Experimental free-energy measurements of kinetic molecular states using fluctuation theorems

Abstract

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...

Mechanical unfolding of a simple model protein goes beyond the reach of one-dimensional descriptions

“…Although extremely simple molecular assays such as DNA or RNA hairpins could fit into a single reaction coordinate description [48], increasing slightly the complexity of the molecule leads to a dramatical rise in the complexity of the actual free energy landscape in the system, requiring more detailed studies. In this sense, molecules such as multiple nucleic-acid hairpins [53], protein-ligand complexes [54] or any mechanically pulled protein [55], appear as potential systems where a one-dimensional description takes the risk of leading to a clear misunderstanding of the actual complexity of their conformational space and the dynamical processes to which they are subject.…”

Detailed Jarzynski equality applied to a logically irreversible procedure

“…This type of fluctuation theorem has already been used to experimentally measure free-energy of kinetic molecular states [17,18]. Here, there are only two subsets j = {00, 10}, defined by the position where the bead starts.…”

Free-energy inference from partial work measurements in small systems

“…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 (1,2). Such measurements have been carried out with laser optical tweezers on different nucleic acid structures such as hairpins (3)(4)(5)(6), G quadruplexes (7,8), and proteins (9)(10)(11)(12) and with atomic force microscopes on proteins (13) and bimolecular complexes (14). An important issue regarding FRs is the correct definition of work, which rests on the correct identification of configurational variables and control parameters.…”