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Depletion Effect on Polymers Induced by Small Depleting Spheres† Jun Soo Kim and Igal Szleifer* Department of Biomedical Engineering and Chemistry of Life Processes Institute, Northwestern UniVersity, EVanston, Illinois 60208, United States ReceiVed: August 11, 2010; ReVised Manuscript ReceiVed: September 21, 2010
Depletion effects on the structure and interactions between polymers induced by the presence of small depleting spheres are investigated by computer simulations. As the separation between two polymers decreases, the polymers repel each other due to the loss of conformational entropy. When the polymers are immersed in a medium crowded with small depleting spheres, however, depletion attractions between the polymer segments are induced. The resulting repulsive interaction is significantly reduced when the polymer segments approach one another closer than the size of the depleting spheres. The distance-dependent potential of mean force shows a highly nonmonotonic behavior reflecting the packing of the small depleting spheres around the polymer segments. We show that the depletion potential, that is, the component of the interactions arising from the presence of the small depleting spheres, between flexible polymers is qualitatively similar to that between two large spheres. However, there are small numerical differences that arise from the connectivity of the polymer chains. We also show that Brownian dynamics simulations of a single polymer chain with depletion potential can predict polymer statistical properties in good agreement with those from molecular dynamics simulations in which depleting spheres are explicitly accounted for. Therefore, we suggest the use of the depletion potentials for computational study of crowding effects on large biopolymers such as chromatin fibers. Introduction In recent years, there has been great interest in the effects of macromolecular crowding in an effort to understand the role of high total concentration of macromolecules in biological environments. Appreciation for the crowding effect is growing as one of the main driving forces of structure formations in biological events.1-9 For instance, it has proven to play an important role in the compaction of chromatin fibers and formations of nuclear bodies such as nucleoli and promyelocytic leukemia (PML) bodies in a cell nucleus.7-9 The crowding effect is often referred to as the depletion effect when large objects experience effective attractions due to the presence of smaller depleting agents. The depletion effect has been extensively studied in the area of colloid-polymer mixtures, where small polymers induce depletion interactions between large colloidal particles.10-15 The origin of this depletion effect is an entropy gain due to the increased accessible volume of the small depleting agents when large objects approach one another and reduce the excluded volume to the depleting agents.1-6 This can also be explained by unbalanced osmotic pressure; when the gap between two large objects is small enough to exclude the depleting agents, the force, equivalent to the osmotic pressure, pushing the large objects toward each other is greater than that pushing them away.6,10,11 At low concentrations of depleting agents, large colloidal particles experience only an effective attraction. At higher concentrations, however, the colloids also have a repulsive barrier due to structuring of the depleting agents within the gap.12-15 In this work, we aim to understand depletion effects on large biopolymers in the presence of a high content of other small †
Part of the “Mark A. Ratner Festschrift”. * To whom correspondence should be addressed. E-mail: igalsz@ northwestern.edu.
macromolecules. Conformational properties of polymers are determined by chain connectivity and interactions between polymer segments. Therefore, it is of great interest to understand how such characteristics of polymers affect the depletion effect on polymer segments as compared with that on colloidal particles. For instance, neighboring polymer segments may affect an arrangement of the depleting agents around each polymer segment and may result in a different extent of the depletion effect compared with that on two large colloids. In particular, depletion effects on chromatin fibers induced by a high content of other macromolecules in a cell nucleus is of our current interest. A chain of nucleosomes, each of which is a complex of DNA and histone proteins wrapped around by DNA, forms a chromatin fiber with a thickness of 30 nm at physiological salt concentrations.16 Therefore, there have been several efforts to understand chromosome structures based on polymer models that consist of segments with a diameter of 30 nm or larger.17-20 To investigate the crowding effect between chromatin fibers induced by a high content of macromolecules in a cell nucleus, we also employ a flexible polymer model with 30 nm segments, while the macromolecules are treated as smaller depleting spheres with a diameter of 6 nm. The size of the depleting spheres is chosen to mimic the proteins in the nucleoplasm. The average molecular weight of these proteins is 67.7 kDa,21 which corresponds to a diameter of about 6 nm, assuming a spherical shape of the proteins and a partial specific volume of 0.73 mL/g.22 This particular choice for the model system will enable the systematic applications of the depletion potential obtained in this work to several existing chromosome models.17-20 Finally, we mention that the depletion effect in the mixture of polymers and proteins when the size of polymers is comparable to or larger than that of proteins was studied in order to understand the role of uncharged polymers in the
10.1021/jp107598m 2010 American Chemical Society Published on Web 10/12/2010
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separation and crystallization of proteins.23,24 The present work is distinguished from earlier work for several reasons; (i) the depletion interactions between polymer segments induced by small macromolecules are studied in contrast to the earlier work where the depletion interactions between proteins induced by polymers are studied; (ii) the size of polymer segments is larger than that of macromolecules, while in the earlier work, the size of the polymer itself was comparable to that of proteins; and (iii) the concentration of polymers is low and that of the crowding macromolecules is significantly higher. Methods and Models Depletion effects on two segments of different polymers can be studied by calculating the potential of mean force. The potential of mean force is the work required to bring two interacting units in a medium to a given distance ξ. In other words, it is the interaction averaged over all of the degrees of freedom of the molecules in the environment. A variety of methods have been proposed to calculate the potential of mean force.25,26 We consider two appropriate calculation methods in the absence and presence of depleting spheres. We first investigate the effect of chain conformational entropy in the absence of depleting spheres by using26
FR(ξ) ) FR(∞) - kBT ln〈exp[-βUinter(ξ)]〉
(1)
where FR(ξ) is the free energy when two middle segments in each polymer are separated by a distance ξ, kB is the Boltzmann constant, T is the absolute temperature, β ) kBT, and Uinter(ξ) is the intermolecular interaction between two polymers at a middle segment distance of ξ (details of interactions are discussed later in this section). 〈...〉 is an average over all possible conformations of the two polymers at the given distance ξ when their intermolecular interactions are turned off. It is noted that the free energy in this work is expressed as a function of the separation between the middle segments of two polymers, while previous work26,27 studied the free energy as a function of the distance between the centers of mass of two polymers. For this method, we prepared one million conformations of a single polymer from molecular dynamics (MD) simulations, from which one million pairs of polymer conformations were selected at random. A segment in the middle of one polymer was located at the origin, and a segment in the middle of another polymer was located at the coordinate (ξ,0,0). For this pair of polymers at a distance of ξ, the intermolecular interaction between two polymers Uinter(ξ) was calculated as described in eqs 3 and 4. The potential of mean force FR(ξ) is obtained from eq 1 using Uinter(ξ). This simple method requires only a sampling of single polymer conformations, and these conformations are used for all of the distances. Albeit fast, this method is not suitable when small depleting spheres exist in addition to polymer chains. Therefore, for the study of depletion effects, we employ the constraint-biased MD simulation method.25,28 In this method, we constrain a distance ξ between the centers of the middle segments of each polymer using the linear constraint solver (LINCS),29 and the potential of mean force is obtained by integration of the constraint force obtained from simulations
FR(ξ) ) FR(ξ0) +
∫ξξ 0
[
〈fc〉s +
]
2kBT ds s
(2)
Figure 1. A snapshot of a simulation system with two polymers of seven segments and small depleting spheres whose size is one-fifth of a diameter of polymer segments, created using VMD.30 Note that only the depleting spheres within a certain distance from polymer segments are shown for clarity, even though the entire simulation box is filled with them according to volume fractions under study.
where fc is the constraint force obtained at a separation of ξ and 2kBT/s is included to remove the entropic contribution due to the rotation of two centers of the middle segments.28 For calculation of the potential of mean force in the constraintbiased MD simulations, a simulation box contains two polymers with seven segments and depleting spheres, as shown in Figure 1. The figure is created using VMD,30 and only the depleting spheres close to polymer segments are shown for visual clarity. Polymer segments and depleting spheres interact with each other by a repulsive Lennard-Jones potential of the form
[(
Ur(r) ) 4ε
σ r - r0
) ( 12
-
σ r - r0
)] 6
+ε
(3)
for r0 < r < r0 + rc and Ur(r) ) 0 for r > r0 + rc. Here, ε is a Lennard-Jones well depth and set to kBT, and rc ) 21/6σ. We choose r0 ) 0 for depleting spheres, r0 ) 4σ for polymer segments, and r0 ) 2σ between a depleting sphere and a polymer segment. This defines a diameter of depleting spheres roughly as σ and that of polymer segments as 5σ. A mass of polymer segments is set to 125 times greater than that of the depleting spheres so as to keep the mass density constant. Bonded segments in a polymer chain interact with a combination of a finite extension nonlinear elastic (FENE) potential
1 Ub(r) ) - kbR2b ln[1 - ((r - r0)/Rb)2] 2
(4)
and the repulsive Lennard-Jones potential given in eq 3, where r0 ) 4σ, kb ) 30kBT/σ2, and Rb ) 1.5σ to prevent bonds in polymers from crossing each other.31 All simulations were performed with GROMACS version 4.0.5 with a use of tabulated interaction functions.32,33 For the constraint-biased MD simulations, the side length of the cubic simulation box was chosen as 40σ greater than the contour length of polymers, and the volume fractions (φ) of depleting spheres under study were 0.10 and 0.20, amounting to 12223 and 24446 depleting spheres in the simulation box. Each polymer consisted of seven segments. Simulations were at a
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Figure 2. Potential of mean force in the absence of depleting spheres (φ ) 0.00) as a function of a surface separation between middle segments of each polymer, ξ′ ) ξ - 5σ, where ξ is a distance between centers of the middle segments.
reduced temperature (T* ) kBT/ε) of 1.0 and run with a time step (δt) of 0.005τMD, where τMD ) σ(m/ε)1/2. Equations of motion were integrated using the leapfrog algorithm, and a total of 4 × 107 time steps (2 × 105τMD) was run for each simulation at a fixed separation. It is confirmed that the depletion potential does not change significantly when the number of time steps is greater than 2 × 107. To calculate the potential of mean force, a distance between the centers of middle segments of each polymer was set to ξ, and constraint forces were calculated every 10 simulation time steps. It was shown that the potential of mean force calculated between two large spheres becomes negligible beyond 7.5σ at volume fractions of small depleting spheres up to 0.20.12-15 Therefore, in our study for polymers, we considered a slightly broader range of ξ that is between 5.1σ and 8.5σ. The effect of the polymer length was investigated by the constraint-biased MD simulations for 15 segment polymers, showing no significant variation of the potential of mean force, as will be discussed in the following section. To test the use of the depletion potentials to mimic the effect by small spheres, we performed Brownian dynamics (BD) simulations of a single polymer with 15 segments using the effective potentials determined from eq 2 and MD simulations of the same polymer in the presence of depleting spheres. For these MD simulations, interactions between polymer segments and depleting spheres were the same as those in the constraintbiased MD simulations given in eqs 3 and 4. The side length of the simulation box was set to 50σ, and then, the volume fractions (φ) of 0.10 and 0.20 corresponded to 23874 and 47748 depleting spheres each. Simulations were run for 5 × 108 and 4 × 108 time steps for φ ) 0.10 and 0.20, respectively. Details of the simulations are the same as those in the constraint-biased MD simulations described above. The BD simulations were performed for a single polymer of 15 segments with a combination of the repulsive Lennard-Jones potential and the depletion potentials determined in eq 2. We assumed that the viscosity was 0.98 cP34 and the size of the time step ∆t was 10-4τBD, where τBD is the time for a polymer segment to move a distance of σ with a diffusion coefficient of D (τBD ) σ2/D) and D is calculated from the viscosity using the Einstein-Stokes equation. The number of simulation steps was 5 × 109 for all volume fractions. Results and Discussion When two polymers approach each other in the absence of depleting spheres (φ ) 0.00), the polymers experience repulsive interactions due to the loss of conformational entropy and, as depicted in Figure 2, show an increase of the potential of mean
Kim and Szleifer
Figure 3. The potential of mean force between segments of flexible polymers induced by small depleting spheres at different volume fractions of 0.00, 0.10, and 0.20. Note that ∆FR(ξ′) is defined in eq 5 such that ∆FR(3.5σ) ) 0. Error estimates are of the same size as the symbols or less. Lines are included as guides to the eye.
force when the separation decreases. Note that the potential of mean force calculated as a function of ξ (the distance between two middle segments of each polymer) from eq 1 is translated horizontally so that it is shown as a function of the surface separation of the two middle segments, that is, ξ′ ) ξ - 5σ. Here, 5σ is the distance between two contacting segments, and therefore, the range of ξ between 5.1σ and 8.5σ corresponds to the surface separation of ξ′ between 0.1σ and 3.5σ. Also note that all of the following data for the potential of mean force are given as a function of ξ′. In Figure 2, it is shown that the potential of mean force is nearly zero when ξ′ is greater than 20σ. As ξ′ decreases, however, the potential of mean force increases, implying the stronger repulsive interactions between the two middle segments. The interpretation becomes easier if we assume that the polymers are hard chains consisting of hard sphere segments since the physical meaning of 〈exp[-βUinter(ξ′)]〉 in eq 1 is, in the case of hard chains, a probability of nonoverlapping conformations at a separation ξ′.35 Then, the increase of the potential of mean force resulting from the decrease of 〈exp[-βUinter(ξ′)]〉 in eq 1 is interpreted as the reduced number of nonoverlapping conformations, or the increased number of overlapping conformations, and therefore as the loss of conformational entropy. A more thorough discussion of the potentials of mean force between polymers in the absence of crowding agents can be found in ref 26. When depleting spheres are added, the potential of mean force changes significantly at small separations. Figure 3 depicts the difference between potentials of mean force at separations ξ′ < 3.5σ and at 3.5σ, that is
∆FR(ξ′) ) FR(ξ′) - FR(3.5σ)
(5)
at different volume fractions of depleting spheres φ ) 0.00, 0.10, and 0.20. Note that ∆FR(ξ′) at ξ′ ) 3.5σ is 0 by definition. In the absence of depleting spheres (φ ) 0.00), the potential of mean force increases as the separation decreases from 3.5σ through 0.1σ, as already discussed above. As depleting spheres are added (φ ) 0.10 and 0.20), the potential of mean force shows an oscillatory behavior which is qualitatively different from the monotonic potential in the absence of depleting spheres (φ ) 0.00). As the separation decreases from 3.5σ, the potential of mean force also increases initially for φ ) 0.10 and 0.20, but the increase is less than that for φ ) 0.00 when the separation is larger than σ. As the separation decreases further, the increase of the potential of mean force for φ ) 0.10 and 0.20 is greater
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than that for φ ) 0.00 near 0.7σ-0.8σ and then drops sharply at smaller separations. The change of the potential of mean force is significantly more pronounced at the higher volume fraction of 0.20. The potential of mean force in Figure 3 at all volume fractions is purely entropic. As discussed above, the repulsive potential at φ ) 0.00 arises from the increased number of overlapping polymer conformations. At volume fractions φ ) 0.10 and 0.20, however, there is another entropic contribution to the potential of mean force, entropic effects induced by small depleting spheres. The depleting spheres are excluded from a spherical volume around polymer segments whose radii are equal to the sum of a radius of polymer segments and that of the depleting spheres. When a separation between polymer segments is smaller than the size of the depleting spheres, the excluded volumes around the polymer segments overlap, and then, the free volume for the depleting spheres increases, resulting in a gain in the configurational entropy. Therefore, polymer segments experience effective attractions in the presence of small depleting spheres,1-6 and as a result, the repulsive interactions arising from the loss of polymer conformational entropy are largely reduced. For φ > 0, we define
∆FR ) ∆FR0 + ∆FRd
(6)
where ∆F0R is the contribution from the polymer conformational entropy in the absence of depleting spheres (∆FR(φ ) 0.00)) and ∆FRd is the contribution from depletion entropy. Therefore, the sole contribution from depletion effects at a given φ is obtained by subtracting ∆FR at φ ) 0.00 from ∆FR at φ > 0, that is, ∆FRd(φ) ) ∆FR(φ) - ∆FR(φ ) 0.00). The contribution of depletion effects to the potential of mean force (∆FRd) is shown for flexible polymers at φ ) 0.10 and 0.20 in Figure 4. ∆FdR(ξ′) ) 0 at ξ′ ) 3.5σ is from the definition in eq 5. Since one of our motivations is to compare the depletion effect on polymer segments with the depletion effect on colloidal particles, we also present in Figure 4 the potential of mean force between two large spheres of the same size as polymer segments. The potential of mean force for large spheres is qualitatively in good agreement with those in earlier research,14 with a slight difference attributed to the difference between hard particle interactions used in earlier works and the (softer) repulsive Lennard-Jones interactions used in this work. The depletion contribution to the potential of mean force (∆FRd) for polymer segments is qualitatively the same as ∆FR between the large spheres shown in Figure 4. In both cases, the attractive local minima and the repulsive maximum are observed, which manifest the structure of the depleting spheres around large objects (both large spheres and polymer segments). Particularly, the presence of the repulsive barrier implies that the depleting spheres arrange themselves between large objects, preventing them from moving closer to each other. The potential of mean force calculated for the interactions between polymers is lower than that of the spheres, with stronger attractions and a reduced repulsive barrier. We believe that this is the result of aligned conformations of two polymers where not only the middle segments but also neighboring segments are located close to each other, as seen in Figure 1. In this situation, depletion effects can be induced for multiple segments at the same time, and as a result, the potentials of mean force for polymers can become more attractive than those between two large spheres. Note, however, that such effect on the depletion potentials is not very significant, as seen in Figure 4,
Figure 4. The depletion potentials between polymer segments at volume fractions (φ) of depleting spheres, (a) 0.10 and (b) 0.20. Those between two large spheres are also included for comparison. The inset in (b) is the comparison of total potential of mean force between polymer segments shown in Figure 3 with that between two large spheres. The errors are of the same size as the symbol or smaller. Lines are included as guides to the eye.
since the potential of mean force is an averaged property not only over the aligned conformations but also over other polymer conformations whose probability distribution is determined by polymer chain entropy as well as the depletion effect. It is noted that the potential of mean force between two polymer segments in Figure 4 is purely from depletion entropy, and the contribution from polymer conformational entropy that depends on the polymer length is already excluded in Figure 4. Therefore, the depletion potential between polymer segments shown in Figure 4 is anticipated to be valid independent of polymer length. The inset in Figure 4b compares the total potential of mean force ∆FR between two polymer segments shown in Figure 3 with that between two large spheres in order to emphasize the difference of the total potentials of mean force and the similarity between the part of the potential induced by the depleting spheres. Again, ∆FR(ξ′) ) 0 at ξ′ ) 3.5σ is from the definition in eq 5. The dependence of the depletion potential on the polymer length is studied by calculating the depletion potential for polymers with 7 and 15 segments, as shown in Figure 5. The potential of mean force at φ ) 0.00 presented in Figure 5a increases with the polymer length at the same separation ξ′, in agreement with previous studies for ring polymers.36,37 The difference between the potentials of mean force ∆FR(ξ′) ) FR(ξ′) - FR(3.5σ), however, becomes identical for both polymer lengths (see the inset in Figure 5a). It is also found that ∆FR at φ > 0 does not depend on the polymer length, and therefore, ∆FRd calculated as their difference as in eq 6 is almost identical for both polymer lengths, as shown in Figure 5b. This finding implies that the results obtained for seven segment polymers in this work can be applied to the study of longer polymers.
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Figure 5. Polymer length dependence on the depletion effect. (a) At φ ) 0.00. FR is depicted as a function of ξ′ and ∆FR(ξ′) ) FR(ξ′) FR(3.5σ) is shown in the inset. Data are for polymers with 7 segments (dashed blue line; Nseg ) 7) and polymers with 15 segments (solid red line; Nseg ) 15). (b) Depletion potentials for different polymer lengths at φ ) 0.10. Data are for polymers with 7 segments (blue square; Nseg ) 7) and polymers with 15 segments (red circle; Nseg ) 15).
The study of the crowding effect on biopolymers such as chromatin fibers has received great interest in recent years.9 Computer simulations in the explicit presence of crowding agents, however, are severely limited by the extremely large number of crowding agents in the simulation box, necessary to accommodate chromosome models. Human chromosomes have ∼100 Mbp of nucleotides, and this corresponds to 34 000 polymer segments based on our model, for which the number of crowding agents with a diameter of 6 nm can exceed ∼107 for volume fractions of 0.10 and 0.20. Therefore, the depletion potentials determined in this work can be utilized to mimic the depletion effect induced by crowding agents in an implicit way. To validate the use of the depletion potentials, we perform BD simulations of a single polymer with 15 segments which interact with each other by ∆FRd (Figure 4) in addition to the repulsive Lennard-Jones potential of eq 3. The results are then compared with those obtained from MD simulations of a single polymer in the explicit presence of small depleting spheres. Figure 6 shows comparisons of results from both BD and MD simulations for φ ) 0.10 and 0.20. As can be seen in Figure 6a, the radius of gyration of a polymer determined from explicit MD simulations is reduced with crowding; that is, the distribution is shifted to a smaller value of the radius of gyration for φ ) 0.10 and 0.20. This indicates the slight compaction of the single polymer induced by the depletion attraction between polymer segments. The compaction is not very large in absolute magnitude due to the short polymer length, but the distributions at φ > 0.00 are still clearly distinguished from that at φ ) 0.00. We expect that the compaction will be more pronounced for complex and long polymers such as chromosomes since they make frequent contacts with themselves by forming loops and
Kim and Szleifer
Figure 6. Comparison of results from the implicit BD simulations with the effective potential in Figure 4 and from the MD simulations with depleting spheres explicitly accounted for. (a) The probability distribution of the square root of the mean-squared radius of gyration, (R2g)1/2, of a polymer chain and (b) the radial distribution function between the ith and the (i + 2)th segments, Ri,i+2, of a polymer.
subcompartments and with other chromosomes in the interchromatin compartments.38,39 The agreement between the distributions of the radius of gyration from both implicit BD simulations and explicit MD simulations is reasonable. It is conceivable that the agreement may improve with longer duration of MD simulations. However, that requires an almost prohibitively long computation time. More interestingly, the radial distribution function between the ith and the (i+2)th segments from BD simulations captures all of the subtle features from MD simulations, as seen in Figure 6b. The peak locations and the amplitudes are well reproduced by BD simulations for both volume fractions. Therefore, the use of depletion potentials ∆FdR can be a very good approximation to mimic depletion effects induced by small depleting spheres, and this can provide an efficient way for computer simulations to study biopolymer conformations in a crowded medium. We emphasize that the depletion potentials obtained in this study can be implemented in existing chromosome models. For instance, the depletion potentials can be easily implemented in the multiloop subcompartment (MLS) model,17 the random loop model,18 and the linear and ring polymer models.19,20 Conclusions Depletion effects have been appreciated as an important factor in determining the structural organization in crowded biological environments. In this work, we investigated the depletion effects induced between segments of flexible polymers commonly used as a model of biopolymers. The polymers repel each other due to the loss of conformational entropy when their separation
Depletion Effects on Polymers by Small Depleting Spheres decreases. When crowding agents are present, attractive interactions between polymer segments are observed at short separations. The change of the potential of mean force for polymers by the depletion effects is qualitatively in good agreement with that between two large spheres and is slightly more attractive due to depletion effects occurring simultaneously to neighboring segments. This effect demonstrates the interesting coupling that exists between conformational degrees of freedom and interactions. Depletion attractions induced by crowding agents estimated in this work are not negligible, with the amplitude close to ∼1kBT at volume fractions of 0.10 and 0.20, and they will be more significant at higher volume fractions. Therefore, the chromatin compaction in in vitro crowding experiments9 can be explained partly by the depletion attractions induced by a high content of macromolecules. In biological environments, the degree of crowding is estimated to be between volume fractions of 0.1 and 0.4 inside of a cell,2,40,41 and hence, there will be a significant extent of depletion effects. The use of MD simulations to study the crowding effects on biopolymer conformations is severely limited by the large number of small depleting spheres. This work suggests that implicit simulations using the depletion potentials can be a good approximation for the study of crowding effects. The validity of the use of the depletion potentials in implicit simulations is confirmed in simulations of a single polymer by comparison with explicit MD simulations. These effective potentials can be incorporated systematically into existing chromosome models to appreciate the role of crowding in the structural organization of nuclear chromosomes. It is important to note, however, that this work focuses on depletion effects due to the excluded volume interactions between monodisperse depleting spheres. The strength and range of depletion potentials depend on the sizes of polymer segments and depleting spheres as well as on the interactions between these particles. Biological environments including the cell nucleus are, indeed, polydisperse media with macromolecules of varying molecular weights, and the macromolecules therein interact with each other via nonspecific interactions such as electrostatic and van der Waals interactions in addition to the excluded volume interactions. Therefore, the study of depletion effects due to the presence of polydisperse depleting spheres interacting via other nonspecific interactions represents possible future directions of this work. Acknowledgment. This work is supported by the National Science Foundation under Grant EFRI CBET-0937987. We thank Mark Uline and Jennifer Campbel for carefully reading the manuscript. References and Notes (1) Zhou, H.-X.; Rivas, G.; Minton, A. P. Annu. ReV. Biophys. 2008, 37, 375–397. (2) Hall, D.; Minton, A. P. Biochim. Biophys. Acta 2003, 1649, 127– 139.
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