Elastic energy storage in unmineralized and mineralized extracellular matrices (ECMs): a comparison between molecular modeling and experimental measurements

In order to facilitate locomotion and limb movement many animals store energy elastically in their tendons. In the turkey, much of the force generated by the gastrocnemius muscle is stored as elastic energy during tendon deformation and not within the muscle. As limbs move, the tendons are strained causing the collagen fibers in the extracellular matrices to be strained. During growth, avian tendons mineralize in the portions distal to the muscle and show increased tensile strength, modulus, and energy stored per unit strain as a result. In this study the energy stored in unmineralized and mineralized collagen fibers was measured and compared to the amount of energy stored in molecular models. Elastic energy storage values calculated using the molecular model were slightly higher than those obtained from collagen fibers, but display the same increases in slope as the fiber data. We hypothesize that these increases in slope are due to a change from the stretching of flexible regions of the collagen molecule to the stretching of less flexible regions. The elastic modulus obtained from the unmineralized molecular model correlates well with elastic moduli of unmineralized collagen from other studies. This study demonstrates the potential importance of molecular modeling in the design of new biomaterials.     

Elastic energy storage in human articular cartilage: estimation of the elastic modulus for type II collagen and changes associated with osteoarthritis.

The viscoelastic mechanical properties of normal and osteoarthritic articular were analyzed based on data reported by Kempson [in: Adult Articular Cartilage (1973)] and Silver et al. (Connect. Tissue Res., 2001b). Results of the analysis of tensile elastic stress-strain curves suggest that the elastic modulus of cartilage from the superficial zone is approximately 7.0 GPa parallel and 2.21 GPa perpendicular to the cleavage line pattern. Collagen fibril lengths in the superficial zone were found to be approximately 1265 microm parallel and 668 microm perpendicular to the cleavage line direction. The values for the elastic modulus and fibril lengths decreased with increased extent of osteoarthritis. The elastic modulus of type II collagen parallel to the cleavage line pattern in the superficial zone approaches that of type I collagen in tendon, suggesting that elastic energy storage occurs in the superficial zone due to the tensile pre-tension that exists in this region. Decreases in the elastic modulus associated with osteoarthritis reflect decreased ability of cartilage to store elastic energy, which leads to cartilage fibrillation and fissure formation. We hypothesize that under normal physiological conditions, collagen fibrils in cartilage function to store elastic energy associated with weight bearing and locomotion. Enzymatic cleavage of cartilage proteoglycans and collagen observed in osteoarthritis may lead to fibrillation and fissure formation as a result of impaired energy storage capability of cartilage.     

Effective energy function for proteins in solution

A Gaussian solvent-exclusion model for the solvation free energy is developed. It is based on theoretical considerations and parametrized with experimental data. When combined with the CHARMM 19 polar hydrogen energy function, it provides an effective energy function (EEF1) for proteins in solution. The solvation model assumes that the solvation free energy of a protein molecule is a sum of group contributions, which are determined from values for small model compounds. For charged groups, the self-energy contribution is accounted for primarily by the exclusion model. Ionic side-chains are neutralized, and a distance-dependent dielectric constant is used to approximate the charge-charge interactions in solution. The resulting EEF1 is subjected to a number of tests. Molecular dynamics simulations at room temperature of several proteins in their native conformation are performed, and stable trajectories are obtained. The deviations from the experimental structures are similar to those observed in explicit water simulations. The calculated enthalpy of unfolding of a polyalanine helix is found to be in good agreement with experimental data. Results reported elsewhere show that EEF1 clearly distinguishes correctly from incorrectly folded proteins, both in static energy evaluations and in molecular dynamics simulations and that unfolding pathways obtained by high-temperature molecular dynamics simulations agree with those obtained by explicit water simulations. Thus, this energy function appears to provide a realistic first approximation to the effective energy hypersurface of proteins.     

Mechanical implications of the domain structure of fiber-forming collagens: comparison of the molecular and fibrillar flexibilities of the alpha1-chains found in types I-III collagen

Fibrillar collagens store, transmit and dissipate elastic energy during tensile deformation. Results of previous studies suggest that the collagen molecule is made up of alternating rigid and flexible domains, and extension of the flexible domains is associated with elastic energy storage. In this study, we model the flexibility of the alpha1-chains found in types I-III collagen molecules and microfibrils in order to understand the molecular basis of elastic energy storage in collagen fibers by analysing the areas under conformational plots for dipeptide sequences. Results of stereochemical modeling suggest that the collagen triple helix is made up of rigid and flexible domains that alternate with periods that are multiples of three amino acid residues. The relative flexibility of dipeptide sequences found in the flexible regions is about a factor of five higher than that found for the flexibility of the rigid regions, and the flexibility of types II and III collagen molecules appears to be higher than that found for the type I collagen molecule. The different collagen alpha1-chains were compared by correlating the flexibilities. The results suggest that the flexibilities of the alpha1-chains of types I and III collagen are more closely related than the flexibilities of the alpha1-chains in types I and II and II and III collagen. The flexible domains found in the alpha1-chains of types I-III collagen were found to be conserved in the microfibril and had periods of about 15 amino acid residues and multiples thereof. The flexibility profiles of types I and II collagen microfibrils were found to be more highly correlated than those for types I and III and II and III. These results suggest that the domain structure of the alpha1-chains found in types I-III collagen is an efficient means for storage of elastic energy during stretching while preserving the triple helical structure of the overall molecule. It is proposed that all collagens that form fibers are designed to act as storage elements for elastic energy. The function of fibers rich in type I collagen is to store and then transmit this energy while fibers rich in types II and III collagen may store and then reflect elastic energy for dissipation through viscous fibrillar slippage. Impaired elastic energy storage by extracellular matrices may lead to cellular damage and changes in signaling by mechanochemical transduction at the extracellular matrix-cell interface.     

Modeling loop reorganization free energies of acetylcholinesterase: a comparison of explicit and implicit solvent models

The treatment of hydration effects in protein dynamics simulations varies in model complexity and spans the range from the computationally intensive microscopic evaluation to simple dielectric screening of charge-charge interactions. This paper compares different solvent models applied to the problem of estimating the free-energy difference between two loop conformations in acetylcholinesterase. Molecular dynamics (MD) simulations were used to sample potential energy surfaces of the two basins with solvent treated by means of explicit and implicit methods. Implicit solvent methods studied include the generalized Born (GB) model, atomic solvation potential (ASP), and the distance-dependent dieletric constant. By using the linear response approximation (LRA), the explicit solvent calculations determined a free-energy difference that is in excellent agreement with the experimental estimate, while rescoring the protein conformations with GB or the Poisson equation showed inconsistent and inferior results. While the approach of rescoring conformations from explicit water simulations with implicit solvent models is popular among many applications, it perturbs the energy landscape by changing the solvent contribution to microstates without conformational relaxation, thus leading to non-optimal solvation free energies. Calculations applying MD with a GB solvent model produced results of comparable accuracy as observed with LRA, yet the electrostatic free-energy terms were significantly different due to optimization on a potential energy surface favored by an implicit solvent reaction field. The simpler methods of ASP and the distance-dependent scaling of the dielectric constant both produced considerable distortions in the protein internal free-energy terms and are consequently unreliable.     

Molecular-mechanics modelling of drug-DNA structures; the effects of differing dielectric treatment on helix parameters and comparison with a fully solvated structural model

This study analyses the influence that the nature of the dielectric constant has on the final structures obtained from in vacuo molecular mechanics calculations on a drug-DNA complex and compares these structures with the energy minimised complex including explicit solvent molecules. Minimisations have been performed on a proflavine-decanucleotide structure, where the drug was intercalated at the d(CpG) site of the d(GpApTpApCpGpApTpApC) decamer duplex, using two expressions for the dielectric constant: a distance-independent, epsilon ij = EPS, and a distance-dependent, epsilon ij = EPS*Rij, form and for values of EPS from 1 to 8. Significantly different structures are obtained for the distance-independent and the distance-dependent expressions of the dielectric constant. The use of a distance-independent dielectric constant leads to distorted structures, which are very sensitive to slight changes in the value of EPS. The use of a distance-dependent dielectric constant leads to less distorted and more stable structures. The effects on helical parameters are analysed in detail. The structures obtained for different values of EPS (within the distance-dependent formalism) seem to converge for values of EPS equal to 4 or greater. Based on these results a distance-dependent form of the dielectric with an EPS value of 4 is recommended in order to produce reliable refined nucleic acid structures by molecular mechanics. These conclusions have been supported by molecular-mechanics minimisation of the same structure with the inclusion of explicit water molecules and counter-ions.     

Collagen self-assembly and the development of tendon mechanical properties

The development of the musculoskeleton and the ability to locomote requires controlled cell division as well as spatial control over deposition of extracellular matrix. Self-assembly of procollagen and its final processing into collagen fibrils occurs extracellularly. The formation of crosslinked collagen fibers results in the conversion of weak liquid-like embryonic tissues to tough elastic solids that can store energy and do work. Collagen fibers in the form of fascicles are the major structural units found in tendon. The purpose of this paper is to review the literature on collagen self-assembly and tendon development and to relate this information to the development of elastic energy storage in non-mineralizing and mineralizing tendons. Of particular interest is the mechanism by which energy is stored in tendons during locomotion. In vivo, collagen self-assembly occurs by the deposition of thin fibrils in recesses within the cell membrane. These thin fibrils later grow in length and width by lateral fusion of intermediates. In vitro, collagen self-assembly occurs by both linear and lateral growth steps with parallel events seen in vivo; however, in the absence of cellular control and enzymatic cleavage of the propeptides, the growth mechanism is altered, and the fibrils are irregular in cross section. Results of mechanical studies suggest that prior to locomotion the mechanical response of tendon to loading is dominated by the viscous sliding of collagen fibrils. In contrast, after birth when locomotion begins, the mechanical response is dominated by elastic stretching of crosslinked collagen molecules.     

Rapid boundary element solvation electrostatics calculations in folding simulations: successful folding of a 23-residue peptide.

Solvation effects play a profound role in the energetics of protein folding. While a continuum dielectric model of solvation may provide a sufficiently accurate estimate of the solvation effects, until now this model was too computationally expensive and unstable for folding simulations. Here we proposed a fast yet accurate and robust implementation of the boundary element solution of the Poisson equation, the REBEL algorithm. Using our earlier double-energy scheme, we included for the first time the mathematically rigorous continuous REBEL solvation term in our Biased Probability Monte Carlo (BPMC) simulations of the peptide folding. The free energy of a 23-residue beta beta alpha-peptide was then globally optimized with and without the solvation electrostatics contribution. An ensemble of beta beta alpha conformations was found at and near the global minimum of the energy function with the REBEL electrostatic solvation term. Much poorer correspondence to the native solution structure was found in the "control" simulations with a traditional method to account for solvation via a distance-dependent dielectric constant. Each simulation took less than 40 h (21 h without electrostatic solvation calculation) on a single Alpha 677 MHz CPU and involved more than 40,000 solvation energy evaluations. This work demonstrates for the first time that such a simulation can be performed in a realistic time frame. The proposed procedure may eliminate the energy evaluation accuracy bottleneck in folding simulations.     

The structure and function of normally mineralizing avian tendons

The leg tendons of certain avian species normally calcify. The gastrocnemius, or Achilles, tendon of the domestic turkey, Meleagris gallopavo, is one such example. Its structure and biomechanical properties have been studied to model the adaptive nature of this tendon to external forces, including the means by which mineral deposition occurs and the functional role mineralization may play in this tissue. Structurally, the distal rounded, thick gastrocnemius bifurcates into two smaller proximal segments that mineralize with time. Mineral deposition occurs at or near the bifurcation, proceeding in a distal-to-proximal direction along the segments toward caudal and medial muscle insertions of the bird hip. Mineral formation appears mediated first by extracellular matrix vesicles and later by type I collagen fibrils. Biomechanical analyses indicate lower tensile strength and moduli for the thick distal gastrocnemius compared to narrow, fan-shaped proximal segments. Tendon mineralization here appears to be strain-induced, the muscle forces causing matrix deformation leading conceptually to calcium binding through the exposure of charged groups on collagen, release of sequestered calcium by proteoglycans, and increased diffusion. Functionally, the mineralized tendons limit further tendon deformation, reduce tendon strain at a given stress, and provide greater load-bearing capacity to the tissue. They also serve as important and efficient elastic energy storage reservoirs, increasing the amount of stored elastic energy by preventing flexible type I collagen regions from stretching and preserving muscle energy during locomotion of the animals.     

Implicit solvent models for flexible protein-protein docking by molecular dynamics simulation

The suitability of three implicit solvent models for flexible protein-protein docking by procedures using molecular dynamics simulation is investigated. The three models are (i) the generalized Born (GB) model implemented in the program AMBER6.0; (ii) a distance-dependent dielectric (DDD) model; and (iii) a surface area-dependent model that we have parameterized and call the NPSA model. This is a distance-dependent dielectric model modified by neutralizing the ionizable side-chains and adding a surface area-dependent solvation term. These solvent models were first tested in molecular dynamics simulations at 300 K of the native structures of barnase, barstar, segment B1 of protein G, and three WW domains. These protein structures display a range of secondary structure contents and stabilities. Then, to investigate the performance of the implicit solvent models in protein docking, molecular dynamics simulations of barnase/barstar complexation, as well as PIN1 WW domain/peptide complexation, were conducted, starting from separated unbound structures. The simulations show that the NPSA model has significant advantages over the DDD and GB models in maintaining the native structures of the proteins and providing more accurate docked complexes.     

Solvation model dependency of helix-coil transition in polyalanine

Helix-coil transitions in polyalanine molecules of length 10 are studied by multi-canonical Monte Carlo simulations. The solvation effects are included by either a distance-dependent dielectric permittivity or by a term that is proportional to the solvent-accessible surface area of the peptide. We found a strong dependence of the characteristics of the helix-coil transition from the details of the solvation model.     

The role of tendon elasticity in hopping in a wallaby (Macropus rufogriseus)

The elastic properties of the major tendons of wallaby feet have been investigated in dynamic tensile tests at a frequency close to normal hopping frequencies. In addition, bending tests have been performed on the intertarsal and tarso-metatarsal joints. The data are used to calculate the strain energy stored temporarily in the tendons and ligaments of the legs during a typical hopping stride. The contribution made by stored strain energy to the work required for hopping is assessed.     

Collagen fibril morphology and organization: implications for force transmission in ligament and tendon.

Connective tissue mechanical behavior is primarily determined by the composition and organization of collagen. In ligaments and tendons, type I collagen is the principal structural element of the extracellular matrix, which acts to transmit force between bones or bone and muscle, respectively. Therefore, characterization of collagen fibril morphology and organization in fetal and skeletally mature animals is essential to understanding how tissues develop and obtain their mechanical attributes. In this study, tendons and ligaments from fetal rat, bovine, and feline, and mature rat were examined with scanning electron microscopy. At early fetal developmental stages, collagen fibrils show fibril overlap and interweaving, apparent fibril ends, and numerous bifurcating/fusing fibrils. Late in fetal development, collagen fibril ends are still present and fibril bundles (fibers) are clearly visible. Examination of collagen fibrils from skeletally mature tissues, reveals highly organized regions but still include fibril interweaving, and regions that are more randomly organized. Fibril bifurcations/fusions are still present in mature tissues but are less numerous than in fetal tissue. To address the continuity of fibrils in mature tissues, fibrils were examined in individual micrographs and consecutive overlaid micrographs. Extensive microscopic analysis of mature tendons and ligaments detected no fibril ends. These data strongly suggest that fibrils in mature ligament and tendon are either continuous or functionally continuous. Based upon this information and published data, we conclude that force within these tissues is directly transferred through collagen fibrils and not through an interfibrillar coupling, such as a proteoglycan bridge.     

Cell orientation determines the alignment of cell-produced collagenous matrix

In healing ligaments and tendons, the cells are not aligned and collagen matrix is not organized as in normal tissues. In addition, the mechanical properties of the tissues are abnormal. We hypothesized that the lack of alignment of the collagen matrix results from random orientation of the cells seen in the healing area. To test this hypothesis, a novel in vitro model was used in which the orientation of cells could be controlled via microgrooves, and alignment of the collagen matrix formed by these cells could be easily observed. It is known that cells align uniformly along the direction of microgrooves; therefore MC3T3-E1 cells, which produce large amounts of collagen, were grown on silicone membranes with parallel microgrooves (10 microm wide x 3 microm deep) in the surface. As a control, the same cells were also grown on smooth silicone membranes. Cells on both the microgrooved and smooth silicone surfaces produced a layer of readily visible collagen matrix. Immunohistochemical staining showed that the matrix consisted of abundant type I collagen. Polarized light microscopy of the collagen matrix revealed the collagen fibers to be parallel to the direction of the microgrooves, whereas the collagen matrix produced by the randomly oriented cells on the smooth membranes was disorganized. Thus, the results of this study suggest that the orientation of cells affects the organization of the collagenous matrix produced by the cells. The results also suggest that orienting cells along the longitudinal direction of healing ligaments and tendons may lead to the production of aligned collagenous matrix that more closely represents the uninjured state. This may enhance the mechanical properties of healing ligaments and tendons.     

Effective energy function for proteins in lipid membranes

A simple extension of the EEF1 energy function to heterogeneous membrane-aqueous media is proposed. The extension consists of (a) development of solvation parameters for a nonpolar phase using experimental data for the transfer of amino acid side-chains from water to cyclohexane, (b) introduction of a heterogeneous membrane-aqueous system by making the reference solvation free energy of each atom dependent on the vertical coordinate, (c) a modification of the distance-dependent dielectric model to account for reduced screening of electrostatic interactions in the membrane, and (d) an adjustment of the EEF1 aqueous model in light of recent calculations of the potential of mean force between amino acid side-chains in water. The electrostatic model is adjusted to match experimental observations for polyalanine, polyleucine, and the glycophorin A dimer. The resulting energy function (IMM1) reproduces the preference of Trp and Tyr for the membrane interface, gives reasonable energies of insertion into or adsorption onto a membrane, and allows stable 1-ns MD simulations of the glycophorin A dimer. We find that the lowest-energy orientation of melittin in bilayers varies, depending on the thickness of the hydrocarbon layer.     

Conformational free energy maps for globobiose (alpha-D-Galp-(1-->4)-beta-D-Galp) in implicit and explicit aqueous solution

Four Ramachandran maps of the conformational potential of mean force (PMF) for the galactose disaccharide globobiose (alpha-D-Galp-(1-->4)-beta-D-Galp) were calculated in vacuum, explicit water, with a simple high dielectric constant and a distance-dependent dielectric coefficient, respectively. This simple model of the galactan alpha-(1-->4)-linkage is shown to be conformationally restricted, with only a small range of syn-phi/syn-psi conformations predominating at standard temperature and pressure. This has implications for the preferred conformation and chain dynamics of alpha-galactosides. In addition, comparison of the relevant PMF surfaces reveals the substitution of a high dielectric constant for explicit water solution to be a valid approximation for reproducing the minimum energy conformation of this glycosidic linkage.     

Changes in the distribution of fibrillar collagens in the collateral and cruciate ligaments of the rabbit knee joint during fetal and postnatal development

Summary The collateral ligaments can be clearly distinguished in the 25-day fetal rabbit knee joint. Types I and V collagens are present in the extracellular matrix between the cells of the lateral and medial collateral ligaments and this distribution persists until the rabbit is skeletally mature. From 8 months onwards type III collagen is also present, particularly around the cells. Type I collagen mRNA is expressed by the cells from the 25-day fetal to 8-month-old adult ligament. The ligament sheath is composed of types III and V collagens. The cruciate ligaments are present between the femur and tibia in the 20-day fetus. The matrix is composed of types I and V collagens from the 25-day fetus until at 12- to 14-weeks postnatal, type III collagen appears in the pericellular regions together with type V. At 8 months and 2 years, the amount of type III collagen has increased. All the cells express the mRNA for type I collagen at 12- to 14-weeks, but only isolated cells express this mRNA at 8 months. Thus, both the collateral and cruciate ligaments undergo changes in their complement of collagens during postnatal development and ageing. The implications of these complex interactions of different types of collagen are discussed in relation to healing and the surgical replacement of torn ligaments by tendons.     

Optimized atomic radii for protein continuum electrostatics solvation forces

Recently, we presented a Green's function approach for the calculation of analytic continuum electrostatic solvation forces based on numerical solutions of the finite-difference Poisson-Botzmann (FDPB) equation [Im et al., Comp. Phys. Comm. 111 (1998) 59]. In this treatment the analytic forces were explicitly defined as the first derivative of the FDPB continuum electrostatic free energy with respect to the coordinates of the solute atoms. A smooth intermediate region for the solute-solvent dielectric boundary needed to be introduced to avoid abrupt discontinuous variations in the solvation free energy and forces as a function of the atomic positions. In the present paper we extend the set of optimized radii, which was previously parametrized from molecular dynamics free energy simulations of the 20 standard amino acids with explicit solvent molecules [Nina et al., J. Phys. Chem. 101 (1997) 5239], to yield accurate solvation free energy by taking the influence of the smoothed dielectric region into account.     

Molecular dynamics studies of a DNA-binding protein: 2. An evaluation of implicit and explicit solvent models for the molecular dynamics simulation of the Escherichia coli trp repressor.

Although aqueous simulations with periodic boundary conditions more accurately describe protein dynamics than in vacuo simulations, these are computationally intensive for most proteins. Trp repressor dynamic simulations with a small water shell surrounding the starting model yield protein trajectories that are markedly improved over gas phase, yet computationally efficient. Explicit water in molecular dynamics simulations maintains surface exposure of protein hydrophilic atoms and burial of hydrophobic atoms by opposing the otherwise asymmetric protein-protein forces. This properly orients protein surface side chains, reduces protein fluctuations, and lowers the overall root mean square deviation from the crystal structure. For simulations with crystallographic waters only, a linear or sigmoidal distance-dependent dielectric yields a much better trajectory than does a constant dielectric model. As more water is added to the starting model, the differences between using distance-dependent and constant dielectric models becomes smaller, although the linear distance-dependent dielectric yields an average structure closer to the crystal structure than does a constant dielectric model. Multiplicative constants greater than one, for the linear distance-dependent dielectric simulations, produced trajectories that are progressively worse in describing trp repressor dynamics. Simulations of bovine pancreatic trypsin were used to ensure that the trp repressor results were not protein dependent and to explore the effect of the nonbonded cutoff on the distance-dependent and constant dielectric simulation models. The nonbonded cutoff markedly affected the constant but not distance-dependent dielectric bovine pancreatic trypsin inhibitor simulations. As with trp repressor, the distance-dependent dielectric model with a shell of water surrounding the protein produced a trajectory in better agreement with the crystal structure than a constant dielectric model, and the physical properties of the trajectory average structure, both with and without a nonbonded cutoff, were comparable.