Hydrodynamic multibead modeling: problems, pitfalls and solutions. 3. Comparison of new approaches for improved predictions of translational properties

European Biophysics Journal, Jul 2013

Modeling simple and complex biopolymers in solution requires the shapes of these molecules to be approximated by bead modeling procedures, primarily for the prediction of hydrodynamic and scattering quantities. Though several bead modeling strategies (strict, shell and filling models) and a variety of computer programs (preferably the HYDRO suite by the García de la Torre group) are available, several subtle questions remain to be answered, in particular concerning the appropriate volume correction for intrinsic viscosity computations. In this context, various versions of the HYDRO programs and different types of volume corrections, as well as the novel, alternative program ZENO of the Mansfield group, were applied to a plethora of thoroughly designed multibead models of spherical, ellipsoidal, cylindrical and prismatic shapes. A critical comparison of the results obtained reveals a variety of new aspects, useful for many future applications. Among these, application of our recently suggested “reduced volume correction” (RVC) together with specially adapted HYDRO versions and use of ZENO turned out to be highly effective, in particular when aiming at filling model strategies and using high bead numbers, a domain not fully supported by the recent HYDRO++ versions. By our approaches, the values of translational properties (diffusion coefficients, D, and intrinsic viscosities, [η]) of all multibead models applied were anticipated correctly.

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Hydrodynamic multibead modeling: problems, pitfalls and solutions. 3. Comparison of new approaches for improved predictions of translational properties

Eur Biophys J Hydrodynamic multibead modeling: problems, pitfalls and solutions. 3. Comparison of new approaches for improved predictions of translational properties Peter Zipper 0 1 Helmut Durchschlag 0 1 0 H. Durchschlag (&) Institute of Biophysics and Physical Biochemistry, University of Regensburg, Universita ̈tsstrasse 31 , 93040 Regensburg , Germany 1 P. Zipper Physical Chemistry, Institute of Chemistry, University of Graz , Heinrichstrasse 28, 8010 Graz , Austria Modeling simple and complex biopolymers in solution requires the shapes of these molecules to be approximated by bead modeling procedures, primarily for the prediction of hydrodynamic and scattering quantities. Though several bead modeling strategies (strict, shell and filling models) and a variety of computer programs (preferably the HYDRO suite by the Garc´ıa de la Torre group) are available, several subtle questions remain to be answered, in particular concerning the appropriate volume correction for intrinsic viscosity computations. In this context, various versions of the HYDRO programs and different types of volume corrections, as well as the novel, alternative program ZENO of the Mansfield group, were applied to a plethora of thoroughly designed multibead models of spherical, ellipsoidal, cylindrical and prismatic shapes. A critical comparison of the results obtained reveals a variety of new aspects, useful for many future applications. Among these, application of our recently suggested ''reduced volume correction'' (RVC) together with specially adapted HYDRO versions and use of ZENO turned out to be highly effective, in particular when aiming at filling model strategies and using high bead numbers, a domain not fully supported by the recent HYDRO?? versions. By our approaches, the values of translational properties (diffusion coefficients, D, and intrinsic viscosities, [g]) of all multibead models applied were anticipated correctly. Multibead models; Novel approaches; Translational diffusion coefficient; Intrinsic viscosity; Parameter predictions; Hydrodynamics - Bead modeling in combination with predictions of scattering, hydrodynamic and other solution properties is gaining increasing importance in the investigation of simple and complex biopolymers. This is partly due to the ever-growing number of structural data available at atomic or near-atomic resolution from crystallography, NMR, or cryo-electron microscopy and deposited in publicly accessible data bases (wwPDB: cf. Berman et al. 2012), the usage of modern ab initio approaches to retrieve low-resolution structural data from solution scattering (cf. Zipper et al. 2005 and references therein; Franke and Svergun 2009), but of course also due to the progress achieved in developing appropriate procedures and programs for deriving the desired solution properties from the given structural information. In the field of hydrodynamic modeling, several bead modeling strategies exist (Carrasco and Garc´ıa de la Torre 1999a; Garc´ıa de la Torre et al. 2000; Byron 2008; Ortega et al. 2011): one is the classical approach of modeling the overall shape and structure of a particle by means of an assembly of relatively few spherical beads of often quite different size (‘‘bead modeling in the strict sense’’); other strategies attempt to simulate the structure of a particle in a more detailed and realistic way, leading to multibead models composed of a large number of beads of similar or even identical size. If only hydrodynamic properties have to be predicted, it is sufficient to take only the beads on the surface of the model into account (‘‘shell model strategy’’), whereas for the prediction of hydrodynamic and scattering properties a ‘‘filling model strategy’’ considering all beads of the multibead model has to be applied (Zipper et al. 2005). The feasibility of the latter strategy may be limited by the number of beads that can be handled by the software used. Thus, for large biomolecules, an appropriate data reduction step may be required prior to the prediction of solution properties (Zipper and Durchschlag 2007; Durchschlag and Zipper 2012a). When performing bead modeling in the strict sense or applying the filling model strategy, one has to cope with the problem of volume corrections in the computation of intrinsic viscosity or rotational properties (Garc´ıa de la Torre and Rodes 1983; Garc´ıa de la Torre 1989; Garc´ıa de la Torre and Carrasco 1998; Carrasco and Garc´ıa de la Torre 1999b). Though, according to experimental and theoretical evidence, such corrections may be quite important, the correction following from the theoretical considerations is often found to be rather inappropriate (Zipper and Durchschlag 1997; Carrasco and Garc´ıa de la Torre 1999b; Garc´ıa de la Torre et al. 2000). The shell model strategy avoids the volume correction problem by a stepwise reduction of the bead radius (rb) of the shell models and a final extrapolation of the computed parameters to rb = 0 (Garc´ıa de la Torre et al. 2000, 2010; Ortega et al. 2011). In view of the ambiguities of appropriate volume corrections and the inherent uncertainty, e.g., of experimental viscosity data of biomolecules, it is plausible that careful tests of the various approaches with simple models ought to precede the application of the approaches to models of real biomolecules such as proteins. In this context the application of alternative procedures that do not require volume corrections would also be very useful. In a previous paper (Zipper and Durchschlag 2010a), we have reported on the application of several programs of the HYDRO family (Garc´ıa de la Torre et al. 1994) to multibead models of spheres and several prolate and oblate ellipsoids of revolution. A main goal of that study had been the comparison of different approaches to improve the prediction of the intrinsic viscosity by modifying the usual volume correction. Our approach of a ‘‘reduced volume correction’’ (RVC) (Zipper and Durchschlag 2007), implemented in our in-house versions of HYDRO based on the previously available source code, turned out to be superior to the novel approach (Garc´ıa de la Torre et al. 2007) of an ‘‘adjusted, intermediate volume correction’’ (AVC; the authors prefer the terminus KRI correction) as implemented in Garc´ıa de la Torre’s HYDRO??8c program (binaries were downloaded from http://leonardo. fcu.um.es/macromol). The results from our RVC approach were found to be quite similar to those obtained by means of the well-known ‘‘cubic-substitution approach’’ (CBS) (Wilson and Bloomfield 1979; Garc´ıa de la Torre and Bloomfield 1981), however, without the inherent restrictions of that approach. Another important outcome of our previous study was the finding that a double-precision implementation of HYDRO was necessary for obtaining accurate results for the translational diffusion coefficient D and the sedimentation coefficient s at bead numbers N [ 2,000. A comparative application of the different approaches to various proteins was presented in an accompanying paper (Zipper and Durchschlag 2010b). Meanwhile binaries of more recent versions of HYDRO?? (versions 9 and 10) became available at http://leonardo.inf.um.es/macromol. These are based on a completely rewritten code and contain a revised implementation of the AVC approach (KRI correction) (Garc´ıa de la Torre et al. 2010). At about the same time, we became aware of an alternative program that opens a promising way to check the hydrodynamic predictions of the various HYDRO/HYDRO?? versions by an independent method. The program, called ZENO after the underlying Zeno algorithm (Mansfield et al. 2001), uses numerical path integration to calculate the translational diffusivity and intrinsic viscosity of macromolecules (Kang et al. 2004; Mansfield and Douglas 2008) and offers some valuable advantages: there is virtually no limitation of the bead number N, overlapping beads do not cause any problems, and volume corrections are not needed. Some pilot tests indicated the applicability of ZENO as an alternative to HYDRO runs (Durchschlag and Zipper 2012a, b). These facts and our own success in improving our in-house versions of HYDRO strongly suggested a resumption and continuation of our previous studies on multibead models. Besides the spherical models already used previously, the present study is based on a gamut of new spherical, ellipsoidal, cylindrical and prismatic bead models. By keeping the volumes of the models nearly constant at all axial ratios, disturbing effects like the previously encountered dependence of practicable bead size on the axial ratio are now avoided. Construction of bead models Several in-house programs were used for building appropriate bead models to be analyzed in this study. The models were constructed by filling a given geometric body of spherical, ellipsoidal, cylindrical or prismatic shape by touching equal-sized spheres of given bead radius rb, arranged according to one of the following lattices: simple cubic (C), body-centered cubic (BCC), face-centered cubic (FCC), and hexagonal close packing (HCP). The bead radii of the models were varied systematically. The same procedure was used to construct models of hollow bodies. In the case of spherical models of lattice type C, two variants of models were created depending on whether the center was occupied by a bead or not. No measures were applied to optimize the curved surfaces of the bead models. For cylindrical and prismatic models, however, the bead radii were tuned according to the bias that the axial radii of gyration along the height of the cylinders or the edges of the prisms ought to match the values calculated for the corresponding whole-body model. Hydrodynamic analysis of the bead models For the calculation of hydrodynamic and structural properties of the bead models several programs of the wellknown HYDRO family and, as an alternative, the program ZENO were used. HYDRO was introduced by Garc´ıa de la Torre et al. (1994) as a versatile computer program for the prediction of hydrodynamic properties of rigid macromolecules. The source code of a later version (file hydrod13.f, subroutine HYDRO, version 13, June 1999, downloaded from http://leonardo.fcu. um.es/macromol) served as the basis for our in-house version ‘‘double-precision HYDRO’’ (DPH) used in this study. To obtain this special version, several changes were applied to the downloaded source code. These are described in detail in the Supplementary Material (SM_Text_1). As a matter of fact, not all calculations presented in this study were performed by using the final version of DPH because the gradual development of the program extended over a longer period of time. However, the changes (i)–(iii) (cf. SM_Text_1) have always been implemented (‘‘first version of DPH’’) and careful tests have shown that all versions of DPH used in this study led to consistent results. The first version was already used in previous studies (Zipper and Durchschlag 2010a, b). The computing time for the final version, resulting from the implementation of all changes, amounts to less than 25 % of the time needed with the first DPH version. Further improvements resulted in mixed-precision HYDRO (MPH) versions which also contain an implementation of a novel reduced volume correction for the prediction of rotational properties. Details will be dealt with in a subsequent paper. While the full volume correction (FVC) for the intrinsic viscosity (Garc´ıa de la Torre 1989; Garc´ıa de la Torre and Carrasco 1998) adds a term 2.5NAV/M (where NA is Avogadro’s number and V and M are the total volume and molar mass of all N beads, respectively) to the uncorrected value [g]NVC, for our reduced volume correction (Zipper and Durchschlag 2007) this term is divided by the cuberoot of the bead number N before it is added to [g]NVC: The idea underlying this approach is that only the beads on the surface are considered to be essential for the volume correction and that their contribution scales with N2/3. For convenience, the output of DPH provides the results for [g]NVC, [g]FVC, and [g]RVC simultaneously and, if overlapping beads were encountered, the volume correction is also computed automatically by applying a corrected volume besides the uncorrected total bead volume. It should be mentioned that our special interaction tensor for handling unequal-sized overlapping beads (Zipper and Durchschlag 1997, 1999; Carrasco et al. 1999) was already implemented in the downloaded source code of HYDRO. In the present study, however, this tensor will not be used since all models considered here consist of equal-sized beads and bead overlaps occur, if at all, in the context of the cubic substitution (CBS) approach only. The most recent authentic versions of HYDRO, named HYDRO??9 and HYDRO??10, were downloaded as 32-bit binaries from http://leonardo.inf.um.es/macromol (Garc´ıa de la Torre et al. 2010). Both programs are very fast due to a completely rewritten code and use of parallel calculation. It is claimed in the HYDRO?? user manual (http://leonardo.inf.um.es/macromol/programs/hydro??/ hydro??10.pdf) that these programs solve definitely some ambiguities related to rotational diffusion and intrinsic viscosity. These quantities should now be evaluated with great accuracy, whereas the results for translational diffusion and sedimentation ought to be practically the same as with the HYDRO program. The programs can be run in different modes. To cope with the problem that the full volume correction for the intrinsic viscosity usually results in too high values for [g], Garc´ıa de la Torre and coworkers have introduced an ‘‘adjusted, intermediate volume correction’’ that is activated by setting the option ICASE = 12 (standard mode). A first version of this correction had already been implemented in the outdated version HYDRO??8c (Garc´ıa de la Torre et al. 2007) and was revised later on (Garc´ıa de la Torre et al. 2010). The present revised correction is based on the ‘‘degree of fragmentation’’ S (calculated from the size distribution of the beads) and the asphericity AG (calculated from eigenvalues of the gyration tensor) of the model under analysis and yields a viscosity value that is called [g]KRI (Kirkwood-Riseman intermediate). On the other hand, by setting the option ICASE = 21 the HYDRO?? programs are run in the cubic substitution mode: each bead of the original model is replaced with eight smaller beads in a cubic arrangement and the intrinsic viscosity [g]CBS is calculated in this case without any volume correction (Wilson and Bloomfield 1979; Garc´ıa de la Torre and Bloomfield 1981). This mode, however, is feasible only for models with bead numbers N well below 1,000. By setting the option ICASE = 20, the programs are run in a special mode taking 3rd-order hydrodynamic interactions explicitly into account, thereby yielding improved results for translational diffusion and rotational properties (Carrasco and Garc´ıa de la Torre 1999b). Since for this mode a 6N 9 6N supermatrix of interaction tensors is used instead of the usual 3N 9 3N supermatrix, application of this mode is restricted to models with N well below 4,000. While the maximum number of beads, Nmax, that can be handled by HYDRO?? (in standard mode) or DPH on 32-bit platforms is restricted by the system to values around 7,500, with the 64-bit versions of DPH Nmax & 10,500 is available already with 4 GB RAM. This feature of 64-bit DPH enhances the range of applicability of the CBS approach to models with N = 1,250 and even more. To enable an independent control of the hydrodynamic properties predicted by means of the programs HYDRO?? and/or DPH, the program ZENO was chosen as a straightforward alternative. This program calculates (by numerical path integration) random walks from arbitrary points on the surface of a ‘‘launch sphere’’, completely surrounding the body under investigation, towards the surface of the body (Mansfield et al. 2001; Kang et al. 2004). The statistics of random walk trajectories reaching the surface (i.e., ending at a distance from the surface smaller than the ‘‘skin thickness’’ e) yields electrostatic properties of a perfect conductor of the same shape and size as the body under analysis. These properties are the electrostatic capacity C, the polarizability tensor ae and the mean polarizability haei = (1/3) Raii. The hydrodynamicelectrostatic analogy relates these electrostatic properties with hydrodynamic properties of the body such as the hydrodynamic radius Rh (and, hence, also the translational diffusion coefficient D) and the intrinsic viscosity [g] (Mansfield and Douglas 2008): m : mass of the body: Since the factor qg depends on the shape of the body (for a sphere: qg = 0.833; for infinitely long rods, infinitely long prolate ellipsoids, and circular plates: qg = 0.8; for a pair of spheres at infinite separation: qg = 0.75), by using a Pade´ approximant based on the eigenvalues of the polarizability tensors of ellipsoids, an individual value for qg can be assigned to each body under analysis in dependence on its polarizability tensor (Mansfield and Douglas 2008). For ellipsoids, this approach is exact, while for other shapes it at least considerably reduces the 5 % uncertainty in qg (and, hence, in [g]) resulting from the above Eq. 6. Two interesting features of ZENO should be mentioned though they are not of importance for the present study: (i) ZENO does not restrict the elements used to model a body to be spheres; ellipsoids, cubes, and several other shapes may be used as well; (ii) ZENO has no problems with overlapping elements; the calculated volume represents the total volume of the body, not of the elements, comparable to the volume corrected for overlaps as provided by DPH. That ZENO can handle up to 65,000 elements is of greatest general importance. The source code of the most recent version (ZENO version 3.3) was obtained directly from Marc Mansfield (). Note that the version posted at http://www.stevens.edu/zeno/downloads. html is outdated and lacking the later improvements in the calculation of [g]. To obtain the hydrodynamic parameters D, s and [g], together with the structural parameters V and RG, the actions ‘‘zeno calculation’’ and ‘‘interior calculation’’ must be chosen. The accuracy of calculations and the computing time depend on the selected number of random walkers (for ‘‘zeno calculation’’) or points (for ‘‘interior calculation’’). In this study, a value of 1,000,000 was used for both quantities. Furthermore, with ZENO the computing time also scales with the bead number N, whereas for HYDRO/HYDRO?? the computing time is proportional to N3. A valuable feature of ZENO is that it performs numerical error estimates on all computations; appropriately rounded results with estimated errors and the raw data without rounding are provided on separate output files. Results and discussion Experience has shown that the factors qh and qg adopt the following values: A sphere (S) of 50 A˚ radius served as the starting point for modeling. It represents per se the simplest whole-body (WB) model and all other models in this study are somehow related to this sphere. As in our previous study (Zipper and Durchschlag 2010a) the sphere was assumed to have a mass of 298.75 kDa. Its structural and hydrodynamic parameters (volume, V, radius of gyration, RG, translational diffusion coefficient, D, and intrinsic viscosity, [g]) can be found in the Supplementary Material (SM_Table_1). We abstain from presenting the sedimentation coefficient additionally, because this parameter is strictly proportional to D. In our previous study, this sphere has been modeled by a series of bead models composed of equal-sized touching spherical beads in a body-centered cubic arrangement, the radii rb of the beads varying systematically from 2 to 10 A˚ in steps of 0.25 and 0.5 A˚ , respectively. Of those bead models, for practical reasons only the fraction corresponding to the rb range from 2.25 to 7 A˚ was chosen for the present study; this selection of S (BCC) models was, however, completed by an additional model with rb = 1.5 A˚ . Similar series of bead models for the sphere, S(FCC) and S(HCP), were created on the basis of closepacked beads in face-centered cubic and hexagonal arrangements, respectively, while the lower packing density of beads arranged in a simple cubic lattice favored the inclusion of S(C) bead models with rb = 2 A˚ in this case. Examples of the various types of bead models for the sphere, all corresponding to rb = 2.25 A˚ , are shown in Fig. 1. The figure also shows a central slab of a bead model for a hollow sphere, HS(BCC) 0.5, with a ratio of 0.5 between inner and outer radius; this and similar models corresponding to hollow spheres of ratios 0.25 and 0.75 were obtained by deleting the appropriate number of interior beads from the S (BCC) model series. The calculated structural and hydrodynamic parameters of the hollow spheres are also given in SM_Table_1. While in our previous study prolate and oblate ellipsoids were derived from the basic sphere by enlarging the length of one or two axes, thereby enlarging the volume too, we preferred a different procedure for the ellipsoids used in this study: the axial ratio was changed, but the volume of the basic sphere was retained. For building bead models to simulate eventually the ellipsoidal whole-bodies (for the theoretical parameters see SM_Table_1) a body-centered cubic arrangement of beads was used again; the same bead radii were selected as for the S (BCC) models. Examples of bead models PE and OE for prolate and oblate ellipsoids of different axial ratios are also shown in Fig. 1. The creation of cylindrical models followed similar principles. First, an isometric circular cylinder (CC) having the same volume as the basic sphere was taken as primary whole-body, and then its axial ratio was systematically changed to generate various prolate and oblate circular cylinders (PCC and OCC) all having the same volume. All these whole-bodies were then modeled by beads assembled in a body-centered cubic lattice. As a consequence of the bias laid upon the axial radius of gyration along the height of the cylinders, the bead radii could not be chosen arbitrarily as was the case with the spherical and ellipsoidal models discussed before; instead they were calculated by the modeling program in dependence on an adjustable input parameter. Examples of cylindrical bead models composed of beads with rb & 2.1 A˚ are shown in Fig. 2. The starting point for prismatic whole-bodies was the cube (CUB) whose faces are tangential to the basic sphere; its volume is, of course, larger that that of the sphere. Changing the axial ratio systematically under retention of the volume led to various prolate and oblate prisms (PP and OP). These and the cube were then modeled by beads assembled in a simple cubic lattice. Due to the bias laid upon the three axial radii of gyration, the bead radii were calculated also in this case by the modeling program in dependence on an adjustable input parameter. Examples of prismatic bead models, corresponding to rb & 2.5 A˚ , are presented in Fig. 2. The calculated structural parameters of the cylindrical and prismatic whole-bodies are also given in SM_Table_1; accurate hydrodynamic parameters of these WB models are not available because exact analytical expressions are lacking for these kinds of models. Hydrodynamic calculations The results from the hydrodynamic calculations ought to serve as a sound basis for a subsequent critical comparison and valuation of the various programs and approaches used in this study. Therefore, a careful check and appropriate consideration of all factors that could influence the results of our calculations was a necessary prerequisite in order to attain our concern. The results from a test applying the various programs and approaches to the basic sphere (S), representing a whole-body and a single-bead model as well, illustrate the success of our measures to rule out systematic errors. These results are tabulated in detail at the top of SM_Table_1: Both the rounded results for the structural and hydrodynamic parameters obtained from a single run of ZENO (SM_Table_1, first line of ZENO data, with the estimated errors in parentheses) and the mean values and standard deviations of the unrounded parameters obtained from 10 subsequent runs of ZENO (SM_Table_1, second line of ZENO data for the single-bead model) are in best agreement with the theoretical values for a solid sphere (e.g., D = 4.286 9 10-7 cm2/s and [g] = 2.639 cm3/g). And the same is true for the data that were calculated by means of DPH and by HYDRO?? in standard mode (ICASE = 12) or in the special 3rd-order mode (ICASE = 20). On the Fig. 1 Images of selected bead models of a sphere (S), a hollow sphere (HS), and prolate and oblate ellipsoids of revolution (PE, OE). The models shown are composed of spherical beads of rb = 2.25 A˚ , arranged according to the following types of packing: body-centered cubic (BCC), face-centered cubic (FCC), hexagonal close packing (HCP), and simple cubic (C) with central bead OE (BCC) 1:1:0.125 N = 7501 contrary, all results obtained by applying the CBS approach to the single-bead model deviate significantly from the theoretical parameters; this finding suggests that the CBS approach is not a good choice for the single-bead model. For the hydrodynamic calculations on the various multibead models, a common procedure scheme was adopted. This scheme was based on the dependence of the feasibility of the various methods and approaches on the bead number N. According to the scheme, ZENO was applied to all bead models generated, regardless of the bead number N (the limiting value Nmax for this program is much higher). The program DPH was applied to all models with N \ 10,000. The range where HYDRO?? could run in standard mode was limited approximately by N \ 7,400, and still lower (about N 4,000) was the limit for HYDRO?? runs in the 3rd-order mode. Accordingly, when DPH was used, the feasibility range for performing the CBS approach extended to higher bead numbers (up to N & 1,200) as compared to the range when using the CBS implementation (ICASE = 21) of HYDRO?? (up to N & 940). The results obtained for the multibead models of the sphere and the other whole-bodies are much too numerous to be presented in detail in a table. This can be done only graphically (Fig. 3 and figures in the Supplementary Material). Mean values and the corresponding standard deviations, obtained by averaging the respective structural or hydrodynamic parameters within the ranges limited by the given N and rb values, are presented in SM_Table_1. Already a first glance on the plotted results reveals a considerable scatter of data with increasing bead radius rb. This scatter appears to be random in the case of all spherical, ellipsoidal and cylindrical models (Fig. 3a–c and SM_Fig_1–SM_Fig_7), probably due to accidental insufficiencies in modeling the curved surface of the underlying Fig. 2 Images of selected bead models of an isometric circular cylinder (CC), prolate and oblate circular cylinders (PCC, OCC), a cube (CUB), and prolate and oblate rectangular prisms (PP, OP). The cylindrical models (packing type BCC) are composed of spherical beads of rb & 2.1 A˚ , the prismatic models (packing type C) consist of beads of rb & 2.5 A˚ bodies by means of large beads (cf. also the horizontal dashed lines representing the theoretical values DWB and [g]WB of the respective whole-bodies). On the other hand, the plots for the prismatic models (Fig. 3d and SM_Fig_8– SM_Fig_12) suggest a functional dependence of D and [g] on rb; this behavior can be explained by the systematically changing angularity of these models. The process of averaging the hydrodynamic parameters D and [g] delivers compressed information about the parameters obtained from the different programs and approaches, but presumably obscures any differences depending on the bead radius rb or the bead number N. Moreover, a direct comparison of the tabulated mean values for D and [g] of a great number of models of different type and axial ratio turns out to be quite cumbersome rather than conclusive. We therefore decided to ground our critical comparison on relative quantities rather than on absolute ones. For convenience, the values DZENO and [g]ZENO obtained by means of ZENO for each of the multibead models were used as a reference for computing the ratios Dx/DZENO and [g]x/[g]ZENO, where the subscript x stands for DPH, H??, H??(3rd) or CBS and for RVC, KRI or CBS, respectively. At last, these ratios were averaged in the same way and range as the D and [g] values themselves had been averaged before. The mean values of Dx/DZENO 2 /)s4.5 m -7 c4.0 0 (13.5 D3.0 Fig. 3 Plots of the predicted values for the translational diffusion coefficient D (circles) and the intrinsic viscosity [g] (diamonds and squares) of bead models of selected geometrical bodies versus the bead radius rb. a sphere (S, packing type BCC), b prolate ellipsoid (PE 1:1:2.82), c oblate ellipsoid (OE 1:1:0.125), d cube (CUB). The data were obtained by means of the programs ZENO (filled blue symbols), DPH (empty black symbols; diamonds: [g]RVC, squares: [g]NVC), and HYDRO?? (versions 9 and/or 10) in standard mode (filled red symbols) and the special mode taking 3rd-order hydrodynamic interactions into account (empty red circles); empty green symbols represent results obtained by applying the cubic substitution (CBS) approach by means of DPH; filled green circles represent values for D obtained by HYDRO?? in CBS mode. The horizontal lines in images a–c symbolize the theoretical values DWB and [g]WB of the corresponding whole-bodies; the colored lines in image d were obtained by 2nd-order regression analysis of the predicted data and [g]x/[g]ZENO thus obtained and the corresponding standard deviations are listed in Table 1. These data already represent a firm basis for a critical comparison and valuation of the programs and approaches applied, and this basis can be widened and strengthened further by inclusion of the information extracted visually from Fig. 3 and from the figures presented in the Supplementary Material (SM_Fig_1–SM_Fig_12). In a previous paper (Zipper and Durchschlag 2010a) we had used the theoretical values of the underlying whole-bodies, DWB and [g]WB, for establishing normalized hydrodynamic parameters of multibead models. In this study, however, we abstain from following that approach because ratios like Dx/DWB and [g]x/[g]WB would be available only for the spherical and ellipsoidal models. Moreover, these ratios would be a priori less informative and conclusive than the ratios based on DZENO and [g]ZENO (Table 1) because of the reasons already discussed above and because any insufficiencies of the bead models would contribute considerably to Dx/DWB and [g]x/[g]WB, whereas the ratios Dx/DZENO and [g]x/[g]ZENO would be affected, if at all, in a much lesser extent. Spherical and ellipsoidal models A close inspection of the plotted data for the multibead models of the sphere S (BCC) (Fig. 3a) reveals a perfect agreement of the values for D obtained by HYDRO?? run in the standard mode and by DPH, whereas the values for [g] obtained concurrently disagree to a large extent, except for the models with rb C 6 A˚ . Below this rb range, the values of [g]KRI obtained by HYDRO?? are always definitely larger than the values of [g]RVC predicted by DPH. The results obtained by means of ZENO, DZENO and [g]ZENO, are throughout smaller than the corresponding data obtained by HYDRO?? or DPH. The differences between [g]RVC and [g]ZENO are, however, much smaller than those between [g]KRI and [g]ZENO and appear to decrease with decreasing rb. A similar decrease can be Table 1 Averages and standard deviations of ratios Dx/DZENO and [g]x/[g]ZENO obtained for multibead models of selected geometrical bodies by various computational approaches using ZENO and different versions of HYDRO S sphere, PE prolate ellipsoid of revolution, OE oblate ellipsoid of revolution, HS hollow sphere, CC circular cylinder, PCC prolate circular cylinder, OCC oblate circular cylinder, CUB cube, PP prolate quadratic prism, OP oblate quadratic prism, WB whole-body model of given shape and axial ratio (r or a:b:c or 2r:H or A:B:C in relative units) a The mean ratios were obtained from series of geometrically similar bead models differing in the number and size of the beads (for details cf. SM_Table_1) bHYDRO?? was run in standard mode cHYDRO?? was run in the special mode taking 3rd-order hydrodynamic interactions into account dThese data are based on results computed by DPH e [g]RVC was obtained by means of DPH applying the reduced volume correction f [g]KRI was obtained by means of HYDRO?? in standard mode, applying the adjusted intermediate volume correction g No volume correction was applied in the computation of [g]CBS observed for the discrepancies between the predicted D values. The latter discrepancies can be reduced effectively either by applying the CBS approach (green circles) or by taking 3rd-order hydrodynamic interactions into account (empty red circles). The data given in Table 1 for S (BCC) summarize these findings in a quantitative manner. In particular, the differences between the averaged ratios of [g]KRI/[g]ZENO (derived from HYDRO?? data) and [g]RVC/[g]ZENO (derived from DPH data) and between the pertinent standard deviations are striking. Similar results are also obtained for the spherical multibead models based on the packing schemes FCC, HCP and C (Table 1; cf. also SM_Fig_1; SM_Table_1): also for these models, the averaged ratios of [g]KRI/[g]ZENO and their standard deviations exceed the corresponding data for the ratios of [g]RVC/[g]ZENO; the effect is most pronounced for S(FCC) and S(HCP), and weakest for S(C) (cf. Table 1). Some of the results obtained for the multibead models of prolate ellipsoids of revolution (Fig. 3b; Table 1; cf. also SM_Fig_2; SM_Table_1) are similar to those of the spherical models. Again, the D values predicted by DPH or by HYDRO?? are virtually identical and slightly larger than the values predicted by ZENO; the mean ratios of DDPH/DZENO and DH??/DZENO are nearly the same as obtained for the models S (BCC), irrespective of the axial ratio. And also the application of the CBS approach or the consideration of 3rd-order hydrodynamic interactions has similar effects on the D values as observed for the models S (BCC). On the contrary, the results obtained for the intrinsic viscosity obviously behave in a different way as follows from an inspection of the averaged ratios of [g]RVC/[g]ZENO and [g]KRI/[g]ZENO (Table 1). While the mean values of [g]RVC/[g]ZENO decrease with increasing anisometry of the models, approaching a value of about 1.0 at an axial ratio of 1:1:2.82, the opposite behavior, an increase with increasing anisometry, can be observed for the mean values of [g]KRI/[g]ZENO. On the other hand, the mean values of [g]CBS/[g]ZENO, resulting from application of the CBS approach, apparently do not depend much on the axial ratio; the mean value obtained at the axial ratio of 1:1:2.82 is not significantly smaller than the values observed for the spherical models. The predictions of hydrodynamic parameters for the multibead models of oblate ellipsoids of revolution by HYDRO??, DPH, and ZENO led to results that show some peculiarities (Fig. 3c; Table 1; cf also SM_Fig_3; SM_Table_1). Most remarkable is the finding that for the same model the programs HYDRO?? and DPH may predict quite different values of D: an inspection of Fig. 3c illustrating the results obtained for the oblate models of axial ratio 1:1:0.125 reveals an obvious underestimation of D by HYDRO?? in the standard mode (filled red circles) at rb = 2.5 A˚ (corresponding to N = 5,481) and in the CBS mode (filled green circles) at nominal rb = 5.0 A˚ (corresponding to effective values of rb = 2.33 A˚ and N = 5,528 for the bead model obtained by the cubic substitution of the original bead model) or at even larger rb (filled green circles in comparison to the empty green circles representing the CBS results for D obtained by means of DPH). According to our data, the divergence of D values obtained by means of HYDRO?? and DPH, respectively, for this type of ellipsoid becomes obvious already at effective bead numbers as low as about 3,150. A similar, however less pronounced underestimation of D by HYDRO?? in CBS mode can also be observed with the oblate ellipsoid of axial ratio 1:1:0.3 (cf. SM_Fig_3) and even the CBS results for the already mentioned prolate ellipsoid of revolution (axial ratio 1:1:2.82) are affected as follows from a close inspection of Fig. 3b. Another peculiarity of the hydrodynamic predictions for the oblate ellipsoidal multibead models concerns the ratios derived from the values predicted for the intrinsic viscosity (Table 1): the mean values of [g]RVC/[g]ZENO, based on the predictions by DPH, decrease gradually with increasing anisometry, reaching a level slightly below 0.99 at an axial ratio of 1:1:0.125; on the other hand, the mean values of [g]KRI/[g]ZENO, derived from the HYDRO?? results, are at low or medium anisometry slightly higher, and at the highest degree of anisometry slightly lower than the mean ratio found for the S (BCC) models; finally, the mean values of [g]CBS/[g]ZENO again decrease gradually, approaching a level of about 1.0 at maximum anisometry. As expected, the results obtained for the translational diffusion coefficient of the bead models of hollow spheres are independent of the ratio of inner and outer radius and, in general, identical with the values predicted for the underlying S (BCC) models. This statement holds both for the D values themselves (SM_Fig_4; SM_Table_1) and for the various ratios derived therefrom (Table 1). The results obtained for the intrinsic viscosity of the bead models, of course, change systematically ([g] increases, the mean values of the ratios [g]RVC/[g]ZENO and [g]KRI/[g]ZENO decrease) with increasing ratio of inner and outer radius; only the mean values of [g]CBS/[g]ZENO appear to be nearly independent of the degree of hollowness of the models. Eventually, a few remarks have to be made concerning the agreement of the parameters predicted by ZENO and the other approaches for the multibead (MB) models with the theoretical parameters of the underlying spherical or ellipsoidal whole-body (WB) models. A comparison of the data in SM_Table_1 reveals that at rb = 1.5 A˚ the theoretically calculated values for RG, D, and [g] and the predictions by ZENO agree within the given limits of error. A close inspection, however, shows that the predicted values DZENO are slightly smaller than the theoretical values, whereas for the predicted values [g]ZENO the opposite is true. A plausible explanation for this finding might be the still rugose surface of the multibead models, even at such large numbers of beads, while the whole-body models, of course, possess a perfectly smooth surface. The predictions for the volume V disagree substantially with the theoretical volume of the whole-body models even at rb = 1.5 A˚ , but the extent of these discrepancies, expressed by the volume ratio VMB/VWB, is close to the theoretical packing density of the beads in the MB models. Since, in general, the insufficiencies of the bead models will increase upon lowering N, it is not surprising that the discrepancies observed between the theoretical values DWB and [g]WB (cf. the horizontal lines in Fig. 3a–c and SM_Fig_1–SM_Fig_4) and the predictions by ZENO and the other approaches increase with increasing rb. Cylindrical models The results obtained for the multibead models of circular cylinders (Table 1; cf. also SM_Fig_5–SM_Fig_7; SM_Table_1) are qualitatively similar to those for the multibead models of spheres and ellipsoids. For instance, striking similarities can be observed in the behavior of the intrinsic viscosity [g]KRI if compared to [g]RVC and [g]ZENO, in particular for prolate cylinders: while the mean values for [g]RVC/[g]ZENO gradually decrease from 1.018 for the isometric cylinder to 0.998 at an axial ratio of 1:4, the mean values for [g]KRI/[g]ZENO change in the opposite direction from 1.061 to 1.175, with a maximum of 1.183 encountered at the axial ratio of 1:3. The behavior of the mean viscosity ratios of the bead models for oblate cylinders is also quite close to that observed with oblate ellipsoids. Discrepancies between the values of D predicted by DPH and HYDRO??, respectively, can be observed both with prolate and oblate cylindrical models, both in the standard mode and when applying the CBS approach. The most pronounced examples are shown in SM_Fig_6 (PCC with axial ratio 1:4) and SM_Fig_7 (OCC with axial ratios 1:0.167 and 1:0.125, respectively). Except for the divergent DH?? values at large N, the D values obtained by DPH and HYDRO?? for the cylindrical models agree better with the values DZENO than in the case of the ellipsoidal models. This follows directly from a comparison of the mean values of the ratios DDPH/DZENO and DH??/DZENO (Table 1). Moreover, taking 3rd-order hydrodynamic interactions into account leads to values DH??(3rd) that are usually slightly smaller than DZENO, contrary to the opposite relation, DH??(3rd) [ DZENO observed for the ellipsoidal models. In general, the results obtained by application of the CBS approach follow the relation DCBS \ DH??(3rd) (cf. SM_Table_1). For generating bead models of a cube and various anisometric prisms, increasing numbers of touching beads were packed in a simple cubic lattice. We always started with a bead model in the strict sense, built from the smallest number of beads (with radius rb between about 26 and 12 A˚ , depending on the shape to be modeled), and ended at multibead models composed of thousands of beads with rb & 1.5 A˚ . Due to our special algorithm, all bead models for a given prism have the same overall and axial radii of gyration as the underlying whole-body. This was achieved by tuning the bead size appropriately. Accordingly, the total bead volume of the models decreases with increasing number of beads. This decrease is most pronounced in the case of the cube, amounting to nearly 15 % of the initial volume, and reduces to about 2.5 % with the oblate prism of axial ratio 1:1:0.33. Regardless of the axial ratio, more than 50 % of the entire change in volume occurs in the step from the model with minimum bead number to the next higher one and the volume at high N is the same for all shapes. Upon increasing the bead number, of course, the rugosity of the plain faces decreases and the angularity of the bead models increases. In contrast to the experiences with the ellipsoidal and cylindrical bead models, the predictions of the hydrodynamic parameters for the prismatic bead models result in values of D and [g] that clearly reveal functional dependencies on the bead radius rb. The dependencies of the viscosity data are shown most impressively by the profiles obtained from the bead models for the cube (Fig. 3d) whereas the dependencies of the diffusion data can be studied in more detail at hand of the profiles obtained from the anisometric bead models (SM_Fig_8, SM_Fig_9). A common feature of all graphs is a cross-over of the profiles for DDPH (or DH??) and DZENO, resulting from a considerable divergence of the diffusion values at large rb (corresponding to the models with smallest N) and a smaller divergence, in the opposite direction, in the region of low rb (high N); usage of the CBS approach or consideration of 3rd-order hydrodynamic interactions led to diffusion data that are at large rb quite similar to DZENO but diverge even more than DDPH or DH?? at medium and low rb. Slight discrepancies between DDPH and DH?? at small rb can also be observed with the highly anisometric prolate and oblate models (axial ratios 1:1:3 and 1:1:0.33, respectively; SM_Fig_8, SM_Fig_9). An inspection of the viscosity data in Fig. 3d, SM_Fig_8, SM_Fig_9 reveals that the profiles for [g]CBS are always quite similar to those for [g]ZENO, and a similar behavior can also be observed for [g]RVC, except for the CUB model at largest rb (Fig. 3d). This model with N = 8 is a classical bead model in the strict sense and such models are a priori somewhat beyond the scope of our approach of reduced volume correction. This may explain the about 10 % difference between the value of [g]RVC found for this model and the value of [g]KRI obtained by means of the approach of adjusted volume correction implemented in HYDRO??. The AVC approach has been developed with special regard to bead models in the strict sense. It is, therefore, not surprising that for this distinguished CUB model [g]KRI and [g]CBS nearly coincide and both values are close to [g]ZENO. However, already with the next higher CUB model (N = 27) the situation changes completely: [g]KRI becomes significantly lower than the value of [g]CBS and the further course of the [g]KRI profile towards lower rb is clearly different from the profiles obtained from the other predictions. In the region of low rb eventually the [g]KRI values exceed all other predicted viscosities. Similar discrepancies between the profiles for [g]KRI and the viscosity profiles obtained by means of ZENO, DPH, and the CBS approach, respectively, can also be observed with the anisometric prismatic bead models (SM_Fig_8, SM_Fig_9). For convenience, the viscosity data [g]NVC, obtained without any volume correction by means of DPH (cf. last column in SM_Table_1), are also presented in these figures as black squares. These data convincingly demonstrate the importance of appropriate volume corrections in the prediction of intrinsic viscosity, in particular for models with small or moderate bead numbers. Especially in the region of small rb (high N) the shape of the profiles for [g]KRI and [g]NVC appears to be similar; they run nearly parallel, however, separated by a gap the width of which is different for the various types of models. Regarding the considerable spread of the values for D and [g] obtained for the prismatic models within the rb ranges covered by our computations, it would not make sense to average these data over the full spectrum of rb values. To obtain results that can be compared to the corresponding data for the ellipsoidal and cylindrical models (Table 1) we selected only prismatic models with rb \ 7 A˚ for calculating averages of the hydrodynamic parameters and of the various ratios derived therefrom. The results obtained from these subsets of prismatic models are also presented in SM_Table_1 and Table 1. They demonstrate once more the exceptional behavior of [g]KRI. This is also reflected by the mean values for [g]KRI/[g]ZENO which are significantly higher than the mean viscosity ratios derived from the other approaches; the most pronounced effect can be observed with the prolate models of axial ratio 1:1:3. The smooth course of the profiles shown in Fig. 3d; SM_Fig_8, SM_Fig_9 facilitates regression analyses of the data. It turned out that second-order polynomials are a good choice for approximating complete profiles, whereas first-order polynomials are sufficient for the approximation of the profiles cut off at rb [ 7 A˚ (cf. SM_Fig_10– SM_Fig_12). The detailed results of the regression analyses, the values for D and [g] extrapolated to rb = 0 and the corresponding standard errors, are compiled in SM_ Table_2. Remarkable features of the results both from firstorder and second-order regression are the close similarity or even coincidence of the extrapolated diffusion values (e.g., for CUB D = 3.18 9 10-7 cm2/s) based on the predictions by DPH, H?? in standard mode and in the special mode considering 3rd-order interactions, and by usage of the CBS approach, and a similarly good agreement of the extrapolated viscosity values (for CUB [g] & 6.20 cm3/g) obtained from the predictions again by DPH, the CBS approach, and even from the data calculated without volume correction. There is also a good agreement between the extrapolated ZENO data for D and [g], calculated by first-order and second-order regression, respectively, but these data (for CUB DZENO = 3.24 9 10-7 cm2/s and [g]ZENO = 6.11 cm3/g) deviate slightly from those mentioned above. Large discrepancies exist, however, between the extrapolated [g]KRI (for CUB [g]KRI & 6.60 cm3/g) and all other extrapolated viscosity data. To complete the data available from our computations, we performed regression analyses also of the total bead volumes as calculated by DPH and by ZENO, respectively. It is plausible that for this purpose usage of third-order polynomials turned out to be best. Averaging the DPHbased results obtained for the cube and the four anisometric prisms yielded for the volume extrapolated to rb = 0 a mean value of 523,596 ± 9 A˚ 3 which is nearly the exact value expected from the theoretical packing fraction of p/6 for a simple cubic lattice. A slightly higher, but due to the larger standard deviation not significantly different, result was obtained from the ZENO-based volumes; the larger standard deviation arises from the fact that ZENO determines the volumes by means of a Monte Carlo approach whereas DPH calculates the volumes directly from the given bead radii. These findings corroborate the assumption that the extrapolated values for D and [g] actually represent reliable predictions of the hydrodynamic parameters for the respective whole-bodies. Evaluation of the results An essential outcome of our study is the compilation of the mean ratios of hydrodynamic parameters presented in Table 1. In these ratios results of hydrodynamic predictions performed by means of approaches based on programs of the HYDRO family (DDPH, DH??, DH??(3rd), DCBS, [g]RVC, [g]KRI, [g]CBS) are compared with the results of predictions by means of the alternative program ZENO. The choice of DZENO and [g]ZENO as reference values was favored by the fact that ZENO data were available without any restrictions regarding the number of beads whereas all other approaches were limited in their feasibility by such restrictions; the most confining limitations (N B 1,250) were those for the CBS approach, followed by the limitations when using the special mode (3rd) of HYDRO?? (N 4,000). HYDRO?? in the standard mode was found to suffer from two different kinds of limitations: the restrictions imposed by the operating system (N \ 7,400) and additionally the beginning loss of reliability of some results (DH??) for anisometric models at large N. Because of this exceptional failure of HYDRO?? at large N, we will exclude results related to DH?? in the averaging procedures discussed below and we do also not use any values for DCBS that were computed by means of HYDRO?? instead of DPH. Already for SM_Table_1, we have neglected the results for the intrinsic viscosity when HYDRO?? was run in the special mode (3rd). According to Garc´ıa de la Torre et al. (2007), the inclusion of 3rd-order terms makes the formalism for the intrinsic viscosity extremely complicated; consequently the HYDRO?? user manual claims that in the 3rd-order mode no intrinsic viscosities are calculated. Nevertheless, the program HYDRO?? provides viscosity data also in this mode and these are identical with the values for [g]KRI obtained in the standard mode of HYDRO??. In the following we will use the tabulated mean ratios of hydrodynamic parameters (Table 1) as basis for a comparative evaluation of the various approaches and for drawing founded conclusions therefrom. For this purpose it appeared useful to begin with the calculation of grand averages of ratios, by averaging all mean values for six types of ratios given in Table 1, however, separately for isometric, prolate, and oblate models. The resulting grand averages and standard errors are compiled in Table 2. As follows from these data, the values for DDPH of the multibead models are on the average by about 1 % larger than the values for DZENO; the values for DH??(3rd) obtained by taking 3rd-order interactions into account are of about the same size as DZENO, whereas the values for DCBS are on the average by about 1 % smaller than DZENO. Thus, on the whole, all four approaches (DPH, H??(3rd), CBS, and ZENO) yield quite similar results for the translational diffusion coefficient D, the similarity being highest between DH??(3rd) and DZENO. The results obtained by these approaches do not depend significantly on the axial ratio of the bead models. On the contrary, the observed occurrence of divergences between DH?? and DDPH at large N is obviously favored by high degrees of anisometry. Presumably these divergences arise from numerical problems in the implementation of the algorithm for the prediction of D in HYDRO??. Far-reaching conclusions can be drawn from the grand averages of the viscosity ratios in Table 2. Here, an influence of the shape (axial ratio) on the ratio [g]KRI/[g]ZENO cannot be overlooked. The grand averages of the ratios [g]RVC/[g]ZENO and [g]CBS/[g]ZENO are for isometric bead models slightly larger than for oblate bead models, while in the case of prolate bead models these ratios show an ambiguous behavior: [g]RVC/[g]ZENO is definitely smaller than in the isometric case, whereas [g]CBS/[g]ZENO is of about the same size as for the isometric models. Due to the mentioned limitations for the CBS approach, the mean values for [g]CBS/[g]ZENO in Table 1 are based on much Table 2 Grand averages and standard errors of various ratios of hydrodynamic parameters for isometric, prolate, and oblate bead models fewer data than the mean values for [g]RVC/[g]ZENO. All grand averages for these two ratios, however, agree within 1 %, demonstrating that our approach of reduced volume correction as implemented in DPH, the CBS approach, and the numerical path integration by ZENO yield quite similar results for the intrinsic viscosity [g] of multibead models. The CBS approach was often claimed to yield nearly exact results for the hydrodynamic parameters D and [g] of bead models (Garc´ıa de la Torre and Carrasco 1998; Garc´ıa de la Torre et al. 2007, 2010). On the other hand, the approach of adjusted volume correction as implemented in HYDRO?? obviously fails to predict similarly reliable values for [g] for the multibead models investigated: as follows from Table 2, the smallest grand average for [g]KRI/[g]ZENO, 1.043, is obtained for isometric models and even slightly higher is the grand average obtained in the case of oblate models. Both values are, however, compatible with the magnitude of errors in experimental determinations of intrinsic viscosity (Garc´ıa de la Torre et al. 2007). On the other hand, the grand average of the ratio [g]KRI/[g]ZENO for prolate bead models is so large that its physical relevance has to be doubted. Considering the fact that the data reported in Table 2 represent averages of averages, it is evident that extreme results have already been leveled off in the computations. For example, in the case of cylindrical bead models with axial ratio 1:3, values as high as 1.20 were indeed observed for the ratio [g]KRI/[g]ZENO at rb \ 2.6 A˚ (N [ 4,900). The inefficiency of HYDRO?? in the prediction of acceptable values for the intrinsic viscosity of multibead models is not surprising because, according to the authors of HYDRO??, the program is intended for bead models in the strict sense, composed of less than 300 beads, and the approach of adjusted, intermediate volume correction (by the authors also called the KRI procedure) was based on calculations made in this range of bead numbers only, using CBS calculations for reference (Garc´ıa de la Torre et al. 2010). On the other hand, HYDRO?? can easily handle bead models composed of thousands of beads, and for such multibead models the AVC approach is obviously not sufficient. Garc´ıa de la Torre et al. (2010) recommend the application of shell model strategies for such models or, if one prefers to use filling models and HYDRO??, to neglect the volume correction at all. Neglecting volume corrections is, however, possible with HYDRO??9 but not with the latest version HYDRO??10 (the corresponding option ICASE = 10 is not accepted as valid by this version). Unfortunately, selection of ICASE = 10 implies the suppression of all volume corrections so that the calculation of the radius of gyration and of rotational properties are affected as well. Apart from these certainly resolvable handicaps, there remains the question whether for bead models beyond the range of feasibility of the AVC approach the volume correction of intrinsic viscosity can really be neglected safely. To answer this question, we calculated the mean ratios [g]NVC/[g]ZENO and [g]CBS/[g]NVC for all bead models used in this study (cf. SM_Table_3). Grand averages of these mean ratios are also presented in Table 2. These results reveal that, within the range of bead radii and bead numbers as specified for DPH in SM_Table_1, the uncorrected intrinsic viscosity [g]NVC is on the average by about 4 % smaller than the viscosity predicted by ZENO (cf. also SM_Fig_13), while, within the limited range accessible by the CBS approach, the intrinsic viscosity predicted by this approach is on the average by more than 6 % larger than [g]NVC. These data do not really support the neglect of any volume correction for the intrinsic viscosity in the range where the AVC approach becomes insufficient. Our concept of reduced volume correction, on the other hand, is a much better alternative to neglecting any volume correction for the intrinsic viscosity, because it takes the physically reasonable dependence of the necessary volume correction on the bead number N appropriately into account: our approach provides the full volume correction at N = 1 and smaller volume corrections, decreasing with 1/N1/3, for all N [ 1. Thus, at N = 1,000 our reduced volume correction corresponds to 10 % of the full volume correction, and for very large N, our correction eventually becomes negligibly small. Presently, our approach does not include any adjustment for shape and fragmentation comparable to that being an essential feature of the AVC approach (Garc´ıa de la Torre et al. 2010). Therefore, our approach may be less efficient for models belonging to the dedicated domain of the AVC approach; but it performs remarkably well when it is applied to multibead models. A possibly simple way to improve the performance of our approach could be a fine tuning achieved by varying the exponent x (presently: 1/3) in the expression 1/Nx. This will be a subject of future investigations. We have thoroughly tested the application of several programs and approaches to multibead filling models of simple geometric bodies of spherical, ellipsoidal, cylindrical or prismatic shape. All bead models were composed of equalsized touching spheres; number N and radius rb of the beads were varied systematically. The programs used in this study comprised DPH (an in-house modified version of HYDRO, based on the source code from 1999), the most recent versions 9 and 10 of HYDRO??, and the program ZENO as an independent alternative. For comparing the various programs and approaches with regard to the hydrodynamic predictions for the translational diffusion coefficient D and the intrinsic viscosity [g], the results obtained by means of ZENO, DZENO and [g]ZENO, were used as reference. The analysis of the ratios D/DZENO obtained by the different approaches at given rb and of the mean ratios calculated for the appropriate range of rb values clearly revealed a very good agreement of the diffusion coefficients obtained by DPH, by HYDRO?? in standard mode and in the special 3rd-order mode, and by the CBS approach (performed by DPH or HYDRO??) with the results obtained by ZENO. On the average, the best agreement was achieved when HYDRO?? was run in the 3rd-order mode. However, in some cases of very anisometric models, composed of a large number of beads, HYDRO?? obviously underestimated the diffusion coefficient significantly. The analysis of the ratios [g]/[g]ZENO and of their mean values exhibited a similarly good agreement also of the intrinsic viscosities obtained by DPH, using our approach of reduced volume correction (RVC), or by the CBS approach with the corresponding data from ZENO. On the contrary, the approach of adjusted, intermediate volume correction (AVC, KRI correction) implemented in HYDRO?? turned out to overestimate the intrinsic viscosity of multibead filling models considerably. The extent of overestimation increases with increasing N. This failure of HYDRO?? is, however, not unexpected if one considers that the AVC approach was explicitly intended for bead models in the strict sense (Garc´ıa de la Torre et al. 2010). Our RVC approach, on the other hand, was intended to cover preferably the range of multibead models (such models are usually obtained, e.g., from crystal structures) but not exclusively: it is applicable also for models composed of fewer beads and even yields the correct intrinsic viscosity at N = 1. Our reduced volume correction for the intrinsic viscosity is mathematically very simple. Therefore [g]RVC can be calculated quite easily, even by means of a pocket calculator, either from [g]NVC or from the full-corrected [g]FVC. Unfortunately, the most recent version of HYDRO?? does not allow us to compute [g]NVC or [g]FVC (in contrast to previous versions), and, to our knowledge, there is also no way to retrieve either of them from [g]KRI. Thus, for users of HYDRO??, the CBS approach is the only way to predict reliable intrinsic viscosities of multibead models, provided the number of beads does not exceed a limiting value of about 940. In this context, we notice that the recent versions of HYDROPRO (HYDROPRO10 and WinHYDROPRO10; Ortega et al. 2011) offer a novel possibility to compute [g]NVC for multibead models if they are composed of equal-sized beads. The very good agreement of the results for D and [g] obtained in this study by means of the CBS approach and by usage of the program ZENO convincingly demonstrates the power of ZENO to serve as a real alternative to the programs of the HYDRO family for the prediction of hydrodynamic parameters of multibead models. Motivated by our experience, ZENO was meanwhile integrated also into the Solution Modeler (SOMO) of the UltraScan analytical ultracentrifugation data analysis suite (Brookes et al. 2010). The excellent results of our reduced volume correction for the intrinsic viscosity suggest using a similar approach also for the volume correction of rotational properties. Indeed, pilot tests using the same models as in this study led to acceptable data. A detailed report is in preparation. Acknowledgments The authors are much obliged to J. Garc´ıa de la Torre for use of various versions of HYDRO and HYDRO??, to M. Mansfield for ZENO, to R. A. Sayle for RASMOL, and to the Intel Corporation for trial and academic licenses of Fortran compilers for Windows and Linux, respectively. Berman HM , Henrick K , Kleywegt G , Nakamura H , Markley J ( 2012 ) The worldwide protein data bank . In: Arnold E, Himmel DM , Rossmann MG (eds) International tables for crystallography , vol F: Crystallography of biological macromolecules, 2nd edn, chap 24 .1. Wiley, pp 827 - 832 Brookes E , Demeler B , Rosano C , Rocco M ( 2010 ) The implementation of SOMO (SOlution MOdeller) in the UltraScan analytical ultracentrifugation data analysis suite: enhanced capabilities allow the reliable hydrodynamic modelling of virtually any kind of biomacromolecule . 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Peter Zipper, Helmut Durchschlag. Hydrodynamic multibead modeling: problems, pitfalls and solutions. 3. Comparison of new approaches for improved predictions of translational properties, European Biophysics Journal, 2013, 559-573, DOI: 10.1007/s00249-013-0905-1