Factors Contributing to the Collision Cross Section of Polyatomic Ions

Jan 10, 2013 - ... Brian C. Bohrer , Doyong Kim , Kelly A. Servage , David H. Russell , David ... to data from ion mobility spectrometry and ab initio...
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Factors Contributing to the Collision Cross Section of Polyatomic Ions in the Kilodalton to Gigadalton Range: Application to Ion Mobility Measurements Thomas Wyttenbach, Christian Bleiholder, and Michael T. Bowers* Department of Chemistry and Biochemistry, University of California Santa Barbara, Santa Barbara, California 93106, United States ABSTRACT: The projected superposition approximation (PSA) method was used to theoretically evaluate the factors contributing to the cross section measured in ion mobility experiments and to study how the significance of these factors varies with ion size from diglycine to a 1 μm oil droplet. Thousands of PSA calculations for ∼400 different molecules in the temperature range from 80 to 700 K revealed that the molecular framework made up of atomic hard spheres is, as expected, a major component of the cross section. However, the ion−buffer gas interaction is almost equally important for very small peptides, and although its significance decreases with increasing ion size, interaction is still a factor for megadalton ions. An additional major factor is the ion shape: Fully convex ions drifting in a buffer gas have a minimal frictional resisting force, whereas the resisting force increases with degree of ion surface concaveness. This added resistance is small for peptides and larger for proteins and increases the ion mobility cross section from 0 to greater than 40%. The proteins with the highest degree of concaveness reach a shape-effected friction similar to, and sometimes larger than that of, macroscopic particles such as oil droplets. In summary, our results suggest that the transition from nanoparticle (with Lennard−Jones-like interaction with the buffer gas) to macroscopic particle (with hard sphere-like interaction) occurs at ∼1 GDa.

I

question is, which factors contributing to the cross section are important for which size of ion? And in particular: Where is the transition from molecule, where interaction with the buffer gas matters, to macroscopic object, where interaction does not matter anymore? Here we use a theoretical model, the recently developed “projected superposition approximation” (PSA) algorithm,16−18 as a tool to address these questions and apply the method to polyatomic ions ranging from small peptides to large protein complexes and beyond. No attempts are made here to further improve the PSA method, it is simply used to evaluate the cross section of a given molecular geometry and decompose it into a number of components: ion projection cross section for given (but temperature-dependent) atomic radii, cross section increase due to the superposition of atom−buffer gas interaction potentials, and cross section increase due to the deviation of the molecular shape from a fully convex body. Furthermore, we will see how these contributions are different for a small molecule compared to an oil droplet on the micrometer scale.

on mobility spectrometry (IMS) was almost exclusively employed either in analytical applications1 or in basic physical chemistry research studying the interaction of small (often monatomic) ions with various buffer gases 20 years ago.2 However, since then IMS has emerged as a powerful tool to probe the structure of polyatomic ions.3−6 With the development of new ionization methods such as matrix-assisted laser desorption/ionization (MALDI) and electrospray ionization (ESI),7−9 IMS has been applied to increasingly larger polyatomic ions. Today IMS applications extend to ions in the megadalton range.10−15 IMS as a tool to get structural information of molecules is strongly coupled to evaluating the ion mobility theoretically for proposed model structures. For macromolecular ions this is a great challenge. The computation involves an accurate evaluation of the collision cross section of the proposed structure for comparison with experiment. Some of the challenges are that molecules are not hard spheres; that the atoms in the molecule are not hard spheres, either; that molecular “size” is therefore an inherently difficult to define quantity; and that the IMS cross section is the result of a complicated scattering process and not a projection cross section. However, the hope is that not all aspects of computing the ion mobility theoretically get more complicated with increasing molecular size. In fact, certain problems should disappear for sufficiently large systems. For example, any complications due to the exact nature of the ion−buffer gas interaction potential will vanish for very large particles. The © 2013 American Chemical Society

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BACKGROUND One of the most fundamental quantities in the kinetic theory of gases is the collision cross section σ. Accurate knowledge of this Received: October 5, 2012 Accepted: January 10, 2013 Published: January 10, 2013 2191

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Figure 1. Cartoon capturing various aspects of an ion−buffer gas collision. (a) Collision cylinder of a spherical ion moving (upward) in a buffer gas. The base of the cylinder (dark gray area) is the collision cross section σ = πa2 in the hard-sphere limit. (b) Schematic representation of a collision of two spherical particles in the center of mass frame (center of mass at rest). Prior to the collision, the two particles approach each other from the top and bottom, respectively. Their approach is off center as indicated by the impact parameter b. After the collision, both particles have changed their direction of motion with the deflection angle θ indicated. (c) Buffer gas particle colliding from the top (parallel to the z-axis) with a larger particle. The point of reflection is projected onto the xy-plane where it is surrounded by a small area element dx dy. (d) Buffer gas trajectories with a scattering angle of 180° for the reflection off a smooth sphere and off a particle with concave surface elements. (e) Specular vs diffuse reflection. The trajectory of an incoming particle (dashed line) continues along the black arrow after specular reflection. After diffuse reflection, the particle is scattered into a random direction with the probability distribution schematically indicated by the length of the gray arrows. (f) Potential energy EPOT showing schematically the Lennard−Jones-like interaction potential curve (fat line) of two particles as a function of distance r between the particles. The potential rises very steeply on the left due to the hard spherelike repulsive wall giving rise to the dark gray area of the projection cross section σ. At longer range, the attractive part of the potential dominates and contributes the fuzzy shaded ring to σ with its thickness depending on the potential shape and collision energy. The momentum transfer cross section Ω is not only given by σ but also depends, for a buffer gas particle approaching from the top, on the surface makeup of the cross-hatched area on top of the particle. For a smooth convex surface, Ω = σ, whereas concave surface elements increase Ω over σ.

section Ω is readily defined for any type and size of particle and is more readily accessible by experiment even for very small particles. The experiment involves evaluation of the resisting force of a particle drifting in a buffer gas under the influence of an accelerating field (gravity, electric field). The magnitude of the momentum transfer in a single collision, given by (1 − cos θ), covers values from 0 (no scattering) up to a maximum of 2 (head-on collision with momentum p of particle X turning into −p in the center of the mass frame).2 Hence, whereas the calculation of σ involves integration over the projection area elements dx dy

quantity is essential in chemical kinetics for the evaluation of the collision frequency, for example. The quantity σ has the dimension of an area and is related to the size of the collision partners. For a selected spherical particle X colliding with buffer gas particles, the collision cross section corresponds to the area of the base of a cylinder aligned along the velocity vector of X. In simple hard-sphere terms, buffer gas particles inside the cylinder will intersect with the present trajectory of X and particles outside the cylinder will not collide with X (Figure 1a). For a macroscopic spherical particle, σ and the collision radius a are essentially given by the particle radius and can readily be measured using microscopy, for example. The size of the buffer gas can be neglected. σ = πa

2

σ=

(1)

∫ ∫ dx dy

(2)

in the calculation of Ω (Figure 1c), each cross section element dx dy (Figure 1c) is scaled by the momentum transfer occurring at the corresponding position (x,y)

For small particles, however, it is much harder to measure σ. The quantity which ultimately determines whether a collision between two particles took place or not is the scattering angle θ, the change in direction of the trajectories of the two particles involved (Figure 1b). For ideal hard spheres, the situation is simple and two particles either hit or miss. However, small real particles on the scale of atoms and small molecules cannot be considered hard spheres. Instead, there is a long-range (Lennard−Jones-like) interaction between the two particles without any sharp cutoff at any distance. Hence, the definition of the size of the collision cylinder becomes less clear. Does a deflection of 10° or 1° or 0.1° constitute a collision? Where is the cutoff? Hence, the lack of a clear cutoff between collision and miss results in a poorly defined quantity σ for real (nonhard sphere) particles. However, the related momentum transfer cross

Ω=

∫ ∫ (1 − cos θ) dx dy

(3)

with θ = θ(x,y) being a function of x and y. Therefore, a near head-on collision resulting in a near maximum change of momentum contributes more to Ω than a glancing collision with a small deflection angle. However, since the number of glancing collisions is generally much larger than that of near head-on collisions, glancing collisions contribute significantly overall. Note that whereas the integration in eq 3 is through the entire space (from −∞ to +∞), the integration in eq 2 requires clear boundaries for σ to take on a finite value. Hence, for an object without clear boundaries, σ can neither be unambigu2192

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ously measured nor calculated without assumptions about the object’s boundaries. For a hard sphere with clear boundaries and with a collision radius a, the scattering angle as a function of impact parameter b (Figure 1b) is given by θ(b) = 2 cos−1(b/a) and eq 3 turns into Ω=

∫ (1 − cos θ)2πb db = πa

2

=σ

is offset by the resisting force due to collisions of the ion with the buffer gas F=

4 2 πa dv⟨c⟩ 3

(4)

K=

⎛4 π⎞ ⎜ + ⎟πa 2dv⟨c⟩ ⎝3 6⎠

(10)

(diffuse reflection)

(11)

with σ = πa . Equation 11 clearly displays how the details of the collision process can lead to a substantial discrepancy between the two quantities σ and Ω, the collision cross section and the momentum transfer cross section, respectively. It should be noted that the mobility-derived experimental measure of the momentum transfer cross section, ΩIMS (eq 10), is an average value with respect to collision energy and ion orientation. Except for hard spheres, θ (and therefore Ω; see eq 3) is a function of collision energy and consequently the IMS experiment delivers an energy distribution-weighted average of Ω values. Furthermore, for nonspherical ions, the experimental value is also an average over all possible orientations in space. Hence, for real molecular systems, eq 3 applies only for one particular collision energy and one particular ion orientation. Therefore attempting a theoretical calculation of the momentum transfer collision cross section (or collision integral as it is sometimes called), ΩCALC, involves evaluation of eq 3 as a function of energy and orientation and the resulting Ω values have to be properly averaged.2,24 Since IMS makes use of the resisting force of drifting ions, it is relevant to explore the limitation of eqs 5 and 6. In the derivation of these equations it is assumed that the resisting force is entirely due to the inertia of the buffer gas particles. This situation applies when the particle dimension a is small compared to the mean free path λ. In the other extreme case, when a is large compared to λ, the resisting force is determined by the viscosity η of the buffer gas. In other words, the retarding force is determined by how well the gas particles are able to get out of the way, how much friction there is between buffer gas particles. In this model, F, the “drag resistance”, is given by Stokes’ law 2

(5)

(6)

F = 6πηav

(7)

(12)

which assumes that a thin layer of buffer gas travels along with the particle X at the same speed v. Note, that F is proportional to the radius a in this case and not to πa2 as in eqs 5 and 6. If there is some slip between X and the first layer of buffer gas particles which is increasingly more probable with increasing λ/a ratio, F becomes slightly smaller than given by eq 12. This

The proportionality constant K is the ion mobility by definition.2 The velocity v is constant because the forward accelerating force of an ion with charge q F = qE

2π 1 μkT Ω IMS

Ω IMS = 1.39σ

A range of careful experiments involving small solid and liquid spheres (∼1 μm diameter) made of a variety of materials, such as glass balls and oil droplets, show diffuse reflection is dominant for these systems in air and other gases. However, very generally all these experiments indicate the presence of a small fraction (approximately 10%) of specular reflections.20,22 The IMS method is a convenient way to explore the resisting force of any charged particle, including a particle as small as an atomic ion, drifting in a buffer gas under the influence of a weak electric field. Under typical IMS conditions, ions travel with a constant velocity v which is given by the electric field strength E v = KE

3q 16N

which is used in the ion mobility community to connect the ion mobility K with the average momentum transfer cross section ΩIMS.2,23 In these equations, N is the buffer gas number density; μ = m × mion/(m + mion) the reduced mass of the collision partners with masses m (buffer gas) and mion (ion); k the Boltzmann constant; and T the buffer gas temperature. Using the relationships d = mN and ⟨c⟩ = (8kT/(πm))1/2 and assuming mion ≫ m, the equivalence of eqs 5 and 9 is apparent: ΩIMS = σ = πa2. The momentum transfer cross section is identical with the collision cross section in this case. However, comparison of eq 9 with eq 6 (diffuse reflections) yields the relationship

where d is the buffer gas density and ⟨c⟩ the average buffer gas particle speed.20 As discussed above, assuming diffuse scattering for the same size sphere results in an increased resisting force of20 F=

(9)

Equations 7−9 lead to the well-known equation

Hence, for a hard sphere the momentum transfer cross section is identical to the projection cross section.19 Ω of a real particle X corresponds to the cross section a hard sphere would have such that the resisting force of the hard sphere drifting in a buffer gas matches that of X. For a fully convex object with a smooth surface, very few collisions lead to near maximum momentum transfer (with θ near 180°). For an object with dents in the surface, on the other hand, there are more possibilities for obtaining large scattering angles (Figure 1d) and the momentum transfer cross section is increased over that of a fully convex object with the same projection cross section. For a very rough surface, the value of θ(x,y) may change very rapidly for very small changes in x and y. If the dimensions of the surface roughness are much smaller than the particle dimension, it may appear as if the buffer gas is scattered randomly in every direction at any given position (x,y). In this description of the scattering process, there is a distribution of reflection angles for every position (x,y). This leads to the phenomenon known as diffuse reflection (Figure 1e).20 The resisting force acting on a particle moving through a buffer gas has been studied experimentally and theoretically more than a century ago by Langevin, Knudsen, Millikan, and others.20−22 For a sphere of collision radius a and drift velocity v, theory indicates a resisting force due to specular scattering given by

F=

4 8kT Ω IMSNv μ 3 π

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Table 1. Mass and Dimensions of Various Particles along with the Air Pressure Pλ=a Where the Mean Free Path Equals the Particle Collision Radius particle

mass (Da) 1.1 × 10

bradykinin sphereb sphereb sphereb ubiquitin sphereb sphereb sphereb GroEL oil dropletc oil dropletc oil dropletc

collision radius (a) (Å) 8.9 12.6 17.8

8.6 × 103 25.2 35.7 50.5 8.0 2.3 2.3 2.3

× × × ×

105 106d 109d 1012d

collision cross section (πa2) (Å2)

ring area 2πa × 0.1 Å (Å2)

ratio ring:πa2

Pλ=a(atm)

6 8 11

2.242% 1.585% 1.121%

>10 >10 >10

16 22 32

0.793% 0.560% 0.396%

>10 >10 >10

63 630 6 300

0.200% 0.020% 0.002%

10 1.0 0.1

a

3

100.0 1 000.0 10 000.0

242 250 500 1 000 1 000a 2 000 4 000 8 000 20 700a 31 400 3 140 000 314 000 000

Experimental ΩIMS values for bradykinin (1+), ubiquitin (7+), and the GroEL tetradecamer complex (69+) in helium, respectively.25−27 Hypothetical sphere with a cross section of 250, 500, 1000, 2000, 4000, and 8000 Å2, respectively. cHypothetical oil droplet of 0.01, 0.1, and 1.0 μm radius, respectively. Oil droplets with a ≥ 0.25 μm have been studied by Millikan.20 dEstimated using a typical oil density of 0.9 kg/L. a b

effect led Millikan20 to summarize the different situations of resisting forces (eqs 5, 6, and 12) in eq 13 F=

thereby correctly accounting for a collective size effect: the more atoms are near x, the larger is P(T,x) and the more likely it is that the area element at position x counts toward the projection area. The physical explanation for this phenomenon is that the interaction potential well gets deeper with increasing number of atoms thereby effectively increasing the size of each atom in the polyatomic system.28 The shape factor ρ is a measure for the concaveness of the molecule relative to a purely convex molecule of the same size. Whereas a lot of details have to be taken care of in the computation of ρ,16−18 the basic concept is simple: ρ is essentially the ratio of the actual molecular surface area Amol of a molecule to the surface area Ace of the convex envelope of the molecule.

6πηav 1+

λ (A a

+ B e−Ca/ λ)

(13)

For appropriate values for the constants A, B, and C, this equation describes F correctly both for small and large λ/a values with a linear a dependence for λ ≪ a and an a2 dependence for λ ≫ a. Table 1 indicates that at 1 atm air pressure the a ≈ λ limit is only reached for gigadalton ions. Hence, the formalism used in eqs 9 and 10 holds under all typical IMS applications even for very massive ions, and the transition to friction following Stokes’ law is not a concern in IMS.

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ρ = A mol /Ace

METHODS Here we use the recently introduced PSA method to obtain a theoretical momentum transfer cross section, ΩCALC, for a given polyatomic ion structure as a function of temperature. The PSA algorithm has previously been described in detail.16−18 Briefly, Ω is approximated by an orientation-averaged PSA projection cross section ⟨ΩPSA⟩ which is scaled by a shape factor ρ ΩCALC ≡ Ω(1,1) PSA = ρ⟨Ω PSA ⟩

Hence, a fully convex particle yields a value of ρ = 1, whereas concave surface elements push ρ to values >1. Obviously a surface with dents is larger than one without the dents. However, in molecules with deep channels connecting to large interior cavities very few buffer gas collisions actually sample this large interior surface. In fact, the number of buffer gas particles interacting with the interior surface is proportional to the area of the opening to the cavity. Therefore, the PSA algorithm also evaluates the “visibility” of surface elements. If surface elements are not visible from the outside they are not counted toward the molecular surface area. In previous work,16−18 it has been shown for a range of molecules that Ω(1,1) PSA agrees well with experimental ΩIMS data and with more sophisticated (but much more time-consuming) theoretical methods24 to compute ΩCALC. Here we employ thousands of PSA computations involving ∼400 different polyatomic ions from 100 Da to 0.5 MDa and covering a temperature range from 80 to 700 K for our analysis. A series of gas-phase peptide geometries for diglycine to hexaglycine (protonated and sodiated), the (protonated) pentapeptides GRGDS and SDGRG, angiotensin II (charge state 2+), bradykinin (1+ and 2+), neurotensin (2+ and 3+), and insulin B chain (3+ and 4+) are the result of theoretical molecular modeling work.26,28 Protein and protein complex structures are downloaded from the Worldwide Protein Data Bank (wwPDB)29 and comprise the test set TS396 previously used.17 This set includes both solution NMR structures and X-

(14)

Both ρ and ⟨ΩPSA⟩ are generally a function of temperature. The calculation of ⟨ΩPSA⟩ is based on evaluating atomic collision probabilities pj(T,x) which equal 1 near the center of an atom j (up to a distance equal to the atomic collision radius qj) and tail off to zero at a larger distance. The atomic collision radius qj is a function of the temperature and the Lennard−Jones potential well depth εLJ and position rLJ for the interaction between buffer gas particle and atom j in the ion qj = qj(T , rLJ , εLJ)

(15)

In the projection of the molecule onto a plane, the collective molecular collision probability at a position x on the plane is assessed as a superposition of all atomic contributions P(T , x) =

∑ pj (T , x) j

(17)

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Å2 at 300 and 80 K, respectively, is considered an interaction effect. The question then is: At which ion size does the interaction effect become insignificant and atoms can be considered hard spheres? To address this question, it is useful to study the effect of a 0.1 Å increase in the atomic collision radius on the molecular cross section as a function of molecule size. A spherical ion with a cross section of 1000 Å2 has a molecular radius of 18 Å (Table 1). An increase of all atomic radii in the ion by 0.1 Å is essentially equivalent to adding a ring of 0.1 Å thickness to the outside of the 1000 Å2 circular area and the resulting cross section is 1.1% larger (Table 1). However, for a smaller ion a 0.1 Å uncertainty in the radius leads to a larger cross sectional error: 1.6% and 2.2% error for a 500 and 250 Å2 ion, respectively. For comparison, a 1000 Å2 ion corresponds about to the size of ubiquitin, an 8.6 kDa protein;25 a 250 Å ion is about as large as bradykinin, a 1.1 kDa peptide with nine residues.26 This effect of a 0.1 Å increase in ion radius on the ion cross section is graphically shown in Figure 2 as a function of ion size

ray crystal structures. Water molecules are removed from and hydrogen atoms added to all PDB X-ray structures before the PSA analysis. Ω(1,1) PSA values are converged to 0.7% accuracy.

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RESULTS AND DISCUSSION IMS is one of a range of methods to obtain information regarding the size of a molecule. However, since atoms and small molecules are not hard spheres and do not have a sharp outer boundary, “size” is a poorly defined quantity for these systems and it depends on the type of experiment used to measure the size. For instance, the proton is an elementary particle and electron−proton scattering experiments yield a proton radius of rp < 1 fm.30,31 The corresponding area πrp2 is 5 kDa molecules. The superposition effect is more pronounced at temperatures below 300 K because the nonhard sphere nature of the interaction potential becomes correspondingly more evident. For diglycine, the 6% effect at 300 K shown in Figure 3b increases to 50% at 80 K; for bradykinin, the values are 1.7% and 14% at 300 and 80 K, respectively. At 80 K, the effect levels off for ions ≥10 kDa. Table 2 compiles ⟨ΩPSA⟩ and ⟨ΩHS PSA⟩ values for a range of molecules and for a number of temperatures. ⟨ΩHS PSA⟩ is a superposition-free cross section calculated based on atomic hard spheres with radii qj (eq 15) identical to those used in the ⟨ΩPSA⟩ calculation. Hence, the difference ⟨ΩPSA⟩ − ⟨ΩHS PSA⟩ is related to the superposition effect. The breakdown in Table 2 suggests that the 300 K diglycine cross section of 59.7 Å2, for instance, is composed of a 49.7 Å2 hard sphere-based core 2196

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area in Figure 1f matters. It makes a difference whether the cross-hatched surface (see top of sphere in Figure 1f) is smooth or rough, flat, convex, concave, or rippled. The scattering of particles hitting the surface from the top is obviously different for different surface makeups (Figure 1d) and average momentum transfer is increased for a rough surface compared to a smooth convex surface. As outlined in the Methods section, in the PSA algorithm the effect of the average surface makeup of a molecule is evaluated in the computation of the shape factor ρ. An object with a smooth convex surface has a shape factor of 1, and for objects with concave features ρ is >1. In practice it is found that ρ assumes values from 1.0 through 1.5. Figure 4 shows a scatter

Figure 4. PSA shape factor ρ (300 K, helium) for a range of molecular structures. Data below 500 Da are based on model structures for protonated and sodiated oligoglycines Gly2−Gly6. Data in the 500− 1400 Da range include calculations for six peptides with dozens of model structures for each peptide giving rise to the spread in the ydirection at a given mass (some peptides overlap with each other in mass). The remaining data capture ∼400 protein and protein complex structures downloaded from the PDB data bank. The limit for spherical macroscopic particles of 1.36 is noted (see text).

Figure 3. Data displaying the superposition effect: (a) 300 K interaction effect shown in Figure 2 and expanded down to 100 Da. The set of green circles includes the data of Figure 2 supplemented by values obtained for a series of peptides ranging from protonated and sodiated diglycine to insulin B chain. Peptide data are averages over a number of molecular mechanics structures for each peptide. The spread in the data is typically ±1%, e.g., the bradykinin result is (7.6 ± 1.2)%. The green curve (relative contribution of a 0.45 Å wide ring matching the mion> 5 kDa data) follows the equation indicated in the figure with a being the approximate particle radius (a ∝ mion1/3). Bradykinin and ubiquitin data are used to relate particle size to particle mass (Table 1). Similar curves for rings with widths of 0.40, 0.35, and 0.30 Å are shown in black which agree better with the low-mass data. (b) Deviation of the interaction data (green circles in panel a) from the 0.45 Å ring data (green curve in panel a). The data indicate that the deviation reaches zero for ions above about 5 kDa. For ions significantly below 5 kDa, the superposition term is smaller (relative to the molecule surface), resulting in a decreased interaction effect and smaller atomic collision radii and therefore a larger deviation from the 0.45 Å ring curve (toward smaller ring widths).

plot of ρ versus molecule size (mass). It can be seen that small molecules in the 100−1000 mass range tend to have a near convex surface with ρ ≈ 1. With increasing molecule size, the range of possible ρ values increases and appears to settle in the 1.15−1.45 region for most molecules above 10 kDa. Hence, concaveness contributes significantly (∼30% on average) to Ω for proteins >10 kDa. The ρ values listed in Table 2 indicate that the surface concaveness decreases slightly with decreasing temperature. The reason for this temperature effect is that atomic collision radii get larger as T gets smaller tending to smear out minor surface variations existing in the molecular framework. An equipotential surface a distance away from the molecule (in the attractive long-range part of the interaction potential curve which is sampled in low-energy collisions)34 is a lot smoother than the surface of a bumpy repulsive wall sampled in highenergy collisions. Despite the trends apparent in Figure 4, the shape contribution to Ω cannot be predicted with any accuracy simply based on molecular mass given the broad distribution in ρ for a given mass in the 10 000 to 150 000 Da range. (This degree of variation probably exists for higher mass, but we have insufficient data points to say so definitively.) The reason for this is that different proteins have different surfaces, some rougher than others. A protein with lots of dents and crevasses

2 (⟨ΩHS PSA⟩700K), a 4.2 Å ring around the core due to interaction of every individual atom of diglycine with helium at 300 K 2 HS (⟨ΩHS PSA⟩300K − ⟨ΩPSA⟩700K), and a second ring of 5.9 Å due to the collective interaction of diglycine with helium (⟨ΩPSA⟩300K − ⟨ΩHS PSA⟩300K), the superposition effect. At this point we understand that the collision cross sections depend on the size of the molecular framework and that an extra layer is added around the skeleton due to interaction with the buffer gas. The thickness of this layer depends primarily on temperature but also on the size of the skeleton due to the superposition effect. However, Ω is in addition affected by the shape of the molecular framework, because the resisting force probed in a drift experiment (such as IMS) depends on the distribution of scattering angles in individual collisions. Hence, the makeup of the molecular surface giving rise to the dark gray

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relevant for σ for molecules in the size range considered here (test set TS396). Therefore and since clear particle boundaries exist only for sizes ≥1 GDa (Figure 2), Ω seems simply to be the best quantity to approximate σ and to describe “size” in the context of collisions for particles smaller than 1 GDa. In summary, the particle shape strongly contributes to the relationship between the collision cross section σ and the momentum transfer cross section Ω, which are related but not identical quantities. σ is the relevant quantity to calculate the collision frequency, but σ can only be measured directly for large (micrometer) particles, whereas Ω is a readily measurable quantity for any size ion. For macroscopic particles, Ω is measured to be a factor of 1.36 greater than σ (eq 18).20 Using particle boundaries defined by the collision radius a = (ΩIMS/π)1/2 results in the obvious identity Ω = σ for any system with a potential of truly spherical symmetry such as a real atomic ion or an ideal hard sphere (eq 4).2,19,20 A PSA shape factor ρ near 1 for very small polyatomic ions (e.g., diglycine) indicates the two quantities are near equal in this case, Ω ≈ σ, provided that realistic (and temperatureappropriate) atomic collision radii (and superposition effects) are used to define the molecule “surface”. Finally, a relatively poorly defined value of ρ for proteins in the 10−150 kDa range indicates a poorly defined relationship between Ω and σ (due to a varying degree of surface scattering in that size range of molecules).

in the surface has a higher resisting force (Figure 1d) and a larger shape factor (eq 17) than a protein of identical mass with a smoother surface. Nevertheless, it is interesting that macroscopic “spherical” molecular assemblies such as liquid oil droplets and solid glass spheres (with radius a ≈ 1 μm) experience a resisting force which appears universally to be a factor of 1.36 ± 0.01 larger than expected for a hard sphere.20 Hence, the momentum transfer cross section is larger than the size of the sphere Ω ≈ 1.36πa 2

(large spheres)

(18)

due to predominantly diffuse reflections of the buffer gas on the sphere surface. Since for the case of a hard sphere, both quantities πa2 and ⟨ΩPSA⟩ correspond to the sphere projection cross section, a comparison of eqs 14 and 18 suggests an expected value of ρ ≈ 1.36 for macroscopic particles. It is interesting that the larger ρ values seen for molecules in the 10 kDa to 1 MDa range (Figure 4) are of the same order as the values 1.39 and 1.36 given in eqs 11 and 18, although there is a spread of ±20% in the values. This indicates that the PSA algorithm captures the physics of the collision process in essence correctly and that scattering from a protein surface will begin to approximate (on average) macroscopic diffuse scattering for protein complexes at some point above the megadalton range, an approximation that has recently been suggested in the literature44 for systems including the 800 kDa GroEL complex.14 Applying this approximation has lead the authors of the recent GroEL study to scale down their IMS data (20 700 Å2) by a factor of 1.36.14,27 However, it is important to note this approximation has been shown experimentally to hold only in the gigadalton range where both Ω and σ can be measured unambiguously in independent experiments, Ω by IMS, and σ by microscopy, for example. We suggest that the ion mobility community refrain from scaling IMS data and continue to report momentum transfer cross sections ΩIMS as defined in eq 10 even for very large ions rather than attempting to convert Ω into σ using a generally ill-defined conversion factor (Figure 4). For protein molecules, an estimate of the Ωto-σ ratio relies on model parameters and computations. In the above-mentioned GroEL study,14 an exact hard-sphere scattering (EHSS) model45 was employed to estimate the scattering from the GroEL surface. In this model, the ion is composed of atomic hard spheres and helium is scattered off the resulting molecular surface. Potential problems with this approach are that constant atomic radii do not account for the superposition effect (see above) and that the roughness of the molecular surface is artificially increased due to the nature of intersecting hard spheres. Our experience with the EHSS model applied to molecules in the kilodalton range confirms these concerns:16 The surface-effected increase of the resisting force is generally overestimated in the EHSS model. In the context of attempting a Ω-to-σ conversion, it is important to realize that not only is it hard to measure σ for submicrometer systems, this quantity is also essentially irrelevant for a particle with ill-defined boundaries. Although the collision frequency, a relevant quantity in reaction kinetics, is ultimately determined by σ (and not Ω), the collision cross section σ is in fact an abstract quantity for systems with an unknown fraction of scattering due to surface roughness. The reason why σ is hard to define is that the collision event itself is hard to define in the absence of clear particle boundaries. Also neither IMS nor X-ray crystallography nor any microscopy methods are in a position to precisely outline the boundaries

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CONCLUSIONS We use the PSA model as a tool to analyze the factors contributing to the cross section ΩIMS measured by IMS methods as a function of ion size from diglycine to the 800 kDa GroEL protein complex to Millikan’s oil droplets. We find four important factors. (1) The size of the molecular skeleton (framework of atomic hard spheres) of a polyatomic ion accounts for a significant fraction of ΩIMS even for small peptides. (2) However, the ion−buffer gas interaction is a near equally important factor for very small peptides such as diglycine even for the most hard spherelike buffer gas helium. For larger molecules, interaction is (in relative terms) less important. Even though the ion−buffer gas interaction gets stronger with increasing molecular size, the relative contribution to ΩIMS decreases with size. Our data indicate the interaction effect is expected to become negligible for ions approaching the gigadalton range. (3) The superposition effect increases the ion−buffer gas interaction with increasing number of atoms in the polyatomic ion. This increased interaction increases ΩIMS by several percent for small peptides drifting in helium. The effect levels off for ions >5 kDa becoming a constant fraction of the interaction at 300 K. (4) The shape of the ion has a significant effect on the magnitude of momentum transfer and therefore on ΩIMS. Average momentum transfer is minimal for fully convex ions and increases with increasing degree of concaveness. Rotationally averaged, small peptides are essentially fully convex and surface concaveness is only a small factor affecting their cross section. The larger protein molecules have more pronounced concave surface elements leading to increased diffuse scattering which is reflected in PSA ρ values in the 1.15−1.45 range. Macroscopic spheres (1 μm diameter) such as oil droplets have been measured20 to have a factor of 1.36 increased friction over that theoretically expected for a smooth sphere. While actual data is very sparse for molecular assemblies with masses above 1 MDa, when it does 2198

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Analytical Chemistry

Article

become available we expect ρ to eventually begin to converge on the macroscopic limit of 1.36 for protein-based assemblies. However, since the ion−buffer gas interaction is still a factor for megadalton ions, their “size” is still a poorly defined quantity. This is especially true for buffer gases other than helium. Preliminary data in our lab for nitrogen buffer gas37 suggest an interaction effect approximately 4 times that of helium and larger effects are expected for larger buffer gases. Therefore the degree of scattering due to surface roughness cannot in general be evaluated experimentally with sufficient accuracy for these ions for it to be predicted based on a general formula. Consequently, we suggest that no attempt be made to correct experimental ΩIMS values by any shape factor to arrive at the “true” size of the ion. Our study indicates that the transition from nanoparticle (where interaction matters) to macroscopic particle (where interaction does not matter) occurs around 1 GDa (106 Å2) corresponding roughly to a 0.1 μm diameter oil droplet.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We acknowledge the National Science Foundation under Grant CHE0909743 and the Air Force Office of Scientific Research under Grant FA9550-11-1-0113 for support of this work.

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