Design and Optimization of Anti-amyloid Domain Antibodies Specific for β-Amyloid and Islet Amyloid Polypeptide

Antibodies with conformational specificity are important for detecting and interfering with polypeptide aggregation linked to several human disorders. We are developing a motif-grafting approach for designing lead antibody candidates specific for amyloid-forming polypeptides such as the Alzheimer peptide (Aβ). This approach involves grafting amyloidogenic peptide segments into the complementarity-determining regions (CDRs) of single-domain (V_H) antibodies. Here we have investigated the impact of polar mutations inserted at the edges of a large hydrophobic Aβ42 peptide segment (Aβ residues 17–42) in CDR3 on the solubility and conformational specificity of the corresponding V_H domains. We find that V_H expression and solubility are strongly enhanced by introducing multiple negatively charged or asparagine residues at the edges of CDR3, whereas other polar mutations are less effective (glutamine and serine) or ineffective (threonine, lysine, and arginine). Moreover, Aβ V_H domains with negatively charged CDR3 mutations show significant preference for recognizing Aβ fibrils relative to Aβ monomers, whereas the same V_H domains with other polar CDR3 mutations recognize both Aβ conformers. We observe similar behavior for a V_H domain grafted with a large hydrophobic peptide from islet amyloid polypeptide (residues 8–37) that contains negatively charged mutations at the edges of CDR3. These findings highlight the sensitivity of antibody binding and solubility to residues at the edges of CDRs, and provide guidelines for designing other grafted antibody fragments with hydrophobic binding loops.     

Multi‐constraint computational design suggests that native sequences of germline antibody H3 loops are nearly optimal for conformational flexibility

The limited size of the germline antibody repertoire has to recognize a far larger number of potential antigens. The ability of a single antibody to bind multiple ligands due to conformational flexibility in the antigen‐binding site can significantly enlarge the repertoire. Among the six complementarity determining regions (CDRs) that generally comprise the binding site, the CDR H3 loop is particularly variable. Computational protein design studies showed that predicted low energy sequences compatible with a given backbone structure often have considerable similarity to the corresponding native sequences of naturally occurring proteins, indicating that native protein sequences are close to optimal for their structures. Here, we take a step forward to determine whether conformational flexibility, believed to play a key functional role in germline antibodies, is also central in shaping their native sequence. In particular, we use a multi‐constraint computational design strategy, along with the Rosetta scoring function, to propose that the native sequences of CDR H3 loops from germline antibodies are nearly optimal for conformational flexibility. Moreover, we find that antibody maturation may lead to sequences with a higher degree of optimization for a single conformation, while disfavoring sequences that are intrinsically flexible. In addition, this computational strategy allows us to predict mutations in the CDR H3 loop to stabilize the antigen‐bound conformation, a computational mimic of affinity maturation, that may increase antigen binding affinity by preorganizing the antigen binding loop. In vivo affinity maturation data are consistent with our predictions. The method described here can be useful to design antibodies with higher selectivity and affinity by reducing conformational diversity. Proteins 2009. © 2008 Wiley‐Liss, Inc.     

Understanding the relevance of local conformational stability and dynamics to the aggregation propensity of an IgG1 and IgG2 monoclonal antibodies

Aggregation of monoclonal antibodies is often a multi‐step process involving structural alterations in monomeric proteins and subsequent formation of soluble or insoluble oligomers. The role of local conformational stability and dynamics of native and/or partially altered structures in determining the aggregation propensity of monoclonal antibodies, however, is not well understood. Here, we investigate the role of conformational stability and dynamics of regions with distinct solvent exposure in determining the aggregation propensity of an IgG1 and IgG2 monoclonal antibody. The temperatures employed span the pre‐unfolding range (10–40°C) and the onset temperatures (T_onset) for exposure of apolar residues (～50°C), alterations in secondary structures (～60°C) and initiation of visible aggregate formation (～60°C). Solvent‐exposed regions were found to precede solvent‐shielded regions in an initiation of aggregation for both proteins. Such a process was observed upon alterations in overall tertiary structure while retaining the secondary structures in both the proteins. In addition, a greater dynamic nature of solvent‐shielded regions in potential intermediates of IgG1 and the improved conformational stability increased its resistance to aggregation when compared to IgG2. These results suggest that local conformational stability and fluctuations of partially altered structures can influence the aggregation propensity of immunoglobulins.     

Conformational Ensembles of Antibodies Determine Their Hydrophobicity

A major challenge in the development of antibody biotherapeutics is their tendency to aggregate. One root cause for aggregation is exposure of hydrophobic surface regions to the solvent. Many current techniques predict the relative aggregation propensity of antibodies via precalculated scales for the hydrophobicity or aggregation propensity of single amino acids. However, those scales cannot describe the nonadditive effects of a residue’s surrounding on its hydrophobicity. Therefore, they are inherently limited in their ability to describe the impact of subtle differences in molecular structure on the overall hydrophobicity. Here, we introduce a physics-based approach to describe hydrophobicity in terms of the hydration free energy using grid inhomogeneous solvation theory (GIST). We apply this method to assess the effects of starting structures, conformational sampling, and protonation states on the hydrophobicity of antibodies. Our results reveal that high-quality starting structures, i.e., crystal structures, are crucial for the prediction of hydrophobicity and that conformational sampling can compensate errors introduced by the starting structure. On the other hand, sampling of protonation states only leads to good results when combined with high-quality structures, whereas it can even be detrimental otherwise. We conclude by pointing out that a single static homology model may not be adequate for predicting hydrophobicity.     

Structure of an IgNAR-AMA1 complex: targeting a conserved hydrophobic cleft broadens malarial strain recognition

Apical membrane antigen 1 (AMA1) is essential for invasion of erythrocytes and hepatocytes by Plasmodium parasites and is a leading malarial vaccine candidate. Although conventional antibodies to AMA1 can prevent such invasion, extensive polymorphisms within surface-exposed loops may limit the ability of these AMA1-induced antibodies to protect against all parasite genotypes. Using an AMA1-specific IgNAR single-variable-domain antibody, we performed targeted mutagenesis and selection against AMA1 from three P. falciparum strains. We present cocrystal structures of two antibody-AMA1 complexes which reveal extended IgNAR CDR3 loops penetrating deep into a hydrophobic cleft on the antigen surface and contacting residues conserved across parasite species. Comparison of a series of affinity-enhancing mutations allowed dissection of their relative contributions to binding kinetics and correlation with inhibition of erythrocyte invasion. These findings provide insights into mechanisms of single-domain antibody binding, and may enable design of reagents targeting otherwise cryptic epitopes in pathogen antigens.     

Aggregation-resistant domain antibodies selected on phage by heat denaturation

We describe a method for selecting aggregation-resistant proteins by heat denaturation. This is illustrated with antibody heavy chain variable domains (dAbs), which are prone to aggregate. The dAbs were displayed multivalently at the infective tip of filamentous bacteriophage, and heated transiently to induce unfolding and to promote aggregation of the dAbs. After cooling, the dAbs were selected for binding to protein A (a ligand common to these folded dAbs). Phage displaying dAbs that unfold reversibly were thereby enriched with respect to those that do not. From a repertoire of phage dAbs, six dAbs were characterized after selection; they all resisted aggregation, and were soluble, well expressed in bacteria and could be purified in good yields. The method should be useful for making aggregation-resistant proteins and for helping to identify features that promote or prevent protein aggregation, including those responsible for misfolding diseases.     

Competition between Intramolecular and Intermolecular Interactions in an Amyloid-Forming Protein

Despite much progress in understanding the folding and the aggregation processes of proteins, the rules defining their interplay have yet to be fully defined. This problem is of particular importance since many diseases are initiated by protein unfolding and hence the propensity to aggregate competes with intramolecular collapse and other folding events. Here, we describe the roles of intramolecular and intermolecular interactions in defining the length of the lag time and the apparent rate of elongation of the 100-residue protein human β₂-microglobulin at pH 2.5, commencing from an acid-denatured state that lacks persistent structure but contains significant non-random hydrophobic interactions. Using a combination of site-directed mutagenesis, quantitative kinetic analysis and computational methods, we show that only a single region of about 10 residues in length, determines the rate of fibril formation, despite the fact that other regions exhibit a significant intrinsic propensity for aggregation. We rationalise these results by analysing the effect of incorporating the conformational properties of acid-unfolded β₂-microglobulin and its variants at pH 2.5 as measured by NMR spectroscopy into the Zyggregator aggregation prediction algorithm. These results demonstrate that residual structure in the precursor state modulates the intrinsic propensity of the polypeptide chain to aggregate and that the algorithm developed here allows the key regions for aggregation to be more clearly identified and the rates of their self-association to be predicted. Given the common propensity of unfolded chains to form non-random intramolecular interactions as monomers and to self-assemble subsequently into amyloid fibrils, the approach developed should find widespread utility for the prediction of regions important in amyloid formation and their rates of self-assembly.     

Conformational changes and flexibility in T-cell receptor recognition of peptide-MHC complexes

A necessary feature of the immune system, TCR (T-cell receptor) cross-reactivity has been implicated in numerous autoimmune pathologies and is an underlying cause of transplant rejection. Early studies of the interactions of alphabeta TCRs (T-cell receptors) with their peptide-MHC ligands suggested that conformational plasticity in the TCR CDR (complementarity determining region) loops is a dominant contributor to T-cell cross-reactivity. Since these initial studies, the database of TCRs whose structures have been solved both bound and free is now large enough to permit general conclusions to be drawn about the extent of TCR plasticity and the types and locations of motion that occur. In the present paper, we review the conformational differences between free and bound TCRs, quantifying the structural changes that occur and discussing their possible roles in specificity and cross-reactivity. We show that, rather than undergoing major structural alterations or 'folding' upon binding, the majority of TCR CDR loops shift by relatively small amounts. The structural changes that do occur are dominated by hinge-bending motions, with loop remodelling usually occurring near loop apexes. As predicted from previous studies, the largest changes are in the hypervariable CDR3alpha and CDR3beta loops, although in some cases the germline-encoded CDR1alpha and CDR2alpha loops shift in magnitudes that approximate those of the CDR3 loops. Intriguingly, the smallest shifts are in the germline-encoded loops of the beta-chain, consistent with recent suggestions that the TCR beta domain may drive ligand recognition.     

Structural Basis of Epitope Recognition by Heavy-Chain Camelid Antibodies

Truncated versions of heavy-chain antibodies (HCAbs) from camelids, also termed nanobodies, comprise only one-tenth the mass of conventional antibodies, yet retain similar, high binding affinities for the antigens. Here we analyze a large data set of nanobody–antigen crystal structures and investigate how nanobody–antigen recognition compares to the one by conventional antibodies. We find that nanobody paratopes are enriched in aromatic residues just like conventional antibodies, but additionally, they also bear a more hydrophobic character. Most striking differences were observed in the characteristics of the antigen's epitope. Unlike conventional antibodies, nanobodies bind to more rigid, concave, conserved and structured epitopes enriched with aromatic residues. Nanobodies establish fewer interactions with the antigens compared to conventional antibodies, and we speculate that high binding affinities are achieved due to less unfavorable conformational and more favorable solvation entropy contributions. We observed that interactions with antigen are mediated not only by three CDR loops but also by numerous residues from the nanobody framework. These residues are not distributed uniformly; rather, they are concentrated into four structurally distinct regions and mediate mostly charged interactions. Our findings suggest that in some respects nanobody–antigen interactions are more similar to the general protein–protein interactions rather than antibody–antigen interactions.     

A Novel Heavy Domain Antibody Library with Functionally Optimized Complementarity Determining Regions

Today a number of synthetic antibody libraries of different formats have been created and used for the selection of a large number of recombinant antibodies. One of the determining factors for successful isolation of recombinant antibodies from libraries lies in the quality of the libraries i.e. the number of correctly folded, functional antibodies contained in the library. Here, we describe the construction of a novel, high quality, synthetic single domain antibody library dubbed Predator. The library is based on the HEL4 domain antibody with the addition of recently reported mutations concerning the amino acid composition at positions critical for the folding characteristics and aggregation propensities of domain antibodies. As a unique feature, the CDR3 of the library was designed to mimic the natural human immune response by designating amino acids known to be prevalent in functional antibodies to the diversity in CDR3. CDR randomizations were performed using trinucleotide synthesis to avoid the presence of stop codons. Furthermore a novel cycle free elongation method was used for the conversion of the synthesized single stranded DNA containing the randomized CDRs into double stranded DNA of the library. In addition a modular approach has been adopted for the scaffold in which each CDR region is flanked by unique restrictions sites, allowing easy affinity maturation of selected clones by CDR shuffling. To validate the quality of the library, one round phage display selections were performed on purified antigens and highly complex antigen mixtures such as cultured eukaryotic cells resulting in several specific binders. The further characterization of some of the selected clones, however, indicates a reduction in thermodynamic stability caused by the inclusion the additional mutations to the HEL4 scaffold.     

Use of 2D NMR, protein engineering, and molecular modeling to study the hapten-binding site of an antibody Fv fragment against 2-phenyloxazolone

Two-dimensional (2D) 1H NMR spectroscopy was used to study the hapten-binding site of a recombinant antibody Fv fragment expressed in Escherichia coli. Point mutations of residues in the CDR loops of the Fv fragment were designed in order to investigate their influence on hapten binding and to make site-specific assignments of aromatic NMR proton signals. Two tyrosines giving NOEs to the ligand 2-phenyloxazolone were identified, residue 33 in CDR1 of the heavy chain and residue 32 in CDR1 of the light chain. The benzyl portion of 2-phenyloxazolone is located between these two residues. The binding site is close to the surface of the Fv fragment. Comparison with a different anti-2-phenyloxazolone antibody, the crystal structure of which has recently been solved, shows that the general location of the hapten-binding site in both antibodies is similar. However, in the crystallographically solved antibody, the hapten is bound farther from the surface in a pocket created by a short CDR3 loop of the heavy chain. In the binding site identified in the Fv fragment studied in this report, this space is probably filled by the extra seven residues of the CDR3.     

Grafting of material-binding function into antibodies Functionalization by peptide grafting

Quite recently, a few antibodies against bulk material surface have been selected from a human repertoire antibody library, and they are attracting immense interest in the bottom-up integration of nanomaterials. Here, we constructed antibody fragments with binding affinity and specificity for nonbiological inorganic material surfaces by grafting material-binding peptides into loops of the complementarity determining region (CDR) of antibodies. Loops were replaced by peptides with affinity for zinc oxide and silver material surfaces. Selection of CDR loop for replacement was critical to the functionalization of the grafted fragments; the grafting of material-binding peptide into the CDR2 loop functionalized the antibody fragments with the same affinity and selectivity as the peptides used. Structural insight on the scaffold fragment used implies that material-binding peptide should be grafted onto the most exposed CDR loop on scaffold fragment. We show that the CDR-grafting technique leads to a build-up creation of the antibody with affinity for nonbiological materials.     

Human Germline Antibody Gene Segments Encode Polyspecific Antibodies

Structural flexibility in germline gene-encoded antibodies allows promiscuous binding to diverse antigens. The binding affinity and specificity for a particular epitope typically increase as antibody genes acquire somatic mutations in antigen-stimulated B cells. In this work, we investigated whether germline gene-encoded antibodies are optimal for polyspecificity by determining the basis for recognition of diverse antigens by antibodies encoded by three V_H gene segments. Panels of somatically mutated antibodies encoded by a common V_H gene, but each binding to a different antigen, were computationally redesigned to predict antibodies that could engage multiple antigens at once. The Rosetta multi-state design process predicted antibody sequences for the entire heavy chain variable region, including framework, CDR1, and CDR2 mutations. The predicted sequences matched the germline gene sequences to a remarkable degree, revealing by computational design the residues that are predicted to enable polyspecificity, i.e., binding of many unrelated antigens with a common sequence. The process thereby reverses antibody maturation in silico. In contrast, when designing antibodies to bind a single antigen, a sequence similar to that of the mature antibody sequence was returned, mimicking natural antibody maturation in silico. We demonstrated that the Rosetta computational design algorithm captures important aspects of antibody/antigen recognition. While the hypervariable region CDR3 often mediates much of the specificity of mature antibodies, we identified key positions in the V_H gene encoding CDR1, CDR2, and the immunoglobulin framework that are critical contributors for polyspecificity in germline antibodies. Computational design of antibodies capable of binding multiple antigens may allow the rational design of antibodies that retain polyspecificity for diverse epitope binding.     

CDR-H3 loop ensemble in solution – conformational selection upon antibody binding

ABSTRACT

We analyzed pairs of protein-binding, peptide-binding and hapten-binding antibodies crystallized as complex and in the absence of the antigen with and without conformational differences upon binding in the complementarity-determining region (CDR)-H3 loop. Here, we introduce a molecular dynamics-based approach to capture a diverse conformational ensemble of the CDR-H3 loop in solution. The results clearly indicate that the inherently flexible CDR-H3 loop indeed needs to be characterized as a conformational ensemble. The conformational changes of the CDR-H3 loop in all antibodies investigated follow the paradigm of conformation selection, because we observe the experimentally determined binding competent conformation without the presence of the antigen within the ensemble of pre-existing conformational states in solution before binding. We also demonstrate for several examples that the conformation observed in the antibody crystal structure without antigen present is actually selected to bind the carboxyterminal tail region of the antigen-binding fragment (Fab). Thus, special care must be taken when characterizing antibody CDR-H3 loops by Fab X-ray structures, and the possibility that pre-existing conformations are present should always be considered.     

Chemical basis for the affinity maturation of a camel single domain antibody

Affinity maturation of classic antibodies supposedly proceeds through the pre-organization of the reactive germ line conformational isomer. It is less evident to foresee how this can be accomplished by camelid heavy-chain antibodies lacking light chains. Although these antibodies are subjected to somatic hypermutation, their antigen-binding fragment consists of a single domain with restricted flexibility in favor of binding energy. An antigen-binding domain derived from a dromedary heavy-chain antibody, cAb-Lys3, accumulated five amino acid substitutions in CDR1 and CDR2 upon maturation against lysozyme. Three of these residues have hydrophobic side chains, replacing serines, and participate in the hydrophobic core of the CDR1 in the mature antibody, suggesting that conformational rearrangements might occur in this loop during maturation. However, transition state analysis of the binding kinetics of mature cAb-Lys3 and germ line variants show that the maturation of this antibody relies on events late in the reaction pathway. This is reflected by a limited perturbation of k(a) and a significantly decreased k(d) upon maturation. In addition, binding reactions and the maturation event are predominantly enthalpically driven. Therefore, maturation proceeds through the increase of favorable binding interactions, or by the reduction of the enthalpic penalty for desolvation, as opposed to large entropic penalties associated with conformational changes and structural plasticity. Furthermore, the crystal structure of the mutant with a restored germ line CDR2 sequence illustrates that the matured hydrophobic core of CDR1 in cAb-Lys3 might be compensated in the germ line precursor by burying solvent molecules engaged in a stable hydrogen-bonding network with CDR1 and CDR2.     

Structural Reorganization of α-Synuclein at Low pH Observed by NMR and REMD Simulations

α-Synuclein is an intrinsically disordered protein that appears in aggregated forms in the brains of patients with Parkinson's disease. The conversion from monomer to aggregate is complex, and aggregation rates are sensitive to changes in amino acid sequence and environmental conditions. It has previously been observed that α-synuclein aggregates faster at low pH than at neutral pH. Here, we combine NMR spectroscopy and molecular simulations to characterize α-synuclein conformational ensembles at both neutral and low pH in order to understand how the altered charge distribution at low pH changes the structural properties of these ensembles and leads to an increase in aggregation rate. The N-terminus, which has a small positive charge at neutral pH due to a balance of positively and negatively charged amino acid residues, is very positively charged at low pH. Conversely, the acidic C-terminus is highly negatively charged at neutral pH and becomes essentially neutral and hydrophobic at low pH. Our NMR experiments and replica exchange molecular dynamics simulations indicate that there is a significant structural reorganization within the low-pH ensemble relative to that at neutral pH in terms of long-range contacts, hydrodynamic radius, and the amount of heterogeneity within the conformational ensembles. At neutral pH, there is a very heterogeneous ensemble with transient contacts between the N-terminus and the non-amyloid β component (NAC); however, at low pH, there is a more homogeneous ensemble that exhibits strong contacts between the NAC and the C-terminus. At both pH values, transient contacts between the N- and C-termini are observed, the NAC region shows similar exposure to solvent, and the entire protein shows similar propensities to secondary structure. Based on the comparison of the neutral- and low-pH conformational ensembles, we propose that exposure of the NAC region to solvent and the secondary-structure propensity are not factors that account for differences in propensity to aggregate in this context. Instead, the comparison of the neutral- and low-pH ensembles suggests that the change in long-range interactions between the low- and neutral-pH ensembles, the compaction of the C-terminal region at low pH, and the uneven distribution of charges across the sequence are key to faster aggregation.     

Ab initio structure prediction of the antibody hypervariable H3 loop

Antibodies have the capability of binding a wide range of antigens due to the diversity of the six loops constituting the complementarity determining region (CDR). Among the six loops, the H3 loop is the most diverse in structure, length, and sequence identity. Prediction of the three‐dimensional structures of antibodies, especially the CDR loops, is an important step in the computational design and engineering of novel antibodies for improved affinity and specificity. Although it has been demonstrated that the conformation of the five non‐H3 loops can be accurately predicted by comparing their sequences against databases of canonical loop conformations, no such connection has been established for H3 loops. In this work, we present the results for ab initio structure prediction of the H3 loop using conformational sampling and energy calculations with the program Prime on a dataset of 53 loops ranging in length from 4 to 22 residues. When the prediction is performed in the crystal environment and including symmetry mates, the median backbone root mean square deviation (RMSD) is 0.5 Å to the crystal structure, with 91% of cases having an RMSD of less than 2.0 Å. When the prediction is performed in a noncrystallographic environment, where the scaffold is constructed by swapping the H3 loops between homologous antibodies, 70% of cases have an RMSD below 2.0 Å. These results show promise for ab initio loop predictions applied to modeling of antibodies. © 2012 Wiley Periodicals, Inc.     

Does conformational free energy distinguish loop conformations in proteins?

Limitations in protein homology modeling often arise from the inability to adequately model loops. In this paper we focus on the selection of loop conformations. We present a complete computational treatment that allows the screening of loop conformations to identify those that best fit a molecular model. The stability of a loop in a protein is evaluated via computations of conformational free energies in solution, i.e., the free energy difference between the reference structure and the modeled one. A thermodynamic cycle is used for calculation of the conformational free energy, in which the total free energy of the reference state (i.e., gas phase) is the CHARMm potential energy. The electrostatic contribution of the solvation free energy is obtained from solving the finite-difference Poisson-Boltzmann equation. The nonpolar contribution is based on a surface area-based expression. We applied this computational scheme to a simple but well-characterized system, the antibody hypervariable loop (complementarity-determining region, CDR). Instead of creating loop conformations, we generated a database of loops extracted from high-resolution crystal structures of proteins, which display geometrical similarities with antibody CDRs. We inserted loops from our database into a framework of an antibody; then we calculated the conformational free energies of each loop. Results show that we successfully identified loops with a "reference-like" CDR geometry, with the lowest conformational free energy in gas phase only. Surprisingly, the solvation energy term plays a confusing role, sometimes discriminating "reference-like" CDR geometry and many times allowing "non-reference-like" conformations to have the lowest conformational free energies (for short loops). Most "reference-like" loop conformations are separated from others by a gap in the gas phase conformational free energy scale. Naturally, loops from antibody molecules are found to be the best models for long CDRs (> or = 6 residues), mainly because of a better packing of backbone atoms into the framework of the antibody model.     

Aggregation risk prediction for antibodies and its application to biotherapeutic development

Aggregation is a common problem affecting biopharmaceutical development that can have a significant effect on the quality of the product, as well as the safety to patients, particularly because of the increased risk of immune reactions. Here, we describe a new high-throughput screening algorithm developed to classify antibody molecules based on their propensity to aggregate. The tool, constructed and validated on experimental aggregation data for over 500 antibodies, is able to discern molecules with a high aggregation propensity as defined by experimental criteria relevant to bioprocessing and manufacturing of these molecules. Furthermore, we show how this tool can be combined with other computational approaches during early drug development to select molecules with reduced risk of aggregation and optimal developability properties.     

Aggregation risk prediction for antibodies and its application to biotherapeutic development

Aggregation is a common problem affecting biopharmaceutical development that can have a significant effect on the quality of the product, as well as the safety to patients, particularly because of the increased risk of immune reactions. Here, we describe a new high-throughput screening algorithm developed to classify antibody molecules based on their propensity to aggregate. The tool, constructed and validated on experimental aggregation data for over 500 antibodies, is able to discern molecules with a high aggregation propensity as defined by experimental criteria relevant to bioprocessing and manufacturing of these molecules. Furthermore, we show how this tool can be combined with other computational approaches during early drug development to select molecules with reduced risk of aggregation and optimal developability properties.