E Kupce

P A Keifer

M Delepierre

Adiabatic TOCSY MAS in Liquids #

The effect of magic angle spinning (MAS) of liquids upon the performance of various isotropic mixing sequences is investigated.

Although the mathematical formalism for isotropic mixing under MAS conditions is similar for both liquids and solids, the mechanism through which the coherence transfer is disturbed is different. In liquids, the use of sample spinning in the presence of both RF and magnetic-field inhomogeneities introduces a modulation of the effective field, which compromises the performance of the conventional mixing sequences. This effect is further amplified by supercycles, which normally improve the performance of the mixing and decoupling experiments. It is demonstrated that adiabatic mixing sequences are less susceptible to such modulations and perform considerably better in TOCSY MAS experiments. The best performance of TOCSY MAS is observed at the rotational resonance condition when the sample appears static in the RF reference frame.

INTRODUCTION

The small active volumes and the high sensitivities of Varian Nanoprobes (1) make them ideal for applications where only a limited amount of sample is available. In such probes high sensitivity is achieved by maximizing the filling factor of the receiver coil. The line broadenings caused by the resulting magnetic susceptibility discontinuities are eliminated by using susceptibility matched materials and by magic angle spinning (MAS), typically at 2 to 4 kHz speed. This use of MAS has the additional advantage of suppressing line broadenings in heterogeneous samples that have a non-uniform magnetic susceptibility across the sample, such as solvent-swollen resins (2,3). High resolution and high sensitivity allows routine liquids NMR experiments to be performed on a single bead of solid phase synthesis (SPS) resins with typical sample quantities of under 500 picomoles (4).

Although the vast majority of liquid-state NMR experiments can be recorded in Nanoprobes with no complications, in some cases interference effects have been observed (5). Particularly, performance of the conventional, composite-pulse 2D TOCSY experiment can be extremely poor and unstable (see Fig. 1a). On the other hand, the adiabatic TOCSY experiment (6) under the same conditions performs considerably better (see Fig. 1b).

In the course of this study we evaluated a number of standard liquidstate experiments at a wide variety of MAS rates. We verified that the signal-to-noise ratio was always practically independent of the MAS rate for those experiments that did not contain a spin-lock, for instance, NOESY, DQF-COSY or simple spin-echo sequences. On the other hand, the performance of experiments employing recursive spin-lock sequences, such as isotropic mixing and spin decoupling, were very sensitive to the sample spinning speed (see Fig. 2a). The use of adiabatic spin-locks typically improved the overall quality of the data (see Fig. 2b). We also found that the magnitude of these effects strongly depended upon the sample volume, the sample geometry, and even the quality of shimming. Surprisingly, experiments that employ continuous wave spin-locks, for instance offresonance ROESY (7) or selective cross polarization experiments (8,9), seemed to be substantially less sensitive to variation of MAS speed.

In this study we explored the effects of various spin-lock sequences on liquid samples spinning at the magic angle. It is demonstrated that the spin dynamics of experiments that employ repetitive sequences of composite pulses, such as isotropic mixing or spin decoupling, is least disturbed when the sample spinning speed is comparable to the spin nutation frequency. This condition is known as rotational resonance (10) in solid-state NMR, where it has found many useful applications.

DISCUSSION

The presence of rotational resonance in fast rotating liquids can be demonstrated by a simple nutation experiment (see Fig 3). As shown in Fig. 3a the homogeneity of both the RF field and the magnetic field in a static sample is rather poor, partially because of somewhat casual sample shimming. This inhomogeneity is then largely removed by sample spinning at the magic angle (see Fig. 3b). However, if the RF field strength matches the spinning speed, a very sharp peak appears in the indirect dimension, as if the RF field was suddenly extremely homogenous (see Fig 3c). In fact, the magnetization does not de-phase during this continuous wave spin lock period because it is locked by the effective field (11), i.e. the magnetization follows the effective field in a manner similar to the adiabatic following [START_REF] Abragham | The Principles of Nuclear Magnetism[END_REF]. The broad component along the RF field axis in Fig. 3c comes from the parts of the sample where the rotational resonance condition is not met.

It is interesting to compare cross polarization (CP) in solids [START_REF] Hartmann | [END_REF](14)(15) with isotropic mixing in liquids (16)(17)(18)(19). In both cases the spin dynamics can be described using the same formalism. The rotating-frame Hamiltonian is derived for an isolated IS spin pair:

II z z z S x S z I x I CP S I S S I I                 1 1 [1]
where  1 represents the RF field strengths as seen by the I and S spins and the  terms are the corresponding resonance offsets. In solids the polarization transfer is achieved via the dipolar coupling ( = b IS ) whereas in liquids this process is mediated by the indirect (scalar) spin-spin coupling ( = J IS ). Assuming that  1 >> and ignoring the homonuclear interactions, the effective Hamiltonian in a tilted reference frame for a static sample is written as:

z z eff x eS x eI eff S I S I         [2]
With the Hartmann-Hahn condition satisfied the first two terms in Eq. [2] vanish and the maximal polarization transfer can be achieved.

When the sample is spun at the magic angle, the heteronuclear dipolar coupling, b IS becomes time dependent:

t ik k k IS r e b t b    ) ( [3]
This introduces the well known "finger pattern" into the CP spectra as the Hartmann-Hahn condition splits into several sub-conditions:

r k    , [ 4
]
where  is the RF mismatch parameter, k is an integer (usually -2  k  2) and  r is the spinning speed. The efficiency of CP in solids can be substantially improved by employing adiabatic B 1 field sweeps (20)(21)(22)(23).

In liquids the scalar J IS coupling is not directly affected by sample spinning. However, the effective fields seen by spins I and S are time dependent, in a manner similar to that for b IS (see Eq. [3]). In order to show that such modulations can indeed affect the quality of spin lock experiments, we simulated the influence of RF field modulation on the ability of various mixing sequences to return the starting magnetization to its initial state at the end of a single cycle. As it appears from Fig. 4, the composite pulse sequences are most vulnerable to sample spinning (24). This is hardly surprising, because the frequency spectrum of composite pulse sequences is inversion pulses may vary substantially. As a result, at the end of the cycle the magnetization is not returned back to its initial state, as it would be in a static sample. For the same reason, the overall performance of the TOCSY experiment becomes increasingly sensitive to MAS speed as the length of a supercycle increases.

Therefore, the optimum performance of isotropic mixing and spin decoupling can be expected if the length of individual inversion pulses, T p matches the rotor period, T r . This is indeed confirmed by experiment (see Fig. 5). Under these circumstances, in the RF reference frame the sample looks as if it is static and the individual contributions from  1 and  modulation can be regarded as local distortions of the mixing field. As long as these distortions can be tolerated by the applied spin-lock sequence, the performance of the experiment is acceptable. Adiabatic pulses are known to be particularly insensitive to RF field distortions (29). Provided the rotational resonance condition is met, the efficiency of spin mixing is further improved by supercycles. The amplitude and frequency of the RF field was modulated according to Eq. [3]. The modulation depth was 10 % for the first harmonic and 5 % for the second harmonic. The contour plots show residual M z magnetization.

CONCLUSIONS

The contour levels were increased linearly from 0 to 1 in steps of 0.1.

Ideally, very little residual magnetization should be observed (see text).

  particularly "noisy"(25) and additional modulation introduced by sample spinning destroys the delicate balance of RF fields created by numerical optimization. Additional simulations (not shown here) indicate that increasing the length of the initial 4 step MLEV-4 supercycle (26) to either MLEV-16 or a 20-step phase cycle(27,28) makes the performance of mixing sequences in rotating samples even worse. Indeed, supercycles have been designed to work with inversion pulses that are all identical, whereas in the presence of sample spinning the performance of individual (composite)

  Although the appearance of interference between the Hartmann-Hahn polarization transfer and sample spinning is similar in solids and in liquids, the mechanisms affecting the experimental performance are quite different.The conventional composite-pulse mixing sequences are particularly vulnerable to such interference effects. The performance of TOCSY experiments can be substantially improved by using adiabatic (WURST-2) mixing sequences. The optimum performance in 2D TOCSY experiments is achieved at the rotational-resonance condition, when the repetition rate of the adiabatic pulses in a mixing sequence matches the MAS speed. Although both the experimental data and simulations indicate existence of other areas where the spin lock performance is close to the optimal (see Figs.4 and 5), they are typically less efficient and the relation to the MAS speed is not as obvious.

Figure 1 .

 1 Figure 1. Comparison of 2D H-H correlated xy-TOCSY spectra of a small

Figure 2 .

 2 Figure 2. Matrix plots of the first trace from H-H z-TOCSY spectra of 6 mg

Figure 3 .

 3 Figure 3. Nutation experiment recorded at 500 MHz in gHX Nanoprobe

Figure 4 .

 4 Figure 4. Simulation of the effect of MAS upon the performance of (a)

Figure 5 .

 5 Figure 5. An illustration of the effect of the spinning speed upon the

ACKNOWLEDGEMENTS

We thank Tom Barbara, Knut Mehr, and Dave Rice from Varian Inc. (Palo Alto) for many useful discussions and suggestions.

 20


Muriel Delepierre
email: murield@pasteur.fr

Anne Lecroisey

Interface between Microbiology and Structural Biology as viewed by NMR

NMR spectroscopy is one of two principal experimental techniques used in structural biology.

It can be used to determine structures at atomic resolution and to investigate the dynamics of macromolecules and intermolecular interactions. We aim to give an overview of the use of modern high resolution NMR methodology in microbiology.

Introduction

NMR and X-ray crystallography are the two most powerful tools available for structure determination and have been extensively used to obtain structural information on biological molecules. The limitations of X-ray crystallography is the necessity to obtain crystals. NMR is limited by the size of the molecule and the low sensitivity of the method. Thus, these two techniques are complementary. NMR is particularly powerful for determining the structure of small molecules, for obtaining information on the dynamics of macromolecules over a wide time scale range and for studying molecular interactions. Thus, it is not surprising that it is widely used in microbiology. Indeed, NMR can be combined with isotope labelling for non-invasive analytical and quantitative analysis of metabolism, in a range of microorganisms. It has long been a critical tool for the identification of compounds and for measuring concentrations and fluxes both in vitro and in vivo.

Firstly, we will give a brief summary for scientists with no prior understanding of this method and then a few applications involving only soluble molecules will be described.

2-1 What is NMR spectroscopy?

Nuclear Magnetic Resonance spectroscopy quantitatively detects spin reorientation for nuclei that possess a magnetic moment in an applied magnetic field by measuring the amount of energy absorbed from a radio-frequency coil surrounding the sample. A radiofrequency field is applied to the sample and a time dependent signal can be detected and measured from the nuclear spins. This time dependent signal is then Fourier transformed to give a frequency dependent amplitude signal, corresponding to the NMR spectrum.

2-2 What are the parameters?

NMR spectra can be defined by five parameters:

(i) chemical shifts () which indicate the precise resonance frequency of the nuclei. These depend on the chemical environment of the nuclei and are expressed in a dimensionless unit, parts per million (ppm), which is independent of the strength of the magnetic field. Their ranges vary with the group and the type of nuclei. Thus, in biological molecules, 1 H chemical shifts are spread over 14 ppm, 13 C shifts are spread over 200 ppm and 15 N shifts are spread over 40 ppm.

(ii) The spin-spin coupling which arises from weak interactions between magnetic nuclei via the electron of the chemical bond. The size of this interaction is defined by the spin-spin coupling constant (J) and is expressed in Hertz.

(iii) The area of a signal, which is usually proportional to the number of nuclei contributing to the signal.

 (iv)The longitudinal or spin-lattice relaxation time, T 1 , which is the time taken for the magnetisation longitudinal component to return equilibrium value after a perturbation.

(v) The transverse or spin-spin relaxation time, T 2 , which is the time taken for the transverse magnetisation to decay. The transverse relaxation rate (1/T 2 ) is related to the linewidth of the signal for molecules freely tumbling in solution.

There are several different mechanisms for relaxation. However, for spin 1/2 nuclei such as proton, the major relaxation mechanism arises from dipolar interactions with other protons or nuclei. For the dipolar relaxation mechanism the relaxation rates, 1/T 1 and 1/T 2 , depend on the c the overall correlation time of the molecule and on molecular motions. c characterises random molecular motions within a sample and is a function of molecular weight, of solution viscosity and of temperature. When the size of the molecule increases, both the correlation time for reorientation and the T 2 relaxation time decrease and the linewidth increases. The nuclear Overhauser effect (NOE) or dipolar cross-relaxation is related to spin relaxation. It is observed for nuclei located at a distance shorter than 5 Å and is proportional to the inverse six power of the distance. A fixed and known reference distance allows one to associate a distance with each observed NOE. Both the spin-spin relaxation and the nuclear Overhauser affect are crucial for structure determination.

Moreover, in NMR the fundamental time scale is defined by the precession frequency of the nuclei, whereas chemical exchange phenomena allow us to define additional time scales. If a chemical group exists in different environments, the NMR parameters of the corresponding nuclei will vary according to the exchange rate. Two extremely different situations can be encountered: the fast exchange regime in which the observed parameters from all the different environments are averaged, and the slow exchange regime in which sets of parameters are observed for each environment. Finally, motions are reflected in all NMR parameters, chemical shifts, coupling constants and relaxation rates, allowing the qualitative detection of flexibility and the quantitative analysis of relaxation data in terms of internal motions. NMR allows us to study a wide range of time scales, monitoring motions for picoseconds, seconds minutes or hours. This dynamical aspect will not be treated here, as it deserves a review on its own.

2-3 What molecule size?

Most NMR parameters are sensitive to molecular conformation. Quantification of these parameters allows structural analysis. However, this requires the resolution and the assignment of NMR signals to specific nuclei. This can be achieved easily for molecules of less than 10 000 Da by two-dimensional NMR, but is more challenging for bigger molecules.

Assignment strategies based on the use of spin-spin interactions between 13 C and 15 N nuclei in isotopically enriched molecules have enabled molecules of up to about 30,000 Da to be studied, because spin-spin interactions are increased ten-fold.

Recent experiments have considerably extended the applications of NMR for structure determination. Instead of being based on distance information, they are based on orientational information. Indeed, when molecules are aligned in the magnetic field, anisotropic magnetic interactions which cannot normally be observed in high-resolution NMR spectra, can be detected [START_REF] Prestegard | New techniques in structural NMRanisotropic interactions[END_REF]. The two parameters measured are residual dipolar couplings and chemical shift anisotropy (CSA). The interference between dipole-dipole and CSA interactions can also be used constructively, like in TROSY experiment [START_REF] Pervushin | Attenuated T 2 relaxation by mutual cancellation of dipole-dipole coupling and chemical shift anisotropy indicates an avenue to NMR structures of very large biological macromolecules in solution[END_REF][START_REF] Riek | Polarization transfer by cross-correlated relaxation in solution NMR with very large proteins[END_REF] which allows molecules of up to several hundred kDa to be studied. Finally, measurement of these new parameters in partially oriented media has provided new tools to increase the size of the molecules that can be studied and also the accuracy with which these structures can be determined.

2-4 Labelling

Signals from 1 H, 13 C, 15 N and 31 P nuclei are observable in the spectra of biological molecules. Spin-spin interactions between protons vary from few Hertz to 16 Hz when established between two or three bonds. One bond spin-spin interactions between 15 N or 13 C and their attached protons are 95 Hz and 150 Hz, respectively. When the size of the molecule increases, the linewidth increases and spin-spin interactions between protons are no longer detectable. Thus, large biomolecules can be studied by NMR by extensive isotopic labelling.

For medium size molecules between 8 and 12 kDa, 15 N labelling is sufficient, whereas higher molecular weight molecules required double labelling, usually 15 N and 13 C. For molecules of over 30 kDa, additional labelling with deuterium further decreases the linewidth and improves the NMR spectra. Deuteriation is either selective or uniform and is often random fractional deuteriation [START_REF] Gardner | The use of 2 H, 13 C, 15 N multidimensional NMR to study the structure and dynamics of proteins[END_REF]. Usually, samples that have been uniformly labelled with 15 N and 15 N/ 13 C are prepared by growing the bacteria in minimum medium enriched with vitamins and with 15 N ammonium chloride and 13 C 6 glucose or glycerol as the only nitrogen and carbon sources. The expression has to be sufficient, that is at least 10 mg per litre as the labelled sources are expensive. Molecules may also be selectively labelled by growing strains auxotroph for given residues in unlabelled medium supplemented with the labelled residue. For chemically synthesised small proteins of less than 80 amino acids, specific positions or amino acid types can be labelled. For other molecules several methods have been developed: RNA is generally obtained by in vitro transcription using T7 RNA polymerases (28) and 13 C and/or 15 N labelled ribonucleoside triphosphate is prepared from the RNA of organisms grown with 15 N and 13 C enriched nutrients [START_REF] Nikonowicz | Three-dimensional heteronuclear NMR studies of RNA[END_REF]. Methods for the large scale preparation of uniformly isotope-labelled DNA have also been developed [START_REF] Louis | Preparation of uniformly isotope-labelled DNA oligonucleotides for NMR spectrocopy[END_REF]. After the molecule is labelled, its quality can be checked rapidly in 10 minutes or less by 15 N or 13 C proton heteronuclear single quantum coherence experiment (HSQC). This usually fixes the experimental conditions (temperature, pH and ionic strength). Well dispersed 15 N and 1 H resonances indicate a folded protein domain with segments involved in stable secondary and tertiary structures (figure 1).

2-5 Sample preparation

The sample has to be very pure and stable over time and at different temperatures because NMR spectroscopy experiments can last for several days. Proteases or nucleases would obviously destroy the sample. Impurities generally give signals sharper than macromolecules, as they are often small molecules.

NMR has intrinsically poor sensitivity and signals have to be averaged many times during an experiment because the signal-to-noise ratio increases as the square root of the number of times the signal is sampled. A few tens of milligrams of compound are necessary for structural studies depending on its molecular weight. Solutions must have a concentration of at least 1 mM and a volume of 400 l to 600 l is required. If Shigemi tubes (tubes with filled bottom) are used this volume decreases to between 360 l and 250 l. Recently, different types of small sample volume probes have also become available (i) commercial microprobes, which use small-diameter vertical sample tubes in conventional geometry probes, with volumes ranging from 120 l to 150 l (ii) commercial nanoprobes which produce high-resolution spectra with sample volumes of about 40 l or less [START_REF] Keifer | A comparison of NMR spectra obtained for solid-phase-synthesis resins using conventional high-resolution, magicangle-spinning, and high-resolution magic-angle-spinning probes[END_REF] (iii) and finally home-made probes which contain nanoliter to microliter-volume capillaries and consist of a solenoidal coil of about 1 mm [START_REF] Novoa | Cloning and mapping of the genetic determinants for microcin C7 production and immunity[END_REF]. However, small sample volume probes (1 to 100 l) are still being developed. Parallel to this technology, the cryoprobe technology has also emerged improving the signal-to-noise ratio by a factor of 3 to 4 leading, thus reducing the amount of sample necessary for analysis and/or data collection times.

2-6 Assignments and structure determination.

Resonance assignment strategies differ significantly for each type of biomolecule, however, the strategies used are very similar and rely on the identification of connectivities via spin-spin interactions and dipolar interactions [START_REF] Wüthrich | NMR of proteins and nucleic acids[END_REF].

Usually, for moderately sized non-labelled molecules, two-dimensional experiments allow all signal to be assigned. For labelled molecules, assignments are obtained by combining three-dimensional 15 N-edited experiments and triple resonance experiments ( 1 H, from one bond and/or two bond coupling [START_REF] Cavanagh | Protein NMR Spectroscopy: principles and practice[END_REF].

Secondary structural elements are identified by analysing several NMR parameters: (i) induced chemical shift (CSI) [START_REF] Wishart | The Chemical Shift Index: a fast and simple method for assignment of protein secondary structure through NMR spectroscopy[END_REF], (ii) coupling constant values (iii) strong dipolar interactions typical of each type of secondary structures, and (iv) Apparent exchange rates of exchangeable protons with deuterium give an indication of hydrogen bond formation and of reduced accessibility to solvent.

The most important parameter used in structural determination is the nuclear Overhauser effect (NOE). All the distances derived from the quantification of NOEs are used as constraints to calculate the structure. As the precision of the calculated structure is directly related to the number of experimental restraints, it is necessary to assign several hundred NOE cross-peaks unambiguously.

3 Structural studies The aim of this review is not to describe NMR extensively but to use a few examples to show the applications for NMR in microbiology. We chose to omit in situ experiments and metabolism pathways together with metabolic fluxes, as this has already been extensively reviewed recently [START_REF] Barbotin | NMR in Microbiology: Theory and Applications[END_REF]. We place particular emphasis on the interface between structural biology and microbiology, where the interplay between the two communities is not always optimal.

3-1 Peptides and proteins

NMR is very effective for the determination of the structure of small proteins and many studies have been published in this field. However, NMR can be used in its full potential for the determination of the structure of the peptide domain. Indeed, peptides of less than 30 residues are often very flexible, and the rapid equilibrium between several conformers in aqueous solution precludes their crystallisation. In structural promoting environments, the number of conformations sampled can be reduced allowing the structure-function relationship to be established. In the field of microbiology high resolution NMR has been used for studies of antimicrobial peptides and of secretion signals peptides.

3-1-1 Antimicrobial peptides

Compounds obtained from natural sources, such as micro-organisms and plants, are a large number of drugs in current use, either directly or after synthetic modifications. The availability of bacterial genomic information and the increasing resistance of pathogenic organisms to conventional antibiotics means that the antimicrobial drugs need to be developed to control infection. A lot of organisms, such as animals and plants, produce offensive and defensive substances. In the invertebrates this efficient defence system often compensates for the lack of circulating immunoglobulins in vertebrates. Antimicrobian peptides are among the substances invertebrates produce to protect themselves against invading pathogenic microorganisms. These peptides usually lyse cell membranes and each has a broad but different spectrum of antimicrobial activity [START_REF] Epand | Diversity of antimicrobial peptides and their mechanisms of action[END_REF][START_REF] Hwang | Structure-function relationships of antimicrobial peptides[END_REF]. However, despite extensive studies on their mode of action, much knowledge is still needed to fully understand the basis of their selectivity. To improve potency, and specificity of antimicrobial peptides and to design cheaper and more efficient analogues we need to understand theirs structure-function relationships.

One family of antimicrobial peptides is the cecropin family, which consists of 35 to 37 amino acid peptides [START_REF] Hultmark | Insect immunity: Attacins a family of antibactericidal proteins from Hyalophora Cecropia[END_REF]. Cecropins are most active against gram-negative bacteria, but they also have activity against some gram-positive bacteria. The three dimensional structure of several of them have been solved by NMR [START_REF] Steiner | Secondary structure of the cecropins: antibacterial peptides from the moth Hyalophora cecropia[END_REF][START_REF] Holak | The solution conformation of the antibacterial peptide cecropin A: a nuclear magnetic resonance and dynamical simulated annealing study[END_REF][START_REF] Boisbouvier | Solution structure of a cecropin-like synthetic peptide, Shiva active on the sporogenic development of Plasmodium berghei[END_REF]. They usually adopt a helix-bend-helix conformation with a highly basic amphipathic N-terminal helix and a hydrophobic C-terminal helix. The importance of amphiphilic helices has been established in a number of antibacterial and lytic peptides, and their helicoidal geometry is usually considered to be the key factor for the formation of transmembrane pores leading to cell lysis.

Cecropins are thought to be selective for prokaryotes due to the lipid composition of their membrane (14). In Shiva-3, in particular, the in vitro lytic effect on protozoa tested resulted in decreased growth and multiplication of Plasmodium falciparum and Trypanosoma Cruzi. These observations suggested the use of Shiva-like peptide genes to engineer malaria resistant vectors as an alternative treatment for the control of malaria [START_REF] Rodriguez | Effect of a cecropin-like synthetic peptide (Shiva-3) on the sporogenic development of plasmodium berghei[END_REF]. To improve the efficiency of this molecule its structure was studied in several media and particularly in media that mimic the membrane environment. It is composed of two helices, one amphiphilic Nterminal helix of about 20 residues long and a short hydrophobic C-terminal helix. In the presence of micelles N-terminal helix of the Shiva-3 moves much more slowly than the Cterminal helix. In particular the movement the side chains of its basic residues is restricted suggesting that the N-terminal helix is in contact with micelles whereas the C-terminal is not (5).

3-1-2 Signal peptides

Most bacterial virulence factors that interact with host cells need to be secreted across the bacterial membrane. Thus, the various secretion systems found in gram-negative bacteria can be used as target for new therapeutics. To understand the secretion pathways we must first understand how the secretion signal function and therefore we must understand their structure.

In gram-negative bacteria, whose envelope is made of two distinct membranes, most extracellular proteins are secreted by the signal peptide-dependent general secretory pathway (GSP) which involves the N-terminal secretion signal. A other group of extracellular proteins without cleavable N-terminal signal peptide is secreted by a distinct secretion pathway involving a C-terminal secretion signal [START_REF] Wandersman | Protein and peptide secretion by ABC exporters[END_REF].

Although studies on isolated signal sequences are controversial, they have been mainly

justified by the fact that signals can be functionally transferred from one protein to an other, emphasising their ability to act independently. For the N-terminal signal sequences that have been studied, conformational differences have been observed. These range from rather unordered or -sheet structures in aqueous solution to -helices in non polar solvents, micellar solution and phospholipid vesicles. This conformational adaptability is thought to be linked to the function of these signals [START_REF] Gierasch | Signal sequences[END_REF].

Consensus structural information has been gathered on N-terminal signal sequences, however little structural information is available for C-terminal secretion signals [START_REF] Yin | Structural analysis and comparison of the C-terminal transport signal domains of hemolysin A and leucotoxin A[END_REF][START_REF] Wolff | C-terminal Secretion Signal of an Erwinia chrysanthemi Protease Secreted by a Signal Peptide-Independent Pathway: Proton NMR and CD Conformational Studies in Menbrane-Mimetic Environments[END_REF][START_REF] Wolff | Spectroscopic studies of the Cterminal secretion signal of the Serratia marcesens haem acquisition protein HasA in various membrane mimetic environments[END_REF].

Like the N-terminal sequences their primary sequence is not highly conserved. Structural information on the C-terminal peptides of Erwinia Chrisantemi protease G and Serratia marcescens hem acquisition protein (HasA), were obtained by use of circular dichroïsm and NMR spectroscopies. C-terminal signal peptides adopt an unstructured and flexible conformation in aqueous buffer solution, that switch to a helical conformation followed by a short fragment unstructured upon addition of trifluoroethanol, detergents and lipids. These studies show that the C-terminal secretion signals are composed of one -helix followed by an unstructured C-terminal region that terminates with an essential secretion motif [START_REF] Wolff | C-terminal Secretion Signal of an Erwinia chrysanthemi Protease Secreted by a Signal Peptide-Independent Pathway: Proton NMR and CD Conformational Studies in Menbrane-Mimetic Environments[END_REF][START_REF] Wolff | Spectroscopic studies of the Cterminal secretion signal of the Serratia marcesens haem acquisition protein HasA in various membrane mimetic environments[END_REF]. This motif is composed of one negatively charged residue followed by three to five hydrophobic residues.

Structural informations could be obtained for the C-terminal peptide of HasA from

Serratia marcescens when in its proteinic context. The last 15 residues following the Cterminal of the HasA protein are highly flexible and unstructured [START_REF] Izadi-Pruneyre | NMR studies of the C-terminal secretion signal of the heme-binding protein HasA[END_REF]. When fragments experience motions much faster than the overall correlation time for reorientation of the macromolecule these can be detected by either homonuclear 2D experiments such as TOCSY or by measuring heteronuclear 15 N 1 H dipolar interactions. The high flexibility and the lack of well ordered structure of the HasA C-terminal extremity might be a prerequisite for the recognition of the protein by its ABC transporter [START_REF] Izadi-Pruneyre | NMR studies of the C-terminal secretion signal of the heme-binding protein HasA[END_REF].

3-1-3 Identification of small compounds

NMR is often the method of choice for identifying small unknown compounds.

Microcin C7 was a "text book case" of structure determination.

Many strains of the Enterobacteriaceae family produce substances with low molecular-weights or secondary metabolites with antibiotic activity [START_REF] Asensio | A new family of low molecular weight antibiotics from enterobacteria[END_REF]. Microcin C7 (MccC7) is a stationaryphase product of E. coli that inhibits protein synthesis both in vivo and in vitro [START_REF] Kolter | Genetics of ribosomally synthesized peptides antibiotics[END_REF]. MccC7 is secreted from cells by specific dedicated machinery [START_REF] Novoa | Cloning and mapping of the genetic determinants for microcin C7 production and immunity[END_REF]. It is a heptapeptide whose N-and Cterminal residues have been modified. The chemical structure of microcin C7 was obtained from complete assignments of 1 H, 13 C, 15 N and 31 P at natural abundance using homonuclear and heteronuclear two-dimensional spectroscopy. It is composed of one heptapeptide chain modified at the N-terminus by a formyl group and the C-terminus substituent consists of a phosphodiester of 5"-adenylic acid and n-aminopropanol (AMPap) linked via a phosphorus atom to an amide group, forming thus a phosphoramide. The substituted phosphoramide group was identified by pH titration of the phosphorous atom by means of 31 P chemical shift variation [START_REF] Guijarro | Microcine C7 : Chemical structure via Two-dimensional NMR[END_REF].

3-2 Oligosaccharides

Bacterial cell walls ensure cell integrity, resist mechanical constraints and have key roles in the regulation of exchanges between the cell and its surrounding environment and in the host immune response. Mycobacterial pathogenecity is often linked to the nature of the glycolipids, a constituent of bacteria cell walls. In gram negative bacteria the cell wall contains ubiquitous glycolipid molecule, the lypopolysaccharide (LPS), which is the major factor distinguishing gram-negative from gram-positive bacteria. LPS is often involved in disease and is a major surface antigen via its O-specific polysaccharide. Consequently, LPS specific antibodies are potential tools for the diagnosis and therapy of some of the bacterial infections such as tuberculosis, shigellosis and leprosy and it is, therefore, important to determine the structure of immunoreactive epitopes.

Although NMR data on the structure of high molecular weight oligassacharides is more difficult to handle than structure determination of proteins of comparable weights, NMR has been particularly useful and successful in the identification of O-specific antigenic polysaccharide chains. The main reason for this is that oligosaccharide protons have a much smaller chemical shift range (3 to 5 ppm) than proteins (about 12 ppm). The use of higher magnetic fields while increasing the signal to noise decreases the spectral overlap and can improve structural studies of these molecules. In studies on primary structure it is necessary to identify (i) the number of sugar residues in the repeating unit, (ii) the monomeric constituents and their anomeric configuration, (iii) the sequential arrangement of monomers as well as glycosidic linkage, and (iv) the presence or absence of non-glycosidic substituents. Finally, it is necessary to establish the tertiary structure and its dynamics. Usually a combination of 2D homonuclear, 1D selective excitation and 2D heteronuclear methods, allow the complete determination of quite complex polysaccharides [START_REF] Bush | Structure and conformation of complex carbohydrates of glycoproteins, glycolipids and bacterial polysaccharides[END_REF].

Recently studies used NMR and molecular modelling to determine the conformational properties of a synthetic polysaccharide of Shigella dysenteria type I, composed of rhamnopyranosyl, galactopyranosyl, acetamido-2deoxy  D glucopyranosyl. The conformation resembles a short, irregular spiral, with the methyl substituents on the exterior.

The fact that rhamnose and galactose residues are major components of the antigenic determinant suggested that the oligosaccharide binds to the antibody via hydrophobic interactions involving three methyl groups [START_REF] Coxon | NMR spectroscopy, molecular dynamics, and conformation of a synthetic octasaccharide fragment of the O-specific polysaccharide of Shigella dysenteria type 1[END_REF].

Molecular interactions

NMR is a sensitive method for detecting the binding of small molecules to macromolecules, and has proven particularly valuable for the rapid identification of ligand binding sites. Indeed, the chemical shift perturbation observed for the amide proton by means of 15 N-1 H heteronuclear single-and multiple-quantum correlations (HSQC or HMQC respectively) rapidly indicates where the ligand binds. A rapid and sensitive method, the saturation transfer difference (STD) or "bioaffinity NMR spectroscopy", has been used to identify sites at which ligands bind to macromolecules with dissociation constants of between 10 -3 and 10 -8 M. This method was successfully used to identify the sugar residues of the Lewishexasaccharide that are in contact with the fucose-binding lectin AAA protein [START_REF] Mayer | Characterization of ligand binding by saturation transfer difference NMR spectroscopy[END_REF]. Although the method is quite powerful and easy to implement it does yield information on the receptor site of the macromolecule or on the conformation of the bound ligand. This information can only be obtained by transferred NOE techniques if the ligand binds relatively weakly (kD >10 - 7 M) and providing that several conditions are met [START_REF] Clore | Theory and applications of the transferred nuclear Overhauser effect to the study of the conformations of small ligands bound to proteins[END_REF][START_REF] Clore | Theory of the time dependent transferred nuclear Overhauser effect: applications to structural analysis of ligand-protein complexes in solution[END_REF]. Although small molecules can adopt a large variety of conformations in rapid exchange, one of these conformations is usually preferred in the bound state. Moreover, when a small molecule binds to a large molecule, even transiently, the numerical sign and magnitude of the NOEs change. Thus, it is possible to obtain an "image" of the bound ligand conformation providing great care is taken to get rid of all possible artefacts especially indirect NOE due to spin-diffusion [START_REF] Bothner-By | Structure determination of a tetrasaccharide: Transient nuclear Overhauser effects in the rotating frame[END_REF][START_REF] Vincent | The conformation of NAD+ bound to lactate dehydrogenase determined by nuclear magnetic resonance with suppression of spin diffusion[END_REF]. This approach was successfully used to monitor the conformation of a disaccharide bound to a Fab" fragment (1).

For tightly bound ligands "editing" experiments are required to identify specifically NOEs between a 13 C and/or 15 N labelled protein and unlabelled ligand [START_REF] Griffey | Proton-detected heteronuclear edited and correlated nuclear magnetic resonance and nuclear Overhauser effect in solution[END_REF]. If sufficient intermolecular NOEs are collected it will then be possible to dock the ligand to the macromolecule.

Enzyme mechanisms

NMR provides both structural and kinetic information and thus can contribute to mechanistic studies, such as substrate equilibrium, stereochemical aspect of enzyme action by use of isotopically labelled substrates, proton transfer and stereochemical aspects of phosphoryl transfer reactions.

Perhaps one of the most successful application was the study of ionisable groups.

Indeed, although X-Ray crystallography does not provide direct information on the charged state of ionisable groups NMR is the only method that can give pK values. Chemical shifts are very sensitive to the charged state of ionisable groups providing a wealth of information on protein structure in solution and on enzyme mechanisms when these residues are involved in the active site. Histidine was the first ionisable group to be studied in detail in proteins and is still the most widely studied (26).

It is possible to use NMR to obtain information on the predominant tautomeric state of histidine residues in 15 N labelled proteins because the chemical shifts of the 15 N resonances are strongly correlated with the protonation state of the imidazole nitrogen atom. Protonated nitrogen resonates approximately 170 ppm whereas deprotonated nitrogen resonates at about 250 ppm. The 1 H- 15 N heteronuclear multibond correlation experiment (HMBC) connected nitrogen with non-exchangeable protons from the histidine imidazole ring (figure 2).

Furthermore, HMBC can also differentiate between the two neutral tautomers because the cross-peak intensities will depend on the coupling constant values [START_REF] Pelton | Tautomeric states of the activesite histidines of phosphorylated and unphosphorylated IIIGLC, a signal-transducing protein from Escherichia coli, using two-dimensional heteronuclear NMR techniques[END_REF][START_REF] Blomberg | Nuclear magnetic resonance investigation of 15 Nlabeled histidine in aqueous solution[END_REF].

Concluding remarks

Although X-Ray cristallography was used to elucidate 81% of the structures in the data bank and NMR only elucidated 15 % of entries (June 2000) some of the unique features of NMR make it a valuable tool for obtaining structural information on molecules that do not crystallise due to their high flexibility and also for specific experiments that connect structural data with biochemical function and mechanism. Furthermore, recent technological advances in the development of rapid mixing systems with highly sensitive "flow" probes for small sample volumes together with automated sample transport open the way for on-flow analysis that should aid mechanistic studies similar to those carried out on protein folding.

Furthermore, the use of Shigemi microcells with cryogenic NMR probes enables the study of kinetic processes and data acquisition for transient species together with characterisation of metabolites [START_REF] Russell | A comparison of inverse-detection heteronuclear NMR performance: conventional vs cryogenic microprobe performance[END_REF]. NMR spectroscopy can now follow the structural and dynamic changes taking place during intermolecular interactions and mechanistic studies in real time.

Figure 1 :

 1 TROSY-HSQC15 N-1 H experiment on a 189 amino acid protein. The spectrum was recorded over 4 min with 1 scan with a 7000 Hz spectral width and 1K data point in the F 2 dimension and 64 T 1 increments with 1800 Hz in the F 1 dimension.

Figure 2 :

 2 Figure 2: Histidine titration. A) HMBC experiment carried out on a 189 amino acid protein

N,[START_REF] Gardner | The use of 2 H, 13 C, 15 N multidimensional NMR to study the structure and dynamics of proteins[END_REF] C). These experiments are very sensitive due to the efficient transfer of magnetisation


Camille Schreck

Damien Rohmer

Stefanie Hahmann

Interactive paper tearing

Keywords: Descriptors (according to ACM CCS): I, 3, 7 [Computer Graphics]: Three-Dimensional Graphics and Realism-Animation

Figure 1: Interactive tearing of pieces of paper. The path of a tear follows a geometrical curve but also presents stochastic details.

Introduction

Almost everyone has already had occasions to bend, crumple, and tear up a sheet of paper. This very common material, which starting from an initially flat geometry, can undergo a complex geometric deformation when manipulated by hand. Indeed, a sheet of paper exhibits strong mathematical and physical properties, such as a non-smooth developable geometry or its microscopical fiber structure. It may furthermore undergo irreversible damages upon deformation ending up in crumpled and/or torn form. Modeling and animating efficiently such a shape in the virtual world is thus challenging mainly because of the following characteristics of paper: First, a sheet of paper can be represented as a developable surface, which can be flattened onto a plane without distortion, i.e. without stretch nor compression. It is said to be locally isometric to its 2D pattern. This strong geometric property translates into non-linear constraints of length preservation which imply the use of costly algorithms. Second, as mentioned above, at macroscopic scale this surface may not be smooth, exhibiting sharp creases and singularities appearing dynamically when being crumpled. These sharp features are noticeably hard to model using standard triangle meshes as a too coarse sampling or badly oriented connectivity are likely to be seen as visual artifacts. As a consequence, very dense meshes or adaptive remeshing are usually required, increasing even more the computational cost.

Proposing an interactive tool enabling to handle deformation behaviors as different as bending, crumpling and tearing could be highly valuable for both digital artists who need to model and animate the virtual paper, or for digital environment appli-cations such as virtual and augmented reality, or video games, where the user being immersed into the scene can manipulate the virtual material. While interactive paper bending and crumpling acting under compressive forces have already been studied [BW07, KZC09, HY14, SRH * 15], interactive tearing behavior of paper under stretching forces has not yet been addressed.

In this work, we propose a new method to efficiently model paper tearing in the context of an interactive modeling session. The method takes as input a flat sheet of paper. Users can then dynamically distort it and tear it up, the deformation being controlled by the compression or tension forces resulting from the movement of their finger tips applied to the sheet. Note that when you tear the actual paper with your hands, you can naturally control the direction of the tear. One of our objectives is to allow the same kind of intuitive control over our virtual tear. Our approach is associated with bending and crumpling of the paper and thus enables to model common real life scenarios of paper manipulation. While other previous approaches usually require either a fine meshing along the tear in order to reduce dependence from the mesh connectivity [START_REF] Pfaff | Adaptive tearing and cracking of thin sheets[END_REF], or require to embed the mesh into a low resolution proxy [KMB * 09], our approach, instead, can efficiently compute the tear in an arbitrary direction on the surface mesh. By assuming the specific scenario where a sheet of paper is manipulated by the user's hands, we derive our geometry-based algorithm to compute the propagation of a tear from observations made in material sciences and fracture mechanics. By using a dedicated coarse, yet accurate, representation of the developable surface of paper we finally achieve our goal of realistic looking tears of paper at interactive time rates. Our contributions further include a geometricallybased criteria to select a small subset of candidate vertices where to start a tear, an efficient computation of the tearing direction, and a texture-based appearance of fine details along the tear.

Related works

Thin sheet tearing, and more generally, thin sheet fracture have been studied in the Computer Graphics field using two main approaches: the physically-based methods in order to simulate the fracture of different material, and the procedural approaches to generate plausible fracture behaviors and geometries.

Physically-based fracture simulation

The early work of Terzopoulos and Fleischer [START_REF] Terzopoulos | Modeling inelastic deformation: Viscolelasticity, plasticity, fracture[END_REF] is the first to propose a tearing model for thin sheets computed using a breakable mass-spring model. While the approach is conceptually simple, the fracture orientation heavily depends on the mesh connectivity. O'Brien and Hodgins [OH99] propose a continuous tetrahedron based finite element model handling explicitly this fracture orientation for brittle objects. It has been extended later to ductile material [START_REF] Bargteil | Graphical modeling and animation of ductile fracture[END_REF]. Solid object fracture has been subject to specific studies such as efficient formulation for FEM meshes [BHTF07], integral formulation [ZBG15], or adapted to point set data [START_REF] Glondu | Real-Time Simulation of Brittle Fracture using Modal Analysis[END_REF]. [START_REF] Hegemann | A level set method for ductile fracture[END_REF] proposes to solve ductile fracture using the so called Griffith's energy with a level set method. However, such optimizations are restricted to non deformable models. Interactive time rates have recently been achieved using modal analysis [START_REF] Glondu | Real-Time Simulation of Brittle Fracture using Modal Analysis[END_REF]. Gingold et al. [GSH * 04] propose a fracture for-mulation dedicated to discrete deformable triangle meshes, which is thus adapted to thin sheets fracture. Pfaff et al. [START_REF] Pfaff | Adaptive tearing and cracking of thin sheets[END_REF] developed a thin sheet fracture model based on a local stress analysis on top of a local mesh adaptation algorithm [NSO12]. It achieves highly detailed results which can be applied to paper tearing as well, but its computational cost prevents the applications within an interactive scenario.

Procedural fracture model Some previous works use procedural methods for fracturing volumetric objects. For example, fracture patterns can be computed using a fast procedural method, although the kinematics are still usually computed using a physical model. A popular method for cheap fracture computing, dedicated to explosion modeling, is to use prefractured objects that break along a pre-defined pattern [MCK13]. The fracture may also be spread from an initial location using a procedural algorithm [START_REF] Neff | A visual model for blast waves and fracture[END_REF]. This kind of approaches enables fast fracture simulation but cannot be applied to paper as the path of the tear explicitly depends on the motion of the hands tearing the paper. Lejemble et al. [LFD * 15] proposed an entire procedural method to compute the tearing of paper in a very specific scenario where only four fingers with fixed positions on the sheet or paper are able to tear it.

Fracture details

Different approaches have been proposed to create realistic details at the edges of a tear. Metaaphanon et al. [START_REF] Metaaphanon | Simulation of tearing cloth with frayed edges[END_REF] model frayed edges of woven fabrics being torn by representing the two layers of interwoven yarn of the cloth. Although paper may be seen as a structure of interwoven fibers, the small size and stochastic position and orientation of the fibers make this kind of approach hardly efficient for tearing paper. Busaryev et al. [BDW13] simulate multilayered thin material by superposing several thin shells. A mixed approach, proposed by Chen et al. [CYFW14], is to refine a 3D fracture animation, computed for example with a coarse physical simulation, by adding procedural details based on a 3D texture representing the strength field of the material (i.e. its ability to resist to fracture). Highly detailed fracture of inhomogeneous material can thus be computed at low cost. Physical accuracy is limited by the coarse initial simulation, and the collision handling of the refined parts may prove to be problematic. Yet this concept seems to be well-adapted to represent small details of the torn fibrous structure. This method inspired the generation of additional 2D details for the procedural method by [LFD * 15]. We use a similar approach to generate detailed tearing paths, but improve it by efficiently employing textures.

Method overview

The main idea of this work is grounded in several observations from studies made in material sciences and fracture mechanics. First, by reviewing a set of experimental and theoretical studies of tearing paths for different set-ups, such as a cylinder being pushed though a sheet [START_REF] Audoly | Cracks in thin sheets: When geometry rules the fracture path[END_REF] or two points pulling on a tear [START_REF] Okeefe R | Modeling the tearing of paper[END_REF], Roman [START_REF] Roman | Fracture path in brittle thin sheets: a unifying review on tearing[END_REF] shows that a tearing path on the 2D pattern of inextensible thin plates is remarkably explained by geometrical criteria. Therefore material properties and history of elastic deformation may have only few impact on the resulting fracture path. Secondly, as explained in [START_REF] Freund | Dynamic fracture mechanics[END_REF][START_REF] Germain | Mécanique de la rupture fragile et ductile[END_REF], the tear propagation is mainly influ-enced by the stress field surrounding the tip of the tear. This leads to the general idea of our method based on describing the tear propagation path through local geometrical criteria around the tip of the tear in order to derive an efficient algorithm.

The Griffith criterion, derived from [Gri21] is used in numerous former and recent works about fracture mechanics, e.g. [SP74, AL85, CFM09] for predicting the propagation of a tear or fracture. Let us suppose a sheet of paper with constant thickness e, this criterion relates the propagation of tears to two quantities: the energy release rate Er (J m -2 ) expressing the amount of energy released by a infinitesimal fracture propagation and divided by e and the crack resistance force, also called fracture energy E f (J m -2 ) expressing the magnitude of the force divided by e required to generate a fracture, and can be seen as expressing the intrinsic resistance of the material to tearing.

Griffith criterion states that a tear is created when

Er ≥ E f .
(1)

Note that the associated computation of these quantities in our scenario will be described in Section 4 and Section 5. Some works propose to compute the direction of propagation as the direction that maximizes the energy released rate Er [BFM08], i.e. the direction which first satisfies the Griffith criterion when forces applied gradually increase.

We derive our geometric propagation criterion from Griffith's work and a specific case studied theoretically and experimentally by O'Keefe [START_REF] Okeefe R | Modeling the tearing of paper[END_REF] and Roman [START_REF] Roman | Fracture path in brittle thin sheets: a unifying review on tearing[END_REF], where a sheet of paper is torn apart when pulled by two fingers on both sides of the tear, see Figure 2(left). O'Keefe notes that the propagation of the tear can be fully described by the positions of the two fingers, Ā and B, and the crack tip p on the 2D pattern of the sheet of paper. More precisely, in the case of isotropic paper, the direction of propagation that maximizes the energy release rate Er is given by the bisector between the two line segments linking the fingers' positions to the tip p of the tear as seen in Figure 2(right). The trajectories are thus hyperbola with Ā and B as focal points. The maximized energy release rate Er can then be formulated as

Er = F 2 cos θ 2 /e, (2)
where F is the amplitude of the force applied on the pulling points, θ ∈ [0, 2π] is the angle between the two segments linking the finger positions to the tip of the tear, and e is the paper thickness, which is supposed to be a constant defined by the user in our implementation.

Our method aims at applying these ideas to a more general scenario, where tearing may be initiated and propagated using more than only two positional constraints, while still seeking to describe the tearing propagation direction within the 2D pattern domain for efficiency reasons. Our insight is to analyze the distribution of forces acting locally on the tip of the tear to extract two main, approximately equivalent opposite forces. These two opposite forces, seen as generated by two virtual fingers, will be used to define the tearing direction following [START_REF] Okeefe R | Modeling the tearing of paper[END_REF] as the bisector of their directions expressed on the 2D pattern. Note that our assumption of locally equivalent forces is restricted to the cases where multiple positional constraints are pulling apart the sheet of paper which is coherent with our interactive tearing scenario, but may no longer be suitable for other scenarios such as volumetric projectiles passing through the sheet of paper, or when a part of the sheet of paper is stuck on an underlying surface.

In order to apply this tearing method for interactive purposes, we need to base our computations on an efficient deformable model of paper material. Our method can be seen as an extension of the interactive model developed by Schreck et al. [SRH * 15]. This model describes the isometric, and thus developable, non-smooth surface of a deformed sheet of paper as a set of generalized cones and planar surface parts. Each generalized cone is defined as a set of rulings having a single point, the so-called apex, in common. This model can handle both smoothly bent and crumpled papers as response to compressive forces. A smooth shape is defined by piecewise generalized cones with the apex outside the surface, see Figure 3. A crumpled shape includes also conical pieces whose apices are inside the surface. Those apices represent so-called interior singular points -the surface is not smooth (C 2 ) anymore-caused by damage done to the fibrous structure while crumpling. This paper model also includes so-called junction points, which designate the junction between curved and planar regions on the boundary of the sheet of paper, see Figure 3. They represent the singular points on the boundary of the surface. The model is constantly updated at each time step using adapted geometrical remeshing ensuring developability of the shape interleaved with a physically based elastic FEM simulation [ARC] guiding the main deformations on top of the coarse mesh.

Our tearing method can be seen as an additional feature of this existing pipe-line, which thus offers now a complete framework for interactive virtual paper manipulation. The deformation developed in [SRH * 15] (both physical simulation step and geometric remeshing step) is used here to compute the interactive deformation of the surface without tearing consideration. Our method then computes the response of paper to extension forces by applying the following four steps, see also Figure 4:

(1) Selection of potential tearing points and fracture energy computation (Section 4) A list of candidates to extend or start the tears propagation called (2) Hybrid model to compute tearing propagation (Section 5) The local forces and the direction of the tear are efficiently computed based on the previously described assumptions. The release energy rate Er is computed with respect to the applied physical forces, and the tear is then potentially propagated on the surface which is remeshed.

(3) Modeling the details along the tear trajectory (Section 6) A high resolution path along the tear is computed using a texture based approach enabling to efficiently mimic detailed tearing appearance on top of a coarse geometrical mesh.

(4) A physical simulation step is finally applied on the mesh after each propagation of a tear, in order to dissipate the energy released by the propagation and let the lips of the tear move apart before treating the next potential tearing point. This is the same physical step than used in [SRH * 15].

Potential tearing points and fracture energy

Finding the location of the starting position of a tear is a problem which has not been explicitly studied in Computer Graphics so far. Existing approaches consider either a predefined position which does not fit to an interactive model [KMB * 09], or consider all vertices of the mesh as potential starting points for the tear leading to high computational cost [START_REF] Pfaff | Adaptive tearing and cracking of thin sheets[END_REF]. We propose instead to extract a limited subset of points where tearing may take place, and sort them with respect to a procedural criteria.

Selecting potential tearing points

Material sciences studies showed that inextensible thin sheets, including paper material, have a very low resistance to torque [START_REF] Roman | Fracture path in brittle thin sheets: a unifying review on tearing[END_REF]. At the opposite, they have a much higher resistance to in-plane stress. For instance, tearing a rectangular sheet of paper in-plane while keeping the sheet flat on a table is much more difficult compared to tearing a piece of paper by twisting one of its borders in two different directions as shown in Figure 5.

Taking into account the low resistance to torque and the fact that inextensible thin sheets concentrate stress in very localized singular regions [START_REF] Freund | Dynamic fracture mechanics[END_REF] we propose to consider the potential tearing points as being the vertices of the boundaries which may be able to concentrate the stress in our tearing scenarios. We identified three types of points -The finger positions (red points in Figure 4 and Figure 5), where the hard positional constraints are enforced on the surface, therefore potentially accumulate large stress.

-The so-called junction points (yellow points in Figure 4) separating two curved regions (see Figure 3 and Figure 5), or a curved and a flat region on the boundary. These points are singular, and therefore concentrate the stress.

-The points exhibiting an inward angle on the boundary (sky blue points in Figure 4). Under deformation, these points separate different conical or planar surface parts and become therefore junction points. The tips of already existing tears fall in this category as well.

Note that following the underlying geometric model, which well approximates deformed paper sheets, all other points on the boundary are necessarily regular surface points belonging to smooth cones or planar regions, and therefore do not accumulate stress. We are therefore able to limit the number of potential tearing points which have to be checked at each time step to this small set of points, usually around 10 points in our examples.

Computing the fracture energy

Remember that the fracture energy E f (1) represents the threshold above which the energy release rate Er is sufficient to generate or propagate a tear. It is therefore related to the intrinsic resistance of the fiber network constituting the sheet of paper and needs to be estimated for each potential tearing point. We thus propose to model the resistance of the fibers by a 2D texture T f ibers over the surface. We use a Perlin noise as fibers' texture. Note that it would be possible to replace the Perlin noise by a more physically accurate representation of the distribution of the fibers as a future work.

In order to take into account the plastic damage occurring at the tip of a tear that creates microcracks propagating ahead of the tear tip, and thus weaken the structure in its neighborhood, We associate a damage parameter dp to each potential tearing point p = (u, v) (u and v being the coordinates of the point in the 2D pattern) which facilitates the propagation of an already existing tear. If p is a tip of a tear, we set dp = D, where D ∈ [0, 1] is a constant value chosen by the user to define how easy the tear can be propagated. For all other points, we set dp = 1. As shown in the accompanying video, the use of the damage parameter helps to ensure a continuous tear propagation in correspondence to the motion of the hands. The fracture energy of p is then defined as

E f (p) = c T f iber (u, v) dp, (3)
where T f ibers (u, v) ∈ [0, 1] is the value of the fibers texture at the position (u, v) and c is a material constant modeling the general resistance of the paper. Combined with the parameter D, c controls how easily a tear will be created and how easily it will propagate.

We use c = 0.005 in our examples.

Hybrid model to compute tearing propagation

Given a potential tearing point p and its fracture energy E f (p) (3) this section now describes how to compute the energy release rate Er(p) enabling to decide whether or not the tear path propagates at p and how to compute the direction of the path of the tear.

We call our approach hybrid (geometric and physical) because on one hand the underlying deformation model (crumpling model) [SRH * 15] is hybrid and on the other hand we also interleave geometric and physical considerations for the tear path computations. The crumpling model has indeed two layers. The physical layer provides us with the forces derived from a standard elastic physicalbased simulation step. The geometric layer consists in the developable surface of the sheet of paper defined by pieces of generalized cones through their set of rulings and planar surface pieces, see Figure 3. The geometric layer includes the tear path computations and is used to create an optimized mesh for the simulation step. This mesh is very coarse but has all the necessary degrees of freedom. Both the physical layer and the geometric layer of the deformation model are updated at each frame. See the overview Figure 4 to remember the big picture of the algorithm.

Our approach aims at integrating efficiently the geometrical and physical considerations noted by material sciences studies, in order to obtain the controllable, smooth paths specific to paper tearing. Despite the fact that the forces act on the tear in the 3D space, the tear trajectory seems to follow geometric rules on the 2D pattern. This is why, instead of choosing a 3D criterion as proposed in [OH99, PNdJO14] or a 2D energy based criterion as used by Gingold et. al. [GSH * 04], we derive a criterion based both on the 3D forces and on 2D geometric rules. In addition, we can thus perform the tear path propagation and the mesh update in the 2D pattern, which simplifies considerably the computations in the geometric layer.

Tearing forces

Let us consider a potential tearing point p and the forces applied on it. These forces ∈ IR 3 are computed from the standard FEM used by the physical simulation step. Note that due to the very low bending forces of paper material, we consider only in-plane stretching forces during the tear propagation step which are the forces related to the deformation of the triangles of the mesh. If p, a vertex of the mesh, has N adjacent triangles, a set of N forces (f k ) k∈[0,N-1] is applied on it, each one corresponding to an adjacent triangle. As stated previously in Section 3, we aim at efficiently approximate this local distribution of stretching forces by two equivalent (nearly) opposite forces called F le f t and F right , that we consider as generated by two imaginary pulling points, so that we can apply the formula (2) derived by O'Keefe [START_REF] Okeefe R | Modeling the tearing of paper[END_REF]. Herein the direction of tearing is determined in the 2D pattern as the bisector of the pre-image of these two forces.

Our first goal is therefore to divide the forces (f k ) k∈[0,N-1] into two clusters based on the similarity of their directions. For this, we look for a plane separating the forces defined by the normal vector n at position p and a "bisecting" vector b. F le f t and F right are then defined by:

F le f t = ∑ f k •(b×n)>0 f k and F right = ∑ f k •(b×n)<0 f k . ( 4
)
Our computation of b relies on an iterative clustering algorithm. First, we initiate b 0 as the bisector of two tangent vectors on each side of p on the surface boundary, see Figure 6 (top). Note that these (generally not parallel) tangent vectors may correspond either to the initial surface boundary when a new tear is created, or the lips of an already existing tear. F 0 le f t and F 0 right are computed using Eq. (4). We then iterate over i > 0 by defining b i as the bisector of F i-1 le f t and F i-1 right and pursue the computation of the forces F i le f t and F i right still using Eq. (4). Finally, we stop the algorithm when b ib i-1 < ε, which usually takes no more than 10 iterations.

Energy release rate and tearing direction

Once we have obtained the two clusters of 3D forces (represented by F le f t and F right ), we use them to compute the energy release rate and the direction of propagation as described before in Section 3.

The forces (f k ) composing the clusters are in-plane forces, so we can map each f f f k into f f f k in the 2D pattern. We then compute Fle f t and Fright -the counterparts in the 2D pattern of F le f t and F rightas the sum of mapped forces of each cluster:

F F F le f t = ∑ f k •(b×n)>0 f f f k and F F F right = ∑ f k •(b×n)<0 f f f k . ( 5
)
We consider Fle f t and Fright as the projections of forces on the pattern applied by the two imaginary pulling points. The direction t t t (shown in Figure 7) in which the tear should propagate to release the maximum of energy is then the bisector of Fle f t and Fright and the corresponding energy release rate (derived from (2)) is given by

Er = ( F le f t + F right ) cos θ 2 /e, ( 6
)
where θ is the angle ∈ [0, 2π] between Fle f t and Fright .

If the energy release rate Er is greater than the fracture energy E f at the point p (1), then the tear propagates and a new tip of the tear is created in the direction t t t. We adjust the speed of propagation according to the energy released by the propagation:

pnext = p + c Er t t t,
where c is a constant parameter enabling the user to tune the speed of propagation of the tear. Finally, the 3D position of the new tear tip is computed by mapping pnext back onto the 3D surface.

Surface remeshing

Once the new tearing tip pnext has been found, the crumpling model, in particular the geometric layer, needs to be updated. As p is a point belonging to the boundary of the torn paper, pnext has to be integrated into the boundary of the surface. We distinguish two cases. When the tear direction in the 2D pattern changes at vertex p (i.e. the angle formed by the tear at p is greater than a threshold that we set at 0.1rad), we split p into two vertices, each belonging to one side of the tear lips, see Fig. 8(left). Otherwise, when the tear goes straight, we remove p from the boundary of the surface, see Fig. 8(right). This enable us to keep the mesh coarse in order to be as computational efficient as possible.

Finally, the coherence of the geometric layer has to be ensured: the generalized cones and flat regions crossed by the new tear between p and pnext have to be updated to integrate the tear. More technical details are given in Appendix A. Let us however notice, that one of two newly created vertices after splitting, p 1 or p 2 , has necessarily an inward angle on the boundary, which means that it becomes a potential tearing point. The tear could thus even branch at this point in the next iterations of the tearing algorithm.

The physical layer is computed from the geometric layer and then automatically integrates the tear. The physical simulation is considered as static, so we put a zero velocity to every point created while remeshing.

Modeling the details along the tear trajectory

The previous section explained how we compute the next tip of a tear when it propagates during a small time step. This enables to obtain at a large scale a piecewise smoothly curved tearing path, such as the hyperbola-generated trajectories from [START_REF] Okeefe R | Modeling the tearing of paper[END_REF]. At a fine scale however, the border of a paper along a tear exhibits small stochastic details, as the ones that can observed on the real torn paper in Figure 9. In order to obtain realistic looking results with our method, these small stochastic details, need to be added. In this section, we explain how compute a detailed path between two successive tips and how to visualize it with a texture.

Figure 9: Torn border of a real paper. The path of a paper tear presents small stochastic details that would be too expensive to precisely represent with the same mesh used for the modeling of deformation.

Stochastic computation of the detailed path

To compute the fine scale geometry of the tear path between two successive tear tips i and i + 1, we adapt the method from Lejemble et al. [LFD * 15] by defining a 2D scalar mapping, which combines a stochastic distribution T f iber representing the resistance of the fibers to break and a fracture-centered Gaussian stress field G i :

T (x, y) = (1 -ω) T f iber (x, y) + ω G i (x, y). ( 7
)
The size of the details are controlled by the standard deviation σ of the stress field G i . T f iber has already been introduced in formula (3) and causes the stochastic appearance of the resulting path. We model T by a 2D texture, which represents the breaking coefficient at each pixel (intuitively it can be seen as the probability of the path to go though a pixel). A low energy path between the positions of the two tips is then computed by iteratively choosing among the neighbor pixels that are closer to the next tip the one with the greatest breaking coefficient.

Representation of the detail path as textures

The detailed path between the two successive tips is described as a path inside a texture mapping the surface. Our objective is to efficiently render the fine detailed tear segments between successive tips following the global tear trajectory. A naive way to proceed would be to compute the position on the 2D pattern corresponding to each pixel of the path and then triangulate them and map them on the 3D surface. But this would require a very fine meshing near the tear path with an large number of triangles. Our idea is to keep the mesh coarse, and rather "cut" the path on the texture of the paper that we use when rendering our results.

Let us call the initial texture used to render the piece of paper Tpaper. To effectively represent a detailed tearing path we modify this texture. When a new tear is created, we compute two textures from Tpaper, T right and T le f t , that are respectively transparent on each side of the detailed path of tear (see Figure 10). The triangles that are adjacent to the tear are textured by either T right or T le f t according to which side of the tear they belong to. When a tear is propagated, the textures T right and T le f t are updated in order to include the new path. When a tear is branching, two new textures are created for the new branch of the tear from the texture corresponding to the parent tear.

The detailed path between two successive tear tips is shared by two triangles. To have all the details of the path represented for both of them, we extend those triangles by a small textured rectangle as shown in Figure 11.

Results

We present here some of our results using simple and easily reproducible scenarios. The animations corresponding to the examples presented in this section can be found in the accompanying video.

Validation of the starting points

To validate our method for computing the tearing starting point we performed an experimental evaluation. We compared the position of the starting point of the tear obtained by our virtual paper with the position for real paper in three cases: (a) the tear begins at a finger position, (b) the tear starts at a junction point and (c) at an inward angle of the boundary. These cases represent our three different types of potential tearing points on the boundary of the paper described in Section 4.1.

Figure 12: A tear is created on virtual paper by flattening of the border of the bent surface. The deformation is generated by two hands represented each by two red points (modeling two fingers) linked by a red line. The tear appears at a junction point between the curved region and one of the flat regions on both, on our virtual sheet paper (top) as well as on the real one (bottom).

Figure 12 shows a virtual piece of paper bent into a locally cylindrical shape by using two hands placed onto the sheet of paper. Each hand is represented on the figure by two red points linked by a red line. Then the two hands perform a rotation in order to flatten one border of the cylinder until creating a tear. The same experiment is realized with a real piece of paper. We can see that with both virtual and real paper the tear appears near one of the fingers on the border between the curved region and one of the parts maintained flat by a hand. Indeed, we assumed that fingers, modeled as hard positional constraint, can potentially accumulate large stress, leading easily to tears. Figure 13 shows an example of a tear created by twisting the border of a piece of paper. Two corners of the paper are hold and rotated in opposite directions until two curved regions are obtained, curved in opposite directions. In both cases, for the virtual and the real paper, we make the same observations. First a singularity appears at the junction between the two curved regions. In our virtual model, this is identified as a junction point (represented in the virtual model as a yellow point in Figure 5). Then a tear starts at this point and continues to propagate inside the paper.

Figure 13: A tear is created by bending two corners of the sheet of paper in two different directions,i.e. by twisting the border. On both our virtual paper (top) and real paper (bottom), the tear appears at the junction between the two cones created by each bending.

In the third scenario, cut out a triangular piece of paper and therefore create an inward angle at the boundary, see Figure 14 and Figure 15. We then tear the paper at this concave border by placing the fingers at the top and by pulling the extremities in opposite directions. In this situation the point at the inward angle easily becomes a junction point and so easily becomes the starting point of a tear. We reproduce this case in Figure 14 and Figure 15.

It is possible the check the validity of our potential tearing points be reproducing the scenarios presented above.

These results suggest that the three types of particular points on the geometric model (finger positions, junction points and concave points), which we have selected with physical and geometric arguments do work well as potential starting tearing points. We note, that real paper can also be torn starting from the inside of the surface when forcing an object though it. However, this kind of scenario is out of scope of the present paper, as we aimed at reproducing controllable tears when manipulating a piece of paper with our fingers. The tears caused by this particular case are difficult to create solely by hand and appear abruptly without being controllable.

Validation of the tear trajectory

As explained in Section 3, and in more details in [START_REF] Roman | Fracture path in brittle thin sheets: a unifying review on tearing[END_REF] and [START_REF] Okeefe R | Modeling the tearing of paper[END_REF], when the propagation of the tear is only caused by two points being pulled apart, the path drawn by the tear on the 2D pattern follows a very particular geometric curve. Locally, at each tear tip p, the path follows the tangent of a hyperbolic curve whose focal points are the fingers positions (see Figure 2) which is the bisector of the two line segments linking the two pulling points to p. In the next example, we have reproduced the experiment reported in [START_REF] Okeefe R | Modeling the tearing of paper[END_REF] in order to validate our more general model, which has been derived from these results. Figure 14 shows the paper being torn by pulling two points (red dots) located at equal distances of the starting point of the tear. To prepare the sheet of paper, we cut out a triangular piece.This enable us to predict exactly where the tear will start and to place the imaginary fingers at equal distance to the tear's starting point. Following the theory [START_REF] Okeefe R | Modeling the tearing of paper[END_REF][START_REF] Roman | Fracture path in brittle thin sheets: a unifying review on tearing[END_REF], the energy release rate is maximized in the direction of the bisector of the 2 line segments. As we artificially created a symmetric situation, it is expected that the tear trajectory goes straight. Indeed, the tear shown in Figure 14 follows the mediator of the line segments between the two pulling points describing therefore the expected straight tear trajectory.

Other examples

In the next example, our intention was to reproduce an asymmetric case leading to a curved path. Figure 15 shows the tear obtained by pulling two points located at different distances from the starting points. The mapping of the tear correspond closely to the expected hyperbolic trajectory and can be observed in the video. By rotating them successively we obtain the same zigzagging curve for the real paper (left) as well as on our virtual paper (middle).

User control was one of our main goals when developing our algorithm. It turns out that the user can indeed intuitively control the path of the tear by choosing the positions for pulling fingers as shown in Figure 16(top). This enables us to create complex tearing paths, such as the zigzag shown in Figure 16(bottom). The zigzag was obtained by putting the thumb and the forefinger of each hand onto the paper and rotating then successively one hand after the other. We can also obtain a tear with several branches as shown in Figure 17, when using a more general scenario where fingers change positions through the deformation. We thus demonstrate, that a fine and direct user control over the tearing path is possible and that our model behaves as expected in these scenarios. Figure 19 gives a closer view on stochastic details and shows that we can obtain small details without requiring a dense mesh. Varying the size of the resolution and the standard deviation of the Gaussian field G i in (7) enables to modify the appearance of the tear.

For instance, with a texture resolution of 2048×1476 corresponding to a dinA4 sheet of paper (21×29.7 cm), and a standard deviation σ = 0.1mm (see Figure 19 (bottom-middle), we can obtain details that are of the same scale than the real paper shown in Figure 9. Figure 20 shows a close view on the details of real and virtual paper, along 2 cm of torn border.

Comparison with the approach of Pfaff et al. [PNdJO14]

Figure 21: Comparison between a tear obtained with our method (left) and one obtained with the method presented in [START_REF] Pfaff | Adaptive tearing and cracking of thin sheets[END_REF] (right). The tear is created from a notch with two fingers at equal distance of the end of the notch (yellow dot).

Although the method presented by Pfaff et. al. [START_REF] Pfaff | Adaptive tearing and cracking of thin sheets[END_REF] shows convincing results for many cases of tearing thin shells, it is not well-adapted for the specific case of paper tearing and in particular for interactive manipulations. It not only requires a dense mesh at the tip of the tear -making it computationally rather expensivebut also it cannot reproduce the predictable, smooth-looking aspect of tears in paper. Figure 21 shows an example of this effect: a tear is created by pulling apart two points placed at equal distance of a notch. The tear obtained with the method from [START_REF] Pfaff | Adaptive tearing and cracking of thin sheets[END_REF] follows a oscillating slightly curved path that is not controlled by the position of the fingers. On the opposite, our method enables to obtain the expected straight smooth tear trajectory, to which details of various aspects may then be added and parameterized.

Performance

The Table 1 shows the computation times obtained on a laptop with 8 cores (2.70GHz). The average and maximal times per frame are computed for the animations presented in our results. We obtain an animation rate between 10 and 3 frames per second, which enables interactive applications. The bottleneck of the computational time is still the physical simulation used to compute the underlying elastic deformation [ARC]. Notably, we use a small time step of simulation (in the order of 1/10 th of the visual time frame) in order to simulate rigid behaviors. Still, we managed with our tearing method to keep the mesh, which is also used in the simulation, very coarse (in the order of a hundred triangles), so that the computational cost of the physical simulation stays reasonable for interactivity.

Discussion and conclusion

We presented a method to interactively model a piece of paper being torn. By separating the generation of details and the global trajectory of the tear, we are able to obtain tears whose general path follows a smooth and predictable tear, and which at the same time presents small, refined details. The geometrical-based tearing paths are consistent with respect to behaviors described in the physics and fracture mechanics literature and validated with experiments on real paper.

In this work, we focused on efficient computation and intuitively controllable scenarios on inextensible thin sheet material, our method has then several limitations in the range of applicability that may be addressed in future works.

We based our algorithm on assumptions that are very specific to paper-like material, notably the ones from O'Keefe's work used for the propagation of the tear. Such behavior may not be directly applicable to elastic material such as cloth for instance. Also the described approach has been designed to handle tears caused by a torque imposed on a boundary of the surface, which are the tears that occur most often while tearing paper in real life. This assumption prohibits our model to start a tear inside the surface, such as caused by a projectile. Initializing a tear by using two perfectly colinear forces is also out of the scope of the possible scenarios. Note that in these cases, the sheet of paper would get brutally torn and the tear would hardly be controllable. Still, once initiated, our general idea for computing the tearing path in clustering forces and computing direction within the 2D pattern may be used to pursue the tearing process.

As visible in the videos, some sudden geometrical changes may occur during animation. They are related to the remeshing of a very coarse mesh structure and could be limited by using dedicated smoothing such as, for instance, the post-remeshing projection proposed by Narain et al. [NSO12].

In the near future, we plan to improve the details appearance, e.g. by generating damages in the fibers' texture around singular regions. It would thus be possible to control where the fibrous structure of the paper is weaker, in order to simulate the effect of broken fibers. In order to model plastic deformation effects such as an existing fold line, we also plan either to modify the fiber texture along the fold to influence the detailed path, or to influence the general path by introducing a bias in the choice of the direction of propagation described in Section 5.2. Another idea would be to use several fiber textures to represent multiple layers of fibers. In this way, the same tear would be associated to several detailed paths and then be represented by layered semi-transparent textures in order to improve realism.

Figure 2 :

 2 Figure 2: (left) Two points on a flat sheet are pulled away from each other. The tip of the tear p propagates when the sheet is pulled by points A and B. (right) On the 2D pattern, the direction of propagation is the bisector of the lines ( Ā, p) and ( B, p). The trajectory is then a hyperbola (in red), with focal points in Ā and B.

Figure 3 :

 3 Figure 3: Crumpled paper model from [SRH * 15]: the geometric layer (left) is composed of curved regions (in purple) and flat regions (in green). Interior singular points (in red) represent damages in the fibrous structure. Singular points on the boundary, called junction points (in yellow), represent the junction between two region. The mesh used by the physical layer (right) is then obtained from the geometric layer to fit the folds

Figure 4 :

 4 Figure 4: Overview of our tearing method in three steps on top on existing bending and crumpling animation model.

Figure 5 :

 5 Figure 5: Junction points (yellow point) in this real case scenario are specific points on the papers boundary from which a tearing path is supposed to start.

Figure 6 :

 6 Figure 6: Top: Initialization of tearing forces clustering algorithm. b b b 0 is initiated as the bisector of the boundary tangents at p. Bottom: An iteration of the clustering algorithm. (left) F F F i+1 le f t is computed as the sum of the forces (in red) of the left side of the plane defined by b b b i and the normal to the surface at p and analogous for the forces on the right side (in blue). (right) b b b i+1 is then computed as the bisector of F F F i+1 le f t and F F F i+1 right .

Figure 7 :

 7 Figure 7: The direction of propagation of a tear depends on the 2D angle θ between Fle f t and Fright . If propagated, the next tip is found in the direction t t t, the bisector of Fle f t and Fright .

Figure 8 :

 8 Figure 8: Depending on the angle made by the tear at p, we either (left) split or (right) remove the vertex p

Figure 10 :

 10 Figure 10: The two textures T le f t (middle) and T right (right) used to render the fine details along the tear path as defined (in red) in the initial texture (left).

Figure 11 :

 11 Figure 11: A support rectangle is used to represent the details outside a triangle bordering a tear.

Figure 14 :

 14 Figure 14: Two pulling points at equal distance of the starting point of the tear create a straight tear. (top) Evolution of the 3D surface as the tear propagates. (bottom) 2D pattern, the tear path is the mediator of the two pulling points.

Figure 15 :

 15 Figure 15: Two pulling points at different distances of the starting point of the tear create a curved tear. (left) The 3D surface of the torn paper. (right) 2D pattern, at each point, the tangent to the path approximately cuts the angle between the directions toward each pulling points into two equal part.

Figure 16 :

 16 Figure 16: Top: By choosing the position of two pulling points, we can control the direction of the tear. Bottom: We use the thumb and forefinger, represented as red dots on the virtual paper and the corresponding 2D pattern (right) of each hand to tear the paper.By rotating them successively we obtain the same zigzagging curve for the real paper (left) as well as on our virtual paper (middle).

Figure 17 :

 17 Figure 17: A branching tear is obtained by tearing the paper successively into different directions using the fingers represented by red dots.

Figure 18

 18 Figure18shows an example of a more complex deformation involving crumpling and tearing. A sheet of paper is bent in a cylindrical shape by two hands (one of each side of the curved part) (left). A border is then pinched by one of the hands (middle-left). The hands move apart of each other by a rotational motion, creating a singular point on the border and then a tear starting from this point (middle-right, right).

Figure 18 :

 18 Figure 18: A more complex example exhibiting crumpling and tearing. Red dots are finger points and black arrows indicate their motion.

Figure 20 :

 20 Figure 20: Close view on a torn border of real paper (left) and one of our virtual paper (right). Each image represents around 2cm of the torn border.

Table 1 :

 1 Time spent in each step (in ms per frame) Average and maximum computation times for the respective steps: physical simulation[ARC] (including both the physical step use in [SRH * 15] and the one after each propagation of a tear), geometric remeshing of the crumpling method [SRH * 15] and for the tearing step described in this article (steps 1, 2, and 3 from Fig4). The second column indicates the average number of triangles through all the frames.

			Physical	Geometrical	Tearing
		#∆	simulation	remeshing	computation
			avg max	avg	max	avg	max
	Fig. 12	99	248 301	7	11	2	12
	Fig. 13	70	119 206	4	14	2	15
	Fig. 14	68	170 244	4	5	3	7
	Fig. 16 102	254 303	8	17	6	17

c 2017 The Author(s) Computer Graphics Forum c 2017 The Eurographics Association and John Wiley & Sons Ltd.

Acknowledgements: This research was partially funded by the ERC advanced grant no. 291184 EXPRESSIVE. We thank Estelle Charleroy for the help on the rendering and video montage.

Appendix A: Geometry update after tear propagation

The geometric structure used by the crumpling model should be preserved. The geometry update can therefore be divided into three cases. For more details the reader is referred to [SRH * 15].

Case 1: pnext is inside a generalized cone (i.e. pnext is inside the 2D pattern of the cone) whose apex is p, in which case we modify the cone such that its apex becomes pnext (see Figure 22). Case 2: pnext is inside a curved region and p is not the apex of a generalized cone. We apply a remeshing scheme similar to the one used when creating a singular point inside a curved region (see Figure 23). Case 3: pnext is inside a flat region. We just update the coarse triangulation representing the flat region in the geometric structure (see Figure 24).

Note also that if p is the apex of a generalized cone, we updated the regions such that pnext becomes the apex of this cone.