We want to understand whether and to which extent the maximal (Carnot) efficiency for heat engines can be reached at a finite power. To this end we generalize the Carnot cycle so that it is not restricted to slow processes. We show that for realistic (i.e. not purposefully-designed) enginebath interactions, the work-optimal engine performing the generalized cycle close to the maximal efficiency has a long cycle time and hence vanishing power. This aspect is shown to relate to the theory of computational complexity. A physical manifestation of the same effect is the Levinthal's paradox in the protein folding problem. The resolution of this paradox for realistic proteins allows to construct engines that can extract at a finite power 40 % of the maximally possible work reaching 90 % of the maximal efficiency. For purposefully designed engine-bath interactions, the Carnot efficiency is achievable at a large power. Reciprocating heat engines extract work operating cyclically between two thermal baths at temperatures T 1 and T 2 (T 1 > T 2 ) [1]. They have two basic characteristics: (i) efficiency, η = W/Q 1 , is the work W extracted per cycle divided by the heat input Q 1 from the hightemperature bath. (ii) Power W/τ , where τ is the cycle duration. Both these quantities have to be large for a good engine: if η is small, lot of energy is wasted; if the power is small, no sizable work is delivered over a reasonable time [1].The second law establishes the Carnot efficiency η C = 1 − T2 T1 as an upper bound for η [1]. The Carnot cycle reaches the bounding value η C in the (useless) limit, where the power goes to zero [1]. Conversely, realistic engines are not efficient, since they have to be powerful, e.g. the efficiency of Diesel engines amounts to 35-40 % of the maximal value. This power-efficiency dilemma motivated a search for the efficiency that would generally characterize the maximal power regime. One candidate for this is the Curzon-Ahlborn efficiency η CA = 1 − T 2 /T 1 [2], which is however crucially tied to the linear regime T 1 ≈ T 2 [3,4]. Beyond this regime η CA is a lower bound of η for a class of model engines [5]. Several recent models for the efficiency at the maximal power overcome η CA with η * = ηC 2−ηC [6]. As argued in [5,7,8], the maximal power regime allows for the Carnot efficiency, at least for certain models. But it is currently an open question whether the maximal efficiency is attained under realistic conditions (see e.g.[9] versus [7]), and how to characterize the very realism of those conditions. Even more generally: what is the origin of the power-efficiency dilemma? We answer these questions by analyzing a generalized Carnot cycle, which in contrast to the original Carnot cycle is not restricted to slow processes. We now summarize our answers.(1) When the N -particle engine operates at the maximal work extracted per cycle, its efficiency reaches the Carnot bound η C for N 1, while the cycle time isgiven by the relaxation time of the engine. The maximal work and the Carnot efficiency are...

SynopsisThe viscoelastic properties of solid samples (crystals, amorphous films) of hen egg white lysozyme, bovine serum albumin, and sperm whale myoglobin were studied in the temperature range of 100-300 K at different hydration levels. Decreasing the temperature was shown to cause a steplike increase in the Young's modulus of highly hydrated protein samples (with water content exceeding 0.3 g/g dry weight of protein) in the temperature range of 237-251 K, followed by a large increase in the modulus in the broad temperature interval of 240-130 K, which we refer to as a mechanical glass transition.Soaking the samples in 50% glycerol solution completely removed the steplike transition without significantly affecting the glass transition. The apparent activation energy determined from the frequency dependence of the glass-transition temperature was found to be 18 kcal/mol for wet lysozyme crystals. Lowering the humidity causes both the change of the Young's modulus in response to the transition and the activation energy to decrease. The thermal expansion coefficient of amorphous protein films also indicates the glass transition at 150-170 K. The data presented suggest that the glass transition in hydrated samples is located in the surface layer of proteins and related to the immobilization of the protein groups and strongly bound water.

BackgroundType I collagen is the most common protein among higher vertebrates. It forms the basis of fibrous connective tissues (tendon, chord, skin, bones) and ensures mechanical stability and strength of these tissues. It is known, however, that separate triple-helical collagen macromolecules are unstable at physiological temperatures. We want to understand the mechanism of collagen stability at the intermolecular level. To this end, we study the collagen fibril, an intermediate level in the collagen hierarchy between triple-helical macromolecule and tendon.Methodology/Principal FindingWhen heating a native fibril sample, its Young’s modulus decreases in temperature range 20–58°C due to partial denaturation of triple-helices, but it is approximately constant at 58–75°C, because of stabilization by inter-molecular interactions. The stabilization temperature range 58–75°C has two further important features: here the fibril absorbs water under heating and the internal friction displays a peak. We relate these experimental findings to restructuring of collagen triple-helices in fibril. A theoretical description of the experimental results is provided via a generalization of the standard Zimm-Bragg model for the helix-coil transition. It takes into account intermolecular interactions of collagen triple-helices in fibril and describes water adsorption via the Langmuir mechanism.Conclusion/SignificanceWe uncovered an inter-molecular mechanism that stabilizes the fibril made of unstable collagen macromolecules. This mechanism can be relevant for explaining stability of collagen.

We measured the Young's modulus at temperatures ranging from 20 to 100 degrees C for a collagen fibril that is taken from a rat's tendon. The hydration change under heating and the damping decrement were measured as well. At physiological temperatures 25 to 45 degrees C, the Young's modulus decreases, which can be interpreted as an instability of the collagen. For temperatures between 45 and 80 degrees C, the Young's modulus first stabilizes and then increases when the temperature is increased. The hydrated water content and the damping decrement have strong maximums in the interval 70 to 80 degrees C indicating complex intermolecular structural changes in the fibril. All these effects disappear after heat-denaturation of the sample at 120 degrees C. Our main achievement is a five-stage mechanism by which the instability of a single collagen at physiological temperatures is compensated by the interaction between collagen molecules.