Recent advances in non-equilibrium statistical mechanics and single molecule technologies make it possible to extract free energy differences from irreversible work measurements in pulling experiments. To date, free energy recovery has been focused on native or equilibrium molecular states, whereas free energy measurements of kinetic states (i.e. finite lifetime states that are generated dynamically and are metastable) have remained unexplored. Kinetic states can play an important role in various domains of physics, such as nanotechnology or condensed matter physics. In biophysics, there are many examples where they determine the fate of molecular reactions: protein and peptide-nucleic acid binding, specific cation binding, antigen-antibody interactions, transient states in enzymatic reactions or the formation of transient intermediates and non-native structures in molecular folders. Here we demonstrate that it is possible to obtain free energies of kinetic states by applying extended fluctuation relations. This is shown by using optical tweezers to mechanically unfold and refold DNA structures exhibiting intermediate and misfolded kinetic states.Kinetic states are observed under non-equilibrium conditions and have higher free energies than native states. Yet, they can be crucial, as shown by the role that misfolded proteins play in numerous severe diseases [1]. The measurement of the free energy of formation of kinetic states is therefore a central question in biophysics. Recent theoretical developments known as fluctuation relations [13, 12, 4, 5, 6] have been applied to extract free energy differences of equilibrium states from irreversible work measurements. Applications include the measurement of the free energy of formation of RNA and DNA hairpins [7]; the determination of the stability of native domains in proteins [8]; the measurement of mechanical torque in rotary motors [9]; the conversion of information into work in systems under feedback control [10]; or the recovery of free energy landscapes from unidirectional work measurements [11, 12].The characterization of kinetic states under non-equilibrium conditions remains a challenging problem. Here we use a recently introduced extended fluctuation relation (EFR) to extract free energies of kinetic states and thermodynamic branches using irreversible work measurements [13, 10]. In the EFR, a kinetic state is a partially equilibrated region of configurational space, meaning that during a finite timescale the system is confined and thermalized within that region [15]. This is mathematically described by a Boltzmann-Gibbs distribution restricted to configurations contained in that region ( Fig. 1a).Let A, B denote any two kinetic states and λ a control parameter. We consider a forward (F) non-equilibrium process, where the system starts in partial equilibrium in A at λ 0 , and its time-reversed (R), where the partial equilibrium condition is required over B at λ 1 . In the F process λ varies from λ 0 to λ 1 during a time τ according to a predetermined proto...

We investigate the thermodynamics and kinetics of DNA hairpins that fold/unfold under the action of applied mechanical force. We introduce the concept of the molecular free energy landscape and derive simplified expressions for the force dependent Kramers-Bell rates. To test the theory we have designed a specific DNA hairpin sequence that shows two-state cooperative folding under mechanical tension and carried out pulling experiments using optical tweezers. We show how we can determine the parameters that characterize the molecular free energy landscape of such sequence from rupture force kinetic studies. Finally we combine such kinetic studies with experimental investigations of the Crooks fluctuation relation to derive the free energy of formation of the hairpin at zero force.

We present a method for determining the free energy of coexisting states from irreversible work measurements. Our approach is based on a fluctuation relation that is valid for dissipative transformations in partially equilibrated systems. To illustrate the validity and usefulness of the approach, we use optical tweezers to determine the free energy branches of the native and unfolded states of a two-state molecule as a function of the pulling control parameter. We determine, within 0:6k B T accuracy, the transition point where the free energies of the native and the unfolded states are equal.

We analyse the dynamics of a two dimensional system of interacting active dumbbells. We characterise the mean-square displacement, linear response function and deviation from the equilibrium fluctuation-dissipation theorem as a function of activity strength, packing fraction and temperature for parameters such that the system is in its homogeneous phase. While the diffusion constant in the last diffusive regime naturally increases with activity and decreases with packing fraction, we exhibit an intriguing non-monotonic dependence on the activity of the ratio between the finite density and the single particle diffusion constants. At fixed packing fraction, the time-integrated linear response function depends non-monotonically on activity strength. The effective temperature extracted from the ratio between the integrated linear response and the mean-square displacement in the last diffusive regime is always higher than the ambient temperature, increases with increasing activity and, for small active force it monotonically increases with density while for sufficiently high activity it first increases to next decrease with the packing fraction. We ascribe this peculiar effect to the existence of finite-size clusters for sufficiently high activity and density at the fixed (low) temperatures at which we worked. The crossover occurs at lower activity or density the lower the external temperature. The finite density effective temperature is higher (lower) than the single dumbbell one below (above) a cross-over value of the Péclet number.

EF-hand calcium sensors respond structurally to changes in intracellular Ca(2+) concentration, triggering diverse cellular responses and resulting in broad interactomes. Despite impressive advances in decoding their structure-function relationships, the folding mechanism of neuronal calcium sensors is still elusive. We used single-molecule optical tweezers to study the folding mechanism of the human neuronal calcium sensor 1 (NCS1). Two intermediate structures induced by Ca(2+) binding to the EF-hands were observed during refolding. The complete folding of the C domain is obligatory for the folding of the N domain, showing striking interdomain dependence. Molecular dynamics results reveal the atomistic details of the unfolding process and rationalize the different domain stabilities during mechanical unfolding. Through constant-force experiments and hidden Markov model analysis, the free energy landscape of the protein was reconstructed. Our results emphasize that NCS1 has evolved a remarkable complex interdomain cooperativity and a fundamentally different folding mechanism compared to structurally related proteins.