Shape Correlations, Structure-Based Predictors and Di

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Bringing Molecular Dynamics and Ion-Mobility Spectrometry Closer Together: Shape Correlations, Structure-Based Predictors and Dissociation Alexander Kulesza, Erik G. Marklund, Luke MacAleese, Fabien Chirot, and Philippe Dugourd J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.8b03825 • Publication Date (Web): 01 Aug 2018 Downloaded from http://pubs.acs.org on August 2, 2018

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The Journal of Physical Chemistry

Bringing Molecular Dynamics and Ion-Mobility Spectrometry Closer Together: Shape Correlations, Structure-Based Predictors and Dissociation Alexander Kulesza

a,b*

d

a,b

c

, Erik G. Marklund , Luke MacAleese , Fabien Chirot , Philippe Dugourd

a,b

a. Université de Lyon, F-69622, Lyon, France. b. CNRS et Université Lyon 1, UMR5306, Institut Lumière Matière c. Univ Lyon, Université Claude Bernard Lyon 1, Ens de Lyon, CNRS, Institut des Sciences Analytiques UMR 5280, F-69100, Villeurbanne, France d. Department of Chemistry – BMC, Uppsala University, Box 577, SE-751 23, Uppsala, Sweden *E-mail: [email protected]

Abstract Unfolding of proteins gives detailed information about their structure and energetics and can be probed as a response to a change of experimental conditions. Ion-mobility coupled to native mass spectrometry is a gas-phase technique that can observe such unfolding in the gas-phase by monitoring the collisional cross section (CCS) after applying an activation, for example by collisions (collision-induced unfolding, CIU). The structural assignments needed to interpret the experiments can profit from dedicated modelling strategies. While predictions of ion-mobility data for well-defined and structurally characterized systems is straightforward, systematic Free-energy calculations or biased molecular dynamics simulations that employ IMS data are still limited. The methods with which CCS values are calculated so far do not allow for analytical gradients needed in biased molecular dynamics (MD) and further, explicit CCS calculations still can pose computational bottleneck – when integrated into MD-bioinformatics workflows. These limitations motivate to revisit known correlations of the CCS with the aim to find computationally cheap, versatile but still at least semi-quantitative descriptions of the CCS by pure structural descriptors. We have therefore investigated the correlation of CCS with the key structural parameter often used in computational unfolding studies – the gyration radius – for several small monomeric and dimeric proteins. We work out the challenges and caveats of the combinations of configurational sampling method and CCS-calculation algorithm. The correlations were found to be sensitive to the generation conditions additionally to the system topology. To reduce the amount of fitting to be undertaken we devise a simple structural model for the CCS that shares some commonalities with the hard-sphere model and the projection algorithm but is designed to take unfolding into account. With this model, we suggest a two-point interpolating function rather than fitting a large dataset, at only little deterioration of the predictive power. We further proceed to a model with composition and structure dependence that builds only upon the gyration radius and the chemical formula to apply the found CCS scaling behaviour – the scaled macroscopic sphere (sMS) predictor. We demonstrate its applicability to describe unfolding and also its transferability for a larger set of structures from the RSCPDB. As we have found for the dimeric systems, that shape correlations with one global descriptor qualitatively break down, we finally suggest a recipe to switch between global and fragment-based CCS prediction, that takes up the ideas of coarsegraining protein complexes. The presented models and approaches might provide a basis to boost the integration of structural modelling with multistage IMS-experiments, especially in the field of large-scale bioinformatics or “on-the-fly” biasing of MD, where computational efficiency is critical.

22–25

is growing in collision-induced unfolding profiles (CIU) 8,26 popularity. This technique (and related techniques, see Ref. ) can provide the unfolding energetics along the pathway from the folded to unfolded state through all accessible intermediates. The obtained energetic signature can be more specific than a mere CCS value. Thus, such methods can serve for in-depth structural analyses and contribute an integral part 27 of the advancement of gas-phase structural biology nowadays.

INTRODUCTION Ion mobility spectrometry (IMS) coupled to mass spectrometry (MS) is a gas-phase technique, that finds a growing number of 1–3 applications to determine biomolecular structures , ranging 4–7 2,8,9 10–12 from peptides , proteins , protein aggregates and 13–17 complex biomolecular assemblies . Through its complementarity, IMS is becoming frequently used as a tool to validate condensed phase results, e.g. size distributions of Amyloid-β oligomers after cross-linking and gel-electrophoresis 18 separations . IMS experiments are often interpreted by the concept of a collisional cross-section (CCS), owing to its roots in 19 collision theory, as a measure for size and shape of an ion . Interpreting Ion-mobility profiles with several features can provide insight into structural heterogeneities like different isomers, conformers or larger-scale structural differences, given that they do not interconvert in course of the measurement. Performing these experiments in several stages, e.g. applying 8 some activation to induce unfolding events can reveal thermodynamic properties of the structural transitions in the 8,20,21 gas-phase unfolding pathway . Indeed, the analysis of

Theory can provide missing atomistically precise structural information. In general, the structural assignment of the IMSprofiles’ features makes frequent use of a computational approach: prediction of IMS cross sections upon structural models. In several cases, the approximate arrangement of units 11 in an aggregate (e.g. Amyloid-β oligomer) can be identified even when coarse structural representations are used. Nevertheless, for many species, more involved modelling of structural differences such as conformational changes at the atomistic level is needed. Thanks to the advances in bioinformatics approaches (i.e homology modeling) and molecular dynamics (MD) simulation

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techniques, full-detail atomistic models of numerous proteins can be created and refined. Likewise, a hierarchy of predictive models for the CCS is available that bases on the calculation of the momentum transfer integral in statistical collision 19,28 theory . The most prominent methods with increasing complexity, in the way this integral is evaluated, are the 29 projection approximation (PA) algorithm , exact hard-spheres 30 31 scattering (EHSS) method and the trajectory method (TJ) , of which the latter two explicitly calculate collision trajectories with the buffer gas atoms. Lately, scaled and optimized PA implementations were devised, streamlined to tackle also big 32,33 systems or high number of models with better accuracy .

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investigation of correlations regarding molecular shape and collisional cross sections was performed by Calvo et al., generating a large structural dataset by adaptively biased MD simulations with two gyration-tensor based collectivecoordinates (squared radius of gyration and the asphericity 34 parameter) . From this dataset, the best numerical description of the CCS was obtained by using an 8-parameter fit of logbased functions including energetic, gyration-tensor and surface representations of the molecules Bradykinin and M2TMP. In the present theoretical contribution, we take up this work and challenge basic shape correlations with the aim to contribute to modelling strategies in gas-phase. Our work aims at establishing a route to use the CCS as collective coordinate in a) guided/restrained MD that may be used to study selected structural transitions and enable the better characterization of a specific partially unfolded state or b) Free Energy explorations that may be used to reproduce the CIU fingerprints. We hereby specifically address two principal challenges:

Despite these advances, severe challenges persist in combining experimental and theoretical work related to IMS studies: especially when the biomolecular structural models bear uncertainties through structural disorder, poor predictability by homology modelling, or simply because the chosen gas-phase conditions (e.g. charge state) give rise to some structural change. Consequently, first-principles approaches to model gasphase structures may fail to agree with the measured populations. Here, dedicated gas-phase modelling of protein structure becomes a key element. Herein, biasing techniques play a crucial role, as they allow to overcome deficiencies in the sampling efficiency, escape from kinetically trapped structures or simply to compensate deficiencies of the underlying 7,34,35 theoretical framework (e.g force-field) . Further, Free36–38 Energy exploration tools like Umbrella sampling or 39–41 metadynamics could provide a theoretical counterpart to experimental unfolding profiles, especially CIU, as they can provide the energetics for a whole unfolding landscape.

• •

The description of the CCS along an unfolding pathway by simple structural parameters The transferability of this description between different proteins

We exercise our work for several examples of systems including dye-labelled peptides used in gas-phase spectroscopy, Amyloidβ dimers used in FRET studies, Ubiquitin (a protein that has 44 been extensively studied by IMS ) and a somewhat less characterized small acid-stress-shock protein dimer, folded mini-protein and an antifreeze-protein. In addition to fitting log-based predictor functions depending upon the global radius of gyration, we devise a simple model that borrows ideas from PA and the macroscopic hard-sphere model aiming at a) describing unfolding in terms of the CCS and b) having a reduced system dependence that require individual fitting procedures. We will demonstrate that this rough predictor is somewhat transferable and straightforwardly applicable in bioinformatics workflows and “on-the-fly biasing”– only using a single structure e.g. contained in a pdb file. Towards more complex systems, in particular protein complexes, one single global shape descriptor is insufficient to describe dissociation that competes with global unfolding. We formulate a top-level predictor that switches between a monomeric and dimeric predictor functions, depending on the aggregation degree. This method could potentially be useful for describing multidomain proteins. We end with wrapping up the conclusions and provide some outlook.

Several computational approaches exist, that already reproduce 42 many of the general features of CIU experimental data . Among the promising advancements in the field, charge hopping combined with coarse-graining and proton-mobility within all43 atom MD simulations have deen developed. However, still, a complete model that can predict the unfolding transitions of protein complexes with high internal energies during CIU remains elusive. Thus, systematic integration with Free energy exploration methods in frame of atomistic MD seems to be a lead priority. Free Energy exploration methods usually operate with collective coordinates (also called order parameters), streamlined to the process of interest. Unfortunately, the CCS, along which the unfolding profile has to be explored, cannot be directly used as such coordinate as it is computed by numerical integration and hence, analytic atomic gradients are unavailable.

COMPUTATIONAL DETAILS

In this regard, it is a widely-acknowledged fact, that the protein structure’s radius of gyration shows some correlation with the collisional cross section and thus, restraining or accelerated sampling can make use of this descriptor (for which analytic gradients are straightforward) at least qualitatively. A detailed

Classical molecular dynamics (MD) simulations were performed 45,46 based on the AMBER99 force field that was completed with the generalized Amber Force Field (GAFF) to describe the non47,48 standard chromophore grafted cysteine residues . For the 49,50 MD simulations, Gromacs was used (version 5.1.2), with ACS Paragon Plus Environment [2]

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Figure 1: Structural representations of compact and/or native structures of a) Rh575-CAAKAAC-QSY2+, b) TrpCage+1 c) [(Rh575-Aβ1228 )(Aβ12-28-QSY7)]3+, d) Ubiquitin6+, e) RD3-3 f) [(HdeA)2]8-. Different colors of ribbons indicate different monomers in dimeric protein, a linker chain in e) is depicted in green. Details: see main text. 51

PLUMED in its version 2.3 plugged in. MD in the gas-phase and in solution was done using the Velocity-Verlet integration 52 algorithm (time-step of 0.75 fs) with velocity rescaling 53 temperature control , in absence of bond-constraints. In the gas-phase, where periodic boundary conditions do not apply, no cut-offs for nonbonded interactions were used. The velocity rescaling method for temperature control was used. All biasing to MD and definition of collective coordinates used the PLUMED code. MetaD in the well-tempered ensemble placed potentials of the initial height of 12.5 kJ/mol and width of 0.05 nm for every 50 steps applying a biasfactor of 20. All fitting procedures were utilizing the SciPy 1.00 suite of tools.

possess a relatively low number of degrees of freedom and welldefined charge-dependent structural properties. Here we focus on a peptide with the sequence CAAKAAC tagged with a Rhodamine575 and QSY7 dye as example for a short peptide. Details of the dye parametrization which bases on the Amber 47 57 GAFF force field can be found in Ref . We use a +2 charge state with a neutral Lysine residue and neutralized termini. b) TrpCage miniprotein TrpCage is a designed 20-residue mini-protein that possesses a particularly stable folded state and well defined folding 58–62 pathway . It thus provides a well-defined test-case for experimental and computational studies of protein folding and unfolding and its landscape. We have constructed a model from pdb: 1L2Y in the -3 charge state and protonation corresponding to neutral pH in water.

RESULTS AND DISCUSSION Modelling protein unfolding in the gasphase monitored by CCS: Test Systems

c) Dye-tagged Amyloid-β 12-28 dimers

It is one of the main aims of this paper to establish connections, to demonstrate differences and to work out best practices in using simple structural proxies for collisional cross sections when investigating the unfolding behaviour of a variety of proteins. We therefore selected model systems, for which we have experimental gas-phase data available, which are well studied test-cases or well highlight challenges for current CCSguided modelling strategies. As we aspire to perform systematic bottom-up testing, we restrict ourselves the relatively small proteins presented in Fig. 1 for this part of the investigation.

Aβ 12-28 dimers are model systems for the smallest neurotoxic 63 aggregate made responsible for neural degradation in 63,64 Alzheimer’s disease . These systems have recently been employed in FRET measurements in the gas-phase in our group (dye-tagging, see before). Here we focus on the 3+ charge state in line with our previous studies where FRET and IMS data is available: Upon their comparison with predicted values from 57 (biased and unbiased) replica-exchange MD as well as 65 metadynamics simulations in solvent and in the gas phase a memory of solution-structure under non-equilibrium gas-phase conditions was hypothesized.

a) Chromophore-tagged peptides

Short chromophore-tagged peptides have been used in our d) Ubiquitin group previously as model systems for gas-phase fluorescence 54 55 56 self-quenching and FRET experiments , (and IMS ) as they ACS Paragon Plus Environment [3]


Ubiquitin is a small regulatory protein that has been extensively 44,66–71 studied in the gas phase . Its gas-phase IMS profiles (and in particular unfolded populations) are drastically influenced by 72 the solution conditions . A similar effect is observed by FRET on 71 73 the gaseous protein . The particularly stable native fold and the possibility to keep solution-memory could be an avenue to link gas-phase and solution environments for the structural preferences of the system. We select Ubiqutin (pdb: 1UBQ) in the 6+ charge state (with a full retention of the protonation pattern expected from solution) in line with our previous work.

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protonation pattern according to pKa considerations in water.

Systematic unfolding simulations by metadynamics Our first aim is to reveal correlations between structural descriptors and the CCSs for the different systems of this study (thereby connecting with previous work), which requires a systematic generation of increasingly unfolded configurations starting from a compact fold. A way to cover a broad range of structures in molecular dynamics simulations is based on 40,78,79 metadynamics , which can be operated on gyration-tensor 80 based collective coordinates . Calvo et al. have already acknowledged that using a Free-energy calculation with gyration-tensor based collective coordinates (using adaptively 81 biased molecular dynamics - similar to the metadynamics method) can generate a broad variety of configurations in the 34 space of CCS . We chose metadynamics simulations in the welltempered regime as this methods allows to avoid sampling regions of unphysically high potential energy and furthermore 41,82,83 its extensibility with replica exchange of various flavours . The latter serves to improve convergence of the Free-energy profile, though e.g. communicating information from trajectories propagated at higher temperatures. This combination of methods thus promises to achieve a representative coverage of gas-phase phase space structural types while tuning from folded to unfolded states. We carried out parallel-tempering WTMetaD (PT-WTMetaD) simulations with the radius of gyration as order parameter (as defined 80 51 in Ref , implemented in PLUMED ), propagating 20 replica with a geometrical series of temperatures between 220 and 850 K. To prevent diffusion of separated parts in simulations of dimeric proteins (Aβ dimer and HdeA dimer), we additionally placed an upper limit for the gyration radius at 60 and 80 Å, respectively. With this setup configurations from highly compacted to entirely unfolded protein or dissociated protein

e) Antifreeze protein RD3 Intramolecular dimers of functional proteins can possess enhanced activity: for example a protective function against 74–76 low-temperatures . Using NMR spectroscopy, the enhanced activity of the antifreeze protein RD3 - found in Antarctic eel pout - was rationalized in terms of a unique translational topology: two homologous ice-binding domains connected by a 75 linker . Species with this type of topology might have unexpected cooperative effects regarding their binding mechanisms. Their heterogeneous constitution, however, is challenging for systematic biased modelling using global structural descriptors, motivating us to take this protein as an example for partial dissociative unfolding behaviour. We set up the model of RD3 (pdb: 1C8A) in a -3 charge state from solutionpKa considerations. f) HdeA chaperone HdeA is a small acid stress chaperone (preventing other proteins from aggregating) whose resting state is a dimer. Some experimental and theoretical work has unravelled aspects of the dimer to monomer dissociation mechanism – a process that is 77 triggered under low pH conditions . We have employed a gasphase model in the 8- charge state, according to the default

Figure 2: Correlations between radii of gyration (rGyr) and explicitly calculated (EHSS method) collisional cross-sections (CCSEHS) for a) Rh575CAAKAAC-QSY2+, b) TrpCage+1 , c) [(Rh575-Aβ12-28)(Aβ12-28-QSY7)]3+, d) Ubiquitin6+, e) RD33- and d) [(HdeA)2]8- (cf FIg. 1) Black circles: configurations generated with PT-WTMetaD at 220K, blue plus characters coorespond to 850K WTMetaD generation conditions for systems where significant sampling of dissociated states needed a higher temperature.. Full lines correspond to least-squares fitted functions of the gyration radius to represent the CCS, eq. 1.


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dimer are sampled – in line with a broad range of values covered. In the SI, Fig S1 and S2, snapshots of a PT-WTMetaD trajectory for Ubiquitin and the Aβ dimer are given, sorted from low to high values are given. By this analysis the coverage of structural families is documented.

CCS values according to eq. 1 and explicitly calculated values, see also standard deviations in Table S1. This agreement would allow for fast and reasonably accurate prediction of a CCS of these systems using the fitted function once the correlation has been established. 34

In Ref. the unphysical nature of the chosen fitting approach using logarithms was underlined which makes it hard to find system-dependent parameters. In turn, there is the need to perform an explicit calculation of a large number of CCS values for a large number of configurations representing all ranges of CCS , i.e. during an unfolding process. In summary, in the present form, the fitting of a predictor does not greatly ameliorate the computational costs for CCS-related protein unfolding simulations. This fact motivates to look for less system-dependent procedures to predict CCS values.

Correlations of CCS with simple structural descriptors: chances and challenges The radius of gyration is frequently used as a measure of a 84 structure’s compactness (see e.g. Ref ) and can be qualitatively used to judge differences in CCS. More quantitative descriptions employ fitted CCS-predictor functions obtained after explicit calculation of CCS for configurations that have been systematically generated (usually by biased MD approaches). Calvo et al. used an 8-parameter fit employing the logarithms of squared gyration radius, asphericity parameter and surface area 34 of the convex hull as well as their cross terms to second order . It was noted that the fitting is system-dependent. We have performed a similar approach in describing the CCS on basis of the gyration radius for the systems presented here. The results of exact hard spheres scattering (EHSS) calculations for configurations obtained by our metadynamics approach are given in Figure 2 in correlation to the systems’ gyration radius . We adopt a fitting approach, restricting ourselves to the gyration radius as single structural proxy but allow for some flexibility through three fitting parameters (eq. 1). The unbound least squares fits are displayed in Figure 2a-f) as full lines (parameters given in the supporting information, Table S1).

A simple model covering unfolding

CCS

descriptor

In fact, in context of structural proteomics studies, the CCS was 2/3 previously found to be approximately proportional m ,v where 33 m Is the mass of the protein , consistent with earlier 85–87 experimental observations and in line to a simple model of a sphere with effective density , written as Ω = =

3 4

/"

(2)

In case of unfolding behaviour – the main issue of this paper – we aspire to formulate a model that shares its simplicity and universality with the hard-spheres model but qualitatively covers protein unfolding.

= + − ( 1 )

The measured CCS reflects the orientationally averaged 19 momentum transfer integral according to collision theory , 30,88 which is calculated as defined in eq. 3. (see Refs ).

Note that due to the choice of EHSS as numerical calculation method, the intrinsic temperature dependence of the CCS, where the lower velocities of the impinging neutral atoms of the 7 colder gas yield higher CCSs than at higher temperatures , is not accounted for, and only enters indirectly via structural changes. Notice also that metadynamics counts to the flat histogram sampling techniques and does not yield canonical ensembles. To highlight differences between methods for sampling and CCS calculation, Fig S3a contains the correlation using CCSEHS versus from T-REMD simulations calculated by EHSS, and Figure S3b depicts WT-MetaD configurations postprocessed with the 31 more accurate trajectory method , that employs the full and explicit calculation of collision trajectories including the kinetic energy of the impinging atoms. Although the general trend of the structure-CCS correlation is preserved across calculations, the marked difference from different choice of structural sampling and intrinsic CCS-calculation method underline the system- and dependence of the obtained fits, rendering the found predictive models highly specific.

( ( ( 1 % &' % &) sin ) % &8 2 × 2 % / 0(', ), -, /) &/ (3)

=

Here, 0 is the probability for a collision, integrated over collision geometries (impact parameter b) and all potential ionbuffer gas orientations (', ), -). For hard-spheres, the collision probability is taken as being 30,88 either one or zero ; consequently the dependence of on originates from the upper integration limit regarding the impact factor. () reduces to 34 where (34 is the radius of the hard sphere). For all objects with non-spherical interaction potential with the probe particle, explicit numerical integration (e.g. using collision trajectories) is usually performed to obtain the averaged value.

Building upon this formalism, we are interested to a) describe ( ) with being the radius of a hypothetical sphere, just big enough to contain all atoms of the protein and b) to distinguish between different structures, i.e. folded vs. unfolded states. If ACS Paragon Plus Environment

For several systems presented here (Fig 1 and 2 a) Rh575CAAKAAC-QSY, b) TrpCage and d) Ubiquitin), the single fitting approach yields satisfactory agreement between fitted

[5]


function (see for example the PSA and LCPA methods ), we decide for the quotient of densities at / normalized to the reference density of a sphere with the radius 34 . To picture this probability; it corresponds to a strong intramolecular interaction potential (e.g between charges and dipoles in the molecule) where application of an unfolding force gives rise to a normalized radial pair distribution function that decays with 1// " as we scan over the radial coordinate / (see eq. 5 and three different protein/unfolding state scenarios in Figure S5). Both the optimization of such interaction potential for different unfolding scenarios as well as their translation into other radiusdependent collision probabilities could serve to further develop the herein proposed method and (better) adapt it to specific experiments. 32,89

the system is folded and compact, a hard, macroscopic sphere is often an acceptable description for the protein, and the r5 dependence of the CCS of this object should still be Ω( ) = 56

2 . As the system unfolds, the hard-sphere picture is less appropriate (an elongated protein conformation cannot be well represented by a hard sphere, see e.g. the bottom of Fig. S2). In the calculation of the CCS, therefore, trajectories have to be taken into account that can pass through the volume of the hypothetical sphere (enclosing the protein) without collision, meaning that collision probabilities other than 0 and 1 may result (see scheme 1).

7 (', ), -, /) We will thus use a generalized collision probability 0 in eq. 3. Here, the collision probability for a trajectory depends on b, the impact parameter (corresponding to the location where we aim at shooting the probe collision particle) in a way that must reflect also a given unfolding state. Combining aspects of projection approximation and collision-trajectory based methods has already been shown to be a useful concept 89 in the local collision probability approximation (LCPA) .

7 (/; 34 , ) in eq. 5 has the The choice for the form of 0 following convenient properties: •

•

As the functional form of such collision probability arising from 89 the exact molecular interaction (depending itself on the folding state) and its integral over all collision geometries and structural arrangements (impact parameter b, protein orientation ' , ) , - ) might be very complex we will use a simple approximation. We regard an average effective probability 7 (/) which we formally obtain by exchanging the integration 0 order, meaning to perform orientationally averaging of 0(', ), -, /) in the first place (eq. 4). This averaging renders 7 (/) spherically symmetric, which means that functions that 0 7 (/). have solely a radial dependence do serve as guesses for 0 7 (/) = 0

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•

( ( ( 1 8 &' 8 &) sin ) 8 &- 0(', ), -, /) (4) 8

If = 34 the densely packed object is well 7 (/ = 34 , ) = 1 represented by a hard-sphere and 0 . If ≫ 34 the collision probability at the boundary of the object comes close to zero. However, the sphere containing all atoms has always a somewhat sharp boundary where the collision probability drops to zero; introducing a well-defined integration limit. For a molecular system, different orientationally averaged geometrical arrangements (molecular structure, folded vs. unfolded protein) of a given number of atoms and composition will result in different possible radii and 34 , defining the regions introduced before. In this way, the explicit structurecomposition dependence of Ω is established.

While in the hard-spheres approach, the collision probability is 30,88 taken as being either one or zero , we hypothesize that this term depends parametrically also on the folding state which we wish to describe by two radii 34 and 9: as in eq. 5. 1; / @ 34 " (/) 34 7 (/; 34 , ) < 0 = " A 1 ; 34 A / A (5) >(34 ) / = 0 ; / C

?

These parameters determine the ranges of / where collision probability is exactly one and zero, as in the original formulation, but allow us to introduce a non-hard-spheres portion of this range (as proposed in a similar way by Bleiholder et al. in context of the projection superposition approximation, 32 89 PSA in the Local Collision Probility Approximation, LCPA ). For this non-hard sphere range, we introduce a function that takes into account a non-unity but also non-zero collision probability that represents the outer region of the unfolded protein (right side of scheme 1). Among several possibilities to choose such

Scheme 1: Representation of protein (blak dots) by a hypothetical sphere. For a compact fold (left part), probababilities M for collision of probe trajectories (small dots and arrows) are 1 when impact parameter b smaller than outer radius rHS. When unfolding the system (right part), probe trajectories have a chance of passing through the

hypothetical sphere of radius without collision (dashed arrow. Thus, the effective average collision probability M representing the region b < is smaller than 1.


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The Journal of Physical Chemistry vs. correlation with a −1/ function – justified by the scaled macroscopic projection model suggested herein.

With eq. 5, we can extend in eq. 3 so as to cover also a more realistic dependence on the outer radius . The integration over impact parameter can now be written in three ranges (eq. 6) (34 , ) = 2

1 1 ΩZ[\ = UU]VW + U G VW − J (9)

( ( ( 1 8 &' 8 &) sin ) 8 &8

U=

× 2 8 / 0(', ), -, /; 34 ; ) &/ (6)

< 2 G8

HI

/ × 1 &/ + 8

6

HI

/ ×

2 34 &/ + 8 / × 0 &/ J /" 6

HI

HI KL

&/ =

integral simply as eq. 7.

we can rewrite this

" Ω(34 , r ) = 34 + 34 P

1 1 − Q (7) 34

We can easily confirm that, if does not exceed the macroscopic hard-sphere limit, the expression reduces again to 34 .

Even though the TPA predictor eq. 9 and 10 might be very useful for the unfolding behaviour of a single molecule, still, two intrinsically different structural families (folded, unfolded) have to be modelled (e.g. by biased, steered MD or metadynamics). For comparing a high number of different systems, for example as part of workflows combining bioinformatics and IMS data, it would be attractive to obtain CCS predictions from a single structure.

introduce corrections, e.g. deviation from a sphere-like conformation to be included into the reference state ΩST . 1

ST

−

1

N (10)

V^_ −

Towards a fast structure- and composition-based CCS estimation

With this -dependence, we can use a reference CCS represented by ST for a compact structure and its corresponding reference radius ST (eq. 8). In this way we can ( ) = ST + U G

N

The performance of both approaches, with standard deviations 2 2 of 31 Å (fitting, eq. 8) and 37 Å (TPA, eq. 9-10) is only slightly worse than the log-fitting approach (eq 1., standard deviation 2 being 25 Å ). The benefit of the TPA predictor persists in the use of only two points instead of a whole dataset generated by e.g. metadynamics. In this work, as two reference points, the extrema are taken, but other, maybe more convenient choices can be made as well. In view of the considerable amount of computing time needed to generate a full range of physically sound structures - with compact fold to being fully unfolded such reduction can speed up rough estimations of unfolding monitored by CCS.

M HI ×(6 NHI )

6

VW

The resulting predictor is plotted for Ubiquitin in Fig. 3a as blue line.

As in the case before, the dependence of on the specific radii originates from the integration limits within which non-zero collision probabilities are found. In contrast to before, this was only the case for / @ 34 , while within our approach there is also a soft portion of the sphere with 34 A / @ . With % 6 / ×

UU]V^_ − UU]VW

J (8)

A major breakthrough in this direction was the previously 33 2/3 mentioned study (Ref ) revealing the m dependence of the CCS for hundreds of thousands of structures.

Eq. 10 can be used (similar to eq. 1) as basis for a 3-parameter fitting function (ST , C, ST , see Table S2 in the SI), that has the advantage over other (e.g. log-based predictor functions) of being derived from a physical model. We demonstrate the usability of this approach for Ubiquitin (cf. Fig. 1d and Fig. 2d) in Fig 3a (red line). Parameters and standard deviation can be found in Table S2 in the SI.

In this contribution, we wish to, taking up the idea of scaling the CCS system-dependently (according to the system’s mass m) and go one step ahead in combining this from our previously described model for predicting each system’s CCS upon unfolding. The combined approach is sought to roughly predict the CCS from any structure model without any major fitting procedures. We achieve such description by reconsidering eq. 9. written as eq. 11, which we denote the scaled macroscopic sphere (sMS) predictor.

VW More importantly, the at well-defined − scaling gives us X Y

further the possibility to use to use only two points (a reference point and a second point) to predict the unfolding behaviour for the CCS in the limits of this simple model. This two-point approximation (TPA) predictor Z[\

becomes eq. 9. Its " 34 scaling pre-factor can be obtained according to eq. 10. ΩT will Ωab4 = χ d234 − e (11) be taken from the CCS of the smallest sampled – denoted as UU]VW – because the most compacted configuration of a The expression can be interpreted as hard-sphere projection metadynamics run should best meet the picture of a hard CCS with some correction though unfolding (soft-sphere regime V^_ sphere. UU]V^_ and are defined in analogy. In essence, described before). In fact, the first term with 34 is responsible this predictor interpolates between the two extrema of the CCS for m2/3 scaling when comparing differently composed systems ACS Paragon Plus Environment [7]


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Figure 3a and 3b): Correlations between explicitly calculated collisional cross-sections (CCS) and radii of gyration (rGyr) for Ubiquitin (compare Fig. 1d and 2d) as green squares faced with different non-logarithmic predictor functions Ω. a) black line: log-predictor function (eq. 1) from Fig1 d for comparison; red line: fitted -1/r CCS scaling behaviour of eq. 8; blue line: TPA approximated predictor (using configurations with smallest and biggest rGyr. in b) the black line corresponds to the scaled macroscopic sphere (sMS) structure-based predictor from eq. 9,10. Blue point: sum of volume occupied by the atomic van der Waals spheres scaled with 0.85, red point: CCSEHSS and rGyr for pdb:1UBQ. Figure 3c) CCS prediction by rGyr for 1300 structure pdb dataset. The correlation between EHSS CCS and the Ωab4 from eq. 11. is shown as red points (main panel zoom to values below 5000 Å2; inset: full range of CCS values).

(different proteins). The second term accounts for the -1/ scaling of the CCS (suggested in this paper) with unfolding the structure. We introduce χ as a structure factor, that modifies the CCS through deviation from a perfect spherical geometry (surface roughness, concaveness). It can be seen in analogy to the shape factor in the projection superposition 32 approximation .

correlations for protein diffusion coefficients in the condensed 91 phase based on the gyration radius . The resulting predictor curve is shown in Fig. 3b as full black line (χ=1.7 and =0.85). In connection to bioinformatics workflows, it seems useful to predict CCS values from 3D structural models directly. We first demonstrate the proposed procedure for Ubiquitin (Fig 3b). The CCS of the minimal-volume macroscopic hard sphere for the molecular composition (blue dot) is corrected by the second term of eq. 7 to describe the CCS by Ωab4 at a given . The gyration radius from pdb 1UBQ is calculated to be 11.78 Å (red vertical line) CCS/ pair marked as red dot. In this way, a prediction of a CCS amounting to 1053 2 Å is obtained that matches the one from the explicit EHSS 2 algorithm (1086 Å ) within 1.4 %. Overall, the predicted (unfitted) curve matches the explicitly calculated dataset with a 2 standard deviation of 47 Å .

We now can connect the properties of the hypothetical macroscopic sphere to molecular parameters. The sphere radius is related to the radius of gyration according to eq. 12. The radius of gyration can be easily calculated from a molecular structure. = f (12) g "

We further need the radius 34 that would correspond to a macroscopic hard sphere for a given system. We use the minimal possible volume for a molecular composition as reference point here, deriving 34 from the sum of volumes occupied by spheres of atomic van der Waals radii , scaled by a parameter , as in eq. 13.

For being more generally useful, the model should be applicable to other systems than Ubiquitin – meaning to be transferable without any major re-parametrization. We demonstrate that this prerequisite can be met by the approach by the calculation of Ωab4 for a bigger dataset of structures detailed in Figure 3c. We have downloaded the first 1500 entries of the RSCPDB database, extracted all ATOM entries and built all-atom models using OpenBabel. These models (the procedure was successful for about 1300 structures) were processed to give and 34 as detailed before. As can be depicted from the plot, apart from a small number of outliers (that most probably originate from physically

. 34 = f∑ " (13) L

Notice that this choice does not reproduce the expected CCS at 90 infinite r – being the sum of atomic cross sections , but is adapted to realistic r values. It interesting to note that the herein proposed approach shares some ingredients with


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Figure 4): Prediction of CCS during unfolding including dissociation in two monomers. Calculated collisional cross-sections (CCS) versus radii of gyration ( ) for a) Amyloid beta dimer (compare Fig 1c Fig 2c) b) HdeA dimer (compare Figure 1f Fig 2f) and c) the RD3 antifreeze protein (Fig 1e, Fig 2e). Top panel: black, configurations generated with PT-WTMetaD at 850K,850K and 220K respectively (compare Fig. 2); green line: single predictor from eq.1 only ; red “+” chars: values approximated by the dissociation-dependent predictor in eq. 14-15 and eq. 1. Bottom panel:

correlation between explicitly calculated CCS and both CCS predictor values: green points, single predictor from eq.1 only; red “+” chars, dissociation-dependent predictor in eq. 14-15 with eq. 1. On the right, different classes of RD3 protein folds identified in different components of CCS- r.

unsound/incomplete all-atom structures from this non-curated conversion procedure), an overall satisfactory agreement. Because of the very broad range of CCS values, we regard the standard deviation in terms of relative errors (about 6 %) as instructive measure of the model’s performance. It is remarkable that this minimal model apparently performs well (notice the systematic trend at very large structures) up to a CCS 2 regime of more than 3000 Å . It must be noticed, that the EHSS algorithm used for correlation of the values could also be a source of error and that the herein presented model still has many chances of further development.

this paper have been chosen to highlight the breakdown of the correlation between the CCS and one global gyration radius. The Amyloid-β dimer (in Fig 2c) and the HdeA dimer (Fig 2f) evidently possess two entirely differently behaving regions. Since these systems are dimeric proteins in the starting configuration for WTMetaD, it is evident that this break-down is due to the transition from a dimeric to the monomeric state, with separate monomers being sampled at high . Including separated configurations into the fitting dataset hampers with reproduction of CCS values for bound, but partially unfolded states with the same function and vice versa. It might be possible to perform a fit bound to dimeric configurations, however, a) the intermediate region where the transition between bound and unbound states occurs is not well covered and b) defining the region for the fit might not be straightforward in all cases.

In addition to its compatibility to bioinformatics workflows, it is important that the predictor presented herein is computationally cheap and can be evaluated “on the fly” during an MD simulation without creating any major bottlenecks. The procedure might therefore be suitable to being included as order parameter in biased simulations and Free-energy calculations themselves in addition to its usability in bioinformatics workflows.

In view of the importance of protein aggregation processes (e.g. followed by IMS investigations through temperature, collisional activation, charge or charge reduction-induced activation) we hereby present an extension to the single gyration-radius Extension of the CCS predictor predictor approach. Each configuration of a dimer might resemble either a bound state best described by a function of approach to cover dissociation Representing a molecular configuration by a single gyration of the full system. It might as well be in a separated form, radius is a drastic simplification and might be insufficient for where the sum of functions depending on each fragments’ describing more complex structural aspects. The examples of gyration radius of the fragments is the best representative (notice that coarse-graining multimeric proteins by e.g. spheres ACS Paragon Plus Environment [9]


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representing the monomers for the description of CCS in multi16 protein architectures is a successful concept ).

following, that is the numerically more accurate description of the CCS so far.

We devise a flexible description to describe both situations (bound and unbound) by introducing a smooth logistic switching function as defined in equations 14-15. The generalized collisional cross section predictor function Ω (eq. 14) is now expressed by contributions from the gyration radius of the full system (which takes e.g. the form of eq. 1 or 11) and the sum of analogous CCS predictors of the fragments Ω, (with

For comparing the performance of the multi-fragment approach with the single gyration-radius-based predictor approach, we show the correspondence of Ω (eq. 14, utilizing eq. 1) and the EHSS-predicted CCS values as a funtion of in FIgure 4a) for Amyloid-β dimer and in Figure 4b for the HdeA dimer. For both systems, the two regimes in the CCS- correlation are clearly reproduced. Further, from the direct correlation of CCS being excplictly calculated and predicted from in the lower panel, a good agreement is achived throughout the whole range. This underlines that our flexible choice of structural descriptors is suitable to cover the transition between bound and unbound states, and that overall a good predictive performance can be achieved by appropriate choice of predictors for the fragments. The standard deviation of rpredicted vs explicitly calculated CCS values could be reduced from 87 (Aβ dimer) and 128 (HdeA dimer) when using eq. 1 only 2 to values of 23 and 43 Å , respectively, when using eq. 14-15 with eq 1.

their gyration radii ) whose contribution is modulated by the logistic function (eq. 15) whose offset I0, and sharpness b are also fitted. (,)

Ω , , , j = ()

()

k(j) ⋅ Ω:mm +

() () n1 − k(j)o ⋅ pΩ + Ω q (14)

The fragments’ predictor can be chosen freely, depending on the system to treat and computational power available. For example the more accurate but unphysical log-fitting approach (see eq. 1) or the fast structure-based predictor presented in eq. 11. The switching itself is invoked by a third structural descriptor I, for which we chose the interface area, as it is a readily obtained measure for the aggregation degree of the systems and takes the form j =

\6 r\M N\stuu

where v:mm denotes the solvent

accessible surface area (SASA) of the full system and denotes A1,2 the ones of the fragments (other measures of the aggregation degree such as the fragments’ centres of mass distance might also be suitable to achieve the switching). The ready availability of SASA calculations in different trajectory post-processing packages, renders this descriptor attractive. k(j) =

1+

1

(15)

w NK(xNxy)

As in the herein presented examples the system dissociates into two identical (or very similar) fragments, we use a single set of fragment-parameters in the following, which reduces the number of parameters when using fitting procedures. The approach, however, is general and could account for different fragments individually.

Relevance beyond Multidomain proteins

dissociation:

Being capable to approximate a CCS by simple structural descriptor despite a dissociation-like behaviour is not only interesting in view of protein aggregation studies (where in the experiment one is limited to the transition states, just before the dissociation into two separate ions), but – more importantly - to a broader class of systems, which are represented by two domains separated a somewhat flexible linker. This criterion is fulfilled by an antifreeze protein RD3. We have performed the CCS- correlation for this system as shown in Figure 4c, thereby applying our dissociation dependent fitting approach (eq. 14-16). The breakdown of the global correlation is less pronounced as for the dissociating systems in Figure 1b) and 1d), but two different populations are clearly distinguishable, starting at > 17.5 Å. An analysis of the configurations corresponding to the two members of populations around =20 Å (visualized in Fig. 4 on the right) reveals that the lowerCCS component essentially possesses the native protein topology with two domains separated by a linker (green), while the higher-CCS component differs completely in its fold (being some overall partially unfolded structure). The approximation according to eq. 14-16 with eq. 1 succeeds to reproduce these two inherently different populations (see red dots in Figure 4c), while the single – based approach (only eq. 1) averages these differences out (green line). The numerical performance for this complex system is comparable with both methods (standard deviation of both, the approach in eq. 14-16 with eq. 2 1 as well as eq. 1 alone, amount 35 Å ). In view of the possibility to enhance the predictive power (by e.g letting the two fragments be non-identically represented), the presented approach seems attractive for future further development. In

We selected the Amyloid-β dimer (see Fig 1c and 2c) for testing the compatibility of the fragment extension with the structurebased predictor (eq. 11) of which the results are shown in Fig S4 in the SI. The trend and the existence of two regimes corresponding to bound and unbound configurations are correctly reproduced. Generally, the CCS of the Amyloid-β dimer is systematically underestimated with eq. 11. It seems that a more adapted structure factor χ could lead to a more correct reproduction of the numerical values for this system. Instead of optimizing the structure factor, we use the original log-fitting approach with eq. 1 for demonstration purposes in the ACS Paragon Plus Environment [10]

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particular, steering protein populations towards either nativelike or partially dissociated topologies by biased MD seems attractive when utilizing the computationally cheap predictor functions of the form in eq. 7 and 11.

partial dissociation found in multidomain proteins. We have demonstrated for the antifreeze protein RD3 that by the switchable description a near-native topology, with two compact domains separated by a linker can be discriminated from a global partially unfolded configuration.

CONCLUSIONS AND OUTLOOK

Overall, this study contributes towards computationally cheap, yet universal and semi-quantitative descriptions of CCS in relation to unfolding experiments followed by IMS. The proposed predictive models are simple to implement and can be directly included into bioinformatics workflows. Soon, also the integration into biased MD seems feasible as the complexity and the computational effort of the related collective coordinates does not exceed the one of the radius of gyration itself. Thus, Free-energy profile explorations might directly be performed in CCS space, which in turn, serve to compare with unfolding profiles directly.

In this paper, we have confronted the correlation of the collisional cross-sections measured in ion-mobility spectrometry setups with structural parameter – and have related these correlations to unfolding processes in the gas-phase. Still, available explicit (more or less) accurate prediction algorithms for molecular structures pose bottlenecks to the integration of IMS data with unfolding experiments by ion-mobility and/or bioinformatics approaches where large structural datasets have to be treated. We have first revisited the predictor functions that are fitted against unfolding simulation datasets. In addition to the knownsystem dependence, we critically evaluated other factors for this correlation as the generation conditions and the explicit CCS calculation engine. As the fitting has to be done for every species individually and probably also for each simulation condition, a good predictive power is accompanied by computational costs of the structural sampling and explicit CCS calculation. Motivated by this persistent limitation, we devised a simple model for CCS during unfolding, that borrows concepts from the projection approximation and the hard-sphere model. We arrive at an expression that scales the CCS with a factor of -1/ , once a certain radius of the sphere enclosing the structure is exceeded. Based on this dependence we present a fitting approach and a two-point interpolation scheme, that perform only slightly worse than the previously as optimal function determined logdependence. More importantly however, this model allows to formulate a simple estimator that bases only on structural parameters (can thus be calculated from a pdb file) and which makes up a reduced predictive accuracy by its universality. First applied to a single protein unfolding, we show that it may also can predict CCS values (in comparison to those calculated by exact hard-spheres scattering) with a standard deviation of 6 %, even for a larger non-curated pdb dataset with CCS values up to 2 5000 Å . From our investigation, the breakdown of CCS-global-shape correlations becomes clear, which might be resulting from (partial) aggregation and dissociation processes. In this case the structures cannot be described with one global structural descriptor. In line with coarse-graining multimeric proteins, we devise an extension to the shape-correlations described earlier, which smoothly switches between using one global or two fragment’s descriptors for a dimer. This model can well reproduce the CCS in bound and unbound regimes demonstrated for an Amyloid-β model dimer and an acid stress shock protein dimer. Further, the model may be applicable to

Using the CCS as collective coordinate by a computationally cheap transformation is a step forward to reproduce CIU profiles. Especially the possibility to propagate a high number of trajectories at different activation states such as elevated temperatures may allow for obtaining a realistic picture of the energy landscape. Still, the convergence of several theoretical approaches is yet to be performed as Free Energy explorations of conformational transitions will not capture all features in an CIU profile. Here, it will be necessary to include charge and 43,92,93 ) and perform systematic proton mobility (see e.g. refs Free-Energy explorations with the CCS as collective coordinate. Supporting information available: Fitted parameters using eq. 1 for the CCS of all systems described in this paper, Fitted parameters using eq. 8 for the CCS of Ubiquitin, Fitted parameters for the dissociation-covering approach in eq. 1,1416 to describe the CCS unfolding for the Aβ dimer, the HdeA dimer and the antifreeze protein RD3, Snapshots of the 220 K PT-WTMetaD trajectory for the Aβ dimer, Snapshots of the 220 K PT-WTMetaD trajectory for Ubiquitin, Correlations between explicitly calculated collisional cross-sections (CCS) and radii of gyration for Ubiquitin comparing TJ and EHSS method, REMD at 220 and 850K with WT-PTMEtaD, alternative eq. 7, CCSEHS vs radii of gyration for the Amyloid-β dimercompared to Ω from the dissociation-capable approach of eq. 14-15 using the simple structure-based predictor of eq. 11, plot of the approximated 7 (/; 34 , ) along the radius-dependent collision probability 0 scanned impact parameter /.

ACKNOWLEDGEMENTS Computer time granted by the P2HPD (Pôle de Calcul Hautes Performances Dédiés, Université Lyon 1) is gratefully acknowledged. A.K. acknowledges funding from the Deutsche Forschungsgemeinschaft DFG (Research Fellowship Ku 3251/11) and support by COST Action BM1403. The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework



Programme (FP7/2007-2013 Grant agreement No. 320659). EGM acknowledges funding from the Swedish Research Council and the European Commission through a Marie Skłodowska Curie International Career Grant, project code 2015-00559.


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