Energy dissipation in slipping biological pumps

Michaelis–Menten kinetics under non-isothermal conditions

We extend the celebrated Michaelis–Menten kinetics description of an enzymatic reaction taking into consideration the presence of a thermal driving force.

Mesoscopic non-equilibrium thermodynamic analysis of molecular motors

We show that the kinetics of a molecular motor fueled by ATP and operating between a deactivated and an activated state can be derived from the principles of non-equilibrium thermodynamics applied to the mesoscopic domain. The activation by ATP, the possible slip of the motor, as well as the forward stepping carrying a load are viewed as slow diffusion along a reaction coordinate. Local equilibrium is assumed in the reaction coordinate spaces, making it possible to derive the non-equilibrium thermodynamic description. Using this scheme, we find expressions for the velocity of the motor, in terms of the driving force along the spacial coordinate, and for the chemical reaction that brings about activation, in terms of the chemical potentials of the reactants and products which maintain the cycle. The second law efficiency is defined, and the velocity corresponding to maximum power is obtained for myosin movement on actin. Experimental results fitting with the description are reviewed, giving a maximum efficiency of 0.45 at a myosin headgroup velocity of 5 × 10(-7) m s(-1). The formalism allows the introduction and test of meso-level models, which may be needed to explain experiments.

On the thermodynamic efficiency of Ca²⁺-ATPase molecular machines

Experimental studies have shown that the activity of the reconstituted molecular pump Ca(2+)-ATPase strongly depends on the thickness of the supporting bilayer. It is thus expected that the bilayer structure will have an impact on the thermodynamic efficiency of this nanomachine. Here, we introduce a nonequilibrium-thermodynamics theoretical approach to estimate the thermodynamic efficiency of the Ca(2+)-ATPase from analysis of available experimental data about ATP hydrolysis and Ca(2+) transport. We find that the entropy production, i.e., the heat released to the surroundings under working conditions, is approximately constant for bilayers containing phospholipids with hydrocarbon chains of 18-22 carbon atoms. Our estimates for the heat released during the pump operation agree with results obtained from separate calorimetric experiments on the Ca(2+)-ATPase derived from sarcoplasmic reticulum. We show that the thermodynamic efficiency of the reconstituted Ca(2+)-ATPase reaches a maximum for bilayer thicknesses corresponding to maximum activity. Surprisingly, the estimated thermodynamic efficiency is very low, ∼12%. We discuss the significance of this result as representative of the efficiency of other nanomachines, and we address the influence of the experimental set-up on such a low efficiency. Overall, our approach provides a general route to estimate thermodynamic efficiencies and heat dissipation in experimental studies of nanomachines.

Kinetic and mesoscopic non-equilibrium description of the Ca(2+) pump: a comparison

We analyse the operation of the Ca(2+)-ATPase ion pump using a kinetic cycle diagram. Using the methodology of Hill, we obtain the cycle fluxes, entropy production and efficiency of the pump. We compare these results with a mesoscopic non-equilibrium description of the pump and show that the kinetic and mesoscopic pictures are in accordance with each other. This gives further support to the mesoscopic theory, which is less restricted and also can include the heat flux as a variable. We also show how motors can be characterised in terms of unidirectional backward fluxes. We proceed to show how the mesoscopic approach can be used to identify fast and slow steps of the model in terms of activation energies, and how this can be used to simplify the kinetic diagram.

Molecular dynamics simulations of the Ca2+-pump: a structural analysis

We report large scale molecular dynamics computer simulations, ∼100 ns, of the ion pump Ca(2+)-ATPase immersed in a 1-palmitoyl-2-oleoyl-sn-glycero-3-phosphocholine (POPC) bilayer. The structure simulated here, E1, one of the several conformations resolved using X-ray diffraction techniques, hosts two Ca(2+)-ions in the hydrophobic domain. Our results indicate that protonated residues lead to stronger ion-residue interactions, supporting previous conclusions regarding the sensitivity of the Ca(2+) behaviour to the protonated state of the amino acid binding sites. We also investigate how the protein perturbs the bilayer structure. We show that the POPC bilayer is ∼12% thinner than the pure bilayer, near the protein surface. This perturbation decays exponentially with the distance from the protein with a characteristic decay length of 0.8 nm. We find that the projected area per lipid also decreases near the protein. Using an analytical model we show that this change in the area is only apparent and it can be explained by considering the local curvature of the membrane. Our results indicate that the real area per lipid near the protein is not significantly modified with respect to the pure bilayer result. Further our results indicate that the local deformation of the membrane around the protein might be compatible with the enhanced protein activity observed in experiments over a narrow range of membrane thicknesses.

A combined zinc/cadmium sensor and zinc/cadmium export regulator in a heavy metal pump

Heavy metal pumps (P1B-ATPases) are important for cellular heavy metal homeostasis. AtHMA4, an Arabidopsis thaliana heavy metal pump of importance for plant Zn(2+) nutrition, has an extended C-terminal domain containing 13 cysteine pairs and a terminal stretch of 11 histidines. Using a novel size-exclusion chromatography, inductively coupled plasma mass spectrometry approach we report that the C-terminal domain of AtHMA4 is a high affinity Zn(2+) and Cd(2+) chelator with capacity to bind 10 Zn(2+) ions per C terminus. When AtHMA4 is expressed in a Zn(2+)-sensitive zrc1 cot1 yeast strain, sequential removal of the histidine stretch and the cysteine pairs confers a gradual increase in Zn(2+) and Cd(2+) tolerance and lowered Zn(2+) and Cd(2+) content of transformed yeast cells. We conclude that the C-terminal domain of AtHMA4 serves a dual role as Zn(2+) and Cd(2+) chelator (sensor) and as a regulator of the efficiency of Zn(2+) and Cd(2+) export. The identification of a post-translational handle on Zn(2+) and Cd(2+) transport efficiency opens new perspectives for regulation of Zn(2+) nutrition and tolerance in eukaryotes.

Heat transfer in protein-water interfaces

We investigate using transient non-equilibrum molecular dynamics simulation the temperature relaxation process of three structurally different proteins in water, namely; myoglobin, green fluorescence protein (GFP) and two conformations of the Ca(2+)-ATPase protein. By modeling the temperature relaxation process using the solution of the heat diffusion equation we compute the thermal conductivity and thermal diffusivity of the proteins, as well as the thermal conductance of the protein-water interface. Our results indicate that the temperature relaxation of the protein can be described using a macroscopic approach. The protein-water interface has a thermal conductance of the order of 100-270 MW K(-1) m(-2), characteristic of water-hydrophilic interfaces. The thermal conductivity of the proteins is of the order of 0.1-0.2 W K(-1) m(-1) as compared with approximately 0.6 W K(-1) m(-1) for water, suggesting that these proteins can develop temperature gradients within the biomolecular structures that are larger than those of aqueous solutions. We find that the thermal diffusivity of the transmembrane protein, Ca(2+)-ATPase is about three times larger than that of myoglobin or GFP. Our simulation shows that the Kapitza length of these structurally different proteins is of the order of 1 nm, showing that the protein-water interface should play a major role in defining the thermal relaxation of biomolecules.

Coefficients for active transport and thermogenesis of Ca2+-ATPase isoforms

Coefficients for active transport of ions and heat in vesicles with Ca(2+)-ATPase from sarcoplasmic reticulum are defined in terms of a newly proposed thermodynamic theory and calculated using experiments reported in the literature. The coefficients characterize in a quantitative manner different performances of the enzyme isoforms. Four enzyme isoforms are examined, namely from white and red muscle tissue, from blood platelets, and from brown adipose mitochondria. The results indicate that the isoforms have a somewhat specialized function. White muscle tissue and brown adipose tissue have the same active transport coefficient ratio, but the activity level of the enzyme in white muscle is higher than in brown adipose tissue. The thermogenesis ratio is high in both white muscle and brown adipose tissue, in agreement with a specific role in nonshivering thermogenesis. Other isoforms do not have this ability to generate heat. A calcium-dependence of the coefficients is found, which can be understood as being in accordance with the role of this ion as a messenger in muscle contraction as well as in thermogenesis. The investigation points to new experiments related to structure as well as to function of the isoforms.

The measurable heat flux that accompanies active transport by Ca2+-ATPase

We present a new mesoscopic basis which can be used to derive flux equations for the forward and reverse mode of operation of ion-pumps. We obtain a description of the fluxes far from global equilibrium. An asymmetric set of transport coefficients is obtained, by assuming that the chemical reaction as well as the ion transports are activated, and that the enzyme has a temperature independent of the activation coordinates. Close to global equilibrium, the description reduces to the well known one from non-equilibrium thermodynamics with a symmetric set of transport coefficients. We show how the measurable heat flux and the heat production under isothermal conditions, as well as thermogenesis, can be defined. Thermogenesis is defined via the onset of the chemical reaction or ion transports by a temperature drop. A prescription has been given for how to determine transport coefficients on the mesocopic level, using the macroscopic coefficient obtained from measurements, the activation enthalpy, and a proper probability distribution. The method may give new impetus to a long-standing unsolved transport problem in biophysics.

Is the Ca2+-ATPase from sarcoplasmic reticulum also a heat pump?

We calculate, using the first law of thermodynamics, the membrane heat fluxes during active transport of Ca(2+) in the Ca(2+)-ATPase in leaky and intact vesicles, during ATP hydrolysis or synthesis conditions. The results show that the vesicle interior may cool down during hydrolysis and Ca(2+)-uptake, and heat up during ATP synthesis and Ca(2+)-efflux. The heat flux varies with the SERCA isoform. Electroneutral processes and rapid equilibration of water were assumed. The results are consistent with the second law of thermodynamics for the overall processes. The expression for the heat flux and experimental data, show that important contributions come from the enthalpy of hydrolysis for the medium in question, and from proton transport between the vesicle interior and exterior. The analysis give quantitative support to earlier proposals that certain, but not all, Ca(2+)-ATPases, not only act as Ca(2+)-pumps, but also as heat pumps. It can thus help explain why SERCA 1 type enzymes dominate in tissues where thermal regulation is important, while SERCA 2 type enzymes, with their lower activity and better ability to use the energy from the reaction to pump ions, dominate in tissues where this is not an issue.

Thermodynamics for single-molecule stretching experiments

We show how to construct nonequilibrium thermodynamics for systems too small to be considered thermodynamically in a traditional sense. Through the use of a nonequilibrium ensemble of many replicas of the system which can be viewed as a large thermodynamic system, we discuss the validity of nonequilibrium thermodynamics relations and analyze the nature of dissipation in small systems through the entropy production rate. We show in particular that the Gibbs equation, when formulated in terms of average values of the extensive quantities, is still valid, whereas the Gibbs-Duhem equation differs from the equation obtained for large systems due to the lack of the thermodynamic limit. Single-molecule stretching experiments are interpreted under the prism of this theory. The potentials of mean force and mean position, now introduced in these experiments in substitution of the thermodynamic potentials, correspond respectively to our Helmholtz and Gibbs energies. These results show that a thermodynamic formalism can indeed be applied at the single-molecule level.

Unifying Thermodynamic and Kinetic Descriptions of Single-Molecule Processes: RNA Unfolding under Tension

We use mesoscopic nonequilibrium thermodynamics theory to describe RNA unfolding under tension. The theory introduces reaction coordinates, characterizing a continuum of states for each bond in the molecule. The unfolding considered is so slow that one can assume local equilibrium in the space of the reaction coordinates. In the quasi-stationary limit of high sequential barriers, our theory yields the master equation of a recently proposed sequential-step model. Nonlinear switching kinetics is found between open and closed states. Our theory unifies the thermodynamic and kinetic descriptions and offers a systematic procedure to characterize the dynamics of the unfolding process.

Nonequilibrium thermodynamics for open systems

We develop the general equation for the nonequilibrium reversible-irreversible coupling (GENERIC) framework of nonequilibrium thermodynamics for open systems. A clear distinction between bulk and boundary contributions to the Poisson and dissipative brackets employed to generate reversible and irreversible contributions to time evolution from energy and entropy allows us to formulate the bulk equations as well as the exchange and interaction with the environment directly. The full brackets keep all the structure and hence the predictive power of the original GENERIC for isolated systems. The straightforward procedure is illustrated for hydrodynamics of open systems. Boltzmann's kinetic equation is discussed as a further example. In the Appendix, the thermodynamic treatment of surface excess variables at walls and their role in boundary conditions for the bulk variables is exemplified for a diffusion cell.