Research
Since a few years ago my research has been focused in molecular biophysics, in particular in the study of single molecule experiments with potential applications to statistical physics. I find particularly exciting the experimental and theoretical research of nonequilibrium thermodynamics of small systems. Other subjects that keep attracting my attention are nonequilibrium fluctuations in glassy and disordered systems. Here I present a short overview of some topics that mostly interest me.
Small biosystems labModeling single miolecule experiments
Nonequilibrium thermodynamics of small systems
Glassy systems
Small biosystems lab
Biophysics at the molecular level is the subject on which I have currently focused my interest. This research has been boosted by the advent of micromanipulation tools and nanotechnologies in the form of single molecule microscopies that allow scientits to follow the behavior of individual molecules one at a time and study their nonequilibrium behavior. Single molecule experiments allow us to measure distributions describing molecular properties, characterize the kinetics of biomolecular reactions or even detect molecular intermediates. These type of experiments provide the additional information about thermodynamics and kinetics of biomolecular processes that complements information obtained in traditional bulk assays. In addition, single molecule experiments make possible to measure small energies and detect large Brownian deviations in biomolecular reactions, thereby offering new methods and systems to scrutinize the basic foundations of statistical mechanics. If you want to learn more about this you can download here an introductory review article I recently wrote on the subject.
A very useful experimental technique commonly used to manipulate single molecules are optical tweezers. Optical tweezers use light momentum conservation to generate forces in small micron sized polystyrene beads. They generate forces from the light deflection generated by the difference in the index of refraction between the beads and the surrounding medium (such as water). The mechanism of deflection is illustrated in Figure 1. Optical tweezers made out of two dual counterpropagating beams can be used to generate forces in the range 1-200 piconewtons (1 piconewton=10^{-12} Newtons). To exert forces on molecules a tether is attached between two beads. One of the beads is held fixed by air suction on the tip of a micropipette, the other is held by the optical trap. When a force is applied on the trapped bead the incoming rays of light are deflected. By collecting the light in a photodetector it is possible to infer the force acting on the bead. Force-extension curves can be obtained as the molecule is pulled by either steering the trap or moving the micropipette with a movable stage. DNA plays a central role in molecular biophysics and many investigations have been carried out to determine its elastic properties.
Figure 1: A single beam optical tweezers setup. A polysterene or silica bead embedded in water deflects the incident laser beam. The total deflection of the incoming light is proportional to the trapping force acting on the bead.
In January 2005 I started my own laboratory in Barcelona to investigate single molecules using optical tweezers. In the framework of a scientific collaboration I had over the past few years with Carlos Bustamante from Berkeley we acquired an optical tweezers instrument (what we call "the minitwwezers" due to its highly compact and optimized size) that has been designed and assembled by Steve Smith. The current laboratory has two separate spaces: a laboratory to synthesize molecules (nucleic acids and proteins) and a room where we have the minitweezers as well as other instrumentation required to build the fluidic chambers. A view of the tweezers lab is shown in Figure 2.
Figure 2: General view of the laboratory with the minitweezers.
The instrument has been operative since January 2007. It has been used to overstrecth and unzip DNA. At present we are carrying out several projects. I just mention two of them here. In one project we are currently investigating the unzipping of double stranded DNA molecules using molecular constructs that provide us with high resolution in both force and extension. In order to pull the two strands apart we must synthesize DNA molecules with one strand biotinylated at one end and digoxygenins on the other strand at the same end of the DNA. One bead binds to one strand at one end and is captured in the optical trap, the other bead binds to the other strand at the same end and remains immobilized on a pipette. By steering the trap it is possible to pull the molecule and measure the force-extension curve. One selected figure is shown in figure 3 corresponding to a 2200 base pairs DNA molecule.
Figure 3: (Top) Schematic picture of the experimental setup. (Bottom) Unzipping force-extension curve for a 2200 base pairs fragment of lambda DNA. The molecule has been hydridized with a loop at one end so the double stranded form is converted into single stranded DNA after separation of both strands. The force-extension curve shows force rips around a value of 15pN which are characteristic of the breakage of the stabilizing interactions (hydrogen bonds and stacking) that bind the two complementary strands. The final part of the force-extension curve corresponds to the elastic response of single stranded DNA after the DNA molecule has been fully unzipped.
Another class of experiments that we are currently carrying out is the study of the cooperative unfolding/folding force kinetics of short DNA hairpins of a few tens of base pairs. We have currently synthesized two type of molecules where a DNA hairpin is inserted between double stranded DNA handles. Either we synthesize hairpins inserted between short handles (a few tens of base pairs) or we synthesize hairpins with long handles (a few hundreds of base pairs). We can also synthesize hybrid short/long handles (short at one side of the hairpin, long at the other side). Some experimental force-extension curves are shown in figure 4 for a 20 base pairs canonical hairpin (i.e. made of complementary Watson-Crick base pairs). Every cycle the molecule is pulled it breaks at a different value of the force. Conversely, every time the molecule folds back into the native state it does at a different value of the force. Therefore, the unfolding/folding process is stochastic due to the random nature of thermal forces impinged by the water molecules in the aqueous environment. The distribution of unfolding/folding forces depends on the pulling rate, a phenomenology commonly known as dynamic force spectroscopy, and it is useful to characterize the kinetic strength of molecular bonds.
Figure 4: Force-extension cycles in a 20 base pairs hairpin pulled at 100nm/s. The curves show a hysteresis between the unfolding (indigo) and the refolding (green) fore extension curves. If you want to to learn more about this behavior you can read this paper where we carried out a theoretical analysis of the breakage unfolding/refolding force distributions in single RNA pulling experiments
With DNA hairpins it is possible to carry out many additional force kinetic measurements. For instance it is possible to investigate the folding/unfolding kinetics at various values of a constant applied force. In these type of experiments the molecule is held at a constant force through a feedback mechanism while the molecular extension hops between the values corresponding to the folded and the unfolded states. From the time traces of the extension it is then possible to infer the folding and unfolding kinetic rates at various values of the force. This allows for a detailed characterization of the free energy landscape along the reaction coordinate (the molecular extension along the pulling axis) that separates the folded and the unfolded states. Typical extension traces for the hairpin shown in figure 4 are shown in figure 5 for different values of the feedback force.
Figure 5: Extension traces measured at different forces for a 20 base pairs hairpin. From these traces it is possible to infer the residence time of the molecule in both the folded and unfolded states at different values of the force. By plotting the logarithm of the kinetic rates (equal to the inverse of the residence times) as a function of the force we can infer the free energy difference between both states, the height of the activation barrier and the location of the transition state along the reaction coordinate (the molecular extension in this case)
Modeling single molecule experiments
An important part of my recent research has been the modeling of single molecule experiments using statistical models that are simple and, at the same time, realistic enough to quantitatively reproduce the experimental results. Theoretical modeling is deeply connected with our current research at the BIOSMALL lab that has been reported above. An important part of this work has been done in collaboration with my former PhD student Maria Manosas who has modeled RNA hairpin pulling experiments. We have been using mesoscopic models for the experimental setup that include the RNA hairpin, the hybrid RNA/DNA handles as well as the bead captured in the trap. In pulling experiments the control parameter is determined by the position of the trap rather than by the force acting on the bead or the position of the trapped bead (these two quantities are not controlled but fluctuate due to the action of thermal forces). The mathematical description of the system requires of defining the proper thermodynamic potential adapted to the experimental conditions. Figure 6 compares some experimental results obtained for the P5ab hairpin (a derivative of the L21 Tetrahymena Termophila rybozyme) pulled by optical tweezers in Berkeley with the numerical simulations of the model we did in Barcelona.
Figure 6: Comparing experimental and theoretical force-extension curves obtained in pulling cycles of the RNA molecule P5ab. The molecule is in the folded (f) state at force values below the force rip whereas the unfolded (u) state is found for forces above the rip. You can learn more in the paper entitled Thermodynamic and kinetic aspects of RNA pulling experiments
In yet another type of models one can also investigate how the different elements of the setup modify the kinetics of hopping experiments. In hopping experiments it is possible to measure kinetic rates in the passive mode (where both the force and the extension change between two values as the molecule hops between the unfolded and folded states) or the force-feedback mode (where the force is kept constant, see above in Figure 5). To reproduce the experimental results it is then necessary to consider more complex models for the RNA hairpin by including as many intermediate configurations as there are base pairs along the sequence. Beyond the hairpin case it is also possible to investigate more complicated RNA secondary structures where different hairpins compete during the formation of the native structure. Figure 7 shows some examples.
Figure 7: (Top panels) Force and extension recordings in the passive mode (experiments versus simulations). You can learn more in the two papers we have recently written that combine experiments and theoretical modeling. (Bottom) Force-extension curves from experiments and simulations obtained for the three way helix junction S15. The native secondary structure of this molecule contains two hairpins and a stem at the base of the fork (bottom left). Upon folding there is competition between the formation of different (native and non-native) structures that lead to misfolding.
Nonequilibrium thermodynamics of small systems
Another subject in which I have been much interested over the past few years is the study of energy fluctuations in small systems, in particular biomolecules and mesoscopic systems in general. Single molecule experiments are capable of measuring tiny forces and small extensions leading to measurable energies that are on the order of a few kcal/mol corresponding to a few kT units (k being the Boltzmann constant and T the temperature of the environment) at room temperature (approx. 300K). Such energies can correspond to either work or heat depending on which type of nonequilibrium experiment we carry out and what we decide to measure. Of particular interest are work measurements. Mechanical work corresponds to the energy delivered to the system by time dependent forces exerted by external agents. The nonequilibrium protocol is determined by the so called control parameters, a set of externally controlled variables that change in time in a reproducible way and which do not fluctuate. Therefore, a given nonequilibrium experiment can be repeated an arbitrary large number of times starting from a given initial state. Because small systems are embedded in a fluctuating environment they are subject to random thermal forces that lead to a different value of the work every time the same experiment is repeated again and again. When we talk about work fluctuations we refer to the fact that the total work exerted upon a small system depends on the trajectory it follows in response to the externally controlled agent.
The reversible or quasistatic process corresponds to a specific protocol where the perturbation is carried out infinitely slowly and the system goes through a succession of equilibrium states. The second law of thermodynamics establishes that the work along a nonequilibrium process is larger than the reversible work. However, if the system is small enough (or the timescales are short enough) the second law inequality is sometimes not obeyed if the work is measured along an individual trajectory (it is sometimes not obeyed even by averaging the work over a finite number of trajectories). This fact does not contradict at all the second law of thermodynamics which establishes that, in a finite system, the average total work is always larger than the reversible work if the average is taken over an infinite number of trajectories.
A remarkable equality was established few years ago by Jarzynski at Los Alamos (New Mexico, USA) who showed how the measurement of the nonequilibrium work along many trajectories in a system that is initially in thermal equilibrium can be related to the free energy change along the reversible process. From a different perspective, the nonequilibrium work relation derived by Jarzynski can be used to recover equilibrium free energy differences fron nonequilibrium experiments. Molecular biophysics offers excellent model systems to address such question. As biomolecules can be individually manipulated it is then possible to follow their individual trajectories and measure work fluctuations with reasonable accuracy. Work fluctuations are observed whenever the energies involved are several times kT (or kcal/mol). Indeed, many molecular processes are performed by enzymes which use ATP hydrolisis to exert mechanical forces that are on the order of a few piconewtons. A lower bound to the minimal amount of work required by one of such tiny machines can be estimated as follows. The typical distance observed during the conformational changes of these enzymes is on the order of a few nanometers (1 nm=10^(-9) m) giving energies that are at least on the order of E=force x distance=1pN x 1nm =1 pN nm=0.24 kT for T=298K at room temperature. This estimate is a lower bound and energies involved in the form of mechanical work are typically many tens of kT.
Work fluctuations have been measured at the Bustamante lab in Berkeley (California) during the unfolding of a small RNA molecule pulled by an external force. These have been also measured by the Evans group in Canberra, Australia for the case of a micron-sized bead confined in an optical trap and dragged through water. In collaboration with Delphine Collin we have extended the measure of work fluctuations to larger RNA molecules to predict free energy differences using the Crooks fluctuation relation. Work fluctuations during the unfolding of RNA molecules are consequence of the stochasticity of the breakage force at which the molecule unfolds. Characteristic unfolding-refolding curves are shown in Figure 8.
Figure 8: Unfolding (left, orange) and refolding (right, blue) curves of a RNA three-helix junction hairpin (a part of the 16S ribosomal RNA in Escherichia Coli that binds the S15 protein at the junction).
The Crooks fluctuation relation establishes that the work distributions for a process and its time-reversed twin (both starting in equilibrium at the initial and final values of the control parameter, respectively) are related by a very simple relation (figure 9). By measuring both (forward and reverse) distributions along many pulling cycles we can combine forward and reverse work data to determine the free energy difference more efficiently that we would do just by applying the Jarzynski equality to the forward and reverse data sets. One consequence of the Crooks fluctuation relation is that the forward and reverse work distributions should be different due to hysteresis effects. Yet they should cross at a value of the work which is equal to the free energy difference between the initial and final states. For example, if a DNA or RNA hairpin is repeatedly pulled back and forth at a given speed then the forward (e.g. unfolding) work distribution will differ from the reverse (e.g. refolding) work distribution. Yet both distributions will cross at a value of the work which is equal to the free energy difference between the folded and the unfolded states. The first experimental results testing the validity of such fluctuation relation were obtained a few years ago while I enjoyed a sabbatical stay at the Bustamante and Tinoco labs in Berkeley. A few selected results are shown in Figure 9.
Figure 9: (Left panel) The work of a given trajectory is given by the area below the force-extension curve. (Right panel) Work distributions measured for the unfolding and refolding of the RNA three-helix junction hairpin shown in Figure 8 (indigo color) and a mutant differing in only one base pair (orange color). The free energy of formation (DG) of both wild type and mutant are given by the crossings of their respective unfolding and refolding work distributions. The difference between both free energies gives the difference in thermodynamic stability between both structures.(Inset) Experimental verification of the Crooks fluctuation relation in the mutant. If you want to learn more about this you can download the article here
(Just in case you want to learn more...) A few years ago I wrote a report article on some of the themes discussed in this section. A more introductory review on the subject can be downloaded here Finally, a more recent and complete theoretical/experimental review article on this fascinating subject can be downloaded here.
Glassy systems
One of my traditional research subjects during the past years has been the theoretical research of glassy behavior in disordered systems and glass forming systems. Glassy behavior is widespread in condensed matter physics. It refers to a nonequilibrium state of matter charactectized by long relaxation times, aging, thermal hysteresis, non-exponential relaxation, intermittency among other features. A key feature that distinguishes glassy systems from other classes of nonequilibrium states (such as for instance, steady state systems) is the absence of time-translational invariance in two-times correlation functions. The dependence of correlation functions on the time ellapsed since the system was prepared in a nonequilibrium state is commonly known as aging. The corresponding nonequilibrium state is referred to as the aging state. Examples of glassy systems are structural glasses (such as window glass) that can be prepared in the aging regime by lowering the temperature below the glass transition temperature. In this regime the glass is stiff like a solid and looks transparent like a crystal. However the glass state is neither one nor the other. In many aspects the glass resembles a liquid rather than a solid. Scientists start to agree that the physics of glasses could be determined by structural rearrangements of regions of typical dimensions extending over a few nanometers that are characterized by a wide distribution of relaxational timescales (heterogeneous kinetics).
An interesting aspect of glassy systems in their aging state is the existence of strong violations of the fluctuation-dissipation theorem (FDT). Despite of many theoretical investigations that have addressed this problem, experimental verifications of these ideas are still scarce. One of the problems on which I have been interested over many years is the study of the aging dynamics and FDT violations in exactly solvable models. These might help to investigate unifying principles underlying the nonequilibrium aging state. Along these lines I have introduced two solvable models in this field, the Backgammon model and the Oscillator model. An important part of the subsequent work in these models has been done in collaboration with Silvio Franz in Trieste and Adan Garriga in Barcelona. If you want to have a general view of the field (from the perspective of theoretical and numerical studies), you can download a review on FDT violations that I wrote a few years ago in collaboration with Andrea Crisanti from the University of Rome "La Sapienza" in Italy. Other kind of models that have seen a renewed interest over past years are kinetically constrained models. In a nutshell these are models with trivial thermodynamics but complicated kinetics due to some kinetic rules that generate dynamical arrest and slow motion. In 2003 I wrote a review paper on this this subject in collaboration with Peter Sollich from Kings College in London. Along this direction there have been attempts to interpret FDT violations in terms of effective temperatures that have some thermometric properties (such as the zero-th law of thermodynamics). Recent experiments by the group of Sergio Ciliberto in Lyon have measured FDT violations in the frequency domain in organic polymers and glasses. Typically these violations are found to be huge making unclear how to interpretat them in terms of effective temperatures with thermometric properties. Related to that, and in collaboration with Andrea Crisanti in Rome, we have developed a statistical (rather than thermometric) interpretation of the effective temperature in terms of an spontaneous relaxational process. This process is observed whenever the system is prepared in a nonequilibrium state (e.g. after quenching a glass former from high to low temperatures below the glass transition). The spontaneous process is characterized by a net release of heat from the system to the bath as the system relaxes towards the equilibrium state. Superimposed to the spontaneous process there is an stimulated process where heat is quickly exchanged between the system and the bath but zero net heat is transferred over observed long enough timescales. A parallel with light-emission process by atoms appears naturally. The spectrum of blackbody radiation is the result of stimulated and spontaneous light emisions. The stimulated process is determined by the density of photons in the cavity (the temperature of the bath in an aging system) while the spontaneous process is determined by the fact that atoms are in an excited state (corresponding to the fact that the aging system has been prepared in a nonequilibrium state). Figure 10 shows a picture of what experiments would predict for the probability distribution during aging associated to a heat-exchange process between the system and the bath. This probability distribution is measured over a timescales associated to the alfa-relaxation process (i.e. of duration greater or comparable to the waiting time tw).
Figure 10: Stimulated and spontaneous processes concurring in the overall relaxation of glassy systems.The stimulated process corresponds to a fast heat exhange between system and bath, the spontaneous process is slow heat exhange process characterized by a net heat transferred from the system to the bath. You can download here a paper discussing these ideas entitled "Stimulated and spontaneous relaxation in glassy systems"
The mechanism of relaxation of the spontaneous process is entropic and described by a fluctuation theorem similar to that developed by Evans, Cohen and Morriss for steady state systems. However, in isothermal aging systems the equivalent role of the entropy production in steady-state systems is played by the heat exchanged between the system and the environment. The fluctuation theorem then predicts the existence of exponential tails for the probability distribution of heat released from the system to the bath. The exponential tails have a width that is proportional to the value of the effective temperature as derived from FDT violations. Future experiments might identify heat exhange events and measure these exponential tails. In Figure 11 I show the emergence of exponential tails that have been measured using numerical simulations for a structural glass model. A discussion of these ideas in connection with the nonequilibrium thermodynamics of small systems can be found in a recent theoretical/experimental review article I wrote on this subject.
Figure 11: Numerical results on the intermittency in a spin-glass model. The three panels correspond to three different quenching temperatures (temperatures decreases going from the left to the right panel). Negative heat corresponds to events where heat is released from the system to the bath. Positive heat corresponds to events where heat is absorbed by the system. Click here to download the published article.