J. Phys. Chem. 1995, 99, 3406-3410
3406
Membrane Lipid Alkyl Chain Heterogeneity Is a Molecular Transform of the Available Thermal Energy Spectrum Rawle I. Hollingsworth Departments of Chemistry and Biochemistry, Michigan State University, East Lansing, Michigan 48824 Received: September I , 1994; In Final Form: November 23, 1994@
The alkyl chain length distribution of a common gram positive bacterium has been examined from the perspective of their intrinsic longitudinal accordian mode frequencies and their potential roles in coupling the thermal energies of cellular processes to the available ambient thermal energy spectrum. The alkyl chain length distribution is such that the corresponding longitudinal accordian mode energies are normally distributed about kT at the growth temperature of the bacteria. The relationship between this distribution and the Maxwellian distribution of energies of molecules indicates that membranes of living systems may be capable of dynamically tuning their low-frequency vibrational modes to interface the cell with its surroundings at the maximum of the available kinetic energy curve. Isotope perturbation studies and analysis of previous published studies on the effect of temperature on membrane alkyl chain length were used to support the latter observation. These results also suggest that biological membranes can control the dynamics of macromolecules inside of the cell since the broad spectrum of translational quanta from the stochastic collisions of the solvent molecules outside of the cell are converted into highly tailored, specific, oriented waves on passing through the membrane into the cell.
1. Introduction Both the bilayer structure and fatty acid heterogeneity are accepted features of membranes of all living systems. Despite this, there is no understanding of how either feature influences or controls energy flow into and out of the cell. From an analytical standpoint, alkyl chain heterogeneity has always been perceived as a nuisance which presents insurmountableobstacles in the structural chemistry of membranes. Since these structures form an enclosed boundary around cells, it is important to know the consequences of this arrangement. Heat or infrared quanta are propagated by molecular vibrations. In biological systems, water plays an important role in propogating such energies through collisions with other molecular species or through solvent lattice vibrations. These quanta interact with any and all molecular species through vibrational modes, and the resulting spectrum of frequencies which is eventually passed is very dependent on the molecular structures with which the quanta interact on the way. Membranes therefore, because of their spatial location around the cell, have the unique opportunity to control the thermal energy spectrum which enters the cell. They have the opportunity to amplify specific infrared quanta and promote specific vibrational modes of macromolecules such as nucleic acids and proteins and hence regulate their activity in a very dynamic fashion. Here we address the question of whether they may actually do this. After all, membranes are found universally in all living cells and couple living systems to their energy source. Is there some special structural feature(s) which qualifies membranes to be the universal energy interface? The answer to this question can, perhaps, be gleaned by defining what special attributes the perfect molecular energy interface should have. From the standpoint of heat transfer, membranes should have the capacity to pass and to condense (focus) infrared quanta of the magnitude of “kT’ over an appropriate distribution function. This would require a molecular vibrational mechanism capable of supporting a very high density of vibrational states
* Abstract published in Advance ACS Abstracts, February 1, 1995. 0022-365419512099-3406$09.00/0
of magnitude kT. Given the fact that biomolecules are largely composed of light atoms which typically give rise to vibrational energies which are much greater than kT, a coupled or collective mode of vibration is necessary. This requirement could be met by a collection of linearly-coupled oscillators with a collective longitudinalmode equal to kT. The distribution of chain lengths of independent oscillators could then be such that the energyfrequency distribution could be tailored to reflect the energy spectrum to be propagated. The long axes of the oscillators would have to be oriented so as to effectively focus the vibrational quanta into the interior of the cell. The vast resources of theory and measurement support the idea that membrane alkyl chains are the biological, molecular structures which are uniquely suited to this function.
2. Determination of Alkyl Chain Frequencies The normal-mode analysis of the behavior of a system of linearly-coupled equal masses joined by equivalent springs of equal force constant (the linearly coupled harmonic oscillator problem) is a well-studied phenomenon in physics. This treatment forms the basis for the calculation of the longitudinal accordian modes (LAMS) of extended hydrocarbon It is now clear that the frequencies of these modes are very sensitive to the physical state of the alkyl chains. For crystalline systems the frequencies are called LAM-k modes, and for disordered systems they are called d-LAM modes.3 These latter modes are especially relevant to membranes since they (membranes) are essentially fluid with small islands of crystallinity (domains) under physiological conditions. In these analytical treatments, the frequencies of interest for hydrocarbon chains are those which result in expansion and reduction of the valence angle of the carbon-carbon bonds along the hydrocarbon chain in addition to the increase and reduction of the actual bond length. This longitudinal expansion and contraction gives rise to a series of acoustical and optical modes which are, in effect, standing waves (Figure 1). It is the Raman-active acoustical branch that most interests us in the present context. In the case of the LAM-k modes, the frequencies of these collective 0 1995 American Chemical Society
Membrane Lipid Alkyl Chain Heterogeneity
J. Phys. Chem., Vol. 99, No. 10, 1995 3407 treatment of the effect on LAM frequencies of weighting the ends of alkyl chains has been carried out by modifying the elastic rod method to include terms for perturbing forces and inertial effects at the ends.1° These calculations also indicate that there should be only a slight shift in LAM-k frequencies of lipid alkyl chains compared to the frequencies of the corresponding hydrocarbon. The frequencies of d-LAM modes for phospholipids compare well with those of the free fatty acids and alkane^.^
3. Experimental Determination of Chain Length Distribution
Figure 1. Conceptual model explaining the longitudinalexpansion and contraction of an extended alkyl chain over time. The ends of the chains
undergo a time-dependent sinusoidal displacement with the LAM frequency. vibrations can be conveniently (and quite accurately) calculated from models in which the extended chains are treated as continuous extended rods.4 If the Young’s elastic modulus (E), the length (L), and the density (e) are known, the vibrational frequency (v) of the vibration of order rn is given (5) by the relation
v = (m/2L)(El~)”~ The frequency of any given order vibration is, therefore, inversely proportional to chain length and approaches zero for very long chains. An extensive collection of calculated and measured frequencies of these bands for hydrocarbon chains of varying lengths can be found in the l i t e r a t ~ r e . ~The - ~ measured and calculated frequencies of the fundamental LAM-k mode for an 18-carbon hydrocarbon chain are 131 and 132.5 cm-’, respectively.6 This represents only a 1% difference. An even more reliable simple chain model which gives excellent results for alkanes as well as helical polymers has since been described? Unlike the frequencies of the LAM-k modes of crystalline hydrocarbon chains, those of the disordered (d-LAM) modes tend toward a specific nonzero value as chain length increases. At room temperature, this value is approximately 200 cm-1.8 The relationship between chain length and frequency obeys the relationship
v = vo 4-B/n2 where v, is the limiting frequency at infinite chain length, n is the number of carbon atoms in the chain, and B is a parameter which is dependent on the energy difference between the gauche and trans configurations of the carbon-carbon bonds.3 The agreement between measured and calculated values is excellent.3%8 Alkyl chains in biological membranes share several common features with disordered chains in liquid hydrocarbons and noncrystalline domains in polyethylene. The major difference between liquid hydrocarbon alkyl chains and those in fluid membranes is that, in the latter, one end of each chain is fixed to the lipid headgroup. Raman spectroscopy measurements show that weighting the end of an alkyl chain has only a small lowering effect 6n LAM frequencies. Measurements were made comparing LAM frequencies for free hydocarbons, long chain alkanoic acids, and lipid^.^ A rigorous generalized analytical
The biological consequences of membrane bilayer structure and of the existence of LAM vibrations in biological systems were studied using the gram positive bacterium Bacillus subtilis as a model system. The distribution of alkyl chain lengths in this organism, when cultured at 37 “C, was determined by gas chromatography and mass spectrometry. The gas chromatograph was fitted with a flame ionization detector, thus allowing the exact relative masses of each component to be determined. From these relative masses, the relative number of moles of each alkyl chain type (and hence the relative number of oscillators of each type) was determined. This experiment was also performed using the same strain of bacterium after adapting it to and culturing it in 99.9% deuterium oxide.
4. Results and Discussion The gas chromatograph which gives the fatty acid alkyl chain length distribution of Bacillus subtilis at 37 “C is shown in Figure 2A. The chromatograph obtained from membranes of cells grown in 99.9% deuterium oxide is shown in Figure 2B. Culturing cells in deuterium oxide was done as a means of perturbing the energetics of the membrane by increasing the resistance of the solvation shell of the lipid head groups to structural reorganization through reducing the ease of hydrogen bond breakage. There is expected to be some influence on these vibrational transitions involving membrane chains by solvent shell hydrogen bond reorganization. One outstanding feature of the fatty acid distribution observed for this and other bacteria we have studied is the very regular (mathematically speaking) distribution. There appears to be a definite, smooth relationship between the abundance of the different species and their chain lengths. This distribution is shifted toward shorter chain lengths when the bacteria are cultured in deuterium oxide. This regular relationship between alkyl chain length and their relative abundance appears to be a common feature in gram positive bacteria. The frequencies of the d-LAM vibrations were calculated for all of the observed alkyl chain species for completeness. Measured values from the literature were also obtained and were actually used in the subseqent analyses. In all cases, there was excellent agreement between the measured and calculated freq~encies.~.~ The conversion of the chain lengths to frequencies immediately yielded one very interesting result: the most abundant chain length corresponded to a frequency of 215.4 cm-’ or “kT’ (in the spectroscopic unit of wavenumbers) at 37 “C. The energies were then distributed about this value. One interesting analysis (Figure 3) is the plot of the relative energies in each mode vs the energy of that mode (NiEJZVi vs Ei ) where subscript i refers to the ith mode, N denotes the number of vibrators, and E is the energy. The ratio of the number of vibrators of each type to total number is readily obtained by converting the gas chromatographic intensities to molar quantities by dividing the area of each peak by the molecular weight of the corresponding species. The ratio of a particular value to
Hollingsworth
3408 J. Phys. Chem., Vol. 99, No. 10, 1995
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Figure 2. (A) Gas chromatography profile of the methyl ester derivatives of the total membrane fatty acids in Bncillus subrilis strain PY79 cultured in liquid medium. The components were detected by a flame ionization detector which has essentially the same response factor for all long-chain fatty acids. This gives the relative mass of each long-chain acid component. The conversion to relative number of moles (and number of oscillators) is done by dividing the area for each component by the corresponding molecular weight. (B) The profile when 99.9% deuterium
oxide is used in the medium. Note the increase in the relative amounts of the C-16 acids. The d-LAM frequencies for C-16 saturated and unsaturated fatty acids are very similar. The analyses described here were carried out for the saturated acids. Essentially the same trend is followed by the unsaturated acids.
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Figure 3. Plot showing the contribution of each oscillator to the total energy in the oscillator system. Note the shape of the distribution and that the greatest contribution is by the C-16 chains, for which the energy in the d-LAM mode is equal to kT.
the total values thus obtained is an accurate measure of N,ENi. This is possible because the flame ionization detector is a total mass detector. It can be clearly seen from this analysis that the distribution is a smooth function which drops sharply at low energies and tends (apparently asymptotically) toward zero at high frequencies. This is reminiscent of the distribution of
molecular energies and speeds as described by the Maxwell distribution law. The connection which is important here is that the spectrum of energies which gives rise to the distribution of alkyl chain lengths should share a similar distribution to that which is eventually encoded in the structures of the chains. This energy distribution should have a low-frequency cutoff at energies which are not high enough to cause vibrations because of the membrane solvation shell. A more direct comparison between the energy distribution found here and the Maxwell distribution law can be obtained if one were to plot only the probability of an energy vs the corresponding energy (NiENi vs EJ and to use the fact that the formalism of the Maxwell distribution law is essentially the same as that of the Gaussian or normal density function (Figure 4A). This is possible because here the frequency values are discrete and not continuous and E, the mean value of E (or p in the usual mathematical formalism), is simply given by the relation
Here AE;) = NiENi or the probability of a vibration having the energy E,. One can then use the actual probabilities instead of probability densities as are used in the usual continuous distribution formalism of the Maxwell equation. This is essentially the same as saying that 6E and 6 N are approaching zero in the probability density function. This is a curious incidence in which we have used the envelope of a discrete
J. Phys. Chem., Vol. 99, No. 10, 1995 3409
Membrane Lipid Alkyl Chain Heterogeneity 0.9
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the increments are vanishingly small. The taller more narrow curve is from data obtained from cells grown in the presence of deuterium oxide. probability function and treated it as a continuous function. The underlying reason for this is that the molecular basis for the quantization is actually the alkyl chain length. Alkyl chains in membranes are synthesized in two-carbon units. A C-16 chain, for instance, is made up of eight such units. The real underlying statistics, therefore, is goverened by a (stochastic) Poisson process since the real variable is really a positive integer. What we are, in fact, measuring or calculating as oscillator frequencies is the result of the filtering and coloring of this very defined, quantized stochastic process by uncontrollable physical effects, such as the uncertainty principle, which limits the accuracy with which we can define the frequency. The validity of the proposed mathematical form of the distribution could be verified by making a point estimate of the expected probability of a given energy and comparing it to the actual measured probability. This was done for the energy corresponding to the most abundant chain length species, the C-16 alkyl chain. This required the knowledge of the mean energy (the energy at the maximum on the curve) and the standard distribution parameter u from the data. This latter value was calculated from the experimentaldata and from points falling on the distribution curve. The point estimate was made from a table of calculated values of the integral of the Gaussian density function using the standard reduced equation for the integral which has the following form, where the parameters have their usual meaning:
F(x> = @((x - pya) = @(z) The calculated value of u,based on the experimental data and by sampling points on the curve, was 0.080, and the calculated value of z was 0.125, giving a value for @(z), and hence F(x), of 0.54. This value agreed exactly with the experimental value of 0.542 for the probability of occurence of the C-16 chain. It is important to note the connection between the parameter u in the Gaussian density function and “kT’ in the Maxwell distribution law. There is a direct correspondence between a* and kT. As temperature decreases, the value of kT decreases, and the distribution of speeds and momenta described by Maxwell’s distribution laws becomes more narrow and the height of the curve increases. This is also true of the effect of o2 (or the variance) of the Gaussian density function on the
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Figure 5. Plot of probability distribution at 25 and 37 “C for Clostridium butyricum membrane chains.
value of F(x). Thus, culturing the same bacteria in deuterium oxide leads to a reduction in the spread of energies in much the same fashion as a reduction in growth temperature (a lowering of kT). The number of chains with the most probable energy also increased (Figure 4). This is quite consistent with the notion that cellular energetics should be slowed down in the presence of deuterium oxide since hydrogen bond breakage is less likely to occur, thus reducing the frequency of solvent collisions. The lower zero-point energy of vibrations involving deuterium, compared to the same vibrations involving hydrogen, effectively increases activation energy, resulting in a narrower, more limited distribution. This is the basis of the primary kinetic isotope effect. The connection is made more clear if one plots the cumulative frequency distribution as a function of energy and compares its form to that of the well-known cumulative plot of the Maxwell distribution law. The characteristic sigmoidal distribution (not shown) is very evident from inspection of the curve in Figure 4. The gradient at lower energies is positive and tends toward some very large number as energy approaches some minimum threshold. With increasing E, it passes through zero at E = kT (the equipartition value of one vibrational energy mode in one dimension) and then goes to a minimum negative value and increases again, finally becoming zero and remaining there as E increases further. Thermal (translational) energy is, therefore, being trapped in and propagated by vibrational modes which have energies which are narrowly distributed about the energy at the probability maximum of the kinetic energy distribution curve. The explanation of the results of the deuterium isotope study and the proposed connection between “kT’,u,and alkyl chain length distribution was supported by the analysis of published results in which the distribution of membrane alkyl chain lengths, in a gram positive bacterium, was measured as a function of temperature. Although experiments along similar lines performed in our laboratories gave the same indications, the prior work of others (11) is presented here because it represents an unbiased, independent source for validating the critical point of this study. This compositional study was performed on Clostridium butyricum at three distinct temperatures, 25, 30, and 37 “C. The probability distribution curves for the lowest and highest temperatures are shown in Figure 5. The curve for the intermediate temperature fell between the other two. The abundances of all chains of a particular length were summed since, as pointed out earlier, functionalization of an alkyl chain by single methyl, cyclopropyl, or alkenyl groups does not result in a significant difference of its d-LAM
3410 J. Phys. Chem., Vol. 99, No. IO, 1995 frequency. The same shift to higher frequency and the narrowing of the distribution curve, observed in the deuterium isotope study, are, as predicted, also observed at lower temperatures. This was a very general result from all of the temperature dependence studies we found in the published literature. Biological membranes, then, select a very narrow range over which vibrational energy is transmitted into the cell. In the case of this and and the vast majority of microorganisms studied thus far, this range is dynamically tuned to the value of “kT’. More specific frequencies with energies of magnitudes much lower than kT should also be transmitted via the LAM-k modes of the dynamic, crystalline lipid domains known to exist inside of membranes. This would lead to the activation of specific vibrational modes of macromolecules inside of the cell and explain why perturbations of membrane structure and dynamics have such dramatic cellular consequences. The cell can, therefore, be envisioned as a cavity or enclosure into which is pumped energy from its walls which is made up of an array of oscillators with frequencies with a precisely defined distribution. Perhaps the most striking conclusion which can be made from this study is that this distribution seems to be determined by a molecular transform or encodement of the available thermal energy spectrum and that organisms might have the capacity to “tune in” at the peak of this energy spectrum and follow it
Hollingsworth dynamically. If this were true, it would represent an extremely powerful adaptative mechanism which would explain how living systems interact with their heat source. The presence of these collective modes would also explain why membranes appear to be the universal interface through which matter dynamically couples to energy to maintain life.
Acknowledgment. This work was supported by Grant DEFG09-89ER14029 from the U.S. Department of Energy and by the Michigan State University Center for Microbial Ecology, a NSF. Science and Technology Center (BIR 912-0006). References and Notes (1) (2) 19, 85. (3) (4) (5) (6) (7)
Kirkwood, J. G. J. Chem. Phys. 1939, 7, 506. Snyder, R. G.; Schachschneider, R. G. Specrrochim. Acta 1963,
Snyder, R. G. J. Chem. Phys. 1982, 76, 3921. Hsu, S. L., Krimm, S. J. Appl. Phys. 1976, 47, 4265. Mizushima, S., Simanouti, T. J. Am. Chem. SOC.1949, 71, 1320. Peticolas, W. L. Biopolymers 1979, 18, 747. Schaufele, R. F.; Shimanouchi, T. J. Chem. Phys. 1967,47,3605. (8) Schaufele, R. F. J. Chem. Phys. 1968, 49, 4168. (9) Lippert, J. L. Peticolas, W. L. Biochim. Biophys. Acta. 1972, 282,
8. (10) Hsu, S. L.; Ford, G. W.; Krimm, S. J. Polym. Sci. 1977, 15, 1769. (11) Khuller, G. K.; Goldfine, H. J. Lipid. Res. 1974, 15, 500. JF’942358L
