Thermodynamics and kinetics of actin filament nucleation
ABSTRACT We have performed computer simulations and free energy calculations to determine the thermodynamics and kinetics of actin nucleation and thus identify a probable nucleation pathway and critical nucleus size. The binding free energies of structures along the nucleation pathway are found through a combination of electrostatic calculations and estimates of the entropic and surface area contributions. The association kinetics for the formation of each structure are determined through a series of Brownian dynamics simulations. The combination of the binding free energies and the association rate constants determines the dissociation rate constants, allowing for a complete characterization of the nucleation and polymerization kinetics. The results indicate that the trimer is the size of the critical nucleus, and the rate constants produce polymerization plots that agree very well with experimental results over a range of actin monomer concentrations.
Although the results of our model appear to agree very well with experimental results, there are several details about the methods that need to be pointed out. The Brownian dynamics simulations assume that the formation of each proteinprotein complex is controlled by diffusion and electrostatic interactions. We know this to be the case for barbed-end actin polymerization, but here this assumption also applies to the nucleation phase. For the binding free energy calculations, we assumed that we could represent the energies using Eq. 2. This is admittedly a simplified representation, but it captures the essential components: electrostatic and hydrophobic interactions, desolvation and configurational entropy. It also has the advantage that it introduces only one free parameter into the model because it is constrained by the binding energy for the polymerization step. Including more terms in the energy expansion (e.g., van der Waals, polar and apolar contributions), could increase the accuracy of our free energy calculations, but it would also introduce additional free parameters in our model, which do not appear to be required. We are also limited by the fact that we must deal with rigid protein structures. There is no doubt that conformational changes occur during the nucleation and polymerization phases, but, currently, we have no information about the differences between G-actin and F-actin structures, or how these structures compare with the actinDNaseI structure used by Holmes et al. (1990). A recently reported structure for G-actin (R. Dominguez, Boston Biomedical Research Institute, personal communication) should offer new insights into how important these difference are in actin nucleation and polymerization.
CONCLUSIONS
Our study of the spontaneous nucleation of actin filaments leads us to the conclusion that the trimer is the critical nucleus size. Through the combination of BD simulations and free energy calculations, we were able to estimate the kinetic rate constants for each of the nucleation steps by scaling with known values for actin polymerization. The predicted time course of polymerization arising from these rate constants agrees very well with experimental results over a range of actin monomer concentrations. Future work combining such calculations with additional factors, such as the Arp 2/3 complex and other associated proteins, could give more insight into nucleation and polymerization within the cell.
Thermodynamics of the Sn-In-Ag solder system
The lead-free solder system Sn-In-Ag was studied both experimentally and by thermodynamic modeling. Thermodynamic descriptions for the phases in Ag-In and In-Sn binary systems were optimized taking into account the available thermodynamic and phase equilibria data. They were combined with the previously assessed Ag-Sn binary system to get a complete thermodynamic description of the Sn-In-Ag system. The thermodynamic functions obtained are used to model the solidifying behavior of the alloys with diagrams of the relative amount of phases (microstructures) as a function of temperature and the effect of diffusional segregation on the solidification path. The solidification of alloys of different compositions and its relevance to the microstructure and mechanical properties of the solidified solder are discussed. In the experimental research, alloys of different compositions were heat treated and analyzed by differential scanning calorimetry (DSC), optical microscopy, and scanning electron microscopy/electron probe microanalysis techniques. Ternary alloys were annealed at 250degC to gain information about the location of the phase boundaries at this rather typical soldering temperature. Diffusion couple experiments were performed to investigate the interfacial reaction between Ag and binary SnIn and to find out the diffusion paths taken by the system. The melting/solidification behavior of the alloys in the Sn corner of the ternary diagram were investigated by DSC. It was discovered that a strong segregation takes place during the cooling of some In-containing solder alloys as the In component is rejected from the solid with an excess of In remaining in the liquid. At high enough In concentrations (>16%), this results in partial melting of the solder at temperatures in excess of 113degC.
NTRODUCTION
The trend toward increasing interconnection densities and sophisticated new packaging technologies like C4, BGA, and multichip modules places additional demands on the joining materials and methods used. A variety of new, preferably lead-free solders with different melting temperatures are needed to be used in conjunction with or as a replacement for the conventional Pb-Sn solders. Despite their many favored properties, such as good wetting, low melting temperature, attractive price, and well-established technology, the Pb-Sn solders have relatively low strength and can suffer from microstructural coarsening. The new lead-free solder alloys should not only substitute for the lead-tin solders but should also offer significantly improved mechanical properties, in particular better resistance to thermal fatigue.
As the development of lead-free solders becomes more and more complex as a result of the ternary and higher order systems studied, thermodynamic modeling is becoming an increasingly popular tool in designing and evaluating the various solder candidates. The main advantage of thermodynamic modeling is that it significantly decreases the amount of experimental work needed. Modeling can predict the expected equilibrium state of the solder system at a given temperature and composition, after which the validity of the model can be checked with a few well selected experiments. If necessary, alterations to the model parameters can then be made according to the new experimental information.
The present research was motivated by a desire to thermodynamically model the Sn-In-Ag system, which is one of the most promising candidate systems when searching for a lead-free replacement for the eutectic Pb-Sn. Two of its binaries have already been successfully used in electronic soldering for some time, the eutectic Ag-Sn at higher temperatures (Tm = 221degC), and the eutectic In-Sn at low temperatures (Tm = 120degC). The Ag-Sn solder has good wetting and mechanical properties, but a somewhat too high melting temperature. An attractive approach to lower its melting point is to use an additive such as In or Bi, if this can be done without loosing the desirable properties of Ag-Sn. The Ag-Sn-Bi system has been modeled by Kattner and Boettinger;1 and according to their assessment, the liquidus temperature can be lowered by Bi additions to slightly below 200degC before reaching the composition region where eutectic solidification takes place. With its low melting temperature, indium could be even more effective in lowering the melting temperature. Since Bi additions can sometimes have an embrittling effect on the solder,2 the ductility of In is another advantage. There already exists a new commercial solder from the Ag-Sn-In system with melting temperature range close to that of eutectic Pb-Sn, and it is plausible to expect that some other alloys from this system (possibly with quaternary alloying elements) could also be developed for soldering.
Phase diagram modeling can help to sort out possible solder compositions, since the melting temperature range and the changes in the metallographic phases present can be modeled as a function of the solder composition. Furthermore, since Ag is also used as a protective plating, ternary phase description can provide information on the reactions involved when Ag-plated conductors are soldered with Sn-In alloys. If the database were extended to include substrate materials such as Cu or Ni, other interesting solder-substrate reactions could be modeled. In the case of Cu, this could be done relatively easily since there are thermodynamic descriptions available for the binary Cu-In, Cu-Ag, and Cu-Sn systems. In a literature search for the Sn-ln-Ag system, it was discovered that despite the considerable interest in this system in soldering technology, no ternary phase diagram has been published and virtually no experimental information is available for the ternary system. Of the three binary subsystems, a thermodynamic description was available only for the Ag-Sn system, which has been modeled by several authors.1,3,4 However, the binary Ag-In and In-Sn phase diagrams have been relatively well established experimentally, and there are also some thermodynamic data available on these systems. In this work, thermodynamic descriptions of the missing binaries are optimized, and the obtained information is extrapolated to yield a ternary description of the Sn-In-Ag system. Experimental work was also carried out to check the validity of the ternary diagram and the location of the phase boundaries.
MODELING
The parameters Go are the partial molar Gibbs energies (lattice stabilities) of the pure components, and they are characteristic for each phase. The temperature functions of Go were taken from the SGTE databank.5 The standard state has been chosen so that the Gibbs energies of the pure elements are zero in their stable room-temperature phases. The following Gibbs energy functions are used for the different phases:
The standard Gibbs energies of formation of the stoichiometric compounds (Ag2In, AgIn2, and Ag3Sn) have expressions of the form
G=A+BT,
where constants A and B are either found in the literature or optimized.
where the unary (G^sup o^) and binary (L^sub ij^) parameters have the same values as in the binary systems and a ternary interaction parameter (L^sub 123^) has been added. The solution phases of the Sn-In system (In), gamma, beta, (Sn) have such small solubilities for Ag that they have been modeled as binary phases with no solubility for Ag.
THE BINARY SYSTEMS
The thermodynamic description of the binary AgSn system was taken from literature. Description for Ag-In system had not been published previously and was optimized in this study based on the available thermodynamic and phase diagram data. Although a thermodynamic description for Sn-In systems has recently been published, it was not available when this work was started. Therefore, also the Sn-In system was optimized, and the results are compared to the earlier description. In the optimization, the goal was to use as simple phase descriptions and as few parameters as possible and still get a reasonable estimate ofthe phase diagram and the thermodynamic properties.
The Ag-Sn System
The Ag-Sn system has the following phases:
Liquid,
(Ag), the Ag-rich face cubic centered (fcc) solid solution,
zeta, the close packed hexagonal solution phase,
Ag3 Sn, an ordered orthorhombic phase exhibiting only a small variation from stoichiometry, and
(Sn), or beta-Sn, the body centered tetragonal (bct) solid solution a with very small solubility for Ag.
The fcc alpha-Sn has been neglected in the phase diagram evaluation.
The binary phase diagram has been optimized by several authors.1,3,4 The evaluation of Kattner and Boettinger, shown in Fig. 1, has been used since it is a part of a ternary Sn-Bi-Ag assessment and will make our description compatible with this ternary system.
The Ag-In System
The Ag-In phase diagram is very similar to that of the Ag-Sn system. In Ag-In system, the (Ag) phase extends to 20% In, compared to the 10% Sn in the AgSn system. In both systems, (Ag) is followed by the Zeta solution phase, which forms peritectically. In an isothermal ternary phase diagram (Fig. 5) these phases will extend from one binary to another.
The following phases are present in the Ag-In phase diagram:
Liquid,
* (Ag), terminal Ag-rich solid solution with 20% maximum In solubility,
* Zeta, close packed hexagonal solution phase,
* Beta, a high-temperature body centered cubic (bcc)-phase,
Nearly stoichiometric Ag2In with an ordered bcc structure,
AgIn2, a stoichiometric compound, and
* (In), the indium-rich phase with face centered tetragonal (fct) structure and a very small solubility for Ag.
The Beta-phase, which exists between In-compositions .25 and .30 in the temperature range 695-660degC, has been omitted to keep the diagram simple. Since the primary interest of this investigation are the temperatures below 250degC, it was not considered necessary to include this high-temperature phase in the assessment.
The most recent experimental Ag-In phase diagram published is the assessment of Baren,9 based mainly on the investigations of Weibke et al.10 and Campbell et al.ll All regions are not yet well established, particularly the phase boundaries involving the zeta and AgIn phases.
There are relatively much thermodynamic data available on the liquid phase. The activities in the liquid have been measured by several authors.12-15 Unfortunately, all these measurements were made at 1000K or above, and a very long extrapolation has to be made on the In side of the diagram, where liquid is present at as low as 180degC. This, incidentally, is the part of the diagram that we are interested in, since the temperature range for solder reflow is around 200degC. The enthalpy of the liquid has been determined at temperatures from 723 to 1280K.16-18 The enthalpy of formation varies with temperature, indicating a change in heat capacity. However, since no enthalpy of formation data was available for the lower temperatures and we wanted to avoid using large logarithmic terms extrapolated from the high temperature data, a linear temperature dependence was used for the excess Gibbs energy.
For the (Ag) phase, there is activity data of In at 1000K19 and at 900K.20 Enthalpies of formation for the (Ag) and zeta phases have also been reported.18 No information was found on the thermodynamic properties of the stoichiometric phases, so values used are results of optimization.
The optimized Ag-In phase diagram and thermodynamic properties are shown in Fig. 2a-c. Most parts of the diagram agree well with the experimental data, with the exception of the region between 4 and Ag2In phases, where it was extremely difficult to obtain a better fit. Our zeta+Ag2In region is too wide on the Ag side, and the maximum Sn solubility in zeta is smaller than the experimentally determined 46%. Since there is little experimental information available on this region and since our primary interest is not in this particular region but on the Sn side of the diagram, we accepted the current results. The parameters used are given in Table I.
The In-Sn System
The In-Sn system has two intermediate phases with a large range of solubility. There is a eutectic reaction in the middle of the diagram (at 52% In, 120degC) and two peritectic reactions. The following phases are present in the phase diagram:
Liquid (L),
* (In), the In-rich fct terminal phase with 12% maximum solubility for Sn
The fct solid solution which belongs to the same crystallographic class as fct (In) phase, but has a smaller c/a ratio,
gamma, the hexagonal Sn-rich solution phase, and
(Sn), the terminal phase bct Sn.
Again, a-Sn has been omitted from the description.
The phase diagram of the system published in a recent review by Okamoto21 is primarily based on the data of Heumann and Alpaut,22 with some modifications made to the In-rich gamma-boundary.23,24
There are several investigations on the thermodynamic properties of liquid In-Sn. The enthalpy of formation measurements at 521, (25) 644, (26) and 723K (27) agree very well with each other, indicating that the enthalpy of formation does not vary significantly with temperature. Activity data at 673-873K are also reported.28,29
The thermodynamic data for the solid phases come mainly from the EMF measurements of Cakir and Alpaut24 at 75-120degC. The enthalpies of formation of the solid phases have been determined also by Alpaut and Heuman.29
The SGTE lattice stabilities5 were used when available. Since there is no SGTE lattice stability data for metastable phases like fct-Sn, bct-In, gamma-Sn, and gamma-In, optimized values had to be used. Lattice stability values for metastable beta-In have previously been optimized in assessments of the In-Bi 31 and In-Pb 32 systems which also contain the beta phase. These were used as starting values, but were allowed to vary in the optimization. Figures 3a-c show the optimized phase diagram and the thermodynamic properties. The parameters used are listed in Table II.
In comparing the recently published Sn-In description of Lee et al.8 to the present work, we notice that they use the same lattice stabilities for (In) and betaphases. This cannot be physically correct, because the two phases have different structures. However, since the thermodynamic properties are the sum of several parameters, it does not make a difference in the optimized phase diagram if some parameters are inaccurate as long as the overall thermodynamic functions are in agreement with the experimentally measured values.
Since there are relatively much experimental data for this system, the two Sn-In descriptions turn out to be rather similar in terms of the thermodynamic properties, even if the numerical values for the parameters are quite different. The most noticeable difference in the calculated phase diagrams is the location of the In-rich gamma boundary. In the present diagram, it is shifted to higher In contents due to a different choice of the preferred experimental data set for this region. Because even slight differences in the binary parameters can be compounded in the ternary diagram, we predict that substituting the Lee et al. description would affect locations of the phase boundaries, although the phase equilibria would most likely remain what they are.
Thermodynamics-Based Metabolic Flux Analysis
A new form of metabolic flux analysis (MFA) called thermodynamics-based metabolic flux analysis (TMFA) is introduced with the capability of generating thermodynamically feasible flux and metabolite activity profiles on a genome scale. TMFA involves the use of a set of linear thermodynamic constraints in addition to the mass balance constraints typically used in MFA. TMFA produces flux distributions that do not contain any thermodynamically infeasible reactions or pathways, and it provides information about the free energy change of reactions and the range of metabolite activities in addition to reaction fluxes. TMFA is applied to study the thermodynamically feasible ranges for the fluxes and the Gibbs free energy change, ?^sub r^G’, of the reactions and the activities of the metabolites in the genome-scale metabolic model of Escherichia coli developed by Palsson and co-workers. In the TMFA of the genome scale model, the metabolite activities and reaction ?^sub r^G’ are able to achieve a wide range of values at optimal growth. The reaction dihydroorotase is identified as a possible thermodynamic bottleneck in E. coli metabolism with a ?^sub r^G’ constrained close to zero while numerous reactions are identified throughout metabolism for which ?^sub r^G’ is always highly negative regardless of metabolite concentrations. As it has been proposed previously, these reactions with exclusively negative ?^sub r^G’ might be candidates for cell regulation, and we find that a significant number of these reactions appear to be the first steps in the linear portion of numerous biosynthesis pathways. The thermodynamically feasible ranges for the concentration ratios ATP/ADP, NAD(P)/NAD(P)H, and H+^sub extracellular^/H+^sub intracellular^ are also determined and found to encompass the values observed experimentally in every case. Further, we find that the NAD/NADH and NADP/NADPH ratios maintained in the cell are close to the minimum feasible ratio and maximum feasible ratio, respectively.
INTRODUCTION
Thermodynamics have been applied to many areas of analysis of biological systems (1-5), but thermodynamics have yet to be applied to a rigorous examination of entire metabolic networks. This has been primarily due to a scarcity of thermodynamic data on metabolic reactions, a lack of rigorous models of metabolic chemistry, and the absence of any extensive databases, which bring all of this information together. However, the availability of thermodynamic data has increased over time, and group contribution methodologies for estimating thermodynamic properties have also been introduced (6-9). Furthermore, several rigorous models of the metabolic chemistry of a variety of microorganisms have been developed including some genome-scale models (10-13). Recently, the application of thermodynamics to study the feasibility of metabolic pathways has been revisited. Beard, Qian, and co-workers have conducted studies on the topic of eliminating internal flux cycles from flux balance analysis solutions (14-16). These are sets of reactions such as A [arrow right] B [arrow right] C [arrow right] A. According to the first law of thermodynamics, the overall thermodynamic driving force through these cycles must be zero, meaning that no net flux is possible through these cycles. Beard and Qian have also used nonlinear thermodynamic and enzyme activity constraints to determine the concentration profiles of metabolites in the central carbon chemistry of a hepatocyte cell (17). Maskow and Stockar used the pathway analysis method of Mavrovouniotis (18,19) to study the thermodynamic feasibility of the lactic acid fermentation pathway, and they found that without careful consideration of ionic strength of solution, uncertainty in thermodynamic data, and cell pH, feasible pathways can be falsely labeled as infeasible or vice versa (20). However, these previous studies were performed on relatively small-scale pathways due to a lack of thermodynamic data for genome-scale models and utilized nonlinear optimization criteria to determine fixed values for the activities of the metabolites under an isolated set of conditions.
In a previous article, we utilized the group contribution method (7,8) to estimate the standard Gibbs free energy change, ?^sub r^G’°, of the reactions in a genome-scale model of Escherichia coli, and we used these estimates to assess the thermodynamic feasibility of the reactions in the model (21 ). We called this model iHJ873, which is based on the iJR904 model developed by Palsson and co-workers. The iHJ873 model was derived from the UR904 model by removing all of the reactions in the iJR904 model that contain compounds for which the standard Gibbs free energy change of formation, ?^sub f^G’°, could not be estimated and replacing these reactions with lumped reactions. The iHJ873 model contains fewer reactions than the UR904 model (873 vs. 931, respectively), but ?^sub r^G’° of every reaction in the iHJ873 can be estimated. The thermodynamic studies of the iHJ873 model focused on the individual reactions in the model that were found to have a large positive ?^sub r^G’° in the direction of flux. We simulated the impact of removing these unfavorable reactions on the growth of the cell, and we considered the biological implications that these particular reactions were thermodynamically unfavorable. In this article, we take this work a significant step forward by examining the metabolite concentrations required for every reaction essential for optimal growth to be simultaneously thermodynamically feasible. We propose a new methodology that we call thermodynamics-based metabolic flux analysis (TMFA) for integrating thermodynamic data and constraints into a constraints-based metabolic model to ensure that flux distributions produced by the model are thermodynamically feasible and to provide data on the thermodynamically feasible metabolite activity ranges for the metabolites in the cell. TMFA can also be used for the analysis of unmodified models that are lacking some thermodynamic data to allow for direct analysis of models such as UR904 without first creating lumped models like the iHJ873. We apply TMFA to analyze the UR904 model using new thermodynamic data estimated from an updated and expanded implementation of the group contribution method (M. D. Jankowski, C. S. Henry, L. J. Broadbelt and V. Hatzimanikatis, unpublished); we assess the sensitivity of TMFA to changes in ArG’0 due to uncertainty and ionic strength; and we examine the thermodynamically feasible ranges for biologically important concentrations ratios such as ATP/ADP, NAD(P)TNAD(P)H, and H^sup +^^sub extracellular^/H^sup +^^sub intracellular^. Finally, we utilize the ?^sub r^G’ ranges calculated for the UR904 reactions with TMFA to identify candidate reactions for cell regulation as it has been previously proposed (2).
METHODS
Metabolic flux analysis (MFA)
The introduction of thermodynamics-based constraints in MFA will enforce the exclusion of thermodynamical infeasibilities from flux distribution solutions. One example of these infeasibilities would be flux distributions involving flux through the thermodynamically infeasible internal flux loops mentioned earlier. In addition, these constraints will allow the quantification of the ranges in the gradients of metabolite activities required to drive reactions in the direction of flux reported in all calculated flux distributions. Knowledge of the permissible ranges of metabolite activities is essential for the development of kinetic models of metabolism and metabolic control analysis (33-38).
Estimation of ?^sub r^G’° of reactions in the IJR904 metabolic model
Formulation of the thermodynumic constraints in TMFA requires knowledge of ?^sub r^G’° of the reactions in the model, and it must either be estimated or measured experimentally. Experimental data is available for only a small fraction of the reactions involved in a genome-scale metabolic model such as UR904. Fortunately, the group contribution method provides a means of estimating ?^sub r^G’° of nearly every reaction (7,8). In a previous article (21), the group contribution method was used to estimate ?^sub r^G’° of 808 of the 931 reactions in the UR904 model. Recent improvement and expansion of the group contribution method (M. D. Jankowski, C. S. Henry, L. J. Broadbelt and V. Hatzimanikatis, unpublished) based on a refitting of the group contribution values using the thermodynamic data gathered in the NIST Standard Reference Database (39) and other literature (40-42) have allowed the estimation of ?^sub r^G’° for 576 (92%) of the compounds and ?^sub r^G’° for 891 (96%) of the reactions in the UR904 model. In addition, we have been able to quantify the ranges of uncertainty in the estimated energy values due to variances in experimental measurements and the fitting method. All new estimated thermodynamic data for the UR904 model are provided in the Supplementary Material.
Thermodynamics of General Anesthesia
It is known that the action of general anesthetics is proportional to their partition coefficient in lipid membranes (Meyer-Overton rule). This solubility is, however, directly related to the depression of the temperature of the melting transition found close to body temperature in biomembranes. We propose a thermodynamic extension of the Meyer-Overton rule, which is based on free energy changes in the system and thus automatically incorporates the effects of melting point depression. This model accounts for the pressure reversal of anesthesia in a quantitative manner. Further, it explains why inflammation and the addition of divalent cations reduce the effectiveness of anesthesia.
More than 100 years ago. Ham Meyer in Marburg (1) and Charles Ernest Overton in Zurich (2) independently found that the action of general anesthetics is related to their partition coefficient between water and olive oil. Overton performed experiments on tadpoles and recorded the critical drug concentration, ED^sub 50^, at which they stopped swimming. Assuming that the solubility of these anesthetics in olive oil is proportional to that in biomembranes, he suggested that this critical concentration corresponded to a fixed concentration in biomembranes. The Meyer-Overton rule can be expressed as [ED^sub 50^] × P = const, where P is the partition coefficient of the anesthetic drug between membranes and water. Small molecules, as different as nitrous oxide, chloroform, octanol, diethylether, procaine, and even the noble gas xenon, all act as anesthetics. Overton noted that this action is completely unspecific, i.e., dependent only on the solubility of the anesthetic in oil and independent of its chemical nature. Surprisingly, this finding is still valid for general and local anesthetics (2-5) but remains unexplained. Overton concluded that this nonspecific! Iy requires a single mechanism based on physical chemistry and not on the molecular structure of the drugs. Although the close relation between anesthetic effect and solubility in lipids led many scientists to believe that anesthetic action is lipid-related, no model was proposed by Meyer and Overton or by later research. It is known, however, that lipid-melting transitions are lowered in the presence of anesthetics. This has been related to the anesthetic function (6,7).
In the absence of a satisfactory physiological membrane mechanism, many others prefer to view the action of anesthetics as due to specific effects on proteins, e.g., sodium channels or luciferase (8-10). Since anesthetics act on nerves and the Hodgkin-Huxley theory for the action potential is based on the opening and closing of ion channels, it seems natural to attribute the action of anesthetics to interactions with these channels. Some anesthetics show a stereospecificity indicating that the effective anesthetic concentration (ED^sub 50^) is different for the two chiral forms even though the partition coefficient is not affected to the same degree ( 11 ). In this regard, however, we note that lipid molecules are also chiral. While it is widely believed that local anesthetics are sodium channel blockers, a satisfactory general model of how anesthetics act on proteins is again lacking. The action of anesthetics is still mysterious. Some lipid and protein theories on anesthesia are reviewed in the literature (8,12).
The general absence of specificity and the strong correlation between solubility in lipid membranes and anesthetic action seems to speak against specific binding and a protein mechanism. On the other hand, there is clear evidence that the action of some proteins is influenced by anesthetics. Data on the influence of anesthetics on luciferase and on Na- and K-channels are summarized in Firestone et al. (13) and suggest that the action of lipids and that of proteins are coupled in some simple manner. Cantor has thus proposed that all membrane-soluble substances alter the lateral pressure in the hydrocarbon region and thereby influence the structure of proteins (14-16). Lee proposed a coupling of protein function to the transition temperature of a lipid annulus at the protein interface ( 17). While such mechanisms may provide a control of protein function, it is nevertheless remarkable that all animals are affected to the same degree by anesthetics, suggesting that anesthetic action is largely independent of the specific protein composition of membranes. (see (2), foreword to the English edition.) In addition to their effect on nerves, anesthetics also change membrane properties such as permeability and/or the hemolysis of erythrocytes (5,13). This indicates the need for a more general view of anesthetic action.
In this article, we focus on a thermodynamic description of general anesthesia based on lipid properties. We recognize that this can seem heretical given the dominance of the ion channel picture. Nevertheless, there are a variety of reasons for considering a macroscopic thermodynamic view. The striking fact that noble gases can act as general anesthetics speaks against specific binding to macromolecules. In particular, the Meyer-Overton rule would require all anesthetics to have exactly the same partition coefficient between lipid membrane and protein binding sites for all relevant proteins. It is difficult to imagine that nature provides binding sites for such a variety of molecules on the same protein in precisely such a manner that binding affinity is independent of chemical nature. (It is unlikely that one protein provides binding sites for all anesthetics. Therefore, if a protein picture was to be maintained one has to abandon a unique mechanism for anesthesia (Keith Miller, Harvard Medical School, private communication, 2006.)) An acceptable description should account for this evident lack of specificity, and this suggests the utility of thermodynamic arguments. Moreover, it is to be emphasized that thermodynamics is not inimical to microscopic (e.g., ion-channel) descriptions of the same phenomena. No one would claim, for example, that the manifest successes of thermodynamics in describing the properties of real gases in any way contradict the tact that they are composed of interacting atoms. Thermodynamics rather recognizes that many macroscopic phenomena are independent of such microscopic details and that a large number of microscopic systems can display features, which are bolh qualitatively and quantitatively susceptible to more generic methods. Precisely the absence of detail means that thermodynamic approaches are often capable of making testable quantitative predictions, which are often inaccessible to or obscured by more microscopic models. Thus, we wish to propose a simple thermodynamic explanation of the MeyerOverton rule based on the well-known physical chemical phenomenon of freezing-point depression. We will show that this picture has the benefit of providing an immediate and intuitive picture for the pressure reversal of anesthesia as a consequence of the pressure-induced elevation of the melting point in lipid membranes and can explain the effects of inflammation and divalent cations on anesthetic action.
Lipids were purchased from AvanlJ Polar Lipids (Birmingham, AL) and used without further purification. Ociunol was purchased from Fluka (Buchs. Switzerland). Multilamellar lipid dispersions (5 mM, buffer: 2 mM HEPES, pH 7,4, oclanol concentration adjusted) were prepared by vortexing the lipid dispersions above the phase transition temperature of the lipid. We also performed experiments with halothane and other anesthetics that yielded results similar to those of octanol. These data are not shown here.
EschertcMa coli bacteria (XLl blue with tetracycline resistance) and Bacillus suMlis were grown in a LB-medium at .170C. The bacterial membranes were then disrupted in a French Press al 1200 bar (Gaulin, APV Homogeniser, Lubeck. Germany) and centrifuged at low speed in a desk centrifuge to remove solid impurities. The remaining supernatant was centrifuged at high speed in a Beckman ultracentrifuge (50,000 rpm) in a Ti70 rotor to separate the membranes from soluble proteins and nucleic acids. This membrane fraction was measured in a calorimeter. Lipid melting peaks and protein unfolding can easily be distinguished in pressure calorimelry due to their characteristic pressure dependences. The pressure dependence of lipid transitions is much higher than that of proteins and nearly independent of the lipid or lipid mixture (19). Further, in contrast to lipid transitions, the heat unfolding of the proteins is not reversible. More details regarding the E. ctili measurements are given in an MSc thesis (20) and will be published elsewhere.
Heal capacity profiles were obtained using a VP-scanning calorimeter (MicroCal, Northampton, MA) at scan rates of 5°/h (lipid vesicles) and 30°/h for E. coli membranes.
THEORY AND RESULTS
The unspecific effect of anesthetics and other small solutes on lipid melting transitions
Biological membranes are known to undergo a phase transition from a low-temperature solid-ordered (SO or gel) phase to a liquid-disordered (LD or fluid) phase at temperatures slightly below physiological temperature. This transition involves a volume change of [asymptotically =]4% and an area change of [asymptotically =]25%. Il is also known empirically thai nerve pulses are accompanied by density and heat (23) changes consistent with forcing the lipid mixture through [asymptotically =]85% of this phase transition (24,25). When supplemented by the empirical observation that the sound velocity in lipid mixtures increases with frequency, this fact leads to the robust prediction that localized piezo-electric pulses (or “solitons”) can propagate stably in biological membranes (26,27). The Hpid-melting transition is essential for the existence of solitons. In the transition from the LD to the SO phase, membranes become more compressible and also permeable for ions and molecules (28-30), The biological membrane thus resembles a spring that becomes softer upon compression. This nonlinearity is necessary for the formation of solitons, which can propagate in cylindrical membranes without distortion even in the presence of significant noise. Such a description can account naturally for the reversible heat and mechanical features of nerve pulses and also predicts a pulse propagation velocity of [asymptotically =]100 m/s, which is comparable to that in myelinated nerves.
Given the existence of a lipid phase transition and its possible biological relevance, it is tempting to speculate thai it plays a functional role in unspecific anesthetic effects and that it is central Io understanding the Meyer-Overton rule. The basis for such speculation is elementary. The introduction of any solute {i.e., anesthetic) into membranes leads to a lowering of the temperature of the melting transition which is proportional to the molar concentration of the solute and largely independent of its chemical nature.
The heat capacity at constant pressure, c^sub p^, can be calculated as a function of temperature for various solute concentrations using ideat solution theory (21) with the assumption of complete insolubility in the solid phase (Fig. 1, top). The peak in this figure corresponds to the phase transition. We have assumed a small amount of the water phase (as used experimentally) and an accumulation of anesthetics in the fluid phase. This leads to the broadening of the profiles, which are remarkably similarity to experimental results obtained for DPPC vesicles in the presence of various anesthetic concentrations as shown in the lower panel. The quantitative agreement between the experimentally obtained heat capacity profiles in the presence of anesthetics and those calculated justifies the assumptions made and supports the overall notion that anesthetics change the thermodynamic properties of membranes in a simple manner. We will make use of this fact below.
The hydrostatic pressure required to reverse the action of anesthetics on the phase transition is 9.6 bar/mol % using the values of ?H and T^sub m^, appropriate for DPPC.
Pressure reversal of anesthesia was first demonstrated by Johnson and Flagler (34). They anesthetized tadpoles in 3-6 vol % ethanol. A hydrostatic pressure of 140-350 bars was found to reverse anesthesia. According to Firestone et al. ( 13), 190 mM of ethanol (1.1 vol %) in the aqueous phase is necessary for tadpole narcosis. This means that ~3-6 times the anesthetic ethanol concentration was used in Johnson and Plagier (34). The concentration of ethanol in the membrane in Johnson and Flagler’s experiments was therefore 7.5-15 mol %. According to Eq. 3, these concentrations correspond to lowering T^sub m^ by 1.8-3.6 K. From Eq. 9, the pressure necessary to reverse this anesthetic effect is 72-148 bars. Considering the uncertainty of the partition coefficient for real biological membranes (which depends on the precise lipid mixture), this is remarkably close to the order of the values found by Johnson and Plagier (34). The fact the pressure increases T^sub m^ may be related to the observation that nerves fire spontaneously at high pressures (40).
Effects of pH and salts
Ions also change the free energy. Some 10% of the lipids of biological membranes are negatively charged, primarily on the inner membrane. At lower pH, some of these charges are protonated, and the electrostatic potential of the lipid membrane is reduced. Complete protonation increases the melting temperature by ~20 K. The effects of pH and ionic strength on melting transitions have been carefully investigated by the literature (41,42). While these effects depend on the precise composition of the membrane and on ionic strength, they can be calculated using Debye-Hückel theory or determined empirically. For example, the temperature of the melting transition in native E. coli membranes (in the pH range between 5 and 9) is raised by ~1.8° if pH is lowered by one unit (Fig. 2). This shift is approximately that which is produced by 72 bars hydrostatic pressure. Interestingly, it is known that inflammation leads to the failure of anesthesia. The related lowering of pH in inflamed tissue, i.e., on the order of 0.5 pH units (43), is widely assumed to be responsible. According to the above, the lowering of pH from 7 to 6.5 leads to ?T^sub m^ = +0.9 K, which is sufficient to reverse the action of anesthetics at the typical critical dose corresponding to ?T^sub m^ = -0.6 K.
Salts can also effect the melting transition through, e.g., the binding of divalent cations such as Mg^sup 2+^ and Ca^sup 2+^. These ions shift the melting temperatures of both charged and uncharged lipids to higher temperatures. The presence of such ions thus lowers the effectiveness of anesthetics, and appropriate functions of pH and salt concentration should be added to the right side of Eq. 8.
Temperature effects
The effect of 2.6 mol % anesthetics is thus reversed by a 0.6 K reduction of the body temperature for the parameters of DPPC membranes. Interestingly, a well-known finding in clinical anesthesiology is hypothermia, i.e., the lowering of body-temperature during narcosis (46). This decrease partially compensates the effect of the transition temperature shift caused by anesthesia and suggests that the body tries to maintain a constant membrane state. Conversely, the same arguments say an equal rise in body temperature, e.g., by fever, should produce the same effects as a critical anesthetic does. Since this is not the case, a rise in body temperature must be accompanied by other thermodynamic changes which tend to counter this increase in the free energy difference (e.g., pH changes) if our thermodynamic picture is to be maintained. Further, a lowering of the temperature below the phase transition temperature (?T > -15 K) would lead to a complete cessation of nerve activity as found in clinical experiments (47). Note that the chemical composition of lipids can also change in response to changes in other thermodynamic variables. It is well documented that the lipid composition and the melting temperatures of bacterial membranes change as a response to changing growth temperature (e.g., (48). E. colt membranes grown at different temperature shift their melting temperatures to maintain a constant distance to growth temperature (unpublished data from our laboratory)).
CONCLUSION
We have proposed an elementary thermodynamic description of the action of general anesthesia according to which constant anesthetic effects are predicted whenever external thermodynamic variables (e.g., solute concentration, pressure, temperature, pH, and salt concentration) are adjusted to maintain constant values of the free energy difference between the liquid and gel phases of lipid membranes. Indeed, the effect of an anesthetic is intimately connected with its ability to depress the melting point of lipid membranes, which depends on its solubility in lipid mixtures but is otherwise independent of its chemical nature. The basis for the familiar Meyer-Overton rule thus lies in the thermodynamics of biological membranes in general and the properties of the lipid phase transition in particular. The lowering of the membrane melting point results in a change of the free energy of the lipid membrane, which is proportional to the difference between body temperature and the melting temperature of the membrane. This temperature difference, which is on the order of 15 K, is to be compared with the shift in melting point temperature of [asymptotically =]-0.6 K at a typical critical anesthetic dose. Anesthetic effect can be reversed in a quantitatively predictable manner by any mechanism that raises the transition temperature and restores the free energy difference to its original value. Such mechanisms include hydrostatic pressure, a decrease of pH, an increase of calcium concentration, or the lowering of the body temperature. (The hydrostatic pressure necessary to reverse anesthesia is on the order of 24 bars, the pH change on the order of 0.4 pH units, and the hypothermie reversal of anesthesia is ~0.6 K.) While these effects are well-documented, they have not previously been placed in common framework. Although we do not question lhe importance of a better understanding of the microscopic mechanisms underlying general anesthesia, these results support the view that the thermodynamics of the lipid liquid-gel transition is important for understanding the macroscopic effects of general anesthetic action. Finally, we note that a variety of biological phenomena, including fusion and membrane permeability, may reasonably be assumed to have a similar connection to this phase transition and that such assumptions can be tested using approaches similar to those presented here.
Voting to Bell the Thermodynamic Cat
here is an eminently simple solution to the purported greenhouse gas problem: abolish the First and Second Laws of Thermodynamics. This would allow the construction of perpetual-motion machines and the generation of electricity by air conditioners (since the heat could easily produce shaft work as it moves spontaneously from the cold to the hot reservoir). The United States could then sign the Kyoto Global Warming Treaty, because our power plants would not emit any carbon dioxide whatsoever.
They would, in fact, convert carbon dioxide and water into coal and oil. Who needs trees and rain forests as carbon sinks once you’ve repealed the laws of thermodynamics? That’s an important consideration, because South American countries that are among Kyoto’s most vocal advocates are burning down rain forests to clear land for agriculture. This suggests that they don’t really take the global warming “threat” very seriously.
Brian Hogan’s “Wake Up, Brie Eaters!”paints a vivid picture of people whose technological and economic ignorance is so profound that we could probably keep them busy with this project for quite a while. All we need to do is float the idea at a few wine-and-brie parties and let the momentum build. The legislative abolition of the laws of thermodynamics would occupy the Kyotoist politicians, and prevent them from doing real damage by regulating carbon dioxide emissions.
The Kyotoists say we can reduce our emissions by making our manufacturing and generation processes “more efficient.” Perhaps they know how to reduce the amount of electricity that is necessary to separate a pound of aluminum from bauxite, or a pound of chlorine from brine. In 2002, high retail energy prices in California actually induced aluminum plants to stop separating aluminum from bauxite and, instead, resell electricity for which they had long-term contracts.
Henry Ford wrote that inexpensive transportation and energy were his key considerations in locating a manufacturing plant and the high-wage jobs that went with it. He didn’t worry about wages. A competitor’s cheap labor advantage could be overcome by lean manufacturing, which the Ford Motor Company developed. Ford would not, however, build a plant where inexpensive power and reliable transportation were not available.
Nothing has happened to change this, as shown by the wholesale destruction of jobs in California because of costly and unreliable energy. “The New Job Reality” (U.S. News and World Report, August 11 2003) shows, in fact, that the country lost more than two million manufacturing jobs during the past two years. The Kyotoists, however, want carbon dioxide regulations that will make power even more costly and unreliable. They are like blackjack players with 22 or more points who tell the dealer to hit them again.
The “Green” movement thinks solar and wind power will solve the world’s problems. The wind and sun are free. What is the net present value of free electricity (discounted by the required rate of return) for the life of the project minus the necessary capital investment? The ability to do this calculation was once a requirement for getting a professional engineer’s license, but it was obviously never a prerequisite for getting into wine-and-brie parties.
Free market forces are already inducing companies to invest in fuel cells, which circumvent the efficiency limits of the Rankine and Otto power cycles by turning chemical energy directly into electricity. As an example, the reaction of coal with steam produces hydrogen that will work in fuel cells. Scrubbing and sequestration of the carbon dioxide, however, adds no value for the electricity customer. There is no reason to do this until the rest of the world’s actions, as opposed to lip service, show that it thinks global warming is a real problem.
There are exactly three ways to produce real wealth: grow it, mine it, or make it. Anything that undermines manufacturing, including misguided efforts to make power more expensive, is a danger to the security of the United States. There is nothing about “ship jobs offshore” that blue-collar workers do not understand, but it is our duty as manufacturing professionals to educate them about the relationship between cheap and reliable energy and manufacturing.
Thermodynamics
Thermodynamics is the study of the relationships between heat, work, and energy. Though rooted in physics, it has a clear application to chemistry, biology, and other sciences: in a sense, physical life itself can be described as a continual thermodynamic cycle of transformations between heat and energy. But these transformations are never perfectly efficient, as the second law of thermodynamics shows. Nor is it possible to get "something for nothing; as the first law of thermodynamics demonstrates: the work output of a system can never be greater than the net energy input. These laws disappointed hopeful industrialists of the early nineteenth century, many of whom believed it might be possible to create a perpetual motion machine. Yet the laws of thermodynamics did make possible such highly useful creations as the internal combustion engine and the refrigerator.
Machines were, by definition, the focal point of the Industrial Revolution, which began in England during the late eighteenth and early nineteenth centuries. One of the central preoccupations of both scientists and industrialists thus became the efficiency of those machines: the ratio of output to input. The more output that could be produced with a given input, the greater the production, and the greater the economic advantage to the industrialists and society as a whole.
At that time, scientists and captains of industry still believed in the possibility of a perpetual motion machine: a device that, upon receiving an initial input of energy, would continue to operate indefinitely without further input. As it emerged that work could be converted into heat, a form of energy, it began to seem possible that heat could be converted directly back into work, thus making possible the operation of a perfectly reversible perpetual motion machine. Unfortunately, the laws of thermodynamics dashed all those dreams.
SNOW'S EXPLANATION.
Some texts identify two laws of thermodynamics, while others add a third. For these laws, which will be discussed in detail below, British writer and scientist C. P. Snow (1905-1980) offered a witty, nontechnical explanation. In a 1959 lecture published as The Two Cultures and the Scientific Revolution, Snow compared the effort to transform heat into energy, and energy back into heat again, as a sort of game.
The first law of thermodynamics, in Snow's version, teaches that the game is impossible to win. Because energy is conserved, and thus, its quantities throughout the universe are always the same, one cannot get "something for nothing" by extracting more energy than one put into a machine.
The second law, as Snow explained it, offers an even more gloomy prognosis: not only is it impossible to win in the game of energy-work exchanges, one cannot so much as break even. Though energy is conserved, that does not mean the energy is conserved within the machine where it is used: mechanical systems tend toward increasing disorder, and therefore, it is impossible A WOMAN WITH A SUNBURNED NOSE . S UNBURNS ARE CAUSED BY THE S UN ' S ULTRAVIOLET RAYS . for the machine even to return to the original level of energy.
The third law, discovered in 1905, seems to offer a possibility of escape from the conditions imposed in the second law: at the temperature of absolute zero, this tendency toward breakdown drops to a level of zero as well. But the third law only proves that absolute zero cannot be attained: hence, Snow's third observation, that it is impossible to step outside the boundaries of this unwinnable heat-energy transformation game.
W ORK AND E NERGY
Work and energy, discussed at length elsewhere in this volume, are closely related. Work is the exertion of force over a given distance to displace or move an object. It is thus the product of force and distance exerted in the same direction. Energy is the ability to accomplish work.
There are many manifestations of energy, including one of principal concern in the present context: thermal or heat energy. Other manifestations include electromagnetic (sometimes divided into electrical and magnetic), sound, chemical, and nuclear energy. All these, however, can be described in terms of mechanical energy, which is the sum of potential;the energy that an object has due to its position—and kinetic energy, or the energy an object possesses by virtue of its motion.
MECHANICAL ENERGY.
Kinetic energy relates to heat more clearly than does potential energy, discussed below; however, it is hard to discuss the one without the other. To use a simple example—one involving mechanical energy in a gravitational field—when a stone is held over the edge of a cliff, it has potential energy. Its potential energy is equal to its weight multiplied by its height above the bottom of the canyon below. Once it is dropped, it acquires kinetic energy, which is the same as one-half its mass multiplied by the square of its velocity.
Just before it hits bottom, the stone's kinetic energy will be at a maximum, and its potential energy will be at a minimum. At no point can the value of its kinetic energy exceed the value of the potential energy it possessed before it fell: the mechanical energy, or the sum of kinetic and potential energy, will always be the same, though the relative values of kinetic and potential energy may change.
CONSERVATION OF ENERGY.
What mechanical energy does the stone possess after it comes to rest at the bottom of the canyon? In terms of the system of the stone dropping from the cliffside to the bottom, none. Or, to put it another way, the stone has just as much mechanical energy as it did at the very beginning. Before it was picked up and held over the side of the cliff, thus giving it potential energy, it was presumably sitting on the ground away from the edge of the cliff. Therefore, it lacked potential energy, inasmuch as it could not be "dropped" from the ground.
If the stone's mechanical energy—at least in relation to the system of height between the cliff and the bottom—has dropped to zero, where did it go? A number of places. When it hit, the stone transferred energy to the ground, manifested as heat. It also made a sound when it landed, and this also used up some of its energy. The stone itself lost energy, but the total energy in the universe was unaffected: the energy simply left the stone and went to other places. This is an example of the conservation of energy, which is closely tied to the first law of thermodynamics.
But does the stone possess any energy at the bottom of the canyon? Absolutely. For one thing, its mass gives it an energy, known as mass or rest energy, that dwarfs the mechanical energy in the system of the stone dropping off the cliff. (Mass energy is the other major form of energy, aside from kinetic and potential, but at speeds well below that of light, it is released in quantities that are virtually negligible.) The stone may have electromagnetic potential energy as well; and of course, if someone picks it up again, it will have gravitational potential energy. Most important to the present discussion, however, is its internal kinetic energy, the result of vibration among the molecules inside the stone.
H EAT AND T EMPERATURE
Thermal energy, or the energy of heat, is really a form of kinetic energy between particles at the atomic or molecular level: the greater the movement of these particles, the greater the thermal energy. Heat itself is internal thermal energy that flows from one body of matter to another. It is not the same as the energy contained in a system—that is, the internal thermal energy of the system. Rather than being "energy-in-residence," heat is "energy-in-transit."
This may be a little hard to comprehend, but it can be explained in terms of the stone-and-cliff kinetic energy illustration used above. Just as a system can have no kinetic energy unless something is moving within it, heat exists only when energy is being transferred. In the above illustration of mechanical energy, when the stone was sitting on the ground at the top of the cliff, it was analogous to a particle of internal energy in body A. When, at the end, it was again on the ground—only this time at the bottom of the canyon—it was the same as a particle of internal energy that has transferred to body B. In between, however, as it was falling from one to the other, it was equivalent to a unit of heat.
In everyday life, people think they know what temperature is: a measure of heat and cold. This is wrong for two reasons: first, as discussed below, there is no such thing as "cold"—only an absence of heat. So, then, is temperature a measure of heat? Wrong again.
Imagine two objects, one of mass M and the other with a mass twice as great, or 2 M. Both have a certain temperature, and the question is, how much heat will be required to raise their temperature by equal amounts? The answer is that the object of mass 2 M requires twice as much heat to raise its temperature the same amount. Therefore, temperature cannot possibly be a measure of heat.
What temperature does indicate is the direction of internal energy flow between bodies, and the average molecular kinetic energy in transit between those bodies. More simply, though a bit less precisely, it can be defined as a measure of heat differences. (As for the means by which a thermometer indicates temperature, that is beyond the parameters of the subject at hand; it is discussed elsewhere in this volume, in the context of thermal expansion.)
MEASURING TEMPERATURE AND HEAT.
Temperature, of course, can be measured either by the Fahrenheit or Centigrade scales familiar in everyday life. Another temperature scale of relevance to the present discussion is the Kelvin scale, established by William Thomson, Lord Kelvin (1824-1907).