- Email: [email protected]
Shape anisotropy of lipid molecules and Voids
J. theor. Biol. (2003) 220, 189–199 doi:10.1006/jtbi.2003.3155, available online at http://www.idealibrary.com on
Shape anisotropy of lipid molecules and Voids Gauri R. Pradhanwz, Sagar A. Pandit*y, Anil D. GangalwO andV. SitaramamzO wDepartment of Physics, University of Pune, Pune 411 007, India yPhysical Research Laboratory, Ahmedabad 380 009, India and zDepartment of Biotechnology, University of Pune, Pune 411 007, India (Received on 13 May 2002, Accepted in revised form on 22 August 2002)
Biological polymers, viz., proteins, membranes and micelles exhibit structural discontinuities in terms of spaces unfilled by the polymeric phase, termed voids. These voids exhibit dynamics and lead to interesting properties which are experimentally demonstrable. In the specific case of phospholipid membranes, numerical simulations on a two-dimensional model system showed that voids are induced primarily due to the shape anisotropy in binary mixtures of interacting disks. The results offer a minimal description required to explain the unusually large permeation seen in liposomes made up of specific lipid mixtures (Mathai & Sitaramam, 1994). The results are of wider interest, voids being ubiquitous in biopolymers. r 2003 Elsevier Science Ltd. All rights reserved.
Introduction A variety of lipid molecules contribute to the structure and barrier function of biological membranes. The amphiphilic nature of lipid molecules leads to a bilayer structure in the presence of water in which hydrophilic head groups have maximum contact with water and hydrophobic tails have minimum contact with water. The process of membrane formation is one of minimizing the free energy and maximizing the stability of the structure (Israelachvili et al., 1980; Stein, 1986). A comprehensive description of diffusion through biomembranes remained elusive for n Corresponding author. Current address: Department of Chemistry, CB# 3290, University of North Carolina, Chapel Hill, NC 27599, U.S.A. E-mail addresses: [email protected] (G.R. Pradhan), [email protected] (S.A. Pandit), [email protected] (A.D. Gangal), [email protected] (V. Sitaramam). zThe author is thankful to DBT (India) and CSIR (India). O The work was funded by DBT (India).
0022-5193/03/$35.00
many reasons. In terms of a non-interactive diffusing molecule, the simplest of the descriptions involved Fick’s law, which takes into account a relevant partition coefficient (e.g. water:oil) for the diffusing molecule. More complex models involved introduction of frictional terms to account for remarkably nonlinear diffusion with respect to the mass of the solutes leading to extraordinary sieving capabilities of these bilayers (Stein, 1986). Both essentially deal with homogeneous models without or with pores. In traditional models of diffusion, the square of the distance varies linearly with time (i.e. x2 pt). However, in real structured media characteristically described as fractal diffusive traps, the heterogeneity plays a central role such that x2 varies as ta (where aa1) defining sub(super)-diffusive behavior. Formal heterogeneous models involving sub-diffusive behavior or diffusion in the presence of traps is a somewhat nascent area for immediate applicability, though in the right direction (Kolwankar & Gangal, 1998). A feasible approach to these r 2003 Elsevier Science Ltd. All rights reserved.
190
G. R. PRADHAN ET AL.
classes of dynamic models restricts itself to the kinematic aspects (i.e. structure-induced) of the models of the medium of diffusion, i.e. the membrane. Unlike solute oriented models, one can propose a static model for the membrane. In the latter, the path of diffusion may vary for reasons of composition as well as due to dynamics of these rather complex structures. Presence of specific proteinaceous channels accounts for diffusion (transport) of specific molecules where the molecular size per se is not a determinant. What remains unanswered are the situations wherein unexpectedly large diffusion arises from otherwise firm barriers (Rajgopal & Sitaramam, 1998). Thus there is an explicit need to identify the physical basis of heterogeneous, dynamically evolving models of membranes to account for the available experimental observations. A major question, concerning the permeation of water and non-electrolytes across membrane, has been whether the bilayer is to be viewed as an isotropic homogeneous phase or as a heterogeneous phase (Lee, 1975). If osmotic contraction of the bilayer vesicles leads to an altered hydraulic conductivity (water flux coefficient), one obviously favors a heterogeneous membrane model. Otherwise a homogeneous, isotropic phase model would be adequate, obviating the need to look for a fine structure within the bilayer (Macey, 1982). Using the erythrocyte as an experimental system (in which the surface area of the biconcave cell does not change when it is osmotically expanded to a spherical shape), it was concluded that hydraulic conductivity was stretch-independent, i.e. in support of the isotropic model (Mlekoday et al., 1983). An alternative way to assess hydraulic conductivity is to use hydrogen peroxide as an analog of water (Mathai & Sitaramam, 1994): since many experimental systems have catalase (an enzyme that degrades hydrogen peroxide to molecular oxygen and water) within the vesicle/cell, an assay of this occluded catalase directly permits one to measure the conductivity to exogenous hydrogen peroxide. Under equilibrium conditions of assay, the initial velocity of rate of degradation would equal the rate of permeation of the peroxide into the vesicle (Sitaramam et al., 1989). Thus, one can directly assess the stretch
sensitivity of the membrane by osmotic titration with osmolytes, using non-electrolytes like hydrogen peroxide as probes of flux. In the course of these experiments, it was found that (Mathai & Sitaramam, 1994): (i) among all the lipid combinations tested, neither phosphatidylcholine (PC) vesicles nor intact erythrocytes showed a decrease in occluded catalase activity on osmotic compression of the membrane; (ii) on the other hand, all other native membrane systems tested, such as peroxisomes, E. coli and macrophages, showed stretch (osmotic) sensitivity; (iii) so did liposomes made from lipids extracted from these cells and organelles; (iv) further, when binary mixtures were investigated, only cardiolipin (CL) and cerebrosides when individually added to PC (5–10% of PC), and no other phospholipid combination, conferred stretch sensitivity in liposomes; (v) these binary mixtures of phospholipids also exhibited enhanced activation volume (osmotic sensitivity) and diminished activation energy for hydrogen peroxide flux; (vi) further, glucose was readily permeable across these membranes of binary mixtures; and (vii) addition of cholesterol to the PC:CL binary mixture abolished the stretch sensitivity of the liposomes as well as lowered activation volume and activation energy for peroxide permeation. These results on biological membranes, native or reconstituted, helped to visualize an essentially physical model to be investigated to account for the diffusive behaviors seen experimentally. Cardiolipin (CL) and phosphatidylcholine (PC) molecules differ dramatically in their cross sections (seen perpendicular to the length of the molecules, wherein PC is half of cardiolipin). A question arose whether a simple mixture of disks of different cross sections, given the right minimal interaction terms, can give rise to structural ensembles that permit unusually large permeation through formation of voids/ spaces between these lipid molecules. The end point of the simulation would be to identify the minimal description that permits permeation of relatively larger molecules compared to PC per se. Structural changes in the membrane are best identified by non-interactive molecules and therefore leaks across bilayers are commonly studied using non-electrolytes (Sitaramam &
191
SHAPE ANISOTROPY
Janardana Sarma, 1981; Stein, 1986). The diagnostic for non-specific permeation is sizedependent such that these hydrated solutes intercalate, penetrate and navigate through such interstices, spaces or voids, stochastically or in files to reach the other side of the membrane (Lee 1975). In order to capture the diverse features in a parsimonious manner, we restrict the simulations to a two-dimensional cross section of an otherwise three-dimensional system. Such a restriction is reasonable since the probe particle permeating across the membrane at any instant of time experiences the effective cross section rather than the three-dimensional obstruction. The permeation across the membrane depends primarily on the availability of free space or voids. Thus the problem reduces to the study of packing of two-dimensional objects at the first approximation. Then one needs to determine which factor(s) determine the appearance and size distribution of voids in such a two-dimensional system. The end point is the minimal description required to arrive at anomalously large permeation experimentally observed. The Model The configuration space of this model system (membrane) is a two-dimensional box with periodic (i.e. toroidal) boundary conditions. The constituents of this two-dimensional box are the circular disks (and/or the rigid combinations of the circular disks as dopants) of unit radii. A circular disk represents the hard-core scattering cross section, seen by the passing particle (a non-electrolyte, a hard disk in the first approximation, which acts as a probe), across the thickness spanned by two lipid molecules, viewed as cylinders. A typical dopant is two or more circular disks rigidly joined in a prespecified geometry. The circular disks are identified by the position coordinates of their centers and the radius, whereas for complete identification, the dopants need, in addition to the two factors above, the angle made by the major axes with the side of the box. In such a system, a structure would be defined as an identifiable pattern that lasts longer than a time-scale in which the molecules are agitating.
Very short-range attractive potentials would not be expected to give any stable structures, which we confirmed by simulations (data omitted). How long range should the attractive potential be is not relevant except in real-life simulations since the actual choice would then require realistic modeling by numerical simulations of actual observations using measured system parameters. However, the present goal is restricted to the identification of minimal and critical ingredients that control sieving. Therefore, it is reasonable to assume that the disks (which represent molecules) experience a longrange attractive interaction and hard-core repulsion near the center. The simplest physically realizable potential (which is a measure of interaction energy) admitting these conditions is the Lennard-Jones potential (Goetz et al., 1999). This pairwise interaction has a form N X N 12 6 X s s ; VLJ ðrij Þ ¼ 4e rij rij i¼1 j¼iþ1
where rij is the distance between the centers of the i-th and j-th disks, s determines the range of hard core part in the potential and e denotes the depth of the attractive part.* While studying the binary mixtures, we consider different shape anisotropic combinations impurities (or dopants) of k number of unit circular disks. We treat these combinations as one unit. for example, rodn denotes a single dopant made of n unit circular disks rigidly joined one after another in a straight line. The impurities interact with constituent circular disks via potential Vðrij Þ ¼
k X
VLJ ðria j Þ;
a¼1
where ria j is the distance between the centers of the a-th disk in the i-th impurity and the j-th circular disk, and the impurities interact among themselves via V ðrij Þ ¼
k1 X k2 X
VLJ ðria jb Þ;
a¼1 b¼1
* The depth e of the potential does not play a significant role in these simulations because the relevant quantity is be; and the choice of b is arbitrary.
192
G. R. PRADHAN ET AL.
Fig. 1. Typical equilibrium configurations for interacting disks. The parameter, s in Lennard-Jones potential is 2 units. The size of the box is 50 50. (a) A pure membrane with 556 unit circular disks. (b) A doped membrane with 464 circular disks and 46 rod2 (shown in gray). For clarity, two disks from the same molecule are drawn with a line joining the two centers.
where, ria jb is the distance between the centers of a-th disk in the i-th impurity and the b-th disk in the j-th impurity while k1 and k2 are the number of circular disks in the i-th and j-th impurities respectively. AN R-VOID
In order to study the arrangement of these constituents and to calculate the spaces available for permeation of molecules, we need to introduce the notion of a void. An r-void is defined as a closed area in a membrane devoid of disks or impurities, and sufficient to accommodate a circular disk of radius r (Pradhan et al., 1999). Clearly, larger voids also accommodate smaller probes, i.e. an r-void is also an r0 -void if r0 or. Similarly, the voids for the particle of size zero are the voids defined in the conventional sense, i.e. a measure of the net space unoccupied by the disks.
The Method The box was filled with circular disks such that they occupy 70% of the area of the box, i.e. loosely packed to facilitate the formation of voids. The temperature parameter, T, was so chosen that the quantity kB Toe: Similar numerical simulations were performed on the model system with anisotropic dopants (e.g. rod2 ). The
number of dopants was chosen to be 10% (cf. Mathai & Sitaramam 1994) of the number of circular disks with an additional constraint that the total occupied area of the box is still 70% as the focus of the study was on the redistribution of voids of different sizes. Figures 1(a) and (b) show typical equilibrium configurations of model systems without and with dopants respectively. These equilibrium configurations of the model system were obtained by the Monte Carlo method (using the Metropolis algorithm) starting with a random placement as well as orientation of the constituent disks (Pradhan et al., 1999; Binder, 1979). In each Monte Carlo step, every circular disk and dopant in the system was translated and rotated randomly. The new position and orientation of the constituent was accepted using the Metropolis algorithm with the potential energy of the system as cost function. The system was thermalized for 10 000 Monte Carlo steps.* Throughout the thermalization procedure we monitored the total energy of the system. The equilibrium was assumed to be reached when the average value of the energy of the system is stable for 2000 steps.
* The equilibrium configurations thus obtained are further confirmed by simulated annealing (Press et al., 1992).
193
SHAPE ANISOTROPY 800
35
Circular disks and rods of size 2 Only circular disks
700
30 Relative difference curve
600
14
25
(d)
20
(c)
15
(b)
500
Number of r-voids
Number of r-voids
12 10 8
400
6 4
300
2
10 (a)
0
200
0.3
5
0.5
100
0 0 0.3
0.4
0.5
0.6
0.7
0.8
r
-5 0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
r
Fig. 2. Distribution of r-voids in two different configurations. The main graph shows the number of r-voids as a function of the relative size (r) of the probe particle. The vertical bars represent the error margins at the corresponding points. The dotted curve gives the distribution in pure membrane while the solid curve shows the same in a membrane doped with rod2 (10:1). The relative difference curve clearly demonstrates the presence of large voids in the doped membrane.
Results In order to find the distribution of voids in each composition of the model system, we generated 500 different equilibrium configurations in each case and calculated the average variation in the number of r-voids as a function of the size of the permeating particle. The calculation of r-void distribution was carried out using the digitization algorithm (Pradhan et al., 1999). Figure 2 shows the graph of the number of r-voids vs. the size of r-void. When only circular disks are present (dotted curve), very few large r-voids (r40:5)* are seen. When mixed with the anisotropic impurities, say, rod2 ; a distinct increase in the number of large r-voids was seen with appropriate redistribution of smaller r-voids (solid curve). The ratio of the number of r-voids formed in the pure membranes to that of the membranes doped with anisotropic dopants (rod2 ; in this case) is shown in the inset box of Fig. 2. It emphasizes the * It may be recalled that glucose offers approximately half the radius of the PC cross section, yielding a relevant definition for a larger void of interest.
Fig. 3. Relative difference curves of distribution of rvoids. Curves are obtained by treating the void distribution for pure membrane as the base. The vertical bars represent the error margins at the corresponding points. The relative difference curves for p membranes: (a) doped with large ffiffiffi circular disks (of radii 2) which occupy the same area as that of rod2 : (these induce smaller vc number of large voids as compared to rod2 ); (b) doped with rod2 (c) doped with small rods comprising two circular disks of radii 0.5 each (Irrespective of their small size, they induce large voids); and (d) doped with rod4 : (significantly large voids are induced).
above result of the distinct increase in the number of large r-voids. This numerical result is consistent with the unexpected increase in the permeation of small molecules as well as the permeation of large molecules through the doped membrane, observed experimentally (Mathai & Sitaramam, 1994). The questions that naturally follow are: (i) is the induction of large voids due to the anisotropy in potential of the impurities; (ii) should the large voids form around the rods, the centers of anisotropy? First, we carried out simulations with large circular disks in place of rod2 as dopants. The radius of large disk is chosen in such a way that the area occupied by each of the large disk is same as that of a rod2 : Figure 3 shows the results of these further simulations. Curve (b) in Fig. 3 represents the ratio curve of r-void distribution of pure membrane and that of membrane doped with rod2 impurity. Curve (a) represents the same when the membrane is doped with large circular disks. It can be clearly
194
G. R. PRADHAN ET AL. 80
0.016
70
0.014
60
0.012
Local permeation probability
No. of r-voids
r = 0.55
50 40 30
0.01 0.008 0.006 0.004
20 0.002 10 0 rods of size 2
0 1
2
3 4 5 6 Size of the rod shaped impuritie s
7
8
Fig. 4. Dependence of the number of r-voids on the length of the rod-shaped impurities. The graph shows a steady increase in the number of r-voids (for r ¼ 0.55) with n. The first expected large jump in the number of voids because of the shape anisotropy is seen clearly when the configuration consists of molecules in the shape of circular disks and rod2 :
seen that the number of larger r-voids is always less in the latter case, thus confirming the role of shape anisotropy in the formation of large r-voids. Further, simulations are carried out with rod2 of smaller size (where each of the two constituent circular disks are of radius 0.5) and rod4 type impurities. (c) and (d) in Fig. 3 are the corresponding ratio curves. The curves show an interesting feature that the peak of the ratio curve shifts with the change in the type of anisotropy. This offers a definitive means to construct a membrane with selective permeability properties. Further, we consider dopants of the type rodn : Figure 4 represents the relation between the length dimension of rodn and the number of r-voids (for r ¼ 0.55). The anisotropy in the potential of rodn increases with n, such that the number of large r-voids should increase with n. Figure 4 indeed shows a jump when the rod2 impurities are added and afterwards, it shows a slow and almost linear increase with increase in n. Since dopants give rise to voids, their influence is most likely to be seen in their own vicinity, enhancing the ‘‘local transport’’. As the dopants
unit disks
Fig. 5. Local permeation probability in a doped model system. The points show the local permeation probability around 46 randomly chosen unit disks and 46 rod20 s. Further, as a guide line, averages are shown by the heights of the boxes, clearly indicating significantly more permeation in the neighborhood of rod2 :*
exhibit different potential in different directions, certain positions of the constituents are preferred from the point of view of energy minimization, eventually giving rise to voids in the vicinity of dopants. To verify this, we examine a binary mixture of circular disks and rod20 s. For an equilibrium configuration, we calculate the local permeation probability for particle of size r, which is the ratio of the area of r-voids and the area of the local neighborhood. We obtain the local permeation probability around all the dopants (46, in this case) in the shape of rod2 and same number of randomly chosen circular disks from a single equilibrium configuration. We repeat the procedure for the same set of molecules in each of the 500 different equilibrium configurations and finally plot the average local permeability probability in Fig. 5. The higher local permeation probability is indeed seen in the neighborhood of rod20 s. We supplement this result by calculating the radial distribution function (RDF), which is the probability of finding two molecules at different * However, some of the randomly chosen unit disks may share local neighborhood with rod20 s resulting in enhanced average value of local permeability probability for unit disks.
195
SHAPE ANISOTROPY 35
RDF for pair of circular disks 70
P1
Pure membrane Doped membrane
P2
30
P2
60 25
P1
P1
50 g(r)
20 g(r)
40 15
30 10 20 5 10 0 0
5
10
15
20
25
r
Fig. 6. Radial distribution function for rod2 and the circular disks. The peaks P1 and P2 indicate the most probable distance between the centers of circular disk and rod2 : The inset shows the corresponding non-overlapping positions of circular disks around rod2 : It clearly indicates the possibility of voids around rod2 :
distances apart (Bernal, 1964), for circular disks and rod20 s in the membrane made up of binary mixture. From Fig. 6, it is clear that the circular disks are not uniformly arranged around rod2 : The inset in the figure displays most probable position of the center of a circular disk around rod2 : This arrangement gives room for the formation of voids around impurity. Apart from induction of large voids, the anisotropic impurities reduce the long-range order in the system. This fact is demonstrated by radial distribution functions RDF for circular disks in pure and doped membranes. In case of binary mixture, the RDF for circular disks indicates the lack of long-range order and prominent local order (see Fig. 7). The formation of large voids near the centers of shape anisotropy indicates the local disorder in the system. This claim can be supported by the calculation of configuration entropy (Andraud et al., 1994) of the digitized model system where we calculate the amount of disorder in the pixel distribution. We cover the system of disks with a fine mesh. A mesh pixel is painted black if it is covered with a disk. The system is scanned through an elementary sliding cell of size l l:
0 0
5
10
15
20
25
r
Fig. 7. Radial distribution function (RDF) for circular disks in pure and doped membranes. It is evident from the figure that the RDF in the doped membrane shows absence of long-range order as opposed to that of the pure membrane.
Here, we choose l to be twice the diameter of the disk. The choice of l is reasonable since the sliding cell smaller than this results in a poor resolution, whereas larger sliding cell requires long time averaging. Following Andraud et al. (1994), we define configuration entropy HðlÞ as HðlÞ ¼
l2 X
pk ðlÞlogðpk ðlÞÞ;
k¼0
where pk ðlÞ ¼ Nk ðlÞ=NðlÞ; and Nk ðlÞ is the number of cells containing k black pixels and NðlÞ is the total number of cells. HðlÞ indicates the amount of disorder in the model system. Consistent with the result of radial distribution function, we find that the configuration entropy for model system doped with rod2 is indeed higher than that of the pure system. The configuration entropy of the pure system is approximately 27.03 (arbitrary units) whereas the configuration entropy of the system doped with rod2 is around 31.25. Discussion The primary purpose of the model is to capture the observed, unexpectedly large
196
G. R. PRADHAN ET AL.
permeation behavior in biological membranes from structural considerations. Obviously, the nature of the potential plays a role in that both attractive and repulsive components are required for the phenomenon to be observed. Clearly, the choice of Lennard-Jones potential is representative rather than arbitrary since a number of potentials would perform similarly insofar as both the attractive and repulsive components are present. Moreover, the structures deployed (such as disks and rods) are not an exact requirement since all that matters is the length to breadth ratio of such components. Induction of voids is not unique to membranes. In a series of experimental studies, initially anticipated on theoretical grounds in this journal (Rajgopal & Sitaramam, 1998), we could show that dehydration per se could initiate void formation in biopolymers, viz., proteins, membranes and micelles (Madhavarao et al., 2000, 2001; Sauna et al., 2001). Hydration and ionic strength appear to share major terms of interaction wherever both were present. For instance, in lauryl maltoside micelles, it being a neutral detergent, ionic strength would not matter but hydration would. Thus, the observed intra-polymeric voids are statistical, dynamic and inducible. The conventional polymer physics (cf. Grosberg & Khokhlov, 1994) as well as much of protein chemistry (Frauenfelder & Gratton, 1986; Brooks & Karplus, 1986) and osmotic studies (Parsegian et al., 1986; Timasheff, 1993) predict only collapse of polymers on removal of the solvent, i.e. collapse of voids, since hard shell atoms cannot disappear while the spaces in-between can. On the contrary, we could experimentally demonstrate the exact opposite in a variety of biopolymers, i.e. induction of voids on dehydration! The reason would be in the nature of the interaction between the solvent, water, and the polymer. Lack of interaction would not induce voids on desolvation, as a trivial case. If repulsive (as seen with most gels), desolvation would lead to collapse of the gel. If attractive (neither described nor demonstrated earlier), desolvation would imply work done on the system; the consequent increase in the free energy of the system would lead to structural evolution and void induction (Rajgopal & Sitaramam, 1998). A major con-
sequence was that solute/solvent/polymer triad could be understood better in terms of dehydration in determining the polymer movements rather than more abstruse mechanisms involving preferential exclusion of the solute from the surface of the protein (Timasheff, 1998). Enhanced free energy and induction of voids within the polymers would all indicate large changes in the intrapolymeric dynamics, providing a useful handle to studying whether the movements within these ordered polymers are correlated between domains. The experimental studies indicated that the movements within domains of polymers, interior and exterior, could be correlated and surface dehydration could play a major role in these concerted movements of voids within and at the surface. The role of free energy in void formation required no explanation in view of the energy requirement for the formation of a void. Indeed, void formation leading to dramatic permeability changes was reported in a variety of situations in which the free energy available to biological membranes enhanced under controlled experimental conditions, e.g. gravitational field (Sitaramam & Janardana Sarma, 1981; Sambasivarao & Sitaramam, 1985), electrostatic field due to unscreened charges at the surface of erythrocytes (Sambasivarao et al., 1986), electrical fields (Teissie & Tsong, 1981), respiration in mitochondria (Sambasivarao et al., 1985), ATP hydrolysis in mitochondria (Sambasivarao et al., 1988) and respiration even in bacteria (Natesan et al., 2000), etc. How do these experimental observations in polymers relate to the structural considerations in void formation simulated here? The biopolymers are internally organized in terms of aggregation/assemblage/folding. The breathing movements of these polymers have been investigated at considerable depth due to their intimate association with functionally important movements in catalysis/diffusion (Frauenfelder & Gratton, 1986). A major contribution to intra-polymeric movements would be the correlation length of the polymer (Grosberg & Khokhlov, 1994). A consequence of enhanced free energy of the chain is increased stiffness, which in turn affects the movements within the polymer. The stiffness in chains is controlled by a variety of parameters (e.g. ionic
SHAPE ANISOTROPY
strength if fixed charges are carried on the chain) and offers an intuitively important variable for chain dynamics. In order to understand the importance of chain rigidity in void formation, the events that lead to collapse of the bilayer at the phase transition are illustrative (Fisher & Stoeckenius, 1983; Lee, 1975). The lowering of temperature leads to trans-configuration of acyl chain, which is maximally formed at the transition temperature, and the rigid chains would be separated by the progressively smaller number of gauchebends. Some voids will persist as long as a gauche-kink persists along the length of the acyl chain. When all the kinks disappear cooperatively, the thickness of the membrane goes up by some 10% while permeability drops precipitously. At the transition itself, the extraordinary density fluctuations result in massive increase in permeation due to the rapid punctuated appearance of flaws mimicking single-channel conductance behavior (Antonov et al., 1980). The permeability of the bilayer at the fully melted configuration would be annealed by the dominance of gauche-forms, which offer a tortuous path of diffusion, leading again to lowered permeability. Thus, at a submolecular level, appearance of voids is consistent with the appearance of rod-like, stiff forms amidst the flexible gauche forms, consistent with the simulations. Development of voids that admit water, hydrogen peroxide and even glucose, on the other hand, is way above the phase transition temperature for these biological membranes. It actually helps understand, if not resolve, a major debate regarding proton permeation across bilayers. The confusion regarding proton permeation is that it requires the Grotthus mechanism; a chain of water molecules should persist long enough for the passage of the dehydrated proton along these wires and across the membranes (Deamer, 1996). The problem was that such wires were never seen in time-scales and probability of incidence to account for proton permeation (Marrink et al., 1996). Molecular dynamics of sufficient number of chains cannot be extended computationally at this stage to time-scales large enough to account for the voids we have observed on dehydration. If micro- to
197
millisecond range large-scale permeations of non-electrolytes were the rule than the exception, the search for an ephemeral molecular wire mechanism could simply be circumvented by invoking sturdy mechanisms larger in time-scale and physical dimensions (as in the case of voids) as opposed to a simulated molecular wire. The virtue of a void-based explanation is that their ubiquitous presence and measurability make them an attractive alternative to more abstruse mechanisms. The ubiquitous presence of voids in varieties of biopolymers argues for commonalities in terms of structural requirements as well as the proximate forces. Structurally it is legitimate to consider these polymer chains, regardless of their actual dimension (from atomic level as in an acyl chain to even secondary structures in a protein), from the point of view of polymer physics; thus, these polymers may well be viewed as consisting of Gaussian blobs of random chains and more rigid components of finite length, the rods. The simulations reported here not only indicate the occurrence of unexpectedly large voids near rigid rods, but they also point out that, given suitably scaled combination of attractive and repulsive forces, it would be worthwhile to investigate if these could recur even in larger scales in polymers. Thus it helps to identify and catalogue places wherein rigid structures tend to be juxtaposed to voids in polymers. For instance, alpha helices as ensembles in proteins could be considered as rods. Therefore, voids and structural heterogeneities are there in all these structures, though their common theoretical description is far from achieved. It appears that there is some acute need for these considerations. It is common knowledge that serious gaps exist in our knowledge in achieving a clear energetic description of catalysis in enzymes (Dill, 1997). The large separation between the computed and actual energetic terms indicates the presence of missing variables. While the problem statement is clear, how to go about solving it is far from clear. One needs newer ways to look into polymeric aspects of catalysis, i.e. information outside the scope of sequence data and active site chemistry (Welch et al., 1982). Entropic contributions offer a major source of missing terms in tallying energy
198
G. R. PRADHAN ET AL.
terms in processes. There appears to be a definitive justification to consider voids in many of these situations. Voids are attractive since they contribute to the total energetic description of the system; their energy can be abstrated to the surface (wall) of the void permitting thermodynamic handling. Further, absence of matter, which defines voids, restricts the description to energy fluxes alone (Rajagopal & Sitaramam, 1998). However, one cannot simply extend the present calculations to account for voids in proteins, say, as spaces between helices or even monomers, since one encounters large number of scaling problems and the justification of relevant theoretical formalisms. However, one can view the present simulations as illustrative progenitors of computations of potential functions and various scales to examine whether and how such simple approaches could account for structural dynamics which are otherwise impossible to derive by ab initio calculations. The essence is one of visualizing anisotropic rigidities in biologically important polymers, arising due to many reasons including electrostatic interactions, hydration, etc. and also the development of spaces next to such ‘rigidities’ that modulate grosser movement over different time-scales. This tuning involving wide extent of scaling needs to be captured in physical models and this is one such effort. Seen from another point of view, these are simulations using granular media, a favorite area of study for interesting pattern generations (Mo¨bius et al., 2001; Umbanhowar, 1997; Shinbrot et al., 1999). It is of some significance to note that results of biological interest have appeared in these modest simulations, well before any discernible structure has been observed in these granular media. While structure/ function relationship has been emphasized time and again, the tenuousness of such a presumption, as universal, is also worth considering, i.e. major functions can manifest well before a discernible structure actually develops!
simulation. The respective RDF calculations support this claim. Also, the number of larger r-voids increases linearly with increase in anisotropy as expected. These simulations were driven by the need for parsimony of interactions to address to a class of problems of polymeric behavior hitherto not addressed. Nevertheless, there are important limitations to this approach, necessitating future work. The use of Monte Carlo technique limits one from studying the true dynamic behavior of the voids. Further, one cannot investigate the effects of temperature, since multiple time-scales were not incorporated. Moreover, in a three-dimensional simulation factors such as additional interactions in the three-dimensional domain (e.g. bending interactions) would contribute to the measured r-void content and their translation into actual permeation. These are however matters of detail and scaling, relevant only when simulations cease and hard-core modeling begins for specific molecules. Even the void formation will have several molecular details that would require further investigations. For instance, a typical transmembrane channel has a pore with the walls consisting of a-helices or b-barrels, a picture consistent with the description of voids herein. The voids would further show dynamics by a variety means including staggering of the barrels all of which require molecular dynamics evaluated for individual instances (Smart et al., 1998). In such cases, a number of considerations will play a major role in the permeation of a molecule interacting with the membrane. Instead, we have emphasized giving a general principle behind the increased permeation of non-electrolytes through membranes composed of binary mixtures, the principle, which in fact can have applications in various systems in diverse disciplines.
Summary and Conclusion
Andraud, C., Beghdadi, A., Lafait, J. (1994). Entropic analysis of random morphologies, Physica A 207, 208–212. Antonov, V. F., Petrov, V. V., Molnar, A. A., Predvoditelev, D. A. & Ivanov, A. S. (1980). The appearance of single ion channels in unmodified lipid
What has been achieved is a minimal description for unusually large voids arising simply from shape anisotropy in a two-dimensional
We thank Dr. C. N. Madhavarao for discussions.
REFERENCES
SHAPE ANISOTROPY
bilayer membranes at the phase transition temperatures. Nature 283, 585–586. Bernal, J. D. (1964). The Bakerian lecture, 1962, the structure of liquids. Proc. Roy. Soc. A 280, 299–322. Binder K., (ed.) (1979). Monte Carlo Methods for Statistical Physics. Berlin: Springer. Brooks, C. L. & Karplus, M. (1986). Theoretical approaches to solvation of biopolymers. Methods Enzymol. 127, 369–400. Deamer, D. W. (1996). Water chains in lipid bilayers. Biophys. J. 71, 543. Dill, K. A. (1997). Additivity principles in Biochemistry. J. Biol. Chem. 272, 701–704. Fisher, K. & Stoeckenius, W. (1983). Membranes. In: Biophysics (Hoppe, W., Lohmann, W., Markl, H. & Ziegler, H., (eds)), pp 412–514. Berlin: SpringerVerlag. Frauenfelder, H. & Gratton, E. (1986). Protein dynamics and hydration. Methods Enzymol. 127, 207–216. Goetz, R., Gompper, G. & Lipowsky, R. (1999). Mobility and elasticity of self-asssembled membranes. Phys. Rev. Lett. 82, 221–224. Grosberg, A. Y. & Khokhlov, A. R. (1994) Statistical Physics of Macromolecules. New York: AIP (American Institute of Physics) Press. Israelachvili, J. N., Marcelja, S. & Horn, R. G. (1980) Physical principles of membrane organization. Q. Rev. Biophys. 13, 121–200. Kolwankar, K. M. & Gangal, A. D. (1998). Local fractional Fokker–Planck equation. Phys. Rev. Lett. 80, 214–217. Lee, A. G. (1975). Functional properties of biological membranes: a physical–chemical approach. Progr. Biophys. Mol. Biol. 29, 3–56. Macey, R. I. (1982). Water transport in red blood cells. In: Membranes and Transport (Martinosi, A. N., (ed.)), Vol. 2, pp. 461–466. New York: Plenum Press. Madhavarao, C. N., Sauna, Z. E. & Sitaramam, V. (2000). Solvent interconnectedness permits measurement of proximal as well as distant phase transitions in polymer mixtures by fluorescence. Biophys. Chem. 90, 147–156. Madhavarao, C. N., Sauna, Z. E., Srivatsava, A. & Sitaramam, V. (2001). Osmotic perturbations induce differential movements in the core and periphery of proteins, membranes and micelles. Biophys. Chem. 90, 231–246. Mathai, J. C. & Sitaramam, V. (1994). Stretch sensitivity of transmembrane mobility of hydrogen peroxide through voids in the bilayer: role of cardiolipin. J. Biol. Chem. 269, 17 784–17 793. Mlekoday, H. J., Moore, R. & Levitt, D. G. (1983). Osmotic water peremeability of human red cell. J. Gen. Physiol. 81, 213–220. Moºbius, M. E., Lauderdale, B. E., Nagel, S. R. & Jaeger, H. M. (2001). Brazil-nut effect: size separation of granular particles. Nature 414, 270. Natesan, S., Madhavrao, C. N. & Sitaramam, V. (2000). The positive role of voids in the plasma membrane in growth and energetics of Eshcerichia coli. Biophys. Chem. 85, 59–78.
199
Parsegian, A. V., Rand, R. P., Fuller, N. L. & Rau, D. C. (1986). Osmotic pressure for direct measurement of intermolecular forces. Methods Enzymol. 127, 401–416. Pradhan, G. R., Pandit, S. A., Gangal, A. D. & Sitaramam, V. (1999). Effect of shape anisotropy on transport in a two-dimensional computational model: numerical simulations showing experimental features observed in biomembranes. Physica A 270, 288–294. Press, W. H., Teukolsky, S. A., Vellerling, W. T. & Flannery, B. P. (1992). Numerical Recipes in C, 2nd Edn. Cambridge: Cambridge University Press. Rajgopal, K. & Sitaramam, V. (1998). Biology of the inner space: voids in biopolymers. J. theor. Biol. 195, 245–271. Sambasivarao, D. & Sitaramam, V. (1983). Studies on the nonlinear osmotic pressure volume relationship in mitochondria and entry of sucrose into the matrix space during centrifugation. Biochim. Biophys. Acta 722, 256–270. Sambasivarao, D. & Sitaramam, V. (1985). Respiration induces variable porosity to polyols in the mitochondrial inner membrane. Biochim. Biophys. Acta 806, 195–206. Sambasivarao, D., Rao N. M. & Sitaramam, V. (1986). Anomalous permeability and stability characteristics of erythrocytes in non-electrolyte media. Biochim. Biophys. Acta 857, 48–60. Sambasivarao, D., Kramer, R., Rao N. M. & Sitaramam, V. (1988). ATP hydrolysis induces variable porosity to mannitol in the mitochondrial inner membrane. Biochim. Biophys. Acta 933, 200–211. Sauna, Z. E., Madhavarao, C. N. & Sitaramam, V. (2001). Large solutes induce structural perturbations in proteins and membranes. Int. J. Biol. Macromol 29, 5–18. Shinbrot, T., Alexander, A. & Muzzio, F. J. (1999). Spontaneous chaotic granular mixing. Nature 397, 675–678. Sitaramam, V., Mathai, J. C., Rao, N. M. & Block, L. H. (1989). Hydrogen peroxide permeation across liposomal membranes: a novel method to assess structural flaws in liposomes. Mol. Cell. Biochem. 91, 91–97. Sitaramam, V. & Janardana Sarma, M. K. (1981). Does sucrose space exist? J. theor. Biol. 90, 317–336. Smart, O. S., Coates, G. M. P., Sansom, M. S. P., Alder, G. M. & Bashford, C. L. (1999). Structure based prediction of the conductance propecties of ions. Faraday Discuss. 111, 185–199. Stein, W. D. (1986). Transport and Diffusion Across Cell Membranes. London: Academic Press. Teissie, J. & Tsong, T. W. (1981). Electric field induced transient pores in phospholipid bilayer vesicles. Biochemistry 20, 1548–1554. Timasheff, S. N. (1993). The control of protein stability and association by weak interactions with water : how do solvents affect these processes? Annu. Rev. Biophys. Biomol. Struct. 22, 67–97. Timasheff, S. N. (1998). In disperse solutions, ‘‘osmotic stress’’ is a restricted case of preferential interactions. Proc. Natl. Acad. Sci. U.S.A. 95, 7363–7367. Umbanhowar, P. (1997). Patterns in the sand. Nature 389, 541–542. Welch, G. R., Somogyi, B. & Damjanovich, S. (1982). The role of protein fluctuations in enzyme action: a review. Progr. Biophys. Mol. Biol. 39, 109–146.
Shape anisotropy of lipid molecules and Voids Gauri R. Pradhanwz, Sagar A. Pandit*y, Anil D. GangalwO andV. SitaramamzO wDepartment of Physics, University of Pune, Pune 411 007, India yPhysical Research Laboratory, Ahmedabad 380 009, India and zDepartment of Biotechnology, University of Pune, Pune 411 007, India (Received on 13 May 2002, Accepted in revised form on 22 August 2002)
Biological polymers, viz., proteins, membranes and micelles exhibit structural discontinuities in terms of spaces unfilled by the polymeric phase, termed voids. These voids exhibit dynamics and lead to interesting properties which are experimentally demonstrable. In the specific case of phospholipid membranes, numerical simulations on a two-dimensional model system showed that voids are induced primarily due to the shape anisotropy in binary mixtures of interacting disks. The results offer a minimal description required to explain the unusually large permeation seen in liposomes made up of specific lipid mixtures (Mathai & Sitaramam, 1994). The results are of wider interest, voids being ubiquitous in biopolymers. r 2003 Elsevier Science Ltd. All rights reserved.
Introduction A variety of lipid molecules contribute to the structure and barrier function of biological membranes. The amphiphilic nature of lipid molecules leads to a bilayer structure in the presence of water in which hydrophilic head groups have maximum contact with water and hydrophobic tails have minimum contact with water. The process of membrane formation is one of minimizing the free energy and maximizing the stability of the structure (Israelachvili et al., 1980; Stein, 1986). A comprehensive description of diffusion through biomembranes remained elusive for n Corresponding author. Current address: Department of Chemistry, CB# 3290, University of North Carolina, Chapel Hill, NC 27599, U.S.A. E-mail addresses: [email protected] (G.R. Pradhan), [email protected] (S.A. Pandit), [email protected] (A.D. Gangal), [email protected] (V. Sitaramam). zThe author is thankful to DBT (India) and CSIR (India). O The work was funded by DBT (India).
0022-5193/03/$35.00
many reasons. In terms of a non-interactive diffusing molecule, the simplest of the descriptions involved Fick’s law, which takes into account a relevant partition coefficient (e.g. water:oil) for the diffusing molecule. More complex models involved introduction of frictional terms to account for remarkably nonlinear diffusion with respect to the mass of the solutes leading to extraordinary sieving capabilities of these bilayers (Stein, 1986). Both essentially deal with homogeneous models without or with pores. In traditional models of diffusion, the square of the distance varies linearly with time (i.e. x2 pt). However, in real structured media characteristically described as fractal diffusive traps, the heterogeneity plays a central role such that x2 varies as ta (where aa1) defining sub(super)-diffusive behavior. Formal heterogeneous models involving sub-diffusive behavior or diffusion in the presence of traps is a somewhat nascent area for immediate applicability, though in the right direction (Kolwankar & Gangal, 1998). A feasible approach to these r 2003 Elsevier Science Ltd. All rights reserved.
190
G. R. PRADHAN ET AL.
classes of dynamic models restricts itself to the kinematic aspects (i.e. structure-induced) of the models of the medium of diffusion, i.e. the membrane. Unlike solute oriented models, one can propose a static model for the membrane. In the latter, the path of diffusion may vary for reasons of composition as well as due to dynamics of these rather complex structures. Presence of specific proteinaceous channels accounts for diffusion (transport) of specific molecules where the molecular size per se is not a determinant. What remains unanswered are the situations wherein unexpectedly large diffusion arises from otherwise firm barriers (Rajgopal & Sitaramam, 1998). Thus there is an explicit need to identify the physical basis of heterogeneous, dynamically evolving models of membranes to account for the available experimental observations. A major question, concerning the permeation of water and non-electrolytes across membrane, has been whether the bilayer is to be viewed as an isotropic homogeneous phase or as a heterogeneous phase (Lee, 1975). If osmotic contraction of the bilayer vesicles leads to an altered hydraulic conductivity (water flux coefficient), one obviously favors a heterogeneous membrane model. Otherwise a homogeneous, isotropic phase model would be adequate, obviating the need to look for a fine structure within the bilayer (Macey, 1982). Using the erythrocyte as an experimental system (in which the surface area of the biconcave cell does not change when it is osmotically expanded to a spherical shape), it was concluded that hydraulic conductivity was stretch-independent, i.e. in support of the isotropic model (Mlekoday et al., 1983). An alternative way to assess hydraulic conductivity is to use hydrogen peroxide as an analog of water (Mathai & Sitaramam, 1994): since many experimental systems have catalase (an enzyme that degrades hydrogen peroxide to molecular oxygen and water) within the vesicle/cell, an assay of this occluded catalase directly permits one to measure the conductivity to exogenous hydrogen peroxide. Under equilibrium conditions of assay, the initial velocity of rate of degradation would equal the rate of permeation of the peroxide into the vesicle (Sitaramam et al., 1989). Thus, one can directly assess the stretch
sensitivity of the membrane by osmotic titration with osmolytes, using non-electrolytes like hydrogen peroxide as probes of flux. In the course of these experiments, it was found that (Mathai & Sitaramam, 1994): (i) among all the lipid combinations tested, neither phosphatidylcholine (PC) vesicles nor intact erythrocytes showed a decrease in occluded catalase activity on osmotic compression of the membrane; (ii) on the other hand, all other native membrane systems tested, such as peroxisomes, E. coli and macrophages, showed stretch (osmotic) sensitivity; (iii) so did liposomes made from lipids extracted from these cells and organelles; (iv) further, when binary mixtures were investigated, only cardiolipin (CL) and cerebrosides when individually added to PC (5–10% of PC), and no other phospholipid combination, conferred stretch sensitivity in liposomes; (v) these binary mixtures of phospholipids also exhibited enhanced activation volume (osmotic sensitivity) and diminished activation energy for hydrogen peroxide flux; (vi) further, glucose was readily permeable across these membranes of binary mixtures; and (vii) addition of cholesterol to the PC:CL binary mixture abolished the stretch sensitivity of the liposomes as well as lowered activation volume and activation energy for peroxide permeation. These results on biological membranes, native or reconstituted, helped to visualize an essentially physical model to be investigated to account for the diffusive behaviors seen experimentally. Cardiolipin (CL) and phosphatidylcholine (PC) molecules differ dramatically in their cross sections (seen perpendicular to the length of the molecules, wherein PC is half of cardiolipin). A question arose whether a simple mixture of disks of different cross sections, given the right minimal interaction terms, can give rise to structural ensembles that permit unusually large permeation through formation of voids/ spaces between these lipid molecules. The end point of the simulation would be to identify the minimal description that permits permeation of relatively larger molecules compared to PC per se. Structural changes in the membrane are best identified by non-interactive molecules and therefore leaks across bilayers are commonly studied using non-electrolytes (Sitaramam &
191
SHAPE ANISOTROPY
Janardana Sarma, 1981; Stein, 1986). The diagnostic for non-specific permeation is sizedependent such that these hydrated solutes intercalate, penetrate and navigate through such interstices, spaces or voids, stochastically or in files to reach the other side of the membrane (Lee 1975). In order to capture the diverse features in a parsimonious manner, we restrict the simulations to a two-dimensional cross section of an otherwise three-dimensional system. Such a restriction is reasonable since the probe particle permeating across the membrane at any instant of time experiences the effective cross section rather than the three-dimensional obstruction. The permeation across the membrane depends primarily on the availability of free space or voids. Thus the problem reduces to the study of packing of two-dimensional objects at the first approximation. Then one needs to determine which factor(s) determine the appearance and size distribution of voids in such a two-dimensional system. The end point is the minimal description required to arrive at anomalously large permeation experimentally observed. The Model The configuration space of this model system (membrane) is a two-dimensional box with periodic (i.e. toroidal) boundary conditions. The constituents of this two-dimensional box are the circular disks (and/or the rigid combinations of the circular disks as dopants) of unit radii. A circular disk represents the hard-core scattering cross section, seen by the passing particle (a non-electrolyte, a hard disk in the first approximation, which acts as a probe), across the thickness spanned by two lipid molecules, viewed as cylinders. A typical dopant is two or more circular disks rigidly joined in a prespecified geometry. The circular disks are identified by the position coordinates of their centers and the radius, whereas for complete identification, the dopants need, in addition to the two factors above, the angle made by the major axes with the side of the box. In such a system, a structure would be defined as an identifiable pattern that lasts longer than a time-scale in which the molecules are agitating.
Very short-range attractive potentials would not be expected to give any stable structures, which we confirmed by simulations (data omitted). How long range should the attractive potential be is not relevant except in real-life simulations since the actual choice would then require realistic modeling by numerical simulations of actual observations using measured system parameters. However, the present goal is restricted to the identification of minimal and critical ingredients that control sieving. Therefore, it is reasonable to assume that the disks (which represent molecules) experience a longrange attractive interaction and hard-core repulsion near the center. The simplest physically realizable potential (which is a measure of interaction energy) admitting these conditions is the Lennard-Jones potential (Goetz et al., 1999). This pairwise interaction has a form N X N 12 6 X s s ; VLJ ðrij Þ ¼ 4e rij rij i¼1 j¼iþ1
where rij is the distance between the centers of the i-th and j-th disks, s determines the range of hard core part in the potential and e denotes the depth of the attractive part.* While studying the binary mixtures, we consider different shape anisotropic combinations impurities (or dopants) of k number of unit circular disks. We treat these combinations as one unit. for example, rodn denotes a single dopant made of n unit circular disks rigidly joined one after another in a straight line. The impurities interact with constituent circular disks via potential Vðrij Þ ¼
k X
VLJ ðria j Þ;
a¼1
where ria j is the distance between the centers of the a-th disk in the i-th impurity and the j-th circular disk, and the impurities interact among themselves via V ðrij Þ ¼
k1 X k2 X
VLJ ðria jb Þ;
a¼1 b¼1
* The depth e of the potential does not play a significant role in these simulations because the relevant quantity is be; and the choice of b is arbitrary.
192
G. R. PRADHAN ET AL.
Fig. 1. Typical equilibrium configurations for interacting disks. The parameter, s in Lennard-Jones potential is 2 units. The size of the box is 50 50. (a) A pure membrane with 556 unit circular disks. (b) A doped membrane with 464 circular disks and 46 rod2 (shown in gray). For clarity, two disks from the same molecule are drawn with a line joining the two centers.
where, ria jb is the distance between the centers of a-th disk in the i-th impurity and the b-th disk in the j-th impurity while k1 and k2 are the number of circular disks in the i-th and j-th impurities respectively. AN R-VOID
In order to study the arrangement of these constituents and to calculate the spaces available for permeation of molecules, we need to introduce the notion of a void. An r-void is defined as a closed area in a membrane devoid of disks or impurities, and sufficient to accommodate a circular disk of radius r (Pradhan et al., 1999). Clearly, larger voids also accommodate smaller probes, i.e. an r-void is also an r0 -void if r0 or. Similarly, the voids for the particle of size zero are the voids defined in the conventional sense, i.e. a measure of the net space unoccupied by the disks.
The Method The box was filled with circular disks such that they occupy 70% of the area of the box, i.e. loosely packed to facilitate the formation of voids. The temperature parameter, T, was so chosen that the quantity kB Toe: Similar numerical simulations were performed on the model system with anisotropic dopants (e.g. rod2 ). The
number of dopants was chosen to be 10% (cf. Mathai & Sitaramam 1994) of the number of circular disks with an additional constraint that the total occupied area of the box is still 70% as the focus of the study was on the redistribution of voids of different sizes. Figures 1(a) and (b) show typical equilibrium configurations of model systems without and with dopants respectively. These equilibrium configurations of the model system were obtained by the Monte Carlo method (using the Metropolis algorithm) starting with a random placement as well as orientation of the constituent disks (Pradhan et al., 1999; Binder, 1979). In each Monte Carlo step, every circular disk and dopant in the system was translated and rotated randomly. The new position and orientation of the constituent was accepted using the Metropolis algorithm with the potential energy of the system as cost function. The system was thermalized for 10 000 Monte Carlo steps.* Throughout the thermalization procedure we monitored the total energy of the system. The equilibrium was assumed to be reached when the average value of the energy of the system is stable for 2000 steps.
* The equilibrium configurations thus obtained are further confirmed by simulated annealing (Press et al., 1992).
193
SHAPE ANISOTROPY 800
35
Circular disks and rods of size 2 Only circular disks
700
30 Relative difference curve
600
14
25
(d)
20
(c)
15
(b)
500
Number of r-voids
Number of r-voids
12 10 8
400
6 4
300
2
10 (a)
0
200
0.3
5
0.5
100
0 0 0.3
0.4
0.5
0.6
0.7
0.8
r
-5 0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
r
Fig. 2. Distribution of r-voids in two different configurations. The main graph shows the number of r-voids as a function of the relative size (r) of the probe particle. The vertical bars represent the error margins at the corresponding points. The dotted curve gives the distribution in pure membrane while the solid curve shows the same in a membrane doped with rod2 (10:1). The relative difference curve clearly demonstrates the presence of large voids in the doped membrane.
Results In order to find the distribution of voids in each composition of the model system, we generated 500 different equilibrium configurations in each case and calculated the average variation in the number of r-voids as a function of the size of the permeating particle. The calculation of r-void distribution was carried out using the digitization algorithm (Pradhan et al., 1999). Figure 2 shows the graph of the number of r-voids vs. the size of r-void. When only circular disks are present (dotted curve), very few large r-voids (r40:5)* are seen. When mixed with the anisotropic impurities, say, rod2 ; a distinct increase in the number of large r-voids was seen with appropriate redistribution of smaller r-voids (solid curve). The ratio of the number of r-voids formed in the pure membranes to that of the membranes doped with anisotropic dopants (rod2 ; in this case) is shown in the inset box of Fig. 2. It emphasizes the * It may be recalled that glucose offers approximately half the radius of the PC cross section, yielding a relevant definition for a larger void of interest.
Fig. 3. Relative difference curves of distribution of rvoids. Curves are obtained by treating the void distribution for pure membrane as the base. The vertical bars represent the error margins at the corresponding points. The relative difference curves for p membranes: (a) doped with large ffiffiffi circular disks (of radii 2) which occupy the same area as that of rod2 : (these induce smaller vc number of large voids as compared to rod2 ); (b) doped with rod2 (c) doped with small rods comprising two circular disks of radii 0.5 each (Irrespective of their small size, they induce large voids); and (d) doped with rod4 : (significantly large voids are induced).
above result of the distinct increase in the number of large r-voids. This numerical result is consistent with the unexpected increase in the permeation of small molecules as well as the permeation of large molecules through the doped membrane, observed experimentally (Mathai & Sitaramam, 1994). The questions that naturally follow are: (i) is the induction of large voids due to the anisotropy in potential of the impurities; (ii) should the large voids form around the rods, the centers of anisotropy? First, we carried out simulations with large circular disks in place of rod2 as dopants. The radius of large disk is chosen in such a way that the area occupied by each of the large disk is same as that of a rod2 : Figure 3 shows the results of these further simulations. Curve (b) in Fig. 3 represents the ratio curve of r-void distribution of pure membrane and that of membrane doped with rod2 impurity. Curve (a) represents the same when the membrane is doped with large circular disks. It can be clearly
194
G. R. PRADHAN ET AL. 80
0.016
70
0.014
60
0.012
Local permeation probability
No. of r-voids
r = 0.55
50 40 30
0.01 0.008 0.006 0.004
20 0.002 10 0 rods of size 2
0 1
2
3 4 5 6 Size of the rod shaped impuritie s
7
8
Fig. 4. Dependence of the number of r-voids on the length of the rod-shaped impurities. The graph shows a steady increase in the number of r-voids (for r ¼ 0.55) with n. The first expected large jump in the number of voids because of the shape anisotropy is seen clearly when the configuration consists of molecules in the shape of circular disks and rod2 :
seen that the number of larger r-voids is always less in the latter case, thus confirming the role of shape anisotropy in the formation of large r-voids. Further, simulations are carried out with rod2 of smaller size (where each of the two constituent circular disks are of radius 0.5) and rod4 type impurities. (c) and (d) in Fig. 3 are the corresponding ratio curves. The curves show an interesting feature that the peak of the ratio curve shifts with the change in the type of anisotropy. This offers a definitive means to construct a membrane with selective permeability properties. Further, we consider dopants of the type rodn : Figure 4 represents the relation between the length dimension of rodn and the number of r-voids (for r ¼ 0.55). The anisotropy in the potential of rodn increases with n, such that the number of large r-voids should increase with n. Figure 4 indeed shows a jump when the rod2 impurities are added and afterwards, it shows a slow and almost linear increase with increase in n. Since dopants give rise to voids, their influence is most likely to be seen in their own vicinity, enhancing the ‘‘local transport’’. As the dopants
unit disks
Fig. 5. Local permeation probability in a doped model system. The points show the local permeation probability around 46 randomly chosen unit disks and 46 rod20 s. Further, as a guide line, averages are shown by the heights of the boxes, clearly indicating significantly more permeation in the neighborhood of rod2 :*
exhibit different potential in different directions, certain positions of the constituents are preferred from the point of view of energy minimization, eventually giving rise to voids in the vicinity of dopants. To verify this, we examine a binary mixture of circular disks and rod20 s. For an equilibrium configuration, we calculate the local permeation probability for particle of size r, which is the ratio of the area of r-voids and the area of the local neighborhood. We obtain the local permeation probability around all the dopants (46, in this case) in the shape of rod2 and same number of randomly chosen circular disks from a single equilibrium configuration. We repeat the procedure for the same set of molecules in each of the 500 different equilibrium configurations and finally plot the average local permeability probability in Fig. 5. The higher local permeation probability is indeed seen in the neighborhood of rod20 s. We supplement this result by calculating the radial distribution function (RDF), which is the probability of finding two molecules at different * However, some of the randomly chosen unit disks may share local neighborhood with rod20 s resulting in enhanced average value of local permeability probability for unit disks.
195
SHAPE ANISOTROPY 35
RDF for pair of circular disks 70
P1
Pure membrane Doped membrane
P2
30
P2
60 25
P1
P1
50 g(r)
20 g(r)
40 15
30 10 20 5 10 0 0
5
10
15
20
25
r
Fig. 6. Radial distribution function for rod2 and the circular disks. The peaks P1 and P2 indicate the most probable distance between the centers of circular disk and rod2 : The inset shows the corresponding non-overlapping positions of circular disks around rod2 : It clearly indicates the possibility of voids around rod2 :
distances apart (Bernal, 1964), for circular disks and rod20 s in the membrane made up of binary mixture. From Fig. 6, it is clear that the circular disks are not uniformly arranged around rod2 : The inset in the figure displays most probable position of the center of a circular disk around rod2 : This arrangement gives room for the formation of voids around impurity. Apart from induction of large voids, the anisotropic impurities reduce the long-range order in the system. This fact is demonstrated by radial distribution functions RDF for circular disks in pure and doped membranes. In case of binary mixture, the RDF for circular disks indicates the lack of long-range order and prominent local order (see Fig. 7). The formation of large voids near the centers of shape anisotropy indicates the local disorder in the system. This claim can be supported by the calculation of configuration entropy (Andraud et al., 1994) of the digitized model system where we calculate the amount of disorder in the pixel distribution. We cover the system of disks with a fine mesh. A mesh pixel is painted black if it is covered with a disk. The system is scanned through an elementary sliding cell of size l l:
0 0
5
10
15
20
25
r
Fig. 7. Radial distribution function (RDF) for circular disks in pure and doped membranes. It is evident from the figure that the RDF in the doped membrane shows absence of long-range order as opposed to that of the pure membrane.
Here, we choose l to be twice the diameter of the disk. The choice of l is reasonable since the sliding cell smaller than this results in a poor resolution, whereas larger sliding cell requires long time averaging. Following Andraud et al. (1994), we define configuration entropy HðlÞ as HðlÞ ¼
l2 X
pk ðlÞlogðpk ðlÞÞ;
k¼0
where pk ðlÞ ¼ Nk ðlÞ=NðlÞ; and Nk ðlÞ is the number of cells containing k black pixels and NðlÞ is the total number of cells. HðlÞ indicates the amount of disorder in the model system. Consistent with the result of radial distribution function, we find that the configuration entropy for model system doped with rod2 is indeed higher than that of the pure system. The configuration entropy of the pure system is approximately 27.03 (arbitrary units) whereas the configuration entropy of the system doped with rod2 is around 31.25. Discussion The primary purpose of the model is to capture the observed, unexpectedly large
196
G. R. PRADHAN ET AL.
permeation behavior in biological membranes from structural considerations. Obviously, the nature of the potential plays a role in that both attractive and repulsive components are required for the phenomenon to be observed. Clearly, the choice of Lennard-Jones potential is representative rather than arbitrary since a number of potentials would perform similarly insofar as both the attractive and repulsive components are present. Moreover, the structures deployed (such as disks and rods) are not an exact requirement since all that matters is the length to breadth ratio of such components. Induction of voids is not unique to membranes. In a series of experimental studies, initially anticipated on theoretical grounds in this journal (Rajgopal & Sitaramam, 1998), we could show that dehydration per se could initiate void formation in biopolymers, viz., proteins, membranes and micelles (Madhavarao et al., 2000, 2001; Sauna et al., 2001). Hydration and ionic strength appear to share major terms of interaction wherever both were present. For instance, in lauryl maltoside micelles, it being a neutral detergent, ionic strength would not matter but hydration would. Thus, the observed intra-polymeric voids are statistical, dynamic and inducible. The conventional polymer physics (cf. Grosberg & Khokhlov, 1994) as well as much of protein chemistry (Frauenfelder & Gratton, 1986; Brooks & Karplus, 1986) and osmotic studies (Parsegian et al., 1986; Timasheff, 1993) predict only collapse of polymers on removal of the solvent, i.e. collapse of voids, since hard shell atoms cannot disappear while the spaces in-between can. On the contrary, we could experimentally demonstrate the exact opposite in a variety of biopolymers, i.e. induction of voids on dehydration! The reason would be in the nature of the interaction between the solvent, water, and the polymer. Lack of interaction would not induce voids on desolvation, as a trivial case. If repulsive (as seen with most gels), desolvation would lead to collapse of the gel. If attractive (neither described nor demonstrated earlier), desolvation would imply work done on the system; the consequent increase in the free energy of the system would lead to structural evolution and void induction (Rajgopal & Sitaramam, 1998). A major con-
sequence was that solute/solvent/polymer triad could be understood better in terms of dehydration in determining the polymer movements rather than more abstruse mechanisms involving preferential exclusion of the solute from the surface of the protein (Timasheff, 1998). Enhanced free energy and induction of voids within the polymers would all indicate large changes in the intrapolymeric dynamics, providing a useful handle to studying whether the movements within these ordered polymers are correlated between domains. The experimental studies indicated that the movements within domains of polymers, interior and exterior, could be correlated and surface dehydration could play a major role in these concerted movements of voids within and at the surface. The role of free energy in void formation required no explanation in view of the energy requirement for the formation of a void. Indeed, void formation leading to dramatic permeability changes was reported in a variety of situations in which the free energy available to biological membranes enhanced under controlled experimental conditions, e.g. gravitational field (Sitaramam & Janardana Sarma, 1981; Sambasivarao & Sitaramam, 1985), electrostatic field due to unscreened charges at the surface of erythrocytes (Sambasivarao et al., 1986), electrical fields (Teissie & Tsong, 1981), respiration in mitochondria (Sambasivarao et al., 1985), ATP hydrolysis in mitochondria (Sambasivarao et al., 1988) and respiration even in bacteria (Natesan et al., 2000), etc. How do these experimental observations in polymers relate to the structural considerations in void formation simulated here? The biopolymers are internally organized in terms of aggregation/assemblage/folding. The breathing movements of these polymers have been investigated at considerable depth due to their intimate association with functionally important movements in catalysis/diffusion (Frauenfelder & Gratton, 1986). A major contribution to intra-polymeric movements would be the correlation length of the polymer (Grosberg & Khokhlov, 1994). A consequence of enhanced free energy of the chain is increased stiffness, which in turn affects the movements within the polymer. The stiffness in chains is controlled by a variety of parameters (e.g. ionic
SHAPE ANISOTROPY
strength if fixed charges are carried on the chain) and offers an intuitively important variable for chain dynamics. In order to understand the importance of chain rigidity in void formation, the events that lead to collapse of the bilayer at the phase transition are illustrative (Fisher & Stoeckenius, 1983; Lee, 1975). The lowering of temperature leads to trans-configuration of acyl chain, which is maximally formed at the transition temperature, and the rigid chains would be separated by the progressively smaller number of gauchebends. Some voids will persist as long as a gauche-kink persists along the length of the acyl chain. When all the kinks disappear cooperatively, the thickness of the membrane goes up by some 10% while permeability drops precipitously. At the transition itself, the extraordinary density fluctuations result in massive increase in permeation due to the rapid punctuated appearance of flaws mimicking single-channel conductance behavior (Antonov et al., 1980). The permeability of the bilayer at the fully melted configuration would be annealed by the dominance of gauche-forms, which offer a tortuous path of diffusion, leading again to lowered permeability. Thus, at a submolecular level, appearance of voids is consistent with the appearance of rod-like, stiff forms amidst the flexible gauche forms, consistent with the simulations. Development of voids that admit water, hydrogen peroxide and even glucose, on the other hand, is way above the phase transition temperature for these biological membranes. It actually helps understand, if not resolve, a major debate regarding proton permeation across bilayers. The confusion regarding proton permeation is that it requires the Grotthus mechanism; a chain of water molecules should persist long enough for the passage of the dehydrated proton along these wires and across the membranes (Deamer, 1996). The problem was that such wires were never seen in time-scales and probability of incidence to account for proton permeation (Marrink et al., 1996). Molecular dynamics of sufficient number of chains cannot be extended computationally at this stage to time-scales large enough to account for the voids we have observed on dehydration. If micro- to
197
millisecond range large-scale permeations of non-electrolytes were the rule than the exception, the search for an ephemeral molecular wire mechanism could simply be circumvented by invoking sturdy mechanisms larger in time-scale and physical dimensions (as in the case of voids) as opposed to a simulated molecular wire. The virtue of a void-based explanation is that their ubiquitous presence and measurability make them an attractive alternative to more abstruse mechanisms. The ubiquitous presence of voids in varieties of biopolymers argues for commonalities in terms of structural requirements as well as the proximate forces. Structurally it is legitimate to consider these polymer chains, regardless of their actual dimension (from atomic level as in an acyl chain to even secondary structures in a protein), from the point of view of polymer physics; thus, these polymers may well be viewed as consisting of Gaussian blobs of random chains and more rigid components of finite length, the rods. The simulations reported here not only indicate the occurrence of unexpectedly large voids near rigid rods, but they also point out that, given suitably scaled combination of attractive and repulsive forces, it would be worthwhile to investigate if these could recur even in larger scales in polymers. Thus it helps to identify and catalogue places wherein rigid structures tend to be juxtaposed to voids in polymers. For instance, alpha helices as ensembles in proteins could be considered as rods. Therefore, voids and structural heterogeneities are there in all these structures, though their common theoretical description is far from achieved. It appears that there is some acute need for these considerations. It is common knowledge that serious gaps exist in our knowledge in achieving a clear energetic description of catalysis in enzymes (Dill, 1997). The large separation between the computed and actual energetic terms indicates the presence of missing variables. While the problem statement is clear, how to go about solving it is far from clear. One needs newer ways to look into polymeric aspects of catalysis, i.e. information outside the scope of sequence data and active site chemistry (Welch et al., 1982). Entropic contributions offer a major source of missing terms in tallying energy
198
G. R. PRADHAN ET AL.
terms in processes. There appears to be a definitive justification to consider voids in many of these situations. Voids are attractive since they contribute to the total energetic description of the system; their energy can be abstrated to the surface (wall) of the void permitting thermodynamic handling. Further, absence of matter, which defines voids, restricts the description to energy fluxes alone (Rajagopal & Sitaramam, 1998). However, one cannot simply extend the present calculations to account for voids in proteins, say, as spaces between helices or even monomers, since one encounters large number of scaling problems and the justification of relevant theoretical formalisms. However, one can view the present simulations as illustrative progenitors of computations of potential functions and various scales to examine whether and how such simple approaches could account for structural dynamics which are otherwise impossible to derive by ab initio calculations. The essence is one of visualizing anisotropic rigidities in biologically important polymers, arising due to many reasons including electrostatic interactions, hydration, etc. and also the development of spaces next to such ‘rigidities’ that modulate grosser movement over different time-scales. This tuning involving wide extent of scaling needs to be captured in physical models and this is one such effort. Seen from another point of view, these are simulations using granular media, a favorite area of study for interesting pattern generations (Mo¨bius et al., 2001; Umbanhowar, 1997; Shinbrot et al., 1999). It is of some significance to note that results of biological interest have appeared in these modest simulations, well before any discernible structure has been observed in these granular media. While structure/ function relationship has been emphasized time and again, the tenuousness of such a presumption, as universal, is also worth considering, i.e. major functions can manifest well before a discernible structure actually develops!
simulation. The respective RDF calculations support this claim. Also, the number of larger r-voids increases linearly with increase in anisotropy as expected. These simulations were driven by the need for parsimony of interactions to address to a class of problems of polymeric behavior hitherto not addressed. Nevertheless, there are important limitations to this approach, necessitating future work. The use of Monte Carlo technique limits one from studying the true dynamic behavior of the voids. Further, one cannot investigate the effects of temperature, since multiple time-scales were not incorporated. Moreover, in a three-dimensional simulation factors such as additional interactions in the three-dimensional domain (e.g. bending interactions) would contribute to the measured r-void content and their translation into actual permeation. These are however matters of detail and scaling, relevant only when simulations cease and hard-core modeling begins for specific molecules. Even the void formation will have several molecular details that would require further investigations. For instance, a typical transmembrane channel has a pore with the walls consisting of a-helices or b-barrels, a picture consistent with the description of voids herein. The voids would further show dynamics by a variety means including staggering of the barrels all of which require molecular dynamics evaluated for individual instances (Smart et al., 1998). In such cases, a number of considerations will play a major role in the permeation of a molecule interacting with the membrane. Instead, we have emphasized giving a general principle behind the increased permeation of non-electrolytes through membranes composed of binary mixtures, the principle, which in fact can have applications in various systems in diverse disciplines.
Summary and Conclusion
Andraud, C., Beghdadi, A., Lafait, J. (1994). Entropic analysis of random morphologies, Physica A 207, 208–212. Antonov, V. F., Petrov, V. V., Molnar, A. A., Predvoditelev, D. A. & Ivanov, A. S. (1980). The appearance of single ion channels in unmodified lipid
What has been achieved is a minimal description for unusually large voids arising simply from shape anisotropy in a two-dimensional
We thank Dr. C. N. Madhavarao for discussions.
REFERENCES
SHAPE ANISOTROPY
bilayer membranes at the phase transition temperatures. Nature 283, 585–586. Bernal, J. D. (1964). The Bakerian lecture, 1962, the structure of liquids. Proc. Roy. Soc. A 280, 299–322. Binder K., (ed.) (1979). Monte Carlo Methods for Statistical Physics. Berlin: Springer. Brooks, C. L. & Karplus, M. (1986). Theoretical approaches to solvation of biopolymers. Methods Enzymol. 127, 369–400. Deamer, D. W. (1996). Water chains in lipid bilayers. Biophys. J. 71, 543. Dill, K. A. (1997). Additivity principles in Biochemistry. J. Biol. Chem. 272, 701–704. Fisher, K. & Stoeckenius, W. (1983). Membranes. In: Biophysics (Hoppe, W., Lohmann, W., Markl, H. & Ziegler, H., (eds)), pp 412–514. Berlin: SpringerVerlag. Frauenfelder, H. & Gratton, E. (1986). Protein dynamics and hydration. Methods Enzymol. 127, 207–216. Goetz, R., Gompper, G. & Lipowsky, R. (1999). Mobility and elasticity of self-asssembled membranes. Phys. Rev. Lett. 82, 221–224. Grosberg, A. Y. & Khokhlov, A. R. (1994) Statistical Physics of Macromolecules. New York: AIP (American Institute of Physics) Press. Israelachvili, J. N., Marcelja, S. & Horn, R. G. (1980) Physical principles of membrane organization. Q. Rev. Biophys. 13, 121–200. Kolwankar, K. M. & Gangal, A. D. (1998). Local fractional Fokker–Planck equation. Phys. Rev. Lett. 80, 214–217. Lee, A. G. (1975). Functional properties of biological membranes: a physical–chemical approach. Progr. Biophys. Mol. Biol. 29, 3–56. Macey, R. I. (1982). Water transport in red blood cells. In: Membranes and Transport (Martinosi, A. N., (ed.)), Vol. 2, pp. 461–466. New York: Plenum Press. Madhavarao, C. N., Sauna, Z. E. & Sitaramam, V. (2000). Solvent interconnectedness permits measurement of proximal as well as distant phase transitions in polymer mixtures by fluorescence. Biophys. Chem. 90, 147–156. Madhavarao, C. N., Sauna, Z. E., Srivatsava, A. & Sitaramam, V. (2001). Osmotic perturbations induce differential movements in the core and periphery of proteins, membranes and micelles. Biophys. Chem. 90, 231–246. Mathai, J. C. & Sitaramam, V. (1994). Stretch sensitivity of transmembrane mobility of hydrogen peroxide through voids in the bilayer: role of cardiolipin. J. Biol. Chem. 269, 17 784–17 793. Mlekoday, H. J., Moore, R. & Levitt, D. G. (1983). Osmotic water peremeability of human red cell. J. Gen. Physiol. 81, 213–220. Moºbius, M. E., Lauderdale, B. E., Nagel, S. R. & Jaeger, H. M. (2001). Brazil-nut effect: size separation of granular particles. Nature 414, 270. Natesan, S., Madhavrao, C. N. & Sitaramam, V. (2000). The positive role of voids in the plasma membrane in growth and energetics of Eshcerichia coli. Biophys. Chem. 85, 59–78.
199
Parsegian, A. V., Rand, R. P., Fuller, N. L. & Rau, D. C. (1986). Osmotic pressure for direct measurement of intermolecular forces. Methods Enzymol. 127, 401–416. Pradhan, G. R., Pandit, S. A., Gangal, A. D. & Sitaramam, V. (1999). Effect of shape anisotropy on transport in a two-dimensional computational model: numerical simulations showing experimental features observed in biomembranes. Physica A 270, 288–294. Press, W. H., Teukolsky, S. A., Vellerling, W. T. & Flannery, B. P. (1992). Numerical Recipes in C, 2nd Edn. Cambridge: Cambridge University Press. Rajgopal, K. & Sitaramam, V. (1998). Biology of the inner space: voids in biopolymers. J. theor. Biol. 195, 245–271. Sambasivarao, D. & Sitaramam, V. (1983). Studies on the nonlinear osmotic pressure volume relationship in mitochondria and entry of sucrose into the matrix space during centrifugation. Biochim. Biophys. Acta 722, 256–270. Sambasivarao, D. & Sitaramam, V. (1985). Respiration induces variable porosity to polyols in the mitochondrial inner membrane. Biochim. Biophys. Acta 806, 195–206. Sambasivarao, D., Rao N. M. & Sitaramam, V. (1986). Anomalous permeability and stability characteristics of erythrocytes in non-electrolyte media. Biochim. Biophys. Acta 857, 48–60. Sambasivarao, D., Kramer, R., Rao N. M. & Sitaramam, V. (1988). ATP hydrolysis induces variable porosity to mannitol in the mitochondrial inner membrane. Biochim. Biophys. Acta 933, 200–211. Sauna, Z. E., Madhavarao, C. N. & Sitaramam, V. (2001). Large solutes induce structural perturbations in proteins and membranes. Int. J. Biol. Macromol 29, 5–18. Shinbrot, T., Alexander, A. & Muzzio, F. J. (1999). Spontaneous chaotic granular mixing. Nature 397, 675–678. Sitaramam, V., Mathai, J. C., Rao, N. M. & Block, L. H. (1989). Hydrogen peroxide permeation across liposomal membranes: a novel method to assess structural flaws in liposomes. Mol. Cell. Biochem. 91, 91–97. Sitaramam, V. & Janardana Sarma, M. K. (1981). Does sucrose space exist? J. theor. Biol. 90, 317–336. Smart, O. S., Coates, G. M. P., Sansom, M. S. P., Alder, G. M. & Bashford, C. L. (1999). Structure based prediction of the conductance propecties of ions. Faraday Discuss. 111, 185–199. Stein, W. D. (1986). Transport and Diffusion Across Cell Membranes. London: Academic Press. Teissie, J. & Tsong, T. W. (1981). Electric field induced transient pores in phospholipid bilayer vesicles. Biochemistry 20, 1548–1554. Timasheff, S. N. (1993). The control of protein stability and association by weak interactions with water : how do solvents affect these processes? Annu. Rev. Biophys. Biomol. Struct. 22, 67–97. Timasheff, S. N. (1998). In disperse solutions, ‘‘osmotic stress’’ is a restricted case of preferential interactions. Proc. Natl. Acad. Sci. U.S.A. 95, 7363–7367. Umbanhowar, P. (1997). Patterns in the sand. Nature 389, 541–542. Welch, G. R., Somogyi, B. & Damjanovich, S. (1982). The role of protein fluctuations in enzyme action: a review. Progr. Biophys. Mol. Biol. 39, 109–146.