Direct observation of the enhancement of noncooperative protein self-assembly by macromolecular crowding: Indefinite linear self-association of bacterial cell division protein FtsZ

Abstract

Recent measurements of sedimentation equilibrium and sedimentation velocity have shown that the bacterial cell division protein FtsZ self-associates to form indefinitely long rod-like linear aggregates in the presence of GDP and Mg²⁺. In the present study, the newly developed technique of non-ideal tracer sedimentation equilibrium was used to measure the effect of high concentrations—up to 150 g/liter—of each of two inert “crowder” proteins, cyanmethemoglobin or BSA, on the thermodynamic activity and state of association of dilute FtsZ under conditions inhibiting (−Mg²⁺) and promoting (+Mg²⁺) FtsZ self-association. Analysis of equilibrium gradients of both FtsZ and crowder proteins indicates that, under the conditions of the present experiment, FtsZ interacts with each of the two crowder proteins essentially entirely via steric repulsion, which may be accounted for quantitatively by a simple model in which hemoglobin, albumin, and monomeric FtsZ are modeled as effective spherical hard particles, and each oligomeric species of FtsZ is modeled as an effective hard spherocylinder. The functional dependence of the sedimentation of FtsZ on the concentrations of FtsZ and either crowder indicates that, in the presence of high concentrations of crowder, both the weight-average degree of FtsZ self-association and the range of FtsZ oligomer sizes present in significant abundance are increased substantially.

It has been predicted that in “crowded” solutions—solutions containing a total volume fraction of inert proteins or other macromolecules comparable to that found in physiological fluid media—volume exclusion will significantly enhance the tendency of dilute as well as concentrated proteins to self-associate (1–6). Indirect evidence for crowding-induced changes in the equilibrium average state of association of several proteins has been provided by measurement of crowding-linked changes in enzyme activity (7), rotational relaxation times (8), and rates of protein aggregation (9). However, direct evidence for such an effect has been difficult to obtain, because conventional methods for direct measurement of equilibrium average molar masses of proteins, such as light scattering, osmotic pressure, and analytical ultracentrifugation, are poorly suited for the study of the properties of a dilute protein in a solution containing high concentrations of other proteins.

A newly developed analytical method, referred to as nonideal tracer sedimentation equilibrium (NITSE), permits the simultaneous measurement of the concentration gradients at sedimentation equilibrium of a labeled dilute tracer macrosolute and an unlabeled, arbitrarily concentrated “background” macrosolute together in solution (10). From these gradients one can calculate apparent equilibrium weight-average molar masses of both tracer and background species as functions of the concentration of background. To determine the true weight-average molar masses of tracer and background from the experimentally measured apparent weight-average molar masses at high background concentration, it is necessary to take into account repulsive as well as attractive interactions between macrosolute molecules, which requires a description of the approximate size and shape of each macrosolute species present in significant abundance. Lacking such information, the initial study of protein self-association carried out by using NITSE (10) used an empirical approximation that, whereas reasonable, adds uncertainty to the final interpretation of experimental results.

In the present work, we present a NITSE study of the self-association in crowded solution of a bacterial protein, FtsZ, whose self-association has recently been well-characterized in dilute solution by using conventional measurements of sedimentation equilibrium and sedimentation velocity (11). FtsZ is a prokaryotic homolog of tubulin (M_r = 40,000) that has been shown to constitute a major protein component of the septation ring in Escherichia coli and is essential for cell division (12). It has been shown that FtsZ self-associates in a manner and to an extent determined by environmental concentrations of guanine nucleotide(s) and Mg²⁺ (13–15). By means of sedimentation equilibrium and sedimentation velocity experiments, it was established that, in the presence of saturating concentrations of GDP, FtsZ reversibly forms linear oligomers of indefinite length, with an uptake of one mole of Mg²⁺ per mole of FtsZ monomer incorporated into oligomer (11). For a given Mg²⁺ concentration, stepwise association equilibrium constants were found to decrease slightly, but significantly, with increasing oligomer size (11).

In the present study, NITSE is used to measure the apparent weight-average molar mass of FtsZ as a function of the concentrations of FtsZ and each of two “inert” crowding proteins, hemoglobin (Hb) and BSA, in the absence and presence of Mg²⁺. With the aid of information provided by the earlier study of FtsZ self-association under ideal conditions, the NITSE results are interpreted in an unambiguous fashion to characterize quantitatively the role of excluded volume in facilitating protein assembly in crowded solutions.

Materials and Methods

Materials.

E. coli FtsZ was purified as described in ref. 11. Iodination of FtsZ was performed essentially according to ref. 16. The iodination reaction was carried out in 0.05 M Tris⋅HCl/0.05 M KCl (pH 7.5) buffer with 10 mM MgCl₂ to keep the protein in an oligomeric form during the labeling (11), to minimize perturbations on the self-association interaction to be explored. Further controls to check that the labeling procedure did not affect FtsZ self-association were performed by sedimentation equilibrium of mixtures of labeled and cold FtsZ. Crowding proteins used in this study (Hb, BSA) were obtained commercially from Sigma. AquometHb was converted to the cyanmet form (HbCN) as previously described (17), and the monomer of BSA was isolated by size-exclusion chromatography. Solutions of both proteins were prepared in 0.05 M Tris⋅HCl/0.1 M KCl (pH 7.5), and experiments were carried out at 20°C.

Preparative Analytical Ultracentrifugation.

Protein solutions contained in small cylindrical tubes were centrifuged to sedimentation equilibrium in swinging bucket rotors, and the contents of the tubes were subsequently fractionated according to radial distance in the centrifuge as previously described (16). The relative concentration of dilute FtsZ in each fraction was determined by counting of radiolabel (16), and the relative concentration of concentrated crowder protein (HbCN or BSA) in each fraction was determined by the bicinchonic acid (BCA) colorimetric assay (18)(Pierce) of an appropriately diluted aliquot. The sample-average apparent buoyant molar mass of protein X, denoted by M^*_X,app, was determined by nonlinear least squares fitting of the following equation to the observed dependence of relative concentration S on radial distance r:

1

where r_o is an arbitrarily selected reference position, ω the angular velocity of the rotor in radians/sec, R is the molar gas constant, and T the absolute temperature (10).

Nonlinear least-squares modeling was performed by using matlab 5.3 (MathWorks, Natick, MA) together with scripts written by A.P.M. and a script by R. Shrager (personal communication) for efficient implementation of the Marquardt-Levenberg function-minimizing algorithm (19). These scripts are available from A.P.M. on request.

Results and Analysis

Dependence of Crowder Gradients on Crowder Concentration.

Sample gradients of BSA and HbCN are plotted in Fig. 1 A and B, along with the respective calculated best-fits of Eq. 1. The dependence of M^*_Hb,app on Hb concentration and the dependence of M^*_BSA,app on BSA concentration are plotted in Fig. 2 (symbols). It has been previously established that, under conditions comparable to those of the present experiments, neither hemoglobin nor BSA self-associates to a significant extent at concentrations up to and including 150 g/liter (3, 20, 21). For a non-associating protein C (“crowder”), the thermodynamic theory of sedimentation equilibrium (22) states that

2

where M_C, v̄_C, w_C, and γ_C, respectively, denote the actual molar mass, partial specific volume, wt/vol concentration, and activity coefficient of C, and ρ_o denotes the solvent (buffer) density.

Figure 1.

Examples of raw experimental data: equilibrium gradients (20°C and 10,000 rpm) of signals proportional to the wt/vol concentration of solute. (A) BSA, loading concentration 50 g/liter. (B) HbCN, loading concentration 100 g/liter. (C) FtsZ, loading concentration 1 g/liter, in the absence (●) and presence (○) of 5 mM Mg²⁺.

Figure 2.

Effect of loading concentration of crowder protein on the apparent buoyant molar mass of crowder. HbCN: data, circles; calculated best-fit of effective hard sphere model described in text, thick line. BSA: data, squares; calculated best-fit, thin line.

To test the hypothesis that, under the conditions of these experiments crowder molecules interact in solution predominantly via steric repulsion (volume exclusion), the solution is modeled as a homogeneous fluid of hard spheres of radius r_C, and Eq. 14, presented in the Appendix, was used to calculate dlnγ_C/dw_C (where C = Hb or BSA) with R = 1, L = 0, and w₁ = w_C. Setting v̄_C equal to the experimentally determined value, 0.73 cm³/g for both Hb and BSA (23), Eqs. 2 and 14 were fit by the method of nonlinear least squares to the experimentally determined dependences of M^*_Hb,app on w_Hb and M^*_BSA,app on w_BSA. The best-fit values and 95% (2σ) confidence limits of the floating parameters M_C and r_C so obtained are given in Table 1. The dependences of M^*_Hb,app on w_Hb and M^*_BSA,app on w_BSA calculated by using Eqs. 2 and 14 with the best-fit hard sphere parameters are plotted together with the data in Fig. 2 (solid curves).

Table 1.

Values of model constants (in parentheses) and best-fit values of model parameters ±2σ

	HbCN	BSA	FtsZ monomer	FtsZ oligomers
M, g/mol	64600  ± 3000	71700  ± 7000	(40000)*
v̄, cm3/g	(0.73)†	(0.73)†	(0.73)‡
rHS, 10−8 cm	27.5  ± 1.7	31.3  ± 3.1	23.3  ± 1.0
Δ				0.495  ± 0.3
log K, M−1§				4.55  ± 0.15

^*Ref. 11. 

^† Ref. 23. 

^‡ Ref. 25. 

^§ [Mg⁺⁺] = 5 mM. 

Dependence of Gradients of Monomeric FtsZ on Crowder Concentration.

The measured dependences of M^*_FtsZ,app on the concentration of each crowding protein in the absence of Mg²⁺, derived as described above from gradients similar to those shown in Fig. 1C, are plotted in Fig. 3 (symbols). According to the theory of nonideal sedimentation equilibrium (10, 24), the dependence of the apparent buoyant molar mass of a single tracer species Z (monomeric FtsZ) on the concentration of a second crowder species C (where C is either Hb or BSA) is given by

3

Eq. 3 was fit simultaneously to the experimental dependence of M^*_Z,app on both w_Hb and w_BSA in the following manner. For each loading concentration of crowder, the value of M^*_C,app was calculated by using Eqs. 2 and 14 as described above, together with the best-fit hard sphere radius of the appropriate crowder given in Table 1. M_Z was set equal to the measured value of 40,000 (11) and v̄_Z set equal to 0.73 cm³/g, a value typical of apolipo- and apoglycoproteins (25). To test the hypothesis that FtsZ interacts with either crowder primarily via volume exclusion, monomeric FtsZ was modeled as a hard sphere of radius r_Z in a fluid of hard spheres of radius r_C. The value of dlnγ_Z/dw_C was calculated by using Eq. 14 with r = r_Z/r_C and L = 0. The best-fit value and 95% confidence limits of r_Z so obtained are given in Table 1. The dependences of M^*_Z,app on w_Hb and w_BSA, calculated according to Eqs. 4 and 14, together with the best-fit hard-sphere radii of all three proteins, is plotted together with the data in Fig. 3 (solid curves).

Figure 3.

Effect of loading concentration of crowder protein on the apparent buoyant molar mass of dilute labeled FtsZ in the absence of Mg²⁺. HbCN: data, circles; calculated best-fit of effective hard sphere model described in text, thick line. BSA: data, squares; calculated best-fit, thin line.

Dependence of Gradients of Self-Associating FtsZ on Crowder Concentration.

The dependence of M^*_FtsZ,app on the concentrations of FtsZ, Hb, and BSA in the presence of 5 mM Mg²⁺, derived as described above from gradients comparable to those shown in Fig. 1C, are plotted in Fig 4 A and B (symbols). According to the theory of nonideal sedimentation equilibrium (10), the dependence of the apparent buoyant molar mass of a self-associating tracer component Z (FtsZ) on the concentration of a second non-associating crowder component C (where C is either Hb or BSA) is given by

4

where the indicated sum is over all i-mers of FtsZ, and w_Z,tot is the total wt/vol concentration of Z.

Figure 4.

Effect of loading concentrations of FtsZ and crowder protein on the apparent buoyant molar mass of dilute labeled FtsZ in the presence of 5 mM Mg. (A) HbCN; (B) BSA. Symbols, data; solid curve, best-fit of effective hard particle (sphere + spherocylinder) model described in text.

The self-association of FtsZ in the absence of crowder was previously reported (11) to be well-described at any fixed Mg²⁺ concentration by an indefinite linear self-association, with the equilibrium association constant for addition of monomer to an (i−1)mer being given by

5

where c_i is the molar concentration of i-mer, J = 3.3, and α = 0.15, independent of [Mg²⁺], and K is the (magnesium-dependent) asymptotic value of the equilibrium constant for association of monomer to i-mer in the limit of large i. It is assumed that the values of K calculated by using Eq. 5 for a given value of [Mg²⁺] are unchanged by the addition of an arbitrary concentration of protein crowder C that interacts with FtsZ by volume exclusion only. If the total volume fraction of FtsZ remains small, and only the concentration (volume fraction) of C may become large, it follows from excluded volume theory (26) that the stepwise association constants K_i will depend on the concentration of C in the following manner:

6

Cumulative association constants providing a more direct relationship between the concentrations of monomer and i-mer may be defined as follows:

7

By analyzing the dependence of both weight-average molar mass and weight-average sedimentation coefficient on total protein concentration, it was established that oligomers of FtsZ formed in the presence of GDP and Mg²⁺ are linear (11). For the purpose of calculating the equilibrium free energy of steric repulsion, each i-mer is therefore modeled as a hard spherocylindrical particle with radius r(i) and cylindrical length l(i), assumed proportional to (i−1). We define the scaled cylindrical length L(i) ≡ l(i)/2r(i) ≡ Δ(i−1). From conservation of volume, it follows that

8

9

The values of ln γ_i are calculated for all i by using Eq. 12, with R = r_i/r_C and L_i = Δ(i−1). Given the values of γ_i, the values of K_i may be calculated by using Eqs. 5 and 6. Conservation of mass is expressed as (11)

10

where c_Z,tot is the total concentration of FtsZ expressed in moles (protomer)/liter. Given the values of c_Z,tot and the Z_i, Eq. 10 may be solved numerically for the value of c₁ and subsequently for each of the c_i. The weight-average molar mass of FtsZ is then given by

11

Given the value of v̄_Z, M^*_Z,app may then be calculated by using Eq. 4 together with values of dlnγ_i/dw_C calculated by using Eq. 14.

Setting the molar masses, partial specific volumes, and effective hard sphere radii of crowder and FtsZ monomer equal to previously determined values (Table 1), Eq. 4, with terms calculated as described above, was fitted by nonlinear least-squares to the experimentally measured dependence of M^*_Z,app on w_Z and w_C plotted in Fig. 4 A and B. The best-fit values of the sole remaining undetermined parameters, Δ and K are presented in Table 1. The dependence of M^*_Z,app on w_Z and w_C, calculated as described above together with the best-fit parameter values given in Table 1, is plotted together with the data in Fig. 4 A and B.

In summary, we have found that a substantial body of experimental data, consisting of the dependence of the apparent buoyant molar mass of each of the crowders on crowder concentration, the dependence of the apparent buoyant molar mass of monomeric FtsZ (−Mg²⁺) on the concentration of each crowder, and the dependence of the apparent buoyant molar mass of self-associating FtsZ (+Mg²⁺) on the concentration of FtsZ and each crowder, may be accounted for to within experimental precision by a simple model. According to this model, HbCN and BSA interact with FtsZ exclusively via steric repulsion; BSA, HbCN, and monomeric FtsZ are represented by hard spherical particles of molecular size; and each oligomer of FtsZ is represented by a hard spherocylindrical particle, the length of which varies linearly with the number of protomers therein.

Discussion

Conventional measurement and analysis of sedimentation equilibrium in a solution containing a single macrosolute component provides a uniquely powerful means of characterizing both attractive and repulsive solute–solute interactions in solution (21, 27, 28). Recently developed tracer methods (10, 16, 29) permit analysis of interactions between molecules of two different solute species, even when one is highly dilute and the other highly concentrated. We have used the technique of nonideal tracer sedimentation equilibrium (10) to characterize the interaction of two different inert “crowder” proteins, HbCN and BSA, with FtsZ and to measure the effect of such interactions at high crowder concentration on the self-interaction of FtsZ.

The measured dependence of the apparent buoyant molar mass of each of the crowder proteins on its own concentration at concentrations of up to 150 g/liter is well accounted for by a simple model of steric repulsion in which each protein is represented by an effective hard spherical particle with a size close to that of the actual molecule (3, 20, 30). The molar masses and effective hard sphere radii of both Hb and BSA obtained by modeling the present data are in reasonable accord with values obtained earlier under comparable experimental conditions (20, 30–33).

Under conditions in which FtsZ does not self-associate (i.e., in the presence of GDP but in the absence of Mg²⁺), the dependence of the apparent buoyant molar mass of dilute FtsZ on the concentration of either crowder at concentrations up to 150 g/liter is well accounted for by a simple model of steric repulsion in which both crowder species and FtsZ are represented by effective hard spherical particles with sizes close to those of the respective molecules. This is a new result that provides firm evidence for the chemically inert nature of both crowding species under the conditions of the present experiments.

Under conditions in which FtsZ has been shown to self-associate to form linear oligomers of indefinite length (i.e., in the presence of GDP and Mg²⁺), the dependence of the apparent buoyant molar mass of dilute FtsZ on the concentration of both FtsZ (up to 1 g/liter) and crowder (up to 150 g/liter) is well accounted for by a simple model for steric repulsion in which both crowder and monomeric FtsZ are represented by the same effective hard spherical particles as in the non-associating (Mg²⁺-free) system, and each oligomer is represented by an effective hard spherocylinder with a cylindrical radius approximately equal to that of the monomer and a length varying linearly with the number of protomers in accordance with conservation of volume. The radius of a spherocylinder, calculated according to Eq. 9 with the best-fit value of Δ, depends slightly on the value of i, and tends toward an asymptotic limit of 1.1 times the best-fit hard sphere radius of monomeric FtsZ in the limit of large i. The difference between the two quantities is within the joint 95% confidence limit and is probably due to experimental scatter. With a confidence level exceeding 95%, the data exclude the hypothesis that oligomers may be represented by compact quasispherical particles (i.e., Δ = 0), in agreement with results obtained previously under ideal conditions (11).

Given the best-fit values of steric exclusion parameters (Table 1), one may calculate the effect of volume exclusion by added hemoglobin on each of the cumulative association constants Z_i by using Eqs. 12 and 7.‖ The results shown in Fig. 5** indicate that the effect of excluded volume on Z_i becomes extremely large with increasing values of i and w_C.‡‡ The effect of volume exclusion by added hemoglobin^∥ on the (true) weight-average degree of oligomerization of FtsZ and on the distribution of FtsZ among the various states of oligomerization, calculated via Eqs. 5-7, are plotted in Figs. 6 and 7. The shift toward higher states of oligomerization at higher crowder concentration is evident in Fig. 7. Decamers and higher oligomers, which are present only in vanishingly small concentrations in the absence of added crowder, account for more than one-third of total FtsZ at a crowder concentration of ca. 300 g/liter [i.e., a concentration comparable to the estimated total concentration of macromolecules in E. coli cytoplasm (34)].††

Figure 5.

Effect of increasing oligomer size and HbCN concentration on Z_i , calculated as described in the text.

Figure 6.

Effect of increasing concentrations of FtsZ and HbCN on the weight-average degree of oligomerization of FtsZ, calculated as described in text.

Figure 7.

Effect of increasing concentration of HbCN on the distribution of oligomeric species of FtsZ (1 g/liter), calculated as described in text.

The effect of excluded volume on the energetics of formation of large protein arrays formed by a highly cooperative (nucleation-controlled) process resembling condensation has been extensively studied both experimentally and theoretically, and is reasonably well understood at the present time (3, 35–38). In contrast, the only prior attempt to characterize the influence of excluded volume on an indefinite noncooperative self-assembly process appears to be a study by Lindner and Ralston of the effect of an inert polymer, dextran, on the self-assembly of spectrin (39). In this study, it was found that the addition of up to 200 g/liter of dextran increased the monomer-dimer association constant§§ by an amount in reasonable accord with the predictions of a structural model for volume exclusion. There are three major differences between the present study and that of Lindner and Ralston. (i) Whereas Lindner and Ralston quantitatively analyzed only the first step of self-association, we take into account all associations leading to oligomers present in significant abundance, i.e., those contributing significantly to the weight-average molar mass. (ii) Lindner and Ralston used the polymer dextran as a volume-excluding agent, whereas we employ two different globular proteins. Theoretical models of excluded volume interaction between polymers and proteins are not as well-founded theoretically as those for excluded volume interaction between globular proteins (3, 5, 26). (iii) Lindner and Ralston used an ad hoc structural model in which the size and shape of the effective hard-particle representation of each species was arrived at by via elimination of alternative models as incapable of fitting the observed effect of dextran on spectrin self-association. In contrast, the approximate size and shape of the effective hard-particle representation of each species in our model was derived from independent measurement of the nonideal behavior of the crowding species, or from characterization of the concentration dependence of sedimentation equilibrium and velocity of FtsZ in the absence of crowder, or both. The extent to which the best-fit values of structure-based parameters presented in Table 1 agree with independent structural information provides strong evidence for the physical resemblance of the model particles to the actual molecular species, and for the quantitative validity of our analysis of excluded volume interactions in this system.

The results and analysis presented here provide the strongest and most direct evidence to date that pure volume exclusion can significantly promote the self-assembly of proteins in macromolecularly crowded solutions (40). We acknowledge that additional nonspecific interactions between macromolecules, yet to be fully explored, are likely to exist in most physiological media (see, for example, refs. 41 and 42), and that it is therefore unrealistic to expect that all deviations between the behavior of macromolecules in dilute solutions and in, say, cytoplasm may be attributed to volume exclusion. Nonetheless, volume exclusion effects will be unavoidably present in such media, and the present work clearly demonstrates that such effects are of sufficient magnitude to require careful consideration before attempting to assign a physiological role to any macromolecular reaction that has been studied only under quasi-ideal (dilute) solution conditions.

Abbreviation

NITSE

nonideal tracer sedimentation equilibrium

Estimation of Thermodynamic Activity Coefficients Arising from Steric Repulsion Between Protein Solutes.

In the present work, we wish to estimate the activity coefficient (or, equivalently, the nonideal part of the chemical potential) of a molecule of solute species 2, modeled as either a sphere of radius r₂ or a spherocylinder of cylindrical radius r₂ and cylindrical length l₂, in a solution containing a volume fraction φ of a single solute (species 1) modeled as a spherical particle of radius r₁. For this case, the scaled particle theory of Cotter (43) yields the following relation:

12

where

R and L are scaled parameters defined by

and the volume fraction φ is given by

13

where N_A is Avogadro's number and M₁ and w₁ denote the molar mass and wt/vol concentration, respectively, of species 1. It follows from Eqs. 12 and 13 that

14