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ScienceWeek
PHYSICAL CHEMISTRY: ON MACROMOLECULE FREE ENERGY DIFFERENCES
The following points are made by Ronald F. Fox (Proc. Nat. Acad. Sci. 2003 100:12537):
1) Since the advent of techniques that enable experimentalists to manipulate individual macromolecules, there has been a need for accurate estimates of free-energy changes between an initial equilibrium state and a second equilibrium state that is arrived at by a nonequilibrium manipulation.
2) Atomic force microscopy and optical laser tweezers, often using macromolecules attached to micron-sized magnetic beads or polystyrene beads, are techniques that enable researchers to perform experiments on individual molecules, such as proteins and the polynucleic acids RNA and DNA.
3) Traditional thermodynamic theory states that an estimate of the Helmholtz free-energy difference between two states, A and B, of a macromolecular system in contact with a thermal reservoir, delta-F = FB - FA, can be achieved by perturbing the system so that a transition between the two states takes place. For repeated trials, the averaged work done satisfies the inequality W less than or equal to delta-F. Only for infinitely slow, quasi-static processes can equality be achieved. Thus, in practice it once seemed that obtaining free-energy differences by macromolecular manipulations was virtually impossible.
4) However, in 1997, Chris Jarzynski (1) derived the identity e^(-beta.W) = e^(-beta.delta-F), in which (beta) is the reciprocal of the product of the absolute temperature and Boltzmann's constant. In effect, the exponential weighting of the work performed emphasizes rare values of work in the work distribution function tail that are less than the free-energy change. Although this weighting may be expected to improve the estimate, the surprise here is the strict equality. The path integral proof of this identity found in Jarzynski's Physical Review E paper (2) is accessible and clear, and a broader setting and more general proof were subsequently published by Crooks (3). A number of numerical simulations and molecular dynamics calculations were done that confirmed the identity.
5) Thinking about atomic force microscopy and laser tweezers techniques, Hummer and Szabo (4) analyzed how an experiment could be done so that rigorous free-energy profiles are obtained. If the experiment is done at constant temperature and pressure, then the free energy obtained from the generalized Jarzynski equality is the Gibbs free energy rather than the Helmholtz free energy in the original identity (4). In fact, this result is what was subsequently obtained in a real experiment involving the unfolding of a single RNA molecule (5).
References (abridged):
1. Jarzynski, C. (1997) Phys. Rev. Lett. 78, 2690–2693
2. Jarzynski, C. (1997) Phys. Rev. E 56, 5018–5035
3. Crooks, G. E. (2000) Phys. Rev. E 61, 2361–2366
4. Hummer, G. & Szabo, A. (2001) Proc. Natl. Acad. Sci. USA 98, 3658–3661
5. Liphardt, J., Dumont, S., Smith, S. B., Tinoco, I., Jr., & Bustamante, C. (2002) Science 296, 1832–1835
Proc. Nat. Acad. Sci. http://www.pnas.org
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ON THE EQUILIBRIUM MECHANICAL PROPERTIES OF INDIVIDUAL MOLECULES
In general, conventional thermodynamics is the systematic study of the relationship between heat, work, temperature, and energy, and the relations of these variables to the general behavior of systems at equilibrium. The term "classical thermodynamics" usually refers to a phenomenological approach that does not involve consideration of individual atoms or molecules. "Statistical thermodynamics" does consider individual atoms or molecules, in the sense of involving a few elementary assumptions concerning atoms or molecules, but the focus in statistical thermodynamics is on the behavior of statistical populations of atoms or molecules.
In general, statistical thermodynamics attempts to express macroscopic thermodynamic properties in terms of the statistics of the behavior of individual particles and their interactions. During the 20th century, there has emerged the field of "nonequilibrium" ("irreversible") thermodynamics. Unlike classical thermodynamics, in which it is assumed that the system is at equilibrium, nonequilibrium thermodynamics investigates systems that are not at equilibrium. There has been much progress in nonequilibrium thermodynamics, particularly for systems close to equilibrium, but in general our understanding of nonequilibrium phenomena is not comparable to our understanding of equilibrium phenomena.
Now suppose one has an individual molecule isolated and under control, for example an individual polymer molecule specifically constrained and contacted so that it can be stretched, and one wants to describe (and understand) the behavior of this single molecule, not in terms of electrons and atomic nuclei and so on, but as a /single system/. A priori, one can say that if the laws of thermodynamics are not constrained by scale, they should in principle be applicable to a single molecule considered as a thermodynamic system. And, in fact, it should be possible to develop statistical considerations for a single molecule if we consider the real fluctuating states of the molecule as a statistical ensemble of states constrained by thermodynamic parameters. This is the essential basis of research applying statistical thermodynamics (both equilibrium and nonequilibrium) to the behavior of individual molecules.
The following points are made by G. Hummer and A. Szabo (Proc. Natl. Acad. Sci. 2001 98:3658):
1) The authors point out that recent advances in the micromanipulation of single molecules have led to new insights into the dynamics, interactions, structure, and mechanical properties of individual molecules. Single-molecule manipulation with an *atomic force microscope, *laser-tweezer stretching, and analogous computer experiments have revealed details about unfolding and unbinding events of individual proteins and their complexes. In an atomic-force-microscope experiment, a single molecule is subjected to a time-varying external force, e.g., by pulling on the end of a linear polymer. The applied force is determined from the time-dependent position of the cantilever tip with respect to the sample. Thus, one can drive rare molecular events, determine their force characteristics, and simultaneously monitor them with atomic resolution. However, both experiments and simulations actively perturb the system, leading to hysteresis and nonequilibrium effects.
2) The authors ask: How can one extract equilibrium properties from such measurements that drive the system away from equilibrium? From the second law of thermodynamics, we know that on average the mechanical work of pulling will be larger than the free energy. Only if the experiment is performed reversibly, i.e., infinitely slowly, will the work equal the free energy. Thus, making rigorous thermodynamic measurements by pulling appears to require an extrapolation to zero pulling speed. However, C. Jarzynski (1997) recently discovered a remarkable identity between thermodynamic free energy differences and the irreversible work. This identity, although not directly applicable to atomic force measurements, suggests that in principle one should be able to extract free energy surfaces from repeated pulling experiments.
3) The authors (Hummer and Szabo) demonstrate, with a quantitative theoretical analysis, how equilibrium free energy profiles can be extracted rigorously from repeated non-equilibrium force measurements on the basis of an extension of Jarzynski's identity between free energies and irreversible work.
In a commentary on the above study, the following points are made by C. Jarzynski (Proc. Natl. Acad. Sci. 2001 98:3636):
1) Jarzynski points out that what Hummer and Szabo propose amounts to a distinctive method of deducing the equilibrium mechanical properties of individual molecules. Hummer and Szabo provide a prescription for combining the data from [repeated pulling] experiments, so that what ultimately emerges is the equilibrium tension as a function of elongation, even if the molecule was driven away from equilibrium during the pulling process. Jarzynski states: "Moreover, they make a solid case --by using simulations as well as analysis of published micromanipulation data -- that their method is experimentally feasible."
2) Concerning the theoretical approach of Hummer and Szabo, Jarzynski points out that when a system is perturbed away from equilibrium by the arbitrary variation of an external parameter, then a particular statistical description of its response -- the description constructed via a weighting procedure involving a Boltzmann distribution factor [*Note #1] -- behaves with remarkable simplicity: it exactly follows the instantaneous equilibrium state associated with the changing value of the parameter. Jarzynski points out that Hummer and Szabo have translated this abstract notion into a concrete proposal for an experimental method of measuring the properties of molecules. "Not only does their method represent a potentially useful laboratory technique, but an experiment along these lines would provide the first direct test of the underlying theory."
Proc. Nat. Acad. Sci. http://www.pnas.org
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Notes:
atomic force microscope: An atomic force microscope is a type of microscope in which a small probe is held on a spring-loaded cantilever in contact with the surface of a sample. In this context, single polymer molecules are anchored between a surface and an atomic force microscope tip and then stretched. until the molecule became detached.
laser-tweezer stretching: (optical-tweezer stretching) The term "laser tweezers" refers to a laser trap used to hold and move microscopic objects. The term "laser trap" refers to a device for confining atoms, molecules, and neutral particles up to 10 microns in diameter, the trap consisting of a focused laser beam tuned to a frequency such that particles are attracted to regions of high laser intensity.
Note #1: The weighting factor is e^(-W/kT), where (W) is the total work performed, (k) is Boltzmann's constant, (T) is absolute temperature.
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