H )W/(k B T )), of the function G(W )(1.145) hG(W := ⇢(W G(W x(⌧ ) = x⌧ . e NowB consider the value e control , variation and, for simplicity, we will assume that th = )i dx (xdW protocol= of the parameter 0 f expectation 0 ; ˙0 , T ) e W = dt = dH = H(x⌧ ; ⌧) H(x ; ) (1.149) only a single0control0parameter from now on. For example, for the toy model in F we have x = (z1 , z2 , z3 , p1 , p2 , p3 ) and can be expressed as @ 0 0 [H(x ; ) H(x ; )]/(k T ) dx0 f (x0 ; 0 , T ) e Z = X p X 30 H(x; (t)) = + u(z z ) + u( z) ( ⌦ W/(k T ) ↵ expectation Z 2m) = e W/(kB T ) , W/(k T) B x(⌧ ) = x⌧ . Now B where consider the value of the function G(W 1 0; 0, T ) e H(x0 ; 0 ) e = dx f (x T) =0 dx0 exp e [H(x ; ) H(x ; )]/(kwhere u is interaction potential and z (t) ⌘ 0 the position of the wall. The work perf Z( 0 , Tas ) kB T which can be Zexpressed during an infinitesimal parameter variation d is defined by Z 1 Z [H(x ; ⌧ )T )H(x0 ; 0 )]/(kB T ) H(x ; ⌧)/(k @H = dx (1.150) W := d (x; ). ( = dx f (x ; , T ) 0 ee 0 0 0 @ ⌦ W/(k T ) ↵ Z( 0, T ) W/(kB T ) B For a given protocol (t) with initial condition (0) = and final (⌧ ) = , the e = dx f (x ; , T ) e Z 0 0 0 21 performed Similar results hold for remains coupled work to the bath on the system is 1 more complex dynamical models H(xwhere ) system 0 ; 0the [H(x ; ) H(x ; )]/(k ⌧ ⌧ 0 0 BT ) Z Z Zdxin0Ref.exp = process; see discussion e throughout the [Jar11] and references therein. @H W = W = dt ˙ (t) (x(t); (t)) ( Z( 0 , T ) kB T @ [H(x⌧ ; ⌧ ) H(x0 ; 0 )]/(kB T ) =Z dx0 f (x0 ; 0 , T ) e where the integral is computed along the trajectory x(t) realized by the system. 1 H(x31 is, for a given realization W depends not only on the protocol but also on the ⌧ ; ⌧ )/(kB T ) = dx0 e (1.150) Z state x of the system and the initial state y of the environment, if we assum Z( 0 , T ) y) > 0 during the process. If we repeat this process many times for the 1 H(x0 ; 0 ) H (x,[H(x ; ) H(x ; )]/(k T ) Z ⌧ ⌧ 0 0 B 3 2 2 i k+1 i=1 ⌧ ⌧ 0 0 k 3 k=0 B 0 ⌧ ⌧ B 0 ⌧ ⌧ 0 0 = dx0 exp = dx0 e ⌧ observe ⌧ 0 values 0 of work B {W , W , . . . , } that will be govern we will di↵erent eprotocol, a certain probability density ⇢(W ). FTs are exact equalities and inequalities for c Z( 0 , T )models where the system kBremains T milar results hold for more complex dynamical coupled to the bath expectation values Z Z hout the process; see discussion in Ref. [Jar11] and references therein. 1 hG(W )i := Z( 0 , T ) H(x⌧ ; 31 21 0 int ⌧ )/(kB T ) 1 2 dW ⇢(W ) G(W ), (1.150) 30 Similar results hold for more complex dynamical models where the system remains coupled to the bath throughout the process; see discussion in Ref. [Jar11] and references therein. 31 ( = as dx0 exp e which can be expressed Z( 0 , T ) kB T Z Z ⌦ W/(k T ) ↵ 1 H(xW/(k ) T) ⌧ ; ⌧ )/(k B B TB = dx e e = dx0 f (x0 ; 0 ,0 T ) e Z( 0 , T ) Z 21 [H(x⌧ ; where ⌧ ) H(x 0 ; system 0 )]/(k Similar results hold=for more complex dynamical models the dx0 f (x0 ; 0 , T ) e throughout the process; see discussion in Ref. [Jar11] and references therein Z 1 H(x0 ; 0 ) [H( = dx exp e 0 this31 Changing the integration variable from x ! 7 x , we can write as 0 ⌧ Z( 0 , T ) kB T Z Z 1 ⌦ W/(k Tvariable ↵ 1 @x Changing the integration from x ! 7 x , we can write this as T ) ⌧ 01 ) ; )/(k e = dx⌧ ⌧ e H(x H(x ⌧ ; ⌧ )/(kB T ) Z( 0 , T ) Z @x0dx = e 1 0 Z ⌦ W/(k T ) ↵ 11 0 , T ) @x⌧ Z( ; )/(k T ) T) e = dx⌧ H(x ; e)/(kH(x B B ⌧ ⌧ B ⌧ ⌧ B = Z( 0 , T ) dx⌧ e@x0 Z( 0 , T ) Z 21 1 , T )complexH(x ⌧ ; ⌧ )/(kB T ) Z( Similar results hold = for more dynamical models where the syste ⌧ dx e ⌧ = Z( 0 , T ) (1.151) Z( , T ) 0 throughout the process; see discussion in Ref. [Jar11] and references therein Z( ⌧ , T ) (1.151) Here, we have used Liouville’s= theorem, which states that the phase volume is conserved Z( 0 , T ) under a purely Hamiltonian evolution x0 7! x(⌧ ), 31volume is conserved Here, we have used Liouville’s theorem, which states that the phase ⌧ x(⌧ ), under a purely Hamiltonian evolution x@x 0 7! =1 (1.152) @x0 @x⌧ =1 (1.152) Rewriting further @x0 ⇢ ⌦ ↵ kB T Z( ⌧ , T ) W/(kB T ) Rewriting efurther = exp ln k T Z( 0 , T ) B ⇢ ⇢k T ⌦ W/(k T ) ↵ Z( ⌧ , T ) B 1 B e = exp ln ⌧ ⌧ B H(x⌧ ; ⌧ )/(kB T ) Z( ⌦ expressed ↵ ⌧, T) which can be as 1 @x = dx e (1.151) e == dx e⌧ Z( Z( Z( , T )0 , 0T,)T ) @x Z Z 1 ⌦ ↵ Z( ⌧ ,which T )e states that the phase = dx Here, we haveW/(k usedBLiouville’s theorem, volume T) W/(k B T )is conserved Z( , T ) e = dx f (x ; , T ) e 0 0 0 under a purely Hamiltonian evolution x ! 7 x(⌧ ), Z( Z( , T ) 00, T ) = (1.151) Z Z( , T ) @x⌧ [H(x ; ⌧ ) H(x =1 (1.152) ⌧phase 0 ; 0 )]/(k Here, weHere, have used Liouville’s theorem, which states that the volume is we have used Liouville’s theorem, which that = dx f (x , T )volume e is conserved @x0 0states 0 ;the0phase under a purely Hamiltonian evolution x 7! x(⌧ under a purely Hamiltonian evolution x0),7! x(⌧ ), Z Rewriting further @x 1 H(x 0; 0) ⇢ = 1 (1.152) @x⌧⌧, T ) dx0 exp ⌦ W/(k T ) ↵ = kB T@x Z( e [H( e = exp ln 0 , T ) = 1 Z( k T B k T Z( , T ) B 0 @x0 Z Rewriting further ⇢ ⇢ 1 1, TB)T ln Z( ⌧ , T ) ( kH(x ⌦ ↵ T) k T Z([ k ⌧ln; Z( ⌧ )/(k = exp= T ) , T )] B B 0 dx e e = exp ln k T 0 Rewriting further B k T Z( Z( , T ) ⇢ 0, T ) ⌧ W/(kB T ) H(x⌧ ; ⌧ 0 ⌧ )/(kB T ) 0 ⌧ H(x⌧ ; ⌧ )/(kB T ) 0 ⌧ 0 0 ⌧ 0 B W/(kB T ) B ⌧ B 0 ⇢ 1 [ kB T ln Z( ⌧ , T ) ( kB T ) ln Z( 0, T )] Z( , T ) kBkTB T ⌧ 21 W/(kB T ) ⌦ more ↵ dynamical models where the syste eSimilar results = hold exp for W/(k ln T complex ) F/(k T ) e = e (1.153a) k T Z( , T ) one thus finds the FT B 0 throughout the process; ⇢ see discussion in Ref. [Jar11] and references therein ⌦ W/(k T ) ↵ 1 = e F/(k T ) where e (1.153a) = exp [ kB T ln Z( ⌧ , T ) ( kB T ) ln Z( 0 , T )] where F =kB FT ( ⌧ , T ) F ( 0, T ) (1.153b) one thus finds the FT= exp ⌦ ↵ B B B B 31 = F ( ⌧ , T one ) Fcan ( 0 ,estimate T) (1.153b)by measurFT (1.153) states that, inFprinciple, free energy di↵erences one thusThe finds the FT ing work W . Instates this context, it should however that,di↵erences in practice, the exponential The FT⌦(1.153) that, in principle, onebe cannoted estimate free energy by measur↵ ⌦ ↵ W/(k T ) W/(k ) F/(k )from average is difficult to sample asT however direct estimators su↵er slow convergence. B B T the ing work⌦eW . In this context, it should be noted that, in practice, exponential ↵ e = e W/(k T ) B where B average e is difficult to sample as direct estimators su↵er from slow convergence. Furthermore, using Jensen’s inequality Furthermore, using Jensen’s inequality x hex ihxi ehxi he i e (1.154) (1.154) e = e ing work⌦ W . In this context, it should be noted however that,Bin practice, the exponential B ↵ average e W/(kB T ) is difficult to sample as direct estimators su↵er from slow convergence. Furthermore, using Jensen’s inequality22 Jensen’s inequality where 22 hex i ehxi (1.154) Jensens’s inequality states that, if (x) is convex then E[ (X)] F = F( (E[X]) ⌧, T) F ( 0, T ) Proof: Let L(x) = a + bx be a line, tangent to (x) at the point x⇤ = E[X]. Since (x) L(x). Hence E[ (X)] E[L(X)] = a + bE[X] = L(E[X]) = (E[X]) is convex, we have The FT (1.153) states that, in principle, one can estimate fr ing work⌦ W . In this context, it should be noted however t ↵ W/(kB T ) average e is difficult to sample as direct estimato 32 Furthermore, using Jensen’s inequality x he i e hxi we find e source: Wiki F/(kB T ) ⌦ = e W/(kB ↵ T) e h W/( Rewriting further = ↵ = e W/(kB T ) = one thus finds the FT one thus finds the FT = ⌦ one thus finds the FT where where Z( 0 , T ) ⇢k T ⇢ B 1 exp⇢ k T1 [ Z( kB T ,ln Z( ( kB T ) ln Z( 0 , T )] ⌧, T) T ) exp [ k T ln Z( , T ) ( kB T ) ln Z( 0 , T )] BkB T B ⌧ ⌧ exp ln k T kB TB Z( 0 , T ) ⇢ 1 exp ⌦ W/(k[ T )k↵B T ln Z( ⌧F/(k , T ) T ) ( kB T ) ln Z( 0 , T )] B k T ⌦ e W/(k ↵ = e F/(kB T ) B (1.153a) BT ) B e = e (1.153a) ⌦ W/(k T ) ↵ BT ) e = F B( ⌧ , T )= Fe ( 0F/(k (1.153a) F ,T) (1.153b) F = F ( ⌧ , T ) F ( 0, T ) (1.153b) where The FT (1.153) states that, in principle, one can estimate free energy di↵erences by measurThe (1.153) that, in principle, onenoted can estimate di↵erences by measuringFT work . Instates this context, it should be however free that,energy in practice, the exponential ⌦ WW/(k ↵ F sample = F (be⌧ as ,noted T direct ) Fhowever (estimators (1.153b) 0 , T ) that, B T ) context, it should ing work⌦ We . In this in practice, theconvergence. exponential average is difficult to su↵er from slow ↵ average e W/(kB T )using is difficult to inequality sample as direct estimators su↵er from slow convergence. Furthermore, Jensen’s The FT (1.153) states that, in principle, one can estimate free energy di↵erences by measurFurthermore, using Jensen’s inequality x noted hxihowever that, in practice, the exponential ing work⌦ W . In this context, it shouldhebe i e (1.154) ↵ W/(kB T ) average e is difficult to sample hex ias direct ehxi estimators su↵er from slow convergence. (1.154) using Jensen’s inequality weFurthermore, find we find ⌦ xW/(k Thxi) ↵ F/(kB T ) (1.154) e = ⌦he e i Be ↵ eh W/(kB T )i e F/(kB T ) = e W/(kB T ) eh W/(kB T )i which we find is equivalent to the Clausius inequality ⌦ W/(k T ) ↵ which is equivalent to the Clausius inequality F/(kB T ) B i, eh W/(kB T )i (1.155) e = e F hW F hW i, (1.155) i.e., the average work provides an upper bound for the free energy di↵erence. which is equivalent to the Clausius inequality i.e., the average work provides an upper bound for the free energy di↵erence. F hW (1.155) 32 i, 32 for the free energy di↵erence. i.e., the average work provides an upper bound 32 which is equivalent to the Clausius inequality F hW i, i.e., the average work provides an upper bound for the free energy di↵erence. Finally, we still note that Z F ✏ P[W < F ✏] := dW ⇢(W ) 1 Z F ✏ dW ⇢(W ) e( F ✏ W )/(kB T ) 1 Z 1 e( F ✏)/(kB T ) dW ⇢(W ) e W/(kB T ) 1 ⌦ ↵ ( F ✏)/(kB T ) W/(kB T ) = e e = e ✏/(kB T ) (1.155) (1.156) That is, the probability that a certain realization W violates the Clausius relation by an amount ✏ is exponentially small. which is equivalent to the Clausius inequality F hW i, i.e., the average work provides an upper bound for the free energy di↵erence. Finally, we still note that Z F ✏ P[W < F ✏] := dW ⇢(W ) 1 Z F ✏ dW ⇢(W ) e( F ✏ W )/(kB T ) 1 Z 1 e( F ✏)/(kB T ) dW ⇢(W ) e W/(kB T ) 1 ⌦ ↵ ( F ✏)/(kB T ) W/(kB T ) = e e = e ✏/(kB T ) (1.155) (1.156) That is, the probability that a certain realization W violates the Clausius relation by an amount ✏ is exponentially small.